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Intracavity generation of high order harmonics Hammond, Thomas John 2011

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Intracavity Generation of High Order Harmonics by Thomas John Hammond  B.Sc., The University of Winnipeg, 2003 M.Sc., The University of British Columbia, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2011 c Thomas John Hammond 2011  Abstract The goal of this work is the generation of extreme ultraviolet (EUV) radiation from a laser based source. To this end, we use high harmonic generation (HHG) to convert the near infrared output of a mode-locked Ti:Sapphire laser oscillator to the EUV. The requirement for HHG is a high peak intensity (> 1013 W/cm2 ), which can be met by external amplification of the laser output. The method of amplification chosen for this work is a femtosecond enhancement cavity (fsEC), which stores and amplifies the output of a femtosecond mode-locked Ti:Sapphire laser by greater than a factor of 900 while maintaining the original repetition rate of 66 MHz. The design, benefits, and limitations of using a fsEC are discussed. The EUV light is created by the interaction of the amplified light with xenon gas delivered to the fsEC focus. The strong intracavity field leads to xenon plasma generation with detrimental effects on the HHG process, where it is shown that HHG is sensitive to the xenon gas and plasma dynamics. Methods of minimizing the plasma density and maximizing the EUV amplitude are discussed. The EUV is coupled out of the cavity, and up to the thirteenth harmonic (61 nm) of the laser is observed. The relative amplitudes of the different quantum trajectories generating the harmonics are calculated theoretically, and compared to experiment. The generated power of the eleventh harmonic (72 nm) is estimated to be 30 µW, with a measured outcoupled power of 1.1 µW. The relative intensity noise is also measured, with a cumulative root-mean-square (RMS) noise of < 1.2% over 100 Hz - 100 kHz bandwidth. In comparison to other laser based HHG systems, while the EUV flux is similar, the cumulative RMS noise is an order of magnitude lower.  ii  Preface The derivation of the resonance map, as discussed in Sec 2.4.2, as well as the experimental results of the dispersion measurement, shown in Sec 3.4, was published as T.J. Hammond, Arthur K. Mills, and David J. Jones. “Simple method to determine dispersion of high-finesse optical cavities.” Optics Express, 17 (11):8998, (2009). I was responsible for the initial effort, as well as the numerical calculations and experimental data. Some of the theoretical interpretations and the writing of the manuscript were carried out with the aide of the co-authors. The calculations shown in Sec 4.2, as well as the theoretical results of Sec 5.5, are published as T.J. Hammond, Arthur K. Mills, and David J. Jones. “Near-threshold harmonics from a femtosecond enhancement cavitybased EUV source: Effects of multiple quantum pathways on spatial profile and yield.” Optics Express, 19 (25):24871, (2011). I was responsible for the initial effort in Sec 4.2, and I carried out all the theory and numerical calculations. I also initiated the theoretical investigation and numerical simulation presented in Sec 5.5. The data acquisition in this section was performed by Arthur K. Mills. The manuscript was prepared in collaboration with the co-authors. I was responsible for the data acquisition and calculations presented in Chapter 6. David J. Jones was responsible for the initial effort. Arthur K. Mills and David J. Jones collaborated on the interpretation of the results.  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Abstract Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  List of Tables  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Abbreviations List of Symbols  ix  . . . . . . . . . . . . . . . . . . . . . . . . . xiii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1  Motivation  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.2  Outline of Thesis  . . . . . . . . . . . . . . . . . . . . . . . .  2 Mode-locked Laser and Femtosecond Cavity Interactions 2.1  2.2  Mode-locked Lasers  . . . . . . . . . . . . . . . . . . . . . . .  2.1.1  Frequency Domain  2.1.2  Time Domain  1 1 6 8 8  . . . . . . . . . . . . . . . . . . .  9  . . . . . . . . . . . . . . . . . . . . . .  13  Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  14  2.2.1  Dispersion and Pulse Duration . . . . . . . . . . . . .  15  2.2.2  Self Phase Modulation  19  . . . . . . . . . . . . . . . . .  iv  Table of Contents 2.3 2.4  Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21  2.3.1  22  CW Enhancement Cavity Characteristics . . . . . . .  Resonance Map  . . . . . . . . . . . . . . . . . . . . . . . . .  26  2.4.1  Mode Alignment . . . . . . . . . . . . . . . . . . . . .  26  2.4.2  Derivation  . . . . . . . . . . . . . . . . . . . . . . . .  27  2.4.3  Cavity Dispersion Effects Pulse Duration . . . . . . .  32  2.4.4  Nonlinear Intracavity Effects . . . . . . . . . . . . . .  35  2.5  Locking ωc  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36  2.6  Locking Ω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .  39  2.6.1  Using the Cavity to Lock Ω0 . . . . . . . . . . . . . .  41  . . . . . . . . . . . . . . . . . . . .  43  . . . . . . . . . . . . . . . . . . . . . .  43  3 Characterizing the Cavity 3.1  Finesse Measurement  3.2  Cavity Ring-down Measurement  3.3  3.4  . . . . . . . . . . . . . . . .  45  3.2.1  Theory of Cavity Ring-down . . . . . . . . . . . . . .  45  3.2.2  The Ring-down Measurement  . . . . . . . . . . . . .  48  . . . . . . . . . . . . . .  49  Laser Output and Cavity Coupling 3.3.1  Spatial Mode Coupling Efficiency  . . . . . . . . . . .  49  3.3.2  Spectral Coupling . . . . . . . . . . . . . . . . . . . .  50  . . . . . . . . . . . . . . . . . . . .  51  3.4.1  Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .  52  3.4.2  Results . . . . . . . . . . . . . . . . . . . . . . . . . .  55  3.4.3  Sources of Error . . . . . . . . . . . . . . . . . . . . .  57  3.4.4  GDD Measurement Summary  . . . . . . . . . . . . .  60  Mirror Damage . . . . . . . . . . . . . . . . . . . . . . . . . .  60  4 Ion Dynamics and Nozzle Design . . . . . . . . . . . . . . . .  64  3.5  4.1  4.2  Dispersion Measurement  Ionization Dynamics . . . . . . . . . . . . . . . . . . . . . . .  65  4.1.1  Ionization  . . . . . . . . . . . . . . . . . . . . . . . .  66  4.1.2  Freed Electron Spectrum . . . . . . . . . . . . . . . .  69  4.1.3  Electron-Ion Recombination and Diffusion  . . . . . .  72  . . . . . . . . . . . . . . .  75  . . . . . . . . . . . . . . . . . . . .  76  Nozzle Design and Flow Analysis 4.2.1  Analytic Solution  v  Table of Contents 4.2.2 4.3  Numerical Solution  . . . . . . . . . . . . . . . . . . .  80  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  92  5 Generation of High Harmonics via an Enhancement Cavity 93 5.1  5.2  The Classical Picture  . . . . . . . . . . . . . . . . . . . . . .  94  5.1.1  Classical Trajectories and Phase . . . . . . . . . . . .  94  5.1.2  Action  97  . . . . . . . . . . . . . . . . . . . . . . . . . .  The Dipole Response 5.2.1  . . . . . . . . . . . . . . . . . . . . . . 100  Time-dependent Schr¨odinger Equation  5.3  Phase Matching  5.4  Harmonic Amplitude Calculation 5.4.1  . . . . . . . . . . . . . . . . . . . . . . . . . 107 . . . . . . . . . . . . . . . 110  Calculation of Atomic Phase . . . . . . . . . . . . . . 113  5.5  Comparison with Experimental Data  5.6  Optimizing the Output 5.6.1  5.7  . . . . . . . . 100  . . . . . . . . . . . . . 116  . . . . . . . . . . . . . . . . . . . . . 123  Improving the Output Coupling Efficiency  . . . . . . 126  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127  6 Relative Intensity Noise of High Harmonics from an Enhancement Cavity  . . . . . . . . . . . . . . . . . . . . . . . . . 128  6.1  Measurement Details  . . . . . . . . . . . . . . . . . . . . . . 128  6.2  Results  6.3  Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131  7 Conclusions  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141  Appendices A Alternate Derivation of Cavity Ring-down . . . . . . . . . . 158 B OpenFOAM  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159  B.1 Thermophysical Properties B.2 BlockMesh  . . . . . . . . . . . . . . . . . . . 160  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161  vi  Table of Contents B.3 Axis-symmetric Approximation . . . . . . . . . . . . . . . . . 162 B.4 3D Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 B.5 Running the Programme B.6 MapFields  . . . . . . . . . . . . . . . . . . . . 164  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165  B.7 Boundary Conditions  . . . . . . . . . . . . . . . . . . . . . . 166  B.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 C Detection and Generation of Noise in Optical Systems . . 169 C.1 Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . 169  C.1.1 Definitions of Terms . . . . . . . . . . . . . . . . . . . 170 C.2 Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . . . 173 C.2.1 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . 173 C.2.2 Johnson Noise . . . . . . . . . . . . . . . . . . . . . . 175 C.3 Sources of Signals  . . . . . . . . . . . . . . . . . . . . . . . . 176  C.3.1 The Photomultiplier . . . . . . . . . . . . . . . . . . . 177 C.3.2 The Semiconductor Photodiode  . . . . . . . . . . . . 179  C.4 Relative Intensity Noise Measurement . . . . . . . . . . . . . 185 C.4.1 RIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186  vii  List of Tables B.1 Table of used gas properties . . . . . . . . . . . . . . . . . . . 161  viii  List of Figures 1.1  Classical cartoon for high harmonic generation . . . . . . . .  3  1.2  A typical high harmonic spectrum . . . . . . . . . . . . . . .  4  1.3  Picture of amplified field, plasma, and harmonics . . . . . . .  6  2.1  Pulse train and spectrum of a mode-locked laser . . . . . . .  9  2.2  Mode-locked laser spectrum showing Ωrep and Ω0 . . . . . . .  10  2.3  Pulse train of a mode-locked laser . . . . . . . . . . . . . . . .  13  2.4  Dispersive material effects on pulse . . . . . . . . . . . . . . .  18  2.5  Self phase modulation . . . . . . . . . . . . . . . . . . . . . .  19  2.6  Self phase modulation and new frequency creation . . . . . .  20  2.7  Sample intracavity beam path for bow-tie cavity . . . . . . .  22  2.8  The cavity photon lifetime and the cavity linewidth of a single mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24  Single mode field amplification in enhancement cavity . . . .  25  2.10 Laser mode and cavity mode alignment . . . . . . . . . . . .  27  2.11 Resonance map with all modes aligned . . . . . . . . . . . . .  29  2.12 Resonance map with modes offset by Ω0 = 20 MHz . . . . . .  30  2.13 Measurement of resonance change due to Ω0 . . . . . . . . . .  32  2.14 Group delay dispersion effects on resonance map . . . . . . .  33  2.15 Group delay dispersion effects on pulse duration . . . . . . .  34  2.16 Third order dispersion effects on resonance map . . . . . . . .  35  2.17 Third order dispersion effects on pulse duration . . . . . . . .  36  2.18 Pound-Drever-Hall error signal . . . . . . . . . . . . . . . . .  37  2.19 Setup for locking ΩF SR of the cavity and Ω0 of the laser . . .  40  2.20 Three PDH error signals for ωc and Ω0 lock . . . . . . . . . .  41  2.9  ix  List of Figures 3.1  Schematic of cavity ringdown measurement setup . . . . . . .  46  3.2  Simulation of ring-down . . . . . . . . . . . . . . . . . . . . .  47  3.3  Experimental cavity ring-down measurements . . . . . . . . .  48  3.4  Cavity mode coupling and alignment . . . . . . . . . . . . . .  50  3.5  Laser output and amplified spectra . . . . . . . . . . . . . . .  51  3.6  Experimental setup for dispersion measurement . . . . . . . .  53  3.7  The Ω0 measurement . . . . . . . . . . . . . . . . . . . . . . .  54  3.8  Dispersion measurement of bare cavity . . . . . . . . . . . . .  56  3.9  Dispersion measurement of sapphire and fused silica . . . . .  58  3.10 The uncertainty in the dispersion measurement, and a mirror with oscillating dispersion . . . . . . . . . . . . . . . . . . . .  59  3.11 Intracavity power decrease due to mirror damage . . . . . . .  62  4.1  Enhancement cavity schematic, discussing the plasma generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  65  4.2  Xenon ionization rate . . . . . . . . . . . . . . . . . . . . . .  67  4.3  Ionization fraction from single pulse . . . . . . . . . . . . . .  69  4.4  Classical electron kinetic energy gained by field . . . . . . . .  70  4.5  Simulated above threshold ionization spectrum . . . . . . . .  71  4.6  Recombination of low energy electrons . . . . . . . . . . . . .  73  4.7  Cartoon of ambipolar diffusion . . . . . . . . . . . . . . . . .  74  4.8  Simulation of ambipolar diffusion and pressure measurement  76  4.9  Pictures of nozzles used . . . . . . . . . . . . . . . . . . . . .  77  4.10 Schematic of 1D nozzle . . . . . . . . . . . . . . . . . . . . . .  78  4.11 Solidworks and OpenFOAM mesh of endfire nozzle . . . . . .  82  4.12 ParaFOAM image from OpenFOAM simulation of endfire nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83  4.13 Endfire OpenFOAM results for 50 µm and 300 µm endfire nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  84  4.14 Magnified view of xenon density from endfire nozzle . . . . .  85  4.15 Xenon density along beam path at various distances from nozzle end . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  86  4.16 Through nozzle cross-section in ParaFOAM . . . . . . . . . .  87 x  List of Figures 4.17 Intracavity power affected by plasma generation . . . . . . . .  88  4.18 Picture of plasma generated . . . . . . . . . . . . . . . . . . .  89  4.19 OpenFOAM results of hybrid nozzle . . . . . . . . . . . . . .  91  5.1  Classical electron trajectories . . . . . . . . . . . . . . . . . .  95  5.2  Energy and phase of returning classical electrons . . . . . . .  97  5.3  The classical electron SFA phase coefficient . . . . . . . . . .  99  5.4  Probability density evolution from a strong field  5.5  Assumed initial ground state for 1D potential . . . . . . . . . 103  5.6  Comparison of length and acceleration dipole response . . . . 105  5.7  The dipole spectrum changes due to driving field intensity . . 106  5.8  The dipole response as a function of driving field intensity . . 107  5.9  Comparison of the 1D potential with that of full 3D model . 108  . . . . . . . 102  5.10 ∆k(r, z) contour plots . . . . . . . . . . . . . . . . . . . . . . 110 5.11 Relative amplitudes of the phase coefficients . . . . . . . . . . 114 5.12 The cavity setup for measuring the harmonic beam shapes and relative amplitudes . . . . . . . . . . . . . . . . . . . . . 117 5.13 Grating diffraction efficiency . . . . . . . . . . . . . . . . . . . 118 5.14 Images of harmonics as nozzle moved through focus  . . . . . 119  5.15 Theory and data comparison for 150µm nozzle as function of position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.16 Theory and data comparison for 300 µm nozzle as function of position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.17 Theory and data comparison for 150 µm and 300 µm nozzles as function of Ipeak . . . . . . . . . . . . . . . . . . . . . . . . 123 5.18 Theory and data comparison for 500 µm nozzle as function of position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.19 Optimized harmonic output . . . . . . . . . . . . . . . . . . . 125 6.1  Schematic of measuring the RIN . . . . . . . . . . . . . . . . 129  6.2  Picture of xenon gas introduced at focus of the fsEC . . . . . 130  6.3  RIN measurement for laser, circulating field and eleventh harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132  xi  List of Figures 6.4  Spectrally resolved reflection signal with and without gas at focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134  6.5  Ω0 lock effect on RIN  . . . . . . . . . . . . . . . . . . . . . . 135  6.6  Cumulative RMS . . . . . . . . . . . . . . . . . . . . . . . . . 136  C.1 The inverting operational amplifier . . . . . . . . . . . . . . . 170 C.2 An example of a noisy signal . . . . . . . . . . . . . . . . . . 173 C.3 The Poisson distribution . . . . . . . . . . . . . . . . . . . . . 174 C.4 Circuit to calculate Johnson noise . . . . . . . . . . . . . . . . 175 C.5 The photomultiplier schematic . . . . . . . . . . . . . . . . . 178 C.6 A pn junction photodiode . . . . . . . . . . . . . . . . . . . . 180 C.7 Biasing a photodiode . . . . . . . . . . . . . . . . . . . . . . . 181 C.8 Schematic for heterodyne detection . . . . . . . . . . . . . . . 185 C.9 Setup for measring low optical power signals . . . . . . . . . . 186 C.10 The transfer function C(f ) for the SRS current amplifier (SRS-780) at various settings . . . . . . . . . . . . . . . . . . 187 C.11 The RIN of the Femtosource Ti:Sapphire laser. . . . . . . . . 188  xii  List of Abbreviations ADK Ademnosov-Delone-Kranov ionization rate calculation. AOM acousto-optic modulator. APD avalanche photodiode. ATI above threshold ionization. CCD charge coupled device. CFD computational fluid dynamics. CPA chirped pulse amplifier. CW continuous wave. EC enhancement cavity. EUV extreme ultraviolet radiation; light of wavelength 10 nm − 100 nm, or 10 eV − 100 eV.  FEL free electron laser. FFC femtosecond frequency comb. fsEC femtosecond enhancement cavity. FSR free spectral range. FWHM full width at half maximum. GDD group delay dispersion. xiii  List of Abbreviations GVD group velocity dispersion; the group delay dispersion per unit length. HHG high order harmonic generation. IC input coupler. LO local oscillator. OC output coupler. OPA optical parametric amplifier. OSA optical spectrum analyser. PD photodetector. PDH Pound-Drever-Hall locking scheme. PZT piezo-electric transducer. RIN relative intensity noise. RMS root-mean-square. RSA radio spectrum analyser. SA spectrum analyser. SAE single active electron. SFA strong field approximation. SPM self phase modulation. SRS Stanford Research Systems. TDSE time dependent Schroedinger equation. TEM transverse electromagnetic; TEM00 is the fundamental gaussian mode. TOD third order dispersion.  xiv  List of Symbols ωm  angular frequency of the mth laser cavity mode  Ω0  the offset frequency, in angular units  Ωrep  the repetition rate, in angular units  Trt  the repetition period between laser pulses, Trt = 2π/Ωrep  ΩF SR  the free spectral range of the enhancement cavity  ωc  carrier frequency  ω1  fundamental driving field frequency  ωq  the q th harmonic of the driving field  ∆ω  optical bandwidth, dependent on the spectral envelope function  ∆ωF W HM  the (power) FWHM bandwidth  τ  pulse duration, dependent on the field envelope function  τcav  cavity photon lifetime  ∆t  the (power) FWHM pulse duration  δt  the time step for solving the time-dependent Schr¨odinger equation  β  group velocity dispersion  F  enhancement cavity finesse  L0  enhancement cavity length when ΩF SR = Ωrep  λ0  central wavelength, λ0 = 2πc/ωc  w0  the minimum beam radius  γ  Keldysh parameter  Up  ponderamotive energy; the kinetic energy transferred  Ip /2Up  from an oscillating field to the electron Ip  ionization potential of an atom/molecule  Ipeak  peak intensity xv  List of Symbols η0  impedance of free space, η0 ≈ 377Ω  η  photodetector quantum efficiency  η(r, z)  ionization fraction  ρ  gas density  T  gas temperature  U  velocity of gas flow  xvi  Acknowledgements As an undergrad, I vacillated between chemistry and physics, eventually settling on a combination of the two. There were three professors who were my inspiration for pursuing further studies after graduation. I would like to thank Dr Randy Kobes and Dr Brian Pettitt for interesting me in their disciplines (physics and chemistry, respectively). Through them, I was introduced to the worlds of nonlinear dynamical modelling and physical chemistry. I would also like to thank Dr Greg Lopinski for supervising me at NRC, giving me my first opportunity to work in an experimental laboratory. Without their guidance and the enthusiasm that they displayed, I would never have had the interest in pursuing a post graduate degree. Before I began grad school, I emailed David Jones, introducing myself and stating my relevant interests: spectroscopy and non-linear dynamics. Chemistry taught me that if you want to know what is going on in matter, then you need to measure the spectrum. My honours project had been in numerical modelling of non-linear systems. Fortunately for me, David had just started at UBC and had a project in mind that would be a combination of those two subjects. His response to my email was, “Well, I’ll need you to build me a laser,” and from then on I was hooked. For supervising me and giving me the opportunity to pursue this research, I am eternally grateful to David. I would also like to thank the post-docs with whom I have worked: Jie Jiang and Arthur Mills. When I first met Jie, I had the Ti:Sapphire laser built, but I could not make the CW laser mode-lock. He said to me, “Hit it”. I had just spent the past several months building the laser, the last thing I wanted to do was hit it. But he insisted that I hit the prism in the laser cavity, and after doing that, it immediately ran in mode-locked xvii  Acknowledgements operation. He always seemed to have a magic touch when it came to making the laser work how he wanted. I would like to thank Jie for helping with my Ti:Sapphire laser struggles. I would also like to thank Art for explaining to me (what feels like many, many times) about loop filters and circuit design, and how to be a much more careful and thorough researcher. Much of my research has required the expertise of an electrical engineer, and Art has never failed to demonstrate what electrical engineers can do at their best. His perseverance and photography skills were invaluable when we first saw plasma, and continued when taking the images for the high harmonic data. I would also like to thank Kirk Madison for guidance in matters of life as a grad student (and what to do after), and always humouring my many questions. I would also like to thank Bruce Klappauf for many useful discussions, and introducing me to the sport of ultimate. I would also like to thank Roman Krems for mentoring me and giving me encouragement when I needed. I would like to give specific thanks to my labmates, who have contributed greatly to my career as a graduate student. Rob Stead, who helped design and build the (functioning) vacuum system after other failed attempts, has also been a good ear when frustration is setting in and he is a great person to bounce ideas off. Egor Chasovskikh and his smile while cursing in Russian always improved morale; thank you for helping design and construct the isolated harmonic photodetector mount. I would also like to thank Matt Lam for helping me with my Matlab questions as I tried to transition myself away from Mathematica to something more useful and engineering-oriented. I would like to thank Marco Turcios for helping me with OpenFOAM, and helping me parse the user manual. Also, I would like to thank Zhenia Shapiro for knowing as much as he does about quantum mechanics, and pointing me towards efficient algorithms that solve the time-dependent Schr¨ odinger equation. I would also like to thank Jason Jones for many useful discussions, and answering many of our questions when we first began to pursue this research. I would also like to thank David Villeneuve for giving us some insight in to the high harmonic generation process. xviii  Acknowledgements Many thanks go to my committee members Mark Halpern, Carl Michal, Tom Tiedje, and Rob Kiefl for their insightful questions and for keeping me on my toes. To Allan, who taught me how to do arithmetic before I got to grade school, thank you for leading the way. I would also like to thank my dad, who humours me when I talk at length about my research and can appreciate my nerdly interests. And to my mum, thank you for always supporting me no matter what I do.  xix  Dedication For Greg, Heather, Allan, Irene, and family and friends.  xx  Chapter 1  Introduction 1.1  Motivation  A laser allows for the control of the amplitude, frequency, and relative phase of light for the precise manipulation and measurement of matter. The invention of the laser quickly led to the study of nonlinear optics [1], where the strong electromagnetic fields produced by a laser can probe the nonlinear response of materials. The nonlinear light-matter interaction has led to the creation of new optical frequencies, extending the laser spectrum from the visible/near infrared (NIR) to the far infrared (FIR) and ultraviolet (UV). In this manner, the coherent properties and the control of the laser have been transferred to new regions of the electromagnetic spectrum. Unfortunately, creating light with energies deeper into the UV becomes increasingly difficult as most materials absorb in this regime. To overcome this challenge, several systems have been developed as sources of coherent light in the extreme ultraviolet (EUV, photon energies from 10 to 100 eV), such as free electron lasers (FELs) and high harmonic generation (HHG). Undulator based FELs that create coherent radiation are often housed in large facilities [2, 3], and are expensive to construct and operate. In addition, the large amplitude noise, frequency instability, and low repetition rates of FELs can limit their usefulness in some experiments [4–6]. Conversely, HHG has successfully created EUV light from a table-top setup [7–9]. The HHG process was shown to transfer the coherent properties of the laser field to the EUV [10]. However, in order to generate the high harmonics, the laser output must be greatly amplified. Although the peak output power directly from an ultrashort femtosecond (fs) laser oscillator can reach the megawatt (MW) levels, the peak intensity 1  1.1. Motivation is still orders of magnitude lower than the HHG requirement (the minimum threshold for most HHG experiments is > 1 × 1013 W/cm2 )1 . The development of chirped pulse amplification (CPA) in the mid 1980’s [11] allowed  for large, ultrashort pulse amplification. Almost simultaneously Ti:Sapphire lasers were developed [12]. This laser proved to be a robust tool in many optics labs, generating stable, ultrashort pulses centred at 800 nm with pulses below 5 fs [13]. Common table-top Ti:Sapphire CPA systems are often limited to an average output power on the order of 1 W, with repetition rates of 10 − 103 Hz, although by sacrificing the pulse power down to the µJ level,  the repetition rate can be increased to > 100 kHz [14]. Cryogenic cooling of the amplifying crystal can increase the average power to > 50 W [15],  and the repetition rate can increase to > 1 MHz [16]. With the pulse duration compressed to 10 fs, the peak power of these pulses is giga- to terawatt (109 − 1012 W) levels [17, 18]. New fibre-laser based CPAs have been able to  use the long amplification lengths provided by the fibre and high available pump power to increase the average power to >100 W. This has allowed for an increase in the repetition rate of the amplified pulses while maintaining the same pulse energy. However, the narrower bandwidth of these fibre based systems limits the pulse duration to >100 fs; consequently, the peak pulse power is limited to gigawatt levels [19, 20]. Another interesting tabletop source for high peak powers is the optical parametric amplifier (OPA). These systems are extremely tuneable and versatile, and can generate high peak powers comparable to CPA systems in the near infrared (NIR) [21, 22]. The output of all of the above mentioned systems can be tightly focussed to create ultrahigh > 1015 W/cm2 intensities (or electric field amplitudes > 1011 V/m). The ultrahigh intensities (or field amplitudes) of these laser-based systems can be used to probe nonlinear effects and generate new frequencies. When the amplified laser light is focussed into a volume of atomic gas, the peak field amplitude is comparable to the electric field experienced by a 1  The peak output intensity from a laser oscillator with pulse energy of 5 nJ, tightly focussed to a minimum radius of 10 µm, with a pulse duration of 20 fs, is about three orders of magnitude too low to generate high harmonics.  2  1.1. Motivation  tunnel ionization  acceleration  recombination and photoemission  �ωq  Figure 1.1: Cartoon of the three-step model of HHG [23] showing the distortion of the atomic potential from a strong, oscillating field. In the first step, the electron has a finite probability of tunnelling out of the Coulombic potential of the atom (blue dashed) due to the laser field (red dashed); the resulting potential is strongly distorted (blue solid). The electron is then accelerated by the field in the second step. Finally, the return of the electron to the parent ion releases a photon (¯hωq ) of the gained energy.  valence electron in an atom [24, 25]. The electron can be manipulated by the strong, oscillating laser field: it can be freed from the atom to produce a plasma, or it can be returned to the parent ion to create a photon - the source for HHG [23] (see Fig 1.1). The created photon has an energy equal to the kinetic energy that the electron gained by the field, plus the ionization potential of the atom. In an atomic gas, the symmetry of the atomic potential leads to the generation of only the odd harmonics of the laser field. A typical HHG spectrum is shown in Fig 1.2. The maximum photon energy is dependent on the laser field wavelength and peak amplitude, as well as the gas species [26–28]. The high pulse energies of CPA systems have allowed for HHG research to proceed in many different directions. To improve the conversion efficiency of the harmonics, research has gone into designing wave-guides to increase the light-gas interaction region [29–31]. The wavelength dependence on the 3  Amplitude  1.1. Motivation  5  10  15  20  25  Harmonic Order Figure 1.2: A sample spectrum of a typical HHG source. Amplitude would be measured in dB (logarithm scale). Only the odd harmonics are generated because of the symmetry of the field and atomic potential. The red frequency is the fundamental field (typically in the near infrared), and the harmonics above the fifth are typically in the extreme ultraviolet.  generated harmonics has led some groups to develop wavelength tuneable laser and OPA sources to increase the maximum harmonic attainable [32]. Other research has used active feedback to manipulate the driving pulse shape to maximize the amplitude of a selected harmonic [33, 34] and shift the harmonic spectrum [35]. Conversely, because the bandwidth of the entire harmonic spectrum can span many orders of magnitude, experiments in the manipulation of the harmonic phase have led to attosecond (1 as = 10−18 s) pulses [36]. As mentioned above, research has also gone into increasing the repetition rate of the CPA systems, thereby increasing the flux of the low order harmonics [37]. All of these research directions have benefits and limitations as a source for EUV light. CPA based HHG systems have low repetition rates and large amplitude noise [33]; amplitude noise sensitive experiments are then required to have long acquisition (averaging) times. Although these systems have been useful in generating coherent EUV light, there is significant room for improvement in their amplitude noise characteristics in order to be sources for sensitive spectroscopic techniques. Recently, femtosecond enhancement cavity (fsEC) amplification has yielded generation of high harmonics while maintaining the 4  1.1. Motivation repetition rate of the laser (10 − 100 MHz) [38–41], which is the basis for the work in this thesis.  Unlike CPA systems, an enhancement cavity (EC) is a passive (without a gain medium) resonator system, relying on the resonance of the cavity for amplification. Laser light is coupled into the EC, where it is stored and circulated within the resonator. As more laser light is coupled, the intracavity field increases until the intracavity losses (such as mirror reflectivity) prevent further amplification. In this manner, the output of the laser can be amplified by factors of 10−103 while maintaining the useful laser light properties such as directionality, polarization, beam shape, as well as frequency and amplitude stability. The EC can also take advantage of the inefficiency of the HHG process, where the conversion efficiency (converting the fundamental driving field power to the harmonic power) of HHG is on the order of 10−9 − 10−5 ,  thus per-pass, most of the fundamental beam is transmitted through the  harmonic generating medium. Improvements in the conversion efficiency and increasing the output power of the harmonics is of great interest. The EC allows for the recycling of the unused light to further generate harmonics upon multiple round trips, increasing the output power of the harmonics with a relatively low initial power from the laser. Another advantage of using an EC for amplification is that it maintains each frequency component of the output of a mode-locked laser [42, 43]. The output of a mode-locked laser is a train of pulses, which, when viewed in the frequency domain, is an evenly spaced comb [44]. Because the EC amplifies the laser output at the same repetition period as the laser, the frequency comb nature is preserved. Generating harmonics of the laser field via an EC thus transfers the frequency comb from the visible up to the EUV [38, 39]. The frequency comb of a mode-locked laser has been used as a stable frequency reference for many applications in the visible and infrared. Cavity based high harmonic generation could be a potential source for metrology in previously unattainable regions of the electromagnetic spectrum.  5  1.2. Outline of Thesis  1.2  Outline of Thesis  In this thesis, we use a high finesse fsEC to amplify the output of a Ti:Sapphire mode-locked laser oscillator by factors approaching 103 . The laser output characteristics such as spectrum, pulse duration, and repetition rate are maintained in the amplification process. Xenon is introduced to the fsEC focus to generate the high harmonics. Parasitic effects, such as ionization, plasma generation, and non-linear phase shifting, are investigated and are shown to limit the harmonic generation efficiency and the stability condition of the cavity. A sample image of the high intracavity field generating the plasma and EUV at the focus is shown in Fig 1.3. The HHG process is modelled in order to maximize the efficiency while minimizing the parasitic effects. Finally, the generated harmonic power and relative amplitude noise are measured using a photodetector. Generated harmonics  Amplified field  Plasma  Figure 1.3: A picture of the amplified fundamental field (pink dots on mirror faces) leading to the generation of xenon plasma (orange plume), and the resulting high order harmonics (blue dots on fluorescent slide). The theory and experiment of the amplified intracavity field is discussed in Chapters 2 and 3, respectively; the plasma is discussed in Chapter 4, and optimizing the harmonic amplitude is discussed in Chapter 5.  In Chapter 2, the theory of amplifying the output of a mode-locked laser in a fsEC is discussed. The Chapter begins by describing the output of a mode-locked laser, the fs pulse train, and how this can be efficiently 6  1.2. Outline of Thesis amplified within a fsEC. We describe how to lock the laser frequencies and the cavity length for long-term amplification of the entire laser spectrum, required for maintaining a stable, high intracavity average power. Chapter 3 discusses the realised fsEC properties. Limitations due to mirror reflectivity, mode-matching and beam coupling efficiency are discussed, as is the amplified spectrum. We measure the fsEC dispersion with the techniques developed in Chapter 2. In Chapter 4, it is found that the strong fields that are attained within the fsEC can lead to unwanted effects such as above threshold ionization (ATI) and plasma generation. We find that the gas and plasma dynamics are on the same timescales as those of the repetition period of the laser. To understand the timescales, we model the ionization rates. To predict the steady state plasma density, we model the ATI spectrum and the recombination processes and rates. We also model the gas flow near the cavity focus in order to change the gas dynamics, thereby minimizing the intracavity plasma density. The inefficiency of HHG is the motivation for improving the harmonic yield. To this end, in Chapter 5, the HHG process is modelled in order to improve the efficiency of generating EUV light with the experimentally available parameters. We use classical and semiclassical models to generate theoretical spectra. Because phase matching and gas densities play important roles in HHG efficiency, we model the harmonic field generation process. We compare this theory to experiment to validate the model. From this, we predict the EUV yield with the changing of experimentally available parameters, such as gas density, light-gas interaction lengths, laser intensity, and pulse duration. Chapter 6 contains a discussion of the measurement of the EUV amplitude noise. The influence of the intracavity HHG process on the stability of the resonator is discussed. The useable power of a harmonic is also measured. In Chapter 7, this thesis concludes with an outlook and a discussion of possible upcoming experiments that could be performed as a result of this work. 7  Chapter 2  Mode-locked Laser and Femtosecond Cavity Interactions This chapter discusses how the output of a femtosecond mode-locked laser interacts with an enhancement cavity (EC). First, the mode-locked laser is described in terms both of the time and frequency domains. Then, the dispersion and nonlinear index of refraction effects are discussed in the context of a pulse of electromagnetic radiation propagating in matter. This chapter introduces the EC using the language of a continuous-wave (CW) laser, which is still useful for some terminology. The EC response to a single CW mode is then expanded to include the many modes excited by a femtosecond mode-locked laser.  2.1  Mode-locked Lasers  In normal CW operation, the longitudinal modes of a laser have random phases. In mode-locked operation, the phases of the modes that make up the output of a laser oscillator are locked together so that the frequency components (modes) constructively interfere. When viewed in the time domain, the output of a mode-locked laser is a series of ultrashort pulses; in the frequency domain, the spectrum is composed of a discrete set of frequencies as shown in Fig 2.1. The two domains emphasize the different features and consequences of the ultrashort pulses and broad spectrum [45].  8  2.1. Mode-locked Lasers Ωrep ∆t  Time (a)  Power  Power  2π/Ωrep  ∆ωF W HM  Frequency (b)  Figure 2.1: (a) The pulse train, and (b) the spectrum of a frequency stabilized mode-locked laser. The pulses are separated by the repetition period of the laser Trt = 2π/Ωrep , resulting in the spectrum being a set of discrete modes separated in the frequency domain by the inverse of the repetition period, called the (angular) repetition rate Ωrep . It will be shown that the pulse envelope and spectrum envelope (black dash) are related. The pulse duration ∆t is inversely proportional to the full width at half maximum (FWHM) spectral width ∆ωF W HM , where the constant of proportionality is dependent on the pulse shape, discussed in text.  2.1.1  Frequency Domain  In the frequency domain, the output of a mode-locked laser is a broad spectrum that is characterized by the carrier frequency ωc (or central wavelength λ0 = 2πc/ωc ) and the full width at half maximum optical bandwidth ∆ωF W HM . The spectrum is determined by various factors such as the gain medium (in this case, the Ti:Sapphire crystal) and the mirror reflectivity bandwidth, and can be tuned by other intracavity elements (prisms or gratings). Within the spectrum, there are many (∼ 104 − 106 ) discrete, coherent modes.  Mode Spacing and Position The coherent sum of the populated modes of the laser leads to a pulse circulating within the laser cavity. However, there can be a difference between the velocity of the envelope of the pulse (the group velocity vg ), and the velocity of each frequency component (the phase velocity vφ ). The phase  9  2.1. Mode-locked Lasers and group velocities are given, respectively, by, vφ = vg =  c ω = n(ω) k(ω) ∂k −1 ∂ω  (2.1a) (2.1b)  where n(ω) is the (frequency dependent) refractive index. The Taylor series expansion of the wavevector k in Eq 2.1a to first order gives, k(ω) = kc + (ω − ωc )  1 vg  (2.2)  where kc = k(ωc ). The phase and group velocities, in this case, are related by, k kc ω − ωc 1 1 = = + . vφ ω ω ω vg  (2.3)  Ωrep  Power  Ω0  ω Figure 2.2: The spectrum of a mode-locked laser is often referred to as a frequency comb because of the even spacing. The spacing of the modes is determined by the repetition rate Ωrep , and the absolute position is given by Ω0 . The repetition rate is determined by the laser cavity length, and is typically in the MHz−GHz range. The offset frequency is less than the mode spacing. The position of the modes of the laser cavity is given by, ωm =  2πc m n(ω)L  (2.4)  10  2.1. Mode-locked Lasers where m is an integer. The total optical path length, n(ω)L, includes the effects of all the laser cavity optics (such as the Ti:Sapphire crystal, prisms, mirrors, air, etc.), where n(ω) is the average index of refraction and L is the total laser cavity length. In general, using Eq 2.1a, the mode position can also be written as, ωm =  2πvφ m. L  (2.5)  In the case of Eq 2.3, the frequencies allowed by the laser cavity are, 1 L L ωc = + kc − . ωm 2πmvg 2πmω vg  (2.6)  Because the frequencies of interest are those resonant in the laser cavity, ω = ωm ; solving for ωm gives, ωm = =  2πvg m − kc vg + ωc L 2πvg 1 1 m + vg ωc . − L vg vφ  (2.7)  The comb structure of the spectrum is now apparent: there is a constant spacing determined by the repetition rate of the laser, Ωrep =  2πvg , L  (2.8)  and an offset from zero frequency, called the offset frequency, Ω0 = vg ωc  1 1 − . vg vφ  (2.9)  Thus, the difference in phase and group velocities leads to Ω0 . The mth optical mode is then, ωm = mΩrep + Ω0 .  (2.10)  Therefore, the optical frequencies of a mode-locked laser are completely determined by the mode number m, and two radio frequencies Ωrep and Ω0 , as shown in Fig 2.2.  11  2.1. Mode-locked Lasers Dispersion The phase evolution of the different frequency components propagating through material is important when discussing ultrashort pulses. All materials exhibit a frequency dependent index of refraction, referred to as dispersion. Dispersion accounts for higher order terms in the Taylor expansion of k about a frequency ω0 , k(ω) =  n(ω)ω c  = k(ω0 ) + (ω − ω0 ) (ω − ω0 )2 ∂ 2 k 2 ∂ω 2  ∂k ∂ω  ω=ω0  ω=ω0  +  +  (ω − ω0 )3 ∂ 3 k 6 ∂ω 3  ω=ω0  + ···  (2.11)  The first term is the wave-vector evaluated at ω0 , while the derivative in the second term is the inverse of the group velocity vg−1 = ∂k/∂ω, as mentioned above. If the group velocity is frequency dependent, then ∂ 1 ∂2k = ∂ω vg (ω) ∂ω 2  (2.12)  is non-zero, and the group velocity dispersion (GVD, also represented as β in literature [46]) is non-zero. Second order dispersion is also referred to as group delay dispersion, GDD = β z, where z is the distance travelled through the dispersive medium. The next term is referred to as third-order dispersion (TOD). As previously mentioned, the group velocity is the speed at which the envelope of the pulse travels. The frequency dependence of the group velocity means that as the pulse propagates through the material, dispersion leads to a spread in the pulse duration. The spreading of the pulse due to dispersion can be an undesired effect. One method of maintaining the pulse duration is to compensate for the normal material dispersion [47] by a prism pair [48], grating pair [49], and chirped mirrors [50].  12  2.1. Mode-locked Lasers  2.1.2  Time Domain  In the time domain, the output of a mode-locked laser is the coherent addition of the discrete modes which make up the spectrum, Em ei(km z−ωm t) eiφ0  Etrain (z, t) =  (2.13)  m  where Em is the amplitude of the mth mode, km = ωm n(ωm )/c, and φ0 is some phase common to all modes. The field Etrain (z, t) is a train of pulses separated by the round trip time Trt = 2π/Ωrep ; however, the pulses are not necessarily identical, as shown in Fig 2.3.  Field  ∆φ  2π/Ωrep  t Figure 2.3: The pulse train of a mode-locked laser emphasizing the pulseto-pulse phase evolution ∆φ. The separation of the pulses is given by the round trip time Trt = 2π/Ωrep . The phase evolution of the pulse, ∆φ, which represents the slipping of the carrier frequency phase with respect to the envelope phase, is dependent on Ω0 . The offset frequency, which affects the pulse train by causing a phase slip of the field relative to the envelope, is demonstrated by investigating the periodicity of the field. Two identical output pulses from the laser are measured at times t and t + Tl , where the latter is given by, Em ei(km z−ωm (t+Tl )) eiφ0  Etrain (z, t + Tl ) = m  Em ei(km z−ωm t) eiφ0 e−iωm Tl  =  (2.14)  m  13  2.2. Pulses The periodicity of Eq 2.14 would require for the field to repeat every p pulses, which occurs every Tl = e−iωm Tl  2π Ωrep p.  Thus,  2π p Ωrep Ω0 = exp −i2πmp − i2πp Ωrep Ω0 = exp −i2πp Ωrep = exp −i(Ωrep m + Ω0 )  (2.15)  because both m and p are integers. Therefore, after one pulse, the phase evolution is simply, ∆φ = 2π  Ω0 . Ωrep  (2.16)  The phase evolution, ∆φ, is modulo 2π. Consequently, Ω0 must lie within the range of 0 and Ωrep . The common phase φ0 is not measurable, so it is normally ignored and the field evolution phase ∆φ is referred to as the carrier envelope phase φCE .  2.2  Pulses  For our experiments, a benefit of using ultrashort pulses is that we can generate very high peak powers from a low average power source. In order to generate the ultrashort pulses, a broad spectrum is required. Dispersion, as mentioned previously, causes a frequency dependent velocity, which means that as a pulse propagates through a material its pulse duration changes. In this section, we calculate the change in peak power and pulse duration of an ultrashort pulse due to GDD. Additionally, we are using the high peak powers to access the nonlinear response of a material. Unfortunately, not all nonlinear responses are desirable, such as self phase modulation (SPM) in our case. The effects of SPM will be briefly described. An approximation for the peak power of an ultrashort pulse from a  14  2.2. Pulses mode-locked laser is given as2 , Ppeak ≈  2πPavg Ωrep ∆t  (2.17)  where Pavg is the average power and ∆t is the full width at half maximum (FWHM) pulse duration. While the average power of a mode-locked laser oscillator is often tens to hundreds of milliwatts, the peak power for femtosecond pulses can be tens of kilowatts. Lower repetition rates and shorter pulses increase the peak power. Laser amplifiers can bring the peak power up to megawatts, or even tens of gigawatts. Although Eq 2.13 represents the pulse train, it is often easier to represent the field of a single pulse E(z, t) by an amplitude envelope with the carrier frequency and phase. In the frame of the pulse, the field is, E(z, t) = E0 (tr /τ )ei(kc z−ωc t−φCE )  (2.18)  where E0 (tr /τ ) is the amplitude function of the pulse, τ determines the pulse duration3 , and tr = t −  z vg  ensures that we are in the pulse frame.  The pulse envelope travels at the group velocity, whereas the carrier phase velocity is ωc /kc .  2.2.1  Dispersion and Pulse Duration  In normal dispersive materials (such as most glasses in the visible region), the longer wavelengths travel faster than the shorter; the result is the pulse spreads out and becomes chirped. Chirp is when there is a time dependence on the frequency of the pulse. The longer pulse duration leads to a lower peak power. The measured spectrum from the laser is I(0, ω), where the field is ˜ E(0, ω) = 2η0 I(0, ω) (η0 ≈ 377Ω is the impedance of free space)4 . As 2  Ωrep = 2πfrep , where frep would be the (natural) frequency repetition rate that is measured in a spectrum analyser. 3  2 − t  Common envelopes are e 2τ 2 for Gaussian pulses and sech(t/τ ) for hyperbolic secant pulses; τ is dependent on the pulse shape and is not the full width at half maximum. 4 ˜ E(z, ω) is the Fourier transform of E(z, t).  15  2.2. Pulses the field propagates a distance z, it evolves as, ˜ ω) = E(0, ˜ ω)eik(ω)z . E(z,  (2.19)  The wave-vector can be expanded in terms of ω; accounting for the group delay dispersion (GDD), ˜ ω) = E(0, ˜ ω) exp i kc + ω − ωc + β (ω − ωc )2 z E(z, vg 2 (2.20) where we have used kc = k(ωc ) and β z is the GDD. The dispersion causes a (parabolic) frequency dependent phase change across the spectrum; however, the spectrum itself does not change. To see the evolution in time, we take the Fourier transform 1 E(z, t) = √ 2π  ∞  ˜ ω)eik(ω)z e−iωt dω. E(0,  (2.21)  −∞  We can also include higher orders of dispersion. Gaussian pulse broadening An analytic solution exists for Gaussian pulses, which gives some insight into the effect of dispersion on pulse duration and the peak amplitude (illustrated in Fig 2.4). The spectrum of the field of a Gaussian pulse is5 , ˜ ω) = E0 e− E(0, 5  (ω−ωc )2 2∆ω 2  .  The full width at half maximum spectrum ∆ωF W HM  (2.22) √ = 2∆ω ln 2.  16  2.2. Pulses The Fourier transform after propagation through a material with GDD becomes, 2  tr i∆ω 2 i∆ω 2 i(kc z−wc t) − 2 i+β z∆ω2 E(z, t) = E0 e e (2.23) i + β z∆ω 2 i ∆ω −1 2 e 2 tan (β z∆ω ) ei(kc z−ωc t) × = E0 4 (2.24) 2 2 1 + (β z∆ω ) ∆ω 2 t2r β z∆ω 4 t2 exp −i exp − r 2 1 + (β z∆ω 2 )2 2 1 + (β z∆ω 2 )2  The above equation shows that, for a pulse with an initial time-bandwidth √ limited FWHM duration (referred to as transform limited) ∆t0 = 2 ln 2/∆ω, becomes, ∆t = ∆t0 1 + β z∆ω 2  2 1/2  .  (2.25)  Also, the prefactor reduces the field amplitude by (1+(β z∆ω 2 )2 )1/4 . Therefore, minimizing the net (normal and anomalous) dispersion allows for the shortest pulses with the greatest amplitude. Dispersion leading to chirp Frequency is calculated by the rate of change of the phase, ω(t) = −  ∂φ(z, t) . ∂t  (2.26)  For the chirped Gaussian pulse in Eq 2.24, φ(z, t) = kc z − ωc t +  1 t2 β z∆ω 4 tan−1 (β z∆ω 2 ) − r . 2 2 1 + (β z∆ω 2 )2  (2.27)  The frequencies are then time dependent, ω(t) = ωc + tr  β z∆ω 4 . 1 + (β z∆ω 2 )2  (2.28)  Therefore, for normal dispersion (in this thesis case β z > 0 or positive dispersion), when tr < 0, ω < ωc ; for tr > 0, ω > ωc , and ω changes linearly 17  Field  2.2. Pulses  z  Phase  (a)  z  Frequency  (b)  ωc  z (c)  Figure 2.4: The effect of a normal dispersive material (blue rectangle) on an ultrashort, initially transform limited pulse. (a) The pulse duration increases and the peak amplitude decreases as a result of dispersion; (b) the group delay dispersion (GDD) causes a quadratic phase in the pulse, and (c) a linear chirp, with the red frequency components leading the blue. Ideally, dispersion of the same magnitude, but opposite sign, can recompress the pulse to its time-bandwidth limit.  18  2.2. Pulses with time. The β z causes a linear chirp as the pulse propagates through the material, where the sign of β z determines the sign of the chirp.  Self Phase Modulation  Phase  Frequency  2.2.2  ωc  z  z  (a)  (b)  Figure 2.5: SPM arises from the intensity dependent index of refraction n2 , affecting the phase in a nonlinear manner and results in the creation of new frequencies. The linear frequency dependence in (b) (the red dotted line) has the same sign as normal GDD.  Self-phase modulation (SPM) occurs when an ultrashort pulse propagates through a medium with a nonlinear index of refraction6 n = n0 + n2 I(t), where n0 is the linear coefficient and I(t) is intensity. The phase of the pulse is, φ(z, t) = kc nz − ωc t  (2.29)  where z is the thickness of the medium. Because the pulse intensity is time dependent, the resulting frequencies are [51], ω(t) = −  ∂φ(z, t) ∂ = −kc z n I(t) + ωc . ∂t ∂t  (2.30)  For example, as illustrated in Fig 2.5, a Gaussian pulse of duration τ gives7 , ω(t) = 2kdn2  t I(t) + ωc . τ2  (2.31)  6  To solve for a medium with both self phase modulation and dispersion, the split-step operator technique can be used. The split-step technique is described for solving the time-dependent Schr¨ odinger equation in Chapter 5. 7 Similarly for a sech2 pulse, we can replace t/τ 2 with tanh(t/τ )/τ .  19  2.2. Pulses When t < 0, ω < ωc ; when t > 0, ω > ωc ; this is also the case of normal dispersion leading to a chirp. However, there is only a linear dependence across the central portion of the pulse, and the chirp becomes significantly nonlinear in the wings of the pulse. The change in the index of refraction, dependent on the square of the electric field, is sometimes referred to as the optical Kerr effect.  ω2  ω2  −ω1 ω3  ω  Figure 2.6: Generation of a new frequency component. Because the laser frequencies of a mode-locked laser ω1 (red) and ω2 (green) are determined by Ωrep and Ω0 , the created frequencies will maintain the comb nature of the spectrum. The combination of −ω1 and 2ω2 through the optical Kerr effect leads to the creation of ω3 (blue). Other frequency components can be created by 2ω1 − ω2 . From Eq 2.31, the created frequencies would appear to be continuous. However, because the contributing frequencies have a well defined spacing and absolute position, so too do the generated frequencies. The new frequencies are generated by the combination of the laser modes, which means that the new spectrum will have the same spacing and position (Ωrep and Ω0 ) as the original pulse, as shown in Fig 2.6. This maintaining of the comb nature of the laser output is important when considering self referencing techniques [52]. Additionally, this picture can be extended to second harmonic generation [45], and high harmonic generation [38, 39]. That is, if the high peak power is able to create new frequencies, the new frequencies will be a part of the original frequency comb.  20  2.3. Cavities  2.3  Cavities  Although the peak intensity of a mode-locked laser oscillator is orders of magnitude higher than its average intensity, there are non-linear effects that can only be investigated by further enhancement of the peak intensity. The method of attaining a high peak intensity in this work is with an enhancement cavity (EC). The EC is similar to the laser in that it uses the cavity resonance to provide feedback leading to coherent addition of the field. However, in contrast to a laser, there is no gain medium in an EC. The enhancement is achieved through constructive interference of the intracavity field stored by the EC mirrors, and the field incident on the cavity. Through this process, the field amplitude can be greatly increased within the cavity. The term ‘enhancement’ or ‘amplification’ is used in this thesis to refer to the ratio of the magnitude square of the intracavity field to the magnitude square of the field incident on the cavity. The advantage of using an EC to increase the field amplitude is that it can maintain the frequency comb nature (Ωrep and Ω0 ) of a mode-locked oscillator. Because the cavity is designed to support ultrashort pulses, it will be referred to in this work as a femtosecond enhancement cavity (fsEC). With a well designed cavity, the power increase can reach two to three orders of magnitude across the entire spectrum and maintain the pulse duration [53]. Furthermore, the transverse mode within an EC is not distorted by an active gain medium; thus, the mode profile tends to be close to that of an ideal Gaussian beam. And finally, the enhancement process - the coherent addition of the incident and stored fields - requires that the intracavity field has low amplitude noise. However, there are also disadvantages. Due to the amplification occurring within the EC, all experiments that require the high intensities must be performed within the EC. Also, in order to maintain the high intracavity power, we must actively stabilize (or lock) the cavity length and laser frequency; higher amplification generally leads to higher sensitivity of the lock to perturbations and noise. Furthermore, there is a limitation on the coupling efficiency of the output of a mode-locked laser into an EC. The phase 21  2.3. Cavities front of the beam incident on the cavity (determined by the laser output beam size and divergence) must match that of the cavity [54]. However, the phase front of the intracavity beam is determined by the cavity geometry [55]. Additionally, the beam size of a mode-locked laser is not necessarily uniform across its spectrum [56, 57], which prevents efficient coupling across the input spectrum. To understand how an EC works, the simplest case of the output of a single-mode, continuous wave (CW) laser incident on a cavity is introduced. The characteristics of the EC such as finesse, linewidth, and enhancement are discussed. The number of modes is then increased to account for a mode-locked oscillator.  2.3.1  CW Enhancement Cavity Characteristics  To calculate the ideal intracavity amplification, we model the single-mode field incident on an input coupler (IC) of the EC as a plane wave with a field amplitude E0 as shown in Fig 2.7. The field that is transmitted to the cavity is itic E0 , where tic is the field transmission of the IC. rM 3  rM 2  E0  φ=  tic  ω d c  rM 1  Figure 2.7: The intracavity beam path for a bow-tie cavity configuration. The input field amplitude E0 is incident on the input coupler with field transmission tic ; each mirror has field reflectance rM i for a total cavity reflectance rcav ; the phase of the field after one round trip through the cavity of length d is φ = ωc d. The beam is modelled as a plane wave to avoid beam profile phases and diffraction effects.  In this analysis, it is assumed that all losses in reflectivity are in transmission such that R + T = 1 where R = |r|2 is the (power) reflectivity and 22  2.3. Cavities T = 1 − R = |t|2 is the (power) transmission.  After one round trip, the field (after reflection from the IC) is itic E0 ric rcav eiφ  (2.32)  n eiΦ is the field reflectivity of the cavity of all n mirrors with where rcav = rM i  identical reflectivities rM i and total mirror phase Φ. The phase induced on the intracavity field by the mirrors can have an optical frequency dependence Φ = Φ(ω), which is important in the discussion of dispersion, presented below. The cavity phase, φ =  ω c d,  is the phase of the field after one round  trip within the cavity of length d. Assuming that the intracavity field travels an infinite number of round trips, the total intracavity field at the input coupler after reflection is [58], ∞  Ecav = itic E0  (ric rcav eiφ )N  (2.33)  N =0  where N is the number of round trips. The sum is a geometric series leading to, Ecav =  itic E0 . 1 − ric rcav eiφ  (2.34)  In the lab, we measure the power P ∝ |E|2 ; therefore, it is often more  convenient to write Eq 2.34 as, Pcav (ω) =  (1 −  √  P0 Tic √ Ric Rcav )2 + 4 Ric Rcav sin2  φ 2  (2.35)  where Pcav is the intracavity power and P0 is the incident power, and Rcav = |rcav |2 . From this equation, we can find the spacing of the modes called the  cavity free spectral range ΩF SR , the linewidth of the cavity modes ∆Ω, the photon lifetime τcav , the cavity finesse F, and the amplification factor Pcav /P0 , as discussed below.  Because the reflectivity of the mirrors in our enhancement cavity is close to unity, it is not practical to estimate the enhancement factor based on a single pass reflection measurement. Instead, it is often more accurate to 23  1  1  0.8  0.8  0.6  0.6  Pcav/P0  Pcav/P0  2.3. Cavities  τcav  0.4  0.2  0 0  1/τcav 0.4  0.2  1  2  3  4  5  0 ï4  ï2  Time (o)  (a)  0 Frequency (1/o)  2  4  (b)  Figure 2.8: The relation of the cavity photon lifetime τcav is related to the cavity linewidth ∆Ω = 1/τcav .  measure the cavity finesse F, which is shown to be (discussed in Sec 3.1)  proportional to the cavity photon lifetime. For a high finesse cavity, this becomes, F  =  ≈  π  √  √ Ric Rcav 2 sin−1 1− 2 4 Ric Rcav √ π 4 Ric Rcav √ . 1 − Ric Rcav  (2.36)  (2.37)  The intracavity power decay for a high finesse cavity, where all intracavity mirrors have reflectivity near unity, can be approximated as an exponential with a time constant τcav . In this limit, the spectrum of the cavity has a Lorentzian lineshape with a full width at half maximum (FWHM) of ∆Ω = 1/τcav , as shown in Fig 2.8. Knowing the cavity free spectral range8 , ΩF SR , we can measure the photon lifetime to calculate the finesse, thereby characterising the intracavity mirror reflectivity. This measurement 8 The resonance condition, for an empty cavity with no dispersion, is satisfied when the intracavity power is a maximum,  φ sin2 ( ) = 0 2  (2.38)  24  2.3. Cavities is discussed in Chapter 3. 100  Ric (%)  80  60  40  20  0 0  2  (a)  4  Pcav/P0  6  8  10  (b)  10 0.3  Phase (/ rad)  Pcav/P0  8 6 4 2  0.2 0.1 0 ï0.1 ï0.2 ï0.3  0  1  2  3  1  d (h )  2  3  d (h )  (c)  (d)  Figure 2.9: (a) The intracavity amplification Pcav /P0 of a single-mode, CW laser as a function of input coupler reflectivity Ric and cavity length d for a cavity reflectivity of 90%; (b) amplification when φ = 2πm or d = λ, 2λ, 3λ . . .; (c) amplification at the impedance matched case of Ric = 90% scanning through the cavity length, and (d) the phase imparted on the field by the cavity.  The maximum amplification factor within the EC occurs when the cavity phase φ = 2πm (m is the enhancement cavity mode number - see Eq 2.42) and the field is resonant (see Fig 2.9). The maximum attainable intracavity power occurs when the total intracavity loss (the sum of losses due to intracavity optics and mirror transmission) equals the input coupler transor when  ω d c  = 2πm , where m is an integer. Thus, the separation of the modes is, ΩF SR ≡ ωm  +1  − ωm =  2πc . d  (2.39)  25  2.4. Resonance Map mission, and is referred to as ‘impedance matched’. The cases where the input coupler transmission Tic is greater or less than the total intracavity loss - referred to as overcoupled and undercoupled, respectively - has less intracavity power as shown in Fig 2.9(b). The ideal amplification is [59], Pcav 1 − Ric √ = . P0 (1 − Ric Rcav )2  (2.40)  To increase the intracavity power enhancement, we use very highly reflective mirrors and input coupler; the consequence is a narrow cavity linewidth. The narrow cavity linewidth places a tight constraint on the tolerance of the cavity length d. Because of this, we must actively stabilize the cavity so that the field remains resonant, discussed in Sec 2.5.  2.4  Resonance Map  The expansion of the single-mode CW case to the mode-locked laser requires the addition of ∼ 105 modes incident on the input coupler in the analysis  above [60]. However, all of the laser modes are related by the repetition rate and the offset frequency9 . The effects of the offset frequency and intracavity dispersion on the amplified spectrum are discussed.  2.4.1  Mode Alignment  The spacing of the modes of the EC is ΩF SR . However, due to dispersion of the intracavity elements and mirror coatings, the cavity length can be frequency dependent, which leads to an uneven mode spacing in frequency [53]. The effect of intracavity dispersion is a misalignment between the modes of the cavity and the femtosecond frequency comb (FFC) of the laser, as shown in Fig 2.10, leading to a limit on the supported bandwidth, pulse duration, average power, and peak power [62]. Here we will derive an expression for the resonance condition of a mode-locked laser interacting with a dispersive cavity in terms of cavity length and optical frequency. 9  The derivation of the mode-locked resonance condition appears in Ref [61]  26  2.4. Resonance Map  ΩF SR  cavity  Ωrep  laser  ω  Figure 2.10: The alignment of the laser and cavity modes with intracavity dispersion. Only the centre modes are aligned; changing the cavity length to align the modes for the low and/or high frequencies prevents the centre of the spectrum from being resonant because of the even spacing of the laser frequency comb, given by Ωrep .  2.4.2  Derivation  In the discussion of the empty CW enhancement cavity case, it was shown that the total phase the cavity induces on the field is the sum of the propagation phase φ = phase  Φ(ω)10 .  ω c d,  where d is the total cavity length, and the mirror  On resonance, the incident light of frequency ω precesses in  multiples of 2π times within the cavity. That is, 2πm =  ωd + Φ(ω) c  (2.42)  where m is an integer indexing the enhancement cavity mode. Recall from a mode-locked laser, ω = mΩrep + Ω0 , or solving for m, m=  ω − Ω0 . Ωrep  (2.43)  10 If an optic were inserted into the cavity, such as a Brewster plate, the propagation phase would become,  φ=  ω [d + (n(ω) − 1)dB ] c  (2.41)  where n(ω) is the index of refraction of the optic, and dB is its thickness.  27  2.4. Resonance Map We can relate the mode numbers of the laser and enhancement cavity by, m  = m+p ω − Ω0 +p = Ωrep  (2.44)  where p is some integer. This can then be substituted into Eq 2.42 and solved for d to get, d(ω) =  2πc 2πc Ω0 Φ(ω) − −p −c . Ωrep ω Ωrep ω  (2.45)  The above equation determines the cavity length d at which the mode-locked laser frequencies ω are resonant. For the simplified case of no mirror-induced phase change (Φ(ω) = 0) and no offset frequency (Ω0 = 0), as shown in Fig 2.11, the modes can be completely aligned when Ωrep = ΩF SR . That is, when the EC cavity length d matches the laser cavity length L, at d ≡ L0 . In this instance,  the entire spectrum is resonant, which is shown in Fig 2.11(b), and the  maximum amplification is achieved. It is possible to simulate the cavity fringes, that is, the intracavity power as a function of EC length d, as shown in Fig 2.11(c). The intracavity power is the sum of all frequency components for the EC length d. When all the spectral components are resonant at L0 , the maximum amplification is attained. Unlike the CW case where the intracavity power repeats at every multiple of the wavelength λ, the broad spectrum of the mode-locked laser prevents the next fringe, at d = L0 ± λ0  (or p = ±1 in Eq 2.45), from attaining the same maximum power because the separation of the resonance position is wavelength dependent. All of  the modes are simultaneously resonant at d = L0 ; thus at p = ±1 not all  of the modes can be resonant, and the higher order fringes have a reduced amplification and increased linewidth. It should be noted that each individual laser mode m, when resonant at EC mode m , is amplified by the amount calculated from Eq 2.40, independent of dispersion, mode number, or Ω0 (i.e. the laser and EC modes  28  820  820  815  815  810  810 Wavelength (nm)  Wavelength (nm)  2.4. Resonance Map  805 800 795  805 800 795  790  790  785  785  780 ï1.5  ï1  ï0.5 0 0.5 Length (L0 + h0)  1  ï0.5 0 0.5 Length (L0 + h0)  1  1.5  (a)  780 0  5  10  Pcav/P0  15  20  (b)  Pcav/P0  20  15  10  5  ï1.5  ï1  (c)  1.5  (d)  Figure 2.11: Simulated resonance map of a laser with ∆λ = 20 nm, λ0 = 800 nm, Ωrep = 66 MHz, Ω0 = 0 output interacting with an ideal, dispersionless cavity (F = 80). At enhancement cavity length L0 , the cavity length matches that of the laser, all of the cavity modes and the laser modes align to attain maximum enhancement as shown in (c). At this cavity length, the spectrum is shown in (b). A 3D representation of (a) is shown in (d), where the black line is the solution to Eq 2.45.  29  2.4. Resonance Map need not be the same). The amplification of a single mode, when a mode is resonant and well coupled into the EC, is purely defined by the intracavity mirror reflectivity. The ideal amplification for a mode-locked laser only occurs when all of the laser modes are simultaneously resonant within the EC, and the entire spectrum is uniformly amplified. The effect of dispersion or Ω0 limits the number of modes simultaneously resonant, thus decreasing the maximum total amplification of the spectrum. 820  820  815  ωL  810  810 Wavelength (nm)  Wavelength (nm)  815  805  ωc  800 795  ωH  790  800 795 790 785  785 780  805  ï1.5  ï1  ï0.5 0 Length (L0 + h0)  0.5  ï0.5 0 Length (L0 + h0)  0.5  1  (a)  780 0  5  10  Pcav/P0  15  20  (b)  Pcav/P0  20  15  10  5  ï1.5  ï1  (c)  1  (d)  Figure 2.12: Simulation of the effect of changing Ω0 in Fig 2.11 to 20 MHz. The position of resonance has shifted and rotated, preventing all the modes from being aligned simultaneously. A measurement of the central fringe at three different optical frequencies (ωL , ωc , and ωH ) can show the change in the resonance due to Ω0 . The peak of (c) leads to a resonant spectrum in (b) (red) which is narrowed from the input spectrum (blue). The total amplification is reduced and the pulse duration is increased.  If the additional phase term Φ(ω) does not contain any dispersive terms, 30  2.4. Resonance Map then d(ω) can be a constant across the spectrum and the cavity mode spacing is regular. This can be seen by taking the phase as, Φ(ω) = d0 k(ω) = d0 k(ω0 ) + (ω − ω0 )  1 vg  (2.46)  where d0 is the length required to yield the total additional phase. The result is that d is independent of ω if, 2π  Ω0 1 − p + d0 k0 − ω0 Ωrep vg  = 0.  (2.47)  Equation 2.47 demonstrates that there is an equivalence between the additional phase of the cavity and the offset frequency in the laser; we can tune the offset frequency such that all the modes are simultaneously resonant. In Fig 2.12, Ω0 causes a change in the cavity length position - and value - of maximum amplification, and can reduce the bandwidth that is simultaneously resonant. The time domain argument of the effect of the offset frequency is that the phase evolution of the pulse is ∆φ = 2πΩ0 /Ωrep . The Ω0 leads to a change in phase of sequential pulses; because we are trying to amplify the pulses through coherent addition, the pulses must be identical. The EC’s offset frequency, calculated from Eq 2.47, is, Ω0 d0 =p+ Ωrep 2π  ω0 − k0 . vg  (2.48)  The cavity offset frequency leads to a phase slip of the envelope of the pulse as it propagates within the EC, and the laser offset frequency must be set to match. In Fig 2.13, experimental data is shown for the measurement of three optical signals from the cavity. The signals were measured as in Fig 2.19, where the dips in the signals were caused by the EC resonance. The laser power was tuned to change Ω0 , changing the EC resonance position. Conversely, when the cavity phase contains higher order terms (when 31  −1  Signal (arb u)  Signal (arb u)  2.4. Resonance Map  −0.5  0 Distance (arb u)  0.5  1  −1  (a)  −0.5  0 Distance (arb u)  0.5  1  (b)  Figure 2.13: Changing Ω0 affected mode alignment on three experimentally measured signals (ωL red, ωc blue, and ωH purple). With the modes aligned in (b) (dotted line) the residual dispersion was apparent.  there are dispersive elements), there can never be complete mode alignment. We can use the cavity resonance condition to map out the nonlinearity of Φ(ω) to find the cavity group delay dispersion, GDD(ω) =  d2 Φ(ω) d2 ω = d(ω) . dω 2 dω 2 c  (2.49)  The intracavity dispersion measurement is performed in Sec 3.4. The value of m can also be chosen such that m = m/M , or ΩF SR = Ωrep /M , which is useful for a reference cavity [63]. Mode-locked oscillators tend to have Ωrep ∼ 100 MHz; whereas, reference cavities are built to be as stable (and hence often as small) as possible (ΩF SR > 1 GHz). The result is that the fringes (p = 1, 2, 3 . . .) are now separated by  2.4.3  2πc ωM  = λ/M .  Cavity Dispersion Effects Pulse Duration  The intracavity dispersion limits the number of modes (the bandwidth) that are simultaneously resonant, and can also affect the pulse duration. The maximum peak intracavity power is achieved by careful dispersion management.  32  820  820  815  815  810  810 Wavelength (nm)  Wavelength (nm)  2.4. Resonance Map  805 800 795  800 795  790  790  785  785  780 ï5  0 Length (L0 + h0)  (a)  780 0  5  200  400  ï3  x 10  600 800 Pcav/P0  1000  1200  (b)  600  0.4  500  0.3 Phase (/ rad)  Pcav/P0  805  400 300  0.2 0.1  200 0 100 ï0.1 ï5  0 Length (L0 + h0)  (c)  5 ï3  x 10  780  790  800 810 Wavelength (nm)  820  (d)  Figure 2.14: The simulated resonance map for ∆λ = 20 nm, λ0 = 800 nm, Ωrep = 66 MHz, Ω0 = 0, and the cavity dispersion GDD = 10 fs2 with F = 2240 (Ric = 0.9975 and Rcav = 0.9997). (a) The intracavity dispersion causes the misalignment of the modes so the entire spectrum cannot be simultaneously resonant, causing a curvature in the resonance map (black line is Eq 2.45); (b) the resulting locked spectrum (red) is narrowed substantially from the input (blue) and the off-resonant frequencies are attenuated; (c) the maximum enhancement is half the ideal (given in Eq 2.40), and (d) the off-resonant frequencies have an additional phase from the cavity. Intracavity dispersion narrows the bandwidth and imparts an additional frequency dependent phase on the circulating pulse.  33  2.4. Resonance Map The GDD, a quadratic dependence of phase on frequency, causes a quadratic change in the resonance condition in the map as shown in Fig 2.14. The resonant, intracavity spectrum is narrower than the input, and the maximum total amplification is not at the cavity length for the wavelengths with highest amplitude. Also, the intracavity power as a function of cavity length has an asymmetric lineshape. 1 1200 0.8  800  Amplitude  Amplitude (Pcav/P0)  1000  600  0.6  0.4  400 0.2  200  780  790  800 810 Wavelength (nm)  820  0 −200  −100  (a)  0 Time (fs)  100  200  (b)  Figure 2.15: (a) The simulated spectra of the input (blue) and the circulating (red) pulse of Fig 2.14; (b) the duration of the initially transform limited input pulse is 32 fs (blue). The circulating FWHM pulse duration ∆t, if it were only limited by the resonant, amplified spectrum, would be 81 fs (black). The additional dispersion by the cavity imparted on the circulating pulse further increases the steady state pulse duration to 108 fs (red).  The phase that the cavity imparts on each comb element is, θ(φ) = tan−1 where φ =  ω c d(ω).  ric rcav sin φ 1 − ric rcav cos φ  (2.50)  Thus, the frequencies that are not resonant have an  additional frequency dependent phase. Accounting for this phase can further increase the intracavity pulse duration as shown in Fig 2.15. Higher order dispersive terms, such as third order dispersion (TOD) can also affect the resonance map. The TOD causes a third-order change in the resonance position as a function of cavity length as shown in Fig 2.16. 34  820  820  815  815  810  810 Wavelength (nm)  Wavelength (nm)  2.4. Resonance Map  805 800 795  800 795  790  790  785  785  780 ï5  0 Length (L + h ) 0  0  (a)  780 0  5  200  400  ï3  x 10  600 800 Pcav/P0  1000  1200  (b)  700  0.4  600  0.3 0.2 Phase (/ rad)  500 Pcav/P0  805  400 300  0.1 0 ï0.1 ï0.2  200  ï0.3 100 ï5  ï0.4 0 Length (L0 + h0)  (c)  5  780  ï3  x 10  790  800 810 Wavelength (nm)  820  (d)  Figure 2.16: Same (simulated) conditions as Fig 2.14, but with GDD = 0 fs2 and TOD = 900 fs3 . The resonant FWHM bandwidth is the same as Fig 2.14.  Although the TOD is chosen in this example so that the resonant spectrum has the same bandwidth as Fig 2.14, it leads to different effects on the peak intracavity power. First, the total amplification is higher, mainly due to the spectral components with initially higher amplitudes being resonant. Also, the symmetry of the phase imparted on the field by the cavity leads to a pulse duration that is not significantly increased as shown in Fig 2.17.  2.4.4  Nonlinear Intracavity Effects  The nonlinear dynamics that can occur within the cavity also change the mode spacing and pulse duration [64]. Additionally, power scaling of ultrashort pulses have been shown to be limited by the high reflectivity mirror 35  2.5. Locking ωc 1 1200 0.8  800  Amplitude  Amplitude  1000  600  0.6  0.4  400 0.2  200  780  790  800 810 Wavelength (nm)  (a)  820  0 −200  −100  0 Time (fs)  100  200  (b)  Figure 2.17: (a) The seed spectrum (blue) and amplified spectrum (red) for the EC of Fig 2.16; (b) the input pulse (blue) duration is 32 fs, whereas the circulating pulse has a FWHM pulse duration of 81 fs, due only to the limited amplified bandwidth (black) and is not further increased by the dispersion imparted on the field by the cavity (red).  damage threshold [65] and high peak power [66]. Although self phase modulation does not change the laser frequency comb spacing or absolute position, the intensity dependent index of refraction creates a time-dependent cavity length shift [67]. These issues will be discussed further in Sec 4.2.2 and Sec 6.2.  2.5  Locking ωc  The maximum enhancement occurs when the incident field is resonant with the cavity. We maintain the resonance by locking the cavity to the laser using a (somewhat modified) Pound-Drever-Hall (PDH) scheme [68, 69]. Normally, the phase of the light incident on the EC is modulated to generate the error signal. We have found that modulating the EC length, thereby modulating the intracavity field phase, is analogous. The EC length is modulated by applying a sinusoidal amplitude to a PZT attached to an intracavity mirror. Although we are amplifying a broad spectrum, the carrier frequency ωc 36  2.5. Locking ωc must first be stabilized. It will then be shown, by the nature of the FFC, that the entire spectrum can be amplified by locking three frequencies. 0.8 0.6  ∼ ∆Ω  Signal (arb u)  0.4 0.2  Ω  0 ï0.2 ï0.4 ï0.6 ï0.8 ï0.02  ï0.01  0 0.01 Length (L0 + h0)  0.02  Figure 2.18: Simulated error signal generated by the Pound-Drever-Hall locking scheme as the cavity is swept through a resonance. The linear portion is dependent on the cavity linewidth ∆Ω; the width of the signal (the sidebands) is determined by the modulation frequency Ω. The resonance occurs when the cavity length is L0 , or at the zero crossing of the error signal. The zero crossing is the desired lock point, where the amplitude of the error signal dictates the locking direction.  The PDH lock requires that the phase of the light is modulated to derive the error signal. The modulation, in essence, provides the derivative of the resonance of the enhancement cavity. The derivative tells the controller the correct direction to re-establish resonance. The modulated light field is given by [69, 70], E = E0 ei(ωc t+β sin Ωt) β iΩt ≈ E0 eiωc t 1 + e − e−iΩt 2  (2.51)  where β and Ω are the modulation strength and frequency, respectively (and β  1). There are now three frequencies: the carrier ωc , and two side-bands  ωc ± Ω. The modulation frequency Ω should be chosen to be much larger than acoustic frequencies (  2π × 50 kHz), but in a range where the PZT 37  2.5. Locking ωc still has sufficient amplitude response (< 2π MHz). The optical signal detected is the sum of the fields transmitted through and reflected from the EC input coupler. The transmitted and reflected fields are (respectively), Etrans = itic Ecav rcav = −E0 Eref  = E0 ric  t2ic rcav 1 − ric rcav eiφ  (2.52a) (2.52b)  where the reflection is assumed to be off the intracavity face of the input coupler (IC). The total field ET (ω) that comes from the IC is the sum of these two fields, ET (ω) =  −E0 2 (rcav − ric ) + ric rcav eiφ − 1 1 − ric rcav eiφ  (2.53)  Because of the modulation, the three detected frequencies are, Edet = ET (ωc )eiωc t +  β β ET (ωc + Ω)ei(ωc +Ω)t − ET (ωc − Ω)ei(ωc −Ω)t .(2.54) 2 2  The power on the photodetector is Pdet ∝ |Edet |2 , yielding, Pdet ∝ |ET (ω)|2 + |ET (ω + Ω)|2 + |ET (ω − Ω)|2 + β [ET (ω)ET∗ (ω + Ω)e−iΩt − ET (ω)ET∗ (ω − Ω)e−iΩt + 2 ET∗ (ω)ET (ω + Ω)eiΩt − ET∗ (ω)ET (ω − Ω)eiΩt ] −  (2.55)  (β/2)2 ET (ω + Ω)ET∗ (ω − Ω)e2iΩt + ET∗ (ω + Ω)ET (ω − Ω)e−2iΩt .  The constant terms are not observed by the photodetector because the signal is AC coupled, and the terms that oscillate at 2Ω are a factor of β/2 smaller than those at Ω and can be ignored in this analysis. The field at the  38  2.6. Locking Ω0 modulation frequency becomes, Pdet ∝ β(R [ET (ω)ET∗ (ω + Ω)] cos Ωt + I [ET (ω)ET∗ (ω + Ω)] sin Ωt  −R [ET∗ (ω)ET (ω − Ω)] cos Ωt − I [ET∗ (ω)ET (ω − Ω)] sin Ωt) (2.56)  where R and I are the real and imaginary parts. The phase of the modulation frequency, Ω, relative to the local oscillator in the mixer determines  whether the real or imaginary terms are the detected signal. We tune the phase so that we take the imaginary part. The observed error signal becomes, c  ∝ βI [ET (ω)ET∗ (ω + Ω) − ET∗ (ω)ET (ω − Ω)] ,  (2.57)  as shown in Fig 2.18. The generation of the error signal requires some electronics, as shown in Fig 2.19. The function generator modulates the EC length at frequency Ω, which modulates the phase of the intracavity field. The signal level needs to be a certain amplitude in order to power the mixer at the local oscillator (LO) port. In this case, the large signal level causes the loop to oscillate, preventing stable locking. Therefore, a variable attenuator is used to attenuate the signal that is fed into the PZT to independently control the modulation amplitude. The frequency of the function generator is controlled so as to have the correct phase so that the detected error signal for the carrier frequency ωc matches that of Fig 2.18. In our experiments, we use the same PZT mirror for modulation as we do for locking ωc . To this end, the loop filter signal is amplified in the PZT driver and added to the modulation signal; both signals are sent to the PZT.  2.6  Locking Ω0  Locking the offset frequency is commonly done in a self-referencing technique [71]. However, the intracavity phase causes the cavity to have its own effective offset frequency as shown in Eq 2.47. Thus, we are motivated to  39  2.6. Locking Ω0  AOM driver  -  loop filter  mixer3  AOM mixer2 mode-locked laser Cavity  PDC  Phase shifter  PDL  PZT PDH IC  +  driver  loop filter e  VA  SPL2  LO  mixer1 RF  SPL1 FG  Figure 2.19: The setup for locking the laser and cavity to maintain high intracavity power. To lock the cavity length, the carrier frequency ωc is detected by the photodetector P Dc , which is the radio frequency (RF) signal for the mixer. The function generator (FG) signal is split (via SPL1 and SPL2 ) and sent to the variable attenuator (VA) and the mixer (mixer1 ), where it is used as the local oscillator (LO) reference signal. The derived error signal, e, is sent to the loop filter, the output of which is amplified by the driver and fed into the adding circuit. The adder (+) combines the reference signal and the cavity length compensation signals to send to the piezo-electric transducer (PZT). The PZT is used to maintain resonance of ωc . Once the ωc lock is engaged, Ω0 can be stabilized. The splitter (SPL1 ) sends the FG signal to the phase shifter, which is necessary because each mixer needs the LO signal to have the correct phase for the error signal. The low and high ends of the spectrum, ωL and ωH respectively, are mixed down from the reference frequency via mixer2 and mixer3 , and are subtracted (-) to generate the signal to lock Ω0 . The Ω0 lock also has a loop filter, where the output is fed into the acousto-optic modulator driver (AOM). 40  2.6. Locking Ω0 lock the offset frequency of the laser to that of the cavity. With ωc already locked, stabilizing the laser offset frequency stabilizes the entire intracavity  Signal  spectrum.  −0.02  −0.01  0 Length (L0 + λ0)  0.01  0.02  Figure 2.20: Simulated PDH error signal generated from three different optical frequencies ωL (red), ωc (blue), and ωH (violet) as the cavity length is swept through the cavity length L0 , that is when ΩF SR = Ωrep . Due to Ω0 (see Fig. 2.13(a)), the resonances (at the large zero crossing) are not aligned at the same cavity length position. Tuning Ω0 can align the resonances of the three optical frequencies.  2.6.1  Using the Cavity to Lock Ω0  Once the cavity length lock has been established, the other laser modes can be resonant in the cavity by tuning Ω0 . As shown in Fig 2.12, Ω0 causes the resonance condition to become frequency dependent, but in a nearly linear manner. This ‘rotation’ of the resonance condition can be measured by the ωL and/or ωH error signals [72], as shown in Fig 2.20. It has been found that the passive stability of Ω0 is dependent on the operating conditions of the mode-locked oscillator [73, 74]. Therefore, although we actively correct for fluctuations in Ω0 , we also tune the laser parameters to improve passive mode-lock stability, which can help lock performance. To lock Ω0 , we actively control the acousto-optic modulator (AOM) controlling the pump laser power [75]. 41  2.6. Locking Ω0 Because of cavity dispersion, it may not be possible to have the three frequencies simultaneously resonant, as shown in Fig 2.14. In this case, only two frequencies can be used for locking. In this example, the cavity length could be locked using ωc = 2πc/795 nm. Because we are not using ωH , the PDH and mixer2 in Fig 2.19 are not used, and the signals are not subtracted (-) to form the error signal. In order to stabilize Ω0 with the cavity, the optical frequency that could be used is ωL = 2πc/805 nm, and the signal from mixer3 is fed directly in to the loop filter controlling the AOM driver. The electronics used for locking ωL (and/or ωH ) is similar to locking ωc in that the PZT has created the modulation necessary for generating the PDH error signal, but first the ωc lock must be engaged. The phase of the function generator at the local oscillator port of the ωL /ωH mixer must still have the correct phase relative to the detector signal. For various reasons, such as the cable lengths used to connect the electronics, the phase of the signal for ωL /ωH will not match that of ωc . The phase of the local oscillator (the function generator) must therefore be adjusted by a phase shifter. If we are locking both ωL and ωH , then the difference of the two signals becomes the Ω0 error signal, and is sent to the loop filter. The output of the loop filter is amplified and sent to the AOM to control Ω0 . In this manner, by engaging both the ωc and the Ω0 loop filters, we can maintain stable, high intracavity power.  42  Chapter 3  Characterizing the Cavity The enhancement cavity passively amplifies the output of the mode-locked Ti:Sapphire laser oscillator; the intracavity power can reach two to three orders of magnitude higher than the output of the laser. In order to predict the peak intracavity power, we must first characterize the cavity. The previous chapter discussed the interaction of the laser with the cavity; this chapter uses that knowledge to measure the photon lifetime, spatial mode coupling efficiency, and the intracavity dispersion. These three factors are required in order to predict the amplification factor of the enhancement cavity.  3.1  Finesse Measurement  As discussed in Chapter 2, the reflectivity of the mirrors used for large amplification is near unity. Thus, after a single pass, the power loss is negligible and difficult to measure, which is not desirable when experimentally determining the mirror reflectivity. However, it is also shown that the enhancement cavity (EC) mirror reflectivity can be measured by the finesse F.  Ideally, the additional scattering losses of the input coupler are small  such that the sum of the transmission and reflection are unity, Tic + Ric = 1. From this, we can separately measure the transmission through the input coupler to estimate its reflectivity. The remaining mirror reflectivities can be combined, using the relationship of the field reflectance to the reflectivity |r2 | = R, such that Rcav = RM1 RM2 · · · RMn (where there are n intracavity  43  3.1. Finesse Measurement mirrors) so that (Eq 2.37), √ π 4 Ric Rcav √ . F≈ 1 − Ric Rcav Unfortunately, we cannot measure the reflectivity of a single intracavity mirror in this manner, but we can measure the average mirror reflectivity. Once the finesse has been determined for a set of cavity mirrors, we can use this to measure any additional losses that may occur (beyond the manufacturer’s specifications). The additional losses can come from degradation due to extended use at high power, dust contaminants, or poor alignment. For high finesse cavities, an accurate measurement of the mirrors’ reflectivities is the photon lifetime, τcav , defined as the number of intracavity round trips, N , taken to reach 1/e of its initial intracavity power Pi , PN Pi  = = =  1 e Ric Rcav  N  Ric Rcav  2N  .  (3.1)  Solving for N , the number of round trips it takes then is, N≈ where the approximations ln  F 1/2 √ ≈ 2π 1 − Ric Rcav  √  Ric Rcav ≈  √  Ric Rcav − 1 and  (3.2) √ 4  Ric Rcav ≈ 1  have been made. Because each round trip takes 2π/ΩF SR , the finesse is [60] F = ΩF SR τcav .  (3.3)  The long photon lifetime in high finesse cavities is measured using cavity ring-down.  44  3.2. Cavity Ring-down Measurement  3.2  Cavity Ring-down Measurement  To make a measurement using the analysis above, a switch would be required to turn off the beam incident on the input coupler. Although such switches do exist [76], we would need to insert this switch only for the measurement of the finesse. An alternative method, requiring little change to the normal cavity operating setup, is to measure the cavity ring-down [77, 78].  3.2.1  Theory of Cavity Ring-down  When sweeping the cavity length through a resonance, the intracavity power quickly builds to a maximum value as the cavity moves on resonance; as the cavity moves off resonance, the power decays. In a high finesse cavity, the resonant linewidth is narrow, and quickly changing the cavity length can move the cavity on and off resonance in much less time than the cavity photon lifetime. Therefore, scanning the cavity length through a resonance can replicate the effects of a switch. To sweep the cavity through a resonance quickly, we have a mirror mounted on a PZT (see Fig 3.1). By linearly ramping the mirror position, the cavity length is a function of time such that d(t) = dT + 2vt where dT is the total cavity length, and v is the velocity of a cavity mirror [79]. The initial phase of the field at t = 0, with optical frequency ω, at the output coupling mirror is  ω c d1 .  From this mirror, the beam travels an addi-  tional distance d2 + d3 , plus the distance that the PZT mirror has moved, vt. Because the beam is nearly at normal angle of incidence on the mirrors, the reflected beam from the PZT mirror becomes Doppler shifted by 1 − 2v c .  For the N th roundtrip, the phase becomes, φN  =  ω ω 2v ω d1 + (d2 + d3 + vt) + 1− (d4 + vt + d1 ) c c c c ω 2v ω 2v 2 + 1− (d2 + d3 + vt) + 1− (d4 + vt + d1 ) + · · · c c c c ω 2v N −1 ω 2v N + 1− (d2 + d3 + vt) + (d4 + vt + d1 ). 1− c c c c (3.4)  45  3.2. Cavity Ring-down Measurement  PZT  v  IC d4  lens  ω  slit  r2 OC PD  d1  d3  r1 grating  d2 (a)  (b)  Figure 3.1: (a) Schematic of the finesse measurement of a bowtie EC with a single input frequency ω. The light enters through the input coupler (IC), and a small amount is transmitted through the output coupler (OC) on to the photodetector (PD); the cavity length is ramped by the PZT at velocity v. The total cavity length is dT = d1 + d2 + d3 + d4 ; the phase induced by the intracavity mirrors (such as a reflection phase shift) is ignored in this derivation as it leads to a constant phase. (b) For the case of a mode-locked laser and an enhancement cavity that can support femtosecond pulses, the bandwidth incident on the PD is limited by a grating and slit.  Ignoring higher orders of v, suitable when φN ≈  v c  1, the phase is simplified to,  ω 2v N −1 d1 + N (dT + 2vt) − N dT + d4 + d1 c c 2  .  The amplitude of the field on the photodetector is attenuated by  (3.5) √  Ric Rcav  after each round trip. The field transmission of the input and output couplers is, respectively, itic and itoc . Therefore, the total field that is incident on the photodetector is, ∞  E(t) = −E0 tic toc  N  (Ric Rcav ) 2 eiφN (t) .  (3.6)  N =0  A simulated cavity ring-down signal is shown in Fig 3.2, demonstrating the exponential decay of the intracavity field. The Doppler shifting of the intra46  3.2. Cavity Ring-down Measurement cavity field causes an oscillation, or a ringing, about the average exponential decay with a time-constant τcav . Experimentally, this exponential decay is simple to fit and accurately measures the finesse. An alternative derivation of the field is given in the Appendix A. 0  1  10  Amplitude (a.u.)  Amplitude (a.u.)  0.8  0.6  0.4  0.2  −1  10  −2  10 0 0  5  10  15 Time (µs)  (a)  20  25  0  5  10  15 Time (µs)  20  25  (b)  Figure 3.2: Simulation of cavity ring-down for a high finesse enhancement cavity, excited by a single-mode CW laser. The total mirror loss is 0.04%, cavity length is 6 m (ΩF SR = 2π × 50 MHz), and input coupler transmission is Tic = 0.3%. The photon lifetime is τcav = 5.8 µs, for a cavity finesse of F = 1800 on (a) linear plot, and (b) log plot. The linear plot shows the exponential decay, but the large oscillations can make the photon lifetime difficult to measure experimentally. In a logarithm plot, the oscillation amplitude is constant and the decay is linear, allowing for an accurate fit to find the decay constant. The oscillation frequency increases with time due to the increasing Doppler shift as the field circulates in the cavity.  The analysis above can include multiple frequencies ωm of a femtosecond (fs) mode-locked laser interacting with an enhancement cavity (EC). Because the cavity is designed to support femtosecond pulses, it is referred to as a femtosecond enhancement cavity (fsEC). The mirror reflectivity is dependent on the optical frequency R → R(ω); therefore, the finesse becomes  a function of frequency. In addition, intracavity dispersion and the offset frequency cause a frequency dependent resonance condition, dT → dT (ω).  The total cavity length dT is set such that ΩF SR (ωc ) = Ωrep . The total field  47  3.2. Cavity Ring-down Measurement can be written as, ∞  M  E(t) = −  E(ωm )tic (ωm )toc (ωm ) m=1  Ric (ωm )Rcav (ωm )  N 2  eiφN (ωm )  N =0  (3.7) where m indexes the laser mode number. In this manner, a high finesse femtosecond enhancement cavity (fsEC) can be used for ring-down spectroscopy because of the increased sensitivity across a broad spectrum [80, 81].  3.2.2  The Ring-down Measurement  The fsEC had a small mirror with a fast PZT (normal operation was for locking the cavity length). This same mirror was used to linearly ramp the cavity length for the ring-down measurement. 0  10  Broad spectrum  Narrowed spectrum  ï1  Increasing ramp  ï1  10  ï2  Decreasing ramp  0  5  10 Time (µs)  (a)  Amplitude (V)  Amplitude (V)  10  10  15  0  5  10 Time (µs)  15  (b)  Figure 3.3: Log plots of the ring-down measurement of the cavity using the (a) full spectrum (∆λ = 8 nm), and (b) spectrum narrowed (∆λ = 1 nm) by spatially dispersing the beam transmitted by the EC OC, performed in air. In (a) dispersion caused a difference in the decay time (fit by straight lines) when applying a decreasing (blue) or increasing (red) ramp; limiting the bandwidth minimized these effects as shown in (b) (both ramp directions shown).  An example of a ring-down measurement is shown in Fig 3.3. The measurement was performed in air so that we had full control over the alignment 48  3.3. Laser Output and Cavity Coupling of the input beam and the intracavity mirrors. The intracavity group delay dispersion (for air, GDD ≈ 20 fs2 /m, much larger than that of the mirrors,  discussed in Sec 3.4) caused a frequency dependent resonant EC length, as discussed in Sec 2.4. As the EC length was swept in two different directions, the different frequency components were resonant at different times, as shown in Fig 3.3(a). The measurement of the decay time requires that all frequencies are simultaneously resonant, meaning that Ω0 and dispersion can affect the result. A grating was used to disperse the beam and limited the optical bandwidth on the photodetector; the ring-down signal was then independent of the cavity length scan direction. The ring-down time constant in Fig 3.3(b) was found to be τcav = 5.5 µs, or for a free spectral range ΩF SR = 2π ×50 MHz, the measured finesse was 1700. The Tic was measured  to be 0.3% by a single-pass measurement, therefore the reflectivity was assumed to be Ric = 99.7%. The average reflectivity of the six intracavity mirrors was then RM > 99.99%, which was the mirror design specification.  3.3 3.3.1  Laser Output and Cavity Coupling Spatial Mode Coupling Efficiency  In order to efficiently couple the laser output to the cavity, the incident beam profile must match the cavity eigenmode. Because of the oblique incident angle on the focussing mirrors, the intracavity mode had a slight astigmatism and the mode orders were non-degenerate. The lowest loss and hence highest power mode was the TEM00 . The incident beam, however, was a non-ideal beam with an M 2 > 1. The coupling efficiency was then a measure of the mode-matching of the incident laser beam to the cavity eigenmode. The signal used to measure the coupling efficiency was the reflection zero-frequency (DC) signal from the input coupler (IC). For a schematic of the experimental setup, see Fig 3.6. As shown in Fig 3.4(a), poor modematching and cavity alignment led to populating higher order modes [82]. By improving mode-matching and cavity alignment, we could couple the  49  3.3. Laser Output and Cavity Coupling  1  1  0.8  HO modes  Amplitude (arb u)  Amplitude (arb u)  0.8  0.6  0.4  T EM00 0.2  0.6  0.4  T EM00 0.2  Ωrep = ΩF SR 0  ï1.5  ï1  ï0.5 0 0.5 Distance (arb u)  1  1.5  (a)  2  0 ï2  ï1.5  ï1  ï0.5 0 0.5 Distance (arb u)  1  1.5  (b)  Figure 3.4: Cavity reflection signals showing (a) poor mode coupling and poor cavity alignment, which led to a shallow cavity absorption signal and shows coupling to higher order (HO) modes (peaks at red arrows); (b) good coupling and alignment to only the TEM00 cavity eigenmode (peak at black vertical arrows). The reflected signal can only reach zero when the cavity is impedance matched. For setup refer to Fig 3.6, where the signal is generated from PD2. The sweep rates of the two measurements differ, which led to different mode spacing.  incident beam almost completely to the TEM00 cavity mode. Because the cavity was not impedance matched (when Rcav = Ric ), the “cavity absorption” signal did not reach zero. The lowest cavity reflectivity is when ΩF SR = Ωrep .  3.3.2  Spectral Coupling  An example of the laser output and intracavity spectra are shown in Fig 3.5. The peak of the laser spectrum was tuned to match the minimum of the cavity dispersion (discussed below) to allow for the maximum supported bandwidth. The laser spectrum was centred at λc = 793 nm with a FWHM bandwidth of ∆λ = 15.8 nm. As shown in Fig 3.5(b), the input spectral shape (blue curve) closely follows a sech2 , expected for the output of a mode-locked laser. The amplified spectrum, when the cavity length and laser offset frequency were locked, was slightly narrowed due to dispersion 50  3.4. Dispersion Measurement  1  5 laser cavity  ï5 Amplitude (dB)  Amplitude (arb u)  0.8  laser cavity sech2 fit  0  0.6  0.4  ï10 ï15 ï20 ï25 ï30  0.2  ï35 0 740  760  780 800 Wavelength (nm)  820  840  ï40 740  (a)  760  780 800 Wavelength (nm)  820  840  (b)  Figure 3.5: The (a) linear, and (b) log plots of laser output spectra (blue) and intracavity spectra (red). The laser bandwidth was (FWHM) ∆λ = 15.8 nm with a near sech2 spectral shape (black), and the amplified cavity bandwidth was ∆λ = 14.8 nm. The spectrum extrema were not amplified due to residual cavity dispersion. There was a > 90% spectral overlap of the initial and amplified spectra.  from the cavity mirrors and the intracavity spectral shape differs from the input spectrum. The peak of the locked spectrum was at λc = 792 nm with a bandwidth of ∆λ = 14.8 nm. The spectral overlap was > 90% that of the incident spectrum. However, outside of the laser spectral FWHM bandwidth, there was poor amplification.  3.4  Dispersion Measurement  In order to store the entire femtosecond (fs) pulse bandwidth in the enhancement cavity, the resonance condition must be independent of the optical frequency across the spectrum. However, intracavity dispersion limits the simultaneously resonant spectrum and increases the pulse duration. The difference in the resonance condition for a reference frequency ωref and mea-  51  3.4. Dispersion Measurement sured frequency ω is11 , ∆d(ω) = 2πc  Ω0 1 1 − Ωrep ω ωref  +c  Φ(ω) Φ(ωref ) − . ω ωref  (3.8)  Because dispersion changes the resonance condition, measuring the resonance condition displacement ∆d(ω) yields the intracavity group delay dispersion (GDD, cf Sec 2.4.2), GDD(ω) =  d2 ω ∆d(ω) . dω 2 c  (3.9)  Other methods for measuring the mirror dispersion have been realised [83, 84], as well as the measurement of nonlinear intracavity processes [67]. This section discusses the measurement setup and results of a simple dispersion measurement [61].  3.4.1  Setup  To measure the enhancement cavity’s GDD through the change in resonance condition ∆d(ω), the cavity was excited with a femtosecond frequency comb (FFC) generated by a mode-locked Ti:Sapphire laser. The laser’s repetition rate was 50 MHz and its spectrum had a full width at half maximum (FWHM) bandwidth of ∆λ ∼ 20 nm12 . The experimental setup is shown in Fig 3.6.  The signal used in the measurement was derived from the cavity reflection off the input coupler (IC) mirror and was separated into two spectrally resolved branches with one being the reference branch and the other the measurement branch. Each branch was spectrally dispersed with a grating at 1200 lines/mm. Following the gratings, a portion of the beam was coupled into an optical spectrum analyzer (OSA) for measurement of the optical frequency. The remaining signals in each branch were individually focused onto two photodetectors. The cavity length was scanned with a 11  Portions of this section appear in [61] The spectrum was measured in wavelength with an optical spectrum analyser (OSA), where the conversion to optical (angular) frequency bandwidth is ∆ω = 2πc ∆λ λ2 12  52  3.4. Dispersion Measurement  APD lens  Locking electronics  f-2f  filter KTP  PCF PCF  fs mode-locked laser  Cavity OSA  IC  PZT PD1  Scope  PD2  Figure 3.6: Experimental setup for dispersion measurement. A mode-locked Ti:Sapphire laser was coupled into a six mirror enhancement cavity that was under vacuum (the cavity pressure Pcav < 2mTorr). Ω0 was stabilized by the prism-based f-2f interferometer: KTP frequency doubling crystal; PCF photonic crystal fibre; filter was bandpass 532nm, and APD the avalanche photodetector used for detection of the heterodyne beat of the fundamental 532 nm and frequency doubled 1064 nm light. The cavity reflection from the IC was separated into two spectrally resolved branches with gratings. PD1 the reference branch photodetector; PD2 the measurement branch; IC cavity input coupler (0.25%); the small mirror attached to the PZT was used to sweep the cavity length. The scope was used to measure the (time) delay between the resonance conditions of ωref and ω as the cavity length was swept. The reference and measured frequencies, ωref and ω respectively, were measured with the optical spectrum analyser (OSA), coupled through fibre. To test the GDD of an optic, it was placed in the cavity beam path.  53  3.4. Dispersion Measurement mirror attached to a piezo electric transducer (PZT) that was driven open loop (with no active feedback to control the mirror’s velocity) with a linear ramp a distance of 2500 nm in one cycle - over three times the free spectral range (FSR) - and the two cavity reflection signals were displayed on an oscilloscope. It was important to ensure the PZT moved linearly through its sweep for proper calibration. The time delay between the reference and measurement resonance peaks appearing on the oscilloscope was converted to a distance ∆d by calibrating the time delay against one FSR (i.e., one optical wavelength) at ωref . In our case, the scan rate results in a calibration factor of 1.728 ± 0.008 nm/µs. −10  24 22  −20  20 Ω0 (MHz)  Signal (dB)  −30 −40 −50  18 16 14 12  −60 −70 0  10 5  10 Frequency (MHz)  (a)  15  20  8 0  100  200  300 400 Time (s)  500  600  (b)  Figure 3.7: (a) The signal used to measure Ω0 was typically > 30dB (at 100kHz resolution bandwidth) for good locking; (b) measurements of Ω0 over the period of 10 minutes. This drift demonstrates that Ω0 can change rapidly over the course of the dispersion measurement. Locked, Ω0 had a standard deviation of a few kHz, unlocked it was on the order of a few MHz. This was the dominant source of error in this experiment; locking the offset frequency greatly improved the experimental results.  As indicated in Eq 3.8, the change in resonant condition was dependent on both Ω0 and Ωrep of the FFC. At first, we simply measured Ω0 and Ωrep , but fluctuations in the free-running Ω0 (over the duration of the measurement) were found to corrupt the GDD measurement, particularly when net cavity GDD was close to zero. A sample measurement of Ω0 is shown in 54  3.4. Dispersion Measurement Fig 3.7. To minimize this error, Ω0 was detected and subsequently stabilized using a prism-based f to 2 f referencing interferometer [57]. A key point is the actual value of Ω0 is not necessary to determine the GDD. If the FFC used to probe the cavity is constructed from a fibre laser oscillator, an f to 2 f interferometer is likely not necessary due to the inherent stability of the offset frequency in these systems [85]. The repetition rate of our FFC was sufficiently stable (with a free running drift of about 10 Hz, or 0.2 ppm) that it did not significantly affect the error in ∆d. After adjusting the repetition rate to locate the cavity on the central fringe (via a PZT-mounted mirror) it remained free running during the measurement. Initially, an “empty” enhancement cavity was measured. It consisted of five highly reflecting (R > 99.99%) and low GDD mirrors in an evacuated (< 2 mTorr) chamber. With an input coupler of T = 0.25% this cavity had a finesse of F ≈ 2000. For this case, a representative data set for  ∆d is shown in Fig 3.8 on the left axis. Also shown in this figure is the  resulting GDD, which is calculated by stitching together polynomial fits (over a 5 nm bandwidth) of overlapping ≈ 0.4 nm intervals via spectral  collocation methods [86]. This method of calculating the GDD is called the limited bandwidth fit, or segmented fit. By design, this cavity gave an extremely low and flat GDD of (≤5 fs2 ) for a bandwidth of ∆λ ∼ 40 nm, centred at λ0 = 790 nm. We measured the GDD over a total bandwidth of  60 nm, or 20 dB down from λ0 . While not necessary in our application, if a larger spectral range needs to be characterized, a broadened comb could be used to excite the cavity.  3.4.2  Results  To confirm the accuracy of our GDD measurement technique we performed an additional set of measurements of optical elements with known values of GDD. First a 0.45 mm piece of sapphire was inserted into the cavity at the Brewster angle. To determine the GDD of the sapphire plate, the difference in ∆d(ω) between the cavity with and without the sapphire plate was measured, which was then fit to a polynomial where the GDD was  55  3.4. Dispersion Measurement calculated via Eq 3.9. As the GDD of the sapphire is a smooth function of λ and has a value significantly greater than the empty cavity, both a third order polynomial fit to the entire data set as well as the previously used spectral collocation technique were employed. These results are shown in Fig 3.9(a). Both data analysis methods show excellent agreement with the Sellmeier equation prediction [87]13 . As a second check, we evaluated the GDD of a 2.2 mm piece of fused silica in a similar fashion and again obtained excellent agreement with predicted values of GDD [88] with results shown in Fig 3.9(b).  Figure 3.8: Group delay dispersion (GDD) measurement of an evacuated six mirror cavity. The measured delay (black points, left axes) represent the raw data collected via an oscilloscope and were converted to a path length distance through a calibration of the free spectral range. The resulting GDD (red curve, right axis) is calculated via Eq. 3.9 and spectral collocation methods in numerical analysis [86]. The spectral limits of the measurement are due to the finite width of the FFC. The cavity mirrors were designed for low GDD centered at 790 nm. The calibration factor of 1.728 nm/µs was used to convert the measured time delay to cavity length.  As with nearly all GDD measurement techniques, the GDD itself is found 13  The Sellmeier equation is given as, n2 (λ) − 1 =  A1 λ2 A2 λ2 A3 λ2 + 2 + 2 − B1 λ − B2 λ − B3  λ2  (3.10)  where the coefficients Ai , Bi are given in the reference.  56  3.4. Dispersion Measurement indirectly from the raw data by curve fitting and/or taking numerical derivatives. With these processing procedures it is important to evaluate the resulting error on the obtained value for GDD. In Fig 3.10 we show the upper and lower confidence intervals of the GDD arising from the uncertainty in the calculation via the spectral collocation (limited bandwidth) fit, equivalent to less than ±2 fs2 over ≈ 50 nm of bandwidth. The Sellemeir prediction falls within the uncertainty interval except at the extreme high wavelength limit of our measurement. As a further demonstration of the capabilities of this dispersion measurement approach, we also measured a cavity mirror with oscillating dispersion and an intracavity optic with large GDD. In the former, we selected one mirror from a pair of oscillation compensated GDD mirrors fabricated by Layertec (P/N102267) and it was used to replace a mirror from the cavity measured in Fig 3.6. Shown in Fig 3.10(b) is our measurement result compared with a conventional GDD single pass measurement, provided by the manufacturer for a normal incidence angle. Accounting for the slight off-normal incident angle (∼ 5◦ ) for our measurement due to the intracavity geometry, this result agrees favourably with the expected value. Moreover, it indicates our technique can resolve rapidly oscillating values of GDD. Lastly, a 6.6 mm thick piece of BK7 glass and a 14 mm piece of SF11 glass with predicted GDD of ≈ 360 fs2 and ≈ 3200 fs2 respectively were measured. Over a bandwidth of 50 nm, our measurement differed from the Sellmeier prediction by at most 5%.  3.4.3  Sources of Error  Once the FFC’s offset frequency is stabilized, the largest source of error in determining the GDD is uncertainty in the measurement of the relative time delay between the cavity resonances at ωref and ω. This includes factors such as our ability to locate the peak of the resonance lineshapes amid residual dispersion due to the finite spectral resolution at the detectors, resolution of the oscilloscope, and proper calibration measurement. The latter could potentially be improved by using an encoded PZT for shorter dis-  57  3.4. Dispersion Measurement  (a)  (b)  Figure 3.9: (a) An intra-cavity measurement of a 0.45 mm thick piece of sapphire placed at the Brewster angle. The GDD is calculated both by a polynomial fit to the entire data set (right axis, red) and a limited bandwidth fit (right axis, green). Both show excellent agreement with the Sellmeier prediction (blue). (b) An intra-cavity measurement of a 2.2 mm thick piece of fused silica (also at the Brewster angle) analysed with similar methods to (a).  58  3.4. Dispersion Measurement  Limited BW fit uncertainty Limited BW fit Sellmeier Polynomial fit  GDD (fs2)  105  100  95  90  780  790 800 Wavelength (nm)  810  (a)  (b)  Figure 3.10: (a) The uncertainty in the limited bandwidth fit for the fused silica measurement; (b) the GDD of a cavity mirror with known oscillations in the GDD. The experimentally verified specifications given by the manufacturer are shown as a blue dashed line, our differential measurement with a segmented bandwidth fit is given by the solid green line.  59  3.5. Mirror Damage tance calibration. As indicated in Fig 3.8, for small GDD the relative delay changes on the order of 500 ns (a corresponding distance of 0.8 nm). Thus even at this minimal extreme, fluctuations in the differential physical path between the two branches in the experimental setup are insignificant. As the GDD increases, larger relative delays can be easily determined. However, once the relative delay reaches 25 µs (a distance of 42 nm), the delay measurements begin to suffer some error due to the finite bandwidth (0.5 nm) that is incident on the photodetectors which is manifested by a broadening of the cavity reflection signal. Aside from the GDD value itself, the cavity’s finesse also influences the delay measurement as a lower finesse causes the cavity reflection signal to broaden thereby making it harder to resolve the reflection peak at ω. For the finesse values considered here (>500), this effect was not an issue.  3.4.4  GDD Measurement Summary  The experimental method of measuring the GDD presented here can accurately measure the frequency dependence of the physical path length change of the resonance condition in an optical cavity. By exciting the optical cavity under test with a FFC, the change in resonance condition can be rapidly measured to provide the GDD over a large spectral region. The measurement is simple, robust, and has a high dynamic range with minimal error. Moreover, it only requires the FFC to have a stable offset frequency (its actual value is not needed) nor does the FCC need to be locked to the cavity under test. We have found it to be a valuable tool in optimizing fs enhancement cavities by minimizing the intracavity GDD. It could also be used in the design and fabrication of dispersion compensating mirrors.  3.5  Mirror Damage  The low dispersion, high finesse cavity allowed for the intracavity enhancement factor of > 900 when an input coupler transmission of Tic = 0.1% was used. The intracavity power was estimated to be > 600 W. However, for  60  3.5. Mirror Damage normal operating conditions of the EC, we typically used a Tic = 0.25%, which limited the intracavity power to 400 W. The lower average energy case was beneficial because the pulse duration was shorter, which allowed for a lower ionization population (discussed in Sec 4.1.1) and a more stable cavity lock when generating the EUV. Nevertheless, the high amplification is sensitive to the intracavity mirror reflectivity, which can degrade due to contaminants. At first, after maintaining high intracavity power for an extended period of time (on the order of 30 mins) - without introducing the gas used to generate EUV - the intracavity power would sometimes suddenly decrease as shown in Fig 3.11. It was found that a possible contaminant had been the use of Corning vacuum grease for the o-rings of the vacuum chamber. A possible mechanism for the degradation is that the vacuum grease outgassed enough to contaminate the mirrors, and once the mirrors had built up a sufficient layer of contaminants, the high intracavity power caused mirror damage. This mirror damage was recoverable by submerging the affected mirrors in acetone for 24 hours. This Corning vacuum grease has since been replaced by Apiezon vacuum grease, and we do not see mirror damage when in vacuum with high intracavity power of the fundamental field. However, we have still observed mirror degradation when creating the high harmonic light. The intracavity generation of the harmonics only allows for ∼ 4% of the EUV light to be coupled out of the cavity (see Sec 5.5),  while the other 96% hits the second intracavity focussing mirror. We have found that we can maintain high power of the fundamental beam and the harmonics for approximately ten minutes before the mirrors degrade. We have tried several types of highly reflecting mirrors in order to maintain high power. The first set of mirrors, by Advanced Thin Films (ATF) were again recoverable by submerging in acetone. However, these mirrors could not maintain high power for an extended period of time. Conversely, mirrors provided by Layertec could maintain a high power (although not as high as the ATF mirrors) for a longer period of time, but the recovery of the mirrors by acetone was not as reliable. Additionally, it was found that after several times of a mirror being damaged, the recovery was not as reliable. 61  Output Amplitude (V)  3.5. Mirror Damage  Figure 3.11: Measurement of the intracavity power. The 400 W average power (blue portion) suddenly decayed due to mirror damage (red portion). Unfortunately, the mirror damage was not optically observable so the damage mechanism is not well understood. It was found that the vacuum contaminants and the generated EUV light play a role in the mirror degradation.  The solution to mirror damage has been an ongoing pursuit in order to have an EC as a viable, long term option for generating high harmonics. One method of avoiding mirror damage by EUV is to use a Brewster plate, which can readily absorb the EUV before the second focussing mirror. The difficulty with inserting the Brewster plate into the beam path near the focus is two fold. In the linear regime, the optic has some dispersion, which we have shown to limit the attainable intracavity enhancement. Also, as will be discussed, the nonlinear index of refraction can inhibit cavity stability. A thin, 0.45 mm sapphire window was placed near the intracavity focus, but the above mentioned detrimental effects were readily observed: the peak intracavity power decreased and the long-term stability of the cavity was compromised. In spite of having a lower intracavity power with the insertion of the sapphire Brewster plate, the location where the beam passed through the sapphire readily glowed white. Long term exposure to high intracavity 62  3.5. Mirror Damage powers caused permanent damage to the sapphire plate. We have also found that oxygen, in combination of having the intracavity field present, can temporarily recover the mirror reflectivity. Although this mechanism is not well understood, it may imply that the oxide layer of the fused silica is susceptible to damage, and the presence of oxygen slows down, and even reverses the mirror damage.  63  Chapter 4  Ion Dynamics and Nozzle Design In this thesis, extreme ultraviolet (EUV) light is created via high harmonic generation (HHG) by focussing an optical driving field, amplified by a femtosecond enhancement cavity (fsEC), through xenon gas. During the generation of the high harmonics, the neutral, atomic gas can become significantly ionized. Because the generation process is strongly dependent on the neutral gas density, and because the generation process is much more efficient with atoms than with ions, this chapter investigates the atom and ion dynamics, including the ionization and electron-ion recombination rates, as well as the gas density and flow rate at the fsEC focus. Initially an atom enters the focus, delivered by a gas jet, and the ions are created by the strong driving field, where the ionization probability due to a single pulse is determined by the peak intensity and pulse duration. In the experimentally attainable conditions of this thesis, the ion density can be a significant fraction of the total gas density. Because the pulses are amplified by an enhancement cavity, the repetition period of the amplified pulses is the same as the laser (15 ns between pulses, or a repetition rate of 66 MHz). The repetition period gives a relevant timescale for the electron-ion recombination process. The recombination timescale is determined by the density of the neutral atoms, the ions, the free electrons, and the temperature of the free electrons. The temperature of the free electrons is found from the above threshold ionization spectrum. It is shown in this chapter that within the parameters of this thesis, a significant fraction of the ions remain after several cavity roundtrips.  64  4.1. Ionization Dynamics  cavity  FM  FM focus  Figure 4.1: The EC setup. The plasma is generated when the xenon (purple circle) is delivered to the EC focus, where the two 10 cm radius of curvature (ROC) mirrors tightly focus the amplified power to a radius of w0 = 12 µm. The neutral gas and ion densities are calculated; methods of delivering the gas to the focus are presented in this chapter.  In order to improve the neutral atom density and remove the remaining ions at the focus, we investigate the gas flow within several designs of the delivery system (the nozzle). A schematic of the setup for the femtosecond enhancement cavity (fsEC) is shown in Fig 4.1, where the gas is delivered by the nozzle to the intracavity focus. An analytic approximation is used for basic understanding, such as calculating the gas density and velocity at the exit of a simple nozzle; however, numerical simulations are needed to analyse our more realistic geometries. We compare the gas flow of three different nozzle designs, as well as variations on the dimensions of the nozzles, in order to optimize the density and minimize the time that xenon atoms remain within the focus.  4.1  Ionization Dynamics  An atom can be excited by a strong laser field, causing it to absorb multiple photons with total energy greater than its ionization energy; this process is referred to as above threshold ionization (ATI). ATI can be thought of in two regimes: perturbative and strong field. In the perturbative regime, an electron (typically a valence electron in its ground state) is excited by N 65  4.1. Ionization Dynamics photons to the continuum. Conversely, in the strong field case, the Coulombic potential is distorted enabling the valence electron to tunnel through the atomic potential with non-zero probability. The rates of these two processes differ, but in both cases the kinetic energy spectrum of the electrons is a series of peaks, separated by the photon energy of the laser field. The energy of the electrons is of interest as it dictates much of the electron-ion recombination and expansion dynamics of the gas, the neutral atom density, and thus the overall production efficiency of the EUV.  4.1.1  Ionization  The ionization rate model typically used is that of Ademnosov, Delone, and Kranov (ADK) [17, 89, 90]. The ADK model is often used because of its mathematical simplicity. However, it is limited to the strong field regime, which is when the Keldysh parameter γ =  Ip /2Up < 1, where Ip is the  ionization potential and Up is the ponderamotive energy of the field14 . The limitation stems from the assumption that the tunnelling time is negligible in terms of the optical cycle time, thus, the tunnelling rate is assumed to be due to the instantaneous electric field. The ADK tunnelling rate is [17], WADK = Cwp  4wp |w(t)|  2n∗ −1  exp −  4wp 3|w(t)|  (4.1)  where, wp = Ip /¯h n∗ = w(t) =  (4.2a)  IpH Ip  (4.2b)  qE(t) 2me Ip  (4.2c) ∗  C = 14  Up =  |E0 |2 2 4me ω1  22n n∗ Γ(n∗ + 1)Γ(n∗ )  (4.2d)  ≈ 0.933 × 10−13 λ2 I with I in W/cm2 , λ in µm  66  4.1. Ionization Dynamics Ip and IpH are the ionization energy of the gas used and of hydrogen, respectively, E(t) is the field, and Γ(n) is the Gamma function. 20  20  10  10  Keldysh parameter γ 4 3 2 1  0.5  Ionization Rate (sï1)  10  10  0  10  8γ  ï10  Yudin1 8�ω  10  Yudin ADK ADK ï20  10  11  10  12  10  13  10 Intensity (W/cm2)  14  10  15  10  Figure 4.2: Ionization rate of xenon. The model of Yudin et al [89] (blue) accounts for both the perturbative and strong field regimes - provided by the 8 photon ionization rate (black) and the ADK rates (red), respectively. The ionization rates in the Yudin model are substantially higher for intensities less than 5 × 1013 W/cm2 (dashed line) than that predicted by the ADK model.  A more general model that can be used in the multiphoton case has been developed by Yudin et al [89]. For the perturbative, multiphoton case where 1, the ionization rate scales as ∼ UpN where it requires N photons to  γ  ionize15 . This model accounts for tunnelling time, and thus can calculate the  instantaneous ionization rate through an optical cycle and arbitrary Keldysh parameter. As shown in Fig 4.2, the rate averaged over an optical cycle of the two models agree in the strong field regime (that is, when γ < 1, or the intensity is > 1 × 1014 W/cm2 for an 800 nm laser source). However, in the 15  For a Ti:Sapphire with λ0 = 800 nm laser ionizing xenon, N ≈ 8.  67  4.1. Ionization Dynamics perturbative regime, the ionization rate is much greater than that of ADK. The main difference between the two models appears in the exponential factor, WYudin ∝ exp −  2η0 e2 I(t) Φ(t) ¯hme ω13  (4.3)  where I(t) is the intensity envelope of the pulse and η0 is the impedance of free space, and, Φ(t) =  1 γ + sin ω1 t + 2 2  2  √ √ b+a 3 b−a √ sin |ω1 t| − γ √ ln c − 2 2 2 2 (4.4a)  a = 1 + γ 2 − sin2 ω1 t b = c =  (4.4b)  a2 + 4γ 2 sin2 ω1 t b+a +γ 2  2  +  (4.4c) b−a + sin |ω1 t| 2  2 1/2  ,  (4.4d)  and now γ can include the pulse envelope. The constant of proportionality can be found from the cycle-averaged result [89, 90]. It should be noted that in traditional, single pass, low repetition rate systems, the effect of ionization on the amplitude of the generated harmonics is known [91]. The active electron used in generating the high harmonics is the valence electron; its ionization leads to diminished harmonic amplitude. The remaining neutral atoms are given by, N (t) = 1 − e  t −∞  WYudin (t )dt  .  (4.5)  Longer duration pulses cause more of the atoms to become ionized within a single pulse (see Fig 4.3), leading to a decreased HHG yield. Ultrashort pulses are used in single-pass harmonic generation experiments to avoid ionization before the peak of the pulse arrives.  68  4.1. Ionization Dynamics  Population Fraction  1 0.8  Xe Xe+  0.6 0.4 0.2 0  −600 −400 −200  0 200 Time (fs)  400  600  Figure 4.3: Calculation of fraction of ionized xenon from a pulse centred at 800 nm, peak intensity 5×1013 W/cm2 . Black curve is gaussian envelope for 100 fs (solid curve) and 500 fs (dashed curve) FWHM pulse duration. Solid curve is approximate experimental condition, leading to 12% ionization per pulse.  4.1.2  Freed Electron Spectrum  The kinetic energy of a classical electron excited by a CW, single-mode field can be derived as follows. The electron is ejected at rest from the atom at time t0 by an applied field E(t) = E0 cos(ω1 t) = − ∂A ∂t where A(t) is the  vector potential. If it does not re-interact with the parent ion, then the momentum of the electron comes directly from the potential at its time of ionization p = eA(t0 ) [92]. The kinetic energy of the electron is, 1 p2 e A(t)2 EK = mv(t)2 = − (pA(t) + A(t)p) + . 2 2me 2me 2me  (4.6)  Averaging over a laser cycle of period 2π/ω1 , then16 , EK = 16  p2 + Up . 2me  (4.7)  The vector potential chosen such that A2 (t) = 12 , A(t) = 0.  69  4.1. Ionization Dynamics Solving the time-dependent Schr¨odinger equation (discussed in Sec 5.2.1) leads to a discretization of the spectrum, EK = N ¯hω − EIP =  p2 + Up 2me  (4.8)  where the atom absorbs N photons, and EIP is the ionization energy of the atom. 10 10 1  0 0  0  ï5ï5  -1  ï10 ï10 0 0  0.50.5 0.25 0.75 0.25 0.75 Time (2//t ) ) 1) Time (2//t Ionization time t0 1(2π/ω 1  Field (E0)  Edrift (Up) Edrift (Up)  5 5  Direct Direct Backscatter Backscatter E(t)E(t)  1 1  Figure 4.4: Classical kinetic energy of electrons (in units of Up ) gained from linearly polarized light, plotted as a function of ionization time t0 (in units of optical cycle). Energy can come directly from field (blue), or from rescattering off the ion (red); the field is shown as reference (black). The most energetic electrons directly ionized by the field is when the field amplitude is at a minimum (and hence the lowest ionization probability); backscatter can only occur at certain t0 when the field returns the electron to the ion.  The ponderamotive energy Up affects the electron motion only when the field is present. If the field is pulsed, then after the pulse leaves the interaction region and the field is zero, EK is the drift energy of the electron Edrif t , which is the kinetic energy that the electron has to drift away from the parent ion. As shown in Fig 4.4, the maximum Edrif t for electrons that gain their energy directly from the field occurs when t0 =  π 3π 2ω1 , 2ω1 ,  or when  the vector potential amplitude is at a maximum and the field amplitude is 70  4.1. Ionization Dynamics at a minimum. The maximum possible drift energy for the electrons directly ionized by the field is 2Up .  2  Amplitude (arb u)  10  0  10  −2  10  −4  10  0  10  20  30 40 Energy (eV)  50  60  Figure 4.5: Simulated spectrum of ATI electrons from an 800 nm source, 40 fs FWHM pulse duration and peak power 1 × 1014 W/cm2 (2Up ≈ 12 eV, 10Up ≈ 60 eV). The typical ATI features are the peaks separated by the photon energy (here, 1.55 eV), with the rapid rolloff at 2Up until a plateau out to 10Up ; this spectrum gives a mean electron kinetic energy of 3 eV. The parameters chosen are due to finite computational memory. The model used for the generation of this spectrum is discussed in Sec 5.2.1.  Electrons can attain a higher energy than 2Up through rescattering. Because the applied field is oscillatory, an electron that leaves the parent ion at t0 can return at time tr (where tr is a function of t0 ); if the electron recombines with the ion, then a photon is released, generating high harmonics. Conversely, an electron rescattering off the ion can change direction up to 180◦ . If the applied field is also changing direction at the moment of impact, then the electron is further accelerated up to 10Up . Not all electrons can return to the parent ion; the possible return times will be discussed further in Sec 5.1.1. The resulting spectrum is a plateau in the number of electrons emitted in energies between 2Up and 10Up [93]. A simulated spectrum is given in Fig 4.5.  71  4.1. Ionization Dynamics  4.1.3  Electron-Ion Recombination and Diffusion  Once the electron has been freed from its parent atom (now ion), the electron density is dependent on its motion. If it can return to the parent atom, then recombination can occur. Conversely, if there is sufficient energy for the electron to escape the focus, then the electron can diffuse out. Thus, the electron density is given by, ∂ρe− = −Γρe− − ∇ · Je− ∂t  (4.9)  where ρe− is the electron density, Γ is the recombination rate, and ∇ · Je−  is the ambipolar diffusion of electrons [94]. This is simply the continuity equation of a non-conserved quantity, that is, the electron density. Recombination Process and Rate In Fig 4.3, the ionized population is initially zero, and ionization occurs only due to the single pulse. The assumption is that the ionized species has completely left the interaction region between pulses; for high repetition rate systems, this is not necessarily the case and the electron-ion dynamics must be understood to predict the neutral gas density. In the context of a chemical reaction, the creation of free electrons is given by, Xe + N ¯hω −→ Xe+ + e− where the ionization rate is given by WYudin . Under the conditions of this thesis’ work, where the neutral xenon density is ∼ 1018 atoms/cm3 and the  ionization fraction is up to ∼ 10%, the dominant recombination process is  from three body recombination stabilized by an electron [95], Xe+ + 2e− −→ Xe∗ + e− .  (4.10)  The three body recombination rate is strongly dependent on the electron temperature because the kinetic energy of the electron must be less than the Coulomb potential energy (from the ion) required for recombination. It 72  4.1. Ionization Dynamics  18  10  Density (cm−3)  0.5 eV 1 eV 1.5 eV  17  10  16  10  0  1  10  2  10  10  3  10  Time (ns)  Figure 4.6: Density of xenon ions recombining with 0.5 eV (blue), 1 eV (red), and 1.5 eV (black) electrons through three body recombination stabilized by an electron. The initial electron/ion density is 3 × 1017 atoms/cm3 . The electrons with energies > 1 eV take many pulse periods (one period is given by vertical dashed line) to recombine with an ion, leading to a higher ionization population than predicted in Eq 4.5 for single pass ionization. Once the electron and ion have recombined, the xenon atom is in an excited state. An excited state has a lower ionization energy, and thus greater ionization probability, than the ground state.  has been found theoretically that the three body recombination rate for a hydrogen-like ion is ΓTBRe− = 1.1 × 10−20 T −9/2 [94, 96] (in units of m6 /s),  where the strong temperature dependence, T −9/2 , has been confirmed experimentally [95]. After the pulse, the ion and free electron densities evolve as, ∂ρXe+ dt ∂ρe− dt  = −ΓTBRe− ρXe+ ρ2e−  (4.11a)  = −ΓTBRe− ρXe+ ρ2e−  (4.11b)  where the initial densities are determined by Eq 4.5. The simulated ATI spectrum shows many electrons have energies > 1 eV. For typical gas densities used in our HHG experiments (∼ 1018 atoms/cm3 ), electrons with 73  4.1. Ionization Dynamics energies > 1 eV have a small probability of recombination with the parent ion within the cavity round trip time, as shown in Fig 4.6. Thus, the number of steady state neutral atoms is decreased. Once the electron has recombined with the ion, it is in an excited state Xe∗ [94, 97]. There are many pathways for the excited atom to decay to the ground state, with their own associated rates. Although the atom may have reformed, its excited state has a lower ionization energy, thus increasing its probability of being ionized relative to the ground state in subsequent pulses. Ambipolar Diffusion Looking at the last term in Eq 4.9, it is the density of the electrons flowing out of the focus where the plasma is generated. Its divergence, ∇ · Je− ,  dictates how fast the electrons leave the focus.  -  +  -  +  + +  (a)  (b)  Figure 4.7: Cartoon of ambipolar diffusion. (a) The linearly polarized electric field (red arrow) ionizes the atoms, allowing the energetic electrons (black -) to leave the focus while the ions (purple +) remain. (b) The resulting electric field from the repulsion of the ions (red) and electron-ion attraction (green) moves the ions out of the focus.  The electrons that have sufficient energy to leave the focus create a charge gradient in the plasma, with the energetic (negatively charged) electrons in the perimeter and the (positively charged) ions remaining at the focus as shown in Fig 4.7. The electric field that results draws the ions away 74  4.2. Nozzle Design and Flow Analysis from the focus. The plasma then diffuses away from the focus given by, ∂ρ± ∂ 2 ρ± = −Da ∂t ∂x2  (4.12)  where ρ± is the density of ions or electrons. The ambipolar diffusion coefficient Da = Di (1 + Te− /TXe+ ) is related to the diffusion coefficient of the ions Di and the ratio of the electron to ion temperatures. Equation 4.12 is the one dimensional diffusion equation. Given some initial plasma density and electron temperature, it is possible to predict how the plasma density evolves. This can give insight into the density at the focus upon subsequent laser pulses. A simulation is presented in Fig 4.8(a), calculating the resulting expansion of the plasma generated with a Gaussian diameter of 25 µm, and electrons with 1 eV kinetic energy. After 15 ns, or one cavity round trip, the density of the plasma reduces by a factor of two [98]. The plasma expansion can be observed by measuring the gas line backing pressure, as presented in Fig 4.8(b). The nozzle used was the through nozzle, as shown in Fig 4.9(b), which only allows for the gas to exit through the two small apertures. Because of the confinement from nozzle geometry, the ambipolar diffusion led to an increase in the gas line pressure. The timescales of recombination and ambipolar diffusion are thus on the order of the repetition period of the amplified intracavity pulse. However, the delivery device - the nozzle - can be designed in such a way as to minimize the time that the gas is in the interaction region, maximizing the number of neutral atoms at the enhancement cavity focus. The remainder of this chapter discusses the analysis of the flow in the nozzle.  4.2  Nozzle Design and Flow Analysis  In this section, fluid mechanics and computational fluid dynamics (CFD) are used to model the gas flow into the enhancement cavity focus. The flow of the gas through the nozzle, supersonically expanding into the vacuum chamber, is difficult to analytically solve except for the most simple 75  4.2. Nozzle Design and Flow Analysis 17  x 10 10  15.5  Density (atoms/cm3)  Radius (µ m)  9  14.5  8 14  7 13.5  6  13 12.5 0  5  Time (ns)  10  (a)  5 15  Intensity (x1013 W/cm2)  4 15  3  2  1  0  7.5  7.6  7.7  Pressure (Torr)  7.8  (b)  Figure 4.8: (a) A simulation of the plasma radius (blue) and density (green) for an initially Gaussian profile of diameter 2r = 25 µm and electron energy of 1 eV. (b) A measurement on the backing pressure of the gas nozzle that was affected by the presence of the high peak intensity at the focus (red line is pressure uncertainty). The nozzle was of the geometry type given in Fig 4.9(b), the through nozzle. The generated plasma quickly expanded, but because of the gas confinement within the nozzle, the plasma also expanded back into the gas line. The result is an increase in pressure.  geometry. It is possible to make approximations such that the density and flow rates can be found in certain regions of interest. The more complicated geometries require numerical solvers, which allow for modelling the gas flow within the nozzle and vacuum chamber. The analytical solution is used to understand the basics of compressible, sub- and trans-sonic flow, and is compared to the numerical solution found using the solvers in OpenFOAM. The experimentally available nozzle geometries of interest are presented in Fig 4.9.  4.2.1  Analytic Solution  The simplest geometry is a one-dimensional case, where the variables (such as density and temperature) only vary along the direction of propagation. Because the gas used is xenon, a monoatomic gas, we are assuming that the flow is inviscid and the expansion is adiabatic (no heat exchanged between 76  4.2. Nozzle Design and Flow Analysis  3mm  (a)  (b)  (c)  Figure 4.9: Pictures of the nozzles used in this thesis: (a) the endfire nozzle has a small hole at the bottom for the gas flow (purple arrow), and the laser skirts just below (red arrow); (b) the through nozzle has the gas exiting the holes through which the laser is focussed, and (c) removing the bottom of the through nozzle allows for gas flow out the bottom, referred to as the hybrid design.  the gas and the walls of the nozzle). Also, in order to have a simple solution the flow must be laminar, which requires smooth, gradual changes in geometry, and thus the nozzle design is assumed to have no sharp edges, as shown in Fig 4.10. Although none of the nozzle designs rigourously meet these approximations, we can get an understanding of the flow rate and atomic densities at the enhancement cavity focus in a simple nozzle design to gain intuition when solving numerically a more complicated geometry.  77  4.2. Nozzle Design and Flow Analysis The gas is assumed to be ideal, where the ideal gas law is, P = ρRg T  (4.13)  where P is the pressure, ρ is the mass density, Rg is the ideal gas constant and T is temperature17 . Another assumption is that there is no heat transfer as the gas travels through the nozzle, which leads to the expression for adiabatic expansion, P = P0  T T0  γ γ−1  (4.14)  where P0 and T0 are the initial (backing) pressure and temperature, respectively, and γ = cp /cv is the ratio of the heat capacities at constant pressure and volume18 . The gas is unreactive within the nozzle, so the continuity equation is, m ˙ = ρU A  (4.15)  where m ˙ is the mass flow rate, U is the velocity of the gas flowing through an area A. Pvac inlet P0  Mach disk  outlet  T0  ρ0  Barrel shock  Zone of silence  P ∗ T ∗ρ∗  Figure 4.10: Schematic of a one-dimensional nozzle. As long as the chamber pressure Pvac < P ∗ , then all outlet variables are the critical values. In this case, the outlet velocity is trans-sonic and the Mach number is unity. 17  SI units are used in analytical and numerical (OpenFOAM) solutions: P in Pa, ρ in kg mol−1 m−3 , and T in K, which leads to Rg = 8.314 kgJ K . 18 For xenon, γ = 1.667  78  4.2. Nozzle Design and Flow Analysis The enthalpy is a measurement of the total internal energy plus the work energy available to a system. The enthalpy (density) of a gas is [99, 100], H0 = H + ρ  U2 2  (4.16)  where H0 is the stagnation, or total, enthalpy, and H is referred to as the static enthalpy. Using the relationship between enthalpy and temperature H = ρcp T with the approximation that cp is constant with temperature (valid for monoatomic gases [101]), then the velocity of the gas is, U=  2cp (T0 − T )  (4.17)  where T0 is the initial temperature. The speed of sound is, cs =  γRT  (4.18)  where R = Rg /mm = cp − cv is the ideal gas constant divided by the molar mass, and is also the difference in heat capacity at constant pressure and volume. Equating Eq 4.17 with 4.18, we solve for the critical temperature, T ∗ = T0  2 γ+1  (4.19)  The critical temperature is the temperature of the gas when its velocity is the speed of sound (when the Mach number M = U/cs is unity). From this relation, the critical pressure and density can also be derived, ∗  P = P0 ∗  ρ = ρ0  T∗ T0 T∗ T0  γ γ−1  (4.20) 1 γ−1  .  (4.21)  The critical pressure is useful in that once the vacuum chamber pressure is below the critical value, then the flow at the nozzle outlet is transsonic (M a = 1) and the flow rate cannot be further increased by lowering the  79  4.2. Nozzle Design and Flow Analysis chamber pressure. In this case, the flow is considered choked19 . From the above relations we can solve for the density and velocity of xenon at the nozzle outlet for the one-dimensional case. Given the initial gas temperature of T0 = 300 K and backing pressure of P0 = 100 Torr, the critical temperature is T ∗ = 225 K which leads to a velocity at the nozzle outlet of U = 154 m/s. For the HHG experiments, the intracavity beam waist w0 = 12 µm, which means that the gas is within 2w0 for a duration of Tf oc ≈ 155 ns, or 10 round trips. To increase the speed while maintaining the  density requires an increase the initial temperature (by heating the nozzle  or the gas line), or to add a gas with a higher heat capacity. The critical density ρ∗ = 0.45 kg/m3 = 2 × 1018 atoms/cm3 . For a nozzle with outlet diameter of d = 300 µm, the mass flow is m ˙ ≈ 5 × 10−6 kg/s.  4.2.2  Numerical Solution  As previously mentioned, in order to predict the gas flow and density for the more complicated, realistic nozzle geometries, numerical solvers are required. The development of numerical solutions to study fluid flow is called computational fluid dynamics (CFD). CFD can accurately simulate the fluid mechanics in turbulent, compressible flow [102]. In this thesis, OpenFOAM is used to solve the gas flow in the nozzle and vacuum chamber, chosen for its versatility and its expense (free) [103]. For compressible, supersonic flow, the most robust solver currently developed is rhoCentralFoam [104]. A description of the use of OpenFOAM is given in Appendix B. The equations of motion of the gas, which are a series of conservation equations, can be simplified if viscosity is not considered. In the case of For xenon, P ∗ /P0 = 0.49, which is easily attainable when the chamber pressure is ∼ 1 mTorr 19  80  4.2. Nozzle Design and Flow Analysis inviscid, compressible flow, the equations that are solved are [105], ∂ρ ∂t ∂U ρ ∂t ∂E ρ ∂t  = −∇ · (ρU )  (4.22)  = −ρU (∇ · U ) − ∇P  (4.23)  = −ρ(U · ∇)E − ∇ · (U P )  (4.24)  where the total energy E = e + 21 U 2 and e is the specific internal energy. Equation 4.22 is the conservation of mass, also known as the continuity equation; Eq 4.23 is the conservation of momentum, or the Euler equation, and Eq 4.24 is the conservation of energy and is equivalent to the first law of thermodynamics. Endfire Nozzle Results The endfire nozzle, shown in Fig 4.9(a), is the simplest case, and the one that can use the knowledge of the analytical solution to check the accuracy of the numerical results. For the analytical solution, the approximation was made that the flow was laminar and that there were no sharp edges in the flow region. However, from Fig 4.11, there are regions where laminar flow is not a good approximation, particularly near the nozzle aperture. Therefore, it is expected that the numerical result will differ slightly from the analytical solution. Because the focussed laser interaction region of the endfire nozzle is outside of the nozzle, the modelled region of the vacuum chamber is large to allow for the full supersonic expansion of the gas [106]. Therefore, in the numerical solution the vacuum system is well represented with a large volume relative to the nozzle aperture (Fig 4.11(b)). In order for OpenFOAM to reach a steady-state solution, the shock wave that occurs from supersonic flow cannot reach the boundaries of the mesh (for the boundary conditions, see Appendix B). The distance from the nozzle end to the shock wave (the  81  4.2. Nozzle Design and Flow Analysis  20mm  nozzle  (a)  250µm  Gas flow Nozzle wall  IR 20mm  150µm  (c)  vacuum chamber  (b)  Figure 4.11: (a) The Solidworks drawing of the endfire nozzle with a 300 µm diameter aperture (all dimensions in mm). (b) The OpenFOAM mesh model for a wedge of ±1◦ of the same dimensions, with the mesh extending 20 mm to the vacuum chamber region of interest (10 mm across). The cells used for fluid flow modelling are represented by the black rectangles. (c) The magnified view of the interaction region (IR) of (b) including the 250 µm nozzle wall thickness (the red circle represents the approximate size and position of the focussed laser beam). Due to cylindrical symmetry, a small wedge can accurately represent a round nozzle aperture. 82  4.2. Nozzle Design and Flow Analysis Mach disk) is given by [107], dM = 0.67d  P0 Pvac  (4.25)  where d is the nozzle diameter and Pvac is the vacuum chamber pressure. To avoid the Mach disk/boundary condition issue, the simulated vacuum pressure is set to be slightly less than two orders of magnitude lower than the backing pressure. In this situation, the flow is choked at the nozzle outlet and there is a rapid expansion of the gas into the vacuum region.  ρ(kg/m3 )  vacuum chamber (a)  U (m/s)  vacuum chamber (b)  Figure 4.12: Refer to Fig 4.11(c) for geometry: purple arrow shows direction of gas flow, and shaded black box is nozzle wall. OpenFOAM simulation of the (a) density (in kg/m3 ), and (b) velocity (in m/s) along the nozzle axis for xenon (perpendicular to laser beam direction), in the interaction region. The nozzle aperture has a diameter of 300 µm; the xenon initial temperature is 300 K and pressure 100 Torr; the vacuum chamber pressure is 750 mTorr.  The three-dimensional results of the OpenFOAM simulation, displayed in the viewer programme ParaFOAM, are shown in Fig 4.12. The geometry 83  4.2. Nozzle Design and Flow Analysis of this figure is the same as shown in Fig 4.11. The gas comes through the nozzle channel at approximately 180 m/s with a density of 0.35 kg/m3 . Once the gas enters the vacuum chamber, it quickly accelerates to 300 m/s as the density drops. 2  10  0.7 50µm 300µm  50µm 300µm  0.6  1  0.5 Density (kg/m3)  Pressure (Torr)  10  0  10  0.4 0.3 0.2  −1  10  0.1 −2  10 −0.5  −0.25  0 0.25 0.5 0.75 Distance from nozzle end (mm)  0 −0.5  1  −0.25  (a)  1  (b)  350  350 50µm 300µm  300  300  250  250 Velocity (m/s)  Temperature (K)  0 0.25 0.5 0.75 Distance from nozzle end (mm)  200 150  200 150  100  100  50  50  0 −0.5  0 −0.5  −0.25  0 0.25 0.5 0.75 Distance from nozzle end (mm)  (c)  1  50µm 300µm −0.25  0 0.25 0.5 0.75 Distance from nozzle end (mm)  1  (d)  Figure 4.13: Results of OpenFOAM simulations for an endfire nozzle with 50 µm (blue) and 300 µm (red) diameter apertures along the gas flow direction: (a) pressure; (b) density; (c) temperature, and (d) velocity perpendicular to the laser. The smaller aperture leads to a faster rate of change of the variables once the gas enters the vacuum chamber.  The OpenFOAM results of the endfire nozzle along the gas direction of flow are shown in Fig 4.13. Here, the results for two endfire designs of 50 µm and 300 µm diameter aperture nozzles are presented. These two  84  4.2. Nozzle Design and Flow Analysis different sizes are used to exemplify the advantages and disadvantages of this nozzle design. In both cases, the backing pressure is 100 Torr and the gas is initially at 300 K and a vacuum chamber pressure of 750 mTorr. For the 50 µm nozzle, the gas reaches its minimum pressure and temperature and maximum velocity in a much shorter distance than the 300 µm nozzle. The sharp change in the variables in the 50 µm case near 0.35 mm is the Mach disk, in agreement with Eq 4.25. From the OpenFOAM results, xenon is travelling at 172 m/s and 187 m/s in the transverse direction to the laser at the outlet of the nozzle for the 50 µm and 300 µm nozzles, respectively; both are slightly higher than predicted in Eq 4.17 (which is 154 m/s).  Density (kg/m3)  0.3 0.25 0.2 0.15 0.1 0.05 0 0  20  40 60 Distance (µ m)  80  100  Figure 4.14: Magnified view of the density of xenon from the tip of the nozzle extending to the chamber for the 300 µm (blue) and 50 µm (red) nozzles. The rapid change in density shows the sensitivity to position for the smaller diameter nozzle geometry.  In Fig. 4.14, the region after the nozzle is magnified. The scale is the approximate region that we can put the nozzle exit relative to the laser focus. If we place the nozzle too close to the beam, then the beam will be clipped by the nozzle causing loss to the intracavity power. A disadvantage 85  4.2. Nozzle Design and Flow Analysis of the small nozzle is that an uncertainty of position of only 30 µm can lead to a density uncertainty of a factor of two. The beam propagates through the gas perpendicular to the flow. In the calculations to predict the generated harmonic amplitudes, the gas density must be well understood; the gas density along the beam axis is shown in Fig 4.15. The gas from the 300 µm diameter nozzle maintains its density profile near the exit, which is useful experimentally in that an uncertainty in the location of the beam relative to the exit has minimal effect on the density profile of the gas. 25 µm 50 µm 75 µm  50µm aperture  0.1  25 µm 50 µm 75 µm  0.3 0.25 Density (kg/m3)  0.15  3  Density (kg/m )  0.2  0.2  300µm aperture  0.15 0.1  0.05 0.05 0  20  40  60  80  z (µm)  (a)  100  0  50  100  150 z (µm)  200  250  300  (b)  Figure 4.15: Density of xenon in the transverse direction of flow (in the beam direction of propagation zˆ) at 25 µm (blue), 50 µm (red), and 75 µm (black) from the end of the nozzle for endfire nozzles with an aperture diameter of: (a) 50 µm, and (b) 300 µm.  Through Nozzle Results A second nozzle geometry is the through nozzle case shown in Fig 4.9(b). In this case, the creation of the high harmonics occurs within the nozzle. As can be seen in Fig 4.16, the variables are nearly constant within the nozzle. For these results, the backing pressure is 7.5 Torr and the initial temperature is 300 K, and the cavity pressure is 75 mTorr. This model is for a through nozzle with a 500 µm inner diameter, and two exit apertures 86  4.2. Nozzle Design and Flow Analysis  0.5mm  T (K)  0.25mm  y  0.25mm  z  ρ(kg/m3 )  (a)  Uy (m/s)  (b)  Uz (m/s)  (c)  (d)  Figure 4.16: Cross-section of the results of OpenFOAM simulations for the through nozzle in the region of interest: (a) the density (kg/m3 ); (b) temperature (K), where the large red arrow is the direction of beam propagation; (c) velocity (m/s) of the gas perpendicular (ˆ y ), and (d) the velocity parallel to the direction of beam propagation (ˆ z ). The density is uniform within the nozzle, beneficial experimentally because the density at the focus can be high relative to that in the vacuum chamber. Also, the pressure at the intracavity focus is equal to the measured pressure in the line, which provides less uncertainty of the gas density than in the endfire case.  87  4.2. Nozzle Design and Flow Analysis  0.5  4  0.4  3  0.3  2  0.2 1  0.1 -2  -1  0 1 Time (10-4 s)  0 2  (a)  9th harmonic (V)  5  0.6  Transmission (V)  9 th harmonic (V)  0.7  0  6  6 0.5  5  0.4  4  0.3  3  0.2  2  0.1 0  Transmission (V)  0.8  1  0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time (s)  (b)  Figure 4.17: The motivation for trying different nozzle geometries came from the high intracavity power and high backing pressure for the through nozzle case. (a) A sweep through a resonance showed no effects of the plasma affecting the resonance condition of the fundamental beam (blue) or the EUV generated (9th harmonic measured, green). (b) Locking the cavity to the laser resulted in strong oscillations in the intracavity power when the backing pressure was > 8 Torr. The EUV signal also showed strong oscillations. The details of the harmonic signal measurement are discussed in Sec 6.1.  of 250 µm diameter. The nozzle, which extends another 1 mm below the apertures, is capped at the bottom. The benefit of the through nozzle is that the gas density is easily tuneable at the focus, and the pressure can be high within the nozzle while maintaining a low vacuum pressure. However, because the gas can only exit through the same apertures as the laser, the interaction region is much longer than just within the nozzle. In terms of the produced harmonics, the EUV copropagates with the gas exiting the nozzle. The absorption cross-section of the harmonics near the ionization energy of the gas (such as the 9th and 11th harmonics for xenon) means that much of the created EUV will be reabsorbed by the gas. This effect can severely limit the efficiency of the generation process. Therefore, a detriment feature of this nozzle design is that the interaction length is not limited by the width of the nozzle. Another limitation of the through nozzle is the slow gas flow at the 88  4.2. Nozzle Design and Flow Analysis  Figure 4.18: Picture of plasma generated from a through nozzle at the focus of the fsEC. The white/purple colour was generated by the xenon ions and excited xenon atoms. The gas travelled along the beam direction. The cavity setup is described in Fig 4.1.  interaction region, as shown in Fig 4.16(c) and (d). As described earlier, the electron-ion recombination times are on the order of the repetition rate of the laser. To avoid an accumulation of plasma at the focus, gas must exit the focus of the beam. However, in the transverse direction to the beam the flow is ∼ 10 m/s (represented by Uy in Fig 4.16(c)), resulting in the gas  remaining at the focus for ∼ 2.5 µs, or ∼ 150 cavity round trips. If, instead  the focus is placed near an exit of the nozzle where the flow is ∼ 200 m/s  (the same flow velocity as the flow from the endfire nozzle), the flow is in the same direction as the beam and therefore must traverse the Rayleigh range to leave the focal volume, which for w0 = 12 µm, is zR ≈ 0.57 mm. The time  taken for the gas to exit this region is still on the order of 3 µs. The result is a strongly intensity dependent plasma accumulation at the focus, which affects the cavity resonance. The rapid change in cavity length leads to an unstable cavity lock, as shown in Fig 4.17. Therefore, the accumulation of plasma from this nozzle design limits its usefulness in the high repetition rate EUV generation experiments. An image of the plasma generated from the through nozzle is shown in Fig 4.18. The highly excited ions fluoresced a bright white/purple colour as they exited the nozzle along the beam direction. This nozzle geometry led 89  4.2. Nozzle Design and Flow Analysis to a large ion population, which caused instabilities in the fsEC resonance. Hybrid Nozzle Results The hybrid nozzle is an attempt to have the benefits of both nozzle designs. By using a through nozzle and drilling out the bottom of the nozzle so that the gas can flow perpendicularly to the beam direction, the duration that the gas is in the focus is similar to that of the endfire. Because the interaction occurs within the nozzle, the pressure density is more uniform than the endfire case, which can make it easier to accurately regulate the pressure. As can be seen in Fig 4.19, the flow within the nozzle is mainly in the zˆ direction (perpendicular to the beam) at ∼ 200 m/s. Also, within the nozzle  there is very little flow in the direction of the laser, with only 6% of the gas  flowing out each aperture. Therefore, the plasma does not accumulate to the extent of the through nozzle case. In Fig 4.19(a) and (b), the density and temperature at the bottom of the join of the transverse apertures and the main channel are much higher than the immediate surrounding areas. This can be explained by the gas coming to a stop in this region because of the sharp edges in the model. An improved design would be a slightly larger main channel below the apertures and a less sharp join to improve gas flow, leading to less gas escaping through the apertures. However, machining this nozzle design to such precision is difficult with stainless steel with a wall thickness of a few hundred microns. Although this nozzle was tested, to compare the generated EUV amplitude and cavity stability with the through nozzle, there were still limitations due to the hybrid nozzle geometry. The limitation of this nozzle was that the interaction region could not be made < 0.5 mm, whereas the current Rayleigh range zR ≈ 0.57 mm. As will be shown in Sec 5.6, a short in-  teraction region is optimal. The hybrid design may prove to be useful if the minimum waist can be increased while maintaining high peak intracavity power (which could be met by improving the laser output power), thus leading to an interaction length that is shorter than zR .  90  4.2. Nozzle Design and Flow Analysis  0.5mm  T (K)  0.25mm  y  0.25mm  z  ρ(kg/m3 )  (a)  Uy (m/s)  (b)  Uz (m/s)  (c)  (d)  Figure 4.19: Cross-section of the results of OpenFOAM simulations for hybrid nozzle in the region of interest (variables same as Fig 4.16): (a) the density; (b) temperature; (c) velocity in the yˆ direction, and (d) the velocity in the zˆ direction. The density is less uniform within the nozzle than the through nozzle case, however the velocity in the yˆ direction is −180 m/s, meaning that the gas is at the focus for a shorter time.  91  4.3. Summary  4.3  Summary  In this chapter, the ionization and recombination of the gas was discussed in the context of a high power, high repetition rate system. It was shown that the ionization rate per pulse for a 100 fs Ti:Sapphire system with peak intensity of 5 × 1013 W/cm2 is 12%. Although for a single pass experiment  this is not a substantial ionization fraction, because of the high repetition rates involved with the enhancement cavity, the plasma can accumulate near the focus. By investigating the recombination processes it was shown that a substantial amount of the plasma remains for subsequent pulses. This led to the investigation of gas flow to understand the possibilities of improving flow rates through the focus to minimize the plasma time within the focus. The theory of gas flow was used to give an estimate of flow rates. Further analysis required CFD simulations to model more complex geometry to optimize the nozzle design. Three nozzle types were modelled, and we found that the through nozzle was not adequate for high repetition rate fsEC use. We did not further investigate this nozzle to quantify the ion density present, although that may be of interest. The hybrid nozzle enabled a stable, high intracavity power and did not lead to high ion production. However, the large aperture and long interaction length due to this nozzle geometry was not optimal for HHG. The endfire nozzle also minimized ion production, and the interaction length could be more easily tuned. It is estimated that for Xenon gas travelling at 180 m/s, passing through the focus of a Gaussian beam with a minimum radius w0 = 12 µm, pulse duration ∆t = 70 fs and peak intensity Ipeak = 4 × 1013 W/cm2 that the maximum ionization probability is approximately 7%, useful for high repetition rate systems. A limitation of the endfire nozzle  is that the smaller nozzle apertures led to a larger gas density uncertainty. When using the endfire nozzle for the generation of high harmonics, the density at the focus must be well characterized for consistent measurements.  92  Chapter 5  Generation of High Harmonics via an Enhancement Cavity The generation of high order harmonics is an inefficient process, and many attempts to improve the efficiency have been made. In the low repetition rate, high pulse energy/peak power regime, Ref [108] attempts to explain what effects the various parameters (including gas pressure and species, absorption and coherence lengths) have on the amplitude of the generated harmonics in the long focus regime. Our experimental conditions are significantly different compared to Ref [108] as the driving field in our case is amplified by a femtosecond enhancement cavity (fsEC) and has a MHz repetition rate with small pulse energy. Within this context, it is not obvious whether we can generate more harmonic power by having a tighter focus to increase the peak power, or have a looser focus to increase the interaction region. Because the generation is highly dependent on the intensity, we also need to investigate what other consequences there are by having a tighter focus. There must be some tradeoff in the fundamental beam between intensity and confocal parameter, leading to a spot size that would maximize the EUV output. This chapter investigates the various parameters in play including spot size, gas pressure, nozzle design, and phase matching to optimize the high harmonic generation process for a high repetition rate, 5 − 10 µJ system20 . 20  The sample beam profile image shown Fig 5.14, and the experimental amplitude data in Figs 5.15, 5.16, 5.17, and 5.18 were collected in collaboration with Arthur K Mills.  93  5.1. The Classical Picture  5.1  The Classical Picture  Based on both experimental [9] and numerical [26] studies, it had been suggested that the maximum photon energy attainable by high harmonic generation is, Eγ ≈ Ip + 3Up  (5.1)  where Ip is the ionization potential for the gas and Up is the ponderamotive energy of the driving field21 . The reason was not well understood until the classical picture was developed in the three step model by Corkum [23]. In this picture, the valence electron of the gas subject to a strong laser field can tunnel out of the Coulomb potential of the atom. Away from the atom, the electron is accelerated by the oscillating driving field; the electron returns to the parent ion because of the change in sign of the field. The electron is now in a higher energy state, and the recombination of the electron with the parent ion releases a photon of the energy difference. The maximum kinetic energy that the electron can receive from the field while still returning to the parent ion is 3.2Up , as shown below. Due to the atomic potential symmetry, the spectrum is a series of odd harmonics. The maximum classically allowed harmonic is referred to as the cutoff harmonic.  5.1.1  Classical Trajectories and Phase  In understanding Eq 5.1, it is helpful to examine the electron motion within a classical picture. For an atom subject to a strong, oscillating electromagnetic field, there is a non-zero probability that the atom will become ionized. The trajectory that a freed electron takes in the continuum is dependent on the ionization time t0 . Following ionization, the motion of the electron is governed by the oscillating applied field (here, the field is monochromatic CW and the atomic potential is ignored), me x ¨ = qE10 cos(ω1 t) 21  (5.2)  Up ≈ 0.933 × 10−13 Iλ2 where I is the intensity in W/cm2 and λ is in µm.  94  5.1. The Classical Picture where me is the mass of the electron freed from the parent atom at time t0 , q = −e is the charge of the electron, E10 is the field strength of the applied  laser and ω1 is the laser frequency. The momentum p and the position x of the electron are then, p(t) = x(t) =  −eE10 sin(ω1 t) − sin(ω1 t0 ) ω1 eE10 cos(ω1 t) − cos(ω1 t0 ) + ω1 (t − t0 ) sin(ω1 t0 ) me ω12  (5.3) (5.4)  where the kinetic energy of the electron is EK = p(t)2 /2me . By solving the equations of motion for the return time tr when x(tr ) = 0, we find the  2  1  1  0  0 0  50  100  150 ω1 t (Degrees)  E (E01)  x (− m  e E 10 2) eω1  released photon energy.  200  250  −1 300  200  250  300  (a)  EKin (Up)  3 2  Long Cutoff Short  1 0 0  50  100  150 t1 t (Degrees)  (b)  Figure 5.1: (a) Classical trajectories x(t) of electrons ionized at ω1 t0 = 5◦ (red), ω1 t0 = 18◦ (blue), and ω1 t0 = 45◦ (green) by the field (black line). of MATLAB The electron returns to the parent ion at x = 0 atStudent the Version return time tr ; (b) only one trajectory has the highest possible kinetic energy (3.2Up ) as the electron returns to zero (blue), defined as the cutoff energy; the electrons released before this time (red) and after (green) both have trajectories that return to the parent ion with 1.75Up , but with differing times in the continuum.  95  5.1. The Classical Picture As shown in Fig 5.1, there are two trajectories that lead to the same return energy p(tr )2 /2me = 1.75Up . If the electron is released near the maximum of the field, then it is accelerated away from the ion for nearly a quarter of the driving field cycle. The field changes sign and the electron changes direction and begins to return to the parent ion. As the electron approaches the ion, the field changes sign again, slowing down the electron. Conversely, an electron can be released near the minimum of the field, where shortly after the electron is released it is accelerated towards the parent ion, spending a shorter time unbounded from the ion. The energies of the two trajectories can be the same, however their return times (and hence photon release times) differ. This phase difference can lead to interference effects in the generated harmonic. In general, with the exception of the maximum photon energy of 3.2Up , there are two possible classical trajectories that the electron can take to return to the parent ion. Electrons released near the maximum of the electric field E0 spend almost ω1 t = 2π in the continuum, and are referred to as the long trajectory electrons. Conversely, electrons released near the minimum of the field return to the parent ion shortly after ionization, and are referred to as the short trajectory electrons. Understanding the phase difference of the trajectories allows us to investigate certain parameters that will enable the optimization of the high harmonic amplitudes. The solution of all possible energies and phases of a single-mode CW field (ignoring atomic potential) is shown in Fig 5.2. The electron energy and the field phase at the electron return time tr are shown as a function of the ionization time t0 . The maximum returning energy of 3.2Up occurs when an electron is ionized at ω1 t0 = 18◦ . The energy is π periodic with respect to the field; electrons released 90◦ < ω1 t0 < 180◦ do not return to the parent ion. This is similar to the discussion regarding the rescattering of electrons in above threshold ionization (ATI, discussed in Sec 4.1.2).  96  5.1. The Classical Picture 4  350  long  short 300  3  t t (Degrees)  2.5  250  2  200  1.5 1  1 r  Returning eï Energy (Up)  3.5  150  0.5 0 0  100 10  20  30  40 50 t t (Degrees) 1  60  70  80  0  Figure 5.2: The returning energy (red, left) and phase (blue, right) of an electron in an oscillating electric field. The peak energy of 3.2Up occurs at ω1 t0 = 18◦ (represented by dotted line); electrons freed before this time (left of dotted line) are referred to as the ‘long trajectory’, while those after (right of dotted line) are referred to as the ‘short trajectory’.  5.1.2  Action  The reason for the action’s importance in our analysis is that the phase of the released EUV photon is dependent on the phase of the electron. The action describes the trajectory of the electron as it is accelerated from the applied field and returns to the parent ion [14], tr  S=  Ldt  (5.5)  t0  where S is the action from the electron leaving the atom at t0 and returning at tr , and the Lagrangian L = T − V (the difference of the kinetic and  potential energy) dictates the trajectory of the system. In the simplest form, we use the classical electron trajectory in the strong field regime and  97  5.1. The Classical Picture ignore the effects of the atomic potential (Up  Ip ),  L ≈ T p2 . = 2me  (5.6)  From Eq 5.3, the action is then, 2  S=  e2 E10 2me ω12  tr t0  (sin(ω1 t) − sin(ω1 t0 ))2 dt.  (5.7)  The kinetic energy of the electron is often thought of in terms of its ponderamotive energy, which is the ‘wiggle’ energy of the electron over a laser cycle, 2  Up = =  e2 E10 4me ω12 e2 2η0 I(r, z) 4me ω12  (5.8)  where η0 is the impedance of free space, and I(r, z) is the intensity of the field driving the electron. The intensity dependence leads to a spatially dependent action22 . The classical action can be used to determine the electron trajectory. The phase of the returning electron is then, S(t0 , tr , E10 , ω1 ) ¯h f (t0 , tr ) e2 η0 = − I(r, z) ¯hω1 2me ω 2 = −αi I(r, z)  Φatom = −  22  (5.10)  The intensity profile of the TEM00 mode of a CW Gaussian beam is, I(r, z) = I0  where w(z) = w0  1+  z2 2 zR  w02 2r2 exp − 2 2 w (z) w (z)  (5.9)  is the radius of the beam.  98  5.1. The Classical Picture where, αi =  f (t0 , tr ) e2 η0 ¯hω1 2me ω12  (5.11)  is referred to as the phase coefficient [109], and has units of inverse intensity. The function f (t0 , tr ) is the integration of the electron trajectory, tr  f (t0 , tr ) = 2ω1 t0  sin(ω1 t) − sin(ω1 t0 )  2  dt.  (5.12)  In the classical, strong field case, its value is 2π for long trajectories and approaches zero for short trajectories. A plot of the classical phase coefficient is shown in Fig 5.3. 25  α (× 10−14cm2/W)  20 15 10 5 0 0  1 2 3 Returning e− Energy (Up)  4  Figure 5.3: The phase coefficient α for a classical electron in the strong field, CW case. The different phase coefficients stem from the different electron trajectories. For an 800 nm source, the long, αlong ≈ 24 × 10−14 cm2 /W; the short, αshort ≈ 0. The relative amplitudes of the phase coefficients are calculated by solving the time-dependent Schr¨odinger equation.  The argument presented above is for the strong-field, monochromatic CW case. It gives us some insight into how the intensity affects the energy  99  5.2. The Dipole Response and phase of the released photon; both are important when optimizing the harmonic signal. Quantitative analysis of the harmonic generation process including driving field pulse duration and amplitude, and atomic potential effects - requires solving the time dependent Schr¨odinger equation.  5.2  The Dipole Response  Critical to predicting the amplitude of the harmonics generated by the fsEC is a quantitative description of the atomic response to the driving field. The harmonic amplitude must be calculated as a function of the pulse duration and the peak fundamental intensity. In this section, we numerically solve the time dependent Schr¨ odinger equation to derive the dipole response.  5.2.1  Time-dependent Schr¨ odinger Equation  The time-dependent Schr¨ odinger equation (TDSE) in one dimension is, i¯ h  ∂ p2 ψ(x, t) = ψ(x, t) + V (x, t)ψ(x, t) ∂t 2me  (5.13)  ∂ is the momentum operator and V (x, t) is the potential. where p = −i¯ h ∂x  In the case of a bound electron subject to a strong electric field (where the peak intensity 1013 < Ipeak < 1015 W/cm2 ), the evolution of the initial wavefunction ψ(x, t0 ) is calculated by, ψ(x, t) = ψ(x, t0 ) exp  −i ¯h  t t0  p2 + Vatom (x) + V (t)dt 2me  (5.14)  where Vatom (x) is the Coulomb potential of the atom, and V (t) is the oscillating potential of the laser. Here, the TDSE is solved using the split-step Fourier method [110, 111]. The split-step operator can be used because for  100  5.2. The Dipole Response small dt23 , exp −  i h ¯  −i p2 −i p2 + V (x, t) dt ≈ e h¯ 2me dt e h¯ V (x,t)dt . 2me  (5.16)  The evolution of the wavefunction ψ(x, t) is then calculated by, −i  φ(x, t + δt) = F T ψ(x, t)e h¯  V (x,t)δt −i¯ h  ψ(x, t + δt) = F T −1 φ(x, t + δt)e 2me k  (5.17) 2 δt  (5.18)  where F T and F T −1 are the Fourier transform and Inverse Fourier transform, respectively (in the momentum domain, the operator p becomes h ¯ k). In this manner, the wavefunction is propagated in time by a small time step δt. An example of the evolution of the wavefunction in an oscillating field is shown in Fig 5.4. The harmonic generation process is due to the dipole response of the valence electron to the oscillating field in an anharmonic potential. An inherent assumption in this calculation is that the harmonic fields are generated by the single active electron (SAE) [28]. We calculate the magnitude of the response by numerically integrating the TDSE using a one-dimensional potential as done in Refs [113] and [114], Vatom (x) = −  1 e2 4π 0 x2 + X 2 0  1/2  (5.19)  where X0 dictates the depth of the potential, and is calculated from the variational principle, as discussed below. The initial, ground state of the electron for the one-dimensional potential 23  The non-symmetric split-step method is valid to dt2 because for some non-commuting operators A and B, exp [(A+ B)dt] ≈ exp(Adt) exp(Bdt) exp −  dt2 [A, B] 2  (5.15)  from the Baker-Campbell-Hausdorff formula [112]  101  5.2. The Dipole Response  (a)  (b)  Figure 5.4: The time evolution of the probability density (a) |ψ(x, t)|2 , in space domain, and (b) |φ(k, t)|2 , in momentum domain, of an electron in a one-dimensional potential and a strong oscillating field; shown here are the log10 of the probability densities. The field parameters are ∆t = 20 fs, Ipeak = 10 × 1013 W/cm2 , λ0 = 800 nm. The radial distance and spacing is chosen such that both ψ(x, t) and φ(k, t) are well bounded and can fully evolve. In (a), the trajectories of the electrons can be seen with different release times t0 , leading to different quantum paths. The ATI spectrum (see Sec 4.1.2) can be taken from (b) when t ∆t.  102  5.2. The Dipole Response is assumed to be, ψ(x, 0) = √  1 sech 2X0  x X0  (5.20)  as shown in Fig 5.5. In this figure, |ψeig (x)|2 is the probability density that  remains after the system has evolved to a steady state, thus ψeig is the eigenstate of the potential. The similarity of the probability distributions shows that Eq 5.20 is a valid approximation. 9  6  x 10  10  10  sech2(x/X0) /X0) |seig(x)|2  5  2  5  10  |s(x)|  |s(x)|2  4  sech2(x/X0) /X0) |seig(x)|2  3 2  0  10 1 0 ï1.5  ï1  ï0.5  0 x (nm)  0.5  1  1.5  ï1.5  ï1  (a)  ï0.5  0 x (nm)  0.5  1  1.5  (b)  Figure 5.5: Comparison of the assumed ground state for the 1-D potential (blue) sech2 (x/X0 ) with the remaining steady-state electron density after strong interaction with a pulse (red) |ψeig (x)|2 (both normalised) for (a) linear, and (b) log plots.  The value of X0 is found from the variational principle such that the ground state energy, Egs , equals the ionization energy, −Ip , of the gas. That is,  −Ip = Egs ≤ ψ(x, 0) −  ¯ 2 ∂2 h + Vatom (x) ψ(x, 0) . 2me ∂x2  (5.21)  Thus, the ionization energy can be varied in the model to study its effect on harmonic cutoff and amplitude. The driving field has the potential, V (t) = −exE(t) cos(ω1 t)  (5.22) 103  5.2. The Dipole Response where E(t) is the pulse envelope and ω1 is the fundamental carrier frequency. The total potential is then the sum of the atomic and laser potentials V (x, t) = Vatom (x) + V (t). In the numerical model, the spatial grid spacing is determined by the trajectory of a returning electron as well as its momentum. The maximum x value, that is the boundary of the spatial grid, is chosen to be five to ten times larger than the maximum distance that the electron classically travels, |xmax | =  2eE10 me ω12  (5.23)  where E10 is the peak field amplitude. The number of grid points is determined from the momentum domain such that Ee− =  2 h2 kmax ¯ 2me  10Up where  hkmax is the maximum value in the momentum domain, kmax = π/∆x24 . To ¯ ensure that the electrons do not reflect from the boundary, we use absorbing boundary conditions to remove energetic electrons that never return to the parent atom [115]. To calculate the harmonic spectrum, the time step δt must satisfy the relation, δt  1 qmax ω1  (5.24)  where qmax is the cutoff harmonic. The dipole response can be calculated from the dipole moment d(t) = −e ψ(t)|x|ψ(t) [116] (referred to as the length form). An alternative form  of calculating the dipole response, which we use in our calculations, is to use the acceleration of the dipole operator [26, 111, 117], dA (t) =  1 ∂V ψ(t) − ψ(t) . me ∂x  (5.25)  The acceleration and length forms have identical spectra up to the cutoff harmonic; the acceleration form has been shown to have a smaller background above the cutoff frequency (the difference in spectra are shown in 24  The maximum classical, kinetic energy Ee− that an electron can attain from an oscillating field is 10Up [92], as discussed in Sec 4.1.2  104  5.2. The Dipole Response  d(t) d (t)  ï5  S(t) (arb u)  10  A  ï10  10  ï15  10  ï20  10  5  10 15 Harmonic Harmonic Order  20  25  Figure 5.6: Comparison of the length (blue) and acceleration (red, dashed) form for calculating the dipole response in a strong field (Ipeak = 3 × 1013 W/cm2 ). The length and acceleration forms give nearly identical results until well above the cutoff harmonic, qmax ≈ 11; the results for the acceleration form follow the monotonic decrease in power for higher harmonics.  Fig 5.6). The spectrum of the dipole response, to be used in the calculations of the field generated, is then calculated by the Fourier transform, ˜ d(ω) =  1 1 T2 − T1 ω 2  T2  dA (t)eiωt dt  (5.26)  T1  where T1 and T2 are the beginning and end points of the integration, respectively (|T2 − T1 |  ∆t, the pulse duration). The power spectrum of the  harmonics are calculated by [118], S(ω) =  2 µ0 e2 ω 4 ˜ d(ω) . 12π 0 c  (5.27)  Sample spectra are shown in Fig 5.7 for three different driving field amplitudes. We can now see the relationship between the harmonic amplitudes as a function of the driving field intensity. As the driving field’s intensity is increased above 3 × 1013 W/cm2 , the amplitude of the lower order har-  monics (≤ 11) saturates, and further increasing the driving field leads to the 105  5.2. The Dipole Response  ï5  S(t) (arb u)  10  ï10  10  13  5×10  13  3×10  13  1×10 ï15  10  5  10 Harmonic Order Harmonic  15  20  Figure 5.7: The change in the harmonic power due to the dipole response for peak driving field intensities varying from 1 (red) to 3 (blue) to 5 (black) ×1013 W/cm2 for a 800 nm laser source with 70 fs pulse duration. The dipole response for harmonics three to eleven shows a strong dependence on the peak intensity until Ipeak = 3 × 1013 W/cm2 , above which the harmonic amplitude is nominally constant. Increasing the peak intensity then mainly leads to the generation of higher harmonics and does not substantially further increase the power of these lower harmonics in the single atom response. However, geometric effects such as phase matching and the volume of high intensity can lead to an increase in harmonic power.  generation of higher harmonics. The peaks of the harmonics are used to generate a plot of the harmonic powers as a function of driving field peak intensity, as shown in Fig 5.8. The driving field pulse duration - long for HHG - leads to a relatively narrow harmonic bandwidth. With our current setup, it is not possible to resolve experimentally the spectrum of each harmonic, and thus only the peak amplitude of the harmonics is used in the calculation. We see that the dipole response is linear (on a log-log plot) at low intensities, and saturates for peak intensities above a certain value, dependent on the harmonic. Once the harmonic power begins to saturate, there is a strong, irregular oscillation. This oscillation is the interference that arises from the different quantum trajectories of the electron. Although we only calculate the response of an electron in a one dimen106  5.3. Phase Matching 0  S(ω) (arb. u)  10  3rd −5  5th  10  7th 9th 11th 13th 15th  −10  10  12  10  13  10 Ipeak (W/cm2)  14  10  Figure 5.8: The dipole power of the odd harmonics of the driving field from a single active electron as a function of the peak driving field intensity Ipeak ; λ0 = 800 nm and Gaussian pulse duration is 70 fs. The dipole response is fairly linear up to Ipeak ≈ 1 × 1013 W/cm2 ; above 5 × 1013 W/cm2 the response begins to plateau on average (for the harmonics shown). The strong oscillations are caused by the electron’s multiple quantum paths.  sional potential, the harmonic spectrum shows good agreement with Ref [111] as shown in Fig 5.9, and Refs [26], [118], [117], and [28], as well as the classical solution of the location of the cutoff harmonic.  5.3  Phase Matching  Phase matching can play an important role in the efficiency of the high harmonic generation process. In order to minimize the phase mismatch of the fundamental and harmonic fields, we need to investigate the phase matching condition in high harmonic generation for a tightly focussed Gaussian beam. The harmonic field is created by the fundamental field, but as the two fields propagate their phases can become mismatched. Because the amplitude of the generated field is maximized when the phase mismatch is zero, the phase mismatch limits the efficiency of harmonic generation. Thus we want the phase of the harmonic field to match that of the fundamental field over 107  5.3. Phase Matching −5  10  |d(ω)|2 (arb u)  1D Tong −10  10  −15  10  −20  10  5  10  15  20 25 Harmonic  30  35  Figure 5.9: Comparison of the 1-D potential dipole response to that of Tong et al [111]. The laser parameters are λ0 = 1064nm, Ipeak = 5 × 1013 W/cm2 and the pulse duration is 100f s. The 1-D potential shows good agreement with the more complicated methods developed in this reference.  the longest possible interaction length, allowing for constructive interference and the maximum harmonic amplitude. The causes of the phase mismatch are dispersion, geometrical effects, and the generation process. A noble gas (in our experiments, xenon) is used to create the harmonics. One reason for the phase mismatch is that all materials exhibit dispersion, a wavelength dependent index of refraction. Not only is the index of refraction different between the fundamental field and the harmonics generated, it is also different between the harmonics. Therefore, the regions of good phase matching will depend on the harmonic. Another cause of material dispersion is the plasma, which is generated by the strong laser field. The index of refraction of the plasma is complicated by the fact that the plasma density depends on the intensity of the fundamental driving field (see Sec 4.1.1). The total index of refraction of the gas due to the combination of neutral atoms and free ions is [119], n(ω, r, z) ≈ 1 + P (r, z) (1 − η(r, z))nδ (ω) − η(r, z)  ωp2 , 2ω 2  (5.28)  where P (r, z) is the (position dependent) gas pressure in atmospheres, nδ is 108  5.3. Phase Matching the difference from unity of the index of refraction at standard temperature and pressure, η(r, z) is the ionization fraction (also position dependent), and ωp =  Ne e2 me 0  with Ne is the density of the electrons in m−3 , is the plasma  frequency [58]. Another cause of phase mismatch is the fundamental field itself. In order to reach the high intensities required for high harmonic generation, we use a tightly focussed Gaussian beam. The phase of a Gaussian beam is, φ(r, z) = kz +  kr2 − ζ(z). 2R(z)  (5.29)  The first term is that of a plane wave, the second is the radial dependence because of the finite width of a Gaussian beam, and the third is the Gouy phase shift. The Gouy phase shift, which comes about due to the spacing of the phase fronts near the focus, results in a change in phase by π as a TEM00 beam passes through a focus. The functions R(z) and ζ(z) are given by, zR z z ζ(z) = arctan zR 2 w zR = k 0 2  R(z) = z 1 +  2  (5.30) (5.31) (5.32)  where w0 is the minimum radius of the beam. A third source for phase mismatch stems from the electron trajectory in the generation process (see Eq 5.10). Because the electron trajectory is intensity dependent, the harmonic phase becomes position dependent within a focussed Gaussian beam. This position dependent phase competes with that of the fundamental beam to create small regions of good phase matching. A small wavevector mismatch, ∆k(r, z), indicates a small phase change as the beam propagates (a small gradient of ∆φ(r, z)). The phase mismatch, ∆φ(r, z), is the phase difference of the driving and the generated fields (derived below). Because of the two trajectories contributing to the  109  5.4. Harmonic Amplitude Calculation  4  15  3.5 3  5 2.5 0 2 −5  10 Radius r (µm)  Radius r (µm)  10  4  15  3  5 2 0 1  −5  1.5 −10  −10  0  1 −15  −15 −0.5  0 Beam direction z (mm)  0.5  (a)  0.5  −0.5  0 Beam direction z (mm)  0.5  (b)  Figure 5.10: Contour plots of the log10 of ∆k (units of 1/m) for the 7th harmonic for the (a) short and (b) long trajectories as a function of position. Here the pressure is assumed to be low with no ionization to emphasize the geometric factors involved in phase matching; the peak intensity is 3 × 1013 W/cm2 . The white ellipse, the region where most of the harmonic is created, is the 1/e intensity region; the black dots are for the beam radius. The short trajectory has no regions of good phase matching along the beam axis near the focus, whereas the long trajectory has good phase matching at z = 0.4 mm on axis, as well as at r = 6 µm, z = 0.2 mm (referred to as off-axis phase matching).  harmonic (short and long), the phase matching has two cases. In the regions of small phase mismatch, the harmonic fields add in phase (thus the intensity grows as the square of propagation distance). The geometric effects of the Gouy phase and the atomic phase are presented in Fig 5.10 (note that ∆k(r, z) = ∇ [∆φ(r, z)]) [120].  5.4  Harmonic Amplitude Calculation  Now that the microscopic (single active electron dipole response to driving field) and macroscopic (phase matching) kernels for the harmonic generation process have been understood, we can calculate the generated harmonic amplitudes. To calculate the generated harmonic amplitudes, we begin with  110  5.4. Harmonic Amplitude Calculation Maxwell’s wave equation [51, 121], ∇2 Eq (r, z, t) −  n2 (r, z, ωq ) ∂ 2 ∂2 E (r, z, t) = µ Pq (r, z, t). q 0 c2 ∂t2 ∂t2  (5.33)  The polarization, the driving term for the field of the q th harmonic Eq , is, Pq (r, z, t) = ρ(r, z)dq (r, z, t)eiqφ1 (r,z,t)  (5.34)  where ρ(r, z) is the density of atoms with single active electrons in the ground state (accounting for ionization) and dq (r, z, t) is the dipole strength of the q th harmonic. The generated field is, Eq (r, z, t) = Eq0 (r, z, t)eiφq (r,z,t)  (5.35)  where the polarization phase φ1 and the generated field phase φq are discussed below. Ultimately, we are interested in the harmonic power that can be directed towards an experiment (the far field beam profile and amplitude). Thus, although the amplitude of the harmonics generated off-axis show interesting features, the field generated near the beam axis is used for quantitative analysis. To this end, we drop the r dependence. In the slowly varying amplitude approximation25 , ∂ ∂2 Eq (z, t) ≈ 2ikq Eq0 (r, z, t) − kq2 Eq0 (r, z, t) eiφq (z,t) . 2 ∂z ∂z  (5.37)  Substituting Eq 5.37 into Eq 5.33 and taking the Fourier transform, 2ikq  n2 (ωq )ωq2 ∂ 0 ˜ q )ei(qφ1 −φq ) Eq = kq2 − Eq0 − µ0 q 2 ω12 ρ(z)d(ω ∂z c2  (5.38)  25  The approximation is that the growth rate of Eq0 (r, z, t) is slow with respect to kq , or more specifically, ∂ 2 Eq0 ∂Eq0 k . (5.36) q ∂z 2 ∂z  111  5.4. Harmonic Amplitude Calculation or, ˜ q) n2 (ωq )ωq2 iµ0 q 2 ω12 ρ(z)d(ω ∂ 0 −i 0 kq2 − E + ei∆φ(z) Eq = q ∂z 2kq c2 2kq  (5.39)  where ∆φ(z) = (qφ1 − φq ).  The first term on the right has side of Eq 5.39 is due to the absorp-  tion of the harmonic by the gas, as may be seen by ignoring the generation process. In this case, this equation follows the Beer-Lambert law [101] E(z) = E(0)e−ρσz/2 if, n2 (ωq )ωq2 i 2 ρ(z)σ(ωq ) = k − kq q c2  (5.40)  where the index of refraction n(ωq ) = nR (ωq ) + inI (ωq ) can be complex. Realising that kq = nR ωq /c, then we see that, ρ(z)σ(ωq ) ≈ 2nI (ωq ) where the approximation nI valid because nI =  cρσ 2ωq  ωq c  (5.41)  nR has been used. This approximation is  nR for gas pressures used in HHG experiments.  The harmonic field amplitude is thus calculated by solving the differential equation, ρ(z)σ(ωq ) 0 ∂ 0 iµ0 q 2 ω12 ˜ q )ei∆φ(z) . Eq (z) = − Eq (z) + ρ(z)d(ω ∂z 2 2kq  (5.42)  This differential equation is composed of two terms: the first is a decaying exponential dependent on gas density and absorption cross-section, and the second term is the generation equation which is dependent on the harmonic order, the gas density, the dipole response, and the phase mismatch of the polarization and the harmonic field. As mentioned above, the phase mismatch is not only from the refractive index change due to the gas and ions, but also due to the Gouy phase shift of the driving field and the electron trajectories. Because we are in the tight 112  5.4. Harmonic Amplitude Calculation focussing regime, the geometrical effects of the focus must be taken into account. On axis, the phase of the fundamental and harmonic fields are, qφ1 (z, t) = qk1 n(z, ω1 )z − q arctan −qω1 t φq (z, t) = kq n(z, ωq )z − arctan  z n(z, ω1 )zR  z n(z, ωq )zR  + Φatom (z)(5.43)  − ωq t  (5.44)  where kq = ωq /c and ωq = qω1 . The index of refraction due to the gas and plasma of the fundamental and q th harmonic, respectively, is, n(z, ω1 ) ≈ 1 +  ωp2 ρ(z) (1 − η(z)) δn − 2 η(z) ρ0 2ω1  ωp2 ρ(z) n(z, ωq ) ≈ 1 + − 2 η(z) ρ0 2ωq where ρ0 =  P0 kB T  (5.45) (5.46)  with P0 = 1 atm, δn = 6.9 × 10−4 at λ1 = 800 nm [122],  η(z) is the ionization fraction, and ωp is the plasma frequency.  5.4.1  Calculation of Atomic Phase  The atomic phase, Φatom (z) = −αi I(z) is the intensity dependent phase of  the active electron whose trajectory creates the harmonic q [123]. As shown in the classical, strong field case (see Fig 5.3), there are two dominating electron trajectories that create the harmonics. It has been shown that even for harmonics near the ionization potential of the generating atom, there may be multiple quantum pathways to harmonic generation [124]. Following the analysis of [125], we calculate the values and relative amplitudes of the phase coefficients as a function of the driving field. The method for calculating the values of α uses the fact that α and I are Fourier transform conjugate variables. Given that we have calculated the dipole response for a given harmonic as a function of the the driving field amplitude, ˜ q , I) (such as in Fig 5.8), then the dipole response for a certain harmonic d(ω  113  5.4. Harmonic Amplitude Calculation  1  1  0.8  0.8  0.6  0.6  0.4  0.4  0.2  0.2  0 0  0.2  0.4  0.6  0.8  1  0  (a)  (b)  (c)  (d)  Figure 5.11: Contour plots of the relative amplitudes of the phase coefficient α as a function of the peak driving field for a 70 fs pulse at 800 nm in xenon. The harmonics presented are (a) 7th , (b) 9th , (c) 11th , and (d) 13th . These values agree well with those of classical model calculated in the strong field approximation in Fig 5.3. The relative amplitudes shown are normalized to the sum over all the trajectories for a harmonic at a given intensity (not to the dipole response at that intensity). The 7th harmonic has only the short trajectory present, whereas the 9th and 11th harmonics have some long trajectory at low intensity, and then are dominated by the short at higher intensity. The 13th harmonic is dominated by the cutoff trajectory for low intensity (approximately 4 × 1013 W/cm2 is the classical prediction for creating the 13th harmonic) and bifurcates, with the long trajectory dominating.  114  5.4. Harmonic Amplitude Calculation trajectory is, ˜ ˜ q , α) = d(ω  ∞  ˜ q , I)eiαI dI. d(ω  (5.47)  −∞  Unfortunately, Eq 5.47 only gives the phase coefficient amplitude of the dipole response over the entire intensity range. Because the dipole response changes as a function of driving field intensity, it is also interesting to understand how the phase coefficients also change. This could be solved by ˜˜ changing the limits of the FT integration, and assume that d(ω q , α) is calculated for the mid-point of the integration. However, as this is a FT, a constraint on the limits of integration over I loses resolution in α. In order to have sufficient resolution in the α domain, the range of I must be sufficiently large. A solution to this problem is to use an envelope function that weights the integration. In this manner, the limits of integration in I can be large to allow for sufficient resolution in α, but the dominant (largest weighted) intensity is centred at the intensity of interest. As the centre of mass of the weighting function is moved through I, the dipole response can be calculated, thus giving the FT of the dipole response an intensity dependence. A suitable weighting function is W (I − I0 ) = H(I − I0 )(I − I0 )2 e−a(I−I0 ) [125], where H(I − I0 ) is the Heaviside function. This weighting function has the  properties that it peaks at I = 2/a + I0 , and has an intensity resolution of ∆IW ≈ 3.4/a, where a is given in units of cm2 /W. Thus, the dipole response for the different trajectories is calculated by, ˜ ˜ q , α|I0 ) = d(ω  ∞ −∞  ˜ q , I)W (I − I0 )eiαI dI. d(ω  (5.48)  In Fig 5.11, the phase coefficients are calculated for a 70 fs pulse at 800 nm interacting with xenon. In order to have the resolution presented, the upper peak intensity was 5 × 1014 W/cm2 . However, at these high inten-  sities and long pulse durations, xenon is completely ionized. Consequently, the dipole response for the tail end of the pulse is noisy and corrupts the calculation; for the presented figures only the first half of the pulse was used 115  5.5. Comparison with Experimental Data to calculate the dipole response. The harmonics presented are (a) 7th , (b) 9th , (c) 11th , and (d) 13th . The values of the phase coefficient α agree well with the classical model results calculated in the strong field approximation in Fig 5.3. The relative amplitudes shown are normalized to the sum over all the trajectories for a harmonic at a given intensity (not to the dipole response at that intensity). The 7th harmonic has only the short trajectory present, whereas the 9th and 11th harmonics have some long trajectory at low intensity, and then are dominated by the short trajectory at higher intensity. The 13th harmonic is dominated by the cutoff trajectory for low intensity (approximately 4 × 1013 W/cm2 is the classical prediction for creating the  13th harmonic) and bifurcates, with the long trajectory dominating. The general trends for the values and amplitudes for the phase coefficients are found to be in agreement with those experimentally determined (discussed below in Sec 5.5).  5.5  Comparison with Experimental Data  The setup for measuring the relative harmonic amplitudes is shown in Fig 5.12. The high harmonics were out-coupled via an intracavity grating mirror [41]. We placed a microscope slide coated with sodium salicylate (sensitive to EUV, peak fluorescence at 425 nm) 7 cm from the grating mirror to view the spectrally resolved harmonics. The sodium salicylate slides were imaged from the back using a CCD camera. The counts (or the measured amplitude) for a harmonic were calculated by summing the grey-scale values of the camera of the harmonic over the whole far-field image, meaning both the off-axis and on-axis components of the generated beam were taken into account. The intracavity grating mirror acted as our spectrum analyser. This mirror was designed to be highly reflective (R > 99.99%) with low dispersion GDD = 0 fs2 for the 800 nm light at a 70◦ angle of incidence. The amplification factor and intracavity spectrum was nominally unchanged by the insertion of the grating mirror. The grating etched on the mirror surface is made by interference lithography, where the idealised grating profile is a 116  5.5. Comparison with Experimental Data  Na:S  focus  7th Xe  13th  GM (a)  (b)  Figure 5.12: The cavity setup for measuring the beam shapes and relative amplitudes. (a) The nozzle (purple circle) delivered the gas to the intracavity focus where the harmonics were generated. (b) A magnified view of the of the focal region; GM grating mirror with relief profile used to outcouple the harmonics was highly reflective for 800 nm light (RGM > 99.99%) at 70◦ angle of incidence; Na:S microscope slide coated with sodium salicylate was sensitive to EUV and fluoresced at 425 nm, and Xe was the xenon gas delivered to the focus by the nozzle. The nozzle was moved through the focus in order to determine the spatial dependence on HHG.  rectangular pattern with a 200 nm pitch and 50 nm depth. The calculated diffraction efficiency of the grating is dependent on the harmonic and is between 3 − 5%, peaking around the 9th harmonic [126, 127] as shown in  Fig 5.13. The EUV light that is not coupled out of the cavity is absorbed by the second (intracavity) focussing mirror. The diffraction efficiency is based on the pitch and shape of the grating; small defects within the manufacturing of the grating can change the output coupling efficiency and therefore the amplitude of a harmonic. A sample series of images used in measuring the harmonic amplitude is shown in Fig 5.14. Although only one exposure time is shown, the camera used in taking the data was set at different exposure times, allowing for a larger dynamic range. The nozzle was moved in 150 µm increments to show the harmonic amplitude dependence on position. The exact location of the focus was not known, but is estimated by the position of the created plasma relative to the nozzle location and the generated beam shapes. The different  117  5.5. Comparison with Experimental Data  4.5  Diffraction Efficiency (%)  9th 7th 4 11th  3.5 5th 3 5  10  13th 15 20 25 30 Diffraction Angle (degrees)  35  40  Figure 5.13: The first order diffraction angles and efficiencies calculated by Ref [126] for harmonics five to thirteen. The beam has a 70◦ angle of incidence, and the grating has a 200 nm pitch, 50 nm depth with a relief (not blazed) profile.  nozzle positions show the strong intensity dependence due to the tight focus on the generation process. The off-axis phase matching is apparent for the 9th and 11th harmonics just before the focus by the large, diffuse ring. The nozzle configuration was an end-fire setup, with the beam propagating just below the tip (the exit aperture) of the nozzle. The diameter of the apertures of the nozzles tested were 150 µm, 300 µm, and 500 µm. Although the pressure in the xenon line was measured, there was some uncertainty in the height of the nozzle above the beam, which led to an uncertainty in the density at the focus. The distance from the nozzle tip to the focus was estimated to be 150 µm. The OpenFOAM calculations showed that for the 500 µm and 300 µm nozzles, the density was fairly uniform in this region. However for a nozzle with a smaller aperture of 150 µm, the gas density was reduced to a third of that of the larger nozzles. Because gas density was important in the harmonic amplitude calculation, we used the results of the OpenFOAM simulations to account for its position dependence (both along the beam direction of propagation as well as distance from the nozzle end). The theoretical harmonic amplitude, as calculated by Eq 5.42, was 118  5.5. Comparison with Experimental Data 150µm  7th  9th 11th 13th  Figure 5.14: Image series of 8 s exposures of the harmonics incident on a sodium salicylate slide; the image is saturated to show the full beam profile of harmonics seven through thirteen. The nozzle had a 150 µm aperture diameter and peak intensity was 3 × 1013 W/cm2 . The nozzle position was moved through the focus (approximate position of focus represented by red line) by steps of 150 µm (white lines), showing the change in amplitude and shape of the generated harmonics. The beam shape - the small, high intense region in the middle, or the large, outer ring - and position of maximum amplitude gave insight into the pathways for generation of the harmonic. The 7th and 9th harmonics have a large spot in the middle that reaches a maximum amplitude near the focus, while the 11th and 13th have a diffuse maximum amplitude ∼ 300 µm after the focus.  scaled by the diffraction efficiency of the intracavity grating. To match the theoretical harmonic amplitudes with the experimentally determined count values, the amplitude of the harmonics was scaled by a set value (constant for all pressures, intensities, harmonic, and gas nozzle size). We obtained excellent agreement between the experimental data and theory, especially for the 150 µm and 300 µm nozzles. The data for the 150 µm and 300 µm nozzles (Figs 5.15 and 5.16, respectively) had been collected for a range of intensities and gas nozzle positions. As the nozzle position was moved from before the focus to after (left to right in figures), it was found that the amplitude of the 7th and 9th harmonics peaked near the focus position. From that information and the fact these beam shapes had a bright central spot, it was found that these harmonics were dominated by the short trajectory. Thus, for the comparison of theory to experiment, the  119  5.5. Comparison with Experimental Data  7  2  7  x 10  9th 11th 13th  1  x 10  3 2.5 Counts  1.5 Counts  3.5  7th  2 1.5 1  0.5  0.5 0 ï600  ï400  ï200 0 200 Position (µm)  400  0 −600  600  −400  −200 0 200 Position (µm)  (a)  600  0.4  0.6  (b)  6  7  x 10  x 10 4  18  3.5  16 14  Amplitude (arb u)  Amplitude (arb u)  400  12 10 8 6  3 2.5 2 1.5 1  4  0.5  2 −0.4  −0.2  0 0.2 Position (mm)  (c)  0.4  0.6  −0.4  −0.2  0 0.2 Position (mm)  (d)  Figure 5.15: Comparison of (a), (b) experiment and (c), (d) theory for the 7th (red), 9th (green), 11th (black), and 13th (blue) harmonics. The nozzle aperture diameter was 150 µm. The theoretical density was 0.093 kg/m3 (measured backing pressure was 80 Torr); (a), (c) the intensities were 2.5 × 1013 W/cm2 ; (b), (d) 3.1 × 1013 W/cm2 . For comparison to experiment, the theoretical 7th and 9th harmonics were generated by the short trajectory, while the 11th and 13th harmonics were by the long. To match the theoretical amplitude to the experimental count value, all harmonic amplitudes were scaled by the grating efficiency and a constant amplitude factor to convert the theoretical amplitudes to counts.  120  5.5. Comparison with Experimental Data short trajectory was chosen. This phase coefficient was in agreement with the theoretical calculations as shown in Fig 5.11(a) and (b). Conversely, for the 11th and 13th harmonics, the position of the maximum amplitude occurred slightly after the focus, and the beam shape was larger and diffuse. This led to the conclusion that these harmonics were dominated by the long trajectory, in agreement with theory shown in Fig 5.11(c) and (d). For consistency, all the theoretical calculations of the generated harmonic amplitudes used the short trajectory for the 7th and 9th harmonics and the long trajectory for the 11th and 13th for all nozzle sizes and intensities. 6  12  2.5  7th 9th 11th 13th  10  x 10  2  Counts  8 Counts  7  x 10  6  1.5 1  4  0.5  2 0 ï600  ï400  ï200 0 Position (µm)  200  0 −600  400  −400  (a) 2.5  10  0.2  0.4  x 10  2 Amplitude (arb u)  Amplitude (arb u)  400  7  x 10  8 6 4  1.5  1  0.5  2 0 −0.6  200  (b)  6  12  −200 0 Position (µm)  −0.4  −0.2 0 Position (mm)  (c)  0.2  0.4  0  −0.4  −0.2 0 Position (mm)  (d)  Figure 5.16: Same as Fig 5.15, but for the 300 µm aperture nozzle. The theoretical density was 0.044 kg/m3 (measured backing pressure 20 Torr); (a), (c) the intensities are 2.6 × 1013 W/cm2 ; (b), (d) 3.2 × 1013 W/cm2 .  121  5.5. Comparison with Experimental Data Because the region of good phase matching is strongly dependent on the intensity for the harmonics dominated by the long trajectory (due to the αlong I phase dependence), one may think that the amplitude of the long trajectories should be strongly dependent on the intensity as well. However, from both theory and experiment, we see that the position of peak amplitude for these nozzles did not significantly change over the range of intensities used. Even though the dipole response for the high harmonics has a complicated dependence on the driving field intensity, we got excellent agreement comparing the amplitude to this parameter as shown in Fig 5.17. The nozzle position (for both the 150 µm and 300 µm nozzle) was taken to be 150 µm after the focus and the intensity was varied over the experimentally attainable values. The largest discrepancy occurred for the 7th harmonic which was systematically too high, which implied that the diffraction efficiency of the grating for the 7th harmonic was calculated to be too high. The monotonically increasing harmonic amplitude as a function of driving field intensity presented here is in contrast to that found in Ref [128]. The shorter wavelength driving field leads to a smaller phase coefficient (see Eq 5.11), and thus the interference of the two trajectories is less sensitive to the driving field intensity. Thus, in order to demonstrate the interference from the two quantum pathways, the peak intensity range must be larger than was experimentally available. The 500µm diameter nozzle theory, shown in Fig 5.18(b), has an odd feature near the focus z = 0 for the 7th and 9th harmonics, but not in the data (Fig 5.18(a)). The feature was the decrease in amplitude near the focus, which was due to long interaction of the high intensity driving field and the gas. The Rayleigh range was zR ≈ 550µm, which meant that there  was a large change in the intensity as the beam propagated through the gas. The Gouy phase shift was large enough over this propagation distance that the phase of the 9th harmonic field at the entrance to the gas was nearly π  out of phase with the exit which led to destructive interference around the focus. We can account for the 9th harmonic amplitude being a maximum near the focus by including slight changes in the intensity, dipole response, 122  5.6. Optimizing the Output  7  10 7  Counts  Counts  10  6  7th 9th 11th 13th  5  10  6  10  10  2  2.5 3 Intensity (× 1013 W/cm2)  3.5  2  2.2  (a)  2.4 2.6 2.8 3 Intensity (× 1013 W/cm2)  3.2  (b)  Figure 5.17: Comparison of experiment (circles) and theory (lines) for the 7th (red), 9th (green), 11th (black) and 13th (blue) harmonics for the (a) 150 µm and 300 µm diameter nozzles. The position of the nozzle was located 150µm after the focus for both cases. Because the short trajectory dominated the 7th and 9th harmonics, they had a different intensity dependence than the 11th and 13th harmonics, which were dominated by the long.  and accounting for harmonic generation off-axis. It should be noted that we estimated the minimum intracavity waist from the cavity stability condition and the focussing mirrors’ radius of curvature, and it was not directly measured. Because of the non-zero angle of incidence on the focussing mirrors, the beam is astigmatic within the enhancement cavity. Consequently, the tangential and sagittal planes had different focal positions and minimum radii. This led to an uncertainty in the peak intracavity intensity.  5.6  Optimizing the Output  Now that we have shown that we can predict the power of the harmonics and how they scale with gas jet size, pressure and intensity, we can use the theory to find the optimal configuration to maximize the harmonic power. One further constraint, which is discussed in Sec 4.2.2, is that the strong dependence of the plasma density on the driving field amplitude can cause 123  5.6. Optimizing the Output 7  7  x 10  7th 9th 11th 13th  Counts  1.5  1  0.5  0 ï400  2  x 10  1.5 Amplitude (arb u)  2  1  0.5  ï200  0 200 Position (µm)  400  600  0 −0.4  (a)  −0.2  0 0.2 Position (mm)  0.4  0.6  (b)  Figure 5.18: Comparison of (a) experiment and (b) theory for the 7th (red), 9th (green), 11th (black), and 13th (blue) harmonics for the nozzle aperture size of 500 µm, with a backing pressure of 20 Torr and peak intensity of Ipeak = 3.1 × 1013 W/cm2 . The low power of the 9th harmonic near the focus in (b) is due to the intensity dependent dipole response and the Gouy phase shift. Because we are in the tight focussing regime, the intensity is strongly dependent on the position (both along the beam and off-axis), and accounting for off-axis beam generation can lead to a maximum harmonic at the focus.  problems with the cavity lock. Noise on the cavity lock can make it difficult to maintain stable high peak power, thus affecting the amplitude and quality of the harmonics generated. Thus, we are interested in increasing the generated harmonic amplitudes while minimizing the generated plasma volume. Additionally, we would like to minimize the gas flow rate as that would reduce the load on the vacuum system. The harmonic amplitudes are plotted in Fig 5.19 as a function of gas pressure (at the focus) for different harmonics and nozzle aperture diameters. The harmonic amplitudes are maximized when the pressure near the focus reaches 250 Torr, almost independent of the nozzle diameter and phase coefficient. The smaller aperture nozzle can create a similar harmonic amplitude for a similar gas density. Increasing the density decreases the absorption and coherence lengths, which limits further increase in the harmonic amplitude [37]. Therefore, to minimize the gas flow and the number 124  5.6. Optimizing the Output  α = 1, q = 7, d = 500µm  α = 25, 500µm 30, q = 11, d = 500µm  (a)  (b)  α = 1, q = 7, d = 150µm  α = 25, 30, q = 11, 11, dd = = 150µm 150µm  (c)  (d)  Figure 5.19: Contour plots of the amplitudes of the (a), (c) 7th and (b), (d) 11th harmonics generated in xenon from a 70 fs pulse at 3 × 1013 W/cm2 for the (a), (b) 500 µm diameter aperture nozzle, and (c), (d) 150 µm nozzle. The phase coefficients chosen were determined by the dominating electron trajectory (α = 1 × 10−14 cm2 /W for 7th ; α = 25 × 10−14 cm2 /W for 11th harmonic). Phase matching affects the amplitude of the 11th harmonic for lower pressures until 200 Torr, where the absorption length begins to dominate. The optimal pressure is ∼ 250 Torr for these harmonics, where it is shown that the harmonic amplitude is independent of the nozzle length.  125  5.6. Optimizing the Output of extraneous ions that can disrupt the cavity stability, the optimal nozzle design is a small diameter nozzle near the focus with a very high backing pressure.  5.6.1  Improving the Output Coupling Efficiency  The low diffraction efficiency predicted for the intracavity grating mirror, as shown in Fig 5.13, leaves much room for improving the power of the useable, outcoupled, EUV light. The benefit of using an intracavity grating mirror is the scalability of the intracavity power. It was shown not to damage from prolonged exposure to high fundamental power and EUV light. Unfortunately, the low coupling efficiency allows for the subsequent intracavity mirror to be damaged by EUV. The mirror damage has been a main limitation of the long-term usability of this system. The grating for this work was created by interference lithography, which has a relief grating profile. The efficiency of the grating can be improved by having a blazed profile [129], which can increase the diffraction efficiency to up to 20%. The original method for separating the harmonics generated by a cavity from the fundamental beam was with a Brewster window [38, 39]. This method, exploiting the different indices of refraction of a thin sapphire window for the fundamental field and harmonics, is placed at the Brewster angle. It is moderately more efficient than the grating (∼ 10% for certain harmonics), but does not scale well with power. The fundamental beam is transmitted through the Brewster window, where the nonlinear index of refraction of the window can limit the intracavity power. The sensitivity of the intracavity field to the nonlinear index of refraction will be true of all methods that separate the harmonics from the fundamental beam by transmission. However, by modifying the surface of the Brewster window, the coupling efficiency was recently greatly improved to ∼ 80% [130]. Another output coupling method that was recently investigated can couple up to 20%, and may be able to scale better with fundamental power [131]. Because the harmonics diverge at a smaller angle than the fundamental, another proposal was to have an aperture in the intracavity mirror immedi-  126  5.7. Summary ately after the harmonics are generated. The fundamental beam could diverge and largely avoid the aperture while the harmonics could pass through. However, it was found experimentally difficult to couple out the high harmonics [132]. Further refinements in mode coupling and cavity design have led to tailoring the beam shape so the fundamental beam is less affected by the aperture, while the hope is that the EUV generated can easily pass through [133]. Each of the methods used for coupling the harmonics out of the cavity have their benefits and limitations. A low dispersion, high damage threshold, and efficient harmonic coupling optic could greatly improve the uses of harmonics generated via an enhancement cavity.  5.7  Summary  In conclusion, we have solved the time-dependent Schr¨odinger equation to calculate the dipole response of a 1-D atom in a strong laser field. We have used this for the driving term in the non-linear polarization to determine the amplitude of the harmonics created. We have compared theory with experiment and shown excellent agreement for the seventh to thirteenth harmonics. We have then used the theory to tune the parameters to maximize the amplitude of the harmonics created. Although many research groups who use a single pass system have been increasing the interaction length to increase the harmonic amplitudes [30, 31], we are limited by the pulse energy and must operate with a tight focus. The tight focus has led to a different set of parameters that can optimize the harmonic amplitudes. We have found, for our experimental conditions, that a short interaction region and high density, with the gas delivered near the focus, gives the maximum harmonic power possible.  127  Chapter 6  Relative Intensity Noise of High Harmonics from an Enhancement Cavity In many techniques of laser spectroscopy, it is highly beneficial to utilize a low amplitude noise source for several reasons including higher sensitivity and significantly decreased acquisition times. In the case of high harmonic generation (HHG), the high harmonics are strongly dependent on the fundamental driving field amplitude. Therefore, fluctuations in the power of the fundamental field can lead to significant amplitude noise in the high harmonics. In order to use the intracavity generated high-order harmonics for spectroscopy, it is important to characterize and minimize the amplitude noise. This chapter discusses the measurement and results of relative intensity noise (RIN) for the laser, the amplified field circulating in the femtosecond enhancement cavity (fsEC), and the intracavity generated 11th harmonic. For a discussion of the sources of noise and the RIN measurement, see Appendix C.  6.1  Measurement Details  A highly reflecting (for 800 nm light) intracavity mirror with a grating etched on its top layer was used to outcouple the extreme ultraviolet (EUV) light [41]. The photodetector used to detect the EUV light was placed 12 cm from the grating mirror. Because of the proximity of the photodetector to the fsEC with a circulating power of several hundred watts, we used a  128  6.1. Measurement Details SA Cavity  PDq focus  PDR PDT  Figure 6.1: Schematic of measuring the RIN. The intracavity power is transmitted through a mirror and measured with the photodetector P DT ; the reflection signal is spectrally resolved and measured with P DR . The harmonic signal is coupled out of the cavity by the intracavity grating mirror and measured with a photodiode that has a directly deposited filter to remove the scattered fundamental light PDq ; the signal is amplified by the SRS current amplifier. The spectra are measured with the spectrum analyser (SA).  photodiode with a directly deposited aluminum filter (IRD part number AXUV100Al, passband 17 − 80 nm) to attenuate the scattered visible light.  While other photodiodes were available to measure different harmonics, this model was chosen because of the strong attenuation of the fundamental light and a high quantum efficiency for the 11th harmonic (72 nm). A microscope slide was coated with sodium salicylate and was placed before the photodetector for alignment, and then removed for the measurements. The measurement setup is shown in Fig 6.1. The benefit of having the photodetector close to the grating mirror was that the 11th harmonic beam profile was smaller than the photodiode face (which measured 1 cm×1 cm), allowing us to measure the entire beam power and was easier to align into the photodetector. However, with this position of the photodetector several factors had to be considered: the need to block the incoherent EUV light generated from the electron-ion recombination of the plasma using an anodized aluminum shield, and care was taken to ensure that free electrons and ions would not contact the photodiode face or 129  6.1. Measurement Details  PD case Wire  7th  9th  13th  Plasma shield  Nozzle FM1  FM2  GM  Xe plasma  Figure 6.2: Picture of xenon gas introduced at focus of the fsEC. GM grating mirror used for coupling out harmonics; PD case is the photodetector in an electrically shielded mount; wire is the photodetector signal wire within copper tubing; FMi are the 10 cm radius of curvature focussing mirrors; Plasma shield used to stop incoherent EUV light emitted by the plasma from hitting the PD. The photodiode face is at the square aperture located between the 9th and 13th harmonics. The PD case was coated in sodium salicylate to show location of other harmonics.  130  6.2. Results wire leads which could cause a non-optical signal, as shown in Fig 6.2. The charged particles and excess neutral gas were directed by the nozzle into a grounded metal cup, which was pumped by the turbo pump. The photodetector was electrically shielded by being placed in an aluminum case. The anode and cathode of the photodetector were connected to a pair of wires (LakeShore part number QL-36), which was enclosed in a 1/8” copper tubing. The vacuum chamber feedthroughs were individual pins so the wires were split a few centimeters before the feedthrough. To prevent the separated wires from picking up electrical noise, aluminum foil was used to electrically shield the separated wires. The Stanford Research Systems current amplifier (model number SRS78) was set to 500 nA/V in high bandwidth mode. This setting allowed for a measurable signal of > 100 mV with a bandwidth of 20 kHz. The data was recorded using an SRS FFT spectrum analyser (model number SRS780).  6.2  Results  Although the ultimate goal was to measure the relative intensity noise (RIN) of the high harmonics generated, we were also interested in the influence of the noise of the laser output and the amplified circulating field in the fsEC on the harmonic signal. The RIN of interest is referred to as the excess RIN, that is, the amplitude fluctuations above the shot noise (see Appendix Sec C.4.1); shot noise has a flat spectral response. As expected, the laser had low RIN, where over the span of 100 Hz to 10 kHz it was less than −90 dBc/Hz, as shown in Fig 6.3. Unfortunately, the laser noise spectrum appeared to be shot noise limited above 10 kHz. This  was due to the limited dynamic range of the photodetector, which saturated above a few Volts. Because the noise scaled linearly with the laser power, the laser RIN was independent of laser power. The RIN of the amplified, circulating field in the enhancement cavity was 20 − 30 dB/Hz higher than that of the laser over the measured spectrum in Fig 6.3. The additional noise stemmed from the imperfect lock of the fsEC  131  6.2. Results  Laser Cavity 300Torr Xe H11 H11 BG  ï40 ï50  RIN (dBc/Hz)  ï60 ï70 ï80 ï90 ï100  servo bump  ï110  mirror resonance  ï120 ï130 2 10  3  4  10  10  5  10  Frequency (Hz)  Figure 6.3: RIN for the laser (red), the amplified circulating field in the bare cavity (black), the ampified field with 300 Torr (backing pressure) xenon at the focus (blue), and the 11th harmonic (green). Also shown is the background of the current amplifier set to 500 nA/V (yellow). The 11th harmonic signal was above the instrument noise floor until nearly 10 kHz. The laser and empty cavity noise scaled with power, thus the RIN was independent of power. Gas introduced to the focus had a (intensity dependent) nonlinear index of refraction, changing the amplified field RIN spectrum. Because the amplified field creates the harmonics, the measured 11th harmonic RIN was also intensity dependent.  to the laser comb26 . The loop bandwidth of the fsEC length lock was limited by the mechanical resonance of the PZT. To lock the cavity length to the laser, we used a small (7.75 mm diameter, 2 mm thick) mirror attached to a PZT on a small, lead-filled copper slug. The mechanical setup of the mirror mount, PZT, and slug, led to a resonance around 55 kHz [134]. In order to avoid the mechanical resonance, the servo loop bandwidth was limited to 26  As mentioned in Sec 2.5, we used a Pound-Drever-Hall (PDH) technique to lock the cavity length. The laser Ω0 was locked, as described in Sec 2.6.1. With these two locks, the intracavity power and spectrum remained constant throughout the duration of the measurements.  132  6.2. Results 35 kHz. As long as the intracavity field had only linear loss mechanisms, the intracavity RIN was independent of intracavity power. Introducing 300 Torr of xenon gas at the fsEC focus changed the noise spectrum of the circulating intracavity field. The “servo bump” shifted to 30 kHz, and there was an increase in the RIN attributed to the nonlinear index of refraction of the gas (both contributed by the neutral gas intensity dependent index of refraction, n2 , as well as the plasma) [135]. The nonlinear refractive index caused fluctuations in the intracavity optical path length, which led to noise on the cavity length. Therefore, when gas is present at the cavity focus, the amplified field RIN is power dependent. Methods to overcome the nonlinear phase change caused by the plasma and maintain a stable lock have been developed [136]. The RIN of the 11th harmonic was within 15 dB of the amplified circulating field, and showed many similar features.  The RIN was below  −50 dBc/Hz over the frequency range shown, which demonstrated that this  was a low amplitude noise source (the root-mean-square noise is discussed below). By 2 kHz, its spectrum was already following that of the circulating field and monotonically decreasing. Unfortunately, the noise gain of the SRS current amplifier led to a background gain peak that artificially affected the signal around 10 − 30 kHz. The oscillation of the servo loop, a prominent  feature in the circulating field, slightly affected the harmonic signal spectrum at 30 kHz. The nonlinear dependence of the harmonic power on the amplified field intensity, as well as the increased noise of the amplified field due to the gas, meant that the harmonic RIN was dependent on the power. The fsEC generated harmonics can be very useful in experiments that require near continuous wave EUV radiation with low amplitude noise. The detected signal for the 11th harmonic was 138 mV with a 6 mV background (due to leakage through the filter of the fundamental scattered light). The uncalibrated quantum efficiency for this photodetector is unity at 72 nm. At the SRS settings of 500 nA/V, this implied that we were detecting 4 ×  1011 photons/second at 72 nm, or 1.1 µW. This useable optical power in  the EUV is comparable to that of other fsEC generated high harmonics [40, 41, 136, 137], high repetition rate (50 kHz − 1 MHz) single pass systems 133  6.2. Results [14, 19, 20, 37], single pass waveguide assisted systems [29, 30], free electron lasers (FELs)[6, 138], and synchrotron sources. −45 no Xe 300Torr Xe  −50  RIN (dBc/Hz)  −55 −60 −65 −70 −75 −80 −85 2 10  3  4  10  10  5  10  Frequency (Hz)  Figure 6.4: The effect of introducing 300 Torr (backing pressure) of xenon near the focus on the optical bandwidth limited cavity reflection signal (red) compared to no xenon (blue). Because of the narrow optical bandwidth detected, this measurement was independent of the offset frequency, thus introducing xenon increased the fluctuations of the cavity optical path length.  In Fig. 6.4, we measured the change in the RIN of the reflection signal from the cavity, used for locking ωc . The reflection signal optical bandwidth was limited by a grating (1200 lines/mm), which spatially dispersed the beam so that only 1 nm of optical bandwidth was detected. This signal was therefore largely independent of the offset frequency fluctuations, and was used to lock the cavity length to the laser. The increase in RIN due to the xenon was attributed to optical path length fluctuations within the cavity discussed previously. As shown in Fig 6.5, locking the offset frequency decreased the low frequency intracavity amplitude noise by 20 dB. Systematic, low frequency drifts in the offset frequency limited the intracavity power. Due to the strong dependence of the harmonic amplitude on the fundamental power, 134  6.2. Results  −40  with Ω0 lock no Ω0 lock  −50  RIN (dBc/Hz)  −60 −70 −80 −90 −100 −110 2 10  3  4  10  10  5  10  Frequency (Hz)  Figure 6.5: Effect of offset lock on the intracavity RIN. The unlocked case (magenta) was ∼ 20 dB noisier than the locked case (black) for low frequencies, but did not affect the noise spectrum above the locking bandwidth of the fast PZT (50 kHz). Without an offset lock, slow, systematic drifts in the laser offset frequency limited the average intracavity power.  they were detrimental to an amplitude measurement of the 11th harmonic. Therefore, although the offset frequency lock did not significantly decrease the noise spectrum above 1 kHz, it was required in order to maintain the average (DC) intracavity power for a consistent measurement of the harmonic signal. The cumulative root-mean-square (RMS) noise of the 11th harmonic was below 1.2% (see Fig. 6.6). This is significantly lower (by one to two orders of magnitude) than the fluctuations observed in traditional, low repetition rate high harmonic systems [33] and FELs [139]. The cumulative RMS of the 11th harmonic closely followed that of the intracavity field, but was about a factor of 3 larger. Because the fsEC effectively adds a low pass filter with a double pole at the cavity linewidth (∆Ω/2π = 30 kHz) [140], we do not expect significant contributions to the cumulative RMS beyond the measured bandwidth. Therefore, the cumulative RMS at 100kHz is a good 135  6.3. Outlook measurement of the total RMS amplitude fluctuations in the 11th harmonic. 1.4  Cumulative RMS (%)  1.2 1  H11 with Xe no Xe Laser  0.8 0.6 0.4 0.2 0 2 10  3  4  10  10  5  10  Frequency (Hz)  Figure 6.6: Cumulative RMS of the laser (red), the bare intracavity field (black), the intracavity field with 300 Torr xenon (backing pressure) introduced at the focus (blue), and the 11th harmonic (green).  6.3  Outlook  The main factors contributing to the 11th harmonic RIN power dependence were not obvious because of the complicated nature of the harmonic generation process. Because the peak intensity was 3 × 1013 W/cm2 and the 11th  harmonic was near the maximum harmonic cutoff, the amplitude of this harmonic was extremely sensitive to the fundamental amplitude fluctuations. Increasing the peak intracavity power such that the measured harmonic is well below the cutoff frequency should decrease the noise because of the decreasing sensitivity of the harmonic power on the driving field. However, increasing the intensity also increases the effect of the nonlinear index of refraction of the xenon and plasma, and can add noise to the circulating field amplitude. Therefore, the intensity dependent noise domain must be well 136  6.3. Outlook understood to minimize the noise on the circulating field and the harmonics generated. We have also made attempts to measure lower order harmonics, but we have found that the optical filters were insufficient for a measurement with a good signal to noise ratio. However, these lower harmonics may be interesting to study because the harmonics below cutoff have multiple pathways to creating the harmonics, and the interference of the multiple pathways may introduce its own noise. Using the results of Sec 5.6, we can design and locate the nozzle so as to maximize the two different trajectories (long and short) separately. Because the short trajectory has a phase that is substantially less dependent on intensity than the long, this trajectory may lead to smaller amplitude fluctuations as well. Thus, it may prove interesting to investigate the amplitude noise of the harmonics for the different quantum pathways.  137  Chapter 7  Conclusions In this thesis, the enhancement cavity was shown to be a viable method for amplifying the output of a mode-locked, femtosecond Ti:Sapphire laser, while maintaining the repetition rate. We used the stable comb spacing of the laser to measure the residual intracavity dispersion of the enhancement cavity. We also used the intracavity photon lifetime in order to predict the mirror reflectivity. With this information, the intracavity power of the empty cavity was able to reliably reach peak intensities up to Ipeak = 4 × 1013 W/cm2 .  The interaction of xenon with the amplified and high repetition rate intracavity field at the focus led to interesting - but parasitic - plasma phenomena. The nozzle geometry often used to deliver a noble gas to the focus of a low repetition rate system (referred to as the through nozzle) proved not to be suitable for enhancement cavities. The theory of plasma dynamics and the instability of the cavity lock implied that there was a cumulative generated plasma. To optimize the neutral gas density and minimize the plasma density at the focus, we modelled the gas flow using computational fluid dynamics of various nozzle geometries. Through modelling the harmonic generation process, taking factors into account such as the dipole response, the plasma and gas densities, and the tight intracavity focus, we were able to accurately predict the amplitudes of harmonics seven to thirteen. We found that in our regime, the methods for optimizing the harmonic amplitudes were different from the high power, low repetition rate systems typically used to generate high harmonics. A short interaction region, with high gas density near the focus, maximized the generated harmonics. With the results of our model, we were able to generate > 30 µW of coherent 72 nm light, and have coupled out > 1 µW for 138  Chapter 7. Conclusions use in an experiment. We predict, through our model, that improvements in the vacuum system design and nozzle geometry may increase the harmonic power by a factor of five. We also measured the relative intensity noise of the eleventh harmonic. Despite improvements on nozzle design to minimize plasma density, introduction of xenon to the intracavity focus increased the amplitude noise spectrum of the amplified, circulating field. Nevertheless, the cumulative root-mean-square noise of 1.2% measured over the 100 Hz − 100 kHz range is still substantially lower than the harmonics generated by traditional amplifier systems. By changing the location of the gas nozzle, thereby changing the phase matching, we were able to observe the different quantum trajectories that the electrons take to generate the harmonics. We would like to further investigate the amplitude noise of the generated harmonics. The phase dependence of the different quantum paths that the harmonic generating electrons take has a different intensity dependence. Therefore, the noise may also depend on the different quantum trajectories. We may also be able to lower the harmonic amplitude noise by lowering the plasma generation rates by changing gas types or using a mixture of gases, or further refining the nozzle design to minimize the interaction lengths. A main limitation of this system is the mirror degradation. In order for this source to be more useable in the long term, the high intracavity power must last as long as the experiment requires. Much of the mirror damage seems to come from the EUV light that is not coupled out of the cavity. Improvements in coupling efficiency may increase the generated EUV power by up to a factor of 30, and may increase the lifespan of the highly reflective mirrors. This system can deliver extreme ultraviolet photon fluxes comparable to other sources such as traditional, low repetition rate, high harmonic generating systems and free electron lasers. However, because of the substantial amplitude noise reduction in this system, it may prove useful for many experiments. 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The field builds up at a rate of E0 tic (power transmission of input coupler is TIC ), √ and dissipates by a factor of RIC Rcav per round trip N (see Eq 3.1). To varying degrees of approximations of the decay factor, this decay rate can be expressed as, −ln RIC Rcav ≈ 1 − RIC Rcav 1 ≈ (1 − RIC Rcav ) 2 1 ≈ (TIC + Lcav ), 2 where Lcav is the total loss of the cavity (not counting the input coupler). The phase imparted by the cavity on the field φ dictates how far off resonance the cavity is. The resulting equation for a single field mode of amplitude A is, 1 dA = E0 tic − (TIC + Tcav ) A + iφA dN 2  (A.1)  To simulate the sweeping of the cavity through a resonance, φ → φ(N ). N can be expressed in units of time since N = tΩrep /2π.  158  Appendix B  OpenFOAM In order to solve for the flow in more complex geometries, we use numerical solvers. Computational fluid dynamics (CFD) is a powerful tool to understand flow in complicated geometries where the fluid may be turbulent and/or compressible. Numerical solutions calculate the flow of a fluid by discretizing space and time. The geometry is modelled by creating a mesh, composed of many cells of finite dimensions (height, width, and length). The fluid traverses the cell dimension ∆l at a certain velocity v within a certain amount of time ∆τ . The discretization of time must satisfy the Courant-Lewy-Friedrichs condition, v  ∆τ <C ∆l  (B.1)  where C is referred to as the Courant number. A typical, viable, Courant number is 0.5 which means that fluid travelling within a cell is sampled twice before leaving the cell. In the case of an axis-symmetric wedge geometry (flow in z direction and symmetric about φ), the Courant number can be much larger because the main direction of flow is mainly in the z direction, whereas the smallest cell dimension is dependent on a small angle φ. Conversely, for complicated 3D objects at an intersection, the Courant number may need to be < 0.1 in order for the solution to converge. In this thesis, I have used OpenFOAM to solve for the gas flow in the nozzle and vacuum chamber, chosen for its versatility and expense (free). OpenFOAM is an open-source CFD software programme. Within it are many solvers created for specific purposes (incompressible fluids, turbulent flow, combustion etc). For this thesis the solver used is designed for compressible flow called “rhoCentralFoam”. 159  B.1. Thermophysical Properties rhoCentralFoam  Each type of CFD problem typically requires its own  CFD solver. For compressible, transsonic flow, the most robust solver currently developed in OpenFOAM is rhoCentralFoam. This solver is shown to be accurate in a variety of problems that deal with subsonic and supersonic flow (and the resulting shockwaves), which is ideal for the purposes of this thesis. As mentioned in Sec 4.2.2, the equations that need to be solved for compressible flow are the continuity equation, the Navier-Stokes equation (or, in the inviscid case, the Euler equation), and the conservation of energy. However, the development of this solver is beyond the scope of this thesis.  B.1  Thermophysical Properties  The gas dynamics depend on the type of gas used. Therefore we need to understand the physical and thermal properties. These can be defined in OpenFOAM in the thermophysicalProperties file. For this thesis, the molar mass, heat capacity (at specific pressure) and viscosity are required. OpenFOAM can model the viscosity properties for fluids, and one method is using the Sutherland transport equation. This is an empirical model of the temperature dependence of the viscosity of a fluid. Because the fluid comes out of the nozzle adiabatically it can be cooled to very low temperatures, and so understanding the properties of the gas over a wide range of temperatures makes for a better model. Recall that for the analytical solution for flow in a 1D pipe, the gas is inviscid. Although this approximation is suitable for flow within the nozzle, as the gas expands and accelerates within the vacuum chamber, the temperature changes. The equation for the Sutherland viscosity is µ=  c1 T 3/2 c2 + T  (B.2)  where T is the temperature in Kelvin, and µ is in units of kg/ms, and c1 and c2 are dependent on the gas. Although the viscosity constants are given for air in OpenFOAM, the viscosity for various fluids can also be found in the CRC Handbook at several temperatures. I then fit these values to a function 160  B.2. BlockMesh Gas Air (cfd) Air Xenon Oxygen Argon  g Mass ( mol ) 28.96 28.97 131.30 32.00 39.95  cp ( kgJK ) 1004.5 1005. 158.3 918.0 520.3  γ 1.4 1.400 1.667 1.395 1.667  √ ) c1 (10−6 m kg s K 1.458 1.495 2.564 1.764 1.998  c2 (K) 110.4 116.3 262.6 139.9 153.6  Table B.1: Table of properties of gases used. Air has two values, one from Ref [141] and a second from Ref [142]. (using Mathematica’s FindFit function) to get the viscosity coefficients. For xenon, the thermophysical properties file used is thermoType  ePsiThermo<pureMixture<sutherlandTransport  <specieThermo<hConstThermo<perfectGas>>>>>; mixture xenon 1 131.3 158.3 0 2.564e-6 262.6; Because rhoCentralFoam cannot solve for a mixture of gases, the mixture is always unity. See Tab B.1 for the other thermophysical properties. The 0 is for the latent heat of fusion, where it is assumed that the gas does not change state. The ratio of heat capacities, γ, is used for the boundary conditions described below.  B.2  BlockMesh  There is a simple programme with OpenFOAM which is used to generate the meshes call blockMesh. This programme lets the user input the vertices of a shape to generate the mesh. Each shape is based on a hexahedron, however we can create shapes with fewer than 8 vertices by duplicating vertices (such as to produce wedges). Each basic hexahedron that the user creates can be divided into a certain number of cells with a grading in size. The file containing the mesh is blockMeshDict, and is created using the blockMesh command. Making the hexahedron is called in the line  161  B.3. Axis-symmetric Approximation hex (0 1 2 3 4 5 6 7) (Nx Ny Nz) simpleGrading (Rx Ry Rz) which attaches the vertices from points 0 through 7 in a right-handed coordinate system. This hexahedron is divided into Nx, Ny, and Nz (all integers) cells in the x, y, and z directions respectively. The simpleGrading defines the ratio of the first cell size to the last (where unity makes them the same size) also in the x, y, and z directions respectively. It is recommended to have the grading from adjacent cells, or the cell expansion ratio, to be 0.833 < di+1 /di < 1.2. To keep the size of adjacent cells within this range, R must be R=  di+1 di  N −1  (B.3)  where R then is any positive (decimal) number.  B.3  Axis-symmetric Approximation  For the end-fire configuration (the high flow nozzle design), we have axial symmetry. In this case we can model the geometry of the nozzle with an azimuthal angle δφ which is a small wedge of the overall nozzle. The geometry is such that the mesh is essentially 2D (and only one cell thick) reducing the number of cells and reducing computation time. This geometry was used to compare the analytical results with the numerical solution to ensure that I was using OpenFOAM correctly and that thermophysical constants and boundary conditions were appropriate. Because the flow is choked within the nozzle, the flow at the nozzle outlet is transsonic. The analytical solution predicts the choked pressure to be about half of the line pressure in a smooth, 1D case. Although for a smooth, converging nozzle this is also the case in the numerical solution, the diameter of the backing line is ∼ 5 mm, and the internal diameter nozzle has  a sharp change to ∼ 0.5 mm. In this case, the numerical solution finds that the pressure in the nozzle is closer to a third. In our experimental setup, we measure the line pressure; our knowledge of the change in pressure from the line to the nozzle outlet allows us to estimate the gas density.  162  B.4. 3D Modelling We can also use the flow of the axis-symmetric case to predict some of the features of the flow for more complicated 3D designs.  B.4  3D Modelling  Developing the full 3D models for the flow of fluid in the nozzle is slightly more difficult because of understanding the geometry involved in creating the mesh. We can create a full 3D tube and vacuum chamber to compare our 2D and 3D numerical solutions. However, the purpose of using 3D is to create more complicated geometries such as the through nozzle, hybrid nozzle, and even phase matched nozzle designs, where the density can be tuned precisely. Creating a full 3D tube from the axis-symmetric wedge is not useful because the cell volume along the middle of the tube is much smaller than those near the edge. To maintain similar cell size and shape throughout the tube, we use the “donut” shape, where the mesh is separated into two distinct regions: the internal hexagonal block which defines the number of cells to be used, and the external ring with rounded edges that gives the tube its shape. In the donut design the blocks are of more uniform size and shape, which is good for minimizing the skewness of the mesh, compared to a wedge design. Skewness is the measurement of the cell away from its symmetrical, equiangled, shape. The benefit of smaller skewness can be understood by realising that the cell represents a small locale of fluid flow. The more a cell is skewed, however, the edges are farther to extrema of the cell from the middle, and so those points of the cell are less representative of the properties of the flow in that cell. In terms of computation solutions, cells of high skew can cause inaccuracies and have stability issues. Likewise, a cell with large aspect ratio will also suffer from the same problems as a highly skewed cell. However, in the example of a tube, cells of large aspect ratio can be used when the flow is nearly one dimensional. OpenFOAM comes with a mesh utility to check the mesh quality called checkMesh. If cells are highly skewed they will be written to a file in the 163  B.5. Running the Programme directory containing the blockMeshDict.  B.5  Running the Programme  The programme is run by calling the solver desired (ie in this case rhoCentralFoam). The file that controls the time steps and file writing is controlDict. This file looks like: application  rhoCentralFoam; The name of the routine  startFrom  latestTime; The start time; programme can be re-run  from the last time stopped startTime  0;  stopAt  endTime;  endTime  1e-3; in seconds  deltaT  1e-9;Initial time step.  writeControl  adjustableRunTime;  writeInterval  1e-6;  purgeWrite  0;  writeFormat  ascii;  writePrecision  15; Precision of variables written in files  writeCompression uncompressed; timeFormat  general;  timePrecision  9; Precision of time folders  runTimeModifiable yes; adjustTimeStep  yes; Adjusts the time step until maxCo reached  maxCo  0.5; Maximum Courant number  maxDeltaT  1; Maximum value for deltaT  The maximum Courant number may be adjusted if the programme is crashing. However, if the programme is still crashing even for a Courant number  0.1 then there is probably a problem with the boundary con-  ditions. The maximum Courant number is calculated for the maximum velocity and minimum cell length, which, for the wedge case is very small. Thus, the maxCo value can be increased to ∼ 10 and still yield accurate  results. This is only possible because the flow is perpendicular to the small 164  B.6. MapFields length scale. I have found that the difference in endTime and deltaT must not exceed about 10 orders of magnitude. I think it is a memory issue.  B.6  MapFields  A method to speed up convergence is to begin with a coarse mesh so that there are fewer cells to calculate, and the larger volume of cells allows for a large time step (as limited by the Courant condition). Because we only can measure the boundary conditions experimentally, I have no way of assigning the fields a priori and so I define the boundary conditions (the inlet pressure and temperature, and the vacuum chamber pressure) and then run rhoCentralFoam until a steady state solution exists. However, beginning with a refined mesh, which can have a mesh of nearly 106 cells for complicated 3D objects, is computationally slow. A coarse mesh can have the structure of of the object but with far fewer cells and can converge very rapidly. To map the fields from a coarser mesh to a more refined mesh, there is a mapFields utility. To use this programme, first the coarse mesh case is run to a steady state solution. In a separate directory, the finer mesh case blockMesh is then run. It should be noted that the fields for the finer case cannot have internalField nonuniform List because the number of entries is then listed, and mapFields recognizes that the two list sizes do not match. Now that there is a coarse mesh case with a steady state solution, and a finer mesh case with some uniform field. To map the fields, use the line mapFields coarse folder name -consistent -sourceTime last time folder of coarse mesh case -case refined folder name If the last time folder is unknown (for example, if this were scripted) any given time higher that the last time will imply the last time folder. The -consistent tag implies that the geometry from the coarse mesh to the refined mesh is consistent; changing the geometry requires a mapFields dictionary, ie mapFieldsDict. 165  B.7. Boundary Conditions  B.7  Boundary Conditions  For the flow of gas from a nozzle into a vacuum chamber, the inlet boundary conditions (BCs) are those that we can measure from the gas line. The gas is at room temperature at several Torr of pressure (up to several atmospheres of pressure); the units are in MKS. The inlet BCs are then defined as type fixedValue; value uniform value ; where value is the inlet temperature or pressure. The outlet pressure is whatever our turbopump can pump the cavity out, which for us is a few milliTorr. As mentioned in the analytical solution case, the flow is choked once the pressure of the outlet is below the critical pressure. Therefore the pressure inside the nozzle will not be affected once the pressure is below the critical pressure. However, the trajectories of the gas can be very different outside of the nozzle, particularly the position of the Mach disks. Because we are interested in the atom density in the laser interaction region, only in the endfire nozzle case does the flow in the vacuum chamber interest us. However, it is found from simulation that the vacuum chamber pressure only affects the gas flow far downstream of the end of the nozzle, while the region of high pressure is only within a few nozzle diameters and independent of vacuum chamber pressure. Therefore, the gas flow within the nozzle (and very close to the outlet of the nozzle) is independent of the chamber pressure as long as it remains below the critical pressure. The walls of the nozzle have their own BCs called wall. A wall prevents the flow of fluid through the boundary, and so the normal component of the velocity is zero. The tangential component depends on friction of the boundary; for compressible, monoatomic gas at transsonic velocities we take this to be frictionless, which is the slip BC. The slip BC allows for the normal component to be zero, and the tangential to be zeroGradient. In our case, the inlet and outlet pressures are known so the inlet is a pressure inlet, and the outlet is a pressure outlet. The velocities are to be calculated and so are specified to be consistent with the rest of the fluid flow, which is done with zeroGradient. Because the outlet pressure is orders of 166  B.8. Summary magnitude lower than the inlet, there will be pressure waves which propagate in the fluid until a steady state has been reached. To compensate for these waves reflecting from the outlet BC, there is a BC called waveTransmissive. This is used as type  waveTransmissive;  value  uniform value ;  field  p; The name of the field  gamma  The ratio of cp /cv ;  phi  phi; The mass flux φ = ρU  rho  rho; The density ρ  psi  psi; ψ =  lInf  The distance to the far-field value, in meters ;  fieldInf  The far-field value ;  B.8  ∂ρ ∂p  Summary  The documentation for OpenFOAM leaves much to be desired, particularly for someone starting off. The website has a few good tutorials, but the case presented here is different from those readily available. Luckily, almost all solvers come with a sample solution which helps when an example is needed. However, many of the more subtle problems with the solvers and routines are not very well documented. To ensure that the programme was working correctly, the results from simple geometries were compared with the analytic solutions. Experience with simple geometries enables confidence in the solutions of the more complicated cases. For the problem at hand, the main issues with OpenFOAM come from the boundary conditions (although it is doubtful whether this is unique to this software). Solving the pressure, temperature, and velocity of a gas injected into a vacuum chamber is a challenging problem because of the 5 orders of magnitude change in pressure. Care must be taken to avoid shockwaves near the boundaries as they would always cause the programme to crash. To avoid this problem, the background (vacuum chamber) pressure 167  B.8. Summary in the simulation was set such that the shockwaves were well within the computational domain; the background pressure could then be lowered to see if there was a change in the variables in the regions of interest.  168  Appendix C  Detection and Generation of Noise in Optical Systems This chapter discusses sources of and sensitivity limitations due to noise in the detection of optical radiation. In our experiments we are detecting coherent optical radiation spanning from approximately 50 nm up to roughly 1 µm, where the purpose of detection is specific to the experiment. The photomultiplier is briefly discussed as a simple photosensitive device, but most of the discussion involving modulation of sources and heterodyne detection will be presented with the photodiode, which is the more common photodetector in our experiments. This chapter will also present a method of measuring the noise spectrum.  C.1  Introduction  When a photon of sufficient energy is incident on a photosensitive material, an electron is excited from a non-conducting to a conducting state. The passage of these electrons across an ammeter leads to a measurable photocurrent. However, in our experiments this photocurrent is converted to a voltage using an operational amplifier (op amp) in the appropriate circuit (see Fig. C.1) known as a transimpedance amplifier. The amplitude of the signal is measured with an oscilloscope. Therefore, I will normally use the photocurrent for the generated signal isig (t), but it is equivalent to the voltage measured vout (t). This chapter uses the semiclassical treatment to discuss light-matter interaction, where the radiation is treated classically as a field (although with liberal use of the word “photons”), but the charge carriers used to gener169  C.1. Introduction  Figure C.1: The inverting operational amplifier (op amp) is used to convert the photocurrent iA (t) to a voltage vout (t) with impedance Rg . This circuit is known as a transimpedance amplifier. ate the signal (i.e. the electrons and in some instances holes) are discrete particles. This is because the particle nature of the charge carriers requires the use of photons to describe the discrete excitation of one electron from a non-conductive to a conductive state. I will now go over some terms that will be used throughout this chapter.  C.1.1  Definitions of Terms  The impedance of free space, η0 , is the ratio of the magnitudes of electric field to the magnetic field, that is, η0 =  E H  (C.1)  and has an approximate value of η0 ≈ 377Ω given by, η0 = µ0 c0 = where µ0 is the permeability,  0  1 = 0 c0  µ0  (C.2)  0  is the permittivity, and c0 is the speed of light  in vacuum. The impedance is useful because we can relate the magnitude  170  C.1. Introduction of the electric field |E| to the intensity as, I=  1 |E|2 2η0  (C.3)  and the optical power that we measure with our radiation detector is the intensity times the area A. Using the idea of photons, the intensity is related to the photon flux Φ, I = Φ¯ hω  (C.4)  where Φ is the number of photons per area per second, h ¯ is Plank’s constant divided by 2π, and ω is the angular frequency of the optical radiation. The probability of a photon exciting a charge carrier in the photodetector is η, which is also known as the quantum efficiency. The measured current is then [143], isig (t) = ΦAeη eη = Popt ¯hω  (C.5)  where e is the charge of an electron, and Popt is the optical power incident on our detector. Consequently, the voltage signal that we measure with our oscilloscope is linearly proportional to the incident optical power (see Fig. C.1). To find the constant of proportionality we need to know the impedance of the circuit, the frequency of radiation, and the quantum efficiency of the photodetector. Signal to Noise Ratio The signal to noise ratio (SNR) is given as the ratio of the mean square of the signal to the sum of the mean square of the noise terms SN R =  i2sig i2N 1 + i2N 2 + · · ·  .  (C.6)  The terms in the denominator are discussed in the following section. They are composed of noise generated by the detection process, the circuits used, 171  C.1. Introduction and other terms involving fluctuations in laser power. The minimum detectable signal occurs when the SNR is unity. When the SNR is unity over a specified electrical bandwidth ∆f , then the power due to the noise is equal to the power of the signal. This is referred to as the noise equivalent power (NEP). The noise level of the system, as discussed below, is dependent on the electrical bandwidth ∆f of the detection system. If the detection system (the photodiode, the photodetector amplifiers, the spectrum analyser etc.) responds uniformly across ∆f then the total signal detected is R0 ∆f where R0 is the responsivity of the system. When the responsivity is a function of the frequency, then the total signal becomes  ∞ 2 0 |R(f )| df .  From this we can define the electrical bandwidth as ∞  ∆f = 0  R(f ) 2 df . R0  (C.7)  When the signal is detected for a time T then the response of the system is R(f ) =  1 T  T /2 −T /2  R0 e−2iπf t dt  sin(πf T ) = R0 . πf T  (C.8)  The electrical bandwidth becomes [144], ∞  ∆f  = 0  =  1 . 2T  sin(πf T ) πf T  2  df (C.9)  Therefore, for a 0.5 second integration time, the resolution bandwidth of the system is 1 Hz.  172  C.2. Sources of Noise  C.2  Sources of Noise  The measurement of the photocurrent isig (t) is composed of the average photocurrent i(t) and the fluctuations about the average δi(t) i(t) = i(t) + δi(t).  (C.10)  The causes of δi(t) in the detection process are due to the discrete nature of the field and current (shot noise), as well as the circuit being at some non-zero absolute temperature (Johnson noise). I will now describe these noise terms and their effects on the detectable signal.  δi(t)  i  Figure C.2: The signal i(t) is composed of the average value i(t) and the noise δi(t). The noise can originate from the source or from the detection process, the latter of which I will focus on in this chapter.  C.2.1  Shot Noise  Shot noise is due to the discrete generation of charge carriers in an electric circuit. The average current measured over an observation time T is i=  eN T  (C.11)  173  C.2. Sources of Noise where N is the average number of charge carriers measured. The fluctuations in this measurement, using equation (C.10), are δi(t) = i(t) − i(t), thus the mean square of the current becomes i2S = δi2 =  e2 (N − N )2 . T2  (C.12)  N  δN 2  Figure C.3: The distribution of finite events is given by a Poisson distribution. This distribution function has the property that (δN )2 = N We assume that the photodetector is ideal, in that when there is no incident radiation the output signal is zero and the noise associated with the signal is only due to the shot noise. In this case, the probability distribution for the case of discrete, random events occurring is given by the Poisson distribution  N  p(N ) =  N −N e . N!  (C.13)  A useful property of the Poisson distribution is that variance (δN )2 = 2  (N − N )2 = N 2 − N is equal to the mean, that is (δN )2 = N .  (C.14)  174  C.2. Sources of Noise Combining equations (C.12) and (C.14) with (C.11), the shot noise is i2S =  e i. T  (C.15)  Instead of defining the shot noise over the integration time T , but rather over the resolution bandwidth ∆f , the shot noise is expressed as i2S = 2ei∆f.  (C.16)  This simple expression shows that the noise associated with the discrete generation of carriers in the detection process is dependent on the signal level (optical power) and the resolution bandwidth of the system. When viewing the RF spectrum of our detected signal, we can decrease our shot noise by decreasing the resolution bandwidth of the spectrum analyser.  C.2.2  Johnson Noise  The thermal motion of charge carriers within a resistor causes fluctuations in charge densities; Johnson noise is the resulting fluctuations of the voltage within the circuit.  Figure C.4: The noise equivalent circuit for calculating Johnson noise. The thermal energy of the resistors causes fluctuations in the voltage of the circuit, resulting in noise. In the impedance matched case, we have R1 = R2 = R. The thermal energy of the resistor is composed of modes given by the 175  C.3. Sources of Signals equipartition function as  hf  E= e  hf kB T R  (C.17) −1  where TR is the resistor temperature and f is the mode frequency. For radio frequencies ≤ 100 GHz with the circuit at room temperature, then  this expression simplifies to E ≈ kB TR . The power then dissipated per unit  time T for the entire circuit is then P =  E = E[2∆f ] T  (C.18)  where ∆f is the resolution bandwidth. The electrical power in the circuit is simply given as Pelec = i(t)v(t) = v(t)2 /Rtotal =  v(t)2 2R  (C.19)  when the impedance of the detector matches that of the line (see Fig. C.4). The thermal energy fluctuations then give rise to fluctuations in voltage to give v 2 = 4kB TR R∆f.  (C.20)  Equivalently, the variance of the current is i2J =  C.3  4 kB TR ∆f. R  (C.21)  Sources of Signals  The detection of radiation with a photodetector leads to an electrical signal (equation (C.5)), isig (t) = Popt  eη . ¯ω h  In this section, we explore how various photodetectors work. We begin with the photomultiplier because its properties can be easily extended to the photodiode and avalanche photodiode.  176  C.3. Sources of Signals  C.3.1  The Photomultiplier  The photomultiplier works by having radiation incident on a cathode (see Fig. C.5). When the electrons are ejected by the cathode due to radiation of sufficient energy (E = ¯hω > Φ where Φ is the work function of the cathode metal), they are attracted to dynodes within the photomultiplier tube. These electrons then accelerate to the dynodes and when they hit the dynodes, they eject more electrons. The process ends at the anode, which collects the electrons and sends the electrical signal. The benefit of the photomultiplier is that the gain is G = δ N where δ is the emission multiplication per dynode and N is the number of dynodes. The signal of the photomultiplier at the anode iA due to the cathode iC is then iA (t) = GiC P (t)eη = G + id ¯hω  (C.22)  where id is the dark current. Dark current is the current that flows in the photomultiplier (or any photodetector), even when there is no radiation incident on the detector. The shot noise of the photomultiplier comes mainly from the cathode; the noise from the dynodes is much smaller. The shot noise is given by i2S = 2GeiA ∆f P eη + id ∆f. = 2eG2 ¯hω  (C.23)  The Johnson noise is, again, given by i2J =  4kTR ∆f RL  where RL is the resistance across the anode.  177  C.3. Sources of Signals  Figure C.5: Schematic for the photomultiplier. The cathode (C) connects to a high voltage (HV) source; the dynodes (D) are at a sequentially lower potential; the impedance of the photomultiplier is the load resistance RL where RL R, the resistance between dynodes. Light is incident on the cathode, which ejects an electron accelerated to a dynode. At each dynode, the number of electrons emitted is multiplied by the emission factor δ. The electrons are collected at the anode (A). The SNR then becomes SN R = =  i2A i2S + i2J (GP eη/¯hω)2 . 2eG2 (P eη/¯hω + id )∆f + 4kB TR ∆f /RL  Now we look at some limiting cases. When the gain G  (C.24)  1, the shot noise  dominates the Johnson noise and SN R =  (P eη/¯hω)2 . 2e(P eη/¯hω + id )∆f  (C.25)  The SNR cannot improve further within the detection process and the signal is called shot noise limited. In the ideal case of negligible dark current, P eη/¯ hω  id , then the result is simply SN R = P ηT , with T = 1/2∆f  being the integration time.  178  C.3. Sources of Signals  C.3.2  The Semiconductor Photodiode  The pn Junction The semiconductor photodiode works by the passage of electrons across a pn junction. The pn junction occurs by joining two semiconductor materials, with one half being doped with positive carriers and the other doped with negative carriers. This position dependence of doping creates a potential across the junction in a region called the depletion layer. In this region, the negatively charged electrons from the n side diffuse across the junction to occupy the holes on the p side, which create positively charged holes on the n side. The charge distribution (negative charges on the p side and positive charge on the n side) now creates a potential across the junction that limits further charge carrier diffusion (see Fig. C.6). When the charge carriers no longer have a net flow, the potential is called the junction potential Vj . Because there are no longer free carriers to move about in the region, it is called the depletion layer. We can change the barrier potential by an externally applied potential Va . Decreasing the potential barrier is called forward bias because this exponentially increases the current through the diode. Conversely, reverse bias increases the potential barrier to Vj + Va , which is the typical mode of operation for detection using a photodiode (see Fig. C.7). Detection with a Photodiode In the reverse bias case, the anode is at a lower potential, and the photocurrent flows from the cathode to the anode. The photodiode can be modelled as a capacitor and resistor in parallel, where the photodetector is placed in parallel with a load resistor. The resistance of the photodetector is much greater than the load resistance, therefore 1 Ztotal  = iωCdiode +  1  Rdiode 1 ≈ iωCdiode + RL  +  1 RL (C.26)  179  C.3. Sources of Signals where Ztotal is the total impedance so we get Ztotal =  RL . 1 + iωCdiode RL  (C.27)  The transfer function has a pole when |ω| = (RL Cdiode )−1 , and the inverse of  this gives the characteristic time constant of the circuit. This time constant will dictate the response time of the photodetector, where decreasing RL and Cdiode will decrease the response time.  Figure C.6: A pn juntion photodiode. The photocurrent flows from the cathode (C) to the anode (A). Due to the difference in electron energies in the p and N regions, when the two materials are brought in to contact, it creates a junction potential Vj . The electrons and holes from the n and p regions respectively diffuse across the depletion region, building up a potential barrier. The capacitance is given by C=  Q V  (C.28)  where Q is the total charge across the capacitor. Because the total charge  180  C.3. Sources of Signals is some charge density σ times the area A C=  σA . Vj + Va  (C.29)  Therefore, the capacitance of the photodiode is affected by the applied bias and the area. To decrease the capacitance we can make the photodiode area as small as possible and increase the applied potential bias. This will decrease the time constant, making the photodetector response faster.  Figure C.7: Biasing a photodiode increases the potential barrier by an amount Va . The photodetector signal can be much faster as described in the text.  Photodiode with Modulation In our experiments we modulate the field frequency for the Pound Drever Hall locking scheme (see section 2.5) in order to lock our cavity to the laser. The modulated signal field becomes E(t) = Esig cos(ωsig t)[1 + m cos Ωm t]  (C.30)  181  C.3. Sources of Signals where ωsig is the optical angular frequency, Esig is the signal field amplitude, Ωm is the angular modulation frequency, and m is the relative modulation depth. The optical power is then |Esig |2 A cos2 (ωt)[1 + m cos Ωm t]2 2Z0 = Psig [1 + m cos(Ωm t)]2  Popt (t) =  (C.31)  where Psig is the time average optical power of the signal. Then the resulting photocurrent becomes isig (t) = =  Psig eη [1 + 2m cos Ωm t + m2 cos2 Ωm t] hω ¯ Psig eη m2 [1 + (1 + cos 2Ωm t) + 2m cos Ωm t] hω ¯ 2  (C.32)  where the relation cos2 θ = 12 (1 + cos 2θ) is used. Because we put the modulation signal into a mixer, we are only interested in the signal around the modulation frequency isig (t) =  Psig eη 2m cos Ωm t. ¯hω  (C.33)  1 2mPsig eη 2 ¯hω  (C.34)  The variance of the signal is i2sig =  2  and i = 0 so that the shot noise is given by i2S = 2eid ∆f . The SNR of the modulated signal at the angular frequency Ωm is then 2 m2 Psig eη SNR = ∆f ¯hω  2  eid + 2kB TR /R  −1  .  (C.35)  The SNR is then proportional to the square of the optical power and the modulation depth, while all terms brackets are constants in the circuit.  182  C.3. Sources of Signals Avalanche Photodiode The avalanche photodiode works on the same principle as the photodiode operating in reverse bias mode, however now the reverse bias is large enough such that the field in the depletion layer accelerates the carriers to have enough energy such that they further release carriers, creating an avalanche in carriers. The probability of a carrier releasing another carrier is p. Thus, for enough instances of these collision events occurring, we see that M = 1 + p + p2 + p3 + ... =  1 1−p  (C.36)  where M is the amount that the signal is multiplied. The signal is then proportional to the square of the multiplication factor. This is very useful for low signals that are not shot noise limited because we can increase the SNR until i2shot > i2Johnson . Increasing the gain of the avalanche photodetector beyond the shot noise limited point does not increase the SNR, because the noise mechanism is slightly different from the square of the detected current (the exponent is between 2 and 3, depending on where within the depletion layer the carriers are released). Heterodyne Detection In detecting the offset frequency (see section 3.4), we use an avalanche photodetector in a heterodyne detection setup. This detection method can be very sensitive to low laser power levels, generating a radio frequency (RF) signal by interfering two optical beams. In this scheme, we have a field from a signal source, and a local oscillator as reference (see Fig. C.8). The resulting field on the photodetector is E(t) = Esig cos ωsig t + ELO cos(ωLO t)  (C.37)  where Esig and ωsig are the signal field amplitude and angular frequency respectively, and likewise LO is the local oscillator. This is better represented  183  C.3. Sources of Signals in complex notation as 1 1 E(t) = Esig eiωsig t + ELO eiωLO t + c.c. 2 2  (C.38)  where the c.c. is the complex conjugate and Esig , ELO are real. The detected power is proportional to the magnitude squared of the field 1 2 1 2 Esig + ELO 2 2 + Esig ELO (ei(ωsig −ωLO )t + e−i(ωsig −ωLO )t )  |E(t)|2 =  + O(2ωsig , 2ωLO , ωsig + ωLO )  (C.39)  where the first two terms are the DC component, the next is the difference frequency between the local oscillator and the signal, and the other higher order terms are components that are still in the optical (and thus not detected in the RF by the photodetector). Often, in heterodyne detection, the local oscillator power can be tuned to be much greater than that of the signal. However, in detection of the offset frequency f0 of the laser using an f-2f setup, the amplitudes of the local oscillator (f) and signal (2f) are of comparable power. The signal current, then, is isig (t) =  eη Psig + PLO + 2 hω ¯  Psig PLO cos[(ωsig − ωLO )t] .  (C.40)  When measuring the offset frequency with the f-2f setup, the signal is often small and needs to be amplified. One method of achieving sufficient signal is to use an avalanche photodetector so that the signal current is amplified by M . This means that  SNR =  eη hω ¯  2  Psig PLO M 2  [M 2 e( ¯heηω (Psig + PLO ) + id ) + 2kB TR /R]∆f  .  (C.41)  We can then make our signal shot noise limited by increasing the gain of the avalanche photodetector until the Johnson noise term becomes negligible.  184  C.4. Relative Intensity Noise Measurement In the case that Psig ≈ PLO and the dark current is small, then SNR =  Psig η . 2¯ hω∆f  (C.42)  λ/2  Figure C.8: Schematic for heterodyne detection. In the f-2f interferometer the polarization of two fields can be orthogonal, so that the two beams can be recombined using a polarizing beam splitter (PBS). Using a halfwaveplate and polarizer (λ/2 and POL respectively), we can get the two fields to be of the same polarization to see the interference required for heterodyne detection. The polarizer can be tuned to maximize the SNR. Typically to lock the offset frequency, we need a SNR of at least 30 dB at 100 kHz resolution bandwidth. Therefore, for 532 nm signal the power requirement is about 150 nW for the local oscillator and signal. In other heterodyne detection schemes, where the power of the local oscillator can be much greater than the signal, the SNR is improved by a factor of 2.  C.4  Relative Intensity Noise Measurement  In order to detect relative intensity noise (RIN), the usual method (and the method we are using) is to measure the DC photocurrent from the photodetector. The AC noise is measured using a radio frequency (RF) spectrum analyser. The RIN spectrum is then given by [145] R(f ) =  PRF (f ) 1 P0 C(f )  (C.43)  185  C.4. Relative Intensity Noise Measurement where PRF is the noise of the electrical power spectrum, P0 is the average electrical power, and C(f ) is the frequency response of the system. Current Amplifier Transfer Function In order to detect the EUV signal using photodiodes with directly deposited filters, we use a Stanford Research Systems (SRS) Current Preamplifier. This transimpedance amplifier converts the small current from the photodetector to a much larger voltage, at the expense of the detection response time. The transfer function of the transimpedance amplifier is the main source of non-unity frequency response at low signal levels.  Figure C.9: Setup for detecting low optical power signal levels using a photodiode (PD) and current amplifier (SRS-CA). The signal out of the SRS-CA is then displayed on a radio spectrum analyser (RSA).  C.4.1  RIN  The RIN spectrum is often given in dB/Hz, where the total RIN given in dB, is calculated by ∞  RINtotal =  R(f )df = 0  δPopt (t)2 P (t)opt  2  (C.44)  where P (t)opt is the optical power incident on the photodetector. The RIN is composed of shot noise (which has a Poisson distribution function), and some excess RIN. The shot noise is the minimum achievable, 186  C.4. Relative Intensity Noise Measurement  +14dBm 100µA/V +14dBm 10µA/V +14dBm 1µA/V -4dBm 1µA/V -4dBm 100nA/V -4dBm 10nA/V -4dBm 1nA/V  Figure C.10: The transfer function C(f ) for the SRS current amplifier (SRS780) at various settings. The source was run at either 14 dBm or −4 dBm and the amplifier was adjusted from 100 µA/V to 1 nA/V.  quantum limit and is given by RINP oisson = =  i2S i2  =  2e i  2¯hω Popt η  (C.45)  which is constant over the RF. The excess RIN is simply the deviation from Poisson statistics, so that the excess RIN can be calculated by 2e [1 − ηtotal ] i 2¯ hω 1 = Rmeasured (f ) − [ − (1 − L)] Popt η  Rexcess (f ) = Rmeasured (f ) −  (C.46)  where ηtotal = η(1 − L). L is the loss between the optical source and the detector (including attenuators, filters, mirror losses etc).  187  C.4. Relative Intensity Noise Measurement  Figure C.11: The RIN of the Femtosource Ti:Sapphire laser.  In this manner, we measure the mean voltage v(t) from the SRS amplifier using the oscilloscope making sure both instruments are at the same input impedance, and we measure the spectrum using the SRS-RSA in the Vpeak setting, taking notice of the frequency resolution ∆f of the RSA. Subsequently, the RIN becomes RIN(f ) =  2 Vpeak 2  .  (C.47)  v(t) ∆f The RIN is then inherently measured in a 1 Hz bandwidth.  188  

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