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The adventures of Nikita and Casper : high power ultraviolet lasers for precision spectroscopy Altiere, Emily Elizabeth 2014

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The Adventures of Nikita and CasperHigh Power Ultraviolet Lasers for Precision SpectroscopybyEmily Elizabeth AltiereB.A. (Hons) Physics, Bryn Mawr College, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)November 2014c© Emily Elizabeth Altiere, 2014AbstractThe optically pumped semiconductor laser (OPSL) offers several advantages as a laser source forprecision spectroscopy. The semiconductor gain bandwidth allows an OPSL to run continuous wave(CW) between 920 - 1154 nm and with a free running linewidth 500 kHz. High powers have beenobserved in OPSL, as high as 70 W. Paired with doubling crystals the wavelength range can beextended down to the ultraviolet (UV) with high power. This research presents an OPSL operatingat 972 nm at 1.7 W sequentially doubled down twice to a wavelength of 243 nm at 150 mW. Thelinewidth is reduced by locking one OPSL to a Fabry-Poret stabilization cavity and then the relativelinewidth was measured between two OPSL’s locked together. The linewidth is determined to be87 kHz, dominated mostly by technical noise. This laser is set to be used for cryogenic hydrogenspectroscopy and precision measurements of the Lamor precessional frequency of 129Xe when it isused as a comagnetometer for measuring the electrical dipole moment (EDM) of the neutron.iiPrefaceChapter 2 is based on the work of Dr. Yushi Kaneda at the University of Arizona. He is responsiblefor the construction of the OPSL source and the original design of the second harmonic generation(SHG) cavities. Arthur Mills and I worked together to produce the UV light through the SHG cavities,as presented in Chapter 3. The power conversion through SHG measurements in Chapter 3 wereperformed by myself and the linewidth measurements were performed by myself and Marc Lejay.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Hydrogen Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Ultra Cold Neutron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Ultraviolet Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 The Optically Pumped Semiconductor Laser . . . . . . . . . . . . . . . . . . . . . 92.2 Frequency Doubling Enhancement Cavities . . . . . . . . . . . . . . . . . . . . . 122.3 Optical and Electronic Components . . . . . . . . . . . . . . . . . . . . . . . . . 133 Enhancement Cavities for Second Harmonic Generation . . . . . . . . . . . . . . . 143.1 Enhancement Cavity Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.1 Cavity Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Impedance Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Spatial Mode of Enhancement Cavities: Mode Matching . . . . . . . . . . . . . . 193.2.1 Establishing the Mode of the Enhancement Cavity . . . . . . . . . . . . . 203.2.2 Coupling the Input Beam into the Cavities . . . . . . . . . . . . . . . . . . 213.3 Locking the Enhancement Cavities using Pound-Drever-Hall Scheme . . . . . . . 25iv4 Nonlinear Optics and Second Harmonic Generation . . . . . . . . . . . . . . . . . . 334.1 Nonlinear Crystal Symmetry Basics . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.1 Boyd-Kleinman Nonlinear Conversion Factor . . . . . . . . . . . . . . . . 384.2 Second Harmonic Generation in an Enhancement Cavity . . . . . . . . . . . . . . 405 Measurement of the OPSL Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2.1 Fabry-Perot Stabilization Cavity . . . . . . . . . . . . . . . . . . . . . . . 485.3 Measuring the Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3.1 Heterodyne Beat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3.2 Locking the Slave Laser to the Master Laser . . . . . . . . . . . . . . . . . 505.3.3 Frequency Noise and the β -line . . . . . . . . . . . . . . . . . . . . . . . 515.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3.5 Approximation of the Linewidth at 243 nm . . . . . . . . . . . . . . . . . 576 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61vList of TablesTable 3.1 Characterizing values of the two enhancement cavities for SHG. . . . . . . . . . 17Table 4.1 The relationship of the indices of refraction of the uniaxial and biaxial crystalsalong the x, y and z axes, for the positive and negative orientations. . . . . . . . 34viList of FiguresFigure 1.1 Schematic of the 1s to 2s transition of Hydrogen. This transition requires two243 nm photons to excite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Bloch sphere and pi/2 rotation of the atomic spin state. . . . . . . . . . . . . . 5Figure 1.3 (a) 199Hg transition scheme and (b) 129Xe transition scheme. . . . . . . . . . . 6Figure 1.4 Block diagram to show the infrared (IR) laser and two enhancement cavities thatproduce SHG at each stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.1 Schematic of experimental setup, including the OPSL, two doubling enhance-ment cavities and locking systems. . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.2 Plot to show all of the currently developed OPSLs at their various wavelengthsand powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.3 Plot of the optical pumping power to the output power of the 972 nm OPSL atfour operating temperatures of the semiconductor chip. . . . . . . . . . . . . . 10Figure 2.4 Schematic of the OPSL head and close up of the semiconductor chip. . . . . . . 11Figure 3.1 Resonator cavity geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 3.2 Normalized intensity profile of the circulating light inside the cavity for a vary-ing round trip phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 3.3 Circulating cavity intensity for impedance matching . . . . . . . . . . . . . . 19Figure 3.4 Schematic of a ring, or bow tie, cavity configuration. . . . . . . . . . . . . . . 20Figure 3.5 Schematic of the spatial mode of the input light coupled to the loose focus anddivergence of the enhancement cavity spatial mode. . . . . . . . . . . . . . . . 22Figure 3.6 Beam radius of the 972 nm laser mode matched into the litium barium octate(LBO) cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 3.7 Picture of the 486 nm light as it propagates far from the LBO SHG cavity. . . . 24Figure 3.8 Beam radius of the 486 nm laser mode matched into the bicarbonate bariumoctate (BBO) cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 3.9 Intensity profile of the reflection off the resonator cavity. . . . . . . . . . . . . 26Figure 3.10 A schematic of the electronics for the Pound-Drever-Hall locking scheme. . . . 27Figure 3.11 Representation of the phase and amplitude under phase modulation. . . . . . . 29viiFigure 3.12 The Pound Drever Hall (PDH) error signal after the reflection signal from thecavity is converted to an radio frequency (RF) signal and mixed with the originalphase modulation signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 4.1 Schematic of biaxial positive and negative symmetries . . . . . . . . . . . . . 34Figure 4.2 Index of refraction curves for changing wavelength, λ , for both LBO and BBOcrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 4.3 Crystal axes with the phase-matching angles, θp and φ , for a given direction ofbeam propagation, k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 4.4 Schematic of uniaxial and biaxial crystal symmetries. . . . . . . . . . . . . . . 37Figure 4.5 Boyd-Kleinman factor as a function of the focusing parameter, ξ , for differentvalues of B, the double refraction parameter. . . . . . . . . . . . . . . . . . . . 40Figure 4.6 Experimental results of the nonlinear power conversion from 972 nm to 486 nmthrough the LBO crystal compared with the KNB theory. . . . . . . . . . . . . 42Figure 4.7 Experimental results of the nonlinear power conversion from 486 nm to 243 nmthrough the BBO crystal compared with KNB theory. . . . . . . . . . . . . . . 42Figure 5.1 Schematic of laser linewidth at 308.64 THz, for an ideal delta function and areal Gaussian with some width, full width at half maximum (FWHM). . . . . . 45Figure 5.2 Schematic of the experimental set-up to measure the linewidth of the OPSL. . . 47Figure 5.3 Evolution of Gaussian line shape to a Lorentzian line shape. . . . . . . . . . . 53Figure 5.4 Illustration the β -separation line for an arbitrary frequency noise spectrum. . . 54Figure 5.5 Measuring the linewidth from frequency noise plot . . . . . . . . . . . . . . . 55Figure 5.6 Close-up view of the frequencies mainly contributing to the laser linewidth. . . 56viiiGlossaryAC alternating currentBBO bicarbonate barium octateCW continuous waveDBR distributed Bragg reflectorDC direct currentEDM electrical dipole momentFSR free spectral rangeFWHM full width at half maximumGPE geometric phase effectIR infraredLBO litium barium octateLO local oscillatorOPSL optically pumped semiconductor laserPDH Pound Drever HallPZT piezoelectric transducerRF radio frequencyROC radius of curvatureSHG second harmonic generationTEC thermoelectric coolingUCN Ultra Cold NeutronUV ultravioletixAcknowledgmentsI would like to use a few words to express my gratitude to those who helped me over the past coupleof years. Most important is Dr. D. Jones, my supervisor, who has given me much support andadvising throughout these years. I would also like to thank Arthur Mills for his daily support inand out of the lab. He significantly helped me generate UV light for the first time. Marc Lejayfor helping out so much with the linewidth measurements and always having the laser warmed upand ready every morning. Lastly, I would like to thank those who have given me abundant supportthroughout my life and given me the opportunities to complete this degree. This is includes myparents, grandparents, brother and cats whom have graciously given nothing but support throughthe good times and bad.xChapter 1IntroductionThis chapter presents the introduction and motivation of the development of a continuous wave (CW)high powered, narrow linewidth (< 1 MHz) optically pumped semiconductor laser (OPSL) forsecond harmonic generation (SHG). The goal is to generate high power ultraviolet (UV) light forthe use in precision spectroscopy. Sections 1.1.1 and 1.1.2 discuss the current experimental projectsthat motivate the work presented. This laser source has the potential to impact both the hydrogenspectroscopy and Ultra Cold Neutron (UCN) experiments, which both require a two photon transitionin the UV. The continual development of this type of laser at new wavelengths will also pose animpact the greater scope of atomic and laser research. Section 1.2 will then present the OPSL sourceas the means for generating high powered infrared (IR) light, compared to other available sources.The OPSL is then paired with two enhancement cavities to produce SHG to achieve a high efficiencyharmonics down to the UV as compared to the single-pass method.1.1 MotivationThe general motivation to this project is to perform precision spectroscopy. Spectroscopy is thestudy of the atomic spectra produced when matter interacts with or emits electromagnetic radiation.In other words, the process by which an atom changes energy levels by absorption or emission ofa photon. Energy transitions in atoms can be probed by electromagnetic radiation, or laser light.There are two spectroscopy experiments that motivate the research presented in this thesis. Thefirst, is precision spectroscopy of the fundamental hydrogen atom. Hydrogen was first discovered in1766 by Henry Cavendish, who described it as “inflammable air from metals” [4]. Since then it hasbeen a major focus for study. This atom is not only the most fundamental atom, composed of oneelectron and one proton, but also has a simple energy transition. This transition consists of neitherhyperfine splitting or spin states and is known as the 1s to 2s transition, as shown in Fig. 1.1. Therequired photon wavelength (or energy) is either a single 121 nm photon or two photons at 243 nm.248 years after the atomic discovery, the energy required for this transition is still being measured inthe effort to determine even more fundamental physics, the radius of the proton [21] . Section 1.1.1will discuss the current efforts to refining the 1s to 2s measurement.1Scientific efforts towards reducing the uncertainties on fundamental constants stretch from theradius of the proton [21] to the electrical dipole moment (EDM) of the neutron [16]. Measurementsof the EDM of the neutron have been studied since the 1950’s [32]. The current experimental meth-ods for determining the neutron EDM uses 199Hg, whereas the new approach will use both 199Hgand and 129Xe. These isotopes are used to reduce systematic uncertainties from long term drifts andinhomogeneities in the applied magnetic field from the measurement of the neutron EDM. In termsof the spectroscopy performed, the 129Xe is excited from the ground state up to the 6p state requiringan energy of two photons at 252 nm. Unlike the hydrogen spectroscopy experiment, the goal is notto determine the exact transitional energy, but rather to simply determine how many 129Xe atomsare in a one of the two ground states. The energy levels of 129Xe have hyperfine splitting and twoground states, with opposite spins. A schematic of the energy levels of interest is shown in Fig. 1.3b.Further discussion of the UCN experiment and spectroscopy are described in Section 1.1.21.1.1 Hydrogen SpectroscopyCurrently, the 1s to 2s transition in hydrogen, shown in the Fig. 1.1.1, is known to within an uncer-tainty of 10−15 Hz as published by Ha¨nsch’s group in 2013 [21]. This is the most precise measure-ment to date. The efforts to improve this measurement and minimize the uncertainty are ongoing.There are several ways to do so, including improving the hydrogen source and the laser used toprobe the transition. Effects from Doppler broadening are minimized for a colder hydrogen source,thus increasing the accuracy of the energy measurement. Another way to improve this measurementis with the laser source. By decreasing the linewidth of the laser source and increasing the intensitywill improve measurement statistics. The energy transition can be achieved with two photons at 243nm. An increase is laser intensity will proportionally increase the transition rate. The intensity canbe increased by increasing the power of the laser or decreasing the beam size. The linewidth of thelaser, which can also be thought of the certainty of wavelength, will also contribute to the uncertaintyin the transition measurement. The more narrow the linewidth, the more certain the wavelength isto be the expected value and thus minimizing the uncertainty in the energy transition measurements.More about the linewidth can be found in Chapter 5, where the linewidth of the OPSL is defined andmeasured. This OPSL system combined with the cold hydrogen efforts by Dr. Lenz Cesar at theFederal University of Rio de Janeiro, has the potential to contribute to the precision measurementsof the 1s to 2s transition of hydrogen [26].1.1.2 Ultra Cold NeutronThe primary goal of the UCN collaboration currently under construction at TRIUMF, the Canadiannational laboratory for particle and nuclear physics, is to measure the EDM of the neutron. In orderfor an elementary particle, like a neutron, to possess an EDM there must be a violation of both parity(P) and time inversion (T) [24]. Although several theories suggest the neutron has a non-zero EDM,the predictions vary significantly over orders of magnitude [11]. Since the 1950’s measurements ofthe neutron EDM have reduced the upper limit on the value by six orders of magnitude, reaching a2243 nm243 nm1s2sFigure 1.1: Schematic of the 1s to 2s transition of Hydrogen. This transition requires two 243nm photons to excite.level of 10−26 in 2006 [3]. One of the biggest sources of uncertainty still comes from systematiceffects. To reduce these uncertainties from effects like a non-uniform field and slow drifts in thefield over time, a source with a well measured EDM is used. The previous experiments by Baker etal. performed the experiment with Mercury, specifically 199Hg [3], which has an EDM with upperbound of |d(199Hg)| < 3.1 x 10−29e cm [14]. The use of the additional atomic source is given thename, co-magnetometer. The use of a co-magnetometer allows the magnetic field in the system tobe calibrated very accurately for a better determination of the neutron EDM by a significant reductionof the uncertainty in the magnetic field. A complete description of this experiment using 199Hg canbe found in the 2013 publication by Baker et al. [3].The new project proposed by the UCN collaboration will use 129Xe and 199Hg as an atomicdual co-magnetometer. 129Xe offers a higher vapour pressure than 199Hg and a lower capture crosssection, meaning more 129Xe can be present in the trap while still avoiding collisions with theneutrons. Using both atomic species as a dual co-magnetometer allows a cross check of geometricphase effect (GPE), due to the opposite sign of the magnetic moments. This reduces some significantsystematic errors in the EDM measurements. Conveniently, both the 199Hg and 129Xe require a UVlaser source within 1.3 nm of each other. The same laser source can be used with wavelengthadjustments that are within the tuning range of the OPSL laser.This first experiment to be done is to precisely measure the EDM of 129Xe using 199Hg as a mag-netometer. Then the next experiment will be to measure the EDM of the neutron using both 129Xeand 199Hg as a dual co-magnetometer. In either case, the general process is the same and detailedby Baker et al. [3], with a short description here. The 129Xe and 199Hg are spin polarized by spinexchange optical pumping, while the neutrons are polarized by passing them through a magnetizedpolarizing foil. The atoms and neutrons are then sent into the detection chamber, where a uniformmagnetic and electric field are applied, B0 and E0, such that the spins are aligned antiparallel to B0.Since the goal of the experiment is to measure the EDM, the strength of the electric field is very3high compared to the magnetic field, where B0 is on the order of 1 µT. A resonant oscillating field,B1 is applied perpendicular to B0. The frequency of B1 is such that it is at the resonant frequencyof either 129Xe, 199Hg or the neutrons. B1 is applied with a strength and time-duration such thatthe spin of the atoms or neutrons will undergo a pi/2 phase shift and become perpendicular to B0.The measurements of the atomic sources and neutrons does not occur simultaneously. Rather theexperiment alternates between the neutrons and atomic sources and continues for up to 26 hours [3].The process of the pi/2 phase shift that the atoms and neutrons undergo is similar. Fig. 1.2shows how the spin of either the atoms or neutrons change from a spin-up state, |+ z >, to shiftingby pi/2, to a superposition of the up and down states, |+ z > and | − z >. Fig. 1.2a is the Blochsphere, where the quantization axis here is the z-axis. When B1 pulse is applied to the atomic, orneutron, source along the x− y plane, the atomic spin will rotate and spiral down by θ = pi/2. Thefull θ = pi/2 occurs when the B1 pulse is at the precession resonance of the atomic source. Fromhere the spin will continue to precess around angle φ , in a superposition between states |± x > and|± y >. The atoms are left here to continue to precess, while the neutrons will undergo another B1pulse to add an additional pi/2 shift. The resulting spin state of the neutrons is measured directlyby passing them through another foil. The remaining discussion here will only focus on the atomicsources and leave the neutron discussion to be read in Baker et al. [3].It is worth writing out the wave function for the spin state after the pi/2 shift following Saku-rai [27]. To start, the initial spin state in the z-basis is given as,|ψ >= 1√2(|+ z > +|− z >). (1.1)The atoms experience a magnetic field in the z direction, giving a HamiltonianH =−( emec)S·B, (1.2)where the magnetic moment is eh¯/2mec and S is the spin. The energy eigenvalues can then bewritten as,E± =∓eh¯B2mec. (1.3)Writing the time dependent form of the wave function, Eq. 1.1 with energy eigenvalues, Eq. 1.3,|ψ(t) >= 1√2(eiE+t |+ z > +eiE−t |− z >). (1.4)It is simple to factor out one of the energy components, providing an overall shift to the entirefunction and leaving a term with the difference in energies. This term is defined as the precessionfrequency [3], ωp, where ωp = |e|B/mec [27]. Now, Eq. 1.4 becomes,|ψ(t) >= eiE+t√2(|+ z > +eiωpt |− z >). (1.5)4zxy! "(a)i)ii)iii)(b)Figure 1.2: Bloch sphere and pi/2 rotation of the atomic spin state. (a) Shows a schematic ofthe Bloch sphere which is a representation of the axis and angles of possible quantumspin states, | ± z >, | ± x > and | ± y >, and every superposition in between. (b) Therepresentation of the rotation of the spin state, from spin up in (i) to a pi/2 rotation in (ii)to (iii) using a pulsed magnetic field at the precession frequency of the particle.After the atoms have gone through a pi/2 shift, they are rotating in the x− y plane. Thus, Eq. 1.5can be written in the x-basis, where |± z >= (|+ x >±|− x >)/√2. Eq. 1.5 becomes,|ψ(t) >= 12[(1+ eiωpt)|+ x > +(1− eiωpt)|− x >](1.6)and simples further to,|ψ(t) >= cos(ωp2t)|+ x >−isin(ωp2t)|− x > . (1.7)At this point, a UV laser is used to probe the atoms along the x-axis to measure their precessionfrequency. Both 129Xe and 199Hg have two ground states, as shown in Fig. 1.3. As the atoms arespinning, they will interact with the laser light as when they appear to be in the -1/2 spin stateand will get excited to a higher energy state. At a rotation of pi later the atoms will appear in the+1/2 spin state relative to the photons, and therefore the photons are invisible. This is the same asapplying a projection operator, Pˆ = |− x ><−x|, to Eq. 1.7 and taking the expectation value,|< ψ(t)|Pˆ|ψ(t) > |2 = sin2(ωp2t). (1.8)Only the sine term remains, oscillating at the precession frequency, ωp between 0 and 1. The |+x >state is known as the dark state because there is no available transition for the atoms to be excitedinto by the available photons. In both Fig. 1.3a and 1.3b the atoms from -1/2 state will absorb thephotons, and in the 1/2 state, or when the atoms have precessed by φ = pi rotation, they are in a5-3/2 -1/2-1/2 1/23/2-1/2 1/2F = 1/2F =3/2F = 1/2XDark State253.7 nm6s2 (1S0)6s6p (3P1)hyperfine structure1/2(a)-3/2-3/2-5/2 -1/2-1/2 1/2 3/23/2 5/2-1/2 1/2F = 3/2F = 5/2F = 1/25p5(2P3/2)6sXDark State252.4 nm x2823.4 nm895.5 nm146.9 nm5p6(1S0)5p5(2P3/2)6phyperfine structuretwo-photon selection  ΔMF=+2 (circularly polarized)1/2(b)Figure 1.3: (a) 199Hg transition scheme and (b) 129Xe transition scheme used as the co-magnetometer in the UCN experiment.dark state and absorb nothing. From this absorption, the precession frequency is measured. For thecase of the 129Xe, the excited atoms decay down in two steps, where in the first step they emit anIR photon, which is then easily detected to determine the precession frequency. Any changes in themagnetic field will also cause changes in the precession frequency. By monitoring the precessionfrequency of the atoms, 129Xe and 199Hg, these small field changes can be detected. The measuredfluctuations and drifts in the field are then applied to the neutron measurements to remove thesystematic errors. Thus reducing the uncertainty from previous measurements of the neutron EDM.Similar to the hydrogen experiment, the energy transition for 129Xe is a two photon transition.The higher power in the UV, gives a higher intensity which leads to a higher interaction rate. Dif-ferent from the hydrogen experiment, the linewidth is not as crucial. The linewidth just needs to benarrow enough to ensure the correct transition is made. The next closest transition for the 129Xe is2 GHz away, so as long as the linewidth is less than 2 GHz, the laser is sufficient.1.2 Ultraviolet LaserDescribed by both Sections 1.1.1 and 1.1.2 a high powered, meaning > 100 mW, CW UV laser witha narrow linewidth, <1 MHz is required. There are no commercially available UV laser sources,so the light must be generated in some way by laser sources at a different wavelength. One optionwould be to use nonlinear optics and start with a visible, 486 nm or 505 nm laser, and double thewavelength down (the frequency up) to the UV. However, this is also not commercially available.IR lasers are commercially available and can be frequency doubled twice down to the UV part of thespectrum. Toptica c© has developed such a laser that includes the two SHG stages, converting IR toUV light. The ratings for this laser system, however, are specified at a UV power of < 100mW and6Figure 1.4: Block diagram to show the IR laser and two enhancement cavities that produceSHG at each stage. The laser source is 972 nm and frequency doubled to 486 nm at thefirst enhancement cavity. Then the light is frequency doubled again to 243 nm using asecond enhancement cavity.only a guaranteed linewidth of < 1 MHz.There is a new type of IR laser available, called an optically pumped semiconductor laser (OPSL).This laser has only been available since the turn of the century. A more complete description of thelaser is detailed in Section 2.1. This laser can produce high CW powers in the IR, up to 3 W, and hasthe potential to produce a narrow linewidth, < 1 kHz, beam. To convert the IR into UV light, thereneeds to be two stages of SHG.In this experiment it is important to get the highest efficiency for the SHG. If the efficiency ofthe single pass is only ∼ 10−4 W−1 [8], then two stages would produce an overall efficiency of∼ 0.01% efficiency. The single pass efficiency varies widely depending on the crystal and the inputpower, as the conversion efficiency has a nonlinear relationship to the optical intensity inside thedoubling element. It is clear, that single pass will not provide the desired power for these precisionspectroscopy experiments. The future for this OPSL system is to provide > 100 mW of UV light.For single-pass, that would require an IR laser of 1,000 W. By placing the nonlinear crystals in anenhancement cavity the light can build up and provide higher input power into the crystal. Since thepower through the crystal is 50-60 times greater than the original laser power, and efficiency of thecrystals increases quadratically with increased input power, the overall efficiency is much higher.A block diagram in Fig. 1.4 shows the laser and two stages of SHG in two enhancement cavity.The total efficiency achieved in this experiment from IR to UV was ∼ 11%. These experimentalresults are shown in Section 4.2, where 1.4 W of 972 nm is converted to 150 mW of 243 nm. Thus,the UV required for the precision spectroscopy experiments of hydrogen and xenon is able to reachsufficient power.This OPSL combined with two stages of nonlinear optics inside enhancement cavities to produceSHG has proved to produce > 100 mW in the UV. Chapter 2 will outline the experimental setupinvolving the OPSL and SHG enhancement cavities to produce 150 mW of 243 nm light.7Chapter 2Experimental MethodThis chapter will examine the entire experimental setup used to generate UV light from the IR OPSL.A schematic of the setup can be seen in Fig. 2.1. Each section of this chapter will focus on one partof the experimental setup. Starting with the laser source in Section 2.1, where a brief descriptionof the OPSL technology is described that makes this laser different from all other laser sources. Anoverview of the enhancement cavities where the SHG takes place is outlined in Section 2.2. Finally,the optical components and the locking system used to lock each cavity to the laser frequency arepresented in Section 2.3.OPSL 972 nm CW 1.7WattsLBO ring cavity!243 nm , 145 mW486 nm , 590 mW972 nm , 1.7 WPumpFeedback beam Feedback beamBeam outLBOBBOPZTPZTCavity Locking SystemCavity Locking SystemChip PZT9 MHz 14 MHz1.4 WFigure 2.1: Schematic of experimental setup, including the OPSL, two doubling enhancementcavities and locking systems.8!"#!#!#!!#!!!$!!! %!!! &!!! '!!! #!!!! #$!!!()*+)*,+(-./,012!-34.5.67*8,0692!VisibleUV IRFigure 2.2: Plot to show all of the currently developed OPSLs at their various wavelengths andpowers. Reproduced with permission from Dr. Yushi Kaneda [18].2.1 The Optically Pumped Semiconductor LaserThe first and most important part of the experimental setup is the laser itself. This type of laser iscommercially available from Coherent c©, where a large range of wavelengths and powers, from 920nm to 1154 nm in the IR range are offered. There are even doubled or tripled frequencies throughsecond and third harmonic generation in a range 355 nm to 639 nm. These ranges are not continuousand there is not an available OPSL at 972 nm and thus, neither at 486 nm. The OPSL used in thisexperiment was designed and built by Dr. Yushi Kaneda from the University of Arizona [18] andoperates at 972 nm with a power of 1.7 W. This OPSL combined with the first stage of SHG is ableto produce up to 600mW of 486 nm light with a linewidth of ∼ 123 kHz. The results of the powershown in Section 3.2.2 and the linewidth in Section 5.3.5.The technology for the OPSL is new and still being developed to provide more available wave-lengths and powers. This type of laser was not even a commercially available product from Coherent c©until 2001. Similar to a diode laser, an OPSL uses a semiconductor as the gain medium. However,instead of using current to invert the medium, as OPSL employs an optical pump. The semicon-ductor chip is made up of layers of semiconductor materials, Gallium Arsenide (GaAs) and IndiumGallium Arsenide (InGaAs), as shown in Fig. 2.4b [34]. When this stack is optically pumped witha diode laser, 808 nm, the photons released are in the range of 972 nm. The wavelength of an OPSLis determined by the quantum wells of the semiconductor materials of the chip. As more semicon-ductor materials are developed, the range of wavelengths will also increase. A table of the OPSL’sthat have been produced in the lab are shown in Fig. 2.2, where each point represents an OPSL withits given power. The OPSL in this experiment is not yet added to this plot, but would add a few newpoints, in particular one at 243 nm with 150 mW.The power the chip is able to produce is controlled by the pumping power [34]. A higher9Output Power [W]0.00.20.40.50.70.91.11.31.41.61.8Pump Power [W]0 4 8 12 16 20 2410C8C 6C4CFigure 2.3: Plot of the optical pumping power to the output power of the 972 nm OPSL at fouroperating temperatures of the semiconductor chip. Reproduced with author’s permis-sion [18].pumping power will increase the power in the laser, until the pumping power reaches a threshold.This is when the pump is too high, which heats up the chip, the power quickly drops off. The chipcan also be cooled with water and thermoelectric cooling (TEC), or also known as a Peltier. Fig. 2.3shows how decreasing the temperature of the chip, increases the laser power for an also increasingpump power. Once again, the point where the laser stops lasing can be seen in Fig. 2.3, when thepower drops off for high pumping powers above 20 W. It is also impossible to continuously cool thechip further down, because the risk of condensation and ice arises.Each time a new OPSL chip is grown, the chip will emit a range of wavelengths with a powerdependent on the wavelength. An example of this can be seen by Wang et al. [34]. Here the authorsalso varying the OPSL chip temperatures to show how the range of wavelengths emitted shifts withtemperature. The central wavelength will have the most power available. If the desired wavelengthis off from the centre wavelength, it is possible to operate the OPSL here, knowing that the possiblepower will be compromised. To choose the wavelength of the OPSL from within the range emittedby the chip at a given chip temperature, there are two filters placed inside the laser cavity. The firstis the birefringent filter placed at Brewster’s angle and the second is the etalon. These filters areused in tandem to select a single longitudinal mode. A laser without filters will lase over a rangeof longitudinal modes, which depends on the net gain bandwidth of the laser. These modes are allseparated by the free spectral range (FSR) of the laser cavity, where FSR is the speed of light dividedby the length of the laser cavity. For a typical semiconductor/diode laser, the two cavity mirrorsare directly attached to either end of the semiconductor chip. This gives a typical cavity length of100 µm, yielding a FSR of 420 GHz [30]. Usually, as is the case with this experiment, a singlemode laser is desired. The etalon is the most narrow filter and selects one longitudinal mode pertransmission peak of the etalon. However, there are many transmission peaks of the etalon, which10972 nm CW1.7 WattsPumpChipPZT and Output CouplerBirefringent FilterEtalon(a)Quantum Wells DBR (High reflector) (Y. Kaneda)6-8!m(b)Figure 2.4: Schematic of the OPSL head and close up of the semiconductor chip. (a) Schematicof the OPSL head. The pump is a diode laser, that optically pumps the semiconductorchip, or the gain medium. The birefringent filter at Brewster’s angle and the etalon arefilters to select the wavelength of the output and the longitudinal mode. The curvedmirror is the output coupler with an attached PZT to control the length of the cavity. ThisOPSL provides 1.7W of CW light at 972 nm. (b) Schematic of the OPSL chip. It is pumpedwith the diode laser and emits photons in the IR. The quantum wells are the layers ofGallium Arsenide (GaAs) and Indium Gallium Arsenide (InGaAs) and the high reflectormaterial is labeled as DBR and acts as a cavity mirror. This schematic was provided byDr. Kaneda [18].can still result in the laser running multi mode. The addition of the birefringent filter, which has abroader transition peak compared to the etalon and more narrow than the laser cavity, allows onlyone longitudinal mode to emit. By inserting the birefringent filter and etalon inside the laser cavity,and tuning the transmission peaks such that they are all aligned, it is possible to select just onelongitudinal mode and run the laser single mode [30]. Both the etalon and birefringent filter aretemperature controlled to fine tune the longitudinal mode or wavelength of the laser, by shifting thealignment of the transmission peaks. A close up schematic of the OPSL can be seen in Fig. 2.4a withall of the above mentioned components in their relative positions. It can also been seen that the OPSLcavity is a simple two mirror standing wave cavity. The obvious mirror is the curved output coupler,where the second mirror is a highly reflective material, called distributed Bragg reflector (DBR),under the stacked semiconductor materials of the gain medium. The overall structure of the laser issimple and similar to some other lasers, like the Ti:Sapphire laser, but the technology of the OPSLchip is the unique component that gives the OPSL a big future in changing laser technology.The OPSL used in this experiment produces 1.7 W of power at 972 nm. The limit of this power isdue to the semiconductor chip. The desired wavelength of 972 nm is not at the peak of the spectrumproduced by the chip, so the maximum power possible is reduced. From Fig. 2.1 the light from thelaser is sent into an optical enhancement cavity where the light undergoes SHG. The experimental11setup of these is described in the next section.2.2 Frequency Doubling Enhancement CavitiesThe motivation to develop this laser system is to provide UV light for precision spectroscopy. TheOPSL is an IR laser at 972 nm. Through the use of nonlinear optics the laser light is converted downto the desired UV light at 243 nm. The process of transforming a fundamental beam into a beamwith half the wavelength, or double the frequency, is called second harmonic generation, SHG.The first observation of SHG was made by Franken, Hill, Peters, and Weinreich (FHPW) andpublished in 1961 [12], where they used a pulsed ruby laser with a crystalline quartz sample. Onlyone year later in 1962, phase-matching was discovered [13] and [20], which increased the interactionlength in the crystal allowing for higher conversion efficiencies. The physics of phase-matching isdescribed in Section 4.1. The introduction of the gas laser in 1963 was the first presentation ofCW SHG [1]. Here 5 x 10−12 W of second harmonic for a 0.7 mW input power, for a maximumefficiency of 7 x 10−9. Ashkin et al. were able to show that the use of CW laser allowed for easieroptimization of phase-matching the crystal and well as an overall higher efficiency [1]. Only a fewyears later in 1966, Ashkin et al. presented results to highlight the stability of SHG in a resonancecavity external to the laser cavity [2]. That is, to place a nonlinear crystal in an enhancement cavitythat is separate from the laser cavity.This approach of enhancement cavities around a nonlinear crystal is the approach used in thisexperiment to double the frequency of the laser twice, from the IR to the UV. The experimentalsetup in Fig. 2.1 shows a schematic of both of the SHG laser cavities. The first enhancement cavityuses a litium barium octate (LBO) nonlinear crystal for conversation from 972 nm to 486 nm andthe second cavity uses a bicarbonate barium octate (BBO) nonlinear crystal for conversion from 486nm to 243 nm. The LBO and BBO are part of the borate crystal family, which was not developeduntil 1979 [8]. Previous nonlinear crystals did not extend transparency down to the deep UV (below200 nm) or with high enough nonlinear coefficient, de f f , for SHG. The significance of de f f can beseen in Section 4.2, where γSH ∝ d2e f f and γSH is the nonlinear conversion factor. Thus, a higher de f fgives a higher SHG conversion factor.In the hunt for funding crystals that could generate light down in the deep-UV, Chen et al. de-veloped the borate compounds [8]. This compound allows for hundreds of different crystal options.The BBO was first developed, exhibiting a transparency range down to 185 nm, followed closely bythe development of the LBO, with a transparency range reaching even deeper into the UV. Today,the LBO and BBO are a very commonly used pair of crystals to convert IR to the UV. The reasonsfor choosing the LBO for the first stage of SHG from the IR to visible is due to the phase-matching,damage threshold and SHG efficiency. The same is true for the BBO and why it is chosen to convertvisible light to the UV [8]. It is from Ashkin et al. work from 1966 and Kozlovsky et al. [19] thatthe theory used to evaluate the SHG in the doubling cavities presented is derived in Chapter 4.122.3 Optical and Electronic ComponentsThe final components in the experimental setup are the optical mirrors and lenses. These compo-nents are used in between the laser and enhancement cavities. They are not simply there for aligningthe beam into the cavities, but are special components for ensuring maximum efficiency in the SHG.Essentially, the beam entering into the optical cavities needs to have the same spatial TEM00 moderequired by the cavities. To do this, lenses are placed along the beam path to focus the beam andcreate the desired beam size to match the desired beam size of the cavity. This process is calledmode-matching. A more complete discussion of mode-matching is given in Section 3.2.2.The experimental setup in Fig. 2.1 also shows the cavity locking systems. There is one for eachof the two SHG cavities. The locking systems use the Pound Drever Hall (PDH) locking scheme [5]as described fully in Section 3.3. This locking system requires the fundamental laser frequency tobe phase modulated. The modulation is applied through a phase modulator, run at two frequen-cies, which are combined through a step-up transformer. The modulator, step-up transformer andmodulation frequencies are shown in the Fig. 2.1 at the bottom of the schematic and denoted by, φ .The next chapters will describe the components and the theory for generating UV light fromIR light through enhancement cavities and SHG. Finally Chapter 5 will separately describe theexperimental setup and results for measuring the linewidth of the OPSL.13Chapter 3Enhancement Cavities for SecondHarmonic GenerationThis chapter presents the theory and experimental methods for enhancement cavity resonators usedfor second harmonic generation, SHG. Two ring cavities, or bow tie cavities, are used in this experi-ment, each to enhance the power of the input beam entering the crystal and to double the frequencythrough SHG by means of the crystal’s nonlinear susceptibility. A full discussion of the SHG fromthese enhancement cavities is the focus of Chapter 4. This chapter starts with the basics of enhance-ment cavities in Section 3.1. This includes an introduction to enhancement cavities and the conceptsof FSR, finesse, and linewidth in Section 3.1.1. Following closely in Section 3.1.2, is the theory ofimpedance matching in order to optimize the enhancement of the light inside the cavity. Section 3.2discusses the experimental method and results for matching the spatial mode of the input beam tothe two enhancement cavities used in this experiment. Finally, the chapter concludes with a dis-cussion of the locking scheme used to lock the enhancement cavities on resonance with the laser inSection 3.3. The locking scheme used is known as Pound-Drever-Hall, (PDH).3.1 Enhancement Cavity BasicsAn optical cavity consists of mirrors placed in a specific geometry to reflect the light between themirrors. If the position of the mirrors are angled just right and the light is aligned to hit the mirrors atthe correct position, the light will reflect around the cavity, over and over on itself. The electric fieldcomponent of the the light after each round trip will add together continuously inside the cavity,enhancing the overall power of the light. These enhancement cavities are a crucial component tothis experiment. The basic properties of these cavities are described throughout this section.3.1.1 Cavity CharacteristicsThere are two main types of optical cavities, a Fabry-Perot and ring cavity. A schematic of bothtypes can be compared in Fig. 3.1, where both types of cavities are used to enhance the light that14r1,t1r2 t Input Beam(a)r1,t1r2 r3r4tInput Beam(b)Figure 3.1: Resonator cavity geometries. (a) Is a two mirror, Fabry-Perot cavity, where thelight gets reflected back and forth between the two mirrors. (b) Is a ring cavity, wherethe lights gets propagated around the cavity in one direction. In both (a) and (b), r1 andt1 are the reflection and transmission field coefficients of the input coupler. rn’s, where n= 2, 3, etc. are the field reflection coefficients from the other mirrors in the cavity and tis the field transmission coefficient through the optical component, i.e. nonlinear crystal.enters into the cavity. To discuss how the light enhances inside the cavity, it is best to consider theintensity of the light. The input light enters the cavity through one of the mirrors, known as theinput coupler, where some light is transmitted and some is reflected. For the cavity in Fig. 3.1a withonly two mirrors, the field inside the cavity reflects back and forth between the mirrors, causing thewaves to overlap. This means that the field inside the cavity is a superposition of two transversewaves that results in a standing wave. For the ring cavity in Fig. 3.1b the field inside the cavityonly travels in one direction around the ring. This means that the field simply adds together in thedirection of propagation.The intensity of light, I is determined by the square of the electric field, E given by,Ir,cI0=∣∣∣∣Er,cE0∣∣∣∣2(3.1)where r and c are related to the reflected and circulating intensity, respectively. Both I0 and E0 arethe intensity and field of the input beam. The reflection intensity discussion will be saved for Sec-tion 3.3, while the circulating intensity is important to define the characteristics of the enhancementcavities. The circulating intensity can also be written in terms of the optical properties of the cavitycomponents as well as the phase of the field inside the cavity. The reflectivity’s of the mirrors aregiven by r1 for the input coupler and rm for the remaining components in the cavity, where rm = rtottand rtot is the product of the reflectivity’s of the remaining mirrors. The term t2 = 1− losses, wherethe losses are from the other optical component(s) in the cavity, like a nonlinear crystal. The liter-ature deals with the transmission and reflectivity of mirrors in different ways, but here, r2 + t2 = 1,where r and t and the electric field reflection and transmission coefficients. After one round trip150 1 2 3 4 5 6 7 8 9 1000.10.20.30.40.50.60.70.80.91Relative Intracavity Intensity, I c/I0Round Trip Field Phase, δFWHMFSR→←← →Figure 3.2: Normalized intensity profile of the circulating light inside the cavity for a varyinground trip phase. The full width at half maximum, FWHM, are shown as the width of theintensity peaks and the spacing between the peaks are labeled as the free spectral range,FSR. The intensity in the cavity remains low until the round trip phase of the field is zeroand the circulating intensity peaks.the field with exhibit a phase with respect the the incoming light and denoted as, δ . The total onedirectional circulating intensity over the intensity of the input light is given by,IcI0=t21(1− r1rm)2 +4r1rmsin2(δ/2). (3.2)The input coupler mirror is defined separately from the rest of the cavity for several reasons. Mainly,the reflectivity of the input coupler will be lower than the other mirrors in the cavity. This is so lightcan enter into the cavity, as well as to satisfy the condition known as impedance matching. Thisconcept of impedance matching will be further discussed in Section 3.1.2.A visual representation of the circulating intensity of the light from 3.2 over the round trip phaseis shown in Fig. 3.2. The most obvious observation is the peaks of intensity at specific values of thephase. This occurs when the cavity is on resonance, or when the phase shift is some integer numberof 2pi after one round trip of the light inside the cavity. When the phase is an integer multiple of2pi , there is constructive interference of the electric field components, resulting in a peak of thecirculating intensity. Experimentally, the phase of the field is varied by changing the length of thecavity, using a PZT attached to one of the cavity mirrors. To experimentally see the peak in power,it is impossible to measure the power in the cavity, without disturbing the system. Rather, a littlelight leaks out of the cavity mirrors, as they at not 100% reflective. This light is monitored on aphotodetector. The leakage will peak in power when the circulating power peaks.The intensity profiles of the circulating light can provide valuable information about the cavity.Two cavity characteristics can be determined straight from the transmission signal. In terms of thelength of the cavity, the peaks occur when a whole number of wavelengths fit inside the cavity, so the16Finesse Free Spectral Range LinewidthLBO 297 1.29 GHz 4.4 MHzBBO 265 0.85 GHz 3.2 MHzTable 3.1: Characterizing values of the two enhancement cavities for SHG.periodicity is every λ for the ring cavity and every λ/2 for the Fabry-Perot cavity. When the signalfrom the photodetector is measured the peaks occur on a time scale, rather than phase or cavitylength. The amount of time it takes the PZT to scan from one resonance to another, or the axialspacing between cavity modes, is called the free spectral range, FSR. Mathematically, FSR = cL ,where c is the speed of light and L is the length of the cavity after one round trip. In practice, FSRis reported as a frequency. The FSR of a cavity can simply to be changed by changing the length ofthe cavity and nothing else. In addition to the FSR, the width of the peaks also contains importantinformation about the cavity. The FWHM of the peak is used as a metric for the linewidth of thecavity. These two cavity parameters can be seen in Fig. 3.2, for an arbitrary linewidth.The FWHM can also be computed mathematically, by using the optical parameters of the cavity,r1 and rm in this case. The full relationship is given asFWHM = ∆v1/2 =2(1− r1rm)√r1rm[22]. (3.3)The definition of FWHM is when the amplitude of the intensity falls to half the intensity. Whencomputing the FWHM Eq. 3.3 is used, but in practice the FWHM is quoted as a frequency. Aseither the reflectivity from the input coupler or other mirrors in the cavity decreases, the FWHM willbecome larger, meaning the width of the peaks in Fig. 3.2 will broaden.When the ratio of the FSR and FWHM is taken, the result is also a physically significant value.This dimensionless ratio is called the finesse, F. Finesse is widely used to describe an optical cavity,where high finesse is on the order of 103−104 and low finesse cavities go down as low as 1−10. Inpractice, to make a high finesse cavity, the mirrors would have very high reflectivity’s of > 99.99%with a negligible amount of additional cavity losses. A low finesse cavity usually has lower mirrorreflectivity’s, especially the input coupler, and also would include additional components in thecavity that add loss, like a nonlinear crystal. From Fig. 3.2, a high finesse cavity would have narrowpeaks that are far apart and a low finesse cavity would have wide peaks that are close together.Similar to FWHM, finesse can also be defined mathematically by the optical parameters in thecavity,F =FSR∆v1/2=pi√r1rm1− r1rm. (3.4)A complete set of value for the finesse, FSR, and FWHM determined for the two cavities used in thisexperiment are given in Table 3.1. The reflectivity’s of r1 and rtot where measured in the lab and theexcess loss through the nonlinear crystals are taken to be 0.5% for the LBO and 0.3% for the BBO.17Finally, the last important quantity in an optical cavity is the enhancement, simply given asthe finesse/pi for the impedance matched case. This means that for a higher finesse cavity, theenhancement will be greater. It seems more desirable to have a higher finesse cavity in this case.However, because the enhancement cavities in this experiment are also creating SHG light from thefundamental, there are limits to the finesse and enhancement in order to optimize on the SHG. Thiswill be examined further in Chapter 4.3.1.2 Impedance MatchingIn this section the concept of impedance matching will be discussed in detail. Essentially, impedancematching means that the loss through the input coupler is equal to the loss from the remaining mirrorand optical component(s) in the cavity, or mathematically speaking, r1 = rm. When a cavity isperfectly impedance matched the reflection dips from the input coupler will reach zero on resonanceas the phase, δ , between the electric fields in and out of the cavity is zero. The circulating power,or enhancement, inside the cavity will also be at a maximum. The representation of the circulatingintensity was given in Eq. 3.2, which simplifies to,IcI0=t21(1− r1rm)2, (3.5)when the δ = 0, 2pi, 4pi, etc. on resonance.For a fixed loss in the cavity, rm, and varying reflectivity of the input coupler, r1, Eq. 3.5 producesFig. 3.3a. Both optical cavities used in this experiment, containing the LBO and BBO crystals, havebeen evaluated separately. The point of highest normalized circulating to input intensity correspondsto the condition when the cavity is impedance matched. The distinct shape of the curve depends onrm as can be seen between the LBO and BBO. The rm value is smaller for the BBO, meaning thereis more loss within this cavity. On both curves when r1 < rm the cavity intensity gradually rises,called under-matching, and fall off very quickly for r1 > rm, over-matching. The vertical dashedlines represent the input coupler values used for each cavity. In this case, the cavities are consideredto both be under-matched. Experimentally there are reasons for under-matching. When mirrors arecoated to reflect certain wavelengths, there is some uncertainty in their reflectivity, up to ±0.5%.Additionally, here the loss from the SHG conversion has not been taken into account, which willshift these plots to be closer to the impedance matched case. If the mirror results in having a higherreflectivity, leading to an over-matched cavity, the circulating intensity falls off very quickly, andthe cavity may no longer produce the desired enhancement.Another approach to consider is what happens to the circulating intensity for a fixed r1 andvarying rm. This relationship can be seen in Fig. 3.3b. Once again both the LBO and BBO have beenconsidered separately, as their input couplers have different reflection coefficients. Here it is clearthat regardless of the input coupler, less loss through the rest of cavity will always result is a higherintensity, since the cavity will leak less light through the mirrors. Comparing these two methodsfor changing optical properties of the cavity, Fig. 3.3a gives the more descriptive representation180.97 0.975 0.98 0.985 0.99 0.995 1020406080100120140Input Coupler Reflectivity, r1I c/I o  LBOBBOLBO cavity rm=99.64%BBO cavity rm=99.27%(a)0.97 0.975 0.98 0.985 0.99 0.995 1050100150Cavity reflectivity, rmI c/I o  LBOBBOLBO cavity r1=98.51%BBO cavity r1=98.63%(b)Figure 3.3: Circulating cavity intensity for impedance matching. (a) For a varying input cou-pler reflectivity and constant cavity loss, where the rm is the reflectivity’s of the othercavity mirrors as well as scattering loss through the nonlinear crystals, 0.3% through theLBO and 0.5% through the BBO. The losses due to SHG are not accounted for here. (b)A varying cavity loss, rm, and constant input coupler, for an r1 = 98.51% for the LBOcavity and r1 = 98.63% for the BBO cavity.of where a cavity lies with respect to the impedance matched case and how much a cavity can beexpected to enhance the light.3.2 Spatial Mode of Enhancement Cavities: Mode MatchingThe basic properties of an enhancement cavity have been discussed in the previous section, includ-ing topics of FSR, linewidth and impedance matching. The next topic to discuss is spatial modematching. This is the process of matching the spatial mode of the input light to the spatial mode ofthe optical cavity. When a laser is mode matched into a cavity most of the input light is coupledinto the fundamental spatial mode, TEM00, resulting in a higher intensity in that mode. When themode matching, or coupling, is poor the intensity will be distributed across many spatial modes. Themaximum possible intensity is determined from impedance matching, as discussed in Section 3.1.2and shown in Fig. 3.3a. To maximize on the available intensity in the TEM00 mode, the cavity needsto be spatially mode matched.There are several pieces to the mode matching puzzle, including the stability of the cavity, thebeam profile throughout the cavity, the beam profile of the laser and then the process of matching thespatial mode of the laser to the spatial mode of the cavity. First, the cavities used in this experimentare symmetric bow tie cavities consisting of four mirrors. A schematic of this configuration can beseen in Fig. 3.4. The bow tie cavity is commonly used for generating one directional SHG usingnonlinear optics and does not create standing waves in the crystal, like the two mirror Fabry-Perotcavity. In the case of this experiment it is desired to have one beam of second harmonic light with19Curved MirrorFlat MirrorFlat MirrorCurved Mirrord1d3d2Reference planeInput BeamFigure 3.4: Schematic of a ring, or bow tie, cavity configuration. Two of the mirrors are curvedand two are flat. The beam travels in one direction around the cavity. The distancebetween the mirrors are labeled as d1, d2 and d3.maximum power, rather than two beams. Therefore, the ring cavity is used and not the Fabry-Perotcavity geometry.3.2.1 Establishing the Mode of the Enhancement CavityAn optical cavity requires that the spatial mode is stable in order for the light to resonate within thecavity. An unstable cavity means the spatial mode inside the cavity is unable to be replicated afterone round trip. Thus the light would eventually leave the cavity and prevent intensity build-up andenhancement of the light in the cavity. Details of stability and how to mathematically determinestability can be found in various textbooks including those by Nagourney and Saleh, [22] and [28]respectively. Stability can also be confirmed using the computer program WinLase c© which allowsthe user to build the optical cavity in the desired geometry. One parameter is varied, for examplethe distance between two of the mirrors, to show where the cavity is stable as a function of thatparameter. WinLase c© then plots the beam radius throughout the cavity for the TEM00 mode. Thisinformation is very important for matching the spatial mode of the laser to the spatial mode of thecavity, as described in more detail in Section 3.2.2. It is also simple to calculate the spatial modeprofile of the cavity using the ABCD matrix for one round trip through the cavity for the desired20TEM00 cavity mode. This is given by,(A BC D)=(1 0−2R 1)(1 d1 +2d20 1)(1 0−2R 1)(1 d30 1), (3.6)where R is the radius of curvature of the mirrors, and d1, d2 and d3 are the distances between cavitymirrors, as shown in Fig. 3.4. The bow tie cavity has two foci, one between the two flat mirrorsand one between the two curved mirrors, where the latter is a tighter focus to optimize the SHG.A complete definition and derivation of the ABCD matrix can be found in any introductory opticstextbook [28].3.2.2 Coupling the Input Beam into the CavitiesIn order to match the input light into a cavity, it is crucial to know the spatial beam profile of thefundamental beam itself. The OPSL is a standing wave cavity with one flat mirror and one concavemirror. The flat mirror is at the back of the cavity and the curved mirror is the output coupler. Thefocus of the beam in the OPSL is at the flat mirror and diverges out to the output coupler. Thebeam continues to diverge as a spherically symmetric Gaussian beam. The basic Gaussian beamproperties can be easily found in an introductory optics textbook [28], however, it is still importantto define a few of the key Gaussian beam properties. First, is the beam radius, w, as a function ofdistance, z, given by,w(z) = w0[1+λ znpiw20]1/2, (3.7)where w0 is the beam radius at the waist when z = 0, λ is the wavelength of the propagating beamand n is the index of refraction. Next is the q parameter, or complex beam parameter, which is usedto describe a beam at any point in space. At the beam waist q is given by,q0 = inpiw20λ (3.8)The q parameter can also be written for the beam as it propagates for some distance, z, simply byadding a real term to q0 to get,q(z) = q0 + z. (3.9)Using the ABCD matrix the q parameter can be easily transformed as the beam propagates alongsome distance or through a lens. This simple transformation equation from q1 to q2 is given by,q2 =Aq1 +BCq1 +D. (3.10)It is important to note that when the light is propagated through a lens, the thin lens approximationis used. This approximation treats the refraction of light through the glass as it enters and exits thematerial as just the light bending once due to the curvature of the lens. Therefore it is said the light21Input CouplerInput beamCavity beamfocusfocusFigure 3.5: Schematic of the spatial mode of the input light coupled to the loose focus anddivergence of the enhancement cavity spatial mode. The focus of the input beam matchesthe focus inside the cavity, and the divergence is the same as the beams meet at the inputcoupler and propagate to the next mirror.bends at the centre axis of the lens.The beam profile for the OPSL can be calculated using theory, but this comes with some assump-tions. Namely, the assumption that the beam is Gaussian. To test if the beam is in fact Gaussian andfollows as the theory predicts, it is simple to measure the profile of the beam with a CCD camera.The light is focused down through a lens, followed by measurements of the beam radius around thefocus. This data is then fit to Eq. 3.7 with one modification,w(z) = w0[1+M2λ znpiw20]1/2. (3.11)An added factor has been included, known as the M2 parameter. This parameter is used to determinehow Gaussian the beam is. For a perfectly Gaussian beam, M2 = 1 and for a non Gaussian, M2 > 1.The OPSL was found to have an M2 = 1.14, which shows that the beam is indeed not perfectlyGaussian, but this is still very good. For comparison, a collimated TEM00 diode-laser usually has aM2 ≈ 1.1−1.7 [28]. It is easy to accommodate for an M2 close to one, by simply multiplying thewavelength by M2 in the formulas for Gaussian beam propagation as in Eq. 3.11.As mentioned, the goal of matching the primary beam to the spatial mode of the LBO and BBOcavities is to couple the light to the TEM00 mode and maximize the intensity in that mode. Thebasics of mode matching include using free space and lenses to change the beam shape and size.There are two foci in the ring cavity, one through the nonlinear crystal between the two curvedmirrors and one between the two flat mirrors. The one that goes through the crystal is a tight focuswith a Rayleigh range that is approximately the length of the crystal to maximize SHG. As a result,220 0.2 0.4 0.6 0.8 1 1.2 1.400.10.20.30.40.50.60.70.80.91f = 175 mmf = 100 mmDistance from OPSL head (m)972 nm Beam Radius (mm)  972 nm beam radiuslensTangential cavity beam radiusSagittal cavity beam radiusFigure 3.6: Beam radius of the 972 nm laser mode matched into the LBO cavity, where thelenses used to mode match the beam are given as diamonds with their focal lengthslabeled. The orange and red lines represent the sagittal and tangential spatial mode of thecavity, from the loose focus to the first curved mirror.the focus between the two flat mirrors is a looser focus. The beam then diverges from the loosefocus and reflects off the input coupler until it reaches the first curved mirror, which causes it tothen tightly focus. For mode matching, the input beam is matched to the loose focus and divergenceof the beam at the input coupler. If the incoming beam has a similar radius right before it entersinto the cavity and diverges at the same angle as the cavity profile, the light should couple well. Aschematic of this can be seen in Fig. 3.5, where only the beam around the input coupler is shown inand out of the cavity.When mode matching the beam into the two enhancement cavities, ABCD matrices are usedto propagate the beam from the laser head through the optics until it reaches the first enhancementcavity. In the following sections the beam radius is plotted before each cavity to illustrate how thebeam is shaped to get good coupling into each of the cavities.Mode Matching to the LBO CavityOnce the beam from the OPSL was determined with the corresponding M2 = 1.14, the beam ispropagated through several optics using ABCD matrices, including lenses to focus the beam intothe first enhancement cavity, containing the LBO nonlinear crystal. Fig. 3.6 shows the beam radiusfrom the OPSL to the LBO cavity.The gap in the beam radius is simply where the optical isolator (OI) is located. The red curvesare the tangential and sagittal beam radius1 of the loose focus inside the LBO cavity between the two1The tangential plane is the plane where the optical plane makes an angle with respect to the beam axis. The sagittalplane is perpendicular to the tangential plane. In the case for both the LBO and BBO enhancement cavities, the tangential23Figure 3.7: Picture of the 486 nm light as it propagates far from the LBO SHG cavity. It canbe seen that the beam is in the TEM00 mode. The beam diverges at a larger angle in thevertical direction than the horizontal, producing the ovular beam shape.flat mirrors. As mentioned, this is what the input beam mode is matched too. Since the tangentialand sagittal beam was not considered separately for the input light and the difference is minimalinside the cavity, it was decided to match the beam in-between the two curves. The mode matchinginto the LBO cavity resulted in 74.0% coupling. This result is further discussed in Chapter 4.Mode Matching to the BBO CavityThe mode matching to the second enhancement cavity is similar to the LBO with a few additionalconsiderations. The light emitted from the LBO cavity is the second harmonic light, so 486 nm fromthe fundamental 972 nm. The light is generated in the LBO crystal and propagates out from thecrystal through the curved mirror, not following the spatial mode of the cavity. The first thing tonotice is that the beam is not spherically symmetric. Fig. 3.7 is a picture of the beam far from thecavity. By focusing this beam through a lens and measuring the beam profile for the horizontal andvertical directions separately and fitting each to Eq. 3.11, the beam is found to be nearly Gaussianin both directions, where M2horizontal = 1.02 and also M2vertical = 1.02.The 486 nm light is nearly Gaussian, but is still not spherically symmetric and is astigmatic, soeach component needs to be mode matched separately. Fig. 3.8 shows the two components, wheregreen is the vertical beam radius and blue is the horizontal. The methods are the same as with thefirst enhancement cavity, by using ABCD matrices through lenses to match to the spatial cavitymode. Except this time cylindrical lenses are used to isolate the beam shaping in the vertical andhorizontal directions. This also allows for matching to the tangential and sagittal conditions of thecavity separately for better mode coupling.Similarly to the LBO, the results of the mode matching to the BBO cavity were found to havea coupling of 83.6%, which is later discussed in Chapter 4 through experimental and theoreticalmethods.plane is parallel to the table.240 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.811.21.4 f = 150 mm f = 100 mmf = 20 mmDistance from LBO cavity (m)486 nm Beam Radius (mm)f = 50 mm f = 75 mm  486 nm Sagittal beam radius486 nm Tangential beam radiuslensSagittal cavity radiusTangential cavity radiusFigure 3.8: Beam radius of the 486 nm laser mode matched into the BBO cavity. The lensesused to mode match the beam are given as diamonds with their focal lengths labeled. Theorange and red lines represent the sagittal and tangential spatial mode of the cavity, fromthe loose focus to the first curved mirror. The green and blue represent the horizontal andvertical spatial modes matching individually using cylindrical lenses.3.3 Locking the Enhancement Cavities using Pound-Drever-HallSchemeThe Pound-Drever-Hall locking schemes description starts with a discussion of the reflection signaloff the input coupler of the enhancement cavity. Similar to the discussion in Section. 3.1 regardingthe circulating intensity inside the cavity, the reflection off the input coupler is now considered. Thefractional intensity is given by,IrI0=(r1− rm)2 +4r1rmsin2(δ/2)(1− r1rm)2 +4r1rmsin2(δ/2), (3.12)where r1 is the field reflection coefficient from the input coupler, rm is the remaining reflectivity’sof the cavity mirrors and also includes extra losses due to scattering in the cavity [22]. The roundtrip phase is given by, δ . Eq. 3.12 plotted as a function of the round trip phase gives Fig. 3.9.The intensity in the reflected signal is high except for specific values of the round trip phase.Just like in Fig. 3.2 where their were peaks in intensity, the intensity in the reflection dips. This iswhen the phase difference between the input field and the field inside the cavity after one round tripis a minimum. The electric field components of the reflected signal destructively interferes with thesmall amount of leakage of the circulating light through in input coupler, causing the intensity todip [5]. In an ideal case, the reflection dips would go to zero on resonance2.The reflection signal can be used to force the laser cavity to stay on resonance, this is called2A zero electric field only results when the laser is perfectly mode and impedance matched. Experimentally the goalis to get the dips as close to zero as possible.250 1 2 3 4 5 6 7 8 9 1000.10.20.30.40.50.60.70.80.91Fractional Reflection, I r/I0Round Trip Field Phase, δFWHMFSR→←← →Figure 3.9: Intensity profile of the reflection off the resonator cavity. The full width at halfmaximum, FWHM, is shown as the width of the absorption lines and the spacing betweenthe peaks are labeled as the free spectral range, FSR. The reflection shows maximumintensity until the cavity is on resonance and then the intensity dips due to the destructiveinterference of the electric field at the input coupler.locking. This means that cavity length is locked such that the circulating power is a maximum andthe reflected power a minimum for an extend amount of time. This is very important to the SHGto maximize the power in the conversion. The locking scheme used in this experiment is knownas the Pound-Drever-Hall, PDH, approach. This method for locking was first developed by Poundin 1946 [23] for microwaves. It was not until 1983 when Drever et al. adapted the method for theoptical domain [10] to provide laser stability. It is also important to mention that there are two waysto perform this PDH locking scheme. The classical way is for the laser to be locked to a Fabry-Perotcavity with a fixed length. This method stabilizes the laser frequency to match the cavity length andis used in Chapter 5 for measuring the linewidth of the laser. The other method, which is used forthe enhancement cavities, is locking the cavity to the laser. In this case there is a PZT inside theenhancement cavities that is locks the length of the cavity to the laser frequency. For any drift in thelaser, the PZT changes the length of the cavity to compensate. Before diving into the full explanationof the PDH scheme, it must be established why the PDH approach was chosen over other approaches.A schematic of the optical and electrical signals used to achieve the PDH locking scheme areshown in Fig. 3.10. The alternating current (AC) signal from the photodiode, where the reflectionintensity is measured, enters the mixer as the RF signal. The modulation frequency is split intotwo signals. One signal goes into the phase modulator to modulate the carrier frequency of thefundamental beam and the other signal is sent through a phase shifter and into the local oscillator(LO) input of the mixer. The phase shifter allows for tuning of the phase between the RF and LOsignals to ensure a zero phase shift. The importance of the phase shifter is explained a little later.This is important to the error signal and to the linearity around the zero-zero crossing, as will bealso be described more later. The mixer generates the error signal seen in Fig. 3.12. This signal is26Mixer RFLOIF PhotodiodeΔ! Modulation frequencyError Signal Loop FilterPZTReflection beam Laser Phase modulator!Phase shifterFigure 3.10: A schematic of the electronics for the Pound-Drever-Hall locking scheme. Thelaser beam is modulated by the phase modulator at a frequency, Ωm. The reflectionbeam off the cavity input coupler is propagated onto a photodiode. This signal is thensent into the mixer as an RF signal. A second branch of the modulation frequency issent through a phase shifter, ∆φ , and into the local oscillator side of the mixer. Theoutput gives the error signal. This is then sent into the controller, or loop filter and usedto control the PZT in the cavity.then sent into a servo, where the control system uses the zero-zero crossing to lock the cavity to thelaser frequency. The output signal from the servo, or loop filter as labeled in Fig. 3.10, is fed backto the enhancement cavity PZT where the cavity length is adjusted to push the phase difference backto zero on the error signal.In the case of this experiment, the reflected light off the cavity is detected on a photodiode.When the intensity of the light changes, the photodiode converts this into a change in voltage. Thissignal is sent from the photodetector to a control system that uses feedback to control the length ofthe cavity. The servo can then simply change the voltage on the PZT to maintain the minimum poweron the photodetector. A change in reflected power could have a number of causes, one because thecavity is off resonance and therefore there is some overall intensity change, or simply because thepower in the laser changed. Assuming it is the first reason, the intensity would change the sameamount on either side of the resonance, above or below. The photodetector sends this higher voltagesignal to the servo, which then determines that the cavity is off resonance. Since the signal appearsthe same above and below resonance the electronics are unable to determine in which direction thecavity length is off. The electronics might push the cavity length in the correct direction, or push iteven farther off resonance. In order for the electronics to accurately determine if the cavity is above27or below resonance, it would have to distinguish the phase from the intensity. Currently there are noelectronics that can do this [5]. Another possible locking scheme involves locking to the side of aresonance peak, however since the lock is not at the peak of the resonance, it is not possible to lockto the highest possible power in the cavity.The ability to use changes in phase is what makes the PDH scheme so stable and a desirableapproach to locking. To determine the phase of the reflected signal, the phase of the carrier, or laser,frequency is modulated. This is known as the modulation frequency, Ωm, and is usually larger thanthe linewidth of the enhancement cavity that is being locked to the laser. For this experiment themodulation frequencies are Ωm = 14 MHz for the LBO cavity and Ωm = 10 MHz for the BBO cavity.This modulation is applied by passing the laser light through a phase modulator. Typically, a laseris only modulated at one frequency, however, for this experiment each enhancement cavity requiresits own modulation frequency. To do this, an adding step up transformer is used to combine the twofrequencies that puts the corresponding phase modulations on the carrier frequency, as can be seenin the original schematic in Fig. 2.1.Next, it is useful to jump ahead to Fig. 3.12 to conceptually understand the benefit and the stabil-ity of the the PDH scheme. This figure shows the PDH error signal at high modulation frequencies,in the range of MHz. The zero-zero crossing of the signal is the position where the cavity is onresonance with the laser. The slope of the line crossing through zero is both very steep and withhigh amplitude. When the cavity is slightly detuned from resonance, the signal on the photodiodewill contain some component of the modulation frequency and therefore a change in the phase. Thiswill cause a corresponding change along the error signal, either in the positive or negative direction.This change in voltage will then cause the servo to apply a feedback voltage back on the cavity PZTto drive the phase shift back to zero.To mathematically describe how to arrive at the high modulation frequency PDH error signal,start with the unmodulated electric field of the input beam,E(t) = E0e−iφ(t) = E0e−iωct , (3.13)where E0 is the amplitude of the field, φ(t) is the time-dependent phase, and ωc is the carrierfrequency. A time-dependent phase modulation, given by ∆φ(t), is then applied to the carrier fre-quency. ∆φ(t) can be written as β sinΩmt, where Ωm is the modulation frequency and the mod-ulation depth is given as β and determines the amplitude of the modulation. For the sake of thisderivation, only one phase modulation will be applied to the carrier frequency. This derivation cansimply be expanded to account for the two modulations that are actually applied in the experiment,one for each enhancement cavity. The modulated field then becomes,E(t) = E0e−i(φ(t)+∆φ(t)) = E0e−i(ωct+β sinΩmt). (3.14)28Im-ΩmΩmRecarrier, !cmodulated carrier"Δ"(a)AmplitudeFrequency Spectrum !c!c - Ωm !c + Ωm(b)Figure 3.11: Representation of the phase and amplitude under phase modulation. (a) Is thephasor diagram on the real and imaginary axes. The carrier frequency, ωc rotates withphase, φ . The modulated carrier is phase shifted by ∆φ from the carrier frequency by amodulation frequency of Ωm. (b) Is the amplitude of the carrier frequency with phaseshifted sidebands at frequencies ωc±Ωm.This can then be expanded using Bessel functions to the first order, with small β ,E(t)≈ E0[J0(β )e−iωct + J1(β )e−i(ωc+Ωm)t − J1(β )e−i(ωc−Ωm)t ], (3.15)where J0(β ) and J1(β ) are the Bessel function coefficients [15]. From Eq. 3.15 it is clear to see thatthere are in fact three terms to the electric field contributed by the phase modulation. These J1(β )terms are knows as the sidebands as shown in Fig. 3.11b. Once again, β is the modulation depth.For β < 1 almost all of the power is in the carrier frequency and the first order sidebands [5]. Dueto the phase modulation, one sideband is 180o out of phase with the other one. This is actually akey component to how PDH lock works, being able to discriminate which direction the resonance isoff by the sign due to the sideband.Taking a closer look, Fig. 3.11 can give more conceptual understanding before evaluating themathematics any further. Fig. 3.11b shows the amplitude plot of the carrier frequency and sidebands.This is not the absolute amplitude so that it can clearly be seen that the two sidebands are 180o outof phase with each other. The two sidebands are rotating in opposite directions, so the sum of theamplitudes remains constant. Fig. 3.11a shows the phasor representation of the carrier frequencyand two sidebands. Here the carrier frequency remains constant, rotating at ωc, while the modulationfrequencies rotate creating a new vector, called the phase modulated carrier frequency. This vectorhas a phase shift of ∆φ from the carrier frequency. This phase shift changes as the modulationfrequencies rotate.Continuing with the mathematical derivation of the error signal, the reflected light can be writtenin terms of the electric field of the incident light and the reflection coefficient, F(ω). The reflection29coefficient is simply the ratio of the reflected electric field to the incident electric field at a givenfrequency, ω ,F(ω) =Ere f lE0=r(eiω/∆νFSR−1)1− r2eiω/∆νFSR. (3.16)The amplitude reflection coefficient is r and ∆νFSR is the FSR [5]. Eq. 3.15 can then be written as,Ere f l = E0[F(ωc)J0(β )e−iωct +F(ωc +Ωm)J1(β )e−i(ωc+Ωm)t−F(ωc−Ωm)J1(β )e−i(ωc−Ωm)t ].(3.17)The next stage is to transform the reflected electric field into a power, as that is what the photodetec-tor measures. The relationship between power, P and electric field, E, is simply P = |E|2. Therefore,Eq. 3.17 can be written as,Pre f l = Pc|F(ωc)|2 +Ps|F(ωc +Ωm)|2 +Ps|F(ωc−Ωm)|2+2√PcPs{Re[F(ωc)F∗(ωc +Ωm)−F∗(ωc)F(ωc−Ωm)]cosΩmt+Im[F(ωc)F∗(ωc +Ωm)−F∗(ωc)F(ωc−Ωm)]sinΩmt}+ higher order terms,(3.18)where Pc is the carrier power and Ps is the sideband power [5]. This equation simplifies dependingon the modulation frequency. For the case of this experiment, the modulation frequency is high,meaning Ωm ∆ν f sr/F, where ∆ν f sr/F is the linewidth of the cavity. Only the sine term remains,while the cosine term vanishes [5].This RF signal is fed into the mixer where it is mixed with the original modulation frequency. Inthe mixer, the inputs are multiplied together. Thus, the product of two sine waves will give a cosineof the sum and difference term of the phase shift between the input waves. Similarly, if one input isa sine and the other a cosine, the product is the sine of the sum and difference of the phase shift [5].When the phase between the RF and LO signal is the same, the result is a direct current (DC) plus ACterm for the two sine waves and only an AC term for the sine and cosine inputs. This AC term is theerror signal. The phase of the modulation frequency can simply be adjusted to make this scenariotrue by running the signal through a phase shifter before the mixer.For high modulation frequency, as used in this experiment, the error signal is given by,ε =−2√PcPsIm[F(ωc)F∗(ωc +Ωm)−F∗(ωc)F(ωc−Ωm)], (3.19)and produces the signal shown in Fig. 3.12. For the point where the cavity is on resonance with thelaser carrier frequency, the error signal crosses zero and is nearly linear. Further simplification canbe done to reduce the analysis around this point.When the carrier frequency is at or near resonance, the sidebands are far enough away that theirpower is completely reflected, or r ≈ 1. This results in the sideband reflection coefficients from30Figure 3.12: The PDH error signal after the reflection signal from the cavity is converted toan RF signal and mixed with the original phase modulation signal. When the phasebetween the two mixed signals is the same, the above error signal is produced. Plotproduced by Black [5].Eq. 3.16, F(ωc±Ωm)≈−1. Thus, reducing Eq. 3.18 to,Pre f l ≈ 2Ps−4√PcPsIm{F(ωc)}sinΩmt + higher order terms. (3.20)The error signal also simplifies to,ε = 4√PcPsIm[F(ωc)]. (3.21)The reflection coefficient from Eq. 3.16 can be approximated for the carrier frequency, so that it doesnot go to zero. Exactly on resonance ω∆νFSR = 2piN, where N is an integer number and ei2piN = 1, soF(ωc) = 0. Thus, ω∆νFSR is instead replaced by,ω∆νFSR= 2piN + δω∆νFSR, (3.22)where δω∆νFSR is some small offset from resonance. The finesse of the cavity can also be approximatedas F≈ pi/1−r2, when r≈ 1, and the linewidth is δν =∆νFSR/F. For an offset frequency, δω , muchless than the linewidth, δν , and after some substitution and simplification, the reflection coefficientof the carrier frequency is approximated to,F(ωc)≈ipiδωδν . (3.23)31The error signal becomes,ε ≈− 4pi√PcPsδωδν . (3.24)Now, it is clear to see that the error signal is linear around resonance for small deviations in thephase, or frequency. This is beneficial to the stability of the lock because a small change in phase,translates directly to a small change in the error signal and then the servo can linearly respond to thePZT. If the error signal was not linearly proportional around zero-zero, then the feedback loop mayover compensate or add additional noise to the lock, resulting in a less stable lock.32Chapter 4Nonlinear Optics and Second HarmonicGenerationThe focus of this chapter is specifically on the second harmonic generation, SHG, process usingnonlinear optics through the two optical enhancement cavities. The last chapter detailed the basicsof the optical cavity and how to experimentally match the fundamental spatial mode to the cavitymode. This chapter will explain the theory and experimental results of using nonlinear optics insidethe enhancement cavities, starting with the basics in Section 4.1. This section will cover the differentcrystal types used in this experiment and their geometrical symmetries. Then Section 4.1.1 presentsthe theory for the nonlinear conversion factor, γSH , and the Boyd-Kleinman parameter. Finally thetheory and experimental results for the conversion from the input light into the enhancement cavityto the second harmonic generated light are presented in Section 4.2. The comparison of the theoryto experimental data gives a measure for how well the cavities where impedance and mode matchedfrom Chapter 3.4.1 Nonlinear Crystal Symmetry BasicsThe uses of nonlinear crystals varies widely across the field of optics, from second and third har-monic generation to frequency mixing. Nonlinear crystals have three main symmetries, uniaxial,biaxial and isotropic. In this experiment only uniaxial and biaxial are used, so the discussion willbe focused on these two types. Uniaxial and biaxial refers to the number of optical axis that are inthe crystal as well as the indices of refraction along the x, y and z axes. The optical axis, or c-axis,is defined as the axis along which a beam would experience no birefringence or double refraction.Table 4.1 shows the relative indices of refraction, n, for the uniaxial and biaxial crystal, for bothpositive and negative types. The uniaxial defines the x, y and z in terms of just two values, theordinary and extraordinary, denoted as o and e respectively. This is simply because two of the axeshave the same index of refraction. The relative values of n for the biaxial crystal remain such thatnz > ny > nx for both the positive and negative types. The positive and negative biaxial types comefrom how the two optical axes are oriented in the x− z plane. Fig. 4.1 illustrates the positive and33Uniaxial BiaxialPositive nz = ne > nx = ny = no nz > ny > nxNegative nx = ny = no < nz = ne nz > ny > nxTable 4.1: The relationship of the indices of refraction of the uniaxial and biaxial crystals alongthe x, y and z axes, for the positive and negative orientations. The uniaxial is also describedby ordinary, o, and extraordinary, e, axes. The relationship of n for the biaxial crystals isthe same for both positive and negative symmetries.zxyc-axisc-axis !(a) Positive Biaxialzxyc-axisc-axis !(b) Negative BiaxialFigure 4.1: Schematic of biaxial positive and negative symmetries. (a) Is the positive biaxialcrystal, where the two optical, c-axes are folded around the x-axis and (b) a negativebiaxial crystal where the two optical, c-axes are folded around the z-axis.negative orientations of the optical axes for the biaxial crystal. The angle between the two opticalaxes, θ , is determined by the indices of refraction and is given by,θ = cos−1√1/n2y−1/n2z1/n2x−1/n2z. (4.1)When θ is large, the optical axes are closer to the x-axis and referred to as positive biaxial. Whenθ is small, this is negative biaxial and the optical axes are closer to the z axis. Table 4.1 shows therelationship of the indices of refraction for the uniaxial and biaxial, positive and negative crystalsymmetries.The discussion about the indices of refraction in both crystal symmetries is extensive and mostof the information is easily available in various nonlinear optics text books, including Boyd [7] andNagourney [22]. A brief overview will be given here to preface the next section of second harmonicgeneration. In 1871 Wilhelm Sellmeier first proposed alternate methods to Cauchy’s theory on theindex of refraction through a material and how it depends on wavelength. Sellmeier found the340.2 0.4 0.6 0.8 1 1.21.51.551.61.651.71.751.81.851.9Index of RefractionWavelength (µm)nxnynzλ = 972 nmλ = 486 nm(a) LBO0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.51.551.61.651.71.751.81.851.9Index of RefractionWavelength (µm)noneλ = 486 nmλ = 243 nm(b) BBOFigure 4.2: Index of refraction curves for changing wavelength, λ , for both LBO and BBOcrystals. (a) The LBO crystal, where the indices for each axis are shown as well as thewavelength for both the fundamental and second harmonic. (b) BBO crystal indices ofrefraction are shown, for the ordinary and extraordinary axes, as well as the fundamentaland second harmonic λ . The coefficients used are the Sellmeier coefficients given inNagourney [22].following relationship,n2 = A+Bλ 2λ 2−C +Dλ 2λ 2−E , (4.2)where A, B, C, D, and E are Sellmeier coefficients and λ is the wavelength. These coefficients areunique for each material and have been listed in Table 13.1 in Nagourney for both the LBO andBBO crystals [22]. Fig. 4.2 shows Eq. 4.2 for the LBO and BBO cavities, with both the fundamentalwavelength and second harmonic wavelength identified. The relationships between the indices ofrefraction and wavelength conversion are important for the concept of phase-matching for SHG.This experiment uses two nonlinear crystals for SHG inside an enhancement cavity. The firstcrystal used is the LBO, which is a negative biaxial crystal, and the second is a BBO, which is anegative uniaxial crystal. The beam propagation through the crystal can be defined in terms of twoangles, θp and φ . These are known as the phase-matching angles. The phase-matching angle θp isthe angle k makes with the z-axis and φ is the angle from the x− z plane. In order to achieve SHGin a nonlinear crystal, the crystal must be phase-matched. This means that the indices of refractionare the same for the fundamental and second harmonic wavelengths. This can only be achievedfor one wavelength if the light is simply propagated long a specific axes, say the z-axis. This is notconvenient for all lasers, as the wavelengths desired varies across experiments and there are a limitednumber of available nonlinear crystals. To remedy this, the fundamental beam can be propagatedat some angle relative to the axes, given as either θp or φ . The angle θp is the angle between theoptical axis and the direction of beam propagation, k, and lies in the x− z plane. The angle φ isthe azimuthal angle from the x− z plane to the direction of propagation. A basic schematic of the35zxy!p"k→Figure 4.3: Crystal axes with the phase-matching angles, θp and φ , for a given direction ofbeam propagation, k.axes, beam propagation direction, k, and phase-matching angles are illustrated in Fig. 4.3. For thespecific types of crystal used, the uniaxial and biaxial, a schematic of the axes and phase-matchingangles can be seen in Fig. 4.4. Each type of crystal has a different critical phase-matching angle.For the uniaxial crystal in Fig. 4.4a, θp, is the critical angle and φ is arbitrary because the indicesof refraction along the x and y axis are the same as defined in Table 4.1. For the biaxial crystal, θpis fixed at 90o so that φ becomes the critical phase-matching angle. However, both are importantangles to specify when growing the crystals.The methods for selecting the phase-matching angle are easily described for the uniaxial crystal,but can be applied to the biaxial case in a similar manner. For the negative uniaxial crystal, the fun-damental light is the ordinary wave, polarized perpendicular to the c-axis, and the second harmonicwould be the extraordinary wave, polarized long the c-axis. When a crystal is phase-matched the in-dex of refraction of the the ordinary wave, no, equals the index of the second harmonic extraordinarywave, ne. From Fig. 4.2 it can be seen that for both the LBO and BBO cases, nνo 6= n2νe . The notationν is used to describe the fundamental and 2ν for the second harmonic of the optical frequency. Thisdiscrepancy in indices can be more clearly seen for the BBO crystal, where the no at the fundamental,λ = 486 nm, does not give the same value in ne at the second harmonic, λ = 243 nm. If the beam ispropagated at some angle, θp, relative to the optical axis, then the index of refraction that the beamsees is different. The relationship to describe how the index depends on θp can be written as,1(n2νe (θp))2=cos2(θp)(n2νo )2+sin2(θp)(n2νe )2, (4.3)There exits some angle for which the phase-matching condition holds, no = n2νe, . By rewriting36!2!"z  (c-axis)#p k→xynone(a) Uniaxial!, 2!" k→yxz  (c-axis)nynxnz(b) BiaxialFigure 4.4: Schematic of uniaxial and biaxial crystal symmetries. (a) The negative uniaxialcrystal, as seen from the side view. The direction of propagation, k, of the fundamentalfrequency, ν lies at an angle θp from the z-axis. phase-matching angle, φ = 0 as theindices of refraction along the x and y axis are the same, and called the ordinary axis, no.The z-axis is the extraordinary axis, ne, also known as the c-axis. The walk-off angle, ρis the angle the second harmonic, 2ν , from the fundamental beam, ν . (b) The negativebiaxial crystal, as seen from the top view. In this crystal the angle θp = 90o, making φthe critical phase-matching angle along which k propagates.Eq. 4.3,sin2(θp) =(nνo )−2− (n2νo )−2(n2νe )−2− (n2νo )−2(4.4)a value for θp can be found for the desired fundamental wavelength. This same relationships canbe used in the biaxial crystal cases, where each of the axes exhibit a different index of refraction tofind both the value for θp and φ .There is another phenomenon that takes place inside the crystal, called double refraction orbirefringence. When the second harmonic photons are produced they will propagate at some angleaway from the fundamental direction of propagation. This is known as the walk-off angle anddenoted by ρ . The expression to find this angle is given by,tan(ρ) = (nνo )22[1(n2νe )2−1(n2νo )2]sin(2θp), (4.5)where it can be seen that ρ depends on θp and on the the indices of refraction seen by the funda-mental and second harmonic [17]. A schematic of the walk-off angle and how the second harmonicseparates from the fundamental is shown in Fig. 4.4a for the uniaxial crystal.The BBO, negative uniaxial crystal if found to have a phase-matching angle of θp = 54.8o, withφ = 0 because the index of refraction along the x and y axes are the same, making φ a non-criticalphase-matching angle. From θp, the walk-off angle is found to be ρ = 4.9o. The LBO crystal is37a negative biaxial crystal, with θp = 90o and φ = 17.5o. In this case, θp is the non-critical phase-matching angle. The walk-off angle is proportional to sin(2θp), so when θp = 90o, ρ = 0, so thereis no walk-off angle for this crystal.Finally, the effective nonlinear coefficient, denoted by de f f , is another coefficient used in thedescription of second harmonic generation. This coefficient is dependent on θp, φ , ρ and λ . Thevarious crystal symmetries have a basic formula to describe de f f in terms of other tensor coefficients,di j and θp and φ [22]. For the BBO,de f f = d15sinθ −d22cosθsin3φ . (4.6)The tensor coefficients, like d15 and d22, are dependant on crystal symmetry, ρ and λ . The bestcomputation of de f f for both the LBO and BBO crystal used at their respective wavelengths is aprogram called SNLO c© [31]. This program considers all of the relevant parameters and correctionsto determine the most accurate value for de f f . For the LBO, de f f = 0.82 pm/V and for the BBO,de f f = 1.58 pm/V.Understanding the basics of the crystal symmetry is important for continuing the discussionof second harmonic generation and optimizing conversion efficiency from the fundamental to thesecond harmonic inside the enhancement cavity.4.1.1 Boyd-Kleinman Nonlinear Conversion FactorThis section will present the theory for the second harmonic conversion factor from the funda-mental to the second harmonic through a nonlinear crystal using the previously developed crystalgeometries. The derivation of the conversion factor begins with the wave equation for the secondharmonic,δE(2(2piν))δ z =−i(2piν)n2νcde f f E(2piν)E(2piν)ei(k(2(2piν))−2k(2piν))z [17], (4.7)where n2ν is the index of refraction seen by the second harmonic and de f f is the effective nonlinearcoefficient as discussed in the previous section. For the phase-matched condition, the wave numbers,(k(2(2piν))− 2k(2piν)) = 0 and the index of refraction is the same for both the fundamental andsecond harmonic, n2ν = nν = n. Simplifying and differentiating Eq. 4.7 for z = l,E(2(2piν),z = l) =− i(2piν)ncde f f E2(2piν). (4.8)In terms of the optical intensity,I(2(2piν), l) =−2(2piν)2d2e f fn3c3ε0L2I2(2piν), (4.9)where ε0 is the permittivity of free space and L is the interaction length, or length of the crystal [17].From Eq. 4.9 the coefficients in front of the l2I2(2piν) can be lumped together as a general form for38the second harmonic conversion factor.Since this experiment deals with a focused Gaussian beam through the nonlinear crystal, thesecond harmonic conversion factor requires an additional component that depends on the beamshape and size through the crystal. This term has been derived from the well-known Boyd andKleinman work from 1968 for a focused Gaussian beam [6]. Additionally, it is more convenient towork in terms of power rather than intensity. Then the second harmonic conversion factors fromEq. 4.9 can be written as a single term, γSH , and given as,γSH =(2(2piν)2d2e f f k(2piν)pin3ε0c3)Lh(B,ξ ). (4.10)The length of the crystal is given by L, n is the refractive index, de f f is the effective nonlinearcoefficient, ν is the fundamental laser frequency, k(2piν) is the fundamental wave number, c is thespeed of light, and ε0 is the permittivity of free space. The parameter, h(B,ξ ) is known as theBoyd and Kleinman focusing factor. It is within this factor that the specifics of the beam interactingwithin the crystal are important. For a complete derivation of this parameter, see Boyd and Kleinmanpaper [6]. Throughout the literature there are other approximations for finding h(B,ξ ). Two of thesemethods will be used in this analysis.It must first be established that h(B,ξ ) depends on two parameters, B and ξ . B is the doublerefraction and ξ is the focusing parameter.B = ρ(piL2λ)1/2(4.11)where λ is the wavelength inside the crystal and L is the length of the crystal [6]. It can be seen thatthis parameter depends on the walk-off angle. In the case of the LBO crystal, ρ = 0, because thenon-critical phase-matching angle is 90o. Thus, B = 0 for conversion from 972 nm to 486 nm. Forthe second conversion stage from 486 nm to 243 nm, using the BBO crystal, ρ 6= 0, rather ρ = 4.9oand thus B = 19.16 for a crystal length of 10 mm.The focusing parameter, ξ , is defined as ξ = Lb . Again, L is the length of the crystal and b isthe confocal parameter, which is simply two times the Rayleigh length. The Rayleigh length is acharacteristic parameter used to describe a Gaussian beam around a focus, and given aszR =npiw20λ . (4.12)In words, the Rayleigh length is the distance it takes the beam to grow by the√2, from the waist,w0. For the fundamental beam focusing through the LBO, ξ = 0.93, and for the BBO, ξ = 0.28.To determine h(B,ξ ), there are two methods. The first is to simply use Fig. 4.5 from the Boydand Kleinman paper, where they plot several curves of h(B,ξ ) as a function of ξ , for a few values ofB. For the LBO, h(B,ξ )≈ 0.8, with de f f = 0.82 pm/V, leading to a γSH = 1.19x10−4. Since Fig. 4.5does not give a curve for B = 19 it is best to use an approximation provided by Nagourney [22].39Figure 4.5: Boyd-Kleinman factor as a function of the focusing parameter, ξ , for various val-ues of B, the double refraction parameter. This plot was taken from Boyd and Kleinman(1968), Fig. 2 [6] and is used to approximate a value of h(B,ξ ) for a given ξ and B.This approximation holds when ξ < 0.4 and B >√6/ξ , and is given by,h(B,ξ )≈√piξ 1/22B. (4.13)From this, h(B,ξ ) = 0.024, and for de f f = 1.58 pm/V, γSH = 8.2x10−5. The next step now isapplying the second harmonic coefficient into the intensity calculations of the enhancement cavityto determine how much intensity is generated in the second harmonic.4.2 Second Harmonic Generation in an Enhancement CavityThe single pass efficiency for a CW laser through a nonlinear crystal in the borate family is around10−4 W−1 or 0.1% [8]. For this experiment and many others, a higher efficiency is desired. Addingan enhancement cavity around the crystal leads to an efficiency of 30-45%. The theory for the non-linear optics and conversion from the fundamental beam to the frequency doubled light uses thesame optical component parameters as those defined in Chapter 3. The basic theory for understand-ing and calculating the nonlinear process while the crystal is in an enhancement cavity follows fromthe work of Kozlovsky, Nabors and Byer (KNB) [19] who nicely compiled the more extensive workof Ashkin, Boyd and Dziedzic (ABD) [2].As previously defined, the transmission and reflection through the input coupler remain thesame, t1 and r1 respectively. The definition of rm is now definedrm = t2tSHr2, (4.14)40where r22 is again the field reflection coefficient from the cavity mirrors, excluding the input coupler,and t2 = 1− losses. The new tSH term accounts for a depletion of the fundamental light as it isconverted to the second harmonic. As long as the loss from the conversion is small,t2SH = (1− γSHPc), (4.15)where Pc is the circulating power and γSH is the second harmonic coefficient. Previously, in Chap-ter 3 intensity was used as the metric for measuring the light in and out of the cavity. Intensity issimply the power per unit area. In this section power will be the metric used.Recall Eq. 3.5 that related the circulating intensity to the input intensity. Writing this equationin terms of power and substituting Eq. 4.15 into Eq. 4.14 gives the new relation,PcP0=t21(1− r1t2r2(1− γSHPc))2. (4.16)From the circulating power there is a simple relationship to find the power produced in the secondharmonic and is given by,PSH = γSHP2c . (4.17)This relationship is only true for the ring cavity configuration, since the second harmonic is onlyproduced in one direction. The second harmonic coefficient, γSH , has previously been derived inSection 4.1.1. The next step is to compare the theoretical SHG for given fundamental powers foreach the LBO and BBO cavities to experimental results.LBOThe experimental measurements were taken by measuring the power of the input light right beforethe cavity and power in the second harmonic as it left the cavity, while the cavity was locked. Ascan be seen in Fig. 4.6a the theory is larger than the experimental results. This is expected, as thereare many factors that contribute to achieve the theoretical SHG. To find how far off the experimentalresults are from the theory is simply determined by applying a coefficient to the theory, called themode overlap. To find the best value for the mode overlap, the mean squared error between the fitand data is minimized. As discussed in Chapter 3 the cavity is not perfectly impedance matched,nor is it perfectly spatially mode matched. The losses in the cavity have also not been accuratelymeasured, as this is a non trivial measurement to isolate scattering loss from the SHG loss. All ofthe extra losses and mode-overlap have been lumped together into the 74.0%, so the actual mode-overlap is better than this. In practice, the best coupling constant expected would be ∼ 90%. Forthe LBO the mode overlap is found to be 74.0% with a cavity loss of ∼ 0.5%. The theory with theapplied mode overlap compared with the data is in Fig. 4.6b.The low mode overlap here could be due to the mentioned factors. Based on more recent cal-culations, the spatial mode is probably contributing significantly to this mis-match and has beenimproved with better placements of the mode matching lenses and coupling into the cavity.410 0.2 0.4 0.6 0.8 1 1.2 1.400.10.20.30.40.50.60.70.80.9972nm Input Power [W]486nm SHG Power [W]  TheoreticalExperimental 24/07/13(a) Without mode overlap0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.9972nm Input Power [W]486nm SHG Power [W] Mode Overlap74.0%  TheoreticalExperimental 24/07/13(b) With mode overlapFigure 4.6: Experimental results of the nonlinear power conversion from 972 nm to 486 nmthrough the LBO crystal compared with the KNB theory. (a) Direct theory to exper-imental and (b) applying a mode overlap factor of 74.0% to the theory to match theexperimental data.BBOSimilar to the LBO cavity, Fig. 4.7 shows the theoretical results to the experimental data before andafter the mode overlap is applied. Here the loss in taken to be 0.3% and the mode overlap is foundto be 83.6%. This result is as expected and so it can be concluded the spatial mode matching andimpedance matching are sufficient.From the plots for both the LBO and BBO it is clear to see the conversion to the SHG are good.0 0.1 0.2 0.3 0.4 0.5 0.6 0.700.020.040.060.080.10.120.140.160.180.2486nm Input Power [W]243nm SHG Power [W]  TheoreticalExperimental 24/07/13(a) Without mode overlap0 0.1 0.2 0.3 0.4 0.5 0.6 0.700.020.040.060.080.10.120.140.160.180.2486nm Input Power [W]243nm SHG Power [W] Mode Overlap83.6%  TheoreticalExperimental 24/07/13(b) With mode overlapFigure 4.7: Experimental results of the nonlinear power conversion from 486 nm to 243 nmthrough the BBO crystal compared with KNB theory. (a) Direct theory to experimentaland (b) applying a mode overlap factor of 83.6% to the theory to match the experimentaldata.42At the maximum input power of 1.4 W of 972 nm, the first stage produces 0.6 W of 486 nm, for anefficiency of 43%, and then second stage produces 150 mW of 243 nm light, for a 25% efficiency.This gives an overall efficiency of 11% from the IR to UV. For comparison, if a single pass methodwas used at both stages the overall efficiency would ∼ 0.01%. From these results it can be seen thatenhancement cavities around the nonlinear crystals is the more effective method for producing UVlight as relatively high powers.43Chapter 5Measurement of the OPSL LinewidthAccurately measuring the linewidth of a CW laser is non-trivial and narrowing the linewidth is evenmore of a challenge. Section 5.1 will present the purpose of measuring the linewidth and why it isexperimentally important to narrow it. In Section 5.2 the new experimental setup will be discussedto highlight the various components that are required to make the measurement, like the Fabry-Perotstabilization cavity. Finally, Section 5.3 is where the theory and experimental results are presentedfor the linewidth measurement. Specifically, 5.3.1 will discuss the process of using a heterodyne beatsignal to measure the frequency noise of a laser and 5.3.3 will develop the mathematical methodsused to determine the linewidth from the frequency noise. Section 5.3.4 will conclude the results ofthe linewidth measurement, as well as look at methods to narrow the linewidth even farther. Finally,Section 5.3.5 will approximate how the linewidth from the 972 nm laser can be propagated throughthe SHG process to determine the linewidth at 243 nm.5.1 Introduction and MotivationAn ideal monochromatic laser source would emit photons at a single frequency. This would looklike a delta function on a frequency vs. amplitude plot, like the one in Fig. 5.1a. In reality thelaser will have a minimum linewidth due to quantum noise fluctuations from spontaneous emissionfrom the atoms in the laser cavity. This minimum limit in the linewidth, ∆νlaser, is given by theSchawlow-Townes formula,∆νlaser =4pihν∆ν2Pout, (5.1)where ∆ν is the resonator linewidth, hν is the photon energy, and Posc is the power of the laser [29].Additionally, there are contributions to the laser linewidth that are due to mechanical vibrations,which usually out weigh the quantum noise. These linewidth contributions create a broader, Gaus-sian line shape centred at the laser frequency, shown in Fig. 5.1b. The width, FWHM, of this peakis known as the linewdith. Another way to think about linewidth, is how much time the laser isspending at a particular frequency, or wavelength. The more narrow the linewidth, the more timethe laser is spending at the central frequency. Whereas with a larger linewidth the laser frequency44300 302 304 306 308 310 312 314 316 318 32000.10.20.30.40.50.60.70.80.91Frequency (THz)Amplitude Delta Function at 308.64 THz→←(a)300 302 304 306 308 310 312 314 316 318 32000.10.20.30.40.50.60.70.80.91Frequency (THz)Amplitude Gausisan with FWHM→ ←(b)Figure 5.1: Schematic of laser linewidth at 308.64 THz, for an ideal delta function and a realGaussian with some width, FWHM. (a) An ideal monochromatic laser with linewidthof zero, appearing as a delta function, and (b) a real laser with a Gaussian line shapeand linewidth given by the FWHM [Note: scale is not representative of a real linewidthmeasurement]is changing over a larger range and spends less time at the central frequency. Laser linewidths varyover a large range, from 1Hz to > 10 GHz. There are many factors the contribute the the largeand small linewidths. A free running laser will have a larger linewidth in the GHz range, where astabilized diode laser can have a linewidth down in the Hz range. A HeNe laser can have a linewidthof about 10 kHz [15].The natural, or free funning, linewidth of a diode laser is ∼1MHz, but can be narrowed down to∼1Hz [21]. Similarly, with the OPSL used in this experiment, the linewidth can be narrowed from theMHz range to <1KHz. Narrowing the linewidth of a laser is as simple as locking it to a stabilizationor Fabry-Perot cavity. Typically, to measure the linewidth, two lasers are locked to independentreferences and compared to one another. In this experiment the two lasers are locked together, inorder to use the heterodyne beat to measure the relative linewidth. Both of these topics are furtherdiscussed in Sections 5.2.1 and 5.3.1, respectively, and the actual linewidth measurements of theOPSL are explained in detail in Section 5.3.The above statements clearly state that it is possible to narrow the linewidth of a laser, butwhy is it important? Mentioned in the initial motivation for the development of the laser system inSection 1.1, this laser source is scheduled to be used in precision spectroscopy experiments. Thereare two reasons to have a narrow linewidth laser source for precision spectroscopy. Either simplyto excite the atoms to a higher energy or to determine the exact energy required to make an energytransition. As described in Section 1.1.2, the use of the UV laser is to excite the 129Xe atoms fromthe ground state with spin -1/2 and hyperfine level F = 1/2, to the 6p, spin 3/2 and F = 3/2. Thelinewidth simply needs to be less than 2 GHz, so as to not excite another energy transition. The firstoption for a laser source for the 129Xe atoms has a linewidth larger than 2 GHz, so cannot be used45unless the linewidth is narrowed. Luckily, the OPSL has a free running linewidth in the MHz range,so satisfies this condition without any narrowing. This is not the case for all lasers, and so narrowingwould need to be done.Precision spectroscopy of Hydrogen is actually similar to the Xenon energy transition in thatthe atoms are excited from one state to another, however the goal is very different. In this case thegoal is to determine the exact energy required for the atom to make this transition. Currently, this1s to 2s is known to within an uncertainty of 10−15 as published by the Ha¨nsch group in 2013 [21].One method for decreasing the uncertainty even further is to decrease the linewidth of the laser thatis probing the transition. Since the transition is a two photon transition a narrower linewidth willgreatly reduce the uncertainty on the measurement. For this reason, narrowing the linewidth of theOPSL is desired as this measurement is a future application of this laser. More about the Hydrogenmeasurements can be found in Section 1.1.1.Now that the motivation has been established for measuring the linewidth of the laser and pos-sibly narrowing it down if needed for future applications. It is not possible to simply propagatethe light onto a photodetector and view the output on a spectrum analyzer and take the determinethe linewidth. This is because the frequency of the light is on the order of hundreds of THz andfast electronics and a good spectrum analyzer only measure up to 30 GHz. Instead, what is doneis called using a heterodyne beat, where two lasers with nearly the same frequency are overlappedonto a photodetector. The fields add such that the frequency measured is the difference between thetwo lasers. This difference can be tuned down to <1GHz so the relative linewidth can be evaluated.A more complete description and mathematical formulation of the heterodyne beat can be foundin Section 5.3.1. The frequency noise is measured on a SRS network signal analyzer and then therelative linewidth between the two laser is deduced from this plot. The methods for determining therelative linewidth are detailed in Section 5.3.Measuring the linewidth is the most important step, but the linewidth also needs to be narrowedfor experiments like the Hydrogen spectroscopy. Typically lasers linewidths are narrowed by lock-ing the laser to a stabilization cavity. This cavity is also known as a reference cavity, or Fabry-Perotcavity. Essentially, the stabilization cavity is fixed and the laser wavelength is locked to the lengthof the cavity. A more complete description of the Fabry-Perot cavity is detailed in Section 5.2.1. Bylocking the laser to this cavity, the frequency cannot drift freely with air circulation and temperaturefluctuations, without the PZT in the laser changing oppositely to keep the frequency constant. Thenext section will outline the experimental setup for the linewidth measurement, followed by thevarious components used to narrow the linewidth and perform measurements.5.2 Experimental SetupThis section will outline the new experimental setup that was used to measure the linewidth of theinfrared OPSL. Fig. 5.2 shows a schematic of the setup. The main components present are the twoOPSL lasers, labeled as the master and slave laser, the stabilization cavity and the mixing of the46!/4 stabilization cavityMaster OPSLSlave OPSLCavity Locking SystemPumpChip PZTPumpChip PZTPhase noiseCavity Locking System FrequencyAmplitudeFigure 5.2: Schematic of the experimental set-up to measure the linewidth of the OPSL. Thereare two OPSL, one is the master laser that is locked to the Fabry-Perot stabilization cavityusing a PDH locking scheme. The second is the slave laser that is locked to the beatbetween the two laser. This phase noise is derived from the lock at used to determine thelinewidth of the slave laser.lasers for the heterodyne beat used to measure the linewidth. There are also two locking systemsindicated. The first is to lock the master laser to the stabilization cavity, and the second to lock theslave laser to the master laser.There are several reasons for using two OPSL lasers. The first one was briefly mentioned inSection 5.1 to create a heterodyne beat, so the signal has a frequency that is detectable within thebandwidth of the electronics. It is required that both the lasers are of the same wavelength, as toget a small difference frequency in the beat signal. It would also be possible to split the beamfrom one laser and recombine it with an added frequency shift, however the two beams need to beuncorrelated, or decoupled from each other. This means that two lasers are required so that theirnoise is independent from one another. If the two lasers were correlated then the sum of the twofields would add and give a delta function.It is also possible to do a self-heterodyne beat. To do this, the laser would first have to be split.One of the beam paths would require a delay, on a time scale longer than the coherence length,relative to the other beam. This will allow the beams to be uncorrelated and thus when the beamsare mixed, the linewidths are added independently to make a larger linewidth. In both the duallaser and self-heterodyne beat cases, the measured linewidth is a relative linewidth between two47uncorrelated beams, and therefore larger than the linewidth of one laser. This larger linewidth islater accounted for in the linewidth calculation in Section 5.3.3. In Fig. 5.2 the heterodyne beat isshown at the bottom. This beat signal is also used to lock the slave laser to the master laser. Thisprocess is described in more detail in Section 5.3.1.The next feature is the presence of the stabilization cavity. As mentioned in Section 5.1, thestabilization cavity is used to stabilize the master laser so that the servo that locks the slave to themaster laser locks more effectively. If a laser is not controlled by any voltage on the PZT, the laserwill naturally drift around and have a larger linewidth, or free running linewidth. Locking the slavelaser to the master laser would force the servo to work much harder to maintain the lock, whichcould make the final linewidth measurement inaccurate. By locking the master laser to the Fabry-Perot cavity, the final relative linewidth measurement between the two lasers will be a more accuratemeasurement.5.2.1 Fabry-Perot Stabilization CavityThe stabilization cavity is a simple two mirror cavity, consisting of two curved mirrors with the sameradius of curvature, radius of curvature (ROC). This geometry is also known as a Fabry-Perot cavityand was developed in the last two years of the 19th century by Charles Fabry and Alfred Perot [33].The purpose of this cavity is very different compared to the two ring cavities used for the SHG.This cavity is purely used to stabilize the laser. The two SHG cavities have PZT’s attached to one ofthe flat mirrors. Those cavities were then locked to the laser. As the laser freely drifts, the cavityPZT’s adjust to change length to remain on resonance with the laser frequency. Here the opposite istrue. The Fabry-Perot cavity is set to a fixed length by glueing the mirrors to a tube that minimizeschanges in cavity length due to temperature fluctuations. The OPSL has a PZT attached to the outputcoupler mirror, which can be used to finely tune the wavelength of the laser to match the length ofthe cavity. Similar to the locking methods used for the SHG cavities, a PDH locking scheme is usedhere. The cavity is aligned and spatially mode matched so that the resonant transverse mode is theTEM00. Once again the reflection signal off the input coupler is used to lock the laser frequency tothe resonant peak.Another difference between the SHG cavities and Fabry-Perot cavity is the resonant peaks ofFabry-Perot cavity are very narrow relative to the FSR, implying the cavity has a high finesse. It isideal to have a very high finesse cavity for the stabilization cavity, because that means the linewidthis narrow. Recall the discussion of linewidth from Section 3.1.1, where the FWHM is the metric usedas the linewidth of the signal. The linewidth is also directly comparable to the stability the cavitycan provide for the laser. If the linewidth is narrow, the PDH lock will keep the laser to within anarrow range as large at the linewidth. The laser will not be able to drift around beyond the lock tothe Fabry-Perot resonance. In the end, this reduces extra noise on the laser linewidth measurementthat comes purely from drifting, rather than the actual linewidth of the laser.To determine the finesse of the Fabry-Perot cavity a different method is used than the one from48Section 3.1.1. Here, finesse is determined by the reflectivity’s of the mirrors by,F =pi√r1r21− r1r2, (5.2)where r1 is the field reflectivity of the input coupler and r2 is the reflectivity of the second mir-ror [22]. By definition, r21 + t21 = 1. The measured values for the reflectivity’s are found to be,r1 = 0.99936 and r2 = 0.99975. Using the reflectivity’s of the mirrors in Eq. 5.2, the finesse isfound to be F = 7058. This is considered a high finesse cavity and will provide significant stabilityto the master laser.Once the master laser is locked to the Fabry-Perot stabilization cavity, the two lasers are beattogether and the slave laser is locked to the master laser. The frequency noise from the slave laser isthen measured to determine the linewidth.5.3 Measuring the LinewidthThe linewidth of the laser is determined by measuring the frequency noise from the Fourier trans-form of the RF beat signal. This measurement involves the discussion of several kinds of frequen-cies. First is the frequency of the laser itself, or known as the optical frequency and denoted as ν .There will be two optical frequencies, once for each laser source and denoted as ν0 and ν1. Whenthese two frequencies are beat together, as further described in the next section, the difference be-tween their frequencies is given as, ν1− ν0 = fc, where fc is the in RF region. It is also usefulto write the carrier beat frequency, fc as an angular frequency, ωc, where ωc = 2pi fc. Finally, anyadditional frequency modulations on the carrier beat frequency will be denoted as Ωm. The modu-lation frequencies will have some size corresponding to small deviations away from the carrier beatfrequency, called frequency excursion and will be denoted by ∆Ω.The next sections will discuss the heterodyne beat frequency, f0, between the two laser sources.Then, this beat frequency is locked to a specific f0 value. Then this signal is Fourier transformed tomeasurable modulation frequencies, Ωm, with some frequency noise amplitude. From this frequencynoise the relative linewidth between the two lasers can be determined and thus the linewidth of thelaser.5.3.1 Heterodyne BeatIt is first important to mathematically show how the heterodyne beat gives the difference betweenthe two frequencies of the lasers interacting. The derivation of the intensity due to electric fieldof a wave is reproduced from Saleh and Teich [28]. First, it is important to start with one simplemonochromatic wave, with an electric field given by,E(t) = E0e−i(2piν0)t−iφ0 , (5.3)49where E0 is the amplitude, φ0 is the phase, and ν0 is the optical frequency. For simplification, thephase can be set to zero, φ0 = 0, givingE(t) = E0e−i(2piν0)t . (5.4)The frequency used in this experiment is 3.08× 1013 Hz (or 972 nm). There are not electronicsthat have a bandwidth large enough to detect frequencies this high. If two waves with differentfrequencies, ν0 and ν1 are added together and written in terms of intensity, I, whereI = |E|2 = |E0 +E1|2, (5.5)then,I = E20 +E21 +2E0E1 cos[2pi(ν1−ν0)t]. (5.6)The term (ν1− ν0) is called the heterodyne beat frequency, and denoted as ∆ν or f . If the twofrequencies are close together and less than the bandwidth of the electronics, meaning < 1 GHz,then a signal can be detected. This is exactly what is done in the lab. The two lasers are overlappedand focused onto a photodiode detector. The intensity in the beams are adjusted so I1 = I2, thusreducing Eq. 5.6 to,I = 2I0[1+ cos[2pi(ν1−ν0)t]]. (5.7)5.3.2 Locking the Slave Laser to the Master LaserThe frequency noise comes directly from this heterodyne beat. Before diving into the full evaluationof the frequency noise and linewidth measure, the methods for narrowing the linewidth must beestablished. The master laser is locked to the stabilization cavity using the PDH method as outlinedin Section 3.3. This time the laser is locked to the cavity in order to take out the long term drift of themaster laser. Since the optical frequency of this laser can be measured the slave laser is beat with themaster laser to make and RF frequency. The slave laser is still free running, so the heterodyne beat isa combination of the stable master laser and the free running slave laser. A linewidth measurementof this beat, would result in a broad linewidth, mostly composed of the drifting of the slave laser. Tominimize this, the slave laser can be locked to the master laser.The heterodyne beat frequency can be manually adjusted by simply tuning the wavelength ofthe slave laser. If the frequency of an external synthesizer is set close to the frequency of the beat,then the slave laser can be locked such that the beat frequency remains the same as the external, orreference frequency. The PZT attached to the output coupler in the laser cavity controls the lengthof the laser cavity in order to maintain the beat frequency. The frequency of the master laser is onlyvarying within the linewidth of the stabilization cavity. Thus, by locking the slave laser to the beatfrequency the linewidth of the slave laser linewidth is reduced. From here, the relative linewidth ofthe beat frequency can be determined. Assuming the two lasers are uncorrelated the linewidth ofone laser can be approximated as linewidth of the beat divided by√2.505.3.3 Frequency Noise and the β -lineOnce the linewidth of the slave laser has been reduced, the RF signal from the heterodyne beat canbe converted into frequency noise. It was established that the heterodyne beat produces a wave witha frequency given by the difference between the two optical frequencies, ν1−ν0. This difference isdecreased until it lies within the RF region and detectable by common electronics, this new frequencywill is denoted as fc and the angular frequency ωc, where ωc = 2pi fc.The derivation of frequency noise starts with frequency modulation theory described by JohnHall and Miao Shu [15]. A simple monochromatic beam with phase-modulation, φ(t), is written asan electric field by,E(t) = E0e−iωct−iφ(t). (5.8)Since the field exhibits a phase modulation in real life, this can be represented byφ(t) = β sinΩmt, (5.9)where Ωm is the modulation frequency and β is the modulation index. The argument of phasemodulation on the centre frequency can also be written in terms of frequency modulation. The re-lationship between frequency and phase is, ω = dφ(t)/dt. If the instantaneous frequency is definedas ω(t), then,ω(t) = ωc +ddtφ(t) (5.10)ω(t) = ωc +βΩm cosΩmt. (5.11)This relationship shows the significance of β , either in the case of phase or frequency modulation.Another way to define β is as the frequency excursion divided by the frequency modulation rate,β = ∆ f/ fm. The excursion frequency is a measure of how far away the frequency is from the carrierfrequency and denoted as ∆ f , where ∆ f = f − fc and ω = 2pi f and ωc = 2pi fc. The modulationfrequency of the excursion frequency is denoted as fm, where Ωm = 2pi fm.The RF signal from the heterodyne beat is sent into an SRS network signal analyzer, where thesignal is Fourier transformed into the various modulation frequency components. To describe howthis signal appears on the SRS network signal analyzer, as shown in Fig. 5.5, it is important to havean understanding of frequency modulation theory and the conceptual idea of β . For a small ∆ f ,meaning a small frequency distance from the carrier frequency, and a fast modulation rate, fm, thanβ  1. The contribution from this type of noise to the linewidth will be very small. Then for large∆ f and slow fm, β > 1, meaning the modulation index is high and this noise contributes significantlyto the linewidth.The frequency variation per unit bandwidth at a frequency ∆ f from the carrier frequency ofthe heterodyne beat signal gives the frequency noise spectral density, denoted by, Sδ f ( fm), over the51modulation frequencies. The spectral density is given by,Sδ f ( fm) =∆ frms( fm)2b[Hz2Hz], (5.12)where b is the bandwidth, or bin size of the modulation frequency, ∆ frms( fm) is frequency excursionfrom the beat carrier frequency, fc, and fm is the Fourier modulation frequency [15].There are several methods presented throughout the literature for evaluating the frequency noiseto determine the linewidth of a laser. John Hall and Miao Zhu [15] give a comprehensive expla-nation of phase and frequency noise for optical sources. As well as one method for determiningthe linewidth of the source, based on the spectral density of the phase and frequency noise. A morerefined evaluation of the spectral density and how the frequency noise contributes to the linewidth ofthe optical source is described by Domenico, Schilt and Thomann (DST) [9]. This second methodis the one chosen to determine the linewidth of the OPSL in this experiment. The DST approachconsiders how the frequency excursions depend on the range of modulation frequencies and thuscontribute to the linewidth.An outline of the DST work will be described here to highlight the key components that are usedto determine the linewidth of the laser. A complete and mathematical description of their study andresults can be found in the DST paper, [9]. As mentioned, various frequency components contributedifferently to the linewidth of the source, for example a low modulation frequency with a largeexcursion from the carrier beat frequency will increase the linewidth of the source. Whereas, fastmodulations at low excursion frequencies do not contribute to the linewidth, but do contribute to theline shape. DST establishes a function that separates the significant linewidth contributions fromthe insignificant ones, by means of the point when the line shape changes. This function is knownas the β -separation line.To establish the β -separation line, a conceptual description of the transition in the line shapeis illustrated in Fig. 5.3a. The line shape is broken down into different regions, from a to d. Inregions a and b the line shape is Gaussian and remains Gaussian, but with an expanding linewidth.Beginning in region c and more clearly in region d, the line shape is Lorentzian and the linewidth isno longer changing. This transition from a Gaussian line shape to a Lorentzian line shape is the pointwhich corresponds to the ratio between the modulation frequency, fm, and the frequency noise level,called ho [Hz2/Hz] for now. DST use a simple model to describe this point where the line shapechanges. They start with a constant frequency noise, ho, that depends on fm as a step function. Soat a given frequency, called the cut-off frequency, fcut , the frequency noise level changes from ho tozero. For a fixed value of ho and as the cut-off frequency increases from zero, there is a fcut whichcorresponds to the point when line shape changes from a Gaussian to a Lorentzian. Mathematicallythis can be modelled as a autocorrelation function, as described in the work of DST [9].To maintain the focus on the conceptual understanding, it is safe to jump ahead to the FWHMrepresentation of the spectral noise density. DST describes the mathematical approach to finding a52(a) (b)Figure 5.3: Evolution of Gaussian line shape to a Lorentzian line shape. (a) Shows the evolu-tion from a Gaussian line shape to a Lorentzian line shape from regions a to d, where thex-axis is the linewidth and the y-axis is relative amplitude [9]. (b) Shows the evolutionof the linewidth for an increasing frequency to noise level ratio, fcut/ho. The indicatedpoint where the linewidth stops growing corresponds to the transition from the Gaussianto the Lorentzian line shape [9].clean result of the FWHM for a changing fcut . This is given by,FWHM = ho√8ln(2) fcut/ho[1+(8ln(2) fcutpi2ho)2]1/4. (5.13)The visual representation is even more useful and shown in Fig. 5.3b. This plot shows how theFWHM evolves for a growing fcut relative to the noise level, ho. At a cutoff frequency to noise ratio,the linewidth increases linearly, until a point and then the linewidth remains constant. This pointoccurs at, fcut/ho = pi2/8ln(2). This transition is directly related to the transition from a Gaussianline shape to the Lorentzian line shape.The ideas presented about how the relationship between the noise level, h0, and the cut-offfrequency contribute to the linewidth can be directly applied to the frequency noise from a laser.High index frequency modulations contribute directly to the Gaussian line shape and therefore tothe linewidth. Whereas, fast modulations with low index modulations do not effect the linewidthand only contribute to the Lorentzian line shape. This metric is the β -separation line. Fig. 5.4shows a schematic of the β -separation line and the various modulation index levels [9]. WhenSδ f ( fm)> 8ln(2) fm/pi2 the modulations have a high index and therefore contribute to the Gaussianlineshape and thus the linewidth. Whereas, when Sδ f ( fm) < 8ln(2) fm/pi2, the modulation index islower therefore only contributes to the wings of the line shape and not the linewidth.Now, that the conceptual understanding of the β -separation line has been established. Themathematical determination of the linewidth can now be implemented. The linewidth of the laser53!"" !"! !"# !"$ !"% !"& !"' !"(!""!"!!"#!"$!"%!"&!"'!"(!")!!*+,-./012345-67-,89/:;+<-/012#=123Low modulation index area contributes to the wingsHigh modulation index area contributes to the linewidthFigure 5.4: Illustration the β -separation line for an arbitrary frequency noise spectrum. In-dicated are the regions where the high modulation at low frequencies contribute to thelinewidth and low modulations at high frequencies lie below the β -separation line as theydo not contribute to the linewidth.source can be described as the FWHM, whereFWHM =√8ln(2)A√2. (5.14)The variable A is the integration under the frequency noise curve above the β -separation line asshown shaded in grey in Fig. 5.4. The√2 is used to account for the two lasers being uncorrelatedwith each other. Since the signal used to measure the linewidth is the beat between the master andslave laser, the linewidth is a combination of the two separate linewidths. This produces a slightlylarger linewidth and therefore needs to be reduced by the factor of√2 to approximate the linewidthfrom just the slave laser. The integration variable, A, can be expressed as follows,A =∫ ∞1/ToH(Sδ f ( fm)−8ln(2) fm/pi2)Sδ f ( fm)d f , (5.15)where H(x) is the Heaviside function, and the integration is over the frequency from 1/To to ∞. Tois the measurement period limit. This means that frequencies lower than 1/To cannot be measured.The To in this linewidth measurement is To = 1 s, or 1/To = 1 Hz. When x ≤ 0, meaning thefrequency noise is below the β -line, then H(x) = 0, and when x > 0 and the frequency noise isgreater than the β -line, H(x) = 1. When H(x) = 1, then Eq. 5.15 simplifies to a simple integral,A =∫ ∞1/ToSδ f ( fm)d f , (5.16)for specific fm values.54100 101 102 103 104 105 10610−1510−1010−51001051010Frequency Noise (Hz2/Hz)β−linenoise floorν noiseFrequency (Hz)Figure 5.5: Measuring the linewidth from frequency noise plot. The frequency noise betweentwo lasers beat together, while one laser is stabilized to a Fabry-Perot cavity, is shownin blue. This data has been reduced by subtracting off the data in black, which is thebackground noise from the electronics. The frequencies with a high enough amplitude infrequency space, denoted as frequency noise, above the labeled β -line gives the linewidthof the laser, at 87 kHz.On a frequency noise plots, integrating over the frequencies with frequency noise above the β -line will give the linewidth of the laser. The results of the linewidth measurement from the OPSLlaser are presented in the next section.5.3.4 Results and DiscussionThe results of the frequency noise from the OPSL, when the slave laser is locked to the master laserand the master laser is locked to the stabilization cavity are presented in Fig. 5.5. It is important tonote that there is a noise floor present in the plot. This was measured by blocking the optical signalon the photodiode and simply measuring the static noise from the electronics. When the phasenoise was analyzed, the noise floor was subtracted off to minimize false additions to the linewidthmeasurement.The β -separation line is also displayed in Fig. 5.5. Above the line, the signal is outlined in pinkto show the part of the frequency noise that contributes to the linewidth. Solving Eq. 5.14, the resultgives a linewidth of 87 KHz.The OPSL has the potential to have a narrow linewidth < 1 kHz. From the results of these initial55103 104100102104106108Frequency Noise (Hz2 /Hz)β−lineFrequency (Hz)Figure 5.6: Close-up view of the frequencies mainly contributing to the laser linewidth. Thehump on the left is known as the servo bump and then highest peak on the right hump,around 1.7 kHz comes from a mechanical resonance in the laser. The official sourceremains unknown. If both of those humps can be minimized, the linewidth of the laserwill narrow.measurements, the linewidth is 87 kHz. This is not as low as expected but, there many componentsthat contribute to the narrowing of this linewidth and all need to be optimized for the best linewidthmeasurement. The lock of the master laser to the stabilization cavity may be a weak lock, so thelinewidth is not narrowed as far as it could be. The lock of the slave laser to the master laser couldalso be a tighter lock, decreasing the added fluctuations to the linewidth. To help with some of thesevariables, a box was placed around the stabilization cavity and the master laser to reduce vibrationsfrom noise. The lasers were placed on separate bread boards to help maintain uncorrelated vibrationsand also decrease vibrations that occur on the main optical table. It was with these adjustments thatthe linewidth was narrowed down to the 87 KHz, from the first measurements of a few hundred KHz.More improvements could be made to continue optimizing the locks and minimizing environmentalvibrations.There are also several features to mention on the frequency noise plot in Fig. 5.5 that havelarge contributions to the linewidth. The first is the servo bump, which is the bump at the higherfrequencies in Fig. 5.6. This bump represents the frequency noise from the electronic componentsused to lock the lasers and measure the noise. It is possibly to minimize this bump, but there willalways be a servo bump present. Changing the impedence values on some of the locking componentswas able to decrease the servo bump to what is shown in Fig. 5.6.56The other feature to note is the peak around 1.7 kHz in Fig. 5.6. This peak is not from either theelectrical noise or from the linewidth, but rather is a peak associated with a mechanical resonanceor technical noise. This peak has been observed in other OPSL lasers, so can be concluded that it isnot an environmental resonance. It is possible this resonance is amplitude noise of the OPSL pump,but this is yet to be confirmed. However, it can be concluded this 1.7 KHz is not truly part of thelaser linewidth and once the source of the noise is located and suppressed, the linewidth of the laserwill reduce.Overall, the measurement of 87 kHz is a good first measurement. But, if the 1.7 kHz bump canbe minimized or eliminated than the linewidth can be narrowed even farther, pushing < 1 kHz at972 nm.5.3.5 Approximation of the Linewidth at 243 nmThe linewidth of the 972 nm was measured, but an approximation can be made for how this linewidthwould translate through the SHG and to the 243 nm light. Through the analysis of the linewidth, theβ -separation line limits the line shape to be a Gaussian given by,L(ν) = Ae−(ν−ν0)22σ2ν , (5.17)where ν is the variable frequency, ν0 is the central frequency, and σν is the FWHM, or linewidthof the line shape. The SHG process means the frequency of the fundamental doubles, ν2 = 2ν andν02 = 2ν0, but how is σν related to σν2?Since the process is a two photon process, each photon has the probability of laying anywherewithin the linewidth distribution. Therefore the linedwiths of the two photons are multiplied to-gether, to account for any combination of energies that give the correct total energy required,L1(ν)L2(ν) = Ae−(ν−ν0)22σ2ν ∗ Ae−(ν−ν0)22σ2ν = A2e−2(ν−ν0)22σ2ν . (5.18)The distribution for the second harmonic can be expressed as a Gaussian as well and equated to thetwo photon Gaussian,L(ν2) = Be−(ν2−ν02)22σ2ν2 = A2e−2(ν−ν0)22σ2ν , (5.19)Where B = A2. The exponents can be equated to find the relationship between the linewidths,−(ν2−ν02)22σ2ν2=−2(ν−ν0)22σ2ν. (5.20)Substituting in the relationship between the two frequencies, i.e. ν2 = 2ν and ν02 = 2ν0, and can-57celling the two’s on the right side of Eq. 5.20 gives,(2ν−2ν0)22σ2ν2=(ν−ν0)2σ2ν. (5.21)Simplifying,42σ2ν2=1σ2ν, (5.22)and2σ2ν = σ2ν2 , (5.23)thus,√2σν = σν2 . (5.24)This approximation shows that the linewidth through second harmonic generation increases by√2from the fundamental linewidth. The same approximation can be used for the second stage of secondharmonic generation. Therefore,σν4 = 2σν , (5.25)and the linewidth at 243 nm is ∼174 kHz for 87 kHz linewidth at 972 nm.Another quantity to consider is the quality factor, Q, and spectral resolution. The quality factoris defined as,Q =νσν. (5.26)Comparing the fundamental to the 4th harmonic,ν4σ4ν→4ν2σν, (5.27)therefore,Q4ν = 2Qν . (5.28)The inverse of the quality factor is the spectral resolution which is an important ratio for precisionspectroscopy. In this case the spectral resolution of the 4th harmonic is half the resolution of thefundamental. A higher spectral resolution will decrease the uncertainty in the overall spectroscopymeasurements. However, even though the laser frequency is quadrupled the spectral resolution onlydrops by half.58Chapter 6Conclusion and Future WorkAn IR OPSL was successfully used to generate UV light for the use in precision spectroscopy ex-periments. The IR light was frequency doubled twice to generate 150 mW of 243 nm light. Thelinewidth of the OPSL was also measured to be ∼ 87 kHz.The process to generate UV light started with 1.7 W of CW light from the OPSL at 972 nm. Twostages of enhancement cavities with nonlinear crystals were used in series with the laser to frequencydouble the light twice down to 243 nm. The enhancement cavities have a bow-tie geometry to allowthe light to propagate in one direction through the cavity and also through the nonlinear crystal,generating a uni- directional beam of second harmonic light. The enhancement cavity allows fora more efficient SHG process, because the buildup of light inside the enhancement cavity is 60-90 times the intensity of the input light. If the single pass process was used, the combinationof the two frequency doubling stages would result in an efficiency of ∼ 0.01%. By placing thenonlinear crystals inside enhancement cavities, an efficiency of 11% was achieved, with a total at150 mW of 243 nm light. The first enhancement cavity used an LBO crystal and a BBO in the secondenhancement cavity.The second part of the experiment was to measure the linewidth of the OPSL. This was done byusing two uncorrelated OPSLs. The first laser was stabilized to a Fabry-Perot cavity. The two laserswere beat together, to generate a measurable heterodyne beat. This signal was first used to lock thesecond laser to the first laser. Then the frequency noise of the beat signal was used to determinethe relative linewidth between the two lasers. A careful analysis and separation of the frequencynoise associated with the true relative linewidth, yielded a linewidth of one OPSL to be 87 kHz.The linewidth was successfully narrowed from the free running linewidth on the order of severalhundred MHz. However, the linewidth is still dominated by frequency noise due to mechanical ortechnical vibrations and has the potential to be narrowed even further.The narrow linewidth and high UV power of the OPSL and frequency doubling cavities haveserious implications in the field of precision spectroscopy. This laser system is currently beingreplicated at the wavelength to drive a transition in 129Xe from the ground state. The future of thislaser system is part of the UCN collaboration to measure the EDM of the neutron. The first measure-59ment will establish the 129Xe two photon transition labeled in Fig. 1.3b. Then, the EDM of 129Xeneeds to be measured with a higher precision than current measurements, which is |d(129Xe)|< 4.1x 10−27e cm [25]. This will be performed using the same methods as described in Section 1.1.2.Finally, 129Xe will be combined with 199Hg as a dual co-magnetometer for the measurement ofneutron EDM.60Bibliography[1] A. Ashkin, G. D. Boyd, and J. M. Dziedzic. 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