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Quantum coherent control and compensation of temporal scattering DeWolf, Timothy 2014

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Quantum Coherent Control andCompensation of Temporal ScatteringbyTimothy DeWolfB.Sc. (Biochemistry), The University of Alberta, 2009B.Sc. (Physics), The University of Alberta, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2014c© Timothy DeWolf 2014AbstractThe experimental work in this thesis is divided into two distinct parts. Inboth parts, broadband femtosecond laser pulses are “shaped” by adjustingthe relative phase and amplitude of spectral components.In the first set of experiments, time-dependent perturbation theory isused to show that the probability of a quantum transition in atomic rubidiumcan be substantially enhanced or suppressed using pulse shaping, comparedto the probability of transition observed when a transform-limited or “flatphase” optical pulse is used. These enhancement or suppression effects arealso demonstrated experimentally. As quantum interference (the materialphase having been transferred from the optical phase) is used to enhance ordiminish a particular final quantum state, this can be classified as a quantumcoherent control experiment.In the second set of experiments, an optical pulse is scattered into a trainof pulses by a layered structure. The layered structure is used to simulate theeffect of optical pulses travelling through certain types of complex media.One consequence of the disruption of a single pulse into a train of pulsesis lower per-pulse peak intensity, and thus a greatly diminished nonlinearsignal. It is shown that spectral pulse shaping (in phase only) is sufficientto pre-compensate for the scattering structure, allowing a single transform-limited pulse to be obtained at the output.iiPrefaceThis thesis is original, unpublished, independent work by the author, Tim-othy DeWolf.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivI Motivation and Introduction . . . . . . . . . . . . . . . . . . . . 11 The Goal of Controlling Matter with Light . . . . . . . . . 21.1 Biological Control with Incoherent Fields . . . . . . . . . . . 31.2 An Early Attempt at Molecular Control with MonochromaticFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Quantum Coherent Control . . . . . . . . . . . . . . . . . . . 41.3.1 Tannor-Rice Coherent Control . . . . . . . . . . . . . 41.3.2 Brumer-Shapiro Coherent Control . . . . . . . . . . . 51.4 Spectral Pulse Shaping . . . . . . . . . . . . . . . . . . . . . 62 The Goal of Controlling Matter with Scattered Light . . . 72.1 The Problem of Optically Scattering Material . . . . . . . . 72.2 Experimental Compensation of Spatial Scattering . . . . . . 92.2.1 Experimental Compensation for Temporal Scattering 112.2.2 The Idea That Scattering Media Can Enhance Control 12ivTable of ContentsII Tools Used and Developed . . . . . . . . . . . . . . . . . . . . . 133 Femtosecond Lasers . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 The Femtosecond Oscillator . . . . . . . . . . . . . . . . . . 143.3 The Regenerative Amplifier . . . . . . . . . . . . . . . . . . . 154 Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1 Nonlinear Polarization and the Introduction of New Frequen-cies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.1 Second Harmonic Generation . . . . . . . . . . . . . . 184.1.2 Sum-Frequency Generation . . . . . . . . . . . . . . . 194.2 The Nonlinear Wave Equation . . . . . . . . . . . . . . . . . 194.3 Sum-Frequency Generation and Phase-Matching . . . . . . . 214.4 Second-Harmonic Generation in a Birefringent Crystal . . . 224.5 Relation to Two-Photon Absorption . . . . . . . . . . . . . . 235 Pulse Shaping and Characterization . . . . . . . . . . . . . . 245.1 The 4f Pulse Shaper . . . . . . . . . . . . . . . . . . . . . . 245.1.1 Before the Spatial Light Modulator (SLM) . . . . . . 245.1.2 After the SLM . . . . . . . . . . . . . . . . . . . . . . 265.2 The Liquid-Crystal SLM . . . . . . . . . . . . . . . . . . . . 275.3 Calibration and Operation of the SLM . . . . . . . . . . . . 285.3.1 Pixel-Wavelength Calibration . . . . . . . . . . . . . 285.3.2 Phase-Voltage Calibration . . . . . . . . . . . . . . . 295.3.3 Shaping in Phase and Intensity . . . . . . . . . . . . 335.3.4 Converting Phases To Drive Counts . . . . . . . . . . 345.4 Frequency-Domain Pulse Shaping . . . . . . . . . . . . . . . 345.4.1 Numerical Fourier Transformation and Pulse Shaping 355.4.2 A Gallery of Shaped Pulses . . . . . . . . . . . . . . . 365.5 Pulse Characterization Techniques . . . . . . . . . . . . . . . 395.5.1 Spectrometer . . . . . . . . . . . . . . . . . . . . . . . 395.5.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . 425.5.3 FROG . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5.4 MIIPS . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Software and Algorithms; The Genetic Search . . . . . . . 476.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Genetic Search . . . . . . . . . . . . . . . . . . . . . . . . . . 47vTable of ContentsIII The Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Quantum Coherent Control of a Two-Photon Transition inRubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.2 Two-Photon Transitions in Rubidium . . . . . . . . . . . . . 527.3 Modeling the Fluorescence Intensity . . . . . . . . . . . . . . 567.4 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . 587.5 Coherent Control of Two-Photon Absorption in Rubidium . 607.5.1 Nonresonant Contributions . . . . . . . . . . . . . . . 607.5.2 Nonresonant and Resonant Term Interference . . . . 647.5.3 Two-Pathway Interference . . . . . . . . . . . . . . . 667.5.4 Putting it All Together—Theory . . . . . . . . . . . . 667.5.5 Putting it All Together—Experiment . . . . . . . . . 697.5.6 Other Experimental Investigations . . . . . . . . . . . 727.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 Regaining Coherent Control after Random 1D Glass Stacks 778.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.2 Glass Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.2.1 Experimentally Constructed Glass Stacks and SampleRealizations . . . . . . . . . . . . . . . . . . . . . . . 778.2.2 Numerically Modeled Glass Stacks and Sample Real-izations . . . . . . . . . . . . . . . . . . . . . . . . . . 828.3 Four Possible Phase Pre-Compensation Methods . . . . . . . 838.3.1 Phase Compensation via XFROG . . . . . . . . . . . 838.3.2 A Prerequisite: Creating Irregularly Spaced Pulse Trainswith a Pulse Shaper . . . . . . . . . . . . . . . . . . . 848.3.3 Phase Compensation using Spectral Information . . . 858.3.4 A Prerequisite: A Feedback Signal that Detects whenScattering has been Compensated . . . . . . . . . . . 898.3.5 Phase Compensation using a Genetic Search . . . . . 908.3.6 Phase Compensation using an Interferometric Auto-correlation-Like Method . . . . . . . . . . . . . . . . 908.4 Results: The Achieved Compensations . . . . . . . . . . . . 958.4.1 Compensated Numerical Model Realizations . . . . . 958.4.2 Compensated Experimental Realizations . . . . . . . 1038.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048.5.1 The Numerical Models . . . . . . . . . . . . . . . . . 1048.5.2 Comparing Theory and Experiment . . . . . . . . . . 107viTable of Contents8.5.3 Pulse Interference . . . . . . . . . . . . . . . . . . . . 1108.5.4 A Brief Comment on the Structure of the Search Land-scape . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115AppendicesA The Electric Dipole Hamiltonian . . . . . . . . . . . . . . . . 123B Frequency Domain Time-Dependent Perturbation Theory 125B.1 Time-Dependent Perturbation Theory . . . . . . . . . . . . . 125B.2 TDPT in the Frequency Domain . . . . . . . . . . . . . . . . 127C Software and Algorithms: QuantumBlackbox and Mathe-matica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131C.1 QuantumBlackbox . . . . . . . . . . . . . . . . . . . . . . . . 131C.1.1 Connecting to LabVIEW . . . . . . . . . . . . . . . . 131C.1.2 Hardware Device Proxy Classes . . . . . . . . . . . . 132C.1.3 Numeric Routines . . . . . . . . . . . . . . . . . . . . 132C.1.4 JavaScript (ECMAScript) Engine and Editor . . . . . 133C.1.5 User Interface . . . . . . . . . . . . . . . . . . . . . . 133C.1.6 Genetic Search Implementation Details . . . . . . . . 134C.2 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . 135viiList of Tables7.1 Transition dipole matrix elements relevant for the calculationof 420 nm fluorescence in rubidium. . . . . . . . . . . . . . . 558.1 Second harmonic compensation ratios obtained for the com-putationally simulated glass stacks, for one to six layer stacks. 968.2 Second harmonic compensation ratios obtained for experi-mentally realized glass stacks, having from one to five layers. 104viiiList of Figures1.1 A simple two pathway Brumer-Shapiro coherent control scheme. 52.1 The effect that multiply scattering material has on a broad-band pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The transmission of light through a thin, strongly scatteringlayer of TiO2, as demonstrated by Vellekoop and Mosk. . . . 103.1 A schematic of the two stage amplification scheme used. Thefemtosecond oscillator creates a train of femtosecond pulses,which seed the regenerative amplifier (RGA). . . . . . . . . . 143.2 The typical power spectrum of the laser light used in thiswork (all parts of the thesis), measured with a 300 lines/mmgrating spectrometer. . . . . . . . . . . . . . . . . . . . . . . . 175.1 A schematic diagram of a spectral pulse shaper. . . . . . . . . 255.2 A representation of the two masks appearing in the spatiallight modulator (SLM). . . . . . . . . . . . . . . . . . . . . . 275.3 The pixel-wavelength calibration of the SLM. . . . . . . . . . 295.4 A sample of the data that is acquired to produce the phase-voltage calibration curve for one of the two SLM masks. . . . 305.5 Steps of the analysis of the SLM calibration data. . . . . . . . 315.6 A sample of the phase-voltage curves used to calibrate theSLM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.7 The Gassian input pulse used for the numerical calculationsperformed in this work, along with its appearance in time andassociated second-harmonic spectrum. . . . . . . . . . . . . . 375.8 The effect of a pi-step phase mask; the phase mask steps from0 to pi at the location of the center of the input pulse spectrum. 385.9 The effect of a sinusoidal phase mask, φ(ω) = α sin(τ(ω−ω0)),α = 1 and τ = 300 ps. A train of pulses separated by τ arises. 405.10 The effect of a binary repeating unit used as an SLM mask. . 415.11 An experimental realization of optical intensity autocorrelation. 43ixList of Figures5.12 An experimental realization of optical interferometric auto-correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.1 A Grotrian diagram for rubidium, showing the most relevanttransitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.2 The relaxation process in rubidium, showing how the levelsinitially excited by the pulse decay towards the visible fluo-rescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.3 The experimental apparatus used to study the quantum co-herent control of two photon absorption in rubidium. . . . . . 597.4 The transmission curve for the Thorlabs FB420-10 bandpassfilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.5 A visual comparison of the behavior of the integrand in Eq. 7.1. 617.6 pi/2 window phase scans near 778 nm. . . . . . . . . . . . . . 627.7 Intensity edge scans near 778 nm. . . . . . . . . . . . . . . . . 647.8 pi and intensity edge scans at 795 nm. . . . . . . . . . . . . . 657.9 Evolution of the genetic search, seeking to maximize broad-band excitation of a two-photon transition in rubidum. . . . . 677.10 The evolution of the components of the transition amplitudethroughout the genetic search appearing in Fig. 7.9(a). . . . . 687.11 Illustration of a method by which narrow groups of pixelsare alternately scanned to the left and right of a rubidiumresonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.12 The resonance-centered pixel-scan method described in thetext, applied to the two transition pathways involving reso-nances at 776 and 780 nm. . . . . . . . . . . . . . . . . . . . . 737.13 The resonance-centered pixel-scan method described in thetext, applied to the two transition pathways involving reso-nances at 762 and 795 nm. . . . . . . . . . . . . . . . . . . . . 748.1 A stack of non-uniform glass slides, also called a random stackor glass stack, along with optics that capture the second har-monic spectrum of the light scattered via reflection. . . . . . 788.2 Experimentally measured spectra and XFROG traces of thereflected pulse train arising from one, two and three layerglass stacks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.3 Numerically computed reflection spectra; these simulate thescenario where a Gaussian beam (shown in Fig. 5.7) illumi-nates one to six layer thick stacks of glass slides. . . . . . . . 818.4 Reflected spectra; continued from Fig. 8.3. . . . . . . . . . . . 82xList of Figures8.5 The two-pulse train is created by shaping in intensity andphase, using H(ω) = 1 + eiω 300 fs (Eq. 8.9). See also Figs. 8.6and 8.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.6 A pulse train is created by shaping in intensity only, using|H(ω)|. H(ω) is given by Eq. 8.9. This is essentially a phasecompensated two pulse train. . . . . . . . . . . . . . . . . . . 878.7 A pulse train is created using phase-only shaping, using φ(ω) =Arg H(ω). H(ω) is given by Eq. 8.9. . . . . . . . . . . . . . . 888.8 An illustration of how second harmonic generation (SHG) isused to provide a measure of how well a random pulse train,as generated by a glass stack, has been compensated. . . . . . 898.9 A sample of the time and pulse amplitude scans used in theinterferometric autocorrelation-like or “methodic” scan method. 928.10 The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after com-pensation. This is realization 1 of the one layer numericallymodeled glass stack; the stack spectrum appears in Fig. 8.3. . 978.11 The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after com-pensation. This is realization 1 of the two layer numericallymodeled glass stack; the stack spectrum appears in Fig. 8.3. . 988.12 The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after com-pensation. This is realization 1 of the three layer numericallymodeled glass stack; the stack spectrum appears in Fig. 8.3. . 998.13 The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after com-pensation. This is realization 1 of the four layer numericallymodeled glass stack; the stack spectrum appears in Fig. 8.3. . 1008.14 The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after com-pensation. This is realization 1 of the five layer numericallymodeled glass stack; the stack spectrum appears in Fig. 8.4. . 1018.15 The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after com-pensation. This is realization 1 of the six layer numericallymodeled glass stack; the stack spectrum appears in Fig. 8.4. . 1028.16 Trends in the second harmonic compensation ratios obtainedusing the numerically computed stacks, from one to ten layers.103xiList of Figures8.17 The experimentally measured second harmonic spectra forthe physical glass stacks, having one to three layers. . . . . . 1058.18 The integrated second harmonic compensation ratio, obtainedfrom physical glass stacks compensated using the methodicscan. See also Table 8.2 and Fig. 8.17. . . . . . . . . . . . . . 1068.19 A numerical demonstration of the effect that limited pulseshaper resolution has on the creation of lengthy pulse trains. 1088.20 The second harmonic spectrum associated with an exact (atthe numerical model resolution) phase compensation of therealization 1, four layer numerically simulated glass stack. . . 1098.21 The effect of variations in air gap and layer thickness, simu-lated computationally using shaper-generated four-pulse trains.Each time point used in constructing these curves effectivelyrepresents a different stack. . . . . . . . . . . . . . . . . . . . 1118.22 A numerical simulation for a four layer thick glass stack, pre-pared using the full numerical model that includes all ordersof reflection, and transmission. . . . . . . . . . . . . . . . . . 112xiiAcknowledgementsI would like to thank my thesis supervisor, Dr. Valery Milner, for his ac-cepting me as a graduate student on rather late notice, and helping me tosucceed in completing my thesis. I believe that I have grown in many ways,including maturity, physical stamina, and academic ability during this briefbut important time, and have grown in ways I otherwise would not havegrown.I thank Thomas Drane, one of Dr. Milner’s Ph.D. students, for hisdetailed explanations on a wide variety of topics relevant to the work in thelab. I thank Martin Bitter for his contribution of a high quality sleepingmat for overnight stays at the lab and office.I thank Dr. Stanislav Koronov, a post-doctoral researcher who has donesome work in the Milner lab, for helping me with the optical setup of FROGand XFROG. I also thank Dr. Sergey Zhdanovich, previously a Ph.D. stu-dent in the Milner group, for helping me learn MIIPS, and for providing mewith his MIIPS analysis code.xiiiDedicationI dedicate this thesis to the Only True and Living God.I also thank my wife Chelsea and my two little sons Aaron and Enoch.I love them.xivPart IMotivation and Introduction1Chapter 1The Goal of ControllingMatter with LightMy personal interest in studying the topics that appear in this thesis stemsfrom a long-time interest in the idea that electromagnetic fields can bothget information from and change the “state” of material systems. I amespecially interested in eventually applying these ideas to complex biologicalsystems.Using electromagnetic fields to read information from molecular systemsis one type of spectroscopy. Distinct ranges of the electromagnetic spec-trum give information about particular characteristics of a target system[1]. For example, infrared (IR) radiation can elicit quantum rotational andvibrational transitions. Infrared absorption spectra allow accurate iden-tification of characteristic functional groups based on rotational and vibra-tional structure; often they facilitate exact identification of simple molecularspecies. Radiofrequency (RF) radiation is capable of causing nuclear spinstate transitions, giving rise to a field of study known as nuclear magneticresonance (NMR) spectroscopy. Advanced NMR techniques (making use ofRF pulse sequences that have been designed using quantum mechanics) canobtain enough information to allow the reconstruction of both the three-dimensional structure and the temporal dynamics of proteins [2]. Often,spectroscopic methods that employ electromagnetic fields collect informa-tion non-invasively. In contrast, some types of spectroscopy (e.g. massspectroscopy) and many other common chemical (e.g. titration) and bio-chemical methods of analysis destroy the sample of interest. Often a com-bination of non-destructive and destructive methods is of use; for example,purification methods that isolate only the protein of interest from a cell areoften used to increase signal-to-noise and reduce spectral congestion in NMRspectroscopy.But what about using light to lastingly change or control the state of aparticular target system, not just read information?21.1. Biological Control with Incoherent Fields1.1 Biological Control with Incoherent FieldsBiologists have discovered a number of systems which can be “controlled”with incoherent light: the specific design of the each system allows it torespond in a certain functionally significant way to electromagnetic fields.One example of such a system is vitamin D3 (cholecalciferol). VitaminD3 is produced [3] in the skin from 7-dehydrocholesterol (provitamin D3).7-dehydrocholesterol is converted to previtamin D3 after the absorption ofa photon of ultraviolet (UV) light. Upon relaxation to the ground state,previtamin D3 is converted to (at least) six different products, among whichis vitamin D3, an essential nutrient. It is an equilibrium process with spectralsensitivity to the wavelengths of incoherent radiation used [3, 4].A more elegant example is that of rhodopsin. This protein containsthe retinal molecule as a cofactor; incoming light induces a conformationalcharge of 11-cis-retinal to all-trans-retinal. This conformational change in-duces a subsequent cascade of signal transduction events in the eye, enablingvision [5]. Many other examples of biological systems functionally responsiveto incoherent electromagnetic fields by design exist.1.2 An Early Attempt at Molecular Control withMonochromatic FieldsIs it possible to optically control an arbitrary chemical (or biological) system,namely one that has not been designed to respond to optical excitation in anyespecial functionally significant way? (This is in contrast to the biologicalexamples given above.)To answer this, consider for example the following early attempt [6] tooptically break a chemical bond in a molecule. Similar to the way that single-photon absorption causes a state transition in an atomic species, a narrow-bandwidth (nearly monochromatic) laser beam is tuned to excite a particularrovibrational mode in a molecule. The idea justifying this approach is thatas long as the energy of excitation remains localized, the resulting productstates will be influenced—breaking, for example, a particular chemical bond.There are a few special cases where this method succeeds, and the successof these cases correctly suggests that one may use optical fields to controlarbitrary chemical and biological systems.But this simple control scheme based on monochromatic excitation failsin the majority of cases. The term intramolecular vibrational redistribution(IVR) describes the fact that such excitations do not remain localized [6].31.3. Quantum Coherent ControlAn excitation intended to break a given chemical bond moves away fromthe original site before product formation is complete. This naive approachto control has ignored the true time evolution of the target, which is definedby the molecular Hamiltonian (including, e.g. in the Born-Oppenheimerapproximation, the potential energy surface). Successful control schemesemploy the full molecular Hamiltonian, correctly treating the temporal evo-lution of the system. In this way, the process of IVR that follows monochro-matic field excitation is correctly predicted.1.3 Quantum Coherent ControlTwo successful control schemes based on the full molecular Hamiltonian areintroduced here. Both rely on coherent broadband optical fields. Such fieldsare readily obtained using pulsed lasers (Chapter 3). A coherent field has awell defined phase relationship across the whole bandwidth, and via Fourieranalysis, can give rise to pulses that are short in time.1.3.1 Tannor-Rice Coherent ControlTannor-Rice or “pump-dump” coherent control [6–8] is a scheme specificto the control of molecular species described using wavepackets. Considerthe following example [6]. A triatomic molecule ABC may dissociate in twoways,ABC→ AB + C (1.1a)ABC→ AC + B (1.1b)An initial pump pulse prepares an excited state wavepacket, moving verti-cally from the ground to the excited state surface. This excited state thenevolves in time, and the key to the method is the timing of the subsequentdump pulse. At certain time, the dump pulse will overlap better with anexit channel for the process of Eq. 1.1a whereas at another time processEq. 1.1b is favored. The method has been confirmed experimentally [6, 9],and it is seen that the production of AB+C or AC+B may have an oscilla-tory character with respect to the pulse separation time. The mathematicaldescription of this method [6, 7] is straightforward but will not be presentedhere.41.3. Quantum Coherent Control 𝑖   𝑓   𝑚1   𝑚2  Figure 1.1: A simple two pathway Brumer-Shapiro coherent control scheme.Two multiphoton paths connect the initial state |i〉 to the final state |f〉.Each path j shown is resonant with intermediate state |mj〉 .1.3.2 Brumer-Shapiro Coherent ControlThe basic principle of Brumer-Shapiro control [6, 10, 11] is to look aroundfor initial and final states of a system connected by two (or more) indepen-dent pathways. One chooses amplitudes and phases of the incoming field,which are transferred to the relative amplitudes and phases in the pathwaysbetween initial and final states in the material system. Control is gainedas the pathways leading to a particular final state interfere constructivelyor destructively, thereby enhancing or diminishing the presence of any ob-servables associated with the system having amplitude in that final state.Thus a particular Brumer-Shapiro control scheme is just the clever use ofquantum interference.Consider the simple two level system in Fig. 1.1. Each path from initialstate |i〉 to final state |f〉 (via intermediate state |m1〉 or |m2〉) involvestwo photons; this allows unique optical phases to be assigned to each pathwhen using a broadband input field. As optical phase is transferred toquantum phase (see [10] or apply time-dependent perturbation theory), thefinal probability of finding the system in state |f〉 is given byPf = |c1 + c2|2 (1.2)where cj is the quantum transition amplitude of each two photon path j. Ifone varies only the phase, control is gained through the interference termthat will appear in the expansion of Pf . Amplitude modulation may also beused.Although Brumer-Shapiro control experiments typically require coherent51.4. Spectral Pulse Shapingfields, it is possible to construct control methods that use optical frequen-cies from two different laser sources that do not have a well-defined phaserelationship between them [6, 12]. The method makes use of virtual transi-tions, constructing a control sequence that effectively cancels out the phasedependence.Optical coherent control schemes are not limited to controlling the dy-namics of simple molecular systems in solution. The technique is very gen-eral, and has been applied to quantum dots and spin systems [13]. Quantumcoherent control schemes have also been successfully applied to biologicalsystems [14]. Some examples in protein systems are the chirped-pulse con-trol of fluorescence in green fluorescent protein [15] and control of energy flowbetween intra- and intermolecular channels in a particular light-harvestingcomplex [16].In this work, Brumer-Shapiro quantum coherent control is applied totwo photon absorption in an atomic system (see Chapter 7).1.4 Spectral Pulse ShapingThe Brumer-Shapiro method described in Section 1.3.2 and depicted inFig. 1.1 requires the ability to adjust the phase and/or amplitude of in-dividual frequency components in a broadband optical field. When suchspectral adjustments are made to an electromagnetic field that is pulsed intime, it is called pulse shaping.Optical pulse shapers typically operate in the frequency domain; a prismor diffraction grating maps spectral components of the optical field onto aspatial plane, where some type of 1D mask is placed. This mask may be fixedor programmable; it may allow the phase, amplitude and/or polarization ofindividual frequency components in the pulse to be altered. As an example,it is possible to develop a static mask which converts optical pulses fromhaving a Gaussian temporal envelope to a square temporal envelope. Thiswas shown to improve the switching behavior of an all-optical switch [17].Two examples of programmable masks in common use are acousto-opticmodulators (AOMs) [18] and liquid-crystal spatial light modulators (LC-SLM). In this thesis, the LC-SLM mask will be used exclusively. It containsa row of finite-width pixel elements.As this optical pulse shaper shapes in the frequency domain, it is alsocalled a spectral pulse shaper (or Fourier pulse shaper). The details ofspectral shaping are presented in Chapter 5.6Chapter 2The Goal of ControllingMatter with Scattered Light2.1 The Problem of Optically Scattering MaterialThe introduction of coherent optical fields has introduced a new challenge inthe case when the light must first pass through optically scattering materialbefore reaching the control target. When a laser beam passes through anoptically scattering material it will be distorted spatially; for example, aspeckle pattern is produced. This speckle pattern results from multi-pathinterference between the different spatial components of the input beam asthey traverse the scattering material; this process is shown schematicallyin the inset in Fig. 2.1. Some non-collinear optical paths at the outputconstructively interfere, producing bright spots in the speckle field; othersdestructively interfere, producing dark regions.If the beam is pulsed (corresponding to a finite spectral bandwidth) thetemporal profile of the pulses will also be altered. The ballistic pulse isdefined as that part of the input pulse that travels straight through withoutscattering. As the effective thickness of the scattering material is increased,less and less of the ballistic pulse survives. The intensity of this ballisticpulse can be described by [19]Ib = I0e−µx. (2.1)I0 is the input beam intensity, 1/µ is the scattering length (a measure of thestrength of scattering), and x is the total thickness of scattering materialthat has been traversed. The meaning of this exponential decay is shown inthe main part of Fig. 2.1; the solid black represents the ballistic pulse, whilegray part represents the scattered light, forming a temporally delayed andstretched pulse. Temporal disruption also means spectral disruption, as thetwo are connected via Fourier transformation.Thus, attempts to do coherent control with light that has passed througha spatially scattering material are hindered in two ways:72.1. The Problem of Optically Scattering MaterialFigure 2.1: The effect that multiply scattering material has on a broadbandpulse. As the unscattered or ballistic pulse passes through the material, lightis scattered along various optical paths. The x coordinate is the thicknessof the scattering material in the direction of the input beam. This figure isfrom [19].• the originally well-collimated beam is broken up into a speckle field,so that the intensity of each speckle is a fraction of the original beamintensity, and,• the spectral phase relationships in the scattered light have been dis-rupted.The first point (loss of peak intensity) means that multiphoton optical pro-cesses, an important part of Brumer-Shapiro coherent control (Section 1.3.2),will be reduced, as these optical processes are intensity dependent (Chap-ter 4). The second item means that the chosen phase relationships in theinput field may not match those in the output field, detrimental for Brumer-Shapiro coherent control where proper phase relationships are critical.One way to cope with such spatial and spectral scattering is to simplytreat it as noise, and do as much as possible with the transmitted ballisticpulse. For example, an experiment [19] was done where a broadband pulsedlaser source was aimed through a thin layer of raw chicken breast onto a pH-sensitive probe molecule capable of two-photon absorption and fluorescence.Two phase optimizing masks were then discovered via an adaptive searchalgorithm (see Section 6.2), each mask maximizing the fluorescence signalin a distinct pH region. These coherent control experiments were successfulsimply because the ballistic pulse was sufficiently strong; the scattered field82.2. Experimental Compensation of Spatial Scatteringcould be treated as noise and ignored.2.2 Experimental Compensation of SpatialScatteringExperiments have shown that the amount of coherent control available afterlight passes through a scattering material is not limited to the availabilityof a ballistic (or unscattered) light. Thus, one can go beyond the ballisticpulse regime discussed above.Consider the following experiment by Vellekoop and Mosk [20]. A stronglyscattering material, a 10.1 µm thick layer of TiO2, is used. Only spatialspeckle (and not spectral) compensation is investigated, as the sample is illu-minated with a monochromatic HeNe laser. The incoming beam is stronglydisrupted, and a spatial speckle pattern is observed (Fig 2.2(a)).Vellekoop and Mosk then introduce a 2D spatial light modulator (SLM)into the beam. A 2D SLM is a programmable 2D array of pixel elements;each pixel modulates the phase of part of the spatial wavefront of the beam.(This is different from the spectral shaper containing a 1D SLM introducedpreviously in Section 1.4 as the pixel elements modulate across the spatialprofile of the beam instead of modulating the spectral components.) Theincoming wavefront is subdivided into N segments, corresponding to theavailable pixels of the 2D SLM (or small blocks of pixels). An and φn arerespectively the amplitude and phase of the incoming optical field in segmentn. tmn is a transfer matrix representing the effect of the scattering materialon the impingent field. The output field is then described byEm =N∑n=1tmnAneiφn . (2.2)When the phases on the 2D SLM are set to zero, φn = 0, the outputfield is again the speckle pattern, shown in Fig. 2.2(a). Compensation isachieved in the following way:• An optical detector sensitive to the total light intensity in a localizedregion is placed, for example, near the center of the speckle field.• The phase φn of a pixel on the 2D SLM is scanned from 0 through 2pi.At each value of phase, the intensity at the detector is measured. Thephase of that pixel that maximizes the amount of light at the detectoris recorded.92.2. Experimental Compensation of Spatial ScatteringFigure 2.2: The transmission of light through a thin, strongly scatteringlayer of TiO2, as demonstrated by Vellekoop and Mosk. a. The transmissionmicrograph of the speckle pattern that results when a flat phase incidentbeam is scattered by the sample. b. and c. Demonstrations of single-spotand five-spot focusing, following the iterative procedure described in the textwhereby the incoming beam is pre-compensated. d. The pre-compensationphase mask used in (c), restricted to the area of the impingent beam. Thisis the original figure, taken from their paper [20].102.2. Experimental Compensation of Spatial Scattering• Subsequent pixels are scanned in the same way, repeating the previousstep. Each time a pixel is scanned, all other pixels are set to zero.• At the end of this process, the optimal phase values found for each pixelare applied to the 2D SLM. This is possible as the transformation inEq. 2.2 is linear, meaning that the global optimum is the superpositionof per-pixel optima.Let this be called the “method of Vellekoop and Mosk,” as it will be referredto several times in what is to follow. The result of following this procedurewith a single point detector and with five point detectors arranged in aring is shown in Fig. 2.2(b) and 2.2(c), respectively. The intensity of thesingle-spot focus found is 1000 times above the intensity of speckle-patternbackground level before compensation. The compensating mask found canbe viewed as being the conjugate or inverse of the phase mask applied bythe scattering material.2.2.1 Experimental Compensation for Temporal ScatteringThe above technique of Vellekoop and Mosk has been repeated with a broad-band pulsed laser [21, 22]. It was found that the temporal disruption of apulse was automatically compensated simply by bringing the beam to a spa-tial focus. This was possible because the temporal disruptions are caused bythe same mechanism as spatial speckle—they are both caused by multiplescattering events inside the medium. The conjugate phase mask that com-pensates the spatial scattering then also mitigates the spectral disruption.However, such temporal compensation depends upon the scattering inthe material being weak. This means for example that the mask that recov-ers a focus at one frequency is effective at all other frequencies. There alsoexists another regime, that of strong scattering. In this case, the temporaldisruption of the pulse by the scattering material is enough that the maskfound via the method of Vellekoop and Mosk for one frequency does notautomatically compensate at all other frequencies. In this second regime,automatic temporal (or spectral) compensation is not possible. (The tran-sition between the two regimes is discussed and quantified in [23].)The individual speckles resulting from a spatially scattering material canalso be temporally pre-compensated using a spectral shaper [24]. In thiswork the possibility of compensating temporal scattering using a spectralshaper will be further explored. Instead of a bulk 3D material, however,a 1D system is used to scatter the incoming pulse. This is the topic of112.2. Experimental Compensation of Spatial ScatteringChapter 8, and contributes the second portion of experimental work doneherein.The method of Vellekoop and Mosk is not able to compensate 1D scat-tering of the type used in this work for two reasons. First, there is no spatialvariation of the scattering in the plane of the 2D SLM. Secondly, the scat-tering studied here falls into the “strong” scattering regime, as introducedabove. Thus even if spatial speckle were present, the spatial compensationwould not ensure automatic temporal compensation.2.2.2 The Idea That Scattering Media Can EnhanceControlUp until this point, scattering has been portrayed as something that must becompensated, something that is a nuisance to coherent control. However,spatially scattering media of various types can actually enhance variousaspects of signal transmission, or in other words, add possibilities beyondwhat was available with free-space propagation.For example, again using the method of Vellekoop and Mosk, a focuswas achieved past a scattering material [25]. The focus achieved with thescattering medium present beats the diffraction limit—by as much as tentimes, in one experiment. This idea, that scattering materials can enhancecontrol, will not be explored in this thesis.12Part IITools Used and Developed13Chapter 3Femtosecond Lasers3.1 OverviewA two-stage laser system is used. In the first stage, a femtosecond oscillatorgenerates seed pulses. Subsequently, these seed pulses are fed into a re-generative amplifier. This two-stage laser seed production and amplificationscheme is illustrated schematically in Fig. 3.1, and described in the followingsections. The resulting high-power beam is split at the output; a portionof this beam is fed into a pulse shaper (Chapter 5), which is then used inthe pulse shaping experiments presented here. The rest of this output beamfeeds other experiments in the lab and also powers two optical parametricamplifiers (OPAs).3.2 The Femtosecond OscillatorThe laser oscillator used here is a “Synergy,” manufactured by FemtosecondProduktions GmbH. It is a Ti-sapphire mode-locked laser. A 532 nm greenpump laser pumps a titanium-sapphire lasing crystal inserted into the cavityat the Brewster angle. Ti-sapphire supports a broad fluorescence bandwidthPumplaserFemtosecondoscillatorRegenerativeampli�ierPumplaserFigure 3.1: A schematic of the two stage amplification scheme used. Thefemtosecond oscillator creates a train of femtosecond pulses, which seed theregenerative amplifier (RGA).143.3. The Regenerative Amplifier[26] and is thus able to support the formation of a broad bandwidth laserpulse. At startup, the laser initially runs in a non-modelocked or continuouswave (CW) mode: the various cavity modes present at this stage do nothave a well-defined phase relationship. By perturbing the length of thecavity after a period of thermal stabilization, the modes can be locked intoa well-defined relationship with each other. This allows the cavity to emita train of femtosecond-duration pulses instead of a continuous wave.Several processes contribute to stable, modelocked operation [27]. Whenthe cavity is initially perturbed to initiate modelocking, a small subset ofmodes can be produced which have a phase relationship between them fa-vorable to the production of temporal spikes in intensity in the laser cavity.In order to create a single, strong pulse from any number of intensity spikes,a process known as Kerr focusing is used. (This is related to the Kerr effect[28]; see also Chapter 4 for a general introduction to nonlinear optics.) Kerrfocusing means that these higher-intensity temporal spikes are brought toa closer focus in the cavity than those weak in intensity. By aligning the(focused) pump beam into the crystal such that it overlaps with the high-intensity Kerr-focused cavity modes, a type of self-amplitude modula-tion (SAM) is implemented: low intensity fluctuations that would disruptthe formation of a stable, equilibrium pulse are trimmed away. The Kerreffect occurring in the crystal due to the presence of high intensity opticalfields is also responsible for self-phase modulation (SPM), whereby allavailable cavity modes can be brought into a modelocked relationship dur-ing the initial stages of pulse formation. Both SAM and SPM contribute tolong term pulse stability, imparting tolerance to intra-cavity noise.Finally, as the laser pulse must pass through the gain medium, it isnecessary to compensate for dispersion. Special layered mirror structures[29] are employed which compensate dispersion, thus shortening the pulsetemporally. This train of ultrashort, broadband pulses is then fed in to theregenerative amplifier. The frequency of pulses in the output train is 78MHz.3.3 The Regenerative AmplifierThe regenerative amplifier (RGA) is a Spitfire Pro, made by Spectra-Physics [30]. It is a laser, capable of running independently (initially seeded,like the femtosecond oscillator, from cavity fluctuations). However, it isspecifically designed to use femtosecond pulse seeds from a laser oscillator.The RGA also uses a Ti-sapphire lasing crystal, and an external pump.153.3. The Regenerative AmplifierThe intra-cavity pulse intensities that are reached in this RGA lasing cav-ity would be above the damage threshold (10 GW/cm2) for Ti-sapphire ifshort femtosecond seed pulses were used unmodified. To prevent this dam-age, a three stage amplification process is instead employed: the pulse is“stretched,” “amplified,” then “compressed.” This is called chirped pulseamplification, and will now be explained.The pulse is stretched in time, so that the same energy is present, butthe peak intensity drops. This allows the stretched pulse to be amplifiedmany times without damaging the crystal. A grating-based pulse stretcheris used to stretch the pulse in time, creating what is termed positive groupvelocity dispersion (GVD) on the pulse; the resulting beam has beenstretched in time and has the same energy but much lower peak intensity.The next stage is amplification. At a regular interval, a pulse from theseed train is allowed to enter the cavity. This pulse then undergoes severalrounds of amplification, causing broadband spontaneous emission from theTi-sapphire gain medium. The pulse is then dumped to the compressor.In order to select (“pick”) a pulse from the seed train and confine it inthe cavity, a Pockels cell will turn on (voltage will be applied), causing itto rotate the polarization as a 1/4 waveplate. This pulse then passes twicethrough a (passive) 1/4 waveplate inside the cavity, and back through theactive Pockels cell. The cavity is built to retain only horizontally polarizedlight. In this way, only the selected pulse enters and remains in the cavity.The Pockels cell then switches off before the next seed pulse; the next seedpulse still enters the cavity, but subsequently has its polarization rotatedto vertical and leaves the cavity. This continues while the selected horizon-tal pulse passes repeatedly through the cavity and is amplified. A secondPockels cell can be used to dump (like a Q-switched laser) the cavity atthe appropriate point (once the gain medium inversion has been depletedsufficiently).The compressor is also a grating-based optical component. It adds neg-ative GVD to the previously stretched pulse, causing the output pulse to bebrought (close) to its minimal temporal length, as dictated by Fourier theory.This minimal temporal length happens when the spectral components of thepulse all have the same phase (i.e., zero phase difference). This is known asthe Fourier transform limit, and such a pulse is called a transform-limited(TL) pulse. The spectrum of the approximately transform-limited outputpulse obtained in the lab is shown in Fig. 3.2. The frequency of pulses inthis final output train from the RGA is 1 kHz.163.3. The Regenerative AmplifierIntensity740 760 780 800 820 8400.00.20.40.60.81.0Wavelength HnmLFigure 3.2: The typical power spectrum of the laser light used in this work(all parts of the thesis), measured with a 300 lines/mm grating spectrometer.17Chapter 4Nonlinear OpticsThe study of the optical phenomena occurring when intense fields interactwith material systems is known as nonlinear optics [28]. Specifically, suf-ficiently intense optical fields (such as the pulses from the RGA) elicit anonlinear polarization response from the medium, which in turn leads tore-radiation of new harmonic frequencies.4.1 Nonlinear Polarization and the Introductionof New FrequenciesThe polarization P (t) of a material is a measure of the dipole moment perunit volume present because an electric field E(t) has been applied. Thespatial dependence has been omitted in this notation. If the polarizationresponse is nonlinear, it can be expressed as a power series,P (t) = 0(χ(1)E(t) + χ(2)E2(t) + · · · ) (4.1)in powers of the incident field. 0 is the permittivity of free space, andχ(n) is the nth order susceptibility. In an anisotropic medium, the χ(n)become n+ 1 rank tensors. Also, the assumption that the medium respondsinstantaneously is implicit in this expression [28].4.1.1 Second Harmonic GenerationIf an intense incident optical field described byE(t) = E˜e−iωt + E˜∗eiωt ≡ E˜e−iωt + c.c. (4.2)is incident on an appropriate material (non-centrosymmetric), Eq. 4.1 saysit will have a second-order polarization responseP (2)(t) = 0χ(2)E2(t)= 20χ(2)E˜E˜∗ + (0χ(2)E˜2e−2iωt + c.c.). (4.3)184.2. The Nonlinear Wave Equationc.c. stands for the complex conjugate of the preceding term(s). The zerofrequency component or DC term is a static electric field, and the processis called optical rectification. The terms at ±2ω represent the second har-monic of the fundamental. Using Maxwell’s equations, this polarizationleads to the generation of an optical field containing these new frequencies(see Section 4.2).Second harmonic generation will be used for two pulse characterizationtechniques known as MIIPS (Section 5.5.4) and FROG (Section 5.5.3); thesewill be introduced later. Second harmonic generation will also be used asthe nonlinear feedback signal in the work on the compensation of temporalscattering in Chapter 8.4.1.2 Sum-Frequency GenerationIf an intense incident optical field described byE(t) = E˜1e−iω1t + E˜2e−iω2t + c.c. (4.4)is incident on an appropriate material (non-centrosymmetric), it will yieldthe following second-order polarization response [28],P (2)(t) =0χ(2)E2(t)=0χ(2)(E˜21e−2iω1t + E˜22e−2iω2t + 2E˜1E˜2e−i(ω1+ω2)t+ 2E˜1E˜∗2e−i(ω1−ω2)t + c.c.) + 20χ(2)(E˜1E˜∗1 + E˜2E˜∗2) (4.5)In this expansion one sees terms corresponding to optical rectificationand second harmonic generation. One also observes components appearingat both the sum of and the difference between of the incident frequencies,called sum- and difference-frequency generation, respectively.In this work, sum-frequency generation is important for a closely relatedvariant of the FROG pulse characterization scheme, known as XFROG (seeSection 5.5.3).4.2 The Nonlinear Wave EquationIt is now shown how Maxwell’s equations may be used with such a nonlinearpolarization response. An important result that will arise is the predictionof what is known as phase-matching. This important concept lays the foun-dation for efficient second-harmonic generation in the lab.Several approximations will be made to simplify this treatment; assumethat the nonlinear response happens in a material that194.2. The Nonlinear Wave Equation• contains no free charges: ρ = 0,• contains no free currents: ~J = 0,• is nonmagnetic, such that ~B = µ0 ~H, and,• is isotropic.The displacement field ~D is ~D = 0 ~E + ~P , with ~P = PL + PNL written toshow the linear and non-linear polarization terms explicitly.Maxwell’s equations in a medium [28],~∇ · ~D = ρ (4.6a)~∇ · ~B = 0 (4.6b)~∇× ~E = −∂ ~B∂t(4.6c)~∇× ~H =∂ ~D∂t+ ~J, (4.6d)give the classical evolution of the electromagnetic fields in the system. Theseequations, with the help of the above approximations, yield the wave equa-tion,~∇× ~∇× ~E +1c2∂2∂t2~E = −10c2∂2 ~P∂t2. (4.7)A substantial simplification follows if ~∇· ~E = 0, which is not true in generalbut is true, for example, if ~E takes the form of an infinite plane wave. Thevector identity~∇× ~∇× ~E = ~∇(~∇× ~E)−∇2 ~E (4.8)allows Eq. 4.7 to be written as−∇2 ~E +(1)c2∂2 ~E∂t2= −10c2∂2 ~PNL∂t2(4.9)where (1) is the scalar relative permittivity of the material. 0 ~E + ~PL ≡0(1) ~E for an isotropic material.This equation shows how frequency components appearing in the nonlin-ear polarization response generate new optical fields at the same frequencies:the nonlinear polarization acts as the driving term in this wave equation forthe electric field E.204.3. Sum-Frequency Generation and Phase-Matching4.3 Sum-Frequency Generation andPhase-MatchingEq. 4.9 can now be used to predict phase-matching in the case of sum-frequency generation. The basic ideas along with the solution are presented[28].First, Eq. 4.9 rewritten in a form capable of describing the evolutionof a single frequency component. One may simply Fourier transform it, ormay follow the approach used by Boyd of substituting in a summation overfrequency components ωn. Either way, the end result is the same; Boyd’smethod yields the equation−∇2 ~En +(1)(ωn)c2∂2 ~En∂t2= −10c2∂2 ~PNLn∂t2(4.10)for frequency component n. (1)(ωn) is the frequency component at fre-quency n, essentially the Fourier transform of the scalar permittivity definedpreviously.For sum-frequency (or second-harmonic generation) one uses this equa-tion to investigate how the sum-frequency component, ω3, generated fromω1 and ω2, behaves. (Broadband nonlinear generation around ω3 is an ex-tension of this treatment.) LetEi(z, t) = Aiei(kiz−ωit) + c.c. (4.11)describe the fields in system, all propagating collinearly in direction of thez coordinate with equal optical polarization, and whereki =√(1)ωic. (4.12)(An detailed derivation including the effects of non-collinear beam propaga-tion is possible [31].) The nonlinear polarization term for this system (~PNL)can be written as (see reference [28])P3 = 20χ(2)E1E2(e−iω3t + c.c.). (4.13)Putting these definitions into Eq. 4.10, one obtainsI3 =2(χ(2))2ω23I1I2n1n2n30c2L2 sinc2(∆kL2), (4.14)214.4. Second-Harmonic Generation in a Birefringent Crystaldescribing the intensity at ω3. Ii = 2ni0c|Ai|2 give the intensity of thewaves, with ni the refractive index at ωi, and L gives the length of themedium, for example, L could give the thickness of a crystal.In order to arrive at Eq. 4.14, A1 and A2 have been taken as constant (ini-tially they could have had z dependence), undepleted input beams. The in-put beams often become significantly depleted; this approximation, however,allows for simplified mathematics which correctly predict phase-matching,even though the expression for I3 as a whole is not valid [28].Note the introduction of∆k ≡ k1 + k2 − k3. (4.15)As an argument of the sinc function, this says that I3 generation is maxi-mized when ∆k = 0. Physically, this says that the input and output waves(which are distinct frequencies) are in-phase with each other. The thicknessof the medium L also contributes to argument of sinc; excessively thick non-linear crystals increase the effect of any residual phase-mismatch ∆k. Thisalso helps in understanding why the thickness must be kept to a minimumwhen a broader range of frequencies must be used, as when doing broadbandsecond-harmonic generation using femtosecond laser pulses.4.4 Second-Harmonic Generation in aBirefringent CrystalHaving theoretically established the need for phase matching, one maybriefly consider its physical realization in an arbitrary medium having nor-mal dispersion. Using Eq. 4.12,∆kc = n1ω1 + n2ω2 − n3ω3 = 0, (4.16)one can compute the difference n3 − n2,n3 − n2 = (n1 − n2)ω1ω3. (4.17)Normal dispersion says ni(ωi) > nj(ωj) for ωi > ωj , and thus Eq. 4.17 doesnot have a solution in this case.Two possibilities for realizing phase-matching in the lab are use of ananomalous dispersion material or a method known as quasi-phase-matching[28]. More commonly, however, one uses a birefringent crystal.In birefringent crystal phase-matching, one can choose the input beampolarization(s) and the crystal angle such that the functions ni(ωi) take on224.5. Relation to Two-Photon Absorptionvalues that allow one to satisfy Eq. 4.17. Commercially prepared crystalsare used in the lab, pre-cut such that phase matching can be achieved viarotation about a single axis.In the experiments done in this work a 40-µm thick crystal of β-bariumborate (BaB2O4 or BBO) is used. BBO is a nonlinear material with prop-erties favorable [32] for use in a wide range of nonlinear optical experimentsand devices, including optical parametric amplifiers (OPA).4.5 Relation to Two-Photon AbsorptionTwo-photon absorption will be investigated later (Chapter 7). This is wherean atomic transition is made by the absorption of two-photons of frequen-cies ω1 and ω2, whose sum is resonant with the energy of the total atomictransition. This process is actually very similar to the generation of second-harmonic light described here. When second-harmonic light is generated,e.g. in a BBO crystal, two-photons at ω are absorbed “virtually,” with theproduction of a photon at 2ω. The process is said to be parametric, as nonet absorption of energy has taken place. The process of two-photon ab-sorption in atom is similar, except that one now replaces one or both of thevirtual levels with real levels (energy eigenstates) of the atom. The processis then non-parametric—a state transition has occurred, and instead of pro-ducing a field at the second-harmonic, relaxation pathways to other (atomiceigenstate) energy levels may dominate.23Chapter 5Pulse Shaping andCharacterizationThe basic idea of spectral pulse shaping was previously introduced onlybriefly (Section 1.4). This chapter is an amalgamation of considerationspertinent to working with broadband pulses, considering both pulse shapingand pulse characterization methods. Pulse shaping means changing thespectral amplitude and/or phase content of a pulse; characterization meansdetermining the amplitude and/or phase content of some optical field.5.1 The 4f Pulse ShaperThe laser amplifier (RGA) is a source of approximately Gaussian-shapedpulses in the time domain. As expected from Fourier theory, the observedfrequency spectrum is also approximately Gaussian. Attempting to shapethe pulses directly in the temporal domain would require optical compo-nents with an extreme optical response times on the order of one femtosec-ond. This cannot be achieved with present day technology. Instead, oneoptically Fourier transforms the incoming pulse using a diffraction grating.The pulse’s spectral components are then manipulated directly, after whicha subsequent optical Fourier transformation yields again a well-collimatedbeam of shaped pulses.A schematic of this type of spectral pulse shaper is shown in Fig. 5.1. Thepurpose of the various components appearing in the figure will be describedin the following subsections.5.1.1 Before the Spatial Light Modulator (SLM)The spectral components of an incoming beam must be spatially spread.A diffraction grating is used. A diffraction grating is a periodic structurewhich, when properly illuminated, leads to an interference pattern mappingspectral components of the incident light to a well-defined angular distribu-245.1. The 4f Pulse Shaperܧinሺݐሻ ܧinሺ߱ሻ ݐݑ݋ܧሺݐሻ Diffraction grating Focusing lens/ mirror Dual-mask SLM ܧoutሺ߱ሻ Figure 5.1: A schematic diagram of a spectral pulse shaper. The frequencycomponents present in the input pulse Ein(t) are presented along the spatialdimension of the dual-mask SLM. The SLM adjusts Ein(ω), yielding Eout(ω),which is then converted back into a well collimated beam of output pulses,Eout(t).tion. Multiple spatial “orders of diffraction” result, according to [33]sin θi + sin θr = 10−6 nλm. (5.1)Here, θi is the angle of incidence and θr the angle of diffraction, both takenwith respect to the normal. n is the spatial periodicity of “lines” in thediffraction grating, in units of lines mm−1. The diffraction order m ∈ Z(m is an integer) and the wavelength λ in units of nm then give the spa-tial structure of the diffracted frequency components, order-by-order. Eachdiffractive order gives another copy of the spectrum.The exact electric field produced in the far field after diffraction off thegrating is given by Fraunhofer diffraction theory; it is the two-dimensionalspatial Fourier transform of the aperture function [34], given byE(~ky,~kz) =∫∫ ∞−∞dy dz A(y, z)ei(~kyy+~kzz) (5.2)For the diffraction grating, the aperture function A(y, z) is a product ofa function specifying the physical bounds of the grating (e.g. “rectangle”functions in y and z) , multiplied by a function which describes the periodicstructure of the grating (e.g. a periodic amplitude or phase modulation). Fora grating with vertical lines (etched along the z coordinate), the integrationin the vertical domain is just the 1-dimensional transform of the beam’s(e.g. Gaussian) spatial profile. The interesting part of the transform is1-dimensional transform of the product of the beam’s spatial profile, thegrating’s physical boundaries in that dimension, and the periodic grating255.1. The 4f Pulse Shaperpattern. In the limit that the contributions to the transform from the firsttwo of these are sufficiently broad and smooth features, one considers theeffect of the grating only in the horizontal dimension, computingE(~ky) =∫ ∞−∞dy A(y)ei(~kyy). (5.3)But A(y) is a unit periodic function on [0,1) (with an appropriate choice ofcoordinate system), and so can be expressed by the Fourier series [35],A(y) =∞∑m=−∞Bmei(2pimy) (5.4a)Bm =∫ 1/2−1/2dy A(y)e−i(2pimy) (5.4b)Bm (Eq. 5.4b) gives the relative amplitude of diffraction order m [36]. Thiscan then guide the design of a grating that contains the desired distributionof amplitudes among different orders. For the purpose of this pulse shaper,it is desirable that the power flowing into the first order (m = 1) diffractionspot is maximized. This is achieved by the use of a blazed grating. Insteadof, e.g., a sinusoidal pattern, a blazed grating chooses a repeating unit whichis a wedge-like shape.As shown in Fig. 5.1, a lens is used to bring the diffracted beam intoa focus at the plane of the SLM. A spherical focusing mirror can also beused in place of the lens (with appropriate changes to the overall shaperlayout). The Fourier plane then appears at 2f , twice the focal length f . Ashown in the schematic diagram, this presents Ein(ω) to the SLM masks.The operation of the SLM will be discussed in Section 5.2.5.1.2 After the SLMThe final lens (or mirror) and diffraction grating bring the spectral com-ponents back to a time-domain pulse, Eout(t), effectively performing theinverse operation.If the properly aligned SLM is set such that Ein(ω) = Eout(ω) (blankmask), then the resulting pulse emerges with no added dispersion. Forthis reason, the pulse shaper setup (the actual arrangement of the opticalcomponents) described here is said to create a zero dispersion line. This isalso called a 4f -line, as the line relies on a total of four focal lengths fromgrating to grating [18, 37].265.2. The Liquid-Crystal SLM߶1 ሺ߱ሻ ߶2 ሺ߱ሻ ܧinሺ߱ሻ ܧoutሺ߱ሻ Figure 5.2: A representation of the two masks appearing in the spatial lightmodulator (SLM), a part of the spectral shaper shown in Fig. 5.1. Each maskis made up of a row of pixel elements. The two thin red arrows on the masksrepresent the fact that the two masks have orthogonal extraordinary axes.The different pixel elements modulate the phase φ1,2(ω) of light at differentfrequencies ω; each pixel element overlaps a narrow band of frequencies. Theoutput field is drawn to emphasize the appearance of vertically polarizedlight, in addition to the horizontal component already present at the input.5.2 The Liquid-Crystal SLMA liquid-crystal SLM, manufactured by Cambridge Research & Instrumen-tation, Inc. (CRi), is used [38]. The orientation of the two masks in the spec-tral shaper was presented schematically in Fig. 5.1. Fig. 5.2 complementsthis figure, showing the individual pixels appearing in the two masks. Two640-pixel masks appear back-to-back in this unit. Each pixel is 100± 0.005µm wide; there exists a small 2.0 µm optical gap between each pixel that isnot controlled. The two masks are aligned with each other to a tolerance of±2.0 µm. The pixels are tall enough (5 mm) that the focused beam easilyfits within the pixel area in this direction.Each pixel is driven to a user-specified drive voltage or count, controllingthe refractive index along the crystal’s extraordinary axis. The ordinary axissees no change in index with the application of the field. The horizontallypolarized input beam (see Fig. 5.1) is directed onto the first pixel mask,which has its extraordinary axis oriented at 45◦ to the input beam. The275.3. Calibration and Operation of the SLMsecond mask has its extraordinary axis oriented orthogonally to the first;if the two masks vary a pixel together, the resulting beam will be phase-modulated by this amount. If the two masks vary a pixel differently, theresulting beam will be modulated by this phase difference in polarization.If a linear polarizer is placed at the output, the resulting beam will bemodulated in amplitude and phase. The intensity modulation due to alinear polarizer acting on a polarized beam is given by Malus’s Law [34],I(θ) = I(0) cos2 θ (5.5)where θ is the angle between a linear polarization component of the inputbeam and the output polarization, and I(0) is the incident intensity. Itfollows from the above considerations that modulation of the field due tothe SLM is given by [18]Eout(ω) = H(ω)Ein(ω), (5.6)with complex transfer functionH(ω) = exp(iφ1(ω) + φ2(ω)2)cos(φ1(ω)− φ2(ω)2). (5.7)φn gives the phase appearing at an SLM pixel on the first or second mask,corresponding to n = 1, 2. Note that the SLM must be calibrated in orderto determine what electrical voltage, applied to a given pixel, produces anoptical phase change φn.5.3 Calibration and Operation of the SLMIn this section I discuss the process of calibration, followed by the procedureby which the SLM is used to display an arbitrary phase and amplitude mask.5.3.1 Pixel-Wavelength CalibrationThe SLM is operated with a polarizer in place at the output, so that itcan shape in amplitude. Amplitude shaping is used to block all but fourpixels, appearing in a sufficiently intense part of the pulse spectrum, andthe spectrum of the pulse past the SLM is acquired from the spectrometer(Section 5.5.1). A linear fit matches (to a good approximation) the SLMpixels to the wavelength locations of the peaks or dips in the spectrum; thewavelength matching each SLM pixel is thus determined. This is illustratedin Fig 5.3.285.3. Calibration and Operation of the SLMIntensity780 790 800 810 8200.00.20.40.60.81.0Wavelength HnmL(a) Observed transmission spectrum af-ter all but 4 pixels are blocked on theSLM. Here, for example, pixel numbers200, 250, 345 and 400 are used.WavelengthHnmL0 100 200 300 400 500 600760780800820840SLM Pixel(b) The open pixels are plotted againstthe peak wavelengths. The fit line indi-cates that pixel 0 (the y-intercept) mod-ulates light at 752 nm and that each sub-sequent pixel is about 0.15 nm wide.Figure 5.3: The pixel-wavelength calibration of the SLM.5.3.2 Phase-Voltage CalibrationThe phase-voltage calibration is simple: one needs to measure the phasemodulation that the SLM produces in response to a particular voltage, forall pixels. The SLM is operated so that it can shape in amplitude and phase,with a polarizer at the output. Then the following procedure is performed:1. All pixels on the first mask are set to the same drive voltage, startingat 500 and continuing to 1400 drive counts (in steps of 1), while thesecond mask is set to the midpoint voltage, namely 950 drive counts.This range of 500 to 1400 drive counts corresponds to a region ofhigh modulation for this particular SLM, meaning that the SLM canproduce relative phase changes between the masks in excess of 2pi.2. Each time a flat voltage mask voltage is sent to the SLM, one must waitfor the data to transfer to the SLM. One also must wait for the maskto become optically active, meaning that the liquid crystal moleculesin each pixel have finished their motion in the newly applied electricfield. A total wait of 500 ms is sufficient, most of which is data transfertime.3. After the mask is active, a spectrum is acquired from the spectrom-eter. This spectrum shows the optical attenuation that the SLM hasproduced. As the voltage is swept the relative phase between masks295.3. Calibration and Operation of the SLMWavelengthHnmL500. 580. 660. 740. 820. 900. 980. 1060. 1140. 1220. 1300. 1380.821.925817.875813.825809.775805.725801.675797.625793.575789.525785.475781.425777.375Voltage Applied to Scanned Mask Harb. unitsLFigure 5.4: A sample of the data that is acquired to produce the phase-voltage calibration curve. Vertical slices correspond to the set of collectedspectra. Horizontally, one sees the alternating patterns of bright (transmis-sion) and dark (no transmission) predicted by Eq. 5.7. A similar data set isacquired when this mask is fixed and the other is scanned.passes between multiples of pi and 2pi, and thus between regions ofbright and dark. The pattern of bright (transmission) and dark (notransmission) is due to Eq. 5.7. Bright bands result when the phasedifference φ1−φ2 is 0. Dark bands result when the phase difference ispi; at these spots, the light coming out from the SLM has been rotatedsuch that it is blocked by the output polarizer.4. The process is repeated with the first mask fixed and the secondscanned. Two very similar sets of data are collected in this way, onefor each mask. In Fig. 5.4 the data collected for one of the two masksis shown.The analysis of this set of acquired spectra yields the desired phase-voltage calibration curves for each mask. The procedure is best illustratedvisually, and thus each step here is accompanied by one of the plots appear-ing in Fig. 5.5, in order. The analysis is:1. The entire data set is normalized to a global maximum of 1.305.3. Calibration and Operation of the SLMIntensity600 800 1000 1200 14000.20.40.60.81.0Voltage Applied to Scanned Mask(a) Step 1. Original horizontal slice fromtrace, normalized.Intensity600 800 1000 1200 14000.20.40.60.81.0Voltage Applied to Scanned Mask(b) Step 2. After application of smooth-ing, correction of the peak (minima andmaxima) heights and then taking thesquare root.Intensity600 800 1000 1200 1400-1.0-0.50.51.0Voltage Applied to Scanned Mask(c) Step 3. After application of an alter-nating step function that multiplies alter-nate maxima by -1.PhaseHscaledL600 800 1000 1200 1400-1.5-1.0-0.50.51.01.5Voltage Applied to Scanned Mask(d) Step 4. After taking the arcsin.PhaseHscaledL600 800 1000 1200 1400-1.5-1.0-0.50.51.01.5Voltage Applied to Scanned Mask(e) Step 5. After application of anotheralternating step function, which multi-plies the negative slope components by -1.PhaseHradL600 800 1000 1200 140051015202530Voltage Applied to Scanned Mask(f) Step 6. After addition with a pi stair-case function, division by pi, multiplica-tion by 2, and cubic spline smoothing.This is the final phase-voltage curve forthis mask at this pixel.Figure 5.5: Steps of the analysis of the SLM calibration data in Fig. 5.4.The analysis gives SLM phase-voltage curves for both masks, and at allpixels with sufficient input beam intensity. The analysis involves Eq. 5.9, asdescribed in the text.315.3. Calibration and Operation of the SLM2. The response curve for a particular wavelength is isolated; this cor-responds to taking one of the horizontal lines in Fig. 5.4. The peakspacing widens from left to right because the amount of SLM modula-tion per drive decreases. The curve is then fit to a smooth cubic splinefunction. This process can contribute to the minima and maxima notgoing to 0 and 1, beyond what is present in the original data. By find-ing the offsets between the minima and 0 and maxima and 1, linearfunctions constructed between the peaks allow the curve to be scaledto fit exactly between at 0 and 1. It is important for the followingsteps that this is the case.3. The relationship between the phase difference across the two masksand the transmitted intensity is given by Eq. 5.7; in terms of intensity,I(ω, V ) = sin2(φ(ω)2); (5.8)inverted, this isφ(ω) = 2 arcsin(√I(ω, V )). (5.9)V is the drive voltage applied; it does not matter what the other maskvoltage is as long as it is fixed. Thus, the first step in retrieving thephase is taking square root.4. The sign lost in taking the square is recovered by multiplying theregion between every other pair of zeros of the function by negativeone.5. The arcsin is now taken, as per Eq. 5.9.6. This step involves the first step in unwrapping the phase, done by mul-tiplying the curve by negative one wherever the curve sloped downward(which corresponds to the regions between every other pair of maximain Step 3).7. The last step in unwrapping the phase requires adding a pi staircasefunction to the curve. The curve is then multiplied by two.Some of the phase curves (for some of the SLM pixels) may come out of thisanalysis shifted by 2pi—this can be corrected. The phase-voltage curves fora few of the SLM pixels (sampled at even intervals across the spectrum) isshown in Fig. 5.6. The phase values in the curves are adjusted so that φ = 0appears roughly in the center.325.3. Calibration and Operation of the SLMPhaseHradL600 800 1000 1200 1400051015202530Voltage Applied to Scanned Mask Harb. unitsLFigure 5.6: A sample of a few of the phase-voltage curves, sampled at evenlyspaced intervals across the spectrum, used to calibrate the SLM. This is fromthe analysis of the data in Fig. 5.4.The phase curves are assembled into a matrix, giving the phase at agiven spectrometer wavelength and drive voltage. This calibration matrixis converted, via bilinear interpolation, into a matrix giving the phase at agiven SLM wavelength and drive voltage.5.3.3 Shaping in Phase and IntensityIn order to realize shaping in intensity and phase, one must calculate thetwo phase masks which will be applied to the SLM. Eqs. 5.7 and 5.9 suggestthat the phases two phase masks, φ1 and φ2 respectively, may be computedusingφ1(ω) = φ(ω) +√arccos(I(ω)) (5.10a)φ2(ω) = φ(ω)−√arccos(I(ω)) (5.10b)(5.10c)335.4. Frequency-Domain Pulse Shapingwhere I and φ are the desired intensity and phase.5.3.4 Converting Phases To Drive CountsThese phase values for each SLM mask must then be converted to drivecount levels via the calibration matrix. In the process, the LabVIEW classmethod that handles this request does several things. First, the requestedphase masks are combined (added) to the phase compensation mask (seeSection 5.5.4). Then, this final requested phase is wrapped at 2pi, ensuringthat no phase mask exceeds the available modulation range (e.g., outsidethe 500-1400 drive counts calibration range). Finally, the appropriate drivecounts are determined and sent to the SLM.5.4 Frequency-Domain Pulse ShapingIn electrical engineering, the concept of a linear, time-invariant filter can beused to describe the relationship between the temporal input and outputsignal in a system. This relationship is given byEout(t) = Ein ? H(t) ≡∫dt′Ein(t′)H(t− t′) (5.11)where Ein,out are respectively the input and output signals in the time do-main, and H(t) is called the impulse response function [39]. Fourier trans-forming both sides and invoking the convolution theorem givesEout(ω) = Ein(ω)H(ω), (5.12)where H is now called the frequency response or complex transfer function.As discussed in Section 5.1, the SLM sits in the Fourier plane of the lens(or spherical mirror), where it directly modulates the phase of the spectralcomponents Ein(ω). Thus H(ω) represents the effect of intensity and phasemodulation applied to the shaper and output is given by simple multiplica-tion. This concept has of course already been introduced in Section 5.2, buthere the relationship between the time and frequency domain is emphasized.Also, unlike Section 5.2, the details of the dual-mask structure are now notimportant; rather H(ω) is simply the total phase and amplitude mask thatwill realized by the SLM upon request. The SLM allows masks in the formH(ω) = A(ω)eiφ(ω) (5.13)where A ∈ [0, 1] represents amplitude modulation and φ represents the de-sired phase.345.4. Frequency-Domain Pulse ShapingAt present, consider phase-only shaping,H(ω) = eiφ(ω). (5.14)Expanding H(ω) as a Taylor series to order n gives [18]φ(ω) = φ(ω0) + φ(1)0 (ω − ω0) + · · ·+1n!φ(n)0 (ω − ω0)n (5.15)with φ(n)0 = (dnφ/dωn)ω=ω0 . The various terms each have distinct, simpleeffects on the resulting pulse in time domain, Eout(t):• The carrier envelope phase, or the global phase of a particular pulse,is controlled by φ(0), and has units of radians.• The pulse is delayed in time according to φ(1), having units of time.This is known as group delay.• The pulse is temporally spread according to φ(2), having units of timesquared. This is known as chirp. This is equivalent to the pulsestretching process described for the RGA (Section 3.3).5.4.1 Numerical Fourier Transformation and Pulse ShapingAll shaping on the SLM has to be done in the frequency domain; it is some-times desirable to predict the time-domain result of that shaping. One mayuse the fast Fourier transform (FFT) for this. However, some subtletiesarise in using the FFT to convert an arbitrary spectrum into its time domainsignal and vice versa. As such, this process is described here.1. If the desired input spectrum is in units of wavelength it must beconverted to units of frequency. This can be done simply via linearinterpolation.2. It is then advantageous to increase the resolution of, and zero pad,the input spectrum. Increasing the resolution can be done via linearinterpolation. The final length of the upsampled, zero padded arraymay be chosen to be a power of 2, as the FFT typically operatesoptimally with such lengths.3. The spectrum is mirrored onto negative frequencies, by taking thecomplex conjugate, as the time domain field is real.355.4. Frequency-Domain Pulse Shaping4. Finally, the FFT is computed using standard methods. The resultingtime domain points contain the real field. The two halves of the outputarray might have to be exchanged, an operation known as an “FFTshift” in some FFT libraries.5. The time coordinates corresponding to these new time-domain fieldamplitudes can be computed using the relationδt =2pi∆ω, (5.16)where δt is the spacing in time between points in the transformed data,and ∆ω gives the total angular frequency extent of the input frequencyarray fed into the FFT.There exist limitations to the types of output fields Eout(t) which canbe realized with a particular SLM mask given by H(ω). Via the Fouriertransform, the temporal features which can be realized are limited becauseof the overall bandwidth. Also, and again via Fourier transformation, thepixel resolution limits the maximum length of the shaped pulse.5.4.2 A Gallery of Shaped PulsesIn this section, a number of frequency-domain masks H(ω) will be pre-sented, along with their resulting effect on the pulse in time. These masksproduce time domain behavior more complicated than the simple constant,linear and quadratic phase terms discussed, and have been computed usingthe numerical strategy outlined in Section 5.4.1. These masks are impor-tant because they are similar to the SLM masks used in the experimentsappearing in Part III.A Gaussian input pulse is used for the numerical pulse shaping compu-tations in this work, as shown in Fig. 5.7. The bandwidth of this numericalinput pulse matches (to a good approximation) that of the laser amplifierused in the lab. The time domain electric field amplitude is also shown,computed via Fourier transformation. Second harmonic spectra are alsoshown in this section as they are important to the work in Chapter 8.The pi StepConsider the phase mask shown in Fig. 5.8,φ(ω) = pi θ(ω − ω0). (5.17)365.4. Frequency-Domain Pulse ShapingIntensity760 780 800 820 840 8600.00.20.40.60.81.0Input Spectrum-1.0-0.50.00.51.0PhaseHradLWavelength HnmLIntensity370 380 390 400 410 4200.00.20.40.60.81.0Second HarmonicWavelength HnmLEHtL-50 0 50-1.0-0.50.00.51.0Time DomainTime, t HfsLFigure 5.7: The Gassian input pulse used for the numerical calculationsperformed in this work, along with its appearance in time and associatedsecond-harmonic spectrum. This is provided for reference, as later resultswill provide only the SLM mask applied to this input.ω0 is the pulse center frequency (ω0 = 2.355 fs−1 corresponds to 800 nm),θ is the Heaviside step function, and φ(ω) is used in Eq. 5.14. This delayshalf of the frequency components in the pulse such that the two “halves” ofthe pulse interfere, producing the shown time domain field.Although the pi step mask shown here is not used in this work, a varietyof masks with sharp step-like features appear in Chapter 7.Pulse TrainsOne may create a closely spaced sequence of pulses (known as a pulse train)from a single input pulse by using a certain type of SLM mask. As pulsetrains will be very important to the work appearing in Chapter 8, they willbe discussed here in some detail.A structure occurring with periodicity ∆ω in the frequency domain (e.g.on the SLM phase mask) will lead to a repeating structure in the time375.4. Frequency-Domain Pulse ShapingIntensity760 780 800 820 8400.00.20.40.60.81.01.2SLM Mask0.00.51.01.52.02.53.0PhaseHradLWavelength HnmLEHtL-1500 -1000 -500 0 500 1000 1500-1.0-0.50.00.51.0Time DomainTime, t HfsLIntensity390 395 400 405 4100.00.20.40.60.81.0Second HarmonicWavelength HnmLFigure 5.8: The effect of a pi-step phase mask; the phase mask steps from 0to pi at the location of the center of the input pulse spectrum.385.5. Pulse Characterization Techniquesdomain (e.g. in the pulse train coming out from the shaper) with periodicity∆t, related by [40]∆ω =2pi∆t. (5.18)The above equation does not address the question of how the intensityis distributed among the pulses appearing in the train. Recall that a blazeddiffraction grating tailors the shape of each groove in order to change thespatial intensity of each diffraction order (see Section 5.1, and Eq. 5.4b).Similarly, varying the structure of the periodic unit appearing on the SLMwill change the relative intensity of pulses appearing in the pulse train. Inthe following, two different periodic units will be introduced.A commonly used periodic structure that can be applied to the SLMmask is the sinusoidal phase function,φ(ω) = α sin(τ(ω − ω0)). (5.19)The result of applying Eq. 5.19 with α = 1 and τ = 300 ps is shown inFig. 5.9. Note that the resultant pulse train contains replicas of the inputpulse with varying amplitude, and are spaced in time by the period τ .Instead of a sinusoidal mask, consider an alternating binary mask; noweach periodic phase unit is set to half zero, and half pi. Applying such amask to the input pulse yields the pulse train shown in Fig. 5.10. Note thatEq. 5.18 correctly relates the 20.9 ps−1 periodicity in the frequency domain(on the SLM mask) to the observed 300 fs pulse spacing. Compare the timedomain distribution of pulses observed in here in Fig. 5.10 with Fig. 5.9.The binary unit leads to a much longer train of pulses; the shape of theperiodic unit is also seen to change the “sharpness” of the fringes appearingin the second harmonic spectrum.5.5 Pulse Characterization TechniquesPulse characterization refers to a method which calculates the amplitude(intensity), phase, or amplitude and phase of an unknown optical field.5.5.1 SpectrometerA spectrometer is a device which takes an unknown field, E(t), and returnsthe intensity spectrum, I(ω) = |E(ω)|2. In the spectrometer used here,light first enters a narrow slit. The slit is used for choosing the resolutionof the spectrometer. Making the slit narrow increases the spectral resolu-tion, but limits the amount of light entering the spectrometer. Basic optics395.5. Pulse Characterization TechniquesIntensity760 780 800 820 8400.00.20.40.60.81.01.2SLM Mask-1.0-0.50.00.51.0PhaseHradLWavelength HnmLEHtL-1000 -500 0 500 1000-1.0-0.50.00.51.0Time DomainTime, t HfsLIntensity390 395 400 405 4100.00.20.40.60.81.0Second HarmonicWavelength HnmLFigure 5.9: The effect of a sinusoidal phase mask, φ(ω) = α sin(τ(ω − ω0)),α = 1 and τ = 300 ps. A train of pulses separated by τ arises.405.5. Pulse Characterization TechniquesIntensity760 780 800 820 8400.00.20.40.60.81.01.2SLM Mask0.00.51.01.52.02.53.0PhaseHradLWavelength HnmLEHtL-3000 -2000 -1000 0 1000 2000 3000-1.0-0.50.00.51.0Time DomainTime, t HfsLIntensity390 395 400 405 4100.00.20.40.60.81.0Second HarmonicWavelength HnmLFigure 5.10: The effect of a binary repeating unit used as an SLM mask. Theperiodicity of the mask is 20.9 ps−1 in the frequency domain, chosen to againyield a 300 fs pulse separation. Note that the second harmonic spectrumreflects the change in repeating unit; all fringe peaks are now “sharp.”415.5. Pulse Characterization Techniquescombined with a diffraction grating map the spectral components onto acharge-coupled device (CCD)-based camera that can detect the intensity ofthe spectral components.The grating is rotated via a computer-controlled motor such that thecurrent spectral window can be centered on an arbitrary wavelength (withina certain optical range). The grating line density determines the width ofthe spectral window observed at a given wavelength.In order to calibrate the spectrometer, an atomic lamp (for example,krypton) with a well-known spectrum is used to illuminate the spectrometer,with a narrow slit width. Four spectral lines at known wavelengths arechosen, and a linear fit is used to determine the relationship between a pixelon the CCD and the wavelength of the light hitting it.5.5.2 AutocorrelationAutocorrelation can give a measure of the pulse length [41]. The PulseScoutAutocorrelator made by Newport is used here.The incoming pulsed laser beam is split with a beamsplitter, and oneof the beams is delayed temporally by τ , by allowing it to travel a longerdistance. The distance is controlled via an internal motor in the PulseScout.Then the two beams are overlapped at a focal point in a nonlinear crystal(Section 4.4). They are focused to maximize the intensity of the nonlinearsignal.The two beams are thus E(t) and E(t − τ). The generated second-harmonic signals for such a field can be found in(E(t) + E(t− τ))2 = E2(t) + E2(t− τ) + 2E(t)E(t− τ). (5.20)Three beams emerge from the interaction. Second-harmonic associated withE2(t) continues to travel in the original direction of E(t), and likewise forthe second-harmonic generated from E(t− τ). The cross term E(t)E(t− τ)travels in a direction given by the vector sum of the wave vectors of the twoincoming pulses, as required by photon momentum conversation [18].Intensity AutocorrelationIf the beams overlap non-collinearly, the cross term travels in a spatiallydistinct direction, and can be isolated onto a detector. This is shown inFig. 5.11. If the detector is a photodiode, it will integrate the ultrashort425.5. Pulse Characterization Techniquesܧሺݐሻ ܧሺݐെ߬ሻ Variable time  d elay , ߬ Lens  SHG Crystal  Beam -  splitter  Detector  Input  beam  Figure 5.11: An experimental realization of optical intensity autocorrelation.The input pulse is split, and one copy is delayed in time. The two pulsesE(t) and E(t− τ) are then brought to a focus non-collinearly in the secondharmonic generation (SHG) crystal. This non-collinear arrangement allowsthe detector to be placed to detect ony the sum-frequency generation (SFG)signal, or the cross term E(t)E(t− τ).pulse over time; the detector thus measures a signal proportional toM(τ) =∫ ∞−∞dt |E(t)E(t− τ)|2=∫ ∞−∞dt I(t)I(t− τ) (5.21)which is exactly the definition [42] of autocorrelation for a signal I.This method is thus known as intensity autocorrelation. The full-widthat half maximum (FWHM) of the autocorrelation envelope, τautocor, com-puted from a Gaussian input pulse that has a FWHM τinput is found accord-ing to:τinput =τautocor√2. (5.22)This numerical factor is different for non-Gaussian pulse envelopes; it isknown as the deconvolution factor (see the measurement in [43] for example).Thus the PulseScout, operating in non-collinear overlap mode, provides asimple way to check the temporal duration of a Gaussian-like input pulse.435.5. Pulse Characterization Techniquesܧሺݐሻ ܧሺݐെ߬ሻ Detector  Variable  ti me  d elay , ߬ Input  beam  SHG Crystal  Lens  Figure 5.12: An experimental realization of optical interferometric auto-correlation. The input pulse is split, and one copy delayed in time. Incontrast to intensity autocorrelation, this time the two pulses are broughtto a collinear overlap in the SHG crystal.Interferometric AutocorrelationIf the two beams are collinear, the photodiode integrates the whole signalappearing in Eq. 5.20; the measurement is thus proportional toM(τ) =∫ ∞−∞dt |(E(t) + E(t− τ))2|2. (5.23)This method is called interferometric autocorrelation.An optical setup which achieves interferometric autocorrelation is shownin Fig. 5.12. Interferometric autocorrelation has been introduced here be-cause a variant of interferometric autocorrelation is used in Chapter 8 (whereit is implemented using only the spectral shaper).5.5.3 FROGFrequency-resolved optical grating (FROG) is a pulse characterizationmethod capable of retrieving the full complex field—amplitude and phase—of an unknown pulse. In a FROG experiment, one uses any method whichcan optically measure the spectrogram, defined as [18, 44]S(ω, τ) =∣∣∣∣∫ ∞−∞E(t)g(t− τ)e−iωtdt∣∣∣∣2. (5.24)The gate field g(t − τ) can be the pulse E(t) itself. If the gate field is nota replica of E(t), the measurement is known as an XFROG (see below).Setting g(t) = E(t), the thing being Fourier transformed in Eq. 5.24 isexactly the cross term of Eq. 5.20.445.5. Pulse Characterization TechniquesThus, to collect what is termed a FROG trace one likewise splits theinput beam and sends one copy to a delay line which allows for the variationof τ . The two beams are overlapped space and in time. Again, as done inthe autocorrelation measurement, the beam represented by E(t)E(t− τ) isisolated spatially. The Fourier transform is done by the spectrometer. Bystepping through offsets of τ one assembles the desired spectrogram.XFROGA variant of FROG called XFROG (for cross-correlation FROG) [45] over-laps two different pulses. The spectrogram isS(ω, τ) =∣∣∣∣∫ ∞−∞Eref(t)Etest(t− τ)e−iωtdt∣∣∣∣2, (5.25)with Eref(t) typically being a Gaussian-like reference pulse, typically discov-ered via FROG. Etest can now be an unknown field. The nonlinear signal isnow a sum-frequency signal.Note that this use of second-harmonic or sum-frequency light is one ofseveral methods available for generating a FROG or XFROG spectrogram;methods not involving nonlinear light generation exist.Field amplitude and phase retrievalOnce a FROG (or XFROG) trace has been collected, it remains a numericproblem to figure out what real field E(t) would give such a trace. TheFROG trace contains enough information that simple pulse shapes can becharacterized. However, theoretically the information present in the trace isinsufficient, such that complex pulse shapes cannot be reliably retrieved [44].In this work, a FROG analysis software package from Femtosoft Technolo-gies, LLC is used, which retrieves the real field (both phase and amplitude)from a FROG trace. The actual algorithms [46, 47] used by this softwareare complex.5.5.4 MIIPSThe method known as the multiphoton intrapulse interference phasescan (MIIPS) [48] retrieves field phase information, using information presentin the spectrum of second harmonic light. This well-known method is usedhere to flatten the phase of laser pulses. The spectral shaper is used tomake the initial field phase measurements, and to subsequently apply the455.5. Pulse Characterization Techniquescompensating (inverse) phase mask. This compensating phase mask staysactive and is added to all other phase shaping performed using the SLM.46Chapter 6Software and Algorithms;The Genetic Search6.1 IntroductionAn extensive set of LabVIEW routines was written to communicate withexternal devices. These LabVIEW routines also do some basic data collec-tion, such as that required for SLM calibration (Section 5.3), FROG (Sec-tion 5.5.3), and MIIPS (Section 5.5.4).A program named QuantumBlackbox was then developed. Originally,this development was motivated by the desire to implement a genetic searchalgorithm in the lab. Due to the inherent difficulty in writing complicatedalgorithms in LabVIEW, and the availability of a comprehensive C++ class-based library that implements a variety of genetic search types and routines,C++ was picked as the language of choice. QuantumBlackbox was subse-quently extended to make it a fully functional data collection and visualiza-tion platform; it communicates directly with the LabVIEW suite of software.Finally, a small C++ module was developed to allow direct communicationbetween a set of Mathematica routines and LabVIEW.A few details related to the design of this software appear in Appendix C.The remainder of this chapter is devoted to the implementation of the geneticsearch.6.2 Genetic SearchIn quantum control, one often doesn’t know the optimal field for directingthe system to a particular target state. There exist a variety of approachesto discovering such an optimal field. In certain cases, the theoretical modelof the system is well known and can be used directly to predict the opticalfield needed to move the system into a target state. In other cases, thetheoretical model may be complex, such that predicting the optimal controlfield is a difficult problem. A mathematical tool known as quantum optimal476.2. Genetic Searchcontrol theory has been successfully applied to a variety of such systemswhere the model is known but complex. Finally, in still other cases it maybe that a theoretical model is poorly known or unknown, in which case onecan simply employ a method which will learn the correct control field via aniterative process (see the early work by Rabitz [49]). Test fields are appliedto the system, and some feedback signal allows the search algorithm beingused to converge on the best control [50].Such a search algorithm may search any parameterization of the optimalcontrol field. For example, the search algorithm may directly search theindividual pixels of the SLM. Or perhaps it is known that an optimal fieldfor controlling the system can be obtained using a sinusoidal phase mask,but the exact frequency, amplitude and phase of the sinusoid function areunknown. One could use the search algorithm to search numeric parametersα, τ , δ and so forth, in the parameterizationφ(ω) = α sin(δ + τ(ω − ω0) + τ(2)(ω − ω0)2 + · · ·), (6.1)where φ is the phase and ω is frequency, afterwards attempt to decipher themeaning of the discovered optimal field [51]. In all cases, the user of thesearch algorithm provides a function mapping the parameterization chosento the pixels on the SLM mask. I call this the mapping function. The cho-sen algorithm operates on the returned feedback signal, called the objectivescore, attempting to minimize or maximize it. The details of how a newparameters are picked each time an objective score is returned is left up tothe chosen algorithm.One type of search algorithm is the genetic search. Genetic algorithms[52] maintain a population of individuals; each individual has its own numericvalues for the set of parameters that a particular search scheme uses. Theset of parameter values that each individual has make up its genome. Atthe start, initial (often random) values for the parameters are chosen. Then,each generation involves these steps:1. The mapping function provided by the user is called for each mem-ber of the population. The mapping function is provided a copy ofthe genome for that individual, from which it calculates the objectivescore.2. The objective scores are used to assign a fitness score to each memberof the population.3. One or more of the best individuals may (if desired) propagate intothe next generation unchanged, called elitism.486.2. Genetic Search4. The genetic algorithm then uses genetic operations to generate indi-viduals with new genomes for the next generation. The most commongenetic operations used here include crossover between two genomes,and point mutation. There is also a possibility for complete replace-ment of certain individuals (e.g. created with random valued param-eters).Objective scores are a key to these genetic operations. For example, aparticular type of selection strategy chooses some of the best individ-uals in a population for crossover. The rationale behind this is that a“re-combination” of these good candidates is likely to produce a betterone.The user allows the algorithm to repeat for many generations—ideally untilthe desired or maximal objective score has been reached, time constraintsmake further searching impractical, or the individuals in the population havebecome identical.One item above may be clarified. A scaling scheme [52] is the methodby which fitness scores are calculated from objective scores. These scale anobjective score in such a way that an individual in the population with agenome that is very unique receives a higher fitness score than an individualwith a comparable objective score but a less-unique genome. It is actuallythe fitness score that is used in selection, not the raw objective score.There are several considerations to be made in using a genetic algorithm,which will govern how good a solution it finds:1. The chosen parameterization. A good parameterization can substan-tially decrease the number of generations (and the population size)needed for the genetic algorithm to find a good solution.2. The frequency of the crossover and mutation, and values optimal fora particular problem, will influence how fast the genetic search findsa solution. The percentage of the population replaced, and the pres-ence of elitism, will also contribute to the rapidity with which a goodsolution is discovered. Good values may be discovered by trial anderror, as well as intuition based on past experience with similar typesof searches.3. The noise in the system. In an experiment, the objective score isprone to experimental errors in measurement, as well as drift over timeand fluctuations. A certain level of noise may necessitate refinementsin the parameterization used; for example, a parameterization that496.2. Genetic Searchsearches SLM pixels directly might need to group pixels together sothat changes are measurable and appear above the noise level. Realenhancements must be accompanied by a change in the objective scorethat is above the level of fluctuations in the score due to the noise, ina given generation.One of the strengths of genetic search is that it is well suited to searchinga space (an N -dimensional hyperspace, defined by the number of parameterspresent) with local extrema without getting stuck at them. It combinesstochastic elements with the concept of fitness, meaning that it can lookfor fit individuals anywhere on the search landscape, without bias. On theother hand, random sampling anywhere on the landscape may not be thebest strategy in a system with a very simple, regular landscape. Anotherstrength of the genetic search is that it routinely compares substantiallydifferent individuals, whose objective scores vary by amounts well abovethe noise level. Thus, it still makes progress in the search, whereas a hill-climbing method might get lost quickly in noise when comparing closelyspaced points in a landscape.50Part IIIThe Experiments51Chapter 7Quantum Coherent Controlof a Two-Photon Transitionin Rubidium7.1 IntroductionOne way to determine how to coherently control a quantum system is to startwith a system that is simple, for which the theory is known. Rubidium issuch a system; it is an atomic system, and electromagnetic field excitations(from a laser pulse) can be treated with time-dependent perturbation theory.Here the transitions of interest will be two-photon transitions, as theyare within the bandwidth of the laser. These transitions, as they appearin rubidium, along with a version of time-dependent perturbation theorywell suited to frequency domain pulse shaping, are presented in Section 7.2.After the two-photon excitation, the rubidium atom will relax back to itsground state via several channels, including a visible fluorescence. This willbe discussed in Section 7.3.Pulse shaping of a broadband pulse is used to substantially enhance ordiminish the two photon absorption in rubidium; this is measured via thefluorescence intensity subsequently observed. The experimental setup usedto do this is given in Section 7.4. Numerical and experimental measurementswill be compared side-by-side in Section 7.5. Finally, Sections 7.5.4 and 7.5.5go beyond the regime of “simple” control for a known system, focusing lesson known theory and more on adaptive search methods, namely geneticsearch (Section 6.2) and single-pixel scan methods.7.2 Two-Photon Transitions in RubidiumThe neutral rubidium atom, studied here, is known in spectroscopic tables[53] as Rb I, and has electronic configuration 1s22s22p63s23p63d104s24p65s1/2.It is the electronic excitations of this single valence electron which will be527.2. Two-Photon Transitions in Rubidiumstudied in this work.A Grotrian diagram for the rubidium atom is shown in Fig. 7.1. It givesthe excitation and decay pathways present in rubidium when illuminatedwith a laser beam with the spectrum shown in Fig. 3.2; the numerical com-putations done for rubidium also use this experimentally measured spectralintensity, with an assumed flat phase. It happens that all of the rubidiumspectral lines within (or near) the bandwidth of this incident optical fieldappear in this Grotrian diagram.The second-order transition amplitude (given by perturbation theory) ofa two-photon transition from initial state |i〉 to final state |n〉 is given byEq. B.26,c(2)n ∼∑mµnmµmi(P∫ ∞−∞dωE(ωni − ω)E(ω)ωmi − ω− ipiE(ωni − ωmi)E(ωmi)).(7.1)This expression is rigorously derived in Appendix B; it uses the electricdipole Hamiltonian, derived in Appendix A. It is equivalent to the perhapsmore familiar time-domain formulation. The allowed transitions are gov-erned by the standard electric dipole selection rules; transitions forbiddenby these rules will have c(2)n = 0. The states |m〉 are resonant intermediatestates, meaning that the frequencies of the two photons involved in the tran-sition are close to the frequencies of the individual transitions |i〉 → |m〉 and|m〉 → |f〉. The summation over m arises when more than one intermediatestate appears between |i〉 and |f〉, such that there a multiple paths from |i〉to |f〉. The electric dipole matrix elements µij are given in Table 7.1.The two terms in the brackets give the part of the amplitude associatedwith the spectral distribution of the electric field, E(ω). (The spectral shaperdoes not shape in polarization, and so the scalar field E may be used, ratherthan ~E.) The first term contains the operator P, which denotes the principalvalue of the integral. It causes the point where ω = ωmi to be excluded,and so this first term gives the off-resonance contribution of E(ω) to thetransition. The second term gives the contribution of the field componentsexactly resonant with the transition, meaning that both photons involvedin the transition match the frequencies of the individual transitions. Thissplitting, as given, into off-resonant and resonant terms removes a divergencewhich would otherwise appear in this expression for c(2)n .There are three possible two-photon transitions from the rubidium groundstate, shown in red in Fig. 7.1. One of these two-photon pathways connects∣∣5s1/2〉and∣∣5d5/2〉, and thus leads to an amplitude c(2)d5/2 in the excited state537.2. Two-Photon Transitions in Rubidium5s1/2 5p1/2 5p3/2 6s1/2 6p1/2 6p3/2 4d5/2 4d3/2 7s1/2 5d5/2 8f7/2 7f7/2 740.82 nm 761.89 nm 775.77 nm 775.94 nm 780.03 nm 8f5/2 792.53 nm 792.55 nm 794.76 nm 827.14 nm 7f5/2 827.17 nm 420.17 nm 421.55 nm 5d3/2 Figure 7.1: A Grotrian diagram for rubidium, showing the most relevanttransitions for this experiment. All wavelength-labelled upward transitionsare either within or reasonably close to the spectral bandwidth of the laser.The highlighted transition pathways are the dominant pathways; other path-ways may be minor contributors. The dotted lines represent spontaneoustransitions. The downward pointing blue arrows give the ultimate fate of themajority of the energy pumped into the two-photon transition: two closelyspaced fluorescence lines. The side channel decay paths leading from the 6pto the 6s and 4d states are drawn as single arrows to simplify the diagram.The energy levels are not drawn exactly to scale. Data taken from varioussources [53–57].547.2. Two-Photon Transitions in RubidiumTransition, i→ j Dipole Element, µij (a.u.)Excitation Pathways5s1/2 → 5p1/2 4.2215s1/2 → 5p3/2 5.9565p1/2 → 5d3/2 1.6165p3/2 → 5d3/2 0.7875p3/2 → 5d5/2 2.334Relaxation Pathways5d3/2 → 6p1/2 18.1065d3/2 → 6p3/2 8.1605d5/2 → 6p3/2 24.4916p1/2 → 6s1/2 9.6846p3/2 → 6s1/2 13.5926p1/2 → 4d3/2 4.7176p3/2 → 4d3/2 2.0556p3/2 → 4d5/2 6.1846p1/2 → 5s1/2 0.3336p3/2 → 5s1/2 0.541Table 7.1: Select transition dipole matrix elements relevant for the calcula-tion of 420 nm fluorescence in rubidium after broadband 2-photon excitationwith a pulse centered near 800 nm. These values come from published ta-bles, computed using a relativistic all-order method [58]. A dipole elementof 1 a.u. (atomic units) is equivalent to 8.478× 10−30 C m in SI units.given by Eq. 7.1 with no summation (a single term in the sum). The othertwo two-photon pathways connect∣∣5s1/2〉and∣∣5d3/2〉, via two different res-onant intermediate states, as shown; the transition amplitude c(2)d3/2 is givenby Eq. 7.1 with the summation being over two pathways.The final probability of transition to a level is given by the squaredmodulus of the transition amplitude,Pd5/2 =∣∣∣c(2)d5/2∣∣∣2and Pd3/2 =∣∣∣c(2)d3/2∣∣∣2. (7.2)No single-photon transitions (or contributions from higher-order perturba-tion theory) are necessary.557.3. Modeling the Fluorescence Intensity7.3 Modeling the Fluorescence IntensityTo understand how the transition probabilities of Eq. 7.2 lead to observ-able fluorescence signals, one considers all the possible decay channels (viaspontaneous emission). These levels∣∣5d5/2〉and∣∣5d3/2〉decay to∣∣6p3/2〉and∣∣6p1/2〉as allowed by dipole selection rules, shown in Fig. 7.1. Thepopulation in these 6p levels then decays via three possible routes. Decay tothe∣∣6s1/2〉level is possible, as is decay to the 4d levels. It is assumed thatany 6p population decaying to either of these side channels is lost and doesnot contribute directly to the observable fluorescence (as this side channel-related fluorescence light is very far from 420 nm). The possibility thatthese side channels feed the 5p intermediate levels, thus facilitating subse-quent single-photon transitions back to 5d is a higher-order process that hasnot been considered. The third possible decay channel from 6p leads theatom back to the ground state and is the channel of most interest here, asit produces a pair of fluorescence lines near 420 nm.The fact that the fluorescence signal arises from such a network of decaychannels means that there is no a priori guarantee that population appearingin a certain ratio Pd5/2 to Pd3/2 after two-photon excitation will appear inthat same ratio in the observable fluorescence level M , calculated asM = wd3/2Pd3/2 + wd5/2Pd5/2 . (7.3)As a fictitious example, consider the two 5d levels equally populated aftera pulse. It is conceivable that although the two levels are initially equallypopulated at 50% each, 25% of one level and 15% of the other actuallyreach the observable fluorescence. This would be because one of the 6plevels decayed more rapidly to the 6s or 4d side channel levels than theother. The weights wi must be calculated.In order to determine these weights, one first writes down the coupled567.3. Modeling the Fluorescence Intensitysystem of rate equations describing this system. This system isN˙5d5/2 =−A5d5/2→6p3/2N5d5/2 (7.4a)N˙5d3/2 =−A5d3/2→6p3/2N5d3/2 −A5d3/2→6p1/2N5d3/2 (7.4b)N˙6p3/2 =A5d5/2→6p3/2N5d5/2 +A5d3/2→6p3/2N5d3/2 −A6p3/2→6s1/2N6p3/2−A6p3/2→4d5/2N6p3/2 −A6p3/2→4d3/2N6p3/2 −A6p3/2→5s1/2N6p3/2(7.4c)N˙6p1/2 =A5d3/2→6p1/2N5d3/2 −A6p1/2→6s1/2N6p1/2 −A6p1/2→4d3/2N6p1/2−A6p1/2→5s1/2N6p1/2 (7.4d)N˙5s1/2 =A6p3/2→5s1/2N6p3/2 +A6p1/2→5s1/2N6p1/2 . (7.4e)N gives the population of a level, and N˙ gives the time derivative or timerate of change of a level. The Einstein coefficients Ai→j give the rate ofspontaneous transitions between levels |i〉 and |j〉, calculated according to[59]Aij =2ω3ij30hc3|µij|2, (7.5)with the dipole matrix elements given in Table 7.1. The frequencies ωijcorrespond to the frequency of each atomic transition from state |i〉 to |j〉.The system is then solved numerically; a solution obtained using initialvalues P5d3/2 = P5d5/2 = 1 is given in Fig. 7.2. Only a fraction of excitedinitial population makes it back to the ground state through these 420 nmfluorescence channels. The approximate lifetimes observed in this calcula-tion are consistent with known values [58].If one then sets P5d3/2 = 1 and P5d5/2 = 0, one obtains wd3/2 = P5s1/2 =0.1995. If one instead sets 5d3/2 = 0 and 5d5/2 = 1, wd5/2 = P5s1/2 = 0.2359.This then answers the question posed above as to whether or not the two 5dlevels’ populations are equally visible in the final observable fluorescence—the initial populations of 5d appear in the observed fluorescence in roughlythe same ratio. The same ratio is observed for any value C comparing5d3/2 = C and 5d5/2 = 0 with 5d3/2 = 0 and 5d5/2 = C.The above considerations motivate calculating the fluorescence intensityM as simplyM ∼ Pd3/2 + Pd5/2 , (7.6)as wd3/2 ≈ wd5/2 . (In Fig. 7.4 one sees that both fluorescence lines aretransmitted equally, and thus no additional weighting to account for filtertransmission is needed.) The values P are given by Eq. 7.2.577.4. Experimental ApparatusPopulation0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.05d LevelsN5 d5 2N5 d3 2Time HΜsLPopulation0.0 0.2 0.4 0.6 0.8 1.00.000.050.100.156p LevelsN6 p3 2N6 p1 2Time HΜsLPopulation0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.45sN5 s1 2Time HΜsLFigure 7.2: The relaxation process in rubidium, showing how the levelsinitially excited by the pulse decay towards the visible fluorescence. Thesegraphs give the solution of Eqs. 7.4. One observes the loss to the 6s and 4dlevels by comparing the initial populations to the final, noting that N5d3/2 +N5d5/2  N5s1/2 .7.4 Experimental ApparatusThere are two naturally abundant isotopes of rubidium [60, 61], 8537Rb and8737Rb (72.17% and 27.83% naturally abundant, respectively). Both isotopesrespond identically to the type of excitations used in this work. Both iso-topes have a melting point of 39.31 ◦C. The solid (at room temperature)rubidium metal is inside a sealed glass cell, having flat windows at bothsides. The cell is held in a metal block, equipped with a heating elementand thermocouples. An external temperature controller connected to a ther-mocouple mounted in the block switches the heating element on and off,maintaining the cell at 200 ◦C. The rubidium gas in the cell interacts withthe incoming laser beam. This apparatus is shown schematically in Fig. 7.3.Fluorescence is a omni-directional incoherent field associated with spon-taneous emission. The interaction with the incoming laser field results in apair of fluorescence lines near 420 nm. In addition to fluorescence, superflu-orescence can also be observed. This is a directional, coherent field, and atype of stimulated emission: the superfluorescent beam comes out of the cellcollinear with the laser source. To avoid collecting superfluorescence light[57] the fluorescence signal is observed via a hole cut on the side of the metalblock, perpendicular to the laser beam.An interference filter with a transmission profile shown in Fig. 7.4 helps587.4. Experimental ApparatusInput beam (from shaper) Focusing lens/ mirror Heater block Rb cell Imaging optics PMT 420 nm IF Superfluorescence light+unabsorbed input Fluorescence light Figure 7.3: The experimental apparatus used to study the quantum coherentcontrol of two photon absorption in rubidium. The shaped pulse is focusedinto a rubidium vapor cell. The two photon absoption subsequently decaysvia several routes, including a visible fluorescence at 420 nm. This signal isimaged onto a photomultiplier tube (PMT), after being filtered to containonly the light near 420 nm with an interference filter (IF). This interferencefilter blocks background light. Unused input beam mixed with superfluores-cence light exits the tube collinear with the input, and is blocked.%Transmission410 415 420 425 430010203040Wavelength HnmLFigure 7.4: The transmission curve for the Thorlabs FB420-10 bandpassfilter [62], using data provided by Thorlabs. The filter has a transmissionwindow centered at 420± 2 nm with a FWHM of 10± 2 nm.597.5. Coherent Control of Two-Photon Absorption in Rubidiumto ensure that only the desired fluorescence lines are observed. This filteredlight is imaged onto a photomultiplier tube (PMT). The signal from thePMT is in turn fed into a Boxcar Integrator from Standford Research Sys-tems. This instrument allows a successive averaging of a gated temporalregion of the PMT signal; the PMT response of 30 to 300 laser pulses isaveraged. This averaged signal is collected via a serial interface, and trans-mitted to QuantumBlackbox via a LabVIEW component. Although theboiling point of rubidium is 688 ◦C, at 200 ◦C there exists enough rubidiumvapor (the solid phase vapor pressure [61] is 67.1 mTorr) in the cell suchthat the fluorescence signal collected here is readily observed, and comes inwell above the noise level.7.5 Coherent Control of Two-Photon Absorptionin RubidiumThe following subsections focus on understanding and demonstrating thecoherent control of two-photon absorption in rubidium. The fluorescenceintensity predicted by Eq. 7.6 can be enhanced because a transform lim-ited (flat phase) Gaussian input pulse is not optimal for two-photon exci-tation [55]. In particular, the non-resonant contribution from a particulartwo-photon pathway can be enhanced [55], the relative phase between theresonant and non-resonant component in a pathway can be tuned [63], and fi-nally, the relative phase between different interfering pathways can be tunedwhen a sufficiently broadband laser pulse is used.These three avenues to coherent control in rubidium are illustrated inthe next three subsections. To simplfy this discussion, in addition to thedefinition of c(2)n given by Eq. 7.1, letc(2)n,PP ≡ µnmµmi P∫ ∞−∞dωE(ωni − ω)E(ω)ωmi − ω(7.7)and letc(2)n,res ≡ µnmµmi ipiE(ωni − ωmi)E(ωmi). (7.8)7.5.1 Nonresonant Contributionsc(2)n,PP gives the contribution of the non-resonant part of the field to the two-photon transition, i.e. when ω 6= ωmi. The structure of the denominatorin this principal value integrand, (ωmi − ω)−1, is visualized in Fig. 7.5(a).Two important facts follow from the structure of this denominator. First,607.5. Coherent Control of Two-Photon Absorption in RubidiumPartialIntegrand740 760 780 800 820 840-1.0-0.50.00.51.0Wavelength HnmL(a) 1ωmi−ω .PartialIntegrand740 760 780 800 820 840-1.0-0.50.00.51.0Wavelength HnmL(b) E(ω)ωmi−ω .Integrand740 760 780 800 820 840-1.0-0.50.00.51.0Wavelength HnmL(c) E(ωni−ω)E(ω)ωmi−ω .Figure 7.5: A visual comparison of the behavior of the integrand in Eq. 7.1describing the off-resonant part of the two-photon transition in rubidiumaccording to second-order time dependent perturbation theory. Note that asthe spectral functions E(ω) are added, they have the effect of modulating theintegrand such that the frequencies close to the resonance become relativelymore important. Here, the 780 nm resonance is used for ωmi.the structure of the field E(ω) around a resonance is weighted very stronglyin its importance to the transition amplitude, relative to far off-resonantcomponents. Second, the denominator introduces a sign change in the func-tion around the intermediate resonance. Fig. 7.5 shows how the differentcomponents of the integrand come together to realize this behavior.As it stands, this roughly anti-symmetric structure of the integrand leadsto a low contribution from the integral to the total two-photon transitionamplitude. One may enhance this off-resonant contribution to the transitionin two ways.Phase ShapingPhase shaping can be used to impose a negative sign in an appropriate regionof the integrand [55]. The integrand shown in Fig. 7.5(c) then becomes en-tirely positive (nearly symmetric, rather than antisymmetric); the presenceof the broadband field is now a substantial benefit to the transition.The resonances near the two pathways at 775.8 nm and 780 nm and775.9 nm and 780 nm are closely spaced (approximately 4 nm apart). Aconvenient way to maximize the integrand in Fig. 7.5(c) is to place a pi/2617.5. Coherent Control of Two-Photon Absorption in RubidiumIntensity760 780 800 820 8400.00.20.40.60.81.01.2SLM Mask0.00.51.01.5PhaseHradLWavelength HnmLTransitionProbabilityRatio770 775 780 785012345TheoryCenter Wavelength of 4 nm Π  2 Window HnmLTransitionProbabilityRatio770 775 780 785012345ExperimentCenter Wavelength of 4 nm Π  2 Window HnmLFigure 7.6: pi/2 window phase scans. A 4 nm wide pi/2 phase window,shown in the upper plot, is scanned across the spectrum (the spectral regionof the scans is shown on the x-axis of the lower plots). Several pi/2 windowscentered at different positions are shown on the upper plot; shades of graysuggest the motion of the phase window during the scan. The bottom plotsshow the ratio of enhancement vs. the center wavelength of the pi/2 win-dow. This ratio, shown in the plots, is the ratio of the fluorescence intensitycalculated with a flat phase pulse to that calculated with a phase window ap-plied. Theoretically, the maximal enhancement appears when this windowis exactly centered between the two resonances, as expected. Experimen-tally, uncompensated phase distortions in the input pulse and interferencewith the pathway at 795 nm mean a departure from the ideal result (seeSection 7.5.5). In both cases, however, a 5 fold enhancement in two-photonabsorption is measured.627.5. Coherent Control of Two-Photon Absorption in Rubidiumwindow (which is approximately 4 nm wide) between the resonances near776 and 780 nm (see Fig. 7.6). Then, the factors E(ωni−ω) and E(ω) bothpick up an extra pi/2 phase in the 776 nm to 780 nm region; as a result,the integrand in this region is multiplied by -1, and the whole integrandbecomes positive. Using the full bandwidth of the input pulse, scanningthis pi/2 window through this region yields the enhancement ratios shownin Fig. 7.6. The numerical result computed here matches closely with apublished result [55]; the optimal enhancement is less than the value of7 predicted in that work due to interference with the transition pathwayappearing at 795 nm, due to the present use of a broader bandwidth pulse.By removing the spectral components near the 795 resonance, the numericalmodel used here reproduces this 7 fold enhancement.The off-resonant contribution associated with the two-photon pathwayinvolving resonances at 762 and 795 nm may be enhanced with a pi windowapplied to, e.g. the right side of, the resonance at 795 nm. This workswell here because the two resonances are now far apart. The numericallycomputed result of scanning this window appears in Fig. 7.8, along with theSLM mask used.Intensity ShapingA second method for enhancing the off-resonant part of the two-photontransition is to simply remove the portion of the spectral content (see theSLM mask in Fig. 7.7) which picks up a negative sign relative to the otherside of the resonance. In other words, enhancement is obtained by removingone half of the anti-symmetric shape shown in Fig. 7.5. The maximumenhancement available is less than that possible using phase shaping, but isstill substantial. The scans shown in Fig. 7.7 demonstrate this enhancementby using a intensity window centered between the two resonant wavelengthsof the transition. The maximum enhancement ratio obtained occurs whenthis window exactly removes the spectral components in the input whichcause the integrand in c(2)n,PP to pick up a negative sign across the positionof the intermediate resonance.One sees that the spectral components near 762 nm and 795 nm have alsobeen removed from the broadband pulse in this intensity scan, as they areset to zero transmission using the SLM. This ensures no interference with theother pathway at 795 nm. Instead of removing spectral components aroundthe pathways at 776 and 780 nm, one may alternatively scan this intensityedge near 761 and 795 nm, as shown in Fig. 7.8. One sees that all threeresonant pathways in rubidium behave similarly, and may be coherently637.5. Coherent Control of Two-Photon Absorption in RubidiumIntensity760 780 800 820 8400.00.20.40.60.81.01.2SLM Mask-1.0-0.50.00.51.0PhaseHradLWavelength HnmLTransitionProbabilityRatio2 4 6 8 100.00.51.01.52.0TheoryIntensity Window Width HnmLTransitionProbabilityRatio2 3 4 5 6 70.00.51.01.52.0ExperimentIntensity Window Width HnmLFigure 7.7: Intensity edge scans. The SLM mask uses shades of blue to showhow a transmission window, centered between the two resonant wavelengths,is varied in width during the course of the scan. Two-photon absorption ismaximized when all cancellation in the integrand of c(2)n,PP is eliminated. Itis seen that two-photon absorption can be enhanced to just over twice thelevel observed when transform-limited pulses are used.controlled in the same manner.7.5.2 Nonresonant and Resonant Term InterferenceThe resonant contribution to the two-photon transition amplitude is givenby c(2)n,res. This equation predicts that these resonant field components are inquadrature to those in the non-resonant contribution given by c(2)n,PP—theyinclude this additional phase shift via the factor of i. The existence of thisresonant term means that another possible avenue for enhancement of thetwo-photon transition amplitude is to manipulate the phases of resonantspectral components such that they become in-phase with the off-resonantterms.647.5. Coherent Control of Two-Photon Absorption in RubidiumIntensity760 780 800 820 8400.00.20.40.60.81.01.2SLM Mask for Π Window Phase Scan0.00.51.01.52.02.53.0PhaseHradLWavelength HnmLIntensity760 780 800 820 8400.00.20.40.60.81.01.2SLM Mask for Intensity Window Scan-1.0-0.50.00.51.0PhaseHradLWavelength HnmLTransitionProbabilityRatio796 798 800 80202468Theory - Π Window Phase ScanCenter Wavelength of 8 nm Π Window HnmLTransitionProbabilityRatio30 35 400.00.51.01.52.02.5Theory - Intensity Window ScanIntensity Window Width HnmLFigure 7.8: pi and intensity edge scans at 795 nm. The top two plots showthe SLM masks used for the corresponding computed scans in the bottomplots. Shades of blue and gray are again used to convey changes to the masksduring the course of each scan. Compare these results of applying pi shapingand intensity edges to the pathway involving the 762 and 795 nm resonancesto the scans involving the pathways at 776 and 780 nm. Both measurementshave been made after removing the spectral components around 776 and780 nm, so those pathways do not contribute to the measured transitionamplitude.657.5. Coherent Control of Two-Photon Absorption in RubidiumOne complication to this approach is that SLM pixel elements are rela-tively wide and so any attempt at controlling the residue term phase will alsoinvariably change the off-resonant field contributions. These off-resonantcomponents nearest the residue are strongly weighted (see again Fig. 7.5and the caption), so that attempting to change a pixel at the spot of theresonance produces substantial modulation. There also exists a chance thatthe resonant spectral component simply overlaps with the small gaps be-tween SLM pixels.7.5.3 Two-Pathway InterferenceOne of the two-photon pathways associated with the resonances near 776and 780 nm is summed with the pathway associated with resonances at 762and 795 nm, according to the prescription for c(2)5d3/2 given by Eq. 7.1. Bycareful manipulation, one thus seeks to not only maximize the magnitude ofthe two pathways leading to∣∣5d3/2〉, but also the relative phase.7.5.4 Putting it All Together—TheoryIt is possible to construct genetic search parameterization (Section 6.2) ofan SLM mask which combines all three of the above methods of achievingcoherent control in rubidium. The mask contains six distinct features. Threeof these are pixel windows of variable widths (not all varying in the sameranges) which search near the location of the resonances, each contributingto a residue term c(2)n,res. The widths of these windows (in pixels) are searchedby the genetic scan. One of the two resonances near 776 nm is included, andboth of the resonances at 762 and 795 nm are included. This inclusion ofboth resonances at 762 and 795 nm is redundant as far as the contribution tothe residue term for the pathway is concerned, but it was found to be useful,providing an extra degree of freedom—the resonant windows shape a smallportion of the off resonant components, too. The other three pixel windowsare designed to provide enhancement for the non-resonant contribution tothe three pathways, again having variable widths which are genetic-searched.All six windows shape in phase; the height of each phase window is searchedby the genetic scan. All search parameters are restricted to take valuesonly within a certain set of specified ranges; for the phases, this is simplybetween zero and 2pi. For the widths of the windows, reasonable limits arechosen such that the windows cannot overlap or come to close to neighboringtransition pathways. Thus resulting SLM mask will look similar to that inFig. 7.9(b).667.5. Coherent Control of Two-Photon Absorption in RubidiumTransitionProbabilityRatio0 200 400 600 8000246810Genetic Search Measurement(a) The variation in the population slowlydisappears as the search runs. The phasemask which produces the maximum two-photon absorption in rubidium is found atthe 738th measurement and gives an en-hancement ratio of 9.86.PhaseHradL760 770 780 790 800-1.0-0.50.00.51.01.52.02.5Wavelength HnmL(b) The non-zero part of the optimal phasemask obtained. One sees the various fea-tures described in the text.Figure 7.9: Evolution of the genetic search, seeking to maximize broadbandexcitation of a two-photon transition in rubidum. Each measurement madeby the search algorithm is plotted, in order. Also shown is the maximallyenhancing SLM mask obtained by genetic search. The transition probabilityratio shown here is computed as it was before; compare both the enhance-ment ratios and the optimal SLM mask obtained here with those shown inFigs. 7.6, 7.7 and 7.8.The evolution of the genetic search, applied to the numerical model, isdepicted in Fig. 7.9(a). The initial population is quite diverse; this canbe seen in, e.g., the first 100 iterations. Although the individual genera-tions are not marked, one sees a steady trend of increasing maximum ratioand diminishing diversity as individuals are tested, genetic operations areperformed and generations pass. At the end, the search has essentially con-verged, having found an SLM mask giving an almost ten times enhancementof the two-photon transition probability calculated for rubidium. This op-timal SLM phase mask is shown in Fig. 7.9(b). This solution is not unique,but it is typical. Other executions of this genetic search obtain masks thatprovide comparable enhancements. The features observed in this mask aretypical, as a center window at a height near pi/2 is present, and two sidewindows are present whose heights sum to pi (approximately).A more detailed understanding of the enhancement process can be gained677.5. Coherent Control of Two-Photon Absorption in RubidiumComplexModulus0 200 400 600 800051015202530Genetic Search Measurement(a) |c(2)m→n,PP|.PhaseHradL0 200 400 600 800-3-2-10123Genetic Search Measurement(b) Arg c(2)m→n,PP.ComplexModulus0 200 400 600 8000246810Genetic Search Measurement(c) |c(2)m→n,res|.PhaseHradL0 200 400 600 800-3-2-10123Genetic Search Measurement(d) Arg c(2)m→n,res.ComplexModulus0 200 400 600 80005101520253035Genetic Search Measurement(e) |c(2)m→n|.PhaseHradL0 200 400 600 800-3-2-10123Genetic Search Measurement(f) Arg c(2)m→n.Figure 7.10: The evolution of the components of the transition amplitudethroughout the genetic search appearing in Fig. 7.9(a). The three linesshown in each plot represent the contribution of a particular pathway to thequantity shown below each figure. The blue line shows the contributionfrom the m → n = 5p3/2 → 5d3/2 pathway, purple gives 5p1/2 → 5d3/2,and olive gives 5p3/2 → 5d5/2.687.5. Coherent Control of Two-Photon Absorption in Rubidiumby inspection of Fig. 7.10. These six plots show how the three pathwayswhich contribute to the total fluorescence intensity M (Eq. 7.6) evolve asthe genetic search runs. Extending the notation c(2)n to c(2)m→n, where mindicates the intermediate state in the two-photon transition, M is nowwritten asM ∼∣∣∣c(2)5p3/2→5d3/2+ c(2)5p1/2→5d3/2∣∣∣2+∣∣∣c(2)5p3/2→5d5/2∣∣∣2. (7.9)(The sum over m originally present in the expression for c(2)n has been writtenout explicitly here; this is what gives rise to the summation of the twoterms c(2)5p3/2→5d3/2 and c(2)5p1/2→5d3/2. This expression is still equivalent to theoriginal expression for c(2)n , being only a change of notation.)Fig. 7.10(a) shows the that the contribution to |c(2)m→5d3/2,PP| from the twointerfering pathways leading to∣∣5d3/2〉is nearly equal. Thus, they can inter-fere strongly in this model. The other pathway to∣∣5d5/2〉is somewhat higherin magnitude, due to the particular electric dipole transition matrix elementsinvolved. Fig. 7.10(c) shows, as it must, that the magnitude of the eachresidue term is constant. The importance of interference to Brumer-Shapiro(Section 1.3.2) quantum control in rubidium is emphasized in Figs. 7.10(b)and 7.10(d). As the search evolves, the phase difference between interferingterms is minimized. One sees in the plot that Arg c(2)5p3/2→5d5/2,res does notvary, as the frequencies associated with this term have not been included inany window appearing on the SLM mask. Figs. 7.10(e) and 7.10(f) summa-rize the total contribution coming from each pathway; they are then insertedinto Eq. 7.9 and used to calculate Fig. 7.9(a).This same genetic search parameterization can also be used to minimizethe two-photon transition probability ratio. The genetic search is instructedto minimize the objective score; doing this, one obtains a two-photon tran-sition probability ratio of about 0.1. Plots similar to those given in Fig. 7.10reveal that this minimization is achieved by minimizing the values |c(2)m→n,PP|and by finding phases that lead to destructive interference.7.5.5 Putting it All Together—ExperimentThe agreement shown in Figs. 7.6 and 7.7 of theory with experiment is notgreat, even though the enhancement, looking at it only as a numerical value,is good.The broadband laser pulse used for rubidium excitation is compensatedusing MIIPS and has also been checked using both FROG and temporal697.5. Coherent Control of Two-Photon Absorption in Rubidiumautocorrelation. These measures all suggest that the pulse is close to beingtransform-limited. Despite this, Figs. 7.6 and 7.7 suggest that transform-limited pulse is not really transform limited. It may be that long-termdrift of the laser phase (e.g. on the order of an hour) prevents proper pulsecompensation or introduces distortions after-the-fact. The spectral intensitynear the resonance at 762 nm is relatively weak, and this causes difficultyfor phase compensation methods such as MIIPs. The shaper is temperaturesensitive, and pulse chirp has been observed to drift appreciably during thecourse of operation.Short-time pulse phase instability is also a known problem not com-pletely solved during the time allotted for this thesis work. Particularly,it can be observed that the second-harmonic intensity spectrum, which isphase sensitive, is unstable on a very short timescale (seconds). Air cur-rents are appreciable in the lab room where these experiments are done;these created variations in the optical path length, critical in the region ofthe shaper where spectral components are dispersed in space. Attemptsmade to enclose both the entire optical table, as well as additionally enclosethe shaper and SLM, improved stability, but did not provide a total remedyto the problem.Despite this, the presence of uncompensated phase features (those notvarying on a short timescale on the order of seconds) provides a uniqueopportunity. This is the ability to explore a method for obtaining quantumcoherent control in an atomic system where unknown phase irregularitiesare present (see also Chapter 8). This method makes use of the fact thatspectral components close to rubidium resonances are strongly weighted intheir importance to the two-photon transition probability relative to otherspectral components. Implemented using the JavaScript functionality inQuantumBlackbox, this method involves the following steps, depicted inFig. 7.11:1. A resonance to scan around is selected, e.g. that at 775.94 nm. Astarting pixel group size ng is selected, typically ng = 2.2. ng SLM pixels at the position of the resonance are scanned in phaseon the interval [0, 2pi), measuring the ratio every pi/4 radians and thusproducing a set of points like those in Fig. 7.12(c). The phase yield-ing the highest ratio is selected and two points pi/8 to the left andright of the maximum are also measured in just in case the additionalphase granularity contributes a significantly higher enhancement. (Nu-merical tests using a transform-limited pulse suggest that pi/4 phase707.5. Coherent Control of Two-Photon Absorption in RubidiumIntensity774 775 776 7770.00.20.40.60.81.01.2-3-2-10123PhaseHradLWavelength HnmLIntensity774 775 776 7770.00.20.40.60.81.01.2-3-2-10123PhaseHradLWavelength HnmLIntensity774 775 776 7770.00.20.40.60.81.01.2-3-2-10123PhaseHradLWavelength HnmLFigure 7.11: Illustration of a method by which narrow groups of pixels arealternately scanned to the left and right of the rubidium resonance of inter-est, in order to find substantial enhancements to the transition amplitudewithout the complexity or slow convergence observed in a genetic search.The light green vertical lines mark single pixels on the SLM. Starting at thetop, a group of pixels to the left of the resonance at 776 nm are scanned inphase, as shown by the shades of gray (not all values scanned are shown).The optimal value (e.g. producing the highest measured value of fluorescenceintensity) is kept and applied to the mask. In the middle plot, a group ofpixels to the right of the resonance is again scanned and the optimal phasevalue recorded. In the bottom plot, the alternating pattern continues, butone sees that the number of pixels involved in the scanned group has in-creased. As this method relies on the heavy weighting of E(ω) close to eachresonance, shown in Fig. 7.5, pixel group sizes must be increased as the dis-tance from a resonance increases in order to produce the same measurableresponse in fluorescence.717.5. Coherent Control of Two-Photon Absorption in Rubidiumgranularity is a sufficient basis for excellent two-photon enhancementin rubidium.)3. The above scans are repeated, alternating scanning ng pixels to theleft or to the right of pixels already scanned. If the enhancementgained is small, ng is increased. In this way, one spends the most timescanning near the resonances, where the weight of the contribution tothe two-photon enhancement pathway is the strongest.4. These ratios are plotted to produce a plot like that of Fig. 7.12(a).One sees that as the distance from the center position (of the reso-nance) is increased, the increase in ratio generally plateaus. Driftsup and down in the ratio are typically due to noise—transient driftson timescales on the order of seconds arise from the pulse instabilitypreviously discussed.5. The maximally enhancing SLM mask is thus obtained.This method, like the genetic scan of the numerical model performed inSection 7.5.4, also effectively combines the three methods (the first threesubsections of this section) by which coherent control of two-photon absorp-tion in rubidium may be achieved.Fig. 7.12 shows the result of applying this method to the two transitionpathways involving resonances at 776 and 780 nm. Spectral componentsfurther than about 8 nm away from these two resonances are removed, sothat this scan focuses on only these pathways. Likewise, Fig. 7.13 scans thetransition pathway involving the 762 and 795 nm resonances, removing theintensity around 776 and 780 nm. These scans are, in principle, capable ofenhancing both the nonresonant contribution to the transition probability(Section 7.5.1) and adjusting the relative phase between resonant and non-resonant field components (Section 7.5.2). Due to phase instability acrossthe pulse, scans employing the full spectral content (Section 7.5.3) were notattempted here.These scans have been kept simple. One may expect that many morepixels, more allowed phase values and averaging may enhance this outcome.This scan method may find utility when attempting to do a quantum coher-ent control through temporally scattering media.7.5.6 Other Experimental InvestigationsSeveral other experiments involving two-photon absorption in rubidium wereperformed. Each experiment used the genetic search functionality in Quan-727.5. Coherent Control of Two-Photon Absorption in RubidiumTransitionProbabilityRatio772 774 776 778 7801.21.41.61.82.02.2Wavelength of Scanned Group HnmL(a) The ratio obtained as the correct phasevalues are picked on the SLM mask, start-ing at the center and alternating left andright. This process has been illustrated inFig. 7.11.PhaseHradL770 772 774 776 778 780 782 784-3-2-1012Wavelength HnmL(b) The optimal SLM mask obtained atthe end of the scan.TransitionProbabilityRatio-3 -2 -1 0 1 21.952.002.052.102.152.202.25Phase HradL(c) A sample of the transition probabilityratio vs. phase data measured as a singlepixel group is scanned.Figure 7.12: The resonance-centered pixel-scan method described in thetext, applied to the two transition pathways involving resonances at 776and 780 nm. Spectral components further than 8 nm from these two reso-nances are removed (the pathway involving resonances at 762 and 795 nmis removed).737.5. Coherent Control of Two-Photon Absorption in RubidiumTransitionProbabilityRatio793.0 793.5 794.0 794.5 795.0 795.5 796.01.11.21.31.41.5Wavelength of Scanned Group HnmL(a) Ratios obtained as correct phase valuesare selected for the resonance @ 795 nm.PhaseHradL792 793 794 795 796 797 798-1.0-0.50.00.51.0Wavelength HnmL(b) The optimal SLM mask for the scan @795 nm.TransitionProbabilityRatio761.0 761.2 761.4 761.6 761.8 762.0 762.21.551.601.651.701.75Wavelength of Scanned Group HnmL(c) Ratios obtained as correct phase valuesare selected for the resonance @ 762 nm.PhaseHradL760 770 780 790 800-2.0-1.5-1.0-0.50.00.51.0Wavelength HnmL(d) The optimal SLM mask for the scan @762 nm.Figure 7.13: The resonance-centered pixel-scan method described in thetext, applied to the two transition pathways involving resonances at 762and 795 nm. Spectral components around the resonances at 776 and 780nm are removed). The scans in c. and d. build upon those in a. and b.,and thus the final phase mask includes both regions. The ratios scanned inc. do not plateau in the typical manner, but no additional enhancement isfound (in this case) by going beyond the wavelengths shown.747.5. Coherent Control of Two-Photon Absorption in RubidiumtumBlackbox with a particular SLM mask parameterization. One such ex-periment parameterized the SLM mask by dividing it up into small groupsof pixels, each of which could take the value 0 or pi/2 (motivated by thesuccess of the pi/2 in Fig. 7.6). This yielded a several-fold enhancementin the transition probability ratio; the optimal SLM mask found by thissearch features pi/2 windows between 776 and 780 nm, as well as aroundthe resonances involved in the two-photon pathway beginning at 795 nm.Using a simple sinusoidal mask (see the discussion of Eq. 6.1 in Section 6.2),other experiments were done where both enhancements and reductions inthe rubidium transition probability ratio were obtained.The observed reductions in transition probability were confirmed usingthe numerical model. For example, a genetic search like that presentedin Section 7.5.4 was used to minimize two-photon absorption (fluorescenceintensity). In so doing, the numerical search finds ways to maximize theamount of destructive interference between the non-resonant and resonantterms contributing to each pathway, and also creates destructive interfer-ence between pathways. These reductions in two-photon absorption couldbe made very strong, such that almost no fluorescence was observed in eitherthe numerical or experimental results; no shaping in intensity was required.“Traditional” electromagnetically induced transparency involves destructiveinterference between coupled levels [64], and can be realized using monochro-matic laser light. The reduction in absorption observed here is also a typeof “induced transparency,” but the lack of absorption has instead been re-alized by shaping the phase relationship between spectral components in abroadband laser source. The reduction of two-photon absorption observedhere happens in a system where the two-photon absorption occurs with thehelp of resonant intermediate states; the use of phase shaping to reduce pho-ton two-photon absorption in a non-resonant two-photon absorption (in asystem where a two-photon transition occurs via only virtual intermediatestates) has been studied elsewhere and requires slightly different theory [65].Along with quantitative measurements of the fluorescence intensity M ,direct observation of the visible superfluorescence light at 420 nm comingfrom the cell (as shown in Fig. 7.3) provided a simple qualitative affirmationof the two-photon transition enhancement and reduction effects reportedhere.757.6. Conclusion7.6 ConclusionIn this chapter, particularly in Section 7.5, it is seen that the amplitude oftwo-photon absorption in rubidium, in the presence of a broadband field,is due to interplay between three distinct mechanisms. Two contributionsto this amplitude are off-resonant part of the field and the contributionof the exactly on-resonant parts; interference between multiple two-photonabsorptions leading to the same final level was also seen to contribute.It is then found that quantum coherent control of these two-photonabsorption pathways in rubidium could be understood in terms of thesethree contributions to the two-photon absorption amplitude. The interplaybetween these contributions has been used to both substantially enhanceor substantially diminish the total two-photon transition probability, whencompared excitation via a transform-limited pulse (Gaussian envelope withflat phase). Further, this enhancement or suppression is achieved via onlyphase or only amplitude shaping, although phase-only shaping yields higherenhancements. A key result appearing in this chapter (Fig. 7.6) is the com-parison of a particular phase only shaping scheme (the pi/2 window) withpreviously published results [55].The numerical and experimental tools used for this investigation also pro-vided an opportunity to explore genetic search (adaptive learning) methodsof finding optimal coherent fields (Section 7.5.4). Additionally, the presenceof unsolved phase instabilities in the excitation field provided motivation forexploring a scan-based method that attempts to find optimal control fieldsby scanning groups of pixels in the immediate vicinity of each resonance(Section 7.5.5). This method finds good solutions in less time than the ge-netic search (faster convergence). In both cases (the genetic search and thepixel scans) it is seen that the availability of even limited information aboutthe structure of the system can provide constraints on the types of searchesthat should be used, and thus substantially decreases the required searchspace.The coherent control of two-photon transitions in simple atomic systemslike rubidium provides an example of the power and utility of quantum co-herent control, and its general applicability. Indeed, rubidium is not anelaborate biological system specifically designed for optical functional con-trol (see Section 1.1). Rather, control has been facilitated by the availabilityof a simple and accurate model of the system, combined with the techniqueof pulse shaping.76Chapter 8Regaining Coherent Controlafter Random 1D GlassStacks8.1 IntroductionThe method of Vellekoop and Mosk allows for compensation of spatial scat-tering (Section 2.2). In the weak scattering regime, this spatial compensa-tion also corrects for temporal scattering, when a broadband laser sourceis used (Section 2.2.1). The theory and experiment appearing in this chap-ter shows how compensation may be achieved for a strongly scattering onedimensional system. This scattering system, realized here as a stack ofglass microscope slides, is described in Section 8.2.1. Partial reflections ofthe incident laser pulse at each air-glass or glass-air interface give rise tonon-uniformly spaced trains of pulses.Section 8.2.2, then presents a numerical model of these glass stacks. Thenumerical models provide a convenient way to explore scattering compensa-tion (Section 8.3), aiding in the development of a scan-based compensationalgorithm. This compensation algorithm is presented in Section 8.3.6.The remaining sections in this chapter report (Section 8.4) and discuss(Section 8.5) numerical and experimental compensations achieved for glassstacks that have between one and ten layers.8.2 Glass Stacks8.2.1 Experimentally Constructed Glass Stacks and SampleRealizationsAs shown in Fig. 8.1, a stack of glass microscope cover slides is a simpleway to implement a one dimensional scattering system. Each glass layer inthe stack is characterized by an average thickness (100 µm), but the exact778.2. Glass StacksInput pulse Output  pulse train Random stack Lens  Spectrometer SHG  Crystal  Beamsplitter  1    2  3   4  5    6  6        5       4       3       2       1  Transmitted beam (not used) Reflected beam Glass  layer  Air gap  Figure 8.1: A stack of non-uniform glass slides, also called a random stack orglass stack. The reflection of the input pulse off the glass stack is captured,using the beamsplitter technique shown. The light scattered from the stackconsists of a train of pulses; pulses arise from reflections at each air-glassor glass-air interface. The reflected train is focused onto a second harmonicgeneration (SHG) crystal, and the spectrum acquired. Fig. 8.8 explains thepurpose of the SHG crystal and spectrometer.thickness varies randomly. The exact thickness of the air gap which appearsbetween the sandwiched glass layers also varies. For this reason, the stackis called “random.”The random stack is a scattering material due to the presence of air-to-glass and glass-to-air interfaces in the material. For the glass used here, theindex of refraction is approximately nglass = 1.52, and the index of air istaken to be nair = 1.0. The percent reflection from such an index mismatchis given by [66]R =(nglass − nairnglass + nair)2= 4.3%. (8.1)A reflected pulse arises at each air-glass interface. The spacing between anytwo reflected pulses is given by∆t =2 d× 103cnglass or air, (8.2)where ∆t is the temporal spacing in femtoseconds, d is the layer (glass orair) thickness in µm, n the refractive index, and c = 299.792458 nm fs−1788.2. Glass StacksIntensity780 790 800 810 8200.00.20.40.60.81.0Wavelength HnmLWavelengthHnmL-665.46 1.66782 668.796 1335.92370.773378.782386.791394.801402.81410.819418.828426.837Time HfsL(a) Sample experimentally realized 1 layer glass stack.Intensity780 790 800 810 8200.00.20.40.60.81.0Wavelength HnmLWavelengthHnmL1.66782 1169.14 2336.62370.773378.782386.791394.801402.81410.819418.828426.837Time HfsL(b) Sample experimentally realized 2 layer glass stack.Intensity780 790 800 810 8200.00.20.40.60.81.0Wavelength HnmLWavelengthHnmL-2321.61 -987.35 346.907370.773378.782386.791394.801402.81410.819418.828426.837Time HfsL(c) Sample experimentally realized 3 layer glass stack.Figure 8.2: Experimentally measured fundamental spectra (left) andXFROG traces (right) of the reflected pulse train arising from one, twoand three layer glass stacks. The power spectrum becomes more complex asthe number of layers increases. The XFROG traces show that an approx-imately 1 picosecond delay, due to the approximately 100 µm thick glasslayers, is observed. One also observes pairs of closely spaced pulses; thesepulse pairs exist because of small air gaps between the layers of glass; theseair gaps appear even when the stack is pressed tightly in a mount.798.2. Glass Stacksis the speed of light in vacuum. Multiple reflections can occur inside thestack, meaning that a pulse can be reflected more than once before it exitsthe stack. A random pulse train exits both sides of the stack.On the right side of the stack shown in Fig. 8.1, the pulse train that exitsthe stack also contains the ballistic pulse—the part of the input pulse thatwas never scattered. This is labeled as the transmitted beam. The otherpulses in this transmitted train have been reflected at least twice, and thusare much less intense (by a factor of R2 ≈ 0.1%) than the ballistic pulse.This huge contrast between the ballistic pulse and other pulses in the trainis problematic for the detection scheme that will be used (Section 8.3.4).To avoid this problem, the reflected beam which comes from the left sideof the stack shown in Fig. 8.1, is used. This train of pulses contains nearlyequal-amplitude pulses (i.e. of the same order of magnitude). The reflectedbeam is captured by inserting a beamsplitter in the input beam, as shownin the figure.In Fig. 8.2, measured spectra and XFROG traces of reflected pulse trainsarising from one, two and three layer stacks are presented. The power spec-tra shown in the figure demonstrate fringing. This fringing arises becausetwo or more pulses are present in the reflected train, which interfere onthe detector (on the CCD inside the spectrometer). The XFROG tracesalso verify that the expected approximately one picosecond spacing betweenreflected pulses (Eq. 8.2 using d =100 µm) is observed.The actual mount used for these experimental realizations of glass stacksconsists of two metal plates. A large hole is drilled in the center of each plate.The glass stack is held tightly between the two plates, with the laser beampassing through the center holes and also the stack. Thickness or othersurface irregularities in the glass slides can cause the glass thickness seenby different spatial components of the propagating beam to vary. Likewise,uneven pressure on different parts of the stack in the mount can cause theair gap seen by different spatial components of the beam to vary. To combatthese problems, the beam is brought to a focus inside the stack using twospherical mirrors so that the beam diameter inside the stack is small. Effortsare also made to keep the pressure applied on the stack by the mount as evenas possible, to eliminate the appearance of spatial speckle. In some cases,a rubber O-ring is also inserted in the mount to aid in applying sufficientregular pressure to stacks with more layers (e.g. stacks with 4 or 5 layers).808.2. Glass StacksIntensity770 780 790 800 810 820 8300.00.20.40.60.81.0Numerical 1 Layer Stack, Realization ð 1: Reflection Spectrum-1.5-1.0-0.50.00.51.01.5PhaseHradLWavelength HnmLIntensity770 780 790 800 810 820 8300.00.20.40.60.81.0Numerical 2 Layer Stack, Realization ð 1: Reflection Spectrum-1.5-1.0-0.50.00.51.01.5PhaseHradLWavelength HnmLIntensity770 780 790 800 810 820 8300.00.20.40.60.81.0Numerical 3 Layer Stack, Realization ð 1: Reflection Spectrum-20-15-10-505PhaseHradLWavelength HnmLIntensity770 780 790 800 810 820 8300.00.20.40.60.81.0Numerical 4 Layer Stack, Realization ð 1: Reflection Spectrum-6-4-202PhaseHradLWavelength HnmLFigure 8.3: Numerically computed reflection spectra; these simulate thescenario where a Gaussian beam (shown in Fig. 5.7) illuminates one to sixlayer thick stacks of glass slides (see also Fig. 8.4). They are calculated by themethod described in Section 8.2.2. Other realizations of glass stacks havingfrom one to ten layers are not shown here, but their compensation, alongwith the compensation of the stacks presented here, appears in Section 8.4.1.818.2. Glass StacksIntensity770 780 790 800 810 820 8300.00.20.40.60.81.0Numerical 5 Layer Stack, Realization ð 1: Reflection Spectrum-20-15-10-50510PhaseHradLWavelength HnmLIntensity770 780 790 800 810 820 8300.00.20.40.60.81.0Numerical 6 Layer Stack, Realization ð 1: Reflection Spectrum-20-15-10-505PhaseHradLWavelength HnmLFigure 8.4: Reflected spectra; continued from Fig. 8.3.8.2.2 Numerically Modeled Glass Stacks and SampleRealizationsA method by which glass stacks are numerically modeled is now introduced.One establishes a list of desired glass layer and air gap thicknesses, alongwith their respective indices of refraction. From these, the coefficients ofreflection and refraction are computed. At each frequency, a transfer matrixis computed [67]. The transfer matrices at all frequencies are then usedto compute reflection amplitudes, which are then multiplied by the inputspectrum. (One can also calculate transmission spectra.) This scheme hasbeen implemented in a set of MATLAB scripts [68]. One advantage of thismethod is that it accounts for all orders of reflections inside the stack.A sample of the computed reflection spectra obtained using this methodare shown in Fig. 8.3 and 8.4. These realizations of the numerical modelhave between one and six layers of glass. Other realizations of these andhigher (seven to ten) layer stacks are prepared in the same way. The list ofglass thicknesses is generated around the value 100.0±7.5 µm. The averageair gap thickness is 140 ± 35 nm. In both cases, the random variations aredrawn from a uniform or flat distribution around the mean (e.g. not from aGaussian distribution). Each glass layer and air gap is chosen independently828.3. Four Possible Phase Pre-Compensation Methods(i.e. each layer in the stack is different).8.3 Four Possible Phase Pre-CompensationMethodsFrom a frequency-domain pulse shaping perspective (see Section 5.4), com-pensation of scattering from glass stacks is simply a matter of finding outwhether it is possible to construct a mask H(ω), that undoes the effect (e.g.H−1(ω)) of the stack.Intensity shaping cannot be used in the compensation: from a practicalperspective, the goal is to bring all of the scattered pulses back togetherwithout power loss. Phase based shaping achieves this. Thus, an efficientphase mask φ(ω), that undoes the phase (e.g. φ−1(ω)) applied by the stack,is sought, and no intensity shaping is done. The compensating phase will beapplied by the pulse shaper before the light reaches the stack. This makesit a pre-compensation: a pulse train prepared by the shaper before it isdirected onto the stack, and it comes out of the stack as a single pulse.8.3.1 Phase Compensation via XFROGXFROG, in principle, should be able to recover the unknown electric fieldamplitude and phase, of the pulse train coming out from the stack. Then,the phase conjugate could be applied directly.However, this method has several disadvantages. First, the positionof each pulse in the XFROG trace (Section 5.5.3) can only be known asaccurately as the step size of the trace, or in other words, by the amount thatthe delay τ between cross-correlated pulses is stepped each time a spectrumis acquired. Second, XFROG is ill-suited to retrieving the phase of complexfields, and these random pulse trains are considered complex for XFROG.This method also requires additional optics, typically taking additional timeset up each time a different random stack must be tested. Lastly, even if onemeasures this conjugate phase, that does not solve the problem of how toapply it. For complex compensating phases, important information can belost in the process of interpolation, e.g., when converting the compensatingphase from the high resolution of a spectrometer to a lower resolution SLM.It will be seen later (see Fig. 8.16) that even if the field phase could beretrieved exactly, other compensation methods outperform it, due to thisinterpolation loss.838.3. Four Possible Phase Pre-Compensation Methods8.3.2 A Prerequisite: Creating Irregularly Spaced PulseTrains with a Pulse ShaperOther methods by which the unknown compensating phase may be discov-ered will be introduced later—all will require the material introduced inthis section. These methods do not directly retrieve the unknown spectralphase function φ(ω). Rather, they focus on ways of retrieving parameterscharacterizing a pulse train, produced by a stack. These parameters are therelative amplitude ai and position ti of each pulse i in the train.Once known, these parameters are used to create a phase function φ(ω)that (ideally) represents the inverse of the phase of the pulse train comingout from the stack. If the ai and ti are discovered correctly, this computedphase should approximately cancel the spectral phase originally introducedby the stack.This phase-only random pulse train is constructed in the following way.The time domain electric field of a random pulse train can be written asE(t) = a0 g(t)e−iωt + a1 g(t− t1)e−iω(t−t1) + · · · . (8.3)The temporal envelope of the input pulse is g(t); for a transform-limitedGaussian input pulse,g(t) = e−t22b2 . (8.4)The temporal full width at half maximum τFWHM is given by τFWHM =2√2 ln 2b, and ω0 gives the carrier frequency. a0 is always taken equal toone from now on; the other pulse strengths are made relative to a0. TheFourier transform of 8.3 isE(ω) = G(ω)(a0 + a1eiωt1 + · · ·)(8.5)withG(ω) = be−b2(ω−ω0)22 . (8.6)G(ω) is provided by the input pulse; factoring this out, the part of interest[69] isH(ω) ≡ 1 + a1eiωt1 + · · · . (8.7)The compensating phase function φ(ω) is thenφ(ω) = Arg H(ω). (8.8)It is this compensating phase which is applied to the spectral shaper topre-compensate glass stack scattering.848.3. Four Possible Phase Pre-Compensation MethodsExample: Creating a Two-Pulse Train with a Spectral ShaperConsider a two pulse train with a0 = a1 = 1 and t1 = 300 fs:H(ω) = 1 + eiω 300 fs. (8.9)Applying amplitude and phase together, one obtains the desired pulsetrain, shown in Fig. 8.5. If only the complex magnitude of this function,|H(ω)|, is used, the pulse train shown in Fig. 8.6 results. Essentially this isa perfectly phase compensated two-pulse train. For comparison, the resultof applying the phase Arg H(ω) only is plotted in Fig. 8.7. A comparisonof Fig. 8.6 and Fig. 8.7 shows that it is the phase and not the intensitymodulation done by the stack that gives the stronger contribution to theproduction of the correct number of pulses with the correct amplitudes.Likewise, it is seen that the depth of the fringes seen in the second-harmonicspectrum due to shaping only in intensity is less than those appearing whenshaping in phase. This suggests that the phase-only compensation may workquite well; this is verified in later sections.The phase-only two pulse train shown here is equivalent to the fixedperiodic unit pulse train introduced previously (Section 5.4.2). TransformingH(ω)→ H(ω)e−iωt2 ≡ H˜(ω) yieldsH˜(ω) = 2 cos(ωτ2). (8.10)The transformed phase, Arg H˜(ω), is exactly the repeating unit phase maskshown in Fig. 5.10. The phase-only train has two equal strength pulsesthat appear in the center; it is the addition of the amplitude shaping thatremoves the outlier pulses.8.3.3 Phase Compensation using Spectral InformationThe reflected intensity spectrum, coming from a glass stack (or equivalentlyfrom its second harmonic) can be Fourier transformed to the time domainto give estimates of the pulse times appearing in the random train. A flattemporal phase across each individual reflected pulse is assumed. The timinginformation is good, but only within the limit of spectrometer resolution.The resolution of the spectrometer used here limits this timing accuracy toabout ± 50 fs; this comparable to the temporal pulse width and is far belowthe level required for good compensation.858.3. Four Possible Phase Pre-Compensation MethodsIntensity760 780 800 820 8400.00.20.40.60.81.0SLM Mask-1.5-1.0-0.50.00.51.01.5PhaseHradLWavelength HnmLEHtL-1000 -500 0 500 1000-1.0-0.50.00.51.0Time DomainTime, t HfsLIntensity390 395 400 405 4100.00.20.40.60.81.0Second HarmonicWavelength HnmLFigure 8.5: The two-pulse train is created by shaping in intensity and phase,using H(ω) = 1 + eiω 300 fs (Eq. 8.9). See also Figs. 8.6 and 8.7.868.3. Four Possible Phase Pre-Compensation MethodsIntensity760 780 800 820 8400.00.20.40.60.81.0SLM Mask-1.0-0.50.00.51.0PhaseHradLWavelength HnmLEHtL-1000 -500 0 500 1000-1.0-0.50.00.51.0Time DomainTime, t HfsLIntensity390 395 400 405 4100.00.20.40.60.81.0Second HarmonicWavelength HnmLFigure 8.6: A pulse train is created by shaping in intensity only, using|H(ω)|. H(ω) is given by Eq. 8.9. This is essentially a phase compensatedtwo pulse train.878.3. Four Possible Phase Pre-Compensation MethodsIntensity760 780 800 820 8400.00.20.40.60.81.0SLM Mask-1.5-1.0-0.50.00.51.01.5PhaseHradLWavelength HnmLEHtL-1000 -500 0 500 1000-1.0-0.50.00.51.0Time DomainTime, t HfsLIntensity390 395 400 405 4100.00.20.40.60.81.0Second HarmonicWavelength HnmLFigure 8.7: A pulse train is created using phase-only shaping, using φ(ω) =Arg H(ω). H(ω) is given by Eq. 8.9.888.3. Four Possible Phase Pre-Compensation Methods׬ ɘ ׬ ׬ Inpu t pulse   Phase compensated  SHG Crystal  Spectrometer  ܧሺ߱ሻ Pulse train  from stack pu lse train from stack (a) An unscattered input pulse generates a smooth second harmonic spectrum. Theintegral of the measured spectrum,∫dωE(ω), is maximum.׬ ɘ ׬ ׬ Inpu t puls   Phase compensated  SHG Cry tal  Spectrometer  ܧሺ߱ሻ Pulse train  from stack pu lse train from stack (b) A pulse train, e.g. coming from a random stack, generates a second harmonic spec-trum with heavy fringing. The fringing in the spectrum leads to a low value of theintegral.׬ ɘ ׬ ׬ Inpu t pulse   Phase compensated  SHG Crystal  Sp ctr meter  ܧሺ߱ሻ Pul  tr i  from stackpu lse train from stack (c) A phase compensated pulse train has most of its energy back in a single pulse, withrelativ ly small outlier pulses. The fringing in the spectrum has been lifted to somedegree, and the numeric integral takes on an intermediate value. An adaptive searchalgorithm can refocus a pulse train by looking for ways to maximize this integral.Figure 8.8: An illustration of how second harmonic generation (SHG) isused to provide a measure of how well a random pulse train, as generatedby a glass stack, has been compensated.8.3.4 A Prerequisite: A Feedback Signal that Detects whenScattering has been CompensatedBoth compensating phase discovery methods previously introduced (XFROGand the use of spectral information) attempt to reconstruct the unknowncompensating phase in a single shot. Two additional methods will be in-troduced that instead employ adaptive search algorithms. These methodsiteratively test and refine a collection of pulse train parameters in order todiscover the best compensation.Before these methods are introduced, this section introduces the feedbacksignal used to guide these adaptive search algorithms. The feedback requiredby a typical adaptive search algorithm is a single numerical value which givesa measure of how good one solution is compared to another.This feedback is implemented as shown in Fig. 8.8. The concept behind898.3. Four Possible Phase Pre-Compensation Methodsthis feedback is as follows: the presence of a multiple-pulse train leads tolower second harmonic intensity, and also leads to fringing (due to multi-pulse interference) in the second harmonic spectrum. When a set of ai and tiare discovered that bring at least one pair of pulses in the train back together(after the pre-compensated pulse train is scattered from the stack), the peakintensity available for the second harmonic generation goes up and fringingin the spectrum is lessened. The area under the curve E(ω) increases. Thisarea of integration is used as the value of the feedback signal. The regionof integration is limited to the region of the spectrum in which the spectralcomponents appear above the noise level.Another way in which this feedback signal could be implemented wouldbe to use a slow photodiode, which integrates the total intensity of thesecond-harmonic train.8.3.5 Phase Compensation using a Genetic SearchIn order to use the genetic search (Section 6.2) to find the compensatingfield phase, a suitable parameterization of the phase must be determined.Among the possibilities for parameterization of the phase mask is the use ofa discrete pixel search, where each SLM pixel (or small groups of pixels) isallowed to take values from 0 to 2pi.Another possibility is to use of the pulse-train parameterization intro-duced above, allowing the genetic search to determine the optimal values ofai and ti. The mapping function (see Section C.1.6) converts these to theappropriate phase mask, via Eq. 8.8.This method works for simple stacks. However, at higher numbers oflayers the convergence of the search is poor: the genetic search fails to findgood compensating values of ai and ti in a reasonable time.8.3.6 Phase Compensation using an InterferometricAutocorrelation-Like MethodOriginal attempts at compensating few-layer stacks in the lab [69] involvediteratively “guessing” at and searching by hand the correct pulse parametersthat would optimize the compensation. In an attempt to automate this, thefollowing method is developed. It will be subsequently referred to as the“methodic” scan, as the scans are made methodically (genetic searches onthe other hand are stochastic in nature).Each of the pulse parameters can be scanned independently, refocusingone pulse in the train at a time. This method of compensation works well908.3. Four Possible Phase Pre-Compensation Methodsand it bypasses convergence issues encountered with the genetic search forhigher-layer systems. This methodic scan performs the following steps:1. Uncompensated stack reference. A flat phase mask and unit(100% transmission) intensity mask are applied to the SLM. The valueof the second harmonic feedback signal (Section 8.3.4) is stored as Aref .It is the reference value for the uncompensated stack.2. Scan range selection. A list of N allowed scan ranges ti,start toti,end and ai,start to ai,end, i an integer from 1 to N , is prepared. Goodcompensation can typically be obtained by including one range foreach first order reflection (those shown in Fig. 8.1); N is thus typicallyequal twice the number of glass layers in the stack. It is hoped thateach chosen range includes the actual values of ti and ai, matching thestack. The chosen ranges may be based on estimates of pulse positionsobtained using the second harmonic spectrum (see Section 8.3.3), aswell as upon experience with past scans.3. Pulse time (position) scan. A fixed value of ai, typically near 0.8,is chosen. Then ti is scanned between ti,start to ti,end. At each valueof ti, a phase mask—computed using Eq. 8.8—is applied to the SLM.This computed mask H(ω) always includes terms for any parametersaj , tj previously determined for pulses j = 1, 2, ..., i−1. At each point,the value of the second harmonic feedback signal A is measured. Thesecond harmonic compensation ratio, defined as A/Aref , is recorded;see Fig. 8.9(a). After the interval has been scanned, the selected valueof ti is the one corresponding to the maximum second harmonic com-pensation ratio.4. Pulse amplitude scan. Using the selected value of ti, the value ofai is scanned from ai,start to ai,end. The value of the second harmoniccompensation ratio is recorded at each point; see Fig 8.9(b). Thisamounts to determining the relative peak intensity of the pulse in thetrain. The selected value of ai is the one that maximizes the secondharmonic compensation ratio.5. Repeat. Steps 3 and 4 are repeated for each pulse i from 1 to N . Thefinal second harmonic compensation ratio, after all pulse terms are inplace, is measured.Each reflection at an glass-air interface, moving from glass to air, is as-sociated with an additional pi phase shift. However, attempting to explicitly918.3. Four Possible Phase Pre-Compensation MethodsS.H.Ratio940 960 980 1000 1020 1040 1060 10800.40.60.81.01.21.4Numerical 1 Layer Stack, Realization ð 1: Time ScanTime, t1HfsL(a) At t1 = 950 fs, the applied phase shaping causes the two pulses in the train to startto overlap. The “fringe” pattern arises as the two refocused Gaussian pulses can overlapboth constructively and destructively, once every optical period. The roughness of thecurve is related to the finite sampling resolution.S.H.Ratio0.0 0.2 0.4 0.6 0.8 1.01.01.11.21.31.4Numerical 1 Layer Stack, Realization ð 1: Amplitude ScanAmplitude, a1(b) After the temporal position is determined for a pulse, the relative amplitude isscanned.Figure 8.9: A sample of the time and pulse amplitude scans used in theinterferometric autocorrelation-like or “methodic” scan method. The storedvalues of t1 and a1 are the ones that correspond to the maximum secondharmonic compensation ratio on each curve. These scans correspond torealization 1 of the 1 layer numerically modelled glass stack shown in Fig. 8.3.The final compensation—using the values determined by the scans shownhere—appears in Fig. 8.10.928.3. Four Possible Phase Pre-Compensation Methodsput this phase shift into Eq. 8.3 unnecessarily complicates the algorithm;one would have to carefully track which terms should have the phase shiftand which should not, and include these while scanning. It will be seen thatexcellent compensation can be achieved without this. The presence of thepi shift improves the second harmonic ratio found for realization 1 of thenumerical 1 layer system by only about 1/10th of one percent.In the time domain, what is happening when the ti are being scanned ismuch like interferometric autocorrelation done through a second harmoniccrystal (see Section 5.5.2). One pulse is being scanned past another in time,and the collection is done in a collinear geometry. The trace obtained in thisstep (e.g. Fig. 8.9) thus looks very similar to that obtained by interferometricautocorrelation. It is a bit different because phase-only shaping is used here(see Fig. 8.7).Refinements to the Methodic MethodThe optical period at 800 nm is 2.67 fs; this period is observed in Fig. 8.9(a).The pulse temporal envelope FWHM is much larger than this. Thus the in-terferometric autocorrelation-like structure contains many peaks and valleys,spaced by 2.67 fs, within the overlap of any two pulses. Following this con-sideration, and in order to reduce the amount of measurement, the temporal(ti) and pulse strength (ai) scans are refined a bit:1. For the ti scan, first scan every e.g. third or fourth optical period.Each scanned period has a minimum and maximum.2. Of these scanned periods, find the one containing the highest secondharmonic ratio. Scan a few periods around the peak value on a coarsegrid (e.g. 0.4 fs spacing). This spacing must be small enough thatwithin the optical period, at least one or two points are likely to appearvery close to a maximum.3. Find the ti for the maximum second harmonic ratio on this coarse grid.Then scan on a finer resolution (e.g. about 0.2 fs) grid in a reducedwindow around the maximum, and choose the final ti from this.4. Scan the ai first on a coarse grid (spaced by 0.1) and then a fine grid(spaced by 0.03) in a reduced window around a maximum on the coarsegrid.This modification reduces total search time, and is especially important inthe experimental scans.938.3. Four Possible Phase Pre-Compensation MethodsFor the numerical model searches, an additional enhancement was madeto the methodic method. A couple of routines were added that (one sys-tematically and one stochastically) test small variations of the pulse param-eters found in the methodic scan, and look for improvements to the secondharmonic compensation ratio. The second harmonic compensation ratioimproves, but total computation time increases.This interferometric autocorrelation-like scan method was first developedusing the numerically modeled stacks; this reduced development time, aspulse shaping and second harmonic spectrum computations are much fasterthan the lab measurements. In the process of numerically compensatingsuccessively higher-layer systems, some important details and nuances wereuncovered and understood.Using the numeric model, it was also possible to directly monitor thetemporal electric field E(t), or the structure of the pulse train. Whileinitially attempting to compensate the numerical three layer stack real-izations, second harmonic compensation ratios significantly lower than ex-pected where obtained. The issue was resolved by understanding that themaximum second harmonic ratio the algorithm was finding was not asso-ciated with moving the second pulse in the train onto the first. This wasbecause the third and second pulses in the train, once put together, yieldeda higher second harmonic ratio than the second and first pulses would have.Only pulses with ti > 0 were used. By starting the refocusing process awayfrom the origin, there was no way for the algorithm to bring the first pulsein the train in the opposite direction, and refocus it.The simple solution was to also allow ti < 0. These negative time termsmove the pulses in the opposite direction. (The reason why the first pulseis not the most intense will be discussed in Section 8.5.3.)SLM-based InterferometrySubsequent to the completion of this work, it was found that SLM-basedinterferometric autocorrelation had been studied before. The Michelson in-terferometer required for SPIDER (which stands for Spectral Phase Interfer-ometry for Direct Electric-field Reconstruction), e.g., has been implementedusing an SLM-generated two pulse train [70]. As both intensity and phaseshaping are used (as in Fig. 8.5), this SLM-based SPIDER scan differs fromthe methodic scan. The final objective of the methodic scan is not field mea-surement, but to produce a usable (properly compensated) field containinga single pulse.948.4. Results: The Achieved Compensations8.4 Results: The Achieved Compensations8.4.1 Compensated Numerical Model RealizationsIn this section, the numerically modeled glass stacks introduced in Sec-tion 8.2.2 are compensated. Four different values are presented for eachrealization of a glass stack, shown in Table 8.1. These values are the secondharmonic compensation ratios (defined in Section 8.3.6), and give a measureof the degree to which a scattered pulse train has been refocused in time.The meaning of the different columns in the table follows. “Exact com-pensation” refers to the case where the phase of the stack has been com-pensated exactly. Typically, the resolution of the computed stack spectrumis approximately 5-7 times higher than the resolution of the SLM (whichhas 640 pixels). But experimentally, the compensation must be applied bythe SLM. The “predicted for SLM” column gives the compensation ratioobtained when the exactly known phase mask from the stack spectrum iscrudely downsampled to the resolution of the (numerically simulated) SLM.The “methodic scan” column gives the result of compensation usingthe interferometric autocorrelation-like scan method introduced previously.The “genetic search refinement” column, when not empty, shows the resultof feeding the pulse parameters ti returned from the methodic scan into agenetic search, and letting these parameters be refined along with findingai. At 5 layers, this method of genetic search refinements was also (inaddition to a less constrained genetic search that does not have the pulsetime estimates fed in) found to be ineffective, and so simple pulse parameterrefinement scheme (described in Section 8.3.6) was instead built into themethodic scan.For one layer, a two pulse train arises (excluding weak higher-order re-flections). A genetic scan was performed using the single-pixel parameteri-zation method (Section 8.3.5) with the single layer numerical model stack.The expected sawtooth-like SLM phase (Fig. 8.5 or Fig. 8.10) was indeedretrieved, after many generations. This method requires no prior knowledgeof the problem, but features much slower convergence because of the largesearch space introduced.For the methodic and genetic compensation methods, it is observed thatthe second harmonic compensation ratio of the system is relatively insen-sitive to variations in the single layer thickness—the ratio varies slightlyfor the different 1 layer stack realizations. The uncompensated and com-pensated second harmonic spectra and time domain electric fields for therealization 1 of the one layer stack are shown in Fig. 8.10.958.4. Results: The Achieved CompensationsExact Predicted Methodic Genetic SearchRealization # Compensation for SLM Scan Refinement1 Layer1† 1.381 1.356 1.358 1.3582 1.381 1.359 1.359 1.3583 1.381 1.356 1.347 1.3482 Layers1† 1.491 1.422 1.415 1.4182 1.784 1.686 1.452 1.5223 2.113 1.968 1.885 2.0043 Layers1† 2.143 1.828 1.838 1.9732 2.541 2.256 2.246 2.3023 2.095 1.716 1.951 1.9274 Layers1† 3.495 2.754 3.095 3.1612 3.843 2.798 3.362 3.3413 3.282 2.393 2.710 2.7085 Layers1† 3.827 3.070 3.2732 4.789 3.379 3.9293 4.734 3.410 3.9126 Layers1† 5.878 3.902 5.0702 5.301 2.984 3.8583 5.342 3.124 4.418Table 8.1: Second harmonic compensation ratios obtained for the computa-tionally simulated glass stacks, for one to six layer stacks. The four columnsto the right of the realization number give the ratio obtained using each ofthe four compensation techniques indicated. †See Fig. 8.10, 8.11, 8.12, 8.13,8.14, and 8.15 for samples of systems having each number of layers presenthere. These results are summarized in Fig. 8.16.968.4. Results: The Achieved CompensationsNumerical 1 Layer StackPhaseHradL760 780 800 820 840-1.0-0.50.00.51.0Compensating PhaseWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.0Uncompensated Second HarmonicWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.0Compensated Second HarmonicWavelength HnmLEHtL-1000 -500 0 500 1000 1500 2000 2500-1.0-0.50.00.51.0Uncompensated Time DomainTime, t HfsLEHtL-4000 -2000 0 2000 4000-1.0-0.50.00.51.0Compensated Time DomainTime, t HfsLFigure 8.10: The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after compensation.This is realization 1 of the one layer numerically modeled glass stack; thestack spectrum appears in Fig. 8.3.978.4. Results: The Achieved CompensationsNumerical 2 Layer StackPhaseHradL760 780 800 820 840-50510152025Compensating PhaseWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.0Uncompensated Second HarmonicWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.0Compensated Second HarmonicWavelength HnmLEHtL-1000 0 1000 2000 3000-1.0-0.50.00.51.0Uncompensated Time DomainTime, t HfsLEHtL-4000 -2000 0 2000 4000-1.0-0.50.00.51.0Compensated Time DomainTime, t HfsLFigure 8.11: The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after compensation.This is realization 1 of the two layer numerically modeled glass stack; thestack spectrum appears in Fig. 8.3.988.4. Results: The Achieved CompensationsNumerical 3 Layer StackPhaseHradL760 780 800 820 840-20-1001020Compensating PhaseWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.0Uncompensated Second HarmonicWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.0Compensated Second HarmonicWavelength HnmLEHtL0 1000 2000 3000 4000 5000 6000 7000-1.0-0.50.00.51.0Uncompensated Time DomainTime, t HfsLEHtL-4000 -2000 0 2000 4000 6000-1.0-0.50.00.51.0Compensated Time DomainTime, t HfsLFigure 8.12: The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after compensation.This is realization 1 of the three layer numerically modeled glass stack; thestack spectrum appears in Fig. 8.3.998.4. Results: The Achieved CompensationsNumerical 4 Layer StackPhaseHradL760 780 800 820 840-505Compensating PhaseWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.01.21.4Uncompensated Second HarmonicWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.01.21.4Compensated Second HarmonicWavelength HnmLEHtL0 2000 4000 6000 8000 10 000-1.5-1.0-0.50.00.51.01.5Uncompensated Time DomainTime, t HfsLEHtL-5000 0 5000 10 000-1.5-1.0-0.50.00.51.01.5Compensated Time DomainTime, t HfsLFigure 8.13: The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after compensation.This is realization 1 of the four layer numerically modeled glass stack; thestack spectrum appears in Fig. 8.3.1008.4. Results: The Achieved CompensationsNumerical 5 Layer StackPhaseHradL760 780 800 820 840-100102030Compensating PhaseWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.01.2Uncompensated Second HarmonicWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.01.2Compensated Second HarmonicWavelength HnmLEHtL-2000 0 2000 4000 6000 8000 10 000-1.5-1.0-0.50.00.51.01.5Uncompensated Time DomainTime, t HfsLEHtL-5000 0 5000 10 000-1.5-1.0-0.50.00.51.01.5Compensated Time DomainTime, t HfsLFigure 8.14: The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after compensation.This is realization 1 of the five layer numerically modeled glass stack; thestack spectrum appears in Fig. 8.4.1018.4. Results: The Achieved CompensationsNumerical 6 Layer StackPhaseHradL760 780 800 820 840-20-15-10-50510Compensating PhaseWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.01.2Uncompensated Second HarmonicWavelength HnmLIntensity390 395 400 405 4100.00.20.40.60.81.01.2Compensated Second HarmonicWavelength HnmLEHtL0 2000 4000 6000 8000 10 000 12 000 14 000-1.5-1.0-0.50.00.51.01.5Uncompensated Time DomainTime, t HfsLEHtL0 5000 10 000-1.5-1.0-0.50.00.51.01.5Compensated Time DomainTime, t HfsLFigure 8.15: The compensating phase, along with second harmonic spectraand time domain electric fields observed before and after compensation.This is realization 1 of the six layer numerically modeled glass stack; thestack spectrum appears in Fig. 8.4.1028.4. Results: The Achieved CompensationsSecondHarmonicCompensationRatio0 2 4 6 8 1002468Numerical Model CompensationsMethodic GeneticPredicted for SLMExactNumber of LayersFigure 8.16: Trends in the second harmonic compensation ratios obtainedusing the numerically computed stacks, from one to ten layers. Models withgenetic search refinements plot the final result of the genetic search, insteadof the methodic scan results. See Table 8.1.The extension to two layers is trivial, provided that one uses enoughterms. The genetic search refinement was actually able find a better solutionwith fewer terms ti by varying the exact values of a subset of the ti and aifound via the methodic scan. See Fig. 8.11.Three, four, five and six layer compensations are presented in Fig. 8.12,Fig. 8.13, Fig. 8.14 and Fig. 8.15, respectively.These ratios are also collected for systems having between seven andten layers. Although the individual realizations are not shown in Table 8.1above, they are included in Fig. 8.16. This figure compares all of the nu-merical model compensation ratios.8.4.2 Compensated Experimental RealizationsExperimentally achieved methodic scan-based compensations will now begiven. One successful scan is shown for each glass stack system from one1038.5. DiscussionNumber of Layers Methodic Scan Ratio1 1.42 ± 0.012 1.99 ± 0.043 2.5 ± 0.24 2.22 ± 0.095 2.2 ± 0.2Table 8.2: Second harmonic compensation ratios obtained for experimen-tally realized glass stacks, having from one to five layers. The correspond-ing compensated and uncompensated second harmonic spectra is shown inFig. 8.17, and the data in this table is plotted in Fig. 8.18.to five layers thick. Table 8.2 gives second harmonic compensation ratio(defined in Section 8.3.6) for each stack, and Fig. 8.17 shows the associ-ated uncompensated and compensated second harmonic spectra. The ra-tio values presented are each from a single compensation, or one executionof the methodic scan. The error bars are estimated using ratio obtainedin other successful experimental compensations; unsuccessful compensationruns (e.g. those having significantly lower compensation ratio) have notbeen included in this estimate. Fig. 8.18 shows the overall trend observedin the ratios.Experimentally, it typically takes several tries with each different layer-number stack to find the right combination of input parameters to give asuccessful compensation using this scan method. On some of these unsuc-cessful attempts, experimental noise also appears to play a role in hinderingthe discovery of a proper compensation. In multiple scans of the samestack, slightly different pulse time parameters and small changes in ai areobserved. The four and five layer systems required a slight revision to themount (added rubber O-ring and a greater applied pressure) in order to holdthe higher number of glass slides without slipping.8.5 Discussion8.5.1 The Numerical ModelsThe second harmonic ratio for the exactly compensated stacks, shown inFig. 8.18, rises with the number of stack layers. This makes sense; as the1048.5. DiscussionIntensity390 395 400 4050.00.20.40.60.81.0Experimental 1 Layer StackWavelength HnmLIntensity390 395 400 4050.00.51.01.5Experimental 2 Layer StackWavelength HnmLIntensity390 395 400 4050.00.51.01.52.0Experimental 3 Layer StackWavelength HnmLIntensity390 395 400 4050.00.51.01.5Experimental 4 Layer StackWavelength HnmLIntensity390 395 400 4050.00.51.01.5Experimental 5 Layer StackWavelength HnmLFigure 8.17: The experimentally measured second harmonic spectra for thephysical glass stacks, having one to three layers. In each figure, the graycurve gives the uncompensated second harmonic spectrum, and the tealcurve gives the second harmonic spectrum measured when the SLM appliesthe compensating phase mask discovered with the methodic scan.1058.5. DiscussionSecondHarmonicCompensationRatio0 1 2 3 4 50.51.01.52.02.53.0Experimental Stack CompensationsMethodicNumber of LayersFigure 8.18: The integrated second harmonic compensation ratio, obtainedfrom physical glass stacks compensated using the methodic scan. See alsoTable 8.2 and Fig. 8.17.pulse trains are get increasingly complex, there is more and more “room forimprovement.” But the phase compensation is made at the resolution of themodel, and so the full improvement can always be made.On the other hand, the “predicted for SLM” result is somewhat surpris-ing. It shows that with a careful choice of the values appearing on eachSLM pixel, one can do better than is possible with a naive nearest-neighborinterpolation and an “exactly” known phase. This means that even if a veryhigh fidelity pulse phase characterization technique is employed, it is likelyto yield a poorer compensation than those discovered using the SLM.The methodic and genetic scans are capable of making increasingly goodcompensations up to 6 layers. After 6 layers, the limited resolution of thespectral shaper prevents additional compensation. The phase features of thestack are too complex to be represented properly (consider the phase shownin the 6 layer numerical stack in Fig. 8.4). This is why the compensationratio drops: as disruption to the input pulse gets increasingly worse, it isnot matched by a corresponding increase in shaper resolution.1068.5. DiscussionTo illustrate this point, consider a numerical simulation of a long pulsetrain, with pulses stretching from −7 ps to 7, spaced by 1 ps and havingequal amplitude ai = 1, as shown in Fig. 8.19. One of the physical 4f pulseshapers used in this work is configured such that each pixel covers 0.139 nmof bandwidth, which by the Fourier transform (Eq. 5.18) corresponds to atemporal length of about 15 ps. Computationally, finite pixel resolution issimulated using a nearest neighbor interpolation scheme, to project the 640flat pixels of the SLM onto the higher resolution input pulse. Fig. 8.19(b)demonstrates that the quality of the pulse train declines as the temporallimits are approached.If one computes the same SLM mask and pulse train for a hypotheticalSLM having eight times smaller pixel width (and eight times more pixels),the noise present in Fig. 8.19(b) disappears: the pulse train in Fig. 8.19(c)is very regular. Indeed, the 0.0173 nm pixel width of this higher resolutionvirtual “SLM” mask corresponds to nearly 123 ps in time. Equivalent tolooking at the time-domain pulse structure, one observes this limitationsimply by looking at the SLM mask in Fig. 8.19(a); the phase mask is almostentirely comprised of sharp pixel-to-pixel jumps. Observe that the use ofboth positive and negative pulse times ti in the train will allow the shaper tocreate longer pulse trains than otherwise possible with only positive or onlynegative time parameters, as higher time (larger |ti|) terms produce sharperspectral features; a more sophisticated version of the methodic scan mightattempt to use this fact to enable the compensation of stacks with highernumbers of layers, by selecting where the refocused t = 0 pulse appears,rather than just allowing it to appear wherever the first time scan picks thehighest strength pulse.The resolution of the spectral shaper thus sets a fundamental limit tothe amount of scattering compensation available. More correctly, it is theproduct of the shaper resolution with the duration of the scattered train thatsets this limit. A scattering system with very small layer thicknesses wouldallow the shaper (used here) to compensate 1D scattering from stacks withmany more layers than those studied here. The synthesis of such a systemwas in fact attempted, using thin layer deposition techniques, but difficultiesencountered in the fabrication of these very thin layers favored the use ofthe glass stack.8.5.2 Comparing Theory and ExperimentThe ratios obtained in these experimental models are not as high as thosefound in computational models with the same number of layers. There are1078.5. DiscussionIntensity760 780 800 820 8400.00.20.40.60.81.0SLM Mask020406080100PhaseHradLWavelength HnmL(a) An SLM mask designed to create an equal amplitude train of pulses from −7 ps to 7,spaced by 1 ps.EHtL-20 000 -10 000 0 10 000 20 000-1.0-0.50.00.51.0Time DomainTime, t HfsL(b) The time domain train computed when the above mask is applied on a numericallysimulated 640 pixel SLM. The resolution of this simulated shaper matches the shaper usedin the lab. The temporal noise continues long past the temporal extents shown and ispresent at lower amplitude throughout the pulse train. The pulse amplitudes are uneven.EHtL-10 000 -5000 0 5000 10 000-1.0-0.50.00.51.0Time DomainTime, t HfsL(c) The same time domain pulse train, now created with a numerically simulated shaperhaving with an eight times higher resolution (an eight times smaller SLM pixel widthmatched by an eight times increase in the number pixels), shaping the same bandwidth.Note that both pulse trains still have pulses appearing at the same times, but the noiseis substantially less and the amplitudes are even.Figure 8.19: A numerical demonstration of the effect that limited pulseshaper resolution has on the creation of lengthy pulse trains.1088.5. DiscussionIntensity390 395 400 405 4100.00.20.40.60.81.0Compensated Second Harmonic:Exact Phase CompensationWavelength HnmLFigure 8.20: The second harmonic spectrum associated with an exact phasecompensation (at the resolution of the numerical model) of the realization 1,four layer numerically simulated glass stack. The uncompensated spectrumand shaper-resolution compensation are found in Fig. 8.13. The importantresult to be noted here is that a low level of fringing is still present (andin fact the spectrum looks very similar to the case where the numericallysimulated shaper is used.)a couple of important issues to appreciate in making this observation.First, while the numerical model used strives to be complete, accuratelysimulating multi-layer reflection and transmission in a stack, it does notattempt to exactly match the experimentally observed air gap thickness,nor does attempt to exactly match the amount of deviation seen in the glasslayer or air gap thicknesses. Systematically increasing or decreasing themagnitude of any of these parameters can place the stack in a regime wherethe second harmonic compensation ratio will be either higher or lower (seeFig. 8.21); this has to do with inter-pulse interference effects (Section 8.5.3).Second, and following from this first point, it is incorrect to think of ahigher second harmonic compensation ratio as indicating a more successfulscan when comparing substantially different stacks. The maximally achiev-able second harmonic ratio increases anytime the input pulse train is more“complex,” meaning that the relative amplitudes of the pulses in the scat-tered train have increased. Making the pulse train more complex can beachieved by adding layers; this is why the second harmonic ratio grows withincreasing numbers of layers.It can be seen that the fringe depth appearing in the experimentallymeasured uncompensated second harmonic spectra (Fig. 8.17) is typicallyless than that appearing in the uncompensated computational model spectra1098.5. Discussion(Figs. 8.10 through 8.15). This observation suggests that there is simplyless compensation available to be done in these experimental systems—theinitial integral of the uncompensated spectrum is larger, making the ratiosmaller. A possible reason for this lessened fringe depth in experiment is thatthe stack is not one-dimensional, as the idealized model predicts. Spatialvariations across the surface of the stack (Section 8.2.1) are thus encounteredby the beam, leading to variation of pulse timings and amplitudes in thereflected trains, and thus a blurring of fringe intensity finally measured bythe spectrometer.When comparing the success of theory and experiment via visual in-spection of the measured compensated second harmonic spectra, recall thateven a perfectly phase compensated reflected pulse train does not have a“fringe-free” second harmonic spectrum. This was seen earlier in the two-pulse intensity shaping demonstration (Fig. 8.6), and can also be seen inFig. 8.20. In this latter figure, the second harmonic spectrum for one of thefour layer numerical stacks is shown for the case where the phase has beencompensated exactly, to the full numerical resolution of the model. It isseen that a small degree of fringing remains.8.5.3 Pulse InterferenceWhen the thickness of the air gap is less than the temporal pulse width, theresulting pair of pulses will interfere. This is not the only way that pulsesfrom the stack can interfere, however. Recall the creation of two pulses,separated by τ = 300 fs, using spectral shaping (Fig. 8.5). The phasecompensated two-pulse train, Fig. 8.6, is still a train of pulses, separatedby τ . Some of these lower-intensity pulses that appear in the compensatedtrain can then interfere with seemingly temporally distant pulses comingfrom other layers of the stack.The criteria for observing this second type of interference is as follows.Let there be a pulse at 1120 fs. According to the two-pulse example above,low-intensity pulses will appear at multiples of this time, e.g. a low intensitypulse will appear at 2240 fs. Thus pulse can then interfere with a pulse com-ing from another stack layer, e.g. at pulse at 2250 fs, even though the twopulses originally coming from the stack shared no temporal overlap! This issimulated using a shaper-generated pulse train, as shown in Fig. 8.21(a). Afour-pulse train is created using intensity and phase shaping; the SLM maskis again made using Eq. 8.7. This four-pulse train is structured to simulatethis two layer stack with air gap. As the thickness of the second layer is var-ied, the available second harmonic compensation ratio varies rapidly. (The1108.5. DiscussionS.H.Ratio0 20 40 60 80 1001.41.61.82.02.22.42.6Simulated Pulse Train with t1= 1120 fs, t2= 1130 fs and t3= 2250 fs+ D t23Variation in Second Layer Thickness, D t23HfsL(a) Air gap fixed, layer thickness of second layer varied.S.H.Ratio0 20 40 60 80 1001.52.02.53.0Simulated Pulse Train with t1= 1120 fs, t2= 1120 fs+ D t12and t3= 2500 fs+ D t12Air Gap Pulse Separation, D t12HfsL(b) Glass layer thickness constant, air gap varied.Figure 8.21: The effect of variations in air gap and layer thickness, simulatedcomputationally using shaper-generated four-pulse trains. Each time pointused in constructing these curves effectively represents a different stack.exact phase compensation of this train is applied at each value of thicknessto measure this ratio.) This confirms the predicted interference between thetwo initially non-overlapping pulses. The second harmonic compensationratio shown here oscillates at the carrier period (2.67 fs). When the layerthickness increases such that multiples of the compensated pulse at 1120 (or1130) fs no longer overlap, the oscillations seen in the second harmonic ratioin Fig. 8.21(a) fade.The interference at the air gap is easier to understand, as the pulsesalready overlap due to the closeness of two reflecting layers. As done forthe layer thickness, the effect of this air gap thickness on the total availablesecond harmonic compensation ratio can be computed. In Fig. 8.21(b), afour-pulse, two-layer stack is simulated using spectral shaping. The air gapthickness is varied by scanning the time ∆t12 offset of one of the pulses atthe gap. When the two pulses no longer over lap, the oscillations in second1118.5. DiscussionS.H.Ratio0 5 10 15 20 25 302345Base Air Gap Spacing, D t HfsLFigure 8.22: A numerical simulation for a four layer thick glass stack, pre-pared using the full numerical model that includes all orders of reflection,and transmission. The reflected second harmonic spectrum is computedat each point as the air gap thickness is varied; the ratio is computed byapplying the exact phase conjugate.harmonic ratio fade.The effect of air gap interference in a four-layer numerically modeledglass stack (as opposed to using a simulated pulse train) is shown in Fig. 8.22.A set of random variations in the layers and in the three air gaps are cho-sen once at the beginning. The actual “base” thickness of each air gap isscanned, added each time to the set of fixed air gap variations chosen beforethe start of the scan. One sees that the result is consistent with the sim-pler pulse train model: oscillations and an overall trend towards increasedvalues of the second harmonic ratio are observed as the air gap distancegrows relative to the temporal width of the pulse. The trend is that asthe air gaps grow, the pulses become better separated, and the maximumachievable second harmonic tends to increase.It is thus seen that the exact compensation ratio achieved in a particularmulti-layer stack can vary widely, and that range of compensation observedwhen small deviations in layer thickness are present can also vary widely.The relatively low variability appearing in the achievable second harmoniccompensation ratio for a given layer system amongst the computationalrealizations studied previously (Fig. 8.3 and 8.4) may be understood byrecalling that the air gaps there chosen range from 0.70 to 1.17 fs, a differenceof less than 20% of the optical period. Thus the potentially huge range ofmodulation predicted here is not seen in these realizations.The presence of inter-pulse interference explains why values ai found in1128.6. Conclusionthe scans typically take values between 0 and 2: values above 1 occur thecompensation encounters a “pulse” is actually the constructive interferencebetween two pulses.8.5.4 A Brief Comment on the Structure of the SearchLandscapeThe genetic search method converges quickly, for few-layer systems, whenthe ai and ti parameterization is used. It does not converge within reasonableexperimental and computational time limits when higher-layer systems arescanned. In contrast, studies have been done [71, 72] to explain why theresults found in genetic search methods applied to quantum systems yieldsuch excellent solutions.In the case of a quantum control experiment where a genetic search isused to maximize the probability of single initial to final state transition,the search landscape possesses the property that all extrema in the spacecorrespond to optimal controls for the system [71]. In other words, everyextrema is a global extrema. This surprising result helps to explain why a ge-netic search can rapidly discover very high quality controls in such a system,even if the system Hamiltonian is complex. On the other hand, the glassstack-scattered pulse trains have many sub-optimal extrema in the searchlandscape. For example, the search may discover various ways to “refocus”one or more pulses onto each other in the time domain, finding local extremacorresponding to sub-optimal control, and then get stuck there. The searchwill thus miss finding the global extremum existing in the search space, herecorresponding to the state where all pulses have been recombined into one.8.6 ConclusionIt has been shown that spectral shaping can be used to successfully compen-sate temporal scattering, even in the strongly scattering regime, in whichthe input pulse structure has been substantially changed. A compensationalgorithm was developed, based on a phase shaping-only variant of inter-ferometric autocorrelation. This algorithm allowed easily and automaticcompensation of both numerically realized and experimentally realized glassstacks. In the numerical compensations the electric field is observed directly,and it is seen that the phase compensations are indeed refocusing the train ofpulses, effectively undoing the effect of the stack. In experimental compen-sations, one observes this compensation indirectly by observing the liftingof fringing from the second harmonic spectra presented.1138.6. ConclusionIn the process of achieving compensation, it is also seen that a simplestack of glass microscope slides has provided occasion to study a number ofinteresting effects. These include a brief study of the limitations that shaperresolution has on the ability to compensate scattering, and the observationthat adaptive search schemes may outperform direct phase characterizationmethods, due to losses incurred when attempting to apply the compensatingphase onto the finite resolution pixels of an SLM. Pulse interference effectsdue to the presence of narrow gaps of air between the glass layers havealso been studied; it has been seen these interference effects can also arisebetween even seemingly distant pulses, as they will be coupled by the trainof pulses produced by the compensating phase, and are responsible in partfor the range of available compensation in different stack realizations.In Chapter 1 Brumer-Shapiro coherent control was introduced as a meansby which an arbitrary atomic or molecular system could be guided to aparticular quantum state. The use of broadband coherent fields, however,gives rise to the problem of optical scattering (introduced in Chapter 2).The carefully chosen phase relationships in the broadband optical field aredisrupted in random ways by the scattering materials. The present chapterhas thus introduced an important concept for further studies. 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RFC:4627.122Appendix AThe Electric DipoleHamiltonianIn quantum electrodynamics (QED), the full Hamiltonian describing asystem of charges and fields in the Coulomb gauge [73] isH(t) =∑α12mα(~pα − qα ~A(~rα)− qα ~Ae(~rα, t))2+ VCoul +∑αqαUe(~rα, t) +∑i~ωi(a†iai +12)(A.1)(This Hamiltonian equation comes from Legendre transformation of theQED Coulomb gauge Lagrangian.) ~pα describes the momentum of parti-cle α, and mα is the mass. qα is the charge and ~A the vector potentialof the fields in the system. The terms qα ~A give the effect of the internalfields on the particles (contributing to their motion); qα ~Ae gives the effect ofan external field on the particles (also contributing to their motion). VCoulgives the all Coulomb effects in the system. Ue is the external scalar poten-tial imposed on the system, with qαUe the effect of the scalar potential onparticle α. The final term describes the state of the internal field, an ensem-ble of quantum harmonic oscillators describing field normal modes havingfrequency ωi.This Hamiltonian can deal with a wide variety of scenarios, including(nonrelativistic) quantum optics experiments where the quantum evolutionof the electromagnetic fields is of vital import. Fortunately, substantialsimplification is possible here. The evolution of the internal field is notimportant, and so one can set ~A = 0. Ue can be ignored, as it describesan the effect of external Coulomb field. In this approximation only the“perturbation” of the system by some external transverse field is treated.The Hamiltonian is thenH(t) =∑α12mα(~pα − qα ~Ae(~rα, t))2+ VCoul. (A.2)123Appendix A. The Electric Dipole HamiltonianIn order to further simplify, one assumes that the charges (for example,those in a rubidium atom) are localized at the origin. If the interacting fieldwavelength is long compared to the dimensions of the localized quantumsystem, the Hamiltonian is written instead asH(t) ≈∑α12mα(~pα − qα ~Ae(0, t))2+ VCoul. (A.3)The time-dependent unitary transformation of a time-dependent Hamilto-nian is given by (see Chapter 4 in [73])H ′(t) = T (t)H(t)T †(t) + i~(dT (t)dt)T †(t). (A.4)IfT (t) ≡ exp(−i~~d · ~Ae(0, t)), and, (A.5)~µ =∑αqα~rα (A.6)then still in the long wavelength approximation,H ′(t) =∑αp2α2mα+ VCoul − ~µ · ~Ee(0, t) (A.7)is the known as the electric dipole Hamiltonian.This simple Hamiltonian can be split into two parts, H = H0 + V . H0includes the particle velocity and Coulomb terms, but the only importantthing here will be to know the energy spectrum of H0. V is the interactionterm, given byV (t) = −µEe(0, t) ≡ −µE(t). (A.8)µ is the electric dipole moment of the interaction, and E is the externalelectric field (e.g. the optical pulse imposed on the atom). This form of theexpression applies when the field E is linearly polarized.124Appendix BFrequency DomainTime-DependentPerturbation TheoryB.1 Time-Dependent Perturbation TheoryIdeally suited for Hamiltonians of the form H = H0 + V , the interactionpicture [74] defines a new a time-dependent quantum state ket in the in-teraction picture, |α(t)〉I . This is related to the ordinary state ket in theSchro¨dinger picture, |α(t)〉S , via|α(t)〉I = eiH0t/~ |α(t)〉S . (B.1)It follows that an observable VI in the interaction picture is related to itsdefinition V in the Schro¨dinger picture byVI(t) = eiH0t/~V (t)e−iH0t/~ (B.2)The effect of doing this is that the quantum wave equation for time-evolutionis nowi~∂∂t|α(t)〉I = VI |α(t)〉I . (B.3)By defining the time-evolution operator|α(t)〉I = UI(t, t0) |α(t0)〉 , (B.4)with UI(t0, t0) = 1, Eq. B.3 leads to the integral equationUI(t, t0) = 1−1~∫ tt0dt′ VI(t′)UI(t′, t0). (B.5)(VI is the quantum propagator or kernel of this integral equation.) By125B.1. Time-Dependent Perturbation Theoryiteration, one obtains the Dyson series [74],UI(t, t0) =1 + · · ·+(−i~)n ∫ tt0dt′∫ t′t0dt′′ · · ·∫ t(n−1)t0dt(n)× VI(t′)VI(t′′) · · ·VI(t(n)) + · · · . (B.6)The Dyson series can be made more useful for the present applicationby transforming it into a form that gives the transition amplitude betweentwo energy eigenkets. The state |α〉 is related to its stationary state energyeigenkets |a〉 by|α(t)〉 =∑a|a〉 〈a| α(t0)〉 e−iEa(t−t0)/~, (B.7)where Ea is here the set of energy eigenvalues of H0. At the initial time,|a(t0)〉I = |a〉 (by appropriate choice of the initial phase). The transitionamplitude between initial state |i〉, evolved in time from t0 to t, and finalstate |n〉, iscn(t) = 〈n| UI(t, t0)| i〉 . (B.8)Then, the terms in the Dyson series (Eq. B.6) gives rise to approximationsc(j)n of order j for cn(t). As UI(t0, t0) = 1,c(0)n (t) = δni. (B.9)The first order coefficient c(1)n is identically zero when the laser bandwidthdoes not include spectral components with sufficient energy for the transi-tion (this can be verified using a procedure analogous to that presented inSection B.2).The second order coefficient isc(2)n (t) = −1~∑m∫ tt0dt′∫ t′t0dt′′ eiωnmt′Vnm(t′)eiωmit′′Vmi(t′′). (B.10)In general, Vij ≡ 〈i| V | j〉 and eiωijt ≡ ei(Ei−Ej)t/~. The sum over m is asum over all energy eigenkets, introduced by the identity∑m|m〉 〈m| = 1. (B.11)Expression Eq. B.10 is known as the second-order transition amplitude intime-dependent perturbation theory (TDPT).126B.2. TDPT in the Frequency DomainThe above perturbative expression c(2)n has a simple interpretation. Roughly,one associates the four “factors” in Eq. B.10, from right to left, with tran-sition amplitude flowing from i to m, evolving in time, transition amplitudeflowing from m to n, and then evolving again. It thus appears as an idealcandidate for understanding two-photon transition. A frequency domaintreatment, now presented, complements this time-domain interpretation,allowing the effects of frequency-domain shaping applied via the SLM to beunderstood directly.B.2 TDPT in the Frequency DomainStarting with Eq. B.10, one uses Eq. A.8 and the Fourier transform of E(ω)to E(t), given byE(t) =∫ ∞−∞dωE(ω)e−iωt, (B.12)to obtainc(2)n (t) ∼∑mµnmµmi∫ tt0dt′∫ t′t0dt′′∫ ∞−∞dω′E(ω′)ei(ωnm−ω′)t′×∫ ∞−∞dω′′E(ω′′)ei(ωmi−ω′′)t′′ . (B.13)Reversing the order of the time and frequency integration yieldsc(2)n (t) ∼∑mµnmµmi∫ ∞−∞dω′∫ ∞−∞dω′′E(ω′)E(ω′′)×∫ tt0dt′ ei(ωnm−ω′)t′∫ t′t0dt′′ ei(ωmi−ω′′)t′′ . (B.14)As E(t) is the temporal representation of a single pulse, the interactioncan be taken to start at −∞: let t0 → −∞. The behavior during the opticalpulse (coherent transients) is not of interest here, but rather the amplitudeafter the process is done. Thus let t → ∞. This of course assumes thatassuming no competing processes are transferring amplitude to or away fromthe involved levels during interaction.With these limits, performing t′′ time integral as is would give an indef-inite oscillatory term, evaluated at −∞. In order to overcome this, let (foran example, albeit with a different integrand, see [74])ω′′ → ω′′ + i. (B.15)127B.2. TDPT in the Frequency DomainThen, after the integration, take → 0. Thus,∫ t′−∞dt′′ ei(ωmi−ω′′)t′′ =ei(ωmi−ω′′)t′i(ωmi − ω′′). (B.16)The subsequent integration over t′ yields a δ function fixing w′:c(2)n ∼∑mµnmµmi∫ ∞−∞dω′∫ ∞−∞dω′′E(ω′)E(ω′′)i(ωmi − ω′′)δ(ωnm + ωmi − ω′ − ω′′)(B.17)This i→ n transition has ωni ≡ ωnm + ωmi, ∀m. Thus,c(2)n ∼∑mµnmµmi∫ ∞−∞dωE(ωni − ω)E(ω)ωmi − w. (B.18)(Compare with [55].) Physically, this says that a given two-photon transi-tion is comprised of all pairs of frequencies ωnm and ωmi in the input pulsethat sum to ωni. The denominator of this expression is such that a simplepole appears at the frequency of the intermediate state or resonance. Theavailable laser bandwidth limits which resonances are involved, as only theresonances appearing in regions of nonzero spectral amplitude contribute.Thus, the sum over m is reduced to a sum over bandwidth-accessible inter-mediate states for the transition.Removing the DivergencesIn order to meaningfully evaluate Eq. B.18, the integrand in Eq. B.18 is ex-tended to the complex plane, promoting ω → z, z ∈ C. Then this integrand,be known as f(z), isf(z) =g(z)ωmi − z=E(ωni − z)E(z)ωmi − z. (B.19)Then the integral along the real line is replaced by an integral along the realaxis that excludes the singularity, plus a semi-circular contour of vanishingradius that goes around the simple pole.The integral along the real axis is defined [75], in general, byP∫ badx f(x) = lim→0+(∫ c−adx f(x) +∫ bc+dx f(x))(B.20)and called the Cauchy principal value of the integral. The singular point isthe point at c. For numerical integration, the problem can be simplified and128B.2. TDPT in the Frequency Domainnumerical stability gained. One constructs out of f(x) functions odd andeven about the point c,fodd(x, c) =12(f(c+ x)− f(c− x)) , and, (B.21a)feven(x, c) =12(f(c+ x) + f(c− x)) (B.21b)respectively. Then a symmetric principal value integration of each of theseis formed about the point x0, with user-specified “radius” r,lim→0+(∫ −−rdx fodd(x, c) +∫ rdx fodd(x, c))= 0, and, (B.22a)lim→0+(∫ −−rdx feven(x, c) +∫ rdx feven(x, c))= lim→0+2∫ rdx feven(x, c)(B.22b)where r is a finite (user chosen) integration radius around the singular point.Only the even contributions to the principal value integral about the singularpoint contribute. Thus Eq. B.20 becomesP∫ badx f(x) =lim→0+(∫ c−radx f(x) +∫ rdx (f(c+ x) + f(c− x)) +∫ bc+rdx f(x))(B.23)Note that the divergence due to the denominator in Eq. B.18 imparts anoddness to the whole integrand, about the singular point, making the nu-merical stability good. This is the integral that Mathematica will performif asked to do a principal value integration [76, 77].The contribution from the semi-circular contour Cρ going around thesimple pole at z = ωmi is given by [78]limρ→0∫Cρf(z)dz = −B0pii (B.24)The residue B0 is given byB0 ≡ Resz=ωmif(z) = g(ωmi) = E(ωni − ωmi)E(ωmi). (B.25)129B.2. TDPT in the Frequency DomainDemoting z back to ω ∈ R, Eq. B.18 becomesc(2)n ∼∑mµnmµmi(P∫ ∞−∞dωE(ωni − ω)E(ω)ωmi − ω− ipiE(ωni − ωmi)E(ωmi)).(B.26)130Appendix CSoftware and Algorithms:QuantumBlackbox andMathematicaC.1 QuantumBlackboxQuantumBlackbox is written using C++. Specifically, the Microsoft VisualStudio 2010 interactive development environment (IDE) and compilers areused, along with Qt [79] version 4.8.3. As a result of the use of Qt, theresulting program is readily cross-platform portable.Both the LabVIEW software and the components of QuantumBlackboxemploy use object-oriented approach. For QuantumBlackbox, each distinctcomponent is a C++ class object. These core components are now described.C.1.1 Connecting to LabVIEWOne of chief purposes of this software is to provide a more flexible andconvenient way to communicate with and control the lab hardware. Inorder to still make full use of the functionality provided by the LabVIEWsoftware, QuantumBlackbox communicates with a special LabVIEW VI.This LabVIEW VI connects with QuantumBlackbox via a TCP/IP [80,81], a standard networking protocol used on the Internet. It then readsfrom and writes to the hardware device, returning the information. On theQuantumBlackbox side, the class that implements this communication iscalled NetworkHardwareProxy.An advantage of using a network protocol such as TCP/IP versus otherforms of inter-process communication (IPC) typically provided by an oper-ating system is that communication can easily happen even when the serversoftware is running on a different machine than the client. And, if runningon the same machine, the communication is extremely fast, so that there isno disadvantage to using TCP/IP.131C.1. QuantumBlackboxC.1.2 Hardware Device Proxy ClassesEach of the important hardware devices with which QuantumBlackbox mustcommunicate are represented by their own C++ class. These C++ classesexpose sets of methods that represent each hardware object; the same is truein the LabVIEW software. Certain methods simply call on the Network-HardwareProxy. For example, the SLMService class which represents theSLM calls the NetworkHardwareProxy when a SLM phase or amplitudemask must be applied to the SLM. Two other proxy classes include theSpectrometerService and the OtherHardwareService. These communi-cate with the spectrometer, and provide an means of returning IEEE 754[82] double-precision floating point numbers (e.g. measurement results), re-spectively.C.1.3 Numeric RoutinesFor convenience, a small collection of basic numeric routines exist in Quan-tumBlackbox. Many of these are also available for use with the geneticsearch component (Section 6.2). They include the following:• A set of basic methods for adding arrays, multiplying them by con-stants, averaging an array of numbers and computing its standarddeviation.• Methods to shift or rotate the elements of an array (for use in, e.g., inFourier transforms).• A set of simple routines to find the minimum or maximum elementsin an array, including the index or indexes of several of the maximumor minimum elements.• A method to perform linear interpolation [40], accepting two arraysspecifying the (xin, yin) coordinates of some data, as well as an arrayof new xout values for which new values of yout will be computed vialinear interpolation.• A method to perform cubic spline interpolation [40], accepting argu-ments similar to those accepted by the linear interpolation routine,and returning the smoothly interpolated points yout.• A set of methods for performing fast Fourier transforms (FFT), basedon the open-source FFTW library [83]. A set of helper methods is also132C.1. QuantumBlackboxprovided, which aid in doing the necessary interpolation and book-keeping involved in computing a time-domain signal from a spectrumprovided on a wavelength scale, and vice-versa. These methods are notused in the experiments done in this thesis, but provide a foundationfor some other functionality that was written, but not used, in theprogram. These include routines to calculate the FROG (or XFROG)trace (Section 5.5.3) given some input field(s), part of the beginningsof an attempt to implement a genetic-search based FROG retrievalalgorithm [84]. The use of the FFT to compute time, frequency andwavelength domain signals related to optical pulses is described inSection 5.4.1.• A method to unwrap the phase in an array of points representing aphase function that has been wrapped modulo 2pi.C.1.4 JavaScript (ECMAScript) Engine and EditorQt, and thus QuantumBlackbox, provide access to a JavaScript [85] engine.JavaScript is a scripting language often used in web pages, but the enginethat actually runs the script is general and can be incorporated in any ap-plication. Qt wraps a JavaScript engine in a special way, such that methodsfrom clases which inherit QObject (and are subsequently registered with theengine) can be called transparently.In this way, all of the hardware device (proxy) classes and many of thenumerical routines included in QuantumBlackbox are automatically madeavailable via JavaScript. A particular experiment can be entirely coded inJavaScript, and will have full access to a wide variety of resources providedboth by QuantumBlackbox and the LabVIEW suite.C.1.5 User InterfaceQuantumBlackbox runs natively as a windowed desktop application. Mostof the graphical user interface provided is coded using web technologies.Then each window displayed to the user simply hosts an instance of WebKit[86] (included as part of the Qt library). Qt provides the necessary bindingswhich allow user actions on these web pages to be passed back to the C++program, and vice-versa. The result is a very professional but easy-to-createinterface.In order to make parts of the interface that are hosted in WebKit, Quan-tumBlackbox makes use of the following open-source components:133C.1. QuantumBlackbox• jQuery [87]. jQuery is an JavaScript library that enhances and sim-plifies that process of scripting elements on a web page.• jQuery UI [88]. jQuery UI is an library that provides enhanced user-interface elements for use on a web page, and uses jQuery.• CodeMirror[89]. CodeMirror is an interactive code editor that runs ona web page and uses JavaScript. QuantumBlackbox uses this com-ponent to expose a graphical user interface to the user where theJavaScripts that access lab hardware and run experiments can beedited.The application also includes a module that can display any numberof plots to the user. This module is accessed via JavaScript, and doesnot use WebKit, but rather makes use of an open-source library known asQCustomPlot [90].C.1.6 Genetic Search Implementation DetailsQuantumBlackbox uses GAlib [52], open-source collection of C++ classesthat implement many of the common genetic algorithms and genetic oper-ators. The genetic search component of QuantumBlackbox exposes a userinterface which allows the user to choose values for many parameters pro-vided by GAlib, including the population size, crossover and mutation rates,the replacement percentage, the selection scheme and scaling scheme. It alsoprovides for the selection of the mapping function.When GAlib is called on to run a genetic search, it repeated calls what itcalls an “objective function.” This user-defined function is called by GAlibwhen it wants to receive an objective score for a particular genome. Quan-tumBlackbox sets this objective function to be a wrapper function that callsthe user-provided JavaScript mapping function. This function is executedin an environment that has access to all same resources that the interactiveJavaScript editor and engine (Section C.1.4) have access to. As such, themapping function can basically do anything, including calling on the SLM,the spectrometer, and other laboratory equipment.Each mapping function is actually defined as part of a JavaScript object,stored in a text file. As well as the map property which holds the mappingfunction, the object also defines a configure property. This latter propertyholds a function that returning the form of the genome which GAlib shoulduse in the search. This entire object, originally loaded from the file, livespersistently in the JavaScript engine throughout the duration of the search.134C.2. MathematicaThus there exists a way to create very intelligent, flexible mapping functionsthat remember their state from one call to a next, if needed.QuantumBlackbox provides a mixed genome, meaning that searches canbe conducted with any number of the following features:• A binary string genome. This is simply a string of bits, the lengthrequested by the configuration function.• A decimal number genome. This genome provides a collection of dec-imal numbers; each number in the genome will be represented by therequested number of bits, the number received in the mapping functiontaking values between some user-specified minimum and maximum.The number of bits in the representation will affect the performanceof the search. The numbers can also, optionally, be Gray coded [40, 52].These numbers are also binary strings, as far as basic genetic operatorsare concerned.• A real number allele genome. This provides another way to representreal numbers in a genome. The user may request one or more of: areal number within a specified range (internally represented differentlythan the decimal number genome), a set of numbers between lowerand upper limits, with a given spacing, or a user-specified set of realnumbers. This genome has its own special genetic operators associatedwith it; for example, mutation may change the number by a Gaussian-random amount, rather than flipping a bit.The mapping function receives a copy of the genome, translated to aJavaScript-friendly form. When it is done it returns a JavaScript objectcontaining the objective score, and any other data that the user would like tostore with this genome evaluation. The score along with all data are stored,in the text-based JavaScript Object Notation (JSON) [91] format, forsubsequent analysis and export.C.2 MathematicaThe numerical simulations related to the coherent control of two photonabsorption in atomic rubidium (appearing in Chapter 7) use Mathemat-ica routines that perform numerical principal value integrals (see also thereference in Section B.2 of Appendix B).To make these Mathematica routines callable from LabVIEW and Quan-tumBlackbox, a small module known as a dynamic-link library (DLL) was135C.2. Mathematicawritten. This DLL uses the interface known as MathLink to communicatedirectly with the Mathematica kernel. LabVIEW loads this DLL, whichallows it to communicate with Mathematica. The TCP/IP channel betweenQuantumBlackbox and LabVIEW then allows the QuantumBlackbox di-rect access to numerical integration results returned from the Mathematicakernel.136

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