NEAR-INFRARED ULTRAFAST DEGENERATE OPTICAL PARAMETRIC AMPLIFICATION

Transcription

1 NEAR-INFRARED ULTRAFAST DEGENERATE OPTICAL PARAMETRIC AMPLIFICATION by Andrew J. Niedringhaus

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3 A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Master of Science (Applied Physics). Golden, Colorado Date Signed: Andrew J. Niedringhaus Signed: Dr. Charles Durfee Thesis Advisor Golden, Colorado Date Signed: Dr. Thomas Furtak Professor and Head Department of Physics ii

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5 ABSTRACT A near-degenerate optical parametric amplifier that takes the pulsed output from a fiber oscillator near 1040nm and generates pulses at 2µm was designed, modeled, constructed, and tested. The optimal nonlinear crystal and system configuration were chosen based on optical transmission, conversion bandwidth, effective nonlinearity, group velocity mismatch, and damage thresholds. A numerical model accounting for both time-dependent optical parametric gain and wavelength-dependent material dispersion was developed. The model results agree reasonably well with the measured output of the OPA pumped at 800nm. Modeling predicts that the optimized OPA with a 1mJ, 1040nm input can produce 250µJ signal pulses with enough bandwidth to compress to about 70fs. The chirp of the seed beam and group velocity walkoff between the pump and seed are the primary factors that limit the signal gain and bandwidth. iii

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7 TABLE OF CONTENTS ABSTRACT iii LIST OF FIGURES vi LIST OF TABLES ix LIST OF ABBREVIATIONS x ACKNOWLEDGMENTS xi CHAPTER 1 INTRODUCTION CHAPTER 2 DEGENERATE OPA DESIGN Phase-Matching Conversion Bandwidth Group Delay and Group Velocity Dispersion Effective Nonlinearity Final Design CHAPTER 3 OPA MODELING Split-Step Method Test Cases CHAPTER 4 SIMULATIONS AND RESULTS Split-Step Model Analysis Conversion Bandwidth for Broadband Seed Chirped Seed and Pump Timing Effect of Back-Conversion on Pulse Quality iv

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9 4.1.4 Effect of Group Velocities on Conversion Efficiency Envelope Phase Information Modeling Limitations Ti:sapphire-Pumped OPA Results Operating Parameters Second-Harmonic Generation Measured Signal Spectra Measured Idler Spectra Fiber-Pumped OPA Simulation Conclusion REFERENCES CITED CD Pocket v

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11 LIST OF FIGURES Figure 1.1 Schematic of current-enhanced SASE system Figure 1.2 Type I and II conversion bandwidths phase-matched near degeneracy for 1040nm pump in 2mm thick BBO Figure 2.1 Noncollinear wavevectors Figure 2.2 Illustration of refractive index ellipsoid for biaxial crystal Figure 2.3 Biaxial phase-matching geometry Figure 2.4 Conversion bandwidths for BBO type I, IIa, and IIb Figure 2.5 k for BBO type I, IIa, and IIb Figure 2.6 Conversion bandwidths for 5mm thick BBO, LBO, BiBO, KTP and KTA Figure 2.7 Relative group delays for BBO type I, 1040nm pump Figure 2.8 Relative group delays for KTA type I x-z plane, 1040nm pump Figure 2.9 Signal/idler group delay vs. signal frequency for various pump wavelengths in BBO Figure 2.10 Signal/idler GVD at degeneracy vs. pump wavelength for various crystals Figure 2.11 Degenerate OPA schematic Figure 2.12 Possible phase-matched type I BBO configurations Figure 3.1 Illustration of symmetric split-step Fourier method Figure 3.2 Figure 3.3 Signal and idler amplitudes vs. position for undepleted pump and split-step model Split-step propagation through 1mm BBO with no nonlinear interaction vi

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13 Figure 3.4 Relative field energies vs. crystal position from split-step propagation in BBO Figure 3.5 Scaled split-step and analytic field intensities for SHG Figure 3.6 Z-step convergence test for split-step model Figure 4.1 Figure 4.2 Figure 4.3 Normalized split-step spectra and theoretical conversion bandwidth at several crystal angles Dependence of conversion bandwidth on timing between short pump pulse and chirped seed pulses Effects of back-conversion on temporal profiles and spectra of pump, signal, and idler Figure 4.4 Normalized energy vs. crystal position with back-conversion Figure 4.5 Dependence of conversion bandwidth on timing between short pump pulse and chirped seed pulses Figure 4.6 Unwrapped spectral phase of positively chirped seed pulse Figure 4.7 Illustration of spatial walkoff between pump and seed through 5mm crystal Figure 4.8 Modified degenerate OPA schematic Figure 4.9 Conversion bandwidth for OPA and signal SHG with 790nm pump phase-matched at 1.5µm Figure 4.10 Measured and predicted signal spectra at different crystal angles with fixed timing and seed amplitude Figure 4.11 Measured and predicted signal spectra for back-conversion split peak Figure 4.12 Measured and predicted signal spectra with different pump/signal timing Figure 4.13 Measured and predicted idler spectra at different spectrometer positions Figure 4.14 Pump/idler angle vs. idler wavelength for several pump/signal angles in BBO with 800nm pump vii

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15 Figure 4.15 Simultaneous effects of pump/chirped seed timing and group velocity mismatch on output signal Figure 4.16 Output signal spectra from 1st and 2nd stage 1040nm-pumped OPA 66 Figure 4.17 Temporal profiles of pump and signal at positions within 2nd stage crystal Figure 4.18 Relative energies of 2nd stage pump, signal, and idler vs. crystal position viii

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17 LIST OF TABLES Table 2.1 Phase-matching angles for BBO (degrees) Table 2.2 Phase-matching angles for LBO (degrees) Table 2.3 Phase-matching angles for BiBO (degrees) Table 2.4 Phase-matching angles for KTP (degrees) Table 2.5 Phase-matching angles for KTA (degrees) Table 2.6 Zero second-order dispersion wavelengths for various crystals (nm). 22 Table 2.7 Nonlinear crystal classifications Table 2.8 Effective nonlinearities at phase-matched orientations Table 4.1 Derivatives with respect to ω of split-step and expected spectral phase ix

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19 LIST OF ABBREVIATIONS β-barium Borate BBO 4th-order Runge-Kutta RK4 Bismuth Triborate BiBO Carrier-Envelope Phase CEP Fast Fourier Transform FFT Frequency-Resolved Optical Gating FROG Full-Width Half-Maximum FWHM Group Velocity Dispersion GVD Group Velocity Mismatch GVM Lithium Triborate LBO Optical Parametric Amplifier/Amplification OPA Potassium Titanium Oxide Phosphate KTP Potassium Titanyle Arsenate Second-Harmonic Generation KTA SHG Undepleted Pump Approximation UPA Yttrium Aluminum Garnet YAG Zero Second-Order Dispersion Point ZDP x

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21 ACKNOWLEDGMENTS I express my appreciation to Dr. Durfee for his guidance, support and insight through each step of this project and my physics education at CSM. Dr. Squier has also been a valuable asset throughout my optics and general physics education. I would also like to thank Amanda Meier, Mike Greco, Marin Iliev and Ben Galloway for their invaluable assistance in the lab. xi

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23 CHAPTER 1 INTRODUCTION Optical Parametric Amplification (OPA) is a nonlinear optical process by which energy is transferred from a pump beam with a fixed central frequency to lowerfrequency signal and idler beams. The signal and idler wavelengths can be tuned by matching the relative phases of the three beams within a nonlinear crystal. In order to achieve significant gain, relatively high pump intensities are required, making OPA well-suited for femtosecond pulses, although nanosecond and continuous-wave parametric systems also exist. OPA is used in a wide variety of spectroscopy applications in which wavelength tunability is desirable. Femtosecond OPA is often used to make measurements on extremely short timescales for a wide range of physics, chemistry and biology experiments. OPA has largely replaced dye lasers as the method of choice for producing widely tunable laser pulses. Tuning an OPA involves simply adjusting the angle of a nonlinear crystal, whereas tuning a dye laser requires significant reoptimization after changing dyes [1]. Producing laser pulses near 2µm is particularly useful for high-harmonic generation. This process can be described in terms of the potential of an electron bound to an atom subjected to an intense oscillating electric field. The field adds a linear component to the potential, allowing the electron to tunnel through the potential barrier. It then accelerates in the electric field until the field changes sign and pulls the electron back toward the atom. The electron is re-absorbed by the atom and emits a single photon whose energy hω includes the kinetic energy that the electron gained in the electric field. Essentially, using a lower frequency laser, such as the output of the OPA designed for this project, gives the electron more time to accumu- 1

24 late kinetic energy before being re-absorbed. The cutoff for the harmonic spectrum is inversely proportional to the frequency of the laser [2]. High-harmonic generation can make coherent attosecond soft x-ray pulses [3] which are pushing the limits of time resolution for dynamic measurements. High-harmonic pulses are especially useful for biological imaging applications. Another important application of OPAs is the production of carrier-envelope phase (CEP) stable pulses. When ultrashort pulses are compressed to within a few optical cycles, the envelope amplitude does not fully describe the actual peak amplitude of the field. If the peak of the carrier does not coincide with that of the envelope, the maximum amplitude of the pulse can become significantly lower than when the two are in phase. This variation can have a significant impact on the effectiveness of high-harmonic and attosecond pulse generation. In the case of parametric three-wave mixing, the phases of the three pulses are related such that the CEP of the idler is proportional to the difference between that of the pump and signal. When the signal is seeded by a white-light continuum generated by the same pump pulse, as the OPA presented here is, the idler has a stable CEP [4]. The OPA designed and built for this project will be one component of a system being built in collaboration between CSM, CSU and KMLabs for a project funded by the U.S. Department of Energy. The tunable output of the OPA near 2µm will be used for an enhanced self-amplified spontaneous emission configuration in a free electron laser, similar to the system described by Zholents [5]. The OPA output pulses should be approximately fs and centered at 2.2µm [6]. Figure 1.1 shows a schematic of the enhanced SASE system, for which the OPA output will be used as the laser source. In order to achieve sufficiently short output pulses, designing the system to have a wide conversion bandwidth is critical, since the bandwidth determines the Fourier transform limit for temporal compression. In general, there are two types of OPA 2

25 ¼ 5-10, where d, but with minimal leration the electron hose chromatic disent of the peak curs enter a long undud SASE process but m. The raced portions will be Ä whose total due upstream, energyominate SASE emisnce, U.S. Dept. of Energy C02-76SF the laser/wiggler energy modulator (EM) at 4.54 GeV, and a line-charge (1D) model of coherent synchrotron radiation (CSR) in and after the bends. Figure Figure 1: 1.1: A schematic Schematic of current-enhanced ESASE as applied SASEto system the LCLS. [6] configurations with which the phases of the three beams can be matched to achieve gain: type I, in which the signal and idler have the same polarization, and type II, in which they have orthogonal polarizations. For OPAs operating near the degeneracy point (where signal and idler have the same wavelength), type I phase-matching provides much wider bandwidth than type II. Section 2.2 explains in detail why this is. Figure 1.2 illustrates this by comparing type I and type II conversion bandwidths near degeneracy for several phase-matched signal wavelengths. Clearly, the type I configuration provides the best bandwidth in this case. conversion factor Type I, 2Μm Type II, 2Μm Type I, 2.1Μm Type II, 2.1Μm Type I, 2.2Μm Type II, 2.2Μm Figure 1.2: Type I and II conversion bandwidths phase-matched near degeneracy for 1040nm pump in 2mm thick BBO 3

26 The OPA design has elements of designs presented in several previously reported OPAs. The configuration from Wilson et al. [7] is a two-stage, NIR, white-light continuum seeded OPA. This design incorporates a similar two-stage amplification system, and the first stage is seeded by a white-light continuum in a similar fashion. Brida et al.[8] presented a single-stage, degenerate OPA producing a 1.6µm output. Due to bandwidth considerations, their degenerate OPA used type I phase-matching, so signal and idler have the same polarization as well as wavelength. In order to separate signal and idler and to avoid interference effects between the two, they introduced a slight angle between the pump and seed beams. This OPA design also uses type I phase-matching in order to achieve the best possible output bandwidth in the 2µm range, so it includes a similar angle between the pump and signal in both stages. Another similar OPA design was reported by Rothhardt et al. [9], which employed a two-stage degenerate, noncollinear configuration to produce amplified 1030nm light. However, they used the second-harmonic of the seed as the pump, which is significantly different from seeding with a white-light continuum and does not offer the same tunability. To our knowledge, this will be the first OPA to output 2µm femtosecond pulses near the degeneracy point. 4

27 CHAPTER 2 DEGENERATE OPA DESIGN In order to choose an ideal nonlinear crystal and orientation to achieve the best conversion bandwidth, the following design criteria were used: Optical transmission Possible phase-matched orientations Conversion bandwidth/group velocity walkoff Effective nonlinearity Damage threshold Firstly, only crystals with high transmission over the range of wavelengths involved were considered. Of those crystals, we calculated the conversion bandwidth for each possible phase-matched orientation. For those orientations with a wide enough bandwidth to support femtosecond output pulses, the effective nonlinearities and damage thresholds were used for the final decision. 2.1 Phase-Matching In general, parametric mixing takes place for all of the frequencies present in the seed beam, regardless of the crystal orientation. However, if the phases of the three beams are not matched properly, the fields generated by oscillating electrons at different points in the crystal interfere with each other destructively. The three beams must be phase-matched in order for the generated fields to add constructively and produce gain. Phase-matching is achieved by forcing the difference between the wavevectors of the pump to the generated signal and idler to be zero: 5

28 k = k 3 k 1 k 2 (2.1) k i = 2πn i λ i (2.2) where the wavevectors shown in Figure 2.1 depend on the refractive index n of the material and the angles α and β of the signal and idler relative to the pump. Signal, idler, and pump beams are indexed 1, 2, and 3 respectively. At the degeneracy point, the signal and idler have the same wavelength, so the usual definitions become ambiguous. In the following analysis, the signal refers to the seeded beam and the idler refers to the unseeded beam. If the refractive index were constant and the beams were collinear, Equation 2.1 would be satisfied automatically by conservation of photon energy. However, under normal dispersion, the refractive index increases with frequency, so the pump wavenumber is greater than the sum of signal and idler. In order to resolve this, birefringent crystals are used, and the polarizations of the three field components are arranged to make phase-matching possible. α β Figure 2.1: Noncollinear wavevectors Setting k equal to zero for the vector components parallel and perpendicular to the pump wave gives the following general phase-matching equations [10]: n p (λ p ) λ p = n s(λ s ) λ s cos(α) + n i(λ i ) β = arcsin [ ns (λ s ) n i (λ i ) cos(β) (2.3) λ i ] λ i sin(α) (2.4) λ s 6

29 where the refractive indices for the three waves depend on their polarizations. These are the conditions used for both uniaxial and biaxial phase-matching. The pump and signal beams are inputs, while the idler beam is generated such that its wavelength satisfies conservation of energy and the angle β satisfies Equation 2.4. This phasematched idler satisfies conservation of total photon momentum from the pump to the generated signal and idler. In birefringent crystals, the refractive index changes depending on the polarization of the light relative to the optical axes of the crystal. This can be visualized in terms of the refractive index ellipsoid, illustrated in Figure 2.2. By taking a plane normal to the propagation direction of the beam and through the origin, and finding its intersection with the ellipsoid, one gets an ellipse. The width of this ellipse along a given line through the origin corresponds to the refractive index which that polarization experiences. By tuning the angle of the crystal, one can adjust the refractive indices that the pump, signal, and idler experience, forcing them to be phase-matched. Figure 2.2: Illustration of refractive index ellipsoid for biaxial crystal For uniaxial phase-matching, the crystal tuning angle is a rotation about an axis in the ordinary-index plane, so the polarization parallel to the rotation axis has a constant refractive index with respect to tuning angle. The other, orthogonal polar- 7

30 ization has an angle-dependent refractive index given by [10]: 1 n 2 e(θ) = cos2 θ + sin2 θ n 2 o n 2 e (2.5) where θ is the angle between the wavevector and the extraordinary axis of the crystal. For biaxial crystals, the index of a particular wave is given by [10]: sin 2 θ cos 2 φ (n 2 n 2 x ) + sin2 θ sin2 φ (n 2 n 2 y ) + cos 2 θ (n 2 n 2 z ) = 0 (2.6) where θ and φ are the axial and azimuthal angles of the wavevector in spherical coordinates respectively, and the value of n depends on both angles. In the case where n x and n y are equal, this equation reduces to the uniaxial case, and there is no φ-dependence. Using these relations, an ideal nonlinear crystal and orientation were chosen from several possible crystals and phase-matching orientations. We examined phasematching bandwidth and group velocity dispersion for several nonlinear crystals with high transmission in the near-ir range of interest, including beta-barium borate (BBO), lithium triborate (LBO), bismuth borate (BiBO), potassium titanium oxide phosphate (KTP), and potassium titanyle arsenate (KTA). BBO is uniaxial, while the rest are biaxial. Potassium dihydrogen phosphate (KDP) and potassium dideuterium phosphate (DKDP) were not considered, since their optical transmissions are low beyond 1800nm [11]. First, the phase-matching angles for possible configurations of each crystal were calculated using the Sellmeier equations, such that k = 0. Both type I and type II phase-matching configurations were examined. For the uniaxial crystals (in which two axes have the same refractive index), the geometry is such that one polarization projects onto the ordinary axis, and the other polarization is orthogonal to the first, with an angle-dependent mixture of ordinary and extraordinary indices. For biaxial crystals, the analysis only included configurations where one polarization is parallel to one of the crystallographic axes, and the other is orthogonal, projecting onto the remaining two indices. With two possible po- 8

31 larizations for each of the three wavelengths, and three choices of ordinary axes, there are 24 geometries to examine for each biaxial crystal. Figure 2.3 illustrates the three principal axis geometries examined for the biaxial crystals. With normal dispersion, k cannot be zero in the case where all three beams have the same polarization (and thus the same index), eliminating 6 configurations. Of the 18 remaining, the majority do not have phase-matched solutions. x-z plane x y-z plane x x-y plane y y ϕ=0 θ z y ϕ=90 θ z z θ=90 ϕ x Figure 2.3: Biaxial phase-matching geometry The axes are defined for biaxial crystals such that n x < n y < n z. The index equations for each of the three geometries, given by Equation 2.6, reduce to a case analogous to uniaxial phase-matching, where the axis independent of the phase-matching angle acts as the constant ordinary index, and the angle-dependent extraordinary index is a mixture of the indices of the other two axes like in Equation 2.5. Thus, for the three geometries shown above, the index equations are: x-z plane: n o = n y, y-z plane: n o = n x, x-y plane: n o = n z, 1 n 2 e(θ) = cos2 (θ) + sin2 (θ) n x n z (2.7) 1 n 2 e(θ) = cos2 (θ) + sin2 (θ) n y n z (2.8) 1 n 2 e(θ) = cos2 (θ) + sin2 (θ) n x n y (2.9) The three indices n x, n y, and n z also depend on wavelength according to the Sellmeier equations for the material. Phase-matched configurations satisfy Equation 2.3, where pump, signal, and idler are each assigned either ordinary or extraordinary po- 9

32 larization. Tables show the collinear (α = β = 0) phase-matched angles for all polarization combinations within each principal-plane geometry for each crystal examined, assuming pump and signal wavelengths of 1040nm and 2000nm respectively. Refractive index values were calculated using Sellmeier equations for BBO [12, 13], LBO [14], BiBO [15], KTP [16], and KTA [17]. The definitions of type I, IIa, and IIb phase-matching for uniaxial crystals follow the convention used in [18]. X s denote configurations with no phase-matched solution. Table 2.1: Phase-matching angles for BBO (degrees) Type I p(e), s(o), i(o) 21.5 Type IIa p(e), s(o), i(e) 31.0 Type IIb p(e), s(e), i(o) 32.4 Table 2.2: Phase-matching angles for LBO (degrees) x-z plane y-z plane x-y plane p(o), s(o), i(e) X X X p(o), s(e), i(o) X 61.0 X p(o), s(e), i(e) X X X p(e), s(o), i(o) 18.3 X 31.0 p(e), s(o), i(e) X X 69.9 p(e), s(e), i(o) X X X Table 2.3: Phase-matching angles for BiBO (degrees) x-z plane y-z plane x-y plane p(o), s(o), i(e) X p(o), s(e), i(o) X p(o), s(e), i(e) 35.2 X X p(e), s(o), i(o) 8.6 X X p(e), s(o), i(e) X X X p(e), s(e), i(o) X X X 10

33 Table 2.4: Phase-matching angles for KTP (degrees) x-z plane y-z plane x-y plane p(o), s(o), i(e) X p(o), s(e), i(o) X p(o), s(e), i(e) X p(e), s(o), i(o) X X X p(e), s(o), i(e) X X X p(e), s(e), i(o) X X X Table 2.5: Phase-matching angles for KTA (degrees) x-z plane y-z plane x-y plane p(o), s(o), i(e) X p(o), s(e), i(o) X p(o), s(e), i(e) X p(e), s(o), i(o) X X X p(e), s(o), i(e) X X X p(e), s(e), i(o) X X X 2.2 Conversion Bandwidth The next most important design constraint for those crystals which can be phasematched is conversion bandwidth, or the range of wavelengths that experience gain from OPA at a particular crystal angle. Conversion bandwidth is important because a wider signal bandwidth allows a shorter Fourier transform-limited output pulse. Thus, the crystal with the widest conversion bandwidth will output the shortest pulses after compressing to compensate for chirp in the seed beam. We will show in Section 2.4 that several of the possible phase-matched orientations, including the type I KTP and KTA configurations, have no nonlinear response, and thus cannot generate any gain. However, these cases offer an interesting contrast to the other crystals in the conversion bandwidth analysis that helps to illustrate the connection between bandwidth and group velocity, so they are still included in the analysis below. 11

34 The intensity of the generated signal field is scaled by a factor with a sinc-squared dependence on the phase mismatch and the length of the crystal, according to Equation 2.10 [19]. k is the same phase mismatch used in the phase-matching calculations, depending on the polarization of the three fields, the crystal angle, and Sellmeier equations, and L is the thickness of the crystal. Note that the conversion bandwidth is mostly constrained by the phases of the fields at different wavelengths, and not by the bandwidth of the pump. The seed contains all of the bandwidth, and the phase-matching determines which wavelengths experience gain. These calculations assume a single, central pump wavelength. I s sin2 ( kl/2) ( kl/2) 2 = sinc 2 ( kl/2) (2.10) The refractive indices are either ordinary (without θ-dependence) or extraordinary depending on the phase-matching geometry, so each configuration found in Section 2.1 has a different conversion bandwidth. The pump wavelength λ 3 is fixed, and from conservation of photon energy, the idler wavelength must be: λ 2 = (λ 1 3 λ 1 1 ) 1 (2.11) In order to calculate the conversion bandwidth for each crystal, we find k as a function of signal wavelength and use Equation 2.10 to plot the sinc 2 conversion factor as a function of signal wavelength under the constraint that the idler wavelength satisfies Equation The expression for k is: k(λ 1, λ 3, θ) = 2πn 3(λ 3, θ) λ 3 2πn 1(λ 1, θ) λ 1 2πn 2(λ 2, θ) λ 2 (2.12) Using Equation 2.12 for k in Equation 2.10, plots of the sinc 2 conversion factor were generated and compared for all of the possible phase-matching configurations and angles derived in Section 2.1. All plots assume a 5mm crystal thickness, collinear beams, and a pump wavelength of 1040nm. The crystal angle is held constant to be phase-matched for a signal wavelength of 2µm. Figure 2.4 shows the phase-matching 12

35 conversion factor for all three BBO geometries Type I Type IIa Type IIb Figure 2.4: Conversion bandwidths for BBO type I, IIa, and IIb Clearly, type I phase-matching results in a much wider conversion bandwidth in this case. The reason for this can be illustrated by examining the plots of k vs. wavelength over the same range, shown in Figure 2.5. The phase-matching angles were calculated such that all three curves are zero at a signal wavelength of 2µm, but their derivatives at that point are significantly different. The slope of k for type I phase-matching is closer to zero than the type II curves, so the phase mismatch at other wavelengths is less for type I than for type II configurations. Thus, for BBO, type I phase-matching provides conversion over a significantly wider range of wavelengths than type II. The type I k curve also equals zero at 2.16µm, which is exactly what the idler wavelength must be when the signal is at 2µm. This is because, in a collinear type I configuration, there is no physical distinction between signal and idler since they both have the same polarization. In these plots, when the signal wavelength crosses the degeneracy point (where signal and idler wavelengths are equal) at 2080nm, the signal becomes the lower-energy field and the idler the higher-energy field. Again, 13

36 Type I Type IIa Type IIb Figure 2.5: k for BBO type I, IIa, and IIb to avoid this ambiguous definition near degeneracy, signal will refer to the seeded beam and idler will refer to the unseeded beam. Since the system is phase-matched for a 2µm signal and 2.16µm idler, and the definitions of signal and idler are arbitrary in the expression for type I k, then it must also be symmetrically phase-matched for a 2.16µm signal. However, this is not true for type II geometries, since the signal and idler have different polarizations and break this symmetry. Thus, the type II phase-matching, k curves only intersect the x-axis at 2µm. This signal/idler symmetry is the reason the type I conversion factor in Figure 2.4 has a second peak at 2.16µm, while the type II configurations do not. The local minimum between the two peaks is the degeneracy point. As the crystal angle is tuned to be phase-matched closer to the degeneracy point, the separation between the peaks decreases, until they eventually overlap. Using an angle that is phasematched slightly off of the degeneracy point, as in Figure 2.4, results in a larger total conversion bandwidth by taking advantage of this double-peak [7]. The conversion bandwidths for the other crystals exhibit similar behaviors, where the type I configurations in which signal and idler have the same polarization have 14

37 wide bandwidths with double peaks, while those with opposite polarizations have narrow, single peaks. Figure 2.6 compares the best bandwidths of all the phase-matched configurations from Section 2.1, excluding the narrow-bandwidth configurations in which signal and idler have different polarizations. KTA has significantly better conversion bandwidth, while the other crystals are only slightly different from each other. conversion factor BBO Type I LBO x z LBO x y BiBO x z Type Ia BiBO x z Type Ib KTP x z KTP y z KTA x z KTA y z Figure 2.6: Conversion bandwidths for 5mm thick BBO, LBO, BiBO, KTP and KTA 2.3 Group Delay and Group Velocity Dispersion Another important consideration in crystal selection is group delay between the three fields, which is closely related to the phase mismatch and conversion bandwidth. As we show below, the conversion bandwidth is proportional to the relative group delay between the signal and idler to first order. At degeneracy with type I phase-matching, the signal/idler group delay is nearly zero, which is one of the main advantages of the degenerate type I configuration. To second order, the bandwidth depends on the group velocity dispersion of the signal and idler, and it is maximized when the signal and idler are approximately equally far from the zero-dispersion point of the crystal. Essentially, matching the group velocities and the dispersions of the 15

38 signal and idler makes k in Equation 2.10 small near the phase-matched wavelength, making the sinc 2 central peak wider for a given crystal length. Group velocity mismatch leads to a relative delay between the pump, signal, and idler pulses. For ultrashort pulses, this limits the length within the crystal in which interactions can take place. In the near-degenerate case, the walkoff of the pump relative to the signal and idler is primarily what limits the interaction length. This effect is discussed in detail in Section Using a two-stage OPA helps to correct for this by adjusting the timing between the first stage output signal and the second stage pump. This allows temporal overlap of the pump and signal over more total crystal length. Group velocity matching and phase-matching can potentially be achieved simultaneously by using a combination of noncollinear phase-matching and pulse-front tilt [20, 21]. The group velocity v g is defined as the partial derivative of the frequency with respect to k: v g = ω k = ω λ λ k (2.13) Using the fact that ω = 2πc/λ to evaluate the first partial derivative, take the reciprocal of Equation 2.13 to get the group delay, in units of time per length. The k in Equation 2.14 is 2πn(λ, θ)/λ, where the refractive index still has wavelength dependence from the Sellmeier equations that must be evaluated in the partial derivative. 1 = λ2 k v g 2πc λ (2.14) To determine the relative delay between any two field components, take the difference between their group delays. Figure 2.7 shows the relative group delays between pump, signal, and idler using type I phase-matching and a 1040nm pump. Note that the signal/idler group delay is zero at the degeneracy point since the signal and idler there are identical. The negative signs on the pump-signal and pump-idler delays indicate that the pump has a lower group delay than the signal and idler, and is thus 16

39 traveling through the material faster. 50 Pump signal Pump idler Signal idler group delay fs mm Figure 2.7: Relative group delays for BBO type I, 1040nm pump Compare this to the group delays for the wider-bandwidth KTA shown in Figure 2.8. The pump/signal and pump/idler delays are now positive, with a significantly greater magnitude than in BBO. This means that the pulses will be temporally overlapped longer in BBO, allowing for a longer interaction time within the crystal. However, the signal/idler delay at a signal wavelength of 2µm is less in KTA than in BBO, and it is this delay that affects the conversion bandwidth. The expansion of k with respect to the signal frequency, as plotted in Figure 2.5, is related to the group velocity dispersion and conversion bandwidth. In this case, the expansion is centered at the phase-matched signal frequency and assumes a fixed pump frequency, so the derivative with respect to k 3 is zero. k(ω 1 ) k 0 + k ω 1 (ω 1 ω 0 ) k ω0 2 2 ω 1 (ω 1 ω 0 ) (2.15) ω0 In Equation 2.15, k = k 3 k 1 k 2 and ω 0 is the phase-matched signal frequency. The indices 1,2 and 3 correspond to signal, idler, and pump respectively. k 0 is zero from the phase-matching condition. The derivative k 1 / ω 1 is the inverse group 17

40 Pump signal Pump idler Signal idler Figure 2.8: Relative group delays for KTA type I x-z plane, 1040nm pump velocity of the signal. For the derivative of k 2, k 2 ω 1 = k 2 ω 2 ω 2 ω 1 = k 2 ω 2 (2.16) since ω 2 / ω 1 = 1. Thus, to first order, the expansion for k with respect to ω 1 is: ( k1 k(ω 1 ) k ) 2 (ω 1 ω 0 ) +... (2.17) ω 1 ω 2 From Equation 2.13, the partial derivatives in Equation 2.17 are the inverse group velocities of the signal and idler fields respectively. Thus, the slope of the k plot vs. ω 1 is proportional to the group delay between signal and idler in Figure 2.7 [1, 8]. For type I phase-matching, the signal/idler group delay must be zero at the degeneracy point, since signal and idler have the same wavelength and refractive index [22]. This is consistent with Figure 2.5, which shows a local maximum for k at the degeneracy point λ s = 2080nm, and with Figure 2.7, where the signal/idler delay crosses zero at the same point. In general, the slope of k will be small close to the degeneracy point for type I phase-matching, since the signal and idler have the same group velocities there. Thus, the second-order term in the expansion, corresponding to the group velocity disper- 18

41 sion (GVD) per unit length, becomes significant [8]. The second-order coefficient in Equation 2.15 is related to the first-order coefficient by: 1 2 k 2 2 ω 1 = 1 ( k2 k ) 1 (2.18) ω0 2 ω 1 ω 2 ω 1 ω 0 The term being differentiated is the difference between signal and idler group velocities plotted in Figure 2.7, so the second-order term in the expansion of k is proportional to the slope of the signal/idler group delay vs. signal frequency. To achieve the best conversion bandwidth, we want to minimize k over the widest range of wavelengths. The zeroth-order term is forced to be zero at the specified phase-matched frequency, and for type I interactions the first-order term is zero at the degeneracy point. To minimize the second-order term, we need to minimize the coefficient in Equation By evaluating the partial derivative, the second-order coefficient becomes: k 2 ω 1 = 1 ω0 2 ( ) 2 k 2 2 k 1 = ω 1 ω 2 2 ω 1 ω0 1 ( ) 2 k k 1 (2.19) 2 2 ω 2 2 ω 1 ω 0 where the sign of the k 2 derivative changes again to make it with respect to ω 2. The second derivative of k with respect to ω is the group velocity dispersion, so the coefficient is just the sum of the GVD s of the signal and idler. To minimize the second-order phase-mismatch, the signal and idler should have GVD s of opposite sign. Thus, for the best bandwidth, the zero second-order dispersion point (ZDP) of the crystal should be between the signal and idler wavelengths. Operating near degeneracy, this means the ZDP should be approximately equal to the degenerate wavelength. Figure 2.9 shows the relative delay between signal and idler beams vs. signal frequency for several pump wavelengths in BBO. Each plot assumes phase-matching at the degeneracy point for its corresponding pump wavelength. The derivative of these curves with respect to signal frequency is proportional to the GVD, as shown 19

42 in Equation Since we want to minimize the total GVD, the pump wavelength which gives the least group delay slope at degeneracy will have the widest bandwidth. It is plotted in terms of frequency instead of wavelength so that the slopes with respect to frequency can be directly compared. Figure 2.9 shows that a pump wavelength of 740nm has the least slope when it crosses the x-axis which, as expected, has a degenerate signal close to the zero-dispersion wavelength for BBO [8] nm 740nm 800nm 1040nm 1200nm signal frequency for various pump wave- Figure 2.9: Signal/idler group delay vs. lengths in BBO Figure 2.10 shows the slopes of the signal/idler group delay evaluated at the degeneracy point as a function of pump wavelength for each of the crystals evaluated in this chapter. The pump wavelength where each curve equals zero indicates where the second-order term in the k expansion is minimal, giving the best bandwidth near degeneracy. This will also be where the degenerate signal and idler are near the ZDP, according to Equation From this, we can conclude that, the closer the ZDP of the material is to the degenerate signal wavelength, the wider the bandwidth will be. Figure 2.10 shows that, for a pump wavelength of 1040nm, KTA has the smallest value GVD, followed by BiBO, KTP, BBO, and LBO. This is in agreement with the bandwidth plot in Figure 2.6. Note that, while the pump wavelength giving zero-dispersion in KTP is closer to 1040nm than that of BiBO, BiBO has a smaller 20

43 GVD value at 1040nm, and thus wider bandwidth. slope of sig idler group delay fs 2 mm BBO LBO x z BiBO x z KTP x z KTA x z Figure 2.10: Signal/idler GVD at degeneracy vs. pump wavelength for various crystals Table 2.6 shows the wavelengths at which there is zero second-order dispersion for the relevant indices of each crystal. These values are in agreement with the crossing points in Figure 2.10, where the pump wavelength is half that of the signal and idler. In the cases plotted in Figure 2.6, BBO, LBO and BiBO do not have angle-dependent indices for the signal and idler, so the values below are what the signal and idler see. For KTP and KTA, however, signal and idler are angle-dependent, so the signal and idler see components of two axes. The ZDP s give further insight as to why certain configurations have more bandwidth. For LBO, the difference between the GVD on the y and z axes is small, so both geometries have roughly the same bandwidth. In KTP, x-z and y-z geometries are phase-matched, so the GVD is a mixture of those tuning axes. The x and y axes have similar GVD, so both of those geometries have similar bandwidth. In KTA, however, the x-axis ZDP is significantly closer to 2µm than the y-axis, so the x-z configuration has wider bandwidth than the y-z configuration. 21

44 Table 2.6: Zero second-order dispersion wavelengths for various crystals (nm) BBO ordinary 1487 LBO y-axis 1203 LBO z-axis 1208 BiBO y-axis 1598 KTP x-axis 1683 KTP y-axis 1688 KTP z-axis 1789 KTA x-axis 1941 KTA y-axis 1912 KTA z-axis Effective Nonlinearity While phase-matching ensures that the generated fields add constructively to allow gain, it does not address the nonlinear polarization of the electrons which actually generates the new frequencies. The nonlinear response is determined by the structure of the crystal, and generally depends on the crystal orientation and the polarization of the beams. The stronger the nonlinear response is, the thinner the crystal needs to be to achieve a particular gain. From Equation 2.10, a thinner crystal results in wider bandwidth, so the nonlinearity affects the conversion bandwidth. Thus, it is important to consider the effects of both the k expansion and the effective nonlinearity on the conversion bandwidth. The nonlinear polarization driving sum frequency generation can be represented in terms of the lower frequency fields by [19]: P x (ω 3 ) P y (ω 3 ) P z (ω 3 ) = 4ɛ 0 d 11 d 12 d 13 d 14 d 15 d 16 d 21 d 22 d 23 d 24 d 25 d 26 d 31 d 32 d 33 d 34 d 35 d 36 E x (ω 1 )E x (ω 2 ) E y (ω 1 )E y (ω 2 ) E z (ω 1 )E z (ω 2 ) E y (ω 1 )E z (ω 2 ) + E z (ω 1 )E y (ω 2 ) E x (ω 1 )E z (ω 2 ) + E z (ω 1 )E x (ω 2 ) E x (ω 1 )E y (ω 2 ) + E y (ω 1 )E x (ω 2 ) 22

45 The 3 6 d matrix couples all of the possible polarization combinations of the two lower frequency fields into the three components of the nonlinear polarization of the generated ω 3. Since OPA is the inverse process of sum frequency generation, the coupling between the three frequencies will be the same for a given configuration of field polarizations, so the effective nonlinearity of both processes will be the same. The effective nonlinearity d eff is defined by a scalar relationship between the magnitudes of the polarization and electric fields: P 3 (ω 3 ) = 4ɛ 0 d eff E(ω 1 )E(ω 2 ) (2.20) This d eff term is a measure of the strength of the nonlinear interaction between the three fields for a given geometry, and depends on the polarizations of the fields and the crystal orientation. For type I interactions, in which the signal and idler fields have the same polarization, d eff will only depend on terms in the first three columns of the d matrix, while type II interactions depend on the last three columns. In general, many of the elements of the d matrix are zero, while others are equal to each other, which simplifies the expressions for d eff. The d matrices of crystals in the same crystallographic point group share common symmetries and expressions for d eff. Table 2.7 shows the point groups of the crystals of interest. Table 2.7: Nonlinear crystal classifications Crystal Crystal Type Point Group BBO uniaxial 3m BiBO biaxial 2 LBO biaxial mm2 KTP biaxial mm2 KTA biaxial mm2 Midwinter [23] gives a comprehensive table of expressions for d eff for uniaxial crystals in type I and II configurations. The expression for type I BBO is given by: d eff = d 31 sin(θ) d 22 cos(θ) sin(3φ) (2.21) 23

46 where θ is the angle between the propagation vector and the optical axis of the crystal (the extraordinary axis). The angle φ measures the orientation of the crystal in the ordinary index plane. This angle has no effect on the phase-matching or bandwidth analysis, but it does affect the effective nonlinearity. In the case of 3m negative uniaxial crystals, d eff is maximized when 3φ = π/2 + nπ. General approaches of how to calculate d eff for biaxial crystals are given by Ito et al. [24] and Yao et al. [25], and the general expressions for the type I d eff for groups mm2 and 2 were taken from [24]. For the mm2 group, there are only five nonzero terms in the d matrix, and due to structural symmetry, only three unique values are needed to fully define it. For group 2, there are eight nonzero terms and four unique values [19, 24]. For the mm2 crystals, all of the x-z and y-z principal plane geometries have a d eff of zero for type I phase-matching, regardless of phase-matching angle. Thus, KTP and KTA cannot be used for this application, since their only phase-matched principal plane configurations were in the x-z and y-z planes. The measured d matrix values for BBO [26], BiBO [27], and LBO [28] were used to calculate d eff. The values are shown in Table 2.8. Table 2.8: Effective nonlinearities at phase-matched orientations crystal d eff (pm/v) BBO type I 2.08 LBO x-z 0 LBO x-y 0.81 BiBO x-z(1) BiBO x-z(2) 1.52 KTP x-z 0 KTP y-z 0 KTA x-z 0 KTA y-z 0 24

47 While KTP and KTA do have phase-matched type II configurations with effective nonlinearities slightly higher than BBO (around 2.5pm/V), the conversion bandwidths are much narrower than the type I configurations. While the stronger nonlinearity allows these crystals to be thinner, effectively increasing the bandwidth, it does not sufficiently compensate for the narrow type II bandwidth. This is partly due to the larger signal/idler group delay near degeneracy (since the signal and idler have different refractive indices), and partly because type II configurations are not phase-matched on both sides of the degeneracy point. BiBO has a slightly wider conversion bandwidth than BBO for the same crystal thickness, but BBO has a larger d eff at these particular phase-matched orientations and can achieve the same yield as BiBO with a thinner crystal. The damage thresholds of BiBO and LBO are also significantly lower than that of BBO [29], so BBO allows for a more intense pump beam, resulting in better gain. BBO was chosen because it was readily available, it has the largest effective nonlinearity and damage threshold; the higher gain in BBO compensates for its narrower conversion bandwidth compared to BiBO. 2.5 Final Design The system consists of two amplification stages and uses supercontinuum generation for the seed, as shown in Figure For the 1040nm fiber oscillator output, the pulses were assumed to be approximately 100fs and 1mJ, with a repetition rate of 100kHz and a collimated spot size entering the OPA of approximately 1cm. An 80% reflective beamsplitter was used to divert the second-stage pump, followed by a 97% for the first-stage pump. This leaves approximately 194µJ for the first stage, 800µJ for the second stage, and 6µJ for the supercontinuum generation. The damage threshold of BBO is approximately 50 GW/cm 2 for 50fs pulses [7]. Using these energies and the approximate pulse duration and spot size, lenses were chosen to downcollimate the two pump beams such that the intensities are below the damage 25

48 threshold of BBO. The intensities at the crystals can be adjusted with small chanages in the lens positions. Due to the high intensity of the downcollimated second-stage pump, dielectric mirrors with a high damage threshold were used for the second-stage, while protected gold mirrors were used for the first-stage and seed generation. The lenses are N-BK7 with anti-reflective coatings from nm. 6 μj R=97% R=80% 194 μj 800 μj 1040 nm 100 fs 1 mj 2mm YAG 2 o BBO 2 o BBO ~ 2 μm Figure 2.11: Degenerate OPA schematic To achieve the best conversion efficiency and a clean spatial profile, the first stage operates in a high-gain regime, while the second stage has higher power but less gain. This helps to avoid the effect of gain narrowing due to the fact that the Gaussian profiles of the beams lead to more gain in the beam centers than the edges [30]. The first stage pump is tightly focused to reach intensities close to the BBO damage threshold and gets high gain. The second-stage pump has a larger spot size, distributing the power more evenly over a larger crystal surface, which gives lower gain but higher total energy conversion. The total gain is optimized by fine-tuning the spot sizes on both crystals to avoid back-conversion. The supercontinuum seed is generated by tightly focusing the lowest-energy beam into either a sapphire or undoped-yag crystal. For the best seed, the intensity should be adjusted so that there is a single, stable filament that extends into the visible range, and the size of the seed beam should approximately match that of the pump beam at the surface of the crystal [7]. An iris is used to fine-tune the pulse energy that 26

49 reaches the sapphire or YAG. Using 800nm pulses, it is possible to generate a white light spectrum that extends up to 2100nm [8]. This should be enough bandwidth to seed both sides of the degeneracy point for a 1040nm pump. A small pump-signal angle of approximately 2 is used to spatially separate the signal and idler beams and to avoid interference effects between them, since they have the same polarization and roughly the same wavelength at degeneracy [8]. Since photon momentum must be conserved, the generated (unseeded) idler beam will be on the opposite side of the pump from the (seeded) signal beam. Given a pump/signal angle α, one can calculate the pump/idler angle β for a given idler wavelength using Equation 2.4. All of the signal wavelength components propagate through the crystal in the same direction as the seed beam, so α is constant. However, since β depends on the wavelengths of the signal and idler, different idler wavelengths will leave the crystal at different angles, making a fan of spatially chirped idler rays. We use the seeded signal beam as the output because it does not experience the same spatial chirp as the idler. Depending on whether the pump polarization is vertical or horizontal, the BBO crystal must be oriented and tuned differently to achieve type I phase-matching. Figure 2.12 illustrates the two ways that type I phase-matching can be achieved. Figure 2.12(a) has a pump with horizontal polarization (in the plane of the table), and vertical polarization for signal and idler. The extraordinary axis of the crystal is in the plane of the table, and the crystal is rotated about the axis perpendicular to the table to tune the output. In this case, the phase-matching analysis in Section 2.1 is changed by adding α 2 to Equations 2.3 and 2.4. Such a small angle has a negligible effect on phase-matching angles and conversion bandwidth, so it was not included in the analysis. The modeling described in Section 3.1 does, however, take this angle into account. Depending on the spot sizes of the pump and seed beams at the crystal, this angle could potentially limit the interaction length due to spatial 27

50 walkoff. However, given that the downcollimated pump spot radius is approximately 1mm and the seed spot size should be slightly larger, a 2 angle will still leave the beams well-overlapped through a 2-3mm crystal. s(o) p(e). e o. o.. i(o) p(e) s(o) (a) Horizontal pump configuration s(o) p(e) o. e.. (b) Vertical pump configuration o i(o) p(e) s(o) Figure 2.12: Possible phase-matched type I BBO configurations Figure 2.12(b) switches the polarizations of each beam, so now the crystal is rotated about the axis in the plane of the table and perpendicular to the pump beam. The exact phase-matching expressions become more complicated in this case, because the signal and idler pick up a small component of the extraordinary index, requiring the full, two angle index ellipsoid treatment of Equation 2.6 for the signal and idler refractive indices. Again, due to the small angles involved, these indices were assumed to be approximately the ordinary index. Since the 1040nm fiber amplifier output was not yet available for testing, this system configuration was tested using 800nm, 60fs pulses from an amplified 1kHz Ti:sapphire laser. Reconfiguring the system for a different pump wavelength only requires minor realignment due to slightly different refraction in the beam splitters and lenses. The 1040nm beam splitters do not have the correct percent reflections for 800nm light, so two 80% reflective 800nm beam splitters were used instead. These changes and their effects on the output are described in more detail in Chapter 4. 28

51 CHAPTER 3 OPA MODELING As the three fields propagate through the nonlinear crystal, they are affected by the dispersion of the crystal in addition to nonlinear interactions. In order to accurately predict the magnitudes and phases of the electric field components, we must account for both nonlinear mixing, which occurs in the time domain, and the spectral phase shift resulting from frequency-dependent dispersion in the material, which occurs in the frequency domain. This model can be used to predict how the durations, amplitudes, bandwidths, and phase characteristics of the pulses change with changes in parameters such as the initial pump and signal amplitude, duration, spectral phase, relative timing, relative angle, incident angle to the crystal, and frequencies. This information helps to optimize these parameters in the design process, and the modeling predictions are compared to measured results in Chapter Split-Step Method The nonlinear mixing process between the signal, idler, and pump fields, indexed 1,2, and 3 respectively, is governed by the set of coupled, first-order nonlinear equations [19]: da 1 dz = 2iω2 1d eff A k 1 c 2 3 A 2e i kz (3.1) da 2 dz = 2iω2 2d eff A k 2 c 2 3 A 1e i kz (3.2) da 3 dz = 2iω2 3d eff A k 3 c 2 1 A 2 e i kz (3.3) where d eff is the term calculated in Section 2.4, having units inverse to the field amplitude, and z is the longitudinal position of the fields as they propagate through the nonlinear crystal. A i is defined in terms of the electric fields such that E i (z, t) = 29

52 E i e iωit + c.c. and E i = A i (z) exp(ik i z), so A i is the field envelope which varies with position. It is complex, containing the phase of the field. k is defined the same as in Equation 2.1. Solving this coupled system of ODEs accounts for the OPA mixing process as the fields propagate through the material. Equations account for the nonlinear mixing process, but they do not account for the frequency-dependent dispersion as the pulses propagate through the crystal. In order to simultaneously account for the nonlinearity and dispersion, a slightly modified version of the split-step Fourier method presented by Agrawal [31] was employed. This method starts by defining each nonlinear field equation in terms of abstract operators of the form: A i z = ( ˆD + ˆN i )A i (3.4) where i is the field index, ˆD is the operator representing the dispersion effects, and ˆN accounts for the nonlinearity. This is a general form of the nonlinear Schodinger equation, which can be solved for a new field position z + h at a fixed time T by exponentiating the operators. A i (z + h, T ) = exp[h( ˆD + ˆN i )]A i (z, T ) (3.5) Note that, by replacing z with t and the operator expression with the Hamiltonian, this is completely analogous to the time-evolution operator for a quantum wavefunction [32]. In order to decouple the dispersion and nonlinearity operators, we can use the commutator identity: ( exp(â) exp(ˆb) = exp â + ˆb [â, ˆb] + 1 ) 12 [â ˆb, [â, ˆb]] +... (3.6) where [â, ˆb] = âˆb ˆbâ. Applying this to the exponential term in Equation 3.5 gives: ) exp(h ˆD) exp(h ˆN i ) = exp (h ˆD + h ˆN i + h2 2 [ ˆD, ˆN i ] + h3 12 [ ˆD ˆN i, [ ˆD, ˆN i ]] +... (3.7) 30

53 Thus, exp( ˆD + ˆN i ) exp( ˆD) exp( ˆN i ) to order h 2. By choosing a small enough step size to make the commutator terms negligible, the dispersion and nonlinearity operators become decoupled and can be applied to the fields one at a time. For OPA modeling, there are still three coupled equations for the fields, but by including the dispersion operator they become: da 1 dz = 2iω2 1d eff A k 1 c 2 3 A 2 + ˆDA 1 (3.8) da 2 dz = 2iω2 2d eff A k 2 c 2 3 A 1 + ˆDA 2 (3.9) da 3 dz = 2iω2 3d eff A k 3 c 2 1 A 2 + ˆDA 3 (3.10) where the explicit phase-matching terms exp(i kz) have been removed, since ˆD will account for phase-mismatch in the spectral domain by changing the field phases. Equations are in the form of Equation 3.4 if we define ˆN i such that: ˆN 1 = 2iω2 1d eff A 3 A 2, k 1 c 2 ˆN2 = 2iω2 2d eff A 3 A 1, k 2 c 2 ˆN3 = 2iω2 3d eff A 1 A 2. (3.11) k 3 c 2 A 3 A 1 When each of these operators acts on its respective field, it returns to Equations (without the k term). Since each equation is now in the form of Equation 3.4, we can make the decoupled approximation for each field: A 2 A i (z + h, T ) exp(h ˆD) exp(h ˆN i )A(z, T ) (3.12) assuming a small enough step size. The nonlinearity acts on A i first, such that: exp(h ˆN i )A i (z, T ) = (1 + h ˆN i + h2 2 ˆN 2 i +...)Ai (z, T ) = A i (z, T ) + h A i(z, T ) z = A i (z + h, T ) + h2 2 2 A i (z, T ) z Since this does not account for dispersion, the A i (z +h, T ) term is just the solution to ˆN i A i = A i / z, which can be iterated by steps of h using a 4th-order Runge-Kutta 31

54 algorithm (RK4) [33]. After performing the nonlinear operation in Equation 3.12 using RK4, we must operate with the exp(h ˆD) term. This can be done in the spectral domain by Fourier transforming A i, operating with exp[h ˆD(ω)], and inverse Fourier transforming. exp(h ˆD)A i = F 1 T {exp[h ˆD(ω)]F } T A i (3.13) Agrawal approximates ˆD as an expansion with respect to T in the time domain and translates this into ˆD(ω). Instead of approximating the dispersion, the exact expression for the spectral phase shift in Equation 3.14 can be used, in terms of the frequency-dependent refractive index. e h ˆD(ω) = e iφ(ω) (3.14) ( ω φ(ω) = c n(ω) ( ω ) ) ω c n(ω) h (3.15) The first term of Equation 3.15 is the phase shift that a particular field accumulates as it propagates through the material. In order to prevent the pump, signal, and idler fields from moving off of the time grid due to their group velocities, the second term of the phase shift in Equation 3.15 compensates for the group velocity of the field with central frequency ω 0 (recall from Equation 2.13 that the inverse group velocity is k/ ω). In this case, the pump was chosen to be the reference field, so group velocities are measured relative to the pump field, and ω 0 = ω p. Thus, the n(ω) being differentiated is that of the pump polarization. The error of the split-step method can be further reduced to O(h 3 ) by symmetrizing each step [31], such that: A i (z + h, T ) exp(h ˆD/2) exp(h ˆN i ) exp(h ˆD/2)A(z, T ) (3.16) This is because symmetrizing the operations makes the first commutator in Equation 3.7 zero. Simply applying 3.16 repeatedly propagates each field through the material, ω 0 ω 32

55 for each time t. Since ˆD commutes with itself, all of the dispersion terms except the first and last can be combined and evaluated as whole steps in h. This effectively offsets the time domain and frequency domain propagation by h/2 in the crystal. Figure 3.1 illustrates this symmetrized propagation. Nonlinearity h h h h h/2 h h h h/2 Dispersion Figure 3.1: Illustration of symmetric split-step Fourier method The green arrows in Figure 3.1 represent fast Fourier transforms (FFT), red arrows are inverse Fourier transforms, black arrows propagate the fields in the time domain, and blue arrows propagate in the frequency domain. The process begins with initial field envelopes for the pump, signal and idler as a function of time, on a grid of time points. The fields are Fourier transformed, and the phase shift in Equation 3.14 is applied to each point on the frequency grids, propagating by h/2. Apply an inverse FFT, then apply the nonlinearity by using the RK4 algorithm on Equations to move by a step h, for each time point on the grid. Fourier transform again, apply a new phase shift, and iterate until the fields have propagated through the length of the crystal in both domains. In the limit as h 0, this effectively accounts for nonlinear and dispersive effects simultaneously. The time grid is defined such that the peak of the pump pulse begins in the center, and it remains centered as the fields propagate, since the grid moves at the pump group velocity. The frequency grid is also centered on the pump, so the frequencies on the grid range from zero to twice the pump frequency. When the signal and idler envelopes are Fourier transformed, their FFT s are rotated so that the zero-frequency components of the envelopes are placed at their carrier frequencies relative to the 33

56 pump. That way, the phase shift in Equation 3.15 can be applied to all three fields using the same frequency grid. After applying the phase shift, the signal and idler grids are rotated back before applying the inverse FFT. The primary advantage of this approach is that it avoids using negative frequencies in the phase shift, allowing the Sellmeier equations to be evaluated directly from the values on the grid. 3.2 Test Cases In order to test whether the split-step model gives reasonable, physical results, the output was compared to several analytical cases, including the frequency-dependent group velocity with no nonlinear effects, the signal and idler field amplitudes with the undepleted pump approximation (UPA), and the response of the system to phase mismatch. The case of second-harmonic generation was also examined. In the UPA case, the analytic field amplitudes for signal and idler are given by [19]: A s (z) = A s (0)cosh(κz) (3.17) ( ) 1/2 ns ω i A p A i (z) = i n i ω s A p A s(0)sinh(κz) (3.18) κ 2 = 4d effω 2 sω 2 i k s k i c 4 A p 2 (3.19) Figure 3.2 shows the analytic and numerical signal and idler peak amplitudes vs. distance through a BBO crystal, using a 1040nm pump, 2µm signal, 500V/µm pump amplitude and 1V/µm signal amplitude. Both fields show their respective cosh and sinh behavior for the first 500µm, then start to diverge. This is because in the splitstep model, the signal and idler drift away from the pump due to the GVM, so the interaction becomes weaker than the ideal case as the fields propagate further through the crystal. Figure 3.3 shows a plot of the final pump (1040nm), signal (2µm), and idler fields after propagating 1mm through BBO with d eff set to zero. For this test case the field 34

57 4 3 Signal UPA Idler UPA Signal model Idler model Figure 3.2: Signal and idler amplitudes vs. position for undepleted pump and splitstep model Pump Signal Idler Figure 3.3: Split-step propagation through 1mm BBO with no nonlinear interaction 35

58 amplitudes were scaled to one and the initial pulses were 100fs Gaussians. The time grid moves so that the pump pulse remains stationary, and the differences between the times of the peaks give the group velocity walk-off per millimeter. The model agrees with the predictions in Figure 2.7 with a 2µm signal wavelength. From Equation 2.14, the predicted group delay between the pump and signal with a 21.5 crystal angle is 55.9 fs/mm, and the pump-idler delay is 66.7 fs/mm. The space between pump and signal peaks in Figure 3.3 is 55.5 fs, and the pump/idler delay is 67.6 fs. These time plots are defined such that the leading edge of the pulse is on the left side, indicating that the pump pulse arrives ahead of the signal and idler at these wavelengths. Figure 3.4 shows the relative energies between pump, signal, and idler in a test case which sets the crystal angle 1 off from the ideal phase-matching angle for a 1040nm pump and 2µm signal. The resulting phase mismatch causes back-conversion before the pump can be fully depleted. This demonstrates qualitatively that the nonlinear interaction behaves as expected, and also proves that the simulation conserves the energy of the fields to within three orders of magnitude using 10µm steps. The curves become distorted since the relative timing between the three fields changes due to GVM. As a final test case, the code was applied to second-harmonic generation (SHG), which is the inverse process to degenerate OPA. This can be done simply by setting the pump wavelength to exactly half that of the signal, in this case 1µm and 2µm respectively, setting the initial pump amplitude to zero, and the initial idler envelope to that of the signal. The sum of the identical signal and idler fields are treated as the fundamental field and the pump as the generated second-harmonic. The phase-matching constraints for SHG are identical to those of OPA. The analytic field equations for SHG in the case of perfect phase-matching are given by Equations [19]. 36

59 Pump Signal Idler Total Figure 3.4: Relative field energies vs. crystal position from split-step propagation in BBO u 1 (z) = sech(z/l) (3.20) u 2 (z) = tanh(z/l) (3.21) l = (n 1n 2 ) 1/2 c 2ω 1 d eff A 1 (0) (3.22) The indices 1 and 2 are the fundamental and second-harmonic fields respectively, and the fields are scaled by energy such that u 2 1 +u 2 2 = 1. From this scaling definition, the initial fundamental amplitude becomes 2 times the amplitude used for the signal and idler fields in the split-step code. Figure 3.5 shows the absolute squares of the scaled field amplitudes for the analytical and numerical cases. The numerical case shows slightly slower conversion than the analytic case, again due to the fact that the fields involved have different group velocities. Based on these comparisons to well-known analytic behavior for second-order mixing processes, we can be confident that the split-step model is producing accurate results. To demonstrate that the step size in z is sufficiently small, the final signal outputs were compared for an OPA test case propagating through 2mm of BBO using 50, 100, 200, 500 and 1000 steps in z. Figure 3.6 shows the maximum magnitude of 37

60 Fund. split step SHG split step Fund. analytic SHG analytic Figure 3.5: Scaled split-step and analytic field intensities for SHG the differences between the final signal field with 1000 steps and the other four. As the number of steps increases, the solution converges toward the 1000-step solution. Using only 100 z-steps (20µm increments), the deviation from the 1000-step solution is four orders of magnitude less than the signal amplitude. Thus, the step size in z is sufficiently small that it does not have a significant effect on the accuracy of the results for our purposes maximum difference Figure 3.6: Z-step convergence test for split-step model 38

61 CHAPTER 4 SIMULATIONS AND RESULTS Using the design presented in Chapter 2, we apply the split-step modeling technique to estimate the ideal pump intensities and crystal thicknesses, and predict the optimal output for the system in Figure First, we explore how factors such as a chirped seed pulse, relative timing and intensities change the output. Then we apply these concepts to the analysis of the two-stage OPA, pumped at both 800nm and 1040nm. The measured output of a one-stage OPA based on Figure 2.11 and pumped at 800nm is consistent with what the modeling predicts. Lastly, we apply the split-step model to the 1040nm system to find parameters that optimize the gain and estimate what the best-case scenario is in terms of signal conversion efficiency and bandwidth. 4.1 Split-Step Model Analysis Before presenting the OPA simulation results for the final design, it is useful to explore the degrees of freedom available in the split-step modeling code and observe the different effects taking place while isolating them from other phenomena. This can provide more insight into the results from simulating a more realistic system, and also reveals which variables can be used to better optimize a real system. Below, we examine the effects of seed bandwidth, chirp, pump duration and timing, and crystal angle on the signal bandwidth. We also demonstrate how back-conversion and pump/seed group velocity walkoff affect the conversion efficiency and output bandwidths. 39

62 4.1.1 Conversion Bandwidth for Broadband Seed In order to ensure that a wide enough range of frequencies is seeded to observe conversion bandwidth effects using the split-step method, the signal field can initially be defined as an extremely short (e.g. 2fs) Gaussian pulse in time, so that the Fourier transform covers a wide range of frequencies. In reality, such a pulse is shorter than its transform limit with a NIR carrier frequency, but it can be lengthened in time by applying quadratic or higher-order phase to make a more physical input seed, as discussed in Section Figure 4.1 compares the split-step output spectrum from a broadband unchirped seed pulse to the ideal conversion efficiency predicted by Equation 2.10 for several crystal angles. The pump and seed field amplitudes were kept low enough to stay well within the undepleted pump regime to minimize the effects of depletion on the conversion efficiency. A 1040nm pump was used, with a FWHM of 50fs. Propagating through 2mm of BBO introduced GVM that broke the symmetry in time and had a slight effect on the bandwidth. This demonstrates that the split-step method effectively deals with the phase-matching of the fields in the time domain without an explicit phase mismatch term when the signal (seed) has a wide enough bandwidth to encompass the tuning range of the crystal Chirped Seed and Pump Timing In reality, the seed cannot have a 2fs envelope. It has temporal and spatial chirp, as well as higher-order spectral phase components. A simple way to model a more realistic seed is to start with an extremely short pulse with a wide FT-limited bandwidth and apply a quadratic phase shift exp [iφ 2 (ω ω 0 ) 2 ] to the spectrum, where ω 0 is the carrier frequency on which the Gaussian spectrum is centered, and φ 2 is the second-order phase, in units of time-squared. Depending on the chirp applied, this effectively broadens the pulse in time, while maintaining the initial bandwidth. 40

63 Analytic 21.6 o Split step 22.4 o Analytic 22.4 o Split step 24.6 o Figure 4.1: Normalized split-step spectra and theoretical conversion bandwidth at several crystal angles If the seed pulse is chirped enough that it is significantly longer than the pump, then only the range of wavelengths that are temporally overlapped with the pump experience gain. This can limit the bandwidth of the generated signal and idler beams, since only a portion of the full seed spectrum is present in the mixing process. The ideal sinc 2 ( kl/2) bandwidth assumes that all frequencies are equally seeded with uniform pump intensity, which does not account for the duration and shape of the pump pulse. By changing the relative timing between the chirped seed and the pump, different portions of the seed spectrum experience gain, so the peak wavelengths of the signal and idler can be tuned within the sinc 2 range by adjusting the pump/seed timing, without any changes in the crystal angle. Figure 4.2 illustrates how the timing of identically chirped seed pulses relative to a shorter pump pulse affects the bandwidth of the amplified signal. In order to minimize effects due to group velocity walkoff, a thin, 300µm crystal was used for the simulation. We used an 800nm pump, with the crystal phase-matched at degeneracy. The seed pulses have a bandwidth ranging from about 1-3µm centered at 1.6µm, with an applied second-order phase of 100fs 2. They have a FWHM of about 120fs, while the short pump is 20fs in this extreme case. Figure 4.2(a) shows the relative timing and 41

64 scaled field intensity fs pump 20 fs pump 0 fs pump 20 fs pump (a) Scaled intensities vs. time of identically chirped seed pulses relative to short pump pulse Long pump pulse 40 fs pump 20 fs pump 0 fs pump 20 fs pump (b) Resulting spectral intensities of amplified signals compared to signal from long pump pulse Figure 4.2: Dependence of conversion bandwidth on timing between short pump pulse and chirped seed pulses 42

65 duration of the pump and seed pulses, and Figure 4.2(b) shows the resulting signal spectra after propagating through the crystal. The black curve in Figure 4.2(b) is the signal bandwidth using the same seed pulse, but a 150fs pump pulse, which allows for nearly equal pumping over the full seed spectrum. Since the seed is positively chirped, low frequencies arrive first in time, so they are on the left side of the seed pulses. When the pump leads the seed in time (in the +40 and +20fs cases), only the low-frequency components are temporally overlapped with the pump, so the amplified spectrum is on the long-wavelength side of degeneracy. Likewise, for the -20 and -40fs seeds where the seed arrives before the pump in time, only the shorter wavelengths see gain Effect of Back-Conversion on Pulse Quality When the pump field becomes depleted at a particular time point, the signal and idler start to generate a field at the pump wavelength through sum-frequency mixing. Sum-frequency mixing between the signal and idler must also be phase-matched since the wavevectors involved are the same. Not only does this back-conversion reduce the OPA conversion efficiency, it also distorts the temporal profile and spectra of the signal and idler pulses. To illustrate the effects of back-conversion, a split-step simulation was run with an 800nm pump intensity close to the BBO damage threshold, and phase-matched for a 1.4µm signal. The 800nm pump gives less group velocity walkoff than 1040nm, which allows a longer interaction and more back-conversion to take place. Figure 4.3 shows the three temporal pulse profiles at several positions within the crystal. The first plot shows the initial pump and signal Gaussian profiles, with no idler present. The signal and idler get the most gain in the middle of the grid, where the pump is most intense, so the pump depletes in the middle first. The second plot shows the depleted pump and maximal signal and idler intensities; this is near the ideal output, before back-conversion takes place. The third plot shows the three profiles after back- 43

66 (a) Normalized intensity vs. time at several crystal positions showing backconversion. Top left: front surface of crystal, top right: 1.2mm, bottom left: 2.3mm, bottom right: 3.4mm (b) Normalized intensity vs. wavelength at same crystal positions Figure 4.3: Effects of back-conversion on temporal profiles and spectra of pump, signal, and idler 44

67 conversion has taken place and the pump intensity is at a local maximum, and the fourth plot is after re-conversion has taken place and the signal and idler intensities are at a local maximum. The spatial profiles are roughly symmetric about the center of the grid; slight group velocity walkoff between the pulses breaks this symmetry. Figure 4.3(b) shows the spectral intensity vs. wavelength for the same crystal positions as Figure 4.3(a). The pump and signal spectra are initially Gaussian. At the point of maximum signal yield in the second plot, the signal and idler have clean, roughly Gaussian peaks, but the pump becomes slightly distorted from the depletion of the center of the pulse. After back-conversion takes place, in the third and fourth plots, the signal and idler spectra also become distorted and have split peaks. While this may in some cases increase the bandwidth after back-conversion, the temporal distortion of the output pulses makes it less ideal. Figure 4.4 compares the total energies of the three pulses at positions through the crystal. The dashed lines correspond to the positions of the four plots in Figure 4.3(a) and Figure 4.3(b). Note that the positions where the peak amplitudes are at local extremes are close to, but not exactly, where the energy extremes are. This is because there is still some conversion taking place on either side of the pulse peaks which contributes to the total energy. In this case, the signal and idler energies after backconversion and re-conversion are fairly close to the energies before back-conversion. This shows that with a real system, tuning it to maximize the signal or idler output power may not achieve the best conversion efficiency, since it may be at a local maximum after back-conversion Effect of Group Velocities on Conversion Efficiency Temporal walkoff between ultrashort pulses can greatly limit the maximum conversion efficiency by limiting the crystal length over which a nonlinear interaction takes place. The conversion bandwidth analysis in Section 2.3 focused primarily on the signal/idler walkoff, but the pump walkoff is also important, and depends on the 45

68 0.8 Idler Figure 4.4: Normalized energy vs. crystal position with back-conversion wavelengths involved and the Sellmeier equations of the crystal. The effect of pump/signal timing on the output signal intensity is illustrated in Figure 4.5, in the case where there is significant GVM between the pump and signal. In this simulation, a 1040nm pump was used, with a 2mm BBO crystal phase-matched for a 2µm signal. Figure 4.5(a) shows the temporal profiles of the output signal intensities for several input signal/pump delays. In this case, the group delay of the 2µm signal is greater than that of the pump (as shown in Figure 2.7 and Figure 3.3), so the pump pulse moves faster through the crystal than the signal. When there is no initial delay between the pump and input signal (as in the 0fs curve), the pump and signal peaks are temporally overlapped in the front of the crystal, but by the end of the crystal the pump arrives before the signal. By timing the pulses so that the signal peak arrives at the crystal slightly before the pump, we can increase the interaction length and improve conversion efficiency, allowing the pump to catch up to the signal and sweep across it. The dashed curve is the temporal profile of the pump pulse, which remains relatively unchanged through the crystal in this undepleted case. 46

69 fs 70 fs 30 fs 0 fs Pump (a) Normalized intensity vs. time for final signal pulses at different initial pump/signal delays. The delay refers to the amount of time the signal peak initially leads the pump peak fs 70 fs 30 fs 0 fs (b) Normalized energy vs. crystal position for signal at the same pump/signal delays Figure 4.5: Dependence of conversion bandwidth on timing between short pump pulse and chirped seed pulses 47

70 Figure 4.5(b) shows the total signal energies for each case in Figure 4.5(a). Clearly, the pump/signal timing has a significant impact on the total yield. The dashed line indicates the position where the temporal profiles were recored, assuming a 2mm crystal. For 2mm, the 70fs delay gives the best conversion, but for a thicker crystal, the 100 or 120fs delays give better results. Due to pump/signal GVM, increasing the crystal thickness beyond 3mm yields diminishing returns on conversion efficiency, since there is poor temporal overlap between the pulses in the extra crystal length Envelope Phase Information In addition to the temporal and spectral intensities, the split-step model also provides the phases of the fields vs. time and frequency, since the amplitude values are complex. The spectral phases of the output pulses can be extracted by simply taking the argument of each point on the spectrum and unwrapping the list to get a continuous curve. In order to test the consistency of the output spectral phase from the model with what we would expect from propagation through BBO, we ran a simulation with a d eff of zero, so that only the dispersion of the BBO affected the output pulses. The simulation used the same chirped input seed as in Section 4.1.2, with the seed spectrum centered at 1.6µm and an 800nm pump, propagating through 3mm of BBO. Figure 4.6 shows the unwrapped spectral phase of the seed pulse at the front and back surfaces of the crystal. The phase values in Figure 4.6 become less precise on the edges of the graph because the magnitude of the field becomes extremely small there, so the phase becomes much more sensitive to small numerical errors. Fortunately, since the field amplitude is many orders of magnitude less than the peak amplitude at the edges, the unstable phase values have very little impact on the accuracy of the model. The input seed has the perfectly quadratic phase corresponding to pure second-order positive chirp, while the output accumulates higher-order phase terms from the split-step propaga- 48

71 Input seed Output seed Figure 4.6: Unwrapped spectral phase of positively chirped seed pulse tion through the crystal. The linear component corresponds to its group velocity relative to the pump. To check whether the phase shift from the split-step method is reasonably accurate, we compared the derivatives with respect to frequency of the unwrapped phase with the actual phase given by: φ = kl = 2πn(λ) L (4.1) λ where L is the length of propagation through the crystal and n is given by the Sellmeier equations of the crystal. Using Equations 2.13 and 2.14, the derivative of φ with respect to ω is related to the derivative with respect to λ by: φ ω = λ2 φ 2πc λ (4.2) The derivative of the actual phase is evaluated with respect to λ since the Sellmeier equations are in terms of wavelength. To get higher-order derivatives of the phase, we simply apply Equation 4.2 multiple times. Table 4.1 compares these derivatives to the derivatives with respect to frequency of the phase in Figure 4.6. The derivatives of the unwrapped phase were evaluated by 49

72 successively applying a simple, symmetric two-point difference method. The modeled and predicted phases for the seed and pump pulses agree well, and small differences can likely be attributed to the inaccuracy of the two-point numerical derivative, especially after repeated applications for the higher-order derivatives. Table 4.1: Derivatives with respect to ω of split-step and expected spectral phase 2 φ/ ω 2 (fs 2 ) 3 φ/ ω 3 (fs 3 ) 4 φ/ ω 4 (fs 4 ) Input seed Output seed Predicted output seed Output pump Predicted output pump Note that the second-order derivative of the input seed is 200fs 2, while the chirp of the input seed was defined as 100fs 2. The chirp here is defined as the number multiplying the (ω ω 0 ) 2 term in the Taylor expansion of φ(ω), which is the second derivative divided by two Modeling Limitations The primary purpose of the model is to allow us to explore the general effects that different parameters have on the OPA output. This helps to predict what parameters should be changed to optimize the system, and provides a deeper understanding of why certain adjustments change the system in a particular way. There is a variety of variables that were not exploited in the optimization below which could potentially be explored using this split-step code. The pump/signal angle was kept at 2 to keep the system behavior reasonably close to the collinear phasematching and bandwidth analysis, but in general, that angle offers another degree of freedom affecting the GVM and GVD, which in turn affect the conversion bandwidth and pump/signal interaction length in the crystal. Controlling the chirp of the pump as well as the seed can further improve the conversion efficiency (as in OPCPA) [34]. 50

73 Also, shaping the pump pulse to have a flat temporal and spatial peak can improve both the total conversion and the bandwidth by providing a wider time window in which the chirped seed experiences significant gain. In our modeling, we approximate the input seed as a Gaussian spectrum with a second-order phase shift applied, centered at a particular wavelength. The seed pulses in the modeling below give fairly good approximations of the effects of linear chirp on the OPA output, but the actual supercontinuum seed has a much more complicated spectrum and spatial profile which depends on several processes simultaneously taking place within the YAG crystal, including self-phase modulation, four-wave mixing, and self-steepening [35]. Accurately modeling the supercontinuum generation in the YAG is beyond the scope of this project, and the intensity of the seed was too low to measure with our spectrometer over much of the IR spectrum. Another important factor that this model does not account for is the spatial profile of the beams. The pump intensity is approximately Gaussian in space as well as time, so the intensity, and thus the gain, is different depending on the transverse position. The split-step model only tracks the pulses at a single transverse position relative to the pump, and the simulations in this chapter are at the spatial peak of the pump. The 2 pump/seed angle also introduces spatial walkoff between the pump, signal, and idler. Figure 4.7 illustrates the spatial walkoff between the pump and seed beams used in Section 4.2. Both beams were downcollimated to spot sizes of about 2mm into a 5mm thick BBO, with an angle of about 2. For that particular configuration, the transverse centers of the pump and seed beams are about 0.18mm apart at the back of the crystal. Assuming the pump intensity is Gaussian throughout the crystal with a FWHM of 2mm, then the pump intensity at the peak of the seed is only about 2% lower due to the spatial walkoff. For the 2 or 3mm cyrstals in the 1040nm system, the effect is even less. 51

74 2mm θ Δh 5mm Figure 4.7: Illustration of spatial walkoff between pump and seed through 5mm crystal With high pump intensities near the damage threshold of the crystal, higherorder nonlinear effects such as self-phase modulation and four-wave mixing in the pump, signal, and idler become more significant. This split-step code only includes difference frequency mixing in the nonlinear equations, although in principle, these higher-order processes could be added by simply adding more terms to the coupled nonlinear Equations Ti:sapphire-Pumped OPA Results Since the 1040nm fiber laser for which the OPA was designed was not available for testing, a slightly modified version of the OPA presented in Section 2.5 was tested using the 790nm output from a mode-locked, amplified Ti:sapphire laser as the pump. As described below, this system as it is presented is not optimized for the best gain using the Ti:sapphire input. These results should be viewed as a proof of concept, demonstrating that this configuration functions and its output is reasonably wellunderstood. Showing consistency between the measured and predicted outputs helps to build confidence in the accuracy of the split-step modeling results in Section Operating Parameters The Ti:sapphire system output approximately 200µJ pulses at a 1kHz repetition rate, and a scanning SHG FROG measured pulse durations of 54fs. Since the system in Figure 2.11 was designed for a first-stage pump energy of 194µJ, the output energy 52

75 of the Ti:sapphire system was only sufficient to test the first stage, and since the beam splitters designed for 1040nm light did not have the desired transmissions at 800nm, 80% reflective 800nm beam splitters were used. The beam used for the supercontinuum generation only requires 1-2µJ while the pump requires more than five times that energy, so a second 80% reflective beamsplitter was used to divert some of the excess seed beam energy away from the YAG crystal. An iris was used to fine-tune the energy focused into the YAG. Focusing 1.5µJ pulses into the YAG with a 200mm focal length lens produced a stable filament with a clean spatial profile. A schematic of the 800nm OPA is shown in Figure 4.8. Since it is pumped at 800nm, this is essentially the same system presented by Brida et al. [8], with a slightly larger pump/signal angle and a YAG crystal instead of sapphire. 2.4 μj 9.6 μj R=80% R=80% 50 μj 800 nm 54 fs 62 mj 2mm YAG 2 o μm 3 mj 5mm BBO Figure 4.8: Modified degenerate OPA schematic Since the available beam splitters were only effective for s-polarized light, the pump polarization had to be vertical, corresponding to the orientation in Figure 2.12(b). The crystal was rotated about the axis in the plane of the table and perpendicular to the pump propagation direction. A 5mm BBO crystal cut at 21 was used because it was readily available and had a large enough aperture over the desired tuning range. Ideally, a thinner 2-3mm crystal would be used since the thick crystal reduces the 53

76 conversion bandwidth and adds extra dispersion to the beams. Depending on the intensity of the seed, the thicker crystal can also potentially allow pump depletion and back-conversion. The pulse duration of the Ti:sapphire output is about half of the duration anticipated from the fiber laser, so less pulse energy is needed to achieve the same peak intensity for a given spot size. The pump beam was downcollimated to a spot radius of approximately 1mm. Sending the full 200µJ pulse energy into the system, the pump beam was intense enough to generate a white-light spectrum in the BBO, so the input pulse energy was tuned down using a waveplate and polarizer until the spectrum vanished, at about 62µJ. The system could be optimized for better gain by using different focal lengths to make a larger pump spot size, allowing more pump energy and conversion over a larger crystal surface while staying below the damage threshold. Using the configuration described above, tuned to near degeneracy, we measured 3mW of total output in the signal direction using a thermal power meter (including the 1.5mW from the beam used to generate the seed), indicating approximately 1.5µJ of amplified signal. As expected, the idler power was also 1.5mW. Since the energy in that direction is generated purely from nonlinear mixing and the idler has nearly the same wavelength as the generated pump, the idler energy should be close to that of the amplified signal Second-Harmonic Generation We also observed visible light in the signal and idler directions from SHG of the signal and idler. At the degeneracy point (and when the beams are collinear), the fields are phase-matched for both difference frequency mixing and SHG of the signal and idler, since the phase-matching equations for the two processes are identical when the signal and idler have the same wavelength. At that point, SHG of the signal or idler beams is like back-conversion, except that mixing takes place between two 54

77 photons from the same beam rather than one from each. When a slight angle between the beams is introduced, the OPA phase-matching is governed by Equations , while the signal and idler SHG wavevectors are collinear. This breaks the symmetry of the two processes and makes SHG phase-matched at different wavelengths than the OPA. Figure 4.9 shows the conversion bandwidths for OPA and SHG, where the crystal angle is tuned for phase-matching at a 1.5µm signal wavelength OPA signal SHG Figure 4.9: Conversion bandwidth for OPA and signal SHG with 790nm pump phasematched at 1.5µm Figure 4.9 assumes a 5mm BBO crystal, and a 2 angle between the pump and signal. The SHG peaks lie on the edges of the OPA conversion bandwidth, so it should not significantly deplete the signal and idler fields, and the majority of the potential output bandwidth is unaffected. The second-harmonic fields can help to roughly align the system visually, but the optimal configuration will not necessarily be where the visible output is brightest. In addition to the second-harmonic of the signal and idler, there were additional faintly visible beams propagating in other directions, including ones at larger angles relative to the pump than the signal and idler. This may be due to diffraction from a 55

78 transient grating created from interference between the pump beam and the residual 800nm beam used to generate the seed. The angles between these diffracted beams were roughly the same as the pump/seed angle. Only first and second-order beams were visible, and their energies were too low to register on the thermal power meter Measured Signal Spectra The peak signal wavelength was tunable from a range of about 1.2µm to the maximum spectrometer wavelength of 1.7µm. The figures below compare measured spectra to those predicted by Equation 2.10 and to the spectra predicted from the split-step model. The measured bandwidths are significantly less than the ideal bandwidths for a 5mm BBO crystal. The ideal bandwidths are calculated assuming equal seeding at all wavelengths, neglecting the effects of back-conversion, seed chirp, and the Gaussian beam profile on the output spectrum. It seems from the modeling that the narrow observed bandwidths are partly due to a temporally chirped seed with a short pump, as discussed in Section Also, the spectral intensity of the seed is not uniform, and the intensity of the seed at a given wavelength affects the gain it receives. First, in order to obtain a suitable input seed to account for the chirp in the actual system, the second-order phase was calculated for a 1.6µm beam propagating through 2mm of YAG and the 4.7mm thick N-BK7 lens that was used to focus the seed into the crystal. Using Equation 4.2 and the Sellmeier equations for YAG [36] and BK7, the second-order phase for the seed is -119fs 2. The negative sign indicates negative chirp, so higher frequencies appear earlier in time than lower frequencies. The ZDP of N-BK7 is about 1.33µm, while that of YAG is about 1.6µm, so the BK7 contributes negative dispersion which dominates the chirp for a 1.6µm beam. The calculated negative chirp was applied to a short Gaussian pulse in the same manner as in Section 4.1.2, giving a seed of approximately 150fs, with a bandwidth extending from 1 to 3µm. 56

79 Figure 4.10 shows the measured and modeled spectra for three different crystal angles. The three measured spectra were recorded on the same intensity scale, and were normalized relative to the highest peak. Likewise, the three predicted spectra were normalized on the same scale. When these measurements were taken, the pump intensity was tuned to just below the point where the BBO generated its own whitelight spectrum, so a pump intensity close to the BBO damage threshold was used in the split-step model, and the FWHM pump pulse duration was 54fs, as measured by the FROG. The model accounts for the 2 pump/seed angle. The initial seed amplitude, pump/seed timing and crystal angle were adjusted in the model to obtain the best fit between the measured and predicted spectra. Figure 4.10: Measured and predicted signal spectra at different crystal angles with fixed timing and seed amplitude Changing the pump/seed timing at a fixed angle and seed intensity shifts the position of the peak within a range limited by the phase-matching angle. Likewise, changing the crystal angle while keeping the timing fixed also shifts the peak within a range limited by the chirp of the seed and the pump pulse duration. For each wavelength, there is an optimal combination of timing and angle which allows a particular wavelength to be both phase-matched and temporally overlapped with the 57