## Supercontinuum generation in quasi-phasematched waveguides |

Optics Express, Vol. 19, Issue 20, pp. 18754-18773 (2011)

http://dx.doi.org/10.1364/OE.19.018754

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### Abstract

We numerically investigate supercontinuum generation in quasi-phase-matched waveguides using a single-envelope approach to capture second and third order nonlinear processes involved in the generation of octave-spanning spectra. Simulations are shown to agree with experimental results in reverse-proton-exchanged lithium-niobate waveguides. The competition between *χ*^{(2)} and *χ*^{(3)} self phase modulation effects is discussed. Chirped quasi-phasematched gratings and stimulated Raman scattering are shown to enhance spectral broadening, and the pulse dynamics involved in the broadening processes are explained.

© 2011 OSA

## 1. Introduction

1. C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. **32**, 2478–2480 (2007). [CrossRef] [PubMed]

4. J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. **13**, 738–749 (2007). [CrossRef]

1. C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. **32**, 2478–2480 (2007). [CrossRef] [PubMed]

2. T. Fuji, J. Rauschenberger, A. Apolonski, V. S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T. W. Hänsch, W. Lehnert, M. Scherer, and F. Krausz, “Monolithic carrier-envelope phase-stabilization scheme,” Opt. Lett. **30**, 332–334 (2005). [CrossRef] [PubMed]

5. C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B **27**, 2687–2699 (2010). [CrossRef]

*χ*

^{(3)}-based supercontinuum generation in optical fibers [3

3. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

4. J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. **13**, 738–749 (2007). [CrossRef]

*χ*

^{(2)}processes in quasi-phasematched (QPM) media, even though very high nonlinearities are readily available in QPM waveguides [6

6. C. Langrock, S. Kumar, J. McGeehan, A. Willner, and M. M. Fejer, “All-optical signal processing using *χ*^{(2)} nonlinearities in guided-wave devices,” J. Lightwave Technol. **24**, 2579–2592 (2006). [CrossRef]

1. C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. **32**, 2478–2480 (2007). [CrossRef] [PubMed]

2. T. Fuji, J. Rauschenberger, A. Apolonski, V. S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T. W. Hänsch, W. Lehnert, M. Scherer, and F. Krausz, “Monolithic carrier-envelope phase-stabilization scheme,” Opt. Lett. **30**, 332–334 (2005). [CrossRef] [PubMed]

7. X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, “Efficient continuous wave second harmonic generation pumped at 1.55 μm in quasi-phase-matched AlGaAs waveguides,” Opt. Express **13**, 10742–10748 (2005). [CrossRef] [PubMed]

*μ*m and therefore offers the possibility of generating a supercontinuum across the mid-IR. In order to reach this goal, a detailed understanding and quantitative modeling of the nonlinear dynamics involved is first required. Progress has been made recently in modeling broadband

*χ*

^{(2)}interactions [8

8. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**, 053841 (2010). [CrossRef]

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

_{3}waveguides, and show good agreement between experiments and simulations. Our analysis reveals the importance of several effects including the

*χ*

^{(2)}and instantaneous

*χ*

^{(3)}nonlinearities, stimulated Raman scattering (SRS;

*χ*), the interaction between multiple waveguide modes, and the dispersion of the waveguides and associated modal overlaps. The required modal properties of the RPE waveguides are determined from a proton concentration profile calculated with the concentration-dependent diffusion model given in Ref. [9

_{R}9. R. V. Roussev, “Optical-frequency mixers in periodically poled lithium niobate: materials, modeling and characterization,” Ph.D. thesis, Stanford University (2006), http://nlo.stanford.edu/system/files/dissertations/rostislav_roussev_thesis_december_2006.pdf.

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*χ*

^{(2)}and

*χ*

^{(3)}terms in reproducing the experimental results, and discuss how the model parameters were estimated in cases where absolute values could not be determined from available literature data. In section 5, our model is compared to the 1580-nm-pumped experiments of [1

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

## 2. Numerical model

8. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**, 053841 (2010). [CrossRef]

10. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E **70**, 036604 (2004). [CrossRef]

*χ*

^{(2)}interactions including second harmonic generation (SHG), sum frequency generation (SFG), and difference frequency generation (DFG) including intrapulse DFG. All

*χ*

^{(3)}interactions can be accounted for as well, but we consider only self and cross phase modulation (SPM and XPM), and SRS. Our analysis is suited to weakly-guided modes such as in RPE LiNbO

_{3}waveguides, but could be generalized to describe tightly-confined modes.

*E*(

*x,y,z,t*) (tilde denotes a frequency-domain field quantity) and the optical frequency

*ω*> 0. The real-valued transverse spatial mode profiles of the bound modes are given by

*B*and the z-dependent envelopes by

_{n}*Ã*. We assume that

_{n}*Ã*(

_{n}*z*,

*ω*< 0) = 0 (i.e.

*Ã*are analytic signals). We have also defined a reference propagation constant

_{n}*β*

_{ref}, a reference group velocity

*v*

_{ref}, and a carrier frequency

*ω*

_{ref}; appropriate choices for these reference quantities are discussed below. From Ref. [14

14. G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B **18**, 534–539 (2001). [CrossRef]

*Ã*, suitable integrations involving

_{n}*P̃*are performed over the transverse spatial dimensions

_{NL}*x*and

*y*of the waveguide. For the

*χ*

^{(2)}nonlinear term, these modal overlap integrals are given by

9. R. V. Roussev, “Optical-frequency mixers in periodically poled lithium niobate: materials, modeling and characterization,” Ph.D. thesis, Stanford University (2006), http://nlo.stanford.edu/system/files/dissertations/rostislav_roussev_thesis_december_2006.pdf.

_{3}waveguides, |

*χ*

^{(2)}(

*x,y*)| is negligible close to the upper surface of the crystal, down to a depth

*h*

_{WG}below the surface [9

9. R. V. Roussev, “Optical-frequency mixers in periodically poled lithium niobate: materials, modeling and characterization,” Ph.D. thesis, Stanford University (2006), http://nlo.stanford.edu/system/files/dissertations/rostislav_roussev_thesis_december_2006.pdf.

χ ¯

^{(2)}(

*x,y*) = 0 for

*y*<

*h*

_{WG}and

χ ¯

^{(2)}(

*x,y*) = 1 for

*y*≥

*h*

_{WG}, where

*y*= 0 denotes the upper surface of the crystal.

*Ã*(

_{n}*z*) can be derived using Eqs. (2) and (3), but involve time-consuming integrals in the frequency domain, particularly for the

*χ*

^{(3)}terms. To derive a simple yet accurate propagation equation, note that the scale of

*B*(

_{n}*x, y,ω*) in Eq. (1) is arbitrary since any frequency-dependent scale factor applied to

*B*can be absorbed into

_{n}*Ã*. Hence,

_{n}*B*can be chosen so as to minimize the frequency-dependence of the overlap integral for the fundamental waveguide mode, Θ

_{n}_{000}, given in Eq. (3). We therefore introduce a frequency dependent normalization for the modes, according to where the form of

*g*(

_{n}*ω*) is chosen to simplify the numerics. To satisfy Eq. (4), we define

*B̄*(

_{n}*x, y,ω*) are numerically-determined, un-normalized mode profiles. The strength of the nonlinear interactions will generally scale inversely with the mode area, which we define as

*n*at frequency. We choose

*g*(

_{n}*ω*) =

*a*(

_{n}*ω*)

^{1/3}, since this choice renders Θ

*dimensionless and Θ*

_{npq}_{000}weakly dispersive. The analogous

*χ*

^{(3)}terms Θ

*, which have a form similar to Eq. (3), are not dimensionless. As such, a single set of functions*

_{npqr}*g*(

_{n}*ω*) cannot simultaneously account for the dependence of

*χ*

^{(2)}and

*χ*

^{(3)}modal overlap integrals on optical frequency. Despite this limitation we find that, with the above choice of

*g*, experiments can be described quantitatively while neglecting the dispersion of both the modal overlap integrals Θ

_{n}*and Θ*

_{npq}*. The dependence of the*

_{npqr}*χ*

^{(2)},

*χ*

^{(3)}, and

*χ*susceptibilities on optical frequency is neglected, but the dependence of

_{R}*χ*on the Raman frequency shift, and hence the Raman response function, is retained.

_{R}10. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E **70**, 036604 (2004). [CrossRef]

11. G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modelling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express **15**, 5382–5387 (2007). [CrossRef] [PubMed]

*(*

_{npq}*ω*,

*ω′*) and Θ

*(*

_{npqr}*ω*,

*ω′*,

*ω″*) as constants

*θ*and

_{npq}*θ*, respectively. The smoothly-chirped QPM grating is expanded in a Fourier series for the various QPM orders with Fourier coefficients varying slowly with

_{npqr}*z*. We make use of the analytic signal formalism for the mode envelopes [8

8. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**, 053841 (2010). [CrossRef]

*β*(

_{n}*ω*) is the modal propagation constant of mode

*n*, and

*α*is the loss coefficient of mode

_{n}*n*, which we assume to be non-dispersive except near the 2.85

*μ*m OH absorption feature of RPE LiNbO

_{3}.

*ℱ*denotes the Fourier transform,

*ℱ*[

*f*(

*t*)](Ω) = ∫

*f*(

*t*)exp(−

*i*Ω

*t*)

*dt*. The envelope phase is given by

*ϕ*(

_{m}*z,t*) =

*ω*

_{ref}

*t*– (

*β*

_{ref}–

*ω*

_{ref}

*/v*

_{ref})

*z*–

*mϕ*(

_{G}*z*), where the grating phase

*K*(

_{g}*z*). The QPM Fourier coefficients are given by

*d*(

_{m}*z*) = 2

*d*sin(

*mπD*(

*z*))/(

*mπ*) [15

15. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **28**, 2631–2654 (1992). [CrossRef]

*D*is the QPM duty cycle and

*m*is the QPM order. In Eq. (5), the summation is performed over all the relevant positive and negative QPM orders, with phases exp(

*iϕ*) for SFG terms (which have form

_{m}*A*), and exp(−

_{p}A_{q}*iϕ*) for DFG terms (form

_{m}*ω*, we usually choose

_{c}*ω*

_{ref}≈ 2

*ω*and

_{c}*β*

_{ref}≈

*β*

_{0}(2

*ω*) (since significant spectral components up to around 3

_{c}*ω*or higher can be generated), and

_{c}*A*(

_{n}*z,t*). This property arises from neglecting (or simplifying, in the case of SRS) the frequency-dependence of the modal overlap integrals and nonlinear susceptibilities.

*χ*

^{(3)}(

*ω*;

*ω*– Ω,

*ω*′, Ω –

*ω*′) =

*χ*+

_{E}*χ*(Ω), where

_{R}*χ*is the instantaneous (frequency-independent) electronic susceptibility, and

_{E}*χ*(Ω) is the Raman susceptibility (which depends only on the Raman frequency shift Ω). We define

_{R}*χ*(Ω) ≡

_{R}*χ*(Ω), where

_{R,pk}H_{R}*χ*is the peak Raman susceptibility, and

_{R,pk}*H*(Ω) is the Raman transfer function;

_{R}*H*(Ω) is the Fourier transform of the Raman (temporal) response function

_{R}*h*(

*t*). The peak Raman frequency Ω

*is defined as the frequency shift for which |ℑ[*

_{pk}*χ*|] is largest;

_{R}*H*is normalized so that ℑ[

_{R}*H*(±Ω

_{R}*)] = ∓1, and the peak Raman susceptibility is defined as*

_{pk}*χ*≡ |ℑ[

_{R,pk}*χ*(Ω

_{R}*)]|. The complex Raman transfer function is discussed in Appendix A. The values we use for the nonlinear susceptibilities are discussed in Appendix A and in section 4.*

_{pk}*χ*

^{(2)}interactions modeled by Eq. (5) are highly phase mismatched. In particular, for a given process (SHG, SFG or DFG involving a particular combination of waveguide modes) usually at most one QPM order, denoted

*m*

_{0}, is close to phasematching. To model a different QPM order

*m*, the grid size required increases approximately in proportion to |

*m*–

*m*

_{0}|, while the contribution to the pulse dynamics is (approximately) proportional to 1/|

*m*–

*m*

_{0}|

^{3}. In appendix B, we discuss these terms in more detail and calculate their contribution to the total SPM of the input pulse. In some cases, instead of including higher order terms explicitly in the simulations, their leading-order contributions to the pulse dynamics can be calculated analytically via the cascading approximation, which yields an effective instantaneous

*χ*

^{(3)}coefficient for each term. These coefficients can then be added to the true instantaneous

*χ*

^{(3)}coefficient

*χ*in Eq. (5), yielding an adjusted and possibly

_{E}*z*-dependent effective value of

*χ*(

_{E}*z*).

## 3. Comparison to 1043-nm-pumped experiments

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*μ*m to 11

*μ*m over this length by a linear chirp (i.e.

*dK*/

_{g}*dz*=constant). Based on waveguide fabrication parameters and the concentration-dependent diffusion model given in Ref. [9

_{00}mode at 1043 nm and 521.5 nm as 16.30

*μ*m

^{2}and 6.37

*μ*m

^{2}, respectively. For the modal overlap integrals,

*θ*

_{000}≈ 0.817, and

*θ*

_{0000}≈ 0.3925

*μ*m

^{−2/3}(evaluated for self phase modulation at 1043 nm). For the TM

_{10}mode (mode “4”), which we also include in the model explicitly,

*θ*

_{400}≈ 0.162 and the mode area at 521.5 nm is 9.80

*μ*m

^{2}. We apply the waveguide cascading approximation described in appendix B to remove all but the first-order QPM interactions in Eq. (5). We assume a sech

^{2}input pulse profile with flat spectral phase.

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*μ*m QPM period case, the simulation captures the main features seen experimentally, namely the high power spectral density (PSD) between 1

*μ*m and 1.2

*μ*m, and the “pedestal” generation between 1.4 and 1.75

*μ*m; the reduction in PSD between 1.2 and 1.4

*μ*m in the experiment is also reproduced (although there is a larger reduction in the simulation). For the 7–8

*μ*m QPM period case, the simulated spectrum falls off at approximately the same rate as in the experiment, reaching 10

^{−6}of its peak at 1.45

*μ*m. The simulated spectra were averaged over five simulations with semiclassical quantum noise seeding on the input pulse.

*χ*

^{(3)}can be seen in Fig. 2(b), which shows the phase of the pulse after numerically filtering out the second harmonic spectral components. An intensity-dependent phase is accumulated near the start of the QPM grating, but after approximately 7 mm, the rate of phase accumulation changes sign, suggesting that

*K*(

_{g}*z*) and the local frequency of the pulse

*ω*(

_{FH}*z*). This latter quantity is defined by

*ω*(

_{FH}*z*) = [∫

*ω*|

*Ã*

_{0}(

*z,ω*)|

^{2}

*dω*]/[∫ |

*Ã*

_{0}(

*z,ω*)|

^{2}

*dω*], where integration is performed only over the first harmonic (FH) region of the pulse. Equation (16) is then evaluated using

*ω*(

_{FH}*z*) for the frequency and

*K*(

_{g}*z*) for the local grating k-vector. The solid blue lines in Fig. 2 show to the position at which

*μ*m, we show in Fig. 3(a) the propagation of the pulse in the frequency domain (plotted on a log scale). The generation of spectral components > 1.4

*μ*m can be seen to occur around 15 mm from the start of the QPM grating. To help explain this and other processes, we show in Fig. 3(b) a simulated cross-FROG spectrogram at the output of the QPM grating, using a 150-fs Gaussian gate pulse. Due to a Raman self frequency shift (SFS) effect [21

21. J. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. **11**, 662–664 (1986). [CrossRef] [PubMed]

19. M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007). [CrossRef] [PubMed]

_{3}); the phase mismatch is discussed in section 4. In Fig. 3 this SH wave is shown over a limited temporal range (with delay-dependent frequencies of around 460 THz), but it extends over approximately 24 ps (with frequency increasing with delay), corresponding to the relative delay accumulated between the SH and FH frequencies over the length of the waveguide.

*μ*m can be generated by at least two processes. One of these processes is optical parametric amplification (OPA) of quantum noise: spectral components between 1.4 and 1.75

*μ*m are generated via OPA between the SH pulse (which acts as the “pump”) and quantum noise components around the input frequency. The generation of these spectral components from the (semiclassical) noise floor is evident in Fig. 3(a); when quantum noise is turned off in the simulation, these spectral components become much weaker (see Fig. 5). Because of the high SH intensities involved, the OPA process can have high gain, provided that phasematching is satisfied. The gain for this process is quite subtle, however, since the frequency and intensity of the generated SH wave depends on position

*z*via the QPM chirp and Raman SFS of the FH pulse, and hence there is a spatially-dependent “pump” frequency and intensity; additionally, there is a spatially-dependent QPM period, and rapid temporal walk-off between pump, signal and idler spectral components. Thus, this situation differs somewhat from the chirped QPM OPA interactions studied elsewhere [22

22. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B **25**, 463–480 (2008). [CrossRef]

24. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO_{3},” Opt. Lett. **35**, 2340–2342 (2010). [CrossRef] [PubMed]

*μ*m are generated. This process involves the generated SH pulse mixing with the FH pulse and its Raman trail according to phasematched DFG, and can be seen in the weaker “outer” crescent-like patter which extends to approximately (0.8 ps, 150 THz) in Fig. 3(b).

## 4. Model calibrations for 1043-nm-pumped experiments

*χ*

^{(3)}nonlinear coefficients.

*χ*

^{(2)}and

*χ*

^{(3)}SPM effects were of opposite sign and comparable magnitude. Here, we will quantify these terms, and hence the initial dynamics, by analyzing the

*χ*

^{(3)}and cascaded

*χ*

^{(2)}contributions to SPM near the input of the QPM grating. The effective low-frequency-shift third-order nonlinear susceptibility,

*χ*, SRS, and from each waveguide mode at each order of the QPM grating via cascaded

_{E}*χ*

^{(2)}interactions. These cascaded

*χ*

^{(2)}contributions are labeled

*m*, SH waveguide mode index

*q*and FH waveguide mode index 0, and are described in appendix B in terms of the phase mismatches Δ

*k*

_{m,q}_{00}[defined in Eq. (14)]. The necessary parameters for calculating

*μ*m), and with

*d*

_{33}= 25.2 pm

^{2}/V

^{2},

_{3},

*χ*+

_{E}*H*(0) = 6365 pm

_{R}^{2}/V

^{2}. Based on this

*χ*

^{(3)}contribution, the effective low-frequency

*χ*

^{(3)}would be very small if only the lowest-order waveguide mode, first-order QPM interaction (

*χ*

^{(2)}terms included,

*χ*

^{(2)}terms alone. For the remaining cascading terms,

*k*

_{0,000}(the phase mismatch for SHG involving the lowest-order waveguide modes and with QPM order 0) to decrease, and hence decreases Δ

*k*

_{1,000}since Δ

*k*

_{0,000}>

*K*; however, the QPM chirp increases Δ

_{g}*k*

_{1,000}with

*z*since

*K*(

_{g}*z*) is decreasing. The net result of these two effects for this particular case is that

*χ*

^{(2)}and

*χ*

^{(3)}terms hindered conventional spectral broadening via SPM, the dominant mechanism for generating spectral components > 1.4

*μ*m was optical parametric amplification of quantum noise. This mechanism is consistent with the results of Ref. [1

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*f*

_{CEO}was reported when using a 1580-nm pump, but not for the 1043-nm-pumping case. For many applications, the coherence of the supercontinuum is important; if the cancellation of SPM effects was reduced, the intensity of the SH could be reduced and the rate of FH SPM increased, thereby allowing for coherent supercontinuum generation.

*χ*

^{(3)}terms,

*χ*+

_{E}*H*(0)

_{R}*χ*, which can be determined quite accurately. However, both

_{R,pk}*χ*

^{(3)}parameters need to be known or estimated for the simulations; in appendix A, we show that it is difficult to use available literature data to absolutely calibrate both

*χ*and

_{E}*χ*simultaneously. To show the importance of these terms beyond the

_{R,pk}*χ*

^{(2)}–

*χ*

^{(3)}competition calculated above, we first show that SRS must be included in the model in order to reproduce the experimental results. The importance of SRS is shown in Fig. 4, where we plot the output spectrum for several values of

*χ*while setting

_{E}*χ*= 0. For the case with

_{R,pk}*χ*= 0 and

_{R,pk}*χ*= 0, there is a long-wavelength pedestal which extends to > 2

_{E}*μ*m; this pedestal extends further than the spectra shown in Figs. 1(a) and 1(b). However, based on the results of appendix A,

*χ*+

_{E}*H*(0)

_{R}*χ*≈ 6365 pm

_{R,pk}^{2}/V

^{2}. When

*χ*is increased towards this value (while still maintaining the false assumption that

_{E}*χ*= 0), the spectral broadening is reduced significantly. For the

_{R,pk}*χ*> 3000 pm

_{E}^{2}/V

^{2}cases in Fig. 4, the bandwidth is much narrower than in the experiments. Furthermore, for each case, there is no spectral “flattening” between 1 and 1.2

*μ*m, and no dip between 1.2 and 1.4

*μ*m; both of these spectral features can be seen in Figs. 1(a) and 1(b).

*χ*and

_{E}*χ*. To estimate these parameters, we performed simulations of the experiments discussed in section (3) for several values

_{R,pk}*χ*at fixed (

_{R,pk}*χ*+

_{E}*H*(0)

_{R}*χ*), and compared the resulting simulated spectra to the experimentally observed spectrum in Fig. 1(a). In order to agree reasonably well, the spectrum should exhibit the spectral flattening between 1 and 1.2

_{R,pk}*μ*m (SFS), and have a supercontinuum “pedestal” between 1.4 and 1.75

*μ*m with a PSD around 10

^{−3}less than the peak. To illustrate this numerical procedure, we show in Fig. 5 example spectra for several representative values of

*χ*at fixed

_{R,pk}*χ*+

_{E}*H*(0)

_{R}*χ*; the spectra are averaged over five simulations for each value of

_{R,pk}*χ*. For the dashed black line in Fig. 5, quantum noise was neglected. This case is plotted in order to indicate the importance of noise amplification in these simulations, as discussed in section 3. In comparing the spectrum as a function of

_{R,pk}*χ*to the experimental results in Fig. 1(a), we find that for values of

_{R,pk}*χ*< 5000 pm

_{R,pk}^{2}/V

^{2}the long-wavelength pedestal is weaker and extends to shorter wavelengths, and the spectral broadening between 1 and 1.2

*μ*m is reduced; for

*χ*> 6000 pm

_{R,pk}^{2}/V

^{2}, the opposite trends hold. Therefore, based on simulations similar to those in Fig. 5, we estimate

*χ*= 5.46 × 10

_{E}^{3}pm

^{2}/V

^{2}and

*χ*= 5.30 × 10

_{R,pk}^{3}pm

^{2}/V

^{2}.

*χ*increases the intensity of the generated SH wave (the part of the SH which extends to > 6 ps in the spectrogram), and hence the OPA gain (since this SH wave acts as the pump for the OPA process), which in turn leads to a stronger > 1.4-

_{R,pk}*μ*m pedestal generation, which corresponds to the trends shown in Fig. 5. Since the Raman SFS process broadens the FH pulse bandwidth (primarily over the 1–1.2

*μ*m range), it might be interpreted as enhancing the non-local cascading response [19

19. M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007). [CrossRef] [PubMed]

*χ*

^{(3)}effects are also important for the 1580-nm-pumped experiments of Ref. [1

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*χ*scaled by a factor close to theoretical predictions [25

_{E}25. R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n_{2} in wide bandgap solids,” IEEE J. Quantum Electron. **32**, 1324–1333 (1996). [CrossRef]

*χ*

^{(3)}parameters estimated here are realistic. However, due to the

*χ*

^{(2)}–

*χ*

^{(3)}competition, our simulations are sensitive to relatively small deviations in the model parameters and input conditions from our assumptions. In particular, such variations could arise from differences in

*χ*

^{(2)}and

*χ*

^{(3)}between protonated and congruent LiNbO

_{3}, the frequency-dependence of the

*χ*

^{(2)}and

*χ*

^{(3)}susceptibilities, and the input pulse chirp. Direct measurements of the

*χ*

^{(3)}susceptibilities in future work would be of great value for modeling supercontinuum generation in QPM media.

## 5. Comparison to 1580-nm-pumped experiments

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*μ*m and losses of approximately 0.1 dB/cm. The area of the TM

_{00}mode at 1580 nm and 790 nm is estimated as 28.48

*μ*m

^{2}and 11.35

*μ*m

^{2}, respectively. The modal overlap integral

*θ*

_{000}≈ 0.793, and

*θ*

_{0000}≈ 0.3266

*μ*m

^{−2/3}(evaluated for self phase modulation at 1580 nm). For the TM

_{02}mode (“mode 2”),

*θ*

_{002}≈ 0.043; for the TM

_{10}mode (“mode 4”),

*θ*

_{004}≈ 0.220; the areas at 790 nm for these modes are 17.55

*μ*m

^{2}and 19.09

*μ*m

^{2}, respectively. The remaining terms

*θ*are relatively small and mainly give rise to additional peaks in the pulse spectrum at short wavelengths without altering the spectral broadening very significantly.

_{npq}**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*χ*, with the TM

_{E}_{00}and TM

_{10}modes and the QPM orders +1, +3, and +5 [index

*m*in Eq. (5)] included in the model, and assuming a 1.2-nJ input pulse. The remaining modes and QPM orders are accounted for via the cascading approximation given by Eq. (16). The simulations capture the spectral broadening observed experimentally, particularly the

*χ*= 4368 pm

_{E}^{2}/V

^{2}case; the evolution of the spectrum of the pulse for that case is shown in Fig. 6(c). Semiclassical quantum noise on the input pulses was included, as in sections 3 and 4, but did not change the output spectra.

*χ*

^{(2)}terms,

*m*and SH mode

*q*) for SHG of the 1580-nm input pulse. With

*d*

_{33}= 19.5 pm/V [26

26. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B **14**, 2268–2294 (1997). [CrossRef]

*χ*

^{(2)}terms are

*χ*+

_{E}*H*(0)

_{R}*χ*determined in appendix A combined with these cascade contributions, the total effective

_{R,pk}*χ*

^{(3)}is

*χ*should decrease with wavelength [25

_{E}25. R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n_{2} in wide bandgap solids,” IEEE J. Quantum Electron. **32**, 1324–1333 (1996). [CrossRef]

*χ*with frequency is included such that

_{E}*χ*(1580 nm) ≈ 4368 pm

_{E}^{2}/V

^{2},

_{3}(and RPE waveguides) at 1580 nm. With this value of

*χ*, the simulations are in good agreement with the experimental results. This scaling corresponds to

_{E}*χ*(1580 nm)/

_{E}*χ*(1043 nm) = 0.8, which is very close to the predictions from the (oversimplified) two-band model. Note, however, that due to the almost complete cancellation of

_{E}*χ*

^{(3)}and

*χ*

^{(2)}contributions to the total SPM,

*χ*

_{total}and hence the output spectrum is sensitive to errors in

*χ*and

_{E}*d*

_{33}of as little as 2%. This sensitivity is illustrated in Fig. 6(b), where the three

*χ*cases shown correspond to values for

_{E}*χ*(1580

_{E}*nm*)/

*χ*(1043

_{E}*nm*) of 0.775, 0.8, and 0.825. For more accurate modeling, it would be necessary to know the model parameters precisely, or to operate in a regime where the

*χ*

^{(2)}–

*χ*

^{(3)}cancellation is not so complete.

*χ*, 1580 nm pulse broadens in spectrum and compresses in time due to the negative

_{E}*μ*m and was of comparable magnitude and wavelength for different QPM periods, as shown in Fig. 6(a). Similarly, we found that the wavelength of the long-wavelength peak in the simulations is only weakly dependent on QPM period (i.e. a peak occurs at the same wavelength for a relatively wide range of QPM periods, as long as there is sufficient spectral broadening). For conventional DFG of frequency

*ω*=

_{DFG}*ω*

_{1}–

*ω*

_{2}, one normally would anticipate Δ

*k*=

*β*(

*ω*

_{1})

*– β*(

*ω*

_{2})

*– β*(

*ω*)

_{DFG}*– K*, and thus a wavelength for the phasematched peak that would depend strongly on the QPM period, in contrast to the observed behavior. However, for a DFG process involving a phase mismatched second harmonic (SH) component as one of the participating waves, the phasematching condition differs from the conventional one.

_{g}*ω*, effective soliton propagation constant

_{FH}*v*

_{FH}≈ (∂

*β*

_{0}(

*ω*)

_{FH}*/∂ω*)

^{−1}. The effective propagation coefficient for the phase mismatched SH pulse at frequencies

*ω*in the vicinity of 2

*ω*is given approximately by

_{FH}*π*/2 radians out of phase with its driving polarization, aligned temporally with the FH pulse at group velocity

*v*. For the DFG process involving such a SH spectral component with frequency

_{FH}*ω*

_{2}, a FH spectral component with frequency

*ω*

_{1}, and a generated wave at frequency

*ω*=

_{DFG}*ω*

_{2}–

*ω*

_{1}, the effective Δ

*k*is given by

*K*, in contrast to the conventional phasematching relation. This type of phase-matching relation was discussed in [27

_{g}27. M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A **82**, 063806 (2010). [CrossRef]

*μ*m: this dip occurs due to OH absorption, which we included in the model as a complex Lorentzian perturbation to the effective index, with a corresponding peak absorption of 3 mm

^{−1}.

*μ*m, longer than observed experimentally. However, this waveguide model is calibrated primarily for wavelengths < 2

*μ*m. Furthermore, small changes in fabrication parameters could also lead to shifts in the effective phase mismatch given by Eq. (7). Therefore, some discrepancy with experiments can be expected for processes involving longer wavelengths. The wavelength of the cascaded DFG peak also depends on the pulse frequency, which changes due to the Raman SFS. This SFS depends on the value of

*χ*(which we only estimate via the approach of section 4) and on the input electric field profile (which is not known for these experiments). For Fig. 6(b) we added a small, smooth and monotonic frequency-dependent offset to the effective index in order to shift the cascaded DFG peak to 2.85

_{R,pk}*μ*m. We chose a functional form

*ω*and

_{L}*ω*corresponding to 2

_{OH}*μ*m and 2.85

*μ*m, respectively, Δ

*ω*= 2

*π*× 5 THz, and

*δn*

_{0}= −8 × 10

^{−4}. This functional form was chosen so that

*δn*≈

*δn*

_{0}at 2.85

*μ*m, and so that

*δn*≈ 0 for wavelengths < 2

*μ*m, where our waveguide dispersion model is well-calibrated. This latter constraint helps to ensure that the pulse dynamics are not artificially altered by the effective index offset. Provided the above constraints are met, we have found that the spectrum is relatively insensitive to the functional form of

*δn*(

*ω*). With improved characterization of the RPE waveguide dispersion and the nonlinear parameters of the model, and the input pulse, this offset might not be required.

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

**81**, 053841 (2010). [CrossRef]

*χ*and

_{E}*χ*terms, only the cascaded-

_{R,pk}*χ*

^{(2)}interactions contribute to the total SPM, and so

*c*= 1 for

_{q}*q*= 0 and is zero otherwise (see appendix B), and hence

*χ*

^{(3)}and waveguide modes yields an effective low-frequency

*χ*

^{(3)}that is an order of magnitude too large for these particular experiments. If

*χ*

^{(3)}and waveguide effects are neglected, the SPM is increased by a similar factor, which will substantially alter the dynamics. For some cases, the measured spectrum can still be recovered by treating the pulse intensity as an unconstrained fit parameter in the simulations, but such an approach would not work in other spectral ranges, such as those considered in section 3.

*χ*

^{(3)}terms according to Eq. (5), and we assumed a pulse energy of 1.2 nJ at the input to the waveguide, the same as the experimental value. In section 4, we also showed that properly-calibrated third-order nonlinear effects, particularly SRS, are required to model the 1043-nm-pumped experiments of [1

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

**81**, 053841 (2010). [CrossRef]

## 6. Discussion

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*χ*

^{(2)}, instantaneous

*χ*

^{(3)}, and stimulated Raman scattering nonlinearities are all essential for accurately modeling the supercontinuum generation process.

*χ*and

_{E}*χ*more accurately via direct measurements will be the subject of future work. The dispersion of the nonlinear susceptibilities is important as well, as shown in section 5. It may also be important to determine the dispersion of the RPE waveguides at long wavelengths with greater accuracy. In addition to the model parameters in Eq. (5), measurements of the complex input electric field are needed; plausible values of the pulse chirp, which we assumed to be zero in our simulations, can have a significant impact on the output spectrum. Measurement of the spatial mode content of the output electric field would also be useful since this would help to indicate which waveguide modes are important and must be included in the model.

_{R,pk}**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*χ*

^{(2)}and

*χ*

^{(3)}susceptibilities were of comparable magnitude but opposite sign, and hence competed with each other. This cancellation of contributions to

*λ*

_{GVD}, is often shifted by waveguide design to be nearby the input wavelength,

*λ*. We have seen from simulations based on Eqs. (5) that shifting

*λ*

_{GVD}to be comparable to the input wavelength is one way to significantly enhance the spectral broadening processes for

*χ*

^{(2)}-based continua as well, since generated spectral components would then remain overlapped temporally. Furthermore, if the GVD at the input wavelength is negative (

*λ*>

*λ*) or negligible (

_{GVD}*λ*≈

*λ*), supercontinuum generation could be achieved with a positive

_{GVD}*k*

_{1,000}< 0 the contributions of

*χ*

^{(2)}and

*χ*

^{(3)}to SPM would add rather than cancel while still being of appropriate sign to support

*χ*

^{(3)}-like bright solitons at the input wavelength.

*λ*

_{GVD}cannot be obtained. However,

*λ*

_{GVD}is conveniently located near 2

*μ*m, making Tm-based laser sources promising candidates for increased spectral broadening in RPE waveguides [28

28. C. R. Phillips, J. Jiang, C. Langrock, M. M. Fejer, and M. E. Fermann, “Self-Referenced Frequency Comb From a Tm-fiber Amplifier via PPLN Waveguide Supercontinuum Generation,” in CLEO:2011 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA5.

*λ*

_{GVD}to the 1.55-

*μ*m and 1-

*μ*m spectral regions. There is also the possibility of using AlGaAs QPM waveguides, which can be tightly confining and are transparent in the mid-IR. With simultaneous engineering of the waveguide dispersion and the QPM grating, supercontinuum generation may be possible across the mid-IR. With the model of nonlinear interactions in QPM waveguides we have developed here, strategies can be developed for reaching spectral regions not accessible to silica-fiber-based supercontinuum sources, and for performing optimizations made possible by the versatility of QPM gratings [14

14. G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B **18**, 534–539 (2001). [CrossRef]

23. C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. **35**, 3093–3095 (2010). [CrossRef] [PubMed]

## A. Material properties

*d*

_{33}= 25.2 pm/V for 1064-nm-SHG [26

26. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B **14**, 2268–2294 (1997). [CrossRef]

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*d*

_{33}= 25.2 pm/V for 1043-nm-pumping and

*d*

_{33}= 19.5 pm/V for 1580-nm-pumping [26

26. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B **14**, 2268–2294 (1997). [CrossRef]

*χ*

^{(2)}(

*x,y*) = 0 close to the upper surface of the crystal, as described in section 2.

*H*], by measuring the spontaneous Raman scattering cross section of congruent LiNbO

_{R}_{3}(CLN). The Raman spectrum of LiNbO

_{3}has been measured previously [29

29. A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO_{3},” Phys. Rev. **158**, 433 (1967). [CrossRef]

30. P. J. Delfyett, R. Dorsinville, and R. R. Alfano, “Spectral and temporal measurements of the third-order nonlinear susceptibility of LiNbO_{3} using picosecond Raman-induce phase-conjugate spectroscopy,” Phys. Rev. B **40**, 1885 (1989). [CrossRef]

29. A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO_{3},” Phys. Rev. **158**, 433 (1967). [CrossRef]

*X̄*configuration and at 295 K. The resulting spectrum is shown in Fig. 7 (blue line). To determine ℜ[

*H*] we fitted the measured ℑ[

_{R}*H*] to a sum of Lorentzians; this reconstructed Raman susceptibility is also shown in Fig. 7. The parameters for the fit are given in Table 1. The terms of the Lorentzian fit have form

_{R}*f*(not angular frequency). The small measured peak at around −4.6 THz (≈ −153 cm

^{−1}) was neglected in the fit since it does not correspond to the e-wave polarization component [31

31. N. Surovtsev, V. Malinovskii, A. Pugachev, and A. Shebanin, “The nature of low-frequency raman scattering in congruent melting crystals of lithium niobate,” Phys. Solid State **45**, 534–541 (2003). [CrossRef]

25. R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n_{2} in wide bandgap solids,” IEEE J. Quantum Electron. **32**, 1324–1333 (1996). [CrossRef]

*n*

_{2}was measured for LiNbO

_{3}with the Z-scan method using 30-ps (FWHM) pulses with a center wavelength of 1.064

*μ*m. Since the reconstructed LiNbO

_{3}Raman spectrum in Fig. 7 does not vary significantly for frequencies < 1 THz, and assuming the 30-ps pulses had bandwidths ≤1 THz, the measured

*n*

_{2}represented the total low-frequency-shift

*χ*

^{(3)}response, given by

*n*

_{2}as which in LiNbO

_{3}is given by

*n*

_{2}= 0.933 × 10

^{−6}cm

^{2}/GW [25

_{2} in wide bandgap solids,” IEEE J. Quantum Electron. **32**, 1324–1333 (1996). [CrossRef]

*χ*

^{(2)}interaction is given, via multiple scale analysis [34

34. C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B **19**, 852–859 (2002). [CrossRef]

_{3}in Ref. [25

_{2} in wide bandgap solids,” IEEE J. Quantum Electron. **32**, 1324–1333 (1996). [CrossRef]

*d*

_{eff}=

*d*

_{33}and Δ

*k*=

*k*(2

*ω*) – 2

*k*(

*ω*) = 0.927

*μ*m

^{−1}for 1.064-

*μ*m pumping [35

35. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, n* _{e}*, in congruent lithium niobate,” Opt. Lett.

**22**, 1553–1555 (1997). [CrossRef]

*d*

_{33}is relatively well-known), we find that

*χ*+

_{E}*χ*(0) = 6365 pm

_{R,pk}H_{R}^{2}/V

^{2}, much larger than

*χ*and

_{E}*χ*. One way to determine

_{R,pk}*χ*separately from

_{R,pk}*χ*is by measuring the SRS gain at Ω

_{E}*. However, tabulated values for the Raman gain coefficient are inconsistent with the*

_{pk}*n*

_{2}measurements described above. Consider the Raman gain coefficient, (

*g*/

_{S}*I*), evaluated at Stokes frequency

_{L}*ω*=

_{S}*ω*– Ω

_{L}*for some pump laser frequency*

_{pk}*ω*. It can be shown from Eq. (5) that where the reduction in gain due to Stokes-anti-Stokes coupling effects are neglected when defining (

_{L}*g*/

_{S}*I*), and

_{L}*n*and

_{S}*n*are the refractive indices at the Stokes and pump wavelengths, respectively. In LiNbO

_{L}_{3}with a 1.064-

*μ*m pump laser, the gain coefficient corresponding to a contribution of 100% from SRS to the value of

*g*/

_{S}*I*)

_{L}*≈ 2.51 cm/GW, using*

_{max}*H*(0) = 0.17 from Fig. 7. However, the gain coefficient tabulated in Ref. [36

_{R}36. R. Boyd, “Stimulated Raman scattering and stimulated Rayleigh-Wing scattering,” in “*Nonlinear Optics*”, R. Boyd (Academic, 2008). [CrossRef]

36. R. Boyd, “Stimulated Raman scattering and stimulated Rayleigh-Wing scattering,” in “*Nonlinear Optics*”, R. Boyd (Academic, 2008). [CrossRef]

*μ*m (by scaling with optical frequency); this value is more than twice as large as the upper bound for (

*g*/

_{S}*I*) provided by the

_{L}*n*

_{2}measurement (assuming

*χ*> 0).

_{E}*g*/

_{S}*I*) are also inconsistent with recent experiments with intense IR pulses. The Raman gain rate

_{L}*g*, in cm

_{R}^{−1}, can be approximated as [36

36. R. Boyd, “Stimulated Raman scattering and stimulated Rayleigh-Wing scattering,” in “*Nonlinear Optics*”, R. Boyd (Academic, 2008). [CrossRef]

*g*is the coupling coefficient between the intensity at the Stokes shifted wave at

_{S}*ω*and the pump laser at

_{S}*ω*=

_{L}*ω*+

_{S}*ω*, and the phase mismatch Δ

_{R,pk}*k*

_{R}*=*2

*k*(

*ω*)

_{L}*– k*(

*ω*)

_{S}*– k*(

*ω*) for anti-Stokes frequency

_{AS}*ω*. In turn,

_{AS}*g*is determined via the Raman gain coefficient (

_{S}*g*/

_{S}*I*). In Refs. [24

_{L}24. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO_{3},” Opt. Lett. **35**, 2340–2342 (2010). [CrossRef] [PubMed]

_{3}crystal was pumped with 1.064-

*μ*m pulses with intensities of > 7 GW/cm

^{2}and durations of 12 ps (FWHM); the corresponding pulse bandwidth was significantly narrower than the linewidth of the main peaks of

*H*(Ω) shown in Fig. 7. Assuming (

_{R}*g*/

_{S}*I*) = 6.1 cm/GW, Eq. (11) predicts an SRS gain in this case of 136 dB (185 dB if Δ

_{L}*k*→ ∝). Despite this high predicted gain, no Stokes wave was observed experimentally. If we instead assume the

_{R}*n*

_{2}-based upper bound on (

*g*/

_{S}*I*) of 2.51 cm/GW, the gain at the Stokes frequency would be approximately 69 dB.

_{L}_{3}(which could also be different from those of RPE LiNbO

_{3}). The peak Raman susceptibility

*χ*can be bounded above by the non-linear refractive index and by the absence of SRS in the high intensity experiments discussed, and

_{R,pk}*χ*

_{R,pk}*H*(0) +

_{R}*χ*can be estimated from the measured value of

_{E}*n*

_{2}. For this paper, we further constrain the susceptibilities to yield output spectra in quantitative agreement with the super-continuum generation experiments of Ref. [1

**32**, 2478–2480 (2007). [CrossRef] [PubMed]

*χ*

^{(3)}values that we use are given by

*χ*= 5.3 × 10

_{R,pk}^{3}pm

^{2}/V

^{2}, and

*χ*(1043 nm) = 5.46 × 10

_{E}^{3}pm

^{2}/V

^{2}; these parameters are discussed further in section 4. For the 1580-nm-pumping case discussed in section 5,

*χ*is scaled according to theoretical predictions [25

_{E}_{2} in wide bandgap solids,” IEEE J. Quantum Electron. **32**, 1324–1333 (1996). [CrossRef]

*χ*(1580 nm)/

_{E}*χ*(1043 nm) ≈ 0.8.

_{E}_{2} in wide bandgap solids,” IEEE J. Quantum Electron. **32**, 1324–1333 (1996). [CrossRef]

*n*

_{2}and a theoretical calculation based on a simplified two-band model: at 532 nm, the measured

*n*

_{2}was approximately 9.1× the value at 1064 nm, while the two-band theory predicted a scaling factor of only 2.5. A possible resolution to this discrepancy is that there is a large negative contribution to

*n*

_{2}from

_{3}bandgap, it has a very large phase mismatch with its second harmonic at 266 nm [38], so the contribution to

*n*

_{2}from

*χ*with frequency, the additional 532-nm data point for

_{E}*n*

_{2}is sufficient, in principle, to determine both

*χ*and

_{E}*χ*at 1.064

_{R,pk}*μ*m. With the measured value of

*n*

_{2}= 8.25 × 10

^{−6}cm

^{2}/GW,

*χ*= 5.3 × 10

_{R,pk}^{3}pm

^{2}/V

^{2}is non-dispersive, then

*χ*(532 nm) = 13.4 × 10

_{E}^{3}pm

^{2}/V

^{2}, and

*χ*(532 nm)/

_{E}*χ*(1064 nm) ≈ 2.5, in good agreement with the (oversimplified) two-band model.

_{E}## B. Cascading approximation for QPM waveguides

*χ*

^{(2)}interactions can be approximated by

*χ*

^{(3)}-like self-and cross-phase-modulation (SPM and XPM) terms; this approach is termed the cascading approximation, and has been discussed extensively [33

33. G. I. Stegeman, D. J. Hagan, and L. Torner, “*χ*^{(2)} cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. **28**, 1691–1740 (1996). [CrossRef]

*χ*

^{(3)}susceptibilities given a known nonlinear refractive index. In this appendix, we determine the cascading approximation for the case of QPM waveguide interactions. This calculation gives a total effective

*χ*

^{(3)}, denoted

34. C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B **19**, 852–859 (2002). [CrossRef]

*A*into first harmonic (FH) and second harmonic (SH) pulse components

_{n}*ω*and

_{FH}*ω*= 2

_{SH}*ω*, respectively. In principle, Eq. (5) can give rise to pulses around carrier frequencies

_{FH}*mω*for all positive integers

_{FH}*m*. However, for pulses with a bandwidth ΔΩ less than an octave, i.e. for ΔΩ ≪

*ω*, often only the components around

_{FH}*ω*and

_{FH}*ω*are relevant, to lowest order in the perturbation. Furthermore, higher order modes around

_{SH}*ω*can often be neglected, for example in the case when the waveguide only supports a single mode at that frequency. With these assumptions, Eq. (5) yields the following simplified time-domain coupled wave equations for SHG,

_{FH}*θ*

_{q}_{00}= Θ

_{q}_{00}(

*ω*,

_{SH}*ω*) and

_{FH}*θ*

_{qq}_{00}= Θ

_{qq}_{00}(

*ω*,

_{SH}*ω*,

_{SH}*ω*). In Eqs. (12) and (13), the spatial mode profiles and coupling coefficients have been evaluated at the optical carrier frequencies. We have also assumed that the intensity of the SH pulse is much lower than that of the FH pulse, and have therefore neglected the

_{FH}*χ*

^{(3)}terms involving |

*A*|

_{n,SH}^{2}. For the purposes of this simplified analysis, we have assumed that the pulse bandwidth is narrow enough that

*H*(Ω) can be approximated as

_{R}*H*(0); this approximation does not apply for supercontinuum generation [in the simulations, we use

_{R}*H*(Ω)], but is useful for estimating the rate of SPM for the FH pulse at the start of the QPM grating. The phase mismatch terms are given by for QPM order

_{R}*m*and waveguide mode

*q*of the SH pulse. If the characteristic length defined by

*L*

_{m1,m2}≡ |Δ

*k*

_{m1,q00}– Δ

*k*

_{m2,q00}|

^{−1}is much shorter than any other characteristic lengths of the problem for all

*m*

_{1}≠

*m*

_{2}, the multiple-scale analysis can be applied. In Ref. [34

34. C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B **19**, 852–859 (2002). [CrossRef]

*L*

_{m1,m2}for all

*m*

_{1}≠

*m*

_{2}, and assuming that there is no SH pulse input at the start of the interaction, multiple scale analysis of Eq. (12) yields

*n*=

_{q,j}*β*(

_{q}*ω*)

_{j}*c*/

*ω*and

_{j}*g*=

_{n,j}*g*(

_{n}*ω*) for wave

_{j}*j*(

*j*=

*FH*or

*j*=

*SH*) and mode normalization coefficient

*g*(

_{n}*ω*) given by Eq. (4). To analyze the different terms, it is convenient to introduce a simpler normalization of the mode profiles than the one used in sections 3 and 5 to analyze broadband pulses. If we choose

*g*(

_{n}*ω*) =

*a*(

_{n}*ω*) instead of

*g*(

_{n}*ω*) =

*a*(

_{n}*ω*)

^{1/3}, the mode profiles

*B*(

_{n}*x, y,ω*) are dimensionless. With this definition of

*g*,

_{n}*θ*

_{0000}/

*g*

_{0}(

*ω*) = 1, and the total effective

_{FH}*χ*

^{(3)}is given by

*c*are given by independent of the choice of normalization of the spatial mode profiles

_{q}*B*;

_{j}*(*χ ¯

*x,y*) is the transverse spatial profile of the second order nonlinear susceptibility, which appears in Eq. (3). The

*χ*

^{(2)}contributions to

*c*and Δ

_{q}*k*

_{0,}

_{q}_{00}are given in Tables 2 and 3 for the 1043-nm and 1580-nm pumped calculations discussed in sections 4 and 5, respectively.

*c*=

_{q}*δ*

_{q}_{0}and

*d*=

_{m}*δ*

_{m}_{0}

*d*

_{33}where

*δ*is the Kronecker delta. To calculate the cascaded phase shifts, we assumed a pulse centered around a particular carrier frequency. However, some care should be taken with this procedure since during the supercontinuum generation process the center frequency of the pulse can shift. This frequency shift can reduce the accuracy of the cascading approximation for terms that are nearly phasematched. For the simulations performed here, we add

_{ij}*χ*, with summation performed over all terms except those which are either explicitly included in Eq. (5) or for which

_{E}*m*= 1.

## Acknowledgments

## References and links

1. | C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. |

2. | T. Fuji, J. Rauschenberger, A. Apolonski, V. S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T. W. Hänsch, W. Lehnert, M. Scherer, and F. Krausz, “Monolithic carrier-envelope phase-stabilization scheme,” Opt. Lett. |

3. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

4. | J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. |

5. | C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B |

6. | C. Langrock, S. Kumar, J. McGeehan, A. Willner, and M. M. Fejer, “All-optical signal processing using |

7. | X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, “Efficient continuous wave second harmonic generation pumped at 1.55 μm in quasi-phase-matched AlGaAs waveguides,” Opt. Express |

8. | M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A |

9. | R. V. Roussev, “Optical-frequency mixers in periodically poled lithium niobate: materials, modeling and characterization,” Ph.D. thesis, Stanford University (2006), http://nlo.stanford.edu/system/files/dissertations/rostislav_roussev_thesis_december_2006.pdf. |

10. | M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E |

11. | G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modelling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express |

12. | M. Conforti, F. Baronio, and C. De Angelis, “Ultrabroadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. |

13. | S. Wabnitz and V. V. Kozlov, “Harmonic and supercontinuum generation in quadratic and cubic nonlinear optical media,” J. Opt. Soc. Am. |

14. | G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B |

15. | M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. |

16. | X. Liu, L. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascade |

17. | J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. |

18. | S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. |

19. | M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. |

20. | M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A |

21. | J. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. |

22. | M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B |

23. | C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. |

24. | C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO |

25. | R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n |

26. | I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B |

27. | M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A |

28. | C. R. Phillips, J. Jiang, C. Langrock, M. M. Fejer, and M. E. Fermann, “Self-Referenced Frequency Comb From a Tm-fiber Amplifier via PPLN Waveguide Supercontinuum Generation,” in CLEO:2011 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA5. |

29. | A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO |

30. | P. J. Delfyett, R. Dorsinville, and R. R. Alfano, “Spectral and temporal measurements of the third-order nonlinear susceptibility of LiNbO |

31. | N. Surovtsev, V. Malinovskii, A. Pugachev, and A. Shebanin, “The nature of low-frequency raman scattering in congruent melting crystals of lithium niobate,” Phys. Solid State |

32. | R. Schiek, R. Stegeman, and G. I. Stegeman, “Measurement of third-order nonlinear susceptibility tensor elements in lithium niobate,” in “ |

33. | G. I. Stegeman, D. J. Hagan, and L. Torner, “ |

34. | C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B |

35. | D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, n 22, 1553–1555 (1997). [CrossRef] |

36. | R. Boyd, “Stimulated Raman scattering and stimulated Rayleigh-Wing scattering,” in “ |

37. | C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “High-power mid-infrared optical parametric chirped-pulse amplifier based on aperiodically poled Mg:LiNbO3,” presented at the Conference on Lasers and Electro-optics (2011). |

38. | E. D. Palik and G. Ghosh, |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(320.6629) Ultrafast optics : Supercontinuum generation

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: May 23, 2011

Revised Manuscript: July 19, 2011

Manuscript Accepted: July 19, 2011

Published: September 12, 2011

**Citation**

C. R. Phillips, Carsten Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and Martin E. Fermann, "Supercontinuum generation in quasi-phasematched waveguides," Opt. Express **19**, 18754-18773 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-18754

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### References

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- T. Fuji, J. Rauschenberger, A. Apolonski, V. S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T. W. Hänsch, W. Lehnert, M. Scherer, and F. Krausz, “Monolithic carrier-envelope phase-stabilization scheme,” Opt. Lett.30, 332–334 (2005). [CrossRef] [PubMed]
- J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006). [CrossRef]
- J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron.13, 738–749 (2007). [CrossRef]
- C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B27, 2687–2699 (2010). [CrossRef]
- C. Langrock, S. Kumar, J. McGeehan, A. Willner, and M. M. Fejer, “All-optical signal processing using χ(2) nonlinearities in guided-wave devices,” J. Lightwave Technol.24, 2579–2592 (2006). [CrossRef]
- X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, “Efficient continuous wave second harmonic generation pumped at 1.55 μm in quasi-phase-matched AlGaAs waveguides,” Opt. Express13, 10742–10748 (2005). [CrossRef] [PubMed]
- M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841 (2010). [CrossRef]
- R. V. Roussev, “Optical-frequency mixers in periodically poled lithium niobate: materials, modeling and characterization,” Ph.D. thesis, Stanford University (2006), http://nlo.stanford.edu/system/files/dissertations/rostislav_roussev_thesis_december_2006.pdf .
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- G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modelling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express15, 5382–5387 (2007). [CrossRef] [PubMed]
- M. Conforti, F. Baronio, and C. De Angelis, “Ultrabroadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J.2, 600–610 (2010). [CrossRef]
- S. Wabnitz and V. V. Kozlov, “Harmonic and supercontinuum generation in quadratic and cubic nonlinear optical media,” J. Opt. Soc. Am.B27, 1707–1711 (2010).
- G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B18, 534–539 (2001). [CrossRef]
- M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron.28, 2631–2654 (1992). [CrossRef]
- X. Liu, L. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascade χ(2) : χ(2) nonlinearity,” Opt. Lett.24, 1777–1779 (1999). [CrossRef]
- J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett.31, 1881–1883 (2006). [CrossRef] [PubMed]
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- M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett.32, 2490–2492 (2007). [CrossRef] [PubMed]
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- C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO3,” Opt. Lett.35, 2340–2342 (2010). [CrossRef] [PubMed]
- R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996). [CrossRef]
- I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B14, 2268–2294 (1997). [CrossRef]
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- C. R. Phillips, J. Jiang, C. Langrock, M. M. Fejer, and M. E. Fermann, “Self-Referenced Frequency Comb From a Tm-fiber Amplifier via PPLN Waveguide Supercontinuum Generation,” in CLEO:2011 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA5.
- A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO3,” Phys. Rev.158, 433 (1967). [CrossRef]
- P. J. Delfyett, R. Dorsinville, and R. R. Alfano, “Spectral and temporal measurements of the third-order nonlinear susceptibility of LiNbO3 using picosecond Raman-induce phase-conjugate spectroscopy,” Phys. Rev. B40, 1885 (1989). [CrossRef]
- N. Surovtsev, V. Malinovskii, A. Pugachev, and A. Shebanin, “The nature of low-frequency raman scattering in congruent melting crystals of lithium niobate,” Phys. Solid State45, 534–541 (2003). [CrossRef]
- R. Schiek, R. Stegeman, and G. I. Stegeman, “Measurement of third-order nonlinear susceptibility tensor elements in lithium niobate,” in “Frontiers in Optics,” (Optical Society of America, 2005), p. JWA74.
- G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996). [CrossRef]
- C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B19, 852–859 (2002). [CrossRef]
- D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett.22, 1553–1555 (1997). [CrossRef]
- R. Boyd, “Stimulated Raman scattering and stimulated Rayleigh-Wing scattering,” in “Nonlinear Optics”, R. Boyd (Academic, 2008). [CrossRef]
- C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “High-power mid-infrared optical parametric chirped-pulse amplifier based on aperiodically poled Mg:LiNbO3,” presented at the Conference on Lasers and Electro-optics (2011).
- E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).

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