## Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides

Optics Express, Vol. 15, Issue 9, pp. 5382-5387 (2007)

http://dx.doi.org/10.1364/OE.15.005382

Acrobat PDF (134 KB)

### Abstract

We describe generalized nonlinear envelope equation modeling of sub-cycle dynamics on the underlying electric field carrier during one-dimensional propagation in fused silica. Generalized envelope equation simulations are shown to be in excellent quantitative agreement with the numerical integration of Maxwell’s equations, even in the presence of shock dynamics and carrier steepening on a sub-50 attosecond timescale. In addition, by separating the effects of self-phase modulation and third harmonic generation, we examine the relative contribution of these effects in supercontinuum generation in fused silica nanowire waveguides.

© 2007 Optical Society of America

## 1. Introduction

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

2. M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express **13**, 6848–6855 (2005). [CrossRef] [PubMed]

3. V. P. Kalosha and J. Herrmann, “Self-phase modulation and compression of few-optical-cycle pulses,” Phys. Rev. A **62**, 011804(R) (2000). [CrossRef]

4. N. Karasawa, “Computer simulations of nonlinear propagation of an optical pulse using a finite-difference in the frequency-domain method,” IEEE J. Quantum Electron. **38**, 626–629 (2002). [CrossRef]

5. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. **25**, 2665–2673 (1989). [CrossRef]

9. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. **87**, 203901 (2001). [CrossRef] [PubMed]

## 2. Generalized Nonlinear Envelope Equation Model

5. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. **25**, 2665–2673 (1989). [CrossRef]

6. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. **78**, 3282–3285 (1997). [CrossRef]

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

5. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. **25**, 2665–2673 (1989). [CrossRef]

*E*͂ (

*z*,ω) =1/(2π) ∫

^{∞}

_{-∞}

*E*(

*z*,

*t*)exp(

*i*ω

*t*)

*dt*, where

*E*(

*z*,

*t*) is the real electric field and

*z*and

*t*are laboratory-frame variables. Considering only a third-order nonlinear response, the operator 𝒩͂ has time-domain definition: 𝒩 = χ

^{(3)}

*E*

^{2}(

*z*,

*t*)/

*n*

^{2}(ω). Parameters β (ω) and

*n*(ω) are the usual wavevector and refractive index.

*z*or a perturbative treatment of the nonlinear response. It is these approximations that restrict the validity of the resulting equation to optical fields with temporal structure slower than an optical cycle [5–7

**25**, 2665–2673 (1989). [CrossRef]

*E*͂

_{+}(

*z*,ω) and backward

*E*͂ -(

*z*,ω) propagating fields [14]. Indeed, analytical techniques based on Green’s functions [15

15. A. Ferrando, M. Zacares, P. F. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E **71**, 016601 (2005). [CrossRef]

16. M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. **89**, 283902 (2002). [CrossRef]

17. P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A **72**, 063807 (2005). [CrossRef]

15. A. Ferrando, M. Zacares, P. F. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E **71**, 016601 (2005). [CrossRef]

*E*͂ (

*z*,ω) =

*E*͂

_{+}(

*z*,ω)+

*E*͂

_{-}(

*z*,ω), we can transform Eq. (1) into a pair of first-order equations:

*E*

_{+}(

*z*,

*t*) = [

*A*

_{+}(

*z*,

*t*)exp(

*i*ω

_{0}

*t*)+c.c.)]/2, but it is crucial to note that the (generalized) envelope here can contain temporal structure on an arbitrarily fast scale. The field spectrum

*E*͂

_{+}(

*z*,ω) thus retains frequency components at many multiples of ω

_{0}, and it is this that allows the reconstructed field waveform

*E*

_{+}(

*z*,

*t*) to accurately model sub-cycle dynamics.

13. Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond pulse compression and carrier envelope phase effect,” Phys. Rev. A **72**, 063802 (2005). [CrossRef]

17. P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A **72**, 063807 (2005). [CrossRef]

*U*(

*z*,

*t*)] is normalized such that |

*U*|

^{2}yields the instantaneous power in watts [18]. After some algebra, we obtain the time domain GNEE:

_{k}’s are the usual dispersion coefficients, although the global dispersion β(ω) can also be directly applied [18]. On the right-hand-side, the terms proportional to |

*U*|

^{2}

*U*and

*U*

^{3}describe self phase modulation and THG respectively, and the nonlinear coefficient γ = ω

_{0}

*n*

_{2}/

*cA*, where the nonlinear refractive index

_{eff}*n*

_{2}and the effective area

*A*are evaluated at ω

_{eff}_{0}. In the absence of a frequency-dependent effective area, the envelope self-steepening timescale τ

_{ss}= 1/ω

_{0}, but this can be readily generalized for a dispersive

*A*[19

_{eff}19. B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B **81**, 337–342 (2005). [CrossRef]

*f*, and the generalized Raman function is given by:

_{R}*g*(

*z*,

*t*,

*U*) = 2/3 (

*h*(

_{R}*t*) ∗ |

*U*(

*z*,

*t*)|

^{2})

*U*(

*z*,

*t*)+ 2/3exp(-

*i*ω

_{0}

*t*)(

*h*´

_{R}(

*t*) ∗

*U*

^{2}(

*z*,

*t*)´Re(exp(-iω

_{0}

*t*)

*U*(

*z*,

*t*)) with

*h*´

_{R}(

*t*) =

*h*

_{R}(

*t*)exp(2iω

_{0}

*t*). Note also that this Raman term includes rapidly oscillating components valid beyond slowly-varying envelope approximations [8

8. P. Kinsler and G.H.C. New, “ Wideband pulse propagation: single-field and multi-field approaches to Raman interactions,”Phys.Rev. A **72**, 033804 (2005). [CrossRef]

*n*

^{2}= 2.6×10

^{-20}m

^{2}W

^{-1}and, when Raman is included, we take

*f*= 0.18 and use the experimentally-measured Raman response function of silica [18].

_{R}## 3. Numerical Simulations

**25**, 2665–2673 (1989). [CrossRef]

*U*are determined from the input field ℰ = [

*U*exp(iω

_{0}

*t*)+ c.c.]/2, and initial conditions on the carrier envelope offset phase can be specified through an additional constant phase term of the type exp(iΔϕ

_{CEO}). Here we use normalization as above so that ℰ has units of W

^{1/2}. Once the resulting output envelope is determined, the corresponding output field is then reconstructed and, with no restriction on the temporal structure that can be incorporated onto the envelope, this naturally allows for the inclusion of sub-cycle dynamics, if present.

*I*

_{0}=

*P*

_{0}/

*A*= 4.6×10

_{eff}^{17}Wm

^{-2}propagating in the absence of dispersion and Raman scattering, and we assume zero initial carrier-envelope offset phase. The initial nonlinear strength is δ

_{n}=

*n*

_{2}

*I*

_{0}= (

*c*/ω

_{0})γ

*P*

_{0}= 1.2×10

^{-2}. GNEE simulations modeled propagation over 2.81 μm where theory predicts carrier shock onset [20–22]. Fig. 1(a) shows the output reconstructed temporal field, with the expected carrier steepening apparent in the exploded view in Fig. 1(b) (solid line). Physically, the shock arises from the generation of multiple odd-order harmonics that are seen in the corresponding spectrum (solid line) in Fig. 1(c). The generation of the oddorder harmonic cascade occurs as a result of the χ

^{(3)}nonlinearity mixing pump and harmonic fields through processes such as ω

_{0}+ω

_{0}+3ω

_{0}→ 5ω

_{0};ω

_{0}+ω

_{0}+5ω

_{0}→ 7ω

_{0}etc.

23. J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. **52**, 973–986 (2005). [CrossRef]

13. Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond pulse compression and carrier envelope phase effect,” Phys. Rev. A **72**, 063802 (2005). [CrossRef]

25. P. Kinsler, S. B. P. Radnor, J. Tyrrell, and G. H. C. New, “Optical carrier wave shocking and the effect of dispersion,” Phys. Rev. E , submitted (2007). [CrossRef]

_{CEO}= π/2 on the input field. Although the effect of the initial phase on the dynamics is minor for this set of parameters, a difference is nonetheless manifested in the depth of the spectral minima, and Fig. 1(d) shows the spectral amplitude in the vicinity of the first minimum from the GNEE simulations for Δϕ

_{CEO}= 0 (solid line), and for Δϕ

_{CEO}= π/2 (dashed line). The PSSD simulation results for the latter case are shown as the circles, and the excellent quantitative agreement again confirms the GNEE validity.

3. V. P. Kalosha and J. Herrmann, “Self-phase modulation and compression of few-optical-cycle pulses,” Phys. Rev. A **62**, 011804(R) (2000). [CrossRef]

_{ss}= 1/ω

_{0}and neglect Raman scattering. GNEE results showing the output reconstructed temporal field after 100 μm propagation are shown in Fig. 2(a), with an exploded view near the pulse centre in Fig 2(b) (solid lines). Fig. 2(c) (solid line) shows the corresponding spectrum. The principle differences in the dynamics here arise from pump-THG walkoff, and cross-phase modulation on the THG field from the pump which displaces it from the ideal 3ω

_{0}value. Despite these additional effects, however, the GNEE simulations are again in excellent agreement with the numerical integration of Maxwell’s equations (circles) shown in Figs. 2(b) and 2(c). We also note that propagation over the 100 μm distance here is convenient to illustrate the broadened pump and THG signatures clearly, but additional simulations confirm agreement between GNEE and Maxwell’s equation simulations over~mm distances when the spectral characteristics are more complex.

2. M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express **13**, 6848–6855 (2005). [CrossRef] [PubMed]

26. M. A. Foster, J.M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B **81**, 363–367 (2005). [CrossRef]

## 4. Conclusions

^{(3)}medium. This is explicitly confirmed by comparisons of GNEE and PSSD simulations. The GNEE possesses a number of particular features when compared to field-based propagation models, notably its expression in terms of the familiar NLSE-based formalism of nonlinear fiber optics, and its convenient numerical implementation through straightforward modifications to existing generalized NLSE solvers used for SC generation modeling. We anticipate that it will become widely used to study few cycle dynamical processes in nonlinear waveguides.

## References and links

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

2. | M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express |

3. | V. P. Kalosha and J. Herrmann, “Self-phase modulation and compression of few-optical-cycle pulses,” Phys. Rev. A |

4. | N. Karasawa, “Computer simulations of nonlinear propagation of an optical pulse using a finite-difference in the frequency-domain method,” IEEE J. Quantum Electron. |

5. | K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. |

6. | T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. |

7. | N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. |

8. | P. Kinsler and G.H.C. New, “ Wideband pulse propagation: single-field and multi-field approaches to Raman interactions,”Phys.Rev. A |

9. | A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. |

10. | A. V. Husakou and J. Herrmann, “Supercontinuum generation, four-wave mixing, and fission of higher-order solitons in photonic-crystal fibers,” J. Opt. Soc. Am. B |

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

12. | M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B |

13. | Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond pulse compression and carrier envelope phase effect,” Phys. Rev. A |

14. | Y. R. Shen, |

15. | A. Ferrando, M. Zacares, P. F. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E |

16. | M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. |

17. | P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A |

18. | G. P. Agrawal, |

19. | B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B |

20. | G. Rosen, “Electromagnetic shocks and the self-annihilation of intense linearly polarized radiation in an ideal dielectric material,” Phys. Rev. A |

21. | R. G. Flesch, A. Pushkarev, and J. V. Moloney, “Carrier wave shocking of femtosecond optical pulses,” Phys. Rev. Lett. |

22. | L. Gilles, J. V. Moloney, and L. Vazquez, “Electromagnetic shocks on the optical cycle of ultrashort pulses in triple-resonance lorentz dielectric media with subfemtosecond nonlinear electronic debye relaxation,” Phys. Rev. E |

23. | J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. |

24. | P. Kinsler, G. H. C. New, and J.C.A. Tyrrell, ”Phase sensitivity of nonlinear interactions”, arXiv.org/physics/0611213. |

25. | P. Kinsler, S. B. P. Radnor, J. Tyrrell, and G. H. C. New, “Optical carrier wave shocking and the effect of dispersion,” Phys. Rev. E , submitted (2007). [CrossRef] |

26. | M. A. Foster, J.M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B |

**OCIS Codes**

(060.7140) Fiber optics and optical communications : Ultrafast processes in fibers

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 5, 2007

Manuscript Accepted: April 2, 2007

Published: April 18, 2007

**Citation**

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, "Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides," Opt. Express **15**, 5382-5387 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5382

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### References

- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78,1135-1184 (2006). [CrossRef]
- M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, "Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires," Opt. Express 13,6848-6855 (2005). [CrossRef] [PubMed]
- V. P. Kalosha and J. Herrmann, "Self-phase modulation and compression of few-optical-cycle pulses," Phys. Rev. A 62,011804(R) (2000). [CrossRef]
- N. Karasawa, "Computer simulations of nonlinear propagation of an optical pulse using a finite-difference in the frequency-domain method," IEEE J. Quantum Electron. 38,626-629 (2002). [CrossRef]
- K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25,2665-2673 (1989). [CrossRef]
- T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78,3282-3285 (1997). [CrossRef]
- N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber," IEEE J. Quantum Electron. 37,398-404 (2001). [CrossRef]
- P. Kinsler and G.H.C. New, " Wideband pulse propagation: single-field and multi-field approaches to Raman interactions,"Phys.Rev. A 72, 033804 (2005). [CrossRef]
- A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87,203901 (2001). [CrossRef] [PubMed]
- A. V. Husakou and J. Herrmann, "Supercontinuum generation, four-wave mixing, and fission of higher-order solitons in photonic-crystal fibers," J. Opt. Soc. Am. B 19,2171-2182 (2002). [CrossRef]
- M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70,036604 (2004). [CrossRef]
- M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, "Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air," Appl. Phys. B 85,531-538 (2006). [CrossRef]
- Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, "Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond pulse compression and carrier envelope phase effect," Phys. Rev. A 72,063802 (2005). [CrossRef]
- Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).
- A. Ferrando, M. Zacares, P. F. de Cordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71,016601 (2005). [CrossRef]
- M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional optical pulse propagation equation," Phys. Rev. Lett. 89,283902 (2002). [CrossRef]
- P. Kinsler, S. B. P. Radnor, and G. H. C. New, "Theory of directional pulse propagation," Phys. Rev. A 72,063807 (2005). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, San Diego, 2006).
- B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81,337-342 (2005). [CrossRef]
- G. Rosen, "Electromagnetic shocks and the self-annihilation of intense linearly polarized radiation in an ideal dielectric material," Phys. Rev. A 139,A539- A543 (1965).
- R. G. Flesch, A. Pushkarev, and J. V. Moloney, "Carrier wave shocking of femtosecond optical pulses," Phys. Rev. Lett. 76,2488-2491 (1996). [CrossRef] [PubMed]
- L. Gilles, J. V. Moloney, and L. Vazquez, "Electromagnetic shocks on the optical cycle of ultrashort pulses in triple-resonance lorentz dielectric media with subfemtosecond nonlinear electronic debye relaxation," Phys. Rev. E 60,1051-1059 (1999). [CrossRef]
- J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, "Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion," J. Mod. Opt. 52,973-986 (2005). [CrossRef]
- P. Kinsler, G. H. C. New and J.C.A. Tyrrell, "Phase sensitivity of nonlinear interactions", arXiv.org/physics/0611213.
- P. Kinsler, S. B. P. Radnor, J. Tyrrell, and G. H. C. New, "Optical carrier wave shocking and the effect of dispersion," Phys. Rev. E, submitted (2007). [CrossRef]
- M. A. Foster, J.M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81,363-367 (2005). [CrossRef]

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