## Spatio-temporal instabilities for counter-propagating waves in periodic media

Optics Express, Vol. 10, Issue 2, pp. 114-121 (2002)

http://dx.doi.org/10.1364/OE.10.000114

Acrobat PDF (169 KB)

### Abstract

Nonlinear evolution of coupled forward and backward fields in a multi-layered film is numerically investigated. We examine the role of longitudinal and transverse modulation instabilities in media of finite length with a homogeneous nonlinear susceptibility *χ ^{(3)}
*. The numerical solution of the nonlinear equations by a beam-propagation method that handles backward waves is described.

© Optical Society of America

## 1. Introduction

1. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. **58**, 160–163 (1987). [CrossRef] [PubMed]

3. M. Scalora, J.P. Dowling, C.M. Bowden, and M. J. Bloemer,“ Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. **73**, 1368–1371 (1994). [CrossRef] [PubMed]

4. M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. **76**, 2023–2026 (1994). [CrossRef]

5. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. **75**, 1896–1899 (1994). [CrossRef]

6. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E **54**, R1078–1081 (1996). [CrossRef]

7. G. G. Luther and C. J. McKinstrie, “ Transverse modulational instability of collinear waves,” J. Opt. Soc. Am. B **7**, 1125–1141 (1990). [CrossRef]

10. B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. **32**, 2341–2342 (1996). [CrossRef]

12. N. M. Lichinitzer, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B **18**, 45–54 (2001). [CrossRef]

13. M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. **108**, 191–196 (1994). [CrossRef]

## 2. Mode-coupled equations and computational approach

### 2.1 Mode-coupled Equations

10. B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. **32**, 2341–2342 (1996). [CrossRef]

*S*(

*x*,

*y*,

*t*) is the pulse input to the sample. The linear equations are analytically solved by taking Fourier transforms of the time and transverse coordinates, using

*exp(-iωt)*for the time-harmonic component. The solutions are

*δ*+

*ω*/

*v*-

*q*

^{2}/

*F*, and Δ

^{2}=Ω

^{2}-

*κ*

^{2}. The solutions have the transversal and temporal coordinate dependence, albeit in Fourier transformed variables.

### 2.2 Numerical Computations for Coupled Mode Equations

13. M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. **108**, 191–196 (1994). [CrossRef]

*V*, is a matrix with diagonal components we denote by

*N*

*K*

*U*= (

*E*,

_{f}*E*)

_{b}^{T}, where the superscript

*T*denotes the transpose of the vector) is

*L*is diagonalized by Fourier transforming the (

*x*,

*z*) variables. The split-step method therefore uses fast Fourier transforms to simplify the linear operator matrices. The central operator in Eq. (7) can be subsequently decomposed into a linear portion and a nonlinear contribution, namely

*z*=

*0*. The relation between the two conditions is

*F*

*(x,z,0)*=

*S(x,0,*

*(z*-

*z*, because the pulse propagates without dispersion in the medium outside the nonlinear medium. We also assume that the boundary is impedance matched, so that in the absence of the coupling

_{0}F)v)*κ*no reflected wave is generated. The initial pulse has been displaced by

*z*

_{0}to a position outside the medium. The accuracy of the code was verified by comparing results with the analytic solutions of the linear equations and by verifying the conservation of energy at each time step.

## 3. Results

*E*=

_{x}*A*exp(

_{x}*iKz*), where the subscript is

*x*=

*f*,

*b*. The result is a quadratic equation, which can be expressed it in the form

*δ*is larger than

*κ*. A band gap separates the two branches. Tuning the laser close to one band or another determines the characteristics of the wave’s propagation. The transmission curve is determined from the coefficient of F(

*q*,

*ω*) on the right hand side of Eq.(3a) when

*z*=

*L*. In our numerical equations the space, time and field amplitude are scaled so that the following parameter values are used:

*κ*=

*1*,

*v*=

*1*and

*η*=

*1*. Only real, positive values of the nonlinear coefficient are discussed here and for simplicity we chose

*F*=

*100*. For a medium length

*L*=6.

*1*, the transmission curve is shown in Figure 1.

*z*, of the pulse is chosen to be several times longer than its pulse width

_{0}*σ*= 1/8

_{z}*π*. This prevents any significant overlap between the medium and the leading edge of the wave. The transverse width was chosen as

*σ*=

_{x}*π*/

*64*. The z-axis spans 20π and the x-axis span is 8π. In our computations

*F*=

*100*so that the transverse coupling is weak and does not noticeably affect the pulse tuning at a selected frequency.

*A*=

*0.2*). The absolute amplitude of the field at each pixel is represented by a color. The vertical axis displays the transverse coordinate and the horizontal axis is the propagation direction. The evolution is stepped through a time sequence beginning with the pulse to the left of the nonlinear PBG medium. The time increment between frames is

*Δt*=

*1*. Three subfigures displayed are displayed in the animation. The top figure is the input Gaussian field. The initial pulse is displaced in the window and propagates to the right. The nonlinear medium begins at the far left of the top panel and is not shown. The middle figure is the continuation of the forward-propagating pulse as it passes into and through the medium. The boundaries of the medium are denoted by the green boxes positioned to the left side on the top and bottom of the figure. The bottom panel shows the propagation of the backward field in the medium. The green boxes on the right side of the figure denote the boundaries of the medium. The pulse width is shorter than the medium; it is reflected inside the medium as contributes to pulse break up along the longitudinal axis. This behavior can be expected by comparing the pulse spectrum with the transmission curve in Figure 1. The pulse is short enough to sample a large portion of the dispersive spectrum in the medium.

*A*=

*0.6*. In this case we do not plot the input field displayed in the top panel of Figure 2. The laser is still detuned from the center of the gap to the first transmission maximum on the high frequency side,

*δ*=

*1.12*. As the field interacts with the medium the pulse begins to break up in both the transverse and longitudinal directions. There is also strong focussing of the pulse as it enters the medium.

*F*also leads to a larger spread of the pulse in passage through the medium and greater divergence of the nonlinearly modulated beam in free space.

*A*=

*0.4*. But a closer look at the transverse spatial spectra of the forward propagating field, defined by

*H*(

*q*) is plotted for three amplitudes in Figure 5. The pulses have propagated through the medium. The spectrum for

*A*=

*0.2*in Figure 5 is Gaussian. However, the spectrum for

*A*=

*0.3*is already deformed from a Gaussian shape; the wings of the initial Gaussian profile acquire a broad shoulder. At threshold the transverse coordinate has relatively low spatial frequencies. Above threshold the spatial frequencies become higher; we find the maximum develops in the nonlinear regime around

*q*=

*2.2*and moves towards larger values.

12. N. M. Lichinitzer, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B **18**, 45–54 (2001). [CrossRef]

*δ*= -

*1.12*. A large field amplitude is chosen for this example

*A*=

*0.7*. The pulse does undergo a pulse focussing effect in the medium due to the dominance of the nonlinearity. However, the pulse break up does not occur. For our chosen parameters the absence of modulations instability is expected based on the linear stability analysis of Lichtinitzer et al. [12

12. N. M. Lichinitzer, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B **18**, 45–54 (2001). [CrossRef]

## 4. Conclusions

*A*=

*0.3*. This compares favorably with the value

*A*=

*0.14*we deduce from the paper of Lichinitzer et al. [12

**18**, 45–54 (2001). [CrossRef]

*k*=

_{t}*16*/

*π*

^{2}, and longitudinal wavenumber,

*k*=

_{p}*0.4*. This comparison is only qualitative though, since the stability analysis is made for an infinite medium. In a finite medium we chose the central frequency at the first transmission maximum, because the field is locally enhanced. Our pulses are short and the spectral width of the input pulse samples a spread of dispersion values. Also the modulation instability gain needs a sufficient path length to grow a local perturbation and our sample was kept short. A stability analysis for the finite system is required for a quantitative comparison with our simulations, but the problem may not be manageable by analytic techniques.

*A*=

*0.7*. On the high frequency branch and tuned to the first transmission maximum the transmission coefficient increases with the input intensity from about 0.4 at low intensity to about 0.6 for

*A*=

*0.6*; however, the diffraction reduces the on-axis energy, so that by using an apertured detector the off-axis portion is eliminated from the system. To put values on the threshold intensity for the modulation instability consider a sample made from two dielectrics with dielectric contrast, 0.01 and a nonlinear optical coefficient n

_{2}= 10

^{-10}cm

^{2}/W, the corresponding threshold intensity in finite systems is about 3 MW/cm

^{2}. At threshold the nonlinear change of the index is about 0.0003.

## Acknowledgements

## References and links

1. | W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. |

2. | C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics |

3. | M. Scalora, J.P. Dowling, C.M. Bowden, and M. J. Bloemer,“ Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. |

4. | M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. |

5. | J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. |

6. | M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E |

7. | G. G. Luther and C. J. McKinstrie, “ Transverse modulational instability of collinear waves,” J. Opt. Soc. Am. B |

8. | L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A |

9. | Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. |

10. | B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. |

11. | B. J. Eggleton, C. M. deSterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B |

12. | N. M. Lichinitzer, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B |

13. | M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 5, 2001

Revised Manuscript: January 17, 2002

Published: January 28, 2002

**Citation**

Joseph Haus, Boon Yi Soon, Michael Scalora, Mark Bloemer, Charles Bowden, Concita Sibilia, and Alexei Zheltikov, "Spatio-temporal instabilities for counterpropagating waves in periodic media," Opt. Express **10**, 114-121 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-2-114

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### References

- W. Chen and D. L. Mills, ?Gap solitons and the nonlinear optical response of superlattices,? Phys. Rev. Lett. 58, 160-163 (1987). [CrossRef] [PubMed]
- C. M. de Sterke and J. E. Sipe, ?Gap Solitons,? Progress in Optics 33, 203-260 (1994).
- M. Scalora, J. P. Dowling, C. M. Bowden and M. J. Bloemer, ?Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,? Phys. Rev. Lett. 73, 1368-1371 (1994). [CrossRef] [PubMed]
- M. Scalora, J. P. Dowling, M. J. Bloemer and C. M. Bowden, ?The photonic band edge optical diode,? J. Appl. Phys. 76, 2023-2026 (1994). [CrossRef]
- J. P. Dowling, M. Scalora, M. J. Bloemer and C. M. Bowden, ?The photonic band edge laser: A new approach to gain enhancement,? J. Appl. Phys. 75, 1896-1899 (1994). [CrossRef]
- M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling and R. P. Leavitt, ?Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,? Phys. Rev. E 54, R1078-1081 (1996). [CrossRef]
- G. G. Luther and C. J. McKinstrie, ?Transverse modulational instability of collinear waves,? J. Opt. Soc. Am. B 7, 1125-1141 (1990). [CrossRef]
- L. W. Liou, X. D. Cao, C. J. McKinstrie and G. P. Agrawal, ?Spatiotemporal instabilities in dispersive nonlinear media,? Phys. Rev. A 46, 4202-4208 (1992). [CrossRef] [PubMed]
- Y. Silberberg, ?Collapse of optical pulses,? Opt. Lett. 15, 1282-1284 (1990). [CrossRef] [PubMed]
- B. J. Eggleton, C. M. deSterke, R. E. Slusher and J. E. Sipe, ?Distributed feedback pulse generator based on nonlinear fiber grating,? Electron. Lett. 32, 2341-2342 (1996). [CrossRef]
- B. J. Eggleton, C. M. deSterke and R. E. Slusher, ?Nonlinear pulse propagation in Bragg gratings,? J. Opt. Soc. Am. B 14, 2980-2993 (1997). [CrossRef]
- N. M. Lichinitzer, C. J. McKinstrie, C. M. de Sterke and G. P. Agrawal, ?Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,? J. Opt. Soc. Am. B 18, 45-54 (2001). [CrossRef]
- M. Scalora and M. Crenshaw, ?Beam propagation method that handles reflections,? Opt. Commun. 108, 191-196 (1994). [CrossRef]

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