## Dynamic cross-waveguide optical switching with a nonlinear photonic band-gap structure

Optics Express, Vol. 3, Issue 1, pp. 28-34 (1998)

http://dx.doi.org/10.1364/OE.3.000028

Acrobat PDF (571 KB)

### Abstract

We present a numerical study of a two dimensional all-optical switching device which consists of two crossed waveguides and a nonlinear photonic band-gap structure in the center. The switching mechanism is based on a dynamic shift of the photonic band edge by means of a strong pump pulse and is modeled on the basis of a two dimensional finite volume time domain method. With our arrangement we find a pronounced optical switching effect in which due to the cross-waveguide geometry the overlay of the probe beam by a pump pulse is significantly reduced.

© Optical Society of America

## 1. Introduction

4. 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]

3. P. Tran, “Optical switching with a nonlinear photonic Crystal: a numerical study,” Opt. Lett. **21**, 1138–1140 (1996). [CrossRef] [PubMed]

*ϵ*

_{h}, the transmission curve is shifted and the probe beam no longer passes the structure but instead is nearly completely reflected.

*ϵ*

_{h}is realized by means of a second sufficiently strong pump pulse. The principle of this optical switching has been demonstrated by Scalora and Tran on the basis of a one-dimensional (1D) model. There, both the probe and the pump pulse are propagating in the same direction such that the strong pump pulse overlays the probe beam. Here, we present an alternative set-up with a cross-geometry and take into account the propagation and dynamic two-dimensional (2D) light field variations. The underlying configuration is shown in Fig. 2. The photonic switch consists of two crossed waveguides with a nonlinear photonic band-gap structure in the center. This cross geometry represents in comparison to the one-dimensional arrangement a more realistic configuration for an optical switch. In particular, the pump beam propagates perpendicular to the direction of the probe beam. In our cross-waveguide optical switch the overlay of the probe beam by the pump beam is significantly reduced - a fact which may become important in potential technical applications of all-optical switching devices. Note that the multilayer is structured periodicly only in the direction of the probe beam and there is no further restriction with respect to the frequency of the pump pulse.

*ω*,

*ω*,

5. S. V. Polstyanko, R. Dyczij-Edlinger, and J. F. Lee Lee, “Full vectorial analysis of a nonlinear slab waveguide based on the nonlinear hybrid vector finite-element method,” Opt. Lett. **21**, 98–100 (1996). [CrossRef] [PubMed]

4. 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]

*t*,

## 2. Basic equations and numerical method

*x*-

*y*plane reduce to2

*c*is the speed of light and

*x*,

*y*,

*z*label the Cartesian components of the fields. The interaction between the optical field and the nonlinear dielectric medium is modeled by a Kerr-type nonlinearity

*ϵ*=

*ϵ*(

*x*,

*y*), the Kerr nonlinearity

*χ*

_{3}=

*χ*

_{3}(

*x*,

*y*), and the nonlinear dielectric function

*ϵ*

_{nl}=

*ϵ*

_{nl}(

*x*,

*y*,

*E*

_{z}) =

*ϵ*(

*x*,

*y*) +

*χ*

_{3}(

*x*,

*y*) ∙

*x*,

*y*,

*t*) represent the nonlinear photonic band-gap structure depicted in Fig. 2. Note that we have assumed possible dispersive effects to have no critical influence on the switching effects studied here. Further, we neglect any magnetic response in the material, i.e.

*B*

*= H*. Assuming the boundaries of the waveguides to be perfectly conducting leads to

*G*and

*c*∙ (

*D*

_{z},

*B*

_{x},

*B*

_{y}),

_{x}= (-

*H*

_{y},0,-

*E*

_{z}) and

_{y}= (

*H*

_{x},

*E*

_{z},0).

_{n+1}at time

*t*=

*t*

_{n+1}=

*t*

_{n}+ Δ

*t*from the existing values

_{n}at time

*t*=

*t*

_{n}. We note that our numerical procedure is similar to the frequently used method of time domain finite differences [8

8. P. M. Goorjian and A. Taflove, “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. **17**, 180 (1992). [CrossRef] [PubMed]

## 3. Dynamic optical switching

*TM*

_{1}modes (or superpositions of these modes) into account. The explicit functional form of these modes is described in the appendix. Besides the finite extension in

*y*-direction, the multilayer sequence is periodically structured only along the

*x*-direction. Thus, a photonic band-gap does not exist for all directions of the wave vector

*k*

_{x},

*k*

_{y}) of the incident probe beam. A well pronounced band gap is easily obtained, however, by choosing

*k*

_{y}sufficiently small with respect to

*k*

_{x}3.

*f*=

*f*

_{probe}, marked by the arrow in Fig. 1, and the amplitude

*A*

_{probe}is set to 1.0. The resulting energy density distributions are shown in Fig. 3 and Fig. 4 for eh = 2.0 and

*ϵ*

_{h}= 2.1, respectively. As expected, the beam passes the multilayer structure in the case of

*ϵ*

_{h}= 2.0 and is exponentially attenuated in the case of

*ϵ*

_{h}= 2.1. In addition, the multilayer structure also serves as an effective waveguide in the center of the photonic switch where the waveguides are absent: In Fig. 3 and Fig. 4 the portion of the intensity which is being scattered towards the top and the bottom only amounts to ≈ 0.1%.

*ϵ*

_{h}, the change of the dielectric properties is now provided by a sufficiently strong pump pulse. For specifity, we assume a Gaussian shaped superposition of the basic

*TM*

_{1}modes (see appendix) resulting in a pulse whose intensity smoothly rises and falls. The frequency of the pulse is chosen as

*f*

_{pump}= 0.483 and its width amounts

*σ*= 5.0 ∙

*T*

_{pump}, where

*T*

_{pump}= 1/

*f*

_{pump}. Its amplitude is set to

*A*

_{pump}= 14.0. In the absence of the probe beam one obtains the results illustrated as snapshots in Fig. 5 and Fig. 6 which show the spatial distribution of the pump pulse as it passes the multilayer structure and the distribution of the induced nonlinear refractive index

*ϵ*

_{nl}, respectively. Besides a small portion which is reflected, the main intensity of the pulse passes the photonic switch. In Fig. 5 one can clearly see that in the photonic band gap structure most of its energy is spatially confined to the center, resembling the initial shape of the incident pulse. As a consequence, the region with a strong induced change of the (nonlinear) dielectric constant

*ϵ*

_{nl}is thus also located at the center of the switch. In addition, the distribution of

*ϵ*

_{nl}is wave shaped, according to the wave length of the pump pulse. In passing we note that this wave length differs between the waveguide and the area of the photonic band-gap structure. Hence we have a dynamically induced 2D photonic band-gap structure. Note further that due to the transverse shape of the pump pulse the number of dielectric layers which will effectively contribute to a switching process is reduced in comparison to the static change of the dielectric constant assumed in Fig. 4. In Fig. 5 the loss due to scattering of the pump pulse in the waveguide amounts to ≈ 0.4% with respect to the incident pump pulse.

*ϵ*

_{nl}. The time scale is given in units of [

*λ*

_{0}/

*c*]. For optical wave lengths, this corresponds to 10

^{-15}to 10

^{-14}seconds.

*t*= 0, the intensity of the pump pulse has not yet reached its maximum, and the change of the

*ϵ*

_{nl}is small. Hence, the probe beam passes the photonic switch. The animation also shows that the multilayer structure also causes a resonator effect reminiscent of a distributed feedback arrangement used e.g. in semiconductor lasers. With increasing time the intensity of the pump pulse rises and

*ϵ*

_{nl}increases. As a consequence, the band edge is dynamically shifted such that the propagation of the probe beam is significantly disturbed. At

*t*= 10 the maximum of the pump pulse is reached an the transmitted intensity of the probe beam is continously reduced until

*t*≈ 20. In the follwing, with the pump pulse having passed the structure, the probe beam returns to his initial intensity.

## 4. Conclusion

## Appendix: Beam and pulse shape

*y*= 0 and

*y*=

*a*, the basic guided

*TM*

_{1}modes have the following form:

*k*

_{x}=

*k*

_{y}=

*π*/

*a*. The amplitude A and phase

*ϕ*

_{x}may be freely chosen. Multiplication of these modes with an envelope exp (0.5 ∙ (x -

*t*-

*ϕ*)/

*σ*

_{2}) results in a Gaussian shaped superposition. The corresponding modes of the waveguide in

*y*-direction are obtained by an appropriate coordinate-rotation.

## Footnotes

1 | With respect to the z-axis. |

2 | In Heaviside-Lorentz units. |

3 | In our case we have k_{y}
/k_{x}
≈ 0.1. |

## References

1. | E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B |

2. | John D. Joannopoulos, R. D. Meade, and Joshua N. Winn, |

3. | P. Tran, “Optical switching with a nonlinear photonic Crystal: a numerical study,” Opt. Lett. |

4. | 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. |

5. | S. V. Polstyanko, R. Dyczij-Edlinger, and J. F. Lee Lee, “Full vectorial analysis of a nonlinear slab waveguide based on the nonlinear hybrid vector finite-element method,” Opt. Lett. |

6. | A. Reineix and B. Jecko, “A new photonic band gap equivalent model using finite difference time domain method,” Ann. Telecommun. |

7. | S. Scholz (Ph. D Thesis, University of Stuttgart, 1999). |

8. | P. M. Goorjian and A. Taflove, “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. |

9. | J. D. Jackson, |

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(230.1150) Optical devices : All-optical devices

**ToC Category:**

Focus Issue: Photonic crystals

**History**

Original Manuscript: April 16, 1998

Revised Manuscript: April 21, 1998

Published: July 6, 1998

**Citation**

Stefan Scholz, Ortwin Hess, and Roland Ruhle, "Dynamic cross-waveguide optical switching
with a nonlinear photonic band-gap structure," Opt. Express **3**, 28-34 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-1-28

Sort: Journal | Reset

### References

- E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993). [CrossRef]
- John D. Joannopoulos, R. D. Meade, Joshua N. Winn, Photonic Crystals, (Princeton University Press, Princeton, NJ, 1995).
- P. Tran, "Optical switching with a nonlinear photonic Crystal: a numerical study," Opt. Lett. 21, 1138-1140 (1996). [CrossRef] [PubMed]
- 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]
- S. V. Polstyanko, R. Dyczij-Edlinger, and J. F. Lee Lee, "Full vectorial analysis of a nonlinear slab waveguide based on the nonlinear hybrid vector finite-element method," Opt. Lett. 21, 98-100 (1996). [CrossRef] [PubMed]
- A. Reineix and B. Jecko, "A new photonic band gap equivalent model using finite difference time domain method," Ann. Telecommun. 51 656-662 (1996).
- S. Scholz (Ph. D Thesis, University of Stuttgart, 1999).
- P. M. Goorjian and A. Taflove, "Direct time integration of Maxwell's equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons," Opt. Lett. 17, 180 (1992). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, Inc., NJ, 1975).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.