## Light-bullet routing and control with planar waveguide arrays

Optics Express, Vol. 18, Issue 11, pp. 11671-11682 (2010)

http://dx.doi.org/10.1364/OE.18.011671

Acrobat PDF (762 KB)

### Abstract

Spatial mode-locking in three dimensions can be achieved in a slab waveguide array architecture. This study focuses on using the resulting robust and self-starting light bullet formation for photonics applications. Specifically, light bullets can be manipulated through a simple electronically addressable spatial gain dynamics. By applying gain ramps in time and/or space via electronics technology, complete control and manipulation of the light bullets can be achieved, thus allowing for the construction of the master logic gates of NAND and NOR. Its robustness, self-starting behavior and easy addressability suggest that the slab waveguide array mode-locking merits serious consideration as a next generation photonics device.

© 2010 Optical Society of America

## 1. Introduction

1. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**(3383–3386) (1998). [CrossRef]

2. D. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides”, Opt. Lett. **13**, 794–796 (1988). [CrossRef] [PubMed]

1. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**(3383–3386) (1998). [CrossRef]

3. A. B. Aceves, C. D. Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E **53**, 1172–1189 (1996). [CrossRef]

1. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**(3383–3386) (1998). [CrossRef]

*intensity discrimination*(also known as saturable absorption) necessary for mode-locking. In this manuscript, higher-dimensional spatial confinement is achieved with a planar WGA architecture, thus producing a mechanism for light-bullet formation [10–12

10. Y. Silberberg, “Collapse of optical pulses”, Opt. Lett. **15**, 1282–1284 (1990). [CrossRef] [PubMed]

**81**(3383–3386) (1998). [CrossRef]

13. M. O. Williams and J. N. Kutz, “Spatial Mode-Locking of Light Bullets in Planar Waveguide Arrays,” Opt. Express **17**(20), 18,320–18,329 (2009). [CrossRef]

14. P. Y. P. Chen, B. A. Malomed, and P. L. Chu, “Trapping Bragg solitons by a pair of defects,” Phys. Rev. E **71**(6), 066,601 (2005). [CrossRef]

10. Y. Silberberg, “Collapse of optical pulses”, Opt. Lett. **15**, 1282–1284 (1990). [CrossRef] [PubMed]

17. A. A. Sukhorukov and Y. S. Kivshar, “Slow Light Bullets in Arrays of Nonlinear Bragg-Grating Waveguides,” in *Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies*, p. JWB82 (Optical Society of America, 2006). [PubMed]

20. Y. V. Kartashov, L. Torner, and D. N. Christodoulides, “Soliton dragging by dynamic optical lattices,” Opt. Lett. **30**(11), 1378–1380 (2005). [CrossRef] [PubMed]

24. S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities”, Nature **419**, 699–702 (2002). [CrossRef] [PubMed]

13. M. O. Williams and J. N. Kutz, “Spatial Mode-Locking of Light Bullets in Planar Waveguide Arrays,” Opt. Express **17**(20), 18,320–18,329 (2009). [CrossRef]

13. M. O. Williams and J. N. Kutz, “Spatial Mode-Locking of Light Bullets in Planar Waveguide Arrays,” Opt. Express **17**(20), 18,320–18,329 (2009). [CrossRef]

## 2. Governing equations

**17**(20), 18,320–18,329 (2009). [CrossRef]

**17**(20), 18,320–18,329 (2009). [CrossRef]

**17**(20), 18,320–18,329 (2009). [CrossRef]

^{2}= ∂

_{x}^{2}+∂

_{y}^{2}and the saturating gain is given by

*A*

_{0},

*A*

_{1}, and

*A*

_{2}are the envelopes of the normalized electric fields in the 0th, 1st and 2nd waveguide. The parameter

*D*is the diffraction coefficient which is scaled to be -1 or 1 depending upon the sign of the index of refraction. The coupling strength between waveguides is given by the parameter

*C*, and the Kerr nonlinearity strength is described by the parameter

*β*. At high-intensities, the parameter

*ρ*models the the nonlinear loss due to three-photon absorption. The saturating gain is described by

*g*, which depends upon the level of charge carrier injection and the energy of all of the bullets in the system. The parameter

*g*(

*t*)∇

^{2}term provides a bandwidth limitation on spatial modes as described by the parameter

*τ*. The presence of charge carrier diffusion in the semiconductor rate equations implies this term should exist [29

29. L. Rahman and H. Winful, “Nonlinear dynamics of semiconductor laser arrays: a mean field model,” IEEE J. Quantum Electron. **30**(6), 1405–1416 (1994). [CrossRef]

**17**(20), 18,320–18,329 (2009). [CrossRef]

*f*(

*x*,

*y*,

*t*) that contains all of the non-uniformity in the applied gain. The ability to control this parameter is what makes the SWGAML an ideal all-optical processing device. Further, the fine control of the applied gain in both space and time can be readily achieved with today’s modern technological tools [28], thus enabling remarkable potential for all-optical, photonic applications.

30. J. N. Kutz and B. Sandstede, “Theory of passive harmonicmode-locking using waveguide arrays,” Opt. Express **16**(2), 636–650 (2008). [CrossRef] [PubMed]

## 3. Bullet routing and control

### 3.1. Bullet stability

**17**(20), 18,320–18,329 (2009). [CrossRef]

*A*, is assumed to have radial symmetry. In practice, radial solutions are the only light bullets observed in simulations. Using the following parameter values [13

_{n}**17**(20), 18,320–18,329 (2009). [CrossRef]

**17**(20), 18,320–18,329 (2009). [CrossRef]

8. J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays”, Opt. Express **16**, 636–650 (2008). [CrossRef] [PubMed]

*g*

_{0}. Indeed, the radially symmetric branch of solutions and the linearized eigenvalues (spectra) can be computed as shown in Fig. 3. Thus the stable regions of light bullet formation can be computed. Further, if the gain is increased, a Hopf bifurcation can occur, leading to the formation of breathing light bullet solutions [13

**17**(20), 18,320–18,329 (2009). [CrossRef]

*g*

_{0}shown in Fig. 3 was obtained using the software package AUTO’s continuation capabilities [31]. The individual spectra were obtained by taking solutions provided by AUTO and computing the linearized spectra using the Chebyshev polynomial representation of the solution [32

32. L. N. Trefethen, *Spectral methods in MATLAB* (Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2000). [CrossRef]

^{−8}, so the elements of the discrete spectrum appear to be quite accurate.

### 3.2. Linearly sloped gain

*m*, the larger the speed (translation) of the bullet. Additionally, the velocity of the bullet is always in the direction o the gradient of the gain. Using this method, it is possible to control both the position and the velocity of the bullet by appropriately selecting the slope of the gain.

*m*value used is

*m*= 0.02. In principle, even larger values of slope could be used. However, large values of

*m*can produce regions where a second bullet forms. The second bullet further saturates the gain and destroys the original bullet. While this may be a physically valid form of movement, it is not the type we are considering here, i.e. the control and manipulation of individual light bullets. If one envisions input and output ports at the edges of the SWGAML, then the gain slopping allows for the ability to move the light bullets to a desired output port for further processing or optical transmission.

### 3.3. Pulse routing

*m*and

*n*vary and control the bullet by creating preferential directions for bullet motion.

*m*= 0.01 and

*n*= 0.01 (output 1), 0 (output 2), and -0.01 (output 3) for routing the pulse up, right, and down respectively. Notice that the photonic wires force the route of the light bullet to pass through the cross shaped junction instead of simply traveling in a straight path.

### 3.4. Time-dependent gain

*α*= 0.001 and

*ω*= 2

*π*/4000. This simple time-dependent profile is a Gaussian that is translated in a circle. The value of

*α*controls how strongly the gain profile traps the bullet. The larger

*α*is, the stronger the trapping of the bullet. However,

*α*must also be small enough so that the majority of the bullet receives gain.

## 4. Multiple bullet interaction

### 4.1. NOR gate

### 4.2. NAND gate

*Therefore, with a very simple external change, this system can toggle between two different master gates*.

## 5. Conclusions and outlook

10. Y. Silberberg, “Collapse of optical pulses”, Opt. Lett. **15**, 1282–1284 (1990). [CrossRef] [PubMed]

17. A. A. Sukhorukov and Y. S. Kivshar, “Slow Light Bullets in Arrays of Nonlinear Bragg-Grating Waveguides,” in *Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies*, p. JWB82 (Optical Society of America, 2006). [PubMed]

## Acknowledgements

## References and links

1. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

2. | D. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides”, Opt. Lett. |

3. | A. B. Aceves, C. D. Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E |

4. | H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. 1. Soliton formation,” J. Opt. Soc. Am. B |

5. | U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamics properties,” J. Opt. Soc. Am. B |

6. | J. N. Kutz, |

7. | J. Proctor and J. N. Kutz, “Theory and Simulation of Passive Mode-locking with Waveguide Arrays,” Opt. Lett. |

8. | J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays”, Opt. Express |

9. | B. Bale, J. Kutz, and B. Sandstede, “Optimizing Waveguide Array Mode-Locking for High-Power Fiber Lasers,” Selected Topics in Quantum IEEE J. Electron. |

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

11. | F. Wise and P. Di Trapani, “The Hunt for Light Bullets Spatiotemporal Solitons,” Optics and Photonics News , |

12. | See the Fundamentals, Functionalities, and Applications of Cavity Solitons (FunFACS) webpage for a complete overview of current and potential methods and realizations of generating localized optical structures: www.funfacs.org. |

13. | M. O. Williams and J. N. Kutz, “Spatial Mode-Locking of Light Bullets in Planar Waveguide Arrays,” Opt. Express |

14. | P. Y. P. Chen, B. A. Malomed, and P. L. Chu, “Trapping Bragg solitons by a pair of defects,” Phys. Rev. E |

15. | S. Chi, B. Luo, and H.-Y. Tseng, “Ultrashort bragg soliton in a fiber bragg grating,” Opt. Commun. |

16. | J. T. Mok, C. M. de Sterke, I. C. M. Liter, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nature Physics |

17. | A. A. Sukhorukov and Y. S. Kivshar, “Slow Light Bullets in Arrays of Nonlinear Bragg-Grating Waveguides,” in |

18. | R. H. Enns and S. S. Rangnekar, “Bistable spheroidal optical solitons,” Phys. Rev. A |

19. | A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, “Light bullets formation in a bulk media,” IEEE J. Quant. Electron. |

20. | Y. V. Kartashov, L. Torner, and D. N. Christodoulides, “Soliton dragging by dynamic optical lattices,” Opt. Lett. |

21. | W. Królikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, “Solitonlike optical switching in a circular fiber array,” Opt. Lett. |

22. | J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Beam interactions with a blocker soliton in one-dimensional arrays,” Opt. Lett. |

23. | Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser”, Phys. Rev. Lett. |

24. | S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities”, Nature |

25. | V. B. Taranenko and C. O. Weiss, “Incoherent optical switching of semiconductor resonator solitons”, Appl. Phys. B |

26. | S. Barbay, Y. Ménesguen, X. Hachair, L. Lery, I. Sagnes, and R. Kuszelewics, “Incoherent and coherent writing and erasure of cavity solitons in an optically pumped semiconductor amplifier”, Opt. Lett. |

27. | X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle, and J. Tredicce, “Cavity-solitons switching in semiconductor microcavities,” Phys. Rev. A |

28. | M. O. Williams, M. Feng, J. N. Kutz, K. Silverman, R. Mirin, and S. Cundiff, “Intensity Dynamics in Semiconductor Laser Arrays”, OSA Nonlinear Optics 2009 Technical Digest JTuB14 (2009). |

29. | L. Rahman and H. Winful, “Nonlinear dynamics of semiconductor laser arrays: a mean field model,” IEEE J. Quantum Electron. |

30. | J. N. Kutz and B. Sandstede, “Theory of passive harmonicmode-locking using waveguide arrays,” Opt. Express |

31. | E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang, “AUTO 97: Continuation And Bifurcation Software For Ordinary Differential Equations (with HomCont),” . |

32. | L. N. Trefethen, |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(140.4050) Lasers and laser optics : Mode-locked lasers

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: March 22, 2010

Manuscript Accepted: April 23, 2010

Published: May 18, 2010

**Citation**

Matthew O. Williams, Colin W. McGrath, and J. Nathan Kutz, "Light-bullet routing and control with planar waveguide arrays," Opt. Express **18**, 11671-11682 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11671

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### References

- H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81(3383-3386) (1998). [CrossRef]
- D. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides", Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
- A. B. Aceves, C. D. Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete self-trapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996). [CrossRef]
- H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, "Optical discrete solitons in waveguide arrays. 1. Soliton formation," J. Opt. Soc. Am. B 19, 2938-2944 (2002). [CrossRef]
- U. Peschel, R. Morandotti, J.M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. 2. Dynamics properties," J. Opt. Soc. Am. B 19, 2637-2644 (2002). [CrossRef]
- J. N. Kutz, Mode-Locking of Fiber Lasers via Nonlinear Mode-Coupling, vol. 661 of Lecture Notes in Physics (Springer Berlin / Heidelberg, 2005).
- J. Proctor and J. N. Kutz, "Theory and Simulation of Passive Mode-locking with Waveguide Arrays," Opt. Lett. 13, 2013-1015 (2005). [CrossRef]
- J. N. Kutz and B. Sandstede, "Theory of passive harmonic mode-locking using waveguide arrays", Opt. Express 16, 636-650 (2008). [CrossRef] [PubMed]
- B. Bale, J. Kutz, and B. Sandstede, "OptimizingWaveguide Array Mode-Locking for High-Power Fiber Lasers," IEEE J. Sel. Top. Quantum 15(1), 220-231 (2009). [CrossRef]
- Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990). [CrossRef] [PubMed]
- F. Wise and P. Di Trapani, "The Hunt for Light Bullets Spatiotemporal Solitons," Opt. Photon. News (2), 28-32 (2002) [CrossRef]
- 12. See the Fundamentals, Functionalities, and Applications of Cavity Solitons (FunFACS) webpage for a complete overview of current and potential methods and realizations of generating localized optical structures: www.funfacs.org.
- M. O. Williams and J. N. Kutz, "Spatial Mode-Locking of Light Bullets in Planar Waveguide Arrays," Opt. Express 17(20), 18,320-18,329 (2009). [CrossRef]
- P. Y. P. Chen, B. A. Malomed, and P. L. Chu, "Trapping Bragg solitons by a pair of defects," Phys. Rev. E 71, 066,601 (2005). [CrossRef]
- S. Chi, B. Luo, and H.-Y. Tseng, "Ultrashort bragg soliton in a fiber bragg grating," Opt. Commun. 206(1-3), 115- 121 (2002). [CrossRef]
- J. T. Mok, C. M. de Sterke, I. C. M. Liter, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nat. Physics 2, 775-780 (2006). [CrossRef]
- A. A. Sukhorukov and Y. S. Kivshar, "Slow Light Bullets in Arrays of Nonlinear Bragg-Grating Waveguides," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, p. JWB82 (Optical Society of America, 2006). [PubMed]
- R. H. Enns and S. S. Rangnekar, "Bistable spheroidal optical solitons," Phys. Rev. A 45(5), 3354-3357 (1992). [CrossRef] [PubMed]
- A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. 27(8), 2060-2065 (1991). [CrossRef]
- Y. V. Kartashov, L. Torner, and D. N. Christodoulides, "Soliton dragging by dynamic optical lattices," Opt. Lett. 30(11), 1378-1380 (2005). [CrossRef] [PubMed]
- W. Krolikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, "Solitonlike optical switching in a circular fiber array," Opt. Lett. 19(5), 320-322 (1994). [CrossRef] [PubMed]
- J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo,M. Sorel, and J. S. Aitchison, "Beam interactions with a blocker soliton in one-dimensional arrays," Opt. Lett. 30(9), 1027-1029 (2005). [CrossRef] [PubMed]
- Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jager, "Realization of a Semiconductor-Based Cavity Soliton Laser," Phys. Rev. Lett. 100, 013907 (2008). [CrossRef] [PubMed]
- S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller and R. Jager, "Cavity solitons as pixels in semiconductor microcavities," Nature 419, 699-702 (2002). [CrossRef] [PubMed]
- V. B. Taranenko and C. O. Weiss, "Incoherent optical switching of semiconductor resonator solitons", Appl. Phys. B 72, 893-895 (2001).
- S. Barbay, Y. M’enesguen, X. Hachair, L. Lery, I. Sagnes and R. Kuszelewics, "Incoherent and coherent writing and erasure of cavity solitons in an optically pumped semiconductor amplifier", Opt. Lett. 31, 1504-1506 (2006). [CrossRef] [PubMed]
- X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle and J. Tredicce, "Cavity-solitons switching in semiconductor microcavities," Phys. Rev. A 72, 013815 (2005). [CrossRef]
- M. O. Williams, M. Feng, J. N. Kutz, K. Silverman, R. Mirin and S. Cundiff, "Intensity Dynamics in Semiconductor Laser Arrays", OSA Nonlinear Optics 2009 Technical Digest JTuB14 (2009).
- L. Rahman and H. Winful, "Nonlinear dynamics of semiconductor laser arrays: a mean field model," IEEE J. Quantum Electron. 30(6), 1405-1416 (1994). [CrossRef]
- J. N. Kutz and B. Sandstede, "Theory of passive harmonicmode-locking using waveguide arrays," Opt. Express 16(2), 636-650 (2008). [CrossRef] [PubMed]
- E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang, "AUTO 97: Continuation And Bifurcation Software For Ordinary Differential Equations (with HomCont),".
- L. N. Trefethen, Spectral methods in MATLAB (Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2000). [CrossRef]

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