1. Coupled resonator slow-wave optical structures Parma, 5/6/2007 Jiří Petráček, Jaroslav Čáp petracek@fme.vutbr.cz

2. all-optical high-bit-rate communication systems - optical delay lines - memories - switches - logic gates -.... “slow” light nonlinear effects increased efficiency

3. Outline Introduction: slow-wave optical structures (SWS) Basic properties of SWS –System model –Bloch modes –Dispersion characteristics –Phase shift enhancement –Nonlinear SWS Numerical methods for nonlinear SWS –NI-FD –FD-TD Results for nonlinear SWS

4. Outline Introduction: slow-wave optical structures (SWS) Basic properties of SWS –System model –Bloch modes –Dispersion characteristics –Phase shift enhancement –Nonlinear SWS Numerical methods for nonlinear SWS –NI-FD –FD-TD Results for nonlinear SWS

5. Slow light the light speed in vacuum c phase velocity v group velocity v g

6. How to reduce the group velocity of light? Electromagnetically induced transparency - EIT Stimulated Brillouin scattering Slow-wave optical structures (SWS) – – pure optical way Miguel González Herráez, Kwang Yong Song, Luc Thévenaz: „Arbitrary bandwidth Brillouin slow light in optical fibers,“ Opt. Express 14 1395 (2006) Ch. Liu, Z. Dutton, et al.: „Observation of coherent optical information storage in an atomic medium using halted light pulses,“ Nature 409 (2001) 490-493 A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. And Quantum Electron. 35, 365 (2003).

7. Slow-wave optical structure (SWS) - chain of directly coupled resonators (CROW - coupled resonator optical waveguide) - light propagates due to the coupling between adjacent resonators

8. coupled Fabry-Pérot cavities 1D coupled PC defects 2D coupled PC defects coupled microring resonators Various implementations of SWSs

9. Outline Introduction: slow-wave optical structures (SWS) Basic properties of SWS –System model –Bloch modes –Dispersion characteristics –Phase shift enhancement –Nonlinear SWS Numerical methods for nonlinear SWS –NI-FD –FD-TD Results for nonlinear SWS

10. A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. And Quantum Electron. 35, 365 (2003). J. K. S. Poon, J. Scheuer, Y. Xu and A. Yariv, “Designing coupled-resonator optical waveguide delay lines", J. Opt. Soc. Am. B 21, 1665-1673, 2004. System model of SWS

12. Relation between amplitudes

13. Transmission matrix

14. For lossless SWS it follows from symmetry: real – (coupling ratio) real

15. Propagation in periodic structure

16. Bloch modes eigenvalue eq. for the propagation constant of Bloch modes A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. And Quantum Electron. 35, 365 (2003). J. K. S. Poon, J. Scheuer, Y. Xu and A. Yariv, “Designing coupled-resonator optical waveguide delay lines", J. Opt. Soc. Am. B 21, 1665-1673, 2004.

17. Dispersion curves (band diagram)

18. Dispersion curves

19. Bandwidth, B at the edges of pass-band

20. Group velocity for resonance frequency

21. GVD: very strongvery strongminimal Group velocity

22. Infinite vs. finite structure dispersion relation Jacob Scheuer, Joyce K. S. Poonb, George T. Paloczic and Amnon Yariv, “Coupled Resonator Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/

23. COST P11 task on slow-wave structures One period of the slow-wave structure consists of one-dimensional Fabry-Perot cavity placed between two distributed Bragg reflectors DBR

24. Finite structure consisting 1, 3 and 5 resonators 3 5

25. Fengnian Xia,a Lidija Sekaric, Martin O’Boyle, and Yurii Vlasov: “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Applied Physics Letters 89, 041122 2006.

26. experiment theory number of resonators Fengnian Xia,a Lidija Sekaric, Martin O’Boyle, and Yurii Vlasov: “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Applied Physics Letters 89, 041122 2006. 1550 nm

27. Fengnian Xia,a Lidija Sekaric, Martin O’Boyle, and Yurii Vlasov: “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Applied Physics Letters 89, 041122 2006.

28. Delay, losses and bandwidth (usable bandwidth, small coupling) loss per unit length Jacob Scheuer, Joyce K. S. Poon, George T. Paloczi and Amnon Yariv, “Coupled Resonator Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/ loss

29. Tradeoffs among delay, losses and bandwidth Jacob Scheuer, Joyce K. S. Poon, George T. Paloczi and Amnon Yariv, “Coupled Resonator Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/ 10 resonators FSR = 310 GHz propagation loss = 4 dB/cm

30. Phase shift...... is enhanced by the slowing factor effective phase shift experienced by the optical field propagating in SWS over a distance d

31. Nonlinear phase shift Total enhancement: J.E. Heebner and R. W. Boyd, JOSA B 4, 722-731, 2002  intensity dependent phase shift is induced through SPM and XPM  intensities of forward and backward propagating waves inside cavities of SWS are increased (compared to the uniform structure) and this causes additional enhancement of nonlinear phase shift

32. Advantage of non-linear SWS: S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap structures,” Opt. Lett. 27 (2002) 613-615. A. Melloni, F. Morichetti, M. Martinelli, „Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,“ Opt. Quantum Electron. 35 (2003) 365. nonlinear processes are enhanced without affecting bandwidth

33. Outline Introduction: slow-wave optical structures (SWS) Basic properties of SWS –System model –Bloch modes –Dispersion characteristics –Phase shift enhancement –Nonlinear SWS Numerical methods for nonlinear SWS –NI-FD –FD-TD Results for nonlinear SWS

34. COST P11 task on slow-wave structures One period of the slow-wave structure consists of one-dimensional Fabry-Perot cavity placed between two distributed Bragg reflectors DBR Kerr non-linear layers

35. Integration of Maxwell Eqs. in frequency domain One-dimensional structure: - Maxwell equations turn into a system of two coupled ordinary differential equations - that can be solved with standard numerical routines (Runge-Kutta). H. V. Baghdasaryan and T. M. Knyazyan, “Problem of plane EM wave self-action in multilayer structure: an exact solution,“ Opt. Quantum Electron. 31 (1999), 1059-1072. M. Midrio, “Shooting technique for the computation of plane-wave reflection and transmission through one-dimensional nonlinear inhomogenous dielectric structures,” J. Opt. Soc. Am. B 18 (2001), 1866-1981. P. K. Kwan, Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures“ Opt. Commun. 238 (2004) 169-174. J. Petráček: „Modelling of one-dimensional nonlinear periodic structures by direct integration of Maxwell’s equations in frequency domain.“ In: Frontiers in Planar Lightwave Circuit Technology (Eds: S. Janz, J. Čtyroký, S. Tanev) Springer, 2005.

36. Maxwell Eqs. Now it is necessary to formulate boundary conditions.

37. Analytic solution in linear outer layers

38. Boundary conditions

39. Admittance/Impedance concept E. F. Kuester, D. C. Chang, “Propagation, Attenuation, and Dispersion Characteristics of Inhomogenous Dielectric Slab Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23 (1975), 98-106. J. Petráček: „Frequency-domain simulation of electromagnetic wave propagation in one-dimensional nonlinear structures,“ Optics Communications 265 (2006) 331-335.

40. new ODE systems for and The equations can be decoupled in case of lossless structures (real n )

41. Lossless structures (real n ) is conserved decoupled

42. ? ? known Technique

43. Advantage Speed - for lossless structures – only 1 equation Disadvantage Switching between p and q formulation during the numerical integration

44. FD-TD

45. FD-TD: phase velocity corrected algorithm A. Christ, J. Fröhlich, and N. Kuster, IEICE Trans. Commun., Vol. E85-B (12), 2904-2915 (2002).

46. FD-TD: convergence corrected algorithm common formulation

47. Outline Introduction: slow-wave optical structures (SWS) Basic properties of SWS –System model –Bloch modes –Dispersion characteristics –Phase shift enhancement –Nonlinear SWS Numerical methods for nonlinear SWS –NI-FD –FD-TD Results for nonlinear SWS

48. Results for COST P11 SWS structure is the same in both layers nonlinearity level F. Morichetti, A. Melloni, J. Čáp, J. Petráček, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, „Self-phase modulation in slow-wave structures: A comparative numerical analysis,“ Optical and Quantum Electronics 38, 761-780 (2006).

49. Transmission spectra

50. 1 period

51. 2 periods

52. 3 periods

53. Transmittance normalized incident intensity λ =1.5505 μm

54. Here incident intensity is about 10 -6 However usually 10 -4 - 10 -3 P. K. Kwan, Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures“ Opt. Commun. 238 (2004) 169-174. W. Ding, “Broadband optical bistable switching in one-dimensional nonlinear cavity structure,” Opt. Commun. 246 (2005) 147-152. J. He and M. Cada,”Optical Bistability in Semiconductor Periodic structures,” IEEE J. Quant. Electron. 27 (1991), 1182-1188. S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap structures,” Opt. Lett. 27 (2002) 613-615. A. Suryanto et al., “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron. 35 (2003), 313-332. 10 -2 L. Brzozowski and E.H. Sargent, “Nonlinear distributed-feedback structures as passive optical limiters,” JOSA B 17 (2000) 1360-1365.

55. Upper limit of the most transparent materials 10 -4 S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap structures,” Opt. Lett. 27 (2002) 613-615. Here incident intensity is about 10 -6 However usually 10 -4 - 10 -3 Are the high intensity effects important? (e.g. multiphoton absorption)

56. Maximum normalized intensity inside the structure normalized incident intensity

57. 2 periods

58. 3 periods

59. Selfpulsing

61. Conclusion SWS could play an important role in the development of nonlinear optical components suitable for all-optical high-bit- rate communication systems.