Coupled resonator slow-wave optical structures

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

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

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 vg

6. How to reduce the group velocity of light? Electromagnetically induced transparency - EIT Ch. Liu, Z. Dutton, et al.: „Observation of coherent opticalinformation storage in an atomic medium using halted light pulses,“ Nature 409 (2001) 490-493 Stimulated Brillouin scattering Miguel González Herráez, Kwang Yong Song, Luc Thévenaz: „Arbitrary bandwidth Brillouin slow light in optical fibers,“ Opt. Express 14 1395 (2006) Slow-wave optical structures (SWS) – – pure optical way A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave opticalstructures,” Opt. AndQuantum 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. Various implementations of SWSs coupled Fabry-Pérot cavities 1D coupled PC defects 2D coupled PC defects coupled microring resonators

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. System model of SWS A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave opticalstructures,” Opt. AndQuantum 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.

11. 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 opticalstructures,” Opt. AndQuantum 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. Group velocity GVD: very strong minimal very strong

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 Oneperiod of the slow-wave structure consists of one-dimensional Fabry-Perot cavity placed between two distributed Bragg reflectors DBR 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 number of resonators theory 1550 nm 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.

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

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

30. Phase shift ... effective phaseshift experienced by the optical fieldpropagating in SWS over a distance d ... is enhanced by the slowing factor

31. Nonlinear phase shift • 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 Total enhancement: J.E. Heebner and R. W. Boyd, JOSA B 4, 722-731, 2002

32. Advantage of non-linear SWS: nonlinear processes are enhanced without affecting bandwidth 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 coupledresonator slow-wave opticalstructures,“ Opt. Quantum Electron. 35 (2003) 365.

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 Oneperiod of the slow-wave structure consists of one-dimensional Fabry-Perot cavity placed between two distributed Bragg reflectors DBR 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 and The equations can be decoupled in case of lossless structures (real n)

41. Lossless structures (real n) is conserved decoupled

42. Technique known ? ?

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 correctedalgorithm A. Christ, J. Fröhlich, and N. Kuster, IEICE Trans. Commun., Vol. E85-B (12),2904-2915 (2002).

46. FD-TD: convergence common formulation correctedalgorithm

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