Ring resonators

1. Ring resonators

2. Ring resonator (no loss) off resonance input output input output at resonance • slow light • intensity enhancement

4. Critical coupling (ring resonator with loss)

5. Add-drop filter drop input through J. Čtyroký, Integrovaná fotonika

7. Photonic crystals “Nanostructured materials containingordered arrays of holes could lead toan optoelectronics revolution, doingfor light what silicon did for electrons.” Eli Yablonovitch, Scientific American472001.

8. Photonic crystals (PhC) 1887 1987 (need a more complex topology) periodic electromagnetic media with photonic band gaps: “optical insulators”

9. Periodic structures (we already know)

10. Interlude: Electromagnetism as an eigenvalue problem + constraint eigen-state eigen-operator eigen-value First task: get rid of this mess 0 dielectric function e(x) = n2(x)

11. Hermitian Eigenproblems + constraint eigen-state eigen-operator eigen-value Hermitian for real (lossless) e well-known properties from linear algebra: w are real (lossless) eigen-states are orthogonal eigen-states are complete (give all solutions)

12. Periodic Hermitian Eigenproblems [ G. Floquet, “Sur les équations différentielles linéaries à coefficients périodiques,” Ann. École Norm. Sup. 12, 47–88 (1883). ] [ F. Bloch, “Über die quantenmechanik der electronen in kristallgittern,” Z. Physik 52, 555–600 (1928). ] if eigen-operator is periodic, then Bloch-Floquet theorem applies: can choose: planewave periodic “envelope” Corollary 1: k is conserved, i.e.no scattering of Bloch wave Corollary 2: given by finite unit cell, so w are discrete wn(k)

13. Band diagram (dispersion relation) ? range of k? w3 map of what states exist & can interact w2 w w1 k Corollary 2: given by finite unit cell, so w are discrete wn(k)

14. Band diagram in 1D band gap e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 Solution is periodic in k : k + 2π/a equivalent to k “quasi-phase-matching” e(x) = e(x+a) a w k 0 –π/a π/a irreducible Brillouin zone

15. Any 1D Periodic System has a Gap a e(x) = e(x+a) e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Splitting of degeneracy: state concentrated in higher index (e2) has lower frequency w band gap 0 π/a

16. mathematical interlude completed… … now consider 1D PhC

17. Making band gaps in 1D w band gap 0 π/a

18. Bragg reflector (consider normal incidence) bandgap band gap 0 π/a

19. Band gaps in 2D and 3D

20. Example: 2D periodicity, ε=12:1 a frequencyw(2πc/a) = a / l irreducible Brillouin zone G G X M M E E TM TE X G H H

21. Example: 2D periodicity, ε=12:1 Ez frequencyw(2πc/a) = a / l Ez G G X M E TM – + H

22. Example: 2D hexagonal lattice, ε=10.5:1 H TM E

23. Example: 2D hexagonal lattice, ε=10.5:1

24. Example: 2D hexagonal lattice, ε=10.5:1 E-field a/l= 0,351 T=0,99 a/l= 0,280 T=0

25. 3D PhC with complete gap, ε=12:1 I. II. [ S. G. Johnson et al., Appl. Phys. Lett.77, 3490 (2000) ]

26. Properties of Bulk Crystals backwards slope: negative refraction dw/dk 0: slow light (e.g. DFB lasers) synthetic medium for propagation strong curvature: super-prisms, … (+ negative refraction) by Bloch’s theorem band diagram (dispersion relation) (cartoon) photonic band gap conserved frequencyw conserved wavevectork

27. Cavity Modes • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Help!

28. Cavity Modes • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • finite region –> discretew

29. Single-Mode Cavity M X G Bulk Crystal Band Diagram frequency (c/a) A point defect can push up a single mode from the band edge G G X M (k not conserved)

30. Tunable Cavity Modes frequency (c/a) Ez: monopole dipole

31. Defect Flavors

32. So what

33. Benefits of a complete gap… effectively one-dimensional broken symmetry –> reflections only

34. Lossless Bends [ A. Mekis et al., Phys. Rev. Lett. 77, 3787 (1996) ] symmetry + single-mode + “1d” = resonances of 100% transmission

35. Waveguides + Cavities = Devices No! Use “coupling-of-modes-in-time” (coupled-mode theory)… [H. Haus, Waves and Fields in Optoelectronics] “tunneling” Ugh, must we simulate this to get the basic behavior?

36. Waveguides + Cavities = Devices FWHM …quality factor Q “tunneling” 1 T = Lorentzian filter w w0

37. Channel-Drop Filters [ S. Fan et al., Phys. Rev. Lett. 80, 960 (1998) ]

38. Enough passive, linear devices… e.g. Kerr nonlinearity, ∆n ~ intensity Photonic crystal cavities: tight confinement (~ l/2 diameter) + long lifetime (high Q independent of size) = enhanced nonlinear effects

39. A Linear Nonlinear Filter Linear response: Lorenzian Transmisson shifted peak + nonlinear index shift in out

40. A Linear Nonlinear “Transistor” semi-analytical numerical Logic gates, switching, rectifiers, amplifiers, isolators, … + feedback Linear response: Lorenzian Transmisson shifted peak Bistable (hysteresis) response Power threshold is near optimal (~mW for Si and telecom bandwidth)

41. 2D and 3D PhC are difficult to fabricate • try a hybrid approach

42. d Air-bridge Resonator: 1d gap + 2d index guiding 5 µm d = 703nm d = 632nm bigger cavity = longer l [ D. J. Ripin et al., J. Appl. Phys.87, 1578 (2000) ]

43. Air-membrane Slabs who needs a substrate? AlGaAs 2µm [ N. Carlsson et al., Opt. Quantum Elec.34, 123 (2002) ]

44. Hollow-core Bandgap Fibers Bragg fiber 1000x better loss/nonlinear limits [ Yeh et al., 1978 ] (from density) 1d crystal + omnidirectional = OmniGuides 2d crystal PCF (You can also put stuff in here …) [ Knight et al., 1998 ] Photonic Crystal

45. Hollow-core Bandgap Fibers Bragg fiber [ Yeh et al., 1978 ] 1d crystal + omnidirectional = OmniGuides 2d crystal PCF [ Knight et al., 1998 ] [ figs courtesy Y. Fink et al., MIT ] [ R. F. Cregan et al., Science285, 1537 (1999) ]

46. Solid-core Holey Fibers solid core holey cladding forms effective low-index material Can have much higher contrast than doped silica… strong confinement = enhanced nonlinearities, birefringence, … [ J. C. Knight et al., Opt. Lett.21, 1547 (1996) ]

47. Photonic Crystals in Nature 3µm Peacock feather Morpho rhetenorbutterfly http://www.bugguy012002.com/MORPHIDAE.html wing scale: [ P. Vukosic et al., Proc. Roy. Soc: Bio. Sci.266, 1403 (1999) ] [J. Zi et al, Proc. Nat. Acad. Sci. USA, 100, 12576 (2003) ] [figs: Blau, Physics Today57, 18 (2004)] [ also: B. Gralak et al., Opt. Express9, 567 (2001) ]

48. High-Q nanocavities

49. Why high Q and small Veff ? required for number of applications - miniature sensors - high resolution fiters - low threshold lasers - switches - frequency converters - ..... účinnost interakce mezi látkou a světlem závisí na Q/Veff

50. Why high Q and small Veff ? Example: photonic sensor with microcavity