Complete Band Gaps: You can leave home without them.

1. Photonic Crystals:Periodic Surprises in Electromagnetism Complete Band Gaps:You can leave home without them. Steven G. Johnson MIT

2. How else can we confine light?

3. rays at shallow angles > qc are totally reflected Snell’s Law: ni sinqi = no sinqo qi qo Total Internal Reflection no ni > no sinqc = no / ni < 1, so qc is real i.e. TIR can only guide within higher index unlike a band gap

4. rays at shallow angles > qc are totally reflected Total Internal Reflection? no ni > no So, for example, a discontiguous structure can’t possibly guide by TIR… the rays can’t stay inside!

5. rays at shallow angles > qc are totally reflected Total Internal Reflection? no ni > no So, for example, a discontiguous structure can’t possibly guide by TIR… or can it?

6. translational symmetry Snell’s Law is really conservation of k|| and w: qi k|| |ki| sinqi = |ko| sinqo qo |k| = nw/c conserved! (frequency) (wavevector) Total Internal Reflection Redux no ni > no ray-optics picture is invalid on l scale (neglects coherence, near field…)

7. no ni > no higher-order modes at larger w, b higher-index core pulls down state w = ck / ni weakly guided (field mostly in no) Waveguide Dispersion Relationsi.e. projected band diagrams w light cone projection of all k in no light line: w = ck / no k|| ( ) (a.k.a. b)

8. Index Guiding Strange Total Internal Reflection a

9. A Hybrid Photonic Crystal:1d band gap + index guiding range of frequencies in which there are no guided modes slow-light band edge a

10. – + increased rod radius pulls down “dipole” mode (non-degenerate) A Resonant Cavity

11. k not conserved so coupling to light cone: radiation w A Resonant Cavity – + The trick is to keep the radiation small… (more on this later)

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

13. Time for Two Dimensions… 2d is all we really need for many interesting devices …darn z direction!

14. How do we make a 2d bandgap? Most obvious solution? make 2d pattern reallytall

15. How do we make a 2d bandgap? If height is finite, we must couple to out-of-plane wavevectors… kz not conserved

16. A 2d band diagram in 3d

17. but this empty space looks useful… projected band diagram fills gap! A 2d band diagram in 3d

18. Photonic-Crystal Slabs 2d photonic bandgap + vertical index guiding [ S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice ]

19. G G X M Rod-Slab Projected Band Diagram M X G

20. mirror plane z = 0 slab: odd (TM-like) and even (TE-like) modes Like in 2d, there may only be a band gap in one symmetry/polarization Symmetry in a Slab 2d: TM and TE modes

21. Slab Gaps TM-like gap TE-like gap

22. Substrates, for the Gravity-Impaired (rods or holes) superstrate restores symmetry substrate breaks symmetry: some even/odd mixing “kills” gap “extruded” substrate = stronger confinement BUT with strong confinement (high index contrast) mixing can be weak (less mixing even without superstrate

23. Extruded Rod Substrate

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

25. effective medium theory: effective e depends on polarization TE “sees” <e> ~ high e TM “sees” <e-1>-1 ~ low e ~ l/2, but l/2 in what material? Optimal Slab Thickness gap size (%) slab thickness (a)

26. Photonic-Crystal Building Blocks point defects (cavities) line defects (waveguides)

27. We cannotcompletely remove the rods—no vertical confinement! (r=0.10a) (r=0.12a) Still have conserved wavevector—under the light cone, no radiation (r=0.14a) (r=0.16a) (r=0.18a) A Reduced-Index Waveguide (r=0.2a) Reduce the radius of a row of rods to “trap” a waveguide mode in the gap.

28. Reduced-Index Waveguide Modes

29. caution: can easily be multi-mode SiO2 GaAs AlO 1µm Experimental Waveguide & Bend [ E. Chow et al., Opt. Lett.26, 286 (2001) ] 1µm bending efficiency

30. Inevitable Radiation Losseswhenever translational symmetry is broken e.g. at cavities, waveguide bends, disorder… coupling to light cone = radiation losses w (conserved) k is no longer conserved!

31. We want: Qr>> Qw 1 – transmission ~ 2Q /Qr Q = lifetime/period = frequency/bandwidth All Is Not Lost A simple model device (filters, bends, …): worst case: high-Q (narrow-band) cavities

32. Make field inside defect small = delocalize mode A low-loss strategy: Make defect weak = delocalize mode Semi-analytical losses far-field (radiation) defect Green’s function (defect-free system) near-field (cavity mode)

33. (e = 12) Monopole Cavity in a Slab Lower the e of a single rod: push up a monopole (singlet) state. decreasing e Use small De: delocalized in-plane, & high-Q (we hope)

34. Delocalized Monopole Q e=11 e=10 e=9 e=8 e=7 e=6 mid-gap

35. Super-defects Weaker defect with more unit cells. More delocalized at the same point in the gap (i.e. at same bulk decay rate)

36. Super-Defect vs. Single-Defect Q e=11.5 e=11 e=11 e=10 e=10 e=9 e=9 e=8 e=7 e=8 e=7 e=6 mid-gap

37. Super-Defect vs. Single-Defect Q e=11.5 e=11 e=11 e=10 e=10 e=9 e=9 e=8 e=7 e=8 e=7 e=6 mid-gap

38. Super-Defect State(cross-section) De = –3, Qrad = 13,000 Ez (super defect) still ~localized: In-plane Q|| is > 50,000 for only 4 bulk periods

39. (in hole slabs, too)

40. excite cavity with dipole source (broad bandwidth, e.g. Gaussian pulse) 1 … monitor field at some point …extract frequencies, decay rates via signal processing (FFT is suboptimal) [ V. A. Mandelshtam, J. Chem. Phys.107, 6756 (1997) ] Pro: no a priori knowledge, get all w’s and Q’s at once Con: no separate Qw/Qr, Q > 500,000 hard, mixed-up field pattern if multiple resonances How do we compute Q? (via 3d FDTD [finite-difference time-domain] simulation)

41. 2 1 1 …measure outgoing power P and energy U Q = w0U / P Pro: separate Qw/Qr, arbitrary Q, also get field pattern Con: requires separate run to get w0, long-time source for closely-spaced resonances How do we compute Q? (via 3d FDTD [finite-difference time-domain] simulation) excite cavity with narrow-band dipole source (e.g. temporally broad Gaussian pulse) — source is at w0 resonance, which must already be known (via )

42. Can we increase Qwithout delocalizing?

43. Semi-analytical losses Another low-loss strategy: exploit cancellations from sign oscillations far-field (radiation) defect Green’s function (defect-free system) near-field (cavity mode)

44. Need a morecompact representation

45. Multipole Expansion [ Jackson, Classical Electrodynamics ] radiated field = dipole quadrupole hexapole Each term’s strength = single integral over near field …one term is cancellable by tuning one defect parameter

46. Multipole Expansion [ Jackson, Classical Electrodynamics ] radiated field = dipole quadrupole hexapole peak Q (cancellation) = transition to higher-order radiation

47. – + increased rod radius pulls down “dipole” mode (non-degenerate) Multipoles in a 2d example as we change the radius, w sweeps across the gap

48. Q = 1,773 Q = 28,700 Q = 6,624 2d multipolecancellation

49. cancel a dipole by opposite dipoles… cancellation comes from opposite-sign fields in adjacent rods … changing radius changed balance of dipoles

50. 3d multipole cancellation? enlarge center & adjacent rods quadrupole mode vary side-rod e slightly for continuous tuning (balance central moment with opposite-sign side rods) (Ez cross section) gap bottom gap top