Download presentation

Presentation is loading. Please wait.

Published byAnnie Bownds Modified over 2 years ago

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

2
How else can we confine light?

3
Total Internal Reflection n i > n o nono rays at shallow angles > c are totally reflected Snell’s Law: ii oo n i sin i = n o sin o sin c = n o / n i < 1, so c is real i.e. TIR can only guide within higher index unlike a band gap

4
Total Internal Reflection? n i > n o nono rays at shallow angles > c are totally reflected So, for example, a discontiguous structure can’t possibly guide by TIR… the rays can’t stay inside!

5
Total Internal Reflection? n i > n o nono rays at shallow angles > c are totally reflected So, for example, a discontiguous structure can’t possibly guide by TIR… or can it?

6
Total Internal Reflection Redux n i > n o nono ray-optics picture is invalid on scale (neglects coherence, near field…) Snell’s Law is really conservation of k || and : ii oo |k i | sin i = |k o | sin o |k| = n /c (wavevector) (frequency) k || translational symmetry conserved!

7
Waveguide Dispersion Relations i.e. projected band diagrams n i > n o nono k || light line: = ck / n o light cone projection of all k in n o (a.k.a. ) = ck / n i higher-index core pulls down state ( ) higher-order modes at larger weakly guided (field mostly in n o )

8
Strange Total Internal Reflection Index Guiding a

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

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

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

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

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 really tall

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

16
A 2d band diagram in 3d

17
projected band diagram fills gap! but this empty space looks useful…

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
Rod-Slab Projected Band Diagram XM X M

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

21
Slab Gaps TM-like gapTE-like gap

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

23
Extruded Rod Substrate

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

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

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

27
A Reduced-Index Waveguide Reduce the radius of a row of rods to “trap” a waveguide mode in the gap. (r=0.2a) (r=0.18a) (r=0.16a) (r=0.12a) (r=0.14a) Still have conserved wavevector—under the light cone, no radiation (r=0.10a) We cannot completely remove the rods—no vertical confinement!

28
Reduced-Index Waveguide Modes

29
Experimental Waveguide & Bend [ E. Chow et al., Opt. Lett. 26, 286 (2001) ] 1µm GaAs AlO SiO 2 bending efficiency caution: can easily be multi-mode

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

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

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

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

34
Delocalized Monopole Q =6 =7 =8 =9 =10 =11 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 =6 =7 =8 =9 =10 =11 =7 =8 =9 =10 =11 =11.5 mid-gap

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

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

39
(in hole slabs, too)

40
How do we compute Q? 1 excite cavity with dipole source (broad bandwidth, e.g. Gaussian pulse) … monitor field at some point (via 3d FDTD [finite-difference time-domain] simulation) …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 ’s and Q’s at once Con: no separate Q w /Q r, Q > 500,000 hard, mixed-up field pattern if multiple resonances

41
How do we compute Q? 2 excite cavity with narrow-band dipole source (e.g. temporally broad Gaussian pulse) (via 3d FDTD [finite-difference time-domain] simulation) — source is at 0 resonance, which must already be known (via ) 1 …measure outgoing power P and energy U Q = 0 U / P Pro: separate Q w /Q r, arbitrary Q, also get field pattern Con: requires separate run to get 0, long-time source for closely-spaced resonances 1

42
Can we increase Q without delocalizing?

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

44
Need a more compact representation

45
Multipole Expansion [ Jackson, Classical Electrodynamics ] radiated field = dipolequadrupolehexapole 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 = dipolequadrupolehexapole peak Q (cancellation) = transition to higher-order radiation

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

48
2d multipole cancellation Q = 1,773 Q = 6,624 Q = 28,700

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? gap topgap bottom enlarge center & adjacent rods vary side-rod slightly for continuous tuning quadrupole mode (E z cross section) (balance central moment with opposite-sign side rods)

51
3d multipole cancellation near field E z far field | E | 2 Q = 408Q = 426Q = 1925 nodal planes (source of high Q)

52
An Experimental (Laser) Cavity [ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ] elongate row of holes Elongation p is a tuning parameter for the cavity… cavity …in simulations, Q peaks sharply to ~10000 for p = 0.1a (likely to be a multipole-cancellation effect) * actually, there are two cavity modes; p breaks degeneracy

53
An Experimental (Laser) Cavity [ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ] elongate row of holes Elongation p is a tuning parameter for the cavity… cavity …in simulations, Q peaks sharply to ~10000 for p = 0.1a (likely to be a multipole-cancellation effect) * actually, there are two cavity modes; p breaks degeneracy H z (greyscale)

54
An Experimental (Laser) Cavity [ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ] cavity quantum-well lasing threshold of 214µW (optically pumped @830nm, 1% duty cycle ) (InGaAsP) Q ~ 2000 observed from luminescence

55
How can we get arbitrary Q with finite modal volume? Only one way: a full 3d band gap Now there are two ways. [ M. R. Watts et al., Opt. Lett. 27, 1785 (2002) ]

56
The Basic Idea, in 2d

57
Perfect Mode Matching …closely related to separability [ S. Kawakami, J. Lightwave Tech. 20, 1644 (2002) ]

58
Perfect Mode Matching (note switch in TE/TM convention)

59
TE modes in 3d

60
A Perfect Cavity in 3d (~ VCSEL + perfect lateral confinement)

61
A Perfectly Confined Mode

62
Q limited only by finite size

63
Q-tips

64
Forget these devices… I just want a mirror. ok

65
k || modes in crystal TE TM Projected Bands of a 1d Crystal (a.k.a. a Bragg mirror) incident light k || conserved (normal incidence) 1d band gap light line of air = ck ||

66
k || modes in crystal TE TM Omnidirectional Reflection [ J. N. Winn et al, Opt. Lett. 23, 1573 (1998) ] light line of air = ck || in these ranges, there is no overlap between modes of air & crystal all incident light (any angle, polarization) is reflected from flat surface needs: sufficient index contrast & n hi > n lo > 1

67
[ Y. Fink et al, Science 282, 1679 (1998) ] Omnidirectional Mirrors in Practice / mid Wavelength (microns) Te / polystyrene Reflectance (%) contours of omnidirectional gap size

Similar presentations

OK

Color of shock waves in photonic crystals Reed, Soljacic, Joannopoulos, Phys. Rev. Lett., 2003 Miguel Antonio D. Sulangi PS 175.2.

Color of shock waves in photonic crystals Reed, Soljacic, Joannopoulos, Phys. Rev. Lett., 2003 Miguel Antonio D. Sulangi PS 175.2.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on fact and opinion Ppt on relations and functions for class 11th Ppt on national education policy 1986 mets Ppt on coal and petroleum management Crt display ppt online Ppt on obesity and nutrition Upload and view ppt online shopping Ppt on network switching systems Ppt on shell structures theory Ppt on electricity for class 10th exercise