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

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How else can we confine light?

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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

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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!

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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?

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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!

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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 )

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Strange Total Internal Reflection Index Guiding a

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

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

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

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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

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

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How do we make a 2d bandgap? Most obvious solution? make 2d pattern really tall

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

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A 2d band diagram in 3d

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

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

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Rod-Slab Projected Band Diagram XM X M

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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

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Slab Gaps TM-like gapTE-like gap

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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

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Extruded Rod Substrate

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

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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

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

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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!

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Reduced-Index Waveguide Modes

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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

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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

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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

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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

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( = 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)

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

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

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

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

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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

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(in hole slabs, too)

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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

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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

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Can we increase Q without delocalizing?

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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

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Need a more compact representation

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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

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

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

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

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

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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)

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

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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

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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)

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

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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) ]

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The Basic Idea, in 2d

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

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

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TE modes in 3d

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

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A Perfectly Confined Mode

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Q limited only by finite size

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Q-tips

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Forget these devices… I just want a mirror. ok

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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 ||

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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

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

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