Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. JohnsonMIT

A “Defective” Lecturecavity waveguide

The Story So Far…a

Waves in periodic media can have:• propagation with no scattering (conserved k)• photonic band gaps (with proper e function)

Eigenproblem gives simple insight:

(r — + i

r k ) ¥

1e

(r — + i

r k ) ¥

È Î Í

˘ ˚ ˙

r H r k =

w n(r k )

cÊ

Ë Á

ˆ

¯ ˜

2 r H r k

Q r k

Hermitian –> complete, orthogonal, variational theorem, etc.

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Photonic Band Gap

TM bands

w

k

r

H = ei(r k ⋅r x -wt ) r

H r k (r x )Bloch form:

band diagram

Properties of Bulk Crystalsby Bloch’s theorem

(cartoon)

cons

erve

d fre

quen

cy w

conserved wavevector k

photonic band gap

band diagram (dispersion relation)

dw/dk Æ 0: slow light(e.g. DFB lasers)

backwards slope:negative refraction

strong curvature:super-prisms, …

(+ negative refraction)

synthetic mediumfor propagation

Applications of Bulk Crystals

Zero group-velocity dw/dk: distributed feedback (DFB) lasers

negative group-velocity ornegative curvature (“eff. mass”):

Negative refraction,Super-lensing source imageobject image

negative refraction medium

super-lensVeselago (1968)

divergent dispersion(i.e.!curvature):Superprisms

[Kosaka, PRB 58, R10096 (1998).]

using near-band-edge effects

[ C. Luo et al.,Appl. Phys. Lett. 81, 2352 (2002) ]

Cavity Modes• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •

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Help!

Cavity Modes• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •

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finite region –> discrete w

Cavity Modes: Smaller Change• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •• • •

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Photonic Band Gap

Cavity Modes: Smaller Change

G X M G

G X

M r k

frequ

ency

(c/a

)

L

Bulk Crystal Band Diagram

Cavity Modes: Smaller Change

J

J

J

J

J

J

J

J

JJ J J

J JJ J

J JJ J J J J

J

J

J

J

J

J

J

J

JJJJJJJ J J

JJJJJJJJ J J

JJ J J

JJJJ

J J J J JJ

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Photonic Band Gap

G X M G

G X

M r k

frequ

ency

(c/a

)

L

Defect Crystal Band Diagram

Defect bands areshifted up (less e)

∆k ~ π / L

with discrete k

# ⋅l2

~ L (k ~ 2p / l)

confinedmodes

k not conservedat boundary, so

not confined outside gap

escapes:

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Photonic Band Gap

Single-Mode Cavity

G X M G

G X

M r k

frequ

ency

(c/a

)

Bulk Crystal Band Diagram

A point defectcan push up

a single modefrom the band edge

(k not conserved)

w0

w

field decay ~ w -w 0

curvature

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Photonic Band Gap

“Single”-Mode Cavity

G X M G

G X

M r k

frequ

ency

(c/a

)

Bulk Crystal Band Diagram

A point defectcan pull down

a “single” mode

(k not conserved)X

…here, doubly-degenerate(two states at same w)

Tunable Cavity Modes

Ez:

Radius of Defect (r/a)0.1 0.2 0.3 0.4

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Air Defect Dielectric Defectair bands

dielect ric bands

0

frequ

ency

(c/a

)

monopole dipole

Tunable Cavity Modes

Ez:

band #1 at M band #2 at X’s

multiply by exponential decay

monopole dipole

Defect Flavors

a

Projected Band Diagrams

conserved k!

1d periodicity

G X

M

r k

conserved

not c

onse

rved

So, plot w vs. kx only…project Brillouin zone onto G–X:

gives continuum of bulk states + discrete guided band(s)

Air-waveguide Band Diagram

any state in the gap cannot couple to bulk crystal –> localized

(Waveguides don’t really need acomplete gap)

Fabry-Perot waveguide:

We’ll exploit this later, with photonic-crystal fiber…

So What?

Review: Why no scattering?

forbidden by gap(except for finite-crystal tunneling)

forbidden by Bloch(k conserved)

Benefits of a complete gap…

broken symmetry –> reflections only

effectively one-dimensional

Lossless Bends

symmetry + single-mode + “1d” = resonances of 100% transmission

[ A. Mekis et al.,Phys. Rev. Lett. 77, 3787 (1996) ]

Waveguides + Cavities = Devices

“tunneling”

Ugh, must we simulate this to get the basic behavior?

No! Use “coupling-of-modes-in-time” (coupled-mode theory)…[H. Haus, Waves and Fields in Optoelectronics]

“Coupling-of-Modes-in-Time”(a form of coupled-mode theory)

[H. Haus, Waves and Fields in Optoelectronics]

ainput outputs1+s1– s2–

resonant cavityfrequency w0, lifetime t |s|2 = flux

|a|2 = energy

dadt

= -iw0 a -2t

a +2t

s1+

s1- = -s1+ +2t

a, s2- =2t

a

assumes only:• exponential decay (strong confinement)• conservation of energy• time-reversal symmetry

“Coupling-of-Modes-in-Time”(a form of coupled-mode theory)

[H. Haus, Waves and Fields in Optoelectronics]

ainput outputs1+s1– s2–

resonant cavityfrequency w0, lifetime t |s|2 = flux

|a|2 = energy

transmission T= | $ s2– |2 / |$ s1+ |2

1

w0

T = Lorentzian filter

=

4t 2

w -w0( )2+

4t 2

w

FWHM1Q

=2

w0t

…quality factor Q

A Menagerie of Devicesl 1.55 microns

Page 4

l

Wide-angle Splitters

[ S. Fan et al., J. Opt. Soc. Am. B 18, 162 (2001) ]

Waveguide Crossings

[ S. G. Johnson et al., Opt. Lett. 23, 1855 (1998) ]

Waveguide Crossings

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thro

ughp

ut

1x10-10

1x10-8

1x10-6

1x10-4

1x10-2

1x100

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cros

stalk

frequency (c/a)

empty

1x1

3x3

5x5

5x5

3x3

empty

1x1

empty

5x5

3x3

1x1

Channel-Drop Filters

[ S. Fan et al., Phys. Rev. Lett. 80, 960 (1998) ]

Channel-Drop Filters

Enough passive, linear devices…

Photonic crystal cavities:tight confinement (~ l/2 diameter)

+ long lifetime (high Q independent of size)

= enhanced nonlinear effects

e.g. Kerr nonlinearity, ∆n ~ intensity

A Linear Nonlinear Filter

in out

Linear response:Lorenzian Transmisson shifted peak

+ nonlinearindex shift

A Linear Nonlinear “Transistor”

Linear response:Lorenzian Transmisson shifted peak

+ feedback

Bistable (hysteresis) responsePower threshold is near optimal

(~mW for Si and telecom bandwidth)

semi-analytical

numerical

Logic gates, switching,rectifiers, amplifiers,

isolators, …

Enough passive, linear devices…

Photonic crystal cavities:tight confinement (~ l/2 diameter)

+ long lifetime (high Q independent of size)

= enhanced nonlinear effects

Photonic crystal waveguides:tight confinement (~ l/2 diameter)

+ slow light (e.g. near band edge)

= enhanced nonlinear effects

Cavities + Cavities = Waveguide

“tunneling”

[ A. Yariv et al., Opt. Lett. 24, 711 (1999) ]

coupled-cavity waveguide (CCW/CROW): slow light + zero dispersion

Enhancing tunability with slow light

[ M. Soljacic et al., J. Opt. Soc. Am. B 19, 2052 (2002) ]

periodicity:light is slowed, but not reflected

Uh oh, we live in 3d…

rod layer

hole layer

(fcc crystal)

A

B

C

2d-like defects in 3d

modify single layerof holes or rods

[ M. L. Povinelli et al., Phys. Rev. B 64, 075313 (2001) ]

3d projected band diagram

wavevector kx ( 2p/a)

2D Photonic Crystal3D Photonic Crystal

J JJ

JJ

J

J

JJ J

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J JJ

J

JJ J

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wavevector kx ( 2p /a)

TM gap

frequ

ency

(c/a

)

frequ

ency

(c/a

)

2d-like waveguide mode

y

-1 1Ez y

z

y

-1 1Ez

x x

3D Photonic Crystal 2D Photonic Crystal

2d-like cavity mode

The Upshot

To design an interesting device, you need only:

symmetry

+ resonance

+ (ideally) a band gap to forbid losses

+ single-mode (usually)

Oh, and a full Maxwell simulator to get Q parameters, etcetera.