Complete Band Gaps:

You can leave home without them.

Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. Johnson

MIT

How else can we confine light?

Total Internal Reflection

ni > no

no

rays at shallow angles > θcare totally reflected

Snell’s Law:

θiθo

ni sinθi = no sinθo

sinθc = no / ni< 1, so θc is real

i.e. TIR can only guidewithin higher indexunlike a band gap

Total Internal Reflection?

ni > no

no

rays at shallow angles > θcare totally reflected

So, for example,a discontiguous structure can’t possibly guide by TIR…

the rays can’t stay inside!

Total Internal Reflection?

ni > no

no

rays at shallow angles > θcare totally reflected

So, for example,a discontiguous structure can’t possibly guide by TIR…

or can it?

Total Internal Reflection Redux

ni > no

no

ray-optics picture is invalid on λ scale (neglects coherence, near field…)

Snell’s Law is reallyconservation of k|| and ω:

θiθo

|ki| sinθi = |ko| sinθo|k| = nω/c

(wavevector) (frequency)

k||

translationalsymmetry

conserved!

Waveguide Dispersion Relationsi.e. projected band diagrams

ni > no

no

k||

ω

light li

ne: ω =

ck / n o

light coneprojection of all k⊥ in no

(a.k.a. β)

ω = ck / ni

higher-index corepulls down state

( ∞)

higher-order modesat larger ω, β

weakly guided (field mostly in no)

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/a)

wavenumber k (2π/a)

light cone

Conserved k and ω+ higher index to pull down state

= localized/guided mode.

Strange Total Internal ReflectionIndex Guiding

a

A Hybrid Photonic Crystal:1d band gap + index guiding

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/a)

wavenumber k (2π/a)

light cone

“band gap”

a

range of frequenciesin which there are

no guided modes

slow-light band edge

A Resonant Cavity

index-confined

photonic band gap

increased rod radiuspulls down “dipole” mode

(non-degenerate)

– +

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uenc

y (c

/a)

wavenumber k (2π/a)

light cone

“band gap” ω

A Resonant Cavity

index-confined

photonic band gap

– +

k not conservedso coupling to

light cone:

radiationThe trick is to

keep theradiation small…

(more on this later)

Meanwhile, back in reality…

5 µm

[ D. J. Ripin et al., J. Appl. Phys. 87, 1578 (2000) ]

d = 703nmd = 632nmd

Air-bridge Resonator: 1d gap + 2d index guiding

bigger cavity= longer λ

Time for Two Dimensions…

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

How do we make a 2d bandgap?

Most obvioussolution?

make2d patternreally tall

How do we make a 2d bandgap?

If height is finite,we must couple to

out-of-plane wavevectors…

kz not conserved

A 2d band diagram in 3d

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TE bandsTM bands

Square Lattice ofDielectric Rods

(ε = 12, r=0.2a)

Let’s start with the 2dband diagram.

This is what we’d liketo have in 3d, too!

wavevector

A 2d band diagram in 3d

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TE bandsTM bands

Square Lattice ofDielectric Rods

(ε = 12, r=0.2a)

Let’s start with the 2dband diagram.

This is what we’d liketo have in 3d, too!

wavevector

3D Structure:

No! When we includeout-of-plane propagation,

we get:

wavevector frequency

ω

ω+δω

projected band diagram fills gap!

but this emptyspace looks useful…

Photonic-Crystal Slabs

2d photonic bandgap + vertical index guiding

[ S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice ]

Rod-Slab Projected Band Diagram

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freq

uenc

y (c

/a)

even (TE-like) bandsodd (TM-like) bands

Square Lattice ofDielectric Rods

(ε = 12, r=0.2a, h=2a)

light cone

The Light Cone:All possible statespropagating in the air

The Guided Modes:Cannot couple tothe light cone… —> confined to the slab

Thickness is critical.Should be about λ/2 (to have a gap & be single-mode)

Γ X M ΓΓ X

M

Symmetry in a Slab

2d: TM and TE modes

slab: odd (TM-like) and even (TE-like) modes

mirror planez = 0

rE

rE

Like in 2d, there may only be a band gapin one symmetry/polarization

Slab Gaps

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freq

uenc

y (c

/a)

even (TE-like) bandsodd (TM-like) bands

even (TE-like) bandsodd (TM-like) bands

Γ X M Γ Γ M K Γ

Square Lattice ofDielectric Rods

(ε = 12, r=0.2a, h=2a)

Triangular Latticeof Air Holes

(ε = 12, r=0.3a, h=0.5a)

light cone light cone

TM-like gap TE-like gap

Substrates, for the Gravity-Impaired

substrate

(rods or holes)

substrate breaks symmetry:some even/odd mixing “kills” gap

BUTwith strong confinement

(high index contrast)

mixing can be weak

superstrate restores symmetry

“extruded” substrate= stronger confinement

(less mixing evenwithout superstrate

Extruded Rod Substrate

S. Assefa, L. A. Kolodziejski

high index

Air-membrane Slabs

[ N. Carlsson et al., Opt. Quantum Elec. 34, 123 (2002) ]

who needs a substrate?

2µm

AlGaAs

Optimal Slab Thickness~ λ/2, but λ/2 in what material?

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even (TE-like) gap odd (TM-like) gap

slab thickness (a)

gap

size

(%

)

TM “sees” -1~ low ε

TE “sees” ~ high ε

effective medium theory: effective ε depends on polarization

Photonic-Crystal Building Blocks

point defects(cavities)

line defects(waveguides)

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0.3 0.35 0.4 0.45 0.5

freq

uenc

y (c

/a)

wavevector k (2π/a)

A Reduced-Index Waveguide

Reduce the radius of a row ofrods to “trap” a waveguide modein the gap.

(r=0.2a)

(r=0.18a)

(r=0.16a)

(r=0.12a)

(r=0.14a)

Still have conservedwavevector—under thelight cone, no radiation

(r=0.10a)

We cannot completelyremove the rods—novertical confinement!

Reduced-Index Waveguide Modes

x →

z →

y →

x →

–1 +1Ez

z →

y →x

→z

→

y →

x →

–1 +1Hz

z →

y →

Experimental Waveguide & Bend

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tran

smis

sion

wavelength (µm)

E experiment

theory

band gap

waveguide mode

[ E. Chow et al., Opt. Lett. 26, 286 (2001) ]

1µm 1µm

GaAs

AlO

SiO2

bend

ing

effi

cien

cy

caution:can easily bemulti-mode

Inevitable Radiation Losseswhenever translational symmetry is broken

e.g. at cavities, waveguide bends, disorder…

k is no longer conserved!

ω(conserved)

coupling to light cone= radiation losses

All Is Not Lost

Qw

A simple model device (filters, bends, …):

Qr

Q1

Qr1

Qw1= +

Q = lifetime/period = frequency/bandwidth

We want: Qr >> Qw

1 – transmission ~ 2Q / Qr

worst case: high-Q (narrow-band) cavities

Semi-analytical losses

r r t r r r r rE x G x x E x x

defect

( ) ( , ) ( ) ( )= ′ ⋅ ′ ⋅ ′∫ ω ε∆

far-field(radiation) Green’s function

(defect-freesystem)

near-field(cavity mode)

defect

A low-lossstrategy:

Make field insidedefect small

= delocalize mode

Make defect weak= delocalize mode

(ε = 12)

Monopole Cavity in a Slab

decreasing ε

Lower the ε of a single rod: push upa monopole (singlet) state.

Use small ∆ε: delocalized in-plane, & high-Q (we hope)

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1,000

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Qr

∆frequency above band edge (c/a)

Delocalized Monopole Q

ε=6

ε=7

ε=8

ε=9

ε=10

ε=11

mid-gap

Super-defects

Weaker defect with more unit cells.

More delocalizedat the same point in the gap

(i.e. at same bulk decay rate)

Super-Defect vs. Single-Defect Q

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∆frequency above band edge (c/a)

ε=6

ε=7

ε=8

ε=9

ε=10

ε=11

ε=7

ε=8

ε=9

ε=10

ε=11

ε=11.5

mid-gap

Super-Defect vs. Single-Defect Q

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∆frequency above band edge (c/a)

ε=6

ε=7

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ε=9

ε=10

ε=11

ε=7

ε=8

ε=9

ε=10

ε=11

ε=11.5

mid-gap

Super-Defect State(cross-section)

∆ε = –3, Qrad = 13,000

(super defect)

still ~localized: In-plane Q|| is > 50,000 for only 4 bulk periods

Ez

(in hole slabs, too)

Hole Slabε=11.56

period a, radius 0.3athickness 0.5a

Reduce radius of7 holes to 0.2a

Q = 2500near mid-gap (∆freq = 0.03)

Very robust to roughness(note pixellization, a = 10 pixels).

How do we compute Q?

1excite 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 viasignal processing (FFT is suboptimal)

[ V. A. Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

Pro: no a priori knowledge, get all ω’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?

2excite cavity with

narrow-band dipole source(e.g. temporally broad Gaussian pulse)

(via 3d FDTD [finite-difference time-domain] simulation)

— source is at ω0 resonance,which must already be known (via )1

…measure outgoing power P and energy U

Q = ω0 U / P

Pro: separate Qw/Qr, arbitrary Q, also get field pattern

Con: requires separate run to get ω0, long-time source for closely-spaced resonances

1

Can we increase Qwithout delocalizing?

Semi-analytical losses

r r t r r r r rE x G x x E x x

defect

( ) ( , ) ( ) ( )= ′ ⋅ ′ ⋅ ′∫ ω ε∆

far-field(radiation) Green’s function

(defect-freesystem)

near-field(cavity mode)

defect

Another low-lossstrategy:

exploit cancellationsfrom sign oscillations

Need a morecompact representation

Cannot cancel infinitely many E(x) integrals

Radiation pattern from localized source…

— use multipole expansion & cancel largest moment

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

Multipole Expansion[ Jackson, Classical Electrodynamics ]

+ + + …

radiated field =

dipole quadrupole hexapole

peak Q (cancellation) = transition to higher-order radiation

Multipoles in a 2d example

index-confined

photonic band gap

increased rod radiuspulls down “dipole” mode

(non-degenerate)

– +

as we change the radius, ω sweeps across the gap

2d multipolecancellation

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10,000

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20,000

25,000

30,000

0.25 0.29 0.33 0.37

Q

frequency (c/a)

r = 0.35a

Q =

1,7

73

r = 0.40a

Q =

6,6

24

r = 0.375a

Q =

28,

700

cancel a dipole by opposite dipoles…

cancellation comes fromopposite-sign fields in adjacent rods

… changing radius changed balance of dipoles

3d multipole cancellation?

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2000

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Q

frequency (c/a) gap topgap bottom

enlarge center & adjacent rods

vary side-rod ε slightlyfor continuous tuning

quadrupole mode

(Ez cross section)

(balance central moment with opposite-sign side rods)

3d multipole cancellationne

ar f

ield

Ez

far

fiel

d |E

|2

Q = 408 Q = 426Q = 1925

nodal planes(source of high Q)

An Experimental (Laser) Cavity[ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ]

elon

gate

row

of

hole

s

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

An Experimental (Laser) Cavity[ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ]

elon

gate

row

of

hole

s

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

Hz (greyscale)

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

How can we get arbitrary Qwith 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) ]

The Basic Idea, in 2d

ε1ε2 < ε1

ε1'

ε2' < ε1'

start with:junction of two waveguides

No radiation at junctionif the modes are perfectly matched

Perfect Mode Matching

ε1ε2 < ε1

ε1'

ε2' < ε1'

requires:same differential equations and boundary conditions

Match differential equations…ε2 − ε1 = ε2' − ε1' …closely related to separability[ S. Kawakami, J. Lightwave Tech. 20, 1644 (2002) ]

Perfect Mode Matching

ε1ε2 < ε1

ε1'

ε2' < ε1'

requires:same differential equations and boundary conditions

Match boundary conditions:field must be TE

(E out of plane, in 2d)

(note switch in TE/TMconvention)

TE modes in 3d

for

cylindrical waveguides,

“azimuthally polarized”

TE0n modes

A Perfect Cavity in 3d

R

N layers

N layers

defect

Perfect indexconfinement

(no scattering)

+1d band gap

=3d confinement

(~ VCSEL + perfect lateral confinement)

A Perfectly Confined Mode

ε1', ε2' = 4, 1

ε1, ε2 = 9, 6λ/2 core

E energy density, vertical slice

Q limited only by finite size

G

G

G

G

G

GG

GG G

J

J

J

J

J

J

J

J

J

J

10

100

1000

10000

100000

1000000

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Q

cladding R / core radius

N = 5

N = 10

~ fixed mode vol.V = (0.4λλλλ)3

Q-tipsThree independent mechanisms for high Q:

Delocalization: trade off modal size for Q

Multipole Cancellation: force higher-order far-field pattern

Qr grows monotonically towards band edge

Qr peaks inside gap

New nodal planes appear in far-field pattern at peak

Mode Matching: allows arbitrary Q, finite V

Requires special symmetry & materials

Forget these devices…

I just want a mirror.

ok

ω

k ||

modesin crystal

TE

TM

Projected Bands of a 1d Crystal(a.k.a. a Bragg mirror)

incident light

k || conserved

(normalincidence)

1d b

and

gap

light

line o

f air

ω = ck

||

ω

k ||

modesin crystal

TE

TM

Omnidirectional Reflection[ J. N. Winn et al, Opt. Lett. 23, 1573 (1998) ]

light

line o

f air

ω = ck

||

in these ω ranges,there is

no overlapbetween modesof air & crystal

all incident light(any angle, polarization)

is reflectedfrom flat surface

needs: sufficient index contrast & nhi > nlo > 1

[ Y. Fink et al, Science 282, 1679 (1998) ]

Omnidirectional Mirrors in Practice

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Inde

x ra

tio,n

2 /n

1

Smaller index, n1

0%

10%

20%

30%

40%

50%

∆λ/λmid

6 9 12 150

5 0

10 0

0

5 0

10 0

0

5 0

10 0

0

5 0

10 0

0

5 0

10 0

normal

450 s

450 p

800 p

800 s

Refle

cta

nc

e (

%)

Wavelength (microns)

Te / polystyrene

Ref

lect

ance

(%

)

contours of omnidirectional gap size