Complete Band Gaps - Ab Initio Physics Researchab-initio.mit.edu/photons/tutorial/L4-slabs.pdf ·...

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 > care totally reflected

Snells Law:

io

ni sini = no sino

sinc = no / ni< 1, so c is real

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

Total Internal Reflection?

ni > no

no


So, for example,a discontiguous structure cant possibly guide by TIR

the rays cant stay inside!

Total Internal Reflection?

ni > no

no


So, for example,a discontiguous structure cant possibly guide by TIR

or can it?

Total Internal Reflection Redux

ni > no

no

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

Snells Law is reallyconservation of k|| and :

io

|ki| sini = |ko| sino|k| = n/c

(wavevector) (frequency)

k||

translationalsymmetry

conserved!

Waveguide Dispersion Relationsi.e. projected band diagrams

ni > no

no

k||

light lin

e: = c

k / no

light coneprojection of all k in no

(a.k.a. )

= ck / ni

higher-index corepulls down state

( )

higher-order modesat larger ,

weakly guided (field mostly in no)

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freq

uenc

y (c

/a)

wavenumber k (2/a)

light cone

Conserved k and + 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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freq

uenc

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/a)

wavenumber k (2/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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freq

uenc

y (c

/a)

wavenumber k (2/a)

light cone

band gap

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 m

[ 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

Time for Two Dimensions

2d is all we really need for many interesting devicesdarn 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

( = 12, r=0.2a)

Lets start with the 2dband diagram.

This is what wed liketo have in 3d, too!

wavevector

A 2d band diagram in 3d

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TE bandsTM bands

Square Lattice ofDielectric Rods

( = 12, r=0.2a)

Lets start with the 2dband diagram.

This is what wed liketo have in 3d, too!

wavevector

3D Structure:

No! When we includeout-of-plane propagation,

we get:

wavevectorfrequency

+

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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0

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freq

uenc

y (c

/a)

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

Square Lattice ofDielectric Rods( = 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 /2 (to have a gap & be single-mode)

X M 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)



X M M K

Square Lattice ofDielectric Rods( = 12, r=0.2a, h=2a)

Triangular Latticeof Air Holes

( = 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?

2m

AlGaAs

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

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0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3

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

slab thickness (a)

gap

size

(%

)

TM sees -1~ low

TE sees ~ high

effective medium theory: effective depends on polarization

Photonic-Crystal Building Blocks

point defects(cavities)

line defects(waveguides)

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.3 0.35 0.4 0.45 0.5

freq

uenc

y (c

/a)

wavevector k (2/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 conservedwavevectorunder thelight cone, no radiation

(r=0.10a)

We cannot completelyremove the rodsnovertical 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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1.2

1200 1300 1400 1500 1600 1700 1800

tran

smis

sion

wavelength (m)

E experiment

theory

band gap

waveguide mode

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

1m 1m

GaAs

AlO

SiO2

be

nd

ing

eff

icie

ncy

caution:can easily bemulti-mode

Inevitable Radiation Losseswhenever translational symmetry is broken

e.g. at cavities, waveguide bends, disorder

k is no longer conserved!

(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

( ) ( , ) ( ) ( )=

far-field(radiation) Greens function

(defect-freesystem)

near-field(cavity mode)

defect

A low-lossstrategy:

Make field insidedefect small

= delocalize mode

Make defect weak= delocalize mode

( = 12)

Monopole Cavity in a Slab

decreasing

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

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

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1,000

10,000

100,000

1,000,000

0.0001 0.001 0.01 0.1

Qr

frequency above band edge (c/a)

Delocalized Monopole Q

=6

=7

=8

=9

=10

=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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Qr

frequency above band edge (c/a)

=6

=7

=8

=9

=10

=11

=7

=8

=9

=10

=11

=11.5

mid-gap

Super-Defect State(cross-section)

= 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 Slab=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 s and Qs 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 0 resonance,which must already be known (via )1

measure outgoing power P and energy U

Q = 0 U / P

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

Con: requires separate run to get 0, 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

( ) ( , ) ( ) ( )=

far-field(radiation) Greens 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 terms strength = single integral over near fieldone 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, sweeps across the gap

2d multipolecancellation

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0

5,000

10,000

15,000

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?

JJ

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J

0

500

1000

1500

2000

0.319243 0.390536

Q

frequency (c/a) gap topgap bottom

enlarge center & adjacent rods

vary side-rod slightlyfor continuous tuning

quadrupole mode

(Ez cross section)

(balance central moment with opposite-sign side rods)

3d multipole cancellationne

ar f

ield

E zfa

r fie

ld |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


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)


cavity

quantum-well lasing threshold of 214W(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

12 < 1

1'

2' < 1'

start with:junction of two waveguides

No radiation at junctionif the modes are perfectly matched

Perfect Mode Matching

12 < 1

1'

2' < 1'

requires:same differential equations and boundary conditions

Match differential equations2 1 = 2' 1' closely related to separability[ S. Kawakami, J. Lightwave Tech. 20, 1644 (2002) ]

Perfect Mode Matching

12 < 1

1'

2' < 1'

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

1', 2' = 4, 1

1, 2 = 9, 6/2 core

E energy density, vertical slice

Q limited only by finite size

G

G

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G

G

GG

GG G

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10

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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 = (0.4)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

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

of a

ir =

ck||

k||

modesin crystal

TE

TM

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

light

line

of a

ir =

ck||

in these 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

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1.8

2

2.2

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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%

/mid

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

Complete Band Gaps - Ab Initio Physics Researchab-initio.mit.edu/photons/tutorial/L4-slabs.pdf ·...

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Transcript of Complete Band Gaps - Ab Initio Physics Researchab-initio.mit.edu/photons/tutorial/L4-slabs.pdf ·...