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Photonic Crystals: Periodic Surprises in Electromagnetism. Complete Band Gaps: You can leave home without them. Steven G. Johnson MIT. How else can we confine light?. rays at shallow angles > q c are totally reflected. Snell’s Law :. n i sin q i = n o sin q o. q i. q o.

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Complete band gaps you can leave home without them l.jpg

Photonic Crystals:Periodic Surprises in Electromagnetism

Complete Band Gaps:You can leave home without them.

Steven G. Johnson

MIT



Total internal reflection l.jpg

rays at shallow angles > qc

are totally reflected

Snell’s Law:

ni sinqi = no sinqo

qi

qo

Total Internal Reflection

no

ni > no

sinqc = no / ni

< 1, so qc is real

i.e. TIR can only guide

within higher index

unlike a band gap


Total internal reflection4 l.jpg

rays at shallow angles > qc

are totally reflected

Total Internal Reflection?

no

ni > no

So, for example,

a discontiguous structure can’t possibly guide by TIR…

the rays can’t stay inside!


Total internal reflection5 l.jpg

rays at shallow angles > qc

are totally reflected

Total Internal Reflection?

no

ni > no

So, for example,

a discontiguous structure can’t possibly guide by TIR…

or can it?


Total internal reflection redux l.jpg

translational

symmetry

Snell’s Law is really

conservation of k|| and w:

qi

k||

|ki| sinqi = |ko| sinqo

qo

|k| = nw/c

conserved!

(frequency)

(wavevector)

Total Internal Reflection Redux

no

ni > no

ray-optics picture is invalid on l scale

(neglects coherence, near field…)


Waveguide dispersion relations i e projected band diagrams l.jpg

no

ni > no

higher-order modes

at larger w, b

higher-index core

pulls down state

w = ck / ni

weakly guided (field mostly in no)

Waveguide Dispersion Relationsi.e. projected band diagrams

w

light cone

projection of all k in no

light line: w = ck / no

k||

( )

(a.k.a. b)


Strange total internal reflection l.jpg

Index Guiding

Strange Total Internal Reflection

a


A hybrid photonic crystal 1d band gap index guiding l.jpg
A Hybrid Photonic Crystal:1d band gap + index guiding

range of frequencies

in which there are

no guided modes

slow-light band edge

a


A resonant cavity l.jpg

– +

increased rod radius

pulls down “dipole” mode

(non-degenerate)

A Resonant Cavity


A resonant cavity11 l.jpg

k not conserved

so coupling to

light cone:

radiation

w

A Resonant Cavity

– +

The trick is to

keep the

radiation small…

(more on this later)


Meanwhile back in reality l.jpg

d

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

Meanwhile, back in reality…

5 µm

d = 703nm

d = 632nm

bigger cavity

= longer l

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


Time for two dimensions l.jpg
Time for Two Dimensions…

2d is all we really need for many interesting devices

…darn z direction!


How do we make a 2d bandgap l.jpg
How do we make a 2d bandgap?

Most obvious solution?

make

2d pattern reallytall


How do we make a 2d bandgap15 l.jpg
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 3d17 l.jpg

but this empty

space looks useful…

projected band diagram fills gap!

A 2d band diagram in 3d


Photonic crystal slabs l.jpg
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 l.jpg

G

G

X

M

Rod-Slab Projected Band Diagram

M

X

G


Symmetry in a slab l.jpg

mirror plane

z = 0

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

Like in 2d, there may only be a band gap

in one symmetry/polarization

Symmetry in a Slab

2d: TM and TE modes


Slab gaps l.jpg
Slab Gaps

TM-like gap

TE-like gap


Substrates for the gravity impaired l.jpg
Substrates, for the Gravity-Impaired

(rods or holes)

superstrate restores symmetry

substrate breaks symmetry:

some even/odd mixing “kills” gap

“extruded” substrate

= stronger confinement

BUT

with strong confinement

(high index contrast)

mixing can be weak

(less mixing even

without superstrate



Air membrane slabs l.jpg
Air-membrane Slabs

who needs a substrate?

AlGaAs

2µm

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


Optimal slab thickness l.jpg

effective medium theory: effective e depends on polarization

TE “sees” <e>

~ high e

TM “sees” <e-1>-1

~ low e

~ l/2, but l/2 in what material?

Optimal Slab Thickness

gap size (%)

slab thickness (a)


Photonic crystal building blocks l.jpg
Photonic-Crystal Building Blocks

point defects

(cavities)

line defects

(waveguides)


A reduced index waveguide l.jpg

We cannotcompletely

remove the rods—no

vertical confinement!

(r=0.10a)

(r=0.12a)

Still have conserved

wavevector—under the

light cone, no radiation

(r=0.14a)

(r=0.16a)

(r=0.18a)

A Reduced-Index Waveguide

(r=0.2a)

Reduce the radius of a row of

rods to “trap” a waveguide mode

in the gap.



Experimental waveguide bend l.jpg

caution:

can easily be

multi-mode

SiO2

GaAs

AlO

1µm

Experimental Waveguide & Bend

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

1µm

bending efficiency


Inevitable radiation losses whenever translational symmetry is broken l.jpg
Inevitable Radiation Losseswhenever translational symmetry is broken

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

coupling to light cone

= radiation losses

w

(conserved)

k is no longer conserved!


All is not lost l.jpg

We want: Qr>> Qw

1 – transmission ~ 2Q /Qr

Q = lifetime/period

= frequency/bandwidth

All Is Not Lost

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

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


Semi analytical losses l.jpg

Make field inside

defect small

= delocalize mode

A low-loss

strategy:

Make defect weak

= delocalize mode

Semi-analytical losses

far-field

(radiation)

defect

Green’s function

(defect-free

system)

near-field

(cavity mode)


Monopole cavity in a slab l.jpg

(e = 12)

Monopole Cavity in a Slab

Lower the e of a single rod: push up

a monopole (singlet) state.

decreasing e

Use small De: delocalized in-plane,

& high-Q (we hope)


Delocalized monopole q l.jpg
Delocalized Monopole Q

e=11

e=10

e=9

e=8

e=7

e=6

mid-gap


Super defects l.jpg
Super-defects

Weaker defect with more unit cells.

More delocalized

at the same point in the gap

(i.e. at same bulk decay rate)


Super defect vs single defect q l.jpg
Super-Defect vs. Single-Defect Q

e=11.5

e=11

e=11

e=10

e=10

e=9

e=9

e=8

e=7

e=8

e=7

e=6

mid-gap


Super defect vs single defect q37 l.jpg
Super-Defect vs. Single-Defect Q

e=11.5

e=11

e=11

e=10

e=10

e=9

e=9

e=8

e=7

e=8

e=7

e=6

mid-gap


Super defect state cross section l.jpg
Super-Defect State(cross-section)

De = –3, Qrad = 13,000

Ez

(super defect)

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



How do we compute q l.jpg

excite cavity with dipole source

(broad bandwidth, e.g. Gaussian pulse)

1

… monitor field at some point

…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 w’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?

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


How do we compute q41 l.jpg

2

1

1

…measure outgoing power P and energy U

Q = w0U / P

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

Con: requires separate run to get w0,

long-time source for closely-spaced resonances

How do we compute Q?

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

excite cavity with

narrow-band dipole source

(e.g. temporally broad Gaussian pulse)

— source is at w0 resonance,

which must already be known (via )


Can we increase q without delocalizing l.jpg
Can we increase Qwithout delocalizing?


Semi analytical losses43 l.jpg
Semi-analytical losses

Another low-loss

strategy:

exploit cancellations

from sign oscillations

far-field

(radiation)

defect

Green’s function

(defect-free

system)

near-field

(cavity mode)


Need a more compact representation l.jpg
Need a morecompact representation


Multipole expansion l.jpg
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 expansion46 l.jpg
Multipole Expansion

[ Jackson, Classical Electrodynamics ]

radiated field =

dipole

quadrupole

hexapole

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


Multipoles in a 2d example l.jpg

– +

increased rod radius

pulls down “dipole” mode

(non-degenerate)

Multipoles in a 2d example

as we change the radius, w sweeps across the gap


2d multipole cancellation l.jpg

Q = 1,773

Q = 28,700

Q = 6,624

2d multipolecancellation


Cancel a dipole by opposite dipoles l.jpg
cancel a dipole by opposite dipoles…

cancellation comes from

opposite-sign fields in adjacent rods

… changing radius changed balance of dipoles


3d multipole cancellation l.jpg
3d multipole cancellation?

enlarge center & adjacent rods

quadrupole mode

vary side-rod e slightly

for continuous tuning

(balance central moment with opposite-sign side rods)

(Ez cross section)

gap bottom

gap top


3d multipole cancellation51 l.jpg
3d multipole cancellation

Q = 408

Q = 1925

Q = 426

near field Ez

far field |E|2

nodal planes

(source of high Q)


An experimental laser cavity l.jpg
An Experimental (Laser) Cavity

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

elongate row of holes

cavity

Elongation p is a tuning parameter for the 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 cavity53 l.jpg
An Experimental (Laser) Cavity

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

elongate row of holes

Hz (greyscale)

cavity

Elongation p is a tuning parameter for the 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 cavity54 l.jpg

Q ~ 2000 observed from luminescence

An Experimental (Laser) Cavity

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

cavity

(InGaAsP)

quantum-well lasing threshold of 214µW

(optically pumped @830nm, 1% duty cycle)


How can we get arbitrary q with finite modal volume l.jpg

Only one way:

a full 3d band gap

Now there are two ways.

[ M. R. Watts et al., Opt. Lett.27, 1785 (2002) ]

How can we get arbitrary Qwith finite modal volume?



Perfect mode matching l.jpg
Perfect Mode Matching

…closely related to separability

[ S. Kawakami, J. Lightwave Tech.20, 1644 (2002) ]


Perfect mode matching58 l.jpg
Perfect Mode Matching

(note switch in TE/TM

convention)



A perfect cavity in 3d l.jpg
A Perfect Cavity in 3d

(~ VCSEL + perfect lateral confinement)





Forget these devices l.jpg
Forget these devices…

I just want a mirror.

ok


Projected bands of a 1d crystal a k a a bragg mirror l.jpg

w

TM

1d band gap

TE

modes

in crystal

k||

(normal

incidence)

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

incident light

light line of air w = ck||

k|| conserved


Omnidirectional reflection l.jpg

all incident light

(any angle, polarization)

is reflected

from flat surface

Omnidirectional Reflection

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

w

in these w ranges,

there is

no overlap

between modes

of air & crystal

light line of air w = ck||

TM

TE

modes

in crystal

k||

needs: sufficient index contrast & nhi > nlo > 1


Omnidirectional mirrors in practice l.jpg

Wavelength (microns)

Omnidirectional Mirrors in Practice

[ Y. Fink et al, Science282, 1679 (1998) ]

Te / polystyrene

contours of omnidirectional gap size

Reflectance (%)

Dl/lmid


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