Photonic Crystals – Chapter 8 Photonic Crystal...

Photonic Modeling and Design Lab.Graduate Institute of Photonics and Optoelectronics &Department of Electrical EngineeringNational Taiwan University

光子晶體光子晶體Photonic Crystals Photonic Crystals –– Chapter 8Chapter 8

Photonic Crystal SlabsPhotonic Crystal Slabs

台大光電所暨電機系

邱奕鵬Room 617, BL Building

(02) 3366-3603

ypchiou@cc.ee.ntu.edu.tw

Photonic Crystal Slabs

Why “SLAB”?

• Full 3D photonic crystals are too expensive

• Difficult to incorporate on a chip

Why “SLAB”?

• In-plane propagation• In-plane 2D photonic crystal

• Vertical index guiding

Why “SLAB”?• Easy to incorporate on a chip and can be

fabricated easier

Total Internal Reflection

ni > no

no

rays at shallow angles > θc

are totally reflected

Snell’s Law:

θi

θo

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 > θc

are 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 > θc

are 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:

θi

θo|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 line: 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)

J

J

J

J

J

J

JJ

JJ J

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

freq

uenc

y (c

/a)

wavenumber k (2?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

J

J

J

J

J

J

JJ

JJ J

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

freq

uenc

y (c

/a)

wavenumber k (2?a)

light cone

and　

ga p�

a

range of frequenciesin which there are

no guided modes

slow-light band edge

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 obvious solution?

make2d pattern really 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

Projection!!

Photonic-Crystal Slabs

2d photonic bandgap + vertical index guiding

[ S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice“Guided modes in photonic crystal slabs,” Phy. Rev. B 60, 5751, (1999)]

Photonic-Crystal Slabs

2d photonic bandgap + vertical index guiding

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

r=0.2a

r=0.45a

Dielectric const. = 12 (both)

Two Dimensional LatticesTriangular Lattice

in z-direction of an length

Draw perpendicular bisectors of any lattice vector startingat origin

spacing

Symmetry in a Slab

2d: TM and TE modes

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

mirror planez = 0

E E

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

purely TM/TE inthe mirror plane

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

Shift of lightcone

n=1 n=1.5

Background n=1.0 Background n=1.0

SANDWICHESBackground n=1.5 Background n=1.5

2D band structure (kz=0, in-plane propagation)Slabs with periodic background

Slabs with symmetry-breaking background

Slabs with symmetry-breaking background

• The guided mode can no longer be classified as even or odd.

• There is no longer any band gap in the guided modes.

Slabs with symmetry-breaking background

• The guided modes are sufficiently localized within the slab.

• Buffer region.

Buffer region

Air-membrane Slabs

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

who needs a substrate?

2µm

AlGaAs

Effects of slab thickness

• Too thick: Higher order modes can be created with little energy cost by adding horizontal nodal plane.

• Too thin: The slab will provide only a weak perturbation on the background dielectric constant.

r=0.2ar=0.45a

Dielectric const. = 12 (both)

Optimal slab thickness• λ/2: half the two-dimensional gap-bottom

wavelength.

• We only know the gap-bottom frequency.

• Need an effective dielectric constant to evaluate the wavelength.

Two dimensional wavefunction

Evaluate the dependence of its

frequency expectation value on the vertical

wavelength

Optimal slab thickness

Wavefunction

Eiganfunction Amplitude

Strongly localized within the slab

Optimal slab thickness

Only for divergenceless fields

Optimal slab thickness

ε

Effective dielectric constant

=5.06

ε=12

εrather than 3.92(uniform-weight mean)

=1.25εrather than 1.13

(uniform-weight inverse mean)

Optimal slab thickness

Photonic-Crystal Building Blocks

point defects(cavities)

line defects(waveguides)

Reduced-index waveguides Increased-index

waveguides

Dielectric-strip waveguides

Three criteria must be satisfied

• Waveguide must support true guided modes.

– Bloch

• The waveguide should be single-mode in the frequency range of interest.

• The guided mode should lie within the band gap of a photonic crystal in order to prohibit radiation losses.

• Conventional waveguides

To achieve optimal performance in many integrated-optical-circuit applications

Photonic-Crystal Building Blocks

point defects(cavities)

line defects(waveguides)

Reduced-index waveguides Increased-index

waveguides

Dielectric-strip 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 conservedwavevector—under thelight cone, no radiation

(r=0.10a)

We cannot completelyremove the rods—novertical confinement!

A Reduced-Index Waveguide

The thickness of the vein is the

critical parameter!

Two Dimensional LatticesTriangular Lattice

in z-direction of an length

Draw perpendicular bisectors of any lattice vector startingat origin

spacing

Reduced-Index Waveguide Modes

y →

x →

? +1Ez

y →

y →

x →

? +1Hz

z →

y →

Increased-index waveguide

• Modes that lie below both the light cone and the slab band continuum.

• A state that is pulled down into the gap from the upper slab bands.

Two sorts of guided modes

Index-guiding both horizontal and vertical

Index-guiding only vertically

higherDegenerate pairM

Increased-index waveguide

Non-degenerate

Single mode

Increased-index waveguide

Degenerate Multi-mode

Increased-index waveguide

Strip waveguides in photonic-crystal slabs

Strip waveguides in photonic-crystal slabs

Fundamental modeSecond

exciting mode

Need for absolute band gap

• Refractive index contrast is as high as possible

• Brillouin zone is close to a circle

• Shape of scattering objects matches the symmetry of Brillouin zone

• The PhC is comprised of isolated dielectric islands connected by narrow veins (this implies a high filling-factor for the low-index material)

Design of photonic crystals

• Thickness

• Size of “air rods”

• Angular orientation with respect to the lattice

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

Quality factor Q

• The number of cycles for its energy to decay by

• The decay rate 1/Q is the bandwidth at half maximum of any filter or other device based on the cavity

π2−e

A Resonant Cavity

index-confined

photonic band gap

increased rod radiuspulls down “dipole” mode

(non-degenerate)

– +

J

J

JJ

JJ

JJ J J J

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

freq

uenc

y (c

/a)

wavenumber k (2?a)

light cone

and　

ga p� 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)

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

Input Waveguide Cavity Output Waveguide

Q

Increase Q⊥

The more spatially delocalized the mode is, the more localized it will be in wave vector space.

A. Mode delocalizationB. Multipole cancellations

Semi-analytical losses

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

Mode frequency

(ε = 12)

Monopole Cavity in a Slab

decreasing ε

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

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

Increase Q

Enlarge the defect size

Weaker defect with more unit cells.

More delocalizedat the same point in the gap

(i.e. at same bulk decay rate)

Super-defects

Super defect v.s. single defect Super-Defect State(cross-section)

Δε = –3, Qrad = 13,000

(super defect)

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

Ez

(in hole slabs, too) 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

Another low-lossstrategy:

exploit cancellationsfrom sign oscillations

far-field(radiation) Green’s function

(defect-freesystem)

near-field(cavity mode)defect

Mode frequency

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

cancel a dipole by opposite dipoles…

cancellation comes fromopposite-sign fields in adjacent rods

… changing radius changed balance of dipoles

3d multipole cancellation

near

fie

ld E

zfa

r fi

eld

|E|2

Q = 408 Q = 426Q = 1925

nodal planes(source of high Q)

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) ]

A Perfect Cavity in 3d

R

N layers

N layers

defect

Perfect indexconfinement

(no scattering)

+1d band gap

=3d confinement

(~ VCSEL + perfect lateral confinement)

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

Smaller index, n

1

0%

10%

20%

30%

40%

50%

Δλ/λmid

6 9 1 2 1 50

5 0

1 0 0

0

5 0

1 0 00

5 0

1 0 00

5 0

1 0 00

5 0

1 0 0

normal

450 s

450 p

800 p

800 s

Refle

cta

nce

(%)

Wavelength (microns)Wavelength (microns)

Te / polystyrene

Ref

lect

ance

(%

)

contours of omnidirectional gap size