Battling imperfections in high index-contrast systems – from Bragg fibers to planar photonic crystals

Maksim SkorobogatiyGénie Physique

École Polytechnique de Montréal

S. Jacobs, S.G. Johnson and Yoel FinkOmniGuide Communications & MIT

Some slides are courtesy of Prof. Steven Johnson

2 All Imperfections are Small for systems that work

• Material absorption: small imaginary

• Nonlinearity: small ~ |E|2

• Acircularity (birefringence): small boundary shift

• Bends: small ~ x / Rbend

• Roughness: small or boundary shift

Weak effects, long distances: hard to compute directly— use perturbation theory

y

xz

Hitomichi Takano et al., Appl. Phys. Let. 84, 2226 2004

• Variations in waveguide size: small boundary shift

3 Perturbation Theoryfor Hermitian eigenproblems

given eigenvectors/values: H u u

…find change & for small u H

Solution:expand as power series in

(1) (2)0

(1)

0u u u &

0(1) 0

0 0

ˆu H u

u u

(first order is usually enough)

H

4 Perturbation Theory for electromagnetism (no shifting material boundries)

2

(1)22

E

E

…e.g. absorptiongives

imaginary = decay!

Dielectric boundaries do

not move

core core+

const

5 Losses due to material absorptionQuickTime™ and aGraphics decompressorare needed to see this picture.1 x 1 0

-51 x 1 0

-41 x 1 0

-31 x 1 0

-21.21.622.42.8

EH11

TE01

TE01 strongly suppressescladding absorption

(like ohmic loss, for metal)

Large differential loss

(m)

Material absorption: small perturbation Im(

2

2

( )Im( )

2

Im

E

E

6 Perturbation formulation for high-index contrast waveguides and shifting material boundaries

Standard perturbation formulation and coupled mode theory in a problem of high index-contrast waveguides with shifting dielectric boundaries generally fail as these methods do not correctly incorporate field discontinuities on the dielectric interfaces.

+ -

Degenerate o of unperturbed fiber

Elliptical deformation lifts degeneracy

"Analysis of general geometric scaling perturbations in a transmitting waveguide. The fundamental connection between polarization mode dispersion and group-velocity dispersion.", M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weisberg, T.D. Engeness, M. Solja¡ci´c, S.A. Jacobs and Y. Fink, J. Opt. Soc. Am. B, vol. 19, p. 2867, 2002

7 Method of perturbation matching

thPosition of the n perturbed dielectric interface:

for every and (0,2 )

x= ( , , )

y= ( , , )

z= ( , , )

n

n

n

s

x s

y s

z s

n

Unperturbed fiber profile

yx

Perturbed fiber profile

•Dielectric profile of an unperturbed fiber o(,,s) can be mapped onto a perturbed dielectric profile (x,y,z) via a coordinate transformation x(,,s), y(,,s), z(,,s).

•Transforming Maxwell’s equation from Cartesian (x,y,z) onto curvilinear (,,s), coordinate system brings back an unperturbed dielectric profile, while adding additional terms to Maxwell’s equations due to unusual space curvature. These terms are small when perturbation is small, allowing for correct perturbative expansions.

•Rewriting Maxwell’s equation in the curvilinear coordinates also defines an exact Coupled Mode Theory in terms of the coupled modes of an original unperturbed system.

Coupled Mode Theory

- modal expansion coefficients, - original propagation constants

ˆ ˆ ˆ

ˆ ˆ, Hermitian

o

o

C

CiB BC HC

s

B H

(x,y,z)o(,,s) mapping

F(,,s) F((x,y,z),(x,y,z),s(x,y,z))

8 Method of perturbation matching, applications

a)

b)

c)

TR

Static PMD due to profile distortions

Scattering due to stochastic profile variations

d)

Modal Reshaping by tapering and scattering (Δm=0)

Inter-Modal Conversion (Δm≠0) by tapering and scattering

"Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates", M. Skorobogatiy, S.A. Jacobs, S.G. Johnson, and Y. Fink, Optics Express, vol. 10, pp. 1227-1243, 2002

"Dielectric profile variations in high-index-contrast waveguides, coupled mode theory, and perturbation expansions", M. Skorobogatiy, Steven G. Johnson, Steven A. Jacobs, and Yoel Fink, Physical Review E, vol. 67, p. 46613, 2003

9

Rs=6.05a Rf=3.05a

L

n=3.0

n=1.0

High index-contrast fiber tapers

Transmission properties of a high index-contrast non-adiabatic taper. Independent check with CAMFR.

th

f s

s

f s

s

Position of the n inter-layer

boundary:

R Rx= Cos( ) (1+ ( ))

R

R Ry= Sin( ) (1+ ( ))

R

z=s

n

n

z

L

z

L

Convergence of scattering coefficients ~ 1/N2.5

When N>10 errors are less than 1%

10 High index-contrast fiber Bragggratings

3.05a

L

n=3.0

n=1.0

w

Transmission properties of a high index-contrast Bragg grating. Independent check with CAMFR.

thPosition of the n inter-layer

boundary:

2x= Cos( ) (1+ sin( ))

2y= Sin( ) (1+ sin( ))

z=s

n

n

z

z

Convergence of scattering coefficients ~ 1/N1.5

When N>2 errors are less than 1%

11 OmniGuide hollow core Bragg fiber

Very high dispersion

Low dispersion

Zero dispersion

[2/a]

[2c

/a]

HE11

B. Temelkuran et al.,Nature 420, 650 (2002)

12 PMD of dispersion compensating Bragg fibers

11 11

11

( ) | 1, | |1, |2HE HE

HE

HPMD

y

x

thPosition of the n inter-layer

boundary:

x= Cos( ) (1+ ( ))

y= Sin( ) (1- ( ))

z=s

n n

n n

f

f

"Analysis of general geometric scaling perturbations in a transmitting waveguide. The fundamental connection between polarization mode dispersion and group-velocity dispersion", M. Skorobogatiy, M. Ibanescu, S.G. Johnson, O. Weiseberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, and Y. Fink, Journal of Optical Society of America B, vol. 19, pp. 2867-2875, 2002

13

ps/n

m/k

m

Find Dispersion

Find PMD

Adjust Bragg mirror layer thicknesses to:

• Favour large negative

dispersion at 1.55m

• Decrease PMD

Iterative design of low PMD dispersion compensating Bragg fibers

h1h2

h3

Optimization by varying layer thicknesses

14 Method of perturbation matching in application to the planar photonic crystal waveguides

GOAL:

Using eigen modes of an unperturbed 2D photonic crystal

waveguide to predict eigen modes or scattering coefficients

associated with propagation in a perturbed photonic crystal

waveguide

Uniform unperturbed waveguide

Uniform perturbed waveguide (eigen problem)

Nonuniform perturbed waveguide

(scattering problem)

20.25

0.2 ; 0.3

3.37

0.5

core reflector

cyl

c

ar a r a

n in air

a m

15 Perturbation matched CMT

Perfect PC Perfect PCScattering region

T

R

1

"Modelling the impact of imperfections in high index-contrast photonic waveguides.", M. Skorobogatiy, Opt. Express 10, 1227 (2002), PRE (2003)

16 Eigen modes of a perfect PC

ˆ ( , )o o

o ii iH x z u u

17 Perturbation matched CMT

( , ) ( ( , ), ( , ))x z x x z z x z

( , )

( ( , ), ( , ))

jy

jy

E x z

E x x z z x z

Perturbation matched

expansion basis

Mapping a perfect PC onto a perturbed one

Regions of field discontinuities are

matched with positions of

perturbed dielectric interfaces

x

z

x

z

18 Perturbation matched CMT

( , ) ( ( , ), ( , ))x z x x z z x z

Mapping system Hamiltonian

onto the one of a perfect PC

+

curvature corrections

Mapping a perturbed PC onto a perfect one

x

z

x

z

ˆ ( , )o oo

o ii iH x z u u

ˆ ˆ ( , )B u H x z uz

ˆ ˆ ˆ( , ) ( , )oB u H x z u H x zz

0( , )

i

iiu C u x z

19 Defining coordinate mapping in 2D

( ) ( )x zx x f x f z

y y

z z

20 Finding the new modes of the uniformly perturbed photonic crystal waveguides

21 Back scattering of the fundamental mode

22 Transmission through long tapers

23 Scattering losses due to stochastic variations in the waveguide walls

Hitomichi Takano et al., Appl. Phys. Let. 84, 2226 2004

24 Scattering losses due to stochastic variations in the waveguide walls

25 Negating imperfections by local manipulations of the refractive index

26

Statistical analysis of imperfections from the images of 2D photonic crystals.

Maksim Skorobogatiy – Canada Research Chair, and Guillaume Bégin

Génie Physique, École Polytechnique de Montréal Canada

www.photonics.phys.polymtl.ca

Opt. Express, vol. 13, pp. 2487-2502 (2005)

Images used in the paper for statistical analysis are courtesy of A. Talneau, CNRS, Lab Photon &

Nanostruct, France

27

By using object recognition and

image processing techniques, one can find and analyze the constituent features

of an image

Once the defects are found and analyzed, one can predict degradation in the performance of a photonic crystal

Image Analysis

28 Characterization of individual features

02

( ) ( ) ( )mN

fit m mm

r R A Sin m B Cos m

0 0 0min( ) , , , ,edge m mQ X Y R A B 2

1

1( ) ( )

edgeN

edge fit i edge iiedge

Q r rN

0 124.5 ;max( ) 1 3.1edgeRm nm Q nm

29 Fractal nature of the imperfections

Self-similar profile of roughness

Standard deviation and mean do not characterize roughness uniquely …

But fractal dimension and correlation length do.

Roughness

Hurst exponent H=0.43

Correlation length =35nm

30 Deviation of an underlying lattice from perfect

1 2 1 2min( ) , , ,i ilatQ a a n n

2

0 1 1 2 21

1( )

holesNi i

latiholes

Q r i a n a nN

21

21 2 2

1/ 2 01( , ) exp( )

2 0 1/ 2

T

x xTx y

y y

R R

0 1 1 2 2( , ) ( ) i ix y r i a n a n

31 Hurst exponent

( ) ( )( ) is not differentiable if :

dos not exists,when 0

f x f xf x

Lipschitz function ( ) :

( ) ( ) , 0, 0<H<1 H

f x

f x f x

Roughness of a hole wall in a planar PC

If H=0, ( ) is discontinuousf x

( ) continuous if : ( ) ( ) 0, 0f x f x f x

If H=1, ( ) is differentiablef x

If 0<H<1, ( ) is continuous but not

differenti FRAa Cbl TAe and is known as

of dimension H

L

D=2-

f x

32 Hurst exponent and structure function

Lipschitz function ( ) :

( ) ( ) , 0, 0<H<1 H

f x

f f x 2

2

Height to height correlation function C( ) :

C( ) ( ) ( ) ~ , 0 H

c

f x f x

Fractal behavior is lost for length scales > 100nm

33 Distribution of parameters characterizing individual features

( ) ( )r fit edger r 2

2

1( ) exp( )

22r

rrr

0 124.3 1.8

ma

. 2.7 1.2

x( ) 2

R nm

Ell m

m

ipt n