Battling imperfections in high index-contrast systems – from Bragg fibers to planar photonic crystals Maksim Skorobogatiy Génie Physique École Polytechnique de Montréal S. Jacobs, S.G. Johnson and Yoel Fink OmniGuide Communications & MIT Some slides are courtesy of Prof. Steven Johnson
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Maksim Skorobogatiy Génie Physique École Polytechnique de Montréal
Battling imperfections in high index-contrast systems – from Bragg fibers to planar photonic crystals. Maksim Skorobogatiy Génie Physique École Polytechnique de Montréal S. Jacobs, S.G. Johnson and Yoel Fink OmniGuide Communications & MIT Some slides are courtesy of Prof. Steven Johnson. x. - PowerPoint PPT Presentation
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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