Maksim Skorobogatiy Génie Physique École Polytechnique de Montréal ( Université de Montréal)
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Transcript of Maksim Skorobogatiy Génie Physique École Polytechnique de Montréal ( Université de Montréal)
Battling imperfections in high index-contrast systems – from Bragg fibers to planar photonic crystals
Maksim SkorobogatiyGénie Physique
École Polytechnique de Montréal (Université de Montréal)
S. Jacobs, S.G. Johnson and Yoel FinkOmniGuide Communications & MIT
Presented at Photonics Europe, SPIE 2004
2Coupled Mode Theory and perturbation formulations for high-index contrast waveguides
Propagation of radiation through a waveguide of generic non-uniform high index-contrast dielectric profile
•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.
Direction of propagation
y
x
z
Other known methods that can solve the problem are:
• Method of crossections (expansion into the instantaneous eigen modes). This method requires recalculation of the local eigen modes at each of the different crossesctions along the direction of propagation, and is computationally intensive.
• Expansion into the eigen modes of a uniform waveguide with smooth dielectric profile (empty metallic waveguide f.e.). Convergence of this method with the number N of expansion modes is slow (linear ~1/N).
• Traditional FDTD, FETD are surprisingly difficult to use for analysis of small variations as one needs to resolve spatially such variations, and the effect of such variations is only observable after long propagation distances.
3 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))
4 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
5
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%
6 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%
7 OmniGuide hollow core Bragg fiber
Very high dispersion
Low dispersion
Zero dispersion
[2/a]
[2c
/a]
HE11
8 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
9
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
10 Method of perturbation matching in application to the planar photonic crystal waveguides
Using the guided and evanescent modes of an unperturbed PxTal
waveguide to predict eigen modes or scattering coefficients for a perturbed PxTal 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
11 Defining coordinate mapping in 2D
( ) ( )x zx x f x f z
y y
z z
12 Finding the new modes of the uniformly perturbed photonic crystal waveguides
13 Back scattering of the fundamental mode
14 Transmission through long tapers
15 Scattering losses due to stochastic variations in the waveguide walls
16 Scattering losses due to stochastic variations in the waveguide walls
17 Negating imperfections by local manipulations of the refractive index