1 sensitivity analysis of narrow band photonic-crystal waveguides and filters ben z. steinberg amir...
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Sensitivity Analysis of Narrow Band Photonic-
Crystal Waveguides and Filters
Ben Z. SteinbergAmir Boag
Ronen LisitsinSvetlana Bushmakin
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Presentation Outline
•Coupled Cavity Waveguide (CCW) (and micro-
cavities)
Filters/routers and waveguides – Optical comm.
Typical length-scale << λ (approaches today’s Fab
accuracy)
•Sensitivity analysis: (Random Structure inaccuracy)
Micro-Cavity
CCW
•Coupling (matching) to outer world
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The coupled micro-cavity photonic waveguide
Goals:
• Create photonic crystal waveguide with pre-scribed:
Narrow bandwidth
Center frequency
Applications:
• Optical/Microwave routing devices
• Wavelength Division Multiplexing components
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The Micro-Cavity Array Waveguides
a1
a2
Intercavity vector:
i ii
mb a
b
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The Single Micro-Cavity
Localized Fields Line Spectrum at ( , )o oE H 0
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Weak Coupling Perturbation Theory
A propagation modal solution of the form:
( ) ( )n nn
A
H r H r ( ) ( )n o n H r H r bwhere
( )oH r - The single cavity modal field
2 ,,
,
H H
c H H
Insert into the variational formulation:
0
1
( )
r
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The result is a shift invariant equation for :
2 2
0 0,m m m nm
h Ac c
mA
Where:
It has a solution of the form:
jkmmA e
00
, ,m m kk
H H
/k b - Wavenumber along cavity array
n
0,m mh H H
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Variational Solution
M
/|a1|/|b|
k
c
M
coso s k
s
c o
1 exp( )b
Wide spacing limit:
Bandwidth:
The isolated micro-cavity resonance
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Transmission & Bandwidh
Transmission vs. wavelength
Bandwidth vs. cavity spacing
Isolated micro-cavity resonance
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Varying a defect parameter tuning of the cavity resonance
Micro-Cavity Center Frequency Tuning
o
Example: Varying posts radius(nearest neighbors only, identically)
Transmission vs. radius
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Interested in: vs.
Then (can show for ! )
Cavity Perturbation Theory
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0 0 0 ,H c H
2
0 0 0 ,H c H
0 0 0
0 002
0 0 0
,
2
E E
H
0 0 0H H H= + ,
- Perfect micro-cavity
- Perturbed micro-cavity
1
1
.
0H O 0
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Example: 2D crystal, with uncorrelated random variation - all
posts in the crystal are varied
Random Structure Inaccuracy
ia
Model: Treat radii variations as perturbations of the reference
cavity.
In a single realization different posts can have different radii.
Cavity perturbation theory gives:
Due to localization of cavity modes – summation can be restricted to closest neighbors
2( )10 0 02
1
( 1) ir i
i
N
oa a
E
1/ 2 1/ 222 2( )1
0 0 021
( 1) ir
N
o ii
a a
E
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Standard Deviation of Resonant Wavelength
• Perturbation theory:
Summation over 6 nearest neighbors
• Statistics results:
Exact numerical results of 40 realizations
All posts in the crystal are RANDOMLY varied
Hexagonal lattice, a=4, r=0.6, =8.41. Cavity: post removal. Resonance =9.06
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CCW with Random Structure Inaccuracy
Mathematical model is based on the physical observations:
1. The microcavities are weakly coupled.
2. The resonance frequency of the i -th microcavity is
where is a variable with the properties studied before.
3. Since depends essentially on the perturbations of the i
-th microcavity closest neighbors, can be
considered as independent for i ≠ j.
0 ,i
ii
i j
0( ) ( ),mm
m
H r A H r Modal field of the (isolated)
m –th microcavity.
Its resonance is 0 .m m
0 ( )mH r
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An equation for the coefficients:
2 2
0, , 0,m n m n m
m
h Ac c
, 0 0 0, 2 ,m nm n k n
k n
H H
, 02 ,m n m n n
Where:
In the limit we obtain 0 0mH H
Random inaccuracy has no effect if 1 0 perfect CCWn n
CanonicalIndependent of specific design
parameters
n
, 0 0,m nm nh H H
,m n m nh h
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Matrix Representation
220 ,A H A
Eigenvalue problem:
- a tridiagonal matrix of the form:
0 1
1 0 1
1 0 1
1 0
0
0
0
0
220 0diag[ ] 2 diag[ ]n kc H
220 0 12 cos ( 1)n nH n N
2 2 2 20c
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Numerical Results – CCW with 7 cavities
n of perturbed microcavities
n of perturbed microcavities
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Matching a CCW to Free Space
Matching Post
R
d
, cavit
perfect match n
y
i g1SWR max
11i
i jj
E
E
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SWR minimization results
0.1
0.05
Crystal ends here
Hexagonal lattice
a=4, r=0.6, =8.41.
Cavity: post removal.
m=2
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Field Structure @ Optimum (r=1.2)
Crystal Matching PostAt 1st optimum
Matching PostAt 2nd optimum
Matching PostAt 3rd optimum
Radiation field is not well
collimated.
Solutions:
• 2D optimization with more than a
single post
• Collect by a lens
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Summary
Cavity Perturbation Theory – effect on the isolated single
cavity
Linear relation between noise strength and frequency shift.
Weak Coupling Theory + above results – effect on the CCW
A novel threshold behavior : noise affects CCW only if it
exceeds certain level.
Matching to free space.
Sensitivity of micro-cavities and CCWs to random
inaccuracy :