theoretical analysis of photonic crystals
DESCRIPTION
Theoretical analysis of photonic crystals. Ph.D. proposal. by Inna Nusinsky-Shmuilov. Supervisor: Prof. Amos Hardy. Department of Electrical Engineering–Physical Electronics. Faculty of Engineering, Tel Aviv University. Outline. Research subject and scientific background. - PowerPoint PPT PresentationTRANSCRIPT
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Theoretical analysis of photonic crystals
Ph.D. proposal
by Inna Nusinsky-Shmuilov
Supervisor: Prof. Amos Hardy
Department of Electrical Engineering–Physical Electronics
Faculty of Engineering, Tel Aviv University
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OutlineResearch subject and scientific background
The main goal and expected significance
Preliminary work and results
Research plan
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Research subject
2D PC 3D PC1D PC
Photonic crystals have a periodic variation in the refractive index in specific directions.
Creation of a periodicity prevents the propagation of electromagnetic waves with certain frequencies
gap
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Point defect
Breaking the periodicity can create new energy levels within the photonic band gap
Research subject
Defect can control the propagation of the light
Linear defect
Applications: sharp band waveguides, filters, low threshold lasers, micro cavities, couplers…
Extended defects
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Scientific background Existing numerical techniques
• Plane wave expansion method
• FDTD
• Transfer matrix method
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Scientific background Analytically solvable structures (2D)
• Asymptotic structure
• Separable structure
x yN Nyx LNyLNxm
r
10
0b bm
ygxfyx
0
,
1
2
3
4
4321
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Main goals and expected significance •To investigate photonic crystals numerically as well as by means of new approximate analytical models
•To employ the new analytical models to investigate photonic crystals properties, to predict and explain their behaviour
Deeper understanding of physical processes in photonic crystals
Practical significance for development new devices
•To find the conditions and photonic crystals' parameters required for their optimal performance
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Preliminary work and resultsOne dimensional photonic crystals (exact analytical model)
•Hill's equation
0 uxau xa -periodic
022
22
2
2
xEk
cxn
dx
xEdz
00
10
1
1
xE
xE 10
00
2
2
xE
xE xECxECxE 2211
Bloch theorem
01212 iKLiKL eLELEe
zkF ,
iKLe
iKLe
9
...3,2,1
coscos
2
2211
mbnan
rmc mm
2
0
mr
Position of the gap edges:
0 mm rr
2, zkF
2, zkF
2, zkF
realK
propagating solutions
K complex
decaying solutions (gap)
LmK
gap edges
Gap closing:
Preliminary work and resultsOne dimensional photonic crystals (exact analytical model)
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Preliminary work and resultsOne dimensional photonic crystals (gap closing points)
Inna Nusinsky and Amos A.Hardy, "Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing", Phys. Rev. B , 73, p.125104 (2006)
Two types of gap closing points:
2 .Identical for TE and TM
1.Brewster closing points
1
21tan
n
n
Exist only for TM polarization
Don’t exist in the first gapFirst gap has only one closing point (Brewster)
Omnidirectional reflection
2211 coscos bqnapn pqim
m gap number
..3,2,1i
pqm,
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Preliminary work and resultsOne dimensional photonic crystals (gap closing points)
Condition for existing M omnidirectional gaps:
1
1
Mnd
nd
HH
LL
Condition for omnidirectional reflection from higher order gaps
2222
2222221sin
LH
LLHH
LL dpdq
dnpdnq
n
LL nn0sin
Light line:
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Preliminary work and resultsOne dimensional photonic crystal (omnidirectional reflection)
Inna Nusinsky and Amos A.Hardy, “Omnidirectional reflection in several frequency ranges of one dimensional photonic crystals", Appl. Opt. 45(15), (2007)
Applications: eye-protection glasses, air-guiding hollow optical fibers, dielectric coaxial waveguides, light-emitting diodes, VCSELs
45.1Ln 5.3Hn 32HL dd nmdL 240 nmdH 360
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Preliminary work and results2D photonic crystals (approximate analytical model)
Assumption:
b is sufficiently small
n1=1 n2=2.1 a=0.85L b=0.15LH polarization E polarization
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Preliminary work and results2D photonic crystals (cont.)
Inna Nusinsky and Amos A.Hardy, “Approximate analytical calculations of two dimensional photonic crystals with square geometry“, in preparation
n1=2.1 n2=1 a=0.85 b=0.15
H-polarization E-polarizationThe band gap edges are located at one of the high symmetry points: Γ, X or M
For H-polarization, the gap between second and third band is easily opened and is wider than the lower gap (between first and second bands)
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Preliminary work and resultsLarge area single mode operation in gain guided fibers
The gain guiding effect is weak
Very large gain coefficient is needed
A.E.Siegman et. al, APL 89,p.251101 (2006)
A.E.Siegman, JOSA A, 20 (8),p.1617 (2003)
ginn
20
1n
2n
2n
gain21 nn
16
g g
Preliminary work and results
Applications: Large area single mode fiber lasers and amplifiers
45.11 n45.1*)01.01(2 n
Large area single mode operation in gain guided fibers
37.6dB/m 20.5dB/m10.3dB/m 4dB/m
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Preliminary work and resultsPublications
1.Inna Nusinsky and Amos A.Hardy, "Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing", Phys. Rev. B , 73, p.125104 (2006)
2.Inna Nusinsky and Amos A.Hardy, “Omnidirectional reflection in several frequency ranges of one dimensional photonic crystals", Appl. Opt. 45 (15), (2007)
3.Inna Nusinsky and Amos A.Hardy, “Approximate analytical calculations of two dimensional photonic crystals with square geometry“, in preparation
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Research plan1One dimensional photonic crystal
2Asymptotic two dimensional model
3Two dimensional PC with square cross section (in plane
propagation(
4Photonic crystal fibers
5Defect analysis in square PC
6Two dimensional PC with circular cross section (in
plane propagation)
7Off-plane propagation
8Asymptotic three-dimensional PC
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Appendix
21
1st gap
2nd gap 3rd gap
missing outside
11,2
both missing
12,3 21,3
both outside
12,3missing 21,3
outside
Appendix
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23
24
25
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