Theoretical analysis of photonic crystals

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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

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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

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25

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