The Dispersion Properties of Coupled Discrete Line DefectsSpeaker: Jia-Wei Hsuundergraduate student

Laser Diagnostics LaboratoryDepartment of PhotonicsNational Chiao Tung University1001 Tahsueh Rd. HsinChu, Taiwan

Co-author: C.H.Huang Advisor: W.F.Hsieh

slideshare.net/shelling

Outline

•Introduction and Motivation

•Theory and Calculation

•Simulation and Results

•Conclusion

Introduction

Extremely simple method to design a photonic crystal (PhCs) structure come with “large linear region in dispersion relation”.

Non-linear Linear

Manipulating Dispersion Relation

Non-dispersion and small group velocity propagation become possible and easy.

Frequency

K-point K-point

Frequency

Plane Wave Expansion Method confirms the coupled

coefficients predicted by TBT could be positive or negative.

Introduction

“Tight-binding Theory for Coupled Photonics Crystal Waveguides” , F.F.Chien, J.B.Tu, W.F.Hsieh, and S.C.Cheng, Physics Review B 75 (2007)

Tight Binding Theoryx

y nearest-neighboring cn (n-1)

next-nearest-neighboring cn (n-2)

d

βγ γ

single line defect

two parallel line defects

MotivationTight Binding Theory (TBT) is

used to calculate coupled coefficients from single defect

mode electric field and to obtain analytic solution of dispersion

relation ω(k).

We could design a structure by modifying cmn to control analytic ω(k) for dispersionless and slow

light propagation.

cp means cn(n+p) later

Single Point Defectslice at 0º and 27º

12

12

slice at 0º and 30º

Determining signs of the coupling coefficients

1

2

Wave function of defect mode of single defect is a standing

wave localized in defect.12

12

The ratio of peak of the wave function is an approximation of ratio of coupling coefficients.

square lattice

triangular lattice

Coupled Equations

“Discrete Temporal Soliton along a Chain of Nonlinear Coupled Microcavities Embedded in Photonics Crystals”, D.N.Christodoulides and N.K.Efremidis, Optics Letters Vol. 27 No. 8 (2002)

isolated point defect line defects

D

Coupled Coefficients

predict ω(k) with localized wave function

Discrete Line Defect

separation = 1, 3, 5 ...➞ En × Em>0 ➞ cmn < 0➞ ω1(k) decreases as k⬆

separation = 2, 4, 6 ...➞ En × Em>0➞ cmn > 0➞ ω1(k) increases as k⬆

separation = 1

separation = 2

separation = 3

separation = 4

MixingWhile we mix two kinds of

separations to form a new space frequencies in crystal, it shows many

useful properties.

1

2

3 New curves always appear in band gap.

It fork two new curves, one’s slope is positive, the other is negative, and

about symmetric.

New curves appear with big linear zone, nonlinear zone just appears at edge.

4 In the combination, it seems that big separations come with big weight to

affect new curves.

MixingWhile we mix two kinds of

separations to form a new space frequencies in crystal, it shows many

useful properties.

1

2

3 New curves always appear in band gap.

It fork two new curves, one’s slope is positive, the other is negative, and

about symmetric.

New curves appear with big linear zone, nonlinear zone just appears at edge.

4 In the combination, it seems that big separations come with big weight to

affect new curves.

Why it ForkSuper Cell 1 Super Cell 2

Even Parity: ψ+

Odd Parity: ψ-

➡ Substituting into the coupled equation again:

ω1+ = ω0+ - c0+ - c1+ 2cos(kL)ω1- = ω0- - c0 - - c1- 2cos(kL)

A mixing PCW contributes two dispersion relation curves.

L=5a

[ω0±]2 = ω02(1±β1)/(1±α1+∆α)α1 = ∫dv ε(r) EΩ(r) EΩ(r-2a)β1 = ∫dv ε0(r-2a) EΩ(r) EΩ(r-2a)∆α = ∫dv [ε(r) - ε0(r)] EΩ(r) EΩ(r)

Coupled Mixing PCWNew interesting properties

New curves still appear in band gap.It forks four new curves now.

Decoupled point now shift to about k = π/2L because new curves is still

about symmetric.

New curves still have large linear zone.

ConclusionWe can manipulate the arrangement of separation rods to control slope of ω(k), the modulus of slope of new

curves is in the range between slope of old curves.

We can create a structure whose ω(k) come with linear zone large enough for dispersionless propagation, and

also with small slope for slowing light.

Simple mathematic analysis shows us why one channel in perfect photonic crystal causes two dispersion

curves, as well as its numeric solution shows us how to draw up a structure with slope we need.

Thank for your attention

References[1] Tight-binding Theory for Coupled Photonics Crystal Waveguides, F.F.Chien, J.B.Tu, W.F.Hsieh, and S.C.Cheng, Physics Review B 75 (2007)

[2] Discrete Temporal Soliton along a Chain of Nonlinear Coupled Microcavities Embedded in photonics crystals, D.N.Christodoulides and N.K.Efremidis, Optics Letters Vol. 27 No. 8 (2002)

[3] Design Equations of Two-Dimensional Dielectric Photonics Band Gap Structure, M.A.El-Dahshory, A.M.Attiya, and E.A.Hashish, PIER 74 319-340 (2007)

[4] Photonics Crystal Structure and Applications: Perspective, Overview, and Development, D.W.Prather, etc, IEEE Journal of Selected Topics in Quantum Electronics Vol. 12 No. 6 (2006)

[5] Propagation in Photonics Crystal Coupled-Cavity Waveguides with Discontinuities in their Optical Properties, B.Z.Steinberg and A.Boag, Journal of Optics Society of America B Vol. 23 No. 7 (2006)

[6] Tight-Binding Description of the Coupled Defect Modes in Three-Dimensional Photonics Crystal, PHYSICAL REVIEW LETTERS, Volume 84, Numver 10 (2000)