The document summarizes research on manipulating the dispersion relation of coupled discrete line defects in photonic crystals. It describes how the separation distance between line defects affects the coupling coefficients and dispersion curves based on tight binding theory. Simulation results show that mixing different separation distances produces new dispersion curves with regions of large linear dispersion ideal for applications like slow light propagation. The slopes of the new curves can be controlled by adjusting the arrangement of separation distances to design structures for specific applications.

1. The Dispersion Properties
of Coupled Discrete Line Defects
Speaker: Jia-Wei Hsu
undergraduate student
Laser Diagnostics Laboratory
Department of Photonics
National Chiao Tung University
1001 Tahsueh Rd. HsinChu, Taiwan
Co-author: C.H.Huang
Advisor: W.F.Hsieh

2. slideshare.net/shelling

3. Outline
•Introduction and Motivation
•Theory and Calculation
•Simulation and Results
•Conclusion

4. 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

5. 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 Theory
x
y nearest-neighboring cn (n-1)
next-nearest-neighboring cn (n-2)
d
βγ γ
single line defect
two parallel line defects
Motivation
Tight 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

6. Single Point Defect
slice at 0º and 27º
1
2
1
2
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.1
2
1
2
The ratio of peak of the wave
function is an approximation of
ratio of coupling coefficients.
square lattice
triangular lattice

7. 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

8. Coupled Coefficients
predict ω(k) with localized wave function

9. 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

10. Mixing
While 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.

11. Mixing
While 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.

12. Why it Fork
Super 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 = ω0
2(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)

13. Coupled Mixing PCW
New 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.

14. Conclusion
We 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.

15. Thank for your attention

16. 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)