The Dispersion Properties of Coupled Discrete Line Defects J.W.Hsu (許家瑋), C.H.Huang (黃至賢), and W.F.Hsieh (謝文峰) Department of Photonics, National Chiao Tung University, HsinChu, 300 Taiwan

The tight binding theory assumes the field distribution of a PhC with defects (the perturbed system) could be expanded as a linear combination of the successive isolated defects. That means, we can say E0’(r,t) = ∑ am(t) E0m and H0’(r,t) = ∑ am(t) H0m[4], where E0’ and H0’ is the field distribution of the perturbed system, E0m and H0m is the field distribution of the isolated defect at lattice site r = ma.

Abstract We introduce an extremely simple method to design a photonics crystal (PhCs) structure come with a large linear region in dispersion relation by manipulating the arrangement of the discrete line defect in photonics crystal waveguides (PCWs). The modulated dispersion relation could mix properties of the dispersion relation of the origin ones, therefore allowing dispersionless and slow light propagation.

By substituting the field distribution E = E0(r)exp(-iω0t), H = H0(r)exp(-iω0t), E’ = E0’(r,t)exp(-iω0t), H’ = H0’(r,t)exp(-iω0t) into Lorentz reciprocity zero equation, ∇·(E0* × H0’ + H0* × E0’) = 0. Applying divergence theorem, the coupled equation is derived.

Introduction PhCs are artificial periodic refractive index structures. These structures could lead to a complete photonic bandgap as modulated the refractive index at the scale around the operating wavelength. Because Maxwell’s equation is not pertinent to scale, complete bandgap allows us choosing suitable wavelength and structure period to confine, control, and route light in photonics circuits. In the last two decades, many devices have been demonstrated, such as filters, switches, etc. [1]

(1) Where an is the field amplitude at site n and cmn is the coupling coefficient between sites m and n.

For confining light, we create defects in perfect PhCs. A line defect would lead to an additional dispersion curve to allow light propagation in the channel. This is socalled the defect mode. For choosing correct wavelength to confine light in those photonic circuits, we should find out where the bandgap and the additional dispersion curve locate.

Being defined as

(2)

There are several methods which are developed for calculating the dispersion relation, such as plane wave expansion method (PWEM)[2] and finite-difference time domain (FDTD). But, we could only obtain numerical solution from theese methods. Tight binding theory is a method developed for obtaining an analytic approximate solution as a powerful properties analysis tool in condensed matter physics and have been extended to deal with photonic crystal waveguides.[3]

If just only neighboring terms is concerned, the equation of a continuous single line defect can be simplified as

(3) By substituting an = A0 exp(ikna-iω1t) into Eq.(3), we obtained the dispersion relation of continuous single line defect.

In this paper, we would investigate the modification of dispersion relation in discrete line defect with different separation rods, and the modulation in structure mixing different separation rods. Calculating dispersion relation curve in PWEM and explaining the phenomena by tight binding theory. A structure guiding light can be designed and well explained by tight binding theory.

(4) Let’s move our perspective upon the isolated electric field of single defect as plotted in Fig. 1 for square and triangular lattices.

Theory Based on the tight binding theory, what we concerned about are the coupling coefficients between two neighboring defects, which will be determined from the structure symmetry and field distribution of the isolated single defect.

We can see the electric field is mainly localized around the dielectric rods, therefore, we can use the maximum values of electric field at each rods to replace the integral values for a simple estimation[3].

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B

A

B

A (a)

(b)

Fig. 2. Different separation periods cause dispersion curve increasing or decreasing. (a)

(b)

Mixing Periods in Single Line Defect Since we have learned the numbers of the separation rods control the coupling coefficients, through control slope of defect mode, we could manipulate the numbers of the separation rods to control slope of defect mode. For these purpose, we want to know what happened in a discrete line defect with mixing different numbers of the separation rods.

Fig. 1. Field distribution and the slice of the distribution at 0º and 27º in square lattice and at 0º and 30º in triangular lattice.

The field distribution also reveals coupling coefficient could be positive or negative which we would discuss in the next section. Because the wave function is an even function, it could be expanded as the following form.

The dispersion relation of the structures mixed one separation rod and two separation rods is shown in Fig. 3a and mixed one separation rod with three separation rods in Fig. 3b.

(5) where An is positive.

Discrete Single Line Defect By substituting Eq.(5) the form into Eq.(2), we found ε’ = ε0, so ε’ - ε < 0 in a reduced-rod defect. The integration will be negative if m-n = 2ha and positive if m-n = (2h-1)a, where h is an integer.

(a)

When one designs a structure with either odd (use 1 here) or even (use 2 here) separation rods, Eq.(3) can be written as

(b)

Fig. 3. There are two defect mode in structure mixing two different periods separation rods, and the dispersion relation always survive in bandgap.

Dispersion curves calculated by MPB show four properties. First, there are two dispersion curve provided by one discrete line defect, the slope of one curve is positive, and the other is negative. Second, the dispersion relation is quite linear except at the two boundaries. Third, new curves are always contained in the bandgap. The defect modes never merge into the band edge. Fourth, the modulus of slope of new curves lay between two original ones.

or

respectively. Thus, the dispersion curve of each structure is

Through mixing different separation rods, we are able to modulate dispersion curves to suit our demand. Large linear region means the EM wave propagate in this structure without dispersion. The small slope of dispersion relation means the wave propagation with the slower group velocity. Those two properties would benefit engineering applications.

The coefficient cp in the form determine the slope of defect mode curve. Because c2 is negative and c3 is positive, we expect slope of defect mode of a structure with odd separation rods (with term of c2, c4, etc) is positive. And negative slope attribute to the even separation rods.

Applying Eq.(3), we could obtain an approximate quantitative analysis. Because the distance to the neighboring defects are asymmetric, there are two situations we should concern. One situation is the nearest neighboring defect (one separation rod) is on the left, and the other is on the right. In structure mixing one

The numerical solutions calculated by MIT Photonic Bands conform our expectation as shown in Fig. 2. -2-

separation rod and two separation rods, we obtained theese two equations.

Substituting an = A0 exp(ikna-iω1t) into the equations, we obtain

This explains why there are two new curves and why two new curves are almost symmetric.

Conclusion Manipulating the combination of separation rods could control the slope of ω(k), the modulus of new dispersion curves is between the slopes of the original dispersion curves based on tight binding theory. It allows us creating a structure whose linear regions of dispersion curves are large enough for dispersionless propagation, and whose small slope one is useful for slowing light. The new curves always completely contain within bandgap completely.

Acknowledgments This work is partially supported by the National Science Council of Taiwan, Republic of China, under grant NSC-97R807. This is my first paper in my life, I want to attribute this paper to my dear mother and father. If I have any strength to contribute anything to this world, I always appreciate their patience.

References [1] D.W.Prather, etc, IEEE Journal of Selected Topics in Quantum Electronics Vol. 12 No. 6, 1416 (2006) [2] F.F.Chien, J.B.Tu, W.F.Hsieh, and S.C.Cheng, Physics Review B 75, 125113 (2007) [3] D.N.Christodoulides and N.K.Efremidis, Optics Letters Vol. 27 No. 8, 568 (2002) [4] M.A.El-Dahshory, A.M.Attiya, and E.A.Hashish, PIER 74, 319 (2007) [5] B.Z.Steinberg and A.Boag, Journal of Optics Society of America B Vol. 23 No. 7, 1442 (2006)

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