Motivations

The tight binding theory (TBT) has been widely used in the CROWs to describe the amplitude of the EM wave propagation in linear or nonlinear system. When a plane wave is incident in these structures, the dispersion of the CROWs can be obtained by PWEM. The numbers of the separation rods extremely influence the sign and the slope of dispersion relation. From the dispersion relation, we can derive the group velocity and various orders of the group velocity dispersion (GVD) which means the difference of the separation rods in the CROWs will also determine the sign and the magnitudes of the

group velocity and GVD. Once, the nonlinear material is added in the waveguides region, the dispersion relation curve of the CROW has a constant frequency shift comparing to the curve of linear waveguides, so the linearly physical properties of the waveguides will be preserved and the properties will influence the performance of the nonlinear materials. Therefore, the properties of the CROWs with different separation rods or structures should initially be investigated and the TBT provides a powerful method for us to realize these properties.

As the defect rods are made of nonlinear materials such as Kerr media, the perturbation superimposed on a plane wave could grow exponentially at a certain condition. It is named as the modulation instability (MI). Because the field evolution in the PCWs with Kerr media can be expressed as a discrete nonlinear Schrödinger (DNLS) equation, the MI region and gain can be derived from this equation. On the other hand, when a pulse is incident into the waveguides with nonlinear materials, the pulse could propagate in the waveguides without distortion which is the so-called soliton. The criterion for soliton propagation under slowly varying envelope approximation (SVEA) can also be derived by the DNLS equation. Therefore, we can use this criterion to discuss the soliton propagation regions in different structures of CROWs. When the width of the pulse becomes shorter, the SVEA is broken. The soliton disperses under propagation due to the high order GVD. We can take the Fourier transform of the amplitudes of the pulse to discuss the pulse broadening caused by high order GVD under soliton propagation condition.

In PCWs, the distance between two defect rods (a) is so close that the next nearest-neighbor coupling is no longer negligible. Under this circumstance, the evolution equation to describe the wave propagation in the PCWs with nonlinear material

should be written as an extended formula. In general, the next nearest-neighbor coupling coefficient is approximately one order smaller than the nearest-neighbor coupling coefficient, such making the properties in the PCWs is different from the properties in the CROWs especially when ka = π/2. Therefore, it is needed to take advanced discussion on the MI and soliton propagation in the PCWs.

When the other identical waveguide is carved into PCs and partitioned with one or several rows of perfect rod(s), another useful device, the directional coupler (DC), is created. The dispersion relation curve of the coupler is usually crossing with triangular lattice but rare crossing in the square lattice. This phenomenon gives some limitation as designing the device. Therefore, we want to create the crossing point in the square lattice and then to move the crossing point at square lattice or triangular lattice which be achieved by moving the defect rods in the waveguides. We also want to use TBT to further realize the trend of shift of the crossing point as moving the defect rods.

On the other hand, when the other waveguide is created asymmetrically, the dispersion relation calculated by PWEM would not cross anymore, but the parity of the eigen mode may switch at a particular point, in which point we named it the decoupling point. At this point, the electric field is only localized at one waveguide of this asymmetric coupler.

These phenomena cannot give a good explanation by the numerical simulation results.

Therefore, we also want to use TBT to derive an analytic description to realize the physical properties of asymmetric PC couplers.

1.5 Organization of the dissertation

In this dissertation, we firstly use TBT to derive the electric field evolution equation in single PCWs and CROWs with or without the nonlinear media in Chapter two. The

coupling equations of double PCWs and the properties are also discussed in this chapter.

By using the derived equation, we discuss the MI when the Kerr media are added in the PCWs or CROWs in Chapter 3. In the Chapter 4, the soliton propagation criterion and pulse broadening at this criterion is discussed. We found the soliton propagation regions agree with those of the MI. In Chapter 5, we investigate the mechanism which causes the movement of the crossing point of the dispersion relation curves by TBT. In Chapter 6, the coupling equations of asymmetric PC coupler are derived to discuss mode switching phenomena and the simulation results by the PWEM.

Chapter 2 Tight binding theory

2.1 Photonic crystal waveguides and coupled resonant optical waveguides

We consider an optical waveguide which consists of a periodic sequence of identical single-mode defects in the PC with lattice constant aL as shown in Fig. 1. The distance between successive defect points or cavities is a. Assuming the isolated point defect is a single mode with eigenfrequency of ω0 and electric field distribution in triangular and square lattices as shown in Fig. 2, we can express the electric and magnetic fields of each point defect as E(r,t) = E0(r)exp(-iω0t) and H(r,t) = H0(r)exp(-iω0t). Let us assume that the presence of the other defects near a particular site perturbs the total permittivity from ε(r) to ε'(r). The fields in the waveguide E r′( , )t =E r₀′( , )t e^−iω0t and waveguide can be expressed as a superposition of the bound states, i.e.,

0( , )t b t_m( ) 0_m

′ = ∑ ′

E r E and ^{H r}₀′( , )t =

∑

0_m = 0( −ma).

H H r Substituting these equations into Eq. (1) and letting

( ) ( ) ^{i t}0 Here the coupling coefficient Pm is defined as

0 0 0 P₀ is a small shift arising from the presence of the neighboring defects. When we consider a plane wave with wave vector k and frequency ω is incident into this waveguide, the dispersion of the waveguide becomes

Fig. 1 The structures of (a) a PCW, (b) a CROW with one separation rod and (c) a CROW with two separation rods, where a is the length of successive defect points and aL is the lattice constant of a PC.

Fig. 2 The electric field distribution (Ez) of a point defect in square lattice for (a) f = 0.364 c/aL

with reduce rod ( rd = 0.05aL) defect and (b) f = 0.333 c/a with rd=0.1aL. (c) (d) The electric field distribution of the blue dash line in (a) (b).

2.2 The properties of coupling coefficients in PCWs and CROWs

The electric field distribution (Ez) of a single point defect, simulated by the PWEM in the square lattice and triangular lattice with the dielectric constant and radii of dielectric rods being 12, 0.2aL is shown in Fig. 2. The radius (rd) of the defect rods and eigen frequency in square lattice are 0.05a_L and 0.364 c/a_L; those in triangular lattice are 0.1a_L and 0.333 c/aL. The field profile along the (blue) dash line in Fig. 2(c) and (d) has the opposite sign when extending to odd nearest-neighbor rod(s) (E₀(0,0)*E₀(xa,0) < 0, x = 1,3,5,…) and has identical sign when extending to even nearest-neighbor rods [18, 38].

To maintain a single mode propagating in the waveguides, the radius or the refraction index of the rods in the waveguides is reduced therefore Δε is negative in the following discussion. Since the electric field is mainly localized around the dielectric rods of the waveguides, we can use the maximum values to replace the integral values for a simple

estimation of Eq. (2.3). Therefore, P1 is positive in even-separated-rod CROWs, in which E0(0,0)*E0(xa,0) < 0, x = 1,3; P1 is negative in odd-separated-rod CROWs; | P2 | is two orders smaller than | P1 | so P2 is negligible for considering the dispersion relation.

In the PCWs, P₁ and P₃ are positive and P₂ and P₄ are negative; P₃ is two orders of magnitude smaller than P1, and thus only P1 and P2 need to take into consideration when calculating the dispersion relation. From the dispersion relation in Eq. (2.4), the frequency increases as k increases in PCWs and CROWs with even separation rods where P1 is positive, but the frequency decreases as k increases in the CROWs with odd separation rods. The self phase modulation (SPM) strength is

4 with n2 being the Kerr coefficient. Let the plane wave with amplitude φ, propagation wave vector k, and frequency ω in site n as bn = φexp(inka-iωt) being the solution of Eq.

(2.5). The dispersion relation of the nonlinear PCW can be derived as

2 2

0 0 1 2

( )ka c 2 cos( ) 2 cos(2 )c ka c ka | | | | .

ω =ω − − − −γ φ =ω γ φ′− (2.7)

Here, ω’ is the dispersion relation of linear waveguides. The SPM will make the dispersion relation a constant frequency shift in all wave vectors. The positive Kerr

coefficient leads a low frequency shift and vice versa.

Fig. 3 Geometric structures of the photonic crystal waveguide couplers of (a) square lattice and (b) triangular lattice with the lattice constant a. Ps and Qs are the coupling coefficients between defects within a single waveguide. α, and β are the coupling coefficients between waveguides.

2.4 Coupling equations in asymmetric photonic-crystal coupler

We consider an asymmetric coupled PCWs in a PC with the lattice constant a, in which a = aL, are formed by two rows of periodic defect rods partitioned by a perfect row of rods, shown as PCW1 and PCW2 in Fig. 3 for the square and the triangular lattices, respectively. The field distribution of the eigenmode of an isolated (point) defect in each PCW can be written as the product of time-varying and spatial-varying functions, i.e., E10(r,t) = u₀(t)E₁₀(r) in PCW1 and E₂₀(r,t) = v₀(t)E₂₀(r) in PCW2, where u₀(t)=U’exp(-iω1t) and v₀(t)= V’exp(-iω2t), with U’ and V’ being the constant amplitudes of electric fields and ω1 and ω2 the frequencies of localized modes of the point defect in each PCW.

Under the TBT, the evolution equation of the isolated PCW1 can be written as

1 0

and P_m=P_m¹¹, where P_m^ij is the coupling coefficient between the site n of the ith PCW

Let k and ω be the wavevector and its corresponding eigenfrequency of PCW1,₁ respectively, we obtain the dispersion relation of PCW1:

1 1 0

Similarly, the evolution equation and dispersion relation of the isolated PCW2 are shown below:

Due to the field distributions of defect modes being not strongly localized around defects, we shall consider the coupling effect of two asymmetric PCWs up to the second nearest-neighboring defects, with coupling coefficient α=C₀¹²=C₀²¹

and β=C¹²_±1=C_±²¹₁ shown in Fig. 3 for the square and the triangular lattices, respectively.

The coupled equations of asymmetric PCWs are given by [39, 40]:

_{1 0 1 1}

When the stationary solutions of coupled Eqs. (2.13) and (2.14) are taken as un = U0

⎣ ⎦ stands for the eigenvector or field amplitudes in two PCWs. The eigenfrequencies (dispersion relations) and eigenvectors (field amplitudes) of Eqs. (2.15) and (2.16) are

1 2 2 2 The existence of g(ka) makes the eigenstates of the coupler be the linear combination of eigenstates of the single waveguides, leading the EM wave coupled from one waveguide from another. When g = 0, the waveguides will be no longer coupled to each other that means the coupling length is infinite.

In this chapter, we used TBT to derive the coupling equations to describe the electric field propagation in nonlinear or linear single waveguides and linear symmetric or asymmetric PC couplers. In the following chapters, these equations will be used to

further discuss pulse propagation in the PCWs and CROWs with nonlinear media and the EM wave coupling between two waveguides.

Chapter 3 Modulation instability in a single PCW and CROW

3.1 Introduction

A pulse experiences serious dispersion in the PCWs and CROWs [42, 43]; therefore, it would hardly propagate within the waveguides without broadening. There are two ideas to improve the situation of allowing the pulse propagation in the waveguides without broadening. The first method is to design a proper structure to create a linear dispersion curve in the range of operating frequency to provide dispersionless propagation; the other method is to add nonlinear Kerr media to provide soliton propagation [37, 44-46].

However, in the latter case, the criteria of forming a soliton is that the wavevector of the incident wave must be located within the modulation instability (MI) regions [46-48], where the MI refers to a process in which a small perturbation upon a uniform intensity beam would grow exponentially [49]. This phenomenon, which is commonly observed in nonlinear optical fibers [50], will also occur in the nonlinear PCWs and CROWs.

3.2 Modulation instability gain

In Section 2.3, we have derived the DNLS equation to describe the EM wave propagating in PCW or CROWs. Now, considering a small perturbation ν_n(t) superimposed on a plane wave with wave vector and frequency being p and ω, shown as [49]

( )

( ( )) ^{i pna t} ,

n n

b = φ+v t e ^−ω (3.1) we can substitute Eq. (3.1) into Eq. (2.5) in which the 1^st and the 2^nd nearest-neighbor coupling coefficients are considered to get

1 1 1 where q and ω are the wavevector and frequency of the modulation perturbation, V₁ and V₂^* represent small perturbation with perturbation wavevectors of q and – q, and substituting v_n(t) into Eq. (3.2), we obtained the dispersion relation of the perturbation:

( , )p q B A A( γ φ| | )2 would become unstable. The intensity growing rate G of MI, also called the MI gain, is related to the imaginary part of Ω (p, q) by

Because of P₂ ≈ 0 for the CROWs, the coefficient A can be rewritten as

2

4 cos(1 )sin ( / 2)

A= P pa qa , in which the sign of A is determined only by pa and it changes sign at pa = π/2. Here the region of pa (or qa) is defined between 0 and π. For positive (negative) A, γ must also be positive (negative) and γ|φ|² > A > 0 (γ|φ|² < A < 0) to support

MI, which can be easily derived from Eq. (3.7); in other words, P1cos(pa)γ must be positive in MI region. Therefore, the boundary of MI must be located at pa = π/2. In odd-separation-rod CROWs, P₁ is negative, therefore A and γ must be both negative when 0 < pa < π/2 and positive as pa > π/2. However, in even-separation-rod CROWs, P1 is positive, therefore A and γ must be both positive when 0 < pa < π/2 and negative as pa

> π/2, shown in Table 1. When the structure of the waveguide (P1) has been chosen, |A|

increases if q increases at constant P₁ and p. When we plot the gain profile as the graph of G vs. q at a given p and define the gain maximum as the maximal values in the graph, from Eq. (3.7), the gain maximum would be located at A = 0.5γ|φ|² and cut off at A = γ|φ|²

In negative (positive) P₁ for an odd-separation-rod (even-separation-rod) case, the slope of dispersion relation is negative (positive) [51] and the frequency dispersion β₂ defined as d²ω/dk² is negative (positive) when pa < π/2 and positive (negative) for pa >

π/2 from Eq. (2.4). Therefore, for negative β₂ (pa < π/2 for the odd-separation-rod case and pa > π/2 for the even-separation-rod case), the negative γ is needed to support MI and positive γ is needed to support MI for positive β2. In other words, the MI regions of the CROWs in pa can also be decided by simply considering the parameters of β₂ and γ.

In PCWs, P1 is positive and P2, which cannot be neglected, is negative. First, we consider the positive Kerr media having positive n₂ (or γ) so the criterion of the MI is γ|φ|²

> A > 0. From the criterion of A=4 cos(P₁ pa)sin (² qa/ 2) 4 cos(2 )sin ( )+ P₂ pa ² qa > 0, since P₂ is an order of magnitude smaller than P₁, this criterion can be further reduced to cos(pa) > -4| P2 / P1 | cos²(qa/2). Under this circumstance, the MI region is determined not only by pa but also by qa, and pa in the MI region can exceed π/2, unlike in CROWs that the MI boundary for pa is located at π/2 and is independent of qa. From the other criterion: γ|φ|² > A, we found A is dominated by the P1 term as pa is located away from π/2, in this case the MI gain is similar to that in the CROWs with even separation rods.

Contrarily, when pa approaches to π/2, the P₁ term is almost zero and A becomes dominated by the P2 term. In this case, A would not increase as increasing qa. From Eq. (3.7), we knew that the maximum of the gain profile, G(q), is located at A = 0.5γ|φ|² or dA/dq = 0. For the latter case, the peak gain would be smaller than that of the former condition. When 4P2 cos(2pa) < 0.5γ|φ|², there would be two gain maxima at a fixed pa and the gain maxima is located at A = 0.5γ|φ|², but there would be only one gain maximum located at dA/dq = 0 as 4P2 cos(2pa) < 0.5γ|φ|².

On the other hand, in the condition of negative γ, the first criterion is cos(pa) < -4 | P2

/ P₁| cos²(qa/2). We found the MI would happen only when pa > π/a. However, when 0 > cos(pa) > -4|P₂ /P₁|, the MI region is located at the higher q rather than the general case in which the perturbation would have gain at qa = 0⁺. The cutoff gain is also decided by A = γ|φ|².

3.4 Simulation results

We consider a square lattice PC with the dielectric constant and radii of the dielectric rods being 12 and 0.2a_L, where a_L is the lattice constant of the PCs. The radii (r_d) of the defect rods are reduced to be 0.05aL and the Kerr media are introduced around the defects between one separation rod to create the CROW and sequentially to create the PCW.

The structures and dispersion relations of the CROW and PCW in TM polarization (the electric field parallels the rod axis) without Kerr media are shown in Fig. 4, which are simulated by the PWEM.

Fig. 4 The dispersion relations of (a) a CROW with one separation rod and (b) a PCW in square lattices, which are simulated by the plane wave expansion method. The dash red lines are the edges of the band gaps.

Fig. 5 (a) The values of A and (b) the gains and regions of the MI of the CROW with γ|φ|²=0.01 (2π c/aL).

First, the properties of the MI in the CROW would be discussed. The coupling coefficient P₁ is -0.00841 (2πc/a_L), where c is the speed of light in the vacuum. Because P1 is negative, the eigenfrequencies will decrease as increasing k. Figure 5(a) shows A vs. qa with different p. Let A’ = γ|φ0|²－A so that G = 2 AA′ . As aforementioned, the MI region is determined by the condition that A lies between 0 and γ|φ|² and the maximum of G appears when A equals (or is the closest) to 0.5γ|φ|². Figure 5(b) shows G(p,a) with γ|φ|²=0.01 (2πc/a_L). It can be seen that there is no MI gain when pa ≤ 0.5π and only a single gain maximum at given pa in the condition of pa > 0.576π.

In PCWs, the coupling coefficients of P1 and P2 are 0.039 and -0.0047(2πc/a), and ω0- P₀ is 0.3632 (2πc/a). The values of A at a given pa were shown in Fig. 6(a). When pa is small, i.e., in [0, 0.4π], A is dominated by P1 term and A increases as qa increases.

Due to P1 is positive, the properties of MI would be similar to the CROWs with even separation rods that possesses a single gain maximum as the solid curve in Fig. 7(a) for pa

= 0.4π. However, as pa is in (0.4π, 0.6π], A is not simple increasing or decreasing function of qa, shown in Fig. 6(b). At a given pa with positive Kerr media (γ > 0), when the values of A(q) is always smaller than 0.5γ|φ|², e.g., γ|φ|² = 0.01 (2πc/a) and pa = 0.6π, there would be a maximal gain as the solid curve in Fig. 7(d). However, when A(q) is larger than 0.5γ|φ|², e.g., γ|φ|²=0.01 (2πc/a) and pa = 0.49π and 0.55π, there would have 2 gain maxima, solid curves shown in Figs. 7(b) and (c). And the MI region with positive γ can extend to pa = 0.6π, as shown in Fig. 6(c). On the other hand, the MI region with negative Kerr media is shown in Fig. 6(d) which is located within π/2 < pa < π but having the MI region located at high qa as pa close to π/2.

Fig. 6 (a) (b)The values of A in the PCW. The region and gains of MI with (c) positive Kerr media (γ |φ |²=0.01*2πc/a) and (d) negative Kerr media (γ|φ |²=-0.01*2πc/a).

Next, we would use the 4^th order Runge-Kutta method to simulate the evolution of the perturbation. A plane wave with 10% initial sinusoidal perturbation is used as the input source in a square-array PCW with γ|φ|² = 0.01 (2πc/a). The perturbation will grow exponentially in the MI region to become a discrete soliton before it splits, as shown in Fig. 8(a), but the perturbation would never grow outside the MI region Fig. 8(b). We plot the gain coefficients with square dots in Fig. 7 by evaluating the growing rate by the Runge-Kutta method then compare with gain profiles (solid curves) calculated by using Eq. (3.7). The results show a quite good agreement.

Fig. 7 The MI gain profiles gotten by analytic solution and the simulation by 4^th order Runge-Kutta method in different qa with γ|φ|² = 0.01 (2πc/a).

Fig. 8 The evolution of the perturbation in the PCW with (a) pa=0.4π and qa=0.1π (b) pa=0.6π and qa=0.1π.

3.5 Summary

We have successfully used the TBT to investigate the MI in both CROWs and PCWs by considering growth of a small perturbation superimposed on a plane wave. The

number of separation rods in the CROWs would decide the signs of the nearest-neighbor coupling coefficients (P₁) and the next nearest-neighbor coefficient (P₂) can be neglected because it is more than 2 orders of magnitude smaller than P1. This leads to positive

number of separation rods in the CROWs would decide the signs of the nearest-neighbor coupling coefficients (P₁) and the next nearest-neighbor coefficient (P₂) can be neglected because it is more than 2 orders of magnitude smaller than P1. This leads to positive