1. Introduction

Photonic crystals (PCs) are artificial structures in which the refractive index is periodically distributed at a length scale comparable to the operating wavelength [1, 2]. A photonic crystal waveguide (PCW) can be created by sequentially changing the radii or dielectric constant of the dielectric rods or changing the radii of periodic air holes in a dielectric slab; on the other hand, the coupled resonator optical waveguide (CROW) is created by arranging the cavities, made of point defects, periodically. The electromagnetic (EM) wave can propagate in these channels, PCWs or CROWs, with a very low loss even through a sharp bend [3-5]. However, a pulse experiences serious dispersion in the PCWs and CROWs [6, 7]; 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; the second method is to add nonlinear Kerr media to provide solitons propagation [8-11]. 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 [12-14], where the MI refers to a process in which small perturbations upon a uniform intensity beam would grow exponentially [14]. This phenomenon, which is commonly observed in nonlinear optical fibers [15], will also occur in the nonlinear PCWs and CROWs.

Mathematical models of these nonlinear systems often lead to the discrete or continuum-discrete evolution equations such as nonlinear Schrödinger (NLS), sine-Gordon, Klein-Gordon, Korteweg-de Vries and Kadomtsev-Petviashvili equations [13]. In CROWs, the amplitudes of the electric field evolution in the cavities or point defects can be expressed as a discrete NLS equation by using the tight binding theory (TBT) [10, 16, 17] in which the field distribution (or wave function) of an individual cavity is localized at this point defect; thus the coupling between two next nearest-neighbor cavities can be neglected due to long distance. By solving the discrete NLS equation, spatiotemporal discrete solitons can propagate undistorted along a series of coupled resonators or defects by balancing of the effects of discrete lattice dispersion with material nonlinearity [10]. However, it is still lack of the criteria for solitons propagation in different structures of CROWs, e.g., different numbers of separation rods between two cavities with positive or negative Kerr media. Moreover, in the PCWs the defect rods are so close that the next nearest-neighbor coupling cannot be neglected [18] and there are rare reports on pulse propagation in nonlinear PCWs. Therefore, it is needed to take the advanced discussion about different kinds of CROWs and to derive the extended discrete nonlinear Schrödinger (EDNLS) evolution equation for describing the nonlinear properties in the PCWs.

In this paper, we first use the TBT to describe the EDNLS equation. By considering a small perturbation superimposed on the plane wave solution, the gain of the perturbation which causes the MI can also be derived. Second, by evaluating the coupling coefficients, we shall discuss the regions and gain coefficients of MI in both CROWs and PCWs. In order to verify the correctness of our equations, a CROW and a PCW with square lattices were proposed to calculate the gain coefficients by using the forth-order Runge-Kutta scheme. The simulated results coincide with our analytic analyses.

2. Theory

We consider an optical waveguide which consists of a periodic sequence of identical single-mode defects in the PCs with lattice constant a_L. The distance between successive defect points or cavities is a, and the Kerr media, in which the refraction index is proportional to the intensity of the incident wave, is put in the defect region, shown in Fig. 1. Assuming the isolated point defect is a single mode with eigenfrequency of ω₀, we can express the mode fields of each point defect as E(r,t) = E ₀(r)exp(-iω₀t) and H(r,t) = H ₀(r)exp(-iω₀t). The electric field E′₀(r,t) and magnetic field H′₀(r,t) of the waveguide can be expressed as a superposition of the bound states, i.e., E′₀(r,t) = ∑b_m(t)E _0m and H′₀(r,t) = ∑b_m(t)H _0m, where E _0m = E ₀(r-ma) and H _0m = H ₀(r-ma).

 

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 a_L is the lattice constant of a PC.

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Under the tight-binding approximation, we consider the couplings up to the next nearest-neighbor defects and obtain the EDNLS equation as [18]

Here the linear coupling coefficient c_m is defined as [10]

with Δε(r) = ε′(r) − ε(r) being the difference of dielectric constants of the waveguide (ε′(r)) and the point-defected PC (ε(r)) and c₀ representing a small shift in the eigenfrequency ω₀ that arises from present of the neighbor defects or cavities. The self-phase modulation strength γ is given by

with n₂ being the Kerr coefficient. Let the plane wave with amplitude ϕ₀, propagation wavevector p, and frequency ω in site n as b_n = ϕ₀ exp(inpa-iωt) be the solution of Eq. (1). The dispersion relation of the nonlinear PCW can be derived as

Considering a small perturbation v _n(t) superimposed on a plane wave, shown as [14]

we can substitute Eq. (5) into Eq. (1) to get

Taking v _n(t) as this form [14]

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. Substituting v_n(t) into Eq.(6), we osbtioned the dispersion relation of the perturbation:

where $A=4c_1\cos \left(\mathit{pa}\right){\sin }^2\left(\frac{\mathit{qa}}{2}\right)+4c_2\cos \left(2\mathit{pa}\right){\sin }^2\left(\mathit{qa}\right)$ and B = 2c ₁ sin(pa)cos(qa) + 2c ₂ sin(2pa)cos(2qa). If the dispersion relation Ω(p,q) is complex as A(A - γ|ϕ₀|²) < 0, the perturbation field 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), i.e.,

 

Fig. 2. (a) The electric field distribution (E_z) of point defect mode simulated by the plane wave expansion method in the square lattice with the dielectric constant, radii of dielectric rods and the radius (r_d) of the defect rods being 12, 0.2a_L and 0.05a_L for frequency f = 0.364 c/a_L. (b) The field distribution of the red dash line in (a).

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3. Analyses and discussion

In this section, we will further discuss about the MI regions and gains in both CROWs and PCWs made of point defects, shown in Fig. 1. The electric field distribution (E_z) of a single point defect, simulated by the plane wave expansion method in the square lattice with the dielectric constant, radii of dielectric rods and the radius (r_d) of the defect rods being 12, 0.2a_L and 0.05a_L for frequency f = 0.364 c/a_L is shown in Fig. 2. And the field profile along the red dash line in Fig. 2(a) is plotted in Fig. 2(b), it has the opposite sign when it extends to the nearest-neighbor defects for the PCW (E ₀(0,0)^* E ₀(a_L,0) < 0) and the CROW (E ₀(0,0)^* E ₀(3a_L,0) < 0) with even (2) separation rods. To maintain a single mode propagating in the waveguides, the radii 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 in Eq. (2). Therefore, c₁ is positive in even-separated-rod CROWs [16]. However, c₂ would be two orders of magnitude smaller than c₁ so that we considered only the nearest-neighbor coupling in the CROWs and let c₂ ≈ 0 [10, 18]. On the other hand, E(0,0)^* E(2a_L,0) is positive in the odd-separation-rod (1) CROWs so c₁ would be negative and c₂ ≈ 0.

Because of c₂ ≈ 0 for the CROWs, the coefficient A can be rewritten as A = 4c ₁ cos(pa)sin² (qa/2), 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|ϕ₀|² > A > 0(γ|ϕ₀|² < A < 0) to support MI, which can be easily derived by Eq. (9); in other words, c₁cos(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, c₁ 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, c₁ 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 (c₁) has been chosen, |A| increases if q increases at constant c₁ and p. When we plot the gain profile as the graph of G vs. q at a given p and defined the gain maximum as the maximal values in the graph, from Eq. (9), the gain maximum would be located at A = 0.5γ|ϕ₀|² and cut off at A = γ|ϕ₀|² when 4|c ₁ cos(pa)|> 0.5 |γ||ϕ ₀|²; otherwise, the gain maximum would be located at qa = π.

Table 1. MI regions of CROWs

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In negative (positive) c₁ for an odd-separation-rod (even-separation-rod) case, the dispersion relation slop is negative (positive) [19] and the frequency dispersion D defined as d²ω/dk² is negative (positive) when pa < π/2 and positive (negative) for pa > π/2 from Eq. (4). Therefore, for negative D (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 D. In other words, the MI regions of the CROWs in pa can also be decided by simply considering the parameters of D and γ.

On the other hand, E _n E _n+1 or E ₀(0,0)^* E ₀(a_L,0) < 0 and E _n E _n+2 > 0 in PCWs with a_L = a, therefore, c₁ is positive and c₂, 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|ϕ₀|² > A > 0. From the criterion of A = 4c ₁cos(pa)sin²(qa/2) + 4c ₂cos(2pa)sin²(qa) > 0, since c₂ is an order of magnitude smaller than c₁, this criterion can be further reduced to cos(pa) > -4|c₂/c₁|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|ϕ₀|² > A, we found A is dominated by the c₁ 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 c₁ term is almost zero and A becomes dominated by the c₂ term. In this case, A would not increase as increasing qa. From Eq. (9), we knew that the maximum of the gain profile, G(q), is located at A = 0.5γ|ϕ₀|² or dA/dq = 0. For the latter case, the peak gain would be smaller than that of the former condition. When 4c₂ cos(2pa) < 0.5γ|ϕ₀|², there would be two gain maxima at a fixed pa and the gain maxima is located at A = 0.5γ|ϕ₀|², but there would be only one gain maximum located at dA/dq = 0 as 4c₂ cos(2pa) < 0.5γ|ϕ₀|².

On the other hand, in the condition of negative γ, the first criterion is cos(pa) < -4|c₂/c₁|• cos²(qa/2). We found the MI would happen only when pa > π/a. However, when 0 > cos(pa) > -4|c₂/c₁|, 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|ϕ₀|².

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 0.05a_L 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. 3, which are simulated by the plane wave expansion method.

 

Fig. 3. 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.

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Fig. 4. (a) The values of A and (b) the gains and regions of the MI of the CROW with γ|ϕ₀|²=0.01 (2πc/a_L).

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First, the properties of the MI in the CROW would be discussed. The coupling coefficient c₁ is -0.00841 (2πc/a_L), where c is the speed of light in the vacuum. Because c₁ is negative, the eigenfrequencies will decrease as increasing k. Figure 4(a) shows A vs. qa with different p. Let A’ be γ|ϕ₀|² - A so that $\mathrm{G}=2\sqrt{\mathrm{AA}\prime }$. As aforementioned, the MI region is determined by the condition that A lies between 0 and γ|ϕ₀|² and the maximum of G appears when A equals (or is the closest) to 0.5γ|ϕ₀|². Figure 4(b) shows G(p,a) with γ|ϕ₀|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 c₁ and c₂ are 0.039 and -0.0047(2πc/a), and ω₀-Δω is 0.3632 (2πc/a). The values of A at a given pa were shown in Fig. 5(a). When pa is small, i.e., in [0, 0.4π], A is dominated by c₁ term and A increases as qa increases. Due to c₁ 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. 6(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. 5(b). At a given pa with positive Kerr media (γ > 0), when the values of A(q) is always smaller than 0.5γ|ϕ₀|², e.g., γ|ϕ₀|² = 0.01 (2πc/a) and pa = 0.6π, there would be a maximal gain as the solid curve in Fig. 6(d). However, when A(q) is larger than 0.5γ|ϕ₀|², e.g., γ|ϕ₀|²=0.01 (2πc/a) and pa = 0.49π and 0.55π, there would have 2 gain maxima, solid curves shown in Fig. 6(b) and (c). And the MI region with positive γ can extend to pa = 0.6π, as shown in Fig. 5(c). On the other hand, the MI region with negative Kerr media is shown in Fig. 5(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. 5. (a) (b)The values of A in the PCW. The region and gains of MI with (c) positive Kerr media (γ|ϕ₀|²=0.01^*2πc/a) and (d) negative Kerr media (γ|ϕ₀|²=-0.01^*2πc/a).

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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|ϕ₀|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. 7(a), but the perturbation would never grow outside the MI region in Fig. 7(b). We plot the gain coefficients with square dots in Fig. 6 by evaluating the growing rate by the Runge-Kutta method and then compare with gain profiles (solid curves) calculated by using Eq. (9). The results show a quite good agreement.

 

Fig. 6. The MI gain profiles gotten by analytic solution and the simulation by 4^th order Runge-Kutta method in different qa with γ|ϕ₀|² = 0.01 (2πc/a).

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Fig. 7. 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.

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5. Conclusion

We have successfully used the TBT to investigate 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 (c₁) and the next nearest-neighbor coefficient (c₂) can be neglected because it is more than 2 orders of magnitude smaller than c₁. This leads to positive dispersion for positive coupling coefficient and vice versa. Although the signs of the coupling coefficient could be different, the criteria: c₁cos(pa)γ > 0 for obtaining modulation instability is the same for incident plane wave of wavevector p. Therefore, the MI region can only be located in either pa < π/2 or pa > π/2 with only one gain maximum. In the air-defect PCWs, c₁ is positive and c₂, which is no longer negligible, is negative. It makes the MI gain of positive Kerr media located at low wavevectors in the first Brilluoin zone and vice versus. The boundary of gain region of pa is not exactly at π/2 due to the MI is mainly dominated by c₂ term as pa approaches π/2 and there could exist two gain maxima. Furthermore, the numerical simulation using the 4^th order Runge-Kutta method reveals exponentially growing perturbation intensity as it propagates and the growing rate matches with the gain coefficient of MI in the analytic solution.