Analytical and Numerical Analyses of Multilayer Photonic Metamaterial Slab Optical Waveguide Structures with Kerr-Type Nonlinear Cladding and Substrate

Abstract

In this paper, we propose the analytical and numerical analyses of multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate. The multiple-quantum-well (MQW) photonic metamaterial optical waveguide structure with Kerr-type nonlinear cladding and substrate was also analyzed. We can use the proposed method to study the multilayer optical metamaterial slab optical waveguide structure with the linear cladding and substrate.

1. Introduction

Photonic metamaterials are electromagnetic metamaterials that interact with light. Some characteristics of metamaterials are contrary to traditional optical materials because of the negative indices of refraction. Photonic materials are, therefore, also called double-negative materials (DNG, $\epsilon <0, \mu <0$) and left-handed materials (LHM). Metamaterials were proposed in 1968 [1,2,3,4]; applications of metamaterials, such as phase holography, phase modulators, invisibility cloaks, super lens, optical sensors, perfect absorbers, and antennas have also been proposed [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. In addition, several numerical and experimental papers on metamaterials have been proposed [28,29,30,31,32,33,34].

Traditional Kerr-type nonlinear optical waveguides are attractive in various signal- processing applications. Many Kerr-type nonlinear waveguide-based all-optical devices have been reported. Kerr-type nonlinear media have spatial optical soliton phenomena [35,36,37]. Multilayer waveguide structures with Kerr-type nonlinear media can be designed as all-optical devices [38,39,40].

In most papers, some features of the LHM are only discussed at single interface [41], but the relationship of the LHM and the nonlinear DSP medium in three-layer or multilayer waveguide is not discussed. Some previous works studied the transverse electromagnetic polarized modes [42,43], but did not discuss the photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate. Other papers reported the different kinds of material in film, cladding, and substrate; for example, the three-layer planar structure of NZPIM waveguide [44], air cladding and LHM film [45], and circular waveguide of metamaterial and linear cladding [46]. The analytical and numerical analyses of multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate are proposed in this paper. The theoretical study of the multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate was not previously reported. The proposed general method was used to investigate the TE-polarized waves in the proposed photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate.

2. Analysis

The conventional transfer matrix method cannot be used to analyze the case of the multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate, proposed in this manuscript. We use our own numerical method to analyze the proposed structure. By properly matching the boundary conditions step by step, we can obtain all the numerical results. When the cladding and substrate of the proposed photonic metamaterial planar optical waveguide structures are Kerr-type nonlinear media, the analyses processes become very complicated and difficult. The difficulty lies in the derivation of the mathematical model and the verification of numerical analyses and simulations. The proposed analytical formulas can be used to calculate the transverse electric field function in each layer of the proposed optical waveguide structures. The analytical solutions are very complicated. It is very difficult to obtain the exact solutions. To the best of our knowledge, available electromagnetic simulation software, such as FDTD, FEM, and BPM, cannot be used to analyze the case of the proposed multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate. In this section, optical waveguide theory was used to analyze the proposed multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate, as shown in Figure 1.

For simplicity, the TE polarized waves were chosen to propagate in the z-direction of the proposed nonlinear photonic metamaterial optical waveguide structures. The wave equation can be simplified to

$${\nabla }^2{E}_{yj}=\frac{n_j}{c^2}\frac{{\partial }^2{E}_{yj}}{\partial t^2},\hspace{1em}j=f,c,s,i$$

(1)

The solutions of Equation (1) can be expressed as the form

$$\begin{array}{c}{E}_{yj}(x,z,t)=E_j(x)\exp [i(\omega t-\beta k_{o}z)],\\ j=f,c,s,i.\end{array}$$

(2)

The parameters $\omega$, $k_0$, and $\beta$ are denoted as the angular frequency, the wave number in the free space, and the effective refractive index, respectively. The index of refraction of the Kerr-type nonlinear cladding and substrate can be expressed as [47,48,49,50]:

$$n_j^2=n_{jo}^2+\alpha {\left|E(x)\right|}^2\hspace{1em}j=c,s$$

(3)

where the parameters α and n_j0 = (ε_j μ_j)^1/2 are denoted as the nonlinear coefficient and the linear refractive index of the nonlinear medium, respectively.

For the analyses, we consider the cases (ε_j μ_j)^1/2 < β < (ε_f μ_f)^1/2 and β > (ε_f μ_f)^1/2 in different modes. For case 1, (ε_j μ_j)^1/2 < β < (ε_f μ_f)^1/2, the solution of the TE wave in each layer is shown as follows:

$$E_c(x)=E_c\left\{\cosh [k_o{q}_c(x-\rho )]+B_c\sinh [k_o{q}_c(x-\rho )]\right\}$$

(4)

$$E_{fn}(x)=A_f(n)\cos [k_0{q}_f(x-{\varphi }_f(n))]$$

(5)

$$E_{in}(x)=A_i(n)\cosh [k_0{q}_i(x-{\varphi }_i(n))]$$

(6)

$$E_s(x)=E_s\left\{\cosh [k_o{q}_s(x+\rho )]-B_s\sinh [k_o{q}_s(x+\rho )]\right\}$$

(7)

For case 2, β > (ε_f μ_f)^1/2, the solution of the TE wave in each layer is shown as follows:

$$E_c(x)=E_c\left\{\cosh [k_o{q}_c(x-\rho )]+B_c\sinh [k_o{q}_c(x-\rho )]\right\}$$

(8)

$$E_{fn}(x)=A_f(n)\sinh [k_0{Q}_f(x-{\varphi }_f(n))]$$

(9)

$$E_{in}(x)=A_i(n)\cosh [k_0{q}_i(x-{\varphi }_i(n))]$$

(10)

$$E_s(x)=E_s\left\{\cosh [k_o{q}_s(x+\rho )]-B_s\sinh [k_o{q}_s(x+\rho )]\right\}$$

(11)

E_fn(x), E_c(x), E_s(x), and E_in(x), are the TE electrical fields in the metamaterial guiding layers, the cladding layer, the substrate layer, and the interaction layer, respectively. The expressions of the parameters $\rho$, $B_c$, $B_s$, $q_c$, $q_i$, $q_s$, $Q_f$ and $q_f$ are shown as follows:

$$q_j=\sqrt{{\beta }^2-{\epsilon }_j{\mu }_j},j=c,i,s$$

(12)

$$Q_f=\sqrt{{\beta }^2-{\epsilon }_f{\mu }_f},\beta >\left({\epsilon }_f{\mu }_f\right){}^{1/2}$$

(13)

$$q_f=\sqrt{{\epsilon }_f{\mu }_f-{\beta }^2},\left({\epsilon }_j{\mu }_j\right){}^{1/2}<\beta <\left({\epsilon }_f{\mu }_f\right){}^{1/2}$$

(14)

$$\rho =\frac{n+1}{2}{d}_f+\frac{n}{2}{d}_i$$

(15)

$$B_k=\sqrt{1-\frac{{\alpha }_k{E}_k}{2q_k^2}}\hspace{1em}k=c,s$$

(16)

The parameters Ec and Es are denoted as the amplitudes of the electric fields at the upper boundary and the lower boundary of the metamaterial guiding film, respectively. To match the boundary conditions, the transcendental equations are shown as follows:

$$\frac{q_f{q}_s\frac{{\mu }_f}{{\mu }_s}{B}_s+q_f{q}_i\frac{{\mu }_f}{{\mu }_i}\tanh {\psi }_i}{q_f{}^2-q_s{q}_i\frac{{\mu }_f}{{\mu }_s}\frac{{\mu }_f}{{\mu }_i}{B}_s\tanh {\psi }_i}=\tan (k_0{q}_f{d}_f),({\epsilon }_j{\mu }_j){}^{1/2}<\mathsf{\beta }<({\epsilon }_f{\mu }_f){}^{1/2}$$

(17)

$$\frac{-Q_f(q_s\frac{{\mu }_f}{{\mu }_s}{B}_s+q_i\frac{{\mu }_f}{{\mu }_i}\tanh {\psi }_i)}{Q_f{}^2+q_s{q}_i\frac{{\mu }_f}{{\mu }_s}\frac{{\mu }_f}{{\mu }_i}{B}_s\tanh {\psi }_i}=\tanh (k_0{Q}_f{d}_f),\mathsf{\beta }>({\epsilon }_f{\mu }_f){}^{1/2}$$

(18)

where

$${\psi }_i=k_0{q}_i[\frac{n}{2}{d}_i+\frac{n-1}{2}{d}_f-{\varphi }_i(1)]$$

The general dispersion Equations (17) and (18) can be solved numerically on a computer. We developed a MATLAB software to calculate the numerical results of the proposed multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate. The proposed general method can also be degenerated to analyze some special cases. When ${\alpha }_c=0$ and ${\alpha }_s=0$, the proposed waveguide structures with Kerr-type nonlinear cladding and substrate can be degenerated into the linear metamaterial waveguide structures. The degenerated equation of the electrical field in each layer and the degenerated transcendental equations are the same as those shown in papers [51,52] and are provided in Appendix A.

The power density can be described by Poynting vector. The Poynting vector is defined as shown below:

$$P=\frac{1}{2}\Re e(E_y\times H^{\ast }{}_x)$$

(19)

The parameters E_y and H_x are denoted the electric field and the magnetic field in the y-direction and in the x-direction, respectively. E_y is defined in Equation (2) and H_x can be defined as:

$$H_x=-i\beta k_0\frac{1}{i\omega {\mu }_0{\mu }_r}E(x)\exp [i(\omega t-\beta k_{0}z)]$$

(20)

By substituting Equations (2) and (20) into Equation (19), the average power density can be written as:

$$P_{av}={\displaystyle {\int }_{-\infty }^{\infty }\frac{\beta }{2}c{\epsilon }_0\frac{1}{{\mu }_r}{E}_y{}^2(x)}$$

(21)

We can solve the transcendental Equations (17) and (18) numerically. When the constant $\beta$ and Es have been obtained, the parameters $q_c$, $q_i$, $q_f$, $Q_f$, $q_s$, $A_f(n)$, $A_i(n)$, $E_c$, ${\varphi }_f(n)$ and ${\varphi }_i(n)$ are also obtained, as shown in the Appendix B.

3. Numerical Analysis Results

We will propose some numerical examples to prove the accuracy of the proposed general method. When n = 0 and 1, the degenerated formulas can be used to analyze the three-layer and the five-layer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate, respectively. The numerical results are shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

For the case of the three-layer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate, the dispersion curve is shown in Figure 2 with the parameters: $u_f=-5$, d_f = 2 µm, ε_fμ_f = 1.57², ε_sμ_s = ε_cμ_c = 1.55², α_c = α_c = 6.3786 µm²/V², and λ = 1.3 µm. When the guided power increases, the effective refractive index of the Kerr-type nonlinear cladding and substrate will also increase, due to the Kerr-type nonlinear effect. The electric field distributions of the case of the photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate is shown in Figure 3, with respect to the six points A, B, C, D, E, and F, which are shown in Figure 2.

Figure 4 shows the dispersion curve of the three-layer metamaterial waveguide with Kerr-type nonlinear cladding and substrate with the parameters: $u_f=-5$, d_f = 5 µm, ε_fμ_f = 1.57², ε_sμ_s = ε_cμ_c = 1.55², α_c = α_c = 6.3786 µm²/V², and λ = 1.3 µm. There are two modes in the proposed three-layer metamaterial waveguide structures with Kerr-type nonlinear cladding and substrate. Points A, B, and C on the same dispersion curve are mode 1. Points D, E, and F on the same dispersion curve are mode 2. The electric field distributions in the case of the photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate are shown in Figure 5, with respect to the six points, A, B, C, D, E, and F, which are shown in Figure 4.

For the case of the five-layer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate, the dispersion curve is shown in Figure 6, with the parameters: $u_f=-5$, d_f = 2 µm, d_i = 6 µm, ε_fμ_f = 1.57², ε_s μ_s = ε_cμ_c = ε_iμ_i = 1.55², α_c = α_c = 6.3786 µm²/V², and λ = 1.3 µm. When the guided power increases, the effective refractive index of the Kerr-type nonlinear cladding and the substrate will also increase, due to the Kerr-type nonlinear effect. The electric field distributions of the case of the photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate are shown in Figure 7, with respect to the eight points, A, B, C, D, E, F, G, and H, which are shown in Figure 6.

Figure 8 shows the dispersion curve of the five-layer metamaterial waveguide with Kerr-type nonlinear cladding and substrate with the parameters: $u_f=-5$, d_f = 5 µm, d_i = 6 µm, ε_fμ_f = 1.57², ε_sμ_s = ε_cμ_c = ε_iμ_i = 1.55², α_c = α_c = 6.3786 µm²/V², and λ = 1.3 µm. There are two modes in the proposed five-layer metamaterial waveguide structures with Kerr-type nonlinear cladding and substrate at d_f = d₁ = d₃ = 5 µm and d_i = d₂ = 6 µm. Points A, B, and C on the same dispersion curve are mode 1. Points D, E, and F on the same dispersion curve are mode 2. The electric field distributions of the case of the photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate are shown in Figure 9, with respect to the six points, A, B, C, D, E, and F, which are shown in Figure 8.

Here, we show some special numerical results to compare to those of a previous paper [43] to prove the accuracy of the proposed results shown in this manuscript. For the case of the parameters: $u_f=-5$, d_f = 5 µm, ε_fμ_f = 1.57², ε_sμ_s = ε_cμ_c = 1.55², α_c = α_c = 6.3786 µm²/V², and λ = 1.3 µm, $\beta =1.56541$, the TE-polarized wave of the proposed Kerr-type nonlinear metamaterial is shown in Figure 10. Also, for the case of the parameters: $u_f=-5$, d_f = 5 µm, ε_fμ_f = 1.57², ε_sμ_s = ε_cμ_c = 1.55², α_c = α_c = 6.3786 µm²/V², and λ = 1.3 µm, $\beta =1.56601$, the TE-polarized wave of the proposed Kerr-type nonlinear metamaterial is shown in Figure 11. As can be seen from the numerical results shown in Figure 10 and Figure 11, the patterns of the TE-polarized waves at the linear and nonlinear interfaces are similar to those shown in Figure 3 in paper [43].

We also use the proposed results shown in the previous sections to analyze the MQW photonic metamaterial optical waveguide structure with Kerr-type nonlinear cladding and substrate. The numerical results are shown in Figure 12. In this case, the TE-polarized waves of the proposed nonlinear multilayer metamaterial waveguide with MQW structure for the parameters: W = 2 µm, $u_f=-5$, $(n+1)d_f+n{d}_i=W$, ε_fμ_f = 1.57², ε_sμ_s = ε_cμ_c = ε_iμ_i = 1.55², α_c = α_c = 6.3786 µm²/V², and λ = 1.3 µm, and n = 7, 10, and 15, respectively. Since the thicknesses of each Kerr-type nonlinear metamaterial guiding layer and the linear interaction layers are very thin, the number of layers only has a small influence on the TE-polarized waves. In this case, it resembles a conventional MQW waveguide structure.

When the parameters ${\alpha }_c=0$ and ${\alpha }_s=0$, the numerical results shown in Figure 13 and Figure 14 are similar to those of the linear multilayer metamaterial optical waveguide structures. For the case of the parameters: ${\alpha }_c=0$, ${\alpha }_s=0$, d_f = 0.198 µm, $u_f=-5$, ${\epsilon }_f{\mu }_f=2.23$, ${\epsilon }_s{\mu }_s={\epsilon }_c{\mu }_c=1$, and λ = 1.55 µm, the TE-polarized wave of the proposed Kerr-type nonlinear metamaterial is shown in Figure 13. Also, for the case of the parameters: ${\alpha }_c=0$, ${\alpha }_s=0$ d_f = 0.593 µm, $u_f=-5$, ${\epsilon }_f{\mu }_f=2.23$, ${\epsilon }_s{\mu }_s={\epsilon }_c{\mu }_c=1$, and λ = 1.55 µm. The TE-polarized wave of the proposed Kerr-type nonlinear metamaterial is shown in Figure 14. Similar to the numerical results shown in Figure 13 and Figure 14, the patterns of the TE-polarized waves are the same as those of Figure 5 shown in paper [51], and Figure 3 shown in paper [52]. As the numerical results above show, we can confirm that the analyses proposed in this manuscript are correct.

4. Conclusions

In this paper, we have proposed the analytical and numerical analyses of multilayer photonic metamaterial slab optical waveguide structures with Kerr-type nonlinear cladding and substrate. We show some special numerical results to compare with those of a previous paper to prove the accuracy of the proposed results shown in this manuscript. We also use the proposed results shown in the previous sections to analyze the MQW photonic metamaterial optical waveguide structure with Kerr-type nonlinear cladding and substrate. When the values of the nonlinear coefficients of the cladding and the substrate are zero, the proposed general method can be degenerated to analyze the linear multilayer metamaterial optical waveguide structures. The numerical results are similar to those of the linear multilayer metamaterial optical waveguide structures. As the numerical results above demonstrate, we can confirm that the analyses proposed in this manuscript are correct.