Analyzing multilayer optical waveguide with all nonlinear layers

Chih-Wen Kuo; Shih-Yuan Chen; Mao-Hsiung Chen; Chih-Fu Chang; Yaw-Dong Wu

doi:10.1364/OE.15.002499

1. Introduction

The nonlinear optical waveguide device is a potential key component in the applications of optical signal processing and communication systems [1–3]. In the past, a number of papers have dealt with the propagation characteristics of TE-polarized waves guided by a three-layer lossless linear, or nonlinear optical waveguide structures [4–19]. Considerable progress has been made by considering only Kerr nonlinearities due to the third-order nonlinearity [20]. Most of the interest has been focused on optical waves guided by a single planar waveguide with linear film bounded by nonlinear medium [4–9] or nonlinear film bounded by linear medium [10–15]. Stegeman et al. [7–8] presented a large number of calculations for three-layer optical planar waveguide and provided many rigorous curves of dispersion relation. Boardman et al. [10–11] have well defined the behavior of TE nonlinear waves in an optically nonlinear film by using the Jacobi elliptic function and boundary field approach. Most studies for optical waveguide with nonlinear guiding film are reviewed by this model. Okafuji [15] expressed for the TE₀ waves supported by a three-layer nonlinear waveguide with a self-focusing nonlinear substrate and a self-defocusing nonlinear film. Sammut et al. [16–17] developed the propagating character in uniform nonlinear medium throughout the cladding, substrate, and guided film. Schürmann [18] used Weierstrass’ elliptic functions to express the field equations in a nonlinear three-layer structure. Shin et al. [21] presented the five-layer optical waveguide with nonlinear cladding and substrate. Sakakibara [22] and Jeong [23] calculated the field profiles and dispersion relations of nonlinear wave in a slab waveguide with two nonlinear films, respectively. The interests in multilayer optical waveguides, which has been growing steadily in recent years, stem from their potential use for ultra-fast all-optical signal processing and optical computing systems. The analysis of TE wave propagating in the multilayer systems has attracted most attention. Radic et al. [24–26] had proceeded serial rigorous studies in nonuniform distributed feedback structures and presented a general transfer matrix method to analyze the optical wave propagating in this structure. Trutschel et al. [27] developed nonlinear matrix formalism and calculated the field profiles and the propagation constant of the nonlinear guide wave in a multilayer system. Ogusu [28] presented a semi-analytical method for calculated the dispersion relations for stationary nonlinear TE waves guided by general multilayer waveguides with Kerr-like permittivity. She et al. [29] analyzed the nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding, both by exaat method and by the method of root mean square approximation. Wu et al. had developed accurate calculations for multilayer systems, as analyzing the multilayer linear planar waveguide [30], multilayer planar waveguide with nonlinear cladding and substrate [31], multilayer planar waveguide with a localized arbitrary nonlinear guiding film [32], and multilayer planar waveguide with all nonlinear guiding film [33].

The analytical formulas of multilayer systems have potential applications in integrated optics. The multiple quantum well waveguide (MQW) structures can be approximated as a sort of multilayer waveguide [28, 31]. The nonlinear graded-index waveguides can be segmented a number of step refractive index layers [31, 34]. These analyses have also been applied to design the multibranch optical waveguide which can be operated as switches [35], logic gates [36], and wavelength auto-router [37]. In this paper, we propose a general method for analyzing the multilayer optical waveguide structure with all nonlinear layers. The general method can also be degenerated into other special cases for analyzing multilayer nonlinear optical waveguide. This method can be used to predict the propagation characteristics in three-layers, seven-layers, or more. It is useful to design all-optical devices. The dispersion relation curves and electric field profiles for multilayer optical waveguides with all nonlinear layers can be described and compared with the finite difference beam propagation method (FD-BPM) [38]. The analytical and numerical results show excellent agreement.

2. Method and analysis

2.1 Analytic formulation

In this section, we use the modal theory [39–40] to derive the general formulas that can be used to analyze the multilayer optical waveguide with all nonlinear layers, as shown in Fig. 1.

The multilayer optical waveguide structure is composed of the guiding films ( $\frac{M - 1}{2}$ layers), the interaction layers ( $\frac{M - 3}{2}$ layers), the cladding layer, and the substrate layer. The total number of layers is M (M=3,5,7,…). The d_i and n_i are used to denote the width and the refractive index of the i-th layer, respectively. The cladding and substrate layers are assumed to extend to infinity in the +x and -x directions, respectively. The major significance of this assumption is that there are no reflections in the x direction to be concerned with, expect for those occurring at interfaces.

For the simplicity, we consider the transverse electric polarized waves propagating along the z direction. The wave equation in the i-th layer can be written as

\nabla^{2} ψ_{yi} = \frac{{n_{i}}^{2}}{c^{2}} \frac{\partial^{2} ψ_{yi}}{\partial t^{2}}, i = 0,1,2, \dots ., M - 1

with solutions of the form

ψ_{yi} (x, z, t) = E_{i} (x) \exp [j (ωt - n_{e} k_{0} z)], x_{i} \leq x \leq x_{i + 1}

where k₀ is the wave number in the free space, and n_e is the effective refractive index, $ω = \frac{c}{k_{0}}$ . For analyzing conveniently, we choose only self-focusing nonlinear medium in following discussions. Similar processes can be applied to the intensity-dependent refractive index with self-defocusing nonlinear medium. For the Kerr-type nonlinear medium [20], the square of the refractive index of each layer can be expressed as

{n_{i}}^{2} = {n_{i 0}}^{2} + α_{i} {∣ E_{i} (x) ∣}^{2}, i = 0,1,2, \dots . M - 1

where n _i0 and α_i are the linear refractive index and the nonlinear coefficient of the i-th layer, respectively. The nonlinear wave equation can be reduced to

\frac{d^{2} E_{i} (x)}{d x^{2}} - {Q_{i}}^{2} E_{i} (x) + {k_{0}}^{2} α_{i} {E_{i}}^{3} (x) = 0, i = 0,1,2, \dots . M - 1

where Q_i ²=k ₀ ²(n_e ²-n _i0 ²). The first integration of Eq. (4) gives

{[\frac{d E_{i} (x)}{dx}]}^{2} - {Q_{i}}^{2} {E_{i}}^{2} (x) + \frac{1}{2} {k_{0}}^{2} α_{i} {E_{i}}^{4} (x) = C_{i}

where the first constant of integration C_i can be expressed as

C_{i} = {[\frac{d E_{i} (x)}{dx}]}^{2} |_{x = x_{i}} - {Q_{i}}^{2} {E_{i}}^{2} (x_{i}) + \frac{1}{2} {k_{0}}^{2} α_{i} {E_{i}}^{4} (x_{i})

Fig. 1 The structure of multilayer optical waveguides with all nonlinear layers.

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For the case C_i ≥ 0 , the Eq. (5) can be rewritten as

\frac{d E_{i} (x)}{dx} = \pm \sqrt{C_{i} + {Q_{i}}^{2} {E_{i}}^{2} (x) - \frac{1}{2} {k_{0}}^{2} α_{i} {E_{i}}^{4} (x)}

Integrating Eq. (7), we obtain

\int_{E_{i} (0)}^{E_{i} (x - x_{i})} \frac{du}{\sqrt{({a_{i}}^{2} + u^{2}) ({b_{i}}^{2} - u^{2})}} = \pm {(\frac{1}{2} {k_{0}}^{2} α_{i})}^{\frac{1}{2}} \int_{0}^{x - x_{i}} dx, x_{i} \leq x \leq x_{i + 1}

where

{a_{i}}^{2} = \frac{{q_{i}}^{2} - {Q_{i}}^{2}}{{k_{0}}^{2} α^{i}}

{b_{i}}^{2} = \frac{{q_{i}}^{2} - {Q_{i}}^{2}}{{k_{0}}^{2} α^{i}}

q_{i} = {({Q_{i}}^{4} + 2 C_{i} {k_{0}}^{2} α_{i})}^{\frac{1}{4}}

The transverse electric field in each layer can be written as

E_{i} (x) = b_{i} cn {q_{i} [(x - x_{i}) + x_{oci}] ∣ m_{i}}, x_{i} \leq x \leq x_{i + 1}

where cn is one of Jacobi elliptic function, x_oci is the second constant of integration, m_i is the Jacobi modulus which can be expressed as

m_{i} = \frac{{q_{i}}^{2} + {Q_{i}}^{2}}{2 {q_{i}}^{2}}

Following the approach described in Ref. [10], the transverse electric field in each layer can be rewritten as

E_{i} (x) = A_{i} \frac{cn [q_{i} (x - x_{i}) + B_{i} ∣ m_{i}]}{cn [B_{i} ∣ m_{i}]}

where A_i is the value of the electric field at the lower boundary in each layer, the constant x_oci has been replaced by the additional unknown variable B_i=q_ix_oci. sn and fnt are the Jacobi elliptic functions. sn [B_i|m_i] and fn [B_i|m_i ] can be given at the lower boundary x=x_i,

fn [B_{i} ∣ m_{i}] = \frac{sn [B_{i} ∣ m_{i}] dn [B_{i} ∣ m_{i}]}{cn [B_{i} ∣ m_{i}]}

cn [B_{i}| m_{i}] = \frac{A_{i}}{b_{i}}

By matching the boundary conditions, the dispersion equation can be expressed as

\frac{q_{0}}{q_{1}} = - \frac{fn [B_{1}| m_{1}]}{fn [B_{0}| m_{0}]}

\frac{q_{i - 1}}{q_{i}} = \frac{fn [B_{i}| m_{i}]}{fn [q_{i - 1} d_{i - 1} + B_{i - 1}| m_{i - 1}]}, 2 \leq i \leq M - 1

where

fn [q_{i} d_{i} ∣ m_{i}] = \frac{sn [q_{i} d_{i} ∣ m_{i}] dn [q_{i} d_{i} ∣ m_{i}]}{cn [q_{i} d_{i} ∣ m_{i}]}

fn [q_{i} d_{i} + B_{i} | m_{i}] = \frac{sn [q_{i} d_{i} + B_{i} | m_{i}] dn [q_{i} d_{i} + B_{i} | m_{i}]}{cn [q_{i} d_{i} + B_{i} | m_{i}]}

Eq. (6) can be rewritten as

C_{i} = {A_{i}}^{2} {{[q_{i} fn (q_{i} d_{i} + B_{i} ∣ m_{i})]}^{2} - {Q_{i}}^{2} + \frac{1}{2} {k_{0}}^{2} α_{i} {A_{i}}^{2}}

The Eqs. (12) to (15) can be solved by a numerical method on a computer. A diagram indicating the computation step is shown in Fig. 2. When the constants n_e and A₁ are determined, all the other constants Q_i, C_i, a_i, b_i, q_i, m_i, and A_i, are also determined.

Fig. 2. Diagram of the computation steps.

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For the case C_i ≤ 0, on the other hand, the transverse electric field and all the other parameters a_ĩ, b_ĩ, q_ĩ, and m_ĩ can be expressed as follows:

E_{i} (x) = \tilde{b_{i}} cn {\tilde{q_{i}} [(x - x_{i}) + x_{oci}] ∣ \tilde{m_{i}}}, x_{i} \leq x \leq x_{i + 1}

{\tilde{a}}_{i}^{2} = \frac{{\tilde{q}}_{i}^{2} - {Q_{i}}^{2}}{{k_{0}}^{2} α_{i}}

{\tilde{b}}_{i}^{2} = \frac{{\tilde{q}}_{i}^{2} + {Q_{i}}^{2}}{{k_{0}}^{2} α_{i}}

\tilde{q_{i}} = i {({Q_{i}}^{4} + 2 C_{i} {k_{0}}^{2} α_{i})}^{\frac{1}{4}} = i q_{i}

\tilde{m_{i}} = \frac{{\tilde{q}}_{i}^{2} + {Q_{i}}^{2}}{{2 \tilde{q}}_{i}^{2}}

We just have to replace the parameters a_i, b_i, q_i, and m_i in Eqs. (12)–(15) with a_ĩ, b_ĩ, q_ĩ, and m_ĩ, respectively, and get these equations with similar expressions. When the constants n_e and A₁ are determined, all the other constants Q_i, C_i, a_ĩ, b_ĩ, q_ĩ, m_ĩ, and A_i, are also determined.

2.2 Degenerated description

The general formulas derived in the preceding section can be degenerated into some kinds of special cases. In this section, we have shown the degenerated processes and compared with the eigenvalue equations for some special case. For special case 1, the general method can be degenerated into the multilayer optical waveguides with nonlinear cladding and nonlinear substrate [31]. The parameters are shown as the followings:

C_{0} = C_{M - 1} = 0, {q_{0}}^{2} = {Q_{0}}^{2}, {q_{M - 1}}^{2} = {Q_{M - 1}}^{2}, m_{0} = m_{M - 1} = 1, c n^{2} [q_{0} x_{c 0}] = \frac{{k_{0}}^{2} α_{0} {A_{1}}^{2}}{2 {Q_{0}}^{2}}

The Eqs. (14a)–(14d) can be degenerated into the eigenvalue equations as shown in Ref. [31]. The degenerated eigenvalue equations are shown in Appendix I. For special case 2, the general method can be degenerated into the multilayer nonlinear optical waveguides with a localized arbitrary nonlinear guiding film [32]. The parameters are shown as the followings:

C_{0} = C_{M - 1} = 0, {q_{0}}^{2} = {Q_{0}}^{2}, {q_{M - 1}}^{2} = {Q_{M - 1}}^{2}, m_{0} = m_{M - 1} = 1, c n^{2} [q_{0} x_{c 0}] = 0

The Eqs. (14a)–(14d) can be degenerated into the eigenvalue equations as shown in Ref. [32]. The degenerated eigenvalue equations are shown in Appendix II. For special case 3, the general method can be degenerated into the multilayer optical waveguides with all nonlinear guiding film [33]. The parameters are shown as the followings:

C_{0} = C_{M - 1} = 0, {q_{0}}^{2} = {Q_{0}}^{2}, {q_{M - 1}}^{2} = {Q_{M - 1}}^{2}, m_{0} = m_{M - 1} = 1, c n^{2} [q_{0} x_{c 0}] = 0

The Eqs. (14a)–(14d) can be degenerated into the eigenvalue equations as shown in Ref. [33]. The degenerated eigenvalue equations are shown in Appendix III.

3. Numerical results

In this section, we use the analytic formulas derived in the preceding section to calculate the transverse electric field function in each layer of the multilayer optical waveguide structure with all nonlinear layers. We can simplify the general formulas to analyze seven-layer optical waveguides with all nonlinear layers. The numerical results are shown in Figs. 3(a) and (b). Figure 3(a) shows the dispersion curves of TE₀ symmetric modes with the constants n₀ = n₂ = n₄ = n₆ = 1.55, n₁ = n₃ = n₅ = 1.57, α ₀ = α ₁ = α ₂ = α ₃ = α ₄ = α ₅ = α ₆ = 6.3786μm ²/V ², the free space wavelength λ = 1.55 μm, the width of guiding film d₁=d₃=d₅=2μm, the width of interaction layer d₂ = d₄ = 3μm. Figure 3(b) shows the electric field distributions for the various input powers with respect to points A-D as shown in Fig. 3(a). As the power of film increases and consequently n_e increases, the field distributions gradually narrow, and the optical wave will be tightly confined in the center nonlinear film.

Fig. 3. (a) Dispersion curve of the seven-layer optical waveguide structure with uniform nonlinearity. (b) The electric field distributions with respect to point A-D as shown in (a).

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We can also use these general equations to analyze a multibranch waveguide with all nonlinear layers, as shown in Fig. 4. The multibranch waveguides are the key component in the applications of integrated optics. The two-branch waveguide structures can be used in the Mach-Zehnder structures [41] and X junction structures [42]. The three-branch or multibranch waveguide structures can be used in the all-optical switches [35] or power divider [43]. A typical application which combined two-branch with three-branch had been studied in Ref. [3]. Such the structures also prove a good guide for all-optical devices. In the following analysis the multibranch waveguide is divided into four sections from bottom to top: the straight-line section (three-layer waveguide), the tapered section (three-layer tapered waveguide), the coupled separating-waveguide section (multilayer waveguides with tapered interaction layers), and the isolated separating-waveguide section (multilayer waveguides isolated from one another). The tapered-waveguide section and the separating-waveguide sect;on can be approximated by straight waveguide segments step by step [39–40]. These step-waveguide segments can be analyzed by the method proposed in the preceding section. First, we display an example of a two-branch waveguide structure with all nonlinear layers, as shown in Fig. 5. The parameters of material are n_c0=n_s0=n_i0= 1.55, n_f0=1.57, α_f = 6.3786μm ²/V ², α_c = α_s = 2α_f, the free space wavelength λ=1.55μm. We discuss the low input power density and the high input power density cases, respectively.

Fig. 4. The multibranch optical waveguide structure.

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Fig. 5. The two-branch optical waveguide structure with all nonlinear layers.

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For the low input power density case, the electric field distributions of the four sections (positions Z₁, Z₂, Z₃, Z₄ in Fig. 5) are shown in Figs. 6(a)–(d), respectively. In Fig. 6(a) the electric field distribution of the straight waveguide section is plotted with the parameters d_f=2μm, n_e=1.5625. In Fig. 6(b) the electric field distribution of the tapered-waveguide section is plotted with the parameters d_f=3μm, n_e=1.5655. In Fig. 6(c) the electric field distribution of the coupled separating-waveguide section is plotted with the parameters d_f=2μm, d_i=3μm, n_e=1.5615. In Fig. 6(d) the electric field distribution of the isolated separating-waveguide section is plotted with the parameters d_f=2μm, d_i=6μm, n_e=1.5609.

Fig. 6. For the low input power density case, electric-field distributions of the two-branch optical waveguide structure with all nonlinear layers at positions (a) Z₁ (b)Z₂ (c) Z₃ (d) Z₄.

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For the high input power density case, the electric field distributions of the four sections (positions Z₁, Z₂, Z₃, Z₄ in Fig. 5) are shown in Figs. 7(a)–(d), respectively. In Fig. 7(a) the electric field distribution of the straight waveguide section is plotted with the parameters d_f=2μm, n_e=1.5799. In Fig. 7(b) the electric field distribution of the tapered-waveguide section is plotted with the parameters d_f=3μm, n_e=1.5808. In Fig. 7(c) the electric field distribution of the coupled separating-waveguide section is plotted with the parameters d_f=2μm, d_i=3μm, n_e=1.5854. In Fig. 7(d) the electric field distribution of the isolated separating-waveguide section is plotted with the parameters d_f=2μm, d_i=6μm, n_e=1.5865.

Fig. 7. For the high input power density case, electric-field distributions of the two-branch optical waveguide structure with all nonlinear layers at positions (a) Z₁ (b)Z₂ (c) Z₃ (d) Z₄.

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Here we show an example of a three-branch waveguide structure with uniform nonlinearity, as shown in Fig. 8. The parameters of material are n_c0=n_s0=n_i0=l.55, n_f0=1.57, α_c =α_s=α_f=6.3786μm ²/V ², the free space wavelength λ=1.55μm. We discuss the low input power density and the high input power density cases, respectively. For the low input power density case, the electric field distributions of the four sections (positions Z₁, Z₂, Z₃, Z₄ in Fig. 8) are shown in Figs. 9(a)–(d), respectively. In Fig. 9(a) the electric field distribution of the straight waveguide section is plotted with the parameters d_f=2μm, n_e=1.5594. In Fig. 9(b) the electric field distribution of the tapered-waveguide section is plotted with the parameters d_f=4μm, n_e=1.5649. In Fig. 9(c) the electric field distribution of the coupled separating-waveguide section is plotted with the parameters d_f=2μm, d_i=3μm, n_e=1.5602. In Fig. 9(d) the electric field distribution of the isolated separating-waveguide section is plotted with the parameters d_f=2μm, d_i=6μm, n_e=1.5594.

Fig. 8. The three-branch optical waveguide structure with uniform nonlinearity.

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Fig. 9. For the low input power density case electric-field distributions of the three-branch optical waveguide structure with all nonlinear layers at positions (a) Z₁ (b)Z₂ (c) Z₃ (d) Z₄.

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For the high input power density case the electric field distributions of the four sections (positions Z₁ Z₂ Z₃ Z₄ in Fig. 8) are shown in Figs. 10(a)–(d) respectively. In Fig. 10(a) the electric field distribution of the straight waveguide section is plotted with the parameters d_f=2μm, n_e= 1.5641. In Fig. 10(b) the electric field distribution of the tapered-wave guide section is plotted with th; parameters d_f=4μm, n_e=1.5688. In Fig. 10(c) the electric field distribution of the coupled separating-waveguide section is plotted with the parameters d_f=2μm, d_i=3μm, n_e=1.5642. In Fig. 10(d) the electric field distribution of the isolated separating-waveguide section is plotted with the parameters d_f=2μm, d_i=6μm, n_e= 1.5641.

Fig. 10. For the high input power density case, electric-field distributions of the three-branch optical waveguide structure with all nonlinear layers at positions (a) Z₁ (b)Z₂ (c) Z₃ (d) Z₄.

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To prove the accuracy of numerical results shown in Figs. 6–10, we use the FD-BPM [38] to simulate the electric field propagating along these structures with all nonlinear layers, from the stem to the branching waveguides. For calculating the two-branch waveguide structure with all nonlinear layers, we choose the following numerical data: the transverse step length Δx = 0.03125μm, the longitudinal step length Δz = 0.05μm , the total propagation distance Z=1100μm, branching angle θ=0.573°. The simulation results for the low input power density and high input power density cases are shown in Figs. 11 and 12. The relevant curves shown in Figs. 6–7 and Figs. 11–12 are superposed on the same graph as shown in Figs. 13–14. For calculating the three-branch waveguide structure with uniform nonlinearity, we choose the following numerical data: the transverse step length Δx = 0.03125μm, the longitudinal step length Δz = 0.05μm, the total propagation distance Z=1500μm, branching angle θ=0.382°. The simulation results for the low input power density and high input power density cases are shown in Figs. 15 and 16. The relevant curves have been shown in Figs. 9–10 and Figs. 15–16 are superposed on the same graph shown in Figs. 17–18. By comparing the results, we confirm that our analyses are correct.

Fig. 11. The typical evolution of a wave propagating along a two-branch optical waveguide structure with all nonlinear layers at the low input power density.

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Fig. 12. The typical evolution of a wave propagating along a two-branch optical waveguide structure with all nonlinear layers at the high input power density.

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Fig. 13. The relevant curves shown in Fig. 6 and 11 on the same graph (a) at position Z₁, (b) at position Z₂, (;) at position Z₃, (d) at position Z₄, (dotted line: the predicted results, dashed line: the numerical simulation results).

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Fig. 14. The relevant curves shown in Fig. 7 and 12 on the same graph (a) at position Z₁, (b) at position Z₂, (c) at position Z₃, (d) at position Z₄, (dotted line: the predicted results, dashed line: the numerical simulation results).

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Fig. 15. The typical evolution of a wave propagating along a three-branch optical waveguide structure with uniform nonlinearity.

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Fig. 16. The typical evolution of a wave propagating along a three-branch optical waveguide structure with uniform nonlinearity.

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Fig. 17. The relevant curves shown in Fig. 9 and 15 on the same graph (a) at position Z₁, (b) at position Z₂, (c) at position Z₃, (d) at position Z₄, (dotted line: the predicted results, dashed line: the numerical simulation results).

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Fig. 18. The relevant curves shown in Fig. 10 and 16 on the same graph (a) at position Z₁, (b) at position Z₂, (c) at position Z₃, (d) at position Z₄, (dotted line: the predicted results, dashed line: the numerical simulation results).

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4. Conclusion

In this paper, we propose a general method for analyzing the multilayer optical waveguide structure with all nonlinear layers. This model can be degenerated into some kinds of special cases. In the future, the multilayer and multibranch waveguide structures are important key components in application of integrated optics. The multilayer structures have been extensively used in the MQW waveguides and the multibranch structures have also been designed to operate as all-optical devices. This method can be used to predict the evolution of waves propagating along the multibranch optical waveguide structures. Similar processes can be extensively used to predict the propagation characteristics of spatial solitons [37, 44]. We give a detail modal analysis of the proposed nonlinear optical waveguide structure. The analytical and numerical results show excellent agreement.

Appendix I: Degenerated into the multilayer optical waveguides with nonlinear cladding and nonlinear substrate

The parameters:

C_{0} = C_{M - 1} = 0, {q_{0}}^{2} = {Q_{0}}^{2}, {q_{M - 1}}^{2} = {Q_{M - 1}}^{2}, m_{0} = m_{M - 1} = 1, c n^{2} [q_{0} x_{c 0}] = \frac{{k_{0}}^{2} α_{0} {A_{1}}^{2}}{2 {Q_{0}}^{2}}

Eigenvalue equations:

\tan (Q_{i - 1} d_{i - 1}) = \frac{Q_{i - 1} [q_{i} \tan (B_{i}) + Q_{0} \tanh (B_{0})]}{{Q_{i - 1}}^{2} - q_{i} q_{0} \tan (B_{i}) \tanh (B_{0})}, n_{e} < n_{i - 1}

\tanh (Q_{i - 1} d_{i - 1}) = \frac{{- Q}_{i - 1} [q_{i} \tan (B_{i}) + q_{0} \tanh (B_{0})]}{{Q_{i - 1}}^{2} + q_{i} q_{0} \tan (B_{i}) \tanh (B_{0})}, n_{e} > n_{i - 1}

Appendix II: Degenerated into the multilayer nonlinear optical waveguides with a localized arbitrary nonlinear guiding film

The parameters:

C_{0} = C_{M - 1} = 0, {q_{0}}^{2} = {Q_{0}}^{2}, {q_{M - 1}}^{2} = {Q_{M - 1}}^{2}, m_{0} = m_{M - 1} = 1, c n^{2} [q_{0} x_{c 0}] = 0

Eigenvalue equation:

\frac{q_{R + 1} b_{R + 2} \cos (B_{R + 2}) - b_{R + 2} q_{R + 2} \sin (B_{R + 2})}{{2 q}_{R + 1} \exp (- q_{R + 1} d_{R + 1})}

Appendix III: Degenerated into the multilayer optical waveguides with all nonlinear guiding film

The parameters:

C_{0} = C_{M - 1} = 0, q_{0}^{2} = Q_{0}^{2}, q_{M - 1}^{2} = Q_{M - 1}^{2}, m_{0} = m_{M - 1} = 1, c n^{2} [q_{0} x_{c 0}] = 0

Eigenvalue equation:

\frac{q_{R + 1} b_{R + 1} \cos (B_{R + 2}) - b_{R + 2} q_{R + 2} \sin (B_{R + 2})}{{2 q}_{R + 1} \exp (- q_{R + 1} d_{R + 1})}

Acknowledgments

This work was partly supported by National Science Council R. O. C. and Ministry of Education R. O. C. under Grant No. 95C9031 and 95TSFC9031.

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Analyzing multilayer optical waveguide with all nonlinear layers

Abstract

1. Introduction

2. Method and analysis

2.1 Analytic formulation

2.2 Degenerated description

3. Numerical results

4. Conclusion

Appendix I: Degenerated into the multilayer optical waveguides with nonlinear cladding and nonlinear substrate

Appendix II: Degenerated into the multilayer nonlinear optical waveguides with a localized arbitrary nonlinear guiding film

Appendix III: Degenerated into the multilayer optical waveguides with all nonlinear guiding film

Acknowledgments

References and Links

Cited By

Figures (18)

Equations (70)

Optics Express