Method for analyzing multilayer nonlinear optical waveguide

Yaw-Dong Wu; Mao-Hsiung Chen

doi:10.1364/OPEX.13.007982

1. Introduction

Kerr-like nonlinear waveguide structures containing one or more media, whose refractive index depends on the local intensity, have stimulated a great deal of theoretical and experimental study [1–13]. The interest in nonlinear waveguide device, which has been growing steadily in recent years, stems from their potential use for ultrafast all-optical signal processing and optical computing systems. The confinement of optical beam in the small core area over long distance increase the efficiency of the nonlinear interaction and permit the use of relatively weak nonlinearities. It has been shown that these nonlinear waveguides possess a variety of novel and exciting features such as power-dependent propagation constants and field profiles leading to novel feasibilities for all-optical signal processing and optical computing.

Until now, the overwhelming majority of the published papers was aimed at analyzing the optical waveguide structures where either linear films were bounded by one or two nonlinear media [14–18] or a nonlinear film was sandwiched between linear films [19–21]. If the guiding dielectric films are all nonlinear, the solution fields are much more complex and consist of Jacobi elliptic functions [22]. Within the past several years, it became evident that the multilayer systems as semiconductor multiple quantum well structures exhibit stronger nonlinearity, low threshold power, very fast response times, and suitability for integration into circuits [23–25].

In the future, the multilayer and multibranch waveguide structures are very important components in the application of integrated optics. The multilayer waveguide structures have been extensively used in the multiple quantum well waveguide structures [26–29] and the multibranch waveguide structures have also been designed to operate as all-optical waveguide devices [30–39]. To the best of our knowledge, the general method for analyzing the multilayer optical waveguide with all nonlinear guiding films that is presented in this paper has not been reported before. This paper gives a detail modal analysis of the multilayer optical waveguide with all nonlinear guiding films. The analytical results are accompanied by numerical examples. This method can also be used to predict the evolution of a wave propagating along the multibranch optical waveguide structure with all nonlinear guiding branches. The numerical results show that our analyses are correct.

2. Analysis process

In this section, we use the modal theory [40–41] to derive the general formulas that can be used to analyze the multilayer optical waveguide structure with all nonlinear guiding films, as shown in Fig. 1. The multilayer optical waveguide structure is composed of nonlinear guiding films $(\frac{m - 1}{2} layers)$ , interaction layers $(\frac{m - 3}{2} layers)$ , cladding, and substrate. The total number of layers is m (m = 3,5,7,…). The cladding and substrate layers are assumed to extend to infinity in the +x and -x direction, respectively. The major significance of this assumption is that there are no reflections in the x direction to be concerned with, except for those occurring at interfaces.

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

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For simplicity, we consider the transverse electric polarized waves propagating along the z direction. The wave equation can be reduced to

\nabla^{2} E_{yi} = \frac{n_{i}^{2}}{c^{2}} \frac{\partial^{2} E_{yi}}{{\partial t}^{2}}, i = 1,2, \dots, m

with solutions of the form

E_{yi} (x, z, t) = ε_{i} (x) \exp [j (ωt - {βk}_{0} z)]

where ω is the angular frequency, k ₀ is the wave number in the free space, and β is the effective refractive index. For a Kerr-type nonlinear medium [42–44], the square of the refractive index of the guiding film can be expressed as

n_{i}^{2} = n_{0 i}^{2} + α_{i} {∣ ε_{i} (x) ∣}^{2}, i = 2,4, \dots, m - 1

where n _0i and α_i are the linear refractive index and the nonlinear coefficient of the i-th layer nonlinear guiding film, respectively. The transverse electric field in each layer can be expressed as:

ε_{1} (x) = E_{s} \exp (p_{1} x) in the substrate

ε_{i} (x) = E_{I (i - 2)} \exp {- p_{i} [x - (\frac{i - 1}{2}) d - (\frac{i - 3}{2}) w]} + E_{I (i - 1)} {\exp p_{i} [x - (\frac{i - 1}{2}) d - (\frac{i - 1}{2}) w]}

ε_{i} (x) = b_{i} cn {A_{i} [x - (\frac{i}{2} - 1) (d + w) + x_{0 i}] ∣ l_{i}}

ε_{i} (x) = \bar{b_{i}} cn {\bar{A_{i}} [x - (\frac{i}{2} - 1) (d + w) + {\overset{̅}{x}}_{0 i}] ∣ \bar{l_{i}}}

ε_{m} (x) = E_{c} \exp {- p_{m} [x - (\frac{m - 1}{2}) d - (\frac{m - 3}{2}) w]} in the cladding

where cn is a Jacobian elliptic function, and the constants d and w are the widths of the guiding film and the interaction layer, respectively.

The constants p_i, b_i, A_i, l_i, b̅_i, A̅_i, and l̅_i. can be expressed as

p_{i} = k_{0} \sqrt{β^{2} - n_{i}^{2}},

where the constants a_i, a̅_i, q_i , Qⁱ, K_i, x _0i and x0_i̅ are shown in the Appendix. For simplicity, the modulus l_i and l̅_i are omitted in the following discussions.

Considering the case β<n_i (i = 2,4,…,m-1) and matching the boundary conditions, we can obtain the following eigenvalue equations:

for m ≤ 7 ,

\frac{[cn (A_{m - 1} d) Δ_{m - 2}^{+} + sn (A_{m - 1} d) dn (A_{m - 1} d) \frac{p_{m - 2} Δ_{m - 2}^{-}}{A_{m - 1}}]}{[1 - l_{m - 1} (1 - \frac{Δ_{m - 2}^{+^{2}}}{b_{m - 1}^{2}}) {sn}^{2} (A_{m - 1} d)] A_{m - 1}} = \frac{{Δ_{m - 2}^{+} [1 - l_{m - 1} (1 - \frac{Δ_{m - 2}^{+^{2}}}{b_{m - 1}^{2}}) sn (A_{m - 1} d) dn (A_{m - 1} d)] - Δ_{m - 2}^{+} l_{m - 1} (1 - \frac{Δ_{m - 2}^{+^{2}}}{b_{m - 1}^{2}}) sn (A_{m - 1} d) {cn}^{2} (A_{m - 1} d) dn (A_{m - 1} d) + \frac{p_{m - 2} Δ_{m - 2}^{-}}{A_{m - 1}} [cn (A_{m - 1} d) {dn}^{2} (A_{m - 1} d) - l_{m - 1} \frac{Δ_{m - 2}^{+^{2}}}{b_{m - 1}^{4}} {sn}^{2} (A_{m - 1} d) cn (A_{m - 1} d)]}}{{[1 - l_{m - 1} (1 - \frac{Δ_{m - 2}^{+^{2}}}{b_{m - 1}^{4}}) {sn}^{2} (A_{m - 1} d)]}^{2} p_{m}},

\frac{E_{c} P_{3}}{A_{2} b_{2}} = \frac{{\frac{E_{s}}{b_{2}} [1 - l_{2} (1 - \frac{E_{s}^{2}}{b_{2}^{2}})] sn (A_{2} d) dn (A_{2} d) - l_{2} \frac{E_{s}}{b_{2}} (1 - \frac{E_{s}^{2}}{b_{2}^{2}}) sn (A_{2} d) {cn}^{2} (A_{2} d) dn (A_{2} d) - \frac{E_{s} p_{1}}{b_{2} A_{2}} [cn (A_{2} d) {dn}^{2} (A_{2} d) - l_{2} \frac{E_{s}^{2}}{b_{2}^{2}} {sn}^{2} (A_{2} d) cn (A_{2} d)]}}{{[\frac{E_{s}}{E_{c}} cn (A_{2} d) + \frac{E_{s} p_{1}}{E_{c} A_{2}} sn (A_{2} d) dn (A_{2} d)]}^{2}},

for m = 5,

and sn and dn are Jacobi elliptic functions. These eigenvalue equations (9)–(11) can be solved by a numerical method on a computer. A diagram indicating the computation step is shows in Fig. 2. When the constants β and E_s are determined, all the other constants q_i, p_i, K_i, A_i, a_i, b_i, l_i, x _0i, E_li, and E_c are also determined (Appendix).

Fig. 2. Diagram of the computation steps (9)–(11)

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For the case β > n_i (n_i = 2,4,…,m-1), on the other hand, we just have to replace the constants A_i, a_i, b_i, l_i, and x _0i in Eqs.(9)–(11) with the constants A̅_i, a̅_i, b̅_i, l̅_i, and x _0i̅, and get these equations with similar expressions. When constants β and E_s are determined, all the other constants Q_i, p_i, K_i, A̅_i, a̅_i, b̅_i, l̅_i, x _0i̅, E_Ii, and E_c are also determined (Appendix).

3. Numerical results and discussion

In this section, we use the formulas derived in the preceding section and in the Appendix to calculate the transverse electric field function in each layer of the multilayer optical waveguide structure with all nonlinear guiding films. Several numerical examples are presented as follows. When m=3, the formulas can be used to analyze the three-layer optical waveguide structure with nonlinear guiding film. The numerical results are shown in Figs. 3(a) and (b). Figure 3(a) shows a dispersion curve of TE ₀ symmetric modes of the three-layer nonlinear optical waveguide structure with the constants d = 3μm, n ₁ = n ₃ =1.55, n ₀₂ =1.57 , α = 6.3786μm ² /V ² (for MBBA Liquid Crystal), and the free space wavelength is λ = 1.3μm . 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). When m = 5, the formulas can be used to analyze the five-layer optical waveguide structure with all nonlinear guiding film. The numerical results are shown in Figs. 4(a) and (b). Figure 4(a) shows a dispersion curve of TE ₀ symmetric modes of the five-layer nonlinear optical waveguide structure with the constants d = 3μm , w = 7μm , n ₁ = n ₃ = n ₅ = 1.55 , n ₀₂= n ₀₄ = 1.57 , α = 6.3786μm ²/V ², and λ = 1.3μm. Figure 4(b) shows the electric field distributions for the various input powers with respect to points A-D as shown in Fig. 4(a). When m = 7 , we can simplify the general formulas to analyze the seven-layer optical waveguide structure with all nonlinear guiding films. The numerical results are shown in Figs. 5(a) and (b). Figure 5(a) shows a dispersion curve of TE ₀ symmetric modes of the seven-layer nonlinear optical waveguide structure with the constants d = 3μm, w = 7μm , n ₁ = n ₃ = n ₅ = n ₇ =1.55 , n ₀₂ = n ₀₄ = n ₀₆ = 1.57 , α = 6.3786μm ² /V ² , and λ = 1.3μm. Figure 5(b) shows the electric field distributions for the various input powers with respect to points A-D as shown in Fig. 5(a). As the results shown in Figs. 3–5, as the guided power increases and consequently β increases, the field distributions gradually narrow, and the optical wave will be tightly confined in the center nonlinear guiding film.

Fig. 3. (a)Dispersion curve of the three-layer optical waveguide structure with the nonlinear central guiding film. (b)The electric field distributions with respect to points A-D as shown in (a).

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Fig. 4. (a)Dispersion curve of the five-layer optical waveguide structure with the nonlinear central guiding film. (b)The electric field distributions with respect to points A-D as shown in (a).

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Fig. 5. (a)Dispersion curve of the seven-layer optical waveguide structure with the nonlinear central guiding film. (b)The electric field distributions with respect to points A-D as shown in (a).

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We can also use these formulas to analyze the multibranch optical waveguide structure with all nonlinear guiding branches, as shown in Fig. 6. In the following analysis the multibranch optical waveguide structure is divided into four sections from bottom to top: the straight-line section (three-layer waveguide), the tapered section (three-layered tapered waveguide), the nonlinear multibranch section (multilayer waveguides with tapered interaction layers), and the isolated separating section (multilayer waveguides isolated from one another). The tapered-waveguide section and the separating-waveguide section can be approximated by straight waveguide segments step by step [40,41]. These step-waveguide segments can be analyzed by the method proposed in the preceding section. For simplicity, the symbol N is used to denoted the nonlinear medium. We can use this method to predict the evolution of a wave propagating along the multibranch waveguide structure with all nonlinear guiding branches. Here we show an example of a three-branch waveguide structure with all nonlinear guiding branches, as shown in Fig. 7. We discuss the low input power density and the high input power density cases, respectively.

Fig. 6. The multibranch optical waveguide structure with all nonlinear guiding branches (N denoted the nonlinear medium).

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Fig. 7. The three-branch optical waveguide structure with all nonlinear guiding branches (N denoted the nonlinear medium).

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For the low input power density case, the electric field distributions of the four sections (positions Z ₁, Z ₂, Z ₃, Z ₄ in Fig. 7) are shown in Figs. 8(a)–(d), respectively. In Fig. 8(a) the electric field distribution of the straight waveguide section is plotted with the constants d =3μm , n_c = n_s = n_I =1.55 , n _f0 = 1.57 , β = 1.5654 , α = 6.3786μm ²/V ² , and λ = 1.3μm. In Fig. 8(b) the electric field distribution of the straight waveguide section is plotted with the constants d =6μm , n_c = n_s = n_I = 1.55 , n _f0 =1.57 , β = 1.5654 , α = 6.3786μm ² /V ² , and λ = 1.3μm. In Fig. 8(c) the electric-field distribution of the coupled separating-waveguide section is plotted with d =3μm , w = 5μm, n_c = n_s = n_I = 1.55 , n _f0 = 1.57, β = 1.5655 , α = 6.3786μm ²/V ², and λ = 1.3μm. And in Fig. 8(d) the electric-field distribution of the isolated separating- distribution of the isolated separating-waveguide section is plotted with d = 3μm, w = 7μm, n_c = n_s = n_I = 1.55 , n _f0 = 1.57 , β = 1.56545 , α = 6.3786μm ²/V ² , and λ = 1.3μm.

Fig. 8. For the low input power density case, electric-field distributions of the three-branch optical waveguide structure with all nonlinear guiding branches at positions (a)Z₁ (d_f = 3μm , β = 1.5654), (b)Z₂ (d_f = 6μm, β = 1.5654), (c)Z₃ (d_f = 3μm , d_I = 5μm , β = 1.5655), (d)Z₄ (d_f = 3μm , d_I = 7μm, β = 1.56545).

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For the high input power density case, the electric field distributions of the four sections (positions Z ₁, Z ₂, Z ₃, Z ₄ in Fig. 7) 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 constants d = 3μm , n_c = n_s = n_I = 1.55 , n _f0 =1.57 , β = 1.56925 , α = 6.3786μm ²/V ² , and λ = l.3μm. In Fig. 9(b) the electric field distribution of the straight waveguide section is plotted with the constants d =6μm , n_c = n_s = n_I = 1.55 , n _f0 = 1.57 , β = 1.56925 , α = 6.3786μm ²/V ² , and λ = 1.3μm. In Fig. 9(c) the electric-field distribution of the coupled separating-waveguide section is plotted with d =3μm , w = 5μm , n_c = n_s = n_I = 1.55 , n _f0 = 1.57, β = 1.5690 , α = 6.3786μm ²/V ² , and λ = 1.3μm . And in Fig. 9(d) the electric-field distribution of the isolated separating-waveguide section is plotted with d = 3μm, w = 7μm , n_c = n_s = n_I = 1.55 , n _f0 = l.51 , β = 1.5693 , α = 6.3786μm ²/V ², and λ = l.3μm.

Fig. 9. For the high input power density case, electric-field distributions of the three-branch optical waveguide structure with all nonlinear guiding branches at positions (a)Z₁ (d_f = 3μm , β = 1.56925), (b)Z₂ (d_f = 6μm, β = 1.56925), (c)Z₃ (d_f = 3μm , d_I = 5μm, β = 1.5690), (d)Z₄ (d_f = 3μm, d_I = 7μm, β = 1.5693).

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To prove the accuracy of the results shown in Figs. 8 and 9, we use the beam propagation method [45] to simulate the electric field propagating along this structure, from the stem to the branching waveguides. For the calculation we choose the following numerical data: the transverse sampling points N = 4096 , the longitudinal step length Δz = 0.05μm, the total propagation distance Z =1500μm , and the branching angle θ = 0.573°. The simulation results for the low input power density and high input power density cases are shown in Figs. 10 and 11. The relevant curves shown in Figs. 8–9 and Figs. 10–11 are superposed on the same graph shown in Figs. 12–13. By comparing the results shown in Figs. 8–9 and Figs. 10–11, we confirm that our analyze are correct.

Fig. 10. The typical evolution of a wave propagating along a three-branch optical waveguide structure with all nonlinear guiding branches at the low input power density.

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Fig. 11. The typical evolution of a wave propagating along a three-branch optical waveguide structure with all nonlinear guiding branches at the high input power density.

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Fig. 12. The relevant curves shown in Figs. 6 and 8 on the same graph (a) at position Z₁, (b) at position Z₂, (c) at position Z₃, (d) at position Z₄. (┄:the predicted results;―: the numerical simulation results).

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Fig. 13. The relevant curves shown in Figs. 7 and 9 on the same graph (a) at position Z₁, (b) at position Z₂, (c) at position Z₃, (d) at position Z₄. (┄:the predicted results;―: the numerical simulation results).

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4. Conclusions

In this paper, we propose a novel method for analyzing the multilayer optical waveguide structure with all nonlinear guiding films. This method can also be used to predict the evolution of waves propagating along the multibranch optical waveguide structure with all nonlinear guiding branches. We give a detail modal analysis of the proposed nonlinear optical waveguide structure. The analytical results are accompanied by numerical examples. The numerical simulation results show that our analyses are correct.

Appendix

a_{i}^{2} = \frac{\sqrt{q_{i}^{4} + {2 α}_{i} k_{0}^{2} K_{i}} + q_{i}^{2}}{α_{i} k_{0}^{2}},

{\overset{̅}{a}}_{i}^{2} = \frac{\sqrt{q_{i}^{4} + {2 α}_{i} k_{0}^{2} K_{i}} - Q_{i}^{2}}{α_{i} k_{0}^{2}},

A_{i} = {[(a_{i}^{2} + b_{i}^{2}) (\frac{α_{i} k_{0}^{2}}{2})]}^{\frac{1}{2}},

{\overset{̅}{A}}_{i} = {[({\overset{̅}{a}}_{i}^{2} + {\overset{̅}{b}}_{i}^{2}) (\frac{α_{i} k_{0}^{2}}{2})]}^{\frac{1}{2}},

q_{i}^{2} = k_{0} \sqrt{(n_{i}^{2} - β^{2})}, for β < n_{i} (n_{i} = 2,4, \dots, m - 1),

Q_{i}^{2} = k_{0} \sqrt{(β^{2} - n_{i}^{2})}, for β > n_{i} (n_{i} = 2,4, \dots, m - 1),

E_{I 1} = \frac{1}{2} {\frac{[E_{s} cn (A_{2} d) + \frac{p_{1} E_{s}}{A_{2}} sn (A_{2} d) dn (A_{2} d)]}{[1 - l_{2} {sn}^{2} (A_{2} d) (1 - \frac{E_{s}^{2}}{b_{2}^{2}})]}}

E_{Ii} = \frac{1}{2} {\frac{[Δ_{i - 2}^{+} cn (A_{i + 1} d) + \frac{p_{i - 2} Δ_{i - 2}^{-}}{A_{i + 1}}] sn (A_{i + 1}) dn (A_{i + 1} d)}{[1 - l_{i + 1} (1 - \frac{Δ_{i - 2}^{+^{2}}}{b_{i + 1}^{2}}) {sn}^{2} (A_{i + 1} d)]} + A_{i + 1} {\frac{Δ_{i - 2}^{+} [1 - l_{i + 1} (1 - \frac{Δ_{i - 2}^{+^{2}}}{b_{i + 1}^{2}})] [1 - l_{i + 1} {cn}^{2} (A_{i + 1})] sn (A_{i + 1} d) dn (A_{i + 1} d) - Δ_{i - 2}^{+} l_{i + 1} (1 - \frac{Δ_{i - 2}^{+^{2}}}{b_{i + 1}^{4}}) sn (A_{i + 1} d) {cn}^{2} (A_{i + 1} d) dn (A_{i + 1} d) - \frac{Δ_{i - 2}^{-} p_{i - 2}}{A_{i + 1}} cn (A_{i + 1} d) [{dn}^{2} (A_{i + 1} d) - l_{i + 1} \frac{Δ_{i - 2}^{+^{2}}}{b_{i + 1}^{2}} {sn}^{2} (A_{i + 1} d)]}}{{[1 - l_{i + 1} {sn}^{2} (A_{i + 1} d) (1 - \frac{Δ_{i - 2}^{+^{2}}}{b_{i + 1}^{4}})]}^{2} p_{i}}}, i = 3,5 \dots \dots, m - 2,

where Δ_{i - 2}^{+} = E_{Ii - 2} + E_{Ii - 1} \exp (- p_{i - 2} w),

{\overset{̅}{x}}_{0 i} = \frac{1}{\bar{A_{i}}} {cn}^{- 1} (\frac{E_{I (i - 3)} \exp (- p_{(i - 1)} w) + E_{I (i - 2)}}{\bar{b_{i}}}),

K_{i} = k_{0}^{2} E_{I (i - 1)}^{2} (n_{i}^{2} - n_{i + 1}^{2} + \frac{α_{i} E_{I (i - 1)}^{2}}{2}) + k_{0}^{2} E_{Ii}^{2} \exp (- 2 p_{(i + 1)} w) [n_{i}^{2} - n_{i + 1}^{2} + \frac{α_{i} E_{Ii}^{2}}{2} \exp (- 2 p_{(i + 1)} w)]

Otherwise,

x_{02} = \frac{1}{A_{2}} {cn}^{- 1} (\frac{E_{0}}{b_{2}}),

x_{0 m - 1} = \frac{1}{A_{m - 1}} {cn}^{- 1} (\frac{E_{0}}{b_{m - 1}}) - d,

{\overset{̅}{x}}_{02} = \frac{1}{\bar{A_{2}}} {cn}^{- 1} (\frac{E_{0}}{{\overset{̅}{b}}_{2}}),

{\overset{̅}{x}}_{0 m - 1} = \frac{1}{{\overset{̅}{A}}_{m - 1}} {cn}^{- 1} (\frac{E_{0}}{{\overset{̅}{b}}_{m - 1}}) - d,

K_{2} = k_{0}^{2} E_{0}^{2} (n_{2}^{2} - n_{1}^{2} + \frac{α_{2} E_{0}^{2}}{2}),

K_{m - 1} = k_{0}^{2} E_{m}^{2} (n_{m - 1}^{2} - n_{m}^{2} + \frac{α_{(m - 1)} E_{m}^{2}}{2}) .

Acknowledgments

The author thanks Jyh-Shiuan Lin and Dong-Heng Cai for his helpful discussions and support in this work.This work was supported by National Science Council, R.O.C. under Grant No. 94-2215-E-151 -001.

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Method for analyzing multilayer nonlinear optical waveguide

Abstract

1. Introduction

2. Analysis process

3. Numerical results and discussion

4. Conclusions

Appendix

Acknowledgments

References and links

Cited By

Figures (13)

Equations (49)

Optics Express