Analyzing multilayer optical waveguide with all nonlinear layers

doi:10.1364/OE.15.002499

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Analyzing multilayer optical waveguide with all nonlinear layers

Chih-Wen Kuo, Shih-Yuan Chen, Mao-Hsiung Chen, Chih-Fu Chang, and Yaw-Dong Wu »View Author Affiliations

Optics Express, Vol. 15, Issue 5, pp. 2499-2516 (2007)
http://dx.doi.org/10.1364/OE.15.002499

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▸ Article Outline
– Abstract
1. Introduction
2. Method and analysis
3. Numerical results
4. Conclusion
– References and links

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Abstract

We propose a general method for analyzing a multilayer optical waveguide with all nonlinear layers. This general method can be degenerated into some special cases, such as symmetric or asymmetric nonlinear optical waveguide structures, the intensity-dependent refractive index with self-focusing nonlinear medium, hollow waveguides, and multilayer systems. Based on this general method, the analysis and calculation of complicated multilayer optical planar waveguides can be achieved easily. The analytical and numerical results show excellent agreement.

1. Introduction

The nonlinear optical waveguide device is a potential key component in the applications of optical signal processing and communication systems [1–3

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, “All-optical efficient wavelength conversion using silicon photonic wire waveguide,” IEEE Photonics Technol. Lett. , 18, pp. 1046–1048 (2006). [CrossRef]

]. 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

Jian-Guo Ma and Ingo Wolff, “Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media,” IEEE Trans. Microw. Theory Tech. 43, 790–795 (1995). [CrossRef]

]. Considerable progress has been made by considering only Kerr nonlinearities due to the third-order nonlinearity [20

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988). [CrossRef]

]. 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

Jian-Guo Ma and Ingo Wolff, “Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media,” IEEE Trans. Microw. Theory Tech. 43, 790–795 (1995). [CrossRef]

] or nonlinear film bounded by linear medium [10–15

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986). [CrossRef]

]. Stegeman et al. [7–8

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985). [CrossRef]

] 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

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986). [CrossRef]

] 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

S. Okafuji, “Propagation characteristics of TE0 waves in three-layer optical waveguides with self focusing and self defocusing nonlinear layers,” IEEE Proceedings-J 140, 127–136 (1993).

] 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

R. A. Sammut, C. Pask, and Q. Y. Li, “Theoretical study of spatial solitons in planar waveguides,” J. Opt. Soc. Am. B 10, 485–491 (1993). [CrossRef]

] developed the propagating character in uniform nonlinear medium throughout the cladding, substrate, and guided film. Schürmann [18

H. W. Schürmann, “On the theory of TE-polarized waves guided by a nonlinear three-layer structure,” Z. Phys. B 97, 515–522 (1995). [CrossRef]

] used Weierstrass’ elliptic functions to express the field equations in a nonlinear three-layer structure. Shin et al. [21

S. Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” J. Lightwave Technol. 6, 977–983 (1988). [CrossRef]

] presented the five-layer optical waveguide with nonlinear cladding and substrate. Sakakibara [22

T. Sakakibara and N. Okamoto, “Nonlinear TE waves in a dielectric slab waveguide with two optically nonlinear layers,” IEEE J. Quantum Electron. 23, 2084–2088 (1987). [CrossRef]

] and Jeong [23

J. S. J;ong, C. H. Kwak, and E. H. Lee, “Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers,” Opt. Commun. 116, 351–357 (1995). [CrossRef]

] 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

S. Radic, N. George, and G. P. Agrawal, “Optical switching in λ/4-shifted nonlinear periodic structures,” Opt. Lett. 19, 1789–1791 (1994). [CrossRef] [PubMed]

] 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

U. Ttrutschel, F. Lederer, and M. Golz, “Nonlinear guided waves in multilayer systems,” IEEE J. Quantum Electron. 25, 194–200 (1989). [CrossRef]

] developed nonlinear matrix formalism and calculated the field profiles and the propagation constant of the nonlinear guide wave in a multilayer system. Ogusu [28

K. Ogusu, “Analysis of nonlinear multilayer waveguides with Kerr-like permittivities,” Opt. Quantum Electron. 21, 109–116 (1989). [CrossRef]

] 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

S. She and S. Zhang, “Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear caldding,” Opt. Commun. 161, 141–148 (1999). [CrossRef]

] 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

M. H. Chen and Y. D. Wu, “A general method for analyzing the multi-layer planar waveguide,” Fiber Integrated Opt. 11, 159–176 (1992). [CrossRef]

], multilayer planar waveguide with nonlinear cladding and substrate [31

Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multilayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B 19, 1737–1745 (2002). [CrossRef]

], multilayer planar waveguide with a localized arbitrary nonlinear guiding film [32

Y. D. Wu, “Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film,” IEEE J. Quantum Electron. 40, 529–540 (2004). [CrossRef]

], and multilayer planar waveguide with all nonlinear guiding film [33

Y. D. Wu and M. H. Chen, “Method for analyzing multilayer nonlinear optical waveguide,” Opt. Express 13, 7982–7995 (2005). http://www.opticsexpress.org/abstract.cfm?id=85750 [CrossRef] [PubMed]

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

K. Ogusu, “Analysis of nonlinear multilayer waveguides with Kerr-like permittivities,” Opt. Quantum Electron. 21, 109–116 (1989). [CrossRef]

, 31

Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multilayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B 19, 1737–1745 (2002). [CrossRef]

]. The nonlinear graded-index waveguides can be segmented a number of step refractive index layers [31

Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multilayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B 19, 1737–1745 (2002). [CrossRef]

, 34

N. Saiga, “Calculation of TE modes in graded-index nonlinear optical waveguides with arbitrary profile of refractive index,” J. Opt. Soc. Am. B 8, 88–94 (1991). [CrossRef]

]. These analyses have also been applied to design the multibranch optical waveguide which can be operated as switches [35

Y. D. Wu, “Nonlinear all-optical switching device by using the spatial soliton collision,” Fiber Integrated Opt. 23, 387–404 (2004). [CrossRef]

], logic gates [36

Y. D. Wu, “Coupled-soliton all-optical logic device with two parallel tapered waveguides,” Fiber Integrated Opt. 23, 405–414 (2004). [CrossRef]

], and wavelength auto-router [37

Y. D. Wu, “New all-optical wavelength auto-router based on spatial solitons,” Opt. Express 12, 4172–4177 (2004). http://www.opticsexpress.org/abstract.cfm?id=81074 [CrossRef] [PubMed]

]. 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

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992). [CrossRef]

]. 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

W. K. Burns and A F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. 11, 32–39 (1975). [CrossRef]

] 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

(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}

(2)

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

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988). [CrossRef]

], 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

(3)

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

(4)

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}

(5)

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})

(6)

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

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)}

(7)

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}

(8)

where

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

(9a)

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

(9b)

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

(9c)

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}

(10)

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}}

(11)

Following the approach described in Ref. [10

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986). [CrossRef]

], 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}]}

= A_{i} cn [q_{i} (x - x_{i}) | m_{i}] \frac{{1 - fn [q_{i} (x - x_{i}) | m_{i}] fn [B_{i} | m_{i}]}}{1 - m_{i} s n^{2} [q_{i} (x - x_{i}) | m_{i}] s n^{2} [B_{i} | m_{i}]}

(12)

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}]}

(13a)

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

(13b)

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}]}

(14a)

\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

(14b)

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}]}

(14c)

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}]}

= \frac{fn [q_{i} d_{i} ∣ m_{i}] {d n^{2} [B_{i} ∣ m_{i}] - m_{i} s n^{2} [B_{i} ∣ m_{i}] c n^{2} [q_{i} d_{i} ∣ m_{i}]}}{{1 - fn [q_{i} d_{i} ∣ m_{i}] ∙ fn [B_{i} ∣ m_{i}]} ∙ {1 - m_{i} s n^{2} [q_{i} d_{i} ∣ m_{i}] s n^{2} [B_{i} ∣ m_{i}]}}

+ \frac{fn [B_{i} ∣ m_{i}] {d n^{2} [q_{i} d_{i} ∣ m_{i}] - m_{i} s n^{2} [q_{i} d_{i} ∣ m_{i}] c n^{2} [B_{i} ∣ m_{i}]}}{{1 - fn [q_{i} d_{i} ∣ m_{i}] ∙ fn [B_{i} ∣ m_{i}]} ∙ {1 - m_{i} s n^{2} [q_{i} d_{i} ∣ m_{i}] s n^{2} [B_{i} ∣ m_{i}]}}

(14d)

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}}

(15)

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.

For the case C_i ≤ 0, on the other hand, the transverse electric field and all the other parameters a_i ̃, b_i ̃, q_i ̃, and m_i ̃ 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}

(16)

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

(17a)

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

(17b)

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

(17c)

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

(17d)

We just have to replace the parameters a_i , b_i , q_i , and m_i in Eqs. (12)–(15) with a_i ̃, b_i ̃, q_i ̃, and m_i ̃, 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_i ̃, b_i ̃, q_i ̃, m_i ̃, 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

Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multilayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B 19, 1737–1745 (2002). [CrossRef]

]. 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}}

, in the nonlinear substrate and nonlinear cladding

C_{i} > 0, {q_{i}}^{2} = {Q_{i}}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 2,4 \dots, M - 3, in the interaction layers

The Eqs. (14a)–(14d) can be degenerated into the eigenvalue equations as shown in Ref. [31

Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multilayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B 19, 1737–1745 (2002). [CrossRef]

]. 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

Y. D. Wu, “Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film,” IEEE J. Quantum Electron. 40, 529–540 (2004). [CrossRef]

]. 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

, in the linear substrate and linear cladding

C_{R} > 0, {q_{R}}^{2} = {({Q_{R}}^{4} + 2 C_{R} {k_{0}}^{2} α_{R})}^{\frac{1}{4}} = 0, R \in {1,3,5, \dots M - 2}

, in the arbitrary nonlinear guiding film

C_{i} > 0, {q_{i}}^{2} = {Q_{i}}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} | 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 2,4 \dots, M - 3, in the interaction layers

The Eqs. (14a)–(14d) can be degenerated into the eigenvalue equations as shown in Ref. [32

Y. D. Wu, “Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film,” IEEE J. Quantum Electron. 40, 529–540 (2004). [CrossRef]

]. 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

Y. D. Wu and M. H. Chen, “Method for analyzing multilayer nonlinear optical waveguide,” Opt. Express 13, 7982–7995 (2005). http://www.opticsexpress.org/abstract.cfm?id=85750 [CrossRef] [PubMed]

]. 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

, in the linear substrate and linear cladding

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

, i = 1,3 \dots, M - 2, in the nonline ar guiding films

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 2,4 \dots, M - 3, in the interaction layers

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

Y. D. Wu and M. H. Chen, “Method for analyzing multilayer nonlinear optical waveguide,” Opt. Express 13, 7982–7995 (2005). http://www.opticsexpress.org/abstract.cfm?id=85750 [CrossRef] [PubMed]

]. 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).

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

M. O. Twati and T.J.F Pavlasek, “A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass,” Opt. Commun. 206, 327–332 (2002). [CrossRef]

] and X junction structures [42

H. Yokota, K. Kimura, and S. Kurazono, “Numerical analysis of an optical X coupler with a nonlinear dieletric region,” IEICE Trans. Electron. E78-C, 61–66 (1995).

]. The three-branch or multibranch waveguide structures can be used in the all-optical switches [35

Y. D. Wu, “Nonlinear all-optical switching device by using the spatial soliton collision,” Fiber Integrated Opt. 23, 387–404 (2004). [CrossRef]

] or power divider [43

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, “Design and analysis of equal power divider using 4-branch waveguide,” IEEE J. Quantum Electron. 41, 1181–1187 (2005). [CrossRef]

]. A typical application which combined two-branch with three-branch had been studied in Ref. [3

Y. D. Wu, “All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 11, 307–312 (2005). [CrossRef]

]. 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

W. K. Burns and A F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. 11, 32–39 (1975). [CrossRef]

]. 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.

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

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.

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.

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.

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.

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.

To prove the accuracy of numerical results shown in Figs. 6–10, we use the FD-BPM [38

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992). [CrossRef]

] 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.

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.

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).

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).

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

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

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).

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).

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

Y. D. Wu, “New all-optical wavelength auto-router based on spatial solitons,” Opt. Express 12, 4172–4177 (2004). http://www.opticsexpress.org/abstract.cfm?id=81074 [CrossRef] [PubMed]

, 44

G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000). [CrossRef]

]. We give a detail modal analysis of the proposed nonlinear optical waveguide structure. The analytical and numerical results show excellent agreement.

Appendices

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}}

, in the nonlinear substrate and nonlinear cladding

C_{i} > 0, {q_{i}}^{2} = {Q_{i}}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 2,4 \dots, M - 3, in the interaction layers

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}

(AI-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}

(AI -2)

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

, in the linear substrate and linear cladding

C_{R} > 0, {q_{R}}^{2} = {({Q_{R}}^{4} + 2 C_{R} {k_{0}}^{2} α_{R})}^{\frac{1}{4}}, R \in {1,3,5, \dots, M - 2}

, in the arbitrary nonlinear guiding film

C_{i} > 0, {q_{i}}^{2} = {Q_{i}}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 1,3 \dots, M - 2, in the guiding films, for n_{e} < n_{i}

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 2,4 \dots, M - 3, in the interaction layers

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})}

= \frac{{q_{R} A_{R} [1 - m_{R} [1 - {(\frac{A_{R}}{b_{R}})}^{2}]] sn [q_{R} d_{R}] dn [q_{R} d_{R}] - q_{R} A_{R} m_{R} [1 - {(\frac{A_{R}}{b_{R}})}^{2}] sn [q_{R} d_{R}] c n^{2} [q_{R} d_{R}] dn [q_{R} d_{R}] + (- A_{R} q_{R - 1}) cn [q_{R} d_{R}] d n^{2} [q_{R} d_{R}] - (- A_{R} q_{R - 1}) {(\frac{A_{R}}{b_{R}})}^{2} m_{R} s n^{2} [q_{R} d_{R}] cn [q_{R} d_{R}]}}{2 {1 - m_{R} [1 - {(\frac{A_{R}}{b_{R}})}^{2} s n^{2} [q_{R} d_{R}]]}^{2}}

+ \frac{{A_{R} cn [q_{R} d_{R} ∣ m_{R}] - \frac{(- A_{R} q_{R})}{q_{R}} sn [q_{R} d_{R} | m_{R}] dn [q_{R} d_{R} | m_{R}]}}{2 {1 - m_{R} [1 - {(\frac{A_{R}}{b_{R}})}^{2}] s n^{2} [q_{R} d_{R}]}}

(AII-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

, in the linear substrate and linear cladding

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

, i = 1,3 \dots, M - 2, in the nonlinear guiding films

C_{i} < 0, {\tilde{q}}_{i}^{2} = {- Q}_{i}^{2}, m_{i} = 0, i = 2,4 \dots, M - 3, in the interaction layers

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})}

\frac{= {q_{R} A_{R} [1 - m_{R} [{1 - (\frac{A_{R}}{b_{R}})}^{2}]] sn [q_{R} d_{R}] dn [q_{R} d_{R}] - q_{R} A_{R} m_{R} [{1 - (\frac{A_{R}}{b_{R}})}^{2}] sn [q_{R} d_{R}] {cn}^{2} [q_{R} D_{R}] dn [q_{R} d_{R}] + (- A_{R} q_{R - 1}) cn [q_{R} d_{R}] {dn}^{2} [q_{R} d_{R}] - (- A_{R} q_{R - 1}) {(\frac{A_{R}}{b_{R}})}^{2} m_{R} {sn}^{2} [q_{R} d_{R}] cn [q_{R} d_{R}]}^{2}}{2 {1 - m_{R} [1 - {(\frac{A_{R}}{b_{R}})}^{2}] {sn}^{2} [q_{R} d_{R}]}^{2}} + \frac{{A_{R} cn [q_{R} d_{R} ∣ m_{R}] dn [q_{R} d_{R} ∣ m_{R}]}}{2 {1 - m_{R} [1 - {(\frac{A_{R}}{b_{R}})}^{2}] {sn}^{2} [q_{R} d_{R}]}}

(AIII-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.

References and Links

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OCIS Codes
(130.2790) Integrated optics : Guided waves
(190.0190) Nonlinear optics : Nonlinear optics
(190.3270) Nonlinear optics : Kerr effect
(230.1150) Optical devices : All-optical devices
(230.4170) Optical devices : Multilayers
(230.7390) Optical devices : Waveguides, planar

ToC Category:
Nonlinear Optics

History
Original Manuscript: September 28, 2006
Revised Manuscript: February 6, 2007
Manuscript Accepted: February 20, 2007
Published: March 5, 2007

Citation
Chih-Wen Kuo, Shih-Yuan Chen, Mao-Hsiung Chen, Chih-Fu Chang, and Yaw-Dong Wu, "Analyzing multilayer optical waveguide with all nonlinear layers," Opt. Express 15, 2499-2516 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2499

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References

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006). [CrossRef]
Y. D. Wu, "New all-optical switch based on the spatial soliton repulsion," Opt. Express 14, 4005-4012 (2006). [CrossRef] [PubMed]
Y. D. Wu, "All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity," IEEE J. Sel. Top. Quantum Electron. 11, 307-312 (2005). [CrossRef]
J.-G. Ma and I. Wolff, "Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media," IEEE Trans. Microw. Theory Tech. 43, 790-795 (1995). [CrossRef]
T. Rozzi, "Modal analysis of nonlinear propagation in dielectric slab waveguide," J. Lightwave Technol. 14, 229-234 (1996). [CrossRef]
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