Complex coupled-mode theory for optical waveguides

Wei-Ping Huang; Jianwei Mu

doi:10.1364/OE.17.019134

1. Introduction

Coupled- mode theory (CMT) has been widely applied as an analytical approach for analysis of wave propagation and field interaction in optical waveguides [1–5]. The classical coupled-mode theory for optical waveguides was developed in early seventies [6–11], followed by a series of advances in both theoretical formulations [12–15] and applications [16–18]. By expanding the total field in an optical waveguide in terms of the field of a reference waveguide structure, a set of coupled ordinary differential equations are derived for the mode amplitudes of the forward and backward propagating fields. Treatment for discrete guided modes has always been an advantage in the coupled-mode theory due to the fact that only those modes close to phase matching play significant roles in the interaction of the modal fields. Also, in practical optical waveguide structures such as optical fibers, an outer cladding region with lower refractive index and high loss is normally utilized. Under this circumstance, one can solve and apply the discrete and complex modes based on the more realistic model [19]. On the other hand, however, for waveguides with infinite cladding (frequently used due to its simplicity) and with finite cladding with index lower than of the outer cladding (widely used for sensing applications), the radiation modes become inevitable. For non-guided radiation fields, the application of coupled-mode theory becomes cumbersome due to the continuous spectrum of radiation modes. For certain waveguide structures such as step-index slab waveguides and circular fibers, the analytical expressions for the radiation modes do exist and can be utilized as illustrated in [17]. The problem is acute for waveguide structures for which analytical expressions for the radiation modes are not readily available. One possible solution to circumvent the problem of radiation modes is to introduce leaky modes to approximate the radiation modes [18, 20, 21]. The leaky modes are, however, not orthogonal and normalizable in real domain. For this reason, it is difficult to deal with leaky mode formulations analytically and even more so numerically for practical applications.

Similar situation occurs for the mode-matching method in which both discrete guided and continuous radiation modes are required in the mode expansion in order to obtain accurate simulation results. Theoretically, the radiation modes may be discretized into a set of box modes by use of perfectly reflecting boundary condition enclosing the transverse structure of the waveguides. Practically, the size of the artificial box must be sufficiently large in order to achieve accurate approximation for the radiation modes. Consequently, a large of number of box modes must be used. Further, this approach is much less appealing to the coupled-mode theory as a large number of box modes with small mode spacing greatly undermine the benefits of the theory in the first place.

Recently, a new computation model was introduced to the mode-matching method in which the waveguide structure is enclosed by a perfectly matched layer (PML) terminated by a perfectly reflecting boundary conditions (PRB) [22–24]. This seemingly paradoxical combination of PML and PRB leads to a somewhat unexpected yet remarkable result: it creates an open and reflectionless environment equivalent to the original physical domain yet in a close and finite computation domain. A set of complex modes can be derived from this waveguide model that are well behaved in terms of orthogonality and normalization and can be readily solved by standard analytical and numerical techniques. By utilizing the complex modes as an orthogonal base functions to represent the radiation fields, the mode-matching method can be applied as if all the modes are discrete and guided. The complex mode-matching method was subsequently applied to simulation and analysis of slab, circular, and channel optical waveguide structures and shown to be highly accurate and versatile. In this work, we employ the same idea in the coupled-mode theory.

In section 2, the waveguide model on which the complex coupled-mode theory is described and the characteristics of the complex modes are examined. Formulations for the complex coupled-mode theory is derived and discussed in Section 3. Solutions for the complex coupled-mode equations for periodic waveguide structures are derived and discussed in 4. In section 5, the theory is applied to slab waveguide Bragg grating with different refractive indices for the surrounding outer cladding. Finally, a summary and conclusion are given in Section 6.

2. Waveguide model and mode characteristics

The waveguide models for which the complex coupled- mode theory is to be formulated in depicted in Fig. 1 .The waveguide structures to be investigated are enclosed transversely by the perfectly matched layer (PML) and terminated by the perfectly reflecting boundary (PRB) condition. The thickness and absorption profile parameters of the PML are chosen such that the reflection generated at the boundary is attenuated in the PML. As such the waveguide domain including the core and the cladding regions will not be affected by the reflection and will work as if it is an open waveguide [25, 26].

Fig. 1 Ideal waveguide models

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The modal solutions for the forward propagating fields on the reference waveguide structures with PML are governed by the modified Maxwell’s equations

\nabla_{t} \times e_{n} - j γ_{n} e_{n} = - j ω μ_{o} [Λ] h_{n}

\nabla_{t} \times h_{n} - j γ_{n} h_{n} = + j ω ε [Λ] e_{n}

where

ε (x, y)

is the refractive index of the unperturbed reference waveguide without PML. The tensor

[Λ]

accounts for the PML and is given as

[Λ] = [\begin{array}{c} S_{y} / S_{x} & 0 & 0 \\ 0 & S_{x} / S_{y} & 0 \\ 0 & 0 & S_{x} S_{y} \end{array}]

where

S_{x}

and

S_{y}

are called the coordinate stretching factors and given by

S_{x} = κ_{x} - j \frac{σ_{x}}{ω ε}

S_{y} = κ_{y} - j \frac{σ_{y}}{ω ε}

and

κ_{x}

(

κ_{y}

) and

σ_{x}

(

σ_{y}

) are the parameters to control the phase shift and absorption of the travelling waves in the PML along x (y). For non-PML media, we have

κ_{x} = κ_{y} = 1

,

σ_{x} = σ_{y} = 0

and hence

S_{x} = S_{y} = 1

.

In practice, we normally set the phase-shift parameter $κ_{x}$ and $κ_{y}$ to unity since they are significant only for dealing with evanescent fields. On the other hand, the profiles of the absorption coefficients $σ_{x}$ and $σ_{y}$ are critical for effectively reducing the reflections from the perfectly reflecting boundary. A commonly used expression for the absorption profile is

σ = σ_{\max} {(\frac{ρ}{T_{P M L}})}^{m}

where

T {}_{P M L}

is the thickness of the PML layer and ρ is the distance measured from the starting position of the PML. The subscript is dropped in Eq. (4) and subsequent equations for the sake of brevity. A good measure for the effectiveness of the PML is the reflection coefficient defined by

R_{P M L} = \exp {- \frac{2 σ_{\max}}{n \sqrt{ε_{o} / μ_{o}}} \int_{0}^{T_{P M L}} {(\frac{ρ}{T_{P M L}})}^{m} d ρ}

In terms of the PML reflection coefficient, the PML coefficient can be conveniently expressed as

S = κ - j \frac{λ}{4 π n T_{P M L}} [(m + 1) \ln (\frac{1}{R_{P M L}})] {(\frac{ρ}{T_{P M L}})}^{m}

It was shown that when $m = 2$ the PML seems to be the most effective [27]. Note that due to the complex nature of the PML coordinate stretching factors, the modes solved from the waveguide models are generally complex. This is true even waveguides made of real refractive indices (lossless and free of gain). The modal propagation constants are expressed as

γ_{n} = β_{n} - j α_{n}

Similarly, the governing equations for the backward propagating modes in the presence of PML are given by

\nabla_{t} \times e_{- n} - j γ_{- n} e_{- n} = - j ω μ_{o} [Λ] h_{- n}

\nabla_{t} \times h_{- n} - j γ_{- n} h_{- n} = + j ω ε [Λ] e_{- n}

By virtue of symmetry, the forward and backward propagation waves are related by

γ_{n} = - γ_{n}

e_{t n} = + e_{t, - n}

e_{_{z n}} = - e_{_{z, - n}}

h_{t n} = - h_{t, - n}

h_{z n} = + h_{z, - n}

It is readily shown that the complex modes obtained from Eqs. (1) are orthogonal in the sense that

\iint_{} (e_{t m} \times h_{t n}) \cdot \hat{z} d a = 0

for

m \neq n

in which the integral is over the entire waveguide model domain including the PML regions. Nevertheless, we may still normalize the complex mode in terms of power such that

\frac{1}{2} ℜ \iint_{} (e_{t n}^{} \times h_{t n}^{*}) \cdot \hat{z} d a = 1

It is interesting to note that for the complex modes, namely, the cladding leaky modes and the PML modes derived from Eqs. (1), the conventional power orthogonality is no longer valid. In other words, the overlap integrals with complex conjugate may not vanish, i.e.,

\iint_{} (e_{t m}^{} \times h_{t n}^{*}) \cdot \hat{z} d a \neq 0

The complex modes may be solved by standard analytical and/or numerical methods once the transverse waveguide structure is given and the PML parameters are selected. In the case of numerical solutions such as the finite-difference scheme, the standard numerical solvers can be applied by simply multiplying the mesh along x (y) direction by the corresponding coordinate stretching factor.

To illustrate the salient features of the complex modes and their dependence on PML parameters, we calculate the mode effective indices and corresponding field patterns for a step-index slab waveguide with refractive indices of the core, inner cladding, and outer cladding equal to $n_{c o}$ , $n_{c l}$ , and $n_{s}$ , respectively. In particular, we vary the value of the refractive index in the outer cladding from below to equal and above the refractive index of the inner cladding so that the modal characteristics of the cladding guided, radiative, and leaky situations can be examined in detail. The waveguide parameters are chosen such that the refractive indices of the core and the inner cladding are $n_{c o}$ =1.458 and $n_{c l} =$ $1.450$ , respectively. The half widths of the core and the inner cladding layers are $d_{c o} =$ $2.5 μ m$ and $d_{c l} =$ $12.5 μ m$ . The operation wavelength is assumed to be $λ =$ $1.550 μ m$ . The thickness of the PML layer is $T_{P M L} =$ $2.5 μ m$ and positioned at $d_{P M L} =$ $38 μ m$ . The PML absorption coefficient is chosen such that the total PML reflection coefficient is $R_{P M L} =$ $10^{- 12}$ .

Figure 2(a) shows the real and imaginary parts of the effective indices for the first twenty (20) modes supported in the waveguide for four different values of the refractive index in the outer cladding. In order to clearly observe the characteristics for the guided and quasi-leaky modes, we further blow up a portion of the graph in the dashed circle in Fig. 2(a) and show it in Fig. 2(b).

Fig. 2 (a) Real and imaginary parts of the effective indices for the first twenty (20) modes (in descent order based on the values of the real part for the mode effective indices) supported in the waveguide for the different outer cladding indices. (b) A blow-up view for the portion of graph in (a) enclosed by the dashed circle.

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It is apparent that, in the case that the index of the outer cladding is lower than that of the inner cladding, all the modes are guided with real effective indices as if the PML and PRB do not exist. In the case of infinite cladding with $n_{c l} = n_{s}$ and leaky waveguide structures with outer cladding index higher than the inner cladding ( $n_{s} > n_{c l}$ ), the cladding modes become complex and are divided into two groups, namely, the cladding quasi-leaky modes mainly confined in the cladding with similar behaviors to leaky modes and the PML modes that are mainly confined in the PML region. The imaginary parts of the quasi-leaky modes represent the radiation loss. It is observed that, as the refractive index of the outer cladding increases from below to above that of the inner cladding, the leakage loss of the cladding modes increases. Further increase of the outer cladding index, however, leads to a decrease in leakage loss due to the increase in reflection from the interface between the inner and outer cladding.

Figure 3 plots the mode leakage loss defined as the power attenuation coefficient in dB per mm for the first five (5) quasi-leaky modes as functions of the outer cladding index. It is observed that the leakage losses of the higher-order complex modes are higher. In addition, we see that, as the refractive index of the outer cladding increase beyond that of the inner cladding, the leakage loss of the corresponding complex modes decreases due to the increase of the reflection at the inner and outer cladding interface.

Fig. 3 Mode leakage loss for the quasi-leaky modes.

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Further, we observe the presence of the PML modes whose imaginary parts of the effective mode indices are much greater than those for the quasi-leaky modes. It is also noted that as the mode order increases, the imaginary parts of the PML modes increase but their real parts decrease, which are fundamentally different from the characteristics of the quasi-leaky modes.

The mode field patterns of the fundamental guided mode, the lowest order quasi-leaky cladding mode, and the lowest order PML mode are depicted in Fig. 4 for the three waveguide structures (i.e., $n_{s} < n_{c l}$ , $n_{s} = n_{c l}$ and $n_{s} > n_{c l}$ ).

Fig. 4 Mode field patterns for the three waveguide structures. (a) the fundamental guided mode; (b) the quasi-leaky cladding modes; (c) the PML modes.

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It is noted from Fig. 4(a) that the guided mode is well confined in the core and hence not affected by the change of refractive index in the outer cladding as expected. On the other hand, fields of the quasi-leaky modes are primarily concentrated in the inner cladding region and hence highly sensitive to the variations of refractive index in the outer cladding. It is observed that more leakage for the cladding modes occurs when the refractive index of the outer cladding is slightly greater than that of the inner cladding, e.g., $n_{s} = 1.455$ . The cladding mode suffers less loss as the index of the outer cladding becomes significantly larger than that of the inner cladding due to the reflection from the interface between the inner and the outer claddings. Finally, the PML modes are mainly present in the PML region and suffer huge mode losses.

The use of PML does introduce additional complexity in the waveguide model with more parameters. Therefore, it is instructive to investigate the effect on PML parameters on the mode characteristics. The field confinement factors in the core and the inner cladding are calculated for the guided core, quasi-leaky cladding and the PML modes for the variations of the PML reflection coefficients (Fig. 5 ) and the starting position of the PML (Fig. 6 ), respectively. It is observed that the core guided mode and the lower order cladding quasi-leaky modes are not much affected by the variation of the PML parameters as shown in Fig. 5(a) and 6(a). On the other hand, the higher-order cladding leaky modes and the PML modes are more sensitive to the computation windows size and the PML absorption coefficient as shown in Fig. 5 (b) and Fig. 6 (b). The use of PML reflection of $10^{- 12}$ and the computation window size of $90.5 μ m$ appears to be sufficient for the calculation of the complex modes of the waveguide under investigation.

Fig. 5 Field confinement factor as a function of PML parameters.

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Fig. 6 Field confinement factors in core and cladding region as functions of computation window ( $n_{s}$ = 1.455).

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Further, we investigate the field overlap integrals defined in Eq. (12) and Eq. (14) for the first forty (40) modes obtained for the waveguide structure with infinite cladding ( $n_{s} = n_{c l} = 1.450$ ) and show the results for the mode order m and n in Fig. 7(a) and Fig. 7(b), respectively. The mode orthogonality defined in Eq. (12) in terms of the overlap integral without complex conjugate is clearly demonstrated in Fig. 7(a) for the complex modes. On the other hand, these complex modes are not necessarily power-orthogonal, i.e., the overlap integral with complex conjugate customarily used in the conventional coupled-mode theory Eq. (13) are not always null, especially for the high-order complex modes as illustrated in Fig. 7(b). Nevertheless, for the coupling among guided and lower-order complex modes with relatively small leakage losses, the power orthogonality relationship is approximately valid as demonstrated in Fig. 7(b).

Fig. 7 Mode field overlap integrals for different modes. (a)Non-conjugate overlapping integral using Eqs. (13); (b) Conjugate overlapping integral using Eqs. (12). ( $n_{s} = 1.45$ .)

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3. Derivation of complex coupled-mode theory

To derive the coupled-mode equations based on the complex modes described in Section 2, we assume that the permittivity function distribution along the waveguide with perturbations can be expressed as

\tilde{ε} (x, y, z) = ε (x, y) + Δ ε (x, y, z)

where the index perturbation

Δ ε

is defined as the difference between the index profiles of the practical waveguides under investigation and the reference waveguides for which the complex modes are known. Maxwell’s equations for the perturbed waveguides are

\nabla \times E (x, y, z) = - j ω μ_{o} H (x, y, z)

\nabla \times H (x, y, z) = + j ω \tilde{ε} (x, y, z) H (x, y, z)

Suppose that the difference between the perturbed and the reference waveguides is sufficiently small so that we can expand the unknown transverse electromagnetic fields of the perturbed waveguides in terms of the transverse modal fields of the reference waveguides, i.e.,

E_{t} (x, y, z) = \sum_{n = 1} [a_{n} (z) + b_{n} (z)] e_{t n} (x, y)

H_{t} (x, y, z) = \sum_{n = 1} [a_{n} (z) - b_{n} (z)] h_{t n} (x, y)

The functions $a_{n} (z)$ and $b_{n} (z)$ are the mode amplitudes for the forward and backward propagating waves, respectively. Substitution of Eqs. (17) into Eqs. (16) leads to the expressions for the longitudinal fields as

E_{z} (x, y, z) = \sum_{n = 1} [a_{n} (z) - b_{n} (z)] \frac{ε}{\tilde{ε}} e_{z n} (x, y)

H_{z} (x, y, z) = \sum_{n = 1} [a_{n} (z) + b_{n} (z)] h_{z n} (x, y)

The next step is to derive the coupled-mode equations governing the mode amplitudes. To do so, we simply substitute Eqs. (17) and (18) into Eqs. (16). With some mathematical manipulations, we derive the following coupled equations:

N_{m} \frac{d a_{m}}{d z} + j γ_{m} a_{m} = - j \sum_{n = 1} κ_{m n} a_{n} - j \sum_{n = 1} χ_{m n} b_{n}

N_{m} \frac{d b_{m}}{d z} - j γ_{m} b_{m} = + j \sum_{n = 1} κ_{m n} b_{n} + j \sum_{n = 1} χ_{m n} a_{n}

in which the coupling coefficients are given by

κ_{m n} = \frac{ω ε_{o}}{4} \int_{A} ({\tilde{n}}^{2} - n^{2}) (e_{t n} \cdot e_{t m} - \frac{n^{2}}{{\tilde{n}}^{2}} e_{z n} \cdot e_{z m}) d a

χ_{m n} = \frac{ω ε_{o}}{4} \int_{A} ({\tilde{n}}^{2} - n^{2}) (e_{t n} \cdot e_{t m} + \frac{n^{2}}{{\tilde{n}}^{2}} e_{z n} \cdot e_{z m}) d a

where the refractive indices are used to replace the permittivity functions, i.e.,

ε = n^{2} ε o

and

\tilde{ε} = {\tilde{n}}^{2} ε o

. The coefficient

N_{m}

is defined as

N_{m} = \frac{1}{2} \iint_{} (e_{t m} \times h_{t m}) \cdot \hat{z} d a

Under the normalization condition (13), $N_{1} = 1$ for the real core guided modes and $N_{m} ≅ 1$ for the complex cladding leaky modes with relatively small mode losses. In general, however, $N_{m}$ may not be equal to unity and can be complex.

The coupled-mode Eqs. (20) and the expressions for the coupling coefficients (21) are formally identical to those derived for guided modes in waveguides made of reciprocal media in presence of loss and/or gain and hence can be solved by the same analytical and numerical techniques as previously done in literature. In the classical coupled-mode theory, to deal with radiation fields based on integral of the continuous radiation modes is possible but extremely cumbersome. On the other hand, however, both guided and radiation fields are considered in the new complex coupled-mode theory in a unified fashion. Also, it is noted that the coupling coefficients for the co- and contra-coupling modes are symmetrical in the sense that

κ_{m n} = κ_{n m}

χ_{m n} = χ_{n m}

Further, we may rewrite the mode amplitudes by separating the slowly varying envelopes with the fast oscillating carriers so as to

a_{n} = A_{n} \exp (- j γ_{n} z)

b_{n} = B_{n} \exp (+ j γ_{n} z)

On substitution of Eqs. (19) by Eqs. (23), we derive

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} κ_{m n} A_{n} \exp [- j (γ_{n} - γ_{m}) z] - j \sum_{n = 1} χ_{m n} B_{n} \exp [+ j (γ_{n} + γ_{m}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} κ_{m n} B_{n} \exp [+ j (γ_{n} - γ_{m}) z] + j \sum_{n = 1} χ_{m n} A_{n} \exp [- j (γ_{n} + γ_{m}) z]

For grating structures whose index perturbations are periodic along the waveguide axis, the coupling coefficients Eqs. (20) are also periodic functions with the same period. We may expand the coupling coefficients in terms of Fourier series as

κ_{m n} = \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp (j l \frac{2 π}{Λ} z)

χ_{m n} = \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp (j l \frac{2 π}{Λ} z)

Substitute Eqs. (25) into Eqs. (24) and we derive

\begin{array}{l} N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [- j (γ_{n} - γ_{m} - l \frac{2 π}{Λ}) z] \\ - j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [+ j (γ_{n} + γ_{m} + l \frac{2 π}{Λ}) z] \end{array}

\begin{array}{c} N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [+ j (γ_{n} - γ_{m} + l \frac{2 π}{Λ}) z] \\ + j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [- j (γ_{n} + γ_{m} - l \frac{2 π}{Λ}) z] \end{array}

It can be shown that the phase factors $γ_{n}$ $- γ_{m}$ $\pm l 2 π / Λ$ and $γ_{n}$ $+ γ_{m}$ ± $l 2 π / Λ$ in the exponential terms in Eqs. (26) are most critical in the determining the strength of interactions between the different modes over distance. Only when these factors close to zero there will be appreciable power exchange between the coupled modes, a condition referred to as the phase-matching conditions. In fact, the role of grating is to facilitate the phase matching between two propagation modes with different propagation constants by providing a grating space harmonic component related to the grating period and profile. Note that the phase matching conditions for the co- and contra-propagating modes are quite distinct and may not be readily realized by the same grating. In practice, we normally design the grating to assist coupling for either contra-propagating (e.g., Bragg gratings) or co-propagating (e.g., long-period gratings) waves only. Under this assumption, the coupled mode equations are further reduced to two groups as described in the following.

For the contra-propagation waves, the coupled-mode equations reduce to

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [+ j (γ_{n} + γ_{m} + l \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [- j (γ_{n} + γ_{m} - l \frac{2 π}{Λ}) z]

and for the co-directional propagation waves, we have

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [- j (γ_{n} - γ_{m} - l \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [+ j (γ_{n} - γ_{m} + l \frac{2 π}{Λ}) z]

Further, for a given grating, the phase matching condition is a function of wavelength through wavelength dependence of the mode propagation constants, i.e., $γ_{n} (λ)$ and $γ_{m} (λ)$ . It also depends on the mode index ( $m, n = 1, 2, ...$ ) and the order of space harmonics in the Fourier expansion ( $l = - \infty, ... + \infty)$ . Any combination of these parameters that lead to a phase matching condition will likely yield a distinct resonant signature in the mode coupling as illustrated later in the transmission and reflection spectra in this paper. By identifying these phase matching conditions in the coupled- mode equations, we can greatly simplify the solutions and also gain great insight into the salient features underlying the interaction of the modes in presence of the gratings.

First of all, we may consider only the largest Fourier expansion coefficient, i.e., $l = \pm 1$ or the 1st-order grating effect and ignore all these other high-order space harmonics. Investigation of the higher-order gratings can be performed in the similarly fashion, but will not be pursued further in this work. The coupled-mode Eqs. (27) and (28) are subsequently decoupled into two separate sets such that

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} B_{n} C_{m n}^{(- 1)} \exp [+ j (γ_{n} + γ_{m} - \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} A_{n} C_{m n}^{(+ 1)} \exp [- j (γ_{n} + γ_{m} - \frac{2 π}{Λ}) z]

for the contra-directional propagation waves and

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} A_{n} D_{m n}^{(- 1)} \exp [- j (γ_{n} - γ_{m} + \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} B_{n} D_{m n}^{(+ 1)} \exp [+ j (γ_{n} - γ_{m} + \frac{2 π}{Λ}) z]

for the co-directional propagation waves.

Further, in many practical situations, the phase matching conditions can only be realized at a distinct wavelength (i.e., $λ {}_{m n}$ ) for a given pair of modes (i.e., m and n). For the sake of simplicity, we suppose that the m-th mode is the forward propagating fundamental guided mode with largest real propagation constant, i.e., $m = 1$ and $γ_{m} = β_{1}$ (real). In the proximity of the phase-matching wavelength $λ ≅ λ {}_{1 n}$ , we may consider only the two modes that are close to the phase match (whenever it is possible!) so that Eqs. (29) and (30) are simplified to

N_{1} \frac{d A_{1}}{d z} = - j C_{1 n}^{(- 1)} B_{n} \exp [+ j (γ_{n} + β_{1} - \frac{2 π}{Λ}) z]

N_{n} \frac{d B_{n}}{d z} = + j C_{n 1}^{(+ 1)} A_{1} \exp [- j (γ_{n} + β_{1} - \frac{2 π}{Λ}) z]

for the contra-directional propagation modes, and

N_{1} \frac{d A_{1}}{d z} = - j D_{1 n}^{(- 1)} A_{n} \exp [- j (γ_{n} - β_{1} + \frac{2 π}{Λ}) z]

N_{n} \frac{d A_{n}}{d z} = - j D_{n 1}^{(+ 1)} A_{1} \exp [+ j (γ_{n} - β_{1} + \frac{2 π}{Λ}) z]

for the co-directional propagating modes.

Note that $γ_{n} = β_{n} - j α_{n}$ and define the phase detuning factors such that

Δ β_{n} = \frac{1}{2} (γ_{1} + γ_{n} - \frac{2 π}{Λ})

for the contra-directional propagation modes and

Δ β_{n} = \frac{1}{2} (γ_{1} - γ_{n} - \frac{2 π}{Λ})

for the co-directional propagation modes. The phase matching conditions happened as

ℜ (Δ β_{n}) = 0

so that the grating period for the phase matching conditions for the contra- and co-directional modes are

Λ = \frac{2 π}{ℜ (γ_{1} + γ_{n})}

and

Λ = \frac{2 π}{ℜ (γ_{1} - γ_{n})}

We may recast Eqs. (31) and (32) into more revealing forms as follows:

N_{1} \frac{d A_{1}}{d z} = - j C_{1 n}^{(- 1)} B_{n} \exp [+ j (2 Δ β_{n}) z]

N_{n} \frac{d B_{n}}{d z} = + j C_{n 1}^{(+ 1)} A_{1} \exp [- j (2 Δ β_{n}) z]

for the contra-directional modes and

N_{1} \frac{d A_{1}}{d z} = - j D_{1 n}^{(- 1)} A_{n} \exp [- j (2 Δ β_{n}) z]

N_{n} \frac{d A_{n}}{d z} = - j D_{n 1}^{(+ 1)} A_{1} \exp [+ j (2 Δ β_{n}) z]

for the co-directional modes.

4. Solutions of complex coupled-mode theory

Computationally, we may solve the full coupled mode Eqs. (29) and (30) (referred to as the full CMT) or the reduced Eqs. (36) and (37) (referred to as the reduced CMT). The former can be readily carried out by a standard numerical algorithm such as Ruga-Kutta method whereas the latter can be solved to yield simple analytical formulas.

Suppose that all the power is initially launched in the forward-propagating fundamental mode, i.e., $a_{1} (0) = 1$ . For the contra-directional modes, we assume that no power is associated with the backward-propagating modes at the other side of the grating, i.e., $b_{n} (L) = 0$ . The analytical solutions are

a_{1} (z) = a_{1} (0) \frac{Δ β_{n} \sin h S (z - L) - j S \cos h S (z - L)}{- Δ β_{n} \sin h S L - j S \cos h S L} e^{- j (β_{1} - Δ β_{n}) z}

b_{n} (z) = - a_{1} (0) \frac{C_{n 1}^{(+ 1)}}{N_{n}} \frac{\sin h S (z - L)}{Δ β_{n} \sin h S L + j S \cos h S L} e^{j (γ_{n} - Δ β_{n}^{}) z}

where

S = \sqrt{κ_{n}^{2} - {(Δ β_{n})}^{2}}

κ_{n} = \sqrt{\frac{C_{1 n}^{(- 1)} C_{n 1}^{(+ 1)}}{N_{1} N_{n}}}

For the co-directional modes, we assume that no power is associated with the secondary mode at the starting point of the grating, i.e., $a_{n} (0) = 0$ ( $n \neq 1$ ). The solutions are

a_{1} (z) = a_{1} (0) \frac{j Δ β \sin (Q_{n} z) + Q_{n} \cos (Q_{n} z)}{Q_{n}} e^{- j (β_{1} - Δ β_{n}^{}) z}

a_{n} (z) = - j e^{j Δ β_{n}^{} z} \frac{D_{n 1}^{(+ 1)}}{N_{n}} \frac{\sin (Q_{n} z)}{Q_{n}} a_{1} (0) e^{- j (γ_{n} + Δ β_{n}^{}) z}

where

Q_{n} = \sqrt{χ_{n}^{2} + {(Δ β_{n})}^{2}}

χ_{n} = \sqrt{\frac{D_{n 1}^{(+ 1)} D_{1 n}^{(- 1)}}{N_{n} N_{1}}}

Note that the analytical solutions of the reduced coupled-mode equations are formally identical to those derived previously for real modes, except that the effective coupling coefficients $κ_{n}$ (or $χ_{n}$ ) and the equivalent phase detuning factor $Δ β_{n}$ may become complex.

Once we obtain the mode amplitudes, we will be able to calculate the guided powers carried by each of the modes. In general, the power flow along the waveguide is given as

P (z) = \frac{1}{2} ℜ \int_{A} [E_{t} (x, y, z) \times H_{t}^{*} (x, y, z)] \cdot \hat{z} d a

By substitution of the field expansions in terms of the complex modes Eqs. (17) into Eq. (44), we obtain

P (z) = \sum_{m = 1} \sum_{n = 1} M_{m n} [a_{m} a_{n}^{*} - b_{m} b_{n}^{*}] - \sum_{m = 1} \sum_{n = 1} N_{m n} [a_{m} b_{n}^{*} - b_{m} a_{n}^{*}]

where the first summation is for the power associated with the co-directional propagating modes in the forward and backward directions whereas the second summation is related to the power of the contra-directional propagating modes. The cross-power coefficients are defined as

M_{m n} = \frac{1}{4} \iint_{A} (e_{t m} \times h_{t n}^{*} + e_{t n}^{*} \times h_{t m}^{}) \cdot \hat{z} d a

N_{m n} = \frac{1}{4} \iint_{A} (e_{t m} \times h_{t n}^{*} - e_{t n}^{*} \times h_{t m}^{}) \cdot \hat{z} d a

Normally, the cross-power associated with the contra-directional modes is negligible in comparison with that with the co-directional modes and hence the second summation in Eq. (45) may be ignored so that

P (z) ≅ \sum_{m = 1} \sum_{n = 1} M_{m n} [a_{m} a_{n}^{*} - b_{m} b_{n}^{*}]

For the complex modes that are significant in the mode coupling process, the transverse fields are almost real and hence these modes are almost power orthogonal so that we may further approximate Eq. (47) as

P (z) ≅ \sum_{n = 1} [{| a_{n} |}^{2} - {| b_{n} |}^{2}]

Assume that the total power is launched into the forward propagating fundamental mode at the input. For the contra-directional mode coupling in Bragg grating, the total power at the input of the grating is therefore

P (0) = 1 - R (λ)

in which the total reflected power is expressed as

R (λ) ≅ \sum_{m = 1} \sum_{n = 1} M_{m n} b_{m} (0) b_{n}^{*} (0)

and the mode indices m and n cannot be equal to unity simultaneously. If we may further neglect the cross-power associated with the co-directional propagating modes, i.e.,

M_{m n} ≅ 0

and

M_{n n} ≅ 1

, we have

R (λ) ≅ \sum_{n = 1} {| b_{m} (0) |}^{2}

Similarly, if assume that at the output of the grating, no backward propagating modes exist. The total power at the output of the grating is

P (L) = {| a_{1} (L) |}^{2} = T (λ)

The power conservation of the waveguide gratings calls for

P (0)

= P (L)

, i.e.,

T (λ) = 1 - R (λ)

5. Applications of complex coupled-mode theory

As an example for the application for the complex coupled-mode theory derived and solved in Section 3 and 4, we consider the volume Bragg gratings structure in Fig. 8 in which the volume grating is placed in the core along the waveguide axis. The index perturbation of the grating is $Δ n$ , which is assumed to be small in comparison with the refractive index difference between the core and the cladding regions ( $Δ < < n_{c o} - n_{c l}$ ).

Fig. 8 Volume Bragg grating structure based on slab waveguides

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Other types of grating such as the corrugated surface gratings widely used in slab and rectangular waveguides can also be simulated by using the same formulations, but will not be examined without loss of generality. We assume that $Δ n_{g r a t i n g} = 9 \times 10^{- 4}$ and $L = 800 μ m$ and consider the following cases in our simulation.

Case A: Bragg gratings with lower index outer cladding ( $n_{s} < n_{c l}$ )

By considering the phase matching condition, we readily identify three distinct wavelengths corresponding to the Bragg conditions between the forward propagating fundamental mode and the first three backward propagating modes, respectively. Note that under this situation all modes are guided with real propagation constants and coupling coefficients so that the conventional CMT does apply. We subsequently calculated the coupling coefficients for these three pairs of mode coupling and show them in Fig. 9(a) . By applying the analytical solutions of the reduced coupled-mode theory around each of the phase matching wavelengths, we obtained the transmission spectra as indicated by the solid, dotted and dash lines in Fig. 9(a). It is observed that each of the phase-matching condition produces a distinct dip in the transmission spectrum with magnitude proportional to the strength of the coupling coefficient $| κ_{n} |$ . The entire transmission spectra may be obtained by first calculating the reflection spectra using the reduced coupled-mode theory near each of the phase matching points followed by Eqs. (51) and (53). The results are shown in Fig. 9(b) as dash lines. Also shown the same figure are the results obtained by solving the full coupled-mode Eqs. (31) numerically with total of three modes and also by applying the rigorous mode-matching method (MMM) with total of 40 modes. The results of the reduced CMT, the full CMT and the rigorous MMM are all in good agreement, indicating that the reduced CMT is sufficient.

Fig. 9 The transmission spectrum for Case A with lower outer cladding index n_s = 1.0. (a) Phase matching wavelengths, corresponding coupling strengths, and the transmission spectrum predicted by the reduced CMT involving only two phase matching modes; (b) The transmission spectrum calculated by the reduced CMT (dash lines), the full CMT (dotted lines) and the rigorous MMM (solid lines).

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Case B: Bragg gratings with equal index outer cladding ( $n_{s} = n_{c l}$ )

For the infinite cladding gratings ( $n_{s} = n_{c l}$ ), the real, guided cladding modes evolve to complex quasi-leaky modes as well as PML modes. The latter plays negligible roles in the interactions with the grating-assisted couplings between the forward propagating guided mode and the backward propagating complex modes due to their huge mode losses and small mode overlaps with the guided modes in the core. For the quasi-leaky modes, however, the spectral spacing between them is too small and hence the phase-matching wavelengths are hardly distinguishable relative to the spectral width of each transmission dip as illustrated in Fig. 8(a). Further, the coupling strengths $| κ_{n} |$ at these phase matching modes are similar. Consequently, we expect that the resonance-like peaks, which are characteristic in Case A, will not be highly visible in Case B. The results by solving the full CMT with consideration for coupling from 1 to 10 backward propagating modes are shown in Fig. 10(a) . A flat overall drop in the transmission spectrum is predicted by considering all the relevant modes. The accuracy of the complex CMT involving 10 modes is verified by comparison with the results from the rigorous MMM with total of 60 modes as evident in Fig. 10(b).

Fig. 10 The transmission spectrum for Case B with equal outer cladding index n_s = 1.450. (a) Phase matching wavelengths, corresponding coupling strengths, and the transmission spectrum predicted by the full CMT involving from 2 up to 11 modes; (b) The transmission spectrum calculated by the full CMT (dotted lines) and the rigorous MMM (solid lines).

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Shown on the same figure with dotted lines are results obtained using the reduced CMT considering only the phase-matched modes near the phase matching wavelengths. It is indeed surprising to see that the simple solutions of the reduced CMT yield remarkably accurate results even under the situation in which the phase matching wavelengths are very close to each other. In comparison with the conventional coupled-mode theory which has to resort to either continuous radiation modes or large number of box modes or tricky leaky modes, the complex coupled mode theory is much more straightforward in dealing with strong radiation fields in this case.

Case C: Bragg gratings with higher index outer cladding ( $n_{s} > n_{c l}$ )

If the refractive index of the outer cladding is higher than that of the cladding, the waveguide structure becomes leaky in the sense that no guided modes exist for mode index lower than the cladding index. In the case that the outer cladding index is close to that of the cladding index (e.g., n_s = 1.455), the situation is somewhat similar to that of Case B except that the mode spacing becomes larger and hence the phase matching condition more distinct as shown in Fig. 12(a) . We identify total of seven (7) phase-matching modes and calculate the transmission spectrum by solving the full CMT considering coupling from 1 to 7 modes, respectively. An overall drop in the transmission spectrum on the shorter wavelength side is predicted similar to Case A except that the signatures of the resonances at the corresponding phase-matching conditions are more visible. Figure 11(b) shows the comparison between the full CMT (total of 7 modes), the reduced CMT (two modes near each phase-matching wavelengths) and the rigorous MMM (total of 60 modes), which are in excellent agreement with each other.

Fig. 12 Transmission spectra with n_s lager than n_cl. (a) n_s = 1.60; (b) n_s = 1.90.

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Fig. 11 Transmission spectra with index of the outer cladding n_s slightly larger than the index of the inner cladding n_cl (n_s = 1.455).

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If we further increase the refractive index of the outer cladding (e.g., n_s = 1.60 and n_s = 1.90), the mode spacing becomes further apart and the phase-matching wavelengths more distinct as illustrated in Figs. 12 (a) and 12 (b), respectively. Also, the mode losses for the complex modes are in fact smaller due to the reflection from the interface of the cladding and outer cladding and coupling strengths vary more significantly from mode to mode. All these lead to the transmission spectra with highly visible resonance dips as evident in Fig. 12 (a) and 12 (b). The comparison between the full complex CMT with total of 4 modes, the reduced CMT with two modes for each of the phase-matching wavelengths and the rigorous MMM with total of 60 modes shows excellent agreement.

6. Summary and conclusion

A complex coupled-mode theory is developed and presented in which the radiation fields are expanded in terms of complex modes solved from a waveguide model with perfectly matched layer (PML) terminated by a perfectly reflecting boundary (PRB) condition. The combination of PML and PRB produce an ideal waveguide model for which the physical domain of the waveguide is not affected by the reflection from the edge of the computation window and yet the model fields are well defined in a finite and closed simulation environment. These complex modes are characterized by the quasi-leaky cladding modes and the PML modes, which are confined mainly in the cladding and the PML regions, respectively. These complex modes are orthogonal and can be normalized in terms of power.

The complex coupled-mode formulations take the same forms as those for waveguides with loss and/or gain and can be solved by employing the same numerical and analytical methods. Simple analytical solutions are derived for both contra- and co-directional coupled modes in waveguide grating structures. We subsequently applied the complex coupled-mode theory to simulation of Bragg grating along the core of a symmetric slab waveguides with different refractive indices for the outer cladding. In particular, we examined the cases where the index of the outer cladding is lower, equal and higher than that of the inner cladding. While the first situation has been studied extensively by the conventional coupled-mode theory involving real guided core and cladding modes, the latter two cases are practically cumbersome to treat due to the involvement of continuous radiation modes. By using the complex coupled-mode theory, all cases can be treated readily in the same fashion and the simulation results of the complex coupled-mode theory are shown to yield results in excellent agreement with the rigorous mode matching method. In comparison with the mode-matching method which is essentially numerical in nature, the coupled-mode theory is more intuitive and insightful. By identifying the distinct phase-matching conditions for different modes at different wavelengths, we can further reduce the coupled-mode equations to a simple two-mode formulation. Analytical solutions obtained from the reduced coupled-mode theory are shown to be highly accurate. We believe that the complex coupled-mode theory extended the scope of the conventional coupled-mode theory based on guided modes so that field radiation and leakage loss in typical optical waveguides can be treated easily, accurately and efficiently. A wide range of applications of this powerful theory can therefore envisaged with the new complex coupled-mode theory.

Acknowledgement

This research is sponsored by NSERC Research Grant. Of of the authors (J.W. Mu) also acknowledges the financial support through Alexander Graham Bell Canada Graduate Scholarship. We wish to thank Dr. Ning Song for insightful discussion on complex mode calculations.

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Complex coupled-mode theory for optical waveguides

Abstract

1. Introduction

2. Waveguide model and mode characteristics

3. Derivation of complex coupled-mode theory

4. Solutions of complex coupled-mode theory

5. Applications of complex coupled-mode theory

Case A: Bragg gratings with lower index outer cladding ( $n_{s} < n_{c l}$ )

Case B: Bragg gratings with equal index outer cladding ( $n_{s} = n_{c l}$ )

Case C: Bragg gratings with higher index outer cladding ( $n_{s} > n_{c l}$ )

6. Summary and conclusion

Acknowledgement

References and links

Cited By

Figures (12)

Equations (81)

Optics Express

Abstract

1. Introduction

2. Waveguide model and mode characteristics

3. Derivation of complex coupled-mode theory

4. Solutions of complex coupled-mode theory

5. Applications of complex coupled-mode theory

Case A: Bragg gratings with lower index outer cladding (ns<ncl)

Case B: Bragg gratings with equal index outer cladding (ns=ncl)

Case C: Bragg gratings with higher index outer cladding (ns>ncl)

6. Summary and conclusion

Acknowledgement

References and links

Cited By

Figures (12)

Equations (81)

Optics Express

Case A: Bragg gratings with lower index outer cladding ( $n_{s} < n_{c l}$ )

Case B: Bragg gratings with equal index outer cladding ( $n_{s} = n_{c l}$ )

Case C: Bragg gratings with higher index outer cladding ( $n_{s} > n_{c l}$ )