Coupled-Wave Theory

This is a continuation from the previous tutorial - Dispersion in Fibers.

The principles of many photonics devices are based on the coupling between optical fields of different frequencies or different spatial modes. In general, the coupling mechanism can be described by a polarization ΔP on top of a background polarization representing the property of the medium in the absence of the coupling mechanism. In this tutorial, we present the general coupled-wave and coupled-mode formalisms, which provide the foundation for understanding the functions of many devices. The coupled-wave formalism deals with the coupling of optical waves of different frequencies, whereas coupled-mode theory applies to the coupling of optical fields of different spatial modes.

In this tutorial, the general formulation of the coupled-wave formalism for coupling of optical waves of different frequencies is presented. For simplicity, we consider only coupling among plane optical waves, but the formulation can be easily extended for nonplane waves, such as optical waves of Gaussian beam profiles.

As discussed in the linear optical susceptibility tutorial, coupling among optical waves of different frequencies is possible only if the optical property of the medium in which the optical waves propagate is either time varying or optically nonlinear. Time-varying optical properties can be induced by time-varying electric, magnetic, or acoustic fields through electro-optic, magneto-optic, or acousto-optic effects, which are discussed in following tutorials. In particular, an acoustic wave always induces time-varying changes in the optical property of a medium, whereas changes induced by electro-optic or magneto-optic effects can be static when they are caused by static electric or magnetic fields. Nonlinear optical properties are discussed in another tutorial. Here we consider the general formulation without specifying the physical mechanism responsible for the coupling of optical waves. Applications of the couple-wave formalism to specific situations are seen in later tutorials, particularly in acousto-optic and optical nonlinear effects.

The time-varying or nonlinear optical property responsible for coupling of optical waves of different frequencies can be generally described by a polarization, ΔP, induced by the underlying effect. In the absence of coupling mechanism, an optical wave propagating in a medium is described by the linear wave equation

\[\tag{1}\pmb{\nabla}\times\pmb{\nabla}\times\mathbf{E}+\mu_0\frac{\partial^2 \mathbf{D}}{\partial t^2}=0\]

where D accounts for only the linear, static property of the medium. For a monochromatic optical wave of constant amplitude at a frequency ω, this equation reduces to

\[\tag{2}\pmb{\nabla}\times\pmb{\nabla}\times\mathbf{E}-\omega^2\mu_0\boldsymbol{\epsilon}(\mathbf{k},\omega)\cdot\mathbf{E}=0\]

where ε(k, ω) describes the linear, time-independent optical property of the medium.

Among the solutions of (2) are monochromatic plane waves and Gaussian waves. Here we consider only the plane waves, but the same concept applies to Gaussian waves as well.

Clearly, an optical wave of frequency ω that is governed by (2) propagates independently of waves of other frequencies. Therefore, optical waves of different frequencies do not couple if each of them is governed by (1) with D characterizing only the linear, static property of the medium. To describe the coupling, a certain polarization ΔP that characterizes the coupling mechanism has to be included in the wave equation:

\[\tag{3}\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\mathbf{E}+\mu_0\frac{\partial^2\mathbf{D}}{\partial t^2}=-\mu_0\frac{\partial^2\Delta\mathbf{P}}{\partial t^2}\]

Because ΔP couples waves of different frequencies, an optical wave at a given frequency ω does not propagate independently of waves of other frequencies any more. A monochromatic wave that is coupled to other frequencies cannot propagate without changing its amplitude, which includes magnitude, phase, and polarization. Consequently, a monochromatic plane wave of constant amplitude is not a solution of (3). In most cases of interest, however, the condition

\[\tag{4}|\Delta \mathbf{P}|\ll|\mathbf{D}|\]

is valid; hence the wave-coupling mechanism can be considered as a perturbation on the linear, static property of the medium. Then, the total field of the waves being coupled can be expressed as a linear combination of plane waves of different frequencies, each of which has a spatially varying amplitude:

\[\tag{5}\mathbf{E}(\mathbf{r}, t)=\sum_q\mathbf{E}_q(\mathbf{r})\exp(-\text{i}\omega_q t)=\sum_q\boldsymbol{\mathcal{E}}_q(\mathbf{r})\exp(\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega_q t)\]

We can also expand ΔP as a linear combination of its various frequency components:

\[\tag{6}\Delta\mathbf{P}(\mathbf{r},t)=\sum_q\Delta\mathbf{P}_q(\mathbf{r})\exp(-\text{i}\omega_q t)\]

Substitution of (5) and (6) in (3) yields the following coupled-wave equation:

\[\tag{7}\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\mathbf{E}_q-\omega_q^2\mu_0\boldsymbol{\epsilon}(\mathbf{k}_q,\omega_q)\cdot\mathbf{E}_q=\omega_q^2\mu_0\Delta\mathbf{P}_q\]

Note that \(\Delta\mathbf{P}_q(\mathbf{r})\) is generally not proportional to \(\mathbf{E}_q(\mathbf{r})\). It contains the fields of other frequencies to facilitate the coupling. Moreover, it does not necessarily contain a spatial phase factor of \(\exp(\text{i}\mathbf{k}_q\cdot\mathbf{r})\). In the special case when the spatial phase factor of \(\Delta\mathbf{P}_q(\mathbf{r})\) is \(\exp(\text{i}\mathbf{k}_q\cdot\mathbf{r})\), the coupling interaction is most efficient and is called phase matched.

Slowly varying amplitude approximation

The coupled-wave equation expressed in (7) is a second-order differential equation. It can be reduced to a first-order differential equation by applying the slowly varying amplitude approximation, which assumes that variation of the wave amplitude \(\boldsymbol{\mathcal{E}}_q(\mathbf{r})\) caused by coupling to other frequencies is negligibly small over the distance of an optical wavelength. This approximation is valid in almost all situations of practical interest.

We first consider the situation in an isotropic medium where ε reduces to a scale ε. From the discussions in the propagation in an isotropic medium tutorial, we find that \(\boldsymbol{\nabla}\cdot\mathbf{E}=0\) in this case. Then, (7) becomes

\[\tag{8}\nabla^2\mathbf{E}_q+\omega_q^2\mu_0\epsilon(\mathbf{k}_q,\omega_q)\mathbf{E}_q=-\omega_q^2\mu_0\Delta\mathbf{P}_q\]

Substitution of the relation \(\mathbf{E}_q=\boldsymbol{\mathcal{E}}_q\exp(\text{i}\mathbf{k}_q\cdot\mathbf{r})\) in (8), followed by application of the condition \(k_q^2=\omega_q^2\mu_0\epsilon(\mathbf{k}_q,\omega_q)\), yields

\[\tag{9}\nabla^2\boldsymbol{\mathcal{E}}_q+2\text{i}(\mathbf{k}_q\cdot\boldsymbol{\nabla})\boldsymbol{\mathcal{E}}_q=-\omega_q^2\mu_0\Delta\mathbf{P}_q\text{e}^{-\text{i}\mathbf{k}_q\cdot\mathbf{r}}\]

Under the slowly varying amplitude approximation, we have

\[\tag{10}|\nabla^2\boldsymbol{\mathcal{E}}_q|\ll|(\mathbf{k}_q\cdot\boldsymbol{\nabla})\boldsymbol{\mathcal{E}}_q|\]

Consequently, the coupled-wave equation in an isotropic medium can be written as

\[\tag{11}(\mathbf{k}_q\cdot\boldsymbol{\nabla})\boldsymbol{\mathcal{E}}_q\approx\frac{\text{i}\omega_q^2\mu_0}{2}\Delta\mathbf{P}_q\text{e}^{-\text{i}\mathbf{k}_q\cdot\mathbf{r}}\]

In the special situation when the amplitudes of all waves being coupled vary only in a particular direction, say the z direction, we can write \(\boldsymbol{\mathcal{E}}_q(\mathbf{r})=\boldsymbol{\mathcal{E}}_q(z)\) even through \(\Delta\mathbf{P}_q(\mathbf{r})\) might have variations in other directions. Then, the coupled-wave equation can be written as

\[\tag{12}\frac{\text{d}\boldsymbol{\mathcal{E}}_q(z)}{\text{d}z}\approx\frac{\text{i}\omega_q^2\mu_0}{2k_{q,z}}\Delta\mathbf{P}_q(\mathbf{r})\text{e}^{-\text{i}\mathbf{k}_q\cdot\mathbf{r}}\]

If, furthermore, the interaction is collinear along the z direction, all participating waves have parallel or antiparallel wavevectors such that \(\mathbf{k}_q=k_q\hat{z}\) for all q. In this situation, \(\Delta\mathbf{P}_q\) can have variations only along the z direction. Then (12) can be further simplified to

\[\tag{13}\frac{\text{d}\boldsymbol{\mathcal{E}}_q(z)}{\text{d}z}\approx\frac{\text{i}\omega_q^2\mu_0}{2k_q}\Delta\mathbf{P}_q(z)\text{e}^{-\text{i}k_qz}\]

For an optical wave propagating in an anisotropic medium, E is not necessarily perpendicular to k and, in general, \(\boldsymbol{\nabla}\cdot\mathbf{E}\ne0\), as discussed in the propagating in an anisotropic medium tutorial. Consequently, (8) and the equations that follow are not valid in an anisotropic medium. In this situation, the field \(\mathbf{E}_q\) propagating in the \(\mathbf{k}_q=k_q\hat{k}_q\) direction can be divided into a transverse and a longitudinal component:

\[\tag{14}\mathbf{E}_q=\mathbf{E}_{q,\text{T}}+\mathbf{E}_{q,\text{L}}\]

where the transverse component is given by

\[\tag{15}\mathbf{E}_{q,\text{T}}=(\hat{k}_q\times\mathbf{E}_q)\times\hat{k}_q\]

and the longitudinal component is given by

\[\tag{16}\mathbf{E}_{q,\text{L}}=(\hat{k}_q\cdot\mathbf{E}_q)\hat{k}_q\]

Clearly, \(\boldsymbol{\nabla}\cdot\mathbf{E}_{q,\text{T}}=0\) but \(\boldsymbol{\nabla}\cdot\mathbf{E}_{q,\text{L}}\ne0\). Therefore, an equation similar to (8) can be written for the transverse component:

\[\tag{17}\nabla^2\mathbf{E}_{q,\text{T}}+\omega_q^2\mu_0[\boldsymbol{\epsilon}(\mathbf{k}_q,\omega_q)\cdot\mathbf{E}_q]_\text{T}=-\omega_q^2\mu_0\Delta\mathbf{P}_{q,\text{T}}\]

where \(\Delta\mathbf{P}_{q,\text{T}}=(\hat{k}_q\times\Delta\mathbf{P}_q)\times\hat{k}_q\). Note that \(\Delta\mathbf{P}_{q,\text{T}}\) can have contributions from the longitudinal components of the interacting waves. Following the same procedure as leads to (11), the coupled-wave equation in an anisotropic medium under the slowly varying amplitude approximation can be written as:

\[\tag{18}(\mathbf{k}_q\cdot\boldsymbol{\nabla})\boldsymbol{\mathcal{E}}_{q,\text{T}}\approx\frac{\text{i}\omega_q^2\mu_0}{2}\Delta\mathbf{P}_{q,\text{T}}\text{e}^{-\text{i}\mathbf{k}_q\cdot\mathbf{r}}\]

In the special situation when (11) can be reduced to (12) or (13), an equation similar to (12) or (13), but expressed in terms of the transverse field components, can be obtained from (18) for wave coupling in an anisotropic medium.

The next part continues with the coupled-mode theory tutorial.