Supplementary Material#
Transverse and Longitudinal Vector Components#
\[\begin{split}\begin{align}
\hat{\mathbf{z}} \times \left(\mathbf{A} \times \hat{\mathbf{z}}\right) &= \mathbf{A}_t
\\
\hat{\mathbf{z}} \cdot \mathbf{A} &= A_z
\end{align}\end{split}\]
Curl#
\[\begin{split}\begin{align}
\hat{\mathbf{z}} \times \left(\left(\nabla \times \mathbf{A}\right) \times \hat{\mathbf{z}}\right) &= \hat{\mathbf{z}} \times \left(\frac{\partial}{\partial z} \mathbf{A}_t - \nabla_t A_z\right)
\\
\hat{\mathbf{z}} \cdot \left(\nabla \times \mathbf{A}\right) &= \hat{\mathbf{z}} \cdot \left(\nabla_t \times \mathbf{A}_t\right)
\\
\hat{\mathbf{z}} \times \left(\left(\nabla \times \left(\nabla \times \mathbf{A}\right)\right) \times \hat{\mathbf{z}} \right) &= -\nabla^2 \mathbf{A}_t + \nabla_t\left( \nabla \cdot \mathbf{A} \right)
\\
\hat{\mathbf{z}} \cdot \left(\nabla \times \left(\nabla \times \mathbf{A}\right)\right) &= \hat{\mathbf{z}} \cdot \left(\frac{\partial}{\partial z} \left(\nabla_t \cdot \mathbf{A}_t\right) \hat{\mathbf{z}} -\nabla_t^2 A_z \ \hat{\mathbf{z}}\right)
\end{align}\end{split}\]
Gradient#
\[\begin{split}\begin{align}
\hat{\mathbf{z}} \times \left(\nabla a \times \hat{\mathbf{z}}\right) &= \nabla_t a
\\
\hat{\mathbf{z}} \cdot \nabla a &= \hat{\mathbf{z}} \cdot \left(\frac{\partial}{\partial z} a \, \hat{\mathbf{z}}\right)
\\
\hat{\mathbf{z}} \times \left(\nabla \left(\nabla \cdot \mathbf{A}\right) \times \hat{\mathbf{z}}\right) &= \nabla_t \left(\nabla \cdot \mathbf{A}\right)
\\
\hat{\mathbf{z}} \cdot \nabla \left(\nabla \cdot \mathbf{A}\right) &= \hat{\mathbf{z}} \cdot \left(\frac{\partial}{\partial z} \left(\nabla \cdot \mathbf{A}\right) \hat{\mathbf{z}}\right)
\end{align}\end{split}\]
Matrix Multiplication#
\[\begin{split}\begin{align}
\epsilon &= \left(
\begin{array}{ccc}
\epsilon_{xx} & \epsilon_{xy} & 0 \\
\epsilon_{yx} & \epsilon_{yy} & 0 \\
0 & 0 & \epsilon_{zz} \\
\end{array}\right)
\\
\epsilon_t &= \left(
\begin{array}{ccc}
\epsilon_{xx} & \epsilon_{xy} & 0 \\
\epsilon_{yx} & \epsilon_{yy} & 0 \\
0 & 0 & 0 \\
\end{array}\right)
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\hat{\mathbf{z}} \times \left(\left(\epsilon \cdot \mathbf{A}\right) \times \hat{\mathbf{z}}\right) &= \epsilon_t \cdot \mathbf{A}_t
\\
\hat{\mathbf{z}} \cdot \left(\epsilon \cdot \mathbf{A}\right) &= \hat{\mathbf{z}} \cdot \left(\epsilon_{zz} \, A_z \, \hat{\mathbf{z}} \right)
\end{align}\end{split}\]
Assuming \(\frac{\partial}{\partial z} \epsilon = 0\),
\[\begin{split}\begin{align}
\hat{\mathbf{z}} \times \left(\epsilon \cdot \left(\mathbf{A}_t \times \hat{\mathbf{z}}\right)\right) &= \mathbf{\tilde{\epsilon}}_t \cdot \mathbf{A}_t
\\
\mathbf{\tilde{\epsilon}} &= \left(
\begin{array}{ccc}
\epsilon_{yy} & -\epsilon_{yx} & 0 \\
-\epsilon_{xy} & \epsilon_{xx} & 0 \\
0 & 0 & \epsilon_{zz} \\
\end{array}\right)
\\
\mathbf{\tilde{\epsilon}}_t &= \left(
\begin{array}{ccc}
\epsilon_{yy} & -\epsilon_{yx} & 0 \\
-\epsilon_{xy} & \epsilon_{xx} & 0 \\
0 & 0 & 0 \\
\end{array}\right) = \left(
\begin{array}{ccc}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}\right) \cdot \epsilon \cdot \left(
\begin{array}{ccc}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}\right)
\\
\mathbf{\tilde{\epsilon}}^{-1}_t &= \left(
\begin{array}{ccc}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}\right) \cdot \epsilon^{-1} \cdot \left(
\begin{array}{ccc}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}\right)
\\
\epsilon \cdot \epsilon^{-1} &= \mathbf{1}
\\
\mathbf{\tilde{\epsilon}} \cdot \mathbf{\tilde{\epsilon}}^{-1} &= \mathbf{1}
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\hat{\mathbf{z}} \times \left(\left(\nabla \times \left(\epsilon^{-1} \cdot \left(\nabla \times \mathbf{A}\right)\right)\right) \times \hat{\mathbf{z}} \right) &= \mathbf{\tilde{\epsilon}}^{-1}_t \cdot \left( -\frac{\partial^2}{\partial z^2} \mathbf{A}_t + \frac{\partial}{\partial z} \nabla_t A_z \right)
\\
& \qquad + \epsilon^{-1}_{zz} \left( -\nabla^2_t \mathbf{A}_t + \nabla_t \left( \nabla_t \cdot \mathbf{A}_t \right)\right)
\\
& \qquad + \left(\nabla_t \epsilon^{-1}_{zz}\right) \times \left( \nabla_t \times \mathbf{A}_t \right)
\\
\hat{\mathbf{z}} \cdot \left(\nabla \times \left(\epsilon^{-1} \cdot \left(\nabla \times \mathbf{A}\right)\right)\right) &= {\mathbf{\tilde{\epsilon}}^{-1}_t}_{ij} \left( \frac{\partial}{\partial z} {\nabla_t}_i {\mathbf{A}_t}_j - {\nabla_t}_i {\nabla_t}_j A_z \right)
\\
& \qquad + \left({\nabla_t}_i {\mathbf{\tilde{\epsilon}}^{-1}_t}_{ij}\right) \left( \frac{\partial}{\partial z} {\mathbf{A}_t}_j - {\nabla_t}_j A_z \right)
\end{align}\end{split}\]
Transverse and Longitudinal Maxwell’s Equations#
Maxwell’s equations are separable into transverse and longitudinal components assuming a source free medium, longitudinal invariance \(\left(\frac{\partial}{\partial z} \epsilon = 0\right)\), and with \(\epsilon\) of the general form given in the previous section. In the frequency-space mixed Fourier domain \(\left(\omega, \mathbf{x}\right)\), and \(\mathbf{D}\) and \(\mathbf{B}\) defined as:
\[\begin{split}\begin{align}
\mathbf{D} &= \epsilon_0 \left(\epsilon \cdot \mathbf{E}\right)
\\
\mathbf{B} &= \mu_0 \mathbf{H}
\end{align}\end{split}\]
Maxwell’s equations#
\[\begin{split}\begin{align}
0 &= \nabla \cdot \left(\epsilon \cdot \mathbf{E}\right)
\\
0 &= \nabla \cdot \mathbf{H}
\\
\mathbf{0} &= \nabla \times \mathbf{E} + i \, \omega \, \mu_0 \, \mathbf{H}
\\
\mathbf{0} &= \nabla \times \mathbf{H} - i \, \omega \, \epsilon_0 \left(\epsilon \cdot \mathbf{E}\right)
\end{align}\end{split}\]
Wave Equations#
\[\begin{split}\begin{align}
\mathbf{0} &= \nabla \times \left(\nabla \times \mathbf{E} \right) - \frac{\omega^2}{c^2} \left(\epsilon \cdot \mathbf{E} \right)
\\
\mathbf{0} &= \nabla \times \left(\epsilon^{-1} \cdot \left(\nabla \times \mathbf{H}\right)\right) - \frac{\omega^2}{c^2} \mathbf{H}
\end{align}\end{split}\]
Divergence Equations#
\[\begin{split}\begin{align}
0 &= \nabla_t \cdot \left(\epsilon_t \cdot \mathbf{E}_t\right) + \epsilon_{zz} \, \frac{\partial}{\partial z} E_z
\\
0 &= \nabla_t \cdot \mathbf{H}_t + \frac{\partial}{\partial z} H_z
\end{align}\end{split}\]
Transverse Curl Equations#
\[\begin{split}\begin{align}
\mathbf{0} &= \hat{\mathbf{z}} \times \left(\frac{\partial}{\partial z} \mathbf{E}_t - \nabla_t E_z\right) + i \, \omega \, \mu_0 \, \mathbf{H}_t
\\
\mathbf{0} &= \hat{\mathbf{z}} \times \left(\frac{\partial}{\partial z} \mathbf{H}_t - \nabla_t H_z\right) - i \, \omega \, \epsilon_0 \left( \epsilon_t \cdot \mathbf{E}_t \right)
\end{align}\end{split}\]
Longitudinal Curl Equations#
\[\begin{split}\begin{align}
\mathbf{0} &= \nabla_t \times \mathbf{E}_t + i \, \omega \, \mu_0 \, H_z \hat{\mathbf{z}}
\\
\mathbf{0} &= \nabla_t \times \mathbf{H}_t - i \, \omega \, \epsilon_0 \, \epsilon_{zz} \, E_z \, \hat{\mathbf{z}}
\end{align}\end{split}\]
Transverse Wave Equations#
\[\begin{split}\begin{align}
\mathbf{0} &= -\frac{\partial^2}{\partial z^2} \mathbf{E}_t - \nabla^2_t \mathbf{E}_t + \nabla_t \left(\nabla \cdot \mathbf{E}\right) - \frac{\omega^2}{c^2} \left(\epsilon_t \cdot \mathbf{E}_t\right)
\\
&= -\frac{\partial^2}{\partial z^2} \mathbf{E}_t - \nabla^2_t \mathbf{E}_t + \nabla_t \left(\nabla_t \cdot \left(\left(\mathbf{1} - \epsilon_{zz}^{-1} \, \epsilon_t \right) \cdot \mathbf{E}_t \right)\right)
\\
& \qquad + \nabla_t \left(\left( \nabla_t \epsilon_{zz}^{-1} \right) \cdot \left(\epsilon_t \cdot \mathbf{E}_t\right)\right) - \frac{\omega^2}{c^2} \left(\epsilon_t \cdot \mathbf{E}_t\right)
\\ \\
\mathbf{0} &= \mathbf{\tilde{\epsilon}}^{-1}_t \cdot \left(-\frac{\partial^2}{\partial z^2} \mathbf{H}_t + \frac{\partial}{\partial z} \nabla_t H_z\right) + \epsilon_{zz}^{-1} \left(-\nabla^2_t \mathbf{H}_t + \nabla_t \left(\nabla_t \cdot \mathbf{H}_t\right)\right)
\\
& \qquad + \left(\nabla_t \epsilon_{zz}^{-1}\right) \times \left(\nabla_t \times \mathbf{H}_t\right) - \frac{\omega^2}{c^2} \mathbf{H}_t
\\
&= -\frac{\partial^2}{\partial z^2} \mathbf{H}_t - \epsilon_{zz}^{-1} \, \mathbf{\tilde{\epsilon}}_t \cdot \left(\nabla_t^2 \mathbf{H}_t\right) - \left(\mathbf{1} - \epsilon_{zz}^{-1} \, \mathbf{\tilde{\epsilon}}_t\right) \cdot \nabla_t \left(\nabla_t \cdot \mathbf{H}_t\right)
\\
& \qquad + \mathbf{\tilde{\epsilon}}_t \cdot \left(\left(\nabla_t \epsilon_{zz}^{-1}\right) \times \left(\nabla_t \times \mathbf{H}_t\right)\right) - \frac{\omega^2}{c^2} \left(\mathbf{\tilde{\epsilon}}_t \cdot \mathbf{H}_t\right)
\end{align}\end{split}\]
where the second step is obtained by applying the divergence equations.
Derivation of the Multi-Mode Propagation Equations#
Unidirectional propagation equations are derived by projecting the curl equations onto the \(\hat{\mathbf{e}}_n\) and \(\hat{\mathbf{h}}_n\) of a waveguide mode and then substituting in the modal expansion of the electric and magnetic fields. If the complete set of modes are included in the expansion, the propagation equations are exact solutions to Maxwell’s equations.
Local modes satisfy a modified form of the source-free, linear, and lossless Maxwell’s equations where all \(z\) derivatives have been replaced by the propagation-constant eigenvalues:
\[\begin{split}\begin{align}
0 &= \left(\nabla_t - i \, \beta_n \, \hat{\mathbf{z}}\right) \cdot \left(\epsilon_0 \, \epsilon^{\prime} \cdot \hat{\mathbf{e}}_n\right) \tag{1a}
\\
0 &= \left(\nabla_t - i \, \beta_n \, \hat{\mathbf{z}}\right) \cdot \hat{\mathbf{h}}_n \tag{1b}
\\
\mathbf{0} &= \left(\nabla_t - i \, \beta_n \, \hat{\mathbf{z}}\right) \times \hat{\mathbf{e}}_n + i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_n \tag{1c}
\\
\mathbf{0} &= \left(\nabla_t - i \, \beta_n \, \hat{\mathbf{z}}\right) \times \hat{\mathbf{h}}_n - i \, \omega \, \epsilon_0 \, \epsilon^{\prime} \cdot {\hat{\mathbf{e}}_n} \tag{1d}
\end{align}\end{split}\]
where \(\epsilon^{\prime}\) represents the relative permittivity with gain/loss removed.
The full electric and magnetic fields satisfy Maxwell’s equations in general:
\[\begin{split}\begin{align}
\rho_{f} &= \nabla \cdot \left(\epsilon_0 \, \mathbf{\epsilon} \cdot \mathbf{E} + \mathbf{P}_\text{NL}\right) \tag{2a}
\\
0 &= \nabla \cdot \mu_0 \mathbf{H} \tag{2b}
\\
\mathbf{0} &= \nabla \times \mathbf{E} + i \, \omega \, \mu_0 \ \mathbf{H} \tag{2c}
\\
\mathbf{J}_{f} + i \, \omega \, \mathbf{P}_\text{NL} &= \nabla \times \mathbf{H} - i \, \omega \, \epsilon_0 \, \mathbf{\epsilon} \cdot \mathbf{E} \tag{2d}
\end{align}\end{split}\]
As seen in the following results, the evolution of the complex spectral amplitudes are driven by the following effects:
\[\begin{split}\begin{gather}
\frac{\partial}{\partial z} a_n = \begin{cases}
\text{linear dispersion} \\
\text{gain or loss} \\
\text{changing mode profiles} \\
\text{free current and nonlinearity}
\end{cases}
\end{gather}\end{split}\]
Projection of 2c#
Projecting the electric curl equation onto a magnetic eigenmode of the waveguide yields:
\[\begin{split}\begin{align}
\mathbf{0} &= \hat{\mathbf{h}}_n^* \cdot \left(\nabla \times \mathbf{E}\right) + i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_n^* \cdot \mathbf{H} \tag{s1}
\\
& = \mathbf{E} \cdot \left(\nabla \times \hat{\mathbf{h}}_n^*\right) + \nabla \cdot \left(\mathbf{E} \times \hat{\mathbf{h}}_n^*\right) + i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_n^* \cdot \mathbf{H} \tag{s2}
\\
& \begin{split} &= \mathbf{E} \cdot \left(-i \, \beta_n \, \hat{\mathbf{z}} \times \hat{\mathbf{h}}_n^* - i \, \omega \, \epsilon_0 \, \epsilon^{\prime*} \cdot \hat{\mathbf{e}}_n^* + \hat{\mathbf{z}} \times \frac{\partial \hat{\mathbf{h}}_n^*}{\partial z}\right)
\\
& \qquad + \nabla \cdot \left(\mathbf{E} \times \hat{\mathbf{h}}_n^*\right) + i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_n^* \cdot \mathbf{H} \end{split} \tag{s3}
\\
& \begin{split} &= \hat{\mathbf{z}} \cdot \left(i \, \beta_n \, \mathbf{E} \times \hat{\mathbf{h}}_n^* - \mathbf{E} \times \frac{\partial \hat{\mathbf{h}}_n^*}{\partial z}\right) - i \, \omega \, \epsilon_0 \, \mathbf{E} \cdot \epsilon^{\prime*} \cdot \hat{\mathbf{e}}_n^*
\\
& \qquad + \nabla \cdot \left(\mathbf{E} \times \hat{\mathbf{h}}_n^*\right) + i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_n^* \cdot \mathbf{H} \end{split} \tag{s4}
\end{align}\end{split}\]
where the steps 2 and 4 were made through application of vector identities and step 3 was accomplished through substitution of eq-1d.
Projection of 2d#
Projecting the magnetic curl equation onto an electric eigenmode of the waveguide yields:
\[\begin{split}\begin{align}
\hat{\mathbf{e}}_n^* \cdot \left(\mathbf{J}_{f} + i \, \omega \, \mathbf{P}_\text{NL}\right) &= \hat{\mathbf{e}}_n^* \cdot \left(\nabla \times \mathbf{H}\right) - i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \epsilon \cdot \mathbf{E} \tag{s1}
\\
& = \mathbf{H} \cdot \left(\nabla \times \hat{\mathbf{e}}_n^*\right) + \nabla \cdot \left(\mathbf{H} \times \hat{\mathbf{e}}_n^*\right) - i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \epsilon \cdot \mathbf{E} \tag{s2}
\\
& \begin{split} &= \mathbf{H} \cdot \left(-i \, \beta_n \, \hat{\mathbf{z}} \times \hat{\mathbf{e}}_n^* + i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_n^* + \hat{\mathbf{z}} \times \frac{\partial \hat{\mathbf{e}}_n^*}{\partial z}\right)
\\
& \qquad + \nabla \cdot \left(\mathbf{H} \times \hat{\mathbf{e}}_n^*\right) - i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \epsilon \cdot \mathbf{E} \end{split} \tag{s3}
\\
& \begin{split} &= \hat{\mathbf{z}} \cdot \left(-i \, \beta \, \hat{\mathbf{e}}_n^* \times \mathbf{H} + \frac{\partial \hat{\mathbf{e}}_n^*}{\partial z} \times \mathbf{H}\right) + i \, \omega \, \mu_0 \, \mathbf{H} \cdot \hat{\mathbf{h}}_n^*
\\
& \qquad + \nabla \cdot \left(\mathbf{H} \times \hat{\mathbf{e}}_n^*\right) - i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \epsilon \cdot \mathbf{E} \end{split} \tag{s4}
\end{align}\end{split}\]
where steps 2 and 4 were made through application of vector identities and step 3 was accomplished through substitution of eq-1c.
Transverse Integration#
The next step in the derivation involves integrating the difference between the projections of 2c and 2d:
\[\begin{split}\begin{align}
-\iint_{A_\infty} \hat{\mathbf{e}}_n^* \cdot \left(\mathbf{J}_{f} + i \, \omega \, \mathbf{P}_\text{NL}\right) dA &= \iint_{A_\infty} \left(\hat{\mathbf{h}}_n^* \cdot \left(\text{eq-2c}\right) - \hat{\mathbf{e}}_n^* \cdot \left(\text{eq-2d}\right)\right) dA \tag{s1}
\\
& \begin{split} =& \iint_{A_\infty} \Biggl(\nabla \cdot \left(\mathbf{E} \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \mathbf{H}\right)
\\
& \qquad + i \, \beta_n \, \hat{\mathbf{z}} \cdot \left(\mathbf{E} \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \mathbf{H}\right)
\\
& \qquad - \hat{\mathbf{z}} \cdot \left(\mathbf{E} \times \frac{\partial \hat{\mathbf{h}}_n^*}{\partial z} + \frac{\partial \hat{\mathbf{e}}_n^*}{\partial z} \times \mathbf{H}\right)
\\
& \qquad + i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \left(\epsilon - \epsilon^{\prime}\right) \cdot \mathbf{E}
\Biggr) dA \end{split} \tag{s2}
\\
& \begin{split} =& \iint_{A_\infty} \Biggl(\hat{\mathbf{z}} \cdot \frac{\partial}{\partial z} \left(\mathbf{E} \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \mathbf{H}\right)
\\
& \qquad + i \, \beta_n \, \hat{\mathbf{z}} \cdot \left(\mathbf{E} \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \mathbf{H}\right)
\\
& \qquad - \hat{\mathbf{z}} \cdot \left(\mathbf{E} \times \frac{\partial \hat{\mathbf{h}}_n^*}{\partial z} + \frac{\partial \hat{\mathbf{e}}_n^*}{\partial z} \times \mathbf{H}\right)
\\
& \qquad + i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \left(\epsilon - \epsilon^{\prime}\right) \cdot \mathbf{E}
\Biggr) dA \end{split} \tag{s3}
\\
& \begin{split} =& \iint_{A_\infty} \Biggl(\hat{\mathbf{z}} \cdot \left(\frac{\partial \mathbf{E}}{\partial z} \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \frac{\partial \mathbf{H}}{\partial z}\right)
\\
& \qquad + i \, \beta_n \, \hat{\mathbf{z}} \cdot \left(\mathbf{E} \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \mathbf{H}\right)
\\
& \qquad + i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \left(\epsilon - \epsilon^{\prime}\right) \cdot \mathbf{E}
\Biggr) dA \end{split} \tag{s4}
\end{align}\end{split}\]
where the third step was made through application of the reciprocity theorem.
Modal Expansion#
The derivation is complete after expanding the electric and magnetic fields in terms of the local modes:
\[\mathbf{E} = \sum_m a_m \, \hat{\mathbf{e}}_m \qquad \mathbf{H} = \sum_m a_m \, \hat{\mathbf{h}}_m\]
\[\begin{split}\begin{align}
-\iint_{A_\infty} \hat{\mathbf{e}}_n^* \cdot \left(\mathbf{J}_{f} + i \, \omega \, \mathbf{P}_\text{NL}\right) dA &= \sum_m \iint_{A_\infty} \Biggl(
\\
& \begin{split} & \qquad + \frac{\partial a_m}{\partial z} \, \hat{\mathbf{z}} \cdot \left(\hat{\mathbf{e}}_m \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \hat{\mathbf{h}}_m\right)
\\
& \qquad + a_m \, \hat{\mathbf{z}} \cdot \left(\frac{\partial \hat{\mathbf{e}}_m}{\partial z} \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \frac{\partial \hat{\mathbf{h}}_m}{\partial z}\right)
\\
& \qquad + i \, \beta_n \, a_m \, \hat{\mathbf{z}} \cdot \left(\hat{\mathbf{e}}_m \times \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* \times \hat{\mathbf{h}}_m\right)
\\
& \qquad + i \, \omega \, \epsilon_0 \, a_m \, \hat{\mathbf{e}}_n^* \cdot \left(\epsilon - \epsilon^{\prime}\right) \cdot \hat{\mathbf{e}}_m
\Biggr) dA \end{split} \tag{s1}
\\
& \begin{split} =& \ 2 \left(\frac{\partial a_n}{\partial z} + i \, \beta_n \, a_n\right)
\\
& + \sum_m a_m \iint_{A_\infty} \left(i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot \left(\epsilon - \epsilon^{\prime}\right) \cdot \hat{\mathbf{e}}_m\right) dA
\\
& + \sum_m a_m \iint_{A_\infty} \hat{\mathbf{z}} \, {\cdot} \! \left(\frac{\partial \hat{\mathbf{e}}_m}{\partial z} {\times} \hat{\mathbf{h}}_n^* + \hat{\mathbf{e}}_n^* {\times} \frac{\partial \hat{\mathbf{h}}_m}{\partial z}\right) dA \end{split} \tag{s2}
\end{align}\end{split}\]
where the second step was made using the orthogonality relationships.
Unidirectional Propagation Equation#
Rearranging the results from above reveals the mode propagation equation:
\[\begin{split}\begin{align}
\frac{\partial}{\partial z} a_n =& - i \, \beta_n \, a_n
\\
& - \sum_{m} a_m \iint_{A_\infty} \frac{1}{2} \left(
i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \cdot
\left(\epsilon - \epsilon^{\prime}\right) \cdot \hat{\mathbf{e}}_m
\right) dA
\\
& - \sum_{m} a_m \iint_{A_\infty} \frac{1}{2} \, \hat{\mathbf{z}} \cdot \left(
\frac{\partial \hat{\mathbf{e}}_m}{\partial z} \times \hat{\mathbf{h}}^*_n + \hat{\mathbf{e}}_n^* \times \frac{\partial \hat{\mathbf{h}}_m}{\partial z}
\right) dA
\\
& -\iint_{A_\infty} \frac{1}{2} \left(\hat{\mathbf{e}}_n^* \cdot \mathbf{J}_{f} + i \, \omega \, \hat{\mathbf{e}}_n^* \cdot \mathbf{P}_\text{NL}\right) dA
\end{align}\end{split}\]
Unit Analysis of the Effective Nonlinear Parameters#
The effective nonlinear parameter combines multiple effects attributable to the waveguide mode. Through unit analysis the effective nonlinearity may be written in terms of the effective mode area, refractive index, and nonlinear susceptibility. The steps taken to arrive at this result are only drawn out for the second-order nonlinearity but are extendible to the third-order through the same procedure.
For mode \(n\) the second-order nonlinearity is written as the product of sums over modes \(r\) and \(s\):
\[\begin{split}\begin{gather}
\frac{\partial}{\partial z} a_n
= \ldots -i \, \omega \sum_{r s} \iint_{-\infty}^{\infty}
g_{n r s}^{\left(2\right)}\!\left[\nu_1, \nu_2\right] a_r\!\left[\nu_1\right] a_s\!\left[\nu_2\right]
\delta\!\left[\nu - \left(\nu_1 + \nu_2\right)\right] d\nu_1 \, d\nu_2
\\
g_{n r s}^{\left(2\right)}\!\left[\nu_1, \nu_2\right] = \iint_{A_\infty} \frac{1}{2} \epsilon_0 \, \chi^{\left(2\right)}_{i j k}\!\left[\nu_1, \nu_2\right] {\hat{\mathbf{e}}_n^*}_i\!\left[\nu_1 + \nu_2\right] {\hat{\mathbf{e}}_r}_j\!\left[\nu_1\right] {\hat{\mathbf{e}}_s}_k\!\left[\nu_2\right] dA
\end{gather}\end{split}\]
where the effective nonlinearity parameter \(g_{n r s}^{\left(2\right)}\) integrates out all transverse spatial dependence.
From the orthogonality relationship, the spatial dependence of the normalized electric field vector scales with the following relationship:
\[\begin{gather}
\hat{e} \sim \frac{1}{\sqrt{\epsilon_0 \, c \, n_\text{eff} \, A_\text{eff}}}
\end{gather}\]
If the effects of spatial overlap between modes can be ignored or absorbed into the effective susceptibility \(\chi^{\left(2\right)}_\text{eff}\), the expression for \(g_{n r s}^{\left(2\right)}\) can be replaced by its unit decomposition:
\[\begin{split}\begin{gather}
\begin{split}
\frac{\partial}{\partial z} a \sim \ldots -i \, \omega &\iint_{-\infty}^{\infty} \frac{1}{2} \epsilon_0 \, \frac{A_\text{eff}}{\left(\epsilon_0 \, c \, n_\text{eff} \, A_\text{eff}\right)^{3/2}} \, \chi^{\left(2\right)}_\text{eff} \, a \, a
\\
& \quad \delta\!\left[\nu - \left(\nu_1 + \nu_2\right)\right] d\nu_1 \, d\nu_2
\end{split}
\\
\begin{split}
\frac{\frac{\partial}{\partial z} a}{\sqrt{\epsilon_0 \, c \, n_\text{eff} \, A_\text{eff}}} \sim \ldots - \frac{1}{2} \frac{i \, \omega}{c \, n_\text{eff}} &\iint_{-\infty}^{\infty} \chi^{\left(2\right)}_\text{eff} \frac{a}{\sqrt{\epsilon_0 \, c \, n_\text{eff} \, A_\text{eff}}} \frac{a}{\sqrt{\epsilon_0 \, c \, n_\text{eff} \, A_\text{eff}}}
\\
& \quad \delta\!\left[\nu - \left(\nu_1 + \nu_2\right)\right] d\nu_1 \, d\nu_2
\end{split}
\end{gather}\end{split}\]
Taken together, the units of the terms about each root-power amplitude \(a\) and of the amplitude itself form a single quantity with units of electric field (see chapter 2 of Nonlinear Optics (Boyd 2020) for propagation equations of the same form). Following this logic, the quantities about each amplitude term are then associated with the quantities of that amplitude’s mode (as they describe a single electric field). Replacing the mode identifiers and manipulating the equation back to its origin form yields a natural decomposition for the effective nonlinearity:
\[\begin{split}\begin{gather}
\begin{split}
\frac{\frac{\partial}{\partial z} a_n}{\sqrt{\epsilon_0 \, c \, {n_\text{eff}}_n \, {A_\text{eff}}}_n} \sim \ldots - \frac{1}{2} \frac{i \, \omega}{c \, {n_\text{eff}}_n} &\iint_{-\infty}^{\infty}
{\chi^{\left(2\right)}_\text{eff}}_{n r s}
\frac{a_r}{\sqrt{\epsilon_0 \, c \, {n_\text{eff}}_r \, {A_\text{eff}}_r}}
\frac{a_s}{\sqrt{\epsilon_0 \, c \, {n_\text{eff}}_s \, {A_\text{eff}}_s}}
\\
& \quad \delta\!\left[\nu - \left(\nu_1 + \nu_2\right)\right] d\nu_1 \, d\nu_2
\end{split}
\\
\begin{split}
\frac{\partial}{\partial z} a_n \sim \ldots -i \, \omega &\iint_{-\infty}^{\infty}
\left(\frac{1}{2} \sqrt{\frac{{A_\text{eff}}_n}{{A_\text{eff}}_r \, {A_\text{eff}}_s}} \frac{{\chi^{\left(2\right)}_\text{eff}}_{n r s}}{\sqrt{\epsilon_0 \, c^3 \, {n_\text{eff}}_n \, {n_\text{eff}}_r \, {n_\text{eff}}_s}}\right) a \, a
\\
& \quad \delta\!\left[\nu - \left(\nu_1 + \nu_2\right)\right] d\nu_1 \, d\nu_2
\end{split}
\\
g_{n r s}^{\left(2\right)} = \frac{1}{2} \sqrt{\frac{{A_\text{eff}}_n}{{A_\text{eff}}_r \, {A_\text{eff}}_s}} \frac{{\chi^{\left(2\right)}_\text{eff}}_{n r s}}{\sqrt{\epsilon_0 \, c^3 \, {n_\text{eff}}_n \, {n_\text{eff}}_r \, {n_\text{eff}}_s}}
\end{gather}\end{split}\]
The frequency dependence of each mode is implicitly assumed.
Likewise, for the third-order nonlinearity:
\[\begin{equation}
g_{n q r s}^{\left(3\right)} = \frac{1}{2} \sqrt{\frac{{A_\text{eff}}_n}{{A_\text{eff}}_q \, {A_\text{eff}}_r \, {A_\text{eff}}_s}} \frac{{\chi^{\left(3\right)}_\text{eff}}_{n q r s}}{\epsilon_0 \, c^2 \, \sqrt{{n_\text{eff}}_n \, {n_\text{eff}}_q \, {n_\text{eff}}_r \, {n_\text{eff}}_s}}
\end{equation}\]
Relationship Between the Kerr Parameter and the Nonlinear Susceptibility#
The nonlinear Kerr parameter \(n_2\) describes the change in refractive index due to the instantaneous, time-averaged optical power. For single-frequency, linearly-polarized light:
\[\begin{split}\begin{align}
n &= n_0 + n_2 \left<I\right>
\\
&= n_0 + n_2 \left( \epsilon_0 \, c \, n_0 \left<E^2\right>\right)
\end{align}\end{split}\]
The third-order nonlinear susceptibility \(\chi^{\left(3\right)}\) can be cast in terms of the Kerr parameter by relating them through the definition of the relative permittivity \(\epsilon\). Following a procedure similar to Nonlinear Optics (Boyd 2020), a single-frequency electric field is written as the sum of its Fourier amplitude \(A\). Its instantaneous time average is then given by twice the absolute square of the Fourier amplitude:
\[\begin{split}\begin{gather}
E\!\left[t\right] = 2 A \cos\left[\omega \, t - \beta \, z\right] = A e^{-i \left(\omega \, t - \beta \, z\right)} + \text{c.c.}
\\
\left<E^2\right> = 2 \bigl|A\bigr|^2
\end{gather}\end{split}\]
The third-order nonlinear polarization may also be expanded in terms of this single-frequency field. This expansion includes a term proportional to the electric field:
\[\begin{gather}
P_{NL} = \epsilon_0 \, \chi^{\left(3\right)} \, E\left[t\right]^3 = \epsilon_0 \, \chi^{\left(3\right)} \left(3 \, \bigl|A\bigr|^2 E\!\left[t\right] + \ldots\right)
\end{gather}\]
The relative permittivity is defined as the collection of terms in the electric displacement \(\mathbf{D}\) directly proportional to the electric field, and alternatively, as the square of the refractive index:
\[\begin{split}\begin{align}
\epsilon &= 1 + \chi^{\left(1\right)} + 3 \, \chi^{\left(3\right)} \bigl|A\bigr|^2 + \ldots
\\ \\
\epsilon = n^2 &= \left(n_0 + n_2 \left<I\right>\right)^2
\\
&= n_0^2 + 2 \, n_0 \ n_2 \left<I\right> + \ldots
\\
&= n_0^2 + 4 \, \epsilon_0 \, c \, n_0^2 \, n_2 \, \bigl|A\bigr|^2 + \ldots
\end{align}\end{split}\]
Equating like terms (those with the same power of \(A\)) yields the following relationships for the electric susceptibilities:
\[\begin{split}\begin{align}
\chi^{\left(1\right)} &= n_0^2 - 1
\\
\chi^{\left(3\right)} &= \frac{4}{3} \epsilon_0 \, c \, n_0^2 \, n_2
\end{align}\end{split}\]