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 Forward and Backward Propagation Equations#
The propagation equations are derived by substituting the modal expansion into the curl equations, with the only additional assumption being that \(\epsilon\) is separable into transverse and longitudinal components.
As seen in the following results, the evolution of the amplitude is driven by the following effects:
\[\begin{split}\begin{gather}
\frac{\partial}{\partial z} a_n = \begin{cases}
\text{linear dispersion} \\
\text{gain or loss} \\
\text{changing mode profiles} \\
\text{free sources and nonlinearity}
\end{cases}
\end{gather}\end{split}\]
Modal Expansion of 4c#
Inserting the expansion of eq-1 into eq-4c of Maxwell’s equations from Nonlinear Optics section yields:
\[\begin{split}\begin{align}
\mathbf{0} &= \sum_m \left(
\nabla \times \left(a_m \, \hat{\mathbf{e}}_m\right)
+ i \, \omega \, \mu_0 \, \left(a_m \, \hat{\mathbf{h}}_m\right)
+ \left(\text{b.p.m.}\right)\right) \tag{s1}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\nabla \times \hat{\mathbf{e}}_m\right)
+ \left(\nabla a_m\right) \times \hat{\mathbf{e}}_m
\\
& \qquad + a_m \left(i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_m\right)
+ \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s2}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(
\hat{\mathbf{z}} \times \left(
\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t
- \nabla_t {\hat{e}_m}_z
\right)
+ \nabla_t \times {\hat{\mathbf{e}}_m}_t
\right)
\\
& \qquad + \left(\frac{\partial}{\partial z} a_m\right) \, \hat{\mathbf{z}} \times \hat{\mathbf{e}}_m
+ a_m \left(i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_m\right)
+ \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s3}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\hat{\mathbf{z}} \times \frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) \hat{\mathbf{z}} \times \hat{\mathbf{e}}_m
\\
& \qquad + a_m \left(
\hat{\mathbf{z}} \times \left( - \nabla_t {\hat{e}_m}_z\right)
+ \nabla_t \times {\hat{\mathbf{e}}_m}_t
+ i \, \omega \, \mu_0 \, \hat{\mathbf{h}}_m
\right)
+ \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s4}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\hat{\mathbf{z}} \times \frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) \hat{\mathbf{z}} \times \hat{\mathbf{e}}_m
\\
& \qquad + a_m \left(\hat{\mathbf{z}} \times \left(i \, \beta_m \, {\hat{\mathbf{e}}_m}_t \right)\right)
+ \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s5}
\end{align}\end{split}\]
where the last step was accomplished by substitution of eq-3c and eq-3e from the Nonlinear Optics section.
Expanding the backward propagating modes, and using the forward/backward mode relationships yields an equation in terms of the spatial distribution of only the forward propagating modes:
\[\begin{split}\begin{align}
\mathbf{0} &= \sum_m \left(
a_m \left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) {\hat{\mathbf{e}}_m}_t
+ i \, \beta_m \, a_m \, {\hat{\mathbf{e}}_m}_t
+ \left(\text{b.p.m.}\right)\right) \tag{s6}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) {\hat{\mathbf{e}}_m}_t
+ i \, \beta_m \, a_m \, {\hat{\mathbf{e}}_m}_t
\\
& \qquad + a_{-m} \left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_{-m}}_t\right)
+ \left(\frac{\partial}{\partial z} a_{-m}\right) {\hat{\mathbf{e}}_{-m}}_t
+ i \, \beta_{-m} \, a_{-m} \, {\hat{\mathbf{e}}_{-m}}_t
\Biggr) \end{split} \tag{s7}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) {\hat{\mathbf{e}}_m}_t
+ i \, \beta_m \, a_m \, {\hat{\mathbf{e}}_m}_t
\\
& \qquad + a_{-m} \left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_{-m}\right) {\hat{\mathbf{e}}_m}_t
- i \, \beta_m \, a_{-m} \, {\hat{\mathbf{e}}_m}_t
\Biggr) \end{split} \tag{s8}
\\
& \begin{split} &= \sum_m \Biggl(
\left(a_m + a_{-m}\right) \left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} \left(a_m + a_{-m}\right)\right) {\hat{\mathbf{e}}_m}_t
\\
& \qquad + i \, \beta_m \, \left(a_m - a_{-m}\right) \, {\hat{\mathbf{e}}_m}_t \Biggr) \end{split} \tag{s9}
\end{align}\end{split}\]
Right multiplying by the cross of \(\hat{\mathbf{h}}^*_n\) allows application of the orthogonality relationships:
\[\begin{split}\begin{align}
& \begin{split} \mathbf{0} &= \iint_{A_\infty} \sum_m \Biggl(
\left(a_m + a_{-m}\right) \left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right) \times \hat{\mathbf{h}}^*_n
\\
& \qquad + \left(\frac{\partial}{\partial z} \left(a_m + a_{-m}\right)\right) \hat{\mathbf{e}}_m \times \hat{\mathbf{h}}^*_n
\\
& \qquad + i \, \beta_m \, \left(a_m - a_{-m}\right) \, {\hat{\mathbf{e}}_m}_t \times \hat{\mathbf{h}}^*_n
\Biggr) \cdot \hat{\mathbf{z}} \, dA \end{split} \tag{s10}
\\
& \begin{split} \phantom{\mathbf{0}} &= \frac{\partial}{\partial z} \left(a_n + a_{-n}\right) + i \, \beta_n \, \left(a_n - a_{-n}\right)
\\
& \qquad + \sum_m \iint_{A_\infty} \left(a_m + a_{-m}\right) \left(\left(\frac{\partial}{\partial z} {\hat{\mathbf{e}}_m}_t\right) \times {\hat{\mathbf{h}}^*_n}_t\right) \cdot \hat{\mathbf{z}} \ dA \end{split} \tag{s11}
\end{align}\end{split}\]
Modal Expansion of 4d#
Inserting the expansion of eq-1 into eq-4d of Maxwell’s equations from the Nonlinear Optics section yields:
\[\begin{split}\begin{align}
\mathbf{J}_{f} + i \, \omega \, \mathbf{P}_\text{NL} &= \sum_m \left(
\nabla \times \left(a_m \, \hat{\mathbf{h}}_m\right)
- i \, \omega \, \epsilon_0 \, \left(\mathbf{\epsilon} \cdot \left(a_m \ \hat{\mathbf{e}}_m\right)\right)
+ \left(\text{b.p.m.}\right)\right) \tag{s1}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\nabla \times \hat{\mathbf{h}}_m\right)
+ \left(\nabla a_m\right) \times \hat{\mathbf{h}}_m
\\
& \qquad - a_m \left(i \, \omega \, \epsilon_0 \, \left(\mathbf{\epsilon} \cdot \hat{\mathbf{e}}_m\right)\right)
+ \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s2}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(
\hat{\mathbf{z}} \times \left(
\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t
- \nabla_t {\hat{h}_m}_z
\right)
+ \nabla_t \times {\hat{\mathbf{h}}_m}_t
\right)
\\
& \qquad + \left(\frac{\partial}{\partial z} a_m\right) \hat{\mathbf{z}} \times \hat{\mathbf{h}}_m
- a_m \left(i \, \omega \, \epsilon_0 \, \left(\mathbf{\epsilon} \cdot \hat{\mathbf{e}}_m\right)\right)
+ \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s3}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\hat{\mathbf{z}} \times \frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) \hat{\mathbf{z}} \times \hat{\mathbf{h}}_m
\\
& \qquad + a_m \left(
\hat{\mathbf{z}} \times \left(-\nabla_t {\hat{h}_m}_z\right)
+ \nabla_t \times {\hat{\mathbf{h}}_m}_t
- i \, \omega \, \epsilon_0 \, \left(\mathbf{\epsilon} \cdot \hat{\mathbf{e}}_m\right)
\right)
\\
& \qquad + \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s4}
\\
& \begin{split} &= \sum_m \Biggl(
a_m \left(\hat{\mathbf{z}} \times \frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) \hat{\mathbf{z}} \times \hat{\mathbf{h}}_m
\\
& \qquad + a_m \left(
\hat{\mathbf{z}} \times \left(i \, \beta_m \, {\hat{\mathbf{h}}_m}_t\right)
+ i \, \omega \, \epsilon_0 \, \left(\mathbf{\epsilon}^{\prime} - \mathbf{\epsilon} \right) \cdot \hat{\mathbf{e}}_m \right)
\\
& \qquad + \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s5}
\end{align}\end{split}\]
where the last step was accomplished by substitution of eq-3d and eq-3f from the Nonlinear Optics section.
Expanding the backward propagating modes, and using the forward/backward mode relationships yields an equation in terms of the spatial distribution of only the forward propagating modes:
\[\begin{split}\begin{align}
& -\hat{\mathbf{z}} \times \left(\mathbf{J}_{f} + i \, \omega \, \mathbf{P}_\text{NL}\right)
\\
& \begin{split} \qquad &= \sum_m \Biggl(
a_m \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) {\hat{\mathbf{h}}_m}_t
\\
& \qquad + a_m \left(
i \, \beta_m \, {\hat{\mathbf{h}}_m}_t
+ i \, \omega \, \epsilon_0 \, \hat{\mathbf{z}} \times \left(
\left(\epsilon_t - \epsilon_t^{\prime}\right) \cdot {\hat{\mathbf{e}}_m}_t
\right)
\right)
\\
& \qquad + \left(\text{b.p.m.}\right)\Biggr) \end{split} \tag{s6}
\\
& \begin{split} \qquad &= \sum_m \Biggl(
a_m \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) {\hat{\mathbf{h}}_m}_t
\\
& \qquad + a_m \left(
i \, \beta_m \, {\hat{\mathbf{h}}_m}_t
+ i \, \omega \, \epsilon_0 \ \hat{\mathbf{z}} \times \left(
\left(\epsilon_t - \epsilon_t^{\prime}\right) \cdot {\hat{\mathbf{e}}_m}_t
\right)
\right)
\\
& \qquad + a_{-m} \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_{-m}}_t\right)
+ \left(\frac{\partial}{\partial z} a_{-m}\right) {\hat{\mathbf{h}}_{-m}}_t
\\
& \qquad + a_{-m} \left(
i \, \beta_{-m} \, {\hat{\mathbf{h}}_{-m}}_t
+ i \, \omega \, \epsilon_0 \, \hat{\mathbf{z}} \times \left(
\left(\epsilon_t - \epsilon_t^{\prime}\right) \cdot {\hat{\mathbf{e}}_{-m}}_t
\right)
\right)
\Biggr) \end{split} \tag{s7}
\\
& \begin{split} \qquad &= \sum_m \Biggl(
a_m \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
+ \left(\frac{\partial}{\partial z} a_m\right) {\hat{\mathbf{h}}_m}_t
\\
& \qquad + a_m \left(
i \, \beta_m \, {\hat{\mathbf{h}}_m}_t
+ i \, \omega \, \epsilon_0 \, \hat{\mathbf{z}} \times \left(
\left(\epsilon_t - \epsilon_t^{\prime}\right) \cdot {\hat{\mathbf{e}}_m}_t
\right)
\right)
\\
& \qquad - a_{-m} \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
- \left(\frac{\partial}{\partial z} a_{-m}\right) {\hat{\mathbf{h}}_m}_t
\\
& \qquad + a_{-m} \left(
i \, \beta_m \, {\hat{\mathbf{h}}_m}_t
+ i \, \omega \, \epsilon_0 \, \hat{\mathbf{z}} \times \left(
\left(\epsilon_t - \epsilon_t^{\prime}\right) \cdot {\hat{\mathbf{e}}_m}_t
\right)
\right)
\Biggr) \end{split} \tag{s8}
\\
& \begin{split} \qquad &= \sum_m \Biggl(
\left(a_m - a_{-m}\right) \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
\\
& \qquad + \left(\frac{\partial}{\partial z} \left(a_m - a_{-m}\right)\right) {\hat{\mathbf{h}}_m}_t
\\
& \qquad + \left(a_m + a_{-m}\right) \left(
i \, \beta_m \, {\hat{\mathbf{h}}_m}_t
+ i \, \omega \, \epsilon_0 \, \hat{\mathbf{z}} \times \left(
\left(\epsilon_t - \epsilon_t^{\prime}\right) \cdot {\hat{\mathbf{e}}_m}_t
\right)
\right)
\Biggr) \end{split} \tag{s9}
\\
\end{align}\end{split}\]
Left multiplying by the cross of \(\hat{\mathbf{e}}_n^*\) allows application of the orthogonality relationships:
\[\begin{split}\begin{align}
& -\iint_{A_\infty} \left(\hat{\mathbf{e}}_n^* \times \left(\hat{\mathbf{z}} \times \mathbf{J}_{f}\right) + i \, \omega \, \hat{\mathbf{e}}_n^* \times \left(\hat{\mathbf{z}} \times \mathbf{P}_\text{NL}\right)\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \qquad \begin{split} &=
\iint_{A_\infty} \sum_m \Biggl(
\left(a_m - a_{-m}\right) \, \hat{\mathbf{e}}_n^* \times \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)
\\
& \qquad + \left(\frac{\partial}{\partial z} \left(a_m - a_{-m}\right)\right) \, \hat{\mathbf{e}}_n^* \times \hat{\mathbf{h}}_m
\\
& \qquad + \left(a_m + a_{-m}\right) \Bigl(i \, \beta_m \, \hat{\mathbf{e}}_n^* \times {\hat{\mathbf{h}}_m}_t
\\
& \qquad + i \, \omega \, \epsilon_0 \, \hat{\mathbf{e}}_n^* \times \left(\hat{\mathbf{z}} \times \left(
\left(\epsilon_t - \epsilon_t^{\prime}\right) \cdot {\hat{\mathbf{e}}_m}_t
\right)\right) \Bigr)
\Biggr) \cdot \hat{\mathbf{z}} \, dA \end{split} \tag{s10}
\\
& -\iint_{A_\infty} \left({\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{J}_{f}}_t + i \, \omega \, {\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{P}_\text{NL}}_t\right) dA
\\
& \qquad \begin{split} &=
\frac{\partial}{\partial z} \left(a_n - a_{-n}\right)
+ i \, \beta_n \, \left(a_n + a_{-n}\right)
\\
& \qquad + \sum_m \iint_{A_\infty} \Biggl(
\left(a_m - a_{-m}\right) \left({\hat{\mathbf{e}}_n^*}_t \times \left(\frac{\partial}{\partial z} {\hat{\mathbf{h}}_m}_t\right)\right) \cdot \hat{\mathbf{z}}
\\
& \qquad + \left(a_m + a_{-m}\right) i \, \omega \, \epsilon_0 \, {\hat{\mathbf{e}}_n^*}_t \cdot \left(
\left(\epsilon_t - \epsilon^{\prime}_t\right) \cdot {\hat{\mathbf{e}}_m}_t
\right)
\Biggr) dA \end{split} \tag{s11}
\end{align}\end{split}\]
Forward Mode Propagation Equation#
The forward mode propagation equation is found by adding the two curl relations (eq-4c + eq-4d) from the Nonlinear Optics section.
\[\begin{split}\begin{align}
& -\iint_{A_\infty} \left({\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{J}_{f}}_t + i \, \omega \, {\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{P}_\text{NL}}_t\right) \, dA
\\
& \qquad = 2 \left(
\frac{\partial}{\partial z} a_n
+ i \, \beta_n \, a_n
\right)
\\
& \qquad \quad + \sum_m \left(a_m + a_{-m}\right) \iint_{A_\infty} \left(
\frac{\partial {\hat{\mathbf{e}}_m}_t}{\partial z} \times {\hat{\mathbf{h}}^*_n}_t
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \qquad \quad + \sum_m \left(a_m - a_{-m}\right) \iint_{A_\infty} \left(
{\hat{\mathbf{e}}_n^*}_t \times \frac{\partial {\hat{\mathbf{h}}_m}_t}{\partial z}
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \qquad \quad + \sum_m \left(a_m + a_{-m}\right) \iint_{A_\infty} \left(
i \, \omega \, \epsilon_0 \, {\hat{\mathbf{e}}_n^*}_t \cdot
\left(\epsilon_t - \epsilon^{\prime}_t\right) \cdot {\hat{\mathbf{e}}_m}_t
\right) dA
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\frac{\partial}{\partial z} a_n =& \text{ Linear Dispersion:}
\\
& - i \, \beta_n \, a_n
\\
& \text{ Gain or Loss:}
\\
& - \sum_{m} \left(a_m + a_{-m}\right) \iint_{A_\infty} \frac{1}{2} \left(
i \, \omega \, \epsilon_0 \, {\hat{\mathbf{e}}_n^*}_t \cdot
\left(\epsilon_t - \epsilon^{\prime}_t\right) \cdot {\hat{\mathbf{e}}_m}_t
\right) dA
\\
& \text{ Changing Mode Profiles (f.p.m.):}
\\
& - \sum_{m} a_m \iint_{A_\infty} \frac{1}{2} \left(
\frac{\partial {\hat{\mathbf{e}}_m}_t}{\partial z} \times {\hat{\mathbf{h}}^*_n}_t + {\hat{\mathbf{e}}_n^*}_t \times \frac{\partial {\hat{\mathbf{h}}_m}_t}{\partial z}
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \text{ Changing Mode Profiles (b.p.m.):}
\\
& - \sum_{m} a_{-m} \iint_{A_\infty} \frac{1}{2} \left(
\frac{\partial {\hat{\mathbf{e}}_{-m}}_t}{\partial z} \times {\hat{\mathbf{h}}^*_n}_t + {\hat{\mathbf{e}}_n^*}_t \times \frac{\partial {\hat{\mathbf{h}}_{-m}}_t}{\partial z}
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \text{ Free Sources and Nonlinearity:}
\\
& -\iint_{A_\infty} \frac{1}{2} \left({\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{J}_{f}}_t + i \, \omega \, {\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{P}_\text{NL}}_t\right) \, dA
\end{align}\end{split}\]
Backward Mode Propagation Equation#
The backward mode propagation equation is found by subtracting the two curl relations (eq-4c - eq-4d) from the Nonlinear Optics section.
\[\begin{split}\begin{align}
& +\iint_{A_\infty} \left({\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{J}_{f}}_t + i \, \omega \, {\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{P}_\text{NL}}_t\right) dA
\\
& \qquad = 2 \left(
\frac{\partial}{\partial z} a_{-n}
- i \, \beta_n \, a_{-n}
\right)
\\
& \qquad \quad + \sum_m \left(a_m + a_{-m}\right) \iint_{A_\infty} \left( \frac{\partial {\hat{\mathbf{e}}_m}_t}{\partial z} \times {\hat{\mathbf{h}}^*_n}_t
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \qquad \quad - \sum_m \left(a_m - a_{-m}\right) \iint_{A_\infty} \left(
{\hat{\mathbf{e}}_n^*}_t \times \frac{\partial {\hat{\mathbf{h}}_m}_t}{\partial z}
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \qquad \quad - \sum_m \left(a_m + a_{-m}\right) \iint_{A_\infty} \left(
i \, \omega \, \epsilon_0 \, {\hat{\mathbf{e}}_n^*}_t \cdot
\left(\epsilon_t - \epsilon^{\prime}_t\right) \cdot {\hat{\mathbf{e}}_m}_t
\right) dA
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\frac{\partial}{\partial z} a_{-n} =& \text{ Linear Dispersion:}
\\
& +i \, \beta_n \, a_{-n}
\\
& \text{ Gain or Loss:}
\\
& + \sum_m \left(a_m + a_{-m}\right) \iint_{A_\infty} \frac{1}{2} \left(
i \, \omega \, \epsilon_0 \, {\hat{\mathbf{e}}_n^*}_t \cdot
\left(\epsilon_t - \epsilon^{\prime}_t\right) \cdot {\hat{\mathbf{e}}_m}_t
\right) dA
\\
& \text{ Changing Mode Profiles (f.p.m.):}
\\
& + \sum_m a_m \iint_{A_\infty} \frac{1}{2} \left(
\frac{\partial {\hat{\mathbf{e}}_m}_t}{\partial z} \times {\hat{\mathbf{h}}^*_{-n}}_t + {\hat{\mathbf{e}}_{-n}^*}_t \times \frac{\partial {\hat{\mathbf{h}}_m}_t}{\partial z}
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \text{ Changing Mode Profiles (b.p.m.):}
\\
& + \sum_m a_{-m} \iint_{A_\infty} \frac{1}{2} \left(
\frac{\partial {\hat{\mathbf{e}}_{-m}}_t}{\partial z} \times {\hat{\mathbf{h}}^*_{-n}}_t + {\hat{\mathbf{e}}_{-n}^*}_t \times \frac{\partial {\hat{\mathbf{h}}_{-m}}_t}{\partial z}
\right) \cdot \hat{\mathbf{z}} \, dA
\\
& \text{ Free Sources and Nonlinearity:}
\\
& + \iint_{A_\infty} \frac{1}{2} \left({\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{J}_{f}}_t + i \, \omega \, {\hat{\mathbf{e}}_n^*}_t \cdot {\mathbf{P}_\text{NL}}_t\right) dA
\end{align}\end{split}\]
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 \(s\):
\[\begin{split}\begin{gather}
\frac{\partial}{\partial z} a_n
= \ldots -i \, \omega \sum_{r s} \iint_{-\infty}^{\infty}
g_{n r s}^{\left(2\right)}\!\left[\nu_1, \nu_2\right] a_r\!\left[\nu_1\right] a_s\!\left[\nu_2\right]
\delta\!\left[\nu - \left(\nu_1 + \nu_2\right)\right] d\nu_1 \, d\nu_2
\\
g_{n r s}^{\left(2\right)}\!\left[\nu_1, \nu_2\right] = \iint_{A_\infty} \frac{1}{2} \epsilon_0 \, \chi^{\left(2\right)}_{i j k}\!\left[\nu_1, \nu_2\right] {{\hat{\mathbf{e}}_n^*}_t}_i\!\left[\nu_1 + \nu_2\right] {\hat{\mathbf{e}}_r}_j\!\left[\nu_1\right] {\hat{\mathbf{e}}_s}_k\!\left[\nu_2\right] dA
\end{gather}\end{split}\]
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:
\[\begin{split}\begin{align}
\chi^{\left(1\right)} &= n_0^2 - 1
\\
\chi^{\left(3\right)} &= \frac{4}{3} \epsilon_0 \, c \, n_0^2 \, n_2
\end{align}\end{split}\]