18 Jun 2025
Planar Dielectric Waveguides and Guided-Wave Power
Total-reflection phase, slab continuity conditions, eigenvalue equations, velocities, field energy, and power.
A symmetric planar dielectric guide has a core of refractive index $n_1$ in $\lvert x\rvert<a$, cladding of index $n_2<n_1$ in $\lvert x\rvert>a$, and is uniform along $y$ and the propagation direction $z$. A guided field must oscillate in the core and decay in the cladding.
Total internal reflection and its phase
Consider a core ray incident on a core-cladding interface at angle $\theta$ to the normal. Snell’s law is
\[n_1\sin\theta=n_2\sin\theta_t.\]When $\theta>\theta_c$, where
\[\boxed{\sin\theta_c=\frac{n_2}{n_1}},\]$\sin\theta_t>1$ and the cladding field is evanescent. Define
\[\gamma=\sqrt{\sin^2\theta-\left(\frac{n_2}{n_1}\right)^2}.\]The Fresnel reflection coefficients have unit magnitude but a nonzero phase. They may be written
\[r_s=\exp(i\phi_s), \qquad \boxed{\phi_s=-2\tan^{-1}\!\left(\frac{\gamma}{\cos\theta}\right)},\] \[r_p=\exp(i\phi_p), \qquad \boxed{\phi_p=-2\tan^{-1}\!\left(\frac{n_1^2\gamma}{n_2^2\cos\theta}\right)}.\]The reflected power is unity, but the phase shift is essential in the guided-mode condition.
Wave equation across the slab
For a TE mode choose
\[\mathbf E=\hat{\mathbf y}\,\psi(x)e^{i(\beta z-\omega t)}, \qquad k_0=\frac{\omega}{c}.\]The Helmholtz equation becomes
\[\boxed{\frac{d^2\psi}{dx^2}+[n^2(x)k_0^2-\beta^2]\psi=0}.\]Define the real positive transverse constants
\[\boxed{h=\sqrt{n_1^2k_0^2-\beta^2}}, \qquad \boxed{q=\sqrt{\beta^2-n_2^2k_0^2}}.\]Both are real only when
\[\boxed{n_2k_0<\beta<n_1k_0}.\]This inequality is the guided-wave condition. It is often written $n_2<n_{\mathrm{eff}}<n_1$, where
\[\boxed{n_{\mathrm{eff}}=\frac{\beta}{k_0}}.\]Continuity and TE eigenvalue equations
For the convention $e^{i(\beta z-\omega t)}$, Faraday’s law gives
\[H_z=\frac{1}{i\omega\mu}\frac{dE_y}{dx} =-\frac{i}{\omega\mu}\frac{dE_y}{dx}.\]For equal magnetic permeabilities, tangential $E_y$ and $H_z$ are continuous. Hence both $\psi$ and $d\psi/dx$ are continuous at $x=\pm a$; the common factor $-i/(\omega\mu)$ does not alter the eigenvalue equation.
For an even mode,
\[\psi(x)=\begin{cases} A\cos(hx),&\lvert x\rvert\le a,\\ A\cos(ha)e^{-q(\lvert x\rvert-a)},&\lvert x\rvert\ge a. \end{cases}\]Derivative continuity at $x=a$ gives
\[-Ah\sin(ha)=-qA\cos(ha),\]so
\[\boxed{h\tan(ha)=q\qquad\text{(even TE)}}.\]For an odd core field $A\sin(hx)$, the same boundary condition gives
\[\boxed{-h\cot(ha)=q\qquad\text{(odd TE)}}.\]These transcendental eigenvalue equations select discrete $\beta$ values at a fixed frequency.
For TM modes it is convenient to use $H_y=\psi(x)e^{i(\beta z-\omega t)}$. Tangential $H_y$ and tangential $E_z\propto(1/\epsilon)dH_y/dx$ are continuous. Therefore
\[\boxed{h\tan(ha)=\frac{\epsilon_1}{\epsilon_2}q \qquad\text{(even TM)}},\] \[\boxed{-h\cot(ha)=\frac{\epsilon_1}{\epsilon_2}q \qquad\text{(odd TM)}}.\]With normalized variables
\[u=ha,\qquad w=qa,\]the definitions give
\[\boxed{u^2+w^2=V^2}, \qquad \boxed{V=k_0a\sqrt{n_1^2-n_2^2}}.\]$V$ is dimensionless and controls how many slab modes can exist.
Phase and group velocities
The longitudinal phase is $\beta z-\omega t$. Therefore
\[\boxed{v_p=\frac{\omega}{\beta}=\frac{c}{n_{\mathrm{eff}}}}.\]The envelope of a narrow frequency band travels at
\[\boxed{v_g=\frac{d\omega}{d\beta}=\left(\frac{d\beta}{d\omega}\right)^{-1}}.\]Both material dispersion $n_j(\omega)$ and waveguide dispersion through the eigenvalue equation contribute to $d\beta/d\omega$. In the simpler nondispersive relation $\beta^2+\kappa^2=n^2\omega^2/c^2$ with fixed transverse eigenvalue $\kappa$,
\[\boxed{v_pv_g=\left(\frac cn\right)^2}.\]Field energy and transmitted power
For peak phasors in a lossless, nondispersive guide, the cycle-averaged energy density is
\[\boxed{\overline u=\frac14\left(\epsilon\lvert\mathbf E\rvert^2+\mu\lvert\mathbf H\rvert^2\right)} \quad[\mathrm{J\,m^{-3}}].\]The time-averaged longitudinal power per unit width in $y$ is
\[\boxed{P'=\frac12\Re\int_{-\infty}^{\infty} (\mathbf E\times\mathbf H^{\ast})\cdot\hat{\mathbf z}\,dx} \quad[\mathrm{W\,m^{-1}}].\]For the TE field above, Faraday’s law gives $H_x=-\beta E_y/(\omega\mu)$. Consequently,
\[\boxed{P'=\frac{\beta}{2\omega\mu} \int_{-\infty}^{\infty}\lvert\psi(x)\rvert^2dx}.\]The energy stored per unit guide length and per unit width is
\[U'=\int_{-\infty}^{\infty}\overline u\,dx \quad[\mathrm{J\,m^{-2}}].\]Their ratio has units of speed. For a lossless guide it equals the group velocity:
\[\boxed{\frac{P'}{U'}=v_g}.\]The normalized-frequency identity and the fixed-$\kappa$ phase-group velocity product are checked in the Unit II Maxima worksheet; every printed residual is zero.
Discussion