#ece1228

Total internal reflection and Brewster’s angles

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Total internal reflection

From Snell’s second law we have

\begin{equation}\label{eqn:brewsters:20} \theta_t = \arcsin\lr{ \frac{n_i}{n_t} \sin\theta_i }. \end{equation}

This is plotted in fig. 3.

fig. 3. Transmission angle vs incident angle.

  For the \( n_i > n_t \) case, for example, like shining from glass into air, there is a critical…

Fresnel angular sum and difference formulas

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In [1] are some sum and angle difference formulations for the Fresnel formulas given a \( \mu_1 = \mu_2 \) constraint. The proof of these trig Fresnel equations is left to an exercise, and will be derived here.

We need a couple trig identities to start with.

\begin{equation}\label{eqn:fresnelSumAndDifferenceAngleFormulas:20} \begin{aligned}

Normal transmission and reflection through two interfaces

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Motivation

In class an outline of normal transmission through a slab was presented. Let’s go through the details.

Normal incidence

The geometry of a two interface configuration is sketched in fig. 1.

  fig. 1. Two interface transmission.

Given a normal incident ray with magnitude \( A \), the respective forward and backwards rays in each…

ECE1228H Electromagnetic Theory. Lecture 10: Fresnel relations. Taught by Prof.\ M. Mojahedi

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Motivation

In class, an overview of the Fresnel relations for a TE mode electric field were presented. Here’s a fleshing out of the details is presented, as well as the equivalent for the TM mode.

Single interface TE mode.

The Fresnel reflection geometry for an electric field \( \BE \) parallel to the interface (TE mode) is sketched in…

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Jackson [1] has an interesting presentation of the transverse gauge. I’d like to walk through the details of this, but first want to translate the preliminaries to SI units (if I had the 3rd edition I’d not have to do this translation step).

Gauge freedom

The starting point is noting that \( \spacegrad \cdot \BB = 0 \) the magnetic field…

Continuity equation and Ampere’s law

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Q:

Show that without the displacement current \( \PDi{t}{\BD} \), Maxwell’s equations will not satisfy conservation relations.

A:

Without the displacement current, Maxwell’s equations are \begin{equation}\label{eqn:continuityDisplacement:20} \begin{aligned} \spacegrad \cross \BE( \Br, t ) &= – \PD{t}{\BB}(\Br, t) \\ \spacegrad \cross \BH(…

Dipole field from multipole moment sum

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As indicated in Jackson [1], the components of the electric field can be obtained directly from the multipole moments

\begin{equation}\label{eqn:dipoleFromSphericalMoments:20} \Phi(\Bx) = \inv{4 \pi \epsilon_0} \sum \frac{4 \pi}{ (2 l + 1) r^{l + 1} } q_{l m} Y_{l m}, \end{equation}

so for the \( l,m \) contribution to this sum the…

Vector wave equation in spherical coordinates

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For a vector \( \BA \) in spherical coordinates, let’s compute the Laplacian

\begin{equation}\label{eqn:vectorWaveEquationSpherical:20} \spacegrad^2 \BA, \end{equation}

to see the form of the wave equation. The spherical vector representation has a curvilinear basis \begin{equation}\label{eqn:vectorWaveEquationSpherical:40} \BA = \rcap A_r…