Electromagnetic Waves

2.1 Properties of EM Waves

From Maxwell’s equations, the following properties hold for plane EM waves:

1. Transversality: $\mathbf{E}$ and $\mathbf{B}$ are perpendicular to $\mathbf{k}$ and to each other.
2. Orthogonality: $\mathbf{E} \perp \mathbf{B}$And $|\mathbf{E}| = c|\mathbf{B}|$.
3. In-phase: $\mathbf{E}$ and $\mathbf{B}$ oscillate in phase.
4. Dispersion relation: $\omega = ck$ in vacuum.

Proof of transversality. For a plane wave $\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}$ Gauss’s law gives $i\mathbf{k}\cdot\mathbf{E}_0 = 0$So $\mathbf{k} \perp \mathbf{E}_0$. Similarly From $\nabla \cdot \mathbf{B} = 0$: $\mathbf{k} \perp \mathbf{B}_0$. $\blacksquare$

Proof of $|\mathbf{E}| = c|\mathbf{B}|$. From Faraday’s law for a plane wave: $\mathbf{k} \times \mathbf{E}_0 = \omega\mathbf{B}_0$. Taking magnitudes: $kE_0 = \omega B_0$So $E_0/B_0 = \omega/k = c$. $\blacksquare$

2.2 Energy and Intensity

The Poynting vector:

$\mathbf{S} = \frac{1}{\mu_0}\mathbf{E} \times \mathbf{B}$

Represents the energy flux (W/m$^2$). The time-averaged intensity for a plane wave:

$I = \langle S \rangle = \frac{1}{2}c\varepsilon_0 E_0^2$

The energy density of an EM field is:

$u = \frac{1}{2}\varepsilon_0 E^2 + \frac{1}{2\mu_0}B^2 = \varepsilon_0 E^2$

(the electric and magnetic contributions are equal for a plane wave). The intensity is related to The energy density by $I = uc$.

Radiation pressure. For a perfectly absorbing surface: $P_{\mathrm{rad} = I/c}$. For a perfectly Reflecting surface: $P_{\mathrm{rad} = 2I/c}$.

2.3 EM Waves in Matter

In a linear, isotropic, non-magnetic medium with refractive index $n$:

$v = \frac{c}{n}, \quad \mathbf{k} = n\frac{\omega}{c}\hat{\mathbf{k}}$

The index of refraction is related to the relative permittivity and permeability:

$n = \sqrt{\varepsilon_r \mu_r}$

For non-magnetic materials ($\mu_r \approx 1$): $n \approx \sqrt{\varepsilon_r}$.

The wavelength inside a medium of refractive index $n$ is $\lambda_n = \lambda_0/n$Where $\lambda_0$ Is the vacuum wavelength. The frequency remains unchanged across the boundary.

2.4 Boundary Conditions and Snell’s Law

At a planar interface between two linear, isotropic media, the tangential components of $\mathbf{E}$ and $\mathbf{H}$ and the normal components of $\mathbf{D}$ and $\mathbf{B}$ are Continuous across the boundary.

Consider a plane wave incident from medium 1 ($n_1$) onto medium 2 ($n_2$), with the interface at $z = 0$ and the plane of incidence the $xz$-plane.

The phase matching condition requires the phases of all three waves (incident, reflected, Transmitted) to match at $z = 0$ for all $x$ and $t$. This gives:

$k_1\sin\theta_i = k_1\sin\theta_r = k_2\sin\theta_t$

From the first equality: $\theta_i = \theta_r$ (law of reflection). From the second equality: $n_1\sin\theta_i = n_2\sin\theta_t$ (Snell’s law).

Proof. The incident, reflected, and transmitted fields are:

$E_i \propto e^{i(k_{1x}x + k_{1z}z - \omega t)}, \quad E_r \propto e^{i(k_{1x}'x + k_{1z}'z - \omega t)}, \quad E_t \propto e^{i(k_{2x}x + k_{2z}z - \omega t)}$

At $z = 0$The tangential field must be continuous for all $x$ and $t$: $k_{1x} = k_{1x}' = k_{2x}$I.e., $k_1\sin\theta_i = k_1\sin\theta_r = k_2\sin\theta_t$. Since $k = n\omega/c$This yields Snell’s law. $\blacksquare$

2.5 Fresnel Equations

Applying the boundary conditions for the tangential fields yields the Fresnel equations for the Amplitude reflection and transmission coefficients.

s-polarisation ($\mathbf{E}$ perpendicular to the plane of incidence, along $\hat{\mathbf{y}}$): The tangential components of $\mathbf{E}$ and $\mathbf{H}$ give:

$r_s = \frac{E_{0r}}{E_{0i}} = \frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}$

$t_s = \frac{E_{0t}}{E_{0i}} = \frac{2n_1\cos\theta_i}{n_1\cos\theta_i + n_2\cos\theta_t}$

p-polarisation ($\mathbf{E}$ parallel to the plane of incidence): The tangential components of $\mathbf{E}$ and $\mathbf{H}$ give:

$r_p = \frac{E_{0r}}{E_{0i}} = \frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}$

$t_p = \frac{E_{0t}}{E_{0i}} = \frac{2n_1\cos\theta_i}{n_2\cos\theta_i + n_1\cos\theta_t}$

Reflectance and transmittance (energy fractions):

$R_s = |r_s|^2, \quad T_s = \frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t_s|^2, \quad R_s + T_s = 1$

$R_p = |r_p|^2, \quad T_p = \frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t_p|^2, \quad R_p + T_p = 1$

At normal incidence ($\theta_i = 0$): $r_s = r_p = (n_1 - n_2)/(n_1 + n_2)$ and $R = [(n_1 - n_2)/(n_1 + n_2)]^2$.

:::caution Common Pitfall The Fresnel coefficients $r_s$ and $r_p$ have different forms. A common error is to swap the $n_1\cos\theta_i$ and $n_2\cos\theta_t$ terms. Remember: for $r_s$The numerator starts with $n_1\cos\theta_i$; for $r_p$The numerator starts with $n_2\cos\theta_i$. Also, $r$ and $t$ are Amplitude coefficients, while $R$ and $T$ are energy coefficients — they are related but not Interchangeable.

2.6 Total Internal Reflection and Evanescent Waves

When light travels from a denser to a rarer medium ($n_1 \gt n_2$) and the angle of incidence Exceeds the critical angle:

$\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right)$

Snell’s law gives $\sin\theta_t = (n_1/n_2)\sin\theta_i \gt 1$So $\theta_t$ becomes complex. Writing $\cos\theta_t = i\sqrt{\sin^2\theta_t - 1}$The Fresnel coefficients become complex with $|r_s|^2 = |r_p|^2 = 1$: all energy is reflected.

The transmitted field becomes an evanescent wave:

$E_t \propto e^{-\kappa z}\, e^{i(k_x x - \omega t)}$

Where:

$\kappa = k_0\sqrt{n_1^2\sin^2\theta_i - n_2^2}, \quad k_x = k_0 n_1\sin\theta_i$

The field decays exponentially with penetration depth $\delta = 1/\kappa$ into the second medium, But propagates without loss along the interface. No net energy is transported across the boundary ($T = 0$).

Frustrated total internal reflection (FTIR). If a third medium (with $n_3 \geq n_2$) is brought Within a distance comparable to $\delta$ of the interface, the evanescent wave can couple into it, Allowing energy transmission across the gap. This is the optical analogue of quantum mechanical Tunnelling.

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