Purdue ME606: EM Field Theory

1 Electromagnetic Field Theory

Electromagnetic Field Theory

Maxwell Equations

$\nabla \cdot \left( {\epsilon \vec E} \right) = \rho_f$

where:

$ε$ is electrical permittivity

$\vec E$ is electric field vector

$ρ f$ is free electron density

$\nabla \cdot \left( {\mu \vec H} \right) = 0$

where:

$μ$ is magnetic permeability

$\vec E$ is magnetic field vector

$\nabla \times \vec E = - \mu \frac{{\partial \vec H}}{{\partial t}}$

$\nabla \times \vec H = \epsilon \frac{{\partial \vec E}}{{\partial t}} + \sigma_e \vec E$

where:

$t$ is time

$σ e$ is electrical conductivity

We will treat as constants $ε$ , $μ$ , $σ e$ , and $ρ f$ .

We expect wave-like solutions to these equations.

Consider a general vector wave field:

$\vec F_c = \vec F_{c0} e^{2\pi i\nu t}$

In our case… for example:

$\vec E = {\rm Re} \left\{ {\vec E_0 e^{ - 2\pi i\left( {\vec w \cdot \vec r - \nu t} \right)} } \right\}$

where $\vec w$ = wave vector = $\vec w' - i\vec w''$

For complex fields:

$\vec E_c = \vec E_0 e^{ - 2\pi \vec w'' \cdot \vec r} e^{ - 2\pi i\left( {\vec w' \cdot \vec r - \nu t} \right)} \vec H_c = \vec H_0 e^{ - 2\pi \vec w'' \cdot \vec r} e^{ - 2\pi i\left( {\vec w' \cdot \vec r - \nu t} \right)}$

If there is no complex part of $\vec w'$ , we have no attenuation… i.e., a perfect dielectric.

Introduce some more familiar terms:

speed of light (in a vacuum): $c_0 = \frac{{\nu}}{{w'}} = \frac{{\nu}}{{\sqrt {\vec w \cdot \vec w} }} = \frac{{1}}{{\epsilon_0 \mu_0}}$

complex index of refraction: $m = n - i k$

where:

$k$ = absorptive index

$n$ = index of refratction

Express $\vec E_c$ , $\vec H_c$ in terms of more familiar variables:

We can deduce a new phase velocity, $c = \frac{{c_0}}{{n}}$

The Poynting vector describes the energy contained in a wave:

$\vec S_{avg} = \frac{{n}}{{2 c_0 \mu}} \left| {E_0^2 } \right| e^{-4 \pi \eta_0 k z } \hat s$

where $\hat s =$ direction of propagation

On Polarization

Consider a point in space, $z = 0$

$\vec E \left( {z=0 , t } \right) = \vec A \cos 2 \pi \nu t + \vec B \sin 2 \pi \nu t$

The resulting time profile would map to an ellipse:

Polarization coordinates can be used to describe an $\vec E$ field as:

$\vec E_0 = E_\parallel \left( t \right)\hat e_\parallel + E_ \bot \left( t \right)\hat e_ \bot$

Interfaces

Gauss’ Theorem:
$\int\limits_\forall {\nabla \cdot \vec Fd\forall } = \int\limits_\Gamma {\vec F \cdot d\vec \Gamma }$

Stokes’ Theorem:
$\int\limits_\forall {\nabla \times \vec Fd\forall } = -\int\limits_\Gamma {\vec F \times d\vec \Gamma }$

In the domain:

Apply the first Maxwell Equation and integrate; then use Gauss’ Theorem. Use a similar approach for the other equations to find:

$\vec E_{c1} \times \hat n = \vec E_{c2} \times \hat n$

$\vec H_{c1} \times \hat n = \vec H_{c2} \times \hat n$

Now consider a plane wave impinging on an interface:

We use some simple geometry to relate $\left| {\overrightarrow {AA'} } \right|$ and $\left| {\overrightarrow {BB'} } \right|$ to $θ 1$ and $θ 2$ to find:

$\frac{{\sin \left( {\theta _2 } \right)}}{{\sin \left( {\theta _1 } \right)}} = \frac{{n_1 }}{{n_2 }}$

Consider the following:

Assume non-absorbing media -> w” = 0

For Medium 1 (incident medium):

$\vec E_{C1} = \vec E_{0i} exp(-2 \pi i(\vec w_i^' \cdot \vec r - \nu t)) + \vec E_{0r} exp(-2 \pi i(\vec w_r^' \cdot \vec r - \nu t))$

similar for $\vec H_{C1}$

For Medium 2 (incident medium): $\vec E_{C2} = \vec E_{0t} exp(-2 \pi i(\vec w_t^' \cdot \vec r - \nu t))$

Convert to $\parallel$ and $\bot$ coordinates

Now define reflection coefficient:

$r_\parallel = \frac{E_{r \parallel}}{E_{i \parallel}}=\frac{n_1 cos(\theta _2)-n_2 cos(\theta _1)}{n_1 cos(\theta _2)+n_2 cos(\theta _1)}$

$r_ \bot = \frac{E_{r \bot}}{E_{i \bot}}=\frac{n_1 cos(\theta _1)-n_2 cos(\theta _2)}{n_1 cos(\theta _1)+n_2 cos(\theta _2)}$

$t_\parallel = \frac{E_{t \parallel}}{E_{i \parallel}}=\frac{2 n_1 cos(\theta _1)}{n_1 cos(\theta _2)+n_2 cos(\theta _1)}$

$t_ \bot = \frac{E_{t \bot}}{E_{i \bot}}=\frac{2 n_1 cos(\theta _1)}{n_1 cos(\theta _1)+n_2 cos(\theta _2)}$

But these coefficients do not express energy ratios, use the Poynting vector to calculate reflectance and transmittance:

$\rho_\parallel = \frac{s_{r \parallel}}{s_{i \parallel}} = \left[\frac{E_{r \parallel}}{E_{i \parallel}} \right]^2 = r_\parallel^2$

$\rho_ \bot = \frac{s_{r \bot}}{s_{i \bot}} = \left[\frac{E_{r \bot}}{E_{i \bot}} \right]^2 = r_ \bot^2$

Finally, for unpolarized and circularly polarized light:

$\rho = \frac{1}{2}(\rho_\parallel+\rho_ \bot) = \frac{1}{2} \left[\frac{tan^2(\theta _1-\theta _2)}{tan^2(\theta _1+\theta _2)}+\frac{sin^2(\theta _1-\theta _2)}{sin^2(\theta _1+\theta _2)} \right]$

note for non-absorbing media, $τ = 1 - ρ$

Back to real surfaces …

Diffuse Reflectances

Bidirectional reflectance represents the ratio of reflected intensity to incident radiative heat flux.

$\rho_{bd}(\theta_i,\phi_i;\theta_r,\phi_r)=\frac{dI_r}{dq''_i}=\frac{dI_r}{I_i cos(\theta_i)d\omega}$

note: $ρ b d$ satisfies reciprocity,meaning that $ρ b d (θ i,φ i;θ r,φ r) = ρ b d (θ r,φ r;θ i,φ i)$

Directional Hemispherical Reflectance

$\rho_{dh}(\theta_i,\phi_i)=\frac{dq''_{r(HEMI)}}{dq''_i}=\frac{reflected~ energy}{incident~ energy}=\int_{HEMIr} \rho_{bd}(\theta_i,\phi_i;\theta_r,\phi_r)cos(\theta_r)d\omega_r$

Hemispherical Directional Reflectance

$\rho_{hd}(\theta_r,\phi_r)=\frac{reflected~ intensity}{average~ incident}=\frac{\int_{HEMIi}dI_r(\theta_i,\phi_i;\theta_r,\phi_r)}{\frac{1}{\pi}\int_{HEMIi}dq''_i}=\frac{\int_{HEMIi}\rho_{bd}I_i cos(\theta_i)d \omega_i}{\frac{1}{\pi}\int_{HEMIi}I_i cos(\theta_i) d\omega_i}$

Hemispherical Hemispherical Reflectance

$\rho_{hh}(\theta_i,\phi_i)=\frac{total~ incident~ energy}{total~ reflected~ energy}=\frac{\int_{HEMIi} \rho_{dh}I_i cos(\theta_i)d\omega_i}{\int_{HEMIi} I_i cos(\theta_i)d\omega_i}$

Last modified on 14 Oct, 2008

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Purdue ME606: EM Field Theory

Table of Contents

Tags

Electromagnetic Field Theory

Maxwell Equations

On Polarization

Interfaces

Diffuse Reflectances

Directional Hemispherical Reflectance

Hemispherical Directional Reflectance

Hemispherical Hemispherical Reflectance