Electromagnetic Field (EMF)

Table of Contents

Maxwell's equations without free charge

\begin{align*} \nabla \times \mathbf{E} &= -\mu \frac{\partial \mathbf{H}}{\partial t} \\ \nabla \times \mathbf{H} &= \epsilon \frac{\partial \mathbf{E}}{\partial t} \\ \nabla \cdot \epsilon\mathbf{E} &= 0 \\ \nabla \cdot \mu\mathbf{H} &= 0 \end{align*}

where

  • \(\mathbf{E}\) and \(\mathbf{H}\) are strengths of electric and magnetic fields respectively;
  • \(\epsilon\) and \(\mu\) are permittivity and permeability respectively.

Wave equation

Both \(\mathbf{E}\) and \(\mathbf{H}\) fulfill wave equation in vector format, i.e.,

\begin{align*} \nabla^2 \mathbf{E} - \frac{n^2}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} &= 0\\ \nabla^2 \mathbf{H} - \frac{n^2}{c^2} \frac{\partial^2 \mathbf{H}}{\partial t^2} &= 0 \end{align*}

where

  • \(n\) is the refractive index of the medium;
  • \(c = \dfrac{1}{\sqrt{\epsilon_0 \mu_0}}\) is the propagation speed of wave in vacuum, with \(\epsilon_0\) and \(\mu_0\) the permittivity and permeability in vacuum respectively.

For any individual component, the wave equation in scalar format applies, i.e.,

\begin{align} \nabla^2 u(\mathbf{p}, t) - \frac{n^2}{c^2} \frac{\partial^2 u(\mathbf{p}, t)}{\partial t^2} = 0, \label{eq:wave-eq} \end{align}

where \(u(\mathbf{p},t)\) can be \(\mathbf{E}_x\), \(\mathbf{E}_y\), \(\mathbf{E}_z\), \(\mathbf{H}_x\), \(\mathbf{H}_y\), or \(\mathbf{H}_z\).

For monochromatic wave, a scalar-field can be written as

\begin{align} u(\mathbf{r}, t) &= a(\mathbf{r}) \cos[2\pi ft + \phi(\mathbf{r})] \nonumber \\ &= \mathcal{R}\left[U(\mathbf{r}, t) \right] \nonumber \\ &= \mathcal{R}\left[U(\mathbf{r}) e^{j2\pi ft}\right] \label{eq:separable} \end{align}

where

  • \(U(\mathbf{r}) \triangleq a(\mathbf{r}) e^{j \phi(\mathbf{r})}\), \(a(\mathbf{r})\) and \(\phi(\mathbf{r})\) are respectively the amplitude and phase at point \(\mathbf{r}\);
  • \(U(\mathbf{r},t) \triangleq U(\mathbf{r})e^{j2\pi ft}\).

Helmholtz equation

By substituting \eqref{eq:separable} into \eqref{eq:wave-eq}, the time-independent \(U(\mathbf{r})\) obey the following Helmholtz equation,

\begin{align} (\nabla^2 + \kappa^2)U(\mathbf{r}) = 0 \label{eq:helmholtz}, \end{align}

where \(κ = 2\pi n \dfrac{f}{c} = \dfrac{2\pi}{\lambda}\) is termed wave number.

Angular spectrum

Without loss of generality, a monochromatic wave can be denoted by

\begin{align} U(\mathbf{r})=e^{j\mathbf{k}\cdot \mathbf{r}}, \label{eq:f-distr} \end{align}

where

  • \(\mathbf{r}\) is a location vector,
  • \(\mathbf{\kappa} = \cfrac{2\pi}{\lambda} (\hat{\mathbf{x}}\cos\alpha + \hat{\mathbf{y}}\cos\beta + \hat{\mathbf{z}}\cos\gamma)\) is wave vector,
  • \(\alpha\), \(\beta\), \(\gamma\) are the angles of \(\mathbf{\kappa}\) relative to \(x\), \(y\), \(z\) axes respectively. Accordingly, their cosine values are direction cosines. Clearly, \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\).

Targetting the field on plane perpendicular to \(z\) axis, 2D Fourier transform can be performed

\begin{align*} A(f_X, f_Y; z) &= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} U(x,y,z)e^{-j2\pi (f_Xx + f_Yy)} dx dy, \\ U(x,y,z) &= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} A(f_X,f_Y; z)e^{j2\pi (f_Xx + f_Yy)} df_X df_Y; \end{align*}

where

  • \(A(f_X, f_Y; z)\) is termed angular spectrum of disturbance \(U(x, y, z)\);
  • \(f_X \triangleq \cfrac{\cos\alpha}{\lambda}\), \(f_Y \triangleq \cfrac{\cos\beta}{\lambda}\) are spatial frequencies.

Based on \eqref{eq:f-distr}, the propagation from \(z=0\) can be expressed by

\begin{align} A(f_X, f_Y; z) = A(f_X, f_Y; 0) e^{j\kappa z \sqrt{1 - \lambda^2f_X^2 - \lambda^2f_Y^2}}. \label{eq:prop} \end{align}
  • For \(f_X^2 + f_Y^2 < \cfrac{1}{\lambda^2}\), relative phase delays are introduced in the propagation over distance \(z\), as shown in \eqref{eq:prop}.
  • For \(f_X^2 + f_Y^2 > \cfrac{1}{\lambda^2}\), \eqref{eq:prop} becomes \(A(f_X, f_Y; z) = A(f_X, f_Y; 0) e^{-\kappa z \sqrt{\lambda^2f_X^2 + \lambda^2f_Y^2 - 1}}\). Accordingly, the component is termed evanescent wave and attenuates quickly. In case of \(z\) larger than several wavelength, the evanescent component can be disregarded.

Equivalently, an effective transfer function for wave propagation can be represented by

\begin{align} H(f_X, f_Y) = \begin{cases} e^{j\kappa z \sqrt{1 - \lambda^2f_X^2 - \lambda^2f_Y^2}} & f_X^2 + f_Y^2 < \cfrac{1}{\lambda^2}; \\ 0 & \mathrm{otherwise}. \end{cases} \label{eq:h} \end{align}

Fresnel diffraction

Conditioned on \(|\lambda f_X| \ll 1\) and \(|\lambda f_Y| \ll 1\), following approximation, a.k.a. Fresnel approximation, paraxial approximation, or near-field approximation holds.

\begin{align} \sqrt{1 - \lambda^2 f_X^2 - \lambda^2f_Y^2} \approx 1 - \dfrac{\lambda^2f_X^2}{2} - \dfrac{\lambda^2f_Y^2}{2}, \label{eq:approx-fresnel} \end{align}

In this case, Huygens principle becomes

\begin{align} U(x, y, z) = \frac{e^{j\kappa z}}{j\lambda z} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}U(\xi, \eta, 0) e^{j \frac{\kappa}{2z}\left[(x - \xi)^2 + (y - \eta)^2\right]} d\xi d\eta, \label{eq:huygens-fresnel} \end{align}

and the transfer function in \eqref{eq:h} becomes

\begin{align} H(f_X, f_Y) = \begin{cases} e^{j\kappa z}e^{-j\pi \lambda z \left( f_X^2 + f_Y^2 \right)} & f_X^2 + f_Y^2 < \cfrac{1}{\lambda^2}; \\ 0 & \mathrm{otherwise}. \end{cases} \label{eq:h-fresnel} \end{align}

Fraunhofer diffraction

Conditioned on \(z \gg \dfrac{\kappa \max\{\xi^2 + \eta^2\}}{2}\), a.k.a. Fraunhofer approximation, or far-field approximation, Huygens principle becomes

\begin{align} U(x, y, z) = \frac{e^{j\kappa z}e^{j\kappa\frac{x^2+y^2}{2z}}}{j\lambda z} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} U(\xi, \eta, 0)e^{-j \frac{\kappa}{z}(x \xi + y \eta)} d\xi d\eta. \label{eq:huygens-fraunhofer} \end{align}

Clearly, besides the coefficient out of integration, the propagation can be is a 2D Fourier transform at frequencies \(f_X = \dfrac{x}{z \lambda}\), \(f_Y = \dfrac{y}{z \lambda}\).

Additionally, there is a relaxed condition \(z > \dfrac{2D^2}{\lambda}\)1 in which Fraunhofer diffraction approximately holds, where \(D\) is the aperture size.

Rectangular aperture

For a rectangular aperture, its amplitude transmittance ratio can be written as

\begin{align} t_{\mathrm{rect}}(\xi, \eta) = \mathrm{rect}\left(\frac{\xi}{A_x}\right) \mathrm{rect}\left( \frac{\eta}{A_y}\right), \label{eq:rect-apert} \end{align}

where \(A_x\) and \(A_y\) are the aperture sizes in the \(x\) and \(y\) axes, respectively. Then its Fraunhofer pattern can be obtained as

\begin{align} U(x, y) &= \frac{e^{j\kappa z}e^{j\kappa\frac{x^2 + y^2}{2z}}}{j\lambda z} \mathcal{F} \left\{ U(\xi, \eta)\right\} \mid_{f_X=\frac{x}{z\lambda}, f_Y=\frac{y}{z\lambda}} \nonumber \\ &= \frac{A_xA_ye^{j\kappa z}e^{jκ\frac{x^2 + y^2}{2z}}}{j\lambda z} \mathrm{sinc}\left( \frac{A_xx}{z\lambda} \right) \mathrm{sinc}\left( \frac{A_yy}{z\lambda} \right). \label{eq:rect-pattern} \end{align}

Circular aperture

For a circular aperture of radius \(R\), its amplitude transmittance ratio can be expressed by

\begin{align} t_{\mathrm{circ}}(q) = \mathrm{circ} \left( \frac{q}{R} \right), \label{eq:circ-apert} \end{align}

where \(q = \sqrt{\xi^2 + \eta^2}\). Accordingly, its Fraunhofer pattern can be computed according to

\begin{align} U(r) &= \frac{e^{j\kappa z}e^{j \frac{\kappa r^2}{2z}}}{j\lambda z} \mathcal{B}\left\{ U(q) \right\} \mid_{\rho = \frac{r}{z\lambda}} \nonumber \\ &= \frac{R e^{j\kappa z}e^{j \frac{\kappa r^2}{2z}}}{jr}J_1 \left( \frac{\kappa Rr}{z} \right), \label{eq:circ-pattern} \end{align}

where \(\mathcal{B}\{\cdot\}\) is the operator of Fourier-Bessel transform, a.k.a. zero-order Hankel transform.

\begin{align} G(\rho) &= \mathcal{B}\{g(r)\} \nonumber \\ &=2\pi \int_0^{+\infty}r g(r)J_0(2\pi r \rho) dr. \label{er:fourier-bessel} \end{align}

Footnotes:

1

The distance is termed Rayleigh distance or Fraunhofer distance.