Maxwell's equations in transform domains

Maxwell's equations (1.1 to 1.4) are a system of eight first-order partial differential equations in four independent variables: three space coordinates and time, whose solution is often quite complicated. It may be advantageous to eliminate the dependence of the field quantities upon one or more of the independent variables by applying a Fourier (or Laplace) transform to (1.1 to 1.4), solving the resulting equations in the transform domain, and then obtaining the desired field quantities by an inverse transformation.

Obviously, the main advantage of a transform technique with respect to an independent variable is to change the dependence of the equations on that variable from a differential one to an algebraic one; thus, a four-fold Fourier transform can change the differential system (1.1-1.4) to an algebraic system in the transform domain.

The Fourier transform pair:

$$\underline{\tilde{E}}(\underline{r},\omega)= \int_{-\infty}^{\infty} \underline{\mathcal{ E}} (\underline{r}, t) e^{- j \omega t} \, dt \space ,$$ (1.23)

$$\underline{\mathcal{ E}}(\underline{r},t) =\frac{1}{2 \pi} \int_{-\infty}^{\infty} \underline{\tilde{E}}(\underline{r},\omega) e^{j \omega t} \, d\omega \space ,$$ (1.24)

allows us to transform the electric field from the time domain, where the appropriate field vector is $\underline{\mathcal{ E}}(\underline{r},t)$, to the frequency domain, where the appropriate field vector is $\underline{\tilde{E}} (\underline{r},\omega)$, and viceversa. Identical transformations can be applied to all field variables in (1.1-1.4); the appropriate symbols are listed in Table 1.1.

Table 1.1: Symbols

time-domain symbol	$\underline{\mathcal{E}}$	$\underline{\mathcal{H}}$	$\underline{\mathcal{D}}$	$\underline{\mathcal{B}}$	$\rho_e$	$\rho_m$	$\underline{\mathcal{J}}_e$	$\underline{\mathcal{J}}_m$
frequency-domain symbol	$\underline{\tilde E}$	$\underline{\tilde H}$	$\underline{\tilde D}$	$\underline{\tilde B}$	$\rho_e$	$\rho_m$	$\underline{\tilde J}_e$	$\underline{\tilde J}_m$

If equation (1.24) and similar formulas are used in (1.1-1.4) we obtain, on equating the integrands:

$$\nabla \times \underline{\tilde{ H}} =\underline{ \tilde J}_e + j \omega \underline{\tilde{D}} \space ,$$ (1.25)

$$\nabla \times \underline{\tilde{E}} =-\underline{ \tilde{J}}_m - j \omega \underline{ \tilde{B}} \space ,$$ (1.26)

$$\nabla \cdot \underline{ \tilde{D}} = \rho_e \space ,$$ (1.27)

$$\nabla \cdot \underline{ \tilde{B}} = \rho_m \space ,$$ (1.28)

similarly, the continuity equations (1.5) and (1.6) become:

$$\nabla \cdot \underline{\tilde{J}}_e + j \omega \tilde{\rho}_e=0 \space ,$$ (1.29)

$$\nabla \cdot \underline{\tilde{J}}_m + j \omega \tilde{\rho}_m=0 \space .$$ (1.30)

Observe that (1.25-1.30) are formally obtained from (1.1-1.6) by replacing the differential operator $\partial/\partial t$ with the multiplicative factor $j \omega$, where $j=\sqrt{-1}$ and $\omega$ is measured in rad/s.

Consider, as an example, the sinusoidal (or time-harmonic) electric field

$$\underline{\mathcal{ E}}(\underline{r},t)=\underline{E}_0 (\underline{r}) \cos (\omega_0 t + \phi ( \underline{r}) )$$ (1.31)

with angular frequency $\omega_0$ and initial phase $\phi(\underline{r})$. The corresponding field in the frequency domain is obtained by substituting (1.31) into (1.23) and using the integral representation

$$\delta(\omega)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-j \omega t} \, dt$$ (1.32)

for the one-dimensional delta-function; we find that

$$\underline{\tilde{E}} (\underline{r},\omega)= \pi \underline{E}_0... ...ega_0) + e^{-j \varphi(\underline{r})} \delta(\omega+\omega_0) \right] \space .$$ (1.33)

In order to express the spatial dependence of the field quantities in (1.1-1.4) in algebraic form, we introduce the three-dimensional Fourier transform pair:

$$\underline{\dot{E}}(\underline{k},t) = \int_{-\infty}^{\infty}\underline{\mathcal{ E}}(\underline{r},t) e^{j \underline{k} \cdot \underline{r}} \, d\underline{r} \space ,$$ (1.34)

$$\underline{\mathcal{ E}}(\underline{r},t) = \frac{1}{( 2 \pi)^3} \int_{-\infty}^{\infty} \underline{\dot{E}... ...ine{k}, t) e^{- j \underline{k} \cdot \underline{r}} \, d\underline{k} \space ,$$ (1.35)

where $\underline{k}=k_x \hat{x} + k_y \hat{y} + k_z \hat{z}$ is in $\mathrm{rad}/\mathrm{m}$, $d \underline{r}=\, dx \, dy \, dz$, $d\underline{k}=dk_x \, dk_y \, dk_z$, and $\underline{k} \cdot \underline{r}=k_x x + k_y y +k_z z$. With similar transformations for the other field quantities, Maxwell's equations in the wavenumber domain (or k-space) become:

$$-j \underline{k} \times \underline{\dot{H}} =\underline{\dot{J}}_e + \frac{\partial \underline{\dot{D}} }{\partial t} \space ,$$ (1.36)

$$j \underline{k} \times \underline{\dot{E}} = \underline{\dot{J}}_m + \frac{\partial \underline{\dot{B}}}{\partial t} \space ,$$ (1.37)

$$- j \underline{k} \cdot \underline{\dot{D}} = \dot{\rho}_e \space ,$$ (1.38)

$$- j \underline{k} \cdot \underline{\dot{B}} =\dot{\rho}_m \space ,$$ (1.39)

where all field quantities are functions of $\underline{k}$ and t. The continuity equations (1.5-1.6) become:

$$\frac{\partial \dot{\rho}_e}{\partial t} - j \underline{k} \cdot \underline{\dot{J}}_e =0 \space ,$$ (1.40)

$$\frac{\partial \dot{\rho}_m}{\partial t} - j \underline{k} \cdot \underline{\dot{J}}_m =0 \space ,$$ (1.41)

Finally, we may elect to work in $k-\omega$ space by subjecting all field quantities to a four-fold Fourier transform. For example, for the electric field we have the transform pair

$$\underline{e}(\underline{k},\omega) = \int_{-\infty}^{\infty} \underline{\mathcal{ E}}(\underline{r},... ...underline{k} \cdot \underline{r} - j \omega t } \,d\underline{r} \, dt \space ,$$ (1.42)

$$\underline{\mathcal{ E}}(\underline{k},t) =\frac{1}{(2 \pi)^4} \int_{-\infty}^{\infty} \underline{e}(\under... ...line{k} \cdot \underline{r} + j \omega t } \,d\underline{k} \, d\omega \space ,$$ (1.43)

According to (1.43), each field quantity may be thought of as the four-fold sum of infinitesimal plane waves propagating in different directions with different amplitudes, wavenumbers, and frequencies. Substitution of (1.43) and similar equations into (1.1-1.4) yields:

$$- j \underline{k} \times \underline{h} = \underline{j}_e + j \omega \underline{d} \space ,$$ (1.44)

$$j \underline{k} \times \underline{e} = \underline{j}_m + j \omega \underline{b} \space ,$$ (1.45)

$$- j \underline{k} \cdot \underline{d} = \rho_e \space ,$$ (1.46)

$$- j \underline{k} \cdot \underline{b} = \rho_m \space ,$$ (1.47)

where all field quantities are functions of $\underline{k}$ and $\omega$. The continuity equations (1.5-1.6) become:

$$\omega \rho_e = \underline{k} \cdot \underline{j}_e \space ,$$ (1.48)

$$\omega \rho_m = \underline{k} \cdot \underline{j}_m \space .$$ (1.49)

Note that eqs. (1.44-1.49) are purely algebraic; the simplification thus achieved is, however, not as significant as might appear at first, because the initial and boundary conditions still must be satisfied.

Whenever a Fourier transform is effected to eliminate the variable t, the charge densities are obtained at once from the current densities, as seen from (1.29-1.30) and (1.48-1.49); it is then necessary to specify only the current densities as sources of the electromagnetic field.