Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume

It is known that statistically stationary, homogeneous, and isotropic source distributions generate, in an unbounded low-loss medium, an electromagnetic field whose electric cross-spectral density tensor is proportional to the imaginary part of the infinite-space Green tensor. Using the recently established electromagnetic theory of coherent modes, we construct, in a finite spherical volume, the coherent-mode representation of the random electromagnetic field having this property. The analysis covers the fundamental case of blackbody radiation but is valid more generally; since a thermal equilibrium condition is not invoked, the electromagnetic field may have any spectral distribution. Within the scalar theory of coherent modes, which has been available for more than two decades, the analogous formulation results in the first explicit three-dimensional coherent-mode representation.

Figure 1

Behavior of the ratio of eigenvalues ${\lambda }_n^{\left(j\right)}\left(k\right)∕{\lambda }_1^{\left(1\right)}\left(k\right)$ as a function of mode number $n$ for spherical volumes of radius $d=1000\lambda$ (lower curve) and $d=1500\lambda$ (upper curve). Dots are for $j=1$, corresponding to both the scalar and the first set of electromagnetic eigenvalues, whereas stars are for $j=2$, corresponding to the second set of electromagnetic eigenvalues. In both curves, the dots and stars are plotted for modes with number $n=\{200,600,1000,\dots \}$ and $n=\{1,400,800,\dots \}$, respectively.

Figure 2

Illustration of functions ${\mid {\psi }_{mn}(\mathbf{r},k)\mid }^2$ for indices $mn=\{00,01,12\}$, when the radius of the sphere is $d∕\lambda =1000$ and $kr=10$. The value of the function is indicated by the distance from the origin, located in the center of each graph. The orientation of the coordinate axes is shown in the inset.

Figure 3

Illustration of functions ${\mid {\mathit{\psi }}_{mn}^{\left(1\right)}(\mathbf{r},k)\mid }^2$ (upper row) and ${\mid {\mathit{\psi }}_{mn}^{\left(2\right)}(\mathbf{r},k)\mid }^2$ (lower row) for indices $mn=\{13,14\}$, when $kr=1$ and $kr=10$. The radius of the sphere is chosen as $d∕\lambda =1000$. The value of the function is indicated by the distance from the origin, which is located in the middle of each graph. The orientation of the coordinate axes is shown in the inset.

Figure 4

Behavior of the quantities $\mu ({\mathbf{r}}_1,{\mathbf{r}}_2,k)$ [Eq. (81), solid line] and $\mid {\mu }_{\mathrm{vis}}({\mathbf{r}}_1,{\mathbf{r}}_2,k)\mid$ [Eq. (82), dashed line] as a function of $R∕\lambda$.