Spectral invariance and the scaling law with random electromagnetic fields

We consider the far-zone spectrum generated by finite, statistically homogeneous, planar electromagnetic sources having three electric field components. We introduce an electromagnetic scaling law which ensures spectral invariance, i.e., that the normalized far-zone spectrum is the same in any paraxial or nonparaxial direction and equal to that of the source. The scaling law for electromagnetic fields presented in this work is more general than that put forward in a previous publication. In particular, we show paraxially that even though the individual field components do not obey the scalar scaling law, spectral invariance may be achieved, and conversely, that spectral invariance may not hold even if the field components separately follow the scalar scaling law.

Figure 1

Geometry and notations. The homogeneous source of area $D$ in plane $z=0$ generates a field in half space $z>0$. Of interest is the frequency dependence of the optical intensity at an arbitrary far-zone point $\mathbf{r}$, i.e., the radiant intensity in direction $\widehat{\mathbf{u}}$.

Figure 2

Modified Mach-Zehnder interferometer. The axial displacement (by distance $a$) of the last mirror creates a transverse shift $a$ between the two replicas of the beam entering at plane $\mathcal{A}$. The shift produces an additional spatial correlation into the recombined beam at plane $\mathcal{B}$. Elements H (half-wave plate) and C (compensator) are described in the text.

Figure 3

Normalized spectra of the $x$ (red) and $y$ (green) source components, as well as the normalized far-zone spectrum of the total field for $u_x=0$ (solid blue), $u_x=0.014$ (dashed blue), and $u_x=0.02$ (dash-dotted blue). The other parameters are $u_y=0$, ${\omega }_{0x}=4\times {10}^{15}$ rad/s, ${\omega }_{0y}=4.1\times {10}^{15}$ rad/s, ${\Omega }_x={\Omega }_y=4\times {10}^{13}$ rad/s, ${\sigma }_x=100$, and ${\sigma }_y=200$.