[4] Since random EM fields are spatially and temporally incoherent, the fundamental statistical quantity is the energy density, rather than the field itself or its magnitude (envelope), and is obtained from the average field intensities 〈∣*E*_{α}(**r**)∣^{2}〉 of the Cartesian complex field components *E*_{α}. These intensities represent self-coherencies, which are a special case of mutual coherencies [*Mehta and Wolf*, 1964]. The latter were derived by *Arnaut* [2006b] for the present configuration. Here, the fact that no separate transverse locations need to be considered in order to obtain first-order statistics considerably simplifies the calculations.

#### 2.1. Incident Plus Reflected Random Field

[5] We consider a semi-infinite isotropic medium with permittivity and permeability *μ* occupying the half-space *z* ≤ 0 (Figure 1). The incident random field at **r** in the region characterized by *z* > 0 is represented by a statistical ensemble (random angular spectrum) of time-harmonic plane waves [*Whittaker*, 1902; *Booker and Clemmow*, 1950; *Hill*, 1998]:

in which an exp(*jωt*) time dependence has been assumed and suppressed, and where ^{i}(Ω) is the delta-correlated random amplitude function.

[6] The incident field **E**^{i} is assumed to be ideal random, i.e., any three complex Cartesian components, in particular (*E*^{i}_{x}, *E*^{i}_{y}, *E*^{i}_{z}), are mutually independent and exhibit identical circular centered Gauss normal distributions. The direction of incidence for each plane wave component (^{i}, ^{i}, **k**^{i}) is arbitrary within the upper hemisphere (Ω_{0} = 2*π* sr) and is specified by azimuthal (_{0}) and elevational (θ_{0}) angles. Since the medium is deterministic, the incident and reflected fields for each individual plane wave are mutually coherent, despite being individually random. Hence their recombination in the region *z* > 0 is governed by superposition of fields, rather than by summing energy densities. The boundary condition does not affect the correlation between field components on either side of the boundary, because ^{i} is itself random and any triplet of Cartesian field components is uncorrelated. As a result, the field components remain mutually independent in the vicinity of the boundary. Since the Cartesian components of the incident field are centered circular Gaussians and because the medium is linear, the incident plus reflected vector field and its components are also circular Gauss normal with zero mean, but now with standard deviations that are different from those of the incident field owing to the boundary conditions. Thus the three Cartesian components no longer exhibit identical parameters for their distributions.

[7] Following *Dunn* [1990] and *Arnaut* [2006a], we perform a transverse electric/transverse magnetic (TE/TM) decomposition for each plane wave component of the angular spectrum with respect to its associated random plane of incidence *ok*^{i}*z*, i.e., ϕ = ϕ_{0}. As is well known, TE and TM polarizations constitute uncoupled eigenpolarizations for stratified isotropic media, hence the polarization of the outgoing wave is completely determined by that of the the incoming wave. As a result, the TE and TM contributions to the overall plane wave spectrum can be calculated independently. Ensemble averaging then yields the TE and TM energy contents of the random field.

[8] Specifically, for the TE components incident at an angle _{0} and with electric field

the incident plus reflected electric field at **r** is

with *ϱ**x*cos _{0} + *y*sin _{0}, and where Γ_{⊥}(θ_{0}) represents the Fresnel reflection coefficient for TE waves for a semi-infinite isotropic medium. In (2) and (3), ψ is an angle of random polarization that is uniformly distributed within the transverse plane. Upon substitution of (3) into an expression similar to (1) for **E**_{⊥}, followed by unfolding and integration, the associated average field intensity is

in which 〈·〉 denotes ensemble averaging of the plane wave spectrum, with

where *u* cos θ_{0}, *C* 〈∣_{0}∣^{2}〉/4, and

Similarly, for the TM components with incident electric field

we obtain

with

where

For the overall (i.e., incident plus reflected) tangential and vector fields,

and

respectively. For a medium for which ∣*μ*_{r}/ε_{r}∣ 0, only the terms *I*_{α1} remain nonzero. Conversely, for ∣ε_{r}/*μ*_{r}∣ 0, only the terms *I*_{α2} survive.

#### 2.2. Refracted Random Field

[9] For the field refracted across the boundary,

with the TE and TM transmission coefficients given by

and

respectively. Thus, unlike for the incident plus reflected field, the intensity of the refracted field is homogeneous because of the absence of interference in the region *z* > 0.