Using a transverse electric/transverse magnetic decomposition for an angular plane wave spectrum of random electromagnetic waves and matched boundary conditions, we derive the probability density function for the energy density of the vector electric field in the presence of a semi-infinite isotropic medium. The theoretical analysis is illustrated with calculations and results for good electric conductors and for a lossless dielectric half-space. The influence of the permittivity and conductivity on the intensity, polarization state, statistical distribution, and standard deviation of the field is investigated, both for incident plus reflected fields and for refracted fields. External refraction is found to result in compression of the fluctuations of the random field. Several applications of the theory are discussed.
 Complex electromagnetic (EM) environments are characterized by the fact that EM fields behave as random or quasi-random quantities. They can be studied in an efficient manner with the aid of statistical electromagnetics methods. With regard to the interaction of random fields with their environment, a fundamental problem of interest is the evolution of statistical properties of the field upon propagation through stratified media, including reflection and refraction at their interfaces. For incidence onto a perfectly electrically conducting (PEC) surface, previous studies of the average value [Dunn, 1990], standard deviation and first-order probability density function (pdf) [Arnaut and West, 2006] of the electric and magnetic energy densities have demonstrated a direction-dependent damped oscillatory behavior of their average value and standard deviation as a function of the distance of the point of evaluation to the interface. This behavior is a consequence of the interference between incident and reflected fields. As a result, unlike for deterministic waves, a boundary zone exists for random fields adjacent to the PEC surface, in which the statistical field properties are inhomogeneous and fundamentally different from those at larger (theoretically infinite) distance. Further insights that were gained from these studies pertain to the statistical anisotropy and polarization state of the field within the boundary zone and, for vector fields, the transitions of the pdf of the energy density from one- or two-dimensionally confined random fields at the interface to fully developed three-dimensional random fields at distances that are large relative to the wavelength. In addition, spatial correlation functions have been obtained previously for unbounded [Bourret, 1960; Sarfatt, 1963; Mehta and Wolf, 1964; Eckhardt et al., 1999; Hill and Ladbury, 2002] and single-interface [Arnaut, 2006a, 2006b] vector EM fields that elucidate the spatial structure of random fields via their two-point coherence properties.
 In the present paper, the methods and results for statistical properties of random fields near a PEC surface are extended to a magneto-dielectric isotropic semi-infinite medium. Having previously analyzed the second-order spatial coherence and correlation properties for an impedance boundary [Arnaut, 2006b], here we are again concerned with local first-order statistical, i.e., distributional properties only. On the basis of previous results for nonlocal spatial coherencies of the electric field 〈Eα(r1) E*β(r2)〉 (α,β = x, y, z), the polarization coefficient and pdf for the local energy density are determined. Because of the single interface and isotropy of the medium, the polarization coefficient is degenerate, whence the pdfs are one-parameter compound exponential (CE-1) distributions [Arnaut, 2002; Arnaut and West, 2006]. However, unlike for a PEC medium, the angular spectra of reflected and refracted random fields exhibit directivity because reflection and transmission coefficients of plane waves for a magneto-dielectric semi-infinite medium depend on the wave polarization and angle of incidence. We shall confine the analysis and results to the electric field E; corresponding results for the magnetic field follow without difficulty. Since we express results in terms of Fresnel reflection and transmission coefficients for an isotropic half-space, analogous results for multilayer strata are easily obtained from the listed integral expressions, on substituting with the appropriate coefficients. In particular, probability distributions of reflected and transmitted fields on either side of a single layer of finite thickness are easily computed.
2. Field Coherencies
 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 〈∣Eα(r)∣2〉 of the Cartesian complex field components Eα. 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
 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.
 The incident field Ei is assumed to be ideal random, i.e., any three complex Cartesian components, in particular (Eix, Eiy, Eiz), are mutually independent and exhibit identical circular centered Gauss normal distributions. The direction of incidence for each plane wave component (i, i, ki) is arbitrary within the upper hemisphere (Ω0 = 2π sr) and is specified by azimuthal (0) and elevational (θ0) 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.
 Following Dunn  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 okiz, i.e., ϕ = ϕ0. 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.
 Specifically, for the TE components incident at an angle 0 and with electric field
the incident plus reflected electric field at r is
with ϱxcos 0 + ysin 0, and where Γ⊥(θ0) 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 θ0, C 〈∣0∣2〉/4, and
Similarly, for the TM components with incident electric field
For the overall (i.e., incident plus reflected) tangential and vector fields,
respectively. For a medium for which ∣μr/εr∣ → 0, only the terms Iα1 remain nonzero. Conversely, for ∣εr/μr∣ → 0, only the terms Iα2 survive.
2.2. Refracted Random Field
 For the field refracted across the boundary,
with the TE and TM transmission coefficients given by
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.
3. Energy Density Distribution
 For a single-interface configuration, the pdf of the electric energy density Se = ε(0) 〈∣E∣2〉/2 can be calculated on the basis of knowledge of the polarization coefficient Pe,i3 for the statistically uniaxial electric field [Arnaut and West, 2006]. For the tangential field, the incoherent superposition 〈∣Et∣2〉 = 〈∣Ex∣2〉 + 〈∣Ey∣2〉 holds, because x and y belong to mutually orthogonal modes. Hence
 For nonmagnetic good electric conductors, i.e., σ ≫ ωε0, ε = ε0, μ = μ0, it follows that η/η0 ≃ (1 ± j), Γ⊥(u) ≃ (ηu − η0)/(ηu + η0) and Γ∥(u) ≃ (η − η0u)/(η + η0u) because θ ≃ 0 irrespective of θ0. As a result, (5)–(7) become
Throughout the range of integration, ∣1 + (η/η0)u∣2 = [1 + u]2 + u2[ωε0/(2σ)] ≃ 1. Therefore, to good approximation, we have
 Before presenting results for the pdfs, we show in Figure 2 the dependencies of 〈〉 and 〈〉 on k0z at selected values of σ/(ωε0), after normalization by the average electric energy density of the incident total (vector) field, 〈〉. The sensitivity to variations of σ/(ωε0) is seen to be significantly higher for the normal component than for the tangential component. For finite σ/(ωε0), the value of 〈(k0z → 0)〉/〈〉 is close to, but different from zero on account of the EM boundary condition. Both asymptotic values 〈(k0z → + ∞)〉/〈〉 and 〈(k0z → + ∞)〉/〈〉 are smaller for a finitely conducting boundary than for a PEC boundary, on account of energy dissipation in the former case. For intermediate values of k0z, finite conductivity also gives rise to a relatively small positive or negative phase shift in the oscillatory behavior of 〈(k0z)〉/〈〉 or 〈(k0z)〉/〈〉.
 The effect of finite conductivity on the statistical polarization state of the field is shown in Figure 3. Unlike for a PEC surface, Pi3(k0z) ≡ Pe,i3(k0z) (i = 1, 2) is no longer oscillating symmetrically with respect to zero, i.e., the random polarization exhibits a conductivity-dependent bias in the normal direction. This bias decreases, on average, with increasing k0z, but persists nevertheless up to asymptotically large distances k0z → +∞.
Figure 4 shows scaled pdfs for selected values of σ/(ωε0) at k0z = π/4. At this intermediate distance, the pdfs rapidly approach the asymptotic distribution of Se near a PEC wall [Arnaut and West, 2006] when σ/(ωε0) is increased. For larger k0z (not shown), the influence of σ/(ωε0) on the pdf was found to be even weaker. On the other hand, very close to the boundary (k0z ≪ 1), the pdf rapidly approaches a distribution close to the χ22 distribution when σ/(ωε0) is increased, as demonstrated in Figure 5 for k0z = 0.01.
 As indicated by Figure 6, the standard deviation (s.d.) for the incident plus reflected total field exhibits increasing oscillations for intermediate distances and lower values upon approaching the surface when conductivity is decreased. The s.d. approaches a location-dependent asymptotic value in the manner shown in Figure 7.
4.2. Lossless Dielectric Medium
 For a lossless medium, η, k, T⊥,∥ and Γ⊥,∥ are real valued. Therefore Iα3 and Iα4 are purely imaginary so that only Iα1 and Iα2 contribute to the intensities 〈∣Eα∣2〉. The following explicit expressions are obtained for the energy density of the fields refracted by a lossless dielectric medium with relative permittivity εr ≡ ε/ε0: for TE wave components,
whereas for the TM components, the integrals
can be substituted into (21) and (22) to yield 〈∣Ex∣2〉 and 〈∣Ez∣2〉 for the refracted field.
Figure 8 shows the intensities of the tangential and normal components as a function of permittivity, for the incident plus reflected field at k0z = 20π, and for the refracted field at any k0z. For the former case (Figure 8a), the minimum in the curves is a result of reduced TM contributions near the Brewster angle for external refraction (ε/ε0 > 1). This effect occurs in both the tangential and normal field components, via 〈∣Ex∣2〉 and 〈∣Ez∣2〉. However, for 〈∣Et∣2〉, it causes only a minor dip in the characteristic because of the dominance of the contribution by 〈∣Ey∣2〉 over that by 〈∣Ex∣2〉. Note the large sensitivity to the permittivity in the vicinity of ε/ε0 = 1. This sensitivity decreases with increasing k0z. For the normal field, the minimum intensity occurs around ε/ε0 ≃ 5.4. The incident energy density for this component is being exceeded only above relatively high permittivities (ε/ε0 ∼ 70). For the refracted field (Figure 8b), the intensities rapidly decrease with increasing ε/ε0 and exhibit an increasingly dominant tangential contribution.
Figure 9 shows the polarization coefficient as a function of ε/ε0. For the incident plus reflected field, an εr-dependent threshold distance exists where Pi3 starts to increase, in an oscillatory manner, toward an (εr-dependent) asymptotic value when k0z → +∞.
Figure 10 shows for the refracted field at selected values of ε/ε0. This pdf is independent of k0z but evolves from a χ26 distribution for ε/ε0 = 1 to a χ24 distribution for ε → +∞.
 From Figure 11, (/ε0) is seen to increase monotonically from its χ26 value, 1/, at ε/ε = 1 to its asymptotic χ24 value, 1/, represented by the dotted line.
Figure 12 shows for the incident plus reflected field close to the surface (k0z = π/128), at selected values of ε/ε0. The pdf is seen to make an excursion as ε/ε0 increases, returning eventually to the asymptotic χ26 pdf when ε → +∞. For larger values of k0z (not shown), it has been found that the excursions are much shorter; for example, for k0z ∼ 2π, there is essentially no longer a discernable parametric dependence of on εr.
5. Effect of Conductivity and Permittivity on the Magnitude of Field Fluctuations
 Since the probability distribution of each Cartesian component of the underlying complex field Eα = E′α − jE″α remains circular Gauss normal, on either side of the interface with an isotropic medium, the standard deviation of Eα can be deduced from the χ22 statistics for ∣Eα∣2 or via
where α = x, y or z, with and /. The dependence of on σ/(ωε0), ε/ε0 and k0z therefore follows immediately from the results shown in Figures 2 and 8. Physically, after rescaling, Figure 8b indicates that an increased permittivity causes the fluctuations of the refracted fields to exhibit a smaller spread compared to the fluctuations of the incident field, where the latter are quantified by with ε/ε0 = 1. By contrast, Figure 8a indicates that the spread of the incident plus reflected field increases, on average, with increasing permittivity.
 Besides its interest as the solution to a fundamental problem in statistical electromagnetics, the above analysis is relevant to several practical applications, of which we now give a few examples.
6.1. Atmospheric and Ionospheric Propagation of EM Waves With Applications, Including Spectroscopy of Stellar Light
 Stellar light propagating through an atmosphere can be represented as a random field radiated by a collection of incoherent point sources that are distributed across one or several narrow solid angle(s) (cf. Arnaut [2006b] for a treatment of random fields produced by a spatially filtered EM beam). Upon refraction by atmospheric layers exhibiting permittivities close to ε0, the light undergoes a change in its distributional and statistical polarization properties; cf. Figures 8b and 9b. Furthermore, recall that the above-mentioned plane wave expansion refers to a harmonic or quasi-harmonic (narrowband) field for a specific central wavelength. Therefore the pdfs of light intensities associated with different spectral lines may undergo different changes upon propagation, as a result of frequency dispersion of the medium, thus producing nonuniform changes of the mean values and standard deviations across the received spectrogram. Similar considerations apply to scattered or reflected fields. In this way, the presented analysis yields corrections that could be significant in astrophysical observations and Earth sciences, including remote sensing.
6.2. Reflection and Refraction of Multipath Signals for Wireless Communications
 An important issue in the accurate prediction of multipath and wideband propagation through a free-space radio channel is the determination of transfer functions for incident and outgoing signals, and how this transfer might affect the properties (statistical and other) of the received signal. By considering the “ratio” of the pdfs in Figures 4, 5, 10, or 12 relative to the χ26 asymptotic distribution of the deep field energy density, one obtains a transfer function for that captures the change in statistical properties, similar to the way in which Fresnel reflection or refraction coefficients relate incident to reflected or refracted fields. It also yields information on the likelihood and extent of signal distortion.
6.3. Measurement of Constitutive EM Properties of Materials Inside a Mode-Stirred Reverberation Chamber, Including Anisotrophy, Absorptivity, Conductivity and/or Permittivity
 Several situations in which reverberant fields near a PEC surface are of practical relevance were already mentioned by Arnaut and West ; here, we focus on media with finite σ/(ωε0) and ε only. Through the measurement of probability distributions of reflected or refracted fields at a fixed and sufficiently close distance from the boundary of a semi-infinite medium or, by extension, from a flat panel of sufficient large electrical thickness, constitutive parameter values can be deduced using Figures 4, 5, 10, and 12 as a statistical inversion problem [Arnaut, 2006b]. Alternatively, the spatial functional k0z dependence of the average electric energy density (particularly its normal component) can be compared with the results in Figures 2 and 6 to deduce these parameters, circumventing the need for determining the pdf.
 In this paper, we have been concerned with the influence of the presence of a deterministic semi-infinite isotropic medium on the local first-order statistical, i.e., distributional properties of an ideal random electromagnetic field for nondirectional (hemispherical) incidence. The EM boundary conditions cause statistical anisotropy of the vector field and a redistribution of the electric energy density between its tangential and normal components, compared to the statistical isotropy and homogeneity of the incident field. This modifies its probability density function. For the reflected plus incident field, this modification changes to a damped oscillatory manner as a function of distance from the interface; for refracted fields, the change is homogeneous within the dielectric because no standing waves occur in this region. It was found that, for the tangential and normal field components, the presence of a dielectric alters their variability (as expressed by their standard deviation) in an opposing manner, viz., for ε/ε0>1 the amplitude fluctuations of the refracted field become more compressed, whereas for the incident plus reflected field the fluctuations expand (for sufficiently large ε/ε0).
 The analytical formulation in terms of numerically solved integrals was made possible by the simplicity of the geometry (single interface) and medium (isotropy). More complex configurations involving multiple, finite, angled or curved interfaces, (e.g., for interior fields inside overmoded cavities or multiple-scattering problems), are unlikely to afford such an semi-analytical approach. In these cases, one may numerically calculate the statistics by using a Monte Carlo simulation for angular spectra of random plane waves (or other suitable random excitation) with specified input statistics. The numerically solved reflected and refracted fields can then be collated and evaluated at a single location to yield the output statistics for these fields [cf. Arnaut and West, 2006, section V].
 This work was sponsored in part by the 2003–2006 Electrical Programme of the UK Department of Trade and Industry National Measurement System Policy Unit (Project E03E54).