## Theoretical model of the shift of the Brewster angle on a rough surface

#### Abstract

A physical mechanism is proposed for the explanation of the angular shift of the Brewster angle induced by a rough surface as recently discovered by Saillard and Maystre [J. Opt. Soc. Am. A 7, 982 (1990)]. An explicit formula giving the angular shift as a function of the correlation length o and the mean-square departure from the flat surface S is derived. It is shown that in the case of a rough surface, a minimum rather than a true zero is obtained for the reflection factor. The Brewster angle OB is the value of the angle of incidence for which the Fresnel reflection factor for p-polarized (TM) light is null. If a plane wave illuminates a flat interface separating two lossless dielectrics, the light is entirely transmitted at the Brewster angle. In the case of an illuminating bounded beam, several angular components have to be used in order to describe it. If the beam is centered around the Brewster angle, the central angular component will be completely transmitted, whereas the other angular components will be partially reflected. The shape of the reflected beam in the far field shows a zero at the Brewster angle. If we put it in mathematical terms, the reflected magnetic field of a bounded beam may be written as H(x, z) = f A(k)exp(ikx-iaz)dk, a =-k2), Im(a) > 0, where k is the component of the wave vector parallel to the interface. A(k) is given by r(k)Aj(k), where r(k) is the Fresnel reflection factor and Ai(k) denotes the amplitude of the incident field. In the far field, the amplitude in the direction 0 is proportional to r(ko sin O)Ai(ko sin 0), where ko = co/c. Hence a reflected beam presents a dip in the direction OB-In a recent paper, Saillard and Maystre 1 reported the numerical observation of an angular shift of the Brewster angle on a rough dielectric surface. This effect has been observed by means of Monte Carlo simulations of the intensity of scattered p-polarized Gaussian beams on a one-dimensional dielectric rough surface. For surfaces with a roughness of A/10, a well-developed speckle is observed, as well as a specular peak that mainly reproduces the Gauss-ian shape of the incident beam. For an angle of incidence close to the Brewster angle OB, Saillard and Maystre have observed a clear minimum in the shape of the specular part of the reflected beam but at an angle slightly lower than the Brewster angle. Thus the effect of the roughness is not to suppress the null reflectivity but to shift the location of the minimum. The absolute value of this angular shift seems to increase with the roughness. However, when the roughness becomes larger than A/5, the features of a flat interface, such as the specular peak and the presence of a minimum of the reflec-tivity, tend to disappear. Several questions are raised by these observations. The first question is what physical mechanism is responsible for this shift. The second question is whether the minimum is a true zero or a simple minimum. It is difficult to obtain the answer to this question from purely numerical results. An interesting point regarding possible metrologic applications is to study the behavior of the angular shift versus the statistical parameters of the surface: the roughness 8 and the length of correlation oc. Let us first address the question of the physical mechanism responsible for the effect. The Brew-ster angle is a property of the Fresnel reflection factor. Thus we study the effect of the roughness on the specular component of the reflected field. Let us first define the model used for the rough surface z = S(x). Throughout this Letter it is assumed that the one-dimensional rough surface is a stationary random process characterized by the following assumptions: (S(x)) = 0, (1) (S(x)S(x')) = 82 exp(-Ix-x112/2). (2) The fact that the correlation function 82 exp (-Ix-x'l 2 /o-2) depends on x and x' only through their difference has the consequence that transla-tional invariance is restored after an averaging process. Thus the average (coherent) reflected field has a specular component only. For this reason one may define an effective reflectivity. A commonly used expression for this reflectivity 2 is given by R(k) =r(k)exp{-2[(co/c)6 cos 0]2}, where r(k) is the 0146-9592/92/040238-03$5.00/0