The first theoretical study of the scattering of light from by the authors [6,7], we address these questions for the case a randomly rough surface was published by Mandel’shtam of one-dimensional diffusers. We illustrate the ideas involved by considering the scattering of s-polarized light from a onein 1913, in the context of the scattering of light from a dimensional, randomly rough, perfectly conducting surface.

liquid surface [1]. In the succeeding years the overwhelming By working within the Kirchhoff approximation, and motimajority of the theoretical work in this field has continued vating the approach by taking the geometrical optics limit to be devoted to the solution of such direct problems, of this approximation, we describe methods for designing namely given the statistical properties of a random surface, and fabricating achromatic, random, uniform diffusers of to calculate the angular and polarization dependence of the light, and test these methods by computer simulations and intensity of the scattered light. In contrast, in this paper we experimentally.

study theoretically and experimentally an inverse problem in rough surface scattering, namely the design and fabrication of a random surface that scatters light in a prescribed way.

1. Light Scattering in the Geometrical For many practical applications it is desirable to have Optics Limit of the Kirchhoff optical elements whose light scattering properties can be Approximation controlled. In particular, a non-absorbing diffuser that scatters light uniformly within a specified range of scattering To motivate the calculations that follow we begin by angles, and produces no scattering outside this range, would considering the scattering of s-polarized light from a onehave applications, for example, to projection systems, where dimensional, randomly rough, perfectly conducting surface it is important to produce even illumination without wasting defined by x3 = (x1). The region x3 >(x1) is vacuum, light. We will call such an element a band-limited uniform the region x3 < (x1) is the perfect conductor. The diffuser.

plane of incidence is the x1x3-plane. The surface profile The design of uniform diffusers has been considered by function (x1) is assumed to be a differentiable, singleseveral authors. The case of binary diffusers has been valued function of x1, and to constitute a random process, studied by Kurtz [2], and work on special cases of one- but not necessarily a stationary one.

The surface is illuminated from the vacuum region. The dimensional diffusers has been reported by Kurtz et al. [3] single nonzero component of the total electric field in this and by Nakayama and Kato [4]. Some work on the region is the sum of an incident wave and of the scattered more general two-dimensional case has been carried out field by Kowalczyk [5]. In addition, diffractive optical elements that scatter light uniformly throughout specified angular E2(x1, x3|) = exp ikx1 - i0(k)xregions have recently become commercially available. These elements, however, are not truly random, and possess the dq desired characteristics over only a relatively narrow range of + R(q|k) exp iqx1+ i0(q)x3, (1.1) wavelengths.

Despite the interest in the problem, at the present time there are no clear procedures for designing and fabricating where 0(q) = (/c)2 - q2 1/2, Re 0(q) > 0, random, band-limited uniform diffusers, and it is unclear 0(q) > 0, and is the frequency of the incident Im what kind of statistics are required for the production of such light. A time dependence of the form of exp(-it) is an optical element. In this paper, that extends earlier work assumed, but explicit reference to it is suppressed.

The Design and Fabrication of One-dimensional Random Surfaces with Specified Scattering Properties In the Kirchhoff approximation, which we adopt here for We focus on the integral in Eq. (1.6). With the change of simplicity, the scattering amplitude R(q|k) is given by variable x 1 = x1 + u it becomes -i R(q|k) = I(q|k) = dx1 du exp i(q - k)u 20(q) - dx1F(x1|) exp -iqx1- i0(q) (x1), (1.2) exp -ia (x1) - (x1 + u). (1.7) where the source function F(x1|) is The geometrical optics limit of the Kirchhoff approximation is obtained by expanding the difference (x1) - (x1 + u) in Eq. (1.7) in power of u and retaining only the leading F(x1|) =2 - (x1) + nonzero term:

x1 x E2 x1, x3|. (1.3) inc I(q|k) dx1 du exp i(q - k)u exp iau (x1).

= x3=(x1) - Substitution of Eq. (1.3) into Eq. (1.2), followed by an (1.8) integration by parts, yields the result that Because we have not assumed (x1) to be a stationary random process, we cannot assume that (x1) is a stationary 2/c2 +0(q)0(k) -qk random process. The average exp[iau (x1)], therefore, R(q|k) = 0(q)[0(q) +0(k)] has to be assumed to be a function of x1, and we cannot out the integral over x1 to yield a factor of L1, as we could if (x1) were a stationary random process.

dx1 exp -i(q - k)x1 - ia (x1), (1.4) 2. Design of a Band-Limited Uniform where, to simplify the notation, we have defined Diffuser a = (q) +0(k).

The mean differential reflection coefficient Rs/s, To evaluate the average in Eq. (1.8) we begin by writing which is defined such that Rs/s ds gives the fraction the surface profile function (x1) in the form of the total, time-averaged, flux incident on the surface that is scattered into the angular interval (s, s +ds), is given in terms of R(q|k) by (x1) = cls(x1 - 2lb), (2.1) l=Rs 1 cos2 s = |R(q|k)|2, (1.5) where the {cl} are independent, positive, random deviates.

s L1 2c cos These properties of the {cl} are dictated by the fabrication where the angle brackets denote an average over the process, described in Section 4. The function s(x1) is ensemble of realizations of the surface profile function defined by (x1), 0 and s are the angles of incidence and scattering respectively, which are related to the wave numbers k and q s(x1) = 0, x1 < -(m + 1)b, by k = (/c) sin 0 and q = (/c) sin s, and L1 is the = -(m + 1)bh - hx1, -(m + 1)b < x1 < -mb, length of the x1-axis covered by the random surface.

With the use of Eq. (1.4) the average |R(q|k)|2 entering = -bh, -mb < x1 < mb, Eq. (1.5) can be written as = -(m + 1)bh + hx1, mb < x1 < (m + 1)b, 1 + cos(0 + s) = 0, (m + 1) b < x1, (2.2) |R(q|k)|2 = cos s(cos 0 + cos s) where m is a positive integer and b is a characteristic length.

The derivative of the surface profile function, (x1), is dx1 dx 1 exp -i(q - k)(x1 - x 1) then given by - (x1) = cld(x1 - 2lb), (2.3) exp -ia (x1) - (x 1). (1.6) l=Физика твердого тела, 1999, том 41, вып. 920 T.A. Leskova, A.A. Maradudin, E.R. Mndez, A.V. Shchegrov where d(x1)= 0, x1 < -(m + 1)b, = -h, -(m + 1)b < x1 < -mb, = 0, -mb < x1 < mb, = h, mb < x1 < (m + 1)b, = 0, (m + 1)b < x1. (2.4) The function s(x1) and d(x1) are shown in Fig. 1.

In what follows the surface will be sampled at the set of equally spaced points {xp} defined by xp = p + b/N p = 0, ±1, ±2,..., (2.5) where N is a large positive integer. None of these values of xp equals an integer multiple of b, at which d(x1) is discontinuous.

When the probability density function (pdf) of cl, f () = ( -cl), (2.6) is known, a long sequence of the {cl} can be generated, e. g.

by the rejection method [7], from which the surface profile function (x1) can be obtained by the use of Eqs. (2.1) Figure 1. The functions s(x1) and d(x1).

and (2.2). We note that since the {cl} are positive random deviates, f () will be nonzero only for positive values of.

When the results given by Eqs. (2.8) are substituted into The average exp iau (x1) can now be written as Eq. (1.8), the latter becomes (2n+1)b exp iau (x1) = exp iau cld(x1 - 2lb) l=I(q|k) = dx1 du exp i(q - k)u n 2nb = exp {iau cld(x1 - 2lb)} l= d f () exp(iahu) = exp iau cld(x1 - 2lb), (2.7) 2nb l=+ du exp i(q - k)u n where the independence of the {cl} has been used in the (2n-1)b last step. With the form of d(x1) given by Eq. (2.4), for any value of x1 chosen from the set of sampling points {xp} d f () exp(-iahu) given by Eq. (2.5) only one factor in the infinite product on the right hand side of Eq. (2.7) is different from unity.

Indeed, we find for m = 2 that when 2nb < x1 < (2n + 1)b L= du exp i(q - k)u (n = 0, ±1, ±2,... ) exp iau (x1) = exp{iauhcn-1} d f () exp(iahu) +exp(-iahu) = d f () exp(iauh), (2.8a) - = L1 d f () (q- k+ ah)+ (q- k- ah) while when (2n - 1)b < x1 < 2nb (n = 0, ±1, ±2,... ) exp iau (x1) = exp{-iauhcn+1} L1 k - q q - k = f + f. (2.9) ah ah ah We note that although Eqs. (2.8) were obtained for the case = d f () exp(-iauh). (2.8b) that m = 2, the result given by Eq. (2.9) is valid for any m.

Физика твердого тела, 1999, том 41, вып. The Design and Fabrication of One-dimensional Random Surfaces with Specified Scattering Properties When the results given by Eqs. (1.7), (1.8) and (2.9) are 3. Computer Simulations substituted into Eq. (2.6), we find that the mean differential The approach to the design of band-limited uniform reflection coefficient is given by diffusers presented in the preceding sections was tested by means of computer simulation calculations. One1 + cos(0 + s) Rs = dimensional random surfaces were generated numerically s 2h cos 0(cos 0 + cos s)on the basis of Eqs. (2.1) and (2.2) with the coefficients {cl} determined by the rejection method with the use of the sin 0 - sin s f pdf (2.15). As an example, we showin Fig. 2 a realization h(cos 0 + cos s) of a sample profile and its derivative, generated in this way.

For a given surface profile the scattering amplitude R(q|k) sin s - sin + f. (2.10) can be calculated in the Kirchhoff approximation, but withh(cos 0 + cos s) out passing to the geometrical optics limit, from Eq. (1.4).

The mean differential reflection coefficient can then be Thus, we find that in the geometrical optics limit of the calculated from Eq. (1.5) by generating a large number Np Kirchhoff approximation the mean differential reflection of surface profiles and averaging over the resulting scattering coefficient is determined by the pdf f () of the coefficient distributions. In Fig. 3 we show an example of a calculated cl entering the expansions (2.1) and (2.3). We also note that mean differential reflection coefficient obtained by averaging it is independent of the wavelength of the incident light.

results obtained for 3000 realizations of the surface profile The result given by Eq. (2.10) simplifies significantly in function. It is seen that the scattering distribution is close the case of normal incidence, 0 = 0:

to the desired result. There is almost no light outside the Rs s range -m

result is normalized to unity, /Rs ds = 1. (2.12) s -/From the result given by Eq. (2.11) we find that if we wish a constant value for R/s for -m

Физика твердого тела, 1999, том 41, вып. 922 T.A. Leskova, A.A. Maradudin, E.R. Mndez, A.V. Shchegrov the result shown in Fig. 4, the main difference being that in the numerical results the two sections of the scattering distribution are not completely separated due to the overlap of their tails, which give rise to a dip in Rs/s. Thus, a value of intermediate between 0 and 0.5 should yield an approximately flat scattering curve. That this is the case is shown in Fig. 5, where Rs/s is plotted for a surface the basis of the pdf (3.1) with = 0.01, and for the same values of 0, b, m, m, and m used in obtaining figs. 3 and 4. Results are presented for three wavelengths of the incident light: a — = 0.6328 µm Figure 3. The mean differential reflection coefficient for normal (He–Ne laser); b — = 0.532 µm (the second harmonic incidence calculated from Np = 3000 realizations of the surface of the YAG laser); c — = 0.442 µm (He–Cd laser). These profile function. The parameters employed are = 0.6328 µm, b = 60 µm, m = 1, m = 1, and m = 5. The sampling interval wavelengths cover the entire visible region of the optical on the surface was x = b/N = 0.2 µm (N = 300), and the spectrum. For each wavelengh the result for Rs/s is length of the surface was L1 = 2000 µm.

seen to consist of a nearly constant scattered intensity for s between -5 and +5, and a zero scattered intensity outside this interval. Moreover, these results confirm the expected independence of the scattering pattern from the wavelength of the incident light over a significant range of wavelengths.

Figure 4. The same as Fig. 3, but with random deviates {cl} drawn from the distribution given by Eq. (3.1) with = 0.05.

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