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OPTICS LETTERS / Vol. 28, No. 2 / January 15, 2003
Two-dimensional random surfaces that act as circular diffusers A. A. Maradudin and T. A. Leskova Department of Physics and Astronomy and Institute for Surface and Interface Science, University of California, Irvine, Irvine, California 92697
E. R. Méndez División de Física Aplicada, Centro de Investigación Científica y de Educación Superior de Ensenada, Apartado Postal 2732, Ensenada, B.C. 22800, Mexico Received August 6, 2002 We propose a method for designing a two-dimensional random Dirichlet surface that, when it is illuminated at normal incidence by a scalar plane wave, scatters the wave with a circularly symmetric distribution of intensity. The method is applied to the design of a surface that acts as a Lambertian diffuser. The method is tested by computer simulations, and a procedure for fabricating such surfaces on photoresist is described. © 2003 Optical Society of America OCIS code: 240.5770.
Optical diffusers that produce a scattered intensity proportional to the cosine of the polar scattering angle are frequently used in the optical industry, e.g., for calibrating scatterometers.1 Such diffusers have the property that their radiance or luminance is the same in all scattering directions. For this reason they are often referred to as Lambertian diffusers. In the visible region of the optical spectrum volumedisordered media, such as compacted powdered barium sulfate and freshly smoked magnesium oxide,2 are used as Lambertian diffusers. However, this type of diffuser is inapplicable in the medium- or far-infrared regions because of its strong absorption and the presence of a specular peak in the scattered light. A different realization of such a diffuser is therefore required. The design of a two-dimensional random surface that acts as a Lambertian diffuser, especially in the medium- or far-infrared region of the optical spectrum, is therefore a desirable goal. In this Letter we propose a method for designing a two-dimensional randomly rough Dirichlet surface that, when it is illuminated at normal incidence by a scalar plane wave, scatters the wave with a prescribed circularly symmetric distribution of intensity. We illustrate this method by applying it to the design of a surface that acts as an achromatic Lambertian diffuser. The physical system that we consider consists of vacuum in the region x3 . z 共xjj 兲, where xjj 苷 共x1 , x2 , 0兲 is a position vector in the plane x3 苷 0, and the scattering medium in the region x3 , z 共xjj 兲. The surface prof ile function z 共xjj 兲 is assumed to be a single-valued function of xjj that is differentiable with respect to x1 and x2 and constitutes a random process. The surface x3 苷 z 共xjj 兲 is illuminated at normal incidence by a scalar plane wave of frequency v, and the Dirichlet boundary condition is satisf ied on it. The mean differential ref lection coefficient 具≠R兾≠Vs 典, where the angle brackets denote an average over the ensemble of realizations of the surface prof ile function, is def ined such that 具≠R兾≠Vs 典dVs is the 0146-9592/03/020072-03$15.00/0
fraction of the total time-averaged incident f lux that is scattered into the element of solid angle dVs about a given scattering direction. In the geometrical optics limit of the Kirchhoff approximation the coeff icient is given by µ ∂ ¿ ø v 2Z 2 1 ≠R 苷 d ujj exp共2iqjj ? ujj 兲 ≠Vs S 2pc Z (1) 3 d2 xjj 具exp关iaujj ? =z 共xjj 兲兴典 , where S is the area of the x1 x2 plane covered by the random surface, qjj 苷 共v兾c兲sin us 共cos fs , sin fs , 0兲, 共us , fs 兲 are the polar and azimuthal angles of scattering, and a 苷 共v兾c兲 共1 1 cos us 兲. The idea underlying our approach to the design of a two-dimensional random surface that scatters in a circularly symmetric fashion a scalar plane wave that is normally incident on it is that the surface itself should have circular symmetry. Consequently, we assume that the surface prof ile function z 共xjj 兲 is a function of the radial coordinate r 苷 jxjj j alone, z 共xjj 兲 苷 H 共r兲, and we choose H 共r兲 to have the form H 共r兲 苷 an r 1 bn ,
nb # r # 共n 1 1兲b , n 苷 0, 1, 2, . . . ,
(2)
where 兵an 其 are independent identically distributed random deviates, b is a characteristic length, and 兵bn 其 are determined in such a way as to make H 共r兲 a continuous function of r: bn 苷 b0 1 共a0 1 a1 1 · · · 1 an21 2 nan 兲b , n $ 1.
(3)
We seek the probability density function (pdf ) of an , f 共g兲 苷 具d共g 2 an 兲典, such that the surface prof ile def ined by Eqs. (2) and (3) yields a mean differential ref lection coefficient of a prescribed form. The average over the ensemble of realizations of the surface © 2003 Optical Society of America
January 15, 2003 / Vol. 28, No. 2 / OPTICS LETTERS
prof ile function in the def inition of f 共g兲 is equivalent to an average over the ensemble of realizations of 兵an 其 and the independence of 兵an 其, and the fact that they are identically distributed has the consequence that this pdf is independent of n. For the form of the surface prof ile function that we have chosen, Eq. (1) becomes µ ∂ ¿ ø µ ∂ qjj . 2p v 2 1 ≠R 苷 f (4) ≠Vs a 2pc qjj a If we make the change of variable g 苷 qjj 兾a 苷 tan共us 兾2兲, we find that f 共g兲 is given by ¿ ø g ≠R f 共g兲 苷 8p 共g兲 , g $ 0 . (5) 共1 1 g 2 兲2 ≠Vs We seek to design a surface that acts as a Lambertian diffuser, for which ¿ ø ≠R 1 cos us , 0 # us # p兾2 . (6) 苷 ≠Vs p
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In Fig. 1 we plot a segment of one realization of the surface prof ile function H 共r兲 and its derivative calculated by the approach proposed here. The parameters used in generating these functions were b 苷 200l and b0 苷 0. Because of the form of f 共g兲 given by Eq. (7), each slope an is positive, and as a result H 共r兲 increases monotonically with increasing r. The average slope of the surface is 具an 典 苷 0.4292, independently of n, with a standard deviation of 0.2078. A plot of 具≠R兾≠Vs 典 as a function of us , calculated for the parameters used in obtaining the surface prof ile function plotted in Fig. 1, is presented in Fig. 2. The results obtained for Np 苷 15, 000 realizations of the surface prof ile function were used in obtaining the average in Eq. (8). The random surface generated by our approach is seen to act as a Lambertian diffuser: The mean differential ref lection coeff icient follows the cosine law Eq. (6) very closely.
We f ind from Eq. (5) that f 共g兲 苷 8
g共1 2 g 2 兲 u共1 2 g兲u共g兲 , 共1 1 g 2 兲3
(7)
where u共z兲 is the Heaviside unit step function. From the result given by Eq. (7), a long sequence of 兵an 其 is obtained, e.g., by the rejection method,3 and the surface profile function H 共r兲 is constructed on the basis of Eqs. (2) and (3). Because rigorous calculations of scattering from two-dimensional random surfaces are time consuming, we illustrate this approach to the design of random surfaces that scatter light in a circularly symmetric fashion by performing the calculations of the mean differential ref lection coefficient in the Kirchhoff approximation. In this approximation we have that µ ∂ ¿ ø v 2 1 ≠R (8) 具jr共qjj 兲2 j典 , 苷 ≠Vs S 2pc
Fig. 1. Segment of a single realization of (a) a numerically generated surface prof ile function H 共r兲 and (b) its derivative for a two-dimensional surface that acts as a Lambertian diffuser. The parameters are b 苷 200l and b0 苷 0.
where r共qjj 兲 苷 2p 3
N21 X
exp共2iabn 兲
n苷0 Z 共n11兲b nb
drrJ0 共qjj r兲exp共2iaan r兲 ,
(9)
J0 共z兲 is a Bessel function, a 苷 共v兾c兲 共1 1 cos us 兲, qjj 苷 共v兾c兲 sin us , and the area S is S 苷 p共Nb兲2 . A large number Np of realizations of the random surface is now generated, and for each realization jr共qjj 兲j2 is calculated. The Np results for jr共qjj 兲j2 that are obtained are summed, and we then divide the sum by Np to yield the average indicated in Eq. (8). The average 具jr共qjj 兲j2 典 must increase quadratically with N to produce a mean differential ref lection coefficient that is independent of N. This is indeed found to be the case for the example studied.
Fig. 2. Mean differential ref lection coeff icient for scattering from a two-dimensional Lambertian diffuser estimated from Np 苷 15, 000 realizations of the surface prof ile function for normal incidence. The parameters are b 苷 200l, b0 苷 0, and N 苷 200. The dashed curve is a plot of the function given by Eq. (6).
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OPTICS LETTERS / Vol. 28, No. 2 / January 15, 2003
Fig. 3. Schematic diagram of the setup proposed for the fabrication of a circularly symmetric random surface.
Several comments need to be made concerning the calculations that led to Fig. 2. If the coefficients 兵an 其 are obtained by a straightforward application of the rejection method to the pdf f 共g兲 given by Eq. (7), a fraction of the surfaces that are generated possess regions in which several consecutive an are very small 共,1022 兲. The presence of such practically horizontal regions on the surface gives rise to a peak in the mean differential ref lection coefficient in the immediate vicinity of the specular direction us 苷 0±. To eliminate this peak, as a particular realization of the function H 共r兲 was being generated, we calculated the corresponding differential ref lection coeff icient at us 苷 0±, and if its value exceeded 2 max具≠R兾≠Vs 典 that surface was discarded. Although this procedure alters the statistics of the resulting surface from the statistics def ined by the pdf f 共g兲, the altered statistics produce a surface with the desired scattering properties and can be used in fabricating surfaces with these properties. The coherent interference of waves scattered from symmetric regions of the surface also contributes a peak in the specular direction. We can eliminate this contribution by making the surface large enough. In addition, a fraction of the surfaces generated possess regions in which several consecutive an are almost equal to unity. The presence of such regions on the surface gives rise to a peak in the immediate vicinity of the grazing directional. This peak, however, can also be eliminated by use of larger surfaces. The form of f 共g兲 given by Eq. (7) allows some segments of the surface to have large enough slopes that when they are illuminated at normal incidence they give rise to scattered rays that, because of the conical bowl shape of the surface, strike the surface a second time before being ref lected back into the vacuum region. Such double-scattering processes, which are not captured by the Kirchhoff approximation, shadow grazing-angle scattering and in principle should cause the mean differential ref lection coeff icient to depart from the desired Lambertian form by decreasing more
rapidly to zero as the scattering angle us approaches 90±. However, estimates of the double-scattering contribution to 具≠R兾≠Vs 典, to be described in the future, show that this departure is negligibly small. Surfaces of the type considered here can be fabricated on photoresist in the following way: Consider the function h共r兲 that constitutes a single realization of the random process constructed according to Eqs. (2), (3), and (7) and satisfies the requirement that the corresponding differential ref lection coeff icient at us 苷 0± does not exceed 2 max具≠R兾≠Vs 典. We def ine the function uh 共r兲 苷2kh共r兲 that, with an appropriate choice of the constant k, can be interpreted as an angle. A disk-shaped mask is then constructed, whose transparent section is defined by the condition uh 共r兲 , u , ug , where ug is a constant greater than 关uh 共r兲兴max . A well-corrected optical system forms a demagnif ied image of the mask on a photoresist-coated plate, as shown in Fig. 3. The rotating photoresist plate is exposed for a time Te , during which it executes a large integer number of revolutions. The total exposure of the plate will then be circularly symmetric, with a radial dependence of the form E共r兲 苷 KTe 关ug 2 uH 共r兲兴兾2p, where uH 共r兲 is a scaled version of the function uh 共r兲, and K is a constant related to the sensitivity of the photoresist and the intensity of the illumination. The exposure of the plate can be put in the final form E共r兲 苷 E0 1 aH 共r兲, where E0 and a are constants that one can adjust by varying the shape of the mask and the exposure time. Assuming that the relation between exposure and resulting height on the plate is linear, the developed surface will have the desired properties. Moreover, the characterized nonlinearities of the photoresist response can be taken into account by prescribed deformations of the mask in Fig. 3. In conclusion, we have proposed a method for generating a two-dimensional random surface that, when it is illuminated at normal incidence by a scalar plane wave, scatters the wave with a circularly symmetric distribution of intensity. We illustrated this method by designing a surface that acts as a Lambertian diffuser. We validated this method by computer simulations and described a procedure for fabricating such surfaces on photoresist. The research of A. A. Maradudin (aamaradu@ uci.edu) and T. A. Leskova (
[email protected]) was supported in part by U.S. Army Research Off ice grant DAAD 19-99-1-0321. The research of E. R. Méndez (
[email protected]) was supported in part by the Consejo Nacional de Ciencia Y Tecnología (Mexico) through grant 32254-E. References 1. J. C. Stover, Optical Scattering (SPIE, Bellingham, Wash., 1995). 2. F. Grum and G. W. Luckey, Appl. Opt. 7, 2295 (1968). 3. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), p. 281.
