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We see from Eqs. (2.11) and (2.15) that in the geometrical optics limit of the Kirchhoff approximation the scattering distribution consists of two tectangular distributions, and it is clear that diffraction effects will smooth these two contributions. The peak observed in the specular direction in the scattering distribution plotted in Fig. 3 is due to the overlap of the tails of the two distributions predicted on the basis of the geometrical optics approximation. To illustrate this point we present, in Fig. 4, a mean differential reflection coefficient for the case in which the random numbers are generated from a drc of the form f () =( -)(m + -)/m, (3.1) where = 0.05. In our approximation the scattering distribution is then given by Rs 1 s s = - - m + + s 4mh 2h 2h s s + - m + -, (3.2) 2h 2h where the smallness of m has been used in obtaining this Figure 5. The same as Fig. 4, but with = 0.01.

a = 0.6328 m; b = 0.532 m; c = 0.442 m.

result. It can be seen that this distribution agrees well with , 1999, 41, . The Design and Fabrication of One-dimensional Random Surfaces with Specified Scattering Properties Figure 6. Schematic diagram of the experimental arrangement employed for the fabrication of the diffusers.

4. Experimental Results we studied these properties in the simpler case of the transmission of s-polarized light through them. Although A schematic diagram of the optical system used in our the theoretical work motivating the method for fabricating efforts to fabricate the kind of surface studied in this paper the uniform diffusers described in the preseding sections is shown in Fig. 6. The illumination is provided by a was based on reflection, an analysis carried out within HeCd laser (wavelength = 442 nm). An optical system the framework of the geometrical optics limit of the thin concentrates the light transmitted through a rotating ground phase screen model [9] shows that surfaces that act as glass on a slit, providing illumination that is effectively band-limited uniform diffusers in reflection also act as incoherent. An incoherent image of the slit is formed by uniform band-limited diffusers in transmission, althought the an X1 (numerical aperture 0.05) microscope objective on a photoresist-coated glass plate.

The width of the slit is approximately l = 180 m, and its incoherent image has a nearly restangular shape (smoothed by diffraction). In order to fabricate grooves with the desired trapezoidal shape on the photoresist, the plate is exposed while executing a scan of length b = l/(2m + 1). This procedure generates, basically, a function s(x1) with the shape defined by Eq. (2.2). The depth of the groove is determined by the time of exposure. An example of such a fabricated groove is shown in Fig. 7, which presents the measured surface profile of a section of a photoresist plate that was exposed in this fashion. Althought the corners are not as sparp as the ones in Fig. 1, a, the result approximates the desired shape quite well.

The photoresist plate is exposed to grooves generated in Figure 7. Measured profile that illustrates the experimental this fashion, with random depths and displaced sequientially realization of the function s(x1). The profile was measured by in steps of 2b. Several hundred uncorrelated random means of a Dektak(st) mecahnical profilometer.

numbers {cl} are generated in the computer with the specified f (). At each position x1 = 2bl, The time of exposure of the groove is proportional to the random number cl generated in the computer [8].

In Fig. 8 we present a profileometric trace of one of the samples fabricated according to Eq. (2.1). The faceted nature of the surface is clearly visible in the figure. In the example displayed we chose m = 0, which produces a function s(x1) of triangular rather than trapezoidal form. The resulting symmetric triangular indentations are clearly visible in the figure. Thus, these preliminary results indicate that the proposed fabrication method is able to produce random uniform diffusers.

In order to study experimentally the scattering properties of these photoresist diffusers in reflection they would have Figure 8. Measured segment of a surface profile for a fabricated had to be coated with a thin metallic layer. Instead, sample. The parameters are b = 60 m, m = 0.

, 1999, 41, . 924 T.A. Leskova, A.A. Maradudin, E.R. Mndez, A.V. Shchegrov References [1] L.I. Mandelshtam. Ann. Physik 41, 609 (1913).

[2] C.N. Kurtz. J. Opt. Soc. 62, 929 (1972).

[3] C.N. Kurtz, H.O. Hoadley, J.J. DePalma. J. Opt. Soc. Am. 63, 1080 (1973).

[4] Y. Nakayama, M. Kato. Appl. Opt. 21, 1410 (1982).

[5] M. Kowalczyk. Opt. Soc. Am. A1, 192 (1984).

[6] E.R. Mndez, G. Martnez-Niconoff, A.A. Maradudin, T.A. Leskova. SPIE 3426 (1998), to appear.

[7] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery.

Figure 9. Experimental result for the anglular dependence of Numerical Recipes, in Fortran. 2nd Edition. Cambridge Unithe intensity of s-polarized light of wavelength = 0.6328 m versity Press, N.Y. (1992). P. 281.

transmitted through a photoresist film. The angle of incidence is [8] E.R. Mndez, M.A. Ponce, V. Ruiz-Corts, Zu-Han Gu. Appl.

0 = 0. The illuminated surface of the film is a one-dimensional Opt. 30, 4103 (1991).

random surface through which light is transmitted within the angle [9] W.T. Welford. Opt. Quant. Electron. 9, 269 (1977).

-5

[11] H.P. Balthes. In: Inverse Scattering Problems in Optics / Ed.

maximum scattering angle m in transmission is different that by H.P. Balthes. Springer-Verlag, N.Y. (1998). P. 1.

it is in reflection [10]. However, the transmission patterns [12] E.R. Mndez, G. Marnez-Niconoff, A.A. Maradudin, obtained with the diffusers fabricated up to now, although T.A. Leskova. Proc. Reunion Iberoamericana de Optica. Cartaband-limited, are not uniform (Fig. 9). Large intensity gena, Columbia (1998), to appear.

fluctuations are present in the angular region in which a constant intensity would be expected. The origin of these fluctuations is the small number of randomly oriented facets that are etched in our surfaces. They represent, simply, statistical noise. For the lengths of the surfaces that we have fabricated only about two hundred random numbers cl are employed. Efforts are currently underway to fabricate surfaces with a larger number of randomly oriented facets.

5. Summary and Conclusions In this paper we have described approaches to designing and fabricating one-dimensional, random, band-limited, uniform diffusers. These approaches are well suited for the generation of such surfaces on photoresist. The results of computer simulations, and some preliminary experimental results, indicate that uniform band-limited diffusers can be fabricated by the method proposed.

The design of band-limited uniform diffusers is but one interesting inverse problem involving the design of random surfaces with specified scattering properties. The design of a Lamberitian diffuser, namely a random surface that produces a scattered intensity proportional to the cosine of the polar scattering angle, is another [11]. Finally, the design and fabrication of two-dimensional random surfaces with specified light scattering properties pose interesting theoretical and experimental challenges. Some first steps in this direction have been taken recently [12], but more remains to be done.

This paper is dedicated to the A.F. Ioffe Physico-technical institute on the occasion of its 80th anniversary, with best wishes for many more years of significant contributions to science. The work reported here was supported in part by Army Research Office grants DAAH 049610187 and DAAG 5598C0034.

, 1999, 41, .

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