Designing and Making Random Diffusers for Depth of Focus Extension: Theory and Experiment, Papers of Spanish Language

Download Designing and Making Random Diffusers for Depth of Focus Extension: Theory and Experiment and more Papers Spanish Language in PDF only on Docsity! Design and fabrication of random diffusers for extending the depth of focus E. E. Garćıa-Guerrero, E. R. Méndez, and H. M. Escamilla División de F́ısica Aplicada - CICESE Km. 107 carretera Tijuana-Ensenada, Ensenada, B. C. 22860 México. T. A. Leskova and A. A. Maradudin Department of Physics and Astronomy, University of California, Irvine, CA 92697 Abstract We present a method for designing diffusers that, when illuminated by a converging beam, produce a specified intensity along the optical axis. To evaluate the performance of the diffusers in an imaging system we also calculate the three-dimensional distribution of the mean intensity in the neighborhood of focus. We find that the diffusers can be used as depth extenders in imaging systems. A method of fabricating the designed diffusers on photoresist-coated plates is discussed. Finally, we present some experimental results obtained with specially fabricated diffusers. 1 Introduction In recent publications, we have described methods for designing and fabricating optical diffusers with specified angular scattering properties [1]-[5]. An important example is the design of diffusers that, on average, produce a uniform intensity distribution over a specified range of angles. Such an element would have applications, for example, in projection systems, where it is important to produce even illumination without wasting light. Based on similar notions, in this paper we consider the design of diffusers that, when illuminated by a converging beam, produce a specified distribution of intensity along the optical axis. The design is evaluated through calculations based on scalar diffraction theory. The results show that the mean intensity produced by these diffusers is indeed fairly constant over the specified region. A common criterion employed in the evaluation of masks for extending the depth-of-focus of optical systems is the slowness of the decay of the axial intensity as one moves away from focus [6, 7]. This suggests the use of diffusers that produce a uniform axial intensity over a specified range as focal depth extenders. We present some preliminary results on the performance of an aberration-free optical system that includes one of these diffusers in the pupil. It is found that, although the image contrast is reduced with respect to that of an aberration-free optical system, the focal depth can be increased substantially. We also propose and implement a method for fabricating the diffusers on photoresist. Finally, experimental results obtained with fabricated diffusers are presented and discussed. 2 Background To explain the design of the diffuser with circular symmetry, it is useful to consider first the scattering of light by a one-dimensional surface illuminated by a plane wave. Adopting, for simplicity, the thin phase screen model [8] to describe the interaction between the incident field and the surface, the scattering amplitude can be written in the form R(q) = A0 L∫ 0 e−iv1x1−iv3ζ(x1)dx1, (1) where A0 is a complex constant, L is the length of the illuminated region, v1 = q−k = (ω/c)[sin θs− sin θ0] and, as usual, ω represents the frequency and c the speed of light in vacuum. Considering only small angles of incidence and scattering v3 = 2(ω/c) for a reflection geometry, and v3 = (ω/c)∆n for a transmission one [9], where ∆n is the difference between the refractive indices of the material from which the diffuser is made and that of the surrounding medium. Neglecting finite-size effects, the mean intensity 〈I(q)〉 = 〈R(q)R∗(q)〉 is given by 〈I(q)〉 = ∞∫ −∞ ∞∫ −∞ dx1dx ′ 1〈exp { −iv3(ζ(x1)− ζ(x′1)) } 〉 exp { −iv1(x1 − x′1) } . (2) The change of variable x′1 = x1 + u yields 〈I(q)〉 = ∞∫ −∞ ∞∫ −∞ dx1dug(u;x1) exp {−iv1u} , (3) where g(u;x1) = 〈exp {−iv3(ζ(x1)− ζ(x1 + u))}〉. The function g(u;x1) represents the autocorre- lation of the amplitude fluctuations introduced by the diffuser. We obtain the geometrical optics limit by expanding the difference ζ(x1)− ζ(x1 + u) in powers of u and retaining only the leading nonzero term. We find g(u;x1) = 〈 exp { −iuv3ζ ′(x1) }〉 (4) where ζ ′(x1) represents the derivative of the surface profile function. Assuming, for a moment, that the diffuser is statistically stationary, we can write 〈I(q)〉 = L ∞∫ −∞ 〈 exp { −iuv3ζ ′ }〉 exp {−iv1u} du, (5) which leads to 〈I(q)〉 = 2πL v3 f ( v1 v3 ) , (6) where f(γ) represents the Probability Density Function (PDF) of surface slopes. 3 Diffusers that produce a prescribed distribution along the opti- cal axis We consider now a circularly symmetric aberration-free imaging system. For a point source object, the system produces a converging spherical wave whose complex amplitude can be expressed as follows [10] ψ(z0, r0) = −ik0 A f2 eik0z0 a∫ 0 P (r)J0 ( k0 r0r a ) exp { −ik0 z0r 2 2f2 } rdr. (12) In this expression, A is a constant amplitude, k0 = (ω/c) is the wavenumber in vacuum, a is the radius of the pupil, f is the distance from the principal plane to the best focus, z0 is the defocus distance, and we have introduced the pupil function P (r) which, in our case, will be a complex function of the form P (r) = e−iv3H(r), (13) where H(r) represents the surface profile function. Let us consider first the complex amplitude along the optical axis: ψ(z0, 0) = −ik0 A f2 eik0z0 a∫ 0 e−iv3H(r) exp { −ik0 z0r 2 2f2 } rdr. (14) With the change of variable t = k0 2f2 r2 = κr2 (15) we can write ψ(z0, 0) = −iAeik0z0 κa2∫ 0 dte−iz0te−iv3h(t). (16) with h(t) = H( √ t/κ). The parallels between Eqs. (16) and (1) are evident. Thus, if we are able to design a surface that produces a specified angular scattering pattern (function of q), we should be able to design a diffuser that produces a specified intensity distribution along z0. Equation (16) represents a particular case of the more general result obtained by McCutchen [11]. Following the procedure described in Sect. 2, and since H(r) = h(κr2), we divide the surface profile function H(r) into segments of the form H(r) = anκr2 + bn, √ nη ≤ r ≤ √ (n+ 1)η, (17) with n = 0, 1, 2, ...., N−1, and η = a/ √ N . The {an} are independent identically distributed random variables and, as before, the {bn} are determined from the condition that the surface profile function H(r) be a continuous function of r. Then, the mean axial intensity is determined directly by f(γ), the PDF of the numbers {an}. In what follows we consider the design of diffusers that produce uniform axial intensity in the interval −zm ≤ z0 ≤ zm, and evaluate the possibilities of employing them to extend the depth of focus of imaging systems. 4 Three-dimensional distribution in the neighborhood of focus If we intend to use the diffuser in an imaging system, not only the axial intensity distribution is important. So, in this section, we calculate the mean intensity in the focal region. We write the expression for the field in the form ψ(z0, r0) = C0 a∫ 0 J0 ( k0 r0r a ) e−iv3H(r)−iκr 2z0rdr. (18) with C0 = −ik0(A/f2)eik0z0 . With the form of H(r) given by Eq. (17), this expression can be written in the form ψ(z0, r0) = C0 N−1∑ n=0 e−iv3bn √ n+1η∫ √ nη J0 ( k0 r0r a ) e−iκ(z0+v3an)r 2 rdr = C0 N−1∑ n=0 e−iv3bnψn(z0, r0; an). (19) The field ψn(z0, r0; an) represents the diffraction pattern of an annular pupil with defocus z0 +v3an, where an is a random quantity. For uncorrelated contributions from the different annular pupils the mean intensity is then given by 〈I(z0, r0)〉 = |C0|2 N−1∑ n=0 〈|ψn(z0, r0; an)|2〉, (20) with ψn(z0, r0; an) = √ n+1η∫ √ nη J0 ( k0 r0r a ) e−iκ(z0+v3an)r 2 rdr. (21) Diffraction integrals like the one represented by Eq. (21) have been well-studied in the past [10, 12]. Here, we evaluate them using the Nijboer expansion [10, 13]. In Figs. 3 to 5, we present maps of the intensity distribution produced by different rings in the focal region of a system with a = 2 cm and f = 25 cm for the particular case an = 0. The pupil was divided into 99 annular apertures and λ = 0.633µm. To facilitate the visualization of the results, the gray-level maps are shown on a logarithmic scale. It can be seen in the figures that the axial distribution is independent of the order of the ring. On the other hand, the transverse distribution decreases in size with the order of the ring. As illustrated by the result of Eq. (20), functions like the ones illustrated in Figs. 3 to 5 are the building blocks of the mean intensity. We consider now, as an example, a diffuser that produces a uniform distribution of intensity in the range −15 cm ≤ z0 ≤ 15 cm. Since the f(γ) is uniform, the average expressed in Eq. (20) can be evaluated by integrating |ψn(z0, r0; an)|2 over this rectangular region. The result is shown in Fig. 6. Again, to facilitate the visualization of the results, the gray-level map is shown on a logarithmic scale. It can be seen that the intensity map is fairly constant in the design region and that it drops quickly outside this region. The decay of the intensity as a function of the transverse coordinate suggests that the diffusers can indeed be used in imaging systems. -0.05 0 0.05 0 0.2 0.4 0.6 0.8 1.0 x 10-4 r [mm]0 In te n si ty (b) -4 -40 -20 0 20 40 0 0.2 0.4 0.6 0.8 1.0 x 10 z [mm]0 In te n si ty (c) -40 -20 0 20 40 0.05 0 -0.05 z [mm]0 r [m m ] 0 (a) Figure 3: Intensity distribution in the focal region produced by the aperture with 0 ≤ r ≤ η. The gray level map is on a logarithmic scale. -0.05 0 0.05 0 0.2 0.4 0.6 0.8 1.0 x 10-4 r [mm]0 In te n si ty (b) -4 -40 -20 0 20 40 0 0.2 0.4 0.6 0.8 1.0 x 10 z [mm]0 In te n si ty (c) -40 -20 0 20 40 0.05 0 -0.05 z [mm]0 r [m m ] 0 (a) Figure 4: Intensity distribution in the focal region produced by the aperture with 9η ≤ r ≤ 10η. The gray level map is on a logarithmic scale. Figure 8: Images of a wheel-like object on the best-focus plane. Figure 9: Images of a wheel-like object on an out-of-focus plane. 6 Summary and conclusions In this paper, we have studied the problem of designing and fabricating optical diffusers that when illuminated by a conveging beam produce a uniform intensity along the optical axis within a specified range. The technique is also suited for the production of such surfaces in photoresist, and we have implemented a procedure for their optical fabrication. The results indicate that good approximations to the desired uniform axial intensity distribution can be obtained with surfaces fabricated with the proposed method. In contrast with the majority of methods proposed for extending the depth of field, our approach is based on refraction, rather than diffraction. The method is verified through calculations based on scalar diffraction theory. The results show that the mean intensity produced by these diffusers is indeed fairly constant over the specified region, and decays as the point of observation moves away from the optical axis. The diffusers can then be used in imaging systems as focal depth extenders. Acknowledgments We would like to express our gratitude to Fabián Alonso and Pedro Leree for help in various aspects of this work. This was supported in part by CONACYT (Mexico), under grant 47712-F. References [1] T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett., 73, 1943-1945 (1998). [2] A. A. Maradudin, I. Simonsen, T. A. Leskova, and E. R. Méndez, “Design of one-dimensional Lambertian diffusers of light”, submitted to Waves in Random Media (2000) [3] E. R. Méndez, E. E. Garćıa-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, “Photofabrication of random achromatic optical diffusers for uniform illumination,” AO 40, 1098-1108 (2001). [4] E. R. Méndez, E. E. Garćıa-Guerrero, T. A. Leskova, A. A. Maradudin, J. Muñoz-López, and I. Simonsen, “Design of one-dimensional random surfaces with specified scattering properties,” APL 81, 798-800 (2002) [5] E. R. Méndez, T. A. Leskova, A. A. Maradudin, M. Leyva-Lucero, and J. Muñoz-Lṕez, “The design of two-dimensional random surfaces with specified scattering properties,” J. Opt. A 7, S141-S151 (2005). [6] J. Ojeda-Castaneda, P. Andrés, and E. Montes, “Phase-space representation of the Strehl ratio: ambiguity function,” J. Opt. Soc. Am. A 4, 313-317 (1987). [7] J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. 30, 1647-1649 (2005). [8] W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269-387 (1977). [9] Z. H. Gu, H. M. Escamilla, E. R. Méndez, A. A. Maradudin, J. Q. Lu, T. Michel, and M. Nieto-Vesperinas, “Interaction of two optical beams at a symmetric random surface,” Appl. Opt. 31, 5878-5889 (1992). [10] M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge University Press, Cambridge, UK, 1999), Sect. 8.8. [11] C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240-244 (1964). [12] W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. 50, 749-753 (1960). [13] J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176-178 (1969).