Hindawi Publishing Corporation Advances in Optical Technologies Volume 2010, Article ID 783206, 5 pages doi:10.1155/2010/783206 Research Article Spherical Aberration Correction Using Refractive-Diffractive Lenses with an Analytic-Numerical Method Sergio Vazquez-M ´ ontiel, Omar Garcıa-Li ´ evanos, ´ and Juan Alberto Hernandez-C ´ ruz Marginal Meridional Rays, Instituto Nacional de Astrof´ısica Optica y Electr´onica, Apdo. Postal 51 y 216, Puebla-72000, Mexico Correspondence should be addressed to Omar Garc´ ıa-Lievanos, ´ ogarcial@ipn.mx Received 19 February 2010; Accepted 14 July 2010 Academic Editor: Michael Fiddy Copyright © 2010 Sergio Vazquez-M ´ ontiel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose an alternative method to design diﬀractive lenses free of spherical aberration for monochromatic light. Our method allows us to design diﬀractive lenses with the diﬀraction structure recorded on the last surface; this surface can be ﬂat or curved with rotation symmetry. The equations that we propose calculate the diﬀraction proﬁles for any substratum, for any f-number, and for any position of the object. We use the lens phase coeﬃcients to compensate the spherical aberration. To calculate these coeﬃcients, we use an analytic-numerical method. The calculations are exact, and the optimization process is not required. 1. Introduction numerical method. The calculations are exact, simple and quick. A process of optimization is not required. Spherical aberration is, in many cases, the most important of The manufacturing problem of diﬀractivelensesisnot all primary aberrations, because it aﬀects the whole ﬁeld of considered here; to solve this problem you can read Castro- the lens, including the vicinity of the optical axis. It is due to Ramos et al. [4]. diﬀerent focus positions for a marginal ray, meridional ray, First, we describe the diﬀractive lenses theory. Also, we and paraxial rays. An alternative to minimize the spherical give a brief derivation of the general grating equation to trace aberration is to use diﬀractive optical elements (DOE). a couple of light rays through a rotationally symmetrical Diﬀractive lenses are essentially gratings with a variable surface. Then, we establish the analytic-numerical method spacing groove which introduces a chromatic aberration to minimize spherical aberration. We propose some heights that is worse than conventional refractive/reﬂective optical to correct the spherical aberration. Finally, we conclude by elements. In some applications, an optical component may providing a design example. require a diﬀractive surface combined with a classic lens element. By using the diﬀractive properties, it is possible to design hybrid elements to obtain an achromatically 2. Theory of the Diffractive Lenses corrected element [1]. In other cases, the requirements Diﬀractive lenses can be described by a polynomial phase can be satisﬁed by just using a diﬀractive element. In function [5] general, iterative methods are used to design these lenses [2]. Also, some people have used analytical third-order and numerical integrator methods to design diﬀractive lenses 2π m n φ x, y = a x y,(1) mn [1, 3]. The diﬀractive lenses we describe in this paper m n are limited to monochromatic applications; however, our proposed method is valid for a wide range of wavelengths. We use lens phase coeﬃcients to compensate spherical where λ is the design wavelength; a are the lens phase mn aberration. To calculate these coeﬃcients, we use an analytic- coeﬃcients; x, y are the coordinates in the diﬀractive lens. 2 Advances in Optical Technologies We will consider that the diﬀractivelensisrotationally symmetrical, so (1) is rerewritten as L , M , N L , M , N 0 0 0 1 1 1 2π 2 4 6 8 φ y = a + a y + a y + a y + a y +··· . (2) 0 2 4 6 8 L , M , N λ 2 2 2 y y 1 2 Here, the longitudinal displacement of the reference sphere is a = 0 because we have assumed it is in the ideal focus. z z d 1 2 2 The coeﬃcient a are implicit lens paraxial properties; it is equal to −1/2 f ,where f is the focal length. The remainder coeﬃcients in (2) give the amount of spherical aberrations of r the ﬁrst- second- and higher-order [6, 7]. Designers usually use some commercial optical design programs to obtain the lens phase coeﬃcients by using an optimization process. We will describe an analytical method Figure 1: Lens parameter. The diﬀractive surface is on the second surface. to obtain these coeﬃcients. To trace a pencil of rays through the diﬀractive optical surface, we use the grating equation. For a ﬂat surface, the Example 1 FBY 0 FBX 0 grating equation is given by Ray-intercept curves WV1 + DY 0.005 DX 0.005 n sin I − n sin I = mλ f,(3) where n and n are refractive indexes for two diﬀerent mediums, I and I are the diﬀractive and incident angles; f is the grating frequency; m is the diﬀracted order. FY FX To analyze the light propagation through a diﬀractive curved surface, we have to change the form of the last equation. After some algebra, we obtain the general grating equation (n M − nM ) cos θ + (nN − n N ) sin θ = mλ cos θ f , 2 1 N 1 2 N N (4) Figure 2: Transversal spherical aberration of the example one. where the direction of refracted and diﬀracted rays is given by the direction cosines M , M , N ,and N as are shown in Table 1: Example 1 data. 1 2 1 2 the Figure 1. This analysis considers that the diﬀractive lens is Radius Thickness Radius aperture rotationally symmetrical, and then the direction cosines L = Surfaces 0 Glass (mm) (mm) (mm) 0, L = 0, L = 0. θ is the angle between the normal at 1 2 N ∞ 200 Air surface and optical axis, it is given by 101.954 8.138 25 BK7 ∂F/∂y, ∂F/∂z DOE −101.954 66.059 25 Air sin θ ,cos θ = ,(5) N N 1/2 2 2 2 (∂F/∂x) + ∂F/∂y + (∂F/∂z) 4 ∞ 0Air here, F is the surface function in which the diﬀractive lens will be recorded, and x and y are the surface coordinates. Using (5)and (7), we can trace n rays through the surface at The grating frequency of (3)and (4) can be calculated in diﬀerent heights on the pupil. Then they can arrange a k × k one dimension by equations system. The number of equations depends of the number of coeﬃcients that we want to ﬁnd. 1 ∂φ f = , Then, if we want to ﬁnd k coeﬃcients, we need to solve 2π ∂y an equation system similar to(8) ⎡ ⎤ (6) 2k−1 ⎡ ⎤ ⎣ ⎦ f = 2ka y , 1 y 2k ⎡ ⎤ ⎡ ⎤ λ w w w ··· w − A 11 12 13 1k 0 ⎢ ⎥ k=1 2 f ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ w w w ··· w A 21 22 23 2k 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ where φ is the phase function given by (2) k is an integer w w w ··· w A ⎢ 31 32 33 3k⎥ ⎢ ⎥ ⎢ 2 ⎥ = ,(8) ⎢ ⎥ ⎢ 6 ⎥ ⎢ ⎥ equalto1,2,3,4, ..., and the diﬀracted order m = 1. With . . . . . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . . . . . ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ . . . ··· . . this, (4)can be rewrittenas ⎣ ⎦ w w w ··· w A k1 k2 k3 kk k 2k [(n M − nM ) cos θ + (nN − n N ) sin θ ] 2 1 N 1 2 N ⎡ ⎤ ∞ where w represents the diﬀerent constants of the right side (7) 2k−1 ⎣ ⎦ = cos θ 2ky a . N 2k of (7), and A are the constants of the left side of the same k=1 equation, all for diﬀerent height rays on the pupil. Advances in Optical Technologies 3 Example 3 Example 2 (our selected points) FBY 0 FBX 0 FBY 0 FBX 0 FOC 0 Ray-intercept curves WV1 + Point spread function-WV1 Y + Xx DY 1e − 05 DX 1e − 05 0.75 0.5 FY FX 0.25 −0.05 −0.0025 0 0.0025 0.005 Position (a) Figure 5: The point spread function of the example 3. Example 2 (Kingslake points) FBY 0 FBX 0 Ray-intercept curves WV1 + DY 2e − 05 DX 2e − 05 Table 2: Coeﬃcients value for the example 1. Coeﬃcients Aperture height (mm) Value −1 a Paraxial −0.005 mm −6 −3 a (25)(1) = 25 1.180409 × 10 mm FY FX −10 −5 a (25)(0.7746) =19.363 −1.270732 × 10 mm Table 3: Coeﬃcients value for the example 2. Coeﬃcients Aperture height (mm) Value (b) −1 a Paraxial −0.005 mm Figure 3: Transversal spherical aberration of the example 2. −6 −3 a (25)(1) =25 1.194430 × 10 mm −10 −5 a (25)(0.9137) =22.842 −1.684803 × 10 mm −14 −7 a (25)(0.7746) =19.363 3.602457 × 10 mm −14 −9 a (25)(0.555) =13.875 −5.856954 × 10 mm Example 3 FBY 0 FBX 0 Ray-intercept curves WV1 + DY 2e − 05 DX 2e − 05 can be represented only for the ﬁrst and second terms, and combining these terms, the spherical aberration of the edge can be corrected. Then, the peak of the spherical aberration residual occurs when y is equal to the marginal y multiplied FY FX by 3/5 = 0.7746. This analysis is similar to Kingslake [8]. The diﬀerence is that the defocus term is not considered here. It is possible to correct the residual spherical aberration by using the third term of the expansion (9), but now the ray aberration curve has two opposite peaks above and below the 0.7746 zone. The zones with maximum and minimum Figure 4: Transversal spherical aberration of the example 3. residuals fall at values of y given by y/y = 0.5550 or 0.9137 (see Figure 2). If we consider f/numbers to be small, we should correct the spherical aberration residual, and its peaks fall at values 3. How Many and What Heights Should y = y (0.5550) or y (0.9137); then we need fourth and m m Be Corrected ﬁfth term to correct these other y’s, now the ray aberration curve has two opposite peaks above and two below of The spherical aberration of the ray in any optical system can 0.5550 and 0.9137 zones. The zones with maximum and be expressed as minimum residuals fall at values of y given by y/y = 0.9681, 0.8505, 0.6661, or 0.3740, Figure 7. This analysis can ∂W 0, y 3 5 7 continue because the expansion (9) is inﬁnity. = 4b y +6b y +8b y +··· . (9) 1 2 3 ∂y The points for Kingslake analysis are y/y = 1, 0.8880, 0.7071, and 0.4597, and for our analysis y/y = 1, 0.9137, Considering only big f/numbers, the spherical aberration 0.7746, and 0.5550. Relative irradiance 4 Advances in Optical Technologies Table 4: Coeﬃcients value for the example 3. Coeﬃcients Aperture height (mm) Value −1 a Paraxial −0.005 mm −7 −3 a (25)(1) =25 9.106091 × 10 mm −10 −5 a (25)(0.9137) =22.842 −1.941142 × 10 mm −14 −7 a (25)(0.7746) =19.363 5.218183 × 10 mm −18 −9 a (25)(0.555) =13.875 −9.916744 × 10 mm The number of y that must be corrected for each The number of rays traced depends on the number of optical system depends on the optical system tolerances, coeﬃcients. In this example, we use two coeﬃcients, and we for example, with one value of y ((1)(y )), we correct get the next equations system a lens with f/number bigger than f/5; with two diﬀerent values of y((1)(y )and (0.7746)(y )), we correct a lens 6.340530E +4 6.130961E +7 a 0.067053 m m 4 = . with f/number bigger than f/2; with four diﬀerent values of 2.924391E +4 1.6731E +7 a 0.032394 y((1)(y ), (0.7746)(y ), (0.5550)(y ), and (0.9137)(y )), m m m m (10) we correct a lens with f/number bigger than f/1, but only the designer should decide the correction that he needs. We have solved (8) to compute the phase coeﬃcients for two diﬀerent pupil positions on the surface; they are shown in Table 2. 4. Results Figure 2 shows the spherical aberration of the refractive- We have proposed a general expression to compute the diﬀractive lens; the graphics were obtained using the com- phase coeﬃcients. Now, we will show how theses coeﬃcients mercial optical design program “OSLO” [9]. minimize the spherical aberration with some numerical We can see in the graphic a maximum transversal examples. All examples considered in this section have the spherical aberration of about 0.004 mm, having zeros on diﬀracted order m = 1. two pupil positions. This is because we had computed two coeﬃcients for the system. The corresponding Strehl Ratio is 4.1. Example 1. In this example, we consider that the of about 0.151. diﬀractive surface is on a spherical surface (the last surface In the Figure 2, FBY and FBX are the fractional object of the system) with 50 mm of diameter aperture, numerical coordinates, and WV1 is the wavelength (λ = 0.587 μm)for aperture 0.375, object distance 200 mm, and λ = 0.587 μm. the evaluation. In Table 1, other characteristics of the refractive- diﬀractive lens are shown. 4.2. Example 2. We consider the same optical system but We must trace light rays until the last surface, and then now using four phase coeﬃcients. Solving the next equations we can calculate all constants of (8). system ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 6.34053E +4 6.130961E +7 5.269617E +10 4.246204E +13 a 0.067053 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 4.821205E +4 3.869253E +7 2.760235E +10 1.84602E +13 a 0.051953 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ , (11) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2.924391E +4 1.6731E +7 8.50856E +9 4.056593E +12 a 0.032394 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 1.071387E +4 3.119059E +6 8.071391E +8 1.958143E +11 a 0.012299 we obtain phase coeﬃcients which are show in Table 3. 4.3. Example 3. Now we consider the same optical system In Figure 3, we can see a maximum traversal spher- but the diﬀractive surface on a hyperbolic surface (last ical aberration of the refractive-diﬀractivelensofabout surface) with conic constant, diameter aperture K = 0.00005 mm, having zeros on four pupil positions. The −4.654, 50 mm, numerical aperture of 0.375, object distance reason is that we had computed four coeﬃcients for this sys- of 200 mm, and λ = 0.587 μm. We must trace rays tem. The corresponding Strehl Ratio is of about 1. Figure 3 to the hyperbolic surface because in this way we can also shows the diﬀerence between the points proposed by calculate all constants of (8) for this example. We use Kingslake [6] and our selected points. It can be seen that four phase coeﬃcients to solve the following equations the points suggested in this paper to correct the spherical aberration are slightly better than the Kingslake points. system: Advances in Optical Technologies 5 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 6.422962E +4 6.209924E +7 5.336845E +10 4.299859E +13 a 0.048792 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 4.86619E +4 3.905076E +7 2.785591E +10 1.862844E +13 a 0.038001 ⎢ ⎥ ⎢ 6 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ (12) ⎣ 2.939127E +4 1.681484E +7 8.550953E +9 4.076689E +12⎦ ⎣ a ⎦ ⎣ 0.023906 ⎦ 1.072902E +4 3.123454E +6 8.082727E +8 1.960884E +11 a 9.203867E − 3 ´ ´ In the Table 4 are the new coeﬃcients for this optical system. [4] J. Castro-Ramos, S. Vazquez-Montiel, J. Hernandez-De-La- Cruz, O. Garc´ ıa-Lievanos, and W. Calleja-Arriaga, “Diﬀractive Figure 4 shows the aberration of this refractive-diﬀractive optics: a review of the optical systems design and construction lens. using diﬀractive lenses,” Revista Mexicana de Fisica, vol. 52, no. We can see again a very small spherical aberration, 6, pp. 479–500, 2006. −5 and its maximum value is of around 2 × 10 mm. It [5] H. P. Herzig, “Design of refractive and diﬀractive micro- has 4 zeros because we have used 4 phase coeﬃcients. optics,” in Micro-Optics: Elements, Systems, and Applications,S. The irradiance distribution corresponding to this system is Martellucci and A. Chester, Eds., pp. 23–33, Plenum Press, New shown in Figure 5. York, NY, USA, 1997. Our proposed method is also for ﬂat surfaces. We only [6] M. Young, “Zone plates and their aberrations,” Journal of the use zero for the angle between the normal to the surface and Optical Society of America, vol. 62, no. 8, pp. 972–976, 1972. optical axis in (4), and then we obtain the grating (3)for a [7] R. W. Meier, “Magniﬁcation and third-order aberrations in ﬂat surface. Then, we can use the procedure that we used in holography,” Journal of the Optical Society of America, vol. 55, the previous examples. pp. 987–992, 1965. [8] R. Kingslake, “Spherical aberration,” in The Lens Design If the designer wants to use the ﬁrst surface, the Fundamentals, chapter 5, pp. 114–115, Academic Press, New conjugates must be changed, and then the method proposed York, NY, USA, 1978. can be applied. [9] Lambda Research Corporation, OSLO Optics Software for Lay- out and Optimization, Optics Reference, Version 6.1, Littleton, Mass, USA, 2001. 5. Conclusions We have established a new exact method to correct the spherical aberration for any optical system using diﬀractive lenses; this method makes use of the general grating equation and exact ray trace. With our method, we can decide how many zeros the spherical aberration should have and ﬁx its position in the exit pupil. The method can only be applied to the ﬁrst and last surface of the optical system. We also have proposed some heights to correct the spherical aberration and how many rays must be traced depending on the f/number. In the ﬁrst and second examples, we have shown that we can have a high control of spherical aberration, minimized at points on the surface where we have wanted. Also, we have shown that our method is valid for any rotationally symmetrical surface. In general, spherical aberration will have as many zeros as the coeﬃcients we calculate. It is very important to see that in order to minimize spherical aberration, we use only as many coeﬃcients as necessary. Finally, to calculate the coeﬃcients, we only use the analytic-numerical method. The calculations are exact, sim- ple, and quick. A process of optimization is not required. References [1] N.Davidson,A.A.Friesem,and E. Hasman,“Analytical design of hybrid diﬀractive-refractive achromats,” Applied Optics, vol. 32, no. 25, pp. 4770–4774, 1993. [2] V. A. Soifer, Methods for Computer Design of Diﬀractive Optical Elements, John Wiley & Sons, New York, NY, USA, 2002. [3] D. A. Buralli and G. M. Morris, “Design of diﬀractive singlets for monochromatic imaging,” Applied Optics, vol. 30, no. 16, pp. 2151–2157, 1991. 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Advances in Optical Technologies – Hindawi Publishing Corporation
Published: Aug 17, 2010
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