Aspherical Lens Design by Using a Numerical Analysis

Transcription

1 Journal of the Korean Physical Society, Vol. 51, No. 1, July 27, pp Aspherical Lens Design by Using a Numerical Analysis Gyeong-Il Kweon Department of Optoelectronics, Honam University, Gwangju Cheol-Ho Kim Department of Information Communication Engineering, Honam University, Gwangju (Received 15 March 27) The surface profiles of four different spherical-aberration-free aspherical lenses are given as conic curves, namely a hyperbola an ellipse. Historically known as Cartesian ovals, these lens profiles can be obtained from Fermat s principle. Alternatively, we have derived the surface profile from Snell s law. The derivational procedure is physically more appealing mathematically elegant. The surfaces profiles of two more spherical-aberration-free aspherical lenses, which cannot be represented by conic curves, are given in terms of a semi-analytic formula. Examples of the lens profiles are given. PACS numbers: E, F Keywords: Lens design, Optical aberration, Geometrical optics, Spherical aberration, Numerical analysis I. INTRODUCTION Aspherical lenses are essential ingredients in reducing the sizes the weights of optical systems. It is said that one aspherical lens can be as good as 3 4 spherical lenses. One rather common reason for employing an aspherical lens is to eliminate spherical aberration. As schematically shown in Figure 1, a rotationally symmetric lens profile can be described well in cylindrical coordinates with the optical axis as the z-axis the coordinate origin coinciding with the vertex. Then a rotationally symmetric surface profile can be parameterized as z = z(ρ), where ρ x 2 + y 2 is the axial radius measured perpendicular to the z-axis z is the distance measured along the optical axis. The coordinate (ρ, z ) of an arbitrary point S on a spherical surface with a radius R is given as ρ 2 R z (ρ) = ρ2 R 2. (1) On the other h, the coordinate (ρ, z) of a point P on an even aspherical surface can be given as z(ρ) = ρ 2 R (1 + k) ρ2 R 2 C i ρ 2+2i, (2) i=1 where R is the vertex radius, k is the conic constant, C i is the (2 + 2i) th aspherical deformation constants [1, 2]. An even aspherical lens profile is given as a sum of a conic surface profile perturbation terms described by even-powered polynomials in ρ. Aspherical lenses are usually designed using dedicated lens design software, such as Code V Zemax. In this method, a trial lens profile is taken has the form of an even-aspheric lens formula given in Eq. (2). Then, a merit function is set-up for quantitatively measuring the quality of the lens. For example, to design kweon@honam.ac.kr; Tel: ; Fax: Fig. 1. Schematic diagram illustrating the difference between a spherical an aspherical surface profile.

2 -94- Journal of the Korean Physical Society, Vol. 51, No. 1, July 27 a spherical-aberration-free lens, the spherical aberration can be taken as the merit function, a smaller value of the merit function means a lens with a better optical quality (i.e., with a lesser spherical aberration). By systematically investigating the effects of various terms, a best combination of coefficients, hence, a best-form lens is obtained. Sometimes, it is necessary to find the exact profile of an aspherical refractive surface having certain characteristics. However, the previous method of lens design based on a multidimensional optimization method cannot mathematically describe the exact shape of an aspherical refractive surface having the desired characteristics, provides an approximate solution that strongly depends on the type of aspherical lens formula, the number of expansion terms, the structure of the merit function, the initial values of the expansion coefficients. Further, when an inappropriate aspherical lens formula is employed, even an approximate solution may not be obtained, even if an approximate solution is obtained, it may be difficult to calculate the error with respect to the mathematically exact solution. For the particular example of spherical-aberration-free refractive surface, the surface profile is either a hyperbola or an ellipse, depending on which of the two refractive indices on both sides of the refractive surface is larger. These curves are special examples of Cartesian ovals have been studied by Descartes Huygens [3 1]. The basic idea is to find a surface that is a collection of points where the sum of the optical path length (OPL) from a source point to a point on the lens surface the OPL from the same point on the lens surface to an image point are identical. However, the necessary mathematics is enervating, this approach cannot be easily generalized to more complex aspherical surfaces having the desirable characteristics. The purpose of present article is to provide a semi-analytical formula for spherical-aberrationfree aspherical lenses that can be readily generalized. II. SPHERICAL-ABERRATION-FREE PLANAR-CONVEX ASPHERIC LENS Aspherical mirror profiles observing the law of specular reflections have been designed using numerical analysis [11]. While the law of specular reflection is observed in reflective lenses, Snell s law of refraction is observed in refractive lenses. Despite this difference, an aspherical refractive surface profile can be similarly obtained using numerical analysis. Figure 2 is a schematic diagram of a spherical-aberration-free planar-convex lens profile. The first surface on the object side is assumed to be a plane surface without any refractive power; therefore, an incident ray having zero field angle (i.e., parallel to the optical axis) is not refracted when entering the first surface. This incident ray is refracted at a point P on the second surface passes through the secondary focal point O Fig. 2. Schematic diagram of a spherical-aberration-free plano-convex lens. of the lens. To analyze the lens profile, a coordinate system different from the one shown in Figure 1 is taken. Although the z-axis coincides with the optical axis, the origin of the coordinate coincides with the secondary focal point of the lens not with the lens vertex. Then, the refractive surface profile can be written as a set of two curvilinear coordinates (r, ) in spherical polar coordinates. In this notation, r is the radial distance from the origin to the point P on the lens surface, is the zenith angle. The two variables in spherical polar coordinates are related to the two variables (ρ, z) in the cylindrical coordinates by z() = r() cos ρ() = r() sin. The tangent plane T to the refractive surface at the point P subtends an angle φ with the x y plane. The angle φ is related to the surface profile by a simple relation given as tan φ = dz dρ. (3) Provided the angle φ is given as a sole function of the zenith angle, the surface profile can be obtained in the form of an indefinite integral given as [ ] sin + tan φ( ) cos r() = r o exp cos tan φ( ) sin d, (4) where r o r() is the distance between the refractive surface the secondary focal point. In other words, r o is the back focal length of the lens. An incident ray is refracted at the second refractive surface according to Snell s law of refraction. The refractive indices of the lens the surroundings are n 1, respectively. As is illustrated in Figure 2, the surface normal N at the point P subtends an angle φ with the optical axis. On the other h, the refracted ray has a zenith angle, the angle between the surface normal the refracted ray is (φ + ). Therefore, Snell s raw of refraction is given as n 1 sin φ = sin( + φ). (5)

3 Aspherical Lens Design by Using a Numerical Analysis Gyeong-Il Kweon Cheol-Ho Kim -95- Rearranging Eq. plane is given as tan φ = (5), the inclination of the tangent sin n 1 cos. (6) Using Eq. (6), the numerator of the integr in Eq. (4) is given as sin + tan φ cos = n 1 sin n 1 cos while the denominator is given as (7) cos tan φ sin = n 1 cos n 1 cos. (8) Therefore, Eq. (4) is reduced to a simple form given as [ ] n 1 sin r() = r o exp n 1 cos d. (9) Except for a sign, the numerator of the integr in Eq. (9) is the exact differential of the denominator; thus, the integral can be readily done as r() = r o n 1 n 1 cos. (1) III. ASPHERIC SURFACE AS A CONIC SURFACE The surface described by Eq. (1) is a conic surface. To see that point, the following substitutions of variables are made: e = n 1 (11) l = r o (e 1). (12) Then, Eq. (1) is reduced to l r() 1 e cos. (13) This is the equation of a conic surface in polar coordinates. When n 1 is larger than, this is the equation of a hyperbolic surface, which in cylindrical coordinates is given as z 2 a 2 ρ2 b 2 = 1. (14) The two constants a b are given as a = l e 2 1 = r o (15) 1 + n1 b = al = r o n1 n 1 +. (16) Fig. 3. Optical path length for a lens with a hyperbolic refractive surface. On the other h, when n 1 is smaller than, this is the equation of an elliptical surface, which in cylindrical coordinate is given as z 2 a 2 + ρ2 b 2 = 1. (17) The two constants a b are given by a = l 1 e 2 = r o 1 + n1 (18) b = al = r o n2 n 1 + n 1. (19) IV. OPTICAL PATH LENGTH In order to function as a spherical-aberration-free lens, the optical path length must be identical for all legitimate rays reaching the secondary focal point of the lens. Figure 3 shows the lens profile schematically shown in Figure 2 in the reverse direction, the coordinate origin is at the middle of the two focal points of the conjugate hyperbola. Assuming n 1 is larger than, the surface profile is given as the hyperbolic surface described by Eq. (1). The coordinates of the first the second focal points are given as F = (,, ae) F = (,, ae). Then, referring to Figure 3, one can see that the back focal length of the lens is given as r o = a(1 + e), the optical path length from the second focal point F to a plane at z = z 2 is given as OP L = r + n 1 (z 2 z). (2) The z-coordinate of the point P is given as z = r cos ae. (21)

4 -96- Journal of the Korean Physical Society, Vol. 51, No. 1, July 27 Fig. 5. Ray aberration plot. Fig. 4. Example of a spherical-aberration-free planoconvex lens with a hyperbolic second surface. Employing Eqs. (2) to (1) (21), one can reduce Eq. OP L = n 1 (z 2 a) + a(1 + e). (22) From Eq. (22), it is clear that the optical path length is independent of the zenith angle of the rays. Therefore, the Cartesian oval can be obtained from Snell s law, as well as from Fermat s principle. Similar arguments for an ellipse can be derived easily. V. NUMERICAL EXAMPLE OF A SPHERICAL-ABERRATION-FREE PLANO-CONVEX LENS If a hyperbolic surface is to be employed in the design of an optical lens, the surface parameters must be given in a form appropriate for a description of the aspherical lens. A conic surface is described by Eq. (2) without the deformation terms. Therefore, the vertex radius R the conic constant k for a hyperbolic surface must be determined. The origin of the coordinate system for Eq. (2) lies at the lens vertex. Therefore, the corresponding hyperbolic surface profile in a cylindrical coordinate system with a shifted origin is given as (z + a) 2 a 2 ρ2 b 2 = 1. (23) Taking the positive solution of the quadratic equation given in Eq. (23) by rearranging terms, we can obtain an equation given as z = 1 + a b ρ 2 2. (24) 1 + ρ2 b 2 Comparing with Eq. (2), we can identify the vertex radius the conic constant as R = b2 a (25) Fig. 6. Spot diagram. k = 1 b2 a 2. (26) With Eqs. (15) (16), the two constants are given in terms of more tangible variables as R = n 1 r o (27) k = ( n1 ) 2. (28) Figure 4 shows an example of a spherical-aberrationfree plano-convex lens. The lens is assumed to be made of BK7 glass. The refractive index of BK7 glass at the sodium d-line (587.6 nm) is , the refractive index of air is taken as 1.. If a back focal length r o = 5. mm is assumed, the vertex radius R is mm, the conic constant k is given as The field angles in the optical layout in Figure 4 are 1, the image space F-number is 1.. As we can see from the figure, incident parallel rays all converge to the secondary focal point. The back focal length the effective focal length are 5. mm. Figure 5 shows the ray aberration plot shows a lot of coma for an offaxis beam. The same conclusion can be drawn from the spot diagram in Figure 6. VI. OTHER SPHERICAL-ABERRATION-FREE LENSES WITH A SINGLE REFRACTIVE SURFACE

5 Aspherical Lens Design by Using a Numerical Analysis Gyeong-Il Kweon Cheol-Ho Kim -97- Fig. 7. Schematic diagram of a spherical-aberration-free positive meniscus lens. There are three more spherical-aberration-free lenses with a single refractive surface. Shown in Figure 7 is a spherical-aberration-free aspherical lens in the form of a positive meniscus lens. In this example, the first surface is an aspherical surface, the second surface is a spherical surface with a radius r B. For the second surface not to refract rays, the back focal length of the lens should be identical to the radius of the second surface. Snell s law of refraction at the point P is given as n 1 sin φ = sin(φ ), (29) where all the variables are similarly defined as those of Figure 2. After a corresponding process of derivation, the surface profile is identical to Eq. (9). The only difference is that in the previous example, n 1 is larger than while in this example, is larger than n 1. Therefore, the same formula can be adopted to design a sphericalaberration-free plano-convex lens or a positive meniscus lens. The shape of the lens is determined by which of the two refractive indices, n 1 or, is larger. Figure 8 schematically shows a cross section of a spherical-aberration-free negative meniscus lens that converts a converging beam into a parallel beam a parallel beam into a diverging beam. The first lens surface is part of a spherical surface having a radius r F around the origin O. Snell s law of refraction for this lens is given as n 1 sin(φ ) = sin φ, (3) where all the variables are similarly defined as those of Figure 2. After a corresponding process of derivation, the surface profile is given as Eq. (31): [ ] sin r() = r o exp cos d. (31) n 1 In Eq. (31), n 1 are the refractive indices of the media located on the left the right sides of the Fig. 8. Schematic diagram of a spherical-aberration-free negative meniscus lens. Fig. 9. Schematic diagram of a spherical-aberration-free concave-planar lens. aspherical refractive surface, respectively. Both of the refractive indices n 1 can take any real number larger than 1. In this regard, the two equations, Eqs. (9) (31), describe the same curve if the refractive indices n 1 in this section are identical to the refractive indices n 1 in the previous section, respectively. Therefore, Eqs. (9) (31) are substantially identical to each other. Figure 9 schematically shows a cross section of an aspherical lens that converts converging rays into parallel rays parallel rays into divergent rays. Snell s law of refraction for this lens is given as n 1 sin(φ + ) = sin φ. (32) After a corresponding process of derivation, the surface profile is identical to Eq. (31). Therefore, from the

6 -98- Journal of the Korean Physical Society, Vol. 51, No. 1, July 27 Fig. 11. Example of an aspherical beam exper composed of two elliptical surfaces. Fig. 1. Schematic diagram of an aspheric beam exper. same equation, either the profile of the aspherical refractive surface shown in Figure 8 or Figure 9 can be obtained, depending on which of the two refractive indices, n 1 or, is larger. VII. ASPHERICAL BEAM EXPANDER In many cases, the beam sizes of collimated beams, such as those emitted from lasers, must be enlarged or reduced. A simple beam exper takes the form of a Galilean telescope composed of one concave lens one convex lens. The focal length of the convex lens is longer than that of the concave lens, the second focal points of the two lenses coincide. Besides this, there exist many kinds of complex beam expers that might include a pair of prisms or a plurality of lenses. As schematically shown in Figure 1, an excellent beam exper can be configured with two aspherical lenses, specifically, as a compound lens including the two lens elements from Figures 7 8. The first lens element is composed of a first aspherical refractive surface functioning as the first lens surface a spherical surface having a radius R B functioning as the second lens surface. The second lens element is composed of a spherical surface having a radius r F functioning as a third lens surface a second aspherical refractive surface functioning as a fourth lens surface. The radius r F of the third lens surface is not larger than the radius R B of the second lens surface. The second lens surface the third lens surface have a common center. A medium with a refractive index n 1 exists at the object side, namely, at the left side of the first aspherical refractive surface. The first lens has a refractive index. A medium with a refractive index n 3 fills the space between the first the second lenses. The second lens has a refractive index n 4. A medium with a refractive index n 5 exists at the image side, namely, at the right side of the second aspherical refractive surface. The shape of Fig. 12. Ray trajectories for the plano-convex lens shown in Figure 4 from the opposite direction. the first lens is identical to that shown in Figure 7, the shape of the second lens is identical to that shown in Figure 8. The first the second lenses share a common optical axis. The second focal points of the first the second lenses, as well as the centers of the second the third lens surfaces, all coincide. In order to use such a compound lens as a beam exper, the refractive index of the first lens should be larger than the refractive index n 1 of the medium at the object side, the refractive index n 4 of the second lens should be larger than the refractive index n 5 of the medium at the image side; that is, > n 1 n 4 > n 5. Furthermore, when = n 4 n 1 = n 5, then the two aspherical surfaces have identical shapes, but on a different scale. Since two spherical surfaces are not essential, a single-piece beam exper can be realized. Figure 11 shows the lens profiles the ray trajectories for a single-piece beam exper made of BK7 glass. Referring to the schematic diagram in Figure 1, the two parameters r o R o are given as 5. mm 5. mm, respectively. The vertex radius R the conic constant k of an elliptical surface are given by the same formulae given in Eqs. (27) (28). The difference is n 1 is smaller than for the elliptical surface. For the single-piece beam exper shown in Figure 11, the conic constants of both surfaces are given as , the vertex radius of the first the second surfaces

7 Aspherical Lens Design by Using a Numerical Analysis Gyeong-Il Kweon Cheol-Ho Kim -99- Fig. 13. Schematic diagram of a spherical-aberration-free convex-planar lens. are given as 1.74 mm mm, respectively. From the figure, it can be seen that this lens functions well as a beam exper that the alignment tolerance is reasonable. VIII. SPHERICAL-ABERRATION-FREE CONVEX-PLANAR ASPHERIC LENS Figure 12 shows the ray trajectories for the lens shown in Figure 4 when the lens is flipped around so that the first the second surfaces are exchanged with each other. As is clear from the figure, incident parallel rays are not focused to the secondary focal point of the lens. For a practical reason, however, it may be desirable to have an aspherical lens that is spherical-aberration-free when the aspherical surfaces are on the object side of the lens, because other lens characteristics, such as coma, can also be improved. All the analyses up to here were about conic surfaces that have been studied by great philosophers such as Descartes. However, the surface profile of a sphericalaberration-free convex-planar lens deviates from a Cartesian oval. Figure 13 shows a schematic diagram of an aspherical lens, which is spherical-aberration free when the aspherical refractive surface faces the object side on the left. A ray incident on the aspherical lens parallel to the optical axis is refracted at a point Q on the aspherical refractive surface, the resultant refracted ray propagates toward a point P on the second lens surface. The refracted ray is refracted again at a point P on the second lens surface propagates toward the second focal point O of the lens. Hereinafter, the refracted ray before being refracted at the point P on the second lens surface is referred to as the first refracted ray, the ray refracted at the point P is referred to as the second refracted ray. The refractive indices of the media on the object the image sides are n o, respectively, the refractive index of the lens is n 1. n o is assumed to be different from, in general. This can be compared to the case of an aquarium with one wall having the shape of the aspherical refractive surface, the inner the outer sides of the aquarium being filled with different media, such as the water the air, respectively. The angle between the incident ray the normal N perpendicular to the tangent plane T at the point Q on the aspherical refractive surface is φ. On the other h, the angle between the first refracted ray the optical axis is δ, the angle between the second refracted ray the optical axis is. Applying Snell s law of refraction at points P Q results in Eqs. (33) (34), respectively: n 1 sin δ = sin (33) n o sin φ = n 1 sin(φ δ). (34) The cylindrical coordinates of the point Q on the aspherical refractive surface are designated as (X, Z), the corresponding coordinates of the point P on the second lens surface as (ρ, z). The back focal length of this lens is f B. Since the second lens surface is a plane, the cylindrical coordinates (ρ, z) of the point P are given as z() = f B (35) ρ() = f B tan. (36) On the other h, the distance from the point P on the second lens surface to the point Q on the aspherical refractive surface (the first lens surface) is designated as L(). If the distance L between the two points Q P is expressed as a function of the zenith angle of the second refracted ray, then the cylindrical coordinates (X, Z) of the point Q on the aspherical refractive surface are given as X() = f B tan + L() sin δ() (37) Z() = f B + L() cos δ(). (38) The following equation can be obtained by differentiating Eq. (37) with respect to : dx d = f B cos 2 + dl sin δ + L cos δ d d. (39) Similarly, the following equation can be obtained by differentiating Eq. (38) with respect to : dz d = dl cos δ L sin δ d d. (4)

8 -1- Journal of the Korean Physical Society, Vol. 51, No. 1, July 27 Using trigonometrical functional relations, the slope tan φ of the tangent plane T at the point Q on the aspherical refractive surface is given as tan φ = dz dx. (41) Since, both the coordinates X Z are functions of the zenith angle, Eq. (41) can be expressed as dz d = tan φdx d. (42) With Eqs. (39) (4), Eq. (42) can be expressed as ( ) dl sin δ tan φ cos δ d L cos δ + tan φ sin δ d = f B cos 2 ( tan φ cos δ + tan φ sin δ ). (43) Before going further to obtain a solution, the functions defined in Eqs. (44) (45) can be used to make the expression in Eq. (43) simpler: ( ) sin δ tan φ cos δ A() cos δ + tan φ sin δ d, (44) B() f ( ) B tan φ cos 2, (45) cos δ + tan φ sin δ dl d + A()L() = B(). (46) Multiplying both sides of Eq. (46) with an unknown function F(), the following relation can be obtained: F ()dl + F ()A()L()d = F ()B()d. (47) The condition for the left side of Eq. (47) to be an exact differential is given as df d = A()F (). (48) Therefore, the unknown function F() must be given as a function of A() as given in [ ] F () = exp A( )d. (49) With Eq. (49), Eq. (47) can be readily integrated to yield [ L() exp A( )d ] L o = Therefore, the function L() is given as L() = 1 F () L o + F ( )B( )d F ( )B( )d.(5) (51) The following equation can be obtained by differentiating Eq. (33) with respect to : d = cos n 1 cos δ. (52) On the other h, the following equation can be obtained by rearranging Eq. (34): tan φ = n 1 sin δ n 1 cos δ n o. (53) With Eq. (53), the numerator of the A() given in Eq. (44) is given as sin δ tan φ cos δ = n o sin δ n 1 cos δ n o. (54) On the other h, the denominator of A() is given as cos δ + tan φ sin δ = n 1 n o cos δ n 1 cos δ n o. (55) With Eqs. (52), (54) (55), the function A() is given as A() = n o sin δ cos n 1 n o cos δ n 1 cos δ. (56) Using Eq. (33), the angle δ can be given as a function of the zenith angle as δ() = tan 1 sin. (57) 1 n2 2 sin2 Therefore, the function A() can be expressed as a sole function of the zenith angle : A() = n o 1 n2 2 sin2 2 sin cos.(58) 1 n o 1 n2 2 sin2 On the other h, the function B() is given as B() = f B n 1 sin δ cos 2 n 1 n o cos δ. (59) The function B() given in Eq. (59) can be expressed also as a sole function of the zenith angle as B() = f B n 1 sin cos 2. (6) 1 n o 1 n2 2 sin2 Accordingly, with Eqs. (49), (58), (6), L() given in Eq. (51) can be obtained. By using the function L() Eqs. (35)-(38), the profile of the aspherical refractive surface can be obtained. Since it involves an evaluation of an integration with one variable, the lens profile can be obtained using primitive techniques of numerical analysis, such as the trapezoidal sum rule.

9 Aspherical Lens Design by Using a Numerical Analysis Gyeong-Il Kweon Cheol-Ho Kim -11- Fig. 14. Surface profile of an exemplary sphericalaberration-free convex-planar lens. Example of a spherical-aberration-free convex- Fig. 16. planar lens. Fig. 17. Ray aberration plot. Fig. 15. Fitting error of the aspherical surface profile shown in Figure 14 for the even aspherical lens formula. Table 1. Fitting coefficients of the surface profile to the even aspheric lens formula. Variable Value ρ max R k e 1 C e 7 C e 11 C e 15 C e 18 C e 21 C e 25 portant design parameter. The lens thickness, as well as the back focal length, is taken as 5. mm. The maximum ray angle for the second refracted ray is taken as 45.. In order to model with a lens design program, we fitted the surface profile to the even aspherical lens formula given in Eq. (2) the fitted parameters are summarized in Table 1. Figure 15 shows the fitting error between the aspherical surface profile the best-fit even aspherical lens surface. As can be seen, the fitting error is less than 1 µm over the entire range of axial radius X. Figure 16 shows the lens profile the ray trajectories modeled in Zemax. The field angles in the optical layout are 1, the image space F-number is 1.. As can be seen from the figure, the incident parallel rays all converge to the secondary focal point. Figure show a ray aberration plot a spot diagram, respectively. The graphs show some spherical aberration coma for the off-axis beam. The presence of remnant spherical aberration is thought to be due to the numerical integration error the fitting error. The coma in this example is much less than the coma in the example shown in Figure 4. IX. NUMERICAL EXAMPLE OF A SPHERICAL-ABERRATION-FREE CONVEX-PLANAR LENS Figure 14 shows the aspherical surface profile of a spherical-aberration-free convex-planar BK7 lens corresponding to the plano-convex lens shown in Figure 4. Note that for this lens, the lens thickness L o is an im- X. SPHERICAL-ABERRATION-FREE PLANAR-CONCAVE ASPHERIC LENS Figure 19 shows a schematic diagram of an aspherical lens with improved aberration characteristics. The aspherical lens is comprised of a plane first lens surface an aspherical second lens surface. The variables in

10 -12- Journal of the Korean Physical Society, Vol. 51, No. 1, July 27 Fig. 18. Spot diagram. Figure 19 have corresponding variables in Figure 13, section VIII can be referred to for the meanings. Applying Snell s law for the refraction at points P Q results in n o sin = n 1 sin δ (61) n 1 sin(φ δ) = sin φ. (62) Since the first lens surface is a plane surface, the height z measured along the optical axis is given as a constant: z() = z() z o. (63) On the other h, the coordinate ρ is a function of the zenith angle is given as ρ() = z o tan. (64) The distance from the point P on the first lens surface to the point Q on the aspherical refractive surface is given as L(). If the distance L between the two points is expressed as a function of the zenith angle of the incident ray, then the cylindrical coordinates (X, Z) of the point Q on the aspherical refractive surface are given as X() = z o tan L() sin δ() (65) Z() = z o L() cos δ(). (66) The following equation can be obtained by differentiating Eq. (65) with respect to : dx d = z o cos 2 dl sin δ L cos δ d d. (67) Again, the following equation can be obtained by differentiating Eq. (66) with respect to : dz d = dl cos δ + L sin δ d d. (68) The slope of the tangent plane T at the point Q on the second lens surface is given as tan φ = dz dx. (69) Fig. 19. Schematic diagram of a spherical-aberration-free planar-concave lens. Since both the coordinates X Z are functions of the zenith angle, Eq. (69) can be expressed as dz d = tan φdx d. (7) With Eqs. (67) (68), Eq. (7) can be expressed as dl() d + A()L() = B(), (71) where Eq. (71) has been expressed in a simpler form using the functions defined in Eqs. (72) (73): ( ) sin δ tan φ cos δ A() (72) cos δ + tan φ sin δ d B() z o ( tan φ ) cos 2 cos δ + tan φ sin δ. (73) Following a similar derivational procedures shown in the previous section, the function L() can be given as L() = 1 F () L o + F ( )B( )d. (74) Here, the function F() is a function of A() is given as [ ] F () = exp A( )d. (75)

11 Aspherical Lens Design by Using a Numerical Analysis Gyeong-Il Kweon Cheol-Ho Kim -13- The following equation is obtained by differentiating Eq. (61) with respect to : d = n o cos n 1 cos δ. (76) The following equation can be obtained by rearranging Eq. (62): tan φ = n 1 sin δ n 1 cos δ. (77) With Eq. (77), the numerator of the function A() defined in Eq. (72) can be given as sin δ tan φ cos δ = sin δ n 1 cos δ. (78) Similarly, the denominator of the function A() can be given as cos δ + tan φ sin δ = n 1 cos δ n 1 cos δ. (79) With Eqs. (76), (78), (79), the function A() is given as A() = sin δ n o cos n 1 cos δ n 1 cos δ. (8) With Eq. (61), the angle δ can be given as a sole function of the zenith angle as δ() = tan 1 n o sin. (81) 1 n2 o si Resultantly, the function A() can be given as a sole function of the zenith angle : A() = 1 n2 o si o sin cos.(82) 1 1 n2 o si On the other h, the function B() is given as B() = z o n 1 sin δ cos 2 n 1 cos δ. (83) Eq. (83) can also be given as a sole function of the zenith angle as B() = z o cos 2 n 1 n o sin. (84) 1 1 n2 o si Accordingly, using Eqs. (75), (82) (84), the function L() given in Eq. (74) can be obtained. Further, by using the function L() Eqs. (63)-(66) (81), the profile of the aspherical refractive surface can be obtained. XI. CONCLUSIONS In this article, analytic semi-analytic equations describing various spherical-aberration-free aspherical lenses have been obtained from Snell s law of refraction, numerical examples have been given, as well. This formalism can be generalized to design more challenging aspherical lenses with desirable characteristics. REFERENCES [1] M. E. Harrigan, Proc. SPIE. 554, 112 (1985). [2] A. W. Greynolds, Proc. SPIE. 4832, 1 (22). [3] H. W. Farwell, Amer. J. Phys. 9, 255 (1941). [4] H. W. Farwell, Amer. J. Phys. 19, 454 (1951). [5] B. Jurek, Czech. J. Phys. 1, 197 (1952). [6] L. Mertz, Appl. Opt. 18, 4182 (1979). [7] N. Cap, B. Ruiz H. Rabal, Optik 114, 89 (23). [8] N. Alamo C. Criado, Inverse Problems 2, 229 (24). [9] O. N. Stavroudis, The Optics of Rays, Wavefronts Caustics (Academic, New York, 1972), Chap. 6. [1] A. Walther, The Ray Wave Theory of Lenses (Cambridge, New York, 1995), Chap. 2. [11] G. Kweon, K. Kim, G. Kim H. Kim, Appl. Opt. 44, 2759 (25).