Aspherical Lens Design by Using a Numerical Analysis
Gyeong-Il Kweon∗Department of Optoelectronics, Honam University, Gwangju 506-714
Cheol-Ho Kim
Department of Information and Communication Engineering, Honam University, Gwangju 506-714
(Received 15 March 2007)
The surface profiles of four different spherical-aberration-free aspherical lenses are given as conic curves, namely a hyperbola and 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 and mathematically elegant. The surfaces profiles of two more spherical-aberration-free aspherical lenses, which cannot be repre-sented by conic curves, are given in terms of a semi-analytic formula. Examples of the lens profiles are given.
PACS numbers: 42.15.E, 42.15.F
Keywords: Lens design, Optical aberration, Geometrical optics, Spherical aberration, Numerical analysis
I. INTRODUCTION
Aspherical lenses are essential ingredients in reducing the sizes and the weights of optical systems. It is said that one aspherical lens can be as good as 3 ∼4 spher-ical lenses. One rather common reason for employing an aspherical lens is to eliminate spherical aberration. As schematically shown in Figure 1, a rotationally sym-metric lens profile can be described well in cylindrical coordinates with the optical axis as the z-axis and the coordinate origin coinciding with the vertex. Then a rotationally symmetric surface profile can be parameter-ized asz=z(ρ), whereρ≡px2+y2 is the axial radius measured perpendicular to the z-axis and z is the dis-tance measured along the optical axis. The coordinate (ρ,z0) of an arbitrary point S on a spherical surface with a radius Ris given as z0(ρ) = ρ2 R 1 + q 1− ρ2 R2 . (1)
On the other hand, the coordinate (ρ,z) of a point P on an even aspherical surface can be given as
z(ρ) = ρ2 R 1 + q 1−(1 +k)Rρ22 +X i=1 Ciρ2+2i, (2)
whereRis the vertex radius,kis the conic constant, and
Ci is the (2 + 2i)thaspherical deformation constants [1,
∗E-mail: [email protected];
Tel: +82-62-940-5540; Fax: +82-62-940-5005
2]. An even aspherical lens profile is given as a sum of a conic surface profile and perturbation terms described by even-powered polynomials inρ.
Aspherical lenses are usually designed using dedicated lens design software, such as Code V and Zemax. In this method, a trial lens profile is taken and has the form of an even-aspheric lens formula given in Eq. (2). Then, a merit function is set-up for quantitatively mea-suring the quality of the lens. For example, to design
Fig. 1. Schematic diagram illustrating the difference be-tween a spherical and an aspherical surface profile.
-93-a spheric-93-al--93-aberr-93-ation-free lens, the spheric-93-al -93-aberr-93-ation can be taken as the merit function, and a smaller value of the merit function means a lens with a better optical quality (i.e., with a lesser spherical aberration). By sys-tematically investigating the effects of various terms, a best combination of coefficients and, hence, a best-form lens is obtained.
Sometimes, it is necessary to find the exact profile of an aspherical refractive surface having certain charac-teristics. However, the previous method of lens design based on a multidimensional optimization method can-not mathematically describe the exact shape of an as-pherical refractive surface having the desired characteris-tics, and provides an approximate solution that strongly depends on the type of aspherical lens formula, the num-ber of expansion terms, the structure of the merit func-tion, and the initial values of the expansion coefficients. Further, when an inappropriate aspherical lens formula is employed, even an approximate solution may not be ob-tained, and even if an approximate solution is obob-tained, 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 and have been studied by Descartes and Huygens [3–10]. 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 and the OPL from the same point on the lens surface to an image point are identical. However, the necessary mathematics is en-ervating, and this approach cannot be easily generalized to more complex aspherical surfaces having the desirable characteristics. The purpose of present article is to pro-vide a semi-analytical formula for spherical-aberration-free 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, Snell0s 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 inci-dent ray having zero field angle (i.e., parallel to the opti-cal axis) is not refracted when entering the first surface. This incident ray is refracted at a point P on the second surface and 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 sys-tem 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 fo-cal point of the lens and not with the lens vertex. Then, the refractive surface profile can be written as a set of two curvilinear coordinates (r, θ) in spherical polar co-ordinates. In this notation,ris the radial distance from the origin to the point P on the lens surface, and θ 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(θ) = r(θ) cosθ and ρ(θ) =
r(θ) sinθ. The tangent plane T to the refractive surface at the point P subtends an angleφwith thex−y plane. The angleφis related to the surface profile by a simple relation given as
tanφ=dz
dρ . (3)
Provided the angleφis given as a sole function of the zenith angleθ, the surface profile can be obtained in the form of an indefinite integral given as
r(θ) =roexp
"
Z θ
0
sinθ0+ tanφ(θ0) cosθ0
cosθ0−tanφ(θ0) sinθ0dθ 0
#
, (4)
where ro ≡ r(0) is the distance between the refractive surface and the secondary focal point. In other words,
ro 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 re-fractive indices of the lens and the surroundings are n1 and n2, 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 hand, the refracted ray has a zenith angleθ, and the angle between the sur-face normal and the refracted ray is (φ+θ). Therefore, Snell’s raw of refraction is given as
Rearranging Eq. (5), the inclination of the tangent plane is given as
tanφ= n2sinθ
n1−n2cosθ
. (6)
Using Eq. (6), the numerator of the integrand in Eq. (4) is given as
sinθ+ tanφcosθ= n1sinθ
n1−n2cosθ
(7) while the denominator is given as
cosθ−tanφsinθ=n1cosθ−n2
n1−n2cosθ
. (8)
Therefore, Eq. (4) is reduced to a simple form given as r(θ) =roexp " Z θ 0 n1sinθ0 n1cosθ0−n2 dθ0 # . (9)
Except for a sign, the numerator of the integrand in Eq. (9) is the exact differential of the denominator; thus, the integral can be readily done as
r(θ) =ro
n1−n2
n1cosθ−n2
. (10)
III. ASPHERIC SURFACE AS A CONIC SURFACE
The surface described by Eq. (10) is a conic surface. To see that point, the following substitutions of variables are made: e=n1 n2 (11) and l=ro(e−1). (12) Then, Eq. (10) is reduced to
r(θ) −l
1−ecosθ . (13)
This is the equation of a conic surface in polar coor-dinates. When n1is larger thann2, this is the equation of a hyperbolic surface, which in cylindrical coordinates is given as
z2
a2 −
ρ2
b2 = 1 . (14)
The two constants aandb are given as
a= l e2−1 = ro 1 + n1 n2 (15) and b=√al=ro r n1−n2 n1+n2 . (16)
Fig. 3. Optical path length for a lens with a hyperbolic refractive surface.
On the other hand, whenn1is smaller thann2, this is the equation of an elliptical surface, which in cylindrical coordinate is given as
z2
a2 +
ρ2
b2 = 1. (17)
The two constantsaandb are given by
a= l 1−e2 = ro 1 +n1 n2 (18) and b=√al=ro r n2−n1 n2+n1 . (19)
IV. OPTICAL PATH LENGTH
In order to function as a spherical-aberration-free lens, the optical path length must be identical for all legiti-mate rays reaching the secondary focal point of the lens. Figure 3 shows the lens profile schematically shown in Figure 2 in the reverse direction, and the coordinate ori-gin is at the middle of the two focal points of the conju-gate hyperbola. Assumingn1 is larger thann2, the sur-face profile is given as the hyperbolic sursur-face described by Eq. (10). The coordinates of the first and the second focal points are given as F = (0, 0, ae) and F0 = (0, 0, −ae). Then, referring to Figure 3, one can see that the back focal length of the lens is given as ro = a(1 +e), and the optical path length from the second focal point F0 to a plane atz =z2 is given as
OP L=n2r+n1(z2−z). (20) Thez-coordinate of the point P is given as
Fig. 4. Example of a spherical-aberration-free plano-convex lens with a hyperbolic second surface.
Employing Eqs. (10) and (21), one can reduce Eq. (20) to
OP L=n1(z2−a) +n2a(1 +e). (22) From Eq. (22), it is clear that the optical path length is independent of the zenith angleθof 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
and 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
a2 −
ρ2
b2 = 1. (23)
Taking the positive solution of the quadratic equation given in Eq. (23) and by rearranging terms, we can ob-tain an equation given as
z= a b2ρ 2 1 + q 1 +ρb22 . (24)
Comparing with Eq. (2), we can identify the vertex radius and the conic constant as
R= b 2
a (25)
Fig. 5. Ray aberration plot.
Fig. 6. Spot diagram.
and
k=−1−b 2
a2 . (26)
With Eqs. (15) and (16), the two constants are given in terms of more tangible variables as
R= n1−n2 n2 ro (27) and k=− n1 n2 2 . (28)
Figure 4 shows an example of a spherical-aberration-free 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 1.51680, and the refractive index of air is taken as 1.0. If a back focal lengthro = 50.0 mm is assumed, the vertex radius R is 25.84 mm, and the conic constantkis given as –2.300682. The field angles in the optical layout in Figure 4 are 0◦ and 1◦, and the image space F-number is 1.0. As we can see from the figure, incident parallel rays all converge to the secondary focal point. The back focal length and the effective focal length are 50.0 mm. Figure 5 shows the ray aberration plot and shows a lot of coma for an off-axis beam. The same conclusion can be drawn from the spot diagram in Figure 6.
VI. OTHER
SPHERICAL-ABERRATION-FREE LENSES WITH A SINGLE REFRACTIVE
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 sur-face is an aspherical sursur-face, and the second sursur-face is a spherical surface with a radius rB. For the second sur-face 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
n1sinφ=n2sin(φ−θ), (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, n1 is larger than n2 while in this example, n2 is larger than n1. Therefore, the same formula can be adopted to design a spherical-aberration-free plano-convex lens or a positive meniscus lens. The shape of the lens is determined by which of the two refractive indices, n1or n2, 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 and a parallel beam into a diverging beam. The first lens surface is part of a spherical surface having a radiusrF around the origin O. Snell’s law of refraction for this lens is given as
n1sin(φ−θ) =n2sinφ , (30) 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):
r(θ) =roexp " Z θ 0 n2sinθ0 n2cosθ0−n1 dθ0 # . (31)
In Eq. (31), n1 and n2 are the refractive indices of the media located on the left and 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 n1 and n2 can take any real number larger than 1. In this regard, the two equations, Eqs. (9) and (31), describe the same curve if the refractive indices
n1 and n2 in this section are identical to the refractive indices n2 and n1 in the previous section, respectively. Therefore, Eqs. (9) and (31) are substantially identical to each other.
Figure 9 schematically shows a cross section of an as-pherical lens that converts converging rays into parallel rays and parallel rays into divergent rays. Snell’s law of refraction for this lens is given as
n1sin(φ+θ) =n2sinφ . (32) After a corresponding process of derivation, the sur-face profile is identical to Eq. (31). Therefore, from the
Fig. 10. Schematic diagram of an aspheric beam expander. same equation, either the profile of the aspherical refrac-tive surface shown in Figure 8 or Figure 9 can be ob-tained, depending on which of the two refractive indices,
n1 orn2, 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 expander takes the form of a Galilean telescope composed of one concave lens and one convex lens. The focal length of the convex lens is longer than that of the concave lens, and the second focal points of the two lenses coincide. Besides this, there exist many kinds of complex beam expanders that might include a pair of prisms or a plurality of lenses.
As schematically shown in Figure 10, an excellent beam expander can be configured with two aspherical lenses, specifically, as a compound lens including the two lens elements from Figures 7 and 8. The first lens ele-ment is composed of a first aspherical refractive surface functioning as the first lens surface and a spherical sur-face having a radius RB functioning as the second lens surface. The second lens element is composed of a spher-ical surface having a radiusrF functioning as a third lens surface and a second aspherical refractive surface func-tioning as a fourth lens surface. The radius rF of the third lens surface is not larger than the radius RB of the second lens surface. The second lens surface and the third lens surface have a common center.
A medium with a refractive indexn1 exists at the ob-ject side, namely, at the left side of the first aspherical refractive surface. The first lens has a refractive index
n2. A medium with a refractive indexn3 fills the space between the first and the second lenses. The second lens has a refractive indexn4. A medium with a refractive in-dexn5exists at the image side, namely, at the right side of the second aspherical refractive surface. The shape of
Fig. 11. Example of an aspherical beam expander com-posed of two elliptical surfaces.
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, and the shape of the second lens is identical to that shown in Figure 8.
The first and the second lenses share a common optical axis. The second focal points of the first and the second lenses, as well as the centers of the second and the third lens surfaces, all coincide. In order to use such a com-pound lens as a beam expander, the refractive indexn2 of the first lens should be larger than the refractive index
n1 of the medium at the object side, and the refractive indexn4of the second lens should be larger than the re-fractive indexn5 of the medium at the image side; that is, n2 > n1 and n4 > n5. Furthermore, when n2 =n4 andn1=n5, then the two aspherical surfaces have iden-tical shapes, but on a different scale. Since two spherical surfaces are not essential, a single-piece beam expander can be realized.
Figure 11 shows the lens profiles and the ray trajecto-ries for a single-piece beam expander made of BK7 glass. Referring to the schematic diagram in Figure 10, the two parametersro andRoare given as 5.0 mm and 50.0 mm, respectively. The vertex radius R and the conic constant k of an elliptical surface are given by the same formulae given in Eqs. (27) and (28). The difference is n1 is smaller than n2 for the elliptical surface. For the single-piece beam expander shown in Figure 11, the conic constants of both surfaces are given as –0.434654, and the vertex radius of the first and the second surfaces
Fig. 13. Schematic diagram of a spherical-aberration-free convex-planar lens.
are given as –1.704 mm and –17.036 mm, respectively. From the figure, it can be seen that this lens functions well as a beam expander and that the alignment toler-ance 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 and 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 spherical-aberration-free convex-planar lens deviates from a Carte-sian oval. Figure 13 shows a schematic diagram of an as-pherical lens, which is sas-pherical-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, and the resultant refracted ray prop-agates toward a point P on the second lens surface. The refracted ray is refracted again at a point P on the sec-ond lens surface and propagates toward the secsec-ond focal point O of the lens. Hereinafter, the refracted ray be-fore being refracted at the point P on the second lens surface is referred to as the first refracted ray, and the ray refracted at the point P is referred to as the second
refracted ray.
The refractive indices of the media on the object and the image sides are no andn2, respectively, and the re-fractive index of the lens is n1. no is assumed to be different from n2, in general. This can be compared to the case of an aquarium with one wall having the shape of the aspherical refractive surface, and the inner and the outer sides of the aquarium being filled with differ-ent media, such as the water and the air, respectively.
The angle between the incident ray and the normal N perpendicular to the tangent plane T at the point Q on the aspherical refractive surface isφ. On the other hand, the angle between the first refracted ray and the optical axis is δ, and the angle between the second refracted ray and the optical axis is θ. Applying Snell’s law of refraction at points P and Q results in Eqs. (33) and (34), respectively:
n1sinδ=n2sinθ (33) and
nosinφ=n1sin(φ−δ). (34) The cylindrical coordinates of the point Q on the as-pherical refractive surface are designated as (X, Z), and the corresponding coordinates of the point P on the sec-ond lens surface as (ρ, z). The back focal length of this lens isfB. Since the second lens surface is a plane, the cylindrical coordinates (ρ,z) of the point P are given as
z(θ) =fB (35)
and
ρ(θ) =fBtanθ . (36) On the other hand, 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(θ). If the distance L between the two points Q and P is expressed as a function of the zenith angleθ of the second refracted ray, then the cylindrical coordinates (X, Z) of the point Q on the aspherical refractive surface are given as
X(θ) =fBtanθ+L(θ) sinδ(θ) (37) and
Z(θ) =fB+L(θ) cosδ(θ). (38) The following equation can be obtained by differenti-ating Eq. (37) with respect toθ:
dX dθ = fB cos2θ + dL dθ sinδ+Lcosδ dδ dθ . (39)
Similarly, the following equation can be obtained by differentiating Eq. (38) with respect toθ:
dZ dθ = dL dθ cosδ−Lsinδ dδ dθ . (40)
Using trigonometrical functional relations, the slope tanφof the tangent plane T at the point Q on the as-pherical refractive surface is given as
tanφ=−dZ
dX . (41)
Since, both the coordinates X and Z are functions of the zenith angleθ, Eq. (41) can be expressed as
dZ
dθ =−tanφ dX
dθ . (42)
With Eqs. (39) and (40), Eq. (42) can be expressed as
dL dθ −L
sinδ−tanφcosδ
cosδ+ tanφsinδ
dδ
dθ
=− fB cos2θ
tanφ
cosδ+ tanφsinδ
. (43)
Before going further to obtain a solution, the functions defined in Eqs. (44) and (45) can be used to make the expression in Eq. (43) simpler:
A(θ)≡ −
sinδ−tanφcosδ
cosδ+ tanφsinδ
dδ dθ , (44) B(θ)≡ − fB cos2θ tanφ
cosδ+ tanφsinδ
, (45)
and
dL
dθ +A(θ)L(θ) =B(θ). (46)
Multiplying both sides of Eq. (46) with an unknown function F(θ), the following relation can be obtained:
F(θ)dL+F(θ)A(θ)L(θ)dθ=F(θ)B(θ)dθ . (47) The condition for the left side of Eq. (47) to be an exact differential is given as
dF
dθ =A(θ)F(θ). (48)
Therefore, the unknown function F(θ) must be given as a function of A(θ) as given in F(θ) = exp " Z θ 0 A(θ0)dθ0 # . (49)
With Eq. (49), Eq. (47) can be readily integrated to yield L(θ) exp " Z θ 0 A(θ0)dθ0 # −Lo= θ Z 0 F(θ0)B(θ0)dθ0.(50)
Therefore, the function L(θ) is given as
L(θ) = 1 F(θ) Lo+ θ Z 0 F(θ0)B(θ0)dθ0 (51)
The following equation can be obtained by differenti-ating Eq. (33) with respect toθ:
dδ dθ =
n2cosθ
n1cosδ
. (52)
On the other hand, the following equation can be ob-tained by rearranging Eq. (34):
tanφ= n1sinδ
n1cosδ−no
. (53)
With Eq. (53), the numerator of the A(θ) given in Eq. (44) is given as
sinδ−tanφcosδ= −nosinδ
n1cosδ−no
. (54)
On the other hand, the denominator of A(θ) is given as
cosδ+ tanφsinδ= n1−nocosδ
n1cosδ−no
. (55)
With Eqs. (52), (54) and (55), the function A(θ) is given as A(θ) = nosinδ n1−nocosδ n2cosθ n1cosδ . (56)
Using Eq. (33), the angleδcan be given as a function of the zenith angleθas
δ(θ) = tan−1 n2sinθ q n2 1−n22sin 2 θ . (57)
Therefore, the function A(θ) can be expressed as a sole function of the zenith angleθ:
A(θ) =q no n2 1−n22sin 2θ n22sinθcosθ n2 1−no q n2 1−n22sin 2θ .(58)
On the other hand, the function B(θ) is given as
B(θ) =− fB cos2θ
n1sinδ
n1−nocosδ
. (59)
The function B(θ) given in Eq. (59) can be expressed also as a sole function of the zenith angleθ as
B(θ) =− fB cos2θ n1n2sinθ n2 1−no q n2 1−n22sin 2 θ . (60)
Accordingly, with Eqs. (49), (58), and (60), L(θ) given in Eq. (51) can be obtained. By using the function L(θ) and Eqs. (35)-(38), the profile of the aspherical refrac-tive surface can be obtained. Since it involves an evalu-ation of an integrevalu-ation with one variable, the lens profile can be obtained using primitive techniques of numerical analysis, such as the trapezoidal sum rule.
Fig. 14. Surface profile of an exemplary spherical-aberration-free convex-planar lens.
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 ρmax 53.78288 R 42.874370 k –6.056876e – 01 C1 1.256774e – 07 C2 1.343433e – 11 C3 5.286947e – 15 C4 –5.974484e – 18 C5 2.379500e – 21 C6 –5.061337e – 25
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 corre-sponding to the plano-convex lens shown in Figure 4. Note that for this lens, the lens thickness Lo is an
im-Fig. 16. Example of a spherical-aberration-free convex-planar lens.
Fig. 17. Ray aberration plot.
portant design parameter. The lens thickness, as well as the back focal length, is taken as 50.0 mm. The maxi-mum ray angle for the second refracted ray is taken as 45.0◦. In order to model with a lens design program, we fitted the surface profile to the even aspherical lens formula given in Eq. (2) and the fitted parameters are summarized in Table 1. Figure 15 shows the fitting er-ror between the aspherical surface profile and 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 and the ray trajecto-ries modeled in Zemax. The field angles in the optical layout are 0◦ and 1◦, and the image space F-number is 1.0. As can be seen from the figure, the incident parallel rays all converge to the secondary focal point. Figure 17 and 18 show a ray aberration plot and a spot diagram, respectively. The graphs show some spherical aberration and coma for the off-axis beam. The presence of remnant spherical aberration is thought to be due to the numer-ical integration error and the fitting error. The coma in this example is much less than the coma in the example shown in Figure 4.
X. SPHERICAL-ABERRATION-FREE PLANAR-CONCAVE ASPHERIC LENS Figure 19 shows a schematic diagram of an aspheri-cal lens with improved aberration characteristics. The aspherical lens is comprised of a plane first lens surface and an aspherical second lens surface. The variables in
Fig. 18. Spot diagram.
Figure 19 have corresponding variables in Figure 13, and section VIII can be referred to for the meanings. Apply-ing Snell’s law for the refraction at points P and Q results in
nosinθ=n1sinδ (61) and
n1sin(φ−δ) =n2sinφ . (62) Since the first lens surface is a plane surface, the height
z measured along the optical axis is given as a constant:
z(θ) =z(0)≡zo . (63) On the other hand, the coordinate ρis a function of the zenith angleθand is given as
ρ(θ) =zotanθ . (64) The distance from the point P on the first lens sur-face to the point Q on the aspherical refractive sursur-face is given as L(θ). If the distanceL between the two points is expressed as a function of the zenith angle θ of the incident ray, then the cylindrical coordinates (X, Z) of the point Q on the aspherical refractive surface are given as
X(θ) =zotanθ−L(θ) sinδ(θ) (65) and
Z(θ) =zo−L(θ) cosδ(θ). (66) The following equation can be obtained by differenti-ating Eq. (65) with respect to θ:
dX dθ = zo cos2θ− dL dθ sinδ−Lcosδ dδ dθ . (67)
Again, the following equation can be obtained by dif-ferentiating Eq. (66) with respect toθ:
dZ dθ =− dL dθ cosδ+Lsinδ dδ dθ . (68)
The slope of the tangent plane T at the point Q on the second lens surface is given as
tanφ=−dZ
dX . (69)
Fig. 19. Schematic diagram of a spherical-aberration-free planar-concave lens.
Since both the coordinates X and Z are functions of the zenith angleθ, Eq. (69) can be expressed as
dZ
dθ =−tanφ dX
dθ . (70)
With Eqs. (67) and (68), Eq. (70) can be expressed as
dL(θ)
dθ +A(θ)L(θ) =B(θ), (71)
where Eq. (71) has been expressed in a simpler form using the functions defined in Eqs. (72) and (73):
A(θ)≡ −
sinδ
−tanφcosδ
cosδ+ tanφsinδ
dδ dθ (72) and B(θ)≡ zo cos2θ tanφ
cosδ+ tanφsinδ
. (73)
Following a similar derivational procedures shown in the previous section, the function L(θ) can be given as
L(θ) = 1 F(θ) Lo+ θ Z 0 F(θ0)B(θ0)dθ0 . (74)
Here, the function F(θ) is a function of A(θ) and is given as F(θ) = exp " Z θ 0 A(θ0)dθ0 # . (75)
The following equation is obtained by differentiating Eq. (61) with respect toθ:
dδ dθ =
nocosθ
n1cosδ
. (76)
The following equation can be obtained by rearranging Eq. (62):
tanφ= n1sinδ
n1cosδ−n2
. (77)
With Eq. (77), the numerator of the function A(θ) defined in Eq. (72) can be given as
sinδ−tanφcosδ= −n2sinδ
n1cosδ−n2
. (78)
Similarly, the denominator of the function A(θ) can be given as
cosδ+ tanφsinδ= n1−n2cosδ
n1cosδ−n2
. (79)
With Eqs. (76), (78), and (79), the function A(θ) is given as A(θ) = n2sinδ n1−n2cosδ nocosθ n1cosδ . (80)
With Eq. (61), the angle δ can be given as a sole function of the zenith angle θas
δ(θ) = tan−1 nosinθ q n2 1−n2osin 2θ . (81)
Resultantly, the function A(θ) can be given as a sole function of the zenith angle θ:
A(θ) = q n2 n2 1−n2osin 2θ n2osinθcosθ n2 1−n2 q n2 1−n2osin 2θ .(82)
On the other hand, the function B(θ) is given as
B(θ) = zo cos2θ
n1sinδ
n1−n2cosδ
. (83)
Eq. (83) can also be given as a sole function of the zenith angleθas B(θ) = zo cos2θ n1nosinθ n2 1−n2 q n2 1−n2osin 2θ . (84)
Accordingly, using Eqs. (75), (82) and (84), the func-tion L(θ) given in Eq. (74) can be obtained. Further, by using the function L(θ) and Eqs. (63)-(66) and (81), the profile of the aspherical refractive surface can be ob-tained.
XI. CONCLUSIONS
In this article, analytic and semi-analytic equations describing various spherical-aberration-free aspherical lenses have been obtained from Snell’s law of refraction, and numerical examples have been given, as well. This formalism can be generalized to design more challenging aspherical lenses with desirable characteristics.
REFERENCES
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[7] N. Cap, B. Ruiz and H. Rabal, Optik114, 89 (2003). [8] N. Alamo and C. Criado, Inverse Problems 20, 229
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[9] O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics(Academic, New York, 1972), Chap. 6. [10] A. Walther,The Ray and Wave Theory of Lenses
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