General formula for bi-aspheric singlet lens design free of spherical aberration

Abstract

In this paper, we present a rigorous analytical solution for the bi-aspheric singlet lens design problem. The input of the general formula presented here is the first surface of the singlet lens; this surface must be continuous and such that the rays inside the lens do not cross each other. The output is the correcting second surface of the singlet; the second surface is such that the singlet is free of spherical aberration.

Full text

General formula for bi-aspheric singlet lens
design free of spherical aberration
RAFAEL G. GONZÁLEZ-ACUÑA1,*AND HÉCTOR A. CHAPARRO-ROMO2,3
1Optics Center, Physics Department, Tecnológico de Monterrey Garza Sada 2501, Monterrey, N.L. 64849, Mexico
2Av. Universidad No. 3000, Universidad Nacional Autónoma de México, C.U., Mexico City, Mexico
3e-mail: moxaika@live.com
*Corresponding author: rafael123.90@hotmail.com
Received 24 August 2018; revised 3 October 2018; accepted 3 October 2018; posted 5 October 2018 (Doc. ID 343158); published 25 October 2018
In this paper, we present a rigorous analytical solution for the bi-aspheric singlet lens design problem. The input
of the general formula presented here is the first surface of the singlet lens; this surface must be continuous and
such that the rays inside the lens do not cross each other. The output is the correcting second surface of the singlet;
the second surface is such that the singlet is free of spherical aberration. © 2018 Optical Society of America
https://doi.org/10.1364/AO.57.009341
1. INTRODUCTION
The design of optical systems with aspheric surface has the goal
to strongly reduce spherical aberration. Spherical aberration on
lenses has been extensively studied by Ref. [1]. Luneberg [2]
established a method for computing the shape of the second
surface from an initial first surface that introduces spherical
aberration, which he described just for special cases. Many
authors proposed a lens design with two aspheric surfaces to
correct spherical aberration [3,4].
The problem of the design of a singlet free of spherical aber-
ration with two aspheric surfaces is also known as the
Wasserman and Wolf problem [5]. The problem has been
solved with a numerical approach by Ref. [6]. Recently, Ref. [7]
has shown a rigorous analytical solution of a singlet lens free of
spherical aberration for the special case when the first surface is
flat or conical. Since its publication, several works inspired by
its solution have emerged [7–12], all of them free of spherical
aberration. The solution has six different signs; therefore, it is a
set of 2664 possible solutions, where only one is right. We
test the formula provided by those in Ref. [7], when the first
surface is not flat or conic, and the equation system does not
give correct answers.
In this paper, we present a rigorous analytical solution for
the design of lenses free of spherical aberration. The solution
presented here has just one sign; therefore, it is a set of just two
possible solutions. Our solution is robust because the set of
solutions is valid for negative and positive refraction indices.
The model allows use of continuous functions, such that
the rays inside the lens do not cross each other. This model
will compute the second surface in order to correct the spherical
aberration produced by the first surface.
2. MATHEMATICAL MODEL
The goal is to determine the shape of the second surface
rb,zb, given a first surface ra,za, in order to correct the
spherical aberration generated by the first surface. Therefore
the objective is to find rb,zbgiven ra,za, where rais the
only independent variable, and zb,rb, and zaare functions of
ra. The origin of the coordinate system is located at the center
of the input surface za00.
We assume that the singlet lens has refraction index nand is
radially symmetric. At the center, the singlet lens has thickness
t. The distance from the object to the first surface is ta. The
distance from the second surface to the image is tb, as can be
seen in Fig. 1.
The first fundamental equation for this model is the vector
form of the Snell’s law:
v21
nna×−na×v1 −naffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−
1
n2na×v1·na×v1
r,
(1)
where v1is the unitary vector of the incident ray, v2is the uni-
tary vector of the refracted ray, and nais the normal vector of
the first surface, as seen in Fig. 2.
The unitary vectors written of the first surface are
8
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>
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>
>
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:
v1ra,za−ta,0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
aza−ta2
p,
v2rb−ra,zb−za,0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rb−ra2zb−za2
p,
naz0
a,−1,0
ffiffiffiffiffiffiffiffiffi
1z02
a
p,
(2)
where z0
ais the derivative with respect to raof the sagitta of the
first surface.
Research Article Vol. 57, No. 31 / 1 November 2018 / Applied Optics 9341
1559-128X/18/319341-05 Journal © 2018 Optical Society of America

Replacing Eq. (2) in Eq. (1) and separating the Cartesian
components, we get the following expressions for the direction
cosines ri,ziof the vector:
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:
ri≡rb−ra
Ψza−taz0
ara
nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
ata−za2
p1z02
a
−z0
aΦ,
zi≡zb−za
Ψraza−taz0
az0
a
nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
ata−za2
p1z02
aΦ,
with,
Ψffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
zb−za2rb−ra2
p
Φffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−raza−taz0
a2
n2r2
ata−za21z02
a
qffiffiffiffiffiffiffiffiffi
1z02
a
p
(3)
and, evidently, z2
ir2
i1. Relations (3) come from the ap-
plication of the Snell law for an arbitrary ray striking the singlet
lens at point ra,za. Note that the expressions in the right sides
of Eq. (3) are fully expressed in terms of the coordinates of the
input surface, i.e., zizira,zaand so on.
Now let us focus on the Fermat’s principle, which is the sec-
ond fundamental equation of our model. We assume for a
spherical aberration-free singlet lens, the optical path of any
non-central ray must be equal to the optical path of the axial ray:
−tant tb−sgntaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
aza−ta2
p
nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rb−ra2zb−za2
p
sgntbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
bzb−t−tb2
q,(4)
where sgntaand sgntbare the sign functions of the variable ta
or tb, respectively.
Now we have a system of three equations: two components
of vector form of Snell’s law, Eq. (3), and Fermat’s principle,
Eq. (4). Furthermore, we have two unknowns, rband zb.
The solution of the system is
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rbrizb−za
zira,
zb
h0s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z2
i−2nfiziza−tbtzi−1raritr2
i
−ri−zatbtrazi2f2
ih1n2
s
1−n2
,
(5)
where s1is the only sign; it comes from the fact that when the
refraction index is positive, the rays are refracted to the opposite
direction when the refraction index is negative. Also, we define
the following auxiliary variables:
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:
fi−sgntaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
ata−za2
pta−tb,
h0nfizi−n2tzizar2
iza−rariziz2
ittb,
h1r2
a2rarittb−za2t2r2
i−1zi2
−2ttb−za−1zi:
(6)
Equation (5) is the most important result in this work. It could
look cumbersome, but it is notorious that it could be expressed
in closed form for an almost arbitrary freeform input surface.
The condition for the validity of Eq. (5) is that the surface nor-
mal should be perpendicular to the tangent plane to the input
surface at the origin, and the rays do not cross each other inside
the lens.
From a topological point of view, since the singlet lens is a
homogeneous optical element, the input and output surfaces
are simply connected sets on R2that can be defined as
Ωafra,za∈R2jza<z
bg,
Ωbfrb,zb∈R2jzb>z
ag,(7)
where Ωais homeomorphic to Ωb, and both surfaces are topo-
logically equivalent. Thus, there exists a continuous and bijec-
tive function f, such that f:Ωa→Ωb, and whose inverse f−1
is continuous.
There are many functions fthat map both sets, but there is
only one that is physically valid and corresponds to the one that
satisfies the variational Fermat principle of minimum optical
length, which is constant. In our case, fis given by Eq. (5).
The uniqueness of fhas as consequence that the Snell law is
automatically fulfilled at the second interface zbas well.
Now, since fis continuous, it means that fmaps open balls
from Ωato Ωb, then the ray neighborhoods are preserved.
Therefore, the validity of Eq. (5) also requires that the rays
do not intersect each other inside the lens because, in the case
when the rays inside cross each other, Ωboverlaps itself, and
then Ωbis no longer a simple connected set; it is not homeo-
morphic with respect to Ωa, the vicinity of the neighborhoods
is not preserved, and finally we do not have a homogeneous
optical element.
-20 -10 10 20 30
-10
-5
5
-30 40
-z z
r
-r
OI
o
Fig. 1. Diagram of a singlet lens free of spherical aberration. The
first surface is given by ra,za, and the second surface is given by
rb,zb. The distance between the first surface and the object is ta,
the thickness at the center of the lens is t, and the distance between
the second surface and the image is tb.
r
z
r
z
Fig. 2. Left: Zoom at point Pof Fig. (1); three unitary vectors can
be seen: v1is the unitary vector of the incident ray, v2is the unitary
vector of the refracted ray, and nais the normal vector of the first sur-
face. Right: Zoom at point Qof Fig. (1); unitary vectors of the second
surface can be seen: v2is the unitary vector of the incited ray, v3is the
unitary vector of the refracted ray, and nbis the normal vector of the
second surface.
9342 Vol. 57, No. 31 / 1 November 2018 / Applied Optics Research Article

3. RESULTS AND EXAMPLES
In this section, we report some illustrative results of Eq. (5), in
six different contexts, with the objective to show the capacity of
our model. We compare our results with Ref. [7], then we in-
ject the model with functions that have never been evaluated,
and we observe that they satisfy the optical path conditions; we
load the system with a fixed input function and change the
distance between the image and the second surface, in order
to show that the model adjusts the second surface and corrects
the spherical aberration. Then, we show an example of negative
refraction index, and an example when the object is at minus
infinity, ta−∞, and finally we compute the efficiency
of Eq. (5).
A. Comparison Between Solutions
Table 1shows the results of the comparison of the solution
given in Ref. [7] with ours, for plane, spherical, parabolic,
and cosine cases. The sagitta for each surface corresponds to
flat za0, parabolic-aspheric case zar2
a∕200, spheric-
aspheric case za100 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1002−r2
a
p, and cosine-aspheric case
zacosra∕3. In addition, the error difference between both
solutions is reported.
The difference is computed by the following formula:
RS Error X
N
zb−zbc2
N,(8)
where Nis the number of samples, zbis the sagitta of Eq. (5),
and zbc is the sagitta of the solution of Ref. [7]. We evaluate the
values of the first quadrant because the solutions provided by
Ref. [7] are not symmetric and fail in the second quadrant. For
the evaluation, we use a sample of N500.
With the data in Table 1, we can argue that our solution in
the first quadrant and for conic surfaces is equal to the solution
of Ref. [7]. Also, it is important to mention that all this
computation was made without the point zbc 0, because in
the method of Ref. [7], that point diverges and for our method
at the same point, we have zb0tza0.
B. Gallery of Lenses with New Solution
This time, the first surfaces are not conic. The model is robust
enough that it can admit nonstandard functions as first surfa-
ces; the sagittae of the first functions of Fig. 3are
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azaexpraJ0rawith,ra<0
exp−raJ0rawith,ra≥0,
bza1
18 r2
acosra
2,
cza5 cosra
5,
dza−1
18 r2
acosra
2,
(9)
where J0is the Bessel function of order zero. Therefore, the case
when the first surface is the first term of Eq. (9) is the plot (a) in
Fig. 3, the case when the first surface is the second term of
Eq. (9) is the plot (b) in Fig. 3and so on.
The shapes presented in Fig. 3are illustrative examples of
the many examples computed in [13]. The general formula still
gives a correct second surface in order to get a singlet lens free of
spherical aberration. Please see Appendix A.
C. Performance of the Formula
The same surface is evaluated for different object distances, and
the system shows the capacity to generate the solution such as
the singlet is free of spherical aberration in the image point. In
Fig. 4, we change tband keep fixed the sagitta zaJ0ra.We
find that the maximum diameter of the lens is proportional
to tb.
Table 1. Benchmark of Resolver Power Between
Analytical Solutions to Design of Singlet Lenses Free of
Spherical Aberration
Solution According
to J.C.V.E
Solution According to
R.G.G.A. and H.A.C.R.
Basic configuration for all cases: n1.5and s11for (1) flat, (2) spherical,
(3) paraboloid, and (4) cosine; we have: (1) ta−70 mm,t10 mm,
tb75 mm,RS 1.20 ×10−29. (2) ta−70 mm,t8mm,
tb80 mm,RS 4.54 ×10−28. (3) ta−80 mm,t10 mm,
tb100 mm,RS 1.75 ×10−28. (4) ta−70 mm,t8mm,
tb80 mm,RS=.
rr
O
z
r
I
I
I
I
O
O
O
z
z
z
0
(a)
(b)
(c)
(d)
Fig. 3. Gallery of singlet lenses bi-aspherical free of spherical aber-
ration. For all cases, the configuration is ta−70 mm,t8mm,
tb75 mm, and n1.5.
Research Article Vol. 57, No. 31 / 1 November 2018 / Applied Optics 9343

D. Example for Negative Refraction Index
In Fig. 5, the ray tracing can be seen for a lens with negative
refraction index n−1.5, for zaJ0ra∕2; therefore, we
take the sign as s1−1. It can be seen that for a negative re-
fraction index, both sets Ωa,Ωbdo not cross each other,
Ωa∩Ωb∅; therefore maximum radius tends to infinity.
E. Example When the Object Is at Minus Infinity
Now, when the object is far away, ta→−∞, we need to com-
pute the limit when ta→−∞for fi,ri,zi, which can be
written as
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lim
ta→−∞
fiza−tb,
lim
ta→−∞
ziffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n2−1z02
an2
n2z02
a1
qffiffiffiffiffiffiffiffiffi
z02
a1
pz02
a
nz 02
an,
lim
ta→−∞
ri−
z0
anffiffiffiffiffiffiffiffiffi
z02
a1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n2−1z02
an2
n2z02
a1
q−1
nz02
a1
:(10)
Then, we can use Eq. (5) with the parameters of Eq. (10)
and test it. For illustration proposes, we choose za
cosra∕2r2∕200 and plot the ray tracing in Fig. 6. From
the figure, it is clear that the output rays converge to the image
point despite ta→∞.
F. Efficiency
To validate the efficiency of Eq. (5) we compare two vectors, v3
and v†
3;−v3comes from the image to the second surface, and v†
3
is computed using Snell’law with the normal vector of the sec-
ond surface, nb, and the unitary vector, v2(Fig. 2); therefore,
nb,v3,v†
3are written as
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nbz0
b,−r0
b,0
ffiffiffiffiffiffiffiffiffiffiffi
z02
br02
b
p,
v3rb,zb−t−tb,0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
bzb−t−tb2
p,
v†
3nnb×−nb×v2 −nbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−n2nb×v2·nb×v2
p
:
(11)
The percentage efficiency of ray is measured how close ends in
the image position, thus percentage efficiency is given by
E100%−
v†
3−v3
v3
×100%:(12)
We compute the efficiency for 550 rays for all the examples
presented in the paper, and the average of all the examples
is 99.999999999999860%≈100%. We believe that the error
is not zero because the equations are very large, and when
evaluated, we get computational errors such as truncation.
Please notice that for all examples, the singlets are free of
spherical aberration even when the incident angles are very large;
this happens because we do not use any paraxial approximation.
4. CONCLUSION
In this paper, we generalized the second surface in a lens in
order to get an image free of spherical aberration for a given
arbitrary first surface. When the first surface is such as the rays
-10 -5 5 10
80
O
I
z
60
40
20
-20
60
z
40
20
-20
5
-5
-10 10
O
I
20
z
r
O
I
-20
-5 5
(a) (b) (c) (d)
Fig. 4. (a) tb75 mm, (b) tb55 mm, (c) tb35 mm,
(d) tb15 mm. The constant configuration of cases (a)–(d) is
ta−30 mm,t5mm,n1.5.
Fig. 5. Configuration of the singlet lens with negative refraction
index: ta−20 mm,t3mm,tb30 mm,n−1.5.
Fig. 6. Configuration of the singlet lens with object at infinity:
t3.5mm,tb50 mm,n1.5.
9344 Vol. 57, No. 31 / 1 November 2018 / Applied Optics Research Article

inside the lens do not cross each other. We exhaustively test the
formula by ray tracing, and we find that the solution presented
works for positive and negative refractive indices. For a negative
refraction index, we take s1−1.
This general formula expands the variety of lenses free of
spherical aberration. The efficiency of the method allows to de-
sign robust optical systems whose first surfaces are not restricted
to the conical family functions.
APPENDIX A
We share the program code in Mathematica language to graph
surfaces free of spherical aberration. The program is composed
of three sections: the first one declares Eq. (5); in the second
section, constants of the optical system are defined; finally, the
graphing process is performed for the functions of the first sur-
face and the second surface simultaneously.
Singlet free of spherical aberration.
1
2 (* Clean register. *)
3Clear[“Global’*”]
4
5 (* Define first surface function *)
6 za[ra_] := Module[{h}, h = Cos[ra 0.5]];
7
8 (* Define variables of Eq. (5)*)
9 Phi[ra_] := Module[{h},
10 h = Sqrt[1 −(ra + (−ta + za[ra]) Derivative[1][za][ra])^2/
11 (n^2 (ra^2 + (ta −za[ra])^2) (1 + (Derivative[1][za][ra])^2))]/
12 Sqrt[1 + (Derivative[1][za][ra])^2]]
13
14 ri[ra_] := Module[{h},
15 h = (ra + (−ta + za[ra]) Derivative[1][za][ra])/(
16 n Sqrt[ra^2+(ta−za[ra])^2] (1 + (Derivative[1][za][ra])^2)) −
17 Derivative[1][za][ra] Phi[ra]]
18
19 zi[ra_] := Module[{h},
20 h = (Derivative[1][za][
21 ra] (ra + (−ta + za[ra]) Derivative[1][za][ra]))/(
22 n Sqrt[ra^2 + (ta −
23 za[ra])^2] (1 + (Derivative[1][za][ra])^2)) +Phi[
24 ra]];
25
26 fi[ra_] := Module[{h}, h = ta −tb −Sign[ta] Sqrt[ra^2 + (ta −za
[ra])^2]];
27
28 h0[ra_] := Module[{h},
29 h = ri[ra]^2 za[ra] + fi[ra] n zi[ra] −
30 ra ri[ra] zi[ra] + (t + tb) zi[ra]^2 −n^2 (za[ra] + t zi[ra])];
31
32 h1[ra_] := Module[{h},
33 h = ra^2 + 2 ra ri[ra] t + (tb −za[ra])^2 +
34 t^2 (ri[ra]^2 + (−1 + zi[ra])^2) −
35 2 t (tb −za[ra]) (−1 + zi[ra])];
36
37 zb[ra_] := Module[{h},
38 h = (h0[ra] +
39 s1 Sqrt[zi[ra]^2 (fi[ra]^2 −
40 2 fi[ra] n (ra ri[ra] + ri[ra]^2 t +
41 zi[ra] (t (zi[ra] −1) −tb + za[ra])) +
42 h1[ra] n^2 −(ra zi[ra] +
43 ri[ra] (t + tb −za[ra]))^2)])/(−n^2 + 1)];
44
45 rb[ra_] := Module[{h}, h = ra + (ri[ra] (−za[ra] + zb[ra]))/zi[ra]];
46
47 (* Define system configuration’s*)
48 s1 = 1;
49 n = 1.5;
50 t = 8;
51 ta = −60;
52 tb = 70;
53 basez = ta;
54 topez = tb + t;
55 rmax = 16.8; (* Iterative value *)
56
57 (* System plot *)
58 g1 = Plot[{za[ra], topez, basez}, {ra, −rmax, rmax},
59 PlotStyle −>{Black, Transparent, Transparent},
60 AspectRatio −>Automatic,AxesLabel −> {r, z}];
61 g2 = ParametricPlot[{rb[ra], zb[ra]}, {ra, −rmax, rmax},
62 AspectRatio −> 1/1, AxesOrigin −> {0, 0}, PlotStyle −>
{Black},
63 Mesh −>None];
64 Show[g1, g2]
Please take care of the copy-paste process to avoid character
coding errors between the pdf reader format and the mathema-
tica work environment. We show the Code 1, Ref. [13].
Funding. Consejo Nacional de Ciencia y Tecnología
(Conacyt) (593740).
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Research Article Vol. 57, No. 31 / 1 November 2018 / Applied Optics 9345