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rspa.royalsocietypublishing.org
Singlet lenses free of all orders
of spherical aberration
Juan Camilo Valencia-Estrada, Ricardo Benjamín
Research
Cite this article: Valencia-Estrada JC,
Flores-Hernández RB, Malacara-Hernández D.
2015 Singlet lenses free of all orders of
spherical aberration. Proc. R. Soc. A 471:
20140608.
http://dx.doi.org/10.1098/rspa.2014.0608
Received: 8 August 2014
Accepted: 22 December 2014
Subject Areas:
optics, mathematical modelling,
mechanical engineering
Keywords:
aspheric lenses, corrective lenses,
spherical aberration
Author for correspondence:
Juan Camilo Valencia-Estrada
e-mail: [email protected]
Flores-Hernández and Daniel Malacara-Hernández
Centro de Investigaciones en Optica A.C., Loma del Bosque 115,
Lomas del Campestre, León de los Aldama, Guanajuato 37150,
Mexico
JCV-E, 0000-0003-2860-1222
This paper describes a method to design families of
singlet lenses free of all orders of spherical aberration.
These lenses can be mass produced according to
Schwarzschild’s formula and therefore one can find
many practical applications. The main feature of
this work is the application of an analysis that can
be extended to grazing or maximum incidence on
the first surface. Also, here, the authors present
some developments that corroborate geometrical
optics results, along with the axial thick lensmaker’s
formula, which can be applicable to any pair of finite
conjugate planes for any lens shape (bending) and
can be used instead of the classical thick lensmaker’s
formula, which always assumes that the object is at
infinity, to attain better accuracy.
1. Introduction
Aspheric lenses are designed and made to reduce optical
system aberrations. Of all the aberrations a lens may
have, the most studied and worked out is the spherical
aberration. The first studied were lenses with refractive
surfaces free of all orders of spherical aberration
corresponding to non-degenerated Cartesian ovals of
revolution, or degenerated, as ellipsoids, or hyperboloids
of revolution. These were described by Descartes [1],
and have recently been described with explicit examples
by Valencia-Estrada et al. [2]. Subsequently, Luneberg
[3] established a method for calculating the geometry
of the second surface from an initial first surface that
introduces spherical aberration, but he only described
two particular cases and did not complete his notes.
An aplanatic system can be designed using aspheric
surfaces, as pointed out by Wassermann & Wolf [4] and
also as described in the book by Born et al. [5].
2015 The Author(s) Published by the Royal Society. All rights reserved.
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A lens is a translucent body with a relative refraction index n, consisting of one or more refractive
surfaces. A simple lens is basically composed of two surfaces: an anterior one, through which
light enters, and a posterior one, from which light emerges. Using different coordinate systems,
the geometry of these surfaces can be mathematically characterized in many ways, which can be
explicit, parametric or implicit.
Common lenses (singlets) are made of two surfaces of revolution with a common axis,
called the optical axis, which also corresponds to rotationally symmetrical lenses described in
a cylindrical coordinate system (r, z). In this document, the anterior and the posterior surfaces are
designated with subscripts a and b respectively, establishing an absolute coordinate origin at the
point of intersection of the optical axis with the first surface (first surface vertex). Thus, the lens
surfaces are explicitly defined by functions za (r) and zb (r), where z is the sagitta measured parallel
to the optical axis, from the tangent plane at the first surface vertex, and r is the cylindrical radius
or the height to the optical axis, at which the sagitta is measured.
Rotationally symmetrical lens surfaces can be flat, spherical or aspherical. Flat surfaces are
quite common and are specified in cylindrical coordinates by a translational constant k as z = k.
...................................................
2. Physical–mathematical model
2
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
Wasserman & Wolf proposed to use two aspheric adjacent surfaces to correct spherical
and coma aberrations, with a solution consisting of two first-order simultaneous differential
equations, which are solved numerically according to Malacara-Hernández et al. [6].
The spherical aberration has been studied by several researchers, trying to get the best image
on the axis with accurate models [7] or using high-order polynomial approximations ([8,9], and
outstanding work done by Buchdahl [10,11]).
Also, many specialized optical systems have been developed to greatly reduce the spherical
aberration, but none has eliminated it completely. Designs with single or multiple lenses were
developed [12–23].
With the invention of optical discs for digital recording, hundreds of designs became available
by Visser et al. [23], Stallinga [24] and Frolov et al. [25] for the optical pick-up head, with a
significant reduction of spherical aberration, allowing the disc capacity to be increased with
technologies like CD, DVD, HD DVD and Blu-ray.
With the advent of intraocular lenses (IOL) to replace the lens in the human eye, multiple
monofocal or multifocal designs [26] have appeared on the market, with reduced or controlled
spherical aberration.
Also, recently there have appeared numerous studies on the design of single aspherical lenses
[27,28], trying to find the best corrective surfaces with an approach that fulfils Schwarzschild’s
formula. Among them the work performed by Avendaño-Alejo [29] and Kiontke et al. stand out
[30], describing a corrective surface with an approximate representation featuring a hyperbolic
meridional section according to Valencia-Estrada [2], when the object is at infinity and the other
surface is flat. Aspheric deformation coefficients are described as functions of the conic constant
and the vertex curvature.
More precision in manufacturing has also been achieved, reaching an RMS roughness and
formal errors of about less than 10 (nm) [30]; it has also been possible to further reduce spherical
aberration, with the design of optical systems using multiple lenses, for photolithography [31].
This is why the dimensions of design tolerances are given with a precision of 1 (nm) in all figures
presented here, almost reaching the limit of interatomic distances.
Finally, no one else has studied, accurately and analytically, how the shape of the second
surface of a lens is free of spherical aberration after refraction at a first smooth rotational
symmetric interface of any kind, since, in many applications, it is practical to completely correct
the spherical aberration aspherizing a single surface. In this paper, novel solutions are presented,
which rigorously calculate, in cylindrical coordinates, the geometry of the second surface zb (r) for
any given first surface preset za (r), through an original method.
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r
3
te
object
I
image
n
dl
Z
Ra
III
Zb(r)
ta
t
II
tb
lf
Figure 1. On axis refraction for a single sphero-aspherical lens, immersed in the same medium. The aspherical back surface was
calculated to obtain an image point free of all orders of spherical aberration. Rays trace with prescribed relative refraction index
n = 1.7, front radius Ra = 10 (mm), object distance ta = −20 (mm), centre thickness t = 6.39993 (mm) and object–image
distance lf = 50 (mm). Image-distance tb = 23.60007 (mm), edge thickness te = 3 (mm) and the maximum lens diameter
dl = 18.85618 (mm) were calculated. Rays are traced from left to right.
Spherical surfaces are the most common and can be specified
in many ways; the representation
used here, in cylindrical coordinates, is z = k + R − sign(R) R2 − r2 , where R is the radius of the
sphere, and k is the translational constant from the origin. The most common aspherical surfaces
are specified with respect to a conic reference surface according to Schwarzschild’s formula as
z=k+
1+
cr2
1 − (K + 1)c2 r2
+
∞
C2j r2j ,
(2.1)
j=2
where c = 1/R is the central curvature, K is a conic constant and C2j are additional even coefficients
of deformation.
The mathematical and physical models normally used in geometrical optics perform many
approximations, i.e. paraxial theory, but they are far from being mathematically accurate.
A lens featuring simple spherical surfaces leads to the well-known ‘spherical aberration’ which
prevents obtaining a stigmatic image of a point-object located on the optical axis.
A physical–mathematical model that obeys rigorous geometrical optics is presented here
for ray tracing by means of calculus and vector analysis in two dimensions, since rotationally
symmetrical lenses are assumed. This ray tracing model applies to a single lens, with axial
thickness t, a prescribed anterior spherical surface za (r), and an unknown posterior zb (r) surface.
The ideal optical system of such a lens has an object-point on the optical axis with an anterior
vertex-object distance ta , positive defined in the positive direction of the optical axis z. From such
point departs a ray that reaches an arbitrary point on the first surface, whose coordinates are
(ra , za (ra )), where it is refracted by the lens due to its relative refraction index n > 1, changing its
propagation direction according to Snell’s Law. This ray arrives at the back surface at a point on
the second surface (rb , zb (rb )), where the ray is again refracted according to Snell’s Law to reach
a pre-established image point on the optical axis. This image point is located at a back vertex to
image distance tb , positively defined in the positive direction of the optical axis z (figure 1).
The purpose of this study was to determine the mathematical prescription of the posterior
surface zb (r), so that the optical system, as shown in figure 1, with any previously defined anterior
surface za (r), and known parameters n, ta , tb and t, to be free of all spherical aberration orders.
This problem has been analytically studied by some authors such as Luneberg [3] and Born
& Wolf [5] with principles that can be applied to an optical system, as shown in figure 1. It is
worth mentioning that Luneburg modelled the problem for two cases: when the first surface is
flat with a finite object-distance and the image towards infinity, and when the first surface is a
convex spherical with the object at infinity and a finite image-distance; however, his analysis was
not extended to other cases.
...................................................
rb(ra)
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ra
IV
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2
za (ra ), za (ra ),zb (rb )
where n is the refraction index of the lens relative to the surrounding environment, θi is the angle
of incidence of the ray with respect to the unit normal vector N̂a , and θr is the angle of the refracted
ray with respect to the unit normal vector −N̂a , v̂1 is a unit vector in the opposite direction of the
incident ray, and v̂2 is a unit vector in the direction of the refracted ray.
The constructed unit vectors correspond to the row vectors
[−ra , −za + ta ] [rb − ra , zb − za ]
,
v̂
=
v̂1 = 2
2 + (z − z )2 za (ra )
2
(r
−
r
)
2
a
a
b
b
ra + (za − ta )
zb (rb )
za (ra )
[z , −1] and
N̂a = a
.
(2.4)
1 + za za (ra )
Replacing the unitary vectors equation (2.4) in equation (2.3) a fundamental equation is
obtained
|ra + (za − ta )za | (rb − ra )2 + (zb − za )2
n=
.
(2.5)
|rb − ra + (zb − za )za | r2a + (za − ta )2
From equations (2.2) and (2.5), the general solution can be deduced, which is the rear surface’s
geometry that corrects the spherical aberration, for a first prescribed planar or non-planar surface.
The object can be at infinity and the image at a finite distance, or both the object and image are
at finite distances, for any prescribed smooth (mathematically) rotationally symmetrical anterior
surface.
The parametric general solution for rb is odd:
B2 FG − Q + s5 |B| (FP − s4 OG)2 + K(P2 − B2 G2 ) rb = r|ra =
(2.6)
R
ra
and the parametric solution for zb is even and meets
zb |ra = [za + D(ra − r) + s1 E|ra − r|]|r(ra )
or
J − Ir − s3 |B| Kr2 + (G − Fr)2 zb |ra = t + tb +
K
(2.7)
,
(2.8)
r(ra )
with dichotomous signs s1−5 , which can be sign functions of the abscissa ra , object distance
ta , image distance tb or some special function. The solution equations (2.6)–(2.8) are valid for
isotropic and homogeneous materials or metamaterials with negative refraction index, for a valid
...................................................
where all sign functions are dichotomic with values ±1, but without a canonical zero.
Snell’s Law at the first surface provides the second fundamental equation required to solve the
problem
1 − v̂1 . N̂a 2 1 − cos2 θi
sin θi
=
=
,
(2.3)
n=
sin θr
1 − cos2 θr
1 − v̂ . N̂a 2 4
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
Considering spherical or flat incoming wavefronts, and using Fermat’s Principle, for a
spherical aberration-free lens, the optical path of any non-central ray (figure 1) must be equal
to the optical path of the axial ray, which travels along the optical axis; therefore,
ta − tb − sign(ta ) r2a + (za − ta )2 + sign(tb ) r2b + (zb − t − tb )2
,
(2.2)
n=
t − (rb − ra )2 + (zb − za )2
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aperture and in the spherical case, for all ta = Ra , with recurrent and parametric variables {A-R}:
(2.9)
Since the parametric function rb is an odd function of ra , with an inflection point at ra = 0, then
r (ra = 0) exists and r (ra = 0) = 0. Moreover, since zb is an even function of ra with either a local
maximum or minimum at ra = 0, then zb (ra = 0) = 0 and zb (ra = 0) exist. So, for the curvature of
the second surface evaluated at the vertex ra = 0, we have
z r − zb r
z cb (0) = b
= 2b .
r ra =0
3
(r2 + z2
b)
(2.10)
If the general solution for r according to equation (2.6) is factorizable by ra , then r can be written
in the form r = ra m, where m is an even function of ra . Thus, the first derivative of r under the chain
rule is equivalent to
dm
dr
= ra
+ m.
dra
dra
(2.11)
If the derivative of m with respect to ra is an odd function of ra , and if the vertex is a regular
point (equivalent to say that if m is expandable in an even power series), its derivative can be
factorizable by ra in the numerator. So the ra that premultiplied the derivative of m does not
vanish; thus, the limit of the first summand vanishes mandatorily. Then, evaluating the curvature
5
...................................................
and
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎞
⎛
⎪
⎪
⎪
2
⎪
n ra + (za − ta )2
⎪
⎪
⎠,
⎪
B = Denominator ⎝
⎪
⎪
|ra + (za − ta )za |
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2 2
2
⎪
⎪
C = A za − B ,
⎪
⎪
⎪
⎪
⎪
2
⎪
A za
⎪
⎪
,
D=
⎪
⎪
⎪
C
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
|B| A2 (z2
+
1)
−
B
a
⎪
⎪
⎪
,
E=
⎪
⎪
C
⎪
⎪
⎪
⎪
⎪
F = nA,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
G = F[ra − (t + tb − za )za ]+s2 B nt − ta + tb + sign(ta ) r2a + (za − ta )2 ,⎬
⎪
⎪
⎪
⎪
H = Fza ,
⎪
⎪
⎪
⎪
⎪
⎪
I = FH,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
J = HG,
⎪
⎪
⎪
⎪
⎪
2
2
⎪
⎪
K=H −B ,
⎪
⎪
⎪
⎪
⎪
⎪
L = K(Dra − t − tb + za ) − J,
⎪
⎪
⎪
⎪
⎪
⎪
M = s1 EK,
⎪
⎪
⎪
⎪
⎪
⎪
N = DK − I,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
O = M + s4 N,
⎪
⎪
⎪
⎪
⎪
⎪
P = L + s4 Mra ,
⎪
⎪
⎪
⎪
⎪
⎪
Q = s4 OP
⎪
⎪
⎪
⎪
⎭
2 2
2
R = B (F + K) − O .
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
⎛ ⎞
n r2a + (za − ta )2
⎠,
A = Numerator ⎝
|ra + (za − ta )za |
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at the limit when ra → 0, we have
6
(2.12)
which is valid, at the central vertex, for every smooth posterior surface of revolution. Equation
(2.13) is particularly fundamental to deduce the axial lensmaker’s formula using equations (2.6)–
(2.9).
3. An example
In the following figures, some results obtained for a front convex spherical surface (Ra > 0) are
shown, whose sagitta is given by
(3.1)
za = Ra − R2a − r2a .
(a) Thick lens solutions without inversion of internal rays, with object point at infinity
and image at a finite distance
Evaluating equations (2.6)–(2.8) with the recurrent variables according to equation (2.9) for
equation (3.1), the simplified general solution, for a first convex spherical surface with an object
at infinite distance (using new recurring lowercase variables a, . . . , m), is
⎫
a = n2 ,
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
b = Ra ,
⎪
⎪
⎪
⎪
⎪
2
⎪
c = a − 1,
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
d = ra ,
⎪
⎪
⎪
⎪
⎪
⎪
e = cd,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
f = R a − t − tb ,
⎪
⎪
⎪
⎪
⎪
⎬
g = b − d,
(3.2)
⎪
⎪
h = nt + tb − Ra + g,
⎪
⎪
⎪
⎪
⎪
⎪
i = ag,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
j = (a + 1) ab − d + i,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
k = −f + g + j − ah,
⎪
⎪
⎪
⎪
⎪
⎪
l = af + h,
⎪
⎪
⎪
⎪
⎪
⎪
2
2
2
⎪
il − jk − sign(tb ) (ik − jl) − e(l − k ) ⎪
⎪
⎭
,⎪
m=
2
2
e+i −j
r = mra ,
zb = Ra − g +
or
zb = t + tb +
ai + j
(1 − m)
c
a(l − im) − sign (tb ) cr2 + (l − im)2
.
c
(3.3)
(3.4)
(3.5)
...................................................
Therefore, the vertex curvature of any second surface correcting the spherical aberration is
reduced to
zb |ra =0
cb (0) =
,
(2.13)
[limra →0 (r/ra )]2
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
dr r
.
=
lim
m
=
lim
dra ra =0 ra →0
ra →0 ra
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(a)
(b)
Z
Z
7
focus
r
r
2Ra
Figure 2. Sphero-aspherical lenses free of spherical aberration with object at infinity and real image, with maximum
diameter 2Ra , sharp knife edges and different vertex-image distances fa . Aspherical back surfaces zb (ra ) were calculated using
equations (3.1)–(3.5), in (a) and (b), to obtain image-points free of all orders of spherical aberration. Rays trace with prescribed
relative refraction index n = 1.7, front radius Ra = 10 (mm), object distance ta = −∞, edge thickness te = 0 (mm) (knife
edge), for the maximum aperture diameter dl = 20 (mm), and front vertex-image distance fa = 40 and 22 (mm), for (a) and
(b), respectively. Centre thickness t = 2.318252 and 5.172142 (mm), back vertex-image distance tb = 37.681748 and 16.827858
(mm) were calculated for (a) and (b), respectively. Rays trace from bottom to top.
(i) With a finite real image
To verify equations (3.1)–(3.5) ray tracing was performed for hundreds of designs. Figure 2
shows two examples of the attainable solutions (rays propagate from the bottom up) evaluating
equations (3.3)–(3.5). Also, to verify the solution, corroborating the paraxial theory, the posterior
vertex curvature of these sphero-aspheric lenses, with an object at infinity, and a real image at tb
behind the lens, is obtained evaluating equation (2.13) with equations (3.2)–(3.5):
cb (0) =
1
1
1
−
=
,
Ra − (n − 1)(t/n) (n − 1)tb Rb
(3.6)
corresponding to a Gaussian central curvature radius Rb
Rb =
(n − 1)[n(Ra − t) + t]tb
.
n[(n − 1)tb + t − Ra ] − t
Rewriting equation (3.7) similar to the classical Gaussian form
1
1
1
1
−
=
= (n − 1)
Fb
Ra − (n − 1)(t/n) Rb
tb
1
n−1
1
1
t
= (n − 1)
−
+
Ra
Rb
n Ra
Ra − (n − 1)(t/n)
(3.7)
(3.8)
we can obtain the classical Gaussian formula for calculating the posterior vertex focal length
Fb = tb for thick lenses that is described by Malacara-Hernández [6], and so
1
1
1
n−1
1
1
t
(n − 1)t
= (n − 1)
= −
−
+
,
(3.9)
f
Ra
Rb
n Ra
Rb
tb
nRa tb
as expected, since f corresponds to the focal distance measured from the principal plane, as
defined for an object located at infinity.
...................................................
Ra
t
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
Z b(ra)
fa tb
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(ii) With a finite virtual image
8
Evaluating equations (2.6)–(2.8) with the recurrent variables according to equation (2.9) for
equation (3.1), the simplified general solution, for a first convex spherical surface with a finite
object distance, is ∀Ra > 0, ta = Ra and tb = 0:
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
a = R2a ,
b = r2a ,
c = a − b,
√
d = c,
e = R a − ta ,
f = (d − e)2 + b,
g = Ra − t − tb ,
h = n2 f ,
i = hf ,
j = i − e2 ,
k = n2 i − e2 ,
(3.10)
l = −sign(ta )e[nt − ta + tb + sign(ta )f ],
⎪
⎪
⎪
⎪
⎪
⎪
d(i − j)
⎪
⎪
ω=
,
⎪
⎪
⎪
j
⎪
⎪
⎪
⎪
⎪
⎪
⎪
o = −sign (ta )dek ai − be2 ,
⎪
⎪
⎪
⎪
⎪
o
⎪
⎪
⎪
p= ,
⎪
⎪
dj
⎪
⎪
⎪
⎪
⎪
⎪
q = hg + l,
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
s = (n − 1)ωi + p,
⎪
⎪
⎪
⎪
⎪
⎪
u = k(g + ω) − hq + p,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
v = dh,
⎪
⎪
⎪
⎪
⎪
⎪
y = bk,
⎪
⎪
⎪
⎪
⎪
2
2
2
2
2
⎪
e qv − su + ŝ|e| (vu − qs) + y(u − e q ) ⎪
⎪
⎪
m=
,
⎭
e2 (n2 ci + y) − s2
r = mra ,
zb = Ra − d +
or
zb = t + tb +
(3.11)
di − sign(ta )e ai − be2 (1 − m)
(3.12)
j
h(q − vm) + s̃ˆ sign (ta ) sign (tb )e ym2 + (q − vm)2
k
.
(3.13)
...................................................
(b) Thick lens solutions without internal ray inversion, with object and image at finite real
or virtual distances
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
To verify equations (3.1)–(3.5) ray tracing was performed for hundreds of designs. Figure 3
illustrates three solutions found through this method evaluating equations (3.3)–(3.5).
Downloaded from http://rspa.royalsocietypublishing.org/ on June 16, 2017
(a)
(b)
Z
Z
(c)
Z
9
Z b(ra)
t
r
fa
r
tb
focus
dmax
Figure 3. Sphero-aspherical lenses free of spherical aberration with object at infinity and virtual image, with negative designs
and different axial thicknesses t. Aspherical back surfaces zb (ra ) were calculated using equations (3.1)–(3.5) for (a–c), to obtain
image-points free of all orders of spherical aberration. Rays trace with prescribed relative refraction index n = 1.7, front radius
Ra = 10 (mm), object distance ta = −∞, front vertex-image distance fa = −20 (mm) and centre thickness t = 2, 3 and 4
(mm) for (a), (b) and (c), respectively. Lens diameter for maximum aperture dl = 16.422513, 15.570748 and 14.709956 (mm),
edge thickness te = 7.008594, 7.886885 and 8.699242 (mm), and back vertex-image distance tb = −22, −23 and −24 (mm)
were calculated for (a), (b) and (c), respectively. Rays travel upwards. Two additional extended emergent rays are shown with
phantom lines in each lens, to indicate the position of the image-point.
If Ra > 0 and ta = Ra , we have
m=
g(h − d) + ((n − 1)/n)hta + sign (tb ) (n2 − 1)bg2 + (ndg + Ra [tb − n(g + tb )])2
(n2 − 1)a
⎪
⎪
⎪
⎪
⎪
⎭
rb = mra
and
⎫
⎪
⎪
⎪
⎪
,⎪
⎬
zb = Ra − md.
(3.14)
ˆ included in the general solution
With the following rules for only two signs (ŝ and s̃)
equations (3.10)–(3.13):
⎧
⎪
ŝ = −sign(tb )
⎪
⎪
⎪
Ra
⎪
⎪
⎪
If ta < 2
⎪
⎪
n
+1
⎪
⎪
⎨ Then ŝ = ŝ sign ([e2 − n4 (a + e2 )]2 − 4n8 ce2 )
End If
⎪
⎪
⎪
⎪
If ta < 0
⎪
⎪
⎪
⎪
⎪
Then ŝ = ŝ sign([e2 − n2 (a + e2 )]2 − 4n4 ce2 )
⎪
⎪
⎩
End If
and
⎧
a
⎪
If ta < 0 and ta > − n2R−1
⎪
⎪
⎪
⎪
Then
⎪
⎪
⎪
⎪
⎨
− m)(di − sign(ta )e ai − be2 )
ˆs̃ = jk(g − d) − hj(q − vm) + k(1 ⎪
sign(ta ) sign(tb )ej ym2 − (q − vm)2
⎪
⎪
⎪
⎪
⎪
ˆ
⎪
Else s̃ = 1
⎪
⎪
⎩
End If
(3.15)
(3.16)
...................................................
r
Ra
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
te
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(a)
Z
(b)
Z
Z
(c)
10
image
tb
Ra
lf
t
r
r
r
ta
object
dl
Figure 4. Positive design of sphero-aspherical convex-inflexed lenses with zero spherical aberration, same real object and real
image, maximum possible diameter dl and different edge thicknesses te . Aspherical back surfaces zb (ra ) were calculated using
equation (3.1) and equations (3.10)–(3.13) with the rules of equations (3.15)–(3.16), in (a), (b) and (c), to obtain image-points
free of all orders of spherical aberration. Rays are traced with several prescribed variables: relative refraction index n = 1.7, front
radius Ra = 10 (mm), object distance ta = −20 (mm), object–image distance lf = 50 (mm) and edge thickness te = 0.1, 1
and 3 (mm) for (a), (b) and (c), respectively. The lens diameter for maximum aperture dl = 18.85618 (mm), a centre thickness
t = 5.06496, 5.555677 and 6.39993 (mm), and a back vertex-image distance tb = 24.93504, 24.44323 and 23.60007 (mm) were
calculated for (a), (b) and (c), respectively. Rays travel upwards.
(i) With a real object and a finite real image
In this case, to verify equation (3.1) and equations (3.10)–(3.13) with the rules established in
equations (3.15)–(3.16), ray tracing was performed for hundreds of configurations. Figure 4 shows
three examples of the solutions found when rays travel upwards. Also, to verify the solution,
corroborating the paraxial theory, the posterior vertex curvature of these sphero-aspheric lenses
with a proximal or distant real object and a real image is obtained by means of the pinching
theorem described by Weisstein [33], at the points of discontinuity when ta = −Ra /(n − 1) and
ta = −Ra /(n2 − 1):
(n − 1)(t + ntb )ta + Ra [t + n(tb − ta )]
1
cb (0) =
,
(3.17)
=
(n − 1)tb [Ra (nta − t) − (n − 1)tta ]
Rb
a solution that corresponds to a Gaussian central curvature radius1,2
Rb =
(n − 1)tb [Ra (nta − t) − (n − 1)tta ]
,
(n − 1)(t + ntb )ta + Ra [t + n(tb − ta )]
which also corresponds to the mnemonic formula:1,2
1
1
1
= (n − 1)
−
+
tb
Rb
Ra − (n−1)t
Ra −
n
R2
a
(n−1) t
Ra −
n
(n−1)t
n
(3.18)
−
Ra t
ta n
,
(3.19)
ta
which is the axial thick lensmaker’s formula with finite conjugate object and image planes, to
calculate the posterior radius Rb for thick lenses, which can also be compared with the best
1
Formulae for the lensmaker with finite conjugate planes, valid for all lens without quadrant inversion within the same (I and
II figure 1).
2
For the formulae equations (3.10)–(3.13) to give valid results, the anterior vertex-object distance ta , of its five irregular values,
should be of at least 1 (nm), either above or below these four critical values, if the object is real when ta = −Ra /(n − 1) and
ta = −Ra /(n2 − 1), or virtual when ta = Ra /(n2 + 1) and ta = Ra /(n + 1); when ta = Ra , it should be below the critical point.
...................................................
te
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
Zb(ra)
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(a)
(b)
Z
(c)
Z
(d)
Z
Z
11
r
r
r
object
Figure 5. Bi-spherical biconvex approximations (b), (c) and (d) of a zero spherical aberration sphero-aspheric lens (a), with
a real object and a real image conjugates. To compare them, the spherical posterior radius Rb for lenses (b), (c) and (d) were
calculated as: for lens (b) using the posterior apical curvature radius of (a) according to the axial thick lensmaker’s formula
equation (3.19). For lens (c) using the paraxial thin lensmaker’s formula equation (3.20) (Assuming a centre thickness t = 0), and
for (d) using the paraxial thick lensmaker’s formula equation (3.20), with the same central thickness t of the sphero-aspherical
lens (a). Based on this result, the convenience of the axial thick lensmaker’s formula over the canonical ones becomes evident
if the lens is going to be designed with finite conjugate planes, if more accuracy is required. Note that for lens (b), the nominal
point-image of lens (a) coincides with the paraxial point-image, but in lens (c), the best approach is slightly lower than the
nominal point if one is using the infinitely thin lensmaker’s formula, and slightly above in lens (d) if one is using the thick
lensmaker’s formula. The difference is evident if the conjugate planes are not far from the lens itself, as shown in this figure.
All lenses were prescribed with relative refraction index n = 1.7, front radius Ra = 10 (mm), edge thickness te = 0 (mm), front
vertex-object distance ta = −20 (mm) and nominal object-image distance lf = 50 (mm). The corrective aspherical surface
in (a) was calculated evaluating equations (3.11)–(3.13) to calculate centre thickness t = 4.92912 (mm), back vertex-image
distance tb = 25.07088 (mm) and back vertex curvature radius Rb = −37.52199 (mm) used to design (b). With the same centre
thickness, lenses (b–d) were designed, and the maximum diameter was calculated for each one. The calculated back vertex
radius for (c) and (d) were Rb = −35.19902 and −39.55698 (mm), respectively. Rays travel upwards.
approach, the Gauss–Gullstrand formula:
1
1
1
1
t (n − 1)
1
t
= (n − 1)
=− + +
−
+
.
f
Ra
Rb
n Ra Rb
ta
tb
nta tb
(3.20)
Figure 5b–d shows the ray tracing for the best bi-spherical approximations to the zero spherical
aberration solution shown in figure 5a.
(ii) With a real object and a finite virtual image
Solving this case, to verify equation (3.1) and equations (3.10)–(3.13) with the set rules of
equations (3.15) and (3.16) ray tracing was once more performed for hundreds of designs. In
figure 6, two typical solutions are shown.
(iii) With a virtual object and a finite real image
To verify equation (3.1) and equations (3.10)–(3.14) with the rules of equations (3.15) and (3.16)
ray tracing was performed for hundreds of designs. In figure 7, two of the solutions found are
illustrated.
...................................................
r
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
image
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(a)
(b)
Z
12
Z
Zb(ra)
t
r
r
ta
tb
object
lf
image
dl
Figure 6. Positive designs of sphero-aspherical convex-inflexed lenses featuring zero spherical aberration with a real object and
a virtual image. Anomalous sectors are observed at the edges, which require grounding and painting of the edge of the lens with
an absorbent material. Both lenses were prescribed with a relative refraction index n = 1.53, a front radius of Ra = 25 (mm),
a front vertex-object distance ta = −10 (mm), a lens diameter dl = 20 (mm) and an object–image distance lf = −10 (mm).
Edge thicknesses were prescribed with te = 2 and 4 (mm) for (a) and (b), respectively. Aspherical back surfaces zb (ra ) to obtain
zero spherical aberration were calculated using equations (3.11)–(3.13). Centre thicknesses t = 4.899318 and 7.026601 (mm),
and back vertex-image distances tb = −24.899318 and −27.026601 (mm), were calculated for (a) and (b), respectively. Rays
travel upwards. Two additional extended emergent rays are shown with phantom lines in each lens, to indicate the position of
the image-point.
(a)
(b)
Z
Z
object
detail A
Zb(ra)
tb
image
lf
Ra
ta
te
r
t
r
detail A
dl
Figure 7. Sphero-aspherical convex-inflexed simple lenses with a virtual object and a real image, and positive design with
different edge thicknesses te . Both lenses were prescribed with relative refraction index n = 1.59, front radius Ra = 10 (mm),
front vertex-object distance ta = 16 (mm), lens diameter dl = 16 (mm) and object–image distance lf = −8 (mm). Edge
thicknesses were prescribed with te = 1 and 2.5 (mm) for (a) and (b), respectively. Aspherical back surfaces zb (ra ) were
calculated to obtain zero spherical aberration using equations (3.11)–(3.13). Centre thicknesses t = 5.578788 and 7.648822 (mm)
and back vertex-image distances tb = 2.421212 and 0.351178 (mm), were calculated for (a) and (b), respectively. Rays travel
upwards. Two additional extended incident rays are shown with phantom lines in each lens, to indicate the position of the
object-point.
(iv) With a virtual object and a finite virtual image
In this case, to verify equation (3.1) and equations (3.10)–(3.14) with the rules of equations (3.15)
and (3.16) ray tracing was performed for hundreds of designs. Figure 8 shows two selected
solutions found.
...................................................
te
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
Ra
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(a)
(b)
Z
Z
13
ta
lf
Ra
te
t
r
r
tb
image
dl
Figure 8. Sphero-aspherical convex–concave simple lenses with a virtual object and a virtual image (negative design), and a
maximum diameter to avoid the back refraction of rays, by refracting near the back edge in lens (a). Both lenses were prescribed
with relative refraction index n = 1.6, front radius Ra = 3 (mm), front vertex-object distance ta = 4 (mm) and object–
image distance lf = −6 (mm). Lens (a) was prescribed with centre thickness t = 0.5 (mm), and lens (b) with t = 1 (mm).
Aspherical back surfaces zb (ra ) were calculated to obtain zero spherical aberration using equations (3.11)–(3.13). Lens diameters
for maximum aperture were calculated: dl = 2.630694 and 2.251933 (mm). Also, edge thicknesses te = 2.095778 and 2.496148
(mm) and back vertex-image distances tb = −2.5 and −3 (mm) were calculated, for (a) and (b), respectively. Rays travel
upwards. Four additional extended (incident and refracted) rays are shown with phantom lines in each lens, to indicate the
position of the object-point and the image-point.
4. Axial thick lensmaker’s formulae
The central back curvature of these sphero-aspherical lenses also satisfies the mnemonic formula
if −∞ < ta < Ra or ta > Ra , but if ta = Ra , another formula may be used (piecewise solution): this
is an example for a first convex spherical surface with finite conjugate planes.
It has been checked above, according to paraxial theory, that the posterior vertex curvature of
these sphero-aspherical lenses, with a convex spherical first surface and a close object, is obtained
by evaluating the general solution according to equations (3.10)–(3.13) when ta = Ra , and with
the particular solution equation (3.14) when ta = Ra . Thus, three cases arise, depending on the
domain of ta , after using the pinching theorem described by Weisstein [32] at the five points of
discontinuity of the apical curvature (irregular points): if the object is real when ta = −Ra /(n − 1)
and ta = −Ra /(n2 − 1), or virtual when ta = Ra /(n2 + 1), ta = Ra /(n + 1) and ta = Ra :
⎫
−∞ < ta < Ra ⎪
⎪
⎬
ta = R a
⎪
⎪
⎭
Ra < ta < ∞.
(4.1)
(a) First case: −∞ < ta < Ra
The vertex curvature is given by equation (3.17) corresponding to a relative or Gaussian
central curvature radius equation (3.18), which also corresponds to the mnemonic formula
equation (3.19).1,2
...................................................
Zb(ra)
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
object
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(b) Second case: ta = Ra
14
(4.2)
which coincides with the limit of curvature formula equation (3.17) when ta → Ra , and
corresponds to the Gaussian radius
Rb =
(n − 1)tb (Ra − t)
,
t + ntb − Ra
(4.3)
and which also corresponds to
1
n
n−1
=
.
−
tb R a − t
Rb
(4.4)
(c) Third case: ta > Ra
The vertex curvature equation (3.17) corresponding to a relative or Gaussian central curvature
radius equation (3.18), which also corresponds to the mnemonic formula equation (3.19),1,2
whose limits, when ta → Ra , converge to the same results of the above second case. It is easy
to demonstrate that the limits of the mnemonic equation (3.19) when the central thickness t
approaches zero converge in the paraxial thin lensmaker’s formula. It is also possible to find
the paraxial thick lensmaker’s formula, computing the limit when ta → −∞, using Gullstrand’s
formula. When the first surface is flat, only the mnemonic formula should be used at the limit
when Ra → ∞:
n−1
1
1
−
=
.
(4.5)
tb ta − (t/n)
Rb
It is also important to establish the conditions to prevent quadrant inversion of the rays while
propagating through the lens. To determine this condition, Snell’s Law can be used at the first
interface, evaluated when rb = 0 and zb = t, while always t < Ra
n=
Ra
−1
tac
−1
Ra
−1
,
t
(4.6)
with a valid solution for a critical object-distance tac > 0 and tac < Ra :
tRa
tac →
;
nRa − (n − 1)t
(4.7)
therefore, ray inversion does not take place if tb < 0 and
ta >
n n−1
−
t
Ra
−1
> 0.
(4.8)
5. Solutions with Schwarzschild’s formula
There exist an infinite number of aspherical approximated surfaces, according to Schwarzschild’s
formula, that permit the minimizing of all orders of spherical aberration, generated by one
spherical surface with radius Ra . The simplest solution may be represented by the canonical form
z̆b = t +
1+
cr2
1 − (1 + K)c2 r2
+
N
C2j r2j ,
(5.1)
j=2
with vertex curvature equation (5.2) when an object is at infinity:
c=
1
1
,
−
Ra − (n − 1)(t/n) (n − 1)tb
(5.2)
...................................................
t + ntb − Ra
1
,
=
(n − 1)tb (Ra − t) Rb
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
cb (0) =
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or when an object is at close distance
z̆b ∼
=t+
∞
Z−1
N
N
c2j−1 (1 + K)j−1 Bin(1/2, j)r2j +
C2j r2j ,
j+1
(−1)
j=1
j=2
c 3
c
z̆b ∼
(1 + K)r4 + O(r)6 +
C2j r2j ,
= t + r2 +
2
2
N
j=2
z̆b ∼
=t+
and
c 2 r +
B2j r2j +
C2j r2j
2
c
z̆b ∼
= t + r2 +
2
N
N
j=2
j=2
N
(B2j + C2j )r2j .
j=2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(5.4)
To calculate the best fitted conic constant K and N aspheric coefficients C2j , it is recommended
to follow one procedure: statistical or analytical. The results may or may not be dependent of the
aperture diameter dl .
(a) Statistical methods
The most commonly used expression is the polynomial least-squares fitting procedure
p
2
(zb − z̆b ) .
minimize Es =
(5.5)
i=1
The best fit can be obtained by solving the nonlinear system equation (5.5) with N + 1
unknowns: K and N coefficients (C2j )
p
dEs
dz̆b
= 2 (zb − z̆b )
= 0,
dK
dK
i=1
.
.
.
p
and
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
dEs
dz̆b
= 2 (zb − z̆b )
=0
⎪
dC2j
dC2j
⎪
⎪
i=1
⎪
⎪
⎪
⎪
.
⎪
⎪
.
⎪
.
⎪
⎪
⎪
⎪
⎪
p
⎪
⎪
dEs
dz̆b
⎪
= 2 (zb − z̆b )
= 0.⎪
⎪
⎭
dC2N
dC2N
i=1
(5.6)
...................................................
⎫
∞
⎪
c(1/z)2
⎪
, z, k r2j +
C2j r2j , ⎪
⎪
⎪
2
⎪
1
+
1
−
(1
+
K)(c/z)
⎪
j=0
j=2
⎪
⎪
⎪
⎪
⎪
∞
∞
⎪
2
⎪
1
−
(1
+
K)(c/z)
1
−
−1
2j
2j ⎪
⎪
z̆b = t +
, z, k r +
Z
C2j r ,⎪
⎪
⎪
c(1 + K)
⎪
⎪
j=0
j=2
⎪
⎪
⎪
⎪
∞
⎪
2
j
2j
∞
⎪
⎪
1 − j=0 [−c (1 + K)] Bin(1/2, j)r
⎪
2j
⎪
⎪
z̆b = t +
+
C2j r ,
⎪
⎪
c(1 + K)
⎪
⎪
j=2
⎪
⎪
⎪
⎪
⎪
∞
∞
⎪
[−c2 (1 + K)]j Bin(1/2, j)r2j ⎪
⎪
2j
⎪
z̆b = t −
+
C2j r ,
⎪
⎪
c(1 + K)
⎪
⎪
j=1
j=2
⎬
z̆b = t +
15
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
(n − 1)(t + ntb )ta + Ra [t + n(tb − ta )]
c=
,
(5.3)
(n − 1)tb [Ra (nta − t) − (n − 1)tta ]
with conic constant K, and a reduced number N of relevant aspheric coefficients C2j according to
the recommendations by Forbes [33].
The exact solution equation (5.1) can be expanded as even powers series around its vertex,
using the Z inverse transform of the conic summand with k = 2j:
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One solution can be obtained if zb can be expanded in even power series around its vertex, as
z̆b = t + A2 r2 +
N
j=2
c
A2j r2j = t + r2 +
(B2j + C2j )r2j ,
2
N
with the simplest solution ∀j = 2, 3, . . . , N
A2 =
⎫
⎪
⎪
⎪
⎪
⎪
⎬
c
2
..
.
and
(5.7)
j=2
⎪
⎪
⎪
⎪
1
j+1 2j−1
j−1
⎭
(1 + K) Bin , j + C2j ,⎪
A2j = B2j + C2j = (−1) c
2
(5.8)
evaluating equation (5.8) for the first five values of j and solving for the unknown aspherical
coefficients C2j :
C4 = A4 −
c 3
(1 + K),
2
c 5
(1 + K)2 ,
2
c 7
(1 + K)3 ,
C8 = A8 − 5
2
c 9
(1 + K)4 ,
C10 = A10 − 14
2
c 11
(1 + K)5
C12 = A12 − 42
2
.
.
.
C6 = A6 − 2
and
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2j−1
2j−1
j−1
⎪
2
Bin(1/2, j)(c/2)
(1 + K)
⎪
⎪
C2j = A2j −
.
⎭
j+1
(−1)
(5.9)
...................................................
(b) Analytical methods
16
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
The data for the ‘experimental’ p-points with coordinates (ra (p), zb (rb (ra (p)))) in the interval
0 < rb (ra (p)) < dl /2 may be calculated using the recurrent exact solutions based on equations
(2.6)–(2.9). At the end, the best-fit result depends on the value of the correlation coefficient
according to Bates & Watts [34]. But this approach is time-consuming and the mental effort will
be greater than by using the exact recurrent solution for fine interpolation with an expert CAM,
with or without tool correction as is recommended by Weck [35] and Valencia-Estrada et al. [38], to
generate ISO codes (G codes) to be executed in computerized numerical control (CNC) machines.
The fine interpolation described by Weck [35] refers to one of the available methods to segment
the trajectory of a tool in a CNC machine. For the machining of a lens it is always better to make a
single stage fine segmentation of the tool path relative to the workpiece, since it generates better
surface quality and better form, rather than a multi-stage algorithm with a first rough stage. A
programmer with basic programming skills can develop all the needed algorithms to create an
expert system for fine interpolation of these corrective surfaces (CAM), pre-setting a very small
sagittal error for each segment, which depends on the resolution of the feed axes and of the
measuring system: i.e. 1 (nm) at current technological achievements. In many instances, current
lens design optimization programmes can find the form of the correcting surface without too
much difficulty, but the required algorithms to perform the tool offset in the CNC machines are
longer and can take a lot of computation time as the number of coefficients increases to achieve
the 1 nm precision. Furthermore, large aperture designs with low working F#, may need a large
number of coefficients, and at the end can present an undesired and considerable residual amount
of spherical aberration, since for steep profiles a 1 nm machining precision might not be enough
to reach the mathematical atomic-level description of the desired surface.
Downloaded from http://rspa.royalsocietypublishing.org/ on June 16, 2017
and
1 d2 zb c
A2 =
= ,
2! dr2 2
r=0
4
Z4
1 d zb cR3
A4 =
+
,
=−
4! dr4 6
24
r=0
7cR23
cR5
R3 Z4
Z6
1 d6 zb A6 =
−
−
+
,
=
6! dr6 72
120
36
720
r=0
5cR33
1 d8 zb cR3 R5
cR7 R23 Z4
R5 Z4
R3 Z6
Z8
A8 =
=−
+
−
+
−
−
+
8
8! dr 72
80
5040
48
720
720
40320
r=0
11cR23 R5
11cR25
143cR43
11cR3 R7
cR9
Z10
1 d10 zb −
+
+
+
+
=
A10 =
10! dr10 2592
720
28800
30240
3628800
3628800
r=0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
3
2
4
6
⎪
. 11R Z
Z
11R
R
Z
Z
Z
Z
11R
R
R
R
3 5 4
7 4
5 6
3 8 ⎪
.
3
3
.−
+
−
+
−
−
,⎭
648
4320
30240
8640
14400
30240
with recursive variables Rk y Zk defined as
dk r/drka r(k) Rk =
= k
(dr/dra )k (r ) ra =0
and
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
dk zb /drka Zk =
(dr/dra )k ra =0
(k)
=
ra =0
(k)
zb = k
(r ) (5.11)
rb |ra =0
[limra →0 (r/ra )]k
, ∀k = 3, 5, 7, . . .
(5.12a)
∀k = 4, 6, 8, . . . ,
(5.12b)
(k)
=
ra =0
zb |ra =0
[limra →0 (r/ra )]k
,
that when they are evaluated and simplified can be replaced in equation (5.9) to determine the
coefficients of asphericity C2j (K) as a function of the conic constant, and therefore there will be
infinite solutions. To find a good solution for the conic constant that determines the best fit, there
are available two practical cases: with the analytical or numerical solution for K in
N
j=2
C2j
dd
2
2j
= 0,
(5.13)
or with a better solution that can be obtained by equalizing the exact and approximated functions
on the edge, with one numerical real solution for K near zero, in
dd
,
(5.14)
zb (rp ) = z̆b
2
where rp is a radius corresponding to the parameter rp = ra when r(ra ) = dd /2 that correspond
to the back aperture. Once a suitable conic constant has been obtained, this can be iteratively
optimized with very small changes of K to minimize the spherical aberration. An example of
the results following this method, asphericity coefficients according to Schwarzschild’s formula
(which determine the best fit of the correcting surface of spherical aberration for a first planar
...................................................
All coefficients A2j can be calculated using the chain rule to implicitly derive the parametric
functions zb and rb . So, in general, performing operations with the same principles to obtain
equation (5.10): using the chain rule, implicitly deriving and vanishing: odd zb derivatives and
even rb derivatives; coefficients A2j of the power series of zb around its vertex, correspond to:
17
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
As a proof, coefficient A2 must correspond to the second derivate of the even parametric
function zb (rb (ra )) with respect to rb , evaluated around its vertex, divided by 2!
zb 1 1 d dzb 1 d
1 zb r − r z 1 A2 =
=
=
2 dr dr r=0 2 dra r r ra =0 2
r ra =0
r2
zb |ra =0
1 zb r − r zb 1 zb c
(5.10)
=
=
=
= .
3
2
2
2
2
2
r
r
2[lim
(r/r
)]
a
r
→0
ra =0
ra =0
a
Downloaded from http://rspa.royalsocietypublishing.org/ on June 16, 2017
interface), obtained by means of equations (2.6) and (2.7) according to (2.9) with signs s1 = −1,
s2 = −sign(ta ) sign(ra ), s4 = −s2 and s5 = sign(tb )s4 , are
− (−1)
c
(1 + K)
j−1
(5.15)
with recursive variable U = nta − t and polynomials P2j that correspond to
⎫
⎪
⎪
⎪
⎪
⎪
⎪
P6 = 4n(n − 1)4 (n + 1)2 t2 t5b + 2n(n − 1)2 [n(3n3 − 9n − 5) + 1]tt5b U + 2n(n + 1)(4(n − 1)2 ⎪
⎪
⎪
⎪
⎪
⎪
4 2
2
× t + n{n(n − 2)[n(n − 1) − 3] − 1}tb )tb U + 2n(n + 1){(n − 1) t + [4n(n − 2) + 1]tb } ⎪
⎪
⎪
⎪
⎪
⎪
3 3
3 4
2 5
6
2 7⎪
× tb U + 2n{n[n(n − 1) − 4] + 2}tb U − 2n(n − 3)tb U + 6n(n + 1)tb U −2(n + 1) U ,⎪
⎪
⎪
⎪
⎪
⎪
⎪
6
3 3 7
4
2
2 7
⎪
P8 = −24n(n − 1) (n + 1) t tb − 24n(n − 1) (n + 1) [2n(n − 2)(n + 1) + 1]t tb U − n
⎪
⎪
⎪
⎪
⎪
2
2
3
⎪
× (n − 1) (n + 1)(56(n − 1) (n + 1)t + {n[n(n − 2)(29n − 106n − 58) − 56] + 5}tb )t ⎪
⎪
⎪
⎪
⎪
⎪
6 2
4
2
2
⎪
× tb U − n(n + 1){8(n − 1) (n + 1)t + 8(n − 1) {n[11n(n − 1) − 20] + 5}ttb + n[n
⎪
⎪
⎪
⎪
⎪
⎪
2 5 3
3
⎪
× (n − 2)(n{n(n − 1)[5n(n − 1) − 34] + 34} − 18) − 4]tb }tb U − 4n(n + 1)[(n − 1)
⎪
⎪
⎪
⎪
⎪
2
5 4
2
⎪
× (3n − 11)t + tb + n(2n{n[4n(n − 3) − 1] + 20} − 13)tb ]tb U − 4n(n − 1)(6(n − 1)t ⎪
⎪
⎪
⎪
⎪
⎪
2
4 5
2
2
3 6
⎪
⎪
+ (n − 2)[n(n − 8) + 2]tb )tb U − 2n(3(n − 1) t + {n[n(12n − 7) − 32] + 7}tb )tb U
⎪
⎪
⎪
⎪
⎪
2
3 7
2
2 8
2
9
⎪
+ 2n{n[n(−3n + 11) + 20] − 8}tb U + 20n(n − 1)tb U − 20n(n + 1) tb U
⎪
⎪
⎪
⎪
⎪
⎪
3 10
⎪
+ 5(n + 1) U
⎪
⎪
⎪
⎪
⎪
8
4 4 9
6
3
3 9
⎪
⎪
and P10 = 176n(n − 1) (n + 1) t tb + 88n(n − 1) (n + 1) [5n(n − 2)(n + 1) + 3]t tb U + 4n
⎪
⎪
⎪
⎪
⎪
4
2
2
⎬
× (n − 1) (n + 1) (120(n − 1) (n + 1)t + {n[n(n{n[95n(n − 2) − 331] + 520} + 332)
P4 = −n(n − 1)2 (n + 1)tt3b − n2 (n − 2)(n + 1)t3b U − 2nt2b U2 − 2ntb U3 + (n + 1)U4 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
− 20n − 9) + 4]ttb + {n[n(n{n[n(n{n[65n(n − 3) − 317] + 1022} + 508) − 1506] − 308} ⎪
⎪
⎪
⎪
⎪
⎪
2 7 3
4
3
2
⎪
+ 506) − 118] + 7}tb )ttb U + 2n(n + 1)[8(n − 1) (n(6n − 27n + 10) + 31)t + 8
⎪
⎪
⎪
⎪
⎪
2
⎪
× (n − 1) {n[n(n{n[41n(n − 2) − 130] + 208} + 122) − 86] + 11}ttb + n(n{n[n(n{n[n(n ⎪
⎪
⎪
⎪
⎪
⎪
2 7 4
⎪
⎪
× {n[7n(n − 5) − 16] + 267} − 175) − 518] + 456} + 260) − 247] + 62} − 5)tb ]tb U
⎪
⎪
⎪
⎪
⎪
4
2
2
+ 2n(n + 1)(2(n − 1) (n + 1)(5n + 42)t + (n − 1) {n[n(n{n[29n(n − 2) − 222] + 634} ⎪
⎪
⎪
⎪
⎪
⎪
⎪
− 34) − 718] + 201}ttb + [n(n{8n[n(n{n[8n(n − 4) − 3] + 119} − 67) − 92] + 485} − 94)⎪
⎪
⎪
⎪
⎪
⎪
2 6 5
4
2 2
2
+ 5]tb )tb U + 2n(n + 1){12(n − 1) (n + 1) t + 4(n − 1) {n[n(n − 1)(4n + 33) − 87] ⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
+ 35}ttb + [n(n{n[n(n{n[5n(n − 4) − 53] + 364} − 444) − 272] + 712} − 284) + 32]tb } ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
× t5b U6 + 4n(n + 1)((n − 1)2 {n[9n(n2 − 4) − 40] + 43}t + {n[n(n{n[3n(n + 5) − 58]
⎪
⎪
⎪
⎪
⎪
5 7
2
2
⎪
⎪
− 78} + 273) − 138] + 23}tb )tb U + 4n{−4(2n − 5)(n − 1) t + [n(n(n − 3){n[3n
⎪
⎪
⎪
⎪
⎪
× (n − 1)(n + 3) − 40] − 14} − 92) + 20]tb }t4b U8 + 4n{5(n − 1)2 (n + 1)3 t + [n(n{n[−4 ⎪
⎪
⎪
⎪
⎪
3 9
2
3 10 ⎪
⎪
× n(2n − 5) + 33] − 25} − 71) + 15]tb }tb U + 20n(n + 1){n[n(n − 1) − 11] + 3}tb U ⎪
⎪
⎪
⎪
⎪
⎭
2
2 11
3
12
4 13
− 10n(n + 1) (11n − 7)tb U + 70n(n + 1) tb U − 14(n + 1) U .
(5.16)
The coefficients calculated C2j with a relative refractive index n = ni /na , in the limit when tb
tends to infinity, match the coefficients recently released by Castillo-Santiago et al. [37].
− 230] + 29}tb )t2 t8b U2 + 2n(n − 1)2 (n + 1)(24(n − 1)4 (n + 1)2 t2 + 72(n − 1)2 [n(7n3
...................................................
(n − 1)j U3j−2 (2tb )2j−1
1
Bin , j ,
2
j+1 2j−1
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
C2j =
P2j
18
Downloaded from http://rspa.royalsocietypublishing.org/ on June 16, 2017
P4 = n(n − 1)3 tt3b + n3 (n − 1)2 t3b U − 2n(n − 1)2 t2b U2 − 2n(n − 1)tb U3 + (n + 1)U4 ,
P6 = 4n(n − 1)5 t2 t5b + 2n(n − 1)4 (2n2 + 1)tt5b U + 2n(n − 1)3 [n4 tb − 4(n − 1)t]t4b U2
− 2n(n − 1)3 [(3n2 + 1)tb + t]t3b U3 − 2n(n − 1)2 [n(3n − 4) + 2]t3b U4
− 2n(n − 3)(n − 1)2 t2b U5 + 6n(n2 − 1)tb U6 − 2(n + 1)2 U7 ,
P8 = 24n(n − 1)7 t3 t7b + 24n(n − 1)6 (n2 + 1)t2 t7b U + n(n − 1)5 [(15n4 + 9n2 + 5)tb − 56
× (n − 1)t]tt6b U2 + n(n − 1)4 [5n6 t2b − 8(n − 1)(6n2 + 5)ttb − 8(n − 1)t2 ]t5b U3 − 4n
× (n − 1)4 {[n2 (5n2 + 2) + 1]tb + [n(7n − 15) + 11]t}t5b U4 − 4n(n − 1)3 [(n{n[5n
× (n − 2) + 7] − 5} + 4)tb − 6(n − 1)t]t4b U5 − 2n(n − 1)3 ({5n[n(n − 5) + 3] − 7}tb
− 3[n + 1]t)t3b U6 + 2n(n − 1)2 {n[5n(n + 3) − 22] + 8}t3b U7 + 20n(n − 1)3 (n + 1)t2b
× U8 − 20n(n − 1)(n + 1)2 tb U9 + 5(n + 1)3 U10
and
P10 = 176n(n − 1)9 t4 t9b + 88n(n − 1)8 (2n2 + 3)t3 t9b U + 4n(n − 1)7 [−120(n − 1)t + (30n4
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
+ 36n
+ 2n(n − 1) [−24(n − 1)t − 72(n − 1)(3n + 4)ttb + (28n ⎪
⎪
⎪
⎪
⎪
⎪
4
2
2 7 3
5
2
⎪
+ 18n + 12n + 7)tb ]ttb U − 2n(n − 1) {8(n − 1)[n(12n − 37) + 31]t + 8(n − 1) ⎪
⎪
⎪
⎪
⎪
⎪
2 2
8 2 7 4
5
2
⎪
× [15n (n + 1) + 11]ttb − 7n tb }tb U − 2n(n − 1) [−2(n − 1)(5n + 42)t + (n{n ⎪
⎪
⎪
⎪
⎪
⎪
6
4
2
2 6 5
⎪
× [17n(5n − 14) + 223] − 242} + 201)ttb + (35n + 15n + 9n + 5)tb ]tb U − 2n
⎪
⎪
⎪
⎪
⎪
4
2
2
⎪
× (n − 1) (−12(n − 1)t + 4(n − 1){n[n(5n − 54) + 47] − 35}ttb + {n[n(n{5n[n(7n ⎪
⎪
⎪
⎪
⎪
⎪
2 5 6
4
⎪
⎪
− 18) + 14] − 58} + 52) − 36] + 32}tb )tb U − 4n(n − 1) {−{n[n(15n + 22) − 62]
⎪
⎪
⎪
⎪
⎪
5 7
3
⎪
+ 43}t + [n(n{5n[n(2n − 13) + 12] − 48} + 39) − 23]tb }tb U + 4n(n − 1) {4(2n − 5) ⎪
⎪
⎪
⎪
⎪
⎪
2
4 8
3
⎪
× (n − 1)t + [n(n{n[n(5n + 53) − 121] + 98} − 49) + 20]tb }tb U − 4n(n − 1) [5
⎪
⎪
⎪
⎪
⎪
⎪
2
3 9
2 4
3
⎪
× (n + 1) t + (n{n[n(2 − 17n) + 70] − 46} + 15)tb ]tb U + 20n(n − 1) (n − 11n
⎪
⎪
⎪
⎪
3 10
2
2 2 11
3
12
4 13 ⎭
+ 9n−3)tb U −10n(11n−7)(n −1) tb U +70n(n−1)(n+1) tb U −14(n + 1) U .
(5.18)
It is noteworthy that the size of the polynomials increases as the order increases and the
number of variables required to represent the first surface. These expressions are too long to be
presented here, but some should be presented in future work.
Figure 10 shows ray tracing for a sphero-aspherical lens with reduced spherical aberration,
with coefficients calculated according to equation (5.17) with polynomials equation (5.18).
2
+ 29)tb ]t2 t8b U2
6
2
2
6
6. Conclusion
The aspheric back surfaces found to correct all orders of spherical aberration, generated by any
kind first surface, can be represented according to general rigorous solution equations (2.6)–(2.9),
...................................................
evaluated with recursive variable U = n(Ra − t) + t and coefficients P2j that correspond to
19
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
Figure 9 shows ray tracing for a flat-aspherical lens with reduced spherical aberration, with
coefficients calculated according to equation (5.15) with polynomial equation (5.16).
Also, we can obtain the best fit of the Schwarzschild’s correcting surface of spherical aberration
for a first convex spherical interface, for an object at infinity obtained by means of the rigorous
solutions according to equations (3.1)–(3.5), with vertex curvature by equation (3.6) and four
aspheric coefficients C2j according to equation (5.9)
P2j
1
j+1 2j−1
j−1
,
j
,
(5.17)
C2j =
−
(−1)
c
(1
+
K)
Bin
2
(n − 1)j U3j−2 (2tb )2j−1
Downloaded from http://rspa.royalsocietypublishing.org/ on June 16, 2017
(a)
(b)
r
diffraction limit
0.00156
Y
image
X
Z
n
200
6.243 × 10–5
geometrical spot size
100
10
Figure 9. Flat-aspherical lens with real object and real image, and maximum diameter, with prescribed variables n = 1.8,
ta = −200 (mm), t = 10 (mm), tb = 100 (mm), lens diameter dl = 50 (mm) and working F# = 1.984718. (a) Shows the
ray trace for the lens, obtained by OSLO without optimization algorithm neither defocus. Aspherical back surface is calculated
with an entrance beam radius of 24.6978 (mm) that correspond to output aperture (stop diameter) dd = 50 (mm), with
vertex curvature c = −0.018581 (mm−1 ) or Rb = −53.818182 (mm), conical constant K = −2.09902, and four coefficients
C4 = −2.03094 × 10−7 (mm−3 ), C6 = 2.45947 × 10−11 (mm−5 ), C8 = −3.45713 × 10−15 (mm−7 ) and C10 = 2.85128 ×
10−19 (mm−9 ) calculated with equations (5.15) and (5.16). (b) Shows the respective spot diagram, also obtained without
optimization algorithm neither defocus.
(a)
(b)
r
50
Zb aspheric surface
object
diffraction limit
0.001531
Y
image
n
X
Z
Ra
10
4.651 × 10–5
100
geometrical spot size
Figure 10. Sphero-aspherical lens with object at infinity and real image, and maximum diameter, with prescribed variables
n = 1.8, Ra = 100 (mm), t = 10 (mm), tb = 100 (mm), lens diameter dl = 50 (mm) and working F# = 2.093023. (a) Shows
the ray trace for the lens, obtained by OSLO without optimization algorithm neither defocus. Aspherical back surface is calculated
with an entrance beam radius of 25 (mm) that correspond to input aperture (stop diameter) dd = 50 (mm), with vertex
curvature c = −0.00203488 (mm−1 ) or Rb = −491.428571 (mm), conical constant K = −0.986008, and four coefficients
C4 = 2.13068 × 10−7 (mm−3 ), C6 = −2.26827 × 10−11 (mm−5 ), C8 = 3.19039 × 10−15 (mm−7 ) and C10 = −5.01481 ×
10−19 (mm−9 ) calculated with equations (5.17) and (5.18). (b) Shows the respective spot diagram, also obtained without
optimization algorithm neither defocus.
but when the first surface is convex-spherical and the object is far away, equations (2.6)–(2.9) can
be reduced to equations (3.2)–(3.5), and when the first surface is convex-spherical and the object
is near, can be reduced to equations (3.10)–(3.14) with signs rules of equations (3.15) and (3.16).
Approximated solutions with reduced spherical aberration can be represented with
Schwarzschild’s formula with deformation coefficients calculated using equations (5.1)–(5.3)
and equation (5.9). Two approximated analytical solutions are described: when the first surface
is flat, and the object is at a finite distance, with deformation coefficients represented with
equations (5.15) and (5.16); and when the first surface is convex-spherical and the object
is at infinity, with deformation coefficients represented by equations (5.17) and (5.18). The
approximated correcting back surface found may be machined using many kinds of approximate
...................................................
object
rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140608
Z b aspheric surface
50
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Nacional de Ciencia y Tecnología de México, CONACYT, for its economic support.
Author contributions. J.C.V.E. developed most of the physical–mathematical model and programming. R.B.F.H.
developed the remaining portion of the model and reviewed and verified all results. D.M.H. reviewed and
verified all results also. All authors gave final approval for publication.
Funding statement. We have no funding or grants.
Conflict of interests. We have no competing interests.
References
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Data accessibility. This work does not have any experimental data.
Acknowledgements. Centro de Investigaciones en Optica A.C. CIO, León, Guanajuato, México, and Consejo
21
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It is important to note that although these corrective surfaces are developed to design lenses
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