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Extended depth-of-focus wavefront design from1
pseudo-umbilical space curves2
SERGIO BARBERO1* , MANUEL RITORÉ,2
3
1Instituto de Óptica (CSIC), Serrano 121, Madrid, Spain.4
2Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 180715
Granada, Spain6
*sergio.barbero@csic.es7
Abstract: Designing extended-depth-of-focus wavefronts is required in multiple optical
8
applications. Caustic location and structure analysis offer a powerful tool to design such
9
wavefronts. An intrinsic limitation of designing extended-depth-of-focus wavefronts is that
10
any smooth surface, with a non-constant mean curvature, unavoidably introduces a separation
11
between caustic sheets, which is proportional to the ratio of change of the mean curvature
12
along a curved embedded in the wavefront. We present a novel method to obtain extended
13
depth-of-focus wavefronts where the mean curvature variation ratio is reduced thanks to using a
14
long circle-involute space curve effectively filling the wavefront surface. Additionally, we present
15
a variant of the method in which the wavefront is modified within a small tubular neighborhood
16
of the circle involute in order to partially meet the umbilical condition along that tubular region.
17
Finally, we provide some numerical results showing the potential of our method in an application
18
example.19
© 2023 Optical Society of America20
1. Introduction21
Multifocal and/or extended-depth-of-focus optical beams are ubiquitous in a wide range of
22
optical engineering applications; to name just a few: ophthalmic corrections for presbyopia [1],
23
microscopy [2], or laser micromachining [3]. Although relatively novel, the analysis or
24
prescription of the caustic structure of such optical beams offers an appealing way of designing
25
them [4–8].26
Here, we restrict our analysis to optical systems that generate smooth, in the sense described by
27
analytical functions, wavefronts; a desirable property in some applications such as visual optics.
28
Then, we exclude the classical cone-shaped-axicon, which contains derivatives discontinuities
29
at the center. Recently, it has been proposed a method to design axially symmetric wavefronts
30
under certain caustic prescriptions [8]. The method is purely within a geometrical optics
31
framework, which is a reasonably good starting point. Once the incoming and outgoing
32
wavefronts are prescribed, it is possible to provide their required coupling by means of, for
33
instance, a refractive/reflective optical surface [9].34
This kind of wavefront provides an extended depth-of-focus by means of a varying mean
35
curvature through the spatial domain. However, any smooth surface with a non-constant mean
36
curvature unavoidable introduces a difference in principal curvatures, which implies the generation
37
of two separated caustic sheets; hence preventing the formation of a perfect geometrical focal
38
line. The amount of cylinder (difference between principal curvatures) is proportional to the
39
ratio of change of one of the principal curvatures along a principal line of curvature [10]. This
40
last property points out one of the drawbacks of axially symmetric wavefronts. Since the
41
change of mean curvature, on these surfaces, is produced along meridians, and meridians are the
42
shortest path from the optical center to the edge, the ratio of mean curvature is maximal, and, in
43
consequence, they presumably induce a non-optimal amount of cylinder. This fact was analyzed
44
in [8].45
Therefore, a natural alternative to axially symmetric wavefronts is to set the mean curvature
46
variation not along meridians but, for instance, along spiral-type curves. Indeed, spiral or helical
47
wavefronts (or phase functions) are intensively used in many different optical applications [11
–
13].
48
Breaking axial symmetry implies that the equations relating caustics with wavefront geometry
49
are no longer ordinary differential equations but partial differential equations (PDE). Therefore,
50
a possible approach would be to handle those PDEs. However, to avoid the complexities and
51
non-uniqueness issues associated with solving PDEs, we propose here a different approach based
52
on differential geometry reasoning.53
Caustics, or focal surfaces as they are known in differential geometry, contain some special
54
curves called focal curves [14]. Given a space curve
𝛾
embedded in a surface, with its associated
55
focal surface, the focal curve
𝐶𝛾
is such that it meets that the points of
𝐶𝛾
are the centers of the
56
osculating spheres of
𝛾
[14]. Setting focal curves and recovering their generating space curves,
57
as we will see, is a way of reducing the dimensionality of the much more complicated problem of
58
recovering surfaces from their focal surfaces. The crux of such simplification is to realize that
59
some space curves can effectively fill in a surface in Euclidean space. Particularly, we considered
60
circle involutes. They are curves optimally sampling a surface, in the sense that the overall length
61
is minimum given the following restriction: the distance of any point on that surface to the circle
62
involute is less than a given magnitude [15].63
A classic result in differential geometry of spatial curves is that curvature and torsion (second
64
curvature) determine the curve up to translation and rotation [16]. Then, the second curvature of
65
the circle involute filling the wavefront can be conferred with an extra convenient optical property;
66
namely, to be close to the umbilical, hence reducing the light dispersion generated by caustic
67
sheet separation. This approach is admissible because, generically, any portion of a smooth
68
surface (at least
𝐶2
) providing spatially varying mean curvature can only contain umbilical lines
69
or isolated umbilical points [10, 17]. In principle, umbilicity or pseudo-umbilicity (close to) are
70
attainable if those curves are not very close to each other on the embedded surface. This is an
71
issue that we have explored experimentally in this paper.72
Inspired by the aforementioned differential geometry properties we propose a method based
73
on several sequential steps. 1) We construct an axial symmetric asphere meeting a sagittal
74
caustic prescription (section 2.1); 2) We build up a uniform arc-length spiral-type circle involute
75
optimally filling that asphere (section 2.2); 3) Those curves are modified such that one of the
76
curvatures meets the imposed extended depth-of-focus target; 4) Small tubular neighborhoods
77
of those curves are constructed such that the umbilical condition (zero astigmatism) is, at least,
78
partially met (subsection 2.3), and, finally, 5) the cloud points are approximated with a surface
79
using Zernike polynomial least squares fitting (subsection 2.4). Once the surface is described
80
analytically by means of a Zernike expansion, caustic sheets can be computed as explained
81
in subsection 2.5. Overall, the method offers an efficient solution to the problem of extended
82
depth-of-focus multifocal wavefront design under some caustic prescriptions.83
2. Method84
In this section, we describe step by step the design method. For a better understanding, we will
85
exemplify it with an example of application; namely, a wavefront aimed at providing prescribed
86
intensities (linear, exponential, or any other type [20]) over a depth of field along a segment of the
87
optical axis, which can be achieved using an axicon or gaxicon [21]. Particularly, we designed a
88
wavefront providing constant intensity distribution. For this purpose, a one-to-one mapping must
89
be established between the generating curve
𝛾
of the wavefront and its associated focal curve
𝐶𝛾
;
90
both curves sampled with points at equal arc-length distances. In doing so, what is ensured is that
91
equal areas of the wavefront are focused on approximately equal-length segments of the optical
92
axis. The example case used in this paper [8] is applicable in visual optics problems, where it
93
has been shown the benefits of including curvature variation not only along the radial coordinate
94
but also along the angular one [1, 27
–
29]. We assumed an eye output wavefront, with a pupil size
95
of
𝜉=3
mm providing 60 D for far vision and 63 D for near vision. This implies a far and near
96
focal distance of 𝑧𝑐=16.67 mm and 𝑧𝑒=15.87 mm, respectively.97
2.1. Axial symmetric wavefront with prescribed sagittal caustic98
Caustic surfaces are the locus of centers of principal curvatures of the wavefront. They can
99
be obtained from the wavefront surface by means of computing distances along the wavefront
100
normals, whose magnitudes are the inverse of the principal curvatures [25]. Each point of the
101
wavefront, except if it is umbilical, generates two caustic points associated with each of the two
102
principal curvatures. Therefore, the caustic manifold comprises two surfaces or sheets. If the
103
wavefront poses axial symmetry, one of the caustic sheets (called sagittal) degenerates into the
104
segment of a straight line along the symmetry axis. Our starting point is an axially symmetric
105
wavefront generating a uniform light concentration along the sagittal caustic. Such wavefront can
106
be obtained by solving the Cauchy problem of a first-order non-linear differential equation [15]:
107
𝑧′(𝑥)=𝑥
𝑔(𝑥) − 𝑧(𝑥),(1)
𝑔(𝑥)=(𝑧𝑒−𝑧𝑐)𝑥
𝜉+𝑧𝑐.
We solved Eq. 1 numerically with the help of ode45 MATLAB function and
𝑧(0)=0
as the
108
initial condition. The numerical solver provides sag data (
𝑧𝑖
) of the wavefront for a set of radial
109
coordinates (
𝑥𝑖
). For subsequent steps, we need to fit those data with an analytical function. For
110
the example case, we experimentally found that the optimal function is an asphere described by a
111
fourth-order power series polynomial:112
𝑧(𝑥)=𝑐1𝑥2+𝑐2𝑥4+𝑐3𝑥6+𝑐4𝑥8.(2)
The least squares coefficients were found using a simplex search optimization method, as
113
implemented with fminsearch MATLAB function. The absolute errors of the fitting, as a function
114
of radial coordinate, are shown in Fig. 1.115
0 0.5 1 1.5 2 2.5 3
Radial coordinate (mm)
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Fitting error ( m)
Fig. 1. Fitting error (absolute errors) of wavefront data when using Eq. 2 in the example
case.
2.2. Circle involutes embedded in an axial symmetric wavefront116
We consider a regular curve
𝛾:𝐽→R2
defined on some interval
𝐽⊂R
, where by regular we
117
mean
¤𝛾1(𝑠)2+ ¤𝛾2(𝑠)2≠0
for every
𝑠∈𝐽
. In this section, the dot indicates the derivative with
118
respect to the parameter
𝑠
. We assume that the coordinates in
R2
are
(𝑥, 𝑧)
, that the curve can be
119
written as 𝛾(𝑠)=(𝛾1(𝑠), 𝛾2(𝑠)) for 𝑠∈𝐽, and that 𝛾1>0at least in the interior of 𝐽.120
We also consider the parameterization
𝜓:𝐽× [0,2𝜋] → R3
of the associated wavefront of
121
revolution (given by Eq. 2) obtained by rotating the curve 𝛾around the 𝑧-axis:122
𝜓(𝑠, 𝜃)=(𝛾1(𝑠)cos 𝜃, 𝛾1(𝑠)sin 𝜃 , 𝛾2(𝑠)).(3)
The tangent plane of the wavefront, at every point (𝑠, 𝜃), is generated by the vectors:
( ¤𝛾1(𝑠)cos 𝜃, ¤𝛾1(𝑠)sin 𝜃 , ¤𝛾2(𝑠)),(−𝛾1(𝑠)sin 𝜃, 𝛾1(𝑠)cos 𝜃, 0).
Hence the Riemannian metric induced in the surface by the Euclidean one is given, in
(𝑠, 𝜃)123
coordinates, by124
©«
𝑓20
0𝑔2ª®¬:=©«
( ¤𝛾1)2+ ( ¤𝛾2)20
0𝛾2
1ª®¬.(4)
A geodesic in this metric is a curve
𝛿(𝑡):=(𝑠(𝑡), 𝜃(𝑡))
satisfying the system of second order
125
ordinary differential equations [26]126
𝑠′′ +¤
𝑓
𝑓(𝑠′)2−𝑔¤𝑔
𝑓2(𝜃′)2=0,
𝜃′′ +2¤𝑔
𝑔𝑠′𝜃′=0.
(5)
In these formulas, the symbol
′
denotes the derivative with respect to
𝑡
. A geodesic is uniquely
127
determined by their initial values
𝛿(0)=(𝑠(0), 𝜃(0))
,
𝛿′(0)=(𝑠′(0), 𝜃′(0))
. It is easily checked
128
that if 𝛿(𝑡)=(𝑠(𝑡), 𝜃 (𝑡)) is a geodesic, then129
𝛿𝜃0(𝑡)=(𝑠(𝑡), 𝜃(𝑡) + 𝜃0)(6)
is also a geodesic since it satisfies Eqs.
(5)
. The curve
𝛿𝜃0
is identified with the geodesic obtained
130
by rotating
𝛿
by an angle
𝜃0
. The initial conditions of the geodesic
𝛿𝜃0
are
𝛿𝜃0(0)=𝛿(0) + (0, 𝜃0)131
and 𝛿′
𝜃0(0)=𝛿′(0).132
Let us fix some
𝑠0∈𝐽
. Then the curve
𝜃↦→ (𝑠0, 𝜃 )
is sent by the parameterization
𝜓
to the
133
circle of revolution134
𝜃↦→ (𝛾1(𝑠0)cos 𝜃, 𝛾1(𝑠0)sin 𝜃, 𝛾2(𝑠0)) (7)
of length
2𝜋𝛾1(𝑠0)=2𝜋𝑔(𝑠0)
. The curve
𝜃↦→ (𝑠0, 𝜃 )
can be parameterized by arc-length
135
defining 𝑐:[0,2𝜋𝑔 (𝑠0)] → R2by:136
𝑐(𝑡):=(𝑠0, 𝑡/𝑔(𝑠0)).(8)
To compute the involute around the curve
𝑐
we consider the geodesics
𝛿𝑡
in the plane with
137
coordinates (𝑠, 𝜃)satisfying the initial conditions 𝛿𝑡(0)=𝑐(𝑡),𝛿′
𝑡(0)=−𝑐′(𝑡). That is138
𝛿𝑡(0)=(𝑠0, 𝑡/𝑔(𝑠0)), 𝛿′
𝑡(0)=(0,−𝑔(𝑠0)−1).(9)
We fix the geodesic
𝛿0(𝑟)=(𝑠(𝑟), 𝜃(𝑟))
corresponding to the value
𝑡=0
. As observed after
Eqs.
(5)
, for any value
𝜃0
, the curve
(𝛿0)𝜃0
defined in Eq.
(6)
is a geodesic. Taking
𝜃0=𝑡/𝑔(𝑠0)
,
the initial conditions of (𝛿0)𝑡/𝑔(𝑠0)are
(𝛿0)𝑡/𝑔(𝑠0)(0)=𝛿0(0) + 0,𝑡
𝑔(𝑠0)=𝑠0,𝑡
𝑔(𝑠0)=𝛿𝑡(0),
(𝛿0)′
𝑡/𝑔(𝑠0)(0)=𝛿′
0(0)=0,−1
𝑔(𝑠0)=𝛿′
𝑡(0).
By the uniqueness of geodesics with the same initial conditions we get (𝛿0)𝑡/𝑔(𝑠0)=𝛿𝑡and so139
𝛿𝑡(𝑟)=(𝛿0)𝑡/𝑔(𝑠0)(𝑟)=𝑠(𝑟), 𝜃(𝑟) + 𝑡
𝑔(𝑠0).(10)
Hence the equation of the involute about the geodesic circle (8) is given by140
𝑡↦→ 𝛿𝑡(𝑡)=𝛿0(𝑡) + 0,𝑡
𝑔(𝑠0)=𝑠(𝑡), 𝜃(𝑡) + 𝑡
𝑔(𝑠0).(11)
Therefore it is enough to compute the geodesic
𝛿0(𝑟)=(𝑠(𝑟), 𝜃(𝑟))
to obtain the involute. The
141
corresponding curve in the wavefront is given by142
𝑡↦−→ 𝜓𝑠(𝑡), 𝜃 (𝑡) + 𝑡
𝑔(𝑠0).
The surface associated with Eq. 2 is:143
𝛾1(𝑠)=𝑠, (12)
𝛾2(𝑠)=𝑐1𝑠2+𝑐2𝑠4+𝑐3𝑠6+𝑐4𝑠8.
Going back to the geodesic (circle involute) curve, eqs. 5 were solved numerically using a
144
built-in MATLAB function: ode45.m, which generates a non-uniform distribution of points:
145
(𝑥𝑖, 𝑦𝑖, 𝑧 𝑗)𝑖=1, ...𝑛
. Next, we interpolate those points imposing equal arc-length separation.
146
The arc-length separation is computed by:147
𝛿𝑖=(𝑥𝑖+1−𝑥𝑖)2+ (𝑦𝑖+1−𝑦𝑖)2+ (𝑧𝑖+1−𝑧𝑖)2.(13)
The total length is
𝐿=Í𝑛
𝑖=1𝛿𝑖
. Then, a uniform arc-length step is given as
𝛿𝑢=𝐿/(𝑛+1)
.
148
We obtained the equal arc-length points of the curve (
(𝑥𝑢, 𝑦𝑢, 𝑧𝑢)𝑢=1, .., 𝑚
) by separately
149
interpolating their
𝑠
and
𝜃
parameters using spline interpolation. Specifically, we used the interp1
150
MATLAB function with the splines option. Finally, the curve points were obtained by applying
151
again Eq. 3.152
For illustrative purposes, Fig. 2 shows a uniform equally arc-length circle involute within the
153
asphere of Eq. 2, as computed with the aforementioned procedure.154
2.3. Pseudo-umbilical tubular neighborhood of the circle involute155
Once we have built the set of points of the circle involute
𝛾:=(𝑥𝑢, 𝑦𝑢, 𝑧𝑢)
, the next step is to
156
perturb them so that the modified curve (say
𝛽
) has a prescribed focal curve. We show a scheme
157
of this procedure in Fig.3. We set that the focal curve is a line along the optical axis at the image
158
space: say, the set of points
𝐶𝛽:=(0,0, 𝜁𝑖)
. This intrinsically prescribes a radius of curvature
159
function given by 𝜁, and discretized by points 𝜁𝑖.160
Then we establish a one-to-one mapping between
𝛽
(red solid line in Fig.3) and
𝐶𝛽
(points
𝜁1
161
and
𝜁2
in Fig.3). For
𝐶𝛽
to be a focal curve of
𝛽
, the points of the former must be the centers of
162
the osculating spheres of the latter (dashed circles in Fig.3). Then we impose that the distance
163
from 𝛽to 𝐶𝛽is given by 𝜁. We transform 𝑐into 𝛽by perturbations along the normal to 𝛾.164
Fig. 2. Equal arc-length spaced circle involute points (blue dots) embedded in asphere
provided by Eq. 2 (colormap yellow-red). Axes scales are not equal for better visualizing
how the circle-involute fills in the surface.
The new curve
𝛽
is not necessarily an umbilical curve embedded in the wavefront. However,
165
we can at least approximate it to umbilicity, building up neighboring points to
𝛽
with some
166
properties. Parametrizing
𝛽
with arc length, the Darboux frame defines a positively oriented
167
orthonormal basis attached to each point
𝑝∈𝛽
along wavefront
𝑆
, which vectors are: 1) the unit
168
tangent:
T(𝑠)=𝛽′(𝑠)
; 2) the unit normal to
𝑆
at
𝑝
:
N(𝑠)
; and 3) the tangent normal given by
169
cross product:
B(𝑠)=T(𝑠) × N(𝑠)
. Now, if
𝛽
is a curve of umbilical points, then it is possible to
170
define a local differentiable
(𝐶∞)
chart in a small tubular neighborhood of
𝛽
as a second-order
171
Taylor series expansion [18,19]:172
𝑊(𝑠, 𝑣)=𝛽(𝑠) + 𝑣B(𝑠) + 𝑘(𝑠)
2𝑣2N(𝑠),(14)
where
𝑘(𝑠)
is the normal curvature at
𝑝
, to which is assigned the value of
𝜁
.
𝑊(𝑠, 𝑣)
provides
173
a parametrization of that small tubular neighborhood close to the point
𝑝∈𝛽
. We computed
T(𝑠)174
by numerically differentiating
𝛾
with respect to arc length (by means of finite differences).
N(𝑠)175
is prescribed as the unitary version of the vector joining each pair of points of the one-to-one
176
mapping between
𝛽
and
𝐶𝛽
. Finally,
B(𝑠)
is obtained, as already mentioned, by the cross product
177
of the previous unitary vectors. Particularly, we computed one point of this chart for each point
178
of 𝛽.179
2.4. Zernike polynomials least squares fitting of the wavefront surface180
The merging of the points of the previous steps, namely the points of
𝛽
and its neighborhood points
181
in the tubular chart, provides a discretization of the wavefront:
𝑆:=(𝑥𝑗, 𝑦 𝑗, 𝑧 𝑗)𝑗=1, ...𝑛
.
182
Next, we fit that point cloud with a polynomial surface representation. Specifically, we used
183
Zernike polynomials, as they are classically used in optics; we followed the polynomial standard
184
Fig. 3. Transformation of circle involute within asphere (
𝛾
depicted with black solid
line) into a space curve with a prescribed focal curve (𝛽depicted with red solid line).
reported in [30]. The fitting scheme we used was the least squares approximation, which sets the
185
following linear system of equations:186
𝑧𝑗=
𝑗
𝐶𝑖𝑍𝑖(𝑥𝑗, 𝑦 𝑗)𝑗=1, ...𝑛, 𝑖 =1, ..𝑚, (15)
where
𝑍𝑖
denotes a Zernike polynomial,
𝐶𝑖
its associated coefficient and the index
𝑚
is the
187
number of Zernike polynomials used in the fitting. Eq. 15 can be expressed in matrix form as:188
z=𝐴∗c,(16)
where
z
is column vector containing
𝑧𝑗𝑗=1, ...𝑛
, and
𝑐
is a column vector containing the
𝑚189
Zernike coefficients. Matrix
𝐴
(dimension:
𝑛 𝑥 𝑚
) is, in general, overdetermined. We computed
190
𝑐
by a two steps procedure. First, we applied the pseudoinverse (Moore–Penrose inverse) to
191
matrix A, i.e. the least squares solution to the system of linear equations given by Eq. 16. The
192
pseudoinverse was computed using a built-in MATLAB function: pinv.m. Second, we refine the
193
solution by a simplex search optimization method as implemented with fminsearch MATLAB
194
function.195
2.5. Caustic construction from Zernike expansion196
We denote the caustic sheets with
𝜒
(sagittal) and
𝜓
(tangential). They can be computed applying
197
the following sequence [16].198
1.
Compute the coefficients of first (
𝐸, 𝐹 , 𝐺
) and second (
𝐿, 𝑀 , 𝑁
) fundamental forms of the
199
Zernike polynomial wavefront with cartesian parametric representation, using first, second
200
derivatives and normals (N) of the surface with respect x and y [16].201
2.
Compute the mean (
𝐻
) and Gaussian (
𝐾
) curvatures from
𝐸, 𝐹 , 𝐺
and
𝐿, 𝑀 , 𝑁
using
202
equations:203
𝐻=𝐸 𝑁 −2𝐹 𝑀 +𝐺 𝐿
𝐸𝐺 −𝐹2(17)
𝐾=𝐿𝑁 −𝑀2
𝐸𝐺 −𝐹2
3. Obtain the two principal curvatures (𝑘 𝑛) from 𝐻and 𝐾applying:204
𝑘1=𝐻+𝐻2−𝐾(18)
𝑘2=𝐻−𝐻2−𝐾
4. Finally, the equations providing caustic curves, are:205
𝜒(𝑥, 𝑦, 𝑧)=𝑊(𝑥, 𝑦, 𝑧) + 𝑘1(𝑥, 𝑦, 𝑧)N(𝑥, 𝑦, 𝑧)(19)
𝜉(𝑥, 𝑦, 𝑧)=𝑊(𝑥, 𝑦, 𝑧) + 𝑘2(𝑥, 𝑦, 𝑧)N(𝑥, 𝑦, 𝑧)
To characterize how the final wavefront deviates from being umbilical at all points, i.e. one
206
provided a single caustic spike, we computed the difference between the two sheet caustic locations,
207
which is itself a surface that from now on we refer to as caustic astigmatism. Additionally, as an
208
average metric evaluation of that condition, we used the root-mean-square of caustic astigmatism,
209
defined as:210
𝑅𝑀𝑆𝑎=Í𝑛
𝑖=1(𝜒𝑥
𝑖−𝜉𝑥
𝑖)2+ ( 𝜒𝑦
𝑖−𝜉𝑦
𝑖)2+ ( 𝜒𝑧
𝑖−𝜉𝑥
𝑖)2
𝑛,(20)
where 𝜒and 𝜉super-indexes denote the 𝑥, 𝑦, 𝑧 components of the caustic location.211
Additionally, we also tracked the difference between the sagittal caustic (
𝜒
) –the caustic sheet
212
closer to the optical axis – and the nominal prescribed depth-of-focus line; this surface is from
213
now on referred to as sagittal error. And, as before, we used the root-mean-square as an average
214
metric:215
𝑅𝑀𝑆𝑥=Í𝑛
𝑖=1(𝜒𝑥
𝑖)2+ ( 𝜒𝑦
𝑖)2+ ( 𝜒𝑧
𝑖−𝑔(𝑥))2
𝑛,(21)
where 𝑔(𝑥)is the z-coordinate of the prescribed focal line as given by Eq. 1.216
3. Simulations and numerical results217
We generated two different wavefronts associated with the example case described in section
218
2. First, a depth-of-focus wavefront, where we applied all steps of the proposed method except
219
the one pursuing the pseudo-umbilical condition (subsection 2.3). Second, a depth-of-focus
220
wavefront where we fully applied the proposed method including the step trying to attain the
221
pseudo-umbilical condition along the embedded circle involute. For reasons of economy of
222
notation, from now on we will refer to the former as the circle-involute generated wavefront
223
(CIGE), and the latter as the circle-involute pseudo-umbilical generated wavefront (CIPU).224
We compared the optical performance of both, algorithms and their output wavefronts (CIGE
225
and CIPU), in terms of caustic astigmatism and sagittal error and their associated metrics:
𝑅𝑀𝑆𝑎
226
and
𝑅𝑀𝑆𝑥
. We did that in order to analyze the always-present trade-off in multifocal wavefront
227
design between astigmatism correction and axial focal prescription (see discussion at [8]). The
228
results depend on several numerical sampling patterns, being the most relevant one the size of
229
the circle generating the circle-involute curve, i.e. the parameter
𝑠0
, which controls how dense
230
the wavefront surface is sampled by the involute circle.231
We empirically found that the most optimal results are obtained for
𝑠0
ranging between
0.7232
and
1.1𝜇𝑚
. We also found that the optimal number of Zernike least-squares polynomial fitting
233
is (fifth-order expansion). This may be due to the fact that using a high number of polynomials
234
sometimes generates ill-conditioned reconstructed matrices, a typical problem in least-squares
235
reconstruction.236
Fig. 4. Caustic astigmatism (mm) of (a-b) CIGE and (c-d) CIPU wavefronts when
𝑠0=0.7𝜇𝑚
. Lateral projections on an (a-c) xz-plane and (b-d) yx-plane. Colormaps
of the surfaces are proportional to the 𝑅 𝑀 𝑆 𝑎of caustic astigmatism.
Table 1 shows comparing results of
𝑅𝑀𝑆𝑎
and
𝑅𝑀𝑆𝑥
for the two wavefronts under test: CIGE
237
and CIPU. The data in the table reveals that while for the CIPU wavefront, both
𝑅𝑀𝑆𝑎
and
238
𝑅𝑀𝑆𝑥are highly dependent on 𝑠0, those values are more stable for the CIGE wavefront.239
Reducing the value of
𝑠0
substantially reduces
𝑅𝑀𝑆𝑎
in the CIPU wavefront, both in absolute
240
terms –a
56.46%
from
𝑠0=1.1
to
𝑠0=0.7𝜇𝑚
– and in comparison to the CIGE wavefront: at
241
𝑠0=1.1
the
𝑅𝑀𝑆𝑎
is
16.04%
smaller but at
𝑠0=0.7
this reduction is improved up to
63.01%
.
242
However, that
𝑅𝑀𝑆𝑎
reduction when reducing
𝑠0
is achieved at the expense of a slight increase
243
in the value of
𝑅𝑀𝑆𝑥
; a
13.3%
from
𝑠0=1.1
to
𝑠0=0.7𝜇𝑚
. On the other hand, for the CIGE
244
there is only a small difference in
𝑅𝑀𝑆𝑎
:
1.15%
between the two limit
𝑠0
values. Significantly,
245
although for CIGE
𝑅𝑀𝑆𝑥
is always bellow
𝑅𝑀𝑆𝑎
, this does not occur with CIPU, where
𝑅𝑀𝑆𝑎
246
is substantially smaller (52.61%) than 𝑅𝑀 𝑆𝑥at 𝑠0=0.7𝜇𝑚.247
𝑅𝑀𝑆𝑥
is always smaller for CIGE with respect to CIPU: at
𝑠0=1.1
the relative difference, in
248
percentage, is 14.7%, and, at 𝑠0=0.7this difference is increased up to 25.15%.249
We analyzed the surface structure of caustic astigmatism and sagittal error when
𝑠0=0.7𝜇𝑚
.
250
The lateral projections on an xz-plane and y-x plane of these surfaces are shown in figures 4
251
and 5, respectively. The colormap mappings of caustic astigmatism and sagittal surfaces are
252
proportional to the 𝑅 𝑀 𝑆𝑎and 𝑅 𝑀 𝑆 𝑥, respectively.253
Caustic astigmatism contains a cusp point (singular point) where both caustic sheets meet
254
(
𝑅𝑀𝑆𝑎=0
). From that point, caustic astigmatism evolves forming a horn-shaped surface.
255
Comparing CIGE and CIPU caustic astigmatism, there is a considerable reduction in the surface
256
size but the horn-shaped morphology is preserved.257
In both, CIGE and CIPU, the separation of the sagittal error from the optical axis is small;
258
𝑠0(𝜇𝑚) 0.7 0.8 0.9 1 1.1
𝑅𝑀𝑆𝑎
(CIGE) 513.72 513.11 512.44 512.99 519.71
𝑅𝑀𝑆𝑎
(CIPU) 190.00 242.20 302.11 365.8 436.34
𝑅𝑀𝑆𝑥
(CIGE) 300.09 300.85 301.27 300.95 301.81
𝑅𝑀𝑆𝑥
(CIPU) 400.91 387.84 375.25 363.95 353.83
Table 1.
𝑅𝑀 𝑆𝑎
and
𝑅𝑀 𝑆𝑜
optical performance values as function of
𝑠0
for CIGE and
CIPU multifocal wavefronts.
always below
1.5𝜇𝑚
. Most of the error is concentrated in differences along the z-coordinate
259
as clearly shown in 5(a-c). In both cases, the sagittal error is highly oscillatory, but the higher
260
values for CIPU, in comparison to CIPU, are due to a higher z-coordinate extension of sagittal
261
error.262
4. Discussion263
We have presented a design method, based on differential geometry, aiming at designing depth-of-
264
focus wavefronts where the optical power variation –associated to mean curvature variation– is
265
produced, not along a meridian as occurs in classical axial axicons, but along a spiral-type circle
266
involute filling the surface. A classical axicon, i.e. a conical surface, has two major drawbacks:
267
first, not equal areas of the refractive surface provide uniform intensity, and, second, the axial
268
symmetry makes the design sensitive to pupil variations. Our proposal overcomes both issues.269
Our method can also, as an alternative, control the separation between caustic sheets (hence
270
improving the energy concentration) by means of pursuing pseudo-umbilicity along the circle
271
involute space curve. The radius of the circle generating the involute is a critical factor to control
272
the separation between caustic sheets. As seen in section 3, reducing its value helps to decrease
273
that difference. However, this has a theoretical threshold, because when the distance between
274
the circle involute loops is too close the umbilicity condition of the tubular neighbors starts
275
to interfere and, as mentioned in section 1, the impossibility of a smooth surface to contain
276
an extended region of umbilical points becomes apparent. It is important to point out that
277
the reduction of the separation between caustic sheets obtained by our method faces a design
278
trade-off; namely, the amount of concentration of light along the prescribed axis segment and the
279
separation between caustics. This trade-off, already discussed in axially symmetric multifocal
280
wavefronts [8], is intrinsic to multifocality because of the general laws of differential geometry in
281
smooth surfaces [10]. However, we comment that the error in sagittal, concentrated along the
282
z-coordinate as analyzed in section 3, could be, in practice, corrected a posteriori by means of
283
readjusting the mean curvature at the apex of the wavefront.284
5. Disclosures285
The authors declare no conflicts of interest.286
6. Data Availability Statement287
Data availability. No data were generated or analyzed in the presented research.288
Fig. 5. Sagittal errors (
𝜇𝑚
) of (a-b) CIGE and (c-d) CIPU wavefronts when
𝑠0=0.7
𝜇𝑚
. Lateral projections on an (a-c) xz-plane and (b-d) yx-plane. Colormaps of the
surfaces are proportional to the 𝑅 𝑀 𝑆 𝑥of sagittal errors.
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