Optics Communications 247 (2005) 11–18
www.elsevier.com/locate/optcom
Three-dimensional control of the focal light intensity
distribution by analytically designed phase masks
Vidal F. Canales *, José E. Oti, Manuel P. Cagigal
Departamento de Fı́sica Aplicada, Universidad de Cantabria, Los Castros S/N, 39005 Santander, Spain
Received 17 September 2004; received in revised form 8 November 2004; accepted 11 November 2004
Abstract
An analytical method for a total control of the light-intensity distribution at the focus of an optical system by simple
pupil-plane phase mask is presented. The masks are composed of three annuli that only reach 0 or p values. A relevant
advantage of this method is that the parameters of the mask are analytically obtained from the figures of merit that
characterize the focal intensity distribution, as the Strehl ratio, the axial and the transverse gains. As a consequence,
the method allows an easy and complete analysis of the masks performance. We demonstrate that these masks allow
us to attain a complete reshaping of the spatial intensity distribution, including apodization or superresolution in the
axial or transverse directions and, especially, three-dimensional superresolution. Finally, we derive the Strehl limit in
the superresolution regime.
Ó 2004 Elsevier B.V. All rights reserved.
PACS: 42.79.Ci; 42.25.Fx; 42.30.Va
Keywords: Filters; Zone plates and polarizers; Diffraction and scattering; Image forming and processing
1. Introduction
The control of the light intensity distribution
near the geometrical focus of an optical system is
an interesting topic in a number of applications.
In recent years, special attention has been paid to
several applications which take advantage of opti*
Corresponding author. Tel.: +34 942 201445; fax: +34 942
201402.
E-mail address: [email protected] (V.F. Canales).
cal superresolution. The design of pupil masks has
been approached from quite different points of
view. Some masks are based on variable transmittance pupils [1–3] or on hybrid amplitude-phase
profiles [4,5] while other use phase-only profiles because these perform better than transmittance
masks [6]. Phase-only masks can be classified in
continuous [7,8] and annular profiles [9–11]. The
latter are more frequently used because of their
in comparison, best performance and simplicity.
Furthermore, they are easy to manufacture for
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2004.11.051
12
V.F. Canales et al. / Optics Communications 247 (2005) 11–18
mass production using available diffractive optics
production methods and replication technologies.
Some examples of annular masks are the diffractive superresolution elements [9], the three zones
binary phase filters designed to increase the data
storage density in DVDs [10] or the N-zones binary filters that provide transverse or axial superresolution [11]. Nonetheless, all the previous
designing methods share a drawback: a fitting process is required to obtain the phase-filter parameters desired for a particular application.
Consequently, the design procedure implies a long
processing time.
Hence, it is of great interest to design pupil
masks whose parameters are analytically derived
from the figures of merit desired for the light intensity distribution [12]. We introduce a fast and easyto-perform procedure for the analytical design of
superresolving binary phase-only pupil masks.
We consider three zones annular phase-masks so
that the phase can only take zero or p value. We
imposed such restrictive condition because these
filters present mathematical properties that can
be fruitfully used. Despite their simplicity, these
masks allow us to attain a great variety of light
intensity distributions, including axial and transverse apodization, axial and transverse superresolution
and,
especially,
three-dimensional
superresolution. Furthermore, these binary phase
masks are particularly interesting since they provide very good transversal and axial resolution
with fairly high Strehl [13].
The only parameters required to describe the
masks are the radii of the boundary between zones.
These radii can be straightforwardly obtained from
the figures of merit required in a particular application. Thus, the main advantage of the technique is
that it allows for the first time the analytical study
of the phase masks behavior in terms of the masks
designing parameters. As a consequence, the technique can be usefully applied to design masks with
any specific behavior in many different fields as for
example in microscopy, in optical tweezers, data
storage, etc. Finally, they allow us the attainment
of three dimensional superresolution, which can
be relevant in 3D imaging [13].
Our designing method consists in the derivation
of the parameters of the phase mask from the fig-
ures of merit that describe the light intensity distribution near the geometrical focus: the Strehl ratio
and the transverse and axial gain factors [8,14,15].
The Strehl ratio S is a relevant parameter for analyzing image quality and is defined as the ratio of
the intensity at the focal point to that corresponding to an unobstructed pupil. The transverse gain
GT (defined as in [8,15]) gives a measure of the
superresolution performance in the transverse
direction. The same applies for the axial gain,
GA. These factors are normalized so that for the
unobstructed pupil they are equal to unity. In a
particular direction, a filter is superresolving when
the corresponding gain is greater than unity and it
is an apodizer when the corresponding gain is
lower than unity.
In Section 2, the theoretical procedure that allows the derivation of the mask radii from the figures of merit is explained. In Section 3, we analyze
the radii values that yield transverse apodization
or superresolution, and in Section 4 those that
yield axial apodization or superresolution. In Section 5, we analyze three dimensional superresolution. In every case, we derive the maximum
Strehl for a fixed superresolution value.
2. Theory for mask design
In this section, the designing method of the
masks that allow us to control the light intensity
distribution near focus is developed. In this theoretical development the mask parameters are derived from the figures of merit that characterize
the diffracted intensity distribution near geometrical focus [8,11]. These figures of merit can be obtained within the framework of the scalar
diffraction theory. Let us consider a general complex pupil function P(q) = T(q) exp [i/q)], where
q is the normalized radial coordinate over the pupil plane, T(q) is the transmittance function and /
(q) is the phase function. For a converging monochromatic spherical wave front passing through
the center of the pupil, the amplitude U in the focal
region may be written as
Z 1
U ðv; uÞ ¼ 2
P ðqÞJ 0 ðvqÞ expðiuq2 =2Þq dq;
ð1Þ
0
V.F. Canales et al. / Optics Communications 247 (2005) 11–18
where v and u are radial and axial dimensionless
optical coordinates with origin at the geometrical
focus, given by v = kNAr and u = kNA2z. NA is
the numerical aperture, k = 2p/k and r and z the
usual radial and axial distances. The intensity distribution provided by this filter is expanded in series near the geometrical focus. The transverse and
axial intensity distributions can be expressed, to
the second order, as:
2
I T ðv; 0Þ ¼ jI 0 j 12ReðI 0 I 1 Þv2 ;
2
2
I A ð0; uÞ ¼ jI 0 j ImðI 0 I 1 Þu 14½ReðI 0 I 2 Þ jI 1 j u2 ;
ð2Þ
here * denotes complex conjugate and In is the nth
moment of the pupil function, defined as
Z 1
2P ðqÞq2nþ1 dq:
ð3Þ
In ¼
0
An annular phase filter consists of a number of
zones, each one determined by its radius and
phase. In this paper, we consider binary filters in
which the phases are p and 0. We choose these values because the pupil moments become real-valued
functions and then the general expressions of the
figures of merit simplify in this form:
uF S ¼ jI 0 j2 Im I 0 I 1 ¼ I 20 ;
2
Re I 0 I 1 uF Im I 0 I 2
I1
GT ¼ 2
¼2 ;
ð4Þ
S
I0
Re I 0 I 2 jI 1 j2
I 0 I 2 I 21
¼ 12
GA ¼ 12
;
S
I 20
Im I 0 I 1
¼ 0;
uF ¼ 2 Re I 2 I 0 jI 1 j2
where uF is the focus displacement and is always
zero for this kind of filters. It is worth noting
that these expressions are equal to those obtained for amplitude filters [15]. Consequently,
the procedure for phase filters shown in this
work could be useful for amplitude filters too.
The only parameters required for describing the
masks we propose are the radii of the boundaries between zones, qi (where q0 = 0 and qM = 1 in a mask
with M zones). The moments of the pupil function,
Eq. (3), can be expressed in terms of these radii,
13
M
2 X
2nþ2
ð1ÞMi qi2nþ2 qi1
2n þ 2 i¼1
!
M 1
X
ð1ÞLZ
Mi 2nþ2
¼
1þ2
ð1Þ qi
;
ð5Þ
nþ1
i¼1
I n ¼ ð1ÞLZ
where LZ = 0 for every filter with 0 phase in the outest annulus, (q 2 qM 1,1), and LZ = 1 for the complementary filters with p phase in that annulus.
The simplest mask of this kind would be a
two-zones mask. However, we choose a threezones mask because it allows us to obtain any
desired intensity distribution (axial and/or transverse superresolution and apodization) while the
two-zones mask does not, as will be shown.
Then, if we introduce Eq. (5) with M = 3 in
Eq. (4), the figures of merit are in terms of the
mask radii, q1 and q2:
2
S ¼ 1 2q22 þ 2q21 ;
GT ¼
GA ¼
1 2q42 þ 2q41
;
1 2q22 þ 2q21
2
4 1 2q62 þ 2q61 1 2q22 þ 2q21 3 1 2q42 þ 2q41
ð1 2q22 þ 2q21 Þ
2
:
ð6Þ
These functions are identical for both complementary masks so it is not necessary to distinguish
between them. The radii that determine the mask
can be analytically derived from these expressions
of the figures of merit. Several cases can be distinguished, and they are analyzed in the following
sections. First, we may be interested on a certain
transverse gain while maintaining an acceptable
Strehl. The derivation of the mask parameters in
such case is studied in Section 3. Second, the analogous case in the axial direction is studied in
Section 4. Finally, the attainment of both axial
and transverse (3D) superresolution is analyzed
in section 5.
3. Mask design from GT and S
A remarkable case is the derivation of the mask
radii from GT and S because transverse superresolution and apodization are of practical interest in
V.F. Canales et al. / Optics Communications 247 (2005) 11–18
many different applications. In such case, the system composed of the two first expressions of Eq.
(6) must be solved. Two different solutions (denoted
by subscripts a and b, respectively) are attained:
pffiffiffi!1=2
1 S 2ðGT 1Þ S
pffiffiffi
q1a ¼
;
4ð1 S Þ
ð7aÞ
pffiffiffi!1=2
3 þ S 2ðGT þ 1Þ S
pffiffiffi
;
q2a ¼
4ð1 S Þ
1
0.75
S lim
14
0.5
0.25
0
q1b ¼
q2b ¼
pffiffiffi!1=2
1 S þ 2ðGT 1Þ S
pffiffiffi
;
4ð1 þ S Þ
pffiffiffi!1=2
3 þ S þ 2ðGT þ 1Þ S
pffiffiffi
:
4ð1 þ S Þ
1
2
1=2
3
4
5
GT
ð7bÞ
These solutions are valid when both normalized
radii, q1 and q2, belong to the interval [0,1]. As a
consequence, only those values of S and GT that
fulfill the previous condition can be attained. Then
if we fix a certain value of transverse superresolution GT, (GT > 1), the maximum possible Strehl is,
S lim ¼ ½½ðGT 1Þ þ 1
2
2
ðGT 1Þ :
ð8Þ
This function is shown in Fig. 1 as a function of
GT. It can be seen that the Strehl decreases for
increasing values of the transverse superresolution.
This Strehl limit is identical for both solutions, a
and b. Furthermore, this limit is attained when
q1a = 0 and when q2b = 1, which means for a
two-zones mask. We have also tried masks with
more than three zones using a fitting procedure
[11] but we have not found a mask with S higher
than Slim. Hence, from this analysis we can conclude that two-zones masks are the best option
for transverse superresolution.
It must be stated that the central core of the
PSF of both solutions, a and b, are identical (be-
Fig. 1. Maximum attainable Strehl as a function of the
transverse gain.
cause S and GT are identical) but the secondary
lobes are generally not. This is a key fact to choose
between both solutions, because it is desirable to
maintain the sidelobe intensity as low as possible
in relation to the peak intensity. Usually, a sidelobe value greater than 10% of the peak intensity
is not acceptable for most applications. In addition, the axial gain for both solutions is also different, which can be relevant when choosing between
them. An example of the application of the procedure is shown in Figs. 2 and 3. We choose
GT = 1.14 and S = 0.45 (the Strehl limit for this
gain is 0.756). For these values the radii of solutions a and b are (q1a = 0.524, q2a = 0.663) and
(q1b = 0.332, q2b = 0.973). Fig. 2 shows the corresponding transverse PSFs. The different height of
the secondary lobe of the two solutions can be
seen. Fig. 3 shows the axial PSFs. In this case,
one of the solutions yields axial superresolution
(GAa = 1.3) while the other does not (GAb = 0.21).
It is also very interesting to study which values
of the radii, (q1 andq2), give transverse superresolution. They can be obtained from Eq. (6), by
imposing GT(q1, q2) P 1:
9
hpffiffiffiffiffiffiffiffiffiffiffiffiffi i
>
2
>
1 q1 ; 1 if q1 < 0:5; >
or q2 2
=
GT ðq1 ; q2 Þ P 1 ()
q
ffiffiffiffiffiffiffiffiffi
h pffiffiffiffiffiffiffiffiffiffiffiffiffii
>
>
1þ2q21
>
>
>
;1
if q1 P 0:5: >
or q2 2
;
: q2 2 q1 ; 1 q21
2
8
qffiffiffiffiffiffiffiffiffi
>
1þ2q21
>
>
< q2 2 q1 ;
2
ð9Þ
V.F. Canales et al. / Optics Communications 247 (2005) 11–18
15
1
PSF
0.75
0.5
0.25
0
0
2
4
6
8
10
v
Fig. 2. Transverse cross-section of the PSF obtained using the
filter with radii q1 = 0.524 and q2 = 0.663 (solid curve) and
using the filter with radii q1 = 0.332 and q2 = 0.973 (circles). In
both cases GT = 1.14 and S = 0.45. The cross-section of the PSF
with no filter (dashed curve) is also shown for comparison.
Fig. 4. Values of the radii, (q1, q2), that yield transverse
superresolution: GT(q1, q2) P 1 (white area).
1
GA and S, and the system to be solved is composed
of the first and the third expressions of Eq. (6).
Once again, two different solutions (denoted by
subscripts a and b, respectively) are attained:
!1=2
3 3S R
pffiffiffi
q1a ¼
;
12ð1 S Þ
ð10aÞ
!1=2
pffiffiffi
9 þ 3S 12 S R
pffiffiffi
;
q2a ¼
12ð1 S Þ
PSF
0.75
0.5
0.25
0
-40
-20
0
20
40
u
Fig. 3. Axial cross-section of the PSF obtained using the filter
with radii q1 = 0.524 and q2 = 0.663 (solid curve) and using the
filter with radii q1 = 0.332 and q2 = 0.973 (circles). The crosssection of the PSF with no filter (dashed curve) is also shown
for comparison. The first filter yields axial superresolution
(GA = 1.3) while the other does not (GA = 0.21).
Fig. 4 shows the regions of transverse superresolution, defined by Eq. (9).
q1b ¼
q2b ¼
3 3S þ R
pffiffiffi
12ð1 S Þ
!1=2
;
!1=2
pffiffiffi
9 þ 3S 12 S þ R
pffiffiffi
;
12ð1 S Þ
ð10bÞ
where
h
i1=2
pffiffiffi
pffiffiffi
R ¼ 3ð1 S ÞðS 2 3S 3=2 þ 3S þ 3 S 4GA SÞ :
ð11Þ
4. Mask design from GA and S
In other applications, we may be interested in
axial superresolution and apodization. In such
case, the radii of the mask must be derived from
The radii given by Eqs. (10a) and (10b) do not
have physical meaning for all the values of S and
GA. Let us analyze the case of axial superresolution, GA > 1. The solutions will be valid provided
R is real, which means that S 2 3S 3=2 þ
16
V.F. Canales et al. / Optics Communications 247 (2005) 11–18
pffiffiffi
3S þ 3 S 4GA S > 0. Then if we fix GA we can
find the maximum Strehl that fulfills such
condition:
pffiffiffi 2
pffiffiffi
U
2GA
ð1 þ 3iÞ ;
S lim ¼ 1 ð1 þ 3iÞ þ
GA
U
ð12Þ
where
U¼
h
48G3A
þ 81G2A
i1=3
1=2
162GA þ 81
54 þ 54GA
;
ð13Þ
pffiffiffiffiffiffiffi
and i ¼ 1. Fig. 5 shows this Strehl limit as a
function of the axial superresolution gain. As expected, superresolution increases at the cost of
the decrease of the Strehl.
Solutions a and b give the same axial PSF core,
but different transverse PSF. When looking for axial superresolution GA > 1, solution a always gives
a superresolved transverse PSF, while solution b
provides GT < 1. Moreover, the gains of both solutions are related by GTa 1 = 1 GTb. To show
an example of the procedure let us choose
S = 0.45 and GA = 1.3. For these values the radii
of solutions a and b are (q1a = 0.524, q2a = 0.663)
and (q1b = 0.748, q2b = 0.851). Fig. 6 shows the
corresponding transverse PSFs. As predicted, the
PSF corresponding to solution a presents transverse superresolution with GTa = 1.14 while the
other solution presents GTb = 0.86. Finally, Fig. 7
shows that the axial PSF is identical for both
solutions.
It is also interesting to study which values of the
radii, (q1 andq2), give axial superresolution. They
can be obtained from Eq. (6), by imposing
GA(q1, q2) P 1,
GA ðq1 ; q2 Þ P 1 () q1 2 ½0:419; 0:888
and
q2 2 ½Maxðq1 ; V Þ; W ;
ð14Þ
where Max is a function whose value is the maximum of the terms inside the parentheses and
pffiffiffi
pffiffiffi
2
1
ð2 18q21 Þ
ð1 þ 3iÞ ;
V ¼ þ q21 þ Zð1 þ 3iÞ þ
3
2
9Z
p
ffiffi
ffi
pffiffiffi
2
1
ð2
18q21 Þ
2
ð1 3iÞ ;
W ¼ þ q1 þ Zð1 3iÞ þ
3
2
9Z
1
Z ¼ 28 216q21 þ 648q41 þ 36 1 20q21 þ 160q41 :
6
1=2 1=3
504q61 þ 324q81
:
ð15Þ
The interval for q1 in Eq. (14) is given by the
conditions: W > q1 and V, W exist. Fig. 8 shows
the regions of axial superresolution, defined by
Eq. (14). It is clear that two-zones masks (q1 = 0
or q2 = 1) do not yield superresolution performance in the axial direction.
1
0.75
PSF
1
0.5
0.75
S lim
0.25
0.5
0
0
0.25
2
4
6
8
10
v
0
1
2
3
4
5
GA
Fig. 5. Maximum attainable Strehl as a function of the axial
gain.
Fig. 6. Transverse cross-section of the PSF obtained using the
filter with radii q1 = 0.524 and q2 = 0.663 (solid curve) and
using the filter with radii q1 = 0.748 and q2 = 0.851 (circles). In
both cases GA = 1.3 and S = 0.45. The cross section of the PSF
with no filter (dashed curve) is also shown for comparison. The
first filter yields transverse superresolution, GT = 1.14, while the
other solution does not, GT = 0.86.
V.F. Canales et al. / Optics Communications 247 (2005) 11–18
1
GA ðq1 ; q2 Þ P 1 and GT ðq1 ; q2 Þ P 1
h
pffiffiffi i
() q1 2 0:419; ð1= 2Þ and
qffiffiffiffiffiffiffiffiffiffiffiffiffi q2 2 Maxðq1 ; V Þ; MinðW ; 1 q21 Þ ;
0.75
PSF
17
0.5
ð16Þ
0.25
0
-40
-20
0
20
40
u
Fig. 7. Axial cross-section of the PSF obtained using the filter
with radii q1 = 0.524 and q2 = 0.663 (solid curve) and using the
filter with radii q1 = 0.748 and q2 = 0.851 (circles). The crosssection of the PSF with no filter (dashed curve) is also shown
for comparison.
where Min is a function whose value is the minimum of the terms inside the parentheses. The three
dimensional superresolution region is shown in
Fig. 9. In addition Fig. 10 shows the axial gain
as a function of the transverse gain for different
values of the Strehl ratio. It can be seen that if
the Strehl ratio is high the superresolution gains
can not be very high. Moreover, for a fixed Strehl,
the axial gain decreases if the transverse gain increases and vice versa.
A mask that yields certain values of the axial
and transverse gains (GA, GT) can be designed
using the theoretical development of the previous
sections. The first step for this task, is to find the
point (GA, GT) in Fig. 10, and to infer the Strehl
ratio of the curve that crosses that point. Then
the radii of the mask can be obtained using Eqs.
(7a) and (7b) with the value of S and GT (or using
Eq. (10a) and (10b) with the value of S and GA).
As an example let us assume that we need
GT = 1.14 and GA = 1.3. From Fig. 10 the Strehl
Fig. 8. Values of the radii, (q1, q2), that yield axial superresolution: GA(q1, q2) P 1 (white area).
5. Three-dimensional superresolution
If we compare Figs. 4 and 8, we can realize that
there is a region of values of (q1, q2) that yield both
transverse and axial superresolution, which is a
very interesting behavior in many applications.
This region of radii corresponds to the intersection
between Eqs. (9) and (14):
Fig. 9. Region of values of (q1, q2) that yield both transverse
and axial superresolution: GT(q1, q2) P 1 and GA(q1, q2) P 1
(white area).
18
V.F. Canales et al. / Optics Communications 247 (2005) 11–18
yield optimum performance while for axial and
three-dimensional superresolution three-zones
masks are required. In every case, we have derived
the maximum Strehl as a function of the corresponding gain. The technique presented here is a
powerful tool for a large variety of applications
using these kind of masks, which are easy to design
and to manufacture.
3
2.5
GA
S = 0.1
S = 0.2
2
S = 0.3
1.5
1
1
1.25
1.5
1.75
2
Acknowledgments
GT
Fig. 10. Axial gain as a function of the transverse gain for
different values of the Strehl ratio. The Strehl S is 0.1 for the
curve which is further from the origin and, as distance from the
origin decreases, S increases 0.1 for each curve (thus, the last
curve corresponds to S = 0.9).
that corresponds to this situation is about 0.45.
Then, introducing these values of S and GT in
Eqs. (7a) and (7b) the radii of the mask can be obtained: q1a = 0.524, q2a = 0.663. The corresponding transverse PSF has already been shown in
Fig. 2 and the axial PSF in Fig. 7.
6. Conclusions
In conclusion, we have studied the performance
of simple binary phase-only masks. We have
shown that masks with only three annuli with 0
or p values can yield any desired behavior as axial
or transverse superresolution and apodization or
even three-dimensional superresolution. Furthermore, we have derived analytical expressions for
the mask parameters from the desired values of
the figures of merit that characterize the system
PSF. These analytical expressions allow a thorough analysis of the masks. We have shown that
for transverse superresolution two-zones masks
This research was supported by Ministerio de
Ciencia y Tecnologı́a AYA 2004-07773-CO2-01.
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