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J. Opt. Soc. Am. A / Vol. 20, No. 11 / November 2003
de Juana et al.
Transverse or axial superresolution in a 4Piconfocal microscope by phase-only filters
Daniel M. de Juana, José E. Oti, Vidal F. Canales, and Manuel P. Cagigal
Departamento de Fı́sica Aplicada, Universidad de Cantabria, Los Castros S/N, 39005 Santander, Spain
Received February 17, 2003; revised manuscript received July 2, 2003; accepted July 14, 2003
A novel procedure to design axial and transverse superresolving pupil filters for the 4Pi-confocal microscope is
presented. The method is based on the use of a series of figures of merit developed to describe the effect of
inserting two identical filters in the two arms of the illumination path of the microscope. As a practical implementation, we have applied our method to obtain superresolving continuous phase-only filters. Different
resolution-improving phase functions are shown for the transverse and the axial direction. These filters provided axial gain up to 1.3 and transverse gain up to 1.4 without an increase in sidelobes. © 2003 Optical
Society of America
OCIS codes: 170.1790, 100.6640, 110.1220, 220.1230, 350.5730.
1. INTRODUCTION
In a 4Pi-confocal microscope the light from a point source
is divided into two beams that are focused into the object
by two identical and opposing high-numerical-aperture
(NA) objectives.1,2 In this case the object is coherently illuminated, resulting in an illumination point-spread
function (PSF) determined by the pattern of two spherical
wave fronts interfering in the common focal volume. As
a consequence, an increase of the effective aperture along
the axial direction is achieved. The light scattered (emitted for fluorescent samples) from the object is collected by
the two objectives and is focused to a pointlike detector.
This setup leads to an axial PSF maximum that is approximately four times narrower than that corresponding
to a conventional confocal microscope. Unfortunately,
the narrowing of the central maximum is accompanied by
high axial sidelobes. These lobes compromise the actual
gain in axial resolution, since they produce periodic artifacts in the reconstructed image. Consequently, postprocessing techniques3 are required to reconstruct the threedimensional image of the scanned object. This data
deconvolution process imposes a restriction on the minimum level of the signals that can be tolerated in the image.
Pupil filtering is a useful tool for improving the resolution of optical systems, and, in fact, different pupil filters
have already been proposed for increasing the axial,4
transverse,5 and three-dimensional6 resolution of confocal
microscopes. Until now, most of the superresolving filters proposed for confocal systems are based on annular,
multiannular, or leaky annular designs.7,8 There are a
small number of publications concerning continuous filters. In this sense, Boyer9 has recently investigated the
application to confocal microscopy of continuously transmitting axially superresolving filters based on prolate
spheroidal wave functions. Furthermore, the use of amplitude annular Toraldo10 filters as axial sidelobe reducers in 4Pi-confocal microscopes has been recently
1084-7529/2003/112172-07$15.00
reported,11 and experimental results of the performance
of this kind of filter have been provided.12
To complete this analysis, we develop a procedure in
the present paper to design general pupil masks for controlling the focal volume in a 4Pi-confocal microscope.
By means of an expansion of the transverse and axial
light intensity distributions in power series near focus, we
define two figures of merit: the transverse and axial gain
factors, G T and G A , which describe the resolution performance in the transverse and axial directions,
respectively.13 We have developed a fitting procedure
that uses the gain factors and the Strehl ratio expressions
to provide a filter able to modify the system PSF at convenience. In particular, we have used it to find simple
continuous phase-only filters providing transverse or
axial superresolution. We have limited the search to a
few phase functions depending on two free parameters.
Section 2 describes the basic theory of the 4Pi-confocal
microscope and provides expressions for the axial and
transverse PSFs of the system. From the series expansion of the light intensity distributions near focus, general
expressions for the axial and transverse gain factors and
Strehl ratio are derived in Section 3. As an example of
application of our method, in Section 4 these figures of
merit are used to find phase-only filters providing superresolution in the axial and the transverse directions. Finally, several examples of different superresolving phaseonly pupil functions obtained by use of our filter design
method are presented.
2. 4PI-CONFOCAL MICROSCOPE THEORY
The analysis of both transverse and axial superresolution
requires a proper description of the system PSF. Since
the 4Pi-confocal microscope is a high-aperture optical system, the commonly used paraxial theory cannot be applied to calculate the PSF of the system. Furthermore,
the low-angle approximation is not justified, and the vec© 2003 Optical Society of America
de Juana et al.
Vol. 20, No. 11 / November 2003 / J. Opt. Soc. Am. A
torial nature of the electromagnetic waves has to be taken
into account. Under these conditions, the electric field in
the focal region for a plane-polarized wave incident on an
aberration-free lens can be expressed as1,11,14
In the axial direction, v ⫽ 0, we find that I 1 (u, v
⫽ 0) ⫽ I 2 (u, v ⫽ 0) ⫽ 0 and that the electric field in
Eqs. (1) reduces to a scalar expression (E y ⫽ E z ⫽ 0).
Consequently, the axial IPSF can be written as
E x ⫽ I 0 共 u, v 兲 ⫹ I 2 共 u, v 兲 cos 2 ␸ ,
H 共 u, v ⫽ 0 兲 ⫽ 兩 I 0 ill共 u 兲 ⫹ I 0 ill共 ⫺u 兲 兩 2
E y ⫽ I 2 共 u, v 兲 sin 2 ␸ ,
⫻ 兩 I 0 det共 ⑀ u 兲 ⫹ I 0 det共 ⫺⑀ u 兲 兩 2 ,
E z ⫽ ⫺2iI 1 共 u, v 兲 cos ␸ ,
(1)
where ␸ is the angle between the polarization direction of
the incident field and the direction of observation and
(u, v) are the axial and transverse optical coordinates
given by
u ⫽ 4nkz sin2 ␣ /2,
(2)
v ⫽ nkr sin ␣ ,
(3)
with ␣ as the semiaperture angle, z and r as the usual
axial and radial coordinates, and n as the refraction index
of the high-NA lens. In Eqs. (1), the functions I n (u, v)
with n ⫽ 0, 1, 2 are integrals over the lens aperture2,14,15:
I 0 共 u, v 兲 ⫽
冕
␣
0
A 共 ␪ 兲 sin ␪ 共 1 ⫹ cos ␪ 兲 J 0
冋
⫻ exp
I 1 共 u, v 兲 ⫽
冕
冕
␣
␣
0
4 sin2 共 ␣ /2兲
A 共 ␪ 兲 sin2 ␪ J 1
0
I 2 共 u, v 兲 ⫽
iu cos ␪
冉
册
冋
sin ␣
4 sin2 共 ␣ /2兲
册
sin ␣
冊
I 0 ill共 u 兲 ⫽
冕
␣
冋
A 共 ␪ 兲共 1 ⫹ cos ␪ 兲 exp
0
冊 冋
册
冉 冊
exp
iu cos ␪
4 sin2 共 ␣ /2兲
d␪ ,
v sin ␪
sin ␣
d␪ .
(4)
In these equations, J 0 , J 1 , and J 2 are Bessel functions
of the first kind, and A( ␪ ) is an apodization function describing the effect of placing radially symmetric pupil filters in the arms of the microscope.
In a 4Pi-confocal microscope the light from a point
source is divided into two beams that are focused into the
object by two identical high-NA objectives. The light
emitted or scattered from the object is collected by the two
objectives and is focused to a pointlike detector. In this
paper we center our attention on a type C confocal microscope, and all the expressions presented here are given
for this geometry. For a fluorescent object the light collected by objectives is due to the emission of the sample,
and hence illumination and detection are mutually incoherent. Consequently, the intensity point-spread function (IPSF) for a fluorescent 4Pi-confocal microscope is
given by the product of two independent PSFs, the illumination PSF and the detection one, and can be expressed
as16
H 共 u, v, ␸ 兲 ⫽ 兩 Eill共 u, v, ␸ 兲 ⫹ Eill共 ⫺u, v, ␸ 兲 兩 2
⫻ 兩 Edet共 ⑀ u, ⑀ v 兲 ⫹ Edet共 ⫺⑀ u, ⑀ v 兲 兩 2 ,
(5)
where ⑀ ⫽ ␭ ill /␭ det is the ratio between the illumination
and the fluorescence (detection) wavelengths.
iu cos ␪
4 sin2 共 ␣ /2兲
册
(6)
sin ␪ d␪ .
(7)
On the other hand, in the transverse direction a useful
expression of the IPSF for the 4Pi-confocal microscope can
be found by use of the scalar high-aperture approximation
instead of the vectorial theory, since the scalar theory provides a good approximation to the focal-plane intensity, as
has been pointed out previously.14 In the scalar highaperture approximation, the field amplitude in the focal
region14 is, omitting a phase factor, which does not contribute to the intensity, the following:
冕
␣
0
d␪ ,
v sin ␪
iu cos ␪
v sin ␪
where
E 共 u, v 兲 ⫽ 2
A 共 ␪ 兲 sin ␪ 共 1 ⫺ cos ␪ 兲 J 2
⫻ exp
冉
2173
A共 ␪ 兲J0
冋
冉
v sin ␪
sin ␣
冊
册
iu sin2 共 ␪ /2兲
⫻ exp ⫺
sin ␪ d␪ .
2 sin2 共 ␣ /2兲
(8)
Thus in the focal plane (u ⫽ 0) the IPSF for the 4Pi
can be expressed as
H 共 u ⫽ 0, v 兲 ⫽ 兩 E 共 u ⫽ 0, v 兲 兩 2 兩 E 共 u ⫽ 0, ⑀ v, A 共 ␪ 兲 ⬅ 1 兲 兩 2 .
(9)
The modification of the three-dimensional intensity distribution by pupil filtering in a 4Pi-confocal microscope
can be done by one’s following different schemes: a filter
in only one arm of the illumination path, two different filters in both arms of the illumination path, etc. We have
selected to place two identical filters in the illumination
arms of the microscope because this election does not introduce additional difficulty to the calculations and does
provide a symmetrical axial PSF even when phase filters
are used. On the other hand, A( ␪ ) ⫽ 1 for the detection
IPSF of the microscope because no filter is inserted in the
detection arms.
3. DERIVATION OF SUPERRESOLUTION
GAIN FACTORS
Since the pupil filtering is performed in the illumination
path of the 4Pi-confocal microscope, only the illumination
part of the IPSF is modified by the filters, whereas the detection part remains unchanged. This fact allows us to
obtain axial and transverse superresolution gain factors
from the second-order approximations of the axial and
transverse illumination PSFs, respectively. By use of a
parabolic approximation of the light intensity distribution, the height and the width of the PSF central lobe can
be described. The zero-order term of the development
gives us the Strehl ratio, and the second-order term provides a measure of the resolution obtained. Since we
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J. Opt. Soc. Am. A / Vol. 20, No. 11 / November 2003
de Juana et al.
have chosen to place two identical filters in the illumination arms, in order to obtain axial symmetry, only the
even terms appear in the axial expansion. Moreover, the
transverse intensity distribution is always symmetrical
when radially symmetric pupil filters are used. This
analysis does not allow us to describe the behavior of the
sidelobes, which require an excessive number of terms in
the development.
Having introduced the variable x ⫽ cos ␪ in Eq. (7), we
performed the series expansion of the axial illumination
PSF providing the following expression (see Appendix A):
兩 C 0兩 2 ⫺
H ill共 u, v ⫽ 0 兲 ⫽
Re共 C 0* C 2 兲
16 sin4 共 ␣ /2兲
3
共 2 ⫺ cos ␣ ⫺
1
2
u2 ⫹ O共 u4兲
cos2 ␣ 兲 2
,
(10)
where * denotes the complex conjugate, O(u 4 ) represents
the terms with powers higher than u 2 , and C n are integrals of the pupil function over the objectives’ apertures
and are defined as
Cn ⫽
冕
1
cos ␣
A 共 x 兲共 1 ⫹ x 兲 x n dx.
(11)
Until now, no assumption has been made about the nature of the filters. We assume a general complex pupil
function A( ␳ ) ⫽ T( ␳ )exp关i␾ (␳)兴, where ␳ is the normalized radial coordinate over the pupil and T( ␳ ) and ␾ (␳)
are the transmittance and phase functions, respectively.
From Eq. (10), expressions for the Strehl ratio S 4Pi and
the axial gain factor G 4Pi
A are obtained as a function of the
semiaperture angle of the objectives and the pupil function, given by (see Appendix A)
S 4Pi ⫽
G 4Pi
A ⫽
兩 C 0兩 2
3
共 2 ⫺ cos ␣ ⫺
1
2
cos2 ␣ 兲 2
,
(12)
6 共 cos ␣ ⫹ 3 兲
Re共 C 0* C 2 兲
3 cos3 ␣ ⫹ 7 cos2 ␣ ⫹ 7 cos ␣ ⫹ 7
兩 C 0兩 2
The axial gain factor has been defined to be unity when
no filter is placed in the illumination path of the microscope. If the extension of the axial PSF central lobe is
lower than the unapodized case, i.e., the filter is superresolving in the axial direction, the axial gain factor is
greater than unity. In contrast, values lower than unity
are found when the filter is apodizing.
Following the same procedure as above, we derived the
expansion in a power series of the transverse illumination
PSF obtained from Eq. (9) as
兩 Q 0兩 2
共 1 ⫺ cos ␣ 兲
⫹ O共 v4兲,
4
2
⫺
Re共 Q 0* Q 2 兲
2 sin ␣ 共 1 ⫺ cos ␣ 兲 2
2
冕
1
cos ␣
A 共 x 兲共 1 ⫺ x 2 兲 n/2dx.
(15)
The transverse gain factor in terms of the semiaperture
angle and the pupil function is obtained from Eq. (14) and
is given by
G 4Pi
T ⫽
1 ⫺ cos ␣
2
3
⫺ cos ␣ ⫹
1
3
cos3 ␣
Re共 Q 0* Q 2 兲
兩 Q 0兩 2
.
(16)
With these three figures of merit (Strehl ratio and axial
and transverse gains), we are able to describe the basic
properties of the illumination PSF central lobe of the 4Piconfocal microscope and its dependence on the pupil function for fluorescent samples. However, for nonfluorescent samples, the expressions for the gain factors remain
valid, since they have been derived from the series expansion of the illumination part of the IPSF of the microscope, which is independent of the nature (fluorescent or
nonfluorescent) of the sample. It should be noted that
the determination of the gain factors of the central lobe
may not be sufficient to describe the imaging properties of
the 4Pi-confocal microscope. It is also necessary to take
into account that if the sidelobes are too high the image
cannot be reconstructed. Hence it would be desirable to
design filters that, in addition to their superresolving performance, maintain or even reduce the height of the secondary axial sidelobes. However, the description of the
sidelobes would require a very high number of terms in
the PSF development
Furthermore, it must be stated that these gain factors
are useful only if the detection PSF is not much smaller
than the illumination one. Finally, it is also necessary to
remark that the given superresolution gain factors are independent of the semiaperture angle ␣, since they have
been defined to be unity for the unapodized case, for any
value of the semiaperture angle. Therefore they reveal
only the effect of the pupil function in the threedimensional intensity distribution near focus.
.
(13)
H ill共 u ⫽ 0, v 兲 ⫽
Qn ⫽
v2
(14)
where O(v ) represents the terms with powers higher
than v 2 and Q n are integrals of the pupil function over
the objectives’ apertures, defined as
4. APPLICATION TO CONTINUOUS PHASE
FILTERS
Once the superresolution gain factors have been properly
defined to describe the behavior of the three-dimensional
intensity distribution near focus for the 4Pi-confocal microscope, we will use them for finding simple continuous
phase filters that achieve transverse or axial superresolution. We have chosen smooth continuous phase-only
functions because they can be easily implemented in a
plate or in dynamical devices such as a LCD spatial light
modulator or a deformable mirror. We have limited the
search to phase functions depending on two free parameters ␾ ( ␳ , a, b), where a and b are the parameters to be
fitted and ␳ is the normalized radial coordinate over the
pupil. Note that the relation between ␳ and coordinate ␪
in Eqs. (4), (7), and (8) is ␳ ⫽ sin(␪)/sin(␣).11 In our
analysis, parameters will be fitted to fulfill desired values
of the Strehl ratio and the superresolution gain factors.
Consequently, the two parameters a and b are obtained
from a best fit of the following system of equations:
de Juana et al.
Vol. 20, No. 11 / November 2003 / J. Opt. Soc. Am. A
Table 1. Axial Superresolution Performance for
Three Phase Filtersa
Phase Function
a
b
GA
GT
S
I lob
a sin(2␲b␳2)
⫺3.592
1.997
1.927
1.804
2.004
2.004
4.759
⫺12.526
⫺0.550
1.318
1.344
1.339
1.310
2.620
3.399
16.181
1.40
1.14
1.09
1.20
1.30
1.14
1.33
1.43
0.44
0.93
0.97
0.81
0.70
0.88
0.70
0.44
0.10
0.08
0.10
0.19
0.12
0.10
0.10
0.25
0.55
0.14
0.15
0.16
0.14
0.15
0.10
0.38
a sin(2␲b␳3)
a␳3 ⫹ b␳5
a
Phase given in radians.
G 0A
0
G 4Pi
A 共 a, b 兲 ⫺ G A ⫽ 0,
(17)
G 4Pi
T 共 a,
⫽ 0,
(18)
S 4Pi共 a, b 兲 ⫺ S 0 ⫽ 0,
(19)
b兲 ⫺
G 0T
G 0T
is the desired axial gain,
is the desired
where
transverse gain, and S 0 is the acceptable Strehl ratio. A
Mathcad mathematical environment was used to perform
a best fit of this system of equations, since it provides
user-friendly functions. The main advantage of this
package is that it selects an appropriate method from a
group of algorithms. The algorithms available for solving nonlinear systems are the conjugate gradient, quasiNewton, and Levenberg–Marquardt algorithms taken
from the public domain MINPACK library. This method
requires a guess value for each unknown at the beginning
of the search process. For systems with more than one
solution, the guess values determine the particular solution that is obtained.
For all the filters presented in this paper, the parameters used to evaluate the related intensity distributions
were ␭ ill ⫽ 350 nm, ⑀ ⫽ 0.8, n ⫽ 1.518, and ␣ ⫽ 67.5°,
giving a NA of 1.4.
A. Superresolving Filters in the Axial Direction
One of the main features of the 4Pi-confocal microscope is
its great optical sectioning capability. Our purpose is to
enhance this feature even more by means of pupil filtering. To increase the axial resolution, we introduced in
Eqs. (17) and (19) different simple phase functions depending on two free parameters. For each target phase
function, different values for the desired axial gain (G 0A )
and the Strehl ratio (S 0 ) were imposed. Table 1 shows
several examples of the values obtained as a result of the
fitting process for three phase functions. For each particular solution determined by the values of the free parameters (a, b), the table shows the values of the corresponding axial and transverse gains (G A and G T ) and the
Strehl ratio (S). Furthermore, it also shows the values
for the axial sidelobe’s relative height (I lob), defined as
the ratio of the maximum intensity in the axial sidelobe to
the maximum intensity in the central peak or, in other
words, the relative height of the axial secondary maximum related to the central peak. This is an important
parameter, since a high axial sidelobe could lead to ambiguity in the image restoration process. For the unapodized case, this parameter takes the value I lob
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⫽ 0.23. All the solutions presented in Table 1 are superresolving in the axial direction, but, to obtain a complete
knowledge of the axial resolution performance of the filters, one needs a further evaluation of the axial intensity
distribution. The procedure provides satisfactory results
for different kinds of phase function with low sidelobe intensities. In those cases with the sidelobe height greater
than in the unapodized case, they still could be used in
combination with the two-photon excitation technique17
that significantly reduces the contribution of the secondary axial sidelobes.
Figures 1, 2, and 3 show the normalized axial intensity
distribution for the 4Pi-confocal microscope compared
with the unapodized case (dashed curve) for three particular cases selected from Table 1 and corresponding to
three different phase functions.
Figure 1 shows the axial intensity distribution given by
the phase filter ␾ ( ␳ , a, b) ⫽ a sin(2␲b␳2) with parameters a ⫽ 1.997 and b ⫽ 1.318. This filter produces a
little narrowing of the axial central peak (G A ⫽ 1.14) and
a significant reduction of the sidelobe height (I lob
⫽ 0.12). Figure 2 corresponds to the phase function
Fig. 1. Normalized axial intensity PSF for the phase filter
␾ ( ␳ , a, b) ⫽ a sin(2␲b␳2) with parameters a ⫽ 1.997 and b
⫽ 1.318 (solid curve) compared with the unapodized case
(dashed curve).
Fig. 2. Normalized axial intensity PSF for the phase filter
␾ ( ␳ , a, b) ⫽ a sin(2␲b␳3) with parameters a ⫽ 2.004 and b
⫽ 1.310 (solid curve) compared with the unapodized case
(dashed curve).
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J. Opt. Soc. Am. A / Vol. 20, No. 11 / November 2003
Fig. 3. Normalized axial intensity PSF for the phase filter
␾ ( ␳ , a, b) ⫽ a ␳ 3 ⫹ b ␳ 5 with parameters a ⫽ 4.759 and b
⫽ 3.399 (solid curve) compared with the unapodized case
(dashed curve).
de Juana et al.
B. Superresolving Filters in the Transverse Direction
The confocal technique is used to obtain threedimensional images of scanned objects. Owing to the diffraction laws, the resolution in the transverse direction
(central peak width) for the 4Pi-confocal microscope is
poorer than its axial counterpart. This difference between axial and transverse resolutions leads to anisotropic images of three-dimensional objects. Our aim in
this subsection is to provide simple superresolving phase
filters for narrowing the transverse PSF central peak.
The increase of resolution in the radial direction results
in a more homogeneous distribution of the focal volume.
To find superresolving filters in the transverse direction, we followed the same method as that explained
above for axial superresolution. Different simple phase
functions depending on two parameters were introduced
in Eqs. (18) and (19). For each phase function, different
values for the desired transverse gain (G 0T ) and the Strehl
ratio (S 0 ) were imposed. Table 2 shows the results obtained for six kinds of phase function. For each particular solution determined by the values of the free paramTable 2. Transverse Superresolution Performance
a
for Six Phase Filters
Phase Function
a
b
GT
GA
S
I lob
a sin(2␲b␳)
22.790
54.542
2.800
25.129
61.453
50.908
207.178
⫺0.993
0.310
0.291
1.425
⫺18.988
⫺42.322
0.193
0.184
1.400
1.30
1.40
1.19
1.30
1.40
1.12
1.14
1.30
0.69
0.60
0.863
0.72
0.63
0.88
0.86
0.72
0.32
0.19
0.11
0.26
0.12
0.74
0.69
0.23
0.39
0.42
0.13
0.27
0.24
0.28
0.30
0.27
⫺1.465
1.462
1.17
0.90
0.10
0.09
a sin(2␲b␳2)
a␳3 ⫹ b␳5
a exp关⫺(␳/b)2兴
Fig. 4. Normalized z response to an infinitely thin fluorescent
layer for the phase filter ␾ ( ␳ , a, b) ⫽ a sin(2␲b␳3) with parameters a ⫽ 2.004 and b ⫽ 1.310 (solid curve) compared with the
unapodized case (dashed curve).
␾ ( ␳ , a, b) ⫽ a sin(2␲b␳3) with parameters a ⫽ 2.004
and b ⫽ 1.310. In this case a greater narrowing of the
axial central peak is achieved (G A ⫽ 1.30) with an axial
sidelobe height similar to the unapodized one. Finally, in
Fig. 3 the axial intensity for the phase filter ␾ ( ␳ , a, b)
⫽ a ␳ 3 ⫹ b ␳ 5 with parameters a ⫽ 4.759 and b ⫽ 3.399
is shown. This filter gives axial superresolution (G A
⫽ 1.33) and a lower axial sidelobe than the corresponding one for the unapodized case (I lob ⫽ 0.16).
To make the study of the influence of the proposed technique on the performance of 4Pi-confocal microscopes
more complete, we show in Fig. 4 the so-called integrated
intensity function (the z response to an infinitely thin
fluorescent
layer2)
for
the
filter
␾ ( ␳ , a, b)
3
⫽ a sin(2␲b␳ ) with a ⫽ 2.004 and b ⫽ 1.310). Note
that also in terms of this figure of merit it is feasible to
obtain axial superresolution.
This series of examples shows that by use of properly
designed phase filters the axial resolution for the 4Piconfocal microscope can be improved without increasing
the secondary axial sidelobe height.
a sin(2␲␳)
⫹ b sin(4␲␳)
a sin(2␲␳2)
⫹ b sin(4␲␳2)
a
Phase given in radians.
Fig. 5. Transverse intensity PSF for three phase filters:
␾ ( ␳ , a, b) ⫽ a sin(2␲b␳) with a ⫽ 22.790 and b ⫽ 0.310 (dotted
curve), ␾ ( ␳ , a, b) ⫽ a sin(2␲␳) ⫹ b sin(4␲␳) with a ⫽ ⫺0.993
and b ⫽ 1.400 (dashed curve), and ␾ ( ␳ , a, b) ⫽ a ␳ 3 ⫹ b ␳ 5 with
a ⫽ 61.453 and b ⫽ ⫺42.322 (dashed–dotted curve) compared
with the unapodized case (solid curve).
de Juana et al.
eters (a, b), Table 2 shows values of the corresponding
axial and transverse gains (G A and G T ), the Strehl ratio
(S), and the axial sidelobe’s relative height (I lob). All the
solutions presented in Table 2 achieve transverse superresolution, since they have values greater than 1 for the
transverse gain factor (G T ). It is also important to note
that a higher superresolution performance in the radial
direction (greater G T ) is accompanied by a lower resolution in the axial direction (smaller G A ). Nonetheless, it
would be possible, in principle, for one to achieve threedimensional superresolution by having the axial and
transverse superresolution gain factors be simultaneously
greater than unity in Eqs. (17) and (18), which would increase the sidelobes. However, we have not found practical examples of simultaneous superresolutions in both
directions. Figure 5 shows the transverse intensity distribution for three particular solutions corresponding to
different phase functions compared with the unapodized
case (solid curve). The intensity has been normalized to
the unapodized case to show the Strehl ratio corresponding to each filter. It can be seen that Strehl ratio values
shown in the figure deviate very slightly from the theoretical exact values presented in Table 2. This small discrepancy is due to the fact that intensity distributions depicted in the figure have been calculated by use of the
approximate high-aperture scalar theory instead of the
exact vectorial theory used in Table 2. In particular, the
dotted curve shows the transverse intensity distribution
corresponding to the filter ␾ ( ␳ , a, b) ⫽ a sin(2␲b␳) with
a ⫽ 22.790 and b ⫽ 0.310. This filter achieves a reduction of the spot size of 8.56% in terms of FWHM (G T
⫽ 1.30) compared with the unapodized case with a
Strehl ratio of 0.32. The dashed curve corresponds to the
phase function ␾ ( ␳ , a, b) ⫽ a sin(2␲␳) ⫹ b sin(4␲␳) with
parameters a ⫽ ⫺0.993 and b ⫽ 1.400, giving a transverse gain G T ⫽ 1.30. Finally, the dashed–dotted curve
shows the transverse intensity distribution corresponding
to the phase filter ␾ ( ␳ , a, b) ⫽ a ␳ 3 ⫹ b ␳ 5 with parameters a ⫽ 61.453 and b ⫽ ⫺42.322 with G T ⫽ 1.40. The
performance of this filter is similar to the preceding filters, which have the slowest variation of the phase inside
the pupil. In addition, these slow-varying filters are
easier to implement than the last one.
Hence it can be seen that pupil filtering with properly
designed phase-only filters can improve the transverse
resolution of a 4Pi-confocal microscope.
5. CONCLUSIONS
A new method for finding axial and transverse superresolving filters for the 4Pi-confocal microscope has been
presented and applied to the case of continuous phaseonly masks. Our method, based on a series expansion of
the illumination PSF near focus, can be applied to both
fluorescence and nonfluorescence 4Pi-confocal microscopes. As a result of the proposed procedure, we have
obtained a different kind of phase-only filter providing
axial superresolution. It has been shown that, by use of
properly designed phase filters, the axial resolution for
the 4Pi-confocal microscope can be improved without increasing the secondary axial sidelobe height. Furthermore, we have found solutions for many different phase
Vol. 20, No. 11 / November 2003 / J. Opt. Soc. Am. A
2177
functions providing transverse superresolution for the
4Pi-confocal microscope. We have also observed that the
number of phase functions that lead to solutions with
transverse superresolution is greater than for the axial
counterpart.
In conclusion, a novel filter design method for improving the axial and transverse resolution of the 4Pi-confocal
microscope has been developed. To check the validity of
our design method, several examples of superresolving
phase-only filters have been obtained, providing good
axial and transverse superresolution values.
APPENDIX A
A. Axial Illumination Point-Spread Function
First, Eq. (10) is derived. This equation represents an
analytical approximate expression for the axial illumination PSF, which is defined as
H ill共 u, v ⫽ 0 兲 ⫽ 兩 I 0 ill共 u 兲 ⫹ I 0 ill共 ⫺u 兲 兩 2 ,
(A1)
ill
where I 0 (u) is given by Eq. (7). Introducing the variable x ⫽ cos ␪ and expanding the exponential function
into a power series, we obtain
冏冕
H ill共 u, 0兲 ⫽ 2
⫻
1
cos ␣
再
A 共 x 兲共 1 ⫹ x 兲
x2
1⫺
2 关 4 sin2 共 ␣ /2兲兴 2
冎冏
2
u 2 ⫹ O 共 u 4 兲 dx ,
(A2)
where O(u 4 ) represents the terms with powers higher
than u 2 . It must be noted that there are no odd terms in
the expansion because two identical filters are used. If
only the first two terms of the expansion are considered,
the PSF remains
冏
H 共 u, 0兲 ⬇ 2C 0 ⫺
ill
C2
u
关 4 sin2 共 ␣ /2兲兴 2
⬇ 4 兩 C 0兩 2 ⫺
4 Re共 C 0* C 2 兲
关 4 sin2 共 ␣ /2兲兴 2
冏
2
2
u 2,
(A3)
where Re means the real part of the term in parentheses
and C n are given in Eq. (11). Finally, the illumination
PSF is normalized so that H ill(0, 0) ⫽ 1 when no filter is
placed in the illumination arm. Hence the PSF is multiplied by a normalization factor 1/兵 4 关 3/2 ⫺ cos ␣
⫺ (cos2 ␣)/2兴 2 其 and Eq. (10) is obtained.
B. Axial Gain
The gain represents the inverse of the PSF width, i.e., the
distance in which the parabolic approximate PSF takes a
null value. Hence
0 ⫽ 4 兩 C 0兩 2 ⫺
⬀
4 Re共 C 0* C 2 兲
1
关 4 sin2 共 ␣ /2兲兴 2 G A
Re共 C 0* C 2 兲
兩 C 0兩 2
冉 冊
.
2
⇒ GA
(A4)
The gain is normalized to 1 when there is no filter in the
illumination arm:
2178
J. Opt. Soc. Am. A / Vol. 20, No. 11 / November 2003
GA ⫽
3
2
7
12
⫺ cos ␣ ⫺ cos2 ␣ /2
Re共 C 0* C 2 兲
⫺ cos3 ␣ /3 ⫺ cos4 ␣ /4
兩 C 0兩 2
.
de Juana et al.
(A5)
With a small amount of algebra and if the common factor
(1 ⫺ cos ␣) is canceled both in the numerator and in the
denominator, Eq. (13) is obtained.
Similar calculations lead to the transverse PSF and
gain.
ACKNOWLEDGMENT
This research was supported by Ministerio de Ciencia y
Tecnologı́a grant AYA2000-1565-C02.
The corresponding author, V. F. Canales, can be
reached by e-mail at [email protected].
6.
7.
8.
9.
10.
11.
12.
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