Download Optimization of axial resolution in a confocal microscope with

Optimization of axial resolution in a confocal
microscope with D-shaped apertures
Wei Gong,1 Ke Si,2 and Colin J. R. Sheppard1,2,3,*
1
Division of Bioengineering, National University of Singapore, Singapore 117576
2
Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 117456
3
Department of Diagnostic Radiology, National University of Singapore, Singapore 119074
*Corresponding author: [email protected]
Received 20 April 2009; revised 19 June 2009; accepted 23 June 2009;
posted 25 June 2009 (Doc. ID 110015); published 7 July 2009
We show theoretically that the axial resolution is improved when two centrosymmetric D-shaped apertures are combined in a confocal microscope with a finite-sized pinhole. The optimum width of a divider
that separates the D-shaped apertures to give the maximum axial resolution for a given pinhole size is
investigated, and the magnitude of the signal level is explored. © 2009 Optical Society of America
OCIS codes:
050.1220, 050.1970, 180.1970, 170.1530, 290.0290.
1. Introduction
D-shaped apertures have been widely used in imaging applications of biological tissue, especially using
the confocal scanning microscope. The scanning slit
confocal microscope with D-shaped apertures was developed for ophthalmological applications by Goldman [1], Maurice [2], and Koester [3,4]. Török et al.
employed D-shaped apertures to achieve dark-field
imaging [5,6]. Later, they further investigated using
D-shaped apertures to achieve both dark-field and
differential phase contrast imaging [7]. Dwyer et al.
successfully applied the confocal scanning microscope with D-shaped apertures to imaging nuclear
and cellular details in human epidermis in vivo [8,9].
The wide applicability of this geometry stems from
the fact that the illumination and detection beams
overlap only in the focal region, resulting in angular
gating and thus improving rejection of scattered light
[10]. In a previous study [11], the amplitude coherent
transfer function indicated that the cutoff frequencies in both axial and transverse directions for a system with a point detector are decreased by using
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APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
D-shaped apertures. This is as expected, as the sizes
of the pupils are decreased. However, in practice, a
finite-sized pinhole is placed in front of the detector,
and in practice the size of the pinhole is sometimes
increased to raise the signal strength. Therefore, our
aim in this paper is to investigate the confocal system
with two centrosymmetrically placed D-shaped apertures with a finite-sized detector and to identify the
optimum geometry for the pupils.
Monte Carlo theories for imaging in scattering
media by confocal systems with off-axis illumination
and detection have been presented [12,13]. However,
although these studies have demonstrated the power
of off-axis geometry, this approach does not include
diffraction effects that are the subject of this study.
Unfortunately, other methods of treating imaging
through scattering media, such as radiative transfer
theory, also neglect diffraction effects. We therefore
base our studies on diffraction theory, using the concepts of signal and background from a volume
scattering object [14,15]. Although this approach is
simplistic in that it neglects depletion of the illuminating beam, it has been shown to give good
agreement with experimental studies in confocal
fluorescence microscopy [16].
For a circular detector, the intensity sensitivity is
2. Confocal System with D-Shaped Apertures
The schematic diagram for the confocal scanning
microscope with two centrosymmetrically placed
D-shaped apertures was illustrated in our previous
paper [11]. Consider a single D-shaped pupil with
outer radius a and distance parameter d (the distance from the center of the circle to the edge of
the D-shaped aperture). The defocused pupil function under paraxial approximation can be expressed
as [17]
Pðρ; θ; uÞ ¼
exp½−iðu=2Þρ2 0
ð2Þ
Here λ and sin α are the incident wavelength and the
numerical aperture of the lens in vacuum, respectively, and z is the defocus distance measured from
the focal plane.
For a confocal scanning microscope with two Dshaped apertures [11], the detected signal intensity
for a perfectly reflecting planar object and a point
source can be expressed as
ZZ
IðuÞ ¼
Dðνx ; νy Þjh1 ðνx ; νy ; uÞ
⊗ h2 ðνx ; νy ; uÞj2 dνx dνy ;
ð3Þ
where ⊗ denotes the convolution operation,
where νd ¼ 2πrd a=λd0 is the normalized radius of the
pinhole, rd is the real radius of the pinhole, and d0 is
the distance between the collector and the detector.
3.
is the transverse optical coordinate, u is the axial
optical coordinate, Dðνx ; νy Þ is the intensity sensitivity of the finite-sized detector, and h1 ðνx ; νy ; uÞ and
h2 ðνx ; νy ; uÞ are the 3-D amplitude point spread
functions of the objective and collection lenses, respectively. For a reflection-mode confocal scanning
microscope with D-shaped apertures, the two point
spread functions are centrosymmetric and can be
written as
h1 ðνx ; νy ; uÞ ¼ h2 ð−νx ; −νy ; uÞ
ZZ
¼
Pðρ; θ; uÞ exp½−iðνx ρ cos θ
ð4Þ
ð1Þ
Axial Resolution
The detected signal intensity can be reduced to
ZZ
Pðρ; θ; 2uÞ exp½−iðνx ρ cos θ
IðuÞ ¼
Dðνx ; νy Þ
2
þ νy ρ sin θÞρdρdθ dνx dνy ;
ð5Þ
ZZ
Half-width u1=2 , which is defined as the defocus distance at which the intensity drops to one-half of the
in-focus intensity [18], can be obtained as a measure
of axial resolution. From the definition of half-width
u1=2 , we can see that a smaller value of half-width
u1=2 indicates better axial resolution. For νd ¼ 0, corresponding to a point detector, the detected intensity
IðuÞ can be expressed as
Z1
IðuÞ ¼
d2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ν2x þ ν2y ¼ ν ¼ kr sin α
þ νy ρ sin θÞρdρdθ:
1; ν < νd
;
0; ν ≥ νd
d ≤ ρ ≤ 1 & −cos−1 ðρ=dÞ ≤ θ ≤ cos1 ðρ=dÞ
otherwise
in cylindrical coordinates, where ρ ¼ r=a denotes the
normalized radial coordinate, r is the real radial
coordinate, and u is the axial optical coordinate defined as
u ¼ ð8π=λÞz sin2 ðα=2Þ:
DðνÞ ¼
d
arccos pffiffiffi expðiuρÞdρ:
ρ
ð6Þ
Figures 1(a) and 1(b) show the axial response of the
intensity for νd ¼ 1 and νd ¼ 6, respectively. It can be
seen that the optical sectioning effect for d ¼ 0 is
stronger than that for d ¼ 0:5 when νd ¼ 1. However,
when νd ¼ 6, for d ¼ 0 it is weaker than that for
d ¼ 0:5. This is somewhat surprising, as we would
usually expect the larger pupil to give a better axial
response. The behavior can be explained qualitatively using geometric optics; for the defocused case a
finite-sized pinhole can be completely in the shadow
region for a big enough value of d.
This property can be further illustrated in numerical plots of half-width u1=2 . Figure 2(a) illustrates
the half-width as a function of νd for a confocal microscope using D-shaped apertures with different distance parameter d, compared with a conventional
confocal microscope with circular apertures. Although for a point detector the confocal microscope
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Fig. 1. (Color online) Axial response of the intensity with different distance parameters d when (a) νd ¼ 1 and (b) νd ¼ 6.
with D-shaped apertures exhibits a poorer axial resolution than in the conventional confocal microscope with circular apertures, the behavior is
different with a finite-sized detector. It is of practical
significance that for a given finite-sized detector
there is an optimum configuration (value of d) for the
confocal microscope with D-shaped apertures. Figure 2(b) shows the variation of the half-width of
the axial response as a function of d for given values
of the detector radii. It can be noted that, for νd ¼ 0,
there are no improvements in the axial resolution for
the confocal microscope with D-shaped apertures.
However, for nonzero values of νd, as d increases the
half-width of the axial response decreases until a
minimum appears. It then increases again and eventually approaches infinity at the nonphysical case of
two point apertures. This indicates an optimum d0
for the confocal system with a finite-sized detector
at the minimum of the half-width. The dashed curve
in Fig. 2(b) shows the variation of the half-width at
optimum point d0 for different detector radii.
Figure 3 shows the variation of the detector radius
at the optimum point compared with that at critical
point dc , where the half-width is equal to that for circular apertures. For the region between zero and the
critical point, the axial resolution is improved compared with that in a conventional confocal micro4000
APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
Fig. 2. (Color online) Half-width u1=2 of the axial response as a
function of (a) detector radius νd and (b) distance parameter d.
scope with circular apertures. When νd is small, d0
and dc are close together and then gradually separate with increasing νd . Eventually both approach infinity, as optical sectioning vanishes for large pinhole
sizes. In practice, for a given value of the detector radius, an improvement in the axial resolution can be
Fig. 3. (Color online) Variations of the detector radius at optimum and critical points.
obtained by altering the distance parameter to an optimum value of d0.
4. Signal Level
From the above analysis it is shown that the axial
resolution improves only when a finite-sized detector
is used. For a finite-sized detector, the detected signal
strength is enhanced, but the amount of unwanted
scattered light is also increased. Therefore, to fully
understand the overall performance, it is necessary
to introduce the signal level defined as the measured
energy divided by that which enters the entrance pupil [19]. The detected intensity on the focal plane can
be expressed as
ZZ
¼
Dðνx ; νy Þ
Pðρ; θ; 0Þ exp½−iðνx ρ cos θ
2
ð7Þ
þ νy ρ sin θÞρdρdθ dνx dνy :
ZZ
I u¼0
The signal level for a given system can be given by
η¼
I u¼0
pffiffiffiffiffiffiffiffiffiffiffiffiffi ;
4π ðarccos d − d 1 − d2 Þ
2
ð8Þ
which is normalized to unity for a large area detector.
For d ¼ −1, it reduces to the conventional confocal
microscope with circular pupils, and η becomes the
well-known result η ¼ 1 − J 21 ðνd Þ − J 20 ðνd Þ:
Figure 4(a) illustrates the signal level as a function
of detector radius νd , with values of d of 0, 0.3, 0.5,
and also for circular pupils. It is seen that for a given
value of d, the signal level increases as the size of the
detector increases. However, for a given radius νd of
the detector, the signal level decreases monotonically
as d increases. For a larger value of detector size, the
signal level changes more slowly when d is small, but
a much steeper curve is obtained as d increases.
Figure 4(b) shows the relationship of the signal
level to half-width u1=2 . At the optimum point, the
signal levels that vary with the detector radius and
the half-width are shown as dashed curves in
Figs. 4(a) and 4(b), respectively. The results are similar to those reported previously for a confocal microscope with an annular pupil and a finite sized
detector [19]. Although for a given pinhole size there
is an optimum value of d to minimize u1=2 , the value
d ¼ 0 always gives the minimum value of u1=2 for a
given η.
5.
Conclusion
We have theoretically analyzed the variations of the
axial response as a function of the radius of the detector and distance parameter of the D-shaped aperture for the confocal microscope with D-shaped
apertures. The results show that, for a given finite
size of detector, by altering distance parameter d, the
axial resolution can be maximized, which is of practical significance in the design and setup of the microscope. For detector size νd less than 2.58, the
optimal value of d is zero; for larger νd, the optimal
value of d increases. When νd ¼ 3:30, the optimal value of d becomes 0.1.
The authors acknowledge support from the
National University of Singapore Life Sciences Institute, R-397-000-615-712 and the Singapore Bioimaging Consortium/Singapore Stem Cell Consortium
(A*STAR), SBIC-SSCC RP C-003/2007.
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Fig. 4. (Color online) Signal level as a function of (a) detector radius νd and (b) half-width u1=2 .
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