Download The effect of detector size on the signal-to

Journal of Microscopy, Vol. 189, Pt 1, January 1998, pp. 12–14.
Received 11 August 1997; accepted 30 September 1997
SHORT COMMUNICATION
The effect of detector size on the signal-to-noise ratio in
confocal polarized light microscopy
T. WILSON, P. TÖRÖK & P. D. HIGDON
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K.
Key words. Confocal microscopy, polarization, signal-to-noise ratio.
Summary
We introduce a signal-to-noise ratio in an attempt to
suggest an optimum pinhole size for confocal polarized light
microscopes. We find that pinhole sizes which are typically
60% greater than those used in nonpolarized light confocal
microscopy are appropriate.
Introduction
A confocal microscope may be converted into a polarization-sensitive instrument merely by placing a polar in the
illumination path and a suitably crossed analyser in the
detection path. The confocal microscope is particularly
suitable for polarised light studies since it possesses an
infinite extinction coefficient if perfect polars and a vanishingly small pinhole are used (Wilson & Juškaitis, 1995). The
extinction coefficient, which is defined as the ratio of the
intensity of the light transmitted between parallel polars to
that transmitted when the polars are crossed (Pluta, 1993),
is typically restricted to 103 in a conventional microscope
whereas values as high at 1·4 × 105 have been obtained in
confocal systems (Higdon et al., 1997). The reason for the
improved performance of the ideal confocal system with
vanishingly small pinhole is that the confocal microscope
essentially measures the field amplitude in the back focal
plane whereas the wide-field instrument measures the
average intensity. In practice, of course, a finite-sized
pinhole must be used and we would expect the value of
extinction coefficient to fall as the pinhole size increases
(Wilson & Tan, 1996). This prediction is borne out in
practice and the extinction coefficient falls smoothly
between the confocal and conventional limit as the pinhole
size is increased (Higdon et al., 1997). This raises the
question of how to choose the optimum pinhole size in
confocal polarized light microscopy. A method which has
been proposed in confocal bright-field and fluorescence
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microscopy is to consider a signal-to-noise ratio and to
select an optimum pinhole size based on maximizing this
ratio (Sandison et al., 1995). We shall adapt this model to
the confocal polarized light microscope.
We consider a subresolution point scatterer to be located
at the geometrical focal point and calculate the detected
signal, I(vp), as a function of pinhole diameter, vp. We elect
to use normalized optical coordinates vp ¼ (2p/l)rpn sina,
where l denotes the wavelength, n sina is the numerical
aperture and rp is the actual pinhole radius. We assume
that Shot noise is the important source of noise which
permits us to define a signal-to-noise ratio as
Iðvp Þ
S
¼ p
N
Iðvp Þ þ B
ð1Þ
where B denotes the background which we will assume to
be proportional to the area of the detector pinhole, thus
B ¼ av2p
ð2Þ
where a is a parameter which determines the relative
strength of the background. In order to proceed further it is
necessary to obtain expressions for I(vp). Since we are
concerned with polarization effects it is necessary to employ
a vector theory rather than a scalar theory. In order to do
this we assume that the subresolution scatterer acts as an
electric dipole whose dipole moment is proportional to the
electric field incident upon it. If we assume that the
scatterer is placed at the geometrical focus then the electric
field E in the back focal plane of the second imaging lens,
Fig. 1, is given by (Wilson et al., 1997)
E ¼ ½ð1 þ cos vÞ ¹ ð1 ¹ cos vÞ cos 2fÿi ¹ ð1 ¹ cos vÞ sin 2fj
ð3Þ
where we have omitted constants of proportionality. It is
now a straightforward matter to find the field in the plane of
the detector, ED, simply by applying the integral formula of
(Richards & Wolf, 1959). The resulting expressions apply
q 1998 The Royal Microscopical Society
S N R I N C O N F O CA L P O L A R I Z ED L I G H T M I C ROS C O P Y
13
Fig. 1. Schematic diagram of the optical system.
equally to both high- and low-aperture systems. If we take
the low-angle, paraxial limit, we find that the detected
intensity, Ic(vp), in the absence of polars is given by
I0 ðvp Þ ,
… 2p … v
p
0
0
jED j2 v dvdf ¼ 2p
… v 2J
p
0
1 ðvÞ
v
2
vdv
,1 ¹ J02 ðvp Þ ¹ J12 ðvp Þ
ð4Þ
where Jn(·) denotes the nth order Bessel function of the first
kind.
In the presence of crossed polars only the j component of
E in Eq. (3) is permitted to pass and hence the detected
intensity in this case, Ip(vp), is given by
Ip ðvp Þ ,
… 2p … v J
p
0
0
3 ðvÞ
v
2
sin2 2f vdv df
,1 ¹ J02 ðvp Þ ¹ 2J12 ðvp Þ ¹ 2J22 ðvp Þ ¹ 2J32 ðvp Þ
ð5Þ
where we have used the Bessel function recurrence relations
to evaluate the integral (Abramowitz & Stegun, 1965) and
have normalized the detected intensities to unity for large
pinholes. Figure 2 shows the form of these functions as a
function of pinhole size. We see that the curves are of
broadly similar shape but that detected intensity rises more
slowly with increasing pinhole size in the crossed polar case.
Fig. 2. The normalized detected intensity as a function of pinhole
radius in optical units for a nonpolarized light confocal microscope
and a polarized light microscope.
q 1998 The Royal Microscopical Society, Journal of Microscopy, 189, 12–14
Fig. 3. The variation of signal-to-noise ratio as a function of pinhole size for a variety of values of a: (a) the nonpolarized light confocal microscope; (b) the crossed polar case.
In order to consider the signal-to-noise ratio it is a simple
matter to substitute Eqs. (2), (4) and (5) into Eq. (1). The
results are shown in Fig. 3. We note that the a ¼ 0 limit
agrees well with the a ¼ 10 –4 curves for the values of vp
considered. A consideration of the curves in Fig. 3(a)
suggests that in the low-noise case a pinhole radius of
slightly more than 3 optical units should be used.
Sanderson et al. (1990) use similar arguments to propose
a value of 3·5 optical units as a general-purpose radius for
biological imaging, although they note that a slightly
smaller value is appropriate for punctate specimens with
little background. The conclusions are intuitively reasonable when we note that the first zero of the Airy disc,
[2J1(v)/v]2, occurs at v ¼ 1·22p ¼ 3·83.
Figure 3(b) shows the signal-to-noise ratio in the crossed
polar case and here we see that a much larger pinhole size
should be used to obtain the optimum signal-to-noise ratio.
Figure 3(b) suggests a value around vp ¼ 5·6 should be
chosen. We note that this is about 60% larger than the value
appropriate to nonpolarized light imaging. Figure 4 shows
14
T. WI L S O N E T A L .
Acknowledgment
This work was funded by the Leverhulme Trust, U.K.
References
Fig. 4. The variation of the optimum pinhole size, vpo, as a function
of a in the confocal polarized light microscope.
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polar case. However, we note that for low values of a a rather
wide choice of pinhole size is suitable and so the exact value
chosen in this case is not critical.
In this short communication we have addressed the issue
of the choice of pinhole size in polarized light confocal
microscopy. We have introduced a signal-to-noise ratio
criterion and have found that pinholes which are roughly
60% larger than the optimum for nonpolarized light
confocal microscopes may be used.
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q 1998 The Royal Microscopical Society, Journal of Microscopy, 189, 12–14