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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 12 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. the optimum pinhole size, vpo, as a function of a in the crossed 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. Abramowitz, M. & Stegun, I.A. (1965) Handbook of Mathematical Functions. Dover, New York. Higdon, P., Juškaitis, R. & Wilson, T. (1997) The effect of detector size on the extinction coefficient in confocal polarization microscopes. J. Microsc. 187, 8–11. Pluta, M. (1993) Advanced Light Microscopy, Vol. 3, Measuring Techniques. Elsevier, Amsterdam. Richards, B. & Wolf, E. (1959) Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system. Proc. Roy. Soc. A253, 358–379. Sandison, D.R., Williams, R.M., Wells, K.S., Strickler, J. & Webb, W.W. (1995) Quantitative fluorescence confocal laser scanning microscopy (CLSM). Handbook of Biological Confocal Microscopy (ed. by J. B. Pawley), pp. 39–53. Plenum, New York. Wilson, T., Juškaitis, R. (1995) On the extinction coefficient in confocal polarisation microscopy. J. Microsc. 179, 238–240. Wilson, T., Juškaitis, R. & Higdon, P. (1997) The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes. Opt. Commun. in press. Wilson, T. & Tan, J.B. (1996) Finite sized coherent and incoherent detectors in confocal microscopy. J. Microsc. 182, 61–66. q 1998 The Royal Microscopical Society, Journal of Microscopy, 189, 12–14