Optimization of axial resolution in a confocal microscope with D-shaped apertures Wei Gong, 1 Ke Si, and Colin J. R. Sheppard 1,,3, * 1 Division of Bioengineering, National University of Singapore, Singapore 117576 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: colin@nus.edu.sg Received 0 April 009; revised 19 June 009; accepted 3 June 009; posted 5 June 009 (Doc. ID 110015); published 7 July 009 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. 009 Optical Society of America OCIS codes: 050.10, 050.1970, 180.1970, 170.1530, 90.090. 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 [], 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 0003-6935/09/03998-05$15.00/0 009 Optical Society of America 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 [1,13]. 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]. 3998 APPLIED OPTICS / Vol. 48, No. 0 / 10 July 009
. 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] For a circular detector, the intensity sensitivity is DðνÞ ¼ 1; ν < νd 0; ν ν d ; where ν d ¼ πr d a=λd 0 is the normalized radius of the pinhole, r d is the real radius of the pinhole, and d 0 is the distance between the collector and the detector. exp½ iðu=þρ Pðρ; θ; uþ ¼ Š 0 d ρ 1 & cos 1 ðρ=dþ θ cos 1 ðρ=dþ otherwise ð1þ 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 ¼ð8π=λÞz sin ðα=þ: ðþ 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 D- shaped apertures [11], the detected signal intensity for a perfectly reflecting planar object and a point source can be expressed as IðuÞ ¼ Dðν x ; ν y Þjh 1 ðν x ; ν y ; uþ h ðν x ; ν y ; uþj dν x dν y ; where denotes the convolution operation, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν x þ ν y ¼ ν ¼ kr sin α ð3þ is the transverse optical coordinate, u is the axial optical coordinate, Dðν x ; ν y Þ is the intensity sensitivity of the finite-sized detector, and h 1 ðν x ; ν y ; uþ and h ðν x ; ν y ; 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 h 1 ðν x ; ν y ; uþ ¼h ð ν x ; ν y ; uþ ¼ Pðρ; θ; uþ exp½ iðν x ρ cos θ þ ν y ρ sin θþšρdρdθ: ð4þ 3. Axial Resolution The detected signal intensity can be reduced to IðuÞ ¼ Dðν x ; ν y Þ Pðρ; θ; uþ exp½ iðν x ρ cos θ þ ν y ρ sin θþšρdρdθ dνx dν y ; ð5þ Half-width u 1=, 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 u 1=, we can see that a smaller value of half-width u 1= indicates better axial resolution. For ν d ¼ 0, corresponding to a point detector, the detected intensity IðuÞ can be expressed as Z 1 dffiffiffi IðuÞ ¼ arccos p expðiuρþdρ: ρ d ð6þ Figures 1(a) and 1(b) show the axial response of the intensity for ν d ¼ 1 and ν d 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 ν d ¼ 1. However, when ν d 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 u 1=. Figure (a) illustrates the half-width as a function of ν d 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 10 July 009 / Vol. 48, No. 0 / APPLIED OPTICS 3999
Fig. 1. (Color online) Axial response of the intensity with different distance parameters d when (a) ν d ¼ 1 and (b) ν d ¼ 6. Fig.. (Color online) Half-width u 1= of the axial response as a function of (a) detector radius ν d and (b) distance parameter d. 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 (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 ν d ¼ 0, there are no improvements in the axial resolution for the confocal microscope with D-shaped apertures. However, for nonzero values of ν d,asd 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 d 0 for the confocal system with a finite-sized detector at the minimum of the half-width. The dashed curve in Fig. (b) shows the variation of the half-width at optimum point d 0 for different detector radii. Figure 3 shows the variation of the detector radius at the optimum point compared with that at critical point d c, 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 microscope with circular apertures. When ν d is small, d 0 and d c are close together and then gradually separate with increasing ν d. 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. 4000 APPLIED OPTICS / Vol. 48, No. 0 / 10 July 009
obtained by altering the distance parameter to an optimum value of d 0. 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 I u¼0 ¼ Dðν x ; ν y Þ Pðρ; θ; 0Þ exp½ iðν x ρ cos θ þ ν y ρ sin θþšρdρdθ dνx dν y : ð7þ The signal level for a given system can be given by I u¼0 η ¼ p 4π ðarccos d d ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d Þ ; 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 1 ðν dþ J 0 ðν dþ: Figure 4(a) illustrates the signal level as a function of detector radius ν d, 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 ν d 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 u 1=. 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 u 1=, the value d ¼ 0 always gives the minimum value of u 1= 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 ν d less than.58, the optimal value of d is zero; for larger ν d, the optimal value of d increases. When ν d 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-397-000-615-71 and the Singapore Bioimaging Consortium/Singapore Stem Cell Consortium (A*STAR), SBIC-SSCC RP C-003/007. Fig. 4. (Color online) Signal level as a function of (a) detector radius ν d and (b) half-width u 1=. References 1. H. Goldman, Spaltlampenphotographie und photometrie, Ophthalmologica 98, 57 70 (1940).. D. M. Maurice, A scanning slit optical microscope, Invest. Ophthalmol. Visual Sci. 13, 1033 1037 (1974). 3. C. J. Koester, Scanning mirror microscope with optical sectioning characteristics: applications in ophthalmology, Appl. Opt. 19, 1749 1757 (1980). 4. C. J. Koester, Comparison of optical sectioning methods: the scanning slit confocal microscope, in Handbook of Confocal Microscopy, J. Pawley, ed. (Plenum, 1990). 5. P. Török, Z. Laczik, and J. N. Skepper, Simple modification of a commercial scanning laser microscope to incorporate darkfield imaging, J. Microsc. 181, 60 68 (1996). 6. P. Török, Z. Laczik, and C. J. R. Sheppard, Effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes, Appl. Opt. 35, 673 6739 (1996). 10 July 009 / Vol. 48, No. 0 / APPLIED OPTICS 4001
7. P. Török, C. J. R. Sheppard, and Z. Laczik, Dark-field and differential phase contrast imaging modes in confocal microscopy using a half-aperture stop, Optik (Jena) 103, 101 106 (1996). 8. P. J. Dwyer, C. A. DiMarzio, J. M. Zavislan, W. J. Fox, and M. Rajadhyaksha, Confocal reflectance theta line scanning microscope for imaging human skin in vivo, Opt. Lett. 31, 94 944 (006). 9. P. J. Dwyer, C. A. Dimarzio, and M. Rajadhyaksha, Confocal theta line-scanning microscope for imaging human tissues, Appl. Opt. 46, 1843 1851 (007). 10. C. J. R. Sheppard, W. Gong, and K. Si, The divided aperture technique for microscopy through scattering media, Opt. Express 16, 17031 17038 (008). 11. K. Si, W. Gong, and C. J. R. Sheppard, Three-dimensional coherent transfer function for a confocal microscope with two D-shaped pupils, Appl. Opt. 48, 810 817 (009). 1. C. Tjokro and C. J. R. Sheppard, Phase space analysis of photon scattering in multiplanes within a microscope system, Proc. SPIE 6163 61630U (006). 13. L. K. Wong, M. J. Mandella, G. S. Kino, and T. D. Wang, Improved rejection of multiply scattered photons in confocal microscopy using dual-axes architecture, Opt. Lett. 3, 1674 1676 (007). 14. X. S. Gan and C. J. R. Sheppard, Detectability: a new criterion for evaluation of the confocal microscope, Scanning 15, 187 19 (1993). 15. D. R. Sandison and W. W. Webb, Background rejection and signal-to-noise optimization in confocal and alternative fluorescence microscopes, Appl. Opt. 33, 603 615 (1994). 16. D. R. Sandison, D. W. Piston, R. M. Williams, and W. W. Webb, Quantitative comparison of background rejection, signal-tonoise ratio, and resolution in confocal and full-field laser scanning microscopes, Appl. Opt. 34, 3576 3588 (1995). 17. T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984). 18. C. J. R. Sheppard and M. Gu, Improvement of axial resolution in confocal microscopy using an annular pupil, Opt. Commun. 84, 7 13 (1991). 19. M. Gu, C. J. R. Sheppard, and H. Zhou, Optimization of axial resolution in confocal imaging using annular pupils, Optik (Jena) 93, 87 90 (1993). 400 APPLIED OPTICS / Vol. 48, No. 0 / 10 July 009