A diffraction analysis is presented for image formation in confocal microscopy using the divided aperture technique, which uses two D-shaped apertures (also called specular microscopy). The effects of increasing the width of a divider, that separates the two D shapes, are investigated. As the width is increased, the resolution degrades. The efficiency of singly-scattered light rejection is not improved with increased width.
© 2008 Optical Society of America
There has been renewed interest recently in the divided aperture technique for imaging through scattering media such as biological tissue [1, 2]. This technique is based on the principle of angular gating, one of several gating mechanisms that can be used to eliminate multiply-scattered light. Other gating mechanisms include confocal, coherence, non-linear and polarization gates. Combinations of gating mechanisms can be used to improve scattered light rejection.
Angular gating had its beginning with the ultramicroscope, in which the sample is illuminated perpendicular to the imaging optical axis . The specular microscope, or divided aperture technique, combines different beam paths for illumination and detection with confocal imaging, so that light scattered other than in the focal region is rejected [4–8]. The ultramicroscope can be regarded as the fore-runner of confocal theta microscopy [9, 10], and selected plane illumination microscopy (SPIM) (also called orthogonal-plane fluorescence optical sectioning, OPFOS) [11, 12], both of which can also be implemented in a fluorescence mode. KEM Equipment Company (Elk Grove Village, IL) and Irvine Optical Corporation (Burbank, CA) manufactured deep field photographic microscope systems in the 1980s. All of these techniques have in common the fact that the illumination and detection pupils do not overlap, so that the illumination and detection beams cross only in the focal region.
In divided aperture microscopy, light scattered in the focal region can be detected, but light scattered by a single scattering event outside of the focal region will not be able to pass through both the collection pupil and the confocal pinhole (or slit). Multiply-scattered light can get through the collection pupil, but is unlikely to pass through the confocal pinhole. Most published work on the divided aperture technique have used two D-shaped apertures (segments of circles), one each for illumination and detection, respectively . The width of the separation between the two Ds can be adjusted in size to reject cross-talk from multiple scattering. For this geometry, light reflected specularly from a surface with its normal parallel to the optical axis can be detected, hence the name specular microscope. It has been shown that the system can be used in surface profiling applications to select the specular reflection from a spherical surface, while rejecting a spurious reflection that comes from the centre of the sphere . Other alternative geometries are possible. It is interesting to note that Dwyer et al.  based their analysis on two offset non-overlapping circles, but they used this only as an approximation to the D-shaped case. Another alternative geometry is to use a circular pupil and a non-overlapping annulus [14, 15]. This arrangement is fundamentally different from the previously mentioned examples, as in this case specular reflections from a normal surface are not detected. It was shown using geometrical optics that such a combination of a circular and an annular pupil with a finite-sized detector pinhole gives an axial response from a planar object that drops identically to zero at a particular defocus distance. So in some cases with a finite sized pinhole the axial resolution can be increased by use of an annular pupil, which is contrary to our normal expectation that the depth of focus in increased for an annular pupil .
Although theory on imaging in a system with D-shaped apertures based on geometrical optics has been presented by Maurice and Koester [6-8], a diffraction theory has not yet been given. In this paper we consider the image of a point object in a confocal system (point source and point detector) with D-shaped apertures. We also present results for the integrated intensity, which can be used to model background from singly-scattered light. We also discuss the axial response from a planar object.
2. Focusing by a D-shaped aperture
The amplitude in the focal region of a lens is calculated from the paraxial scalar Fresnel diffraction formula :
Here the optical coordinates are related to the true distances from the focal point x, y, z by νx=2 πxnsin α/λ, νy=2π y nsinα/λ, u=8πz nsin2(α/2)/λ, with λ the wavelength, α the semi-angular aperture of the lens, and n the refractive index of the immersion medium. The coordinates ρ x, ρy are distances in the pupil plane, normalized by the pupil radius a. We expect the scalar paraxial theory to be valid for values of α < 30° and to give qualitatively correct behavior for larger values of α. The geometry of the system is shown in Fig. 1. The pupil is a D-shaped segment of a circle, with boundaries at ρx=d and ρ2=ρ 2 x+ρ 2 y=1.
Figure 2 (Media 1) shows a density plot of the variation with defocus of the intensity point spread function for a D-shaped pupil for d=0.1. As defocus increases, the illuminated region becomes closer in shape to that of the pupil, but a bright Poisson spot begins to form on the axis. Figure 3 shows contour plots of x-z and y-z cross-sections through the intensity point spread function. The behavior in the x-z plane agrees with the results of Török , while in the y-z plane the behavior is very similar to that for a plain circular aperture .
The intensity along the axis can be calculated by transforming to polar coordinates ρ, ϕ, putting ρ 2=t, and performing the integration in ϕ first, to give
Then the amplitude at the focus is
and the amplitude along the axis is
Figure 4 shows the pupil function P(t) for different values of d, and Fig. 5 shows a loglog plot of the normalized intensity along the axis for a D-shaped pupil. As d increases, the width of the central lobe in the axial intensity also increases. For d=0, corresponding to a semicircular pupil, the intensity along the axis decays asymptotically as 1/u 2, but as d increases, the asymptotic behavior of the intensity for intermediate values of u is found to fall off more quickly. A maximum value for the absolute value of the slope of about 2.2 is achieved at d ≈0.07. For large u (>2000), the absolute value of the slope becomes two, for any value of d, as is expected from the theory of asymptotic values of Fourier transforms , but then the intensity is negligible anyway.
3. Image of a point object in a confocal system
Figure 6 (Media 2) shows the variation with defocus of the intensity in the image of a point object in a confocal system consisting of two D-shaped pupils and a point detector. As the defocus increases, the illumination and detection point spread functions tend to the pupil functions, which do not overlap, but the Poisson spot of each point spread function coincides and hence the product remains strong. The intensity along the axis is just the square of that for a single D-shaped pupil. For d=0, the intensity along the axis decays as 1/u 4, but as d increases, the asymptotic behavior of the intensity for intermediate values of u again tends to fall off more quickly. A maximum value for the absolute value of the slope of about 4.4 is achieved at d ≈ 0.07. For large u (>2000) the absolute value of the slope becomes four, independent of the value of d. Figure 7 shows a contour plots of x-z cross-section through the intensity image of a point object. This can be compared with the diamond shape predicted by Koester for a theory based on geometrical optics . In the y-z plane, the intensity is just the square of that from a single D-shaped pupil. Figure 8 shows the half-widths at half-maximum (HWHM) ν x 1/2, ν y 1/2, u for the intensity image of a point object. The value d=-1 corresponds to a full circular pupil. We see that v y1/2 and u are virtually unchanged for small positive values of d.
4. Integrated intensity
In confocal microscopy, the concept of the integrated intensity was introduced to quantify the background produced by a scattering medium [20, 21]. If the intensity image of a point object is I (νx, νy, u), then the contribution to the background in the focal plane from a distribution of particles a normalized distance u away is I int(u)=∫ I(νx, νy, u)dνx dνy. Figure 9 shows the integrated intensity I int (u) for a confocal microscope with two D-shaped pupils and a point detector as a log-log plot. For a confocal microscope with two circular pupils, I int (u) falls off as 1/u 2, and for one circular and one narrow annular pupil it falls off roughly as 1/u . For the divided aperture system, I int (u) falls off more quickly, for d=0 as 1/u 2.54, becoming close to 1/u 3.2 for larger values of d. This demonstrates an advantage of the divided aperture technique for imaging through scattering media.
The total contribution from single scattering by a thick scattering medium is I volume=∫ I int (u)du. The signal to background ratio is then defined as S/B=I(0,0,0)/I volume, and the detectability as the signal-to-noise ratio of a point object observed in a uniform background, [22, 23]. Note that the value of S/B is independent of the absolute strength of the signal and background. The signal S is proportional to the fourth power of the area of the pupils and hence decreases as d is increased. The background B also decreases with increasing d. Figure 10 shows the variation of S/B with d, showing that it decreases monotonically with the value of d. Thus there is no advantage from the point of view of rejection of singly-scattered background in increasing d above a value of zero.
5. Axial response from a planar object
The axial response from a plane reflector in a confocal microscope with two D-shaped pupils and a point detector is just the intensity along the axis for a single D-shaped pupil, compressed by a factor of two along the axis. Again the intensity falls off as 1/u 2 for large u (>1000). For intermediate values of u, again an optimum value for the absolute value of the slope of about 2.2 is achieved at d ≈ 0.07.
The effect of using non-overlapping D-shaped pupils in a confocal microscope has been investigated. The transverse resolution in one direction, and also the axial resolution, is degraded because the pupil is smaller. If the pupils are separated by a strip of width 2d, as d increases the resolution further decreases. But there is a Poisson spot on the axis that degrades the confocal performance. As the value of d is increased, the rejection of out-of-focus light from single scattering does not increase.
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