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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 3, Iss. 12 — Dec. 1, 2008
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The divided aperture technique for microscopy through scattering media

Colin J. R. Sheppard, Wei Gong, and Ke Si  »View Author Affiliations


Optics Express, Vol. 16, Issue 21, pp. 17031-17038 (2008)
http://dx.doi.org/10.1364/OE.16.017031


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Abstract

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

1. Introduction

There has been renewed interest recently in the divided aperture technique for imaging through scattering media such as biological tissue [1

1. P. J. Dwyer and C. A. DiMarzio, “Confocal reflectance theta line scanning microscope for imaging human skin in vivo,” Opt. Lett. 31, 942–944 (2006). [CrossRef] [PubMed]

, 2

2. P. J. Dwyer, C. A. DiMarzio, and M. Rajadhyaksha, “Confocal theta line-scanning microscope for imaging human tissues,” Appl. Opt. 46, 1843–1851 (2007). [CrossRef] [PubMed]

]. 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 [3

3. H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Annalen der Physik 10, 1–39 (1903).

]. 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

4. H. Goldman, “Spaltlampenphotographie und -photometrie,” Ophthalmologica 98, 257–270 (1940). [CrossRef]

8

8. C. J. Koester, “Comparison of optical sectioning methods: The scanning slit confocal microscope,” in Handbook of Confocal Microscopy , J. Pawley, ed. (Plenum Press, New York, 1990).

]. The ultramicroscope can be regarded as the fore-runner of confocal theta microscopy [9

9. E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens - confocal theta-fluorescence microscopy,” J. Microsc. 179, 1–10 (1995). [CrossRef]

, 10

10. T. D. Wang, M. J. Mandella, C. H. Contag, and G. S. Kino, “Dual-axis confocal microscope for highresolution in vivo imaging,” Opt. Lett. 28, 414–416 (2003). [CrossRef] [PubMed]

], and selected plane illumination microscopy (SPIM) (also called orthogonal-plane fluorescence optical sectioning, OPFOS) [11

11. A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: threedimensional imaging of macroscopic biological specimens,” J. Microsc. 170, 229–236 (1993). [CrossRef] [PubMed]

, 12

12. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004). [CrossRef] [PubMed]

], 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 [7

7. C. J. Koester, “A scanning mirror microscope with optical sectioning characteristics: Applications in ophthalmology,” Appl. Opt. 19, 1749–1757 (1980). [CrossRef] [PubMed]

]. 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 [13

13. J. F. Aguilar, M. Lera, and C. J. R. Sheppard, “Imaging of spheres by confocal microscopy,” Appl. Opt. 39, 4621–4628 (2000). [CrossRef]

]. Other alternative geometries are possible. It is interesting to note that Dwyer et al. [2

2. P. J. Dwyer, C. A. DiMarzio, and M. Rajadhyaksha, “Confocal theta line-scanning microscope for imaging human tissues,” Appl. Opt. 46, 1843–1851 (2007). [CrossRef] [PubMed]

] 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

14. M. Gu, C. J. R. Sheppard, and H. Zhou, “Optimization of axial resolution in confocal imaging using annular pupils,” Optik 93, 87–90 (1993).

, 15

15. M. Gu, T. Tannous, and C. J. R. Sheppard, “Effect of numerical aperture, pinhole size and annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1996). [CrossRef] [PubMed]

]. 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 [16

16. E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953). [CrossRef]

].

Although theory on imaging in a system with D-shaped apertures based on geometrical optics has been presented by Maurice and Koester [6

6. D. M. Maurice, “A scanning slit optical microscope,” Invest. Ophthalmol. 13, 1033–1037 (1974). [PubMed]

-8

8. C. J. Koester, “Comparison of optical sectioning methods: The scanning slit confocal microscope,” in Handbook of Confocal Microscopy , J. Pawley, ed. (Plenum Press, New York, 1990).

], 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 [17

17. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), p. 440.

]:

U(vx,vy,u)=1πpupilP(ρx,ρy)exp{i[vxρx+vyρy12u(ρx2+ρy2)]}dρxdρy.
(1)

Fig. 1. Schematic diagram of a confocal system with D-shaped pupils.

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 [18

18. P. Török, C. J. R. Sheppard, and Z. Laczik, “The effect of half-stop lateral misalignment on imaging of darkfield and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996). [CrossRef] [PubMed]

], while in the y-z plane the behavior is very similar to that for a plain circular aperture [17

17. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), p. 440.

].

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

P(t)=02πP(ρ,ϕ)dϕ=2arccos(dt).
(2)

Then the amplitude at the focus is

U(0)=1π[arccosdd(1d2)1/2]
(3)

and the amplitude along the axis is

U=12πd21P(t)eiut/2dt.
(4)
Fig. 2. The intensity point spread function for a D-shaped pupil for d=0.1 (Media 1). As defocus is increased a Poisson spot develops on the optical axis.
Fig. 3. Cross-sections through the intensity point spread function for a D-shaped pupil for d=0.1: (a) x-z section, (b) y-z section.

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 [19

19. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 253.

], but then the intensity is negligible anyway.

Fig. 4. The pupil function P(t) for different values of d.
Fig. 5. The intensity along the axis for a D-shaped pupil shown as a log-log plot.

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 [7

7. C. J. Koester, “A scanning mirror microscope with optical sectioning characteristics: Applications in ophthalmology,” Appl. Opt. 19, 1749–1757 (1980). [CrossRef] [PubMed]

]. 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.

Fig. 6. The intensity point spread function for a confocal microscope with two D-shaped pupils and a point detector, d=0.1 (Media 2). The Poisson spots from the two point spread functions overlap.
Fig. 7. Cross-section x-z through the intensity point spread function for a confocal microscope with two D-shaped pupils and a point detector, d=0.1.
Fig. 8. Half-width at half-maximum (HWHM) of the intensity point spread function for a confocal microscope with two D-shaped pupils and a point detector in the vx, vy and u directions.

4. Integrated intensity

In confocal microscopy, the concept of the integrated intensity was introduced to quantify the background produced by a scattering medium [20

20. C. J. R. Sheppard and T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978). [CrossRef] [PubMed]

, 21

21. C. J. R. Sheppard and M. D. Sharma, “Integrated intensity, and imaging through scattering media,” J. Mod. Opt. 48, 1517–1525 (2001).

]. 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)x y. 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 [20

20. C. J. R. Sheppard and T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978). [CrossRef] [PubMed]

]. 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.

Fig. 9. The integrated intensity I int(u) for a confocal microscope with two D-shaped pupils and a point detector shown as a log-log plot.

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, D=I(0,0,0)/Ivolume [22

22. X. S. Gan and C. J. R. Sheppard, “Detectability: A new criterion for evaluation of the confocal microscope,” Scanning 15, 187–192 (1993). [CrossRef]

, 23

23. C. J. R. Sheppard, X. Gan, M. Gu, and M. Roy, “Noise in confocal microscopes,” in The Handbook of Biological Confocal Microscopy , 2nd edition ed., J. Pawley, ed. (Plenum Press, New York, 1995), pp. 363–370.

]. 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.

Fig. 10. Signal/Background for a confocal microscope with two D-shaped pupils and a point detector as a function of d.

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.

6. Discussion

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.

References and links

1.

P. J. Dwyer and C. A. DiMarzio, “Confocal reflectance theta line scanning microscope for imaging human skin in vivo,” Opt. Lett. 31, 942–944 (2006). [CrossRef] [PubMed]

2.

P. J. Dwyer, C. A. DiMarzio, and M. Rajadhyaksha, “Confocal theta line-scanning microscope for imaging human tissues,” Appl. Opt. 46, 1843–1851 (2007). [CrossRef] [PubMed]

3.

H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Annalen der Physik 10, 1–39 (1903).

4.

H. Goldman, “Spaltlampenphotographie und -photometrie,” Ophthalmologica 98, 257–270 (1940). [CrossRef]

5.

D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 15, 1094–1095 (1968). [CrossRef]

6.

D. M. Maurice, “A scanning slit optical microscope,” Invest. Ophthalmol. 13, 1033–1037 (1974). [PubMed]

7.

C. J. Koester, “A scanning mirror microscope with optical sectioning characteristics: Applications in ophthalmology,” Appl. Opt. 19, 1749–1757 (1980). [CrossRef] [PubMed]

8.

C. J. Koester, “Comparison of optical sectioning methods: The scanning slit confocal microscope,” in Handbook of Confocal Microscopy , J. Pawley, ed. (Plenum Press, New York, 1990).

9.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens - confocal theta-fluorescence microscopy,” J. Microsc. 179, 1–10 (1995). [CrossRef]

10.

T. D. Wang, M. J. Mandella, C. H. Contag, and G. S. Kino, “Dual-axis confocal microscope for highresolution in vivo imaging,” Opt. Lett. 28, 414–416 (2003). [CrossRef] [PubMed]

11.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: threedimensional imaging of macroscopic biological specimens,” J. Microsc. 170, 229–236 (1993). [CrossRef] [PubMed]

12.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004). [CrossRef] [PubMed]

13.

J. F. Aguilar, M. Lera, and C. J. R. Sheppard, “Imaging of spheres by confocal microscopy,” Appl. Opt. 39, 4621–4628 (2000). [CrossRef]

14.

M. Gu, C. J. R. Sheppard, and H. Zhou, “Optimization of axial resolution in confocal imaging using annular pupils,” Optik 93, 87–90 (1993).

15.

M. Gu, T. Tannous, and C. J. R. Sheppard, “Effect of numerical aperture, pinhole size and annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1996). [CrossRef] [PubMed]

16.

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953). [CrossRef]

17.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), p. 440.

18.

P. Török, C. J. R. Sheppard, and Z. Laczik, “The effect of half-stop lateral misalignment on imaging of darkfield and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996). [CrossRef] [PubMed]

19.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 253.

20.

C. J. R. Sheppard and T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978). [CrossRef] [PubMed]

21.

C. J. R. Sheppard and M. D. Sharma, “Integrated intensity, and imaging through scattering media,” J. Mod. Opt. 48, 1517–1525 (2001).

22.

X. S. Gan and C. J. R. Sheppard, “Detectability: A new criterion for evaluation of the confocal microscope,” Scanning 15, 187–192 (1993). [CrossRef]

23.

C. J. R. Sheppard, X. Gan, M. Gu, and M. Roy, “Noise in confocal microscopes,” in The Handbook of Biological Confocal Microscopy , 2nd edition ed., J. Pawley, ed. (Plenum Press, New York, 1995), pp. 363–370.

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(170.0110) Medical optics and biotechnology : Imaging systems
(170.0180) Medical optics and biotechnology : Microscopy
(180.1790) Microscopy : Confocal microscopy
(260.1960) Physical optics : Diffraction theory

ToC Category:
Microscopy

History
Original Manuscript: August 6, 2008
Revised Manuscript: September 10, 2008
Manuscript Accepted: September 11, 2008
Published: October 10, 2008

Virtual Issues
Vol. 3, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Colin J. Sheppard, Wei Gong, and Ke Si, "The divided aperture technique for microscopy through scattering media," Opt. Express 16, 17031-17038 (2008)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-21-17031


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References

  1. P. J. Dwyer and C. A. DiMarzio, "Confocal reflectance theta line scanning microscope for imaging human skin in vivo," Opt. Lett. 31, 942-944 (2006). [CrossRef] [PubMed]
  2. P. J. Dwyer, C. A. DiMarzio, and M. Rajadhyaksha, "Confocal theta line-scanning microscope for imaging human tissues," Appl. Opt. 46, 1843-1851 (2007). [CrossRef] [PubMed]
  3. H. Siedentopf and R. Zsigmondy, "Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser," Annalen der Physik 10, 1-39 (1903).
  4. H. Goldman, "Spaltlampenphotographie und -photometrie," Ophthalmologica 98, 257-270 (1940). [CrossRef]
  5. D. M. Maurice, "Cellular membrane activity in the corneal endothelium of the intact eye," Experientia 15, 1094-1095 (1968). [CrossRef]
  6. D. M. Maurice, "A scanning slit optical microscope," Invest. Ophthalmol. 13, 1033-1037 (1974). [PubMed]
  7. C. J. Koester, "A scanning mirror microscope with optical sectioning characteristics: Applications in ophthalmology," Appl. Opt. 19, 1749-1757 (1980). [CrossRef] [PubMed]
  8. C. J. Koester, "Comparison of optical sectioning methods: The scanning slit confocal microscope," in Handbook of Confocal Microscopy, J. Pawley, ed. (Plenum Press, New York, 1990).
  9. E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, "A new tool for the observation of embryos and other large specimens - confocal theta-fluorescence microscopy," J. Microsc. 179, 1-10 (1995). [CrossRef]
  10. T. D. Wang, M. J. Mandella, C. H. Contag, and G. S. Kino, "Dual-axis confocal microscope for high-resolution in vivo imaging," Opt. Lett. 28, 414-416 (2003). [CrossRef] [PubMed]
  11. A. H. Voie, D. H. Burns, and F. A. Spelman, "Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens," J. Microsc. 170, 229-236 (1993). [CrossRef] [PubMed]
  12. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004). [CrossRef] [PubMed]
  13. J. F. Aguilar, M. Lera, and C. J. R. Sheppard, "Imaging of spheres by confocal microscopy," Appl. Opt. 39, 4621-4628 (2000). [CrossRef]
  14. M. Gu, C. J. R. Sheppard, and H. Zhou, "Optimization of axial resolution in confocal imaging using annular pupils," Optik 93, 87-90 (1993).
  15. M. Gu, T. Tannous, and C. J. R. Sheppard, "Effect of numerical aperture, pinhole size and annular pupil on confocal imaging through highly scattering media," Opt. Lett. 21, 312-314 (1996). [CrossRef] [PubMed]
  16. E. H. Linfoot and E. Wolf, "Diffraction images in systems with an annular aperture," Proc. Phys. Soc. B 66, 145-149 (1953). [CrossRef]
  17. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), p. 440.
  18. P. Török, C. J. R. Sheppard, and Z. Laczik, "The effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes," Appl. Opt. 35, 6732-6739 (1996). [CrossRef] [PubMed]
  19. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 253.
  20. C. J. R. Sheppard and T. Wilson, "Depth of field in the scanning microscope," Opt. Lett. 3, 115-117 (1978). [CrossRef] [PubMed]
  21. C. J. R. Sheppard and M. D. Sharma, "Integrated intensity, and imaging through scattering media," J. Mod. Opt. 48, 1517-1525 (2001).
  22. X. S. Gan and C. J. R. Sheppard, "Detectability: A new criterion for evaluation of the confocal microscope," Scanning 15, 187-192 (1993). [CrossRef]
  23. C. J. R. Sheppard, X. Gan, M. Gu, and M. Roy, "Noise in confocal microscopes," in The Handbook of Biological Confocal Microscopy, 2nd edition ed., J. Pawley, ed. (Plenum Press, New York, 1995), pp. 363-370.

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