Abstract

In the confocal microscope, tightly focused illumination and spatially filtered detection are combined to reduce out-of-focus background and to produce high-quality images that display thin optical sections within thick fluorescent specimens. We define background as the detected light that originates outside a resolution volume and signal as the detected light that originates within the same volume. Background rejection is measured by the signal-to-background ratio (S/B) and is calculated for confocal, spinning-disk, line-illumination, slit-detection, and conventional fluorescence microscopes as a function of both the spatial filter size and the specimen thickness. Spatial filter sizes that reject background and optimize the signal-to-noise ratio (S/N) are calculated for each microscope. These calculations are normalized so that the time-averaged illumination at each point in the specimen is the same for each microscope. For thick specimens, we show that the S/B obtained with a confocal microscope can be more than 100 times greater than the S/B available with a conventional microscope, and we find that the optimal confocal S/N can be a factor of 10 greater than the S/N in the conventional microscope.

© 1994 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. K. S. Wells, D. R. Sandison, J. Strickler, W. W. Webb, “Quantitative fluorescence imaging with laser scanning confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 3.
    [CrossRef]
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    [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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1991

C. J. R. Sheppard, C. J. Cogswell, M. Gu, “Signal strength and noise in confocal microscopy: factors influencing selection of an optimum detector aperture,” Scanning 13, 233–240 (1991).
[CrossRef]

C. J. R. Sheppard, “Stray light and noise in confocal microscopy,” Micron Microsc. Acta 22, 239–243 (1991).
[CrossRef]

1990

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

1989

S. Kimura, C. Munakata, “Calculation of three-dimensional optical transfer function for a confocal scanning fluorescent microscope,” J. Opt. Soc. Am. A 6, 1015–1019 (1989).
[CrossRef]

J. W. Lichtman, W. J. Sunderland, R. S. Wilkinson, “High-resolution imaging of synaptic structure with a simple confocal microscope,” New Biologist 1, 75–82 (1989).
[PubMed]

1987

F. Johnson, “An improved method for computing a discrete Hankel transform,” Comput. Phys. Commun. 43, 181–202 (1987).
[CrossRef]

C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

1986

1980

1974

1967

M. D. Egger, M. Petràň, “New reflected light microscope for viewing unstained brain and ganglion cells,” Science 157, 305–307(1967).
[CrossRef] [PubMed]

Agard, D. A.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

Bendinelli, M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 437.

Brakenhoff, G. J.

G. J. Brakenhoff, K. Visscher, “Novel confocal imaging and visualization techniques,” in Micro 90: Proceedings of the Royal Microscopical Society Conference, H. Y. Elder, ed. (Hilger, London, 1990), Chap. 9.

Carrington, W. A.

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional imaging on confocal and wide-field microscopes,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 14.
[CrossRef]

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, M. Gu, “Signal strength and noise in confocal microscopy: factors influencing selection of an optimum detector aperture,” Scanning 13, 233–240 (1991).
[CrossRef]

Consortini, A.

Cox, I. J.

Egger, M. D.

M. D. Egger, M. Petràň, “New reflected light microscope for viewing unstained brain and ganglion cells,” Science 157, 305–307(1967).
[CrossRef] [PubMed]

Fay, F. S.

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional imaging on confocal and wide-field microscopes,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 14.
[CrossRef]

Fogarty, K. E.

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional imaging on confocal and wide-field microscopes,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 14.
[CrossRef]

Frieden, B. R.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), integral 6.554.3, p. 682.

Gu, M.

C. J. R. Sheppard, C. J. Cogswell, M. Gu, “Signal strength and noise in confocal microscopy: factors influencing selection of an optimum detector aperture,” Scanning 13, 233–240 (1991).
[CrossRef]

Hiraoka, Y.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

Johnson, F.

F. Johnson, “An improved method for computing a discrete Hankel transform,” Comput. Phys. Commun. 43, 181–202 (1987).
[CrossRef]

Kimura, S.

Kino, G. S.

G. S. Kino, “Intermediate optics in Nipkow disk microscopes,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 10.
[CrossRef]

Koester, C. J.

Lichtman, J. W.

J. W. Lichtman, W. J. Sunderland, R. S. Wilkinson, “High-resolution imaging of synaptic structure with a simple confocal microscope,” New Biologist 1, 75–82 (1989).
[PubMed]

Lifschitz, L.

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional imaging on confocal and wide-field microscopes,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 14.
[CrossRef]

Matthews, H. J.

Minsky, M.

M. Minsky, “Microscopy apparatus,” U.S. patent3,013,467 (19December1961).

Munakata, C.

Petràn, M.

M. D. Egger, M. Petràň, “New reflected light microscope for viewing unstained brain and ganglion cells,” Science 157, 305–307(1967).
[CrossRef] [PubMed]

Piston, D. W.

D. R. Sandison, D. W. Piston, W. W. Webb, “Background rejection and optimization of signal-to-noise in confocal microscopy,” in Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Specimens, J. K. Stevens, L. R. Mills, J. E. Trogadis, eds., (Academic, New York, 1994), Chap. 2.

Ronchi, L.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), integral 6.554.3, p. 682.

Sandison, D. R.

D. R. Sandison, “Fluorescence confocal laser scanning microscopy for three-dimensional imaging of living biological specimens,” Ph.D. dissertation (Cornell University, Ithaca, N. Y., 1993), p. 97.

K. S. Wells, D. R. Sandison, J. Strickler, W. W. Webb, “Quantitative fluorescence imaging with laser scanning confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 3.
[CrossRef]

D. R. Sandison, D. W. Piston, W. W. Webb, “Background rejection and optimization of signal-to-noise in confocal microscopy,” in Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Specimens, J. K. Stevens, L. R. Mills, J. E. Trogadis, eds., (Academic, New York, 1994), Chap. 2.

W. W. Webb, K. Wells, D. R. Sandison, J. Strickler, Optical Microscopy for Biology (Wiley-Liss, New York, 1990), pp. 73–108.

Sedat, J. W.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, “Stray light and noise in confocal microscopy,” Micron Microsc. Acta 22, 239–243 (1991).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, M. Gu, “Signal strength and noise in confocal microscopy: factors influencing selection of an optimum detector aperture,” Scanning 13, 233–240 (1991).
[CrossRef]

C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

I. J. Cox, C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3, 1152–1158 (1986).
[CrossRef]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 25.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 33.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 3.

Strickler, J.

K. S. Wells, D. R. Sandison, J. Strickler, W. W. Webb, “Quantitative fluorescence imaging with laser scanning confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 3.
[CrossRef]

W. W. Webb, K. Wells, D. R. Sandison, J. Strickler, Optical Microscopy for Biology (Wiley-Liss, New York, 1990), pp. 73–108.

Sunderland, W. J.

J. W. Lichtman, W. J. Sunderland, R. S. Wilkinson, “High-resolution imaging of synaptic structure with a simple confocal microscope,” New Biologist 1, 75–82 (1989).
[PubMed]

Visscher, K.

G. J. Brakenhoff, K. Visscher, “Novel confocal imaging and visualization techniques,” in Micro 90: Proceedings of the Royal Microscopical Society Conference, H. Y. Elder, ed. (Hilger, London, 1990), Chap. 9.

Webb, W. W.

W. W. Webb, K. Wells, D. R. Sandison, J. Strickler, Optical Microscopy for Biology (Wiley-Liss, New York, 1990), pp. 73–108.

K. S. Wells, D. R. Sandison, J. Strickler, W. W. Webb, “Quantitative fluorescence imaging with laser scanning confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 3.
[CrossRef]

D. R. Sandison, D. W. Piston, W. W. Webb, “Background rejection and optimization of signal-to-noise in confocal microscopy,” in Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Specimens, J. K. Stevens, L. R. Mills, J. E. Trogadis, eds., (Academic, New York, 1994), Chap. 2.

Wells, K.

W. W. Webb, K. Wells, D. R. Sandison, J. Strickler, Optical Microscopy for Biology (Wiley-Liss, New York, 1990), pp. 73–108.

Wells, K. S.

K. S. Wells, D. R. Sandison, J. Strickler, W. W. Webb, “Quantitative fluorescence imaging with laser scanning confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 3.
[CrossRef]

Wilkinson, R. S.

J. W. Lichtman, W. J. Sunderland, R. S. Wilkinson, “High-resolution imaging of synaptic structure with a simple confocal microscope,” New Biologist 1, 75–82 (1989).
[PubMed]

Wilson, T.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 3.

T. Wilson, Confocal Microscopy (Academic, San Diego, Calif., 1990).

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 25.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 33.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 437.

Appl. Opt.

Biophys. J.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

Comput. Phys. Commun.

F. Johnson, “An improved method for computing a discrete Hankel transform,” Comput. Phys. Commun. 43, 181–202 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Micron Microsc. Acta

C. J. R. Sheppard, “Stray light and noise in confocal microscopy,” Micron Microsc. Acta 22, 239–243 (1991).
[CrossRef]

New Biologist

J. W. Lichtman, W. J. Sunderland, R. S. Wilkinson, “High-resolution imaging of synaptic structure with a simple confocal microscope,” New Biologist 1, 75–82 (1989).
[PubMed]

Scanning

C. J. R. Sheppard, C. J. Cogswell, M. Gu, “Signal strength and noise in confocal microscopy: factors influencing selection of an optimum detector aperture,” Scanning 13, 233–240 (1991).
[CrossRef]

Science

M. D. Egger, M. Petràň, “New reflected light microscope for viewing unstained brain and ganglion cells,” Science 157, 305–307(1967).
[CrossRef] [PubMed]

Other

G. S. Kino, “Intermediate optics in Nipkow disk microscopes,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 10.
[CrossRef]

M. Minsky, “Microscopy apparatus,” U.S. patent3,013,467 (19December1961).

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

T. Wilson, Confocal Microscopy (Academic, San Diego, Calif., 1990).

G. J. Brakenhoff, K. Visscher, “Novel confocal imaging and visualization techniques,” in Micro 90: Proceedings of the Royal Microscopical Society Conference, H. Y. Elder, ed. (Hilger, London, 1990), Chap. 9.

W. W. Webb, K. Wells, D. R. Sandison, J. Strickler, Optical Microscopy for Biology (Wiley-Liss, New York, 1990), pp. 73–108.

K. S. Wells, D. R. Sandison, J. Strickler, W. W. Webb, “Quantitative fluorescence imaging with laser scanning confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 3.
[CrossRef]

D. R. Sandison, D. W. Piston, W. W. Webb, “Background rejection and optimization of signal-to-noise in confocal microscopy,” in Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Specimens, J. K. Stevens, L. R. Mills, J. E. Trogadis, eds., (Academic, New York, 1994), Chap. 2.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 3.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 437.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 25.

Documentation and FORTRAN code for Hankel Transform Library kindly supplied and supported by R. F. Loane, Department of Applied and Engineering Physics, Cornell University, Ithaca, N. Y.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), p. 33.

D. R. Sandison, “Fluorescence confocal laser scanning microscopy for three-dimensional imaging of living biological specimens,” Ph.D. dissertation (Cornell University, Ithaca, N. Y., 1993), p. 97.

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional imaging on confocal and wide-field microscopes,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990), Chap. 14.
[CrossRef]

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), integral 6.554.3, p. 682.

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Figures (13)

Fig. 1
Fig. 1

3-D images of a point source. Isointensity contours that represent sections through the image of a point source (perpendicular to the focal plane) are presented for (a) conventional and (b) ideal confocal microscopes. The intensity at the origin is normalized to 1. The contours in (a) describe the PSF. The PSF also determines the excitation intensity profile in a sample illuminated by a single diffraction-limited spot. The image of a point in confocal microscopy (b) is given by the square of the PSF. In the conventional microscope, all xy planes within the sample thickness contribute equal background, as indicated by contours extending far from the central region. The 0.1 contour in (a) approximates the 1/e 2 resolution volume.

Fig. 2
Fig. 2

Definitions of signal and background. (a) Discrete sample: signal S d is the fluorescence detected from a single in-focus point source; background B d is fluorescence detected from a uniformly fluorescent sample of thickness t, including the focal region. (b) Continuous sample: signal S c is the detected fluorescence originating from within the ellipsoidal resolution volume centered in a uniformly fluorescent sample and bounded by the 1/e 2 isointensity contour of the PSF [the ellipse in Fig. 1(a)]; background B c is the detected fluorescence that originates in the fluorophore that is outside the same volume.

Fig. 3
Fig. 3

Schematic of a generalized epifluorescence microscope. An excitation power P 0 is imaged to the u = 0 plane of a fluorescent sample of thickness t by objective lens L o . The excitation source could be a single point source, an array of point sources, or a line source. The fluorescence generated in the sample is collected by L o and transmitted to image plane I. Image plane photons passing through a circular detector aperture of radius υ D or a slit aperture of half-width υ D are detected and used to form an image. For simplicity, the scanning system is not shown.

Fig. 4
Fig. 4

Fraction of confocally detected fluorescence that originates from inside various resolution volumes. The total fluorescence detected from a thick (t > 100 ou), continuous sample by an ideal confocal microscope is given by n T . The fraction of that fluorescence that originates from within a resolution volume of lateral radius υ R is given by n R /n T [see Eqs. (5) and (6)]. Radii associated with common resolution criteria are identified as full width at half-maximum (■), 1/e (●), 1/e 2 (▲), and the Rayleigh criterion (⋆).

Fig. 5
Fig. 5

Dependence of signal on detector aperture size. In the lower curves the continuous sample is illuminated by a single diffraction-limited spot, and signal (S c [υ D ]) is detected through both a circular aperture of radius υ D and a slit aperture of half-width υ D . The value of S c at υ D = ∞ is normalized to 1 for the continuous sample. In the upper curves the discrete sample is also illuminated by a diffraction-limited spot, and S d [υ D ] is plotted for the circular and the slit apertures. The number of signal fluorophores in the discrete sample is set to the continuous sample value of N d = V e C 0. Because all N d discrete sample fluorophores are illuminated at the peak excitation intensity, the total signal generated is 2.7 times higher than that of the continuous sample.

Fig. 6
Fig. 6

Dependence of continuous sample background on detector aperture radius and sample thickness. Continuous samples of thickness t are illuminated by a diffraction-limited spot, and background B c [υ D , t] is detected through a circular aperture of radius υ D . Note the linear dependence of generated background on t; the prefactor γC 0(4π)2 is set to 1. The inset shows B c [υ D , t] for 0 < υ D < 10 ou. The dashed line through υ D = t indicates the minimum radius that passes all the collected fluorescence to the detector, which yields conventional imaging properties.

Fig. 7
Fig. 7

Dependence of the continuous sample background on sample thickness for various microscope geometries. The background B c [υ D , t] detected through an infinitesimal aperture (υ D < 0.5 ou) is calculated for each illumination geometry; B c [∞, t] is calculated for the conventional microscope. The prefactor “γC 04π is normalized to 1 for all curves. In addition, confocal, spinning-disk, and line-illumination microscopes are normalized by the circular aperture area πυ D 2. The slit detection curve is normalized by the slit width 2υ D , and the conventional microscope curve is normalized by 4π.

Fig. 8
Fig. 8

Dependence of the continuous sample S/B on microscope geometry and sample thickness. (S/B) c [0, t] is plotted for each illumination geometry and is valid for υ D < 0.5 ou. (S/B) c [∞, t] is plotted for the conventional microscope. The vertical dotted line indicates the length of the 1/e 2 resolution volume (17.4 ou).

Fig. 9
Fig. 9

Dependence of the S/B on detector aperture size. For a continuous sample, (S/B) c [υ D , 1000 ou] is plotted for circular apertures of radius υ D in confocal, spinning disk, and line scanning microscopes and for a slit aperture of half-width υ D in the slit detection microscope. (S/B) c [∞, 1000 ou] = 0.0078 in the conventional microscope. The filled circles mark the aperture size where (S/B) c is one-half its peak value; they are at υ D = 3 ou for the circular aperture and at υ D = 9.8 ou for the slit aperture.

Fig. 10
Fig. 10

Dependence of the S/N on image background. In the upper panel the image of an on-axis point source of peak intensity 2.6 is calculated in the presence of a constant background of intensity of 1.0, so that S/B = 2.6. The image plane distance from the optic axis is υ I . In the lower panel the image of the same point source is calculated with 26 times more background, and the Airy pattern is lost in the shot noise of the high background.

Fig. 11
Fig. 11

Dependence of the S/N on the detector aperture radius and the S/B. (S/N) c [υ D , t] is calculated for a thick (t > 100 ou), continuous sample and an aperture of radius υ D for selected values of (S/B) c [0, t]. For each (S/B) c [0, t] there is an optimal radius υ D * that maximizes the S/N, as shown by the dashed curve. The peak value of the (S/B) c [0, t] = 2.6 curve is normalized to 1.

Fig. 12
Fig. 12

Dependence of optimal S/N ratio performance on sample thickness. (S/N) c *[t] for each microscope is normalized by the conventional microscope value (S/N) c [∞, t] and plotted as a function of sample thickness (t).

Fig. 13
Fig. 13

Geometric optics calculation of the background intensity from an out-of-focus plane I B [υ I , u]. The PSF h 2[u, υ] is plotted in (a) for u = 100 ou along with its geometric approximation, which is the constant intensity extending to υ = u. In this approximation, the convolution in expression (B2) becomes the overlap area of two circles, each with a radius u and center-to-center separation υ I , as shown in (b). This overlap area is shown in (c) as the solid curve, and the actual convolution is shown as the barely visible dashed curve.

Tables (4)

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Table 1 Comparison of Continuous Sample Signal for t > 17.4 ou a

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Table 2 Continuous Sample Background for Thick Specimens (t > 100 ou) a

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Table 3 Discrete Sample Signal-to-Background Ratio (S/B) d as a Function of Sample Thickness and Microscope Geometry a

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Table 4 Continuous Sample (S/B)c as a Function of Sample Thickness and Microscope Geometry a

Equations (19)

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v = k ( NA ) r ,
u = k ( NA ) 2 z / n ,
h 2 [ u , v ] = | 2 0 1 ρ d ρ J 0 [ v ρ ] exp ( i u ρ 2 / 2 ) | 2 .
{ S B } = γ D d v I 2 t / 2 t / 2 d u [ ( { F S [ u , v ] F B [ u , v ] } i e [ u , v ] ) h 2 [ u , v ] ] ,
n T = γ A D C 0 sample d u v d v d θ h 4 [ u , v ] .
n R = γ A D C 0 V R d u v d v d θ h 4 [ u , v ] .
F s [ u , v ] = { C 0 if [ v 2 + ( 0 . 3 u ) 2 ] 1 / 2 < 2 . 6 ou 0 otherwise .
S d [ v D ] = γ N d 4 π ( 1 J 0 2 [ v D ] J 1 2 [ v D ] ) ,
B c [ v D , t ] = [ γ C 0 D d v I 2 t / 2 t / 2 d u ( i e [ u , v ] h 2 [ u , v ] ) ] S c [ v D , t ] .
( S / N ) [ v D , t ] = ( S [ v D ] ) 1 / 2 [ ( S / B ) [ v D , t ] 1 + ( S / B ) [ v D , t ] ] 1 / 2 .
h l [ u , v x , v y ] = C d v y i h [ u , v x , ( v y v y i ) ] ,
h l [ u , v x ] = C d v Y 0 1 ρ d ρ × exp ( i u ρ 2 / 2 ) J 0 { v x ρ [ 1 + ( v Y v x ) 2 ] 1 / 2 } .
h l [ u , v x ] = C 0 1 exp ( i u ρ 2 / 2 ) ρ d ρ 1 v x z J 0 [ v x ρ z ] d z z 2 1 .
1 x J 0 [ y x ] d x x 2 1 = cos y y
h l [ u , v x ] = 0 1 exp ( i u ρ 2 / 2 ) cos [ ρ v x ] d ρ .
h 2 [ u , v ] = { 4 u 2 if v < u 0 otherwise .
I B [ v I , u ] ( h 2 [ u , v ] h 2 [ u , v ] )
I B [ v I , u ] ( 16 u 4 ) 2 [ u 2 cos 1 [ v I 2 u ] v I 2 ( u 2 v I 2 4 ) 1 / 2 ] .
I B [ v I , t ] 32 { ( 8 3 v I + 4 v I 3 t 2 ) [ 1 ( v I t ) 2 ] 1 / 2 4 t cos 1 [ v I t ] } .

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