Abstract

Imaging models for differential-interference-contrast (DIC) microscopy are presented. Two- and three-dimensional models for DIC imaging under partially coherent illumination were derived and tested by using phantom specimens viewed with several conventional DIC microscopes and quasi-monochromatic light. DIC images recorded with a CCD camera were compared with model predictions that were generated by using theoretical point-spread functions, computer-generated phantoms, and estimated imaging parameters such as bias and shear. Results show quantitative and qualitative agreement between model and data for several imaging conditions.

© 1999 Optical Society of America

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References

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    [PubMed]
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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]
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1997

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

1993

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and applications in cell, developmental, and marine biology,” Opt. Eng. 32, 3223–3228 (1993).
[CrossRef]

1992

C. J. Cogswell, C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

S. F. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 9, 154–166 (1992).
[CrossRef] [PubMed]

1989

1988

1987

1985

1982

W. Galbraith, “The image of a point of light in differential interference contrast microscopy: computer simulation,” Microsc. Acta 85, 233–254 (1982).

1980

1969

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

W. Lang, “Nomarski differential interference contrast microscopy II. Formation of the interference image,” Zeiss Inf. 17, 12–16 (1969).

1955

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Allen, R. D.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Aten, J. A.

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1964).

Cogswell, C. J.

C. J. Cogswell, C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

A. E. Dixon, C. J. Cogswell, “Confocal microscopy with transmitted light,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 479–490.

C. J. Cogswell, N. I. Smith, K. G. Larkin, P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Conchello, J.-A.

C. Preza, D. L. Snyder, J.-A. Conchello, “Imaging models for three-dimensional transmitted-light DIC microscopy,” in Three-Dimensional Microscopy:Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 245–257 (1996).
[CrossRef]

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

Dana, K.

K. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” M.S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).

David, G. B.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Dixon, A. E.

A. E. Dixon, C. J. Cogswell, “Confocal microscopy with transmitted light,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 479–490.

Galbraith, W.

W. Galbraith, “The image of a point of light in differential interference contrast microscopy: computer simulation,” Microsc. Acta 85, 233–254 (1982).

Gibson, S. F.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1984).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gordon, R. L.

Gundlach, H.

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and applications in cell, developmental, and marine biology,” Opt. Eng. 32, 3223–3228 (1993).
[CrossRef]

Hariharan, P.

C. J. Cogswell, N. I. Smith, K. G. Larkin, P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Hartman, J. S.

Holmes, T. J.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Inoue, S.

S. Inoue, “Ultrathin optical sectioning and dynamic volume investigation with conventional light microscopy,” in Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Systems, J. Stevens, L. Mills, J. Trogadis, eds. (Academic, San Diego, Calif., 1994), pp. 397–419.

Lang, W.

W. Lang, “Nomarski differential interference contrast microscopy II. Formation of the interference image,” Zeiss Inf. 17, 12–16 (1969).

Lanni, F.

Larkin, K. G.

C. J. Cogswell, N. I. Smith, K. G. Larkin, P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Lessor, D. L.

Levy, W. J.

Mao, X. Q.

Markham, J.

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

Nakagawa, K.

K. Nakagawa, “Cross-linked acrylic polymers for optical waveguide applications,” M.S. thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1994).

Nemoto, I.

Nomarski, G.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Pluta, M.

M. Pluta, Advanced Light Microscopy: Specialized Methods (Elsevier, Amsterdam, 1989), pp. 146–197.

M. Pluta, Advanced Light Microscopy: Specialized Methods (Elsevier, Amsterdam, 1989).

Preza, C.

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

C. Preza, D. L. Snyder, J.-A. Conchello, “Imaging models for three-dimensional transmitted-light DIC microscopy,” in Three-Dimensional Microscopy:Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 245–257 (1996).
[CrossRef]

C. Preza, “Phase estimation using rotational diversity for differential interference contrast microscopy,” D.Sc. dissertation (Sever Institute of Technology, Washington University, St. Louis, Mo., 1998).

Rosenberger, F. U.

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Sheppard, C. J. R.

C. J. Cogswell, C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
[CrossRef]

Smith, N. I.

C. J. Cogswell, N. I. Smith, K. G. Larkin, P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Snyder, D. L.

C. Preza, D. L. Snyder, J.-A. Conchello, “Imaging models for three-dimensional transmitted-light DIC microscopy,” in Three-Dimensional Microscopy:Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 245–257 (1996).
[CrossRef]

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

Streibl, N.

van Munster, E. B.

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

van Vliet, L. J.

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1964).

Appl. Opt.

J. Microsc.

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

C. J. Cogswell, C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Microsc. Acta

W. Galbraith, “The image of a point of light in differential interference contrast microscopy: computer simulation,” Microsc. Acta 85, 233–254 (1982).

Opt. Eng.

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and applications in cell, developmental, and marine biology,” Opt. Eng. 32, 3223–3228 (1993).
[CrossRef]

Proc. R. Soc. London Ser. A

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Z. Wiss. Mikrosk.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Zeiss Inf.

W. Lang, “Nomarski differential interference contrast microscopy II. Formation of the interference image,” Zeiss Inf. 17, 12–16 (1969).

Other

M. Pluta, Advanced Light Microscopy: Specialized Methods (Elsevier, Amsterdam, 1989), pp. 146–197.

M. Pluta, Advanced Light Microscopy: Specialized Methods (Elsevier, Amsterdam, 1989).

A. E. Dixon, C. J. Cogswell, “Confocal microscopy with transmitted light,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 479–490.

J. B. Pawley, ed., Handbook of Biological Confocal Microscopy (Plenum, New York, 1995).

C. Preza, D. L. Snyder, J.-A. Conchello, “Imaging models for three-dimensional transmitted-light DIC microscopy,” in Three-Dimensional Microscopy:Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 245–257 (1996).
[CrossRef]

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

K. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” M.S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).

C. Preza, “Phase estimation using rotational diversity for differential interference contrast microscopy,” D.Sc. dissertation (Sever Institute of Technology, Washington University, St. Louis, Mo., 1998).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1964).

J. W. Goodman, Statistical Optics (Wiley, New York, 1984).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

K. Nakagawa, “Cross-linked acrylic polymers for optical waveguide applications,” M.S. thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1994).

C. J. Cogswell, N. I. Smith, K. G. Larkin, P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

S. Inoue, “Ultrathin optical sectioning and dynamic volume investigation with conventional light microscopy,” in Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Systems, J. Stevens, L. Mills, J. Trogadis, eds. (Academic, San Diego, Calif., 1994), pp. 397–419.

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Figures (17)

Fig. 1
Fig. 1

Schematic of a DIC microscope with two typical Wollaston prisms (the Nomarski DIC system uses modified Wollaston prisms, referred to as Nomarski prisms5). Note that the angular wave splitting and the shear distance are exaggerated for clarity, and they should be very small relative to the size of the field of view.

Fig. 2
Fig. 2

Calculated 2-D DIC PSF (of a 10×/0.3-NA lens) with bias equal to 1.57 rad, shear distance equal to 1.1 μm along the horizontal axis, and illumination light wavelength equal to 550 nm: (a) real part of the complex amplitude and (b) imaginary part of the complex amplitude. The direction of shear is along the horizontal axis. The scale bar is approximately 1.64 μm. The black regions in the images represent negative values. Profiles through the center of the images are shown in Fig. 3.

Fig. 3
Fig. 3

Effect of the bias value on the DIC PSF. Horizontal profiles are shown from the DIC PSF of a 10×/0.3-NA dry lens. The profiles plot (a) the real part and (b) the imaginary part of h(x, 0) along the direction of shear (x axis). The shear distance is 1.1 μm. The bias values are shown by the curves and they are in radians. The PSF was normalized so that the real part has a peak equal to 1.0 for bias equal to 0.0 rad, and the imaginary part has a peak equal to -1.0 for bias equal to 3.14 rad.

Fig. 4
Fig. 4

xy- and xz-section images from a 3-D calculated DIC PSF of a 10×/0.3-NA lens with bias equal to 0.0 rad: (a) real part of the complex amplitude, (b) imaginary part of the complex amplitude, (c) squared magnitude. In the images black represents the minimum negative value in (a) and (b) and a zero value in (c), while white represents the maximum positive value of each image. For the xy images, z=1.7 μm away from focus. The shear distance is equal to 0.68 μm, and the direction of shear is along the x axis. The scale bar is approximately 3 μm.

Fig. 5
Fig. 5

Effect of the bias on the Fourier transform of the PSF of a 10×/0.3-NA dry lens. Vertical (top) and horizontal (bottom) profiles through the center of the imaginary part of H(f, g) are shown. The bias values are shown by the curves, and they are in radians. We note that the edges of the curves are sloped because the sampling is coarse.

Fig. 6
Fig. 6

Fourier transform H(f, g, j) of the 3-D DIC PSF of a 10×/0.3-NA lens with bias=0.0 rad, shear=1.0 μm, and 550-nm illumination light wavelength: (a) real part of the complex amplitude, (b) imaginary part of the complex amplitude, (c) squared magnitude. The fg-section images (top row) are cuts through the 3-D image at the lines shown in the fj-section images, while the fj-section images (bottom row) are through the center of the volume as shown by the lines in the fg-section images. The direction of shear is along the f axis. The spatial-frequency cutoffs are fc=gc=0.545 μm-1 and jc=0.083 μm-1. Black regions in (a) and (b) represent negative values, while in (c) they represent values equal to zero.

Fig. 7
Fig. 7

Köhler illumination. An incoherent light source is focused in the front-focal plane of the condenser lens, and thus the specimen is illuminated by plane waves with normal vectors (kx, ky, kz), where kx=2πξ/λfcon, ky=2πη/λfcon, kz=(2π/λ)2-kx2-ky2, which make angles θx=sin-1 (ξ/fc) and θy=sin-1 (η/fc) with the yz and xz planes, respectively (see Ref. 17, p. 49). The direction of each plane wave depends on the illuminating point (ξ, η) in the front-focal plane of the condenser lens. The DIC components are not shown here for simplicity.

Fig. 8
Fig. 8

Schematic of the 3-D phantom specimen with crossed bars. n1 and n2 are the refractive indices of the surrounding medium and of the bars, respectively.

Fig. 9
Fig. 9

Comparison of synthetic DIC images of the groove phantom computed with (a) the point-aperture model and (b) the general model with an aperture of normalized radius r equal to 0.3, assuming a 10×/0.3-NA objective lens, a 0.55-NA condenser lens with focal distance fc=24 mm, light wavelength λ=550 nm, bias=-1.16 rad, and shear=1.5 μm. Horizontal profiles through the images are compared in (c) (r=0.0 corresponds to the point-aperture model).

Fig. 10
Fig. 10

Comparison of DIC images of the groove phantom: (a) measured DIC image from the physical phantom with the use of a 25×/0.5-NA lens (shear=1.0 μm), a 550-nm illumination wavelength, and an open condenser aperture; (b) synthetic DIC image generated by using the general model with an aperture of normalized radius equal to 0.1 and an estimated bias equal to 1.178 rad; (c) horizontal profiles through the measured image (data) and the synthetic image (model).

Fig. 11
Fig. 11

Measured and synthetic DIC images of a phantom with crossed bars. The measured DIC image is acquired with a 10×/0.3-NA lens and a 0.55-NA condenser lens with the aperture closed (a). Model predictions are generated with the 3-D model that assumes (b) superposition of amplitudes and a point condenser aperture and (c) superposition of intensities and a point condenser aperture. A DIC bias equal to 0.0 rad was used for the model predictions. The section images are cuts through the 3-D images, and they are approximately at the best focus of the objective lens.

Fig. 12
Fig. 12

(a) Measured and (b) synthetic DIC images of a cross phantom imaged with a 10×/0.3-NA objective lens and a 0.55-NA condenser lens with the condenser aperture closed. The synthetic images were computed with the point-aperture model. The xy images are cuts through the 3-D image along the z axis at z=0 μm (i.e., near the best focus). The xz-section images are cuts through the top of the xy images. For the model prediction, an estimated bias of 0.05 rad and a shear distance equal to 0.68 μm were used. The xz-section images were scaled (by linear interpolation) to a 1:1 aspect ratio. The scale bar is approximately 9 μm.

Fig. 13
Fig. 13

(a) Measured and (b) synthetic DIC images of a 210-nm-diameter bead imaged with a 10×/0.3-NA objective lens and a 0.55-NA condenser lens with the condenser aperture closed. The synthetic images were computed with the point-aperture model. For the model prediction, an estimated bias of 0.0015 rad and a shear distance equal to 1.2 μm were used. The scale bar is approximately 4.3 μm.

Fig. 14
Fig. 14

Measured DIC images of the crossed-bars phantom (top row) and model predictions (bottom row) of the DIC image of the phantom specimen shown in Fig. 8, obtained with a 3-D model that assumes superposition of amplitudes and a point condenser aperture, with DIC bias equal to -0.001 rad, at different orientations of the phantom: (a) the phantom is oriented so that the bars are aligned with the x and y axes; (b) the phantom is rotated by 13° clockwise from position (a); (c) the phantom is rotated by 32° from position (a); (d) the phantom is rotated by 47° from position (a).

Fig. 15
Fig. 15

Meridional (xz) sections from synthetic bead images computed with the general model for a 40×/0.55-NA objective lens and a 0.55-NA condenser lens with an aperture of normalized radius equal to (a) 0.06, (b) 0.12, (c) 0.25, and (d) 0.44. A DIC bias equal to 0.0 rad and a 550-nm illumination wavelength were used. The shear distance is 1.0 μm, and the direction is along the 135° axis as in Fig. 14. The vertical and horizontal scale bars represent 25.7 μm and 4.8 μm, respectively.

Fig. 16
Fig. 16

Meridional (xz) sections of the bead phantom images measured with a 40×/0.55-NA dry lens and a 0.55-NA condenser with the aperture (a) closed, (b) 25% open, (c) 50% open, (d) 75% open, and (e) open. Because the condenser aperture iris can be opened and closed with an uncalibrated slider, the aperture size reported here is the position of the slider with respect to its closed and open positions. The vertical and horizontal scale bars represent 25.7 μm and 4.8 μm, respectively. The shear direction is along the 135° axis as in Fig. 14.

Fig. 17
Fig. 17

xz-section image from a synthetic bead image computed with the point-aperture model for a 40×/0.55-NA objective lens with a spherically aberrant PSF. A DIC bias equal to 0.0 rad and a 550-nm illumination wavelength were used. The shear distance is 1.0 μm, and the direction is along the 135° axis as in Fig. 14. The vertical and horizontal scale bars represent 25.7 μm and 4.8 μm, respectively.

Equations (25)

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h(x, y)=(1-R)exp(-jΔθ)k(x-Δx, y)-R exp(jΔθ)k(x+Δx, y),
p(x, y, Δz)=|p(x, y, Δz)|exp[j2πw(x, y, Δz)/λ],
H(f, g)=-j sin(2πfΔx+Δθ)K(f, g),
Uc(xo)=-+Us(ξ)hc(ξ; xo)dξ,
Ui(x)=-+Uo(xo)h(x-xo)dxo.
i(x)=-+α(ξ)-+f(xo)h(x-xo)hc(ξ; xo)dx02 dξ.
i(x)=a1-+f(xo)h(x-xo)dxo2,
c(x)=0.5a1-+f(xo)k(x-xo)dxo=|c(x)|exp[-jϕc(x)].
i(x, y)=|exp(-jΔθ)c(x-Δx, y)-exp(jΔθ)c(x+Δx, y)|2=|c(x1, y)|2+|c(x1+2Δx, y)|2-2|c(x1, y)||c(x1+2Δx, y)|cos[ϕc(x1, y)-ϕc(x1+2Δx, y)+2Δθ],
h(x, y)=0.5 exp(-jΔθ)δ(x-Δx, y)-0.5×exp(jΔθ)δ(x+Δx, y),
i(x, y)=a1 sin2{0.5[ϕ(x-Δx, y)-ϕ(x+Δx, y)]+Δθ}.
i(x, y)=a1 sin2Δx ϕ(x, y)x+Δθ.
i(x)=-+-+f(xo)f*(xo)js(xo-xo)h(x-xo)h*(x-xo)dxodxo,
js(xo-xo)=E[Uc(xo)Uc*(xo)]=1/(λfcon)2-+α(ξ)×exp{j2π[(xo-xo)tξ]/λfcon}dξ
i(x)=-+-+TDIC(f; f)F(f)F*(f)×exp[j2π(f-f)tx]dfdf,
TDIC(f; f)=-+H(f+f)H*(f+f)Js(f)df
I(f)=-+TDIC(f+f; f)F(f+f)F*(f)df.
MDIC(f)=H(f)H*(-f),
MDIC(f; f)=sin(2πfΔx+Δθ)×sin[2π(-f)Δx+Δθ]M(f; f),
izf(x, zf-zo)=-+α(ξ)-+f(xo, zo)×h(x-xo, zf-zo)hc(ξ; xo)dxo2 dξ,
i(x, z)=-+izf(x, z-zo)dzo.
i(x, z)=-+α(ξ)-+-+f(xo, zo)h(x-xo, z-zo)×hc(ξ; xo)dxodzo2 dξ.
ur=rcλfcon=r tan αλ,
f(x)=exp[-jϕ(x)]=cos ϕ1-j sin ϕ1,x=(0, 0)1otherwise.
i(x)=a1(b1+jb2)-+δ(xo)h(x-xo)dxo+-+h(x-xo)dxo2=a1|(b1+jb2)h(x)+H(0, 0)|2=a1{(b12+b22)|h(x)|2+|H(0, 0)|2+2 Re[(b1+jb2)h(x)H(0, 0)*]},

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