Abstract

A methodology is described for phase restoration of an object function from differential interference contrast (DIC) images. The methodology involves collecting a set of DIC images in the same plane with different bias retardation between the two illuminating light components produced by a Wollaston prism. These images, together with one conventional bright-field image, allows for reduction of the phase deconvolution restoration problem from a highly complex nonlinear mathematical formulation to a set of linear equations that can be applied to resolve the phase for images with a relatively large number of pixels. Additionally, under certain conditions, an on-line atomic force imaging system that does not interfere with the standard DIC illumination modes resolves uncertainties in large topographical variations that generally lead to a basic problem in DIC imaging, i.e., phase unwrapping. Furthermore, the availability of confocal detection allows for a three-dimensional reconstruction with high accuracy of the refractive-index measurement of the object that is to be imaged. This has been applied to reconstruction of the refractive index of an arrayed waveguide in a region in which a defect in the sample is present. The results of this paper highlight the synergism of far-field microscopies integrated with scanned probe microscopies and restoration algorithms for phase reconstruction.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. S. Inoue, K. R. Spring, Video Microscopy: the Fundamentals (Plenum, New York, 1986), pp. 410–412.
  4. T. J. Holmes, “Signal-processing characteristics of differential-interference-contrast microscopy. 2: Noise considerations in signal recovery,” Appl. Opt. 27, 1302–1309 (1988).
    [CrossRef] [PubMed]
  5. 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. (Oxford) 188, 149–157 (1997).
    [CrossRef]
  6. 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 Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
    [CrossRef]
  7. C. Preza, D. L. Snyder, J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
    [CrossRef]
  8. R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 65–69.
  9. J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust. Speech Signal Process. 29, 188–197 (1981).
    [CrossRef]
  10. 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. (Oxford) 165, 81–101 (1992).
    [CrossRef]
  11. P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. (Oxford) 193, 50–61 (1999).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 434–441.
  13. A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
    [CrossRef]
  14. N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
    [CrossRef]
  15. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  16. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 539.
  17. R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 71–79.
    [CrossRef]

2001 (1)

J. M. Kahn, K. P. Ho, “Communications technology: A bottleneck for optical fibres,” Nature 411, 1007–1010 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (2)

C. Preza, D. L. Snyder, J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
[CrossRef]

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. (Oxford) 193, 50–61 (1999).
[CrossRef]

1997 (1)

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. (Oxford) 188, 149–157 (1997).
[CrossRef]

1995 (1)

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

1992 (1)

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. (Oxford) 165, 81–101 (1992).
[CrossRef]

1988 (1)

1985 (1)

1981 (1)

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust. Speech Signal Process. 29, 188–197 (1981).
[CrossRef]

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. (Oxford) 188, 149–157 (1997).
[CrossRef]

Axelrod, N.

N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
[CrossRef]

Ben-Ami, N.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Ben-Ami, U.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Ben-Yosef, N.

N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
[CrossRef]

Biemond, J.

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 65–69.

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 71–79.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 434–441.

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. (Oxford) 165, 81–101 (1992).
[CrossRef]

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 Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Conchello, J. A.

Fish, G.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Gemkow, M. J.

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. (Oxford) 193, 50–61 (1999).
[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 Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Ho, K. P.

J. M. Kahn, K. P. Ho, “Communications technology: A bottleneck for optical fibres,” Nature 411, 1007–1010 (2001).
[CrossRef] [PubMed]

Holmes, T. J.

Ingle, V. K.

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust. Speech Signal Process. 29, 188–197 (1981).
[CrossRef]

Inoue, S.

S. Inoue, K. R. Spring, Video Microscopy: the Fundamentals (Plenum, New York, 1986), pp. 410–412.

Jovin, T. M.

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. (Oxford) 193, 50–61 (1999).
[CrossRef]

Kahn, J. M.

J. M. Kahn, K. P. Ho, “Communications technology: A bottleneck for optical fibres,” Nature 411, 1007–1010 (2001).
[CrossRef] [PubMed]

Khachatryan, E.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Khatchatouriants, A.

N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
[CrossRef]

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 539.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 539.

Lagendijk, R. L.

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 71–79.
[CrossRef]

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 65–69.

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 Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Lewis, A.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
[CrossRef]

Lieberman, K.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Preza, C.

Radko, A.

N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
[CrossRef]

Shalom, S.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

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. (Oxford) 165, 81–101 (1992).
[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 Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

Snyder, D. L.

Spring, K. R.

S. Inoue, K. R. Spring, Video Microscopy: the Fundamentals (Plenum, New York, 1986), pp. 410–412.

Streibl, N.

Treinin, M.

N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
[CrossRef]

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. (Oxford) 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. (Oxford) 188, 149–157 (1997).
[CrossRef]

Verveer, P. J.

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. (Oxford) 193, 50–61 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 434–441.

Woods, J. W.

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust. Speech Signal Process. 29, 188–197 (1981).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust. Speech Signal Process. 29, 188–197 (1981).
[CrossRef]

J. Microsc. (Oxford) (3)

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. (Oxford) 165, 81–101 (1992).
[CrossRef]

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. (Oxford) 193, 50–61 (1999).
[CrossRef]

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. (Oxford) 188, 149–157 (1997).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature (1)

J. M. Kahn, K. P. Ho, “Communications technology: A bottleneck for optical fibres,” Nature 411, 1007–1010 (2001).
[CrossRef] [PubMed]

Ultramicroscopy (1)

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Other (7)

N. Axelrod, A. Radko, N. Ben-Yosef, A. Khatchatouriants, M. Treinin, A. Lewis, “Near-field optical and atomic force constraints for superresolution 3D deconvolution in far-field optical microscopy,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII, J. A. Conchello, C. J. Cogswell, A. G. Tescher, T. Wilson, eds., Proc. SPIE3919, 161–171 (2000).
[CrossRef]

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 65–69.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 539.

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 71–79.
[CrossRef]

S. Inoue, K. R. Spring, Video Microscopy: the Fundamentals (Plenum, New York, 1986), pp. 410–412.

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 Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 72–81 (1997).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 434–441.

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Figures (9)

Fig. 1
Fig. 1

Diagram of the experimental system used for phase restoration and measurement of the refractive index. The system consists of a conventional upright microscope equipped with a Wollaston prism for DIC imaging. An avalanche photodiode (APD) was used. A confocal aperture of 20-μm diameter was placed before the detector to allow for confocal imaging. We positioned a MultiView 1000 NSOM/SPM confocal system (Nanonics Imaging, Ltd., Jerusalem, Israel) on the microscope stage.

Fig. 2
Fig. 2

AFM image of the chrome mask with a chrome-aluminum coating. The region, which is highlighted by a boxed area, is the region in which phase restoration was completed.

Fig. 3
Fig. 3

(a) Confocal image of the light emitted from the NSOM tip. The images were acquired with a Nikon 50× 0.45 NA objective. (b) Amplitude in-focus PSF obtained from the image in (a).

Fig. 4
Fig. 4

(a) Bright-field and (b) and (c) DIC optical images of the test sample obtained with a Nikon 50× objective with a NA of 0.45. The DIC images were obtained with a phase retardation of ±30° relative to the DIC dark image. Arrows in (a) and (c) indicate the regions that differ significantly from the surrounding regions.

Fig. 5
Fig. 5

(a) Processed image F(x; Δx) obtained from Eq. (7). (b) The phase restored image ϕ c (x) defined in Eq. (3) was obtained by deconvolving image (a) with the PSF ζΔx (x) defined in Eq. (8). The black boxes in (b) and in Fig. (2) correspond to the same region of the object.

Fig. 6
Fig. 6

(a) Phase restored image ϕ f (x), where ϕ f (x) is the phase of object function f, corresponding to the black box highlighted region in Fig. 5(b). This image was obtained by deconvolving function c(x) defined in Eq. (3) with the PSF shown in Fig. 3(b). (b) The AFM image corresponding to the black box highlighted region in Fig. 2. (c) Comparison of the cross sections of the image restored by the algorithm and the simultaneously obtained AFM image. The line scan corresponds to the black lines in (a) and (b). The y axis is in radians, which is obtained directly in the case of a DIC phase restored image and is calculated for the AFM image with the formula phase= 2 × 2π/λ × z(x, y), where z(x, y) represents topography variations. (d) Image obtained by convolution of the AFM image in (b) with the PSF ζΔx (x) defined in Eq. (8) and multiplied by the factor 2 × 2π/λ. In this image the two highlighted regions indicate the regions in which the phase gradient of function ϕ c (x) is more than π/2 and less than -π/2. The color bar added to this image is in radians. The arrows in all the images indicate the region that corresponds to the regions in (a) in which phase unwrapping was problematic.

Fig. 7
Fig. 7

Panoramic DIC image of an arrayed waveguide captured with a video camera. The boxed area is the region that was imaged with confocal DIC. The image was acquired with a Nikon 10× 0.2 NA objective.

Fig. 8
Fig. 8

Confocal DIC optical images of the region indicated in the boxed area in Fig. 7. The DIC images were obtained with a phase retardation of ±30° relative to the DIC dark image. All the images were acquired with a Nikon 50× 0.45 NA objective. Only two DIC images among a set of confocal images acquired at optical section z f corresponding to the middle of the waveguide are shown. The region chosen indicates a discontinuity (see circled regions), which is related to an index of refraction discontinuity as also seen in the restored index of refraction in Fig. 9(b).

Fig. 9
Fig. 9

(a) Restored image a(x, z f ) from the two DIC images shown in Fig. 8. The a(x, z f ) was first obtained by calculation of the function G(x, z f ), from Eq. (28) at the optical section z f corresponding to the middle of the waveguide, and then by deconvolving the function G(x, z f ) with the PSF ζΔx (x) defined in Eq. (8). (b) Image of refractive-index variations δn(x, z f ) obtained from (a) by deconvolving the function a(x, z f ) with the PSF s(x) defined in Eq. (30). (c) Refractive-index profile corresponding to the white line in the image in (b).

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

ix=- fx0hx-x0dx02.
hx=1/2 exp-jΔθkx-Δx-1/2 expjΔθkx+Δx,
cx=- fx0kx-x0dx0|cx|×expjϕcx,
ix; Δθ=1/4 |cx-Δx|2+|cx+Δx|2-2|cx+Δxcx-Δx|cosϕcx+Δx-ϕcx-Δx+2Δθ.
Δi12x; Δθ1, Δθ2=|cx-Δxcx+Δx|×sinΔθ1 -Δθ2×sinϕcx-Δx-ϕcx+Δx+Δθ1+Δθ2.
Δi12x; Δθ1=|cx-Δxcx+Δx|sin2Δθ1×sinϕcx-Δx-ϕcx+Δx.
Fx; Δxϕcx-Δx-ϕcx+Δx.
Fx; Δx=sin-1Δi12x; Δθ1|cx-Δxcx+Δx|sin2Δθ1.
Fx; Δx=- ϕcx0ζΔxx-x0dx0,
|δnx, z|  1.
Ex, z0=E0x, z0+Esx, z0.
E0x, z0+Δz=E0x, z0expjk0nx, z0Δz,
E0x, z0+Δz=E0x, z0+Δz+δEsx, z0+Δz,
δEsx, z0+ΔzE0x, z0+Δzjk0δnx, z0Δz.
Uscx, z=jk0E0 -- δnx0, z0hx-x0, z-z0dx0dz0,
hx, z=1/2 exp-jΔθkx-Δx, z-1/2 expjΔθkx+Δx, z,
kx, z=exp-jk0nczkx, z.
U0=- E0hx-x0dx0=-jE0 sinΔθ,
Ux, z=-jE0 sinΔθ+jk0E0δnx, z * hx, z,
Ix, z=I0sin2Δθ-2k0 sinΔθ×Reδnx, z * hx, z+k02|δnx, z * hx, z|2,
Ix, z=I0sin2Δθ-2k0I0 sinΔθReδnx, z * hx, z.
Ix, z=I0sin2Δθ+δnx, z * Hpx, z+δnx, z * HAx, z,
HPx, z=-2k0 sinΔθRehx, z,
HAx, z=2k0 sinΔθImhx, z.
Ix, z=I0k02|δnx, z * hx, z|2.
Ix, z; Δθ=I0sin2Δθ-k0I0 sinΔθ×Reexp-jΔθ· χx-Δx, z-expjΔθ·χx+Δx, z,
ΔI12x, z=I0k0 sin2Δθ·ax+Δx, z-ax-Δx, z,
ax, zReχx, z=δnx, z*Rekx, z.
Gx, zk0ax+Δx, z-ax-Δx, z.
Gx, z=ΔI12x, zI0 sin2Δθ.
Gx, z=k0- ax0, zζΔxx-x0dx0,
ax, zdf · - δnx0, zsx-x0dx0,
df=--Rekx, zdxdz
sx=1df-Rekx, zdz.

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