Abstract

Analytical and experimental results of a new type of optical scanning microscope, which uses a phase conjugate mirror and pinholes to achieve superresolution, are presented. The phase conjugate scanning microscope has a higher Rayleigh resolution limit and better sectioning discrimination than conventional, single pass, and double pass scanning microscopes. It also can reduce the effect of static and dynamic aberrations on the imaging process, is very easy to align, and has the potential of introducing optical power gain into the system.

© 1990 Optical Society of America

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References

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  1. O. C. Wells, Scanning Electron Microscopy (McGraw-Hill, New York, 1974).
  2. C. F. Quate, “Vacuum Tunneling: A New Technique for Microscopy,” Phys. Today26–33 (Aug.1986).
    [CrossRef]
  3. P. Sulewski, D. J. Bishop, R. C. Dynes, “A Description of the Bell Laboratories Scanned Acoustic Microscope,” Bell. Syst. Tech. J. 61, 2167–2183 (1982).
  4. P. Davidovits, M. D. Egger, “Scanning Laser Microscope for Biological Investigations,” Appl. Opt. 10, 1615–1619 (1971).
    [CrossRef] [PubMed]
  5. H. J. B. Marsman, R. W. Strikker, “Mechanical Scan System for Microscopic Applications,” Rev. Sci. Instrum. 54, 1047–1052 (1983).
    [CrossRef]
  6. Y. Ichioka, T. Kobayashi, H. Kitagawa, T. Suzuki, “Digital Scanning Laser Microscope,” Appl. Opt. 24, 691–696 (1985).
    [CrossRef] [PubMed]
  7. U. C. Fisher, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B 3, 386–390 (1985).
    [CrossRef]
  8. J. S. Ploem, “Laser Scanning Fluorescent Microscopy,” Appl. Opt. 26, 3226–3231 (1987).
    [CrossRef] [PubMed]
  9. C. J. R. Sheppard, A. Choudhury, “Image Formation in the Scanning Microscope,” Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  10. E. Betzig, M. Isaacson, A. Lewis, “Collection Mode Near-Field Scanning Microscopy,” Appl. Phys. Lett. 51, 2088–2090 (1987).
    [CrossRef]
  11. Z. S. Hegedus, V. Sarafis, “Superresolving Filters in Confocally Scanned Imaging Systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]
  12. C. J. R. Sheppard, T. Wilson, “Multiple Traversing of the Object in the Scanning Microscope,” Opt. Acta 27, 611–623 (1980).
    [CrossRef]
  13. K. M. Johnson, W. T. Cathey, C. C. Mao, “Image Formation in a Phase Conjugate Scanning Microscope,” Appl. Phys. Lett. 55, 1707–1079 (1989).
    [CrossRef]
  14. J. Feinberg, R. W. Hellwarth, “Phase-Conjugating Mirror with Continuous-Wave Gain,” Opt. Lett. 5, 519–521 (1980).
    [CrossRef] [PubMed]
  15. J. Feinberg, “Continuous-Wave Self-Pumped Phase Conjugation with Wide Field of View,” Opt. Lett. 8, 480–482 (1983).
    [CrossRef] [PubMed]
  16. 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]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 333–334.
  18. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 41.
  19. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 64.
  20. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 68.
  21. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 69.
  22. C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115–117 (1978).
    [CrossRef] [PubMed]
  23. C. J. R. Sheppard, T. Wilson, “Image Formation in Scanning Microscope with Partially Coherent Source and Detector,” Opt. Acta 25, 315–325 (1978).
    [CrossRef]
  24. I. Abdulhalim, G. Moddel, K. M. Johnson, “High Speed Analog Spatial Light Modulator Using an a-Si:H Photosensor and an Electroclinic Liquid Crystal,” Appl. Phys. Lett. 55, 1603–1605 (1989).
    [CrossRef]
  25. K. M. Johnson, C. C. Mao, G. Moddel, “High-Speed, Low-Power Optical Phase Conjugation Using Optically Addressed Chiral Smectic Liquid Crystal Spatial Light Modulator,” submitted to Ferroelectrics.

1989 (2)

K. M. Johnson, W. T. Cathey, C. C. Mao, “Image Formation in a Phase Conjugate Scanning Microscope,” Appl. Phys. Lett. 55, 1707–1079 (1989).
[CrossRef]

I. Abdulhalim, G. Moddel, K. M. Johnson, “High Speed Analog Spatial Light Modulator Using an a-Si:H Photosensor and an Electroclinic Liquid Crystal,” Appl. Phys. Lett. 55, 1603–1605 (1989).
[CrossRef]

1987 (2)

J. S. Ploem, “Laser Scanning Fluorescent Microscopy,” Appl. Opt. 26, 3226–3231 (1987).
[CrossRef] [PubMed]

E. Betzig, M. Isaacson, A. Lewis, “Collection Mode Near-Field Scanning Microscopy,” Appl. Phys. Lett. 51, 2088–2090 (1987).
[CrossRef]

1986 (3)

1985 (2)

Y. Ichioka, T. Kobayashi, H. Kitagawa, T. Suzuki, “Digital Scanning Laser Microscope,” Appl. Opt. 24, 691–696 (1985).
[CrossRef] [PubMed]

U. C. Fisher, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B 3, 386–390 (1985).
[CrossRef]

1983 (2)

H. J. B. Marsman, R. W. Strikker, “Mechanical Scan System for Microscopic Applications,” Rev. Sci. Instrum. 54, 1047–1052 (1983).
[CrossRef]

J. Feinberg, “Continuous-Wave Self-Pumped Phase Conjugation with Wide Field of View,” Opt. Lett. 8, 480–482 (1983).
[CrossRef] [PubMed]

1982 (1)

P. Sulewski, D. J. Bishop, R. C. Dynes, “A Description of the Bell Laboratories Scanned Acoustic Microscope,” Bell. Syst. Tech. J. 61, 2167–2183 (1982).

1980 (2)

C. J. R. Sheppard, T. Wilson, “Multiple Traversing of the Object in the Scanning Microscope,” Opt. Acta 27, 611–623 (1980).
[CrossRef]

J. Feinberg, R. W. Hellwarth, “Phase-Conjugating Mirror with Continuous-Wave Gain,” Opt. Lett. 5, 519–521 (1980).
[CrossRef] [PubMed]

1978 (2)

C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115–117 (1978).
[CrossRef] [PubMed]

C. J. R. Sheppard, T. Wilson, “Image Formation in Scanning Microscope with Partially Coherent Source and Detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

1977 (1)

C. J. R. Sheppard, A. Choudhury, “Image Formation in the Scanning Microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

1971 (1)

Abdulhalim, I.

I. Abdulhalim, G. Moddel, K. M. Johnson, “High Speed Analog Spatial Light Modulator Using an a-Si:H Photosensor and an Electroclinic Liquid Crystal,” Appl. Phys. Lett. 55, 1603–1605 (1989).
[CrossRef]

Betzig, E.

E. Betzig, M. Isaacson, A. Lewis, “Collection Mode Near-Field Scanning Microscopy,” Appl. Phys. Lett. 51, 2088–2090 (1987).
[CrossRef]

Bishop, D. J.

P. Sulewski, D. J. Bishop, R. C. Dynes, “A Description of the Bell Laboratories Scanned Acoustic Microscope,” Bell. Syst. Tech. J. 61, 2167–2183 (1982).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 333–334.

Cathey, W. T.

K. M. Johnson, W. T. Cathey, C. C. Mao, “Image Formation in a Phase Conjugate Scanning Microscope,” Appl. Phys. Lett. 55, 1707–1079 (1989).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image Formation in the Scanning Microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Cox, I. J.

Davidovits, P.

Dynes, R. C.

P. Sulewski, D. J. Bishop, R. C. Dynes, “A Description of the Bell Laboratories Scanned Acoustic Microscope,” Bell. Syst. Tech. J. 61, 2167–2183 (1982).

Egger, M. D.

Feinberg, J.

Fisher, U. C.

U. C. Fisher, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B 3, 386–390 (1985).
[CrossRef]

Hegedus, Z. S.

Hellwarth, R. W.

Ichioka, Y.

Isaacson, M.

E. Betzig, M. Isaacson, A. Lewis, “Collection Mode Near-Field Scanning Microscopy,” Appl. Phys. Lett. 51, 2088–2090 (1987).
[CrossRef]

Johnson, K. M.

K. M. Johnson, W. T. Cathey, C. C. Mao, “Image Formation in a Phase Conjugate Scanning Microscope,” Appl. Phys. Lett. 55, 1707–1079 (1989).
[CrossRef]

I. Abdulhalim, G. Moddel, K. M. Johnson, “High Speed Analog Spatial Light Modulator Using an a-Si:H Photosensor and an Electroclinic Liquid Crystal,” Appl. Phys. Lett. 55, 1603–1605 (1989).
[CrossRef]

K. M. Johnson, C. C. Mao, G. Moddel, “High-Speed, Low-Power Optical Phase Conjugation Using Optically Addressed Chiral Smectic Liquid Crystal Spatial Light Modulator,” submitted to Ferroelectrics.

Kitagawa, H.

Kobayashi, T.

Lewis, A.

E. Betzig, M. Isaacson, A. Lewis, “Collection Mode Near-Field Scanning Microscopy,” Appl. Phys. Lett. 51, 2088–2090 (1987).
[CrossRef]

Mao, C. C.

K. M. Johnson, W. T. Cathey, C. C. Mao, “Image Formation in a Phase Conjugate Scanning Microscope,” Appl. Phys. Lett. 55, 1707–1079 (1989).
[CrossRef]

K. M. Johnson, C. C. Mao, G. Moddel, “High-Speed, Low-Power Optical Phase Conjugation Using Optically Addressed Chiral Smectic Liquid Crystal Spatial Light Modulator,” submitted to Ferroelectrics.

Marsman, H. J. B.

H. J. B. Marsman, R. W. Strikker, “Mechanical Scan System for Microscopic Applications,” Rev. Sci. Instrum. 54, 1047–1052 (1983).
[CrossRef]

Moddel, G.

I. Abdulhalim, G. Moddel, K. M. Johnson, “High Speed Analog Spatial Light Modulator Using an a-Si:H Photosensor and an Electroclinic Liquid Crystal,” Appl. Phys. Lett. 55, 1603–1605 (1989).
[CrossRef]

K. M. Johnson, C. C. Mao, G. Moddel, “High-Speed, Low-Power Optical Phase Conjugation Using Optically Addressed Chiral Smectic Liquid Crystal Spatial Light Modulator,” submitted to Ferroelectrics.

Ploem, J. S.

Quate, C. F.

C. F. Quate, “Vacuum Tunneling: A New Technique for Microscopy,” Phys. Today26–33 (Aug.1986).
[CrossRef]

Sarafis, V.

Sheppard, C. J. R.

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]

C. J. R. Sheppard, T. Wilson, “Multiple Traversing of the Object in the Scanning Microscope,” Opt. Acta 27, 611–623 (1980).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115–117 (1978).
[CrossRef] [PubMed]

C. J. R. Sheppard, T. Wilson, “Image Formation in Scanning Microscope with Partially Coherent Source and Detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image Formation in the Scanning Microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 41.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 64.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 68.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 69.

Strikker, R. W.

H. J. B. Marsman, R. W. Strikker, “Mechanical Scan System for Microscopic Applications,” Rev. Sci. Instrum. 54, 1047–1052 (1983).
[CrossRef]

Sulewski, P.

P. Sulewski, D. J. Bishop, R. C. Dynes, “A Description of the Bell Laboratories Scanned Acoustic Microscope,” Bell. Syst. Tech. J. 61, 2167–2183 (1982).

Suzuki, T.

Wells, O. C.

O. C. Wells, Scanning Electron Microscopy (McGraw-Hill, New York, 1974).

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Multiple Traversing of the Object in the Scanning Microscope,” Opt. Acta 27, 611–623 (1980).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Image Formation in Scanning Microscope with Partially Coherent Source and Detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115–117 (1978).
[CrossRef] [PubMed]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 41.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 69.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 68.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 64.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 333–334.

Appl. Opt. (3)

Appl. Phys. Lett. (3)

E. Betzig, M. Isaacson, A. Lewis, “Collection Mode Near-Field Scanning Microscopy,” Appl. Phys. Lett. 51, 2088–2090 (1987).
[CrossRef]

K. M. Johnson, W. T. Cathey, C. C. Mao, “Image Formation in a Phase Conjugate Scanning Microscope,” Appl. Phys. Lett. 55, 1707–1079 (1989).
[CrossRef]

I. Abdulhalim, G. Moddel, K. M. Johnson, “High Speed Analog Spatial Light Modulator Using an a-Si:H Photosensor and an Electroclinic Liquid Crystal,” Appl. Phys. Lett. 55, 1603–1605 (1989).
[CrossRef]

Bell. Syst. Tech. J. (1)

P. Sulewski, D. J. Bishop, R. C. Dynes, “A Description of the Bell Laboratories Scanned Acoustic Microscope,” Bell. Syst. Tech. J. 61, 2167–2183 (1982).

J. Opt. Soc. Am. A (2)

J. Vac. Sci. Technol. B (1)

U. C. Fisher, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B 3, 386–390 (1985).
[CrossRef]

Opt. Acta (3)

C. J. R. Sheppard, A. Choudhury, “Image Formation in the Scanning Microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Multiple Traversing of the Object in the Scanning Microscope,” Opt. Acta 27, 611–623 (1980).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Image Formation in Scanning Microscope with Partially Coherent Source and Detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

Opt. Lett. (3)

Phys. Today (1)

C. F. Quate, “Vacuum Tunneling: A New Technique for Microscopy,” Phys. Today26–33 (Aug.1986).
[CrossRef]

Rev. Sci. Instrum. (1)

H. J. B. Marsman, R. W. Strikker, “Mechanical Scan System for Microscopic Applications,” Rev. Sci. Instrum. 54, 1047–1052 (1983).
[CrossRef]

Other (7)

O. C. Wells, Scanning Electron Microscopy (McGraw-Hill, New York, 1974).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 333–334.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 41.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 64.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 68.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984), p. 69.

K. M. Johnson, C. C. Mao, G. Moddel, “High-Speed, Low-Power Optical Phase Conjugation Using Optically Addressed Chiral Smectic Liquid Crystal Spatial Light Modulator,” submitted to Ferroelectrics.

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Figures (18)

Fig. 1
Fig. 1

Double pass scanning microscope.

Fig. 2
Fig. 2

Phase conjugate scanning microscope.

Fig. 3
Fig. 3

Unfolded phase conjugate scanning microscope.

Fig. 4
Fig. 4

Nonzero regions of the transfer functions: (a) for the phase conjugate scanning microscope; (b) for the conventional microscope.

Fig. 5
Fig. 5

Images of a single point object imaged by various optical microscopes. Images using the PCSM with one and two annular lenses are also plotted.

Fig. 6
Fig. 6

Images of a two-point object, which is just resolved by the phase conjugate scanning microscope, imaged by various optical microscopes.

Fig. 7
Fig. 7

Images of an amplitude straight edge object.

Fig. 8
Fig. 8

Images of a phase edge imaged by optical microscopes: (a) in the phase conjugate scanning microscope, (b) in the conventional microscope.

Fig. 9
Fig. 9

Imaging of the phase conjugate scanning microscope: (a) the diagram of the PCSM with a defocused pinhole; (b) the unfolded PCSM showing an optical beam traversing through the system.

Fig. 10
Fig. 10

Integrated intensity variations as the function of the defocused distance for various optical microscopes.

Fig. 11
Fig. 11

Halfwidth of the image of a single point object as the function of the pinhole size.

Fig. 12
Fig. 12

Schematic diagram of depth discrimination of the phase conjugate scanning microscope with large pinhole PH2 before the phase conjugator.

Fig. 13
Fig. 13

Experimental apparatus for the phase conjugate scanning microscope.

Fig. 14
Fig. 14

Images of a single point object obtained from experiments.

Fig. 15
Fig. 15

Images of two lines obtained from experiments.

Fig. 16
Fig. 16

Images of a straight edge obtained from experiments.

Fig. 17
Fig. 17

Images of a phase edge obtained from experiments.

Fig. 18
Fig. 18

Intensity variations as the function of the defocused distance in various optical microscopes.

Tables (1)

Tables Icon

Table I Normalized Rayleigh Resolution Limits for Various Optical Microscopes

Equations (76)

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U p ( x s ) = - + δ ( x 0 ) h 1 ( x 0 / M + x 1 ) × t ( x s - x 1 ) h 2 ( x 1 + x 2 / M ) exp ( i k φ x 2 2 ) × δ ( x 2 ) h 3 ( x 2 / M + x 3 ) t ( x s - x 3 ) h 4 ( x 3 + x 4 / M ) × δ ( x 4 ) d x 0 d x 1 d x 2 d x 3 d x 4 ,
U p ( x s ) = - + h 1 ( x 1 ) t ( x s - x 1 ) h 2 ( x 1 ) h 3 ( x 3 ) × t ( x s - x 3 ) h 4 ( x 3 ) d x 1 d x 3 .
h ( x ) = - + P ( λ d ζ ) exp ( - i 2 π x ζ ) d ζ ,
T ( m ) = - + t ( x ) exp ( - i 2 π m x ) d x .
U p ( x s ) = - + C p ( m , m ) T ( m ) T ( m ) × exp [ - i 2 π ( m + m ) x s ] d m d m ,
C p ( m , m ) = [ P ( m ^ ) P ( m ^ ) ] [ P ( m ^ ) P ( m ^ ) ] ,
C c ( m , m ) = - + P ( x ) P ( m ^ + x ) P ( m ^ + x ) d x .
u = 2 π ( x s 2 + y s 2 ) 1 / 2 ( N . A . ) / λ ,
v = π a ( N . A . ) λ ,
t ( x s - x ) = δ ( x s - x ) ,
U p ( x s ) = h 1 ( x s ) h 2 ( x s ) h 3 ( x s ) h 4 ( x s ) .
I p ( x s ) = h ( x s ) 8 .
h ( u ) = 2 J 1 ( u ) u ,
I p ( u ) = | 2 J 1 ( u ) u | 8 .
I c ( u ) = | 2 J 1 ( u ) u | 2 ,
I s ( u ) = | 2 J 1 ( u ) u | 4 ,
I d ( u ) = | 4 J 1 2 ( u ) J 1 ( 2 u ) u 3 | 2 ,
I p 1 ( u ) = | 2 J 1 ( u ) J 0 ( u ) u | 4 ,
I p 2 ( u ) = J 0 ( u ) 8 .
t ( x s - x ) = δ ( x s - a 2 - x ) + δ ( x s + a 2 - x ) ,
I p ( x s ) = h 4 ( x s - a / 2 ) + h 4 ( x s + a / 2 ) + 2 h 2 ( x s - a / 2 ) h 2 ( x s + a / 2 ) 2 .
h ( u ± v ) = 2 J 1 ( u ± v ) u ± v .
I p ( u , v ) = | [ 2 J 1 ( u - v ) u - v ] 4 + [ 2 J 1 ( u + v ) u + v ] 4 + 2 [ 4 J 1 ( u - v ) J 1 ( u + v ) ( u - 1 ) ( u + v ) ] 2 | 2 .
I u = 0 / I u = v 0.811 ,
I u = 0 / I u = v 0.735.
I c ( u , v ) = [ 2 J 1 ( u - v ) u - v ] 2 + [ 2 J 1 ( u + v ) u + v ] 2 + 2 [ 4 J 1 ( 2 v ) J 1 ( u - v ) J 1 ( u + v ) u ( u - v ) ( u + v ) ] ,
I s ( u , v ) = | [ 2 J 1 ( u - v ) u - v ] 2 + [ 2 J 1 ( u + v ) u + v ] 2 | 2 ,
I d ( u , v ) = | 4 J 1 [ 2 ( u - v ) ] J 1 2 ( u , v ) ( u - v ) 3 + 4 J 1 [ 2 ( u + v ) ] J 1 2 ( u + v ) ( u + v ) 3 + 2 [ 4 J 1 ( 2 u ) J 1 ( u - v ) J 1 ( u + v ) u ( u - v ) ( u + v ) ] | 2 .
I p ( u , v ) = I s ( u , v ) 2 .
I p ( x s ) = | - + C ( m ) T ( m ) exp ( - i 2 π m x s ) d m | 4 ,
C ( m ) = P ( m ^ ) P ( m ^ ) .
t ( x s ) = 1 2 - 2 π n = 1 ( - 1 ) n 2 n - 1 cos ( 2 n - 1 ) 2 π ν x s ,
T ( m ) = 1 2 δ ( m ) - 1 π n = 1 ( - 1 ) n 2 n - 1 { δ [ m - ( 2 n - 1 ) ν ] + δ [ m + ( 2 n - 1 ) ν ] }
I p ( x s ) = ( 1 2 - 2 π s 1 ) 4 ,
S 1 = n = 1 ( - 1 ) n 2 n - 1 C [ ( 2 n - 1 ) ν cos ( 2 n - 1 ) ξ ,
ξ = 2 π ν x s ,
C ( x ) = { 2 π [ cos - 1 x - x ( 1 - x 2 ) 1 / 2 ] , if x 1 ; 0 , otherwise .
S 1 = n = 1 ( - 1 ) n 2 n - 1 C [ ( 2 n - 1 ) σ ] cos ( 2 n - 1 ) ξ ,
σ = ν / 2 ν 0 ,
ξ = 2 σ u .
S 1 = n = 1 ( - 1 ) n 2 n - 1 C [ 2 ( 2 n - 1 ) σ ] cos [ ( 2 n - 1 ) 2 σ u ] .
I c = 1 4 - 2 π S 1 + 2 π S 2 + 2 π S 3 + 4 π S 4 + 4 π S 5 ,
I s = ( 1 4 - 2 π S 1 ) 2 ,
I d = I c 2 ,
S 2 = n = 1 C [ ( 2 n - 1 ) σ ] ( 2 n - 1 ) 2 ,
S 3 = n = 1 C [ 2 ( 2 n - 1 ) σ ] ( 2 n - 1 ) 2 cos [ 4 ( 2 n - 1 ) σ u ] ,
S 4 = n = 1 r = n + 1 ( - 1 ) r + n C [ ( 2 r - 1 ) σ ] ( 2 r - 1 ) ( 2 n - 1 ) × cos [ 2 ( 2 n - 1 ) σ u - 2 ( 2 r - 1 ) σ u ] ,
S 5 = n = 1 r = n + 1 ( - 1 ) r + n C [ ( 2 n - 1 ) σ + ( 2 r - 1 ) σ ] ( 2 r - 1 ) ( 2 n - 1 ) × cos [ 2 ( 2 n - 1 ) σ u + 2 ( 2 r + 1 ) σ u ] .
t ( x s ) = exp ( i ϕ 1 ) [ 1 2 ( 1 + exp ( i Δ ϕ ) ] - 2 π [ 1 - exp ( i Δ ϕ ) ] n = 1 ( - 1 ) n 2 n - 1 cos [ ( 2 n - 1 ) ξ ] ,
I p ( x s ) = | 1 2 [ 1 + exp ( i Δ ϕ ) ] - 2 π [ 1 - exp ( i Δ ϕ ) ] S 1 | 2 ,
I c ( x ) = ( 1 + cos Δ ϕ 2 ) - 4 sin Δ ϕ π Im { S 1 } - 4 π 2 ( 1 - cos Δ ϕ ) [ S 2 + S 3 + 2 ( S 4 + S 5 ) ] ,
I p i ( δ z ) = 0 + h ( δ z , x ) 8 x d x ,
I p i ( α ) = 0 + h ( α , u ) 8 u d u ,
α = 2 π δ z ( N . A . ) 2 / λ ,
h ( α , u ) = 0 1 exp ( 1 2 i α ρ 2 ) J 0 ( u ρ ) ρ d ρ .
D = 0 1 J 0 ( u ρ ) cos ( 1 2 α ρ 2 ) ρ d ρ ,
E = 0 1 J 0 ( u ρ ) sin ( 1 2 α ρ 2 ) ρ d ρ .
I p i ( α ) = 0 + ( D 2 + E 2 ) 4 u d u .
I s i ( α ) = 0 + ( D 2 + E 2 ) 2 u d u ,
I d i ( α ) = 0 + ( D 2 + E 2 ) 2 ( D 1 2 + E 1 2 ) u d u ,
D 1 = 0 1 J 0 ( 2 u ρ ) cos ( 1 2 α ρ 2 ) ρ d ρ ,
E 1 = 0 1 J 0 ( 2 u ρ ) cos ( 1 2 α ρ 2 ) ρ d ρ .
U ( x s ) = - + h ( x 1 ) t ( x s - x 1 ) h ( x 1 + x 2 / M ) P 2 ( x 2 ) × h ( x 3 + x 2 / M ) t ( x s - x 3 ) h ( x 3 + x 4 / M ) P 3 ( x 4 ) d x 1 d x 2 d x 3 d x 4 ,
t ( x s - x ) = δ ( x s - x ) .
I ( x s ) = h ( x s ) 2 [ h ( x s ) 2 P 2 ( M x s ) ] 2 [ h ( x s ) P 3 ( M x s ) ] 2 ,
u = 2 π r ( N . A . ) / λ ,
u p = 2 π b ( N . A . ) / λ M ,
I ( u ) = h ( u ) 2 [ 2 π 0 u p - u h ( t ) 2 t d t + 2 u p - u u p + u h ( t ) 2 cos - 1 ( t 2 + u 2 - u p 2 2 t u ) t d t ] 2 × [ 2 π 0 u p - u h ( t ) t d t + 2 u p - u u p + u h ( t ) cos - 1 ( t 2 + u 2 - u p 2 2 t u ) t d t ] 2
I ( u ) = 4 h ( u ) 2 { u - u p u + u p h ( t ) 3 × [ cos - 1 ( t 2 + u 2 - u p 2 2 t u ) ] 2 t d t } 2
U w ( x s ) = - + h ( x 1 ) t ( x s - x 1 ) h ( x 1 + x 2 / M ) × h ( x 3 + x 2 / M ) t ( x s - x 3 ) h ( x 3 ) d x 1 d x 2 d x 3 .
U w ( x s ) = h ( x s ) 2 - + h ( x s + x 2 / M ) 2 d x 2 = const h ( x s ) 2 .
U w ( x s ) = - + [ h 2 ( x s - a / 2 ) h 2 ( x s - a / 2 + x 2 / M ) + h 2 ( x s + a / 2 ) h 2 ( x s + a / 2 + x 2 / M ) + 2 h ( x s - a / 2 ) h ( x s + a / 2 ) × h ( x s - a / 2 + x 2 / M ) h ( x s + a / 2 + x 2 / M ) ] d x 2 = c o n s t [ h 2 ( x s - a / 2 ) + h 2 ( x s + a / 2 ) + 2 R h ( x s - a / 2 ) h ( x s + a / 2 ) ] .
R = - + h 2 ( x s - a / 2 + x 2 / M ) d x - + h ( x s - a / 2 + x 2 / M ) h ( x s + a / 2 + x / M ) d x .
U w ( x s ) = - + C ( m , m ) T ( m ) T ( m ) × exp [ - i 2 π ( m + m ) x s ] d m d m ,
C ( m , m ) = - + P ( x ) P ( m ^ + x ) P ( m ^ + x ) d x ,
I p i ( α ) = 0 + h ( α , u ) 4 h ( u ) 4 u d u ,

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