Abstract

In-line holography of particle fields suffers from image deterioration caused by intrinsic speckle noise. This is not accounted for by previous theoretical treatments based on the scattering of a single particle, nor has there been any quantitative description of this noise except for an empirical criterion by Royer [ Nouv. Rev. Opt. 5, 87 ( 1974)] based on geometrical obscuration. We develop a theoretical model for in-line holography of multiple particles, using diffraction theory and statistical analysis, and show that the virtual image of the particle ensemble is the dominant source of speckle in reconstruction. We quantify the effect of speckle with a signal-to-noise ratio (SNR). The SNR is found to depend on a speckle parameter (which embodies particle diameter, concentration, and sample depth) and on the film gamma. Experimental results show reasonably good agreement with our model. The SNR equation provides prediction of image quality and thence application limits of in-line holography for particle fields. The fundamental understanding obtained here points not only to constraints but also to possible improvements in experimental procedures.

© 1993 Optical Society of America

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References

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  1. D. Gabor, “A new microscope principle,” Nature (London) 161, 777 (1948).
    [CrossRef]
  2. B. A. Silverman, B. J. Thompson, J. H. Ward, “A laser fog disdrometer,”J. Appl. Meteorol. 3, 792–801 (1964).
    [CrossRef]
  3. B. J. Thompson, G. B. Parrent, J. H. Ward, B. Justh, “A readout technique for the laser fog disdrometer,”J. Appl. Meteorol. 5, 343–348 (1966).
    [CrossRef]
  4. J. D. Trolinger, R. A. Belz, W. M. Farmer, “Holographic techniques for the study of dynamic particle fields,” Appl. Opt. 8, 957–961 (1969).
    [CrossRef] [PubMed]
  5. B. J. Thompson, “Holographic particle sizing techniques,”J. Phys. E 7, 781–788 (1974).
    [CrossRef]
  6. R. A. Belz, R. W. Menzel, “Particle field holography at Arnold Engineering Development Center,” Opt. Eng. 8, 256–265 (1979).
  7. B. C. R. Ewan, “Fraunhofer plane analysis of particle field holograms,” Appl. Opt. 19, 1368–1372 (1980).
    [CrossRef] [PubMed]
  8. S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle sizing using far-field holography: new developments,” Opt. Eng. 19, 727–733 (1980).
    [CrossRef]
  9. P. R. Hobson, “Precision coordinate measurements using holographic recording,”J. Phys. E 21, 139–145 (1988).
    [CrossRef]
  10. R. Menzel, F. M. Shofner, “An investigation of Fraunhofer holography for velocimetry application,” Appl. Opt. 9, 2073–2079 (1970).
    [CrossRef] [PubMed]
  11. B. C. R. Ewan, “Particle velocity distribution measurement by holography,” Appl. Opt. 18, 3156–3160 (1979).
    [CrossRef] [PubMed]
  12. P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
    [CrossRef]
  13. L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” publ. 85–0526 (American Institute of Aeronautics and Astronautics, New York, 1985).
  14. H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
    [CrossRef]
  15. A. R. Jones, M. Sarjeant, C. R. Davis, R. O. Denham, “Application of in-line holography to drop size measurement in dense fuel sprays,” Appl. Opt. 17, 328–330 (1978).
    [CrossRef]
  16. H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouv. Rev. Opt. 5, 87–93 (1974).
    [CrossRef]
  17. W. L. Anderson, “Particle identification and counting by Fourier-optical pattern recognition,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), Chap. 3.
    [CrossRef]
  18. R. Meynart, “Digital image processing for speckle flow velocimetry,” Rev. Sci. Instrum. 53, 110 (1982).
    [CrossRef]
  19. G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 688–700 (1976).
    [CrossRef]
  20. J. D. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.
  21. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 752.
  22. A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 207, 210.
  23. A. A. Friesem, J. S. Zelenka, “Effects of film nonlinearities in holography,” Appl. Opt. 6, 1755–1759 (1967).
    [CrossRef] [PubMed]
  24. A. Kozma, “Photographic recording of spatially modulated coherent light,”J. Opt. Soc. Am. 56, 428–432 (1966).
    [CrossRef]
  25. J. W. Goodman, G. R. Knight, “Effects of film nonlinearities on wavefront-reconstruction images of diffuse objects,”J. Opt. Soc. Am. 58, 1276–1283 (1968).
    [CrossRef]
  26. P. F. Panter, Modulation, Noise and Spectral Analysis (McGraw-Hill, New York, 1965), p. 350.
  27. I. Prikryl, C. M. Vest, “Holographic imaging of semitransparent droplets or particles,” Appl. Opt. 21, 2541–2547 (1982).
    [CrossRef]
  28. Z.-K. Lu, H. Meng, “Holography of semi-transparent spherical droplets,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 19–25 (1986).
    [CrossRef]
  29. D. M. Robinson, “A calculation of edge smear in far-field holography using a short-cut edge trace technique,” Appl. Opt. 9, 496–497 (1970).
    [CrossRef] [PubMed]

1991 (1)

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

1988 (1)

P. R. Hobson, “Precision coordinate measurements using holographic recording,”J. Phys. E 21, 139–145 (1988).
[CrossRef]

1984 (1)

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

1982 (2)

R. Meynart, “Digital image processing for speckle flow velocimetry,” Rev. Sci. Instrum. 53, 110 (1982).
[CrossRef]

I. Prikryl, C. M. Vest, “Holographic imaging of semitransparent droplets or particles,” Appl. Opt. 21, 2541–2547 (1982).
[CrossRef]

1980 (2)

B. C. R. Ewan, “Fraunhofer plane analysis of particle field holograms,” Appl. Opt. 19, 1368–1372 (1980).
[CrossRef] [PubMed]

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle sizing using far-field holography: new developments,” Opt. Eng. 19, 727–733 (1980).
[CrossRef]

1979 (2)

R. A. Belz, R. W. Menzel, “Particle field holography at Arnold Engineering Development Center,” Opt. Eng. 8, 256–265 (1979).

B. C. R. Ewan, “Particle velocity distribution measurement by holography,” Appl. Opt. 18, 3156–3160 (1979).
[CrossRef] [PubMed]

1978 (1)

1976 (1)

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 688–700 (1976).
[CrossRef]

1974 (2)

H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouv. Rev. Opt. 5, 87–93 (1974).
[CrossRef]

B. J. Thompson, “Holographic particle sizing techniques,”J. Phys. E 7, 781–788 (1974).
[CrossRef]

1970 (2)

1969 (1)

1968 (1)

1967 (1)

1966 (2)

B. J. Thompson, G. B. Parrent, J. H. Ward, B. Justh, “A readout technique for the laser fog disdrometer,”J. Appl. Meteorol. 5, 343–348 (1966).
[CrossRef]

A. Kozma, “Photographic recording of spatially modulated coherent light,”J. Opt. Soc. Am. 56, 428–432 (1966).
[CrossRef]

1964 (1)

B. A. Silverman, B. J. Thompson, J. H. Ward, “A laser fog disdrometer,”J. Appl. Meteorol. 3, 792–801 (1964).
[CrossRef]

1948 (1)

D. Gabor, “A new microscope principle,” Nature (London) 161, 777 (1948).
[CrossRef]

Anderson, W. L.

W. L. Anderson, “Particle identification and counting by Fourier-optical pattern recognition,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), Chap. 3.
[CrossRef]

Beeler, G. B.

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” publ. 85–0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

Belz, R. A.

R. A. Belz, R. W. Menzel, “Particle field holography at Arnold Engineering Development Center,” Opt. Eng. 8, 256–265 (1979).

J. D. Trolinger, R. A. Belz, W. M. Farmer, “Holographic techniques for the study of dynamic particle fields,” Appl. Opt. 8, 957–961 (1969).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 752.

Cartwright, S. L.

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle sizing using far-field holography: new developments,” Opt. Eng. 19, 727–733 (1980).
[CrossRef]

Davis, C. R.

Denham, R. O.

Dunn, P.

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle sizing using far-field holography: new developments,” Opt. Eng. 19, 727–733 (1980).
[CrossRef]

Ennos, A. E.

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 207, 210.

Ewan, B. C. R.

Farmer, W. M.

Friesem, A. A.

Gabor, D.

D. Gabor, “A new microscope principle,” Nature (London) 161, 777 (1948).
[CrossRef]

Goodman, J. D.

J. D. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

Goodman, J. W.

Hobson, P. R.

P. R. Hobson, “Precision coordinate measurements using holographic recording,”J. Phys. E 21, 139–145 (1988).
[CrossRef]

Hussain, F.

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

Jones, A. R.

Justh, B.

B. J. Thompson, G. B. Parrent, J. H. Ward, B. Justh, “A readout technique for the laser fog disdrometer,”J. Appl. Meteorol. 5, 343–348 (1966).
[CrossRef]

Knight, G. R.

Kozma, A.

Linderman, A. M.

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” publ. 85–0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

Lu, Z.-K.

Z.-K. Lu, H. Meng, “Holography of semi-transparent spherical droplets,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 19–25 (1986).
[CrossRef]

Malyak, P. H.

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

Meng, H.

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

Z.-K. Lu, H. Meng, “Holography of semi-transparent spherical droplets,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 19–25 (1986).
[CrossRef]

Menzel, R.

Menzel, R. W.

R. A. Belz, R. W. Menzel, “Particle field holography at Arnold Engineering Development Center,” Opt. Eng. 8, 256–265 (1979).

Meynart, R.

R. Meynart, “Digital image processing for speckle flow velocimetry,” Rev. Sci. Instrum. 53, 110 (1982).
[CrossRef]

Panter, P. F.

P. F. Panter, Modulation, Noise and Spectral Analysis (McGraw-Hill, New York, 1965), p. 350.

Parrent, G. B.

B. J. Thompson, G. B. Parrent, J. H. Ward, B. Justh, “A readout technique for the laser fog disdrometer,”J. Appl. Meteorol. 5, 343–348 (1966).
[CrossRef]

Prikryl, I.

Robinson, D. M.

Royer, H.

H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouv. Rev. Opt. 5, 87–93 (1974).
[CrossRef]

Sarjeant, M.

Shofner, F. M.

Silverman, B. A.

B. A. Silverman, B. J. Thompson, J. H. Ward, “A laser fog disdrometer,”J. Appl. Meteorol. 3, 792–801 (1964).
[CrossRef]

Thompson, B. J.

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle sizing using far-field holography: new developments,” Opt. Eng. 19, 727–733 (1980).
[CrossRef]

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 688–700 (1976).
[CrossRef]

B. J. Thompson, “Holographic particle sizing techniques,”J. Phys. E 7, 781–788 (1974).
[CrossRef]

B. J. Thompson, G. B. Parrent, J. H. Ward, B. Justh, “A readout technique for the laser fog disdrometer,”J. Appl. Meteorol. 5, 343–348 (1966).
[CrossRef]

B. A. Silverman, B. J. Thompson, J. H. Ward, “A laser fog disdrometer,”J. Appl. Meteorol. 3, 792–801 (1964).
[CrossRef]

Trolinger, J. D.

Tyler, G. A.

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 688–700 (1976).
[CrossRef]

Vest, C. M.

Ward, J. H.

B. J. Thompson, G. B. Parrent, J. H. Ward, B. Justh, “A readout technique for the laser fog disdrometer,”J. Appl. Meteorol. 5, 343–348 (1966).
[CrossRef]

B. A. Silverman, B. J. Thompson, J. H. Ward, “A laser fog disdrometer,”J. Appl. Meteorol. 3, 792–801 (1964).
[CrossRef]

Weinstein, L. W.

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” publ. 85–0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 752.

Zelenka, J. S.

Appl. Opt. (8)

Fluid Dyn. Res. (1)

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

J. Appl. Meteorol. (2)

B. A. Silverman, B. J. Thompson, J. H. Ward, “A laser fog disdrometer,”J. Appl. Meteorol. 3, 792–801 (1964).
[CrossRef]

B. J. Thompson, G. B. Parrent, J. H. Ward, B. Justh, “A readout technique for the laser fog disdrometer,”J. Appl. Meteorol. 5, 343–348 (1966).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. E (2)

P. R. Hobson, “Precision coordinate measurements using holographic recording,”J. Phys. E 21, 139–145 (1988).
[CrossRef]

B. J. Thompson, “Holographic particle sizing techniques,”J. Phys. E 7, 781–788 (1974).
[CrossRef]

Nature (London) (1)

D. Gabor, “A new microscope principle,” Nature (London) 161, 777 (1948).
[CrossRef]

Nouv. Rev. Opt. (1)

H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouv. Rev. Opt. 5, 87–93 (1974).
[CrossRef]

Opt. Acta (1)

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 688–700 (1976).
[CrossRef]

Opt. Eng. (3)

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle sizing using far-field holography: new developments,” Opt. Eng. 19, 727–733 (1980).
[CrossRef]

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

R. A. Belz, R. W. Menzel, “Particle field holography at Arnold Engineering Development Center,” Opt. Eng. 8, 256–265 (1979).

Rev. Sci. Instrum. (1)

R. Meynart, “Digital image processing for speckle flow velocimetry,” Rev. Sci. Instrum. 53, 110 (1982).
[CrossRef]

Other (7)

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” publ. 85–0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

P. F. Panter, Modulation, Noise and Spectral Analysis (McGraw-Hill, New York, 1965), p. 350.

Z.-K. Lu, H. Meng, “Holography of semi-transparent spherical droplets,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 19–25 (1986).
[CrossRef]

W. L. Anderson, “Particle identification and counting by Fourier-optical pattern recognition,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), Chap. 3.
[CrossRef]

J. D. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 752.

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 207, 210.

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Figures (15)

Fig. 1
Fig. 1

Scattering of multiple particles in in-line hologram recording with collimated illumination.

Fig. 2
Fig. 2

Coordinates for recording. (a) Single-particle case: a particle is centered in the origin of plane (ξη), and its diffraction pattern is centered in the origin of plane (xy) at a distance z away; (b) multiple-particles case: the kth particle is on the plane (ξη)k, with location rk, and its diffraction pattern, centered at rk, is observed on plane (xy) at a distance zk.

Fig. 3
Fig. 3

The Airy cone containing scatterers contributing to speckle at observation point P.

Fig. 4
Fig. 4

In-line hologram reconstruction. (a) Single-particle case, in which the virtual image is seen only as a weak, near-spherical wave background in the real-image plane (xy′); (b) multiple-particle case: speckle arises from interference of waves from the virtual image of the particle ensemble.

Fig. 5
Fig. 5

(a) tE curve and (b) D-log(E) curve obtained by use of Agfa-Gevaert 8E56 plates and D-19 developer for 2-min developing. Region II gives a higer γ and is more suitable for recording in-line holograms of small particles than Region I.

Fig. 6
Fig. 6

Experimental data for film γ as a function of developing time and exposure region defined as in Fig. 5, with 8E56 plates and D-19 developer.

Fig. 7
Fig. 7

Experimental setup for SNR study of in-line particle holography: (a) hologram recording, (b) hologram reconstruction.

Fig. 8
Fig. 8

Reconstructed images of particles at SNR of 19, 14, and 5 dB, corresponding to particle concentrations of 1.5, 6, and 40 mm−3 (top to bottom, respectively). Particle diameter, 21 μm; suspension depth, 19 mm. (a) Pictures as viewed by the CCD camera, (b) intensity plot for subsections of the three pictures in (a).

Fig. 9
Fig. 9

SNR versus film γ. Increase of SNR with γ occurs only up to a point (γ ≈ 4.5). Circles are measured data. The continuous curve is a cubic spline fitted to the data. Curves A and B correspond to maximum and minimum CCD thresholding, respectively. Particle diameter is 21 μm; concentration is 10 mm−3; depth of the suspension is 15 mm; recording distance is 32 mm.

Fig. 10
Fig. 10

Dependence of SNR on particle concentration ns for three particle diameters. Symbols are measured data. Solid curves are calculated with Eq. (44a). Displacements of measured data to calculated curves are related to the deviation in SNR versus d curve (see Fig. 12). Depth of the particle suspension is 19 mm; recording distance z is 32 mm.

Fig. 11
Fig. 11

Dependence of SNR on depth L of the particle suspension for three particle concentrations. Symbols are measured data. Solid curves are calculated with Eq. (44a). Particle diameter is 21 μm; recording distance is 50 mm.

Fig. 12
Fig. 12

Dependence of SNR on particle diameter d. Filled circles are measured data. The solid curve is calculated with Eq. (44a). Deviation of measured data from the calculated curve is due to the aperture effect imposed by fringe smearing, as is explained in Appendix B. Depth L of the particle suspension is 19 mm; particle concentration ns is 3/mm3; recording distance z is 50 mm.

Fig. 13
Fig. 13

Dependence of SNR on recording distance z. Circles are measured data. The solid line is calculated with Eq. (44a). The fall of measured SNR at large z is attributed to loss of modulation fringes in recording. Particle diameter d is 21 μm. Particle concentration ns is 3 mm−3. Depth L of the particle suspension is 19 mm.

Fig. 14
Fig. 14

Secondary scattering at the kth particle Pk. The complex sum of primary scattered waves incident upon Pk is wk; the total wave scattered by Pk is (1 + wk)ok.

Fig. 15
Fig. 15

Numerical simulation of the average intensity Isig of a reconstructed particle image versus particle diameter d, showing monotonic increase of Isig with d as a result of aperture effects based on frequency cutoff within the central lobe.

Equations (73)

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U H ( x , y ) = R + O ( x , y ) = R + k = 1 N o k ( x , y ) ,
I H ( x , y ) = U H U H * = 1 + O * ( x , y ) + O ( x , y ) + O O * ( x , y ) .
U H ( x , y ) = - i λ z exp [ i 2 π z / λ ] - - [ 1 - A ( ξ , η ) ] × exp { i π [ x - ξ ) 2 + ( y - η ) 2 ] / λ z } d ξ d η .
C k = π d 2 / ( 4 λ z k ) ,
Q k = { 2 J 1 ( π d r - r k ) / λ z k ( π d r - r k ) / λ z k } ,
Φ k = ( π r - r k 2 ) / λ z k + π / 2.
O ( x , y ) = C k Q k exp ( i Φ k )
I H ( r ) = 1 + 2 C k Q k cos ( Φ k ) + C k 2 Q k 2 .
O ( x , y ) = k = 1 N o k ( x , y ) = k = 1 N C k Q k exp ( i Φ k ) ;
O + O * = 2 k = 1 N C k Q k cos Φ k .
I N O O * = j ; k = 1 N o j o k * = j ; k = 1 N C j C k Q j Q k exp [ i ( Φ j - Φ k ) ] .
Q j Q k exp [ i ( Φ j - Φ k ) ] = Q j Q k exp [ i ( Φ j - Φ k ) ] = { Q k 2 if j = k ; 0 if j k .
I N = k = 1 N C 2 Q k 2 = N C 2 Q k ¯ 2 ,
σ N = I N .
θ 0 = 1.22 λ / d λ / d ,
N = ( π / 3 ) n s θ 0 2 [ ( z + L / 2 ) 3 - ( z - L / 2 ) 3 ] = π n s ( λ 2 z 2 / d 2 ) ( L + L 3 / 12 z 2 ) ,
N = π n s λ 2 z 2 L / d 2 .
G N C 2 ,
G = ( π 3 / 16 ) d 2 n s L .
I = 2 π 3 n s [ ( z + L / 2 ) 3 - ( z - L / 2 ) 3 ] × 0 θ 0 [ 2 J 1 ( π d λ θ ) / ( π d λ θ ) ] 2 θ d θ .
Q k ¯ 2 = 2 0 1 [ 2 J 1 ( π α ) π α ] 2 α d α = 0.335 1 / 3.
σ N = I N = G / 3 = ( π 3 / 48 ) d 2 n s L .
σ H 2 = σ N 2 + 2 σ N ,
σ H = σ N ( 1 + 2 / σ N ) 1 / 2 = ( G / 3 ) ( 1 + 6 / G ) 1 / 2 .
t ( x , y ) = t b - K [ E ( x , y ) - E b ] ,
t ( x , y ) = t b - K E b Δ I H ( x , y ) ,
U t ( x , y ) B - [ O * ( x , y ) + O ( x , y ) + O O * ( x , y ) ] ,
B t b / K E b .
U R ( x , y ) = - i λ z exp ( i 2 π z / λ ) - - U t ( x , y ) × exp { i π [ ( x - x ) 2 + ( y - y ) 2 ] / λ z } d x d y .
U sig = k circ ( r - r k d / 2 ) ,
r - r k = [ ( x - a k ) 2 + ( y - b k ) 2 ] 1 / 2 ,
U s p 1 = k C ˜ Q ˜ k exp ( i Φ ˜ k ) ,
U s p 2 = i λ z - - O O * ( x λ z , y λ z ) × exp { i π z / λ [ ( x - x ) 2 + ( y - y ) 2 ] } d x d y - 1 4 O O * ( x λ z , y λ z ) .
U R ( x , y ) = B + U sig + U s p 1 + U s p 2 .
I s p = U s p 1 + U s p 2 2 = | k C ˜ Q ˜ k exp ( i Φ ˜ k ) | 2 - k C ˜ Q ˜ k 1 4 O O * × cos Φ ˜ k + 1 16 ( O O * ) 2 .
I s p = I s p 1 + I s p 2 ,
I s p 1 = k = 1 M C ˜ 2 Q ˜ k 2 ,
I s p 2 = 1 / 16 ( O O * ) 2 .
I s p 1 = M C ˜ 2 / 3 = G / 3.
( O O * ) 2 = C 4 j k l m N Q j Q k Q l Q m exp [ i ( Φ j + Φ k - Φ l - Φ m ) ] ,
( O O * ) 2 = N C 4 Q k ¯ 4 + 2 ( N 2 - N ) C 4 ( Q k ¯ 2 ) 2 .
I s p 2 = 0.014 G 2 .
I s p = I s p 1 = G / 3.
I B N = I s p + I c G / 3 + B 2 .
σ B N = I s p ( 1 + 2 β ) 1 / 2 ,
β I c I s p = B 2 G / 3 .
SNR = I sig / σ B N .
SNR = { ( G / 3 ) [ 1 + 2 B 2 / ( G / 3 ) ] 1 / 2 } - 1 ,
t ( E ) = c E - γ / 2 .
K = - t ( E b ) = ( γ / 2 ) c E b - γ / 2 E b - 1 ;
B = 2 / γ .
SNR = { ( G / 3 ) [ 1 + 8 γ 2 ( G / 3 ) ] 1 / 2 } - 1 ,
Sup ( SNR ) = ( G / 3 ) - 1 = ( 0.646 d 2 n s L ) - 1 .
SNR = { ( G / 3 ) [ 1 + 8 γ e 2 ( G / 3 ) ] 1 / 2 } - 1 .
O ( x , y ) = k = 1 N ( 1 + w k ) o k ( x , y ) .
I H ( x , y ) = 1 + k = 1 N ( o k + o k * ) + j ; k = 1 N o j o k * + k = 1 N ( w k o k + w k * o k * ) + j ; k = 1 N ( w j o j o k * + w j * o j * o k + w j o j w k * o k * ) = I H ( x , y ) + I M ( x , y ) ,
I M = k = 1 N ( w k o k + w k * o k * ) ,
I N = I N + I M .
σ N 2 = I N 2 - I N 2 = σ N 2 + σ M 2 ,
σ M 2 = 4 k = 1 N w k 2 o k 2 ,
o k 2 = C 2 Q k ¯ 2 = C 2 / 3.
w k = i = 1 m k o i ,
w k 2 = i = 1 m k o i 2 = m k C 2 / 3.
m k = ( π / 3 ) n s θ 0 2 ( z p , k ) 3 .
w k 2 = ( π / 3 ) n s θ 0 2 L 3 ɛ k 3 C 2 / 3 = N ( L / z ) 2 ɛ k 3 C 2 / 3 ,
σ M 2 = ( N C 2 / 3 ) 2 ( L / z ) 2 k = 1 N ɛ k 3 / N .
σ M 2 = σ N 2 ( L / z ) 2 / 4.
σ N = σ N [ 1 + ( 1 / 4 ) ( L / z ) 2 ] 1 / 2 .
sin ( α r 2 ) [ 2 J 1 ( α d r ) / ( α d r ) ] ,
ρ c = π f c / α .
f ( r , ρ c ) = 2 0 J 1 ( α d ρ ) ( α d ρ ) J 0 ( 2 α r ρ ) H ( ρ , ρ c ) ρ d ρ ,
g ( r , ρ c ) = 2 0 J 1 ( ρ ) J 0 ( r ρ ) H ( ρ , ρ c ) d ρ ,
I sig = 2 0 g 2 ( r ) r d r .

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