Abstract

The partial coherent illumination of the specimen, which is required for white-light interferometric measurements of optically rough surfaces, directly leads to speckle. The electric field of such speckle patterns strongly fluctuates in amplitude and phase. This spatially correlated noise influences the accuracy of the measuring device. Although a variety of noise sources in white-light interferometry has been studied in recent years, they do not account for spatial correlation and, hence, they cannot be applied to speckle noise. Thus, we derive a new model enabling quantitative predictions for measurement uncertainty caused by speckle. The model reveals that the accuracy can be attributed mainly to the degree of spatial correlation, i.e., the average size of a speckle, and to the coherence length of the light source. The same parameters define the signal-to-noise ratio in the spectral domain. The model helps to design filter functions that are perfectly adapted to the noise characteristics of the respective device, thus improving the accuracy of postprocessing algorithms for envelope detection. The derived expressions are also compared to numerical simulations and experimental data of two different types of interferometers. These results are a first validation of the theoretical considerations of this article.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775-3783 (1990).
    [CrossRef] [PubMed]
  2. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919-925 (1992).
    [CrossRef] [PubMed]
  3. L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334-7338 (1994).
    [CrossRef] [PubMed]
  4. G. Häusler, “Three-dimensional sensors--potentials and limitations,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 485-506.
  5. G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Photonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, 1999), pp. 328-342.
  6. J. Schmit and A. Olszak, “High-precision shape measurement by white-light interferometry with real-time scanner error correction,” Appl. Opt. 41, 5943-5950 (2002).
    [CrossRef] [PubMed]
  7. P. Pavlicek and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
    [CrossRef] [PubMed]
  8. A. Pförtner and J. Schwider, “Dispersion error in white-light Linnik interferometers and its implications for evaluation procedures,” Appl. Opt. 40, 6223-6228 (2001).
    [CrossRef]
  9. A. Harasaki and J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101-2106 (2000).
    [CrossRef]
  10. M. Fleischer, R. Windecker, and H. J. Tiziani, “Theoretical limits of scanning white-light interferometry signal evaluation algorithms,” Appl. Opt. 40, 2815-2820 (2001).
    [CrossRef]
  11. P. Pavlicek and J. Soubusta, “Theoretical measurement uncertainty of white-light interferometry on rough surfaces,” Appl. Opt. 42, 1809-1813 (2003).
    [CrossRef] [PubMed]
  12. T. E. Carlsson and B. Nilsson, “Measurement of distance to diffuse surfaces using non-scanning coherence radar,” J. Opt. 29, 146-151 (1998).
    [CrossRef]
  13. M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
    [CrossRef]
  14. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827-832 (1990).
    [CrossRef]
  15. G. Chan and A. T. A. Wood, “An algorithm for simulating stationary Gaussian random fields,” Appl. Stat. 46, 171-181(1997).
    [CrossRef]
  16. J. C. Dainty, Laser Speckle and Related Phenomena, Topics in Applied Physics (Springer-Verlag, 1975).
  17. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 531688-1700(1965).
    [CrossRef]
  18. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Topics in Applied Physics (Springer-Verlag, 1975), pp. 9-76.
    [CrossRef]
  19. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Info. Theory 8, 194-195 (1962).
    [CrossRef]
  20. E. Ochoa and J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. 73, 943-949 (1983).
    [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics, 1st ed., Physical and Quantum Electronics (McGraw-Hill, 1968).
  22. H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” J. Mod. Opt. 22, 15-24 (1975).
    [CrossRef]
  23. H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” J. Mod. Opt. 22, 523-535 (1975).
    [CrossRef]
  24. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1980).
  25. P. de Groot and L. Deck, “Surface profiling by frequency-domain analysis of white light interferograms,” Proc. SPIE 2248, 101-104 (1994).
    [CrossRef]
  26. D. Middleton, “Processes derived from the normal,” in An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 396-437.
  27. D. Middleton, “Special functions and integrals,” in An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 1071-1081.
  28. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford U. Press, 2005).
  29. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

2006

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
[CrossRef]

2004

2003

2002

2001

2000

1998

T. E. Carlsson and B. Nilsson, “Measurement of distance to diffuse surfaces using non-scanning coherence radar,” J. Opt. 29, 146-151 (1998).
[CrossRef]

1997

G. Chan and A. T. A. Wood, “An algorithm for simulating stationary Gaussian random fields,” Appl. Stat. 46, 171-181(1997).
[CrossRef]

1994

P. de Groot and L. Deck, “Surface profiling by frequency-domain analysis of white light interferograms,” Proc. SPIE 2248, 101-104 (1994).
[CrossRef]

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334-7338 (1994).
[CrossRef] [PubMed]

1992

1990

1983

1975

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” J. Mod. Opt. 22, 15-24 (1975).
[CrossRef]

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” J. Mod. Opt. 22, 523-535 (1975).
[CrossRef]

1965

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 531688-1700(1965).
[CrossRef]

1962

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Info. Theory 8, 194-195 (1962).
[CrossRef]

Banyay, M.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
[CrossRef]

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Bohn, G.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Photonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, 1999), pp. 328-342.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1980).

Carlsson, T. E.

T. E. Carlsson and B. Nilsson, “Measurement of distance to diffuse surfaces using non-scanning coherence radar,” J. Opt. 29, 146-151 (1998).
[CrossRef]

Chan, G.

G. Chan and A. T. A. Wood, “An algorithm for simulating stationary Gaussian random fields,” Appl. Stat. 46, 171-181(1997).
[CrossRef]

Chim, S. S. C.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena, Topics in Applied Physics (Springer-Verlag, 1975).

de Groot, P.

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334-7338 (1994).
[CrossRef] [PubMed]

P. de Groot and L. Deck, “Surface profiling by frequency-domain analysis of white light interferograms,” Proc. SPIE 2248, 101-104 (1994).
[CrossRef]

Deck, L.

P. de Groot and L. Deck, “Surface profiling by frequency-domain analysis of white light interferograms,” Proc. SPIE 2248, 101-104 (1994).
[CrossRef]

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334-7338 (1994).
[CrossRef] [PubMed]

Dresel, T.

Ettl, P.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Photonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, 1999), pp. 328-342.

Fleischer, M.

Goodman, J. W.

E. Ochoa and J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. 73, 943-949 (1983).
[CrossRef]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 531688-1700(1965).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 1st ed., Physical and Quantum Electronics (McGraw-Hill, 1968).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Topics in Applied Physics (Springer-Verlag, 1975), pp. 9-76.
[CrossRef]

Grimmett, G. R.

G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford U. Press, 2005).

Harasaki, A.

Häusler, G.

T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919-925 (1992).
[CrossRef] [PubMed]

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Photonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, 1999), pp. 328-342.

G. Häusler, “Three-dimensional sensors--potentials and limitations,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 485-506.

Hering, M.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
[CrossRef]

Herrmann, S.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
[CrossRef]

Jähne, B.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
[CrossRef]

Kino, G. S.

Kirchner, M.

Körner, K.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
[CrossRef]

Laszlo, I.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Photonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, 1999), pp. 328-342.

Leushacke, L.

Middleton, D.

D. Middleton, “Processes derived from the normal,” in An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 396-437.

D. Middleton, “Special functions and integrals,” in An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 1071-1081.

Nilsson, B.

T. E. Carlsson and B. Nilsson, “Measurement of distance to diffuse surfaces using non-scanning coherence radar,” J. Opt. 29, 146-151 (1998).
[CrossRef]

Ochoa, E.

Olszak, A.

Pavlicek, P.

Pedersen, H. M.

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” J. Mod. Opt. 22, 523-535 (1975).
[CrossRef]

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” J. Mod. Opt. 22, 15-24 (1975).
[CrossRef]

Pförtner, A.

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Info. Theory 8, 194-195 (1962).
[CrossRef]

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Schenk, M.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Photonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, 1999), pp. 328-342.

Schmit, J.

Schwider, J.

Soubusta, J.

Stirzaker, D. R.

G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford U. Press, 2005).

Tiziani, H. J.

Venzke, H.

Windecker, R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1980).

Wood, A. T. A.

G. Chan and A. T. A. Wood, “An algorithm for simulating stationary Gaussian random fields,” Appl. Stat. 46, 171-181(1997).
[CrossRef]

Wyant, J. C.

Appl. Opt.

Appl. Stat.

G. Chan and A. T. A. Wood, “An algorithm for simulating stationary Gaussian random fields,” Appl. Stat. 46, 171-181(1997).
[CrossRef]

IRE Trans. Info. Theory

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Info. Theory 8, 194-195 (1962).
[CrossRef]

J. Mod. Opt.

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” J. Mod. Opt. 22, 15-24 (1975).
[CrossRef]

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” J. Mod. Opt. 22, 523-535 (1975).
[CrossRef]

J. Opt.

T. E. Carlsson and B. Nilsson, “Measurement of distance to diffuse surfaces using non-scanning coherence radar,” J. Opt. 29, 146-151 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. IEEE

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 531688-1700(1965).
[CrossRef]

Proc. SPIE

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, “One-shot line-profiling white light interferometer with spatial phase shift for measuring rough surfaces,” Proc. SPIE 6188, 61880E (2006).
[CrossRef]

P. de Groot and L. Deck, “Surface profiling by frequency-domain analysis of white light interferograms,” Proc. SPIE 2248, 101-104 (1994).
[CrossRef]

Other

D. Middleton, “Processes derived from the normal,” in An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 396-437.

D. Middleton, “Special functions and integrals,” in An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 1071-1081.

G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford U. Press, 2005).

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1980).

J. W. Goodman, Introduction to Fourier Optics, 1st ed., Physical and Quantum Electronics (McGraw-Hill, 1968).

G. Häusler, “Three-dimensional sensors--potentials and limitations,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 485-506.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Photonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, 1999), pp. 328-342.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Topics in Applied Physics (Springer-Verlag, 1975), pp. 9-76.
[CrossRef]

J. C. Dainty, Laser Speckle and Related Phenomena, Topics in Applied Physics (Springer-Verlag, 1975).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Setup of a spatially scanning white-light interferometer: Cyl, cylindrical lens; L, spherical lens; MO, microscope objective; BS, beam splitter; ND, neutral density filter; CP, compensation plate; f, focal plane.

Fig. 2
Fig. 2

Propagation geometry of the collimating optical setup.

Fig. 3
Fig. 3

Interferograms acquired by spatially scanning WLI: (a) smooth metallic surface and (b) metallic surface with a RMS roughness of 0.45 μm .

Fig. 4
Fig. 4

Influence of speckle noise N ( k ) on the undisturbed frequency component S ( k ) . With increasing variance of the phase angle ϑ of the resulting component H ( k ) the measurement accuracy is decreased.

Fig. 5
Fig. 5

Phase angle variance ϑ 2 of H ( k ) due to speckle noise N ( k ) .

Fig. 6
Fig. 6

Phase angle variance ϑ 2 of H ( k ) , which originates from the superposition of the undisturbed signal S ( k ) and the complex random variable N ( k ) .

Fig. 7
Fig. 7

Distribution density f ϑ ( ϑ ) for different SNR r due to speckle noise (after Middleton [26]).

Fig. 8
Fig. 8

Phase angle variance ϑ 2 of white-light interferometric signals influenced by fully developed speckle patterns.

Fig. 9
Fig. 9

Measurement error due to speckle, which are approximately three times larger then the coherence length of the source, i.e., σ 0 = 3 σ r .

Fig. 10
Fig. 10

White-light fringes I ( z ) due to the coherent addition of a virtual reference wavefront and a simulated speckle pattern with amplitude | A ( z ) | and phase ϕ ( z ) . The scale of the upper graph refers to the phase distribution in units of 2 π ; intensity and amplitude are depicted in arbitrary units.

Fig. 11
Fig. 11

Standard deviation of the phase angle θ caused by fully developed speckle patterns with size σ 0 = 4 σ r and σ 0 = 8 σ r , respectively.

Fig. 12
Fig. 12

Measurement uncertainty σ z , 2 caused by intensity and phase fluctuations due to the second-order statistics of speckle.

Fig. 13
Fig. 13

RMS roughness determined by tactile and optical measurement techniques. The discrepancies in the interferometric measurements directly reflect the different sensitivity to speckle noise of both devices.

Fig. 14
Fig. 14

Semilogarithmic representation of experimentally determined and theoretically predicted measurement uncertainty σ z , 2 .

Fig. 15
Fig. 15

Dependence of the phase angle variance ϑ 2 on the SNR due to speckle noise.

Equations (50)

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

E ( q , t ) = A ( q ) exp { i 2 π ν t } ,
A ( q ) = 1 N m N | a m | exp { i ρ m } .
A ( q ) = h ( q , p ) α ( p ) d ξ d η .
R I I ( q 1 , q 2 ) = I ( q 1 ) I ( q 2 ) + | R A A * ( q 1 , q 2 ) | 2 ,
R A A * ( q 1 , q 2 ) = A ( q 1 ) A * ( q 2 ) .
R r r ( q 1 , q 2 ) = R i i ( q 1 , q 2 ) = 1 2 Re { R A A * ( q 1 , q 2 ) } ,
R r i ( q 1 , q 2 ) = R i r ( q 1 , q 2 ) = 1 2 Im { R A A * ( q 1 , q 2 ) } .
δ t = 1.22 λ z D 0 ,
δ l = 8 λ ( z D 0 ) 2 .
A P ( x P , 0 ) = i λ f exp { i k 2 f ( 1 d f ) x P 2 } α ( ξ , d ) exp { i k f ξ x P } d ξ ,
A ( x , z ) = i λ z exp { i k 2 z x 2 } A P ( x P , 0 ) exp { i k 2 z ( x P 2 2 x x P ) } d x P .
α ( ξ , 0 ) α * ( ξ , 0 ) κ P ( ξ ) P * ( ξ ) δ ( ξ ξ ) ,
R A A * ( x 1 , x 2 ) | P ( ξ ) | 2 exp { i k f ( x P x P ) ξ } d ξ · exp { i k b 2 ( x P 2 x P 2 ) } · exp { i k z ( x 1 x P x 2 x P ) } d x P d x P ,
b = 1 f ( 1 d f ) + 1 z .
R A A * ( x 1 , x 2 ) exp { ( k σ L 2 f ) 2 1 z 2 b 2 ( x 1 x 2 ) 2 } .
R I I = I ( x 1 ) I ( x 2 ) [ 1 + exp { 1 2 ( k σ L f ) 2 1 z 2 b 2 ( x 1 x 2 ) 2 } ] .
δ x ( z ) = 2 f k σ L z b .
I ˜ ( q , τ ) = | U 0 ( q , t ) + r ˜ ( τ ) U 0 ( q , t + τ ) | 2 t = | U 0 ( q , t ) | 2 t + | r ˜ ( q , τ ) | 2 | U 0 ( q , t + τ ) | 2 t + [ r ˜ ( q , τ ) * U 0 ( q , t ) U 0 * ( q , t + τ ) t + r ˜ ( q , τ ) U 0 * ( q , t ) U 0 ( q , t + τ ) t ] = ( 1 + | r ˜ ( q , τ ) | 2 ) | U 0 ( q , t + τ ) | 2 t + 2 Re { r ˜ ( q , τ ) U 0 * ( q , t ) U 0 ( q , t + τ ) t } .
I ( z ) = ( 1 + | r ( z ) | 2 ) I 0 + 2 Re { r ( z ) γ ( z z 0 ) } I 0 .
γ = | γ | exp { i k 0 ( z z 0 ) }
I ^ ( k ) = G R ( k ) * S ( k ) exp { i Θ ( k ) } .
R 1 R 2 = I π 4 n = 0 [ ( 1 / 2 ) n n ! exp { n δ r 2 ( z 1 z 2 ) 2 } ] 2 ,
G R ( k ) = 1 2 I π [ δ ( k ) + a exp { k 2 4 σ r 2 + i φ ( k ) } ] .
a = ( 4 2 π σ r ) 1 / 2
I ^ ( k ) = 1 2 π I exp { i k 0 Δ z } [ S ( k ) + a exp { k 2 4 σ r 2 + i φ ( k ) } * S ( k ) ] .
S ( k ) = exp { ( k k 0 ) 2 2 σ 0 2 } ,
N ( k ) = a exp { ( k k 0 ) 2 2 σ 0 2 + 4 σ r 2 } exp { κ 2 2 σ 2 + i φ ( c ( k ) κ ) } d κ ,
σ 2 = 2 σ r 2 σ 0 2 2 σ r 2 + σ 0 2
c ( k ) = 2 σ r 2 2 σ r 2 + σ 0 2 ( k k 0 ) .
N r ( k ) 2 = π σ × a 2 exp { ( k k 0 ) 2 2 σ 0 2 + 4 σ r 2 } [ 1 + exp { c ( k ) 2 σ 2 } ] ,
N i ( k ) 2 = π σ × a 2 exp { ( k k 0 ) 2 2 σ 0 2 + 4 σ r 2 } [ 1 exp { c ( k ) 2 σ 2 } ] .
x = exp { c ( k ) 2 σ 2 } ,
f r , i ( H r , H i ) = 1 π N 2 exp { ( S ( k ) H r ) 2 + H i 2 N 2 } .
f | H | , ϑ ( | H | , ϑ ) = | H | π N 2 exp { S ( k ) 2 + | H | 2 2 S ( k ) | H | cos ϑ N 2 } .
f ϑ ( ϑ ) = 1 2 π [ 1 + m = 1 2 r m / 2 m ! Γ ( m 2 + 1 ) F 1 1 ( m 2 ; m + 1 ; r ) cos ( m ϑ ) ] .
r ( k ) = S ( k ) 2 N 2 = 2 2 σ r 2 + σ 0 2 σ 0 2 exp { 2 σ r 2 2 σ r 2 + σ 0 2 · ( k k 0 ) 2 σ 0 2 } .
ϑ 2 1 2 r + 3 16 r 2 .
ϑ 2 = ϑ 2 × ( 1 x 2 ) .
[ θ / k ] 2 | k = k 0 lim k k 0 ϑ ( k ) 2 / ( k k 0 ) 2 .
σ 0 = 2 ln 2 2 π × l .
κ 1 , κ 2 < c exp { κ 1 2 + κ 2 2 2 σ 2 } cos ( φ ( c ( k ) κ 1 ) ) cos ( φ ( c ( k ) κ 2 ) ) d κ 1 d κ 2 + c < κ 1 , κ 2 exp { κ 1 2 + κ 2 2 2 σ 2 } cos ( φ ( c ( k ) κ 1 ) ) cos ( φ ( c ( k ) κ 2 ) ) d κ 1 d κ 2 + 2 c c exp { κ 1 2 + κ 2 2 2 σ 2 } cos ( φ ( c ( k ) κ 1 ) ) cos ( φ ( c ( k ) κ 2 ) ) d κ 1 d κ 2 .
cos ( φ ( x 1 ) ) cos ( φ ( x 2 ) ) = { δ ( x 1 x 2 ) for     x 1 · x 2 > 0 δ ( x 1 + x 2 ) for     x 1 · x 2 < 0
exp { κ 2 σ 2 } d κ + exp { 2 c ( k ) 2 σ 2 } exp { κ 2 2 κ c ( k ) σ 2 } d κ .
N r ( k ) 2 = π σ × a 2 exp { ( k k 0 ) 2 2 σ r 2 + σ 0 2 } [ 1 + exp { c ( k ) 2 σ 2 } ] .
sin ( φ ( x 1 ) ) sin ( φ ( x 2 ) ) = { δ ( x 1 x 2 ) for     x 1 · x 2 > 0 δ ( x 1 + x 2 ) for     x 1 · x 2 < 0 ,
N i ( k ) 2 = π σ × a 2 exp { ( k k 0 ) 2 2 σ r 2 + σ 0 2 } [ 1 exp { c ( k ) 2 σ 2 } ] ,
cos ϑ = r 1 / 2 Γ ( 3 / 2 ) F 1 1 ( 1 / 2 ; 2 ; r ) .
F 1 1 ( α ; β ; r ) Γ ( β ) r α Γ ( β α ) n = 0 ( α ) n ( α β + 1 ) n n ! r n .
cos ϑ = n = 0 ( 1 / 2 ) n ( 1 / 2 ) n n ! r n 1 1 4 r 3 32 r 2 .
cos ϑ 1 1 2 ϑ 2 .

Metrics