Abstract

We introduce the singular-optics approach for classification of rough surfaces with large-scale inhomogeneities into random and fractal surfaces. The maps of amplitude zeros of a field versus the parameters of the rough surfaces and the position of the observation zone are obtained and analyzed. It is shown that the local density of amplitude zeros in the scattered field serves as an appropriate parameter with which to classify the surface of interest into a surface with a height distribution that can be described as a random or a fractal process.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).
  2. O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of random phase objects,” Appl. Opt. 29, 2894–2898 (1990).
    [CrossRef] [PubMed]
  3. O. V. Angelsky, P. P. Maksimyak, “Optical correlation devices for measuring randomly phased objects,” Opt. Eng. 32, 3235–3243 (1993).
    [CrossRef]
  4. O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
    [CrossRef]
  5. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Bochan, ed. (North-Holland, Amsterdam, 1981), pp. 453–543.
  6. G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  7. J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  8. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999), p. 328.
  9. I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation network in highly random media,” Opt. Commun. 101, 247–264 (1993).
    [CrossRef]
  10. M. Soskin, M. Vasnetsov, “Singular optics as new chapter of modern photonics: optical vortices, fundamentals and applications,” Photon. Sci. News 4(1), 21–27 (1999).
  11. R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms in Computer Graphics, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 13–16 and 805–835.
  12. J. W. Goodman, Statistical Optics, (Wiley, New York, 1985).
  13. S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin1989).
  14. M. S. Soskin, M. V. Vasnetsov, I. V. Basisty, “Optical wavefront dislocations,” in International Conference on Holography and Correlation Optics, O. V. Angelsky, ed. Proc. SPIE2647, 57–62 (1995).
    [CrossRef]
  15. N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
    [CrossRef]
  16. I. V. Basisty, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [CrossRef]
  17. N. R. Heckenberg, R. McDuff, C. P. Smith, M. J. Wegener, “Optical Fourier transform recognition of phase singularities in optical fields,” in From Galileo’s Occhialino to Optoelectronics, P. Mazzoldi, ed. (World Scientific, Singapore, 1992), pp. 848–852.
  18. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  19. A. Arneodo, “Wavelet analysis of fractals: from the mathematical concept to experimental reality,” in Wavelets: Theory and Application, M. Y. Hussaini, ed. (Oxford U. Press, New York, 1996), pp. 352–497.

2002 (1)

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

2001 (1)

1999 (1)

M. Soskin, M. Vasnetsov, “Singular optics as new chapter of modern photonics: optical vortices, fundamentals and applications,” Photon. Sci. News 4(1), 21–27 (1999).

1995 (1)

I. V. Basisty, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

1993 (2)

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation network in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Optical correlation devices for measuring randomly phased objects,” Opt. Eng. 32, 3235–3243 (1993).
[CrossRef]

1992 (1)

1990 (1)

1983 (1)

N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

1974 (1)

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Optical correlation devices for measuring randomly phased objects,” Opt. Eng. 32, 3235–3243 (1993).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of random phase objects,” Appl. Opt. 29, 2894–2898 (1990).
[CrossRef] [PubMed]

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

Arneodo, A.

A. Arneodo, “Wavelet analysis of fractals: from the mathematical concept to experimental reality,” in Wavelets: Theory and Application, M. Y. Hussaini, ed. (Oxford U. Press, New York, 1996), pp. 352–497.

Baranova, N. V.

N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

Basisty, I. V.

I. V. Basisty, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, I. V. Basisty, “Optical wavefront dislocations,” in International Conference on Holography and Correlation Optics, O. V. Angelsky, ed. Proc. SPIE2647, 57–62 (1995).
[CrossRef]

Berry, M.

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Bochan, ed. (North-Holland, Amsterdam, 1981), pp. 453–543.

Dogariu, A.

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Freilikher, V.

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation network in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Freund, I.

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation network in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics, (Wiley, New York, 1985).

Hanson, S. G.

O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
[CrossRef]

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

Heckenberg, N. R.

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, M. J. Wegener, “Optical Fourier transform recognition of phase singularities in optical fields,” in From Galileo’s Occhialino to Optoelectronics, P. Mazzoldi, ed. (World Scientific, Singapore, 1992), pp. 848–852.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin1989).

Maksimyak, P. P.

O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Optical correlation devices for measuring randomly phased objects,” Opt. Eng. 32, 3235–3243 (1993).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of random phase objects,” Appl. Opt. 29, 2894–2898 (1990).
[CrossRef] [PubMed]

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

Mamaev, A. V.

N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, M. J. Wegener, “Optical Fourier transform recognition of phase singularities in optical fields,” in From Galileo’s Occhialino to Optoelectronics, P. Mazzoldi, ed. (World Scientific, Singapore, 1992), pp. 848–852.

Nye, J. F.

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999), p. 328.

Pilipetskii, N. F.

N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

Popescu, G.

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin1989).

Ryukhtin, V. V.

Shkunov, V. V.

N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

Shvartsman, N.

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation network in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, M. J. Wegener, “Optical Fourier transform recognition of phase singularities in optical fields,” in From Galileo’s Occhialino to Optoelectronics, P. Mazzoldi, ed. (World Scientific, Singapore, 1992), pp. 848–852.

Soskin, M.

M. Soskin, M. Vasnetsov, “Singular optics as new chapter of modern photonics: optical vortices, fundamentals and applications,” Photon. Sci. News 4(1), 21–27 (1999).

Soskin, M. S.

I. V. Basisty, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, I. V. Basisty, “Optical wavefront dislocations,” in International Conference on Holography and Correlation Optics, O. V. Angelsky, ed. Proc. SPIE2647, 57–62 (1995).
[CrossRef]

Tatarsky, V. I.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin1989).

Vasnetsov, M.

M. Soskin, M. Vasnetsov, “Singular optics as new chapter of modern photonics: optical vortices, fundamentals and applications,” Photon. Sci. News 4(1), 21–27 (1999).

Vasnetsov, M. V.

I. V. Basisty, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, I. V. Basisty, “Optical wavefront dislocations,” in International Conference on Holography and Correlation Optics, O. V. Angelsky, ed. Proc. SPIE2647, 57–62 (1995).
[CrossRef]

Voss, R. F.

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms in Computer Graphics, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 13–16 and 805–835.

Wegener, M. J.

N. R. Heckenberg, R. McDuff, C. P. Smith, M. J. Wegener, “Optical Fourier transform recognition of phase singularities in optical fields,” in From Galileo’s Occhialino to Optoelectronics, P. Mazzoldi, ed. (World Scientific, Singapore, 1992), pp. 848–852.

White, A. G.

Zeldovich, B. Y.

N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

N. V. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkunov, B. Y. Zeldovich, “Wavefront dislocations: topological limitations for adaptive system with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

Opt. Commun. (2)

I. V. Basisty, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation network in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Opt. Eng. (1)

O. V. Angelsky, P. P. Maksimyak, “Optical correlation devices for measuring randomly phased objects,” Opt. Eng. 32, 3235–3243 (1993).
[CrossRef]

Opt. Lett. (1)

Photon. Sci. News (1)

M. Soskin, M. Vasnetsov, “Singular optics as new chapter of modern photonics: optical vortices, fundamentals and applications,” Photon. Sci. News 4(1), 21–27 (1999).

Phys. Rev. Lett. (1)

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Other (9)

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999), p. 328.

N. R. Heckenberg, R. McDuff, C. P. Smith, M. J. Wegener, “Optical Fourier transform recognition of phase singularities in optical fields,” in From Galileo’s Occhialino to Optoelectronics, P. Mazzoldi, ed. (World Scientific, Singapore, 1992), pp. 848–852.

A. Arneodo, “Wavelet analysis of fractals: from the mathematical concept to experimental reality,” in Wavelets: Theory and Application, M. Y. Hussaini, ed. (Oxford U. Press, New York, 1996), pp. 352–497.

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms in Computer Graphics, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 13–16 and 805–835.

J. W. Goodman, Statistical Optics, (Wiley, New York, 1985).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin1989).

M. S. Soskin, M. V. Vasnetsov, I. V. Basisty, “Optical wavefront dislocations,” in International Conference on Holography and Correlation Optics, O. V. Angelsky, ed. Proc. SPIE2647, 57–62 (1995).
[CrossRef]

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Bochan, ed. (North-Holland, Amsterdam, 1981), pp. 453–543.

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 (19)

Fig. 1
Fig. 1

Simulated (a) rough, random and (b), (c) fractal surfaces smoothed following the normal law over (a), (b) 3 and (c) 5 pixels.

Fig. 2
Fig. 2

Distributions of the scattered field’s (a) amplitude, (b) intensity, and (c) phase.

Fig. 3
Fig. 3

Distributions of the scattered field’s (a) amplitude, (b) intensity, and (c) phase with a superimposed reference wave.

Fig. 4
Fig. 4

Simulated interferogram at the boundary field of a rough surface.

Fig. 5
Fig. 5

Intensity distribution at the caustic zone of a rough surface.

Fig. 6
Fig. 6

Interferogram of a field exhibiting phase singularities; the areas of most interest are outlined by white parallelograms.

Fig. 7
Fig. 7

Examples of interference patterns corresponding to amplitude zeros with various topological charges.

Fig. 8
Fig. 8

Illustration of the field transformation: (a) spatial intensity distribution and (b) an interferogram of the field scattered by a rough surface, computed at a distance of 10 μm from the object (the number of amplitude zeros is 8); (c) spatial intensity distribution and (d) an interferogram of the field, computed at a distance of 25 μm, just behind the caustic zone (the number of amplitude zeros is 293).

Fig. 9
Fig. 9

(a) Spatial intensity distribution and (b)–(e) interferograms of the scattered field for a rough surface at a distance of 35 μm from the object for several interference angles. The areas of interest demonstrating the field transformations are outlined by white squares.

Fig. 10
Fig. 10

(a) Intensity distribution of the field scattered at a random rough surface at a distance of 10 μm from the object; (b) zerogram of this field.

Fig. 11
Fig. 11

Optical fields scattered at rough surfaces with various smoothings of microirregularities at a distance of 10 μm from the surfaces: (a) intensity distribution of the field from a random rough surface smoothed by a Gaussian window with of 5-pixel dispersion; (b) interference of the field shown in (a) with a plane reference wave (amplitude zeros are absent); (c) intensity distribution of the field from a random rough surface smoothed by a Gaussian window with 1-pixel dispersion; (d) interference of the field shown in (c) with a plane reference wave (here the number of amplitude zeros is 532).

Fig. 12
Fig. 12

(a) Example of a zerogram and (b) the corresponding histogram of local density of amplitude zeros, K p /K 0 versus p [μm-2].

Fig. 13
Fig. 13

Zerogram of the field with statistically homogeneous, uniform spatial distribution of amplitude zeros and (b) histogram of local density of amplitude zeros, K p /K 0 versus p [μm-2].

Fig. 14
Fig. 14

(a) Zerogram of the field with nonuniform distribution of amplitude zeros and (b) histogram of the local density of amplitude zeros, K p /K 0 versus p [μm-2].

Fig. 15
Fig. 15

(a), (d), (g) Intensity distributions, (b), (e), (h) zerograms, and (c), (f), (i) histograms of distributions of local density of amplitude zeros, K p /K 0, versus p [μm-2] at a distance of 10 μm from the rough surfaces: (a)–(c) random and (d)–(f) fractal surfaces smoothed following a normal law over 3 pixels; (g)–(i) fractal surface smoothed following a normal law over 5 pixels.

Fig. 16
Fig. 16

(a), (d), (g) Intensity distributions, (b), (e), (h) zerograms, and (c), (f), (i) histograms of distributions of local density of amplitude zeros, K p /K 0, versus p [μm-2] at a distance of 20 μm from the rough surfaces: (a)–(c) random and (d)–(f) fractal surfaces smoothed following a normal law over 3 pixels; (g)–(f) fractal surface smoothed following a normal law over 5 pixels.

Fig. 17
Fig. 17

(a), (d), (g) Intensity distributions, (b), (e), (h) zerograms, and (c), (f), (i) histograms of distributions of local density of amplitude zeros, K p /K 0, versus p [μm-2] at a distance of 50 μm from the rough surfaces: (a)–(c) random and (d)–(f) fractal surfaces smoothed following a normal law over 3 pixels; (g)–(i) fractal surface smoothed following a normal law over 5 pixels.

Fig. 18
Fig. 18

(a), (d), (g) Intensity distributions, (b), (e), (h) zerograms, and (c), (f), (i) histograms of distributions of local density of amplitude zeros, K p /K 0, versus p [μm-2] at a distance of 100 μm from the rough surfaces: (a)–(c) random and (d)–(f) fractal surfaces smoothed following a normal law over 3 pixels; (g)–(i) fractal surface smoothed following a normal law over 5 pixels.

Fig. 19
Fig. 19

(a), (d), (g) Intensity distributions, (b), (e), (h) zerograms, and (c), (f), (i) histograms of distributions of local density of amplitude zeros, K p /K 0, versus p [μm-2] at a distance of 500 μm from the rough surfaces: (a)–(c) random and (d)–(f) fractal surfaces smoothed following a normal law over 3 pixels; (g)–(i) fractal surface smoothed following a normal law over 5 pixels.

Equations (9)

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

σn2=1/22nHσ02,
Uξ, ζ=ziλ  Fx, yR2x, y, z, ξ, ζ×exp-ikRx, y, z, ξ, ζ+n-1hx, ydxdy,
Uξ, ζ=Re Uξ, ζ+i Im Uξ, ζ,
Aξ, ζ=Re2 Uξ, ζ+Im2 Uξ, ζ1/2,
φξ, ζ=arctanIm Uξ, ζRe Uξ, ζ,
Iξ, ζ=Re2 Uξ, ζ+Im2 Uξ, ζ.
|Fy-Fx0|<A0|y-x0|hFx0,
ReAx, y=0, ImAx, y=0.
pi, j=ni, jsi, j.

Metrics