Abstract

The pseudo-imaging process in a system consisting of two periodic gratings and illuminated by an incoherent polychromatic and finite extension source placed at a finite distance from the gratings is studied. An analytical expression of the irradiance distribution on a plane, also located at a finite distance from the gratings, has been obtained from previous results on monochromatic illumination. In the analysis presented, different imaging regimes are found and related to the parameters which characterize the double grating system. The pseudo-imaging phenomenon strongly depends on both the spatial and temporal coherence of the incident illuminating field. Certain pseudo-images are observed with polychromatic and incoherent incident light under some restrictions. On the other hand, pseudo-image process analogous to Talbot effect appears only by monochromatic and plane illuminating wavefront.

© 2004 Optical Society of America

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  1. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).
  2. J. M. Cowley and A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
    [CrossRef]
  3. E. Keren and O. Kafri, “Diffraction effects in moire deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]
  4. S. Teng, L. Liu, J. Zu, Z. Luan, and De’an Liu, “Uniform theory of the Talbot effect with partially coherent light illumination,” J. Opt. Soc. Am. A 20, 1747–1754 (2003).
    [CrossRef]
  5. E. Lau. “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1985).
  6. J. Jahns and A.W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  7. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  8. R. Sudol and B.J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
    [CrossRef]
  9. K.H. Brenner, A.W. Lohmann, and Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
    [CrossRef]
  10. L. Liu, X. Liu, and L. Ye, “Joint Talbot effect and logic operated moire patterns,” J. Opt. Soc. Am. A 7, 970–976 (1990).
    [CrossRef]
  11. L. Liu, “Interferometry based on the partially coherent effect lying between the Talbot and Lau effects,” J. Mod. Opt. 35, 1605–1618 (1988).
    [CrossRef]
  12. L. Liu, “Partially coherent diffraction effect between Lau and Talbot Effects,” J. Opt. Soc. Am. A 5, 1709–1716 (1988).
    [CrossRef]
  13. K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175–178 (1997).
    [CrossRef]
  14. M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Las. Tech. 31, 127–134 (1999).
    [CrossRef]
  15. D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
    [CrossRef]
  16. J. Pomarico and R. Torroba, “Colour image operations based on white light diffraction experiments (Lau effect)”, Eur. J. Phys. 14, 114–120 (1993).
    [CrossRef]
  17. L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28, 4668–4677 (1989).
    [CrossRef] [PubMed]
  18. K. Patorski “The self-imaging phenomenon and its applications,” in Progress in Optics,E. Wolf, ed. (North Holland, Amsterdam1989)  vol. 27, pp. 3–108.
  19. K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringes explanation,” Opt. Acta 30, 745–758 (1983).
    [CrossRef]
  20. G. J. Swanson and E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  21. G. J. Swanson and E. N. Leith, “Analisys of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
    [CrossRef]
  22. S.C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
    [CrossRef]
  23. L. Liu, “Ambiguity function and general Talbot-Lau effects,” Acta Opt. Sin. 7, 501–510 (1987).
  24. J. Tu and L. Zhan, “Two-dimensional theory of the Lau-Talbot-Moiré effect under partially coherent illumination,” Opt. Commun. 82, 229–235 (1991).
    [CrossRef]
  25. J. Tu and L. Zhan, “Analysis of general double periodic structure diffraction phenomena based on the ambiguity function,” J. Opt. soc. Am. A 9, 983–995 (1992).
    [CrossRef]
  26. A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
    [CrossRef]
  27. D. Crespo, J. Alonso, and E. Bernabéu, “Generalized imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231–1240 (2000).
    [CrossRef]
  28. D. Crespo, J. Alonso, and E. Bernabéu, “Experimental measurements of generalized grating images,” Appl. Opt. OT 41, 1223–1228 (2002).
    [CrossRef]

2003 (1)

2002 (1)

D. Crespo, J. Alonso, and E. Bernabéu, “Experimental measurements of generalized grating images,” Appl. Opt. OT 41, 1223–1228 (2002).
[CrossRef]

2000 (2)

D. Crespo, J. Alonso, and E. Bernabéu, “Generalized imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231–1240 (2000).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
[CrossRef]

1999 (1)

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Las. Tech. 31, 127–134 (1999).
[CrossRef]

1997 (2)

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175–178 (1997).
[CrossRef]

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

1993 (1)

J. Pomarico and R. Torroba, “Colour image operations based on white light diffraction experiments (Lau effect)”, Eur. J. Phys. 14, 114–120 (1993).
[CrossRef]

1992 (1)

1991 (1)

J. Tu and L. Zhan, “Two-dimensional theory of the Lau-Talbot-Moiré effect under partially coherent illumination,” Opt. Commun. 82, 229–235 (1991).
[CrossRef]

1990 (2)

S.C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

L. Liu, X. Liu, and L. Ye, “Joint Talbot effect and logic operated moire patterns,” J. Opt. Soc. Am. A 7, 970–976 (1990).
[CrossRef]

1989 (2)

L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28, 4668–4677 (1989).
[CrossRef] [PubMed]

K. Patorski “The self-imaging phenomenon and its applications,” in Progress in Optics,E. Wolf, ed. (North Holland, Amsterdam1989)  vol. 27, pp. 3–108.

K. Patorski “The self-imaging phenomenon and its applications,” in Progress in Optics,E. Wolf, ed. (North Holland, Amsterdam1989)  vol. 27, pp. 3–108.

1988 (2)

L. Liu, “Interferometry based on the partially coherent effect lying between the Talbot and Lau effects,” J. Mod. Opt. 35, 1605–1618 (1988).
[CrossRef]

L. Liu, “Partially coherent diffraction effect between Lau and Talbot Effects,” J. Opt. Soc. Am. A 5, 1709–1716 (1988).
[CrossRef]

1987 (1)

L. Liu, “Ambiguity function and general Talbot-Lau effects,” Acta Opt. Sin. 7, 501–510 (1987).

1985 (3)

1983 (2)

K.H. Brenner, A.W. Lohmann, and Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringes explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

1982 (1)

1979 (3)

J. Jahns and A.W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol and B.J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

1957 (1)

J. M. Cowley and A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

1836 (1)

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Alonso, J.

D. Crespo, J. Alonso, and E. Bernabéu, “Experimental measurements of generalized grating images,” Appl. Opt. OT 41, 1223–1228 (2002).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabéu, “Generalized imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231–1240 (2000).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
[CrossRef]

Angel Toro, L.

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Las. Tech. 31, 127–134 (1999).
[CrossRef]

Avudainayagam, K. V.

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175–178 (1997).
[CrossRef]

Bernabéu, E.

D. Crespo, J. Alonso, and E. Bernabéu, “Experimental measurements of generalized grating images,” Appl. Opt. OT 41, 1223–1228 (2002).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabéu, “Generalized imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231–1240 (2000).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
[CrossRef]

Bolognini, N.

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Las. Tech. 31, 127–134 (1999).
[CrossRef]

Brenner, K.H.

K.H. Brenner, A.W. Lohmann, and Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Chitralekha, S.

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175–178 (1997).
[CrossRef]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

Crespo, D.

D. Crespo, J. Alonso, and E. Bernabéu, “Experimental measurements of generalized grating images,” Appl. Opt. OT 41, 1223–1228 (2002).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabéu, “Generalized imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231–1240 (2000).
[CrossRef]

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Jahns, J.

J. Jahns and A.W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Kafri, O.

Keren, E.

Lau, E.

E. Lau. “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1985).

Leith, E. N.

Liu, De’an

Liu, L.

Liu, X.

Lohmann, A.W.

K.H. Brenner, A.W. Lohmann, and Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

J. Jahns and A.W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Luan, Z.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

Morlanes, T.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
[CrossRef]

Ojeda-Castaneda,

K.H. Brenner, A.W. Lohmann, and Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Olszak, A.

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

Patorski, K.

K. Patorski “The self-imaging phenomenon and its applications,” in Progress in Optics,E. Wolf, ed. (North Holland, Amsterdam1989)  vol. 27, pp. 3–108.

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringes explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Pomarico, J.

J. Pomarico and R. Torroba, “Colour image operations based on white light diffraction experiments (Lau effect)”, Eur. J. Phys. 14, 114–120 (1993).
[CrossRef]

Satpathi, A.

S.C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

Som, S.C.

S.C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

Sudol, R.

R. Sudol and B.J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Swanson, G. J.

Talbot, F.

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Tebaldi, M.

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Las. Tech. 31, 127–134 (1999).
[CrossRef]

Teng, S.

Thompson, B.J.

R. Sudol and B.J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Torroba, R.

J. Pomarico and R. Torroba, “Colour image operations based on white light diffraction experiments (Lau effect)”, Eur. J. Phys. 14, 114–120 (1993).
[CrossRef]

Tu, J.

J. Tu and L. Zhan, “Analysis of general double periodic structure diffraction phenomena based on the ambiguity function,” J. Opt. soc. Am. A 9, 983–995 (1992).
[CrossRef]

J. Tu and L. Zhan, “Two-dimensional theory of the Lau-Talbot-Moiré effect under partially coherent illumination,” Opt. Commun. 82, 229–235 (1991).
[CrossRef]

Wolf, E.

K. Patorski “The self-imaging phenomenon and its applications,” in Progress in Optics,E. Wolf, ed. (North Holland, Amsterdam1989)  vol. 27, pp. 3–108.

Wronkowski, L.

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

Ye, L.

Zhan, L.

J. Tu and L. Zhan, “Analysis of general double periodic structure diffraction phenomena based on the ambiguity function,” J. Opt. soc. Am. A 9, 983–995 (1992).
[CrossRef]

J. Tu and L. Zhan, “Two-dimensional theory of the Lau-Talbot-Moiré effect under partially coherent illumination,” Opt. Commun. 82, 229–235 (1991).
[CrossRef]

Zu, J.

Acta Opt. Sin. (1)

L. Liu, “Ambiguity function and general Talbot-Lau effects,” Acta Opt. Sin. 7, 501–510 (1987).

Ann. Phys. (Leipzig) (1)

E. Lau. “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1985).

Appl. Opt. (1)

Appl. Opt. OT (1)

D. Crespo, J. Alonso, and E. Bernabéu, “Experimental measurements of generalized grating images,” Appl. Opt. OT 41, 1223–1228 (2002).
[CrossRef]

Eur. J. Phys. (1)

J. Pomarico and R. Torroba, “Colour image operations based on white light diffraction experiments (Lau effect)”, Eur. J. Phys. 14, 114–120 (1993).
[CrossRef]

in Progress in Optics (1)

K. Patorski “The self-imaging phenomenon and its applications,” in Progress in Optics,E. Wolf, ed. (North Holland, Amsterdam1989)  vol. 27, pp. 3–108.

J. Mod. Opt. (3)

L. Liu, “Interferometry based on the partially coherent effect lying between the Talbot and Lau effects,” J. Mod. Opt. 35, 1605–1618 (1988).
[CrossRef]

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175–178 (1997).
[CrossRef]

S.C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringes explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Opt. Commun. (5)

J. Jahns and A.W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol and B.J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

K.H. Brenner, A.W. Lohmann, and Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

J. Tu and L. Zhan, “Two-dimensional theory of the Lau-Talbot-Moiré effect under partially coherent illumination,” Opt. Commun. 82, 229–235 (1991).
[CrossRef]

Opt. Eng. (2)

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
[CrossRef]

Opt. Las. Tech. (1)

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Las. Tech. 31, 127–134 (1999).
[CrossRef]

Philos. Mag. (1)

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

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

J. M. Cowley and A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

Optical system studied.

Fig. 2.
Fig. 2.

Colormap used thorough this work for all the color plots of C(Z 1, Z 2).

Fig. 3.
Fig. 3.

Pseudo-image contrast for z 0=250 µm and S=0. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 4.
Fig. 4.

Pseudo-image contrast for z 0→∞, S=0 and Δx=0. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 5.
Fig. 5.

Pseudo-image contrast for z 0=250 µm and S→∞. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 6.
Fig. 6.

Pseudo-image contrast for z 0→∞, S→∞ and S/z 0=0.1. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 7.
Fig. 7.

Pseudo-image indexing. z 0=250 µm, S=300µm, Δλ=0 and λ=400 nm.

Fig. 8.
Fig. 8.

(a) Contrast of the pseudo-image (1, 2) with monochromatic (blue) and polychromatic (red) illumination. Parameters: p 1=p 2=10 µm, z 0=250 µm, S=300 µm, λ0=400 nm. The spectral width of the polychromatic source is 50 nm. (b) The same for the pseudo-image (1, 3).

Fig. 9.
Fig. 9.

Pseudo-images obtained with periods p 1=30 µm and p 2=10 µm. (a): Monochromatic light, parameters z 0=250 µm, S=300 µm, λ0=400 nm. (b): Same parameters than (a) except for Δλ=400 nm.

Fig. 10.
Fig. 10.

Effect of the source size. p 1=p 2=10 µm, z 0=250 µm, λ0=400 nm and Δλ=60 nm. (a): S=0; (b): S=10 µm; (c): S=18 µm; (d): S=116 µm.

Fig. 11.
Fig. 11.

Effect of the source location. p 1=p 2=10 µm, S=300 µm, λ0=400 nm and Δλ=60 nm. (a): z 0=1 mm; (b): z 0=10 mm; (c): z 0=20 mm; (d): z 0=104 mm.

Fig. 12.
Fig. 12.

Graphical summary of the distribution of pseudo-imaging into different regimes, and its relation with the coherence of the incident field.

Equations (17)

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

t 1 ( x ) = n a n exp ( i 2 π nx p 1 ) and t 2 ( x ) = n b n exp ( i 2 π nx p 2 )
I λ ( x ) = nlmk A nlmk exp ( i B nlmk λ ) ,
A nlmk = a n a l * b m b k * exp { i 2 π x p 1 z t [ z 0 ( n l ) + Rz 01 ( m k ) ] }
× sinc { 2 π S p 1 z t [ z 12 ( n l ) + Rz 2 ( m k ) ] }
B nlmk = π p 1 2 z t [ z 0 z 12 ( n 2 l 2 ) + R 2 z 2 z 01 ( m 2 k 2 ) + 2 Rz 0 z 2 ( nm lk ) ] .
I ( x ) = + g ( λ ) I λ ( x ) d λ
= nlmk [ + g ( λ ) A nlmk ( λ ) exp ( iB nlmk λ ) d λ ] .
I ( x ) = nlmk [ A nlmk G ( B nlmk ) ] ,
g ( λ ) = 1 2 π ( Δ λ ) 2 exp [ ( λ λ 0 ) 2 2 ( Δ λ ) 2 ] .
I ( x ) = nlmk A nlmk exp [ B nlmk ( i λ 0 B nlmk ( Δ λ ) 2 2 ) ] .
C = max [ I ( x ) ] min [ I ( x ) ] max [ I ( x ) ] + min [ I ( x ) ] .
S z T > n p 2 2 z 2 and S z T > m p 1 2 ( z 1 + z 2 ) .
z 2 = ( m n p 1 p 2 1 ) 1 z 1 ,
( Z 1 ) nm min = 1 Rnm ( k + 1 ) ,
( Z 1 ) nm max = 1 2 Rnm ( 2 k + 1 ) , k = 0 , 1 , 2 ,
( Z 1 ) nm min = 1 2 Rnm ( 2 k + 1 ) ,
( Z 1 ) nm max = 1 Rnm k , k = 0 , 1 , 2 ,

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