Abstract

The nonlinear Talbot effect, in which self-images are formed by generated parametric light from the nonlinear optical process, is presented in this paper. The comparison is made between the conventional Talbot effect and the newly observed nonlinear case. The essence of such nonlinear self-images is provided and future work on this effect is also discussed. The conceptional extension achieved here not only opens a door for broader scopes of applications in imaging techniques but also offers a way for other research fields such as visualizing various ferric domains and subwavelength lithography.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  22. T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  28. H. Jin and P. Xu, Department of Physics, National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China, email pingxu520@nju.edu.cn. (personal communication, 2010).
  29. M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
    [CrossRef]

2010

Y. Zhang, J. -M. Wen, S. -N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104, 183901 (2010).
[CrossRef] [PubMed]

2009

A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051–1129 (2009).
[CrossRef]

K. -H. Luo, J. -M. Wen, X. -H. Chen, Q. Liu, M. Xiao, and L. -A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

2007

2006

2005

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

2002

H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
[CrossRef]

M. Fiebig, Th. Lottemoser, D. Fröhlich, A. V. Goltsev, and R. V. Pisarev, “Observation of coupled magnetic and electric domains,” Nature (London) 419, 818–820 (2002).
[CrossRef]

2001

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

1999

M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

1996

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349–5460 (1996).
[CrossRef] [PubMed]

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

C. Leichtle, I. S. Averbukh, and W. P. Schleich, “Generic structure of multilevel quantum beats,” Phys. Rev. Lett. 77, 3999–4002 (1996).
[CrossRef] [PubMed]

1995

T. B. Pittman, Y. -H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

1988

M. V. Berry and J. Goldberg, “Renormalisation of curlicues,” Nonlinearity 1, 1–26 (1988).
[CrossRef]

1979

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

1972

1967

1965

1963

G. L. Rogers, “Calculations of intermediate Fourier images of a finite line grating on a digital computer,” Br. J. Appl. Phys. 14, 657–661 (1963).
[CrossRef]

1957

J. M. Cowley and A. F. Moodie, “Fourier images,” Proc. Phys. Soc. London Sect. B 70, 486–513 (1957).
[CrossRef]

1913

M. Wolfke, “Uber die Abbildung eines Gitters Auβerhald der Einstellebene,” Ann. Phys. (Leipzig) 345, 194–200 (1913).
[CrossRef]

1910

H. Weisel, “Uber die nach Fresnelscher Art Beobachteten Beugungserscheninungen der Gitter,” Ann. Phys. (Leipzig) 338, 995–1031 (1910).
[CrossRef]

1908

A. Winkelmann, “Uber einige Erscheinungen, die bei der Beugung des Lichtes durch Gitter Auftreten,” Ann. Phys. (Leipzig) 332, 905–954 (1908).
[CrossRef]

1881

L. Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205(1881).

1836

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Averbukh, I. S.

C. Leichtle, I. S. Averbukh, and W. P. Schleich, “Generic structure of multilevel quantum beats,” Phys. Rev. Lett. 77, 3999–4002 (1996).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

M. V. Berry and J. Goldberg, “Renormalisation of curlicues,” Nonlinearity 1, 1–26 (1988).
[CrossRef]

Bienert, M.

H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
[CrossRef]

Bodenschatz, E.

M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

Chen, X. -H.

K. -H. Luo, J. -M. Wen, X. -H. Chen, Q. Liu, M. Xiao, and L. -A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Christodoulides, D. N.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

Coppola, G.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images,” Proc. Phys. Soc. London Sect. B 70, 486–513 (1957).
[CrossRef]

Cronin, A. D.

A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051–1129 (2009).
[CrossRef]

David, C.

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

De Natale, P.

De Nicola, S.

Diaz, A.

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

Eason, R. W.

Ferraro, P.

Fiebig, M.

M. Fiebig, Th. Lottemoser, D. Fröhlich, A. V. Goltsev, and R. V. Pisarev, “Observation of coupled magnetic and electric domains,” Nature (London) 419, 818–820 (2002).
[CrossRef]

Freyberger, M.

H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
[CrossRef]

Fröhlich, D.

M. Fiebig, Th. Lottemoser, D. Fröhlich, A. V. Goltsev, and R. V. Pisarev, “Observation of coupled magnetic and electric domains,” Nature (London) 419, 818–820 (2002).
[CrossRef]

Gioffrè, M.

Goldberg, J.

M. V. Berry and J. Goldberg, “Renormalisation of curlicues,” Nonlinearity 1, 1–26 (1988).
[CrossRef]

Goltsev, A. V.

M. Fiebig, Th. Lottemoser, D. Fröhlich, A. V. Goltsev, and R. V. Pisarev, “Observation of coupled magnetic and electric domains,” Nature (London) 419, 818–820 (2002).
[CrossRef]

Guo, C. -S.

Haug, F.

H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
[CrossRef]

Hong, Z. -P.

Iodice, M.

Iwanow, R.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

Jin, H.

H. Jin and P. Xu, Department of Physics, National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China, email pingxu520@nju.edu.cn. (personal communication, 2010).

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Leichtle, C.

C. Leichtle, I. S. Averbukh, and W. P. Schleich, “Generic structure of multilevel quantum beats,” Phys. Rev. Lett. 77, 3999–4002 (1996).
[CrossRef] [PubMed]

Liu, Q.

K. -H. Luo, J. -M. Wen, X. -H. Chen, Q. Liu, M. Xiao, and L. -A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Lottemoser, Th.

M. Fiebig, Th. Lottemoser, D. Fröhlich, A. V. Goltsev, and R. V. Pisarev, “Observation of coupled magnetic and electric domains,” Nature (London) 419, 818–820 (2002).
[CrossRef]

Luo, K. -H.

K. -H. Luo, J. -M. Wen, X. -H. Chen, Q. Liu, M. Xiao, and L. -A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Mack, H.

H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
[CrossRef]

Mailis, S.

Marzoli, I.

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

May-Arrioja, D. A.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

Min, Y.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

Montgomery, W. D.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier images,” Proc. Phys. Soc. London Sect. B 70, 486–513 (1957).
[CrossRef]

Nöhammer, B.

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

Patorski, K.

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1–108.
[CrossRef]

Paturzo, M.

Pisarev, R. V.

M. Fiebig, Th. Lottemoser, D. Fröhlich, A. V. Goltsev, and R. V. Pisarev, “Observation of coupled magnetic and electric domains,” Nature (London) 419, 818–820 (2002).
[CrossRef]

Pittman, T. B.

T. B. Pittman, Y. -H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Pritchard, D. E.

A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051–1129 (2009).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205(1881).

Rogers, G. L.

G. L. Rogers, “Interesting paradox in Fourier images,” J. Opt. Soc. Am. 62, 917–918 (1972).
[CrossRef]

G. L. Rogers, “Calculations of intermediate Fourier images of a finite line grating on a digital computer,” Br. J. Appl. Phys. 14, 657–661 (1963).
[CrossRef]

Rubin, M. H.

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349–5460 (1996).
[CrossRef] [PubMed]

Schleich, W. P.

H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
[CrossRef]

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

C. Leichtle, I. S. Averbukh, and W. P. Schleich, “Generic structure of multilevel quantum beats,” Phys. Rev. Lett. 77, 3999–4002 (1996).
[CrossRef] [PubMed]

Schmiedmayer, J.

A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051–1129 (2009).
[CrossRef]

Sergienko, A. V.

T. B. Pittman, Y. -H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Shih, Y. -H.

T. B. Pittman, Y. -H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Sohler, W.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

Stegeman, G. I.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

Strekalov, D. V.

T. B. Pittman, Y. -H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Szapiel, S.

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Weisel, H.

H. Weisel, “Uber die nach Fresnelscher Art Beobachteten Beugungserscheninungen der Gitter,” Ann. Phys. (Leipzig) 338, 995–1031 (1910).
[CrossRef]

Weitkamp, T.

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

Wen, J. -M.

Y. Zhang, J. -M. Wen, S. -N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104, 183901 (2010).
[CrossRef] [PubMed]

K. -H. Luo, J. -M. Wen, X. -H. Chen, Q. Liu, M. Xiao, and L. -A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Winkelmann, A.

A. Winkelmann, “Uber einige Erscheinungen, die bei der Beugung des Lichtes durch Gitter Auftreten,” Ann. Phys. (Leipzig) 332, 905–954 (1908).
[CrossRef]

Winthrop, J. T.

Wolfke, M.

M. Wolfke, “Uber die Abbildung eines Gitters Auβerhald der Einstellebene,” Ann. Phys. (Leipzig) 345, 194–200 (1913).
[CrossRef]

Worthington, C. R.

Wu, L. -A.

K. -H. Luo, J. -M. Wen, X. -H. Chen, Q. Liu, M. Xiao, and L. -A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Xiao, M.

Y. Zhang, J. -M. Wen, S. -N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104, 183901 (2010).
[CrossRef] [PubMed]

K. -H. Luo, J. -M. Wen, X. -H. Chen, Q. Liu, M. Xiao, and L. -A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Xu, P.

H. Jin and P. Xu, Department of Physics, National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China, email pingxu520@nju.edu.cn. (personal communication, 2010).

Yin, X.

Zhang, Y.

Y. Zhang, J. -M. Wen, S. -N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104, 183901 (2010).
[CrossRef] [PubMed]

Zhu, L. -W.

Zhu, S. -N.

Y. Zhang, J. -M. Wen, S. -N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104, 183901 (2010).
[CrossRef] [PubMed]

Ann. Phys. (Leipzig)

A. Winkelmann, “Uber einige Erscheinungen, die bei der Beugung des Lichtes durch Gitter Auftreten,” Ann. Phys. (Leipzig) 332, 905–954 (1908).
[CrossRef]

H. Weisel, “Uber die nach Fresnelscher Art Beobachteten Beugungserscheninungen der Gitter,” Ann. Phys. (Leipzig) 338, 995–1031 (1910).
[CrossRef]

M. Wolfke, “Uber die Abbildung eines Gitters Auβerhald der Einstellebene,” Ann. Phys. (Leipzig) 345, 194–200 (1913).
[CrossRef]

Appl. Phys. Lett.

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

Br. J. Appl. Phys.

G. L. Rogers, “Calculations of intermediate Fourier images of a finite line grating on a digital computer,” Br. J. Appl. Phys. 14, 657–661 (1963).
[CrossRef]

J. Mod. Opt.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

J. Opt. Soc. Am.

Nature (London)

M. Fiebig, Th. Lottemoser, D. Fröhlich, A. V. Goltsev, and R. V. Pisarev, “Observation of coupled magnetic and electric domains,” Nature (London) 419, 818–820 (2002).
[CrossRef]

Nonlinearity

M. V. Berry and J. Goldberg, “Renormalisation of curlicues,” Nonlinearity 1, 1–26 (1988).
[CrossRef]

Opt. Acta

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

Opt. Lett.

Philos. Mag.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

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Figures (1)

Fig. 1
Fig. 1

Schematic of the second-harmonic self-imaging from a PPLT nonlinear crystal. The incident fundamental pump laser induces the SH field generation. The CCD camera is used to record the SH self-images behind the crystal.

Equations (16)

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A ( R ) = e i k ( d 1 + d 2 ) i λ d 1 d 2 d r s S ( r s ) d r T ( r ) e i k | r r s | 2 2 d 1 e i k | R r | 2 2 d 2 ,
A G ( X , Y , z ) d x d y e x 2 + y 2 w 2 T ( x , y ) e i k s ( z + X 2 + Y 2 2 z x X + y Y z + x 2 + y 2 2 z ) ,
T ( x ) = n = + b n e 2 π i n x / a ,
d t e α t 2 + i β t = π α e β 2 4 α ,
A G ( X , Y , z ) A 0 e k s 2 ( X 2 + Y 2 ) 4 α z 2 n = + b n e π 2 n 2 α a 2 e π n k s X α a z ,
α = 1 w 2 i k s 2 z .
A G ( X , Y , z ) n = + b n e n 2 λ s 2 w 0 2 z 2 a 2 w z 4 e 2 n λ s w 0 2 z X a w z 4 e i π λ s n 2 w 0 2 z a 2 w z 2 e i 2 π n w 0 2 X a w z 2 .
e i π λ s n 2 w 0 2 z a 2 w z 2
z = 2 m a 2 λ s ( w z w 0 ) 2 ,
M G = ( w z w 0 ) 2 .
e i ( k s c 1 2 x 2 + c 2 x ) ,
A G ( X , Y , z ) n = + b n e ( π n a + c 2 2 ) 2 4 z 2 k s 2 w z 2 e ( π n a + c 2 2 ) 4 z X k s w z 2 e i ( c 1 + 1 z ) ( π n a + c 2 2 ) 2 λ s w 0 2 z 2 π w z 2 e i ( π n a + c 2 2 ) ( c 1 + 1 z ) 2 w 0 2 z X w z 2 ,
2 m a 2 λ s = z ( c 1 z + 1 ) ( w 0 w z ) 2 .
M G = ( c 1 z + 1 ) ( w z w 0 ) 2 .
0 > c 1 λ s 8 m M a 2
c 2 = 2 π J a ,

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