Abstract

Whenever the natural modes of the modal expansion of the cross-spectral density have a common waist, the wave equation in the waist plane has the form of a two-dimensional Schrödinger equation. Thus the results of quantum mechanics and quantum statistics, including the quantized Schrödinger field, can be transferred to partially coherent light. Such conceptions as temperature, entropy, and energy are used advantageously. A subclass of radiation, radiation in thermal equilibrium, is introduced, and, as examples, the Gaussian Schell-model beam and the quasi-rectangle model beam are investigated. The M2 factor is strongly related to the mean value of energy.

© 1994 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  2. See, for instance, S. Flügge, “Problem 42. Circular oscillator,” in Practical Quantum Mechanics (Springer-Verlag, New York, 1974), pp. 107–110.
  3. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  4. S. J. van Enk, “Geometric phase transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
    [CrossRef]
  5. M. Nazarethy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  6. K. Wittig, A. Giesen, H. Hügel, “An algebraic approach to characterize paraxial optical systems,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez- Herrero, A. Gonzales-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 185–196.
  7. G. Nienhuis, L. Allen, “Paraxial optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
    [CrossRef] [PubMed]
  8. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  9. R. Gase, “Dichteoperator-Formalismus für MultimodeLaserstrahlung,” Verh. Dtsch. Phys. Ges. 3, 441 (1993).
  10. R. Gase, “Density operator formalism for partially coherent beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (SEDO, Madrid, 1993), pp. 89–98.
  11. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal- mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993);K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A10, 2017–2023 (1993).
    [CrossRef]
  12. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  13. A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo- Opt. Instrum. Eng.1224, 2–14 (1990).
    [CrossRef]
  14. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  15. R. Gase, “The Wigner distribution function applied to laser radiation,” Opt. Quantum Electron. 24, 1109–1118 (1992).
    [CrossRef]
  16. See, for instance, M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986), Chaps. 2 and 5 and Appendix A.
  17. F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  18. E. Wolf, “New theory of partial coherence in the space- frequency domain. I. Spectra and cross spectra of steady state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  19. R. Barakat, Ch. Brosseau, “Von Neumann entropy of N interacting pencils of radiation,” J. Opt. Soc. Am. A 10, 529–532 (1993).
    [CrossRef]
  20. M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 65–87.
  21. R. P. Feynman, Statistical Mechanics (Addison-Wesley, Redwood City, Calif., 1988).
  22. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1983).
    [CrossRef]
  23. B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
    [CrossRef]
  24. P. De Santis, F. Gori, M. Santarsiero, M. Guattari, “Sources with spatially sinusoidal modes,” Opt. Commun. 82, 123–129 (1991).
    [CrossRef]
  25. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  26. R. L. Phillips, L. C. Anrews, “Spot size and divergence for Laguerre–Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983).
    [CrossRef] [PubMed]
  27. See, for instance, G. Heber, G. Weber, Fundamentals of Modern Quantum Physics, Part II, Quantum Field Theory (Teubner, Leipzig, 1957) (in German).
  28. See, for instance, S. Goldman, Information Theory (Prentice- Hall, New York, 1955).
  29. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distribution in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
    [CrossRef] [PubMed]

1993

S. J. van Enk, “Geometric phase transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

G. Nienhuis, L. Allen, “Paraxial optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

R. Gase, “Dichteoperator-Formalismus für MultimodeLaserstrahlung,” Verh. Dtsch. Phys. Ges. 3, 441 (1993).

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

R. Barakat, Ch. Brosseau, “Von Neumann entropy of N interacting pencils of radiation,” J. Opt. Soc. Am. A 10, 529–532 (1993).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal- mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993);K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A10, 2017–2023 (1993).
[CrossRef]

1992

R. Gase, “The Wigner distribution function applied to laser radiation,” Opt. Quantum Electron. 24, 1109–1118 (1992).
[CrossRef]

1991

P. De Santis, F. Gori, M. Santarsiero, M. Guattari, “Sources with spatially sinusoidal modes,” Opt. Commun. 82, 123–129 (1991).
[CrossRef]

1989

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1988

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

1987

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distribution in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

1984

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983

1982

1980

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1976

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Allen, L.

G. Nienhuis, L. Allen, “Paraxial optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

Anrews, L. C.

Barakat, R.

Bastiaans, M. J.

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 65–87.

Brosseau, Ch.

Cai, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

De Santis, P.

P. De Santis, F. Gori, M. Santarsiero, M. Guattari, “Sources with spatially sinusoidal modes,” Opt. Commun. 82, 123–129 (1991).
[CrossRef]

Feynman, R. P.

R. P. Feynman, Statistical Mechanics (Addison-Wesley, Redwood City, Calif., 1988).

Flügge, S.

See, for instance, S. Flügge, “Problem 42. Circular oscillator,” in Practical Quantum Mechanics (Springer-Verlag, New York, 1974), pp. 107–110.

Gase, R.

R. Gase, “Dichteoperator-Formalismus für MultimodeLaserstrahlung,” Verh. Dtsch. Phys. Ges. 3, 441 (1993).

R. Gase, “The Wigner distribution function applied to laser radiation,” Opt. Quantum Electron. 24, 1109–1118 (1992).
[CrossRef]

R. Gase, “Density operator formalism for partially coherent beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (SEDO, Madrid, 1993), pp. 89–98.

Giesen, A.

K. Wittig, A. Giesen, H. Hügel, “An algebraic approach to characterize paraxial optical systems,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez- Herrero, A. Gonzales-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 185–196.

Goldman, S.

See, for instance, S. Goldman, Information Theory (Prentice- Hall, New York, 1955).

Gori, F.

P. De Santis, F. Gori, M. Santarsiero, M. Guattari, “Sources with spatially sinusoidal modes,” Opt. Commun. 82, 123–129 (1991).
[CrossRef]

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Guattari, M.

P. De Santis, F. Gori, M. Santarsiero, M. Guattari, “Sources with spatially sinusoidal modes,” Opt. Commun. 82, 123–129 (1991).
[CrossRef]

Heber, G.

See, for instance, G. Heber, G. Weber, Fundamentals of Modern Quantum Physics, Part II, Quantum Field Theory (Teubner, Leipzig, 1957) (in German).

Hügel, H.

K. Wittig, A. Giesen, H. Hügel, “An algebraic approach to characterize paraxial optical systems,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez- Herrero, A. Gonzales-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 185–196.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lü, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Mandel, L.

Mukunda, N.

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal- mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993);K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A10, 2017–2023 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distribution in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Nazarethy, M.

Nienhuis, G.

G. Nienhuis, L. Allen, “Paraxial optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

Phillips, R. L.

Santarsiero, M.

P. De Santis, F. Gori, M. Santarsiero, M. Guattari, “Sources with spatially sinusoidal modes,” Opt. Commun. 82, 123–129 (1991).
[CrossRef]

Schubert, M.

See, for instance, M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986), Chaps. 2 and 5 and Appendix A.

Shamir, J.

Siegman, A.

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo- Opt. Instrum. Eng.1224, 2–14 (1990).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Simon, R.

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal- mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993);K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A10, 2017–2023 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distribution in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Starikov, A.

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distribution in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sundar, K.

van Enk, S. J.

S. J. van Enk, “Geometric phase transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

Weber, G.

See, for instance, G. Heber, G. Weber, Fundamentals of Modern Quantum Physics, Part II, Quantum Field Theory (Teubner, Leipzig, 1957) (in German).

Wilhelmi, B.

See, for instance, M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986), Chaps. 2 and 5 and Appendix A.

Wittig, K.

K. Wittig, A. Giesen, H. Hügel, “An algebraic approach to characterize paraxial optical systems,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez- Herrero, A. Gonzales-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 185–196.

Wolf, E.

Yang, C.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Zhang, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

S. J. van Enk, “Geometric phase transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

P. De Santis, F. Gori, M. Santarsiero, M. Guattari, “Sources with spatially sinusoidal modes,” Opt. Commun. 82, 123–129 (1991).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Opt. Quantum Electron.

R. Gase, “The Wigner distribution function applied to laser radiation,” Opt. Quantum Electron. 24, 1109–1118 (1992).
[CrossRef]

Optik (Stuttgart)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Phys. Rev. A

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distribution in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

G. Nienhuis, L. Allen, “Paraxial optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Verh. Dtsch. Phys. Ges.

R. Gase, “Dichteoperator-Formalismus für MultimodeLaserstrahlung,” Verh. Dtsch. Phys. Ges. 3, 441 (1993).

Other

R. Gase, “Density operator formalism for partially coherent beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (SEDO, Madrid, 1993), pp. 89–98.

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo- Opt. Instrum. Eng.1224, 2–14 (1990).
[CrossRef]

See, for instance, S. Flügge, “Problem 42. Circular oscillator,” in Practical Quantum Mechanics (Springer-Verlag, New York, 1974), pp. 107–110.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

K. Wittig, A. Giesen, H. Hügel, “An algebraic approach to characterize paraxial optical systems,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez- Herrero, A. Gonzales-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 185–196.

M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 65–87.

R. P. Feynman, Statistical Mechanics (Addison-Wesley, Redwood City, Calif., 1988).

See, for instance, G. Heber, G. Weber, Fundamentals of Modern Quantum Physics, Part II, Quantum Field Theory (Teubner, Leipzig, 1957) (in German).

See, for instance, S. Goldman, Information Theory (Prentice- Hall, New York, 1955).

See, for instance, M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986), Chaps. 2 and 5 and Appendix A.

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Figures (3)

Fig. 1
Fig. 1

Intensity distribution function of the QRM beam for T/E1 = 0, 3, 10, and 50 (from the innermost to the outermost curve). The spatial coordinate x is in units of 2a.

Fig. 2
Fig. 2

Radiant intensity function of the QRM beam for T/E1 = 0, 3, 10, and 50 (from the innermost to the outermost curve). The angular coordinate u is in units of 1/ka.

Fig. 3
Fig. 3

M2 and M E 2factors for the QRM beam as a function of temperature T.

Equations (96)

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

r = ( x , y ) .
Δ ψ ( r , z ) 2 i k z ψ ( r , z ) = 0
Δ = 2 x 2 + 2 y 2
[ 1 4 k 2 Δ + V ( r ) E ] ψ ( r , z ) + [ i 2 k z V ( r ) + E ] ψ ( r , z ) = 0 ,
ψ ( r , z ) = φ ( r ) [ 1 + f 1 ( r ) z + f 2 ( r ) z 2 + ] .
d ψ ( r , t ) = z 2 i k Δ ψ ( r , z = 0 ) ,
f 1 ( r ) = 1 2 i k Δ φ ( r ) φ ( r ) .
[ 1 4 k 2 Δ + V ( r ) E ] φ ( r ) + [ i 2 k f 1 ( r ) V ( r ) + E ] φ ( r ) = 0 ,
[ 1 4 k 2 Δ + V ( r ) E ] φ ( r ) = 0
[ i 2 k f 1 ( r ) V ( r ) + E ] φ ( r ) = 0
H = 1 4 k 2 Δ + V ( r ) ,
f 1 ( r ) = 2 i k [ V ( r ) E ] .
z | ψ ( r , t ) | z = 0 2 = | φ ( r ) | 2 Re [ f 1 ( r ) ] ,
φ ( r ) = A ( r ) exp [ i χ ( r ) ] ,
1 2 A ( r ) Δ χ ( r ) + x A ( r ) x χ ( r ) + y A ( r ) y χ ( r ) = 0 .
φ ( r ) = A exp ( i k u 0 r ) .
V ( r ) E = 1 4 k 2 Δ φ ( r ) φ ( r )
φ 0 ( r ) = A exp ( r 2 / w 0 2 ) ,
V ( r ) E 0 = r 2 k 2 w 0 4 1 k 2 w 0 2 .
ψ ( r , z ) = A exp ( r 2 / w 0 2 ) [ 1 2 i k ( r 2 / k 2 w 0 4 1 / k 2 w 0 2 ) z ] for z 0 .
ρ ̂ = i p i | φ ( i ) φ ( i ) | .
0 p i 1 , i p i = 1 .
ρ n n = n | ρ ̂ | n .
ρ n n = p n δ n n .
ρ ( r 1 , r 2 ) = r 1 | ρ ̂ | r 2 .
ρ ( r 1 , r 2 ) = n , n r 1 | n n | ρ ̂ | n n | r 2 .
φ n ( r ) = r | n ,
ρ ( r 1 , r 2 ) = n φ n * ( r 1 ) p n φ n ( r 2 ) .
W ( r 1 , r 2 ) = φ * ( r 1 ) φ ( r 2 ) av .
W ( r 1 , r 2 ) = n Φ n * ( r 1 ) λ n Φ n ( r 2 ) .
d 2 r 1 W ( r 1 , r 2 ) Φ n ( r 1 ) = λ n Φ n ( r 2 ) ,
λ n 0 .
W ( r 1 , r 2 ) = ρ ( r 1 , r 2 ) .
Φ n ( r ) = φ n ( r ) ,
S = n p n ln p n
Tr ( ρ ̂ Ĥ ) = n p n E n ,
p n = exp ( β E n ) / n exp ( β E n ) .
β = 1 / T ,
φ ( x , y ) n x , n y = C n x , n y H n x ( x 2 / w 0 ) H n y ( y 2 / w 0 ) × exp ( r 2 / w 0 2 ) ,
φ ( r , ϕ ) p , l = C p , l ( r 2 / w 0 ) l × exp ( i l ϕ ) L p l ( 2 r 2 / w 0 2 ) exp ( r 2 / w 0 2 ) ,
E n x , n y = ( n x + n y + 1 ) / ( k w 0 ) 2 ,
E p , l = ( 2 p + l + 1 ) / ( k w 0 ) 2 .
p n x , n y = exp [ ( n x + n y + 1 ) E 0 , 0 / T ] / n x , n y exp [ ( n x + n y + 1 ) E 0 , 0 / T ] ,
p n x , n y = p n x p n y ,
p n x = exp [ ( n x + 1 / 2 ) E 0 , 0 / T ] / n x exp [ ( n x + 1 / 2 ) E 0 , 0 / T ] ,
W ( r 1 , r 2 ) = W 0 exp ( r 1 2 + r 2 2 w ¯ 0 2 ) exp [ ( r 1 r 2 ) 2 2 σ 2 ] ,
w ¯ 0 2 = w 0 2 coth ( E 0 , 0 / 2 T ) ,
2 σ 2 = w 0 2 sinh ( E 0 , 0 / T ) .
M 2 = δ r δ u / ( δ r δ u ) 0 .
M 2 = ( 1 + w ¯ 2 / σ 2 ) 1 / 2 .
M 2 = coth ( E 0 , 0 / 2 T ) ,
exp [ ( E 0 , 0 / T ) ] = ( M 2 + 1 ) / ( M 2 1 ) .
λ n x / λ 0 = [ ( M 2 1 ) / ( M 2 + 1 ) ] n x .
p n x / p 0 = [ exp ( E 0 , 0 / T ) ] n x .
V ( x ) = { 0 for 0 < x < 2 a elsewhere ,
φ n ( x ) = { C n sin ( n π 2 a x ) for 0 < x < 2 a 0 elsewhere
E n = E 1 n 2 ,
E 1 = π 2 / ( 4 k a ) 2 .
p n = [ exp ( E 1 / T ) ] n 2 / n [ exp ( E 1 / T ) ] n 2 ,
W ( x 1 , x 2 ) = n p n sin ( n π 2 a x 1 ) sin ( n π 2 a x 2 ) .
I ( x ) = n p n sin 2 ( n π 2 a x )
J ( u ) = n p n [ sinc ( kua n π / 2 ) ( 1 ) n sinc ( kua + n π / 2 ) ] 2 ,
Tr ( ρ ̂ Ĥ ) = n x , n y p n x , n y E n x , n y ,
Tr ( ρ ̂ Ĥ ) = E 0 , 0 coth ( E 0 , 0 / 2 T ) .
M 2 = Tr ( ρ ̂ Ĥ ) / E 0 , 0 .
M 2 = x 2 u x 2 / ( x 2 u x 2 ) 0 ,
x 2 = n x , n y p n x n y d x d y I n x , n y ( x , y ) x 2 ,
( M 2 ) 2 = n x , n x p n x p n x x 2 n x u x 2 n x ( x 2 u x 2 ) 0 .
Tr ( ρ ̂ Ĥ ) = 2 n x p n x E n x .
[ Tr ( ρ ̂ Ĥ ) / E 0 , 0 ] 2 = 4 n x , n x p n x p n x E n x E n x / E 0 , 0 2 .
x 2 n x u x 2 n x ( x 2 u x 2 ) 0 = 4 E n x E n x / E 0 , 0 2 .
M x ; n x , n y 2 = 2 E n x / E 0 , 0 .
M 2 = n x p n x M x ; n x , n y 2 .
M p , l 2 = E p , l / E 0 , 0 .
M E 2 ̂ = Ĥ / E 0 , 0
M E 2 = Tr ( ρ ̂ M E 2 ̂ ) .
M E 2 = n p n n 2 .
d 2 r φ * ( r ) φ ( r ) = 1
d 2 r φ ̂ + ( r ) φ ̂ ( r ) = N ̂ ,
H ̂ = d 2 r [ 1 4 k 2 φ ̂ ( r ) φ ̂ ( r ) + V ( r ) φ ̂ + ( r ) φ ̂ ( r ) ] .
φ ̂ + ( r ) = n â n + φ n * ( r ) ,
φ ̂ ( r ) = n â n φ n ( r ) .
[ â n + , â n ] = δ n n ,
[ â n + , â n + ] = [ â n , â n ] = 0 ,
N ̂ = n N ̂ n ,
N ̂ | N ( N 1 , , N n , ) = N | N ( N 1 , , N n ) .
H ̂ = n N ̂ n E n
φ ̂ + ( r 1 ) φ ̂ ( r 2 ) q = Tr [ ρ ̂ φ ̂ + ( r 1 ) φ ̂ ( r 2 ) ] ,
ρ ̂ = ( 1 / N ) | N ( N 1 , , N n , ) N ( N 1 , , N n , ) | ,
φ + ( r 1 ) φ ( r 2 ) q = ( 1 / N ) N ( N 1 , , N n , ) | φ ̂ + ( r 1 ) φ ̂ ( r 2 ) | × | N ( N 1 , , N n , ) ,
φ ̂ + ( r 1 ) φ ̂ ( r 2 ) q = n φ n * ( r ) ( N n / N ) φ n ( r ) .
Ŵ ( r 1 , r 2 ) = φ ̂ + ( r 1 ) φ ̂ ( r 2 ) ,
W ( r 1 , r 2 ) = Tr [ ρ ̂ Ŵ ( r 1 , r 2 ) ] .
Tr [ ρ ̂ φ ̂ ( r ) ] = 0 .
ρ ̂ = n p n | a n a n | ,
Tr [ ρ ̂ Ŵ ( r 1 , r 2 ) ] = n φ n * ( r 1 ) p n φ n ( r 2 ) ,

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