Abstract

Expressions are derived for the cross-spectral density matrix of an electromagnetic Gaussian Schell-model beam propagating through a paraxial ABCD system. Using the recently developed unified theory of coherence and polarization of electromagnetic beams and the ABCD matrix for gradient-index fibers, we study the changes of the spectral density, of the spectral degree of polarization, and of the spectral degree of coherence of such a beam as it travels through the fiber. Effects of material dispersion are also considered.

© 2006 Optical Society of America

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References

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  1. D. F. V. James, "Change of polarization of light beam on propagation through free space," J. Opt. Soc. Am. A 11, 1641-1643 (1994).
    [CrossRef]
  2. S. R. Seshadri, "Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams," J. Appl. Phys. 87, 4084-4087 (2000).
    [CrossRef]
  3. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
    [CrossRef]
  4. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  5. H. Roychowdhury, S. Ponomarenko, and E. Wolf, "Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
    [CrossRef]
  6. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beam propagating through the turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
    [CrossRef]
  7. T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
    [CrossRef]
  8. G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, "Propagation of partially coherent beam through selfoc fibers," Opt. Commun. 12, 333-337 (1974).
    [CrossRef]
  9. G. P. Agrawal, "Application of angular spectrum to optical coherence and nonlinear optics," Ph.D. thesis, (Indian Institute of Technology, Delhi, 1974).
  10. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982), p. 267.
  11. A. Gamliel and G. P. Agrawal, "Wolf effect in homogenous and inhomogenous media," J. Opt. Soc. Am. A 7, 2184-2192 (1990).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science Books, 1986), Chap. 20, Sec. 20.1. Additional references are mentioned at the end of that chapter.
  13. B. Lü and Liuzhan Pan, "Propagation of vector Gaussian-Schell-model beams through a paraxial optical ABCD system," Opt. Commun. 205, 7-16 (2002).
    [CrossRef]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press), 1995.
  15. A. T. Friberg and J. Turunen, "Imaging of Gaussian Schell-model sources," J. Opt. Soc. Am. A 5, 713-720 (1988).
    [CrossRef]
  16. A. T. Friberg and R. J. Sudol, "The spatial coherence properties of Gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
    [CrossRef]
  17. E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996)
    [CrossRef]
  18. G. P. Agrawal and E. Wolf, "Propagation-induced polarization changes in partially coherent optical beams," J. Opt. Soc. Am. A 17, 2019-2023 (2000). The vector GSM source introduced in this paper had the additional restrictions σx=σy,Ix=Iy, and δxx=δyy.
    [CrossRef]
  19. H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 259, 379-385 (2005).
    [CrossRef]
  20. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
    [CrossRef]
  21. T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
    [CrossRef]
  22. O. Korotkova, M. Salem, and E. Wolf, "Beam-conditions for radiation generated by an electromagnetic Gaussian Schell-model sources," Opt. Lett. 29, 1173-1175 (2004).
    [CrossRef] [PubMed]
  23. The validity of the paraxial approximation is justified in the case under consideration [see T. Okoshi, Optical Fibers (Academic Press, New York, 1982), Sec. 5.2.2].
  24. A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), Chap. 7, Sec. 7.9. See also Chap. 9.

2005

H. Roychowdhury, S. Ponomarenko, and E. Wolf, "Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 259, 379-385 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

2004

2003

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

2002

B. Lü and Liuzhan Pan, "Propagation of vector Gaussian-Schell-model beams through a paraxial optical ABCD system," Opt. Commun. 205, 7-16 (2002).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

2001

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

2000

1996

E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996)
[CrossRef]

1994

1990

1988

1983

A. T. Friberg and R. J. Sudol, "The spatial coherence properties of Gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

1974

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, "Propagation of partially coherent beam through selfoc fibers," Opt. Commun. 12, 333-337 (1974).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and E. Wolf, "Propagation-induced polarization changes in partially coherent optical beams," J. Opt. Soc. Am. A 17, 2019-2023 (2000). The vector GSM source introduced in this paper had the additional restrictions σx=σy,Ix=Iy, and δxx=δyy.
[CrossRef]

A. Gamliel and G. P. Agrawal, "Wolf effect in homogenous and inhomogenous media," J. Opt. Soc. Am. A 7, 2184-2192 (1990).
[CrossRef]

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, "Propagation of partially coherent beam through selfoc fibers," Opt. Commun. 12, 333-337 (1974).
[CrossRef]

G. P. Agrawal, "Application of angular spectrum to optical coherence and nonlinear optics," Ph.D. thesis, (Indian Institute of Technology, Delhi, 1974).

Borghi, R.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beam propagating through the turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Friberg, A. T.

A. T. Friberg and J. Turunen, "Imaging of Gaussian Schell-model sources," J. Opt. Soc. Am. A 5, 713-720 (1988).
[CrossRef]

A. T. Friberg and R. J. Sudol, "The spatial coherence properties of Gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

Gamliel, A.

Ghatak, A.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), Chap. 7, Sec. 7.9. See also Chap. 9.

Ghatak, A. K.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, "Propagation of partially coherent beam through selfoc fibers," Opt. Commun. 12, 333-337 (1974).
[CrossRef]

Gori, F.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

James, D. F.

E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996)
[CrossRef]

D. F. V. James, "Change of polarization of light beam on propagation through free space," J. Opt. Soc. Am. A 11, 1641-1643 (1994).
[CrossRef]

Korotkova, O.

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 259, 379-385 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "Beam-conditions for radiation generated by an electromagnetic Gaussian Schell-model sources," Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beam propagating through the turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Lü, B.

B. Lü and Liuzhan Pan, "Propagation of vector Gaussian-Schell-model beams through a paraxial optical ABCD system," Opt. Commun. 205, 7-16 (2002).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press), 1995.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982), p. 267.

Mehta, C. L.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, "Propagation of partially coherent beam through selfoc fibers," Opt. Commun. 12, 333-337 (1974).
[CrossRef]

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Okoshi, T.

The validity of the paraxial approximation is justified in the case under consideration [see T. Okoshi, Optical Fibers (Academic Press, New York, 1982), Sec. 5.2.2].

Pan, Liuzhan

B. Lü and Liuzhan Pan, "Propagation of vector Gaussian-Schell-model beams through a paraxial optical ABCD system," Opt. Commun. 205, 7-16 (2002).
[CrossRef]

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Ponomarenko, S.

H. Roychowdhury, S. Ponomarenko, and E. Wolf, "Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury, S. Ponomarenko, and E. Wolf, "Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 259, 379-385 (2005).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, "Beam-conditions for radiation generated by an electromagnetic Gaussian Schell-model sources," Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beam propagating through the turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Santarsiero, M.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Seshadri, S. R.

S. R. Seshadri, "Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams," J. Appl. Phys. 87, 4084-4087 (2000).
[CrossRef]

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 20, Sec. 20.1. Additional references are mentioned at the end of that chapter.

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, "The spatial coherence properties of Gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), Chap. 7, Sec. 7.9. See also Chap. 9.

Turunen, J.

Wolf, E.

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

H. Roychowdhury, S. Ponomarenko, and E. Wolf, "Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beam propagating through the turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "Beam-conditions for radiation generated by an electromagnetic Gaussian Schell-model sources," Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

G. P. Agrawal and E. Wolf, "Propagation-induced polarization changes in partially coherent optical beams," J. Opt. Soc. Am. A 17, 2019-2023 (2000). The vector GSM source introduced in this paper had the additional restrictions σx=σy,Ix=Iy, and δxx=δyy.
[CrossRef]

E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996)
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press), 1995.

J. Appl. Phys.

S. R. Seshadri, "Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams," J. Appl. Phys. 87, 4084-4087 (2000).
[CrossRef]

J. Mod. Opt.

H. Roychowdhury, S. Ponomarenko, and E. Wolf, "Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

J. Opt. A

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

A. T. Friberg and R. J. Sudol, "The spatial coherence properties of Gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

Opt. Commun.

B. Lü and Liuzhan Pan, "Propagation of vector Gaussian-Schell-model beams through a paraxial optical ABCD system," Opt. Commun. 205, 7-16 (2002).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 259, 379-385 (2005).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, "Propagation of partially coherent beam through selfoc fibers," Opt. Commun. 12, 333-337 (1974).
[CrossRef]

Opt. Lett.

Phys. Lett. A

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Rep. Prog. Phys.

E. Wolf and D. F. V. James, "Correlation-induced spectral changes," Rep. Prog. Phys. 59, 771-818 (1996)
[CrossRef]

Waves Random Media

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beam propagating through the turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Other

G. P. Agrawal, "Application of angular spectrum to optical coherence and nonlinear optics," Ph.D. thesis, (Indian Institute of Technology, Delhi, 1974).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982), p. 267.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 20, Sec. 20.1. Additional references are mentioned at the end of that chapter.

The validity of the paraxial approximation is justified in the case under consideration [see T. Okoshi, Optical Fibers (Academic Press, New York, 1982), Sec. 5.2.2].

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), Chap. 7, Sec. 7.9. See also Chap. 9.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press), 1995.

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Figures (9)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Normalized spectral shift Δ ω ω 0 of the field generated by a scalar Gaussian Schell-model source, as a function of the propagation distance z, within a GRIN with a parabolic refractive-index profile. The frequency shifts are calculated for points at a fixed off-axis distance 10 k 0 , for a source with σ = 20 k 0 , δ = 4 k 0 . The dashed curve shows the frequency shifts when the frequency dependence of α is ignored, taking α ( ω 0 ) = ( 4.8 ) 10 4 k 0 .

Fig. 3
Fig. 3

Normalized spectral shift Δ ω ω 0 for the field generated by an electromagnetic Gaussian Schell-model source as function of the propagation distance z within a dispersive graded-index fiber with a parabolic refractive-index profile. The frequency shifts are calculated for points at a fixed off-axis distance 10 k 0 from the fiber axis for a source with σ x = σ y = 50 k 0 , I x = I y = 0.5 . The correlation parameters for the source for the dashed curve are δ x x = 2 k 0 , δ y y = 4 k 0 , and those for the solid curve are taken to be δ x x = 2 k 0 , δ y y = 18 k 0 .

Fig. 4
Fig. 4

The region around the second zero crossing of the normalized spectral shift of Fig. 2 shown in greater detail.

Fig. 5
Fig. 5

Normalized spectrum of the field generated by an electromagnetic Gaussian Schell-model source for different values of the propagation distance within the fiber. The spectral density is calculated for points at a fixed off-axis distance 10 k 0 from the fiber axis with a source for which σ x = σ y = 50 k 0 , I x = I y = 0.5 . The correlation parameters for the source are taken to be δ x x = 2 k 0 , δ y y = 4 k 0 . The dashed curve shows the source spectrum ( z = 0 ) . Spectral density of the field at a propagation distance z = 1200 μ m within the fiber shows a red shift (A), and spectral density of the field at a propagation distance z = 1165 μ m within the fiber shows a blue shift (B).

Fig. 6
Fig. 6

Spectral degree of polarization of the field on-axis as a function of propagation distance z within a dispersive GRIN fiber with a parabolic refractive-index profile. The spectral and correlation parameters of the EGSM source are given by σ x = σ y = 50 k 0 , I x = I y = 0.5 , δ x x = 2 k 0 , δ y y = 2 k 0 , I x y = 0.1 . (A) δ x y = 3.0811 k 0 , (B) δ x y = 4.1623 k 0 , (C) δ x y = 5.2434 k 0 , (D) δ x y = 6.3246 k 0 .

Fig. 7
Fig. 7

Illustration of the notation.

Fig. 8
Fig. 8

Spectral degree of coherence as a function of propagation distance z for a pair of points located radially symmetrically at a distance 1 k 0 from the fiber axis. The different curves show the variation as the spectral density of the x component of the field is changed while that of the y component is kept fixed. The parameters of the source are taken to be σ x = σ y = 50 k 0 , I y = 1 , δ x x = 40 k 0 , δ y y = 30 k 0 , ω = ω 0 . (A) I x = 1 , (B) I x = 0.5 , (C) I x = 0.25 .

Fig. 9
Fig. 9

Spectral degree of coherence as a function of the propagation distance z for a pair of radially symmetric points at a distance 1 k 0 from the fiber axis. The different curves show the variation of the spectral degree of coherence as the parameter that characterizes the correlations of the x component of the field is changed, keeping that of the y component fixed. The source is specified by the following parameters: σ x = σ y = 50 k 0 , I x = 0.5 , I y = 1 , δ y y = 4 k 0 , ω = ω 0 . (A) δ x x = 3 k 0 , (B) δ x x = 18 k 0 , (C) δ x x = 25 k 0 .

Equations (51)

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

W ( r 1 , r 2 , ω ) [ W i j ( r 1 , r 2 , ω ) ] = [ E i * ( r 1 , ω ) E j ( r 2 , ω ) ] , ( i = x , y ; j = x , y ) ,
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = S i ( 0 ) ( ρ 1 , ω ) S j ( 0 ) ( ρ 2 , ω ) μ i j ( 0 ) ( ρ 2 ρ 1 , ω ) , ( i = x , y ; j = x , y ) .
S i ( 0 ) ( ρ , ω ) = A i 2 exp ( ρ 2 2 σ i 2 ) , ( i = x , y ) ,
μ i j ( 0 ) ( ρ 2 ρ 1 , ω ) = B i j exp ( ρ 2 ρ 1 2 2 δ i j 2 ) , ( i = x , y ; j = x , y ) .
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = A i A j B i j exp ( ρ 1 2 4 σ i 2 ρ 2 2 4 σ j 2 ) exp ( ρ 2 ρ 1 2 2 δ i j 2 ) , ( i = x , y ; j = x , y ) .
n ̃ 2 ( x , y ; ω ) = { n 0 2 ( ω ) [ 1 α 2 ( ω ) ( x 2 + y 2 ) ] , x 2 + y 2 R 0 2 n 0 2 ( ω ) [ 1 α 2 ( ω ) R 0 2 ] , x 2 + y 2 R 0 2 } .
α ( ω ) = 1 R 0 [ 1 n 2 2 ( ω ) n 1 2 ( ω ) ] 1 2 .
[ A B C D ] = [ cos ( α z ) sin ( α z ) n α n α sin ( α z ) cos ( α z ) ] .
E ( ρ , z ; ω ) = i k exp ( i k z ) 2 π z E ( 0 ) ( ρ ; ω ) exp [ i k 2 B ( D ρ 2 2 ρ ρ + A ρ 2 ) ] d 2 ρ .
W i j ( ρ 1 , ρ 2 , z ; ω ) = ( k 2 π B ) 2 d 2 ρ 1 d 2 ρ 2 W i j ( 0 ) ( ρ 1 , ρ 2 ; ω ) × exp [ i k 2 B ( A ( ρ 1 2 ρ 2 2 ) ) 2 ( ρ 1 ρ 1 ρ 2 ρ 2 ) + D ( ρ 1 2 ρ 2 2 ) ] .
S ( r ; ω ) Tr W ( r , r ; ω ) = W x x ( r , r ; ω ) + W y y ( r , r ; ω ) ,
η ( r 1 , r 2 , ω ) Tr W ( r 1 , r 2 , ω ) Tr W ( r 1 , r 1 , ω ) Tr W ( r 2 , r 2 , ω ) = W x x ( r 1 , r 2 , ω ) + W y y ( r 1 , r 2 , ω ) S ( r 1 , ω ) S ( r 2 , ω ) .
P ( r , ω ) = 1 4 Det W ( r , r , ω ) [ Tr W ( r , r , ω ) ] 2 ,
W i j ( ρ 1 , ρ 2 , z ; ω ) = ( k 2 π B ) 2 A i A j B i j exp [ i k 2 B ( D ) ( ρ 1 2 ρ 2 2 ) ] d 2 ρ 1 d 2 ρ 2 exp ( ρ 1 2 4 σ i 2 ρ 2 2 4 σ j 2 ) exp ( ρ 2 ρ 1 2 2 δ i j 2 ) exp [ i k 2 B ( A ( ρ 1 2 ρ 2 2 ) 2 ( ρ 1 ρ 1 ρ 2 ρ 2 ) ) ] .
ξ = ρ 1 + ρ 2 2 ; ξ = ρ 2 ρ 1 ;
ρ = ρ 1 + ρ 2 2 ; ρ = ρ 2 ρ 1 ,
W i j ( ρ , ρ , z ; ω ) = ( k 2 π B ) 2 A i A j B i j exp [ i k B ( D ρ ρ ) ] d 2 ξ exp [ ξ 2 ( 1 4 σ i 2 + 1 4 σ j 2 ) ] exp ( i k B ξ ρ ) d 2 ξ exp [ ξ 2 ( 1 16 σ i 2 + 1 16 σ j 2 + 1 2 δ i j 2 ) ] exp [ ξ ( ξ 4 σ i 2 ξ 4 σ j 2 + i k A B ξ i k B ρ ) ] .
W i j ( ρ , ρ , z ; ω ) = A i A j B i j 1 Δ i j exp [ 4 α i j ρ 2 Δ i j ] exp { [ i k D B ( β i j + i k A B ) 1 Δ i j ] ρ ρ } exp [ γ i j ρ 2 Δ i j ] ,
α i j = 1 16 ( 1 σ i 2 + 1 σ j 2 ) , β i j = 1 4 ( 1 σ i 2 1 σ j 2 ) , γ i j = α i j + 1 2 δ i j 2 ,
Δ i j = ( B k ) 2 [ 16 α i j γ i j ( β i j + i k A B ) 2 ] .
Δ i i = ( B k ) 2 [ 16 α i i γ i i + ( k A B ) 2 ] ,
W i j ( ρ , ρ , z ; ω ) = A i A j B i j 1 Δ i j exp [ 4 α i j ρ 2 Δ i j ] exp { [ i k cot ( α z ) n α ( β i j + i k cot ( α z ) n α ) 1 Δ i j ] ρ ρ } exp [ γ i j ρ 2 Δ i j ] ,
Δ i j = ( sin ( α z ) k n α ) 2 { 16 α i j γ i j [ β i j + i k n α cot ( α z ) ] 2 } ,
W i i ( ρ , ρ , z ; ω ) = A i 1 Δ i i exp ( 4 α i i ρ 2 Δ i i ) , ( i = x , y )
Δ i i = [ cos 2 ( α z ) + sin 2 ( α z ) ( 2 n k α σ i 2 ) 2 ( 1 + 4 σ i 2 δ i i 2 ) ] ,
α i i = 1 8 σ i 2 , ( i = x , y ) .
A i I i s ( 0 ) ( ω ) , ( i = x , y ) ,
s ( 0 ) ( ω ) = S ( 0 ) ( ρ , ω ) S ( 0 ) ( ρ , ω ) d ω .
s ( 0 ) ( ω ) = 1 1 + [ 4 ( ω ω 0 ) 2 ( δ ω ) 2 ] ,
n 2 ( ω ) = 1 + j = 1 3 B j ω j 2 ω j 2 ω 2 .
W x y ( ρ , ρ , z ; ω ) = A i A j B x y 1 Δ x y exp [ 4 α x y ρ 2 Δ x y ] ,
Δ x y = ( B k ) 2 [ 16 α x y γ x y ( β x y + i k A B ) 2 ]
α x y = 1 16 ( 1 σ x 2 + 1 σ y 2 ) , β x y = 1 4 ( 1 σ x 2 1 σ y 2 ) , γ x y = α x y + 1 2 δ x y 2 .
β i i = 0 , α x y = 1 8 σ 2 , γ x y = 1 8 σ 2 + 1 2 δ x y 2 ,
Δ x y = [ cos 2 ( α z ) + sin 2 ( α z ) ( 2 n k α σ 2 ) 2 ( 1 + 4 σ 2 δ x y 2 ) ] .
max { δ x x , δ y y } δ x y min { δ x x Q x y , δ y y Q x y } .
ρ = ρ 1 + ρ 2 2 = 0 and ρ = ρ 2 ρ 1 2 ρ 2 .
W i i ( ρ , ρ , z ; ω ) = A i 1 Δ i i exp ( 4 γ i i ρ 2 2 Δ i i ) ,
Δ i i = [ cos 2 ( α z ) + sin 2 ( α z ) ( 2 n k α σ i 2 ) 2 ( 1 + 4 σ i 2 δ i i 2 ) ]
γ i i = 1 8 σ i 2 + 1 2 δ i i 2 .
η ( ρ , ρ , z ; ω ) = A x 1 Δ x x exp ( 4 γ x x ρ 2 Δ x x ) + A y 1 Δ y y exp ( 4 γ y y ρ 2 Δ y y ) , A x 1 Δ x x exp ( α x x ρ 2 Δ x x ) + A y 1 Δ y y exp ( α y y ρ 2 Δ y y ) .
W i j ( ρ , ρ , z ; ω ) = ( k 2 π B ) 2 A i A j B i j exp ( i k B ( D ρ ρ ) ) d 2 ξ exp [ ξ 2 ( 1 4 σ i 2 + 1 4 σ j 2 ) ] exp ( i k B ξ ρ ) d 2 ξ exp [ ξ 2 ( 1 16 σ i 2 + 1 16 σ j 2 + 1 2 δ i j 2 ) ] exp [ ξ ( ξ 4 σ i 2 ξ 4 σ j 2 + i k A B ξ i k B ρ ) ] .
α i j = 1 16 ( 1 σ i 2 + 1 σ j 2 ) , β i j = 1 4 ( 1 σ i 2 1 σ j 2 ) , γ i j = α i j + 1 2 δ i j 2 ,
I ( ξ , ρ ) = d 2 ξ exp ( γ i j ξ 2 ) exp { ξ [ ( β i j + i k A B ) ξ i k B ρ ] } .
exp ( p 2 x 2 ± q x ) d x = π p exp ( q 2 4 p 2 ) ,
I ( ξ , ρ ) = π γ i j exp { [ ( β i j + i k A B ) ξ i k B ρ ] 2 4 γ i j } .
I ( ξ , ρ ) = π γ i j exp ( k 2 4 γ i j B 2 ρ 2 ) exp [ i k 2 γ i j B ( β i j + i k A B ) ξ ρ ] exp [ 1 4 γ i j ( β i j + i k A B ) 2 ξ 2 ] .
I ( ρ , ρ , z ; ω ) = d 2 ξ exp { [ 4 α i j 1 4 γ i j ( β i j + i k A B ) 2 ] ξ 2 } exp { [ i k B ρ + i k 2 γ i j B ( β i j + i k A B ) ρ ] ξ } ,
I ( ρ , ρ , z ; ω ) = π Δ i j ( 4 γ i j B 2 k 2 ) exp [ 1 4 Δ i j γ i j ( β i j + i k A B ) 2 ρ 2 ] exp ( γ i j Δ i j ρ 2 ) exp [ 1 Δ i j ( β i j + i k A B ) ρ ρ ] ,
Δ i j = ( B k ) 2 [ 16 α i j γ i j ( β i j + i k A B ) 2 ] .
W i j ( ρ , ρ , z ; ω ) = A i A j B i j 1 Δ i j exp ( 4 α i j ρ 2 Δ i j ) exp { [ i k D B ( β i j + i k A B ) 1 Δ i j ] ρ ρ } exp ( γ i j ρ 2 Δ i j ) .

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