Abstract

By adopting a new tensor method, we derived an analytical propagation formula for the cross-spectral density of partially coherent twisted anisotropic Gaussian Schell-model (GSM) beams through dispersive and absorbing media. Using the derived formula, we studied the evolution properties and spectrum properties of twisted anisotropic GSM beams in dispersive and absorbing media. The results show that the dispersive and absorbing media have strong influences on the propagation properties of twisted anisotropic GSM beams and their spectrum evolution. Our method provides a simple and convenient way to study the propagation of twisted anisotropic GSM beams in media with complex refractive index.

© 2002 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  6. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [CrossRef]
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    [CrossRef]
  9. G. Ding, X. Yuan, B. Lu, “Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems,” J. Mod. Opt. 47, 1483–1499 (2000).
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    [CrossRef]
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    [CrossRef]
  17. C. Palma, P. D. Santis, C. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
    [CrossRef]
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    [CrossRef]
  23. Q. Lin, L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
    [CrossRef]
  24. E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
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    [CrossRef]
  27. Z. Dacic, E. Wolf, “Changes in the spectrum of partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
    [CrossRef]
  28. F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
    [CrossRef]
  29. M. Santarsiero, F. Gori, “Spectral changes in a Young interference pattern,” Phys. Lett. A 167, 123–128 (1992).
    [CrossRef]
  30. H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
    [CrossRef]
  31. H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
    [CrossRef]
  32. F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

2002 (1)

2001 (1)

G. Ding, X. Yuan, B. Lu, “Propagation of twisted Gauss-ian Schell-model beams through a misaligned first-order optical system,” J. Mod. Opt. 48, 1617–1621 (2001).
[CrossRef]

2000 (3)

G. Ding, X. Yuan, B. Lu, “Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems,” J. Mod. Opt. 47, 1483–1499 (2000).

R. Simon, N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[CrossRef]

Q. Lin, L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[CrossRef]

1999 (1)

1998 (2)

1997 (1)

1995 (4)

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E 52, 5532–5539 (1995).
[CrossRef]

C. Palma, P. D. Santis, C. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

S. B. Cavalcanti, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef] [PubMed]

1994 (2)

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental decomposition of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

1993 (1)

1992 (2)

W. Wang, E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
[CrossRef]

M. Santarsiero, F. Gori, “Spectral changes in a Young interference pattern,” Phys. Lett. A 167, 123–128 (1992).
[CrossRef]

1991 (1)

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

1990 (3)

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized gaussian beams,” Opt. Commun. 80, 350–352 (1990).
[CrossRef]

G. P. Agrawal, “Wolf effect in homogeneous and inhomogeneous media,” J. Opt. Soc. Am. A 7, 2184–2192 (1990).
[CrossRef]

1989 (1)

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

1988 (1)

1987 (2)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987).
[CrossRef]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

1986 (1)

E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

1985 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1983 (1)

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

1982 (1)

1970 (1)

1965 (1)

Agrawal, G. P.

Alda, J.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized gaussian beams,” Opt. Commun. 80, 350–352 (1990).
[CrossRef]

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bernabeu, E.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized gaussian beams,” Opt. Commun. 80, 350–352 (1990).
[CrossRef]

Cai, Y.

Cavalcanti, S. B.

S. B. Cavalcanti, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef] [PubMed]

Cincotti, C.

C. Palma, P. D. Santis, C. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

Collins, S. A.

Dacic, Z.

Ding, G.

G. Ding, X. Yuan, B. Lu, “Propagation of twisted Gauss-ian Schell-model beams through a misaligned first-order optical system,” J. Mod. Opt. 48, 1617–1621 (2001).
[CrossRef]

G. Ding, X. Yuan, B. Lu, “Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems,” J. Mod. Opt. 47, 1483–1499 (2000).

Faklis, D.

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

Friberg, A. T.

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

M. Santarsiero, F. Gori, “Spectral changes in a Young interference pattern,” Phys. Lett. A 167, 123–128 (1992).
[CrossRef]

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Guattari, G.

C. Palma, P. D. Santis, C. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Hardy, A.

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Kandpal, H. C.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Kogelnik, H.

Lin, Q.

Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
[CrossRef]

Q. Lin, L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Lu, B.

G. Ding, X. Yuan, B. Lu, “Propagation of twisted Gauss-ian Schell-model beams through a misaligned first-order optical system,” J. Mod. Opt. 48, 1617–1621 (2001).
[CrossRef]

G. Ding, X. Yuan, B. Lu, “Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems,” J. Mod. Opt. 47, 1483–1499 (2000).

Marcopoli, G. L.

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

Mehta, D. S.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

Morris, G. M.

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

Mukunda, N.

Nayyar, V. P.

Nazarathy, M.

Palma, C.

C. Palma, P. D. Santis, C. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

M. Santarsiero, F. Gori, “Spectral changes in a Young interference pattern,” Phys. Lett. A 167, 123–128 (1992).
[CrossRef]

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

Santis, P. D.

C. Palma, P. D. Santis, C. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

Saxena, K.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

Shamir, J.

Simon, R.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Tervonen, E.

Turunen, J.

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Wang, L.

Q. Lin, L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[CrossRef]

Wang, S.

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized gaussian beams,” Opt. Commun. 80, 350–352 (1990).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Wang, W.

W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E 52, 5532–5539 (1995).
[CrossRef]

W. Wang, E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
[CrossRef]

Wolf, E.

W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E 52, 5532–5539 (1995).
[CrossRef]

W. Wang, E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
[CrossRef]

Z. Dacic, E. Wolf, “Changes in the spectrum of partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

Yuan, X.

G. Ding, X. Yuan, B. Lu, “Propagation of twisted Gauss-ian Schell-model beams through a misaligned first-order optical system,” J. Mod. Opt. 48, 1617–1621 (2001).
[CrossRef]

G. Ding, X. Yuan, B. Lu, “Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems,” J. Mod. Opt. 47, 1483–1499 (2000).

Appl. Opt. (1)

J. Mod. Opt. (6)

W. Wang, E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
[CrossRef]

C. Palma, P. D. Santis, C. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

G. Ding, X. Yuan, B. Lu, “Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems,” J. Mod. Opt. 47, 1483–1499 (2000).

G. Ding, X. Yuan, B. Lu, “Propagation of twisted Gauss-ian Schell-model beams through a misaligned first-order optical system,” J. Mod. Opt. 48, 1617–1621 (2001).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

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

Nature (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987).
[CrossRef]

Opt. Acta (1)

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Opt. Commun. (5)

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized gaussian beams,” Opt. Commun. 80, 350–352 (1990).
[CrossRef]

Q. Lin, L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Phys. Lett. A (1)

M. Santarsiero, F. Gori, “Spectral changes in a Young interference pattern,” Phys. Lett. A 167, 123–128 (1992).
[CrossRef]

Phys. Rev. A (2)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

S. B. Cavalcanti, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E 52, 5532–5539 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Evolution of the transverse spot width matrix element σI112 of the twisted anisotropic GSM beams (a) in free space, (b) in dispersive and absorbing media with ωr=4.00001×1015 s-1, (c) in dispersive and absorbing media with ωr=4.0×1015 s-1.

Fig. 2
Fig. 2

Evolution of the transverse spot width matrix element σI112 of the twisted anisotropic GSM beams with different twist factor μ in media. (a) μ=0, (b) μ=0.00005 (mm)-1, (c) μ=0.00008 (mm)-1.

Fig. 3
Fig. 3

Evolution of the transverse coherence width matrix element σg112 of the twisted anisotropic GSM beams (a) in free space, (b) in dispersive and absorbing media with ωr=4.00001×1015 s-1, (c) in dispersive and absorbing media with ωr=4.0×1015 s-1.

Fig. 4
Fig. 4

Evolution of the transverse coherence width matrix element σg112 of the twisted anisotropic GSM beams with different twist factor μ in media. (a) μ=0, (b) μ=0.00005 (mm)-1, (c) μ=0.00008 (mm)-1.

Fig. 5
Fig. 5

Evolution of the modulus of the coherence degree D of the twisted anisotropic GSM beams (a) in free space, (b) in dispersive and absorbing media with ωr=4.00001×1015 s-1, (c) in dispersive and absorbing media with ωr=4.0×1015 s-1.

Fig. 6
Fig. 6

Evolution of the modulus of the coherence degree D of the twisted anisotropic GSM beams with different twist factor μ in media. (a) μ=0, (b) μ=0.00005 (mm)-1, (c) μ=0.00008 (mm)-1.

Fig. 7
Fig. 7

Evolution of the twist factor of the twisted anisotropic GSM beams (a) in free space, (b) in dispersive and absorbing media.

Fig. 8
Fig. 8

Evolution of the twist factor of the twisted anisotropic GSM beams with different transverse coherence width matrix element σg112 in media. (a) σg112=0.1 (mm)2 (b) σg112=0.2 (mm)2, (c) σg112=0.3 (mm)2.

Fig. 9
Fig. 9

Normalized on-axis spectrum S(ω) in the plane of (a) z=0, (b) z=5000 mm, (c) z=50,000 mm.

Fig. 10
Fig. 10

On-axis relative central frequency shift of twisted anisotropic GSM beams along the propagation axis z (a) in free space, (b) in dispersive and absorbing media.

Equations (30)

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E(ρ1)=-ik2πzE(r1)exp(-ikl)dr1,
l=l0+12z(x12+y12-2x1ρ1x-2y1ρ1y+ρ1x2+ρ1y2),
W(r)=E(r1)E*(r2),W(ρ)=E(ρ1)E*(ρ2),
W(ρ)=exp(ikl0-ik*l0)π2[det(B˜)1/2]W(r)exp(-L)dr,
L=rTB˜-1r-2rTB˜-1ρ+ρTB˜-1ρ
B˜=2ikz00002ikz0000-2ik*z0000-2ik*z
W(r1, r2)=S0(ω)exp-14[r1T(σI2)-1r1+r2T(σI2)-1r2]-12(r1-r2)T(σg2)-1(r1-r2)-iω2c(r1-r2)T(R-1+μJ)(r1+r2),
(σI2)-1=σI11-2σI12-2σI12-2σI22-2,(σg2)-1=σg11-2σg12-2σg12-2σg22-2,
R-1=R11-1R12-1R21-1R22-1.
J=01-10.
W(r1, r2, ω, 0)=S0(ω)exp(-rTM˜-1r),
M˜1-1=M˜11-1M˜12-1(M˜12-1)T(M˜11-1)*=iω2cR-1+14(σI2)-1+12(σg2)-1-12(σg2)-1+iω2cμJ-12(σg2)-1+iω2cμJT-iω2R-1+14(σI2)-1+12(σg2)-1.
μJ=ciω[M˜12-1-(M˜12-1)T],
(σg2)-1=-[M˜12-1+(M˜12-1)T],
R-1=ciω[M˜11-1-(M˜11-1)*],
(σI2)-1=2[M˜11-1+(M˜11-1)*+M˜12-1+(M˜1-1)T].
W(ρ)=S0(ω)exp(ikl0-ik*l0)π2[det(B˜)]1/2×exp{-ρT[B˜-1-B˜-1T(B˜-1+M˜1-1)-1B˜-1]ρ}× exp[-|(B˜-1+M˜1-1)1/2r-(B˜-1+M˜1-1)-1/2B˜-1ρ|2]dr=S0(ω)exp(ikl0-ik*l0)[det(B˜)]1/2[det(B˜-1+M˜1-1)]1/2×exp{-ρT[B˜-1-B˜-1T(I+B˜M˜1-1)-1]ρ}=S0(ω)exp(ikl0-ik*l0)[det(I+B˜M˜1-1)]1/2×exp[-ρT(B˜-1-B˜-1T+M˜1-1)×(I+B˜M˜1-1)-1ρ]=S0(ω)exp(ikl0-ik*l0)[det(I+B˜M˜1-)]1/2exp(-ρTM˜2-1ρ),
M˜2-1=(M˜1+B˜)-1.
n(ω)=1+b2ωr2-ω2-iΓrω1/2,
b2=1025 s-2,ωr=4.0×1015 s-1,Γr=108 s-1.
(σI2)-1=10.10.10.5 (mm)-2,
(σg2)-1=10113.3 (mm)-2,
R-1=0000 (mm)-1,
μ=0.00001 (mm)-1,ω=3.99999×1015 s-1.
D=W(r1, r2, ω, z)[W(r1, r1, ω, z)W(r2, r2, ω, z)]1/2.
S(ρ1, ω)=W(ρ1=ρ2, ω).
S0(ω)=S0δ2(ω-ω0)2+δ2,
μ=0.00001(mm)-1,(σI2)-1=10.10.10.5(mm)-2,
(σg2)-1=10113.3(mm)-2,R-1=0000(mm)-1.
Δω/ω0=(ωm-ω0)/ω0.

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