Abstract

A new class of partially coherent pulses of Schell type with cosine-Gaussian temporal degree of coherence is introduced. Such waves are termed the Cosine-Gaussian Schell-model (CGSM) pulses. The analytic expression for the temporal mutual coherence function of the CGSM pulse in dispersive media is derived and used to study the evolution of its intensity distribution and its temporal degree of coherence. Further, the numerical calculations are performed in order to show the dependence of the intensity profile and the temporal degree of coherence of the CGSM pulse on the incident pulse duration, the initial temporal coherence length, the order-parameter n and the dispersion of the medium. The most important feature of the novel pulsed wave is its ability to split into two pulses on passage in a dispersive medium at some critical propagation distance. Such critical distance and the subsequent evolution of the split pulses are shown to depend on the source parameters and on the properties of the medium in which the pulse travels.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
    [CrossRef]
  4. F. Gori, M. Santarsiero, R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
    [CrossRef] [PubMed]
  5. S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [CrossRef] [PubMed]
  6. O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [CrossRef] [PubMed]
  7. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [CrossRef] [PubMed]
  8. Z. R. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [CrossRef] [PubMed]
  9. Z. R. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [CrossRef] [PubMed]
  10. Z. R. Mei, E. Shchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
    [CrossRef] [PubMed]
  11. H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [CrossRef] [PubMed]
  12. Z. S. Tong, O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [CrossRef] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  14. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
    [CrossRef]
  15. Q. Lin, L. G. Wang, S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
    [CrossRef]
  16. L. G. Wang, Q. Lin, H. Chen, S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
    [CrossRef] [PubMed]
  17. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
    [CrossRef] [PubMed]
  18. H. Lajunen, J. Tervo, P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21(11), 2117–2123 (2004).
    [CrossRef] [PubMed]
  19. H. Lajunen, P. Vahimaa, J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
    [CrossRef] [PubMed]
  20. J. Lancis, V. Torres-Company, E. Silvestre, P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005).
    [CrossRef] [PubMed]
  21. V. Torres-Company, G. Mínguez-Vega, J. Lancis, A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007).
    [CrossRef] [PubMed]
  22. V. Torres-Company, H. Lajunen, J. Lancis, A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
    [CrossRef]
  23. C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
    [CrossRef] [PubMed]
  24. L. Mokhtarpour, S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett. 37(17), 3498–3500 (2012).
    [CrossRef] [PubMed]
  25. H. Lajunen, T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013).
    [CrossRef] [PubMed]
  26. C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
    [CrossRef]
  27. F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [CrossRef] [PubMed]
  28. S. P. Dijaili, A. Dienes, J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
    [CrossRef]

2014 (1)

2013 (5)

2012 (4)

2011 (2)

2008 (2)

V. Torres-Company, H. Lajunen, J. Lancis, A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[CrossRef] [PubMed]

2007 (2)

2005 (2)

2004 (1)

2003 (3)

Q. Lin, L. G. Wang, S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[CrossRef]

L. G. Wang, Q. Lin, H. Chen, S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
[CrossRef] [PubMed]

2002 (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[CrossRef]

1996 (1)

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[CrossRef]

1990 (1)

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[CrossRef]

Andrés, P.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[CrossRef] [PubMed]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[CrossRef]

Cai, Y. J.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[CrossRef]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[CrossRef] [PubMed]

Chen, H.

L. G. Wang, Q. Lin, H. Chen, S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[CrossRef]

Dienes, A.

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[CrossRef]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[CrossRef]

Ding, C. L.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[CrossRef]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[CrossRef] [PubMed]

Friberg, A. T.

V. Torres-Company, H. Lajunen, J. Lancis, A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[CrossRef]

V. Torres-Company, G. Mínguez-Vega, J. Lancis, A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[CrossRef]

Gori, F.

Korotkova, O.

Lajunen, H.

Lancis, J.

Lin, Q.

L. G. Wang, Q. Lin, H. Chen, S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Q. Lin, L. G. Wang, S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[CrossRef]

Mei, Z. R.

Mínguez-Vega, G.

Mokhtarpour, L.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[CrossRef]

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[CrossRef]

Pan, L. Z.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[CrossRef]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[CrossRef] [PubMed]

Ponomarenko, S. A.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Silvestre, E.

Smith, J. S.

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[CrossRef]

Tervo, J.

Tong, Z. S.

Torres-Company, V.

Turunen, J.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[CrossRef]

Vahimaa, P.

Wang, H. X.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[CrossRef]

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Q. Lin, L. G. Wang, S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[CrossRef]

Wyrowski, F.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[CrossRef]

Zhang, Y. T.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[CrossRef]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[CrossRef] [PubMed]

Zhao, Z. G.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[CrossRef]

Zhu, S. Y.

Q. Lin, L. G. Wang, S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[CrossRef]

L. G. Wang, Q. Lin, H. Chen, S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[CrossRef]

Q. Lin, L. G. Wang, S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[CrossRef]

Opt. Express (3)

Opt. Lett. (12)

H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[CrossRef] [PubMed]

Z. S. Tong, O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

J. Lancis, V. Torres-Company, E. Silvestre, P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005).
[CrossRef] [PubMed]

V. Torres-Company, G. Mínguez-Vega, J. Lancis, A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[CrossRef] [PubMed]

Z. R. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Z. R. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[CrossRef] [PubMed]

L. Mokhtarpour, S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett. 37(17), 3498–3500 (2012).
[CrossRef] [PubMed]

Phys. Lett. A (1)

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[CrossRef]

Phys. Rev. A (1)

V. Torres-Company, H. Lajunen, J. Lancis, A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

L. G. Wang, Q. Lin, H. Chen, S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of pulse durations (a) T0 = 15ps, (b) T0 = 25ps, (c) T0 = 35ps and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are Tc = 10ps, n = 2, β2 = 20ps2km−1.

Fig. 2
Fig. 2

Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of temporal coherence lengths (a) Tc = 10ps, (b) Tc = 20ps, (c) Tc = 30ps and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, n = 2, β2 = 20ps2km−1.

Fig. 3
Fig. 3

Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of parameters (a) n = 2, (b) n = 4, (c) n = 6 and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, β2 = 20ps2km−1.

Fig. 4
Fig. 4

Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of second-order dispersion coefficients (a) β2 = 20ps2km−1, (b) β2 = 35ps2km−1, (c) β2 = 50ps2km−1 and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2.

Fig. 5
Fig. 5

The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td at some propagation distances z. The calculation parameters are (a) n = 0, (b) n = 2, (c) n = 4, and the other parameters are T0 = 15ps, Tc = 10ps and β2 = 20ps2km−1.

Fig. 6
Fig. 6

(a) The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td for different values of pulse durations T0 = 15ps, T0 = 25ps, T0 = 35ps, respectively, and Tc = 10ps. (b) The temporal degree of coherence γ (t1, t2, z) as a function of separation between two time points td for different values of temporal coherence lengths (b) Tc = 10ps, Tc = 20ps, Tc = 30ps, respectively, and T0 = 15ps. The other parameters are n = 2 and β2 = 20ps2km−1.

Fig. 7
Fig. 7

The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td for different values of the second-order dispersion coefficients β2 = 20ps2km−1, β2 = 35ps2km−1, β2 = 50ps2km−1, β2 = 123ps2km−1, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2.

Fig. 8
Fig. 8

The functions I+(t, z), I-(t, z) and I (t, z) versus the time t for different values of (a,d) z = 4km, (b,e) z = 2km and (c,f) z = 0, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2, β2 = 20ps2km−1.

Equations (16)

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

Γ( t 1 , t 2 )= p(v) H ( t 1 ,v)H( t 2 ,v)dv,
p(v)= T c 2π cosh( n 2π T c v )exp( T c 2 v 2 +2 n 2 π 2 ),
H(t,v)=exp( t 2 4 T 0 2 )exp( ivt ),
Γ( t 1 , t 2 )=exp( t 1 2 + t 2 2 4 T 0 2 )γ( t 1 , t 2 ),
γ( t 1 , t 2 )=cos[ n 2π ( t 2 t 1 ) T c ]exp[ ( t 2 t 1 ) 2 2 T c 2 ].
γ( t 1 , t 2 )=exp[ ( t 2 t 1 ) 2 2 T c 2 ].
Γ( t 1 , t 2 ,z )= 1 2π β 2 z Γ( t 10 , t 20 ) exp{ i 2 β 2 z [ ( t 10 2 t 20 2 )+( t 1 2 t 2 2 )2( t 10 t 1 - t 20 t 2 ) ] }d t 10 d t 20 ,
Γ( t 1 , t 2 ,z )= T 0 2 2 β 2 zΔ(z) exp[ ( t 1 t 2 ) 2 R(z) + i 2 β 2 z ( t 1 2 t 2 2 ) ][ exp( η + 2 Δ 2 (z) )+exp( η 2 Δ 2 (z) ) ],
1 R(z) = T 0 2 2 β 2 2 z 2 ,
Δ 2 (z)= 1 R(z) + 1 2 ξ 2 ,
1 ξ 2 = 1 4 T 0 2 + 1 T c 2 ,
η ± = 1 R(z) ( t 1 t 2 ) i 4 β 2 z ( t 1 + t 2 )± in 2π 2 T c .
I( t,z )=Γ( t,t,z ),
γ( t 1 , t 2 ,z )= Γ( t 1 , t 2 ,z ) Γ( t 1 , t 1 ,z ) Γ( t 2 , t 2 ,z ) .
I ( t , z ) = T 0 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 { e x p [ - 1 2 ( t n 2 π T c β 2 z ) 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 ] + e x p [ - 1 2 ( t + n 2 π T c β 2 z ) 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 ] } ,
I ± ( t , z ) = T 0 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 e x p [ - 1 2 ( t ± n 2 π T c β 2 z ) 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 ] ,

Metrics