Abstract

Light pulse group velocity manipulations due to the specific dispersion of a medium (so-called “slow” and “fast” light phenomena) can be obtained on the basis of several mechanisms. One of these techniques is two-wave mixing in a photorefractive crystal. This work presents a modification of this method, exploiting the strong beam fanning in Sb-doped Sn2P2S6 crystals. Our experimental results demonstrate a “fast light” behavior of Gaussian pulses transmitted through a Sn2P2S6:Sb sample. The phenomenon is due to the beam fanning (i.e., the self-diffraction of the incident beam on self-induced noisy photorefractive gratings) that ensures a significant depletion of the input beam. Due to the relatively fast photorefractive response of the Sn2P2S6:Sb crystals this depletion occurs with times in the range of 10–100 ms, depending on the beam intensity, and the “fast light” feature is observed. The temporal and amplitude characteristics of the output beam are measured in function of the intensity and polarization azimuth of the incident beam. Besides, a negative phase shift of the periodical output beam relative to a sinusoidal intensity-modulated input beam is also obtained experimentally. It is shown that the phase and amplitude relation between the input and output periodic signals are described by a simple analytical expression that takes into account the beam fanning strength (depletion factor) and its dynamics (depletion time constant). It is also demonstrated that the pulse advance (or phase shift of the modulated signal) can be regulated by the light polarization azimuth. The advantages of the proposed method are discussed.

© 2014 Optical Society of America

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2013 (3)

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

A. Volkov, A. Shumelyuk, S. Odoulov, and M. Imlau, “Polarization structure of beam fanning in low-symmetry photorefractive crystals,” J. Opt. Soc. Am. B 30, 1102–1108 (2013).
[CrossRef]

U. Bortolozzo, S. Residori, and J. C. Howell, “Precision Doppler measurements with steep dispersion,” Opt. Lett. 38, 3107–3110 (2013).
[CrossRef]

2012 (1)

2011 (3)

2010 (2)

A. Shumelyuk and S. Odoulov, “Light pulse manipulation in Sn2P2S6,” J. Opt. 12, 104015 (2010).
[CrossRef]

J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010).
[CrossRef]

2009 (2)

O. M. Shumelyuk, A. I. Hryhorashchuk, and S. G. Odoulov, “Optical forerunners in crystals with photorefractive dynamic gratings,” Ukr. J. Phys. 54, 33–37 (2009).

U. Bortolozzo, S. Residori, and J.-P. Huignard, “Slow-light through nonlinear wave-mixing in liquid crystal light-valves,” C. R. Phys. 10, 938–948 (2009).
[CrossRef]

2008 (1)

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

2006 (1)

F. Bo, G. Zhang, and J. Xu, “Ultraslow Gaussian pulse propagation induced by a dispersive phase coupling in photorefractive bismuth silicon oxide crystals at room temperature,” Opt. Commun. 261, 349–352 (2006).
[CrossRef]

2005 (1)

2004 (1)

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

2003 (5)

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light pulse slowing down up to 0.025  cm/s by photorefractive two-wave coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef]

2001 (2)

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

A. A. Grabar, M. I. Gurzan, I. V. Kedyk, I. M. Stoika, and Yu. M. Vysochanskii, “Optical properties and applications of photorefractive Sn2P2S6,” Ferroelectrics 257, 245–254 (2001).
[CrossRef]

2000 (1)

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

1997 (1)

M. J. Miller, G. L. Wood, and G. J. Salamo, “Photorefractive beam fanning optical limiter,” MRS Proc. 479, 193–198 (1997).

1995 (2)

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

H. Rehn, R. Kowarschik, and K. H. Ringhofer, “Beam-fanning novelty filter with enhanced dynamic phase resolution,” Appl. Opt. 34, 4907–4911 (1995).
[CrossRef]

1994 (1)

1993 (1)

S. Residori, P. L. Ramazza, and M. Zhao, “Dynamics of beam fanning in Cu-doped KNSBN,” Opt. Commun. 102, 100–104 (1993).
[CrossRef]

1982 (1)

J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. A 72, 46–51 (1982).
[CrossRef]

1970 (1)

R. Nitsche and P. Wild, “Crystal growth of metal-phosphorus-sulfur compounds by vapor transport,” Mater. Res. Bull. 5, 419–423 (1970).
[CrossRef]

Anderson, D. Z.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Antón, M. A.

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Bo, F.

F. Bo, G. Zhang, and J. Xu, “Ultraslow Gaussian pulse propagation induced by a dispersive phase coupling in photorefractive bismuth silicon oxide crystals at room temperature,” Opt. Commun. 261, 349–352 (2006).
[CrossRef]

F. Bo, G. Zhang, and J. Xu, “Transition between superluminal and subluminal light propagation in photorefractive Bi12SiO20 crystals,” Opt. Express 13, 8198–8203 (2005).
[CrossRef]

Bortolozzo, U.

U. Bortolozzo, S. Residori, and J. C. Howell, “Precision Doppler measurements with steep dispersion,” Opt. Lett. 38, 3107–3110 (2013).
[CrossRef]

U. Bortolozzo, S. Residori, and J.-P. Huignard, “Slow-light through nonlinear wave-mixing in liquid crystal light-valves,” C. R. Phys. 10, 938–948 (2009).
[CrossRef]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef]

R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics, E. Wolf, ed. (Elsevier2002), Vol. 43, pp. 497–530.

Buse, K.

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

Caimi, G.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

Calderón, O. G.

Carreño, F.

Czaia, L.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Dorfler, U.

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Egatz-Gómez, A.

Eyal, A.

Fedyo, K. V.

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

Feinberg, J.

J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. A 72, 46–51 (1982).
[CrossRef]

Fischer, B.

Fry, E. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Gadret, G.

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

K. Shcherbin, P. Mathey, and G. Gadret, “Slow light with photorefractive four-wave mixing,” Phys. Rev. A 84, 063802 (2011).
[CrossRef]

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics, E. Wolf, ed. (Elsevier2002), Vol. 43, pp. 497–530.

Goulkov, M. Y.

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

Grabar, A. A.

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

A. A. Grabar, M. I. Gurzan, I. V. Kedyk, I. M. Stoika, and Yu. M. Vysochanskii, “Optical properties and applications of photorefractive Sn2P2S6,” Ferroelectrics 257, 245–254 (2001).
[CrossRef]

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications, P. Günter and J.-P. Huignard, eds. (Springer, 2006), pp. 327–359.

Granzow, T.

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

Grunnet-Jepsen, A.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, 1996).

Gunter, P.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

Günter, P.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications, P. Günter and J.-P. Huignard, eds. (Springer, 2006), pp. 327–359.

Gurzan, M. I.

A. A. Grabar, M. I. Gurzan, I. V. Kedyk, I. M. Stoika, and Yu. M. Vysochanskii, “Optical properties and applications of photorefractive Sn2P2S6,” Ferroelectrics 257, 245–254 (2001).
[CrossRef]

Guyard, R.

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

Haertle, D.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

Haldi, A.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

Ham, B. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Hemmer, P. R.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Hollberg, L.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Horowitz, M.

Howell, J. C.

Hryhorashchuk, A. I.

O. M. Shumelyuk, A. I. Hryhorashchuk, and S. G. Odoulov, “Optical forerunners in crystals with photorefractive dynamic gratings,” Ukr. J. Phys. 54, 33–37 (2009).

Huignard, J.-P.

U. Bortolozzo, S. Residori, and J.-P. Huignard, “Slow-light through nonlinear wave-mixing in liquid crystal light-valves,” C. R. Phys. 10, 938–948 (2009).
[CrossRef]

Imlau, M.

A. Volkov, A. Shumelyuk, S. Odoulov, and M. Imlau, “Polarization structure of beam fanning in low-symmetry photorefractive crystals,” J. Opt. Soc. Am. B 30, 1102–1108 (2013).
[CrossRef]

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

Jauslin, H. R.

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

Jauslin, H.-R.

Jazbinsek, M.

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications, P. Günter and J.-P. Huignard, eds. (Springer, 2006), pp. 327–359.

Kash, M. M.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Kedyk, I. V.

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

A. A. Grabar, M. I. Gurzan, I. V. Kedyk, I. M. Stoika, and Yu. M. Vysochanskii, “Optical properties and applications of photorefractive Sn2P2S6,” Ferroelectrics 257, 245–254 (2001).
[CrossRef]

Khurgin, J. B.

Kowarschik, R.

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef]

Lukin, M. D.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Mathey, P.

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

K. Shcherbin, P. Mathey, and G. Gadret, “Slow light with photorefractive four-wave mixing,” Phys. Rev. A 84, 063802 (2011).
[CrossRef]

B. Sturman, P. Mathey, and H.-R. Jauslin, “Slowdown and speedup of light pulses using the self-compensating photorefractive response,” J. Opt. Soc. Am. B 28, 347–351 (2011).
[CrossRef]

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

Melle, S.

Miller, M. J.

M. J. Miller, G. L. Wood, and G. J. Salamo, “Photorefractive beam fanning optical limiter,” MRS Proc. 479, 193–198 (1997).

Montemezzani, G.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications, P. Günter and J.-P. Huignard, eds. (Springer, 2006), pp. 327–359.

Musser, J. A.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Nitsche, R.

R. Nitsche and P. Wild, “Crystal growth of metal-phosphorus-sulfur compounds by vapor transport,” Mater. Res. Bull. 5, 419–423 (1970).
[CrossRef]

Odoulov, S.

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

A. Volkov, A. Shumelyuk, S. Odoulov, and M. Imlau, “Polarization structure of beam fanning in low-symmetry photorefractive crystals,” J. Opt. Soc. Am. B 30, 1102–1108 (2013).
[CrossRef]

A. Shumelyuk and S. Odoulov, “Light pulse manipulation in Sn2P2S6,” J. Opt. 12, 104015 (2010).
[CrossRef]

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light pulse slowing down up to 0.025  cm/s by photorefractive two-wave coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef]

Odoulov, S. G.

O. M. Shumelyuk, A. I. Hryhorashchuk, and S. G. Odoulov, “Optical forerunners in crystals with photorefractive dynamic gratings,” Ukr. J. Phys. 54, 33–37 (2009).

Okamoto, A.

Pankrath, R.

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

Podivilov, E.

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light pulse slowing down up to 0.025  cm/s by photorefractive two-wave coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef]

Prits, I. P.

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

Ramazza, P. L.

S. Residori, P. L. Ramazza, and M. Zhao, “Dynamics of beam fanning in Cu-doped KNSBN,” Opt. Commun. 102, 100–104 (1993).
[CrossRef]

Rehn, H.

Residori, S.

U. Bortolozzo, S. Residori, and J. C. Howell, “Precision Doppler measurements with steep dispersion,” Opt. Lett. 38, 3107–3110 (2013).
[CrossRef]

U. Bortolozzo, S. Residori, and J.-P. Huignard, “Slow-light through nonlinear wave-mixing in liquid crystal light-valves,” C. R. Phys. 10, 938–948 (2009).
[CrossRef]

S. Residori, P. L. Ramazza, and M. Zhao, “Dynamics of beam fanning in Cu-doped KNSBN,” Opt. Commun. 102, 100–104 (1993).
[CrossRef]

Ringhofer, K. H.

Rostovtsev, Y.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Salamo, G. J.

M. J. Miller, G. L. Wood, and G. J. Salamo, “Photorefractive beam fanning optical limiter,” MRS Proc. 479, 193–198 (1997).

Sato, K.

Sautenkov, V. A.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Scully, M. O.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Shahriar, M. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Shcherbin, K.

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

K. Shcherbin, P. Mathey, and G. Gadret, “Slow light with photorefractive four-wave mixing,” Phys. Rev. A 84, 063802 (2011).
[CrossRef]

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

Shumelyuk, A.

A. Volkov, A. Shumelyuk, S. Odoulov, and M. Imlau, “Polarization structure of beam fanning in low-symmetry photorefractive crystals,” J. Opt. Soc. Am. B 30, 1102–1108 (2013).
[CrossRef]

A. Shumelyuk and S. Odoulov, “Light pulse manipulation in Sn2P2S6,” J. Opt. 12, 104015 (2010).
[CrossRef]

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light pulse slowing down up to 0.025  cm/s by photorefractive two-wave coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef]

Shumelyuk, A. N.

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications, P. Günter and J.-P. Huignard, eds. (Springer, 2006), pp. 327–359.

Shumelyuk, O. M.

O. M. Shumelyuk, A. I. Hryhorashchuk, and S. G. Odoulov, “Optical forerunners in crystals with photorefractive dynamic gratings,” Ukr. J. Phys. 54, 33–37 (2009).

Snowbell, M.

Solymar, L.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, 1996).

Stoika, I. M.

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

A. A. Grabar, M. I. Gurzan, I. V. Kedyk, I. M. Stoika, and Yu. M. Vysochanskii, “Optical properties and applications of photorefractive Sn2P2S6,” Ferroelectrics 257, 245–254 (2001).
[CrossRef]

Sturman, B.

B. Sturman, P. Mathey, and H.-R. Jauslin, “Slowdown and speedup of light pulses using the self-compensating photorefractive response,” J. Opt. Soc. Am. B 28, 347–351 (2011).
[CrossRef]

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light pulse slowing down up to 0.025  cm/s by photorefractive two-wave coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef]

Sudarshanam, V. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Takayama, Y.

Tur, M.

Turukhin, A. V.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Volkov, A.

Vysochanskii, Y. M.

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

Vysochanskii, Yu. M.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

A. A. Grabar, M. I. Gurzan, I. V. Kedyk, I. M. Stoika, and Yu. M. Vysochanskii, “Optical properties and applications of photorefractive Sn2P2S6,” Ferroelectrics 257, 245–254 (2001).
[CrossRef]

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications, P. Günter and J.-P. Huignard, eds. (Springer, 2006), pp. 327–359.

Webb, D. J.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, 1996).

Welch, G. R.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Wild, P.

R. Nitsche and P. Wild, “Crystal growth of metal-phosphorus-sulfur compounds by vapor transport,” Mater. Res. Bull. 5, 419–423 (1970).
[CrossRef]

Woike, T.

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

Wood, G. L.

M. J. Miller, G. L. Wood, and G. J. Salamo, “Photorefractive beam fanning optical limiter,” MRS Proc. 479, 193–198 (1997).

Xu, J.

F. Bo, G. Zhang, and J. Xu, “Ultraslow Gaussian pulse propagation induced by a dispersive phase coupling in photorefractive bismuth silicon oxide crystals at room temperature,” Opt. Commun. 261, 349–352 (2006).
[CrossRef]

F. Bo, G. Zhang, and J. Xu, “Transition between superluminal and subluminal light propagation in photorefractive Bi12SiO20 crystals,” Opt. Express 13, 8198–8203 (2005).
[CrossRef]

Yoshida, T.

Zadok, A.

Zgonik, M.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Zhang, G.

F. Bo, G. Zhang, and J. Xu, “Ultraslow Gaussian pulse propagation induced by a dispersive phase coupling in photorefractive bismuth silicon oxide crystals at room temperature,” Opt. Commun. 261, 349–352 (2006).
[CrossRef]

F. Bo, G. Zhang, and J. Xu, “Transition between superluminal and subluminal light propagation in photorefractive Bi12SiO20 crystals,” Opt. Express 13, 8198–8203 (2005).
[CrossRef]

Zhao, M.

S. Residori, P. L. Ramazza, and M. Zhao, “Dynamics of beam fanning in Cu-doped KNSBN,” Opt. Commun. 102, 100–104 (1993).
[CrossRef]

Zibrov, A. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Zozulya, A. A.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (3)

Appl. Phys. B (2)

M. Y. Goulkov, T. Granzow, U. Dorfler, T. Woike, M. Imlau, and R. Pankrath, “Study of beam-fanning hysteresis in photo- refractive SBN:Ce: light-induced and primary scattering as functions of polar structure,” Appl. Phys. B 76, 407–416 (2003).
[CrossRef]

I. V. Kedyk, P. Mathey, G. Gadret, A. A. Grabar, K. V. Fedyo, I. M. Stoika, I. P. Prits, and Y. M. Vysochanskii, “Investigation of the dielectric, optical and photorefractive properties of Sb-doped Sn2P2S6 crystals,” Appl. Phys. B 92, 549–554 (2008).
[CrossRef]

C. R. Phys. (1)

U. Bortolozzo, S. Residori, and J.-P. Huignard, “Slow-light through nonlinear wave-mixing in liquid crystal light-valves,” C. R. Phys. 10, 938–948 (2009).
[CrossRef]

Ferroelectrics (1)

A. A. Grabar, M. I. Gurzan, I. V. Kedyk, I. M. Stoika, and Yu. M. Vysochanskii, “Optical properties and applications of photorefractive Sn2P2S6,” Ferroelectrics 257, 245–254 (2001).
[CrossRef]

J. Opt. (1)

A. Shumelyuk and S. Odoulov, “Light pulse manipulation in Sn2P2S6,” J. Opt. 12, 104015 (2010).
[CrossRef]

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

J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. A 72, 46–51 (1982).
[CrossRef]

J. Opt. Soc. Am. B (4)

Mater. Res. Bull. (1)

R. Nitsche and P. Wild, “Crystal growth of metal-phosphorus-sulfur compounds by vapor transport,” Mater. Res. Bull. 5, 419–423 (1970).
[CrossRef]

MRS Proc. (1)

M. J. Miller, G. L. Wood, and G. J. Salamo, “Photorefractive beam fanning optical limiter,” MRS Proc. 479, 193–198 (1997).

Nature (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Opt. Commun. (3)

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–343 (2003).
[CrossRef]

S. Residori, P. L. Ramazza, and M. Zhao, “Dynamics of beam fanning in Cu-doped KNSBN,” Opt. Commun. 102, 100–104 (1993).
[CrossRef]

F. Bo, G. Zhang, and J. Xu, “Ultraslow Gaussian pulse propagation induced by a dispersive phase coupling in photorefractive bismuth silicon oxide crystals at room temperature,” Opt. Commun. 261, 349–352 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

K. Shcherbin, P. Mathey, and G. Gadret, “Slow light with photorefractive four-wave mixing,” Phys. Rev. A 84, 063802 (2011).
[CrossRef]

K. Shcherbin, P. Mathey, G. Gadret, R. Guyard, H. R. Jauslin, and S. Odoulov, “Slowing down of light pulses using photorefractive four-wave mixing: nontrivial behavior with increasing coupling strength,” Phys. Rev. A 87, 033820 (2013).
[CrossRef]

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Günter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[CrossRef]

Phys. Rev. Lett. (5)

A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, and K. Buse, “Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).
[CrossRef]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef]

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light pulse slowing down up to 0.025  cm/s by photorefractive two-wave coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Science (1)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Ukr. J. Phys. (1)

O. M. Shumelyuk, A. I. Hryhorashchuk, and S. G. Odoulov, “Optical forerunners in crystals with photorefractive dynamic gratings,” Ukr. J. Phys. 54, 33–37 (2009).

Other (3)

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effects in Sn2P2S6,” in Photorefractive Materials and Their Applications, P. Günter and J.-P. Huignard, eds. (Springer, 2006), pp. 327–359.

R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics, E. Wolf, ed. (Elsevier2002), Vol. 43, pp. 497–530.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, 1996).

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Figures (9)

Fig. 1.
Fig. 1.

Optical scheme of the experimental setup. M, spherical mirror; HWP, half-wave plate; EOM, electro-optic modulator; BS, beam splitter; PMT1, PMT2, PMT3, photomultiplier tubes. Inset: picture of the fanning on a screen placed behind the sample.

Fig. 2.
Fig. 2.

(a) Temporal dependence of the output beam intensity reduced by beam fanning for different input intensities. (b) Exponential depletion factor Gd calculated with Eq. (1).

Fig. 3.
Fig. 3.

Dependences of fitting parameters A1, A2, τ1, and τ2 for the curves presented in Fig. 2 by the expression (2).

Fig. 4.
Fig. 4.

Output pulses (red solid line) and corresponding input pulses (black dashed line) normalized on their peak values, for various input pulse durations (a) 0.1 s, (b) 1 s, (c) 10 s, and (d) 100 s. The input beam intensity at the maximum of the pulse is 0.67W/cm2.

Fig. 5.
Fig. 5.

(a) Output pulse intensity relative to the undistorted pulses (circles) as function of the input pulse width and (b) output pulse advance (squares) and output pulse width (triangles) as function of the input pulse width for an input intensity of 0.67W/cm2.

Fig. 6.
Fig. 6.

Advance-bandwidth product (ratio of the output pulse advance to its width) as function of the input pulse width for an input intensity of 0.67W/cm2.

Fig. 7.
Fig. 7.

Parameters of the pulse speeding up as functions of the polarization azimuth relative to the X-axis of the crystal. The input intensity is set at 0.67W/cm2 and the input pulse width is 0.23 s. (a) Output pulse advance and (b) relative output pulse advance and relative output peak intensity.

Fig. 8.
Fig. 8.

Oscillograms of the sinusoidal intensity-modulated input (dash) and output (solid) signals, illustrating the phase shift due to the PRBF, for modulation frequency of (a) 0.25 Hz and (b) 1 Hz.

Fig. 9.
Fig. 9.

Spectra of I(d,t)/I(0) (squares) and ϕ (circles). The solid lines represent the best fits by the relations (3).

Equations (3)

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ΔII(0)=1exp(Gd).
I(t)=I(t)+ΔI[A1exp(tτ1)+A2exp(tτ2)].
I(d,Ω)I(0)=exp(Gd1+Ω2τ2);ϕ(Ω)=Ωτ·Gd1+Ω2τ2,

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