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Numerical simulation of the propagation of a single laser pulse in a photorefractive medium
Optical Materials 18 (2001) 81±84
www.elsevier.com/locate/optmat
Numerical simulation of the propagation of a single laser pulse in a photorefractive medium D. Wolfersberger a,b,*, N. Fressengeas a, J. Maufoy a, G. Kugel b a
b
Equipe de Recherche en Photonique et Opto electronique, Sup elec, Technopole METZ 200, 2 Rue Edouard Belin, Metz Cedex 57070, France Laboratoire MOPS-CLOES, Universit e de Metz et Sup elec, 2 Rue Edouard Belin, Metz Cedex 57070, France
Abstract The paper presents a numerical study of the self-focusing time behaviour of a ns single laser pulse propagating in a biased photorefractive sillenite crystal. Assuming particular hypotheses in the case of short pulse illumination, a nonlinear partial dierential equation linking the refraction index and the beam intensity has been derived from a monodimensional Kukhtarev band transport model. The numerical resolution of this equation has been coupled to the simulation of the propagation of a beam in a nonlinear medium using a Beam Propagation Method. We thus simulate the self-focusing phenomena of a single laser pulse on the nanosecond time-scale. These calculations let us describe the output beam pro®le evolution during the laser pulse and study theoretically the in¯uence of dierent parameters on selffocusing such as the applied electric ®eld, the beam intensity and waist. A successful comparison to previous experimental measurements is reported. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Self-focusing; Photorefractive eect; Pulsed laser; Nanosecond scale
1. Introduction Beam self-focusing in photorefractive media has been the focus of many studies, as far as low power level continuous lasers are concerned. Most of theoretical studies deal with photorefractive steady-state; in particular, bright spatial solitons have been predicted [1±4]. The time behaviour of the self-focusing phenomena has been studied later theoretically [5,6], showing that a beam can be selffocused in a time as short as desired if its intensity is high enough [7]. The recent literature reports
*
Corresponding author. Tel.: +33-3-87-76-47-04; fax: +33-387-76-47-00. E-mail address: [email protected] (D. Wolfersberger).
that self-focusing leading to spatial solitons occurs in photorefractive media under pulsed illumination [8], in accordance with previously developed theoretical predictions [9±11]. In the present paper, we propose a numerical study of the build-up mechanisms of this fast photorefractive phenomena. By achieving timeresolved systematic simulations of a single laser pulse self-focusing in a sillenite crystal, we show the possibility for a laser pulse to be self-focused and evidenced the in¯uence of the external applied electric ®eld. 2. Wave propagation equation Within the electric ®eld wave equation in the paraxial approximation and assuming negligible
0925-3467/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 0 1 ) 0 0 1 3 7 - 9
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D. Wolfersberger et al. / Optical Materials 18 (2001) 81±84
absorption and low refractive index variation [1,2], we can determine in the case of a (1 1) model (z for propagation, x for diraction), a time dependent nonlinear wave propagation equation given by: i
oA 1 o2 A 1 2 2 2 k x n reff E x; tA 0 oZ 2 oX 2 2 0
1
with A the amplitude of the beam electric ®eld, Z z=kx20 , X x=x0 (x0 is an arbitrary length); n is the index of refraction of the medium, reff is the eective electro-optic coecient. The space charge ®eld E x; t is determined using the model developed on the basis of Kukhtarev band transport model [12] as reported in [10,11]. It is important to note that A is a vector ®eld with two complex transverse components, which allow to take in account the gyratory power exhibited by sillenite crystals such as Bi12 SiO20 , Bi12 GeO20 and Bi12 TiO20 . This time dependent nonlinear wave propagation equation is valid for pulse durations shorter than the recombination time. To study the temporal evolution of the pulse propagation through the crystal, we integrated numerically Eq. (1) using the Beam Propagation Method also called ``SplitStep Fourier'' method [13].
waist we 20 lm. We consider an incident ¯uence de 5 mJ=cm2 , which corresponds to a maximum beam local intensity Iem equal to 1 MW=cm2 for a pulse duration of 5 ns. The simulations are made considering the physical parameters of a Bi12 SiO20 crystal. The electric ®eld applied Eext is equal to 6.25 kV/cm. Fig. 1 illustrates the beam propagation using level curves corresponding to the dierent intensity levels. At t 0, we observe the natural diraction of the beam (Fig. 1(a)). Fig. 1(b) and (c) illustrate the self-focusing phenomena during the pulse duration. These calculations allow one to analyze theoretically ``the diraction coecient a t'', which is de®ned as the ratio of the output beam diameter over the input one. Fig. 2 presents the comparison between the corresponding theoretical result and experimental measurements of the diraction
3. Numerical simulation One case of simulation is presented in Fig. 1 for a crystal length of 6 mm. The initial pro®le used at the entrance face of the crystal to calculate the evolution is the pro®le of a Gaussian beam with a
Fig. 2. Comparison of experimental and theoretical diraction coecients of a 5 ns laser pulse propagating through a sillenite crystal.
Fig. 1. Numerical simulation of a pulse propagating during its duration: (a) t 0; (b) t 3 ns; (c) t 5 ns.
D. Wolfersberger et al. / Optical Materials 18 (2001) 81±84
83
Fig. 3. Evolution of the diraction coecient a t (a) versus time for dierent applied voltages, (b) versus the external applied electric ®eld Eext .
coecient that we have measured in a BTO crystal and as detailed in [14].
4. In¯uence of the external applied electric ®eld In order to study the in¯uence of the external applied electric ®eld Eext on the pulse self-focusing, we simulate the propagation of a single laser pulse of 20 ns duration of power density I0 1 MW= cm2 in a 3 mm long crystal for dierent ®elds between 0 and 6.25 kV/cm and for a beam entrance waist we 20 lm. The result is presented in Fig. 3. Fig. 3(a) presents the temporal evolution of the diraction coecient for a pulse duration around 20 ns. a t decreases from its natural initial value 1.15 (due to the shorter crystal length) when no electric ®eld is applied, showing the pulse is selffocused, to a minimum reached around 7 ns: this minimum value correspond to what is called the ``photorefractive saturation''. After this time, the phenomena relaxes to a less focused state, as was found for continuous beams with another model [6]. The minimum value of a becomes lower when the electric ®eld is more important. The self-focusing phenomena increases with the external electric ®eld applied amplitude. More simulations showed that the diraction coecient evolution a t at saturation versus Eext is quasi-linear (Fig. 3(b)): the more intense Eext is, the more the pulse is self-focused.
5. Conclusion We have demonstrated theoretically the possibility for a single laser pulse to be self-focused when it passes through a Bi12 TiO20 crystal. The (1 1) model developed allows doing good quantitative comparisons with experimental measurements. The in¯uence of dierent parameters such as the external ®eld applied, the beam intensity, the entrance waist have been shown both theoretically and experimentally. Acknowledgements The authors wish to thank D. Rytz, from FEE Idar-Oberstein (Germany) for his Bi12 TiO20 crystal. The theoretical calculation has been developed with the support of the Centre Charles Hermite (Nancy, France) on their 64 processor Origin 2000. This work was supported in part by the Region Lorraine. References [1] M. Segev, B. Crosignani, A. Yariv, Phys. Rev. Lett. 68 (1992) 923. [2] B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, G. Salamo, J. Opt. Soc. Am B 10 (1993) 446. [3] M. Segev, B. Crosignani, P. Di Porto, G.C. Duree, G. Salamo, E. Sharp, Opt. Lett. 19 (1994) 1296. [4] D.N. Cristodoulides, M.I. Carvalho, Opt. Lett. 19 (1994) 1714.
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D. Wolfersberger et al. / Optical Materials 18 (2001) 81±84
[5] A.A. Zozulya, D.Z. Anderson, Opt. Lett. 20 (1995). [6] N. Fressengeas, J. Maufoy, G. Kugel, Phys. Rev. E 54 (1996). [7] N. Fressengeas, D. Wolfersberger, J. Maufoy, G. Kugel, Opt. Commun. 145 (1998). [8] M. Segev, M. Shih, G.C. Valley, J. Opt. Soc. Am. 13 (1996). [9] K. Kos, G. Salamo, M. Segev, Opt. Lett. 23 (1998) 1001. [10] D. Wolfersberger, N. Fressengeas, J. Maufoy, G. Kugel, J. Korean Phys. Soc. 32 (1998).
[11] D. Wolfersberger, N. Fressengeas, J. Maufoy, G. Kugel, Proc. SPIE 3491 (1998). [12] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, V.L. Vinetskii, Ferroelectrics 22 (1979) 949. [13] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, London, 1989. [14] D. Wolfersberger, N. Fressengeas, J. Maufoy, G. Kugel, Ferroelectrics 238 (2000) 273.
www.elsevier.com/locate/optmat
Numerical simulation of the propagation of a single laser pulse in a photorefractive medium D. Wolfersberger a,b,*, N. Fressengeas a, J. Maufoy a, G. Kugel b a
b
Equipe de Recherche en Photonique et Opto electronique, Sup elec, Technopole METZ 200, 2 Rue Edouard Belin, Metz Cedex 57070, France Laboratoire MOPS-CLOES, Universit e de Metz et Sup elec, 2 Rue Edouard Belin, Metz Cedex 57070, France
Abstract The paper presents a numerical study of the self-focusing time behaviour of a ns single laser pulse propagating in a biased photorefractive sillenite crystal. Assuming particular hypotheses in the case of short pulse illumination, a nonlinear partial dierential equation linking the refraction index and the beam intensity has been derived from a monodimensional Kukhtarev band transport model. The numerical resolution of this equation has been coupled to the simulation of the propagation of a beam in a nonlinear medium using a Beam Propagation Method. We thus simulate the self-focusing phenomena of a single laser pulse on the nanosecond time-scale. These calculations let us describe the output beam pro®le evolution during the laser pulse and study theoretically the in¯uence of dierent parameters on selffocusing such as the applied electric ®eld, the beam intensity and waist. A successful comparison to previous experimental measurements is reported. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Self-focusing; Photorefractive eect; Pulsed laser; Nanosecond scale
1. Introduction Beam self-focusing in photorefractive media has been the focus of many studies, as far as low power level continuous lasers are concerned. Most of theoretical studies deal with photorefractive steady-state; in particular, bright spatial solitons have been predicted [1±4]. The time behaviour of the self-focusing phenomena has been studied later theoretically [5,6], showing that a beam can be selffocused in a time as short as desired if its intensity is high enough [7]. The recent literature reports
*
Corresponding author. Tel.: +33-3-87-76-47-04; fax: +33-387-76-47-00. E-mail address: [email protected] (D. Wolfersberger).
that self-focusing leading to spatial solitons occurs in photorefractive media under pulsed illumination [8], in accordance with previously developed theoretical predictions [9±11]. In the present paper, we propose a numerical study of the build-up mechanisms of this fast photorefractive phenomena. By achieving timeresolved systematic simulations of a single laser pulse self-focusing in a sillenite crystal, we show the possibility for a laser pulse to be self-focused and evidenced the in¯uence of the external applied electric ®eld. 2. Wave propagation equation Within the electric ®eld wave equation in the paraxial approximation and assuming negligible
0925-3467/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 0 1 ) 0 0 1 3 7 - 9
82
D. Wolfersberger et al. / Optical Materials 18 (2001) 81±84
absorption and low refractive index variation [1,2], we can determine in the case of a (1 1) model (z for propagation, x for diraction), a time dependent nonlinear wave propagation equation given by: i
oA 1 o2 A 1 2 2 2 k x n reff E x; tA 0 oZ 2 oX 2 2 0
1
with A the amplitude of the beam electric ®eld, Z z=kx20 , X x=x0 (x0 is an arbitrary length); n is the index of refraction of the medium, reff is the eective electro-optic coecient. The space charge ®eld E x; t is determined using the model developed on the basis of Kukhtarev band transport model [12] as reported in [10,11]. It is important to note that A is a vector ®eld with two complex transverse components, which allow to take in account the gyratory power exhibited by sillenite crystals such as Bi12 SiO20 , Bi12 GeO20 and Bi12 TiO20 . This time dependent nonlinear wave propagation equation is valid for pulse durations shorter than the recombination time. To study the temporal evolution of the pulse propagation through the crystal, we integrated numerically Eq. (1) using the Beam Propagation Method also called ``SplitStep Fourier'' method [13].
waist we 20 lm. We consider an incident ¯uence de 5 mJ=cm2 , which corresponds to a maximum beam local intensity Iem equal to 1 MW=cm2 for a pulse duration of 5 ns. The simulations are made considering the physical parameters of a Bi12 SiO20 crystal. The electric ®eld applied Eext is equal to 6.25 kV/cm. Fig. 1 illustrates the beam propagation using level curves corresponding to the dierent intensity levels. At t 0, we observe the natural diraction of the beam (Fig. 1(a)). Fig. 1(b) and (c) illustrate the self-focusing phenomena during the pulse duration. These calculations allow one to analyze theoretically ``the diraction coecient a t'', which is de®ned as the ratio of the output beam diameter over the input one. Fig. 2 presents the comparison between the corresponding theoretical result and experimental measurements of the diraction
3. Numerical simulation One case of simulation is presented in Fig. 1 for a crystal length of 6 mm. The initial pro®le used at the entrance face of the crystal to calculate the evolution is the pro®le of a Gaussian beam with a
Fig. 2. Comparison of experimental and theoretical diraction coecients of a 5 ns laser pulse propagating through a sillenite crystal.
Fig. 1. Numerical simulation of a pulse propagating during its duration: (a) t 0; (b) t 3 ns; (c) t 5 ns.
D. Wolfersberger et al. / Optical Materials 18 (2001) 81±84
83
Fig. 3. Evolution of the diraction coecient a t (a) versus time for dierent applied voltages, (b) versus the external applied electric ®eld Eext .
coecient that we have measured in a BTO crystal and as detailed in [14].
4. In¯uence of the external applied electric ®eld In order to study the in¯uence of the external applied electric ®eld Eext on the pulse self-focusing, we simulate the propagation of a single laser pulse of 20 ns duration of power density I0 1 MW= cm2 in a 3 mm long crystal for dierent ®elds between 0 and 6.25 kV/cm and for a beam entrance waist we 20 lm. The result is presented in Fig. 3. Fig. 3(a) presents the temporal evolution of the diraction coecient for a pulse duration around 20 ns. a t decreases from its natural initial value 1.15 (due to the shorter crystal length) when no electric ®eld is applied, showing the pulse is selffocused, to a minimum reached around 7 ns: this minimum value correspond to what is called the ``photorefractive saturation''. After this time, the phenomena relaxes to a less focused state, as was found for continuous beams with another model [6]. The minimum value of a becomes lower when the electric ®eld is more important. The self-focusing phenomena increases with the external electric ®eld applied amplitude. More simulations showed that the diraction coecient evolution a t at saturation versus Eext is quasi-linear (Fig. 3(b)): the more intense Eext is, the more the pulse is self-focused.
5. Conclusion We have demonstrated theoretically the possibility for a single laser pulse to be self-focused when it passes through a Bi12 TiO20 crystal. The (1 1) model developed allows doing good quantitative comparisons with experimental measurements. The in¯uence of dierent parameters such as the external ®eld applied, the beam intensity, the entrance waist have been shown both theoretically and experimentally. Acknowledgements The authors wish to thank D. Rytz, from FEE Idar-Oberstein (Germany) for his Bi12 TiO20 crystal. The theoretical calculation has been developed with the support of the Centre Charles Hermite (Nancy, France) on their 64 processor Origin 2000. This work was supported in part by the Region Lorraine. References [1] M. Segev, B. Crosignani, A. Yariv, Phys. Rev. Lett. 68 (1992) 923. [2] B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, G. Salamo, J. Opt. Soc. Am B 10 (1993) 446. [3] M. Segev, B. Crosignani, P. Di Porto, G.C. Duree, G. Salamo, E. Sharp, Opt. Lett. 19 (1994) 1296. [4] D.N. Cristodoulides, M.I. Carvalho, Opt. Lett. 19 (1994) 1714.
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D. Wolfersberger et al. / Optical Materials 18 (2001) 81±84
[5] A.A. Zozulya, D.Z. Anderson, Opt. Lett. 20 (1995). [6] N. Fressengeas, J. Maufoy, G. Kugel, Phys. Rev. E 54 (1996). [7] N. Fressengeas, D. Wolfersberger, J. Maufoy, G. Kugel, Opt. Commun. 145 (1998). [8] M. Segev, M. Shih, G.C. Valley, J. Opt. Soc. Am. 13 (1996). [9] K. Kos, G. Salamo, M. Segev, Opt. Lett. 23 (1998) 1001. [10] D. Wolfersberger, N. Fressengeas, J. Maufoy, G. Kugel, J. Korean Phys. Soc. 32 (1998).
[11] D. Wolfersberger, N. Fressengeas, J. Maufoy, G. Kugel, Proc. SPIE 3491 (1998). [12] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, V.L. Vinetskii, Ferroelectrics 22 (1979) 949. [13] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, London, 1989. [14] D. Wolfersberger, N. Fressengeas, J. Maufoy, G. Kugel, Ferroelectrics 238 (2000) 273.