Abstract

We present experimental results on the self-focusing of light beams in a photorefractive, optically active Bi12TiO20 crystal. Using circular polarization of the input HeNe laser beam we succeeded to create a stable self-focused, asymmetric light-beam over several hours. Depending on external parameters, like the external electric field and the intensity ratio, a transient symmetric state is observed.

© 2004 Optical Society of America

1. Introduction

The self-focusing and self-trapping of a single laser beam in photorefractive media have been studied thoroughly in the last two decades. Photorefractive solitons have the advantage to build up at low powers and they have numerous possible applications in the photonic and communications field. For example, they can be used either for adaptive or fixed structuring of nonlinear materials to produce guiding or routing devices. A stable and reproducible self-focusing or spatial soliton formation process is therefore mandatory.

Photorefractive solitons have been investigated experimentally and theoretically to a large extend, especially in strontium-barium niobate (SBN) [1, 2, 3]. Only a few papers exist for the sillenite type crystals, namely for Bi12SiO20 (BSO) [4] and Bi12TiO20 (BTO) [5, 6]. The temporal evolution of the beam width towards spatial photorefractive solitons has been studied theoretically [7] as well as experimentally (BTO [8], SBN [9], KNbO3 [10]). To the best of our knowledge, stable soliton structures in BTO have not been reported until now.

In this paper we present experimental verifications of stable self-focusing in a Bi12TiO20 crystal at 633 nm. Crystals of the sillenite type, to which BTO belongs, exhibit optical activity which rotates the plane of linear polarization of the wave while propagating through the material. In our case the rotary power is 6.3°/mm at λ = 633 nm. Compared to the other sillenites (up to 20°/mm in BSO for λ = 633 nm [12]) it is relatively low, but as we deal with a very long sample (21.24 mm, total rotation of 134° inside the crystal), the influence of the polarization on the effective electro-optic coefficient r eff and hence the self-focusing effect cannot be neglected. Also “pre-rotating” the linear polarization of the input wave does not compensate this effect because there are regions along the propagation axis z for all angles of the input polarization where the product of r eff and the space-charge electric field E⃗ becomes zero or negative and no self-trapping of light is possible [11]. Former investigations with our sample and using linear polarizations revealed that the observed light distribution at the exit face of the crystal undergoes strong fluctuations, both in the intensity itself and also in the coordinates of the intensity maximum. We therefore use circular polarization of the laser beam. This assures an effective electro-optic coefficient which is indeed smaller than the maximal one for linear polarization but which is almost constant over the whole propagation distance. Thus, the beam experiences approximately the same strength of nonlinearity throughout the crystal length. We show that this way it is possible to produce stable self-focused structures over several hours.

2. Experimental arrangement

The set-up for verifying the self-trapping in the BTO is shown in Fig. 1. Our BTO sample is cut along the (11¯1), (1¯12), and (110) crystallographic faces and measures 0.89 × 5.64 × 21.24mm3 (fiber-like geometry). The external field is applied in the direction of its smallest dimension, x. The light propagates along the z ([110]) direction. We use a HeNe laser beam at λ = 633 nm which is focused onto the entrance face of the crystal to a waist of about 30 μm. Background illumination is provided by a conventional halogen lamp in order to raise artificially the dark irradiance inside the crystal. The exit face of the crystal is monitored onto a CCD camera using an imaging system of a magnification ratio equal to 20. We preferred a CCD to a photodiode in order to observe qualitatively the evolution of the beam cross-section and also to avoid problems that occur with beam bending which takes place in the red region of the spectrum, even if this effect is very small. The output beam peak intensity as well as the beam widths parallel and perpendicular to the external field are measured while the electric field is switched on.

 

Fig. 1. Experimental set-up for the verification of self-focusing in BTO. F - filter, P - polarizer, BS - beam splitter cube, L - focusing lens, U 0 - external voltage, I - imaging system, CCD - CCD camera.

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The distance between the focusing lens (f′ = 6 cm) and the crystal was chosen so as to have the focus just a little bit outside the crystal (≤ 1 mm between the crystal’s entrance face and the focus). Our recent experiments have shown that better results can be obtained if the focus of the beam does not lie inside the crystal.

The polarity of the external field applied to the crystal determines if there is self-focusing or defocusing provided that all other parameters rest unchanged (see, for example [13]). This corresponds to inverting the polarization of the beam while keeping the polarity of the electric field.

3. Experiments, results, and discussion

We perform measurements using right circular polarization of the signal beam. The polarization rotation direction is therefore parallel to the direction of the specific rotation. It is chosen in order to avoid the disturbing effects of the optical activity. Left circular polarization, on the other hand, results in a very slight, but not measurable defocusing of the beam. We determine the beam diameter at the crystal’s end facet in regular intervals to observe its temporal development.

As an example, a whole series of measurements of the beam width evolution is shown in Fig. 2. The inset depicts the quasi-steady state in which the self-focusing power is most pronounced. This is a transient state of which the time scale is related to the dielectric relaxation time in the dark. There exist two points in time where the beam widths in x and y direction are equal, see Fig. 3(a). This behaviour is characteristic of the quasi-steady state. After about 45s, the beam diameter in the direction parallel to the external field is always smaller than that perpendicular to it, Fig. 3(b). The steady state is completely formed after about 90 s. Then the process of screening of the external electric field by the internal field is completed, and the total space charge field is fully established. It is interesting to note that in the quasi-steady state the component perpendicular to the external field experiences a stronger focusing than the parallel component, but then relaxes into the less focussed steady state whereas the parallel component is most narrowed in the steady state.

 

Fig. 2. Temporal development of the beam width both parallel (horizontal) and perpendicular (vertical) to the externally applied electric field E 0. r = 27, E 0 = 28.1 kV/cm.

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Fig. 3. Intensity profiles of the output beam in (a) the quasi-steady state (after 7 s) and (b) the steady state (after 2 min), both parallel (horizontal) and perpendicular (vertical) to the externally applied electric field, with the parameters from Fig. 2. (The profiles in (a) are shifted against each other for better viewing.)

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The anisotropic nature of self-focusing in photorefractive media in the steady state has been treated experimentally as well as theoretically, for example, in [14, 15, 16]. Note that therein, the authors dealt with linear polarization. But, of course, an initial asymmetry comes into play by the direction of the externally applied electric field E 0.

In Fig.4 we present the temporal development of the beam width from t = 0 to t = 60 s (a) and from t = 3.3 h to t = 6.5 h (b). Movie (a) shows the quasi-steady process with the circular self-focused beam and the transition to the asymmetric beam cross-section, whereas movie (b) gives an impression of the fully established steady state. A slight beam displacement in the direction of the applied electric field can be observed.

 

Fig. 4. Movies of the temporal development of the beam width at the exit face of the crystal, with the parameters from Fig. 2. (a) Quasi-steady state, t = 0…60 s (1.25 frames/s, 168 kB), (b) parts of the steady state, t = 3.3…6.5 h (1 frame/160 s, 120 kB).

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The whole process directly points to a self-focusing of the beam inside the crystal. The cross-section of the beam, i. e. the intensity distribution at the exit face of the crystal in the steady state remains nearly constant over almost seven hours. No statements can be made of the overall lifetime of the self-focused state because our measurement system had to be switched off after 6.5 hours. But, as Fig. 2 suggests, the half-life time has not yet been reached within the experimentation period.

The interruptions in the curves w(t) (Fig. 2) have their origin in the limited storage capacity of our acquisition system. In particular, the exact observation of the quasi-steady state could only be realized at the expense of (less important) parts of the steady state.

The dependence of the temporal development of the beam width w(t) on two parameters is investigated: the intensity ratio r, which is the quotient of the beam peak intensity in the steady state and the artificially raised dark irradiance I d, and the parameter N2=12k2n2reffX02E0 [11]. Here, n is the unperturbed refractive index, k is the wave number, r eff is the effective electro-optic coefficient, X 0 is an arbitrary length (of the order of the input beam width), and E 0 is the externally applied electric field. In N 2, all important experimental quantities are combined, if one does not consider photovoltaic and diffusion effects, which is true for the BTO crystal and high applied field strengths.

Consequently, two kinds of experiments are carried out: First, the external field E 0 is kept constant (and therefore the parameter N 2) while the intensity ratio r is changed. Second, the external field is varied and the intensity ratio kept constant.

Fig. 5 shows the temporal development of the beam width w (t) in the direction parallel to the external field for different intensity ratios r. This is realized by changing the beam intensity of the steady state. The respective input intensities of the beam are 2.8, 11.3, and 21.2 mW/cm 2. The given ratios r are always approximate values because the beam peak intensity changes according to the beam width during the measurements. The experimental error of r is mainly due to the error made by determining the dark intensity I d. This latter is introduced by measuring I d only in a small field around the signal beam and therefore possible small inhomogeneities outside the observed field cannot be taken into account. Also, we do not determine its absolute value but compare it to the beam peak intensity in the steady state in the plane of the crystal’s exit face. The error is estimated to about 20%. The electric field has a value of E 0 = 28.1 kV/cm. With n = 2.58, r eff = 3.4 pm/V, and X 0 = 10μm, this gives N 2 = 2.077. It can be clearly seen that the first minimum in the beam width is reached earlier for larger intensity ratios. Only for r = 27 a quasi-steady state is observed. This means that the other two intensity ratios are too small, the minimum in the beam width is reached at steady state. It is, of course, also possible that the resolution of our acquisition system (max.25Hz) was not sufficient to catch the transient processes.

 

Fig. 5. Temporal development of the beam width parallel to the external field E 0 for different intensity ratios r. E 0 = 28,1 kV/cm. (The curve for r = 4 is shifted downwards for about 35 μm for better viewing.)

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The results of the second part of the experiment are shown in Fig.6. Here, the dependencies w (t) are measured for the parameter E 0. N 2 varies from 1.353 to 2.338. The intensity ratio is set to r ≈ 11 by adjusting the input beam intensities so that the output beam peak intensities in the steady state are nearly equal. According to the results obtained in [8, 9], the first minimum in the beam width is reached at nearly the same time for all electric fields and therefore for all N. No quasi-steady state is observed. Also, within the limited performance and resolution of our acquisition system, for larger electric fields the beam width in the steady state is slightly smaller. However, the chosen values of the electric fields are not distinctively different, which is due to the lower (efficiency and onset of the self-focusing process) and upper (electric discharges between the electrodes) limitations of the experiment.

 

Fig. 6. Temporal development of the beam width parallel to the external field for different values of E 0. r = 11.

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To succeed in obtaining stable self-trapped beams it is mandatory to adjust the background illumination such that its intensity distribution is uniform over the whole crystal sample. This assures that no trapping takes place outside the illuminating spot.

4. Conclusion

In this work, we presented experimental results concerning self-focused structures in a photorefractive, optically active Bi12TiO20 crystal which belongs to the sillenites. We succeeded in evading the disturbing effects of the optical activity on the self-focusing process by using circular polarization of the laser beam. Therefore, we could observe the existence of the self-focused beam over several hours without considerable loss of quality. The temporal development of the beam width for different electric fields and intensity ratios was experimentally investigated. The beam cross-section is circular in the quasi-steady state of which the existence depends on the external parameters. In the steady state a strong asymmetry appears which is due to the anisotropy of the photorefractive effect.

Acknowledgments

This study was partially supported by the DFG within the Forschergruppe “Nonlinear spatiotemporal dynamics in dissipative and discrete optical systems”.

References and links

1. M. Shih, P. Leach, M. Segev, M. H. Garret, G. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett. 21, 324–326(1996). [CrossRef]   [PubMed]  

2. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993). [CrossRef]   [PubMed]  

3. C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar-and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig) 11, 573–629 (2002). [CrossRef]  

4. E. Fazio, W. Ramadan, M. Bertolotti, A. Petris, and V. I. Vlad, “Complete characterization of (2+1)D soliton formation in photorefractive crystals with strong optical activity,” J. Opt. A: Pure Appl. Opt. 5S119–S123 (2003). [CrossRef]  

5. M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity,” Appl. Phys. Lett. 64, 408–410 (1994). [CrossRef]  

6. N. Fressengeas, D. Wolfersberger, J. Maufoy, and G. Kugel, “Experimental study of the self-focusing process temporal beahvior in photorefractive Bi12TiO20,” J. Appl. Phys. 85, 2062–2067 (1999). [CrossRef]  

7. A. A. Zozulya and D. Z. Anderson, “Nonstationary self-focusing in photorefractive media,” Opt. Lett. 20, 837–839 (1995). [CrossRef]   [PubMed]  

8. N. Fressengeas, D. Wolfersberger, J. Maufoy, and G. Kugel, “Build up mechanisms of (1+1)-dimensional pho-torefractive bright spatial quasi-steady-state and screening solitons,” Opt. Commun. 145, 393–400 (1998). [CrossRef]  

9. M. Wesner, C. Herden, R. Pankrath, D. Kip, and P. Moretti, “Temporal development of photorefractive solitons up to telecommunication wavelengths in strontium-barium niobate waveguides,” Phys. Rev. E 64, 036613- – 036613–4 (2001). [CrossRef]  

10. C. Hesse, N. Fressengeas, D. Wolfersberger, and G. Kugel, “Self-focusing of a continuous laser beam in KnbO3 in the presence of an externally applied electric field,” Opt. Mat. 18, 187–190 (2001). [CrossRef]  

11. N. Fressengeas, J. Maufoy, and G. Kugel, “Temporal behavior of bidimensional photorefractive bright spatial solitons,” Phys. Rev. E 54, 6866–6875 (1996). [CrossRef]  

12. W. Królikowski, N. Akhmediev, D. R. Andersen, and B. Luther-Davis, “Effect of natural optical activity on the propagation of photorefractive solitons,” Opt. Commun. 132, 179–189 (1996). [CrossRef]  

13. M. Segev, M. Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13, 706–718 (1996). [CrossRef]  

14. G. S. García Quirino, M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, S. Stepanov, and V. Vysloukh, “Intrfer-ometric measurements of the photoinduced refractive index profiles in photorefractive Bi12TiO20 crystal,” Opt. Commun. 123, 597–602 (1996). [CrossRef]  

15. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1531 (1995). [CrossRef]   [PubMed]  

16. N. Korneev, P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311–321 (1996). [CrossRef]  

References

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  • |

  1. M. Shih, P. Leach, M. Segev, M. H. Garret, G. Salamo, and G. C. Valley, �??Two-dimensional steady-state photorefractive screening solitons,�?? Opt. Lett. 21, 324�??326 (1996).
    [CrossRef] [PubMed]
  2. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, �??Observation of self-trapping of an optical beam due to the photorefractive effect,�?? Phys. Rev. Lett. 71, 533�??536 (1993).
    [CrossRef] [PubMed]
  3. C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, �??Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media,�?? Ann. Phys. (Leipzig) 11, 573-629 (2002).
    [CrossRef]
  4. E. Fazio, W. Ramadan, M. Bertolotti, A. Petris, and V. I. Vlad, �??Complete characterization of (2+1)D soliton formation in photorefractive crystals with strong optical activity,�?? J. Opt. A: Pure Appl. Opt. 5 S119�??S123 (2003).
    [CrossRef]
  5. M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, �??Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity,�?? Appl. Phys. Lett. 64, 408�??410 (1994).
    [CrossRef]
  6. N. Fressengeas, D. Wolfersberger, J. Maufoy, and G. Kugel, �??Experimental study of the self-focusing process temporal beahvior in photorefractive Bi12TiO20,�?? J. Appl. Phys. 85, 2062�??2067 (1999).
    [CrossRef]
  7. A. A. Zozulya, D. Z. Anderson, �??Nonstationary self-focusing in photorefractive media,�?? Opt. Lett. 20, 837�??839 (1995).
    [CrossRef] [PubMed]
  8. N. Fressengeas, D. Wolfersberger, J. Maufoy, and G. Kugel, �??Build up mechanisms of (1+1)-dimensional photorefractive bright spatial quasi-steady-state and screening solitons,�?? Opt. Commun. 145, 393�??400 (1998
    [CrossRef]
  9. M. Wesner, C. Herden, R. Pankrath, D. Kip, and P. Moretti, �??Temporal development of photorefractive solitons up to telecommunication wavelengths in strontium-barium niobate waveguides,�?? Phys. Rev. E 64, 036613- �??036613-4 (2001).
    [CrossRef]
  10. C. Hesse, N. Fressengeas, D. Wolfersberger, and G. Kugel, �??Self-focusing of a continuous laser beam in KnbO3 in the presence of an externally applied electric field,�?? Opt. Mat. 18, 187�??190 (2001).
    [CrossRef]
  11. N. Fressengeas, J. Maufoy, and G. Kugel, �??Temporal behavior of bidimensional photorefractive bright spatial solitons,�?? Phys. Rev. E 54, 6866�??6875 (1996).
    [CrossRef]
  12. W. Królikowski, N. Akhmediev, D. R. Andersen, and B. Luther-Davis, �??Effect of natural optical activity on the propagation of photorefractive solitons,�?? Opt. Commun. 132, 179�??189 (1996).
    [CrossRef]
  13. M. Segev, M. Shih, and G. C. Valley, �??Photorefractive screening solitons of high and low intensity,�?? J. Opt. Soc. Am. B 13, 706�??718 (1996).
    [CrossRef]
  14. G. S. García Quirino, M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, S. Stepanov, and V. Vysloukh, �??Intrferometric measurements of the photoinduced refractive index profiles in photorefractive Bi12TiO20 crystal,�?? Opt. Commun. 123, 597�??602 (1996).
    [CrossRef]
  15. A. A. Zozulya, D. Z. Anderson, �??Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,�?? Phys. Rev. A 51, 1520�??1531 (1995).
    [CrossRef] [PubMed]
  16. N. Korneev, P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, M. Klein, and B. Wechsler, �??Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,�?? J. Mod. Opt. 43, 311�??321 (1996).
    [CrossRef]

Ann. Phys. (1)

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, �??Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media,�?? Ann. Phys. (Leipzig) 11, 573-629 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, �??Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity,�?? Appl. Phys. Lett. 64, 408�??410 (1994).
[CrossRef]

J. Appl. Phys. (1)

N. Fressengeas, D. Wolfersberger, J. Maufoy, and G. Kugel, �??Experimental study of the self-focusing process temporal beahvior in photorefractive Bi12TiO20,�?? J. Appl. Phys. 85, 2062�??2067 (1999).
[CrossRef]

J. Mod. Opt. (1)

N. Korneev, P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, M. Klein, and B. Wechsler, �??Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,�?? J. Mod. Opt. 43, 311�??321 (1996).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

E. Fazio, W. Ramadan, M. Bertolotti, A. Petris, and V. I. Vlad, �??Complete characterization of (2+1)D soliton formation in photorefractive crystals with strong optical activity,�?? J. Opt. A: Pure Appl. Opt. 5 S119�??S123 (2003).
[CrossRef]

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

Opt. Commun. (3)

G. S. García Quirino, M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, S. Stepanov, and V. Vysloukh, �??Intrferometric measurements of the photoinduced refractive index profiles in photorefractive Bi12TiO20 crystal,�?? Opt. Commun. 123, 597�??602 (1996).
[CrossRef]

W. Królikowski, N. Akhmediev, D. R. Andersen, and B. Luther-Davis, �??Effect of natural optical activity on the propagation of photorefractive solitons,�?? Opt. Commun. 132, 179�??189 (1996).
[CrossRef]

N. Fressengeas, D. Wolfersberger, J. Maufoy, and G. Kugel, �??Build up mechanisms of (1+1)-dimensional photorefractive bright spatial quasi-steady-state and screening solitons,�?? Opt. Commun. 145, 393�??400 (1998
[CrossRef]

Opt. Lett. (2)

Opt. Mat. (1)

C. Hesse, N. Fressengeas, D. Wolfersberger, and G. Kugel, �??Self-focusing of a continuous laser beam in KnbO3 in the presence of an externally applied electric field,�?? Opt. Mat. 18, 187�??190 (2001).
[CrossRef]

Phys. Rev. A (1)

A. A. Zozulya, D. Z. Anderson, �??Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,�?? Phys. Rev. A 51, 1520�??1531 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (2)

N. Fressengeas, J. Maufoy, and G. Kugel, �??Temporal behavior of bidimensional photorefractive bright spatial solitons,�?? Phys. Rev. E 54, 6866�??6875 (1996).
[CrossRef]

M. Wesner, C. Herden, R. Pankrath, D. Kip, and P. Moretti, �??Temporal development of photorefractive solitons up to telecommunication wavelengths in strontium-barium niobate waveguides,�?? Phys. Rev. E 64, 036613- �??036613-4 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, �??Observation of self-trapping of an optical beam due to the photorefractive effect,�?? Phys. Rev. Lett. 71, 533�??536 (1993).
[CrossRef] [PubMed]

Supplementary Material (2)

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Figures (6)

Fig. 1.
Fig. 1.

Experimental set-up for the verification of self-focusing in BTO. F - filter, P - polarizer, BS - beam splitter cube, L - focusing lens, U 0 - external voltage, I - imaging system, CCD - CCD camera.

Fig. 2.
Fig. 2.

Temporal development of the beam width both parallel (horizontal) and perpendicular (vertical) to the externally applied electric field E 0. r = 27, E 0 = 28.1 kV/cm.

Fig. 3.
Fig. 3.

Intensity profiles of the output beam in (a) the quasi-steady state (after 7 s) and (b) the steady state (after 2 min), both parallel (horizontal) and perpendicular (vertical) to the externally applied electric field, with the parameters from Fig. 2. (The profiles in (a) are shifted against each other for better viewing.)

Fig. 4.
Fig. 4.

Movies of the temporal development of the beam width at the exit face of the crystal, with the parameters from Fig. 2. (a) Quasi-steady state, t = 0…60 s (1.25 frames/s, 168 kB), (b) parts of the steady state, t = 3.3…6.5 h (1 frame/160 s, 120 kB).

Fig. 5.
Fig. 5.

Temporal development of the beam width parallel to the external field E 0 for different intensity ratios r. E 0 = 28,1 kV/cm. (The curve for r = 4 is shifted downwards for about 35 μm for better viewing.)

Fig. 6.
Fig. 6.

Temporal development of the beam width parallel to the external field for different values of E 0. r = 11.

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