Three-dimensional spatial solitons in CS
2E. L. Falcão-Filho1,*, C. B. de Araújo1, G. Boudebs2, H. Leblond2, and V. Skarka2
1Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil
2Laboratoire de Photoniques d’Angers, Université d`Angers, 49045 Angers, France [email protected]
Abstract: Three-dimensional spatial solitons excited by near-infrared femtosecond pulses in CS2
are demonstrated. The propagation of such stable solitons is allowed due to the presence of the fifth-order nonlinearity which prevented the catastrophic collapse.
OCIS codes: (190.4420) Transverse effects in nonlinear optics; (190.6135) Spatial solitons
Propagation of spatial solitons may occur when there is a balance between diffraction and self-focusing, resulting in a self-trapped optical beam [1]. Bright solitons were observed few decades ago in liquid carbon disulfide, CS2, only for the two dimensional case (the propagation direction, z, and one transverse coordinate) [2]. Indeed, in general, three-dimensional solitons (z and the two transverse dimensions) are not stable suffering catastrophic beam collapse unless high-order NL mechanisms intervene on its propagation. In such case stability limits for three-dimensional solitons can be identified investigating the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity, including refractive and absorptive contributions [3].
In the present paper we demonstrate the successful excitation of three-dimensional solitons in liquid CS2 due to the simultaneous contribution of χ (3), the third-order susceptibility, and χ (5), the fifth-order susceptibility. Stable solitons propagating for more than 10 Rayleigh lengths were obtained and characterized, where the high-order nonlinear mechanism plays a crucial role on the soliton stabilization.
The experiments were carried out using rectangular quartz cells of 1.0 cm and 2.0 cm length containing liquid CS2 at room temperature. The excitation beam at 920 nm was obtained from an optical parametric amplifier pumped by an amplified Ti: sapphire laser delivering pulses of 100 fs at 1 kHz. The Rayleigh length was 0.1 cm and the beam waist at the focal plane was 16 μm. The focal point was determined imaging the beam at different positions and considering a Gaussian dependence for the beam waist. The experiments were performed with the cell in two different positions with respect to the focal point. In one case the entrance face of the cell was located in the focal point to guarantee that the input beam in the cell entrance can be described as a plane wave. In the other case the entrance face of the cell was 1.4 mm beyond the focus where the incident beam is diverging. The results discussed here refer to the second case. After the cell a telescope with magnification equal to 2.9 was used to enlarge the beam diameter and to exploit a large sensitive area of the CCD camera. The CCD camera was mounted in a translation stage with two lenses and a filter, in order to obtain images in different positions.
Fig. 1: (a) Absorbance of CS2 mixed with ethanol: red line is for pure ethanol; blue line (green line) corresponds to 50μl
(5 μl) of CS2 in 2ml of ethanol. (b) Transmittance of the CS2 as a function of laser intensity. Cell length: L=1.0 mm.
Figure 1(a) shows the absorbance spectra of CS2 diluted in ethanol in order to avoid high absorption preventing the detector to be underexposed in the near ultraviolet region. The band centered in ≈ 317 nm has been previously identified [4] and can be excited by the simultaneous absorption of three laser photons with wavelength at 920 nm.
The negligible absorbance of pure ethanol is also shown to confirm that the band at ≈ 317 nm is due to the CS2
molecule. The two-photon absorption is negligible at 920 nm but the three-photon absorption coefficient, α4, is an important parameter to consider. In order to infer α4 an optical limiting experiment was performed, as shown in
200 400 600 800 1000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Absorbance
Wavelength (nm)
(b)
0.0 1.0 2.0 3.0 4.0 5.0 6.0
0.0 1.0 2.0 3.0 (a)
Iout(1015 W/m2 )
Iin(1015 W/m2)
(a) L = 1.0 cm
(b) L=1.0 mm
Fig.1 (b), and the value of α4=5.8×10-29m3/W2 was obtained, which has the same order of magnitude than it was published in ref. [5]. On the other hand, the nonlinear refractive indices, n2∝Re[χ(3)] equal to +3.1×10-19m2/W, and n4∝Re[χ(5)] equal to –2.0×10-35m4/W2 were published for CS2 in refs. [6,7], respectively. Notice the opposite signs of n2 and n4 that is an important condition to have a stable three-dimensional soliton. Therefore, due the cubic- quintic nonlinearity, CS2 may support the propagation of stable three dimensional solitons if lasers with adequate intensities and proper wavelengths are employed.
Fig. 2: The beam images for different laser intensities and positions of the 1.0 cm long cell. The image is presented in grey scale; the blue and the magenta lines means the raw data and a fitting using a Gaussian function centered in the green lines, respectively. (a) Beam at the entrance face of the cell; (b) beam at the output face of the empty cell; (c-e) beam profiles after propagation in CS2 for input intensities of (c) 0.5×1011 W/cm2, (d) 1.8×1011 W/cm2, (e) 2.1×1011 W/cm2. The entrance face of the cell was positioned 1.4 mm beyond the focus, where the beam waist was about 28 μm. (f) Beam waist at the exit face of the cell as a function of the input intensity.
Figures 2(a) - 2(e) show images of the laser spatial profile for different conditions. Figure 2(a) show the laser profile at the entrance of the cell when it is 1.4 mm beyond the focus where the incident beam is diverging. The beam profile at the output face of the empty cell is shown in Fig. 2(b). Figures 2(c) - 2(e) show beam images after propagation in CS2 for input intensities of (c) 0.5×1011 W/cm2, (d) 1.8×1011 W/cm2, and (e) 2.1×1011 W/cm2. Figure 2(f) presents a summary of the output beam waist results as a function of the laser intensity when the entrance face of the cell is located at 1.4 mm beyond the focus. For intensities above 2.5 x 1015 W/m2 the beam profile is distorted due to a high nonlinear phase shift which leads to filamentation that will be studied in more details in the near future. Analogous behavior was observed when the entrance face of the cell was positioned at the focus and when a 2.0 cm thick cell was used. In general, the experimental results suggest a stable and robust solution for the pulse propagation in CS2.
The data of Fig.2 were used for computer simulations with the Complex Ginzburg-Landau equation and results taking into account the NL parameters presented here are in agreement with the experimental results.
References
[1] R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479-482 (1964).
[2] A. Barthélémy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201-206 (1985).
[3] See for instance: B. A. Malomed, “Complex Ginzburg-Landau equation” in Encyclopedia of Nonlinear Science, A. Scott, ed. (Routledge, New York, 2005).
[4] S. P. McGlynn, J. W. Rabalais, J. R. McDonald, and V. M. Scherr, Chem. Rev. 71, 73108 (1971).
[5] R. A. Ganeev et al., Opt. Commun. 231, 431-435 (2004).
[6] S. Couris et al., Chem. Phys. Lett. 369, 318-324 (2003).
[7] D. G. Kong et al., J. Phys. B: At. Mol. Phys. 42, 065401 (2009).
( a ) (b) (c)
( d ) ( e )
0.3 0.6 0.9 1.2 1.5 1.8 2.1 20
30 40 50 60 70 80 90 100 110
Beam waist at the entrance of 28±3 μm
Beam waist at the output (μm)
Input Intensity (1011W/cm2)
