## Abstract

The evolutions of polarization and nonlinearities in an isotropic medium induced by anisotropy of third-order nonlinear susceptibility were studied experimentally and theoretically. The anisotropy of imaginary part of third-order susceptibility was verified to exist by the change of ellipticity of polarization ellipse in the isotropic nonlinear medium CS_{2}. The changes of nonlinear refraction and nonlinear absorption depending upon the ellipticity of polarization ellipse are also presented. The numerical simulations based on two coupled nonlinear Schrödinger equations (NLSE) provide an excellent quantitative agreement with experimental results.

©2008 Optical Society of America

## 1. Introduction

Since the pioneering experimental work of Maker, *et al.*, [1], the dynamical evolution of polarization state of light due to the anisotropy of real part of third-order nonlinear susceptibility (*χ*
^{(3)}=*χ*
^{(3)}
_{xxyy}+*χ*
^{(3)}
_{xyxy}+*χ*
^{(3)}
_{xyyx}) has been studied extensively in isotropic media as well as anisotropic media [2–7]. The existence of *χ*
^{(3)}
_{xyyx} can induce a rotation of axis position as a polarization ellipse propagates through the medium. Some nonlinear effects relating to nonlinear polarization rotation, such as bistable, unstable, and chaotic behaviors were realized [8, 9]. A fundamental elliptically polarized vector soliton was also observed in the spatial domain in a CS_{2} liquid slab planar waveguide [10–12]. However, reports on the anisotropy of imaginary part of *χ*
^{(3)} are few. Although several theoretical analyses [3–5] predicted that the anisotropy of imaginary part of *χ*
^{(3)} can lead to the ellipticity change of an elliptically polarized beam in isotropic media, to the best of our knowledge no experimental observation has supported such a prediction, and most of experimental reports were only involved in anisotropic media [13, 14].

The anisotropy of *χ*
^{(3)} can lead to dynamical evolution of the polarization state of light, meanwhile, the change of the polarization state also has a drastic effect on third-order susceptibility. However, most of reports on polarization dependent nonlinear refraction and nonlinear absorption were concentrated on anisotropic media [13, 14]. In this letter, we present the anisotropy of imaginary parts of *χ*
^{(3)} and the ellipticity dependent nonlinear refraction and nonlinear absorption in the isotropic medium CS_{2}. Like the anisotropy of real part of *χ*
^{(3)}, the anisotropy of imaginary part is very important to the evolution of nonlinear polarization dynamics.

## 2. Experimental details

Our experimental setup is shown in Fig. 1. A commercial optical parametric oscillator (Continuum Panther Ex OPO) pumped by the third harmonic (355 nm) from Continuum Surelite-II is used to generate 4–5 ns pulses with a repetition rate of 10 Hz and tunable in the range of 420–2500 nm. A Glan prism (G1) was used to generate a linearly polarized light. The elliptic polarization state of the input beam was adjusted by angle (*φ*) between G1 and a quarter-wave plate. The input beam has nearly Gaussian transverse shape and was focused by a 150 mm focal length lens to form a beam waist of 19 µm. To determine the axis position and *e* of the global polarization state, we directly measured the transmitted energy as the analyzer G2 rotates. The experiment is carried out on the isotropic nonlinear medium CS_{2}. This molecule has been thoroughly studied in nonlinear ellipse rotation and exhibits a large molecular reorientation nonlinearity. The nonlinear susceptibility tensor of CS_{2} at 440 nm, 470 nm, and 532 nm was studied because CS_{2} exhibits a large nonlinear absorption at 440 nm and 470 nm in the nanosecond regime [15]. The 5 mm length CS_{2} cell was fixed on the focus.

## 3. Results and discussions

Figures 2(a)–2(c) show the experimental normalized transmittance T as a function of orientation of G2 (*ϕ*) for nonlinear output and linear output at 440 nm, 470 nm, and 532 nm. The values of *e* of polarization ellipse can be obtained by using the relationship of *e*
^{2}=T_{min}/T_{max}, where T_{min} and T_{max} are the minimum and maximum of transmittance T, respectively. The x-coordinates relative to T_{min} and T_{max} represent the positions of major and minor axis of polarization ellipse. Relative to the case of linear output, an obviously shift of axis position of polarization ellipse in nonlinear output can be observed at these wavelengths, which indicates the existence of the anisotropy of (Re(*χ*
^{(3)}). This is consistent with large molecular reorientation nonlinearity of CS_{2} observed in the subnanosecond and nanosecond regimes [16]. The Re(*χ*
^{(3)}
_{xyyx}) mainly contributes to the shift of axis position, i.e., the rotation of polarization ellipse. Additionally, the change of T_{max} is much larger than that of T_{min} as shown in Figs. 2(a) and 2(b) at 440 nm and 470 nm. Therefore, the ellipticity *e* changes as an elliptical polarized beam propagates through the medium, and nonlinear absorption is anisotropic due to Im(*χ*
^{(3)}
_{xyyx})≠0.

To picture the evolutions of polarization state more clearly, the ellipticity *e* and the rotation angle *θ* as a function of input intensity *I*
_{0} at 440 nm, 470 nm and 532 nm are given in Figs. 2(d), 2(e), and 2(f), respectively. Rotation angle *θ* increases with *I*
_{0} at these wavelengths due to Re(*χ*
^{(3)}
_{xyyx})≠0. At 440 nm and 470 nm, the value of *e* also increases with *I*
_{0}, and the slope of curve *θ*(*I*
_{0}) at 440 nm is larger than that at 470 nm. However, the value of *e* almost keeps unchanged at 532 nm since CS_{2} has no obvious nonlinear absorption. The increasing *e* with *I*
_{0} further verifies the existence of the anisotropy of Im(*χ*
^{(3)}) of CS_{2} at 440 nm and 470 nm.

To model the evolutions of polarization and nonlinearities as an elliptically polarized beam propagates in a nonlinear medium, the following coupled NLSEs are employed:

where *r* is the radial coordinate, *z* is the longitudinal coordinate, *n*
_{0} is the linear refractive index, *k*=*2πn*
_{0}/λ is the wave vector and λ is the wavelength. *E*
_{+}, *E*
_{-} are the left- and right-hand circularly polarized components of the electric field. Following the notation of nonlinear polarization of Maker, *et al.*, [1], the effective nonlinear susceptibilities of two circular components can be written as [16]:

where *A*=3*χ*
^{(3)}
_{xyxy}+3*χ*
^{(3)}
_{xxyy}, *B*=6*χ*
^{(3)}
_{xyyx}. The solid lines in Fig. 2 are the results of numerical simulations using Eqs. (1) and (2). The parameters used in the simulations are Re(*A*)=13, 8, 3.5×10^{-20} m^{2}/V^{2}, , Im(*A*)=7, 2.5, 0×10^{-20} m^{2}/V^{2}, Re(*B*)=27, 20, 14×10^{-20} m^{2}/V^{2}, and Im(*B*)=19, 6, 0×10^{-20} m^{2}/V^{2} at 440 nm, 470 nm, and 532 nm, respectively. The total third-order nonlinear susceptibility |*χ*
^{(3)}| (*χ*
^{(3)}=*A*/3+*B*/6) are 10.1, 6.3, and 3.5×10^{-20} m^{2}/V^{2} at 440 nm, 470 nm, and 532 nm, respectively. The value of |*χ*
^{(3)}| at 532 nm agrees well with that in previous report [17].

As mentioned above, nonlinear susceptibility component *B* induces the evolution of ellipticity and axis position of polarization ellipse. Meanwhile, different polarization state also affects the change of nonlinear refraction and nonlinear absorption [18]. Using the relationship between the dielectric constant *ε* and nonlinear susceptibility *χ*
^{(NL)} with *ε*=*ε*
_{0}+4*πχ*
^{(NL)}, where *ε*
_{0} is the linear dielectric constant, we can write the differences in refractive index (Δ*n*
_{±}) and absorption (Δ*α*
_{±}) due to nonlinear reaction as follows:

Note that the differences of refraction and absorption depend upon only the coefficient *B* but not the coefficient *A*.

Be different from anisotropic medium, the changes of nonlinear refraction and absorption in an isotropic medium are dependent on only the ellipticity of polarization ellipse, but not the polarization orientation. First, for circularly polarized light with *e*=1, only one of two circular components is present, and the changes in refractive index and absorption can be given by Δ*n*=2*π*/*n*
_{0} Re(*A*)|*E*|^{2}, and Δ*α*=4*πk*/*n*
_{0} Im(*A*)|*E*|^{2}. Second, for linearly polarized light with *e*=0, we can see that the changes of refractive index and absorption can be given by Δ*n*=2*π*/*n*
_{0} Re(*A*+*B*/2)|*E*|^{2}, and Δ*α*=4*πk*/*n*
_{0} Im(*A*+*B*/2)|*E*|^{2}, since linearly polarized light is a combination of equal amounts of left-and right-hand circular components (i.e. |*E*
_{+}|^{2}=|*E*
_{-}|^{2}), where *E* denotes the total field amplitude of the linearly polarized radiation with |*E*|^{2}=2|*E*
_{+}|^{2}=2|*E*
_{-}|^{2}.

Open and closed aperture Z-scan [17] experiments were carried out to determine the ellipticity dependent nonlinear refraction and absorption. The experimental results at 440 nm are shown in Fig. 3. The on-axis intensity *I*
_{0} used in our Z-scan experiments is 3.2×10^{8} W/cm^{2}. For a linearly polarized light as a light source in our Z-scan measurements, nonlinear refraction coefficient *n*
_{2lin} and absorption coefficient *β _{lin}* were determined to be 13.5×10

^{-14}cm

^{2}/W and 17.4×10

^{-9}cm/W, respectively, which are over one time larger than those of circular polarization with

*n*

_{2cir}=6.1×10

^{-14}cm

^{2}/W and

*β*=6.5×10

_{cir}^{-9}cm/W. Moreover, from the results of Z-scan with circularly polarized light, one can obtain the value of complex nonlinear susceptibility component

*A*because the changes of refractive index and absorption depend on only

*A*in the case of circular polarization. The real and imaginary parts of

*A*are 13×10

^{-20}m

^{2}/V

^{2}and 8.0×10

^{-20}m

^{2}/V

^{2}, respectively. And then, the coefficient

*B*can be determined from the Z-scan experimental results of linear polarization or elliptical polarization, and the values of Re(

*B*) and Im(

*B*) are 24×10

^{-20}m

^{2}/V

^{2}and 18×10

^{-20}m

^{2}/V

^{2}respectively, which agree well with the results obtained from nonlinear polarization experiments shown in Fig. 2.

Once *n*
_{2lin} and *β _{lin}* of linearly polarized light and

*n*

_{2cir}and

*β*of circularly polarized light are determined, from Eqs. (4) and (5) one can obtain the expressions of

_{cir}*n*

_{2ell}and

*β*

_{ell}of elliptical polarized light as a function of ellipticity as follows:

where *q*=(1-*e*)/(1+*e*). The Z-scan curves of nonlinear refraction and absorption with *e*=0.41 are shown in Fig. 3 and we can get *n*
_{2ell}=8.4×10^{-14} cm^{2}/W and *β _{ell}*=9.5×10

^{-9}cm/W. Other fitting parameters are the same as those of linearly polarized light. Figure 4 gives experimental and theoretical results of the changes of

*n*

_{2}and

*β*as a function of

*e*at 440 nm, 470 nm, and 532 nm, respectively. The symbols represent the experimental results, and agree well with the solid lines obtained by theoretical simulations using Eqs. (6) and (7). The change of nonlinearities indicates that nonlinear refraction and nonlinear absorption are tunable by controlling the ellipticity of elliptically polarized beam.

The relative magnitude of *A* and *B* depends upon the nature of the physical process of optical nonlinearities. For molecular orientation nonlinearities, the ratio of the real part of *B* to that of *A* is 6, this is the case of optical nonlinearities of CS_{2} in the nanosecond and picosecond regimes. However, Re(*B*)/Re(*A*) and Im(*B*)/Im(*A*) obtained in our nanosecond experiments at 440 nm are 2.1 and 2.7, respectively. Re(*B*)/Re(*A*)=2.5 and Im(*B*)/Im(*A*)=2.4 were obtained at 470 nm, and Re(*B*)/Re(*A*)=4 was obtained at 532 nm. The decreasing ratio of *B*/*A* indicates that other nonlinear mechanism should exist in the nanosecond regime besides molecular orientation. The origin of the different physical characters of the two contributions (*A* and *B*) to nonlinear susceptibility can be understood in terms of the energy level [16]. One-photon-resonant processes contribute only to the coefficient *A*, while two-photon- resonant processes contribute to both the coefficients *A* and *B*. In Ref. [15] we reported that the large nonlinear absorption of CS_{2} in a short wavelength region and the nanosecond regime can arise from a combination of two-photon absorption and the excited-state absorption induced by two-photon absorption. Excited state nonlinearity can cause the decrease of *B*/*A* since effective third-order nonlinearities are sequential one-photon process and independent upon the change of polarization state.

## 4. Conclusion

In summary, we present the evolutions of polarization and nonlinearities in an isotropic medium CS_{2}. In the early sixties, Maker, *et a.,l* planed to simultaneously study the polarization dependence of the intensity-induced absorption and the intensity-induced rotation in order to obtain accurate relative values of Im(*A*), Re(*A*), Im(*B*) and Re(*B*). In our work the complex third-order susceptibility tensors of CS_{2} at 440 nm, 470 nm, and 532 nm were measured. To our knowledge, our results offer the first experimental evidence of Im(*B*) induced nonlinear polarization dynamics and ellipticity-dependent nonlinearities in an isotropic medium. Further experiments aiming at studying influence of spatial-temporal effects on self-induced polarization changes due to complex third order nonlinear susceptibility are expected to sharpen this analysis. Many interesting extensions are possible, including the tuning of optical limiting, optical switching, and photonic crystal by controlling polarization state.

## Acknowledgments

This work is supported by the Natural Science Foundation of China (No. 60708020, 10574075), Chinese National Key Basic Research Special Fund (No. 2006CB921703), and the Program for Changjiang Scholars and Innovative Research Team in University (IRT0149).

## References and links

**1. **P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. **12**, 507–509 (1964). [CrossRef]

**2. **P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. **137**, A801–818 (1965). [CrossRef]

**3. **P. X. Nguyen and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta. **25**, 233–246 (1978). [CrossRef]

**4. **P. X. Nguyen, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of picosecond light pulses in nonlinear isotropic media,” Opt. Commun. **46**, 329–333 (1983). [CrossRef]

**5. **A. J. van Wonderen, “Influence of transverse effect on self-induced polarization changes in an isotropic Kerr medium,” J. Opt. Soc. Am. B **14**, 1118–1130 (1997). [CrossRef]

**6. **M. Lefkir and G. Rivoire, “Influence of transverse effects on measurement of third-order nonlinear susceptibility by self-induced polarization state changes,” J. Opt. Soc. Am. B **14**, 2856–2864 (1997). [CrossRef]

**7. **M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. I. The single-pulse equations,” Phys. Rev. A **35**, 2965–2975 (1987). [CrossRef] [PubMed]

**8. **D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. **187**, 281–367 (1990). [CrossRef]

**9. **A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A **48**, 1610–1624 (1993). [CrossRef] [PubMed]

**10. **M. Delqué, G. Fanjoux, and T. Sylvestre, “Polarization dynamics of the fundamental vector soliton of isotropic Kerr media,” Phys. Rev. E **75**, 016611 (2007). [CrossRef]

**11. **M. Delqué, T. Sylvestre, H. Maillotte, C. Cambournac, P. Kockaert, and M. Haelterman, “Experimental observation of the elliptically polarized fundamental vector soliton of isotropic Kerr media,” Opt. Lett. **30**, 3383–3385 (2005). [CrossRef]

**12. **C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, “Symmetry-Breaking Instability of Multimode Vector Solitons,” Phys. Rev. Lett. **89**, 083901 (2002). [CrossRef] [PubMed]

**13. **R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. **18**, 194–196 (1993). [CrossRef] [PubMed]

**14. **Sean J. Wagner, J. Meier, A. S. Helmy, J. Stewart Aitchison, M. Sorel, and D. C. Hutchings, “Polarization-dependent nonlinear refraction and two-photon absorption in GaAs/AlAs superlattice waveguides below the half-bandgap,” J. Opt. Soc. Am. B **24**, 1557–1563 (2007). [CrossRef]

**15. **Z. B. Liu, Y. L. Liu, B. Zhang, W. Y. Zhou, J. G. Tian, W. P. Zang, and C. P. Zhang, “Nonlinear absorption and optical limiting properties of carbon disulfide in short-wavelength region,” J. Opt. Soc. Am. B **24**, 1101–1104 (2007). [CrossRef]

**16. **R. W. Boyd, *Nonlinear Optics*, second edition (Academic Press, San Diego, 2003).

**17. **M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron **26**, 760–169 (1990). [CrossRef]

**18. **Z. B. Liu, X. Q. Yan, J. G. Tian, W. Y. Zhou, and W. P. Zang, “Nonlinear ellipse rotation modified Z-scan measurements of third-order nonlinear susceptibility tensor,” Opt. Express **15**, 13351–13359 (2007). [CrossRef] [PubMed]