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 CS₂ 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 CS₂. 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 CS₂. This molecule has been thoroughly studied in nonlinear ellipse rotation and exhibits a large molecular reorientation nonlinearity. The nonlinear susceptibility tensor of CS₂ at 440 nm, 470 nm, and 532 nm was studied because CS₂ exhibits a large nonlinear absorption at 440 nm and 470 nm in the nanosecond regime [15]. The 5 mm length CS₂ cell was fixed on the focus.

 

Fig. 1. Schematic of experimental Setup. G1 and G2 are Glan prisms, D1 and D2 are detectors.

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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 CS₂ 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.

 

Fig. 2. Experimental and theoretical results: (a), (b), (c): transmittance for nonlinear output (open diamonds) and linear output (open circles) at 440 nm, 470 nm, and 532 nm, respectively. (d), (e), (f): the ellipticity e (open circles) and the rotation angle θ (open squares) as a function of input intensity I ₀ at 440 nm, 470 nm and 532 nm. The solid lines are theoretical simulations with Re(A)=13, 8, 3.5×10^-20 m²/V², Im(A)=7, 2.5, 0×10^-20 m²/V², Re(B)=27, 20, 14×10^-20 m²/V², and Im(B)=19, 6, 0×10^-20 m²/V² for 440nm, 470nm, and 532nm, respectively.

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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 CS₂ has no obvious nonlinear absorption. The increasing e with I ₀ further verifies the existence of the anisotropy of Im(χ ⁽³⁾) of CS₂ 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=3χ ⁽³⁾ _xyxy+3χ ⁽³⁾ _xxyy, B=6χ ⁽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 m²/V², , Im(A)=7, 2.5, 0×10^-20 m²/V², Re(B)=27, 20, 14×10^-20 m²/V², and Im(B)=19, 6, 0×10^-20 m²/V² 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 m²/V² 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+4πχ ^(NL), where ε ₀ is the linear dielectric constant, we can write the differences in refractive index (Δn _±) and absorption (Δα _±) 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 Δn=2π/n ₀ Re(A)|E|², and Δα=4πk/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 Δn=2π/n ₀ Re(A+B/2)|E|², and Δα=4πk/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 3.2×10⁸ W/cm². 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 cm²/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 cm²/W and β_cir=6.5×10^-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 m²/V² and 8.0×10^-20 m²/V², 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 m²/V² and 18×10^-20 m²/V² respectively, which agree well with the results obtained from nonlinear polarization experiments shown in Fig. 2.

 

Fig. 3. Z-scan curves of nonlinear refraction and nonlinear absorption in the cases of linearly, elliptical and circular polarized input laser at 440nm. The solid lines are the theoretical fits with n _2lin=13.5×10^-14 cm²/W, n _2ell=8.4×10^-14 cm²/W, n _2cir=6.1×10^-14 cm²/W, and β_lin=17.4×10^-9 cm/W, β _ell=9.5×10^-9 cm/W, β _cir=6.5×10^-9 cm/W.

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Fig. 4. The n ₂ (open squares) and β (open circles) as a function of e at 440nm, 470nm, and 532nm. The solid lines are the theoretical results.

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Once n _2lin and β_lin of linearly polarized light and n _2cir and β_cir of circularly polarized light are determined, from Eqs. (4) and (5) one can obtain the expressions of 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 cm²/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 CS₂ 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 CS₂ 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 CS₂. 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 CS₂ 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.