1. Introduction

There are a variety of experimental methods for measurements of third-order nonlinear susceptibility, such as degenerate four-wave mixing [1], nonlinear imaging techniques [2], nonlinear ellipse rotation [3,4] and Z-scan method [5]. Among the methods used for determination of third-order nonlinear susceptibilities, the very efficient technique is degenerate four-wave mixing, which allows not only to determine the different tensor components, but also effectively to separate the electronic and nuclear contribution for isotropic materials including the CS2. However, this method is much complicated compared with single beam Z-scan method. The well-known Z-scan method has become a standard tool for determining nonlinear parameters of various materials because of its simplicity, sensitivity, accuracy, and the ease of separation between nonlinear refraction (NLR) and nonlinear absorption (NLA) [5-7]. Since the pioneering experimental work of M. Sheik-Bahae et al [5], there have been various theoretical and experimental modifications of this method, such as thick sample Z-scan [8,9], eclipsing Z-scan [10], pump-probe Z-scan [11,12], two-color Z-scan[13], non-Gaussian beams Z-scan [14], top-hat beam Z-scan[15], etc. However, among these Z-scan methods the linearly polarized lights are always used as incidence beams, and there is no theoretical and experimental work to study the Z-scan method with elliptically and circularly polarized lights.

For the effect of polarization on Z-scan measurements, R. DeSalvo et al presented the work on polarization dependence of nonlinear refractive index in crystals [16], in which they measured the transmittance by altering the linear polarization direction of incidence beam relative to crystallographic axis when crystal was stationary in the Z direction. Sean J. Wagner et al studied the polarization-dependent nonlinear refraction and two-photon absorption in GaAs/AlAs superlattice waveguides [17]. However, the incidence lights still were linearly polarized in their studies. Recently, Liang et al used different polarized beams to study the polarization dependence effects of refractive index change (RIC) associated with photoisomerization in poly (methyl methacrylate) (PMMA) film by Z-scan method [18], in which the peak-valley difference Δ_Tp-v with linearly polarized light was larger than those with circularly and elliptically polarized lights. They tried to explain it by decomposing the polarized light into two perpendicular linear polarized components, but they did not quantitatively analyze the effect of the RIC due to the appearance of perturbation of polarization ellipse and did not give the relationship between normalized transmittance and nonlinear phase shift as in Ref. 5.

In an isotropic medium, the third-order susceptibility tensor has only two independent third-order susceptibility components, namely χ_xxyy ⁽³⁾ and χ_xyyx ⁽³⁾. For molecular orientation and nonresonant electronic nonlinearities, RIC is greatly dependent on polarization state due to the anisotropy of third-order nonlinear susceptibility [19]. By contraries, for thermal induced nonlinearity, excited state nonlinearity and electrostriction, RIC is non-dependent on polarization state. Therefore, in order to obtain more information of third-order nonlinear susceptibility tensor and understand the nonlinear mechanisms from RIC, elliptically and circularly polarized lights should be used in Z-scan measurements. We have demonstrated a method that combines the Z-scan technique with nonlinear ellipse rotation to measure third-order nonlinear susceptibility components [20,21]. However, to the best of our knowledge, there is no report on analytical expression of Z-scan normalized transmittance under the condition of elliptically or circularly polarized lights. In this paper, we theoretically and experimentally study the polarization dependence of Z-scan measurements. The normalized transmittance formula of closed-aperture Z-scan is obtained for linearly, elliptically and circularly polarized incidence beams. The normalized transmittances of CS₂ for linearly, elliptically and circularly polarized incidence beams were measured and analyzed theoretically. The experimental results agree well with theoretical analyses. Further, the elliptically polarized light Z-scan was used to study intensity dependence of normalized transmittance.

 

Fig. 1. Experimental arrangements for polarization changeable Z-scan

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2. Theory

The modified Z-scan experimental arrangement is shown in Fig.1. Compared with normal Z-scan method, a quarter-wave plate is placed between the polarizer and the sample to create different polarized lights. When a linearly polarized beam passes the λ/4 plate with angle φ ₁( -π/2≤φ ₁</2) which is the angle between the linear polarization direction and the slow axis of the λ/4 plate, it can be converted into a polarized beam (φ ₁=-π/2, 0 for linearly polarized beam, φ ₁=±π/4 for circularly polarized beam, and others for elliptically polarized beam).

Assuming a TEM₀₀ Gaussian beam of waist radius w ₀ traveling in the +z direction, we can write E as

where w_z ² = w ² ₀(1 + z ²/z ² ₀) is the beam radius, R(z) = z(1 + z ₀ ²/z ²) is the radius of curvature of the wave front at Z, z ₀ = κw ^{2 0}/2 is the diffraction length of the beam, and κ = 2π/λ is the wave vector. E→₀(t) denotes the radiation electric field vector at the focus and contains the temporal envelope of the laser pulse. The exp[-iϕ(z,t)] term contains all the radically uniform phase variations.

If we define the slow axis of the λ/4 plate as x-axis, after passing the λ/4 plate, the electric field can be written as

where δ ₁ is the phase retardation, x̂ and ŷ are unit vectors. δ ₁=π/2 for a λ/4 plate. The electric field vector of such a beam can always be decomposed into a linear combination of left- and right-hand circular components as [19]

where σ̂₊ = (x̂+iŷ)√2 and σ̂_- = (x̂-iŷ)√2 are the circular-polarization unit vectors, E ₊=(E_x-iE_y)/√2, E _-=(E_x+iE_y)/√2. The other two parameters can be expressed as

The ellipticity used to describe the polarization geometry is defined as e = ∥E ₊∣ -∣E _-∥/(∣E ₊∣ + ∣E _-∣) , i.e. $e=\mid \sin {\delta }_1\sin 2{\phi }_1\mid /\left(1+\sqrt{1-{\sin }^2{\delta }_1}{\sin }^{2}2{\phi }_1\right).e$ is 1 for circularly polarized light, and 0 for linearly polarized light.

Because of the nonlinear effect in medium, the refractive indexes of the two circular components in the medium are different and given by [19]:

where n ₀ is the linear refractive index, A = 6χ_xxyy ⁽³⁾ and B = 6 χ_xyyx ⁽³⁾. Substituting Eq. (4) into Eq. (5), we can rewrite Eq. (5) as follows:

Due to the change in refractive index δ _n± is different for two components, the left- and right-hand circular components propagate with different phase velocities, thus, the polarization direction will rotate as the elliptically polarized beam propagates through the nonlinear medium. This is called as nonlinear ellipse rotation [19,3].

Considering a thin medium and using the slowly varying envelope approximation, we can separate the beam propagation equations with the two circular components into a couple of equations, one for the phase and another for the irradiance, as follows:

where z′ is the propagation depth in the sample. If nonlinear absorption of the medium is negligible, the phase shifts at the exit surface for the two circular components can be obtained by substituting Eq. (6) into Eq. (7) and solving the equations:

with

where Δϕ _+,0(t) and Δϕ _-,0(t) are on-axis phase shift at the focus of left- and right-hand circular components, respectively. They are defined as

where L_eff=[1-exp(-αL)]/α: is the effective length, L is the sample length, α is the linear absorption coefficient and Δn _±0(t) = 2π[A + (1±sinδ ₁sin2φ ₁)B/2]∣E ₀(t)∣²/n ₀ are the on-axis RICs of the two circular components at the focal plane.

Ignoring Fresnel reflection losses and diffraction effect, we can obtain the resultant complex electric field pattern at the exit surface of the medium as [5]

We use Gaussian decomposition (GD) method to calculate the complex field pattern at the aperture plane. The nonlinear phase term exp[-iΔϕ _± (z,r,t)] in Eq. (11) can be expanded as

The complex field pattern at the aperture is expressed as

Defining d as the propagation distance in free space from the medium to the aperture plane, and g =1 + d / R(z), the other parameters in Eq. (13) are expressed as

The normalized transmittance can be calculated as

where r _a is the radius of the aperture, S = 1- exp(-2r ² _a/w ² _a) is the aperture linear transmittance, w _a is the beam radius at the aperture in the linear regime. Under the far-field condition of z ₀≪d, the normalized transmittance of Eq. (19) can be calculated as:

with

where Y _a=r _a/Dω ₀ is a dimensionless aperture radius, x=z/z ₀ is the dimensionless sample position, and D=d/z ₀ is the dimensionless distance from the sample to the aperture plane. From Eq. (21), it is seen that there is no nonlinear phase shifts coupling terms of left- and right- hand lights, which means the two circular components independently propagate and reach the detector. The transmittance at the detector plane is the weighted average of the two components. By using L’Hôpital’s rule, the normalized transmittance for closed-aperture (CA) case can be obtained from Eq. (20) as

Under first-order approximation, T_CA can be written as

where Δϕ_eff =kπ[2A + (1-sin² δ ₁sin² 2φ ₁)B]∣E ₀(t)∣² L_eff/n ₀ is effective nonlinear phase shift at the focus on the Z axis. If the first quarter-wave plate is taken away or the angle φ ₁ is set at 0 deg, above formula will return to the normal Z-scan transmittance formula in Ref. 5. Only the weighted average RICs of the two components can be obtained if Eq. (27) is used to analyze the experimental data no matter what polarization state of the incidence beam is. If we want to separate A and B values from one experiment, one possible way is to abandon first-order approximation by taking more terms in Eqs. (26) and (20) as normalized transmittance fitting formula because the coefficients of high-order terms are non-correlation. According to Chen et al’s works [22], normalized transmittance formulas of Eq. (20) obtained by GD method can also be available for large nonlinear phase shifts.

As in Ref. 5, the steady-state (the same as cw situation) results can be extended to transient effects induced by pulsed radiation. For a Gaussian pulse, the average refractive index is

where Δn _±,0 is the peak-on-axis index change at the focus.

3. Experimental result and discussions

In our experiments, a frequency-doubled Continuum Model PY61 mode-locked pulsed laser was used to generate 35 ps pulses at 532 nm with a repetition rate of 10 Hz. The laser pulse focused by a lens of 150 mm focal length to produce a beam waist w ₀ of 18 μm was incident to a 1 mm quartz cell containing CS₂. The on-axis peak intensity I ₀ was 5.23 GW/cm². The extensively studied isotropic nonlinear medium, CS₂, was chosen as the sample. CS₂ is transparent and has very small linear absorption coefficients α ₀ (<10^-3 cm^-1) in the visible region [23], and the nonlinear absorption coefficient is less than 10^-11 cm/W, so we can ignore the nonlinear absorption in our experimental conditions. The physical mechanism leading to nonlinear refraction of CS₂ in subnansecond regime is molecular orientation, and the ratio of B to A should be 6 [19]. In order to make the reference light be synchronous with the signal light, we placed the beam splitter between the λ/4 plate and the lens. Polarization states could be changed by altering the λ/4 plate. These experiments were carried out at room temperature.

First, polarization states dependence of Z-scan were examined. From Eq. (26), it can be seen that the normalized transmittance T of Z-scan is dependent on the angle φ ₁, i.e., the ellipticity e. Figure 2 gives the CA Z-scan curves of linear, circular, and elliptical polarizations of CS₂. The angle φ ₁ was set 78 degree (i.e. e=0.2126) to create an elliptically polarized light. It can be seen that the NLR is largest for a linearly polarized beam and smallest for a circularly polarized beam, which is consistent with Liang’s experimental result [18]. The lines are least-squares fit of Eq. (26) (terms were taken until m+n=3) to experimental results, the theoretical fits are in good agreement with experimental results.

 

Fig. 2. Normalized transmittance curves for linearly, circularly and elliptically polarized (φ ₁=78deg) incidence beams.

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Since the RIC of linearly polarized light is given by Δn = π(2A + B)∣E∣²/n₀ [19], we can not distinguish the B and A values (only 2A+B is obtained) from linearly polarized light Z-scan. The (A + B/2) from the fit of linear polarization component was 9.232×10^-12 esu. For circular polarization case, from Eq. (5) the RIC equals to 2πA∣E∣²/n ₀ for both left- and right-hand lights, thus, only one value A can be obtained from the fit. The A from the fit in Fig. 2 is 2.241×10^-12 esu (i.e. χ_xxyy ⁽³⁾ = 3.735×10^-13esu ). Comparing the two cases, we could obtain B = 1.398×10^-11 esu (i.e. χ_xyyx ⁽³⁾ = 2.330×10^-12esu). The ratio B to A is 6.240, which agrees very well with theoretical value for molecular orientation. The ratio of B to A is 6 for molecular orientation, 1 for nonresonant electronic response, and 0 for electrostriction [19].

However, the RICs of elliptically polarized light are different for the two circular components, inducing different nonlinear phase shifts. The nonlinear phase shifts of the two components could be separately obtained by using Eq. (26) to fit the experimental data, further the B and A values could be determined from Eq. (10). The fitting values of B and A for elliptically polarized case is 1.313×10^-11 and 2.182×10^-12 esu respectively (i.e. χ_xyyx ⁽³⁾ =2.189×10^-12esu, and χ_xxyy ⁽³⁾, =3.637×10^-13esu). The ratio B to A was 6.019. The A and B values from these fits agree well with theoretical simulation [24] and previous measurements [3, 26].

Although the total effective RIC obtained by using first-order approximation Eq. (27) is different for three kinds of polarized lights, the third-order nonlinear susceptibility χ ⁽³⁾ = 2 χ_xxyy ⁽³⁾ + χ_xyyx ⁽³⁾) obtained from any polarized light experiment is identical. Indeed the polarization state of the light just changes the contribution of B to nonlinear refractive index as shown in Eq. (6) since both polarization state and nonlinear refractive index are determined by angle φ ₁.

As shown in above, if the incidence light is elliptically polarized, the A and B values could be simultaneously obtained from single experiment by taking more terms in Eq. (26) during experimental data analysis. Furthermore the ratio of B to A can be used to determine the nonlinear mechanism. However the values of A and B can not be simultaneously obtained from circularly or linearly polarized light Z-scan no matter how many terms are taken for the transmittance fitting formula. Thus, circularly and linearly polarized light have to be used together in Z-scan experiments as in Ref. 20 if one wants to measure χ_xxyy ⁽³⁾ and χ_xyyx ⁽³⁾. It can be easily understood from Eq. (21), the condition for determining A and B is that Δϕ _+,0(t) and Δϕ _-,0(t) should be simultaneously obtained from fitting and they must be different, only elliptically polarized light can satisfy this requirement. Compared with circularly and linearly polarized lights Z-scans, the advantage of elliptically polarized light Z-scan is that two nonlinear susceptibility components can be simultaneously determined in a single Z-scan measurement.

 

Fig. 3. Normalized transmittance curves for different peak on-axis radiation intensities: (a) 4.26, (b) 7.93, (c) 12.13, (d) 17.59, (e) 20.02, (f) 24.83 GW/cm² . The polarizer angle φ ₁ is 78 deg. Owing to the blue line covers the red dashed line we can not watch red lines in (a) and (b).

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Figure 3 gives the normalized transmittance curves for different peak on-axis irradiance intensities (4.26, 7.93, 12.13, 17.59, 20.02, 24.83 GW/cm²). The angle φ ₁ was set 78 deg for these experiments, and the linear transmittance S is 15%. It is seen that the anti-symmetry of the peak and valley has been destroyed by large intensities. Blue lines are the fits using Eq. (20) (terms are taken to m+n≤30 in order to eliminate the errors from missing terms [22]) when the B value is identical for all the intensities and the ratio B to A is 6. B=1.372×10^-11 esu chosen for this fitting was from best fit result of Fig. 3a. The B value is set to be identical for all the intensities because it is usually assumed to be independent of intensity in the χ_xyyx⁽³⁾ measurements [3].

Note that the scattering and nonlinear absorption are ignored in the fitting. The theoretical simulations agree well with experimental data for two smaller intensity cases (i.e, Fig. 3(a), Fig. 3(b)), but the fits are not appropriate for the latter four intensities. And the larger the intensity is, the more evident the discrepancy between fitting and experimental results is. These discrepancies are more obvious from the comparison of the peak-valley transmittance differences (ΔT_p-v) as shown in Fig. 4(a). The experimental data do not match the fitting results for higher intensities. It can be seen that the peak-valley difference for fixed B is not proportional to the light intensity but is reaching its saturation value. According to Eq. (27), the effective nonlinear phase shift at the focus is proportional to the intensity, so the ΔT_p-v should also be proportional to the intensity as the green dash line shown, which has been widely used as a criterion to judge the nonlinear type [18]. The appearance of nonlinear relationship between ΔT_p-v and intensity means that the first-order approximation cannot be used for large nonlinear phase shift case. For large nonlinear phase shift case, although ΔT_p-v ~ I ₀ may present a saturation, the nonlinear phase shift (so does the RIC) still increases with intensity. In order not to wrongly regard large nonlinear phase shift case as saturated nonlinear effect, the proportional relationship between ΔT_p-v and intensity under first-order approximation could not simply treated as criteria to determine whether it is a Kerr nonlinear effect or not. In this case, more terms in Eqs. (20) and (26) should be taken to analyze the experimental data and get the nonlinear phase shifts, and then judge the nonlinear type of the medium.

 

Fig. 4. Peak-to-valley transmittance difference (a) and B value used in above fits (b) as a function of peak on-axis radiation intensity.

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From the last four fits shown as blue lines in Fig. 3, it is seen that fitting results which did not take into account scattering and nonlinear absorption have lower valley and higher peak than experimental curves, so the discrepancy could not mainly come from scattering or nonlinear absorption. If scattering played an important role in higher intensity Z-scan experiments, there would be less energy into the detector. Both the valley and the peak would become lower than the case without scattering effect, and the valleys of experimental curves should be deeper than the valleys of fitting curves. But, the valleys of experimental curves are shallower than the valleys of our fitting results shown in Fig. 3. Therefore, the contribution of scattering to the discrepancy between fixed B fitting and experimental results can be ignored. Moreover, during our experiments, no obvious scattering was observed. According to the literatures [7,25,27], the nonlinear absorption is negligible for intensity less than 100 GW/cm². Since the intensities (<25 GW/cm²) used in our experiments are much smaller than the critical intensity, the discrepancy between fixed B fitting and experimental results should not mainly arise from nonlinear absorption. No nonlinearity for the quartz cell was observed in our experiments. In addition, the self-defocusing effects of acoustic generation and heat accumulation can also be excluded, because the repetition rate of picosecond pulses in our experiments was too low to bring this type self-defocusing effects [27].

One origin of the discrepancy between fixed B fitting and experimental results may be the temperature changes because of absorption. According to references [19,24], the B value is inverse to temperature. The temperature was higher for larger radiation intensity, so the B value should be smaller for larger radiation intensity. In order to check whether the discrepancy could come from the decreasing of B value, the fits with different B were carried out. In the new fitting (terms in Eq. (20) is taken until m+n=30), the B values are altered to get the best fit for different intensities. The criterion of best fit is that the peak-valley transmittance difference from fit is nearly identical to the experimental ΔT_p-v, when the ratio of B to A was fixed at the theoretical value 6. The new fitting curves are shown as red dashed lines in Fig. 3, and the ΔT_p-v and B values obtained from the new fits are shown in Fig. 4. Comparing the two fitting results in Fig. 3, we find that the new fits are much better than the former ones (fixed B fits), and the B value varies with intensity. The B value used in new fits decreases with the intensity as shown in Fig.4b, which is consistent with theoretical prediction [19,24]. Note the analysis of the B value is just qualitative, from which we can conclude that the B value decreases with intensity. If one wants to quantitatively determine the relationship between B value and intensity, the temperature change of the liquid should be studied fully.

If the discrepancy is completely induced by the decreasing of B value, the temperature of liquid at 24.83 GW/cm² should be larger than 400 K [19], which is much higher than the boiling point 319 K of CS₂ [28]. Thus, it is probable that there are other processes contributing to the discrepancy. Another origin for the discrepancy may be higher-order nonlinearities. If CS₂ has a negative fifth-order nonlinear refraction, the valley will be less deep and the peak be lower than pure third-order nonlinearity [29,30], which are consistent with the discrepancies for higher intensity cases in Fig. 3. It is possible that there is an effect of fifth-order nonlinearity refraction of Z-scan experiments at high intensity cases. Additional source of the discrepancies may be related to simultaneous multi-photon processes and saturation absorption processes, multi-photon processes make the peak lower and saturation absorption processes make the valley shallower. In summary, from the two fitting results, It can be confirmed that the Eqs. (20) and (26) can also be available for a large nonlinear phase shift.

4. Conclusion

We have presented the theoretical and experimental analyses of Z-scan measurements by arbitrary polarized lights in isotropic medium. The normalized transmittance formulas for arbitrary aperture radius were obtained. The CA case of Z-scan was used to analyze the linearly, circularly, elliptically polarized lights Z-scan data in CS₂ medium. From the analysis, we found that the elliptically polarized light Z-scan could be used to simultaneously measure the two third-order nonlinear susceptibility components χ _xyyx ⁽³⁾ and χ _xxyy ⁽³⁾. Furthermore, the ratio of B to A was determined to be about 6, which agrees very well with theoretical value for molecular orientation. The normalized transmittance formulas of elliptically polarized light Z-scan were also examined for large nonlinear phase shifts.