The normal elliptically polarized light Z-scan method is modified by adding a quarter-wave plate and an analyzer before the detector. The normalized transmittance formulas of modified elliptically polarized light Z-scan are obtained for media with negligible nonlinear absorption. Compared with normal linearly and elliptically polarized light Z-scan methods, an increase of sensitivity by a factor of larger than 4 is achieved for the real part of third-order susceptibility component’s measurements using this modified elliptically polarized light Z-scan method. The analytical results are verified by studying the real part of independent susceptibility components of CS2 liquid. Moreover, the potential application for cross-polarized wave generation is discussed.
©2010 Optical Society of America
There are great interests in measuring third-order nonlinear susceptibility, χ (3), for finding appropriate materials for optical switching, optical limiter and so on. The techniques most often used to determine components of third-order susceptibility include degenerate four-wave mixing (DFWM) , nearly degenerate three- and four-wave mixing , nonlinear ellipse rotation [3–5], Z-scan method , eclipsing Z-scan  and so on.
The single beam Z-scan technique in which linearly polarized light is usually utilized has been widely used for studying nonlinear refraction and nonlinear absorption. Nonlinear refractive index from the normal Z-scan measurements, however, is a combination of third-order susceptibility components. Although its real and imaginary parts can be separated, their individual tensor components cannot be distinguished. To measure the real part of third-order susceptibility components using Z-scan method, DeSalvo et al. suggested measuring the normalized transmittance difference ΔT p-v as a function of angle between incident polarization direction and crystal orientation while the crystal is stationary on the Z direction, but this method can be only used to study crystalline materials whose orientations can be ascertained , and it cannot be used to measure the susceptibility components of liquids. Krauss et al. extended the Z-scan method to determine the sign and the magnitude of independent tensor elements of third-order susceptibility for isotropic and cubic-symmetrical materials by making appropriate measurements of nonlinear index change for linearly and circularly polarized lights .
In our previous works [10–12], we found that the susceptibility component χxyyx (3) can be determined by nonlinear ellipse rotation (NER) modified Z-scan method without an aperture before the far-field detector, and this method is very sensitive [10,11]. Further, we also found that the elliptically polarized light could be used to simultaneously measure the real part of two susceptibility components, namely χxxyy (3) and χxyyx (3). Unfortunately its measurement sensitivity is the same as that of normal Z-scan method . In this paper, we suggest a modified Z-scan method, which is obtained by adding a λ/4 plate and an analyzer before the detector in the normal elliptically polarized light Z-scan (NEZ-scan) method, and the modified method is used to study independent susceptibility components of CS2 liquid. Results show that its sensitivity can be increased by a factor of larger than 4 for the case with the second λ/4 plate and 1.5 for the case without the second λ/4 plate while the analyzer is oriented near the extinction position, and the sensitivity enhancement depends on the angle of analyzer. Additionally, the dependency of cross-polarized wave (XPW) generation  efficiency curve on the product of intensity and material length is also analyzed for isotropic medium.
The geometry arrangement of the modified NEZ-scan is shown in Fig. 1. The second λ/4 plate is enclosed by a dashed box, which means that the second λ/4 plate can be removed. Two modified cases without and with the second λ/4 plate are discussed below.
A TEM00 Gaussian beam is used in our work, and its electric field E⃑ can be written as
where wz 2 = w 0 2(1 + z 2/z 0 2) is the beam radius, w 0 is waist radius, R(z) = z(1 + z 0 2/z 2) is the radius of curvature of the wave front at z, z 0 = κw 0 2/2 is the diffraction length of the beam, and κ = 2π/λ is the wave vector. E⃑ 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 radially uniform phase variations.
After the beam passes the first λ/4 plate whose slow axis oriented an angle φ 1 to the polarizer, an elliptically polarized beam is created. If the slow axis of the λ/4 plate is defined as x axis and the fast axis as y axis as shown in Fig. 2, the electric field can be rewritten as
where δ 1 is the phase retardation, x̂ and ŷ are unit vectors. δ 1 = π/2 establishes for a λ/4 plate.
The electric field vector of such a beam can be decomposed into a linear combination of left- and right-hand circular components 
where σ̂̂ = (x̂+iŷ)/√2 and σ̂-=(x̂-iŷ)/√2 are the left- and right- hand circular-polarization unit vectors, E + = (Ex - iEy)/√2, E - = (Ex + iEy)/√2, and E ±,0(t) = E 0(t)(cosφ 1 ∓ie -iδ1sinφ 1)/√2.
Considering a thin and not optically active medium, by using the slowly varying envelopes and stationary regime approximations, the coupled nonlinear Schrödinger equations for the two polarized lights in the medium can be derived as follows:
where ∈eff ± =1 + 4π(χLin + χNL ±) = 1 + 4πχ (1) + 4π[A|E ±|2 + (A + B)|E ∓|2] are the effective susceptibilities, A = 6χxyyx (3), B = 6χxyyx (3), and κ ± =n ±,0 κ = n 0 ω/c are the wave vector of the two circular components . From above equation, we can find that nonlinear interactions can occur between the two circular components in the nonlinear medium for linearly and elliptically polarized lights. According to the works of Nguyen [16–18] and Wonderen , the polarization state of a linearly or circularly polarized beam can conserve after propagating through the nonlinear medium [16, 17]. For elliptically polarized light, both the orientation (the polarization ellipse rotation depends on the real part of the component χxyyx (3) ) and the shape of the polarization ellipse (i.e., ellipticity, the ellipticity change can happen if the imaginary part of χxyyx (3) is nonzero ) are not conserved . Therefore, owing to the presence of the imaginary part of Χxyyx (3), it is difficult to obtain the analytical complex electrical field at the exit surface of the medium from Eq. (4) as done by Sheik-Bahae . So, here we limit our medium to have negligible nonlinear absorption compared with nonlinear refraction. For those medium with nonlinear absorption, we recommend to use both linearly and circularly polarized lights Z-scan method proposed by Krauss  to study the susceptibility components.
According to our previous work , for the case without the second λ/4 plate and the analyzer, the complex electric field at the aperture plane after passing the nonlinear medium is given as
where L is medium length, α is linear absorption coefficient. Δϕ ±,0(z, t) = Δϕ ±,0(t)/(1 + z 2/z 2 0), Δϕ ±,0(t) = κΔn ±,0(t)Leff are the on-axis phase shifts at focus for two circular components. Δn ±,0(t) = 2π[A + (1 ± sinδ 1sin2φ 1)B/2]|E 0(t)|2/n 0 are the on-axis refractive index changes at focus for two circular components. The other parameters in Eq. (5) are defined as in Ref. 6.
After passing the second λ/4 plate, the phase retardation of the wave plate is δ 2, the complex electric field becomes
where γ is defined as the angle between the fast axis of the second λ/4 plate and x axis. The complex electric field after passing the analyzer which is rotated at an angle φ2 to the x axis, can be written as
So, the normalized transmittance can be calculated as
where ra is the radius of aperture, S =1-exp(-2ra 2/wa 2) is the aperture linear transmittance, wa is the beam radius at the aperture in linear regime. For steady-state case, the normalized transmittance Eq. (8) can be completed as
where Ya=ra/Dw 0 is a dimensionless aperture radius, x = z/z 0 is the sample dimensionless position, and D = d/z 0 is the dimensionless distance from the medium to the aperture plane. The coefficients in Eqs. (10) and (11) are defined as:
When the aperture is a pinhole (i.e., ra → 0), the normalized transmittance can be calculated as
Since the coefficients in Eqs. (9) and (20) are non-correlation, the values of Δϕ +,0 and Δϕ -,0 can be obtained from experimental data fittings. Further, the third-order nonlinear susceptibility components can be determined.
3. Experimental results and discussions
In our experiments, a Q-switched mode-locked Nd:YAG laser (continuum Model PY61) was used to produce 35-ps (FWHM) TEM00-mode pulses at a repetition rate of 10-Hz, the wavelength was 532 nm, and the beam waist w 0 of the beam focused by a lens (focal length was 150 mm) was 18 ± 1 μm. The on-axis peak intensity I 0 was 4.90 ± 0.15 GW/cm2. The extensively studied liquid CS2 was chosen as the nonlinear medium, the linear absorption coefficients α 0 of CS2 in the visible region is negligible (<10-3 cm-1) . The quartz cell length was 1 mm. the angle φ 1 was set 78 degrees, (i.e. ellipticity e = 0.2126 ) in order to compare the modified method with NEZ-scan in our previous study .
The experiments were carried out for two cases with and without second λ/4 plate. When the second λ/4 plate was placed, it was crossed to the first λ/4 plate (i.e. γ = 0). The linear transmittance S was 15.93%. The angle φ 2 of the analyzer was chosen to be 14°, 130°, 145° and 167° respectively since their Z-scan curves at these angles may be very different from the NEZ-scan curves according to our theoretical prediction. These Z-scan curves are shown in Fig. 3. It is seen that the shapes of these curves change greatly with analyzing angle φ 2 no matter whether the second λ/4 plate was placed or not. At the same analyzing angle, the curves are quite different for the two cases with and without second λ/4 plate. When angle φ 2 is 14°, the normalized transmittance curves given in Fig. 3a are not symmetric or antisymmetric around the position of the beam waist (x = 0). When the second λ/4 plate is placed, the curve before focus is steeper than that behind focus. However, when the second λ/4 plate is removed, the curve only has a valley, which is much broader than the peak, and the curve behind focus is steeper than that before focus. For φ 2 = 130° and 145° (Figs. 3b and 3c), Z-scan curves exhibits a peak-valley structure. But they are not exactly anti-symmetric around the beam waist. The valley is deeper but the peak is a little lower for the case with the second λ/4 plate when φ 2 is 130° (Fig. 3b), and the peaks are much higher than the valley for the case without the second λ/4 plate. When the analyzing angle φ 2 is 167° (Fig. 3d) which is close to the extinction angle of 168°, the Z-scan curves have a peak around beam waist, but they are very different, especially for the case with the second λ/4 plate, the peak value is larger than 2.5. So the sensitivity at this analyzing angle is very high. Compared with NEZ-scan  and NER modified Z-scan curves , these curves exhibit some unusual characters, especially for the two cases: φ 2 = 14° and 167°. While the medium is moving along Z direction the normalized transmittance changes for all the three Z-scan methods, but their mechanisms are different. In NEZ-scan, the transmittance change arises from the distribution change of the light field at the aperture plane as normal Z-scan , and in NER modified Z-scan the transmittance change results from change of the light field amplitude along the analyzer direction (i.e., the change of angle between ellipse polarization direction and analyzer direction) . However, our modified NEZ-scan combines the structures of NEZ-scan and NER modified Z-scan, thus its normalized transmittance change arises from the combined effect of above two mechanisms, so its Z-scan curve is different from theirs and exhibits new characters.
The solid lines in Fig. 3 are the least-squares fittings using Eq. (9). For the case φ 2 = 167°, the extinction ratios have been taken into account during the fittings (the measured extinction ratio of the polarizer-analyzer pair is ~ 2.5 × 10-4 without λ/4 plates and ~ 7.4 × 10-3 with λ/4 plates). From the fittings, the nonlinear phase shifts of left- and right- hand components can be determined, then, the susceptibility components were calculated, and the results are listed in Table 1. From the table it is seen that the values obtained from fittings are very close within experimental errors for all analyzing angles no matter whether the second λ/4 plate is placed or not. It is consistent with expectation since the susceptibility components are not a function of above measurement conditions (the second λ/4 plate and analyzing angle). The ratio of B to A is about 6 for all situations, which confirms the mechanism of the nonlinear process for this temporal pulse width is molecular reorientation . The susceptibility components values obtained from our fittings are identical to those of other works [5,10,20].
To compare the measuring sensitivity with that of NEZ-scan method, normalized transmittance difference between peak and valley ΔT p-v as a function of analyzing angle φ 2 is given in Fig. 4. The B value used for calculating theoretical curves is set to be 1.286×10-11 esu and the ratio of B to A is 6 . Results show that the experimental results agree well with theoretical results except that a discrepancy exists around the extinction angle for the case with the second λ/4 plate. The discrepancy arises mainly from non-complete extinction of the polarizers, better results can be obtained by using achromatic wave-plate and better polarizers. The difference between the NEZ-scan data and theoretical calculations arises from the different values of susceptibility components obtained from our experimental results and Lefkir’s . It can be seen that the theoretical curves firstly decrease and then increase, and there is a peak around extinction angle for the two curves. The peak values are larger than the one of NEZ-scan (without analyzer), which means the sensitivity of modified methods is higher for analyzing angle near extinction position than that of NEZ-scan. The sensitivity enhancements against φ 2 near extinction position compared to NEZ-scan are listed in Table 2. Thus, for enhancing measuring sensitivity, it is useful to add an analyzer which is nearly crossed to the polarizer when we use elliptically polarized light in Z-scan measurements to measure the susceptibility components. The theoretical peak value for the case with the second λ/4 plate is infinity because there is no energy into detector at extinction angle in linear regime, so the sensitivity can be very high when the analyzing angle is close to extinction angle. Compared with the case without second λ/4 plate, the advantage for using the second λ/4 plate is obvious. However, as shown in Fig. 4 and Table 2 the peak for the case with the second wave-plate is very steep. And the steeper the peak is, the closer the analyzer to extinction angle, which means the accuracy decreases. The sensitivity enhancement conflicts with accuracy requirement. As shown in the case of φ2 = 167°, the actual sensitivity obtained in experiment was greatly suppressed by the quality of polarizer and alignment. And the more obvious suppressing effect is, the closer the analyzer to extinction position. A compromise way is to set the analyzing angle several degrees away from the extinction position (such as φ 2 = 163° or 173° for φ 1 = 78°) in order to get good measuring accuracy and sensitivity enhancement. The use of achromatic λ/4 plates and better polarizer with precise rotation control may make the realization of higher sensitivity enhancement with appropriate accuracy easier, and this way should be a good choice to simultaneously measure χxxyy (3) and χxyyx (3) of the medium with very small third-order nonlinearity. Another problem is that when the closed aperture is very small there will be little energy into the detector, especially near the extinction position. Thus, the noise effect will be enhanced in experimental results. So we propose that the linear transmittance S should be not to too small when we use this method to measure the susceptibility components near the extinction position. From the simulation result of Eq. (9) we can see that the Z-scan curve for S ≤ 0.25 is quite similar to pinhole case when the analyzer is within 5 degrees around extinction position for with the second λ/4 case, so as a simplified data dealing way, the data from large aperture experiment (S ≤ 0.25) can still be analyzed by Eq. (20) within ± 5% error for this case.
The experimental arrangement in Fig. 1 is quite similar to ones in Refs. 21 and 22, where the authors measured the efficiency of XPW generation in BaF2 crystal while the crystal was placed at the focus produced by a 30 cm focal-length lens. According to the work of Jullien et al., the efficiency of XPW generation in isotropic nonlinear medium was identical by rotating the medium, and much smaller than anisotropic medium . From the variation of the transmittance traces in Fig. 3(d), we can see that the XPW generation efficiency should be a function of medium position in the Z direction (i.e. intensity distribution). Obviously, the efficiency of open aperture case is largest due to more rotated polarized light into the detector. From Eq. (9), the formula for XPW generation efficiency (which means two λ/4 plates and two polarizers are exactly crossed, respectively) in isotropic nonlinear medium can be written as:
where H =2πωBLeffE 0 2 sin 2φ 1/cn. Using above equation, the efficiency as a function of medium position is calculated for different products of intensity and medium length I 0 L (Leff = L if the linear absorption is negligible). CS2 is taken as an example of our simulation. The Gaussian pulse (both spatial and temporal) is used in our calculation and the values of susceptibility components are from Ref. 5, the φ 1 is set 22.5° to obtain optimal efficiency [21,23], the other parameters are the same as above experimental conditions.
The efficiency traces from the simulation are shown in Fig. 5(a), the dependence of maximum efficiency and efficiency at beam waist on I 0 L is shown on left-side of Fig. 5(b), and the average angle of ellipse rotation against I 0 L is shown on right-side, where the medium is at the beam waist. From these figures, we can see that for small I 0 L the largest efficiency appears at the beam waist, and decreases as the medium moves away from the beam waist. But, for large I 0 L, the maximum efficiency does not appear at the waist but symmetrically appears on two sides of the beam waist. And the distance from maximum efficiency position to the beam waist increases with I 0 L, maximum efficiency also becomes saturated with I 0 L for large I 0 L, but the efficiency at the beam waist decreases first and then goes up. It is can be easily understood from Eq. (21), there is a turning point at about H = 4.4931 which is calculated from dηopen(0)/dH = 0. The physical process of XPW generation in Eq. (21) comes only from nonlinear ellipse rotation because there is no anisotropy in the simulation . So when the medium is at the beam waist, although the rotated angle of the polarization direction increases with I 0 L, the angle of ellipse rotation of light beam near the light axis may be larger than 90° for large I 0 L, thus, the average efficiency of the whole pulse may decrease with I 0 L, and is smaller than the efficiency when the medium is placed near the beam waist for larger I 0 L. A larger efficiency can be obtained by moving the medium small distance from the beam waist if the average angle of rotation is larger than 45.5°. The above simulation was completed under the hypothesis that there is no other nonlinear effects for a larger I 0 L. From the simulation, it can be found that the efficiency at the beam waist is not always the largest. In fact this result can be extended to anisotropic medium since ellipse rotation is also significant for XPW generation when the light focused into medium is elliptically polarized [21,24–26]. If the rotated angle is very large (the threshold of rotated angle is about 45.5° for isotropic medium, it may be different for anisotropic medium), we should think about whether the optimal position of medium is still at focus or not.
We have presented a modified way to enhance the sensitivity of elliptically polarized light Z-scan method for simultaneously determining the third-order nonlinear susceptibility components of mediums with negligible nonlinear absorption by adding a second λ/4 plate and an analyzer before detector. The normalized transmittance trace changes greatly with analyzing angle, and the sensitivity can be enhanced at appropriate analyzing angle no matter whether the second λ/4 plate exists or not. The increased factor of sensitivity is larger than 4 and 1.5 for the cases with and without second λ/4 plate respectively when the analyzer is near extinction position, and it is a function of analyzing angle. The susceptibility components of CS2 measured by this method are accurate within experimental errors compared with others’ results. We discussed the dependence of cross-polarized wave generation on medium position with respect to the focus and find that the maximum efficiency is no longer at the beam waist if the average angle of ellipse rotation is larger than 45.5° for Gaussian pulse in isotropic nonlinear media.
This research was supported by the Natural Science Foundation of China (grant 60708020), Chinese National Key Basic Research Special Fund (grant 2006CB921703), and 111 Project (B07013).
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