1. Introduction

A variety of experimental methods are used for measurements of third-order nonlinear susceptibility, such as four-wave mixing [1,2], nonlinear imaging techniques [3], nonlinear ellipse rotation (NER) [4,5], and Z-scan [6–8]. Among these methods, NER and Z-scan use only one optical wave and measure either spatial profile changes or polarization changes of the wave. The NER method permits the determination of the nonlinear susceptibility component χ ⁽³⁾ _xyyx , and its history goes back to the classic work of Maker et al. [4]. The Z-scan method has been employed extensively because of its simplicity, sensitivity, accuracy, and the ease of separation of nonlinear refraction (NLR) and nonlinear absorption (NLA). However, it is practically impossible for any single technique or method to unambiguously separate the different nonlinearities at once. Every technique or method has advantages and disadvantages. For example, degenerate four-wave mixing (DFWM) can measure different tensor components of χ ⁽³⁾ in a single experimental setup for isotropic materials. It is not necessary for the beams employed in the experiment to be perfect TEM₀₀ Gaussian modes as long as they are well-characterized. The time dependence of the nonlinearity can be studied readily. However, the disadvantages of this technique include the fact that only the modulus of χ ⁽³⁾ (i.e., |χ⁽³⁾|) can generally be measured. A far more complicated experimental apparatus is needed; in general, the technique must be supplemented with another measurement to extract the real and imaginary parts of χ ⁽³⁾. Hence, different measurements are usually required to unravel the underlying physical mechanism by varying parameters such as irradiance and polarization state, or even the use of different measurement methods [1].

For an isotropic medium, the third-order susceptibility tensor has three independent components, χ ⁽³⁾ _xxyy, χ ⁽³⁾ _xyxy, and χ ⁽³⁾ _xyyx [9], which have not been taken into account in Z-scan measurements as yet. NER and other nonlinear polarization dynamics are owing to the existence of χ ⁽³⁾ _xyyx . Moreover, polarization-dependent NLR can be observed in the isotropic medium based on the effect of χ ⁽³⁾ _xyyx [9]. Therefore, it is expected that different tensor components of χ ⁽³⁾ can be obtained by Z-scan measurements using linearly, circularly, and elliptically polarized light. In this work, the polarization dependence of NLR in CS₂ is studied. This molecule has been studied thoroughly in NER and presents large molecular reorientation nonlinearities in the subnanosecond regime. We combine the Z-scan technique and NER to carry out a sensitive and simple measurement of χ ⁽³⁾ _xyyx . The theoretical analysis is made and the obtained transmittance formulae allow direct estimation of the component χ ⁽³⁾ _xyyx .

2. NER modified Z-scan technique

In Z-scan measurements, the sample is moved along the propagation direction (z-axis) of a tightly focused beam, and the variation of the far-field intensity is used to determine the NLR. Although Z-scan has the simplicity of both experimental setup and data analysis, a high-quality Gaussian TEM₀₀ beam for absolute measurements is required. Sample distortions or wedges, or a tilting of the sample during translation, can cause the beam to walk off the far-field aperture. This produces unwanted fluctuations in the detected signal. Even if these are kept under control, beam jitter will produce the same effect. The technique cannot be used to measure off-diagonal elements of the susceptibility tensor except when a second nondegenerate frequency beam is employed.

Another widely used technique, NER, is a single-beam experiment also using an elliptically polarized wave. A linearly polarized wave is incident on a quarter-wave plate, with its direction of polarization making an angle α with respect to the slow axis of the wave plate. This creates an elliptically polarized beam with left-circularly and right-circularly polarized components E₊ and E_- , respectively. Without NLA or scattering, when the elliptically polarized beam propagates through the nonlinear medium, the orientation of the polarization ellipse will rotate an angle θ as a function of input intensity under a given sample length d. The beam going through the nonlinear medium passes through the second quarter-wave plate oriented crosswise to the first and then a polarizer oriented for extinction of the beam in the absence of the nonlinear sample. NER can determine only one component χ ⁽³⁾ _xyyx , but not the total third-order nonlinear susceptibility χ ⁽³⁾ .

 

Fig. 1. (a) Experimental arrangement for NER modified Z-scan. D1 and D2 are the detectors. (b) and (c) are the geometry of incident and the transmitted polarization ellipse, respectively. The slow axis of the first λ/4 plate is taken along the x-axis. α is the angle between the first polarizer direction and the x-axis. θ is the rotating angle of the polarization ellipse induced by nonlinearity. φ is the angle of the analyzer.

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Since the rotation angle θ of the polarization ellipse depends upon the intensity of the input beam, the tight-focus geometry in Z-scan can be also used in NER. It is possible to combine the advantages of Z-scan and NER to measure the component χ ⁽³⁾ _xyyx simply and sensitively. Figure 1(a) gives the configuration of the NER modified Z-scan method, which is the same as that of an open-aperture Z-scan except that two paralleled polarizers and two crossed quarter-wave plates are used. When the sample is far away from focus, the beam irradiance is low and NLR is negligible; the polarization state remains unchanged for incident and transmitted beams. Hence, all of the transmitted irradiance through the sample is collected into detector D2, and the transmittance [D2/D1, in Fig. 1(a)] remains relatively constant. As the sample is brought closer into focus, the beam irradiance increases, thus leading to the rotation of the polarization ellipse. The rotation of the polarization ellipse permits only part of the transmitted irradiance to pass the second polarizer, and a decrease in the measured transmittance occurs. Such Z-scan traces with NER are expected to be symmetric with respect to the focus (z = 0); they have minimum transmittance at focus. The coefficient of χ ⁽³⁾ _xyyx can be calculated easily from such transmittance.

3. Theory

Let us consider a linearly polarized beam propagating along the z-axis. The electric field can be written as E_lin = E₀(r,z)exp[i(kz-ωt)], where k = nω/c is the wave vector, ω is the frequency of light, and n is the linear refractive index of the medium. We assume a stationary regime, and thus the amplitude E ₀(r,z) does not depend on the time t. When the linearly polarized beam passes the λ/4 plate with angle α [with -π/2≤α≤π/2 being the angle between the linear polarization direction and the λ/4 plate slow axis—see Fig. 1(b)], it can be converted into an elliptically polarized beam. The electric field E of such an elliptically polarized beam can always be decomposed into a linear combination of the x- and y-direction components (E_x and E_y), or left- and right-hand circular components (E ₊ and E ₋) with the unitary transformation E₊ = (E_x - iE_y)√2 and E ₋ =(E_x+iE_y)/√2. If we define the λ/4 plate slow axis as the x-axis, the electric field of x- and y-direction components can be written as

where δ is the phase retardation. δ = π/2 for the λ/4 plate.

For the choice of frequencies given by χ ⁽³⁾ _ijkl (ω=ω+ω−ω) , the condition of intrinsic permutation symmetry requires that χ ⁽³⁾ _xxyy be equal to χ ⁽³⁾ _xyxy . Hence, there are only two independent elements of the susceptibility tensor describing the NLR of an isotropic medium. Following the notation of Maker et al. [4], the nonlinear polarization P ^NL can be written as

where A = 6χ ⁽³⁾ _xxyy and B = 6χ ⁽³⁾ _xyyx. Hence, the total refractive indexes of two circular components is given by [9]

We see from Eq. (3) that the change in refractive index δn _± is different for two circular components and depends upon ellipticity e, where e=||E ₊|-|E ₋||/(|E ₊| + |E ₋|) . First, we consider two specific cases. One case is for a circularly polarized beam with e=1; this means that only one of two circular components is present. Thus, the change of the refractive index can be given by δn=(2π/n ₀)A|E|², which clearly depends on A but not B. The NLR coefficient n ₂ of the total beam should reach a minimum in this case. Another case is for a linearly polarized beam. Since a linearly polarized beam is a combination of equal amounts of left-and right-hand circular components (i.e. |E|² = 2|E ₊|²=2|E ₋|²), the change of the refractive index can be given by δn = (2π/n ₀)(A+B/2)|E|² , and n ₂ reaches a maximum. For the case of arbitrary ellipticity, n ₂ can be written as

where q = (1-e)/(1+e), and n _2cir and n _2lin are the NLR coefficients in the case of circularly and linearly polarized beams, respectively.

We can see from Eq. (3) that the left- and right-hand circular components of the beam propagate with different phase velocities because of different δn _± and cause the rotation of the polarization ellipse of the transmitted wave [see Fig. 1(c)]. The angle of rotation can be written as

where Q = -(2πω/cn)Bd sinα cosα and d is the sample length. In new x́-ý coordinates taken along the major and minor axes of the ellipse, we can write the electric field E as

where x $\acute{̂}$ and y $\acute{̂}$ are the polarization unit vectors in the new coordinates system with x $\acute{̂}$ = x̂cosθ-ŷsinθ and y $\acute{̂}$ = x̂sinθ + ŷcosθ. Therefore, by transforming the electric field vector from x́-ý coordinates to x-y coordinates, E can be written as

The output beam is then directed to another λ/4 plate crossing to the input λ/4 plate. Its phase retardation is Δ. If the orientation of the analyzer (second polarizer) has an angle of φ (-π/2≤φ≤π/2) relative to the x-axis, the output electric field E _out through the analyzer can be written as

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

where E ₀₀ is the on-axis electric field at focus, w_z ² = w ₀ ²(1 + z ²/z ₀ ²) is the beam radius, R_z = z(1 + z ₀ ²/z ²) is the radius of curvature of the wavefront at z, and z ₀=kw ₀ ²/2 is the diffraction length of the beam. For the spatial Gaussian beam, the influence of transverse effects on self-induced polarization changes must be considered [10,11]. Hence, the transmitted power through the analyzer is obtained by spatially integrating E _out up to infinity:

Substituting Eqs. (8) and (9) into Eq. (10) and completing the spatial integration, we can obtain an expression of transmitted power as a function of z as follows:

where ε ₀ is the permittivity of vacuum and P = E ₀₀ ² w ² ₀/w_z ². In the case of linear propagation (i.e., θ = 0), the transmitted power can be written as

The normalized Z-scan transmittance can be calculated by

We first consider the case in which the analyzer is set parallel to the first polarizer, and meanwhile the second λ/4 plate exists, i.e., α=φ and Δ = π/2. The normalized transmittance T(z) can be written as

Another case is where the second λ/4 plate is removed, and the sensitivity of the Z-scan measurement is enhanced as analyzed below. The removal of the second λ/4 plate makes Δ = 0. Hence, T(z) can be written as

where R = 1/(sin² αsin² φ + cos² αcos² φ).

Now we can easily extend the steady-state results to the transient effects induced by pulsed radiation by using the time-averaged rotation angle 〈θ(t)〉:

For a nonlinearity having instantaneous response, we obtain for a temporally Gaussian pulse

4. Experimental results and discussions

Our experiments are carried out on an isotropic nonlinear medium, CS₂ This molecule has been studied extensively by many experimental methods and now is widely used as a reference sample. As the nonlinearities of CS₂ mainly rely on the molecular reorientation effect in the subnanosecond regime, the high ratio of B to A attains 6 [9] and the NLR coefficient n ₂ =3.4×10^-18 m²/W for linearly polarized light [6,7]. In our experiments, a mode-locked Nd:YAG laser (Continuum Model PY61) was used to generate 30 ps pulses at 532 nm with a repetition rate of 10 Hz. The pulse laser focused by a lens to form a beam waist w ₀ of 22 μm was incident to a 1 mm cell containing CS₂. The on-axis peak intensity I ₀ was 5.93GW/cm². To keep the intensities of input beams fixed for different polarization states, linear, elliptical, and circular polarizations were realized by altering only the angle between the first polarizer and the λ/4 plate with α = 0, α = -22.5°, and α = -45°, i.e., the ellipticity e = 0, e = 0.4142 , and e = 1, respectively. The negative sign of α implies that the beam is a left-handed elliptical polarization after passing through the λ/4 plate.

First, we examined the dependence of NLR on the polarization ellipse. Figure 2 gives the normalized transmittance of NLR for linear, circular, and elliptical polarization obtained by a traditional closed aperture Z-scan in which an aperture was used instead of the second λ/4 plate and the analyzer in Fig. 1(a). Among those, the NLR is largest for an elliptically polarized beam and smallest for a circularly polarized beam. The values of n ₂, fitted by using the theoretical model of Ref[6], are 3.0×10^-18, 1.6×10^-18, and 0.78×10^-18 m²/W for linear, circular, and elliptical polarization, respectively. From Eq. (3), one can get the ratio of n _2lin to n _2cir to be 1 + B/2A. For molecular reorientation nonlinearities, n _2lin/n _2cir=4 owing to B/A = 6. It is obvious that n _2lin/n _2cir = 3.8 obtained experimentally agrees well with the theoretical one. Note that the change of NLR for different polarization states depends upon B but not A. It is identical to NER in which the rotation of the elliptical axis is caused only by B.

 

Fig. 2. Z-scan curves for linear, circular, and elliptical polarization obtained by traditional closed aperture Z-scan.

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The real part of third-order nonlinear susceptibility χ⁽³⁾ is related to the NLR coefficient n ₂ through [12]

Therefore, from the results of the Z-scan with a circularly polarized light, we can obtain A = 2.16×10^-20m²/V², since the changes of refractive index and absorption only depend on A in the case of circular polarization. And then, B can be determined to be 12.8×10^-20m²/V² by the Z-scan experimental results of linear polarization or elliptical polarization, which is 5.9 times as large as that of A. Therefore, B/A is closer to that of theoretical analysis.

 

Fig. 3. Z-scan curves of NER modified Z-scan with and without the second λ/4 plate. The polarization directions of the polarizer and the analyzer parallel each other (α=φ=-22.5°). The solid lines are the theoretical fittings with B = 12.6×10^-20m²/V² and I ₀ = 5.93 GW/cm².

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To measure the coefficient B directly, we used the experimental setup combining Z-scan and NER as shown in Fig. 1(a), and two paralleled polarizers and two crossed quarter-wave plates were additionally used. The circular symbols in Fig. 3 represent the experimental results of the Z-scan. Using Eq. (14) to fit the experimental data, we obtain coefficient B to be 12.6×10^-20m²/V², which is identical to the results of closed aperture Z-scan measurements within errors. If the second λ/4 plate is removed and other experimental conditions are kept unchanged, the transmittance change of the polarization ellipse through the analyzer due to the rotation of axis position can be observed directly. The valley of the Z-scan curve has a larger magnitude than that with the second λ/4 plate, indicating that the removal of a λ/4 plate can enhance the sensitivity of Z-scan measurements.

 

Fig. 4. (a) Normalized transmitted power through the analyzer at linear output and nonlinear output obtained by rotating the analyzer. (b) The ratio of nonlinear transmitted power to linear transmitted power as a function of rotation angle of the analyzer.

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Using the experimental setup without the second λ/4 plate and fixing the sample on focus, the information of the ellipse axis and ellipticity e can be obtained by measuring the transmitted power as the analyzer rotates. The normalized transmitted power as a function of φ are shown in Fig. 4(a) for linear output at low input power (I ₀ < 10⁶ W/cm²) and nonlinear output at high input power (I ₀ = 5.93GW/cm²). A 19° rotation of the polarization ellipse at nonlinear output relative to that of linear output can be observed, while no obvious change of ellipticity occurs. Figure 4(b) gives the ratio of transmitted power at nonlinear output to that of linear output. The solid line is the theoretical fit with B= 12.6×10^-20m²/V², which is easily obtained by assuming z = 0 in Eq. (15). In other words, the ratio in Fig. 4(b) also shows the valley change of the NER-modified Z-scan without the second λ/4 plate as the analyzer rotates. Therefore, if we change the orientation of the analyzer φ, Z-scan curves will give different profiles as shown in Fig. 5. First, φ=90° means that the analyzer is along the minor axis of the polarization ellipse; hence, the rotation of the polarization ellipse causes the increase of transmitted power through the analyzer, and the Z-scan curve exhibits a peak structure as shown in Fig. 5(a). For φ=80° and 76°, the normalized transmittance first decreases and then increases as the sample moves toward focus [see Figs. 5(b) and 5(c)]. The decrease of normalized transmittance is caused by the relative rotation of the analyzer to the minor axis of the polarization ellipse and is terminated while the analyzer is along the minor axis. Thereafter, the polarization ellipse continues to rotate as the sample moves continually, since the maximum rotation angle of polarization ellipse attains 19° (80° + 19° = 99° and 76° + 19° = 95° are larger than 90°). Therefore, after the analyzer reaches the minor axis, the analyzer will be away from the minor axis as the input intensity further increases, leading to the increase of transmittance. In the case of φ = 55° (55° + 19° < 90°), the Z-scan curve exhibits a valley structure as shown in Fig. 5(d), since the analyzer rotates towards but does not reach the minor axis of the polarization ellipse.

As mentioned above, although the nonlinear coefficient measured by the NER modified Z-scan is a real part of the nonlinear susceptibility component χ ⁽³⁾ _xyyx, the curves of the NER modified Z-scan are similar to those of an open-aperture Z-scan with multiphoton absorption or saturable absorption. Furthermore, the experimental setup of the NER modified Z-scan is also similar to an open-aperture Z-scan and has less strict conditions on the beam profile than a closed-aperture Z-scan. This is because the nonlinear effect measured by an open-aperture Z-scan depends upon the amplitude change but not the phase distortion of the beam.

 

Fig. 5. Z-scan curves of NER modified Z-scan without second λ/4 plate at (a) φ=90°, (b) φ = 80°, (c) φ=76°, and (d) φ=55°. The solid lines are the theoretical fittings with B=12.6×10^-20m²/V² and I ₀=5.93 GW/cm².

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5. Conclusion

We present here a method that combines the Z-scan technique with NER to study third-order nonlinear susceptibility tensor of isotropic media. It can obtain complete data on third-order susceptibility tensor properties of a material. Our results show that NLR is dependent on polarization ellipticity. It is expected that this method can be extended to anisotropic media.