## Abstract

We report on the accurate measurement of nonlinear ellipse rotation (NER) by means of a phase-sensitive method employing a dual-phase lock-in. The magnitudes and signs of pure refractive electronic nonlinearities of silica and BK7 were determined with this new method using 150 femtosecond (fs) laser pulses at 775 nm. Experimental and theoretical analyses of the NER signal were carried out and the results were compared to those obtained with the *Z*-scan technique.

© 2014 Optical Society of America

## 1. Introduction

The measurement of third-order nonlinearities still attracts great interest due to the continuous need of characterizing new materials with simpler and more sensitive techniques. There are a number of methods already developed and amongst them, the *Z*-scan technique [1] became a standard tool because of its simplicity, precision and ability to measure nonlinear refraction (magnitude and sign) and absorption with a single laser beam. However, since it probes variations in the beam profile due to the Kerr lens effect, it can be influenced by spurious factors such as the laser transverse mode quality and sample inhomogeneities. In addition, there are cases where nonlinear effects of different origins contribute to the Kerr lens and, consequently, the standard single beam *Z*-scan may not discriminate them. On the other hand, an interesting phenomenon associated to the refractive nonlinearity that appears when an elliptically polarized beam is employed is the nonlinear ellipse rotation (NER) [2,3] which samples only the *χ*_{1221} third-order nonlinear susceptibility component of an isotropic medium. This particular characteristic of the NER measurement can be exploited to discriminate the origin of the nonlinearity and thus, improvements of such technique are welcome.

Usually, the ellipse rotation angle in NER is detected by decomposing the elliptically polarized beam in two orthogonal linearly polarized components [4–6] and measuring the change in their ratio. However, this approach is not very precise for small rotations and has usually been employed just to study strong nonlinearities such as that related to the molecular orientation that appears, for instance, when nanosecond (ns) and picosecond (ps) pulses are applied to CS_{2} molecules. There are new methods proposed in the literature where NER is combined with the *Z*-scan technique to explore the potential that the use of elliptically polarized light offers [7,8]. For example, it is possible to easily discriminate between thermal and orientational nonlinear effects because NER is insensitive to the thermal nonlinearity [9]. More recently, NER were also studied with ultrafast fs pulses [10,11], and some preliminary measurements with femtosecond and picoseconds pulses at 532 nm were carried out to distinguish between electronic and orientational nonlinearities [12].

The present work aims at improving the precision of the refractive nonlinearity determination by NER with a phase sensitive method based on a dual phase lock-in amplifier [13]. As we shall see later, the NER can be related to a phase change when an elliptically polarized laser beam is modulated by a rotating polarizer. In this way, we can take advantage of the high sensitivity and precision of lock-in amplifiers to measure this phase change. The method has great simplicity and is highly sensitive for refractive nonlinearity determination using an experimental configuration similar to that of the *Z*-scan technique. The sensitivity and precision introduced by the use of a lock-in amplifier allowed to measure small electronic refractive nonlinearity of silica and BK7, with pulses with 150 fs duration at 775 nm. *Z*-scan measurements were also performed for calibration purposes and to compare with our results.

## 2. Theoretical considerations

The NER is a nonlinear optical process that occurs when an intense elliptically polarized laser beam interacts with an isotropic medium and changes the refractive indices for left- and right-circular polarizations. In order to explain this effect one has to consider the tensor nature of the third-order nonlinear susceptibility. In an isotropic medium, there are only two independent tensor components which can be represented by coefficients *A* and *B* [14]. The ratio between these two coefficients depends on the physical origin of the nonlinearity. For example, for molecular orientation, *B*/*A* = 6, while for a pure nonresonant electronic process, *B*/*A* = 1 and for electrostriction or thermal nonlinearity, *B*/*A* = 0. Also, depending on the experimental technique and laser regime (pulse duration, polarization state, wavelength, etc.), different coefficients can be excited. In the *Z*-scan technique, the signal obtained with linearly polarized light depends on both *A* and *B* coefficients while for circular polarization it depends only on *A*. On the other hand, NER depends only on the coefficient *B*. Therefore, the study of the NER signal as a function of the laser polarization and its comparison with different techniques may improve the understanding of refractive nonlinearities.

In the tensorial model for the third-order nonlinearity in isotropic materials, the coefficients *A* and *B* are related to susceptibility coefficients according to [14]:

*Z*-scan signal

*ΔT*, is proportional to

_{pv}*Δn*and, depending of nonlinearity origin we have:

*E*|

^{2}=

*I*/(2

*n*

_{0}

*ε*

_{0}

*c*), for a beam with linear polarization and frequency

*ω*passing through a sample of thickness

*L*, one can find the phase change,

*Δϕ*, for

*A*=

*B*(nonresonant electronic process) as:

*Z*-scan signal for linear polarization. In the case of NER, there is an anisotropy when right-hand ( + ) and left-hand circular (-) components of the beam propagate with different phase velocity. This produces an index change giving by:

*E*and degree of ellipticity of the polarization

*θ*as [10]:

*θ*= tan

*ϕ*, and

*ϕ*is the angle between the axis of the input linear polarized laser beam and the fast axis of a quarter-waveplate (positive angle is for counter-clockwise waveplate rotation if you look back at the source). Frequently, a quarter-waveplate is used to control the polarization state of the laser beam from linear to elliptical polarization. For instance,

*ϕ*= 0° (

*θ*= 0) and

*ϕ*= 45° (

*θ*= 1) for linear and circular polarizations, respectively. Equation (5) predicts no index anisotropy for linear polarization (|

*E*

_{+}| = |

*E*

_{-}| =

*E*/√2), but for elliptical polarization, there is an anisotropy that produces a nonlinear phase change,

*Θ*, given by:

This phase change rotates the ellipse and the effect is dependent only on the *B* coefficient. The above expression was obtained for the case of a plane wave, however, all measurements are carried out with lasers with a Gaussian profile presenting a transverse intensity gradient. In this case, the original plane wave irradiance of Eq. (7) must be divided by 2 (*I*(*z* = 0) = *I*/2) [4] and the rotation angle for beams with transversal Gaussian mode is:

The maximum rotation is achieved for *θ* = 1 (circular polarization, (*θ* /(1 + *θ*^{2}) = 0.5)), and the maximum *ΔT _{pv}* in the

*Z*-scan technique is obtained for a linearly polarized beam. It is interesting to point out that

*θ*/(1 +

*θ*

^{2}) = 0.5 sin(2

*ϕ*). Using Eqs. (4) and (8) one can link

*α*and

*Δϕ*(

*Z*-scan).

In a practical point of view, the maximum ellipse rotation (*α _{max}*) for a nonresonant electronic nonlinearity can be related to Δ

*T*obtained with the

_{pv}*Z*-scan method for small apertures (

*S*≈0) and linearly polarized light as:

*Δϕ*> =

*Δϕ*/√2,

*<α*/√2, for nonlinear response faster than the pulse duration) [1]. It is also important to point out that in the NER measured using a rotating analyzer and the lock-in amplifier, the angle measured by the lock-in is twice that of the real angle due to the Malus’s law. From the experimental point of view, one can work directly with the angle given by the lock-in:

_{max}> = α_{max}It is also possible to determine the sign of the nonlinearity because according to the theory, right-elliptically (left-) polarized beam tends to rotate to right (left) for a positive nonlinearity and the opposite for the negative nonlinearity. Experimentally, in order to find it is necessary to know all parameters used in the experiments, especially the signal of the phase given by lock-in and the sign of the elliptical polarization defined by the orientation of the fast and slow axes of waveplate.

## 3. Experimental setup

The rotation of an elliptically polarized beam is generally determined by using a polarizer (analyzer) to select the two orthogonal linear polarizations to be measured. This method is not sensitive to small rotations [4,5] but can be easily improved by passing the elliptically polarized beam through a rotating linear analyzer. In this case, a modulation proportional to the ellipticity is produced and the ellipse rotation can be associated to a phase change that can be measured with a dual-phase lock-in amplifier [13]. The maximum and minimum in the modulated signal correspond to the major and the minor axis of the ellipse, respectively.

Using a setup similar to that of the *Z*-scan technique one measures the nonlinear
polarization rotation of a laser beam as a function of *z* position. The
schematic diagram of the experimental setup is shown in Fig.
1.We have used 150 fs pulses at a 1 kHz repetition rate from a Ti:Sapphire amplified system
(CPA 2001, Clark MXR) centered at 775 nm. As in the standard *Z*-scan method, the
sample translation along the *z*-direction is used to change irradiance on the
nonlinear medium. Using thin samples (thicknesses smaller than the Rayleigh range,
*z*_{0}) and starting the scan from a distance far away from focus, the
beam irradiance is initially low and negligible NER signal is observed (zero phase). As the
sample is moved closer to the focal point, the irradiance increases and the NER becomes visible,
reaching its maximum at the focal point. As the sample moves away from the focus, the NER signal
decreases again and vanishes for *z* >> *z*_{0}. As
we are interested in the average rotation along the transversal Gaussian profile of the laser
beam, we use a lens to collect the entire beam, similar to the open-aperture
*Z*-scan method, and consequently, this method is less sensitive to sample
inhomogeneity.

In Fig. 1, an appropriate zero-order quarter-waveplate is used to control the polarization state of the ultrafast laser. A lens with *f* = 12 cm is used to focus the beam into the nonlinear sample. As in the open-aperture *Z*-scan, we have used a lens to direct the laser beam into a large area Si-PIN photodetector. A rotating standard polaroid filter is used to modulate the laser beam at the photodetector, whose output is fed into an oscilloscope.

As we explained earlier, a high intensity elliptically polarized laser beam suffers a rotation after passing through a nonlinear medium. Such nonlinear rotation corresponds to a phase change in the modulated signal which can be easily measured by the lock-in amplifier. The dual-phase lock-in measures the *In Phase* (*X*) and the *In Quadrature* (*Y*) signals, and the angle change can be determined according to: *α* = atan(*Y*/*X*). Also, one can use two single phase lock-in amplifiers instead of the dual phase lock-in, if necessary. This method allows measurements of angle changes with high precision. According to the theory, no nonlinear polarization rotation is expected for a linearly polarized beam. In addition, no modulation can be produced for circular polarization, preventing any phase detection in this case. For any polarization different from linear and circular the NER signal can be obtained. The rotation observed in thin samples is proportional to laser irradiance at each *z*-position and since the irradiance is inversely proportional to the square of the beam waist, the rotation angle will depend on *z* according to:

*α*is the maximum phase (angle) achieved for a particular ellipticity of the laser beam. This gives a Lorentzian curve as the one obtained for typical absorptive processes.

## 4. Results and discussion

In order to test the method, we carried out *Z*-scan and NER measurements in
silica (thickness = 1.22 mm) and BK7 glass (thickness = 0.93 mm), where the chief nonlinearity
is purely electronic at the femtosecond (fs) regime. First, we performed *Z*-scan
measurements in silica with the purpose of calibrating the laser beam parameters. We measured
normalized transmittances as function of *z* for different laser ellipticities
that were controlled by the angle *ϕ* of the quarter-waveplate. Figure 2(a) shows the refractive *Z*-scan
signature for linear and near circular polarization beam. By using the well-known electronic nonlinearity of silica (*n*_{2}
= 2.5 × 10^{−20} m^{2}/W, *B* = 3.74 ×
10^{−22} m^{2}/V^{2}) [15] for linear polarization (*ΔT _{pv} =* 0.11), we obtain
the laser beam irradiance (

*I*

_{0}≈155 × 10

^{9}W/cm

^{2}), Rayleigh range (

*z*= 0.08 cm) and beam waist (

_{R}*w*

_{0}≈14 μm). Figure 2(b) depicts

*ΔT*as function of the ellipticity achieved for different

_{pv}*ϕ*s. It shows a ratio of 3/2 between

*ΔT*obtained for linear and circular polarization, indicating a pure electronic nonlinearity. In order to adjust the experimental points with the expected theoretical curve, we have considered 8% of error in the experimental data, which is the typical uncertainty of the

_{pv}*Z*-scan method.

Next, we performed NER measurements as function of the ellipticity using the same laser
conditions of Fig. 2. The results are shown in Fig. 3 together with a fit carried out with the Lorentzian
Eq. (12), which allowed to confirm the Rayleigh
range and the beam waist obtained with the *Z*-scan technique. Carrying out these measurements for different ellipticities, different maximum phase
changes are obtained at *z* = 0. Figure
3(b) shows these results and the fitting to Eq.
(8), where 8% error bars was also considered. From *ϕ* = 0°
to 90° and *ϕ* = 90° to 180°, we have right-hand and
left-hand elliptical polarization, giving positive and negative NER signals, respectively, which
is expected for the positive nonlinearity of silica. The maximum phase amplitude was observed
for *ϕ* near 45° (or 135°) where
*<α _{max}*>

*= 0.08 rad (~2° rotation of the elliptical polarization).*

_{lock-in}Using the laser irradiance determined by the *Z*-scan measurement, together with Eq. (8) (with *θ* = 1) and Eq. (12), gives *n*_{2} = 2.2 × 10^{−20} m^{2}/W, which is slightly smaller than the value known for silica. Another approach is to use Eq. (11) to directly relate the maximum phase read in the lock-in with *ΔT _{pv}*. In this case, the expected maximum phase would be

*<α*>

_{max}*= 0.09 rad for*

_{lock-in}*ΔT*= 0.11 (linear polarization), which is slightly larger than 0.08 rad, but in good agreement with what is expected due to the imprecision of each technique.

_{pv}We also performed measurements in BK7 optical glass using the same laser conditions as for
silica. Similar results (Fig. 4) were obtained in this
case for *Z*-scan and NER methods, including the same error in the experimental
data.

Using the refractive theory for *Z*-scan and the laser irradiance determined in the previous experiment with silica, we determined the BK7 nonlinearity. For linear polarization, we observed *ΔT _{pv}* = 0.11 and, consequently, the

*n*

_{2}= 3.3 × 10

^{−20}m

^{2}/W (

*B*= 5.3 × 10

^{−22}m

^{2}/V

^{2}), which is about 1.3 times higher than in silica, which agrees with the values already determined for BK7 [16,17]. Like silica, we have observed the ratio of 3/2 between

*ΔT*obtained for linear and circular polarization, indicating a pure electronic nonlinearity. No perceptible nonlinear absorption was observed in both samples for the irradiance used here.

_{pv}For BK7 (Fig. 5), the maximum rotation was also observed
near 45° (or 135°) where
*<α _{max}*>

*= 0.08 rad. The maximum phase predicted by Eq. (11) using the*

_{lock-in}*Z*-scan result (

*ΔT*= 0.11) is

_{pv}*<α*>

_{max}*= 0.09 rad, which is also close to that measured by the NER method. In this case, the magnitude of the phase measured for BK7 was also 1.3 times higher than silica.*

_{lock-in}Based on the results achieved with these two samples, we notice that the nonlinearities obtained
with the *Z*-scan technique are slightly larger than those obtained by the NER
technique. We believe that this happened because the laser beam used here was not purely
Gaussian, as required in the *Z*-scan method. Usually, in order to improve the
transverse mode quality of the laser beam, one uses a spatial filter (pinhole) set inside a
proper telescope, and in this situation it is possible to get a spatial mode that increases the
amplitude of the *Z*-scan refractive signal [18]. Such situation is not a problem for measuring different samples if all
measurements are carried out with the same experimental conditions and a well-known nonlinear
medium is used as reference. However, to compare two distinct methods which explore different
nonlinear effects, such absolute value comparison can be a big issue. Nevertheless, the results
obtained by these two methods show very good agreement considering that there are an uncertainty
of about 8% in the magnitude obtained by them. In fact, the noise of 8% estimated by the
dispersion of NER signal obtained for different waveplate angles do not reflect that the NER
signal has low noise at some particular ellipticity. In the NER measurement using a dual phase
lock-in and rotating polarizer there is a relation between ellipticity, NER amplitude and noise.
When a near linear polarization is used, the laser beam is well modulated when the analyzer
rotates improving the signal amplitude reading in the lock-in, but in this case, the NER
magnitude is very small. On the other hand, using a near circular polarization, the expected NER
signal is higher but, the laser beam can be barely modulated by rotating analyzer leading to a
poor signal in the lock-in. In this way, a good signal-to-noise is obtained at elliptical
polarization near to *θ* = 0.414 (quarter-waveplate set at
*ϕ* ≈22.5°). As seen in Fig. 6, using *ϕ* = 20° (*θ* =
0.364) we obtain good NER signal with very low noise.

In Fig. 6(b) we observe a maximum noise of about 0.008 rad which is equivalent to a phase distortion of about *λ*/800. Such a resolution is considerable better compared to the 8% estimated from the dispersion of all data obtained with multiple ellipticities. This is a clear indication that there are good potential to improve the determination of the magnitude of the material’s nonlinearity using NER measurements, especially when a standard nonlinear material is used as reference. In other words, we can make measurements with different samples, including the reference, at the same laser parameters at a fixed ellipticity, *θ* = ± 0.414, where the signal-to-noise reaches its best condition.

## 5. Conclusions

In summary, we proposed to use a dual-phase lock-in to improve the measurement of NER angles in order to determine the magnitude and sign of the third-order material’s nonlinear refractive indices. Pure electronic nonlinearities of silica and BK7 optical glass were measured using laser pulses at 775 nm with 150 fs duration. The well-known *Z*-scan method was used to demonstrate that the absolute nonlinearities obtained by this new method are appropriates. We believe that this method can be a useful tool for measuring nonlinearities of different materials due to high sensitivity, low noise and simplicity, especially, when a reference sample is used because such quality features improve at some particular beam ellipticity conditions.

## Acknowledgments

This research was supported by the Brazilian agencies: Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

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