1. Introduction

The field of Nonlinear Optics has provided many techniques to characterize photonic materials. The nonlinear refractive index can be determined by spectroscopic methods exploiting nonlinear interferometry [1], wave mixing [2] and beam distortion measurements [3]. One well established method, known as Z-scan (hereafter called conventional Z-scan), was introduced in 1989 by Sheik-Bahae and coworkers [3]. Since then, several variations of the method have been developed, including improvement of its sensitivity [4], allowing time-resolved measurements [5] and thermal and non-thermal analysis [6, 7].

The Z-scan technique [3] is the most widely used method to measure the nonlinear index of refraction and the nonlinear absorption of organic materials [8], optical materials [9], nanostructured materials [10], etc. The nonlinear characterization is important to describe effects such as second harmonic generation [11] and multiphoton fluorescence [12]. The Z-scan method exploits the wavefront distortion of a focused beam, analyzing the transmittance value of a sample measured through a finite aperture in the far field as the sample is moved along the propagation path (Z)[3]. Theoretically, it has a wavefront sensitivity of better than λ/300 to determine the nonlinear properties of optical materials [3]. However, for laser systems having pulsewidth instabilities, fluctuations of the irradiance or sample imperfections it may lead to a loss in resolution. As reported in Ref. [13] a 5% fluctuation of laser intensity is enough to decrease the wavefront resolution to approximately λ/10.

Here we propose two different techniques to characterize the nonlinear refractive index of materials. The methods use a Hartmann-Shack wavefront sensor (HS) with near-IR light (800nm) to determine the nonlinear optical properties of materials in the femtosecond regime. The first technique uses a collimated setup, where the beam transmitted through a sample is analyzed with a HS sensor. The second technique is a new variation of the conventional Z-scan method, where the wavefront sensor measures simultaneously, the wavefront distortion and the transmittance changes along the focalized laser beam (hereafter called Hartmann Shack Z-scan or HS Z-scan). As the beam radius of curvature is not sensitive to intensity fluctuations, measuring the wavefront instead of the transmittance makes the HS Z-scan technique highly sensitive. The two techniques are demonstrated measuring the n ₂ value of CS ₂ and Quartz, transparent materials that are commonly used as reference materials in nonlinear optics [3]. The usefulness of the techniques is under evaluation with samples having strong (nonlinear) absorption.

A HS wavefront sensor is commonly used for adaptive optics systems in astronomy [14], visual optics and retinal imaging [15, 16], optical components testing and laser beam characterization [17]. It consists of an array of microlenses of equal focal length, focusing an incident wavefront onto a CCD array. The local tilt of the wavefront across each lenslet can be calculated from the displacement of each focal spot on the sensor. By sampling an array of lenslets all of these local tilts can be measured and the whole wavefront determined. The wavefront can be expressed in terms of a set of Zernike polynomials as W(x,y)=∑^∞ ₀ c_nZ_n(x,y), where the wavefront W(x,y) is derived from a summation of Zernike polynomials Z_n(x,y) weighted by the Zernike coefficients c_n[18]. Different wavefront aberrations such as Tilt, Astigmatism and Defocus can be described using the Zernike functions. Common low-order Zernike polynomials in the Malacara index scheme are shown in Table 1 as a function of polar coordinates (r,θ) [14].

Table 1. Malacara Zernike index scheme

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If a sufficiently energetic laser pulse propagates through a nonlinear medium, it will induce an intensity-dependent change in the index of refraction given by n(r, t)=n ₀+n ₂ I(r, t), where I is the irradiance of the laser beam, and n ₀ and n ₂ are the linear and nonlinear indices of refraction, respectively [19]. When n ₂>0 or n ₂<0 the induced wavefront distortion results in a self-focusing or self-divergence effect of the beam.

Assuming a Gaussian beam travelling in the z direction we can write the complex electric field exiting the sample as E_out (z,r,t)=E_in(z,r,t)e ^-αL/2 e ^{iΔϕL(z,r,t)} where Ein is the incident electric field, L is the sample length and α is the linear absorption. The phase change inside of the sample is exclusively due to linear refraction given by Δϕ_L(z,r,t)=(2π/λ)n_iL where n_i is the linear refractive index of the sample. For a high intensity beam a new phase change Δϕ_NL(t) is introduced due to self-phase modulation effects, being directly proportional to n2 and the (ideally) Gaussian beam intensity $I\left(r,t\right)=t_0\left(t\right)e^{[-2r^2⁄w^2]}$. Therefore, the nonlinear phase change is given by [3] :

where L_eff is given by L_eff=(1-e ^-αL)/α. Therefore, the total wavefront at the end of the sample is given by W_total=W _(z=0)+W _L(z=L)+W _NL(z=L), where W _(z=0) contains the wavefront profile including the aberrations of the experimental setup, W_{L (z=L)} the aberration induced by the linear refraction and sample imperfections, and WNL(z=L) the nonlinear phase change accumulated in the sample. The nonlinear phase change Δϕ_NL(t) can be expressed in terms of Zernike polynomials.

2. System configurations

The two methods were performed using an infrared (800nm) femtosecond laser, Coherent Libra (mJ, 80fs) delivering optical pulses at 1kHz repetition rate. In the first technique (see Fig. 1), we also used a Coherent Mira (nJ, 150fs) at 76MHz, ideal to create thermo-optical effects inducing nonlinear negative refraction effects in the sample [7]. The wavefront was acquired using a H-S wavefront analyzer (Thorlabs WFS150C) with a 15Hz frame rate (i.e. 66ms).

2.1. Beam collimated configuration

As shown schematically in Fig. 1, the laser intensity was controlled with a polarizer and a half-wave plate (λ/2) and the beam diameter was collimated and reduced to 1.1mm using a keplerian telescope (T1) such that the beam Rayleigh length (L_R~5m) remained much larger than the sample thickness (L≤1cm). At the exit of the sample the laser beam was magnified with a second telescope (T2) to obtain a beam waist about 3mm and imaged on a HS wave-front analyzer. The wavefront data was acquired using the Thorlabs software and the National Instruments DataSocket server integrated with the LabView platform.

 

Fig. 1. Experimental set-up for HS sensing. A half-wave plate (λ/2) and a polarizer are used to adjust the laser intensity. T1 and T2 are Keplerian telescopes.

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2.2. HS Z-scan configuration

The experimental setup for HS Z-scan is shown schematically in Fig. 2. The optical beam was focused using a 15cm focal length lens (L1), and after exiting the sample the beam were captured by the HS sensor. The HS sensor is located at a distance such that the laser beam overfills its sensitive area (diameter=4mm). A computer-controlled motor translation stage is used to scan the sample along the beam while the Zernike coefficients and the CCD array intensities are recorded. The beam waist measured at the sample position was 20µm.

When a convergent Gaussian beam passes through a self-focusing absorbing medium, the radius of curvature of the wavefront and the beam intensity changes. Due to the HS sensor characteristics, it is possible to measure simultaneously the sample transmittance on the CCD array and the wavefront distortion. Therefore, just one branch is necessary to measure the wave-front distortions due to the nonlinear self-focusing effect as well as transmittance changes due to the nonlinear absorption during the axial scan [20].

 

Fig. 2. HS Z-scan experimental set-up.

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3. Experimental results and discussion

3.1. Beam collimated configuration

A wavefront reference was taken using lower intensities with the sample placed in the collimated beam, W_ref=W _(z=0)+W _L(z=L). In that case the measured wavefront changes are based on this reference to allow accurate measurements of further wavefront distortions W _NL(z=L) induced by higher intensities.

We have used the initial 6 Zernike modes indicated in Table 1 to fit the measured wavefront within the pupil area (the piston term, not measurable with the HS sensor has been excluded). Figure 3 shows the Zernike coefficients used to describe the wavefront aberrations induced by the CS2 sample to different intensities (I_a=3.0GW/cm ², I_b=4.5GW/cm ², I_c=5.5GW/cm ² and I_d=6.0GW/cm ²). While Zernike coefficients corresponding to tilt and astigmatism (C ₂, C ₃, C ₄, C ₆) remain approximately constant there is a dependence of the defocus coefficient C ₅ with the laser intensity. In accordance with the Zernike orthogonality properties [14], the defocus Zernike is thus enough to describe the wavefront distortions caused by nonlinear effects.

 

Fig. 3. Zernike coefficients used to describe the wavefront aberrations induced for different intensities: (a) I_a=3.0GW/cm ², (b) I_b=4.5GW/cm ², (c) I_c=5.5GW/cm ² and (d) Id=6.0GW/cm ².

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Due to the self-focusing effect the initial wavefront reference becomes defocused with a negative or positive radius of curvature according to the sign of n ₂. Neglecting the evolution of the beam diameter within the sample, the nonlinear index of refraction can be estimated as

where C ₅ can be described in waves (wavelength) or radians.

Figure 4 shows the defocus coefficient C ₅ dependence on laser intensity for CS ₂ (L=2mm) and Quartz (L=10mm) samples. In Fig. 4(a), the evolution ofC ₅ for different irradiances is plotted with a positive linear behavior for a low rate repetition laser excitation (1KHz). Figure. 4(b) shows the positive and negative dependence for 1kHz and 76MHz rate repetition lasers in a CS ₂ sample. From the measured C5 coefficient and the laser beam power at the sample, the n2 value of the given sample could be calculated from Eq. 2. The measured values for CS ₂ and Quartz are: $n_2^{{\mathrm{CS}}_2}\simeq (2.7\pm 0.2)\times {10}^{-15}{\mathrm{cm}}^2⁄W$ respectively, in agreement with previously reported values [3, 7]. To demonstrate that the method measures not only the magnitude but also the sign of n ₂, a high rate repetition laser was used to induce negative nonlinearities in the CS ₂ sample due to thermal effects [7]. The thermal nonlinearity value measured for CS ₂ was $n_{2\left(\mathrm{thermal}\right)}^{{\mathrm{CS}}_2}\simeq (-4.4\pm 0.2)\times {10}^{-15}{\mathrm{cm}}^2⁄W$, in accordance with the values obtained under similar conditions using the TMEZ-scan technique [7].

In conclusion, the collimated beam technique is suitable to measure the n ₂ value and sign of a transparent material. Though the wavefront sensitivity is an order of magnitude less than reported values using the Z-scan technique [3] larger areas can be evaluated (focalization is not necessary), and sample thickness limitations are negligible. These properties make the method suitable for many applications.

 

Fig. 4. Evolution of the defocus Zernike coefficient: (a) induced in a 2mm-thick cuvette CS2 sample (squares) and a 10mm-thick quartz sample (circles) at 1KHz, (b) in a 2mm CS2 sample using 1KHz (Non-thermal) and 76MHz (Thermo-optical) pulse repetition.

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3.2. HS Z-scan.

In the HS Z-scan sensor configuration the wavefront of the transmitted beam with the sample far away from the focal point was used as reference wavefront with low beam intensity and negligible nonlinear refraction. The measurements were based on this reference and allow wavefront distortions at the pre- and post-focal positions due to self-focalization to be determined. Figure 5 shows the results of the HS Z-scan compared to Z-scan performed at 800nm in CS ₂ (2mm thick cuvette). Figure 5(a) shows the HS Z-scan curves, obtained for different intensities (I_b=6.8GW/cm ², I_c=0.5GW/cm ² and I_d=0.2GW/cm ²)at a repetition rate of 1KHz. The changes in radius of curvature can be described using the defocus Zernike coefficient C5. To demonstrate the method, we measured under the same conditions the Z-scan curves shown in Fig. 5(b). The n ^CS2 ₂ is positive, therefore the self-lensing effect prior to focus will tend to increase the beam divergence, causing an increase in the curvature radius (Zernike C5) and a decrease of the transmittance detected by the conventional Z-scan setup. As the scan continues the same self-lensing increase the beam divergence, collimating the beam and decreasing the radius of curvature. As a consequence, the HS Z-scan curve looks inverted compared with the Z-scan curve.

The experimental relation between the normalized peak and valley transmittance ΔT_p-v and the phase distortion Δϕ (using Z-scan and HS Z-scan, respectively) is ΔT_p-v⇍0.4Δϕ, in agreement with the relation obtained using a Gaussian decomposition [3]. The HS Z-scan method has a wavefront distortion resolution of approximately λ/50. Although the HS Z-scan sensitivity is lower than reported by Bahae [3], the fact that it measures the wavefront slope of the entire laser beam instead of the transmittance laser intensity makes it potentially more sensitive. This could be explored with a higher precision wavefront sensor than used in the present work. In turn, a higher resolution with the conventional Z-scan technique depends strongly on the aperture size and would eventually become signal limited. As shown in the inset figure in Fig. 5(a), using a lower irradiance I_d=0.2GW/cm ², we measured the wavefront change using the HS Z-scan for CS ₂. Nevertheless, for intensities lowers than I_c=0.5GW/cm ², it was not possible to distinguish the transmittance variance from laser fluctuations with the Z-scan technique as shown in Fig. 5(b). Naturally, there are other experimental ways to overcome power fluctuations which can be implemented in a laboratory set-up.

 

Fig. 5. (a) HS Z-scan and (b) Z-scan signature for a CS ₂ sample (2mm thick cuvette) using different intensities: I_a=120GW/cm ² (▫), I_b=6.1GW/cm ² (○), I_c=0.5GW/cm ² (+) and I_d=0.2GW/cm ² (★). The case of I_a has been excluded from plot (a) since the beam diverges beyond the entrance pupil of the HS.

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4. Conclusions

We have demonstrated that the HS wavefront sensor can be used to characterize nonlinear properties of materials using two techniques. In the first one, no scanning is required, the method is insensitive to misalignments, linear scattering and sample imperfections. A second technique, referred to as HS Z-scan is insensitive to fluctuations during the measurement. The HS based methods proposed here are useful low-cost techniques to characterize nonlinear optical properties of biological samples, films and others materials.