## Abstract

Using an electrically focus tunable lens (EFTL) in an F-scan setup and a tunable femtosecond-pulse laser (Mai Tai HP), we were able to measure the degenerated two-photon absorption coefficient (in transmission) of CdS and ZnSe in an extended range of wavelengths (690-1040 nm), with a 5 nm resolution. The process of measuring takes less than 30 minutes. We compared our results with theoretical approaches for the dispersion relations of the nonlinear properties of semiconductors and found excellent agreement with the experimental results. We also compare our results with those reported in the literature. We derive the nonlinear refraction using a Kramers-Kronig relation and compare it with the values reported in the literature. The system has no moving parts, is highly compact, and is fully automated.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Characterization of nonlinear optical properties of solids is based primarily on the use of experimental techniques based on single wavelength lasers, with only a few examples on the use of broadband lasers. Balu et al. [1–3] proposed two techniques to use the broadband spectrum from a femtosecond Ti:Saphire pulsed laser to measure nonlinear absorption (NLA) and nonlinear refraction (NLR) of solids: the first one uses a water cell to generate a broadband white-light continuum (WLC) ranging from 560 nm to 710 nm, and the second one uses high-pressure Krypton gas. In both cases, to avoid the occurrence of nondegenerate nonlinearities, they used narrowband filters before the sample. Another approach to produce the WLC was proposed by Dey et al. [4] by using a photonic crystal fiber instead of the water cell. In all cases, the techniques are based on the popular and straightforward Z-scan technique [5], where the NLA is obtained in transmission mode using an open-aperture architecture and the mechanical motion of the sample through a high irradiance point. As pointed out recently by Steiger et al. [6], the problem with WLC systems is that they require complex optical paths. Instead, Steiger et al. used a broadband tunable femtosecond laser for the open aperture Z-scan setup to measure the two-photon absorption (TPA) in photoinitiator materials. To achieve this, they emphasize the necessity of having, for each wavelength, all the experimental parameters (beam waist, laser power, pulse duration, among others) fully characterized in order to avoid inaccuracies in the determination of the TPA values. In this work, we have accomplished this in an F-scan setup [7]. As complementary information, we have measured the nonlinear refraction index using Kramers-Kronig relations. We compared our results with the values reported in the literature, finding, in general, an excellent agreement.

## 2. System characterization

The broadband laser source (Spectra Physics Mai Tai HP, USA) is tunable from 690 nm to 1040 nm with a 1 nm resolution. We fully characterized the wavelengths in steps of 5 nm. The pulses’ temporal widths were measured using autocorrelation in a BBO crystal (where angle phase matching produced a measurable signal) by looking at the pulses’ spectral width and inferring from there the pulse duration assuming a transform-limited pulse.

Figure 1 shows a typical autocorrelation trace and the corresponding spectral width at 788 nm. The beam's irradiance was scanned with a 10 µm pinhole and modeled as a Gaussian beam profile to measure the spot size at the focus for each wavelength. The Electrically Focus-Tunable lens (EFTL) (Optotune, EL10-30, Switzerland) generates high and low irradiance points, as described in [7]. In this setup, there is no need to characterize the EFTL because the measurement is based on high and low irradiance points used to extract the TPA coefficient. To eliminate the focusing and defocusing of the EFTL as the lens focal distance is changing, we used a set of photo-detectors (Thorlabs, In-FGAP71 150nm-550nm, Si-IDAS015 400nm-1100nm, and Ge-FD605 800nm-1800nm, USA) mounted in an integrating sphere (Newport, 819C-Sl-2, USA). In this way, the system is adapted to work in a broader range of wavelengths.

## 3. Experimental setup

Figure 2 shows the experimental setup. The tunable laser generates, on average, 75 fs pulses at 80 MHz. The light passes through a continuous neutral density filter mounted on a computer-controlled rotational stage, in a closed-loop with a feedback mechanism, to maintain the average power of the laser at the sample almost constant (in our case at values ranging from 44 mW to 56 mW). After passing through the neutral density filter NF, the beam reaches the EFTL, focusing or defocusing it at the sample plane. The light transmitted through the sample is focused into the integrating sphere D1 by the lens L1 and corresponds to the NLA open-aperture architecture. The signal is sent to the Lock-in amplifier that averages each of the data points 40 times to reduce the noise in the output. The computer changes the wavelength of the laser, and the process is repeated until it reaches the highest wavelength of the oscillator (1040 nm).

## 4. Two-photon absorption measurement technique

Figure 3(a) shows a typical trace for transmission. The position of the sample holder is such that minimum transmission takes place for a focal distance corresponding to 150 mA (programmed EFTL current for the case of 690 nm). The following paragraph describes the procedure of how the technique works and the experimental results for the case of two semiconductors, ZnSe, and CdS. Because all the experimental parameters have to be well characterized in order to use the tunable laser, it is reasonable to think that for the case of Z-scan, in order to determine the TPA, for each wavelength only two values are needed, the highest transmittance *T _{H}* at $z \to \infty$ and the lowest transmittance

*T*at

_{L}*Z*= 0. Thus, using the well-known expression for Z-scan open aperture architecture [8],

*β*is the TPA coefficient,

*I*is the maximum irradiance of the beam at

_{0}*z*= 0,

*R*is the reflection coefficient at normal incidence, ${L_{eff} = (1 - e^{ -\alpha L})/\alpha}$ is the effective length, α is the linear absorption,

*L*is the sample thickness, $x = {z \mathord{\left/ {\vphantom {z {z_0 }}} \right.} {z_0}}$, z is the position of the sample with respect to the beam waist, and

*z*is the Rayleigh range. By taking the highest and lowest transmittance, using Eq. (1), the relation

_{0}*T*will be equal to

_{L}/T_{H}*λ*can be found from the relation Although we use an F-scan setup instead of a Z-scan setup, because we only need the highest and lowest transmittance for each wavelength, it is true that

*T*(see Fig. 3(a)) and we can use Eq. (3) to determine the TPA. This idea is implemented experimentally by taking a trace where the current is change between 140 mA – 200 mA. After taking the trace, the system stores in memory the values for HI and LI. In Fig. 3(b) we have plotted the HI and LI for each wavelength.

_{L}/T_{H}= LI/HIFrom the above experimental analysis in Fig. 4 and Fig. 5 the value of β as a function of energy is plotted for ZnSe and CdS, respectively, together with the theoretical prediction based on a two-band model [9] where the TPA coefficient is given by

*E*is the energy gap of the semiconductor. A and

_{g}*E*are the fitting parameters, and

_{g}*H*corresponds to the Heavyside step function. From the above model, we have found an excellent agreement within the experimental error of the bandgap.

For ZnSe, our results are in good agreement with the results reported by Dabbico et al. [15] and approximately one standard deviation from the values reported by Balu et al. [13]. In particular, we are in complete agreement with the value we had reported at 790 nm [7]. For energies greater than 1.8 eV, where we do not have experimental data, the obtained theoretical curve is approximately one standard deviation above the values reported in the literature. For CdS, we found much fewer values reported in the literature. Interestingly, we are in excellent agreement with the values reported by Krauss et al. at 780 nm [10], and by Van Stryland et al. at 532 nm [11], but we are approximately at three standard deviations from the value we had reported at 790 nm [7]. Nevertheless, we are more confident in the value we have reported in this work due to the higher quality and reliability of the Mai-Thai tunable laser.

## 5. Nonlinear refraction using K-K relations

The linear absorption for the two semiconductors was measured using an Ocean-Optics spectrometer (USB-400, USA) (see Fig. 6). The semiconductors bandgaps determined from Fig. 6 are 2.63 eV for ZnSe and 2.36 eV for CdS, which are in agreement with the values found with Eq. (4). It is worth mentioning that the TPA absorption can be used to measure the bandgap of semiconductors, and this is of particular relevance to semiconductors, where selection rules may forbid the direct gap transition.

Because of the broad range of experimental data for the nonlinear degenerated TPA dispersion curve, that allows obtaining a theoretical dispersion curve with confidence, in principle, the nonlinear refraction *n _{2}* can be obtained through a Kramers-Kronig relation [17] using the following equation.

*c*is the speed of light in vacuum, and

*C*is a constant that accounts for unknown contributions to the nonlinear refraction of the unknown TPA part of the spectrum, and that can be adjusted from already known values of

*n*. In particular, we have chosen C for the nonlinear refractive index at the frequency where the TPA is at its maxima, and the nonlinear refractive index

_{2}*n*is equal to zero. Another advantage compared to the linear case is that in the nonlinear case, the NLA can be known for all frequencies because only in a very definite gap of the spectrum it is different from zero (this is not necessarily true for the linear case). In Fig. 7, we show the nonlinear refraction for ZnSe and CdS using Eq. (5) and the

_{2}*β(ω*) fitted with Eq. (4) and the experimental NLA data. For ZnSe (Fig. 7a), the agreement with the experimental data reported in the literature is remarkable, supporting the idea of using K-K relation. For CdS (Fig. 7b), there is not sufficient experimental data reported in the literature, but based on the results for ZnSe, it is expected that the NLR found for CdS is reliable.

## 6. Conclusions

We have determined the two-photon absorption for CdS and ZnSe using a compact, fully automated system and with a broad spectral range (690 nm- 1040 nm) nonlinear absorption spectrometer. The system has no moving parts, and we obtained each spectrum in less than 30 minutes. The resolution of the system can be reduced down to 1 nm. One critical element is the use of an Electrically Focus Tunable Lens (EFTL) that eliminated the need for a translation stage, reducing the vibrational noise typical in Z-scan dramatically. The results are in excellent agreement with the predictions of a two-band model and with the experimental values reported in the literature. We have also extended the analysis for the case of nonlinear refraction using Kramers-Kronig relations and found good agreement with ZnSe experimental. For CdS, there are not enough NLR experimental results, yet, based on the results for ZnSe, we are confident with the NLR found for CdS. Nevertheless, it will be important in future work to measure NLR experimental data for CdS. We are in the process of using the same system to measure the nonlinear refraction in an open aperture reflection mode for thin films where bulk TPA contributions can be neglected. The goal is to implement this system to measure the nonlinear absorption in transmission mode and the nonlinear refraction in reflection mode for thin films; in this way, we obtain the full nonlinear spectrum of the film. This technique will open the doors for the development of a *nonlinear spectrometer*.

## Acknowledgments

We acknowledge Dr. Jack Glassman for critical reading of the manuscript. Dr. E. Rueda thanks Universidad de Antioquia (U de A) for financial support. Dr. J. Serna acknowledges the support from Universidad Pontificia Bolivariana (UPB)., and Dr. H. Garcia thanks Southern Illinois University, Edwardsville (SIUE), for financial support.

## Disclosures

The authors declare no conflicts of interest.

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