1. Introduction

The z-scan technique proposed by Sheik-Bahae et al. [1] has become very popular for determining optical nonlinearities owing to its experimental simplicity. Some modifications have been made to the z-scan technique to improve its sensitivity in order to extend its application [2,3]. In the original z-scan technique, the sample is displaced around the focal plane of a focused Gaussian beam. The refractive nonlinearity of the material expands or compresses the cross-section of the beam in the far field. These variations in the cross-section are analyzed by measuring the optical power of light transmitted by a circular aperture, whose center coincides with the beam propagation axis (z-axis). The theoretical treatment is easier when the beam and the aperture have the same circular shape than for cases where they are dissimilar, such as for elliptic Gaussian beams. The inherent astigmatism in most laser systems causes most laser beams to be elliptic. In some papers, the use of an aperture for measuring the nonlinear refraction index has been eliminated, for example by Tsigaridas et al. for elliptic Gaussian beams [4] and Fischer et al. for highly scattering samples [5].

For elliptic Gaussian beams, the theoretical treatment can be performed by the Gaussian decomposition (GD) method [6, 7]. However, there are few studies regarding the experimental application to elliptic beams owing to their asymmetrical shape. Some of these studies use charge-coupled devices (CCD) [8, 9], nevertheless, the experimental arrangement is not very easy to implement because a CCD detector and a laser beam profiler are necessary. Here, a much simpler way to approach the problem is implemented.

In this study, the chopper z-scan technique has been applied to elliptic Gaussian beam profiles by direct measurement of beam widths [10], which involves measuring the time it takes for a rotating slotted disk (hereafter called a chopper) to overshadow a beam that is transmitted through a nonlinear sample. Thus, when time is measured instead of intensity, immunity to nonlinear absorption is obtained, which improves the signal to noise ratio. In addition, it has been demonstrated that the chopper z-scan technique can be applied to moderated scattering samples [10] and as in [5] allows simultaneous measurements of both nonlinear refractive index and absorption coefficients.

In this modified technique, it was assumed that the chopper rotates at a constant frequency (or with a constant period), and that the beam width is smaller than the dimensions of the chopper and detection area of the photodiode used to detect the beam. This modified technique presents some difficulties because it was found that small fluctuations in the rotation frequency degrade the z-scan curves obtained.

This paper presents a solution to decrease the dependence of the z-scan curve on the rotation frequency and its application to elliptic Gaussian beams.

2. Decreasing the influence of rotation period

2.1. Duty cycle measurement

From reference [10] the elapsed time (τ) to eclipse the beam is

This dependency is undesirable, because generally a commercial chopper presents an uncertainty of about 2% [11] in its rotation frequency. This causes samples with weak nonlinearity to generate poorly defined z-scan curves, thus making their interpretation difficult.

In order to mitigate this dependency with the period, the physical variable τ has been exchanged by the duty cycle δ, which is defined as $\delta =\frac{\tau }{T}$ obtaining the relation

The variable δ is less straightforward to measure; however, there are devices such like oscilloscopes that are capable of measuring the duty cycle of a square wave. Therefore, if an oscilloscope is to be used to measure δ, it is necessary to transform the detected signal (V_ph) into a square signal (V_sq). This can be accomplished by a circuit that generates a positive signal when V_ph is between two threshold voltages V_th1 and V_th2 and zero in any other case, as shown in Fig. 1. An electronic circuit that performs this transformation is presented in [12]. The choice of threshold voltages is adjusted according to the measurement of the beam width criterion e⁻² [13].

 

Fig. 1 Processing of the detected signal (V_ph) into a square signal (V_sq).

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Another method is to virtually implement the aforementioned circuit through software such as Labview. This is more advisable because if the sample presents nonlinear absorption, its influence on V_ph amplitude can be easily taken into account by dynamically adjusting the threshold levels required for calculating δ according to the amplitude of V_ph. The nonlinear absorption effect can also be removed by normalizing the signal. However, this is not recommended because the amplitude also contains the information that allows the measurement of the nonlinear absorption coefficient [10].

After the method for measuring δ and τ was established, we proceeded to experimentally investigate the influence of rotation period (T) on δ and τ. For this, T was varied by adjusting the frequency control of the chopper. The setup of the chopper z-scan is shown in Fig. 2. This setup is comprised of 633nm HeNe laser of 10mW, whose power was attenuated to 100μW through a Glan Thompson polarizer (P), a lens of 150mm focal length (L1), a chopper (Ch) that according to the manufacturer has a 2% frequency drift, a photodiode (PD) with an active area of 94mm², a nonlinear sample of bacteriorhodopsin (S) from Munich Innovative Biomaterials GmbH (WT1N3), an electronic circuit (EC) to transform the signal (see Ref. [12]), an oscilloscope TDS 1012C-EDU, a data acquisition card (DAQ), and a personal computer (PC) running LabView software.

 

Fig. 2 Experimental setup of the chopper z-scan technique, P: Polarizer, L1: Lens, S: Sample, Ch: Chopper, PD: Photodetector, EC: Electronic circuit, DAQ: Data acquisition card and PC: Personal computer.

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The values obtained without the sample, by the electronic circuit-oscilloscope (black square) and through the data acquisition card-software (in green circle), for the eclipsing time τ and duty cycle δ versus rotation period T are presented in Figs. 3 and 4, respectively.

 

Fig. 3 Eclipsing time as a function of the rotation period of the chopper.

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Fig. 4 Duty cycle versus rotation period.

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As shown by a linear fit, both techniques provide similar results for δ and τ against rotation period T. However, it should be noted that points in the graphs obtained with the oscilloscope correspond to a single measurement, unlike those obtained with Labview, where each point corresponds to an average of 20 measurements. The use of Labview significantly decreases the time required to obtain a point on the graph. For example, in this study, a chopper with a 10-slot disk was used, rotating at a frequency such that 20 measurements per second were generated, therefore the time to make an average of 20 measurements is just one second, and hence its graph exhibits less fluctuations.

Finally, it can be concluded by comparing the slopes of the data in Fig. 3 (slope m_τ = 0.057) and Fig. 4 (slope m_δ = −8.29 × 10⁻⁴) that the duty cycle δ is approximately 69 times less sensitive than τ with respect to the period T. This means that the curve z-scan obtained by measuring δ shows less fluctuations than that obtained with τ.

3. Application to elliptic Gaussian beams

3.1. Experimental setup

In order to demonstrate the usefulness of the chopper z-scan technique for the case of elliptic Gaussian beams, the HeNe laser in the experimental setup shown in the Fig. 2 was substituted by a laser diode (DL) from a simple laser pointer emitting at 635nm center wavelength with elliptic symmetry. The knife-edge technique was used to measure the ratio between the major and minor axis at the exit of the laser: $\frac{r_M}{r_m}=2.7$, where r_M corresponds to the major axis and r_m the minor axis. The emission power was attenuated to 100μW through a Glan-Thomson polarizer (P).

To characterize the elliptic Gaussian beam by the knife-edge method it is necessary to use a platform displacement along the minor axis and major axis of the elliptic spot. This is difficult compared to the chopper technique, which can simply be placed as shown in Figs. 5(a) or 5(b) such that its blades travel along each axis as the case (the movement is illustrated with the red arrow). The beam width can then be measured along the corresponding axis using Eq. (3).

 

Fig. 5 Colocation of the chopper in order to sweep along (a) minor axis and (b) major axis.

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In order to characterize the focused beam along each axis, the chopper is placed around the focus of the lens L1 (see Fig. 2) and according to Fig. 5. The result of this characterization is shown in Fig. 6 where the astigmatism of the elliptic beam can be seen. The origin coordinate has been chosen as the position of the beam focus along the minor axis, Z_0m and Z_0M correspond to the distance where the duty cycle increases by a factor of $\sqrt{2}$ of its minimum value, and the ratio of Rayleigh distances on each axis $(\frac{Z_{0M}}{Z_{0m}})$ are given by Eq. (3).

 

Fig. 6 Characterization of the elliptic beam to establish the origin where the minimum waist along the minor axis occurs.

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Subsequently, the chopper was placed at the position shown in Fig. 2 so that the chopper blades travel the minor or major axis according to Fig. 5(a) or 5(b) in order to obtain the z-scan curves along the main axis. The results of these measurements are shown in Fig. 7.

 

Fig. 7 z-scan curves obtained along major and minor axes. The Rayleigh range and astigmatism differences can be seen in both curves.

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These curves show that the normalized peak-valley difference $\frac{\mathrm{\Delta }{\delta }_{p-v}}{{\delta }_L}$ (where δ_L is the horizontal dashed line in Fig. 7) remains practically equal for both axis. The value of $\frac{\mathrm{\Delta }{\delta }_{p-v}}{{\delta }_L}$ along the minor axis was equal to 0.1724 whereas that for the major axis was 0.1718. In fact, similar values of $\frac{\mathrm{\Delta }{\delta }_{p-v}}{{\delta }_L}$ were obtained for z-scan curves when the major axis was 23.5°, 45° and 68° from its horizontal direction.

In Fig. 7, the difference between peak and valley positions was Δz_m = 47.06mm for the minor axis z-scan curve and Δz_M = 17.66mm for the major axis z-scan curve. Therefore,

3.2. n₂ calculus using chopper z-scan theory

Considering a cubic nonlinearity according to [1], the phase change Δϕ₀ is related with the nonlinear index n₂ through

Using the fact that the rising time ratio $\frac{\mathrm{\Delta }{\tau }_{p-v}}{{\tau }_L}$ is equal to the duty cycle ratio $\frac{\mathrm{\Delta }{\delta }_{p-v}}{{\delta }_L}$, we obtain an expression for n₂ in terms of the chopper z-scan curve obtained by duty cycle measurements:

Our experimental data were: λ = 635nm, P = 100μW, W_om = 69μm, W_oM = 58μm, L_eff = 4μm, P is the laser optical power, and W_om and W_oM are the minimum and maximum waists along the minor and major axes. The values obtained for n₂ along the minor and major axes, respectively, were $n_{2m}=1.34\times {10}^{-8}\frac{m^2}{W}$ and $n_{2M}=2.25\times {10}^{-8}\frac{m^2}{W}$, with the average being

It was reported in [14] that the nonlinear refractive index depends significantly on the intensity, and can even change its sign at low power. For much lower powers, such as in our case, n₂ is in the range of 6.4 × 10⁻⁴ to $4.4\times {10}^{-1}\frac{c{m}^2}{W}$ at a wavelength of 647.1nm, and thus it can be seen that our results agree well with the values reported in literature. It should be mentioned that the n₂ value was obtained at different irradiance and wavelength that in [14].

4. Conclusion

We have demonstrated that changing physical variables reduces the dependence of the measured signal on the period of the rotating disk, and thus improves its sensitivity. It is easy to implement the chopper z-scan algorithm using specialized software to obtain more accurate measurements by averaging, because it allows more samples per unit of time than the electronic circuit method. We demonstrated that the chopper z-scan is a useful technique to resolve problems caused by an elliptic beam during z-scan optical nonlinear characterization. It was found that the positions of the peak-valley difference of the amplitude of the z-scan curve relative to its linear value do not dependent on the axis along which the scan is made. In addition, the z-scan curves along the minor and major axes are displaced and the degree of displacement corresponds to the difference between the positions of the minor and major focus caused by the astigmatism of elliptic beams. The z-scan curve is compressed along the major axis, and the compression ratio corresponds to ratio of the beam radius of minor axis to major axis. This technique is therefore suitable for solving the problems of the inherent astigmatism in most laser systems that results from most laser beams being elliptic.

Funding

Consejo Nacional de Ciencia y Tecnología (CONACyT) (51757).