Two-photon absorption coefficient determination using the differential F-scan technique

Optics and Laser Technology 119 (2019) 105584

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Two-photon absorption coefficient determination using the differential Fscan technique

T

Edgar Ruedaa, Juan H. Sernab, Abdullatif Hamadc, Hernando Garciac,

⁎

a

Grupo de Óptica y Fotónica, Instituto de Física, U de A, Calle 70 No. 52-21, Medellín, Colombia Grupo de Óptica y Espectroscopía, Centro de Ciencia Básica, Universidad Pontificia Bolivariana, Ca. 1 No. 70-01, Campus Laureles, Medellín, Colombia c Department of Physics, Southern Illinois University, Edwardsville, IL 60026, USA b

HIGHLIGHTS

in scan systems reduces sensitivity to laser fluctuations, simplifies setups. • EFTLs is more accurate than TF-scan. • DF-scan • A new fitting protocol assures more accurate determination of the TPA parameter. ARTICLE INFO

ABSTRACT

Keywords: Nonlinear optics Two-photon absorption Z-scan Electronic focus-tunable lens

In this paper we present a modification to the recently proposed transmission F-scan technique, the differential F-scan technique. In the differential F-scan technique, the programmed focal distance in the electronic-tunable lens is modulated, allowing the light detector in the setup to record a signal proportional to the derivative of the signal recorded with the F-scan. As with the differential Z-scan a background-free signal is obtained, but in this case the optical setup is simplified and the available laser power is twice that for the F-scan. We also present and validate a new fitting-procedure protocol that increases the accuracy of the technique. Finally, we show that fitting a signal from a differential F-scan or the derivate of the signal of a transmission F-scan is more accurate than simply fitting the signal from the F-scan directly. Results for two-photon absorption at 790 nm of CdS, ZnSe and CdSe are presented.

1. Introduction In the past decades, several optical techniques have been proposed to measure nonlinear optical properties, such as the two-photon absorption (TPA) coefficient, for different types of materials, especially metals, and organic and inorganic semiconductors [1–4]. In particular, the Z-scan technique is widely used due to its relatively simple optical setup and data analysis [5]. In the Z-scan as well as the technique described here, the TPA coefficient can be measure in semiconductors when the energy of the laser is lower than the bandgap of the semiconductor. The technique described here therefore focuses on the contribution to the nonlinear absorption coming only from TPA. For other materials, like organic compounds where the nonlinear absorption may come from different mechanism, the theory could be extended. Z-scan is based on the scanning of spatial beam modifications experienced by a laser beam interacting with a sample while it is focused or defocused. When the sample is near the focal point of the beam ⁎

the high intensity generated produces nonlinear phenomena such as variations in the refractive index and multi-photon absorption. However, some problems, related to laser fluctuations, beam alignment, and mechanical vibrations can influence the results obtained, compromising the sensitivity of the technique. To overcome these limitations, modifications to the basic Z-scan setup were proposed. For example, to enhance the sensitivity of the technique Xia et al. [6] replaced the farfield aperture in the standard Z-scan by an obscuration disk that blocks most of the beam, while Zhao et al. [7] used a top-hat beam instead of a Gaussian beam, and Martinelli et al. [8] measured the reflected beam from the sample in a Brewster angle configuration. Also, to improve the signal to noise ratio, some researchers have used balance-detection systems [8,9], and Ménard et al. [10] introduced the differential Z-scan. In differential Z-scans, a piezo-transducer device generates an oscillatory motion to induce a periodic modulation of the beam intensity at the sample, which in turn produces a modulation of the transmitted light proportional to the spatial derivative of the transmitted light and

Corresponding author. E-mail addresses: [email protected] (E. Rueda), [email protected] (H. Garcia).

https://doi.org/10.1016/j.optlastec.2019.105584 Received 6 December 2018; Received in revised form 17 February 2019; Accepted 19 May 2019 Available online 26 May 2019 0030-3992/ © 2019 Elsevier Ltd. All rights reserved.

Optics and Laser Technology 119 (2019) 105584

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therefore provides a background-free measurement. More recently, a new technique, which is a variation of Z-scan, was proposed. This technique, called transmission F-scan (TF-scan) [11,12], uses an electrically focus-tunable lens (EFTL) instead of a lens with fixed-focal length. The focal distance in the EFTL is a function of the applied current to the lens. Since the EFTL generates different focal points, the sample is fixed in space and the use of a translation stage to move the sample along the laser beam is not necessary. This eliminates any issues that might occur from mechanical movements in the setup. In this paper, we present an open aperture differential F-scan technique (DFscan) to determine the nonlinear absorption coefficient of materials. The DF-scan is analogous to the modified Z-scan introduced by Ménard et al. [10]. The experimental setup has been markedly simplified by using an EFTL, and the sensitivity of the system has been increased by modulating the focal length of the EFTL with a rectangular low frequency signal that can be detected with a PSD (phase sensitive detector), in our case a lock-in amplifier. The lock-in amplifier also effectively eliminates effects due to laser fluctuations.

Fig. 2. OPTOTUNE-EL-10-30-C electrically focus-tunable lens. This type of lens changes its shape (curvature) due to an optical fluid sealed off by a polymer membrane, when a current is applied.

sphere with a large area Si-photodiode (PDA 50 THORLABS) to measure the transmitted laser light. This modification to the typical setup compensates for any lens-divergence and eliminates signal losses due to scattering from the sample-surface. The current generated by the photodiode is sent to a STANFORD RESEARCH 830 dual channel lock-in amplifier, controlled through a GPIB interface and processed with a data acquisition system.

2. TF-scan and DF-scan experimental setup The open aperture TF-scan and DF-scan experimental setup depicted in Fig. 1 was used to determine the two-photon absorption (TPA) coefficient β. We used a Ti:Sapphire laser with repetition rate of 90.9 MHz, pulse width of 71 fs, and laser emission centered at 790 nm. The average power at the entrance surface of the sample was 145 mW, with a maximum fluence of approximately 57 µJ/cm2. In the TF-scan a laser with a Gaussian cross-sectional profile modulated with a chopper (not shown in Fig. 1) impinges on the EFTL. The EFTL is a lens that has the capability to vary its focal distance over a specific range when an electric current is applied to it (see Fig. 2), thus focusing the Gaussian beam at different positions. The sample is placed at a fixed position inside the focal length range of the EFTL. In DF-scans, the focal length is dithered by modulating the lens current, eliminating the need for the optical chopper. The EFTL is an OPTOTUNE-EL-10-30-C controlled by an OPTOTUNE lens-driver that gives a maximum current of 300 mA with a resolution of 0.1 mA, delivering a focal length resolution of 0.017 mm. The beam diameter at the EFTL was measured by a laser beam profiler and found to be 2.0 mm (at the 1/ e 2 positions). The Gaussian beam is focused at quasi-normal incidence in order to eliminate etalon effects and multiple reflections. We used an integrating

2.1. EFTL characterization Fig. 3 shows the dependence of the EFTL, focal length on the applied current and illustrates the disagreement between the experimental data and the data reported by the manufacturer. In the TF-scan technique it is crucial to know the focal length with high accuracy in order to obtain correct values of β. Therefore, the characterization of the EFTL focal length response to the applied current has to be done with high-precision. To measure the correct dependence of the EFTL focal power on the applied current (circles in Fig. 3) we make use of TPA phenomena by placing a sample with a nonzero TPA coefficient at different distances from the EFTL and find the values of the applied current that produce the lowest intensity for each of the locations: the distance from the EFTL to the sample corresponds to the EFTL focal length f. In our case the experimental data was fitted obtaining the following expression (continuous line in Fig. 3):

f=

Fig. 1. Open aperture DF-scan experimental setup for determination of TPA coefficients. The experimental setup for TF-scan includes a chopper before the EFTL.

1 0.0444J + 1.52

(1)

Fig. 3. EFTL focal length f as a function of the applied current. (circles) Experimental data; (continuous line) fit of the experimental data (Eq. (1)); (dashed line) data provided by OPTOTUNE Inc. 2

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here J is the applied current measured in mA and f has units of meters. In the denominator of Eq. (1), the factor multiplied by J has an uncertainty of 0.0005 m−1/mA and the constant has an uncertainty of 0.08 m−1. Another important experimental parameter for the correct determination of the nonlinear optical parameters is the beam-waist radius w0. Typically, for spherical lenses, and assuming that the beam has a spatial Gaussian profile, the radius of the beam waist is determined as a function of the focal length using the following equation [13]:

w0 (f ) =

2 f D

(2)

Where λ is the wavelength of the incident beam, and D is the beam diameter at the EFTL. However, a deficiency of the EFTL is a gravity induced coma, which depends on the mechanical properties of the lens membrane and the density of the low dispersion liquid contained in it. To compensate for this, a correction factor Cf is needed in order to correctly calculate the radius of the beam waist. Thus, Eq. (2) is replaced by

w0 (f ) =

2 f Cf D

Fig. 5. Beam diameter at the sample plane as a function of the EFTL focal length using Eq. (4). The continuous and dashed lines are obtained using the corrected (Eq. (3)) and not corrected (Eq. (2)) beam waist radius w0, respectively. We used ds = 10.7 cm.

(3)

transmitted intensity relative to the intensity corresponding to the shortest focal length used in the experiment.

To determine the correction factor, we used a laser beam profiler to measure the radius of the beam waist at each focal plane. Then, the correction factor Cf was found by fitting the experimental data using Eq. (3). For the special case of our EFTL, Cf = 1.36 ± 0.04. Fig. 4 shows the difference between the corrected and non-corrected beam-waist diameter values as a function of the EFTL focal length. Once the beam-waist radius is correctly determined, it is possible to calculate with precision the beam radius w(f) at the sample plane for every programed EFTL focal length using Eq. (4),

w (f ) = w0 (f ) 1 +

ds f z 0 (f )

3. Theoretical background (TF-scan) The laser beam in our experimental setup has a Gaussian spatial profile and a hyperbolic secant temporal profile. Therefore, at the front surface of the sample, the incident beam intensity as a function of the EFTL focal length f is given by:

2

Iin (r , f , t ) = I0 (f ) exp

(4)

2

r w (f )

2

sech2

t 0

(5)

In the above equation r is the radial position with respect to the optical axis, t is time, 0 = /2 ln(1 + 2 ), where τ is the full width at half-maximum pulse duration and I0(f) is the peak intensity of the beam at sample position as a function of the EFTL focal length, given by

In Eq. (4), z 0 (f ) = is the Rayleigh range, and ds is the distance between the EFTL and the sample. Fig. 5 shows the dependence of the beam radius at the sample location as a function of the EFTL focal length. Notice the difference between the results obtained with and without the corrected beam-waist radius. Also notice the asymmetry relative to the sample location. For f values smaller than ds the radius at the sample increases faster than those for f values larger than ds. This will cause the shape of the TF-scan to be asymmetric around ds. Therefore, experimentally, we must normalize the

w02 (f )/

I0 (f ) =

2 ln(1 +

2 ) Pavg (6)

w 2 (f )

here Pavg is the average power of the incident laser beam at the sample, and ν is the laser pulse repetition rate. The intensity inside a thin sample can be modeled as [2].

dI (z ) = dz

I 2 (z )

I (z )

(7)

where α is the linear absorption coefficient, and β is the TPA coefficient. Then, the intensity at the exit surface of the sample can be written as:

Iout (r , f , t ) =

(1 R)2Iin (r , f , t ) e L 1 + (1 R) Iin (r , f , t ) Leff

(8)

where L is the thickness of the sample, R is the reflection coefficient of the sample, and Leff = (1 e L)/ is the effective sample thickness. By defining the normalized optical transmittance 0

T (f ) =

(1

0

Iout (r , f , t ) rdrdt R) 2Iin (r , f , t ) e

Lrdrdt

(9)

the transmittance at the detector plane can be express as [14]. Fig. 4. Beam-waist diameter as a function of EFTL focal length. (circles) Experimental data measured with a laser beam profiler; (dashed line) beam waist diameter calculated with Eq. (2); (continuous line) beam waist diameter calculated with Eq. (3), Cf = 1.36.

T (f ) =

1 B (f )

0

ln[1 + B (f )sech2 ( )]d

(10)

where B (f ) = (1 R) I0 (f ) Leff and = 2 ln(1 + 2 ) t / . The transmittance given by Eq. (10) can be simplified when B(f) < 1 [14], 3

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T (f ) = m=0

[ B (f )]m m+1

m n=0

2(m n) + mn 2(m n) + 1

This background-free technique reduces the laser fluctuation noise and improves the sensitivity of the technique. One feature of the EFTL is that its focal length can be modulated with different types of signal profiles (rectangular, triangular, or sinusoidal) and can be tuned in the frequency range of 0.2–2000 Hz. The modulating signal is provided by the EFTL driver. This feature makes it easy to modify the TF-scan into a DF-scan without using piezoelectric actuators. In addition, no chopper is needed when using the modulated frequency of the EFTL as the reference for the lock-in amplifier. Thus, for the case of the DF-scan technique, z in Eq. (11) is redefined in terms of the focal distance f and the distance between the sample and the EFTL ds, and is given by z(f,ds) = f − ds. When z is substituted in Eq. (11) and taken the derivative with respect to f we obtain, after some algebra, the following expression:

(11)

where δmn is Kronecker’s delta function. As B(f) gets closer to one, more terms of the sum are needed. Thus, one must use at least N = 11 in order to guarantee that the obtained value of β is not underestimated. Otherwise, the obtained value of β will be smaller than its actual value. To determine the TPA coefficient β we measured the transmittance of the nonlinear medium as a function of the focal length f by collecting all light transmitted through the sample. When the distance |ds – f| is large, the normalized transmittance has a value close to unity because linear optical effects are produced in the sample. In contrast, small values of |ds – f| imply that the laser beam is focused near the sample, thus increasing the optical intensity and generating nonlinear optical phenomena such as TPA. The TPA coefficient is obtained by fitting Eq. (11) to the experimental data, using β as the fitting parameter and under the assumption that all other experimental parameters are known. A typical experimental curve is shown in Fig. 9(a).

T (f ) f

f+ =

4. Differential F-scan (DF-scan)

T (z 0 ; 0) + S

T (z; 0) z

sin(2 Ft ) z =z0

T (z ) z

z=z0

(f

k2

where k1 =

ds )

N m=0

2m [ B (f )]m + 1 (m + 1)

m n=0

2(m n) + mn 2(m n) + 1

4 C 2f , D2

and k2 = 2 ln(1 +

2)

(1

R) Leff Pavg k1

.

4.1. “Normalized” DF-scan In order to fit the data of the TF-scan by the analytical model, the data must be normalized. The normalization is simply done by dividing the data by the lock-in amplifier signal A for the shortest EFTL focal length (at this focal length there is no nonlinear response from the sample). In a DF-scan experiment this is not as straightforward because the signal detected by the lock-in amplifier at shortest focal length is zero since it is the derivative of a constant signal. Thus, the signal detected for any programmed-focal length of the EFTL can be expressed as

(12)

where t is time. In the DF-scan, the first term corresponds to the transmission of unmodulated focal distance of the EFTL, and the second term corresponds to small variations of the transmission value due to the modulated focal distance of the EFTL. Note that Eq. (12) is valid as far as the modulation amplitude S is comparable to the Rayleigh range of the laser beam. If a lock-in amplifier is used with a reference signal corresponding to the frequency F coming from the piezoelectric actuator [10], only the amplitude of the signal given by Eq. (13) is detected by the lock-in amplifier because it only detects signals that are modulated with a frequency F:

S

ds k12 f 3

(14)

To reduce noise in the TF-scan, or in any intensity scanning technique, due to laser fluctuations where the change in the transmission is small compared to these fluctuations, Ménard et al. [10] proposed a method where the sample was mounted on an oscillating-actuator in a Z-scan setup. Thus, for an oscillating-actuator with amplitude S and frequency F, the transmission signal around (and near) position z0 will be

T (z ; t )

f = f0

AS

T (f ) f

f = f0

(15)

In order to obtain data that can be fitted using Eq. (14), which is independent of the modulation parameters A and S and that will be dubbed “normalized” DF-scan signal, the experimental signal represented by Eq. (15) has to be divided by the factor AS, which has to be known in advance. To show this, in our experiment, we programed the EFTL to vary its focal length using a square signal of frequency F = 739 Hz and 50% duty cycle, and for S amplitudes corresponding to

(13)

Fig. 6. (a) Raw-data of DF-scan signals for two oscillating amplitudes S: 0.5 mA and 1.0 mA. (b) “Normalized”-data of DF-scan signals after division by amplitude S, voltage A and 2 (50% duty cycle). The corresponding first derivative of the normalized TF-scan signal (NT-fscan derivative) is also shown. 4

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driving currents of 0.5 and 1.0 mA (see Fig. 6(a)). The voltage detected by the lock-in amplifier for the shortest focal length when it is not being modulated was 46.23 mV, corresponding to A = 46.23/2 mV. Division by two is necessary because of the 50% duty cycle of the square signal. Fig. 6(b) shows the corresponding “normalized” DF-scan signals for both of the amplitudes used and for the corresponding derivative of the normalized TF-scan. As can be seen from Fig. 6(b) they are in excellent agreement. These “normalized” DF-scan signals can now be fitted using Eq. (14).

Table 1 Comparison of β values, in cm/GW, retrieved from the different data-fitting approaches used with the simulated DF-scan signal. Error % is calculated with respect to the “target” value. Set 1

Target Direct Multiple 5 Multiple 3 Protocol

5. Experimental data-fitting protocol A correct experimental data fitting is crucial in order to obtain a reliable value of the parameter of interest, in this case, the two-photon absorption coefficient. It is desirable to know the values of all the experimental parameters with the highest precision possible in order to use only the parameter of interest as the fitting variable, especially if the technique requires the knowledge of a large number of parameters (for our case 10 experimental parameters must be known: sample thickness L, distance from EFTL to the sample ds, laser wavelength λ, beam-spot diameter at EFTL D, beam correction factor Cf, laser average power Pavg, laser pulse temporal width τ, laser pulse repetition rate ν, sample linear absorption α, and sample reflection coefficient R). But in many cases, this may not be possible and the experimental parameters are only known with considerable uncertainty. As a first approach, one can use the measured experimental parameters ignoring their corresponding uncertainties (named here “Direct” approach), but this probably will result in an inaccurate fitting and thus a wrong value of the parameter of interest (see for example Fig. 7(a) and set 1 in Table 1). Another approach will be to use more than one parameter as fitting variables (named here “Multiple” approach). However, for this case the model does not have a unique solution and one can end with a perfect data fitting but with the wrong value of the parameter of interest, as shown, for example, in Fig. 7(b) (see also set 1 in Table 1). Based on the assumption that the uncertainties of the experimental parameters correspond to random fluctuations that can be modeled with a Gaussian distribution, we have implemented a protocol with the goal to obtain a range that contains the real value of the parameter of interest, from a statistical viewpoint (named here “Protocol” approach). The protocol is explained below:

Set 2

Set 3

β

Error %

β

Error %

β

Error %

3.4 2.2 33.9 3.3 3.3

36 874 2 2

3.4 3.8 14.5 5.2 3.0

13 325 53 11

3.4 5.3 46.4 3.8 3.5

53 1265 11 2

satisfies the criteria keep the value of the parameter of interest. If not, discard it (Information about the metric used in this work can be found in Appendix A). 4. Repeat steps 1–3 until a distribution with a good sample size of acceptable values is obtained. 5. With the parameters of interest that were accepted, calculate the average-weighted value, using each corresponding metric value as weights. Calculate the corresponding standard error. The averageweighted value corresponds to the parameter of interest best estimate. 5.1. Comparison between approaches: Example Below we present an example to show the effectiveness of our approach (the “Protocol” approach). In this example we are going to retrieve β from a simulated data for a sample with a TPA coefficient that has a value of 3.4 cm/GW (“Target” value). The central values of the various parameters used to create the simulated DF-scan signal are presented in Table A.1 in Appendix A. We assume that the value of each parameter follows a Gaussian distribution. Also, we assume that these values are measured with 10% uncertainty. Then, we pick up randomly three sets of parameters from their corresponding Gaussian distributions and used them to fit the simulated DF-scan signal. Set 1 and 2 have 300 sample points, and set 3 has 50 sample points. Finally, the data is fitted using each set of parameters and β is obtained using the “Direct” approach, our “Protocol” approach, and the “Multiple” approach. For “Multiple 5” β, D, τ, Pavg, and ds were used as fitting variables. For “Multiple 3” β, D, and τ were used as fitting variables (Appendix A contain the information related to the fitting methods). To compare the results obtained with the different approaches the percentage of error defined by the expression 100·|Vt Vf |/Vt is calculated. Vt is the “Target” β value and Vf is the β

1. For each experimental parameter pick a random value from the normal distribution of the possible values. 2. Fit the experimental data and obtain the corresponding value for the parameter of interest. 3. Calculate a metric to evaluate the quality of the fit. If the metric

Fig. 7. Fitting results of the simulated data for Set 1 in Table 1. (a) “Direct” approach, (b) “Multiple 5” approach, (c) “Protocol” approach. The error on top of the plots corresponds to the mistake made in the prediction of β. 5

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Fig. 8. Differences between the data and the fit for set 1. (a) “Direct” approach, (b) “Multiple 5” approach, (c) “Protocol” approach. The error on top of the plots corresponds to the mistake made in the prediction of β.

value obtained by the “Direct”, “Multiple” or “Protocol” approach. The results are presented in Table 1. It is clear that the “Multiple 5” approach is not desirable because it mostly gives a wrong β value while giving an almost perfect fitting, see Fig. 7(b) and Fig. 8(b). The reason for this is the fact that we are using multiple degrees of freedom that will allow different results for the same fitting. The “Direct” approach depends directly on the parameters chosen, retrieving values close to the correct β with 13% of error (Table 1: set 2), or far from the correct β with 53% of error (Table 1: set 3). The “Multiple 3” approach is more accurate than “Multiple 5”, but like the “Direct” approach the result depends on the chosen set, 2% of error for set one but 53% of error for set 2. Finally, our proposal, “Protocol” approach, has a good accuracy, but more important, the results were more robust (less variable) with differences ranging from 2% to 11% of error (see sets 1, 2 and 3 in Table 1). In conclusion, when

the measured experimental parameters that are going to be used to fit a signal have an appreciable experimental uncertainty, the “Protocol” approach is the best option for the fitting due to its robustness under uncertainties. In Fig. 7 we plotted the fitting results and Fig. 8 we plotted the difference between the curve obtained with the fit and the simulated data curve, all for set 1. Results for sets 2 and 3 are almost identical to set one and are not shown. 6. Experimental results We have determined the values for the TPA coefficients for ZnSe, CdS and CdSe at 790 nm. The experimental parameters for both techniques and all materials are listed in Appendix A, and the values for the TPA coefficients obtained from fitting our data are listed in Table 2. In Fig. 9 experimental and fitting curves are presented for the case of CdSe. From Table 2 it is evident that there exists a discrepancy between TF-scan and DF-scan results, in particular, DF-scan always gives a bigger value. This discrepancy will be analyzed and explained in the following subsection. For the value reported by Krauss et al. [15] for ZnSe, and from a statistical point of view, there is a probability of 82% and 35% that the difference is due to a random fluctuation with respect to TF-scan and DF-scan, respectively. For the case of CdS the probabilities are 3% and 8%, respectively. Random deviations of this magnitude are not considered rare due to the high experimental uncertainties, especially for the case of ZnSe. For CdSe we were not able to find a value for wavelengths close to 790 nm.

Table 2 Comparison of β values, in cm/GW at 790 nm, for ZnSe, CdS and CdSe. The relative error (Rerr) corresponds to the experimental relative standard error. ZnSe

DF-scan TF-scan Krauss [15]

CdS

CdSe

β

Rerr %

β

Rerr %

β

Rerr %

5.1 3.2 3.5*

23 16 > 35

2.4 1.5 6.4*

22 13 > 35

4.6 1.8

12 15

* at 780 nm.

Fig. 9. Fitting result for CdSe. (a) TF-scan, and (b) DF-scan. 6

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Fig. 10. Wherrett ratio for (a) ZnSe, (b) CdS, and (c) CdSe. The experimental values at 790 nm correspond to the values measured in this work for each of the techniques, the values at the other wavelengths are taken from Krauss et al. [15] and V. Stryland et al. [17].

To compare our results in a broader way with the values that have been reported for the three materials, we used the expression derived by Wherrett [16] for TPA, 3/2

2hc Eg

1

/

2hc Eg

6.1. Difference between TF-scan and DF-scan results We believe the discrepancy between the values obtained with DFscan and TF-scan is due to fitting robustness directly related to curve shape. In our fitting criteria, the metric minimizes the value of the sum of the distance between the calculated and experimental peaks of the curves (see Appendix A). The existence of two peaks in DF-scan signals reduces the spectrum of possible fitting values, making the process more robust and accurate than with TF-scan signals. Then, the same result must be obtained for both, TF-scan derivative and the DF-scan signal. To validate this idea, first we performed a simulation with the values of Table A.1, and secondly, we performed the fitting for TF-scan derivative signal for the three materials and compared it with the results of Table 2. Results are presented in Fig. 11 for the simulation and in Fig. 12 for the materials. From the simulation it is evident that fitting the derivative of the signal from the TF-scan or the signal from the DF-scan produces a more accurate value for the TPA coefficient. This behavior is also corroborated by the experimental data results shown in Fig. 12. It is also interesting to point out that the TPA coefficient values obtained by fitting

5

(16)

where Eg is the energy of the bandgap, c is the speed of light in vacuum, and h is Planck’s constant. In particular, the ratio r790,λ between the TPA at 790 nm with respect to the TPA value for the other wavelengths is given by

r790, =

790

7/2

2hc 2hc

790 Eg

Eg

3/2

(17)

In Fig. 10 the ratio is plotted for the three materials. Except for the case of CdSe, there is a probability of 5% or more that the differences are due to random fluctuations. For the case of CdSe the probability is lower than 5%, but the available data is also limited.

Fig. 11. Fitting curves of simulated data using parameters of Table A.1 and a TPA coefficient of 3.4 cm/GW. (a) TF-scan, (b) DF-scan and TF-scan derivative, (c) comparison of the values obtained with 95.5% of confidence.

7

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Fig. 12. (a) CdSe TF-scan derivative fit: “Experimental” corresponds to the derivate of the experimental TF-scan data. (b) Comparison of all the fitting results for the three materials, with 95.5% of confidence. The colors are green for CdSe, blue for CdS, and red for ZnSe. (▾) TF-scan derivative, (•) DF-scan, and (♦) TF-scan. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the TF-scan signal were always lower than those from DF-scan and TFscan derivative. Also, its intervals of confidence did not coincide with the others.

TPA coefficient. The new protocol assures more accurate determination of the TPA parameter. The effectiveness of the proposed protocol has been validated with respect to other curve-fitting procedures. Finally, it was shown that it is more reliable to fit the derivative of the experimental signal than the signal itself, either by obtaining the derivative directly (DF-scan) or by calculating the derivative of the TF-scan signal.

7. Conclusions The DF-scan is a modification to the F-scan technique in which the scanning is done over the rate of change of the transmission signal with respect to the focal distance of the EFTL. The DF-scan reduces drastically the sensitivity to laser fluctuations, increases the available laser power, and simplifies the optical setup by eliminating the need for a chopper to modulate the signal. We proposed a new protocol for fitting the experimental data to an analytical expression in order to obtain the

Acknowledgements E. Rueda thanks Universidad de Antioquia U de A for financial support. J. Serna acknowledges the support from Universidad Pontificia Bolivariana. H. Garcia and A. Hamad thank Southern Illinois University, Edwardsville for financial support.

Appendix A. Supplementary material Simulated parameters for the comparison of the fitting approaches Table A.1 corresponds to the parameters used to create the simulated DF-scan signal in the example of Section 5, were the different fitting approaches are compared.

Table A.1 Parameters used to create the simulated DF-scan signal for the example. L (mm)

ds (mm)

λ (nm)

D (mm)

Cf

0.8 Pavg (mW) 145

116.0 τ (fs) 71

790 ν (MHz) 90.9

2.0 α (1/m) 2.64 × 10−11

1.36 R 0.1567

Experimental parameters for ZnSe TableA.2 corresponds to the experimental parameters used to determined TPA for ZnSe. Table A.2 Experimental parameters for TF-scan and DF-scan, for ZnSe. ν (MHz)

ds (mm)

λ (nm)

D (mm)

Cf

90.9 Pavg (mW) 145 ± 5

116.0 ± 0.5 τ (fs) 71.0 ± 0.3

790 ± 1 L (mm) 0.80 ± 0.01

2.0 ± 0.8 α (1/m) 4.772

1.36 ± 0.04 R 0.182 ± 0.005

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Experimental parameters for CdS Table A.3 corresponds to the experimental parameters used to determined TPA for CdS. Table A.3 Experimental parameters for TF-scan and DF-scan, for CdS. ν (MHz)

ds (mm)

λ (nm)

D (mm)

Cf

90.9 Pavg (mW) 145 ± 5

116.0 ± 0.5 τ (fs) 71.0 ± 0.3

790 ± 1 L (mm) 0.85 ± 0.01

2.0 ± 0.8 α (1/m) 2.64 × 10−11

1.36 ± 0.04 R 0.157 ± 0.005

Experimental parameters for CdSe Table A.4 corresponds to the experimental parameters used to determined TPA for CdSe. Table A.4 Experimental parameters for TF-scan and DF-scan, for CdSe. ν (MHz)

ds (mm)

λ (nm)

D (mm)

Cf

90.9 Pavg (mW) 145 ± 5

116.0 ± 0.5 τ (fs) 71.0 ± 0.3

790 ± 1 L (mm) 0.79 ± 0.01

2.0 ± 0.8 α (1/m) 369 ± 37

1.36 ± 0.04 R 0.185 ± 0.005

Metric used in the different fitting approaches Because in DF-scan the signal is characterized by a maximum and a minimum (see for example Fig. 11(b)) our metric to be minimized corresponds to the sum of the square differences between positions and values of experimental and fitted values:

(Vmi,e

Vmi, f )2 + (Vma,e

Vma, f ) 2 + (Pmi,e

Pmi, f )2 + (Pma,e

(A.1)

Pma, f ) 2

where Vmi and Vma are the minimum and maximum values of the DF-scan signal, respectively, Pmi and Pma are the focal distance position of the minimum and maximum of the signal, respectively. e and f subindices corresponds to experimental and fit data, respectively. Fitting methods For the “Direct” approach we used the function fmin() of the python library scipy.optimize. This function uses the downhill algorithm for the minimization. For the “Multiple” and “Protocol” approach we used the function minimize() of the python library scipy.optimize. As a solver the function used the “Nelder-Mead” method. Damage laser thresholds for ZnSe, CdS, and CdSe The optical damage threshold fluence is approximately 0.47 J/cm2 for CdS [18], 1.03 J/cm2 for ZnSe [19], and 1.06 J/cm2 CdSe [20].

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