Two-photon absorption coefficient determination with a differential F-scan technique

Transcription

1 Two-photon absorption coeicient determination with a dierential F-scan technique E RUEDA, 1 J H SERNA, A HAMAD AND H GARCIA 3,* 1 Grupo de Óptica y Fotónica, Instituto de Física, U de A, Calle 70 No. 5-1, Medellín, Colombia Grupo de Óptica y Espectroscopía, Centro de Ciencia Básica, Universidad Pontiicia Bolivariana, Ca. 1 No , Campus Laureles, Medellín, Colombia 3 Department o Physics, Southern Illinois University, Edwardsville, Illinois, 6006, USA hgarcia@siue.edu Abstract: In this paper we present a modiication to the recently proposed transmission F-scan technique, the dierential F-scan technique. In dierential F-scan technique the programmed ocal distance in the electronic-tunable lens oscillates, allowing the light detector o the setup to record a signal proportional to the derivative o the signal recorded with an F-scan. As or the dierential Z-scan a background-ree signal is obtained, but also the optical setup is simpliied and the available laser power is double. We also present and validate a new ittingprocedure protocol that increments the accuracy o the technique. Finally, we show that itting a signal rom dierential F-scan or the derivate o the signal o transmission F-scan is more accurate than simply itting the signal rom F-scan directly. Results rom two-photon absorption at 790 nm o CdS, ZeSe and CdSe are presented. 018 Optical Society o America under the terms o the OSA Open Access Publishing Agreement OCIS codes: ( ) Nonlinear optics; (10.010) Instrumentation, measurement, and metrology; ( ) Multiphoton processes. Reerences and links 1. J. H. Bechtel and W. L. Smith, "Two-photon absorption in semiconductors with picosecond laser pulses," Phys. Rev. B 13(8), (1976).. E. W. Van Stryland, M. a. Woodall, H. Vanherzeele, and M. J. Soileau, "Energy band-gap dependence o twophoton absorption," Opt. Lett. 10(10), 490 (1985). 3. 1M. Sheik-Bahae, A. a. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement o optical nonlinearities using a single beam," IEEE J. Quantum Electron. 6(4), (1990). 4. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement o lambda/10^4 wave-ront distortion," Opt. Lett. 19(5), (1994). 5. W. Zhao and P. Paly-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63(1), 1613 (1993). 6. M. Martinelli, S. Bian, J. R. Leite, and R. J. Horowicz, "Sensitivity-enhanced relection Z-scan by oblique incidence o a polarized beam," Appl. Phys. Lett. 7(1), (1998). 7. M. Martinelli, S. Bian, J. R. Leite, and R. J. Horowicz, "Sensitivity-enhanced relection Z-scan by oblique incidence o a polarized beam," Appl. Phys. Lett. 7(1), (1998). 8. J. Serna, E. Rueda, and H. García, "Nonlinear optical properties o bulk cuprous oxide using single beam Z- scan at 790 nm," Appl. Phys. Lett. 105(19), (014). 9. J.-M. Ménard, M. Betz, I. Sigal, and H. M. van Driel, "Single-beam dierential z-scan technique," Appl. Opt. 46(11), 119 (007). 10. R. Kolkowski and M. Samoc, "Modiied Z -scan technique using ocus-tunable lens," J. Opt. 16(1), 150 (014). 11. J. Serna, A. Hamad, H. Garcia, and E. Rueda, "Measurement o nonlinear optical absorption and nonlinear optical reraction in CdS and ZnSe using an electrically ocus-tunable lens," in International Conerence on Fibre Optics and Photonics, 014 (014). 1. R. L. Sutherland, Handbook o Nonlinear Optics (CRC Press, 003). 13. T. Krauss and F. Wise, "Femtosecond measurement o nonlinear absorption and reraction in CdS, ZnSe, and ZnS," Appl. Phys. Lett. 65(March 1994), (1994). 14. B. S. Wherrett, "Scaling rules or multiphoton interband absorption in semiconductors," J. Opt. Soc. Am. B 1(1), 67 (1984).

2 1. Introduction In the past decades, several optical techniques have been proposed to measure nonlinear optical properties such as two-photon absorption (TPA) coeicient or dierent types o materials, especially metals, organic and inorganic semiconductors [1,]. Among them, Z-scan is a particularly widely used technique due to its relatively simple optical setup and data treatment [3]. It is based on the scanning o spatial beam modiications suered by a laser beam ater interacting with a sample, while it is ocused and deocused in time: when the sample is near the ocal point o the beam the high intensities generated produce nonlinear phenomena such variations in the reractive index and multi-photon absorption. However, some problems, related to laser luctuations, beam alignment and mechanical vibrations can inluence the results obtained, compromising the sensitivity o the technique. To overcome these limitations, modiications to the basic Z-scan setup were proposed in the ollowing years. To enhance the sensitivity o the technique Xia et al. [4] replaced the ar-ield aperture in the standard Z-scan by an obscuration disk that blocks most o the beam, while Zhao et al. [5] used a top-hat beam instead o a Gaussian beam, and Martinelli et al. [6] measure the relected beam rom the sample in a Brewster angle coniguration. To improve the signal to noise ratio some authors have used balance-detection systems [7,8], while Ménard et al. [9] introduce the dierential Z-scan where a piezo-transducer device generates an oscillatory motion to induces a periodic modulation o the beam intensity at the sample, which in turn produces a modulation o the transmitted light proportional to the spatial derivative o the transmitted light, and thereore provides a background-ree measurement. More recently, a new technique, that is a variation o Z-scan, was proposed. This method, which is called F-scan [10,11], use an electrically ocus-tunable lens (EFTL) instead o a ixed lens to generate dierent ocal points, allowing to replace the translation stage and leaving the sample ixed in space, i.e. eliminating mechanical movements rom the setup. The ocal distance in the EFTL is a unction o the applied current to the lens. Analogously to Ménard et al. or the case o Z-scan, we present in this paper an open aperture dierential F-scan technique (DF-scan) to determine the nonlinear absorption optical properties o materials. The setup has been oversimpliied by using an EFTL and at the same time modulating the ocal length o the EFTL with a rectangular low requency signal that can be detected with a PSD (phase sensitive detector), in our case a lock-in ampliier, increasing the sensitivity o the system and reducing or eliminating laser luctuations.. Transmission F-scan (TF-scan) The F-scan experimental setup depicted in Fig. 1. is used or the determination o the twophoton absorption (TPA) coeicient (open aperture architecture).

3 Fig. 1. TF-scan experimental optical setup or determination o TPA coeicients (open-aperture architecture). A laser Gaussian beam modulated with a chopper impinges on an EFTL which is a lens that has the capability to vary its ocal distance over a speciic range when an electric current is applied to it (see Fig. ), ocusing the Gaussian beam at dierent positions. The sample is placed at a ixed position inside the range o the EFTL. The light transmitted through the d s sample is collected by a photodetector PD1. The output signal is then iltered with a Lock-in ampliier and processed with an acquisition data system. Fig.. OPTOTUNE C electrically ocus-tunable lens. This type o lens changes its shape (curvature) due to an optical luid sealed o by a polymer membrane, when a current is applied. To determine the TPA coeicient β we measured the transmittance o the nonlinear medium as a unction o the ocal length,, (see Fig. 1), collecting all light transmitted through the sample. When the distance d s is large the normalized transmittance has a value close to unity because linear optical eects are produced in the sample. In contrast, small values o d s imply that the laser beam is ocused near the sample, thus increasing the optical intensity and generating nonlinear optical phenomena such as TPA. The TPA coeicient is obtained by itting a theoretical curve (Eq. (9)) to the experimental data, using β as the itting parameter, and under the assumption that all the experimental parameters are known. A typical experimental curve is shown in Fig. 10(let). EFTL characterization Fig. 3 shows the dependence o the EFTL optical power on the applied current and expose the disagreement between the experimental data and the data reported by the manuacturer. In the T-Fscan technique it is crucial to know the ocal length with high accuracy in order to obtain correct values o. Thereore, the characterization o the EFTL optical power on the applied current has to be done with high-precision.

4 Fig. 3. The optical power,, o the EFTL as a unction o the applied current. (circles) Experimental data; (continuous line) it o the experimental data using Eq. (1); (dashed line) data provided by Optotune Inc. Dierent techniques can be used to obtain the correct dependence o the EFTL optical power on the applied current (circles in Fig. 3) by measuring the ocal length as a unction o current. We used a laser beam proiler to measure as a unction o the applied current. Another way is to use the TPA phenomena: by placing a sample with a nonzero TPA coeicient at two dierent distances rom the EFTL and inding the values o the applied current that produce the lowest intensity or each o the locations. The sample distance rom the EFTL will correspond to the EFTL ocal length that produced a minimum transmission. Then, using the relation 1/, the EFTL optical power is obtained as a unction o current. In our case the experimental data was itted obtaining the ollowing expression (continuous line in Fig. 3): where J 0.045J 1.5 (1) is the applied current measured in ma. Another important experimental parameter or the correct determination o the nonlinear optical parameters is the beam-waist radius Typically, or spherical lenses and assuming that the beam has a spatial Gaussian proile, the radius o the beam at the beam waist is determined as a unction o the ocal length with the equation: w ( ) () 0 D where is the wavelength o the incident beam with spot diameter D. But being aware o the existence o optical aberrations on the optical system that distort the waveront o the beam, a correction actor Eq. () is replaced by C is needed in order to correctly calculate the beam waist. Thus, w ( ) 0 C (3) D w 0.

5 To determine the correction actor we used a laser beam proiler to measure the beam waist at each ocal plane. Then, by using and Eq. (3), a correction actor C C 1.36 as the itting parameter between the experimental data was obtained or the special case o our EFTL. Fig. 4 shows the dierence between the corrected and non-corrected beam-waist diameter value as a unction o the EFTL ocallength. Fig. 4. Beam-waist diameter as a unction o EFTL ocal length. (circles) Experimental data measured with a laser beam proiler; (dashed line) beam waist diameter calculated with Eq. (); (continuous line) beam waist diameter calculated with Eq. (3) and C Once the beam-waist radius is correctly determined, it is possible to calculate with precision the beam radius w( ) (Eq. (4)) at the sample surace or every programed EFTL ocal length, where ( ) w ( ) / 0 0 ds w( ) 1 z0( ), (4) z is the Rayleigh range. Fig. 5 shows the dependence o the beam radius at the sample location as a unction o the EFTL ocal length. Notice the dierence between the results obtained with and without the corrected beam-waist radius. Also notice the asymmetry relative to the sample location. For values smaller than the radius at the sample increases aster than those or values larger than d s d s. This will cause the shape o the TF-scan to be asymmetric around d. Thereore, experimentally, we must normalize the s transmitted intensity relative to the intensity corresponding to the shortest ocal lengths used in the experiment.

6 Fig. 5. Beam radius at the sample as a unction o the EFTL ocal length. The continuous and dashed lines are the corrected (Eq. (3)) and not corrected (Eq. ()) beam waist radius respectively. For this plot we used Theoretical background D.0 mm and ds cm The laser beam in our experimental setup has a Gaussian spatial proile and a hyperbolic secant temporal proile. Thereore, at the ront surace o the sample, the incident beam intensity as a unction o the EFTL ocal length is given by: r t I r,, t I ( ) exp seh c in 0 w( ) 0 In the above equation r is the radial position with respect to the optical axis,. w 0 t (5) is time, / ( ln(1 )), where is the ull width at hal-maximum pulse duration and I ( ) 0 0 is the peak intensity o the beam at sample position as a unction o the EFTL ocal length: Here P avg ln(1 ) Pavg I ( ). (6) 0 w ( ) is the average power o the incident laser beam at the sample, and is the laser pulse repetition rate. The intensity at the exit surace o the sample can be written as: I out L (1 R) I ( r,, t) e in ( r,, t), (7) 1 (1 R) I ( r,, t) L where L is the thickness o the sample, R is the relection coeicient o the sample, is the linear absorption coeicient, is the two-photon absorption coeicient (TPA), and L 1 e L / is the eective sample thickness. Thus, the transmittance at the detector e plane can be express as: ( ) ln 1 B( )sech ( ) d T 0 in 1, (8) B( ) e

7 where B( ) (1 RI ) ( ) L 0 e and ln(1 ) t /. The transmittance given by Eq. (8) can be simpliied when As B( ) B( ) 1 [1]: m N m B( ) ( m n) mn T( ). (9) m0 m 1 n0 ( mn) 1 gets closer to one, more terms o the sum are need it. Thus, one must use at least is not underestimated. Otherwise, the obtained value o will be smaller than its actual value. N 11 in order to guarantee that the obtained value o 3. Dierential F-scan (DF-scan) To reduce noise in TF-scan, or in any intensity scanning technique, due to laser luctuations where the change in the transmission is small compared to these luctuations, Ménard et al. [9] proposed a method where the sample was mount on an oscillating-actuator in a Z-scan setup, thus, or an amplitude and requency F the transmission signal around (and near) position z 0 will be S t T T T ( z ) S sin( Ft) 0, (10) z where is time. Eq. (10) is valid as ar as the oscillating amplitude is comparable to the Rayleigh range o the beam. I a lock-in ampliier is used with a reerence signal F coming rom the piezoelectric actuator, then only the amplitude o the signal given by Eq. (11) is detected by the lock-in ampliier: T S z zz 0 z z0 S. (11) This background-ree technique reduces the laser luctuation noise and improves the sensitivity o the technique. One eature o the EFTL is that its ocal length can be modulated with dierent types o signal proiles (rectangular, triangular or sinusoidal) with a requency range in its modulation between 0. up to 000 Hz. This allowed us to modiy the TF-scan into a DF-scan without using piezoelectric actuators, and also using the modulated signal as the reerence or a lock-in ampliier; being the signal proportional to the derivative o the transmitted signal. Thus, or the case o the DF-scan technique, is replaced by the ocal distance, and using Eq. (9) the derivative in Eq. (11) becomes d s d 3 s m1 N m T k ( ) ( ) 1 m B mn mn k n 0 ( ) 1 m m m n where k 1 Normalized DF-scan z, (1) 4C (1 RL ) P e avg, and k ln(1 ). D k For the case o TF-scan, normalization o the experimental data in order to it it by using the analytical model is simply done by dividing the data with respect to the lock-in ampliier signal 1

8 A or the shortest ocal length. In a DF-scan setup this is not so direct because the signal detected by the lock-in ampliier at this same ocal length is null. Knowing that or any ocal length, programmed in a EFTL, the signal detected is AS T 0, (13) in order to obtain the normalized DF-scan signal o Eq. (1), which is independent o the modulation parameters A and S, the signal represented by Eq. (13) has to be divided by the actor AS, which is known in advance. To show this, in our experiment we programed the EFTL to vary its ocal length using a square signal o requency and 50% duty cycle, and or S amplitudes corresponding to driven currents o 0.5 and 1.0 ma (see Fig. 6(let)). The voltage detected by the lock-in ampliier or the shortest ocal length, when it is not being modulated, was 46.3 mv, corresponding to A = 46.3/ mv. Division by two is because o the 50% duty cycle o the square signal. Fig. 6(right) shows the corresponding normalized DF-scan signals or both used amplitudes and or the corresponding derivative o the normalized TF-scan; they are in excellent agreement. These normalized DF-scan signals can know be itted using Eq. (1). F 739 Hz Fig. 6. (let) Un-normalized DF-scan signals or two oscillating amplitudes S, 0.5 ma and 1.0 ma. (right) Normalized DF-scan signals ater dividing by amplitude S, voltage A and due to the 50% duty cycle. The corresponding Normalized TF-scan signal derivative is also shown. 4. Experimental data itting protocol A correct experimental data itting is crucial in order to obtain a reliable value o the parameter o interest; in this case: the two-photon absorption coeicient. Is then desirable to know the values o all the experimental parameters with the highest precision possible in order to use only the parameter o interest as the itting variable, especially i the technique requires the knowledge o a great number o parameters (or our case 10 experimental parameters have to be known). But in occasions this is not possible, and the experimental parameters are only known with a considerable uncertainty. As a irst approach, one can use only the central values o the experimental parameters, ignoring the uncertainties, but this must probably will end in an inaccurate itting and thus a wrong value o the parameter o interest (see or example Fig. 7(let) and set 1 in table ). One naive approach will be to use more than one parameter as itting variables, but because the model used does not have a unique solution, as is the case or TF-scan and DF-scan, one can end with a wrong value o the parameter o interest, although with a perect curve itting, as it is shown or example in Fig. 7(center). Based on the assumption that the uncertainties o the experimental parameters correspond to random luctuations that can be model with a Gaussian distribution, we implemented a protocol with the goal to obtain an interval that contains the real value o the parameter o interest, under a statistical approach. The protocol is the ollowing:

9 1. For each experimental parameter pick a random value rom the Gaussian distribution o the possible values.. Fit the experimental data an obtain the corresponding value or the parameter o interest. 3. Calculate a metric to evaluate the quality o the it. I the metric satisies a criteria keep the value o the parameter o interest. I not, discard it. 4. Repeat steps 1-3 until a distribution with a good sample size o acceptable values is obtained. 5. With the parameters o interest that were accepted calculate the average-weighted value, using each corresponding metric value as weights. Calculate the corresponding standard error. The average-weighted value corresponds to the parameter o interest. To show the eectiveness o this approach a simulated experimental data Real rom a sample with nonlinear TPA is presented. In table 1 the parameters used or the simulated data are presented. Table 1. Simulation: experimental parameters. L P avg d s (nm) D (mw) (s) (MHz) To retrieve we supposed that the experimental parameters are known with a 10% uncertainty. Then, three sets o experimental parameters are pick rom their normal distributions: set 1 and correspond to a ocal distance range rom 8 to 16 cm with a total o 300 sample points, and set 3 correspond to the same range but 50 sample points. Finally, the data is itted and is obtained using the central values Direct approach, our approach Protocol, and the multiple itting-parameters Multiple approach. For Multiple approach D,, P avg, and d s are also use as itting parameters. In table the results are presented. (1/m) C R Table. Comparison o the values, in cm/gw, retrieved rom the dierent data-itting techniques. Set 1 Set Set 3 Error % Error % Real Direct Multiple Protocol It is clear that the Multiple approach is not desirable because it most probably return a wrong value while giving an almost perect data itting curve, see Fig. 7(center), Fig. 8(center) and Fig. 9(center). The Direct approach depends directly on the experimental parameters chosen, and due to the uncertainty, this can mean that the value retrieved is close to the correct one (Fig. 8(let)) or ar (Fig. 7(let) and Fig. 9(let)). Finally, our proposal, Protocol, gives the certainty that it will always retrieve the value with the greatest possible accuracy (see Fig. 7(right), Fig. 8(right) and Fig. 9(right)). Error %

10 Fig. 7. Fitting results o the simulated data or Set 1 in table. (Let) Direct approach, (center) Multiple approach, (right) Protocol approach. Fig. 8. Fitting results o the simulated data or Set in table. (Let) Direct approach, (center) Multiple approach, (right) Protocol approach. Fig. 9. Fitting results o the simulated data or Set 3 in table. (Let) Direct approach, (center) Multiple approach, (right) Protocol approach. 5. Experimental results For the experimental implementation o the TF-scan and DF-scan we used a Ti:Sapphire oscillator laser with repetition rate o 90.9 MHz, pulse width o 71 s, and laser emission centered at 790 nm. The average power at the entrance surace o the sample was 145 mw. The beam diameter at the EFTL was D =.0 mm, and was measured by a laser beam proiler. The EFTL is an OPTOTUNE-1030, controlled by an OPTOTUNE lens-driver that gives a maximum current o 300 ma with a resolution o 0.1 ma, delivering a ocal length resolution o mm. The laser Gaussian beam is ocused at quasi-normal incidence in order to eliminate Fabry-Perot eects and multiple relections. We used an integrating sphere with a large area Si-photodiode (PDA 50 THORLABS) to measure the transmitted laser light. This

11 modiication to the common setup compensates any lens-divergence and eliminates signal losses due to scattering rom the sample-surace roughness. The current generated by the photodiode is sent to a STANFORD RESEARCH 830 dual channel Lock-in ampliier, controlled through a GPIB interace. We have measured the TPA coeicient at 790 nm or ZnSe, CdS and CdSe. The experimental parameters or both techniques and all materials are listed in table 3, and the obtained TPA coeicients are listed in table 4. In Fig. 10 the experimental curves and curve itting is presented or the case o CdSe. Table 3. Experimental parameters or TF-scan and DF-scan, or CdS, ZnSe and CdSe. (MHz) d s (nm) D ± ± 1.0 ± ± 0.01 P avg (mw) (s) L ZnSe ZnSe (1/m) C R ZnSe 145 ± ± ± L CdS CdS (1/m) R CdS L CdSe CdSe (1/m) 0.85 ± 0.01 R CdSe ± Table 4. Comparison o values, in cm/gw at 790 nm, or ZnSe, CdS and CdSe. ZnSe CdS CdSe Relative error % Relative error % DF-scan TF-scan Krauss [13] 3.5 * > * >35 Relative error % *at 780 nm. From table 4 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 allowing subsection. For the value reported by Krauss et al. [13] or ZnSe, and rom a statistical point o view, there is a probability o 8% and 35% that the dierence is due to a random luctuation with respect to TF-scan and DF-scan, respectively. For the case o CdS the probabilities are 3% and 8%, respectevely. A random deviation o this magnitude in the values is not consider rare. For CdSe we were not able to ind a value or wavelengths close to 790 nm. Fig. 10. Fitting result or CdSe. (Let) TF-scan, (right) DF-scan.

12 To compare our results with the values that have been reported, in a broader way, or the three materials, we used the expression derived by Wherrett [14] or TPA, where E g ( hc / ( E ) 1) g ~ 5 ( hc / ( E )) g 3/, (14) is the energy o the bandgap, c is the speed o light in vacuum, and h is the Planck s constant. In particular, the ratio r 790, between the TPA at 790 nm with respect to the TPA value or the other wavelengths is given by r 790, hc E ( ) ( ) 790 7/ 790 g 3/ hc E In Fig. 11 the ratio is plot or the three materials. Except or the case o CdSe, there is a probability o 5% or more that the dierences are due to random luctuations. g (15) Fig. 11.Wherrett ratio. The values or the other wavelengths are taken rom Krauss et al. [13] and V. Stryland et al. [13]. Dierence between TF-scan and DF-scan results We believe the discrepancy between the values obtained with DF-scan and TF-scan is due to itting robustness directly related to curve shape. In our itting criteria the metric minimizes the value o the sum o the distance between the calculated and experimental peaks o the curves. The existence o two peaks in DF-scan signals reduce the spectrum o possible itting values, making the process more robust and accurate than with TF-scan signals. Then, the same result must be obtained or both, the derivative o TF-scan signal and the DF-scan signal. To validate this idea, irst we perorm a simulation with the values o table 1, and secondly we perorm the itting or the derivative o the TF-scan signal o CdSe and compare it with the results o table 4. Results are presented in table 5 and Fig. 1, showing, without doubt, that itting the derivative o the signal o TF-scan or the signal o DF-scan outcomes a more accurate value. Table 5. Comparison o the values, in cm/gw, retrieved rom the simulated data o table 1. Simulated 3.4 TF-scan.6 4 DF-scan TF-scan derivative Error %

13 Fig. 11. Fitting curves o table 5 simulated-experimental data results. (Let) TF-scan, (right) DFscan and TF-scan derivative. 6. Conclusions DF-scan is a modiication to the F-scan technique where the scanning is done over the rate o change o the transmission signal with respect o the ocal distance o the EFTL, reducing drastically the sensibility to laser luctuations, increasing the available laser power and simpliying the optical setup by eliminating the need o a chopper to modulate the signal. For the curve itting step, where the experimental signal is itted to an analytical model in order to obtain the TPA coeicient, a new protocol is proposed in order to secure the correct determination o the TPA parameter. The eectiveness o the proposal has been validated with respect to other curve itting procedures. Finally, it was shown that it is more reliable to it the derivatives o the experimental signals than the signal itsel, either o the DF-scan signal or the derivative o the TF-scan signal. Acknowlegments E. Rueda thanks Universidad de Antioquia or inancial support. J. Serna acknowledges the support rom Universidad Pontiicia Bolivariana. H. Garcia and A. Hamad thanks Southern Illinois University, Edwardsville, or inancial support.