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A model to characterize the response of a thin media that can exhibit more than one nonlocal nonlinear response when it is illuminated with a Gaussian beam in a z-scan experiment is proposed. The model considers that these nonlocal contributions can be treated as independent contributions in the refractive or absorptive nonlinear response. Numerical results for two nonlocal nonlinear contributions with different magnitudes between them are presented. Experimental results obtained from a hydrogenated amorphous silicon sample are used to corroborate this model.
© 2016 Optical Society of America
1. Introduction
The adequate characterization of the nonlinear response of a material allows to determine its possible applications. Several techniques are routinely used to characterize materials with an intensity dependent refractive index. One of the most popular and powerful is the z-scan technique proposed by S. Bahae et al [1], it is used to determine the sign and magnitude of the nonlinear refractive and absorptive coefficient. Initially the technique was proposed considering a Gaussian beam, a thin nonlinear samples and small photoinduced phase shifts. The theory to obtain the behavior of the z-scan curves was based on the Gaussian decomposition (GD) method [2,3]. Later, the theoretical predictions were extended to different kind of samples: for thin media with large phase shifts [4], for thick samples [5–7], for systems that present two or more simultaneous nonlinear refractive effects at different order [8–11]. In particular, a theoretical model and numerical results of the influence of two-photon absorption on the z-scan traces when a material possess simultaneous third- and fifth-order nonlinear refraction effects was presented by B. Gu et al in [12] and in [13] they introduced a coupling function. In [14] were presented values of the third and fifth order by a single crystal (p-toluene sulfonate) with nonlinear phase change and nonlinear absorption were considered as a sum of the third and fifth order. In [15], was discussed a way to separate the third and fifth order in semiconductor doped glasses (SDGs). In [16], high-order contributions in the z-scan experiments were analyzed, where the authors presented the non linear phase as a sum of third, fifth, seventh and ninth order terms and the absorptive nonlinear coefficients were also considered as a sum. In [17] an analytic solution and numerical simulation were provided to analyze Z-scan curves of layers of nonlinear medium in cascaded. In [18] a general Z-scan theory by Pálfalvi et al was introduced. This theory is valid for thin and thick samples that present small or large local nonlinearities. A model that considers the nonlocal feature of the nonlinear response by a thin sample with nonlinear refraction was presented in [19], where the nonlocal response was depicted by an m parameter. The model calculates the electric field profile at the exit of the nonlinear media and then numerically calculate its Fast Fourier Transform [20], to obtain the far field intensity distribution. In order to complete this model in [21] was discussed the influence of the nonlocality in materials with simultaneous nonlinear refraction and absorption; finally analytical expressions were presented for the normalized intensity of z-scan curves for arbitrary phase changes. In [22] it was demonstrated that the nonlocality affects the transverse spatial extension of the far field intensity distributions. However, a study that takes into account a material with two simultaneous nonlocal nonlinear optical responses has not been presented. In this work we propose a model to describe the response of a thin media that presents two simultaneous nonlocal nonlinear effects when it is illuminated with a Gaussian beam. The nonlocal response is characterized by a nonlinear refractive and a nonlinear absorptive phase changes of arbitrary magnitude. Numerical simulations are obtained for open/close z-scan configuration. The influence of the two nonlocalities in the z-scan curve is analyzed for different combinations of the nonlinear contributions. Z-scan experimental results are obtained, for a hydrogenated amorphous silicon sample under CW illumination, and fitted with this model with a good correspondence.
2. Theoretical model
Consider a thin nonlocal nonlinear material of length L, illuminated with a Gaussian beam of wavelength λ and beam waist, that propagates along the z-axis with field amplitude. The field, at the output face of this thin nonlocal nonlinear media, is given by [22]:
where , , , , is the nonlinear refractive coefficient, is the nonlinear absorptive coefficient, is the linear absorption coefficient, I0 is the on-axis intensity at the waist, k is the wavevector, m is a real number associated to the nonlocality of the media and , where .We propose that for a media that presents simultaneously two nonlinear, nonlocal responses, m1 and m2, the output field can be calculated as:
where and , with j = 1 or 2, are the maximum on-axis refractive and absorptive changes for each nonlocality. This expression can be understood as that obtained when the output field, transmitted through a thin nonlinear media with nonlocal response characterized by m1 is used to illuminate immediately a second thin nonlinear media with nonlocal response characterized by m2. Then the far field intensity distribution is calculated from the Fourier transform of [Eq. (2)].3. Numerical results
In this section we present numerical results calculated from the fast Fourier transform (FFT) of the field expression given by [Eq. (2)], for different nonlocalities to obtain curves for the close- and open-aperture z-scan configurations.
Initially, z-scan curves were numerically calculated without nonlinear absorption, ΔΨ01 = ΔΨ02 = 0, for the case m1 = 2 (local) with ΔΦ01 = 0.4 rad and m2 = 4 for ΔΦ02 = 0 with positive values of 0.4π rad, 1.2π rad and 2π rad in [Fig. 1(a)] and negative values of −0.4π rad, −1.2π rad and −2π rad in [Fig. 1(b)]. When both contributions have the same sign an increasing in the amplitude and a shifting of the peak occurs as the phase increases. However, when the contributions present opposite signs the shape of the z-scan curves were inverted with the increasing in magnitude of the second contribution. Note that the same type of curves were obtained in [12] for a material with simultaneous third- and fifth-order refractive nonlinear contributions.
Fig. 1 Numerical close-aperture z-scan curves without nonlinear absorption for m1 = 2 with ΔΦ01 = 0.4π rad and m2 = 4 for ΔΦ02 = 0 (black) with (a) positive and (b) negative values of: 0.4π rad (blue), 1.2π rad (red) and 2π rad (green).
In Fig. 2 z-scan curves are plotted for the same values of on-axis nonlinear phase changes than in the previous case with m2 = 1. The qualitative behavior is the same as that obtained in Fig. 1. but in this case the second nonlocality dominates the shape of the z-scan curves in [Fig. 2(b)] faster than in [Fig. 1(b)].
Fig. 2 Numerical close-aperture z-scan curves without nonlinear absorption for m1 = 2 with ΔΦ01 = 0.4π rad and m2 = 1 for ΔΦ02 = 0 (black) with (a) positive and (b) negative values of: 0.4π rad (blue), 1.2π rad (red) and 2π rad (green).
As a next step the influence of the nonlinear absorption was considered only for the second nonlocal contribution. Keeping the first contribution as local, m1 = 2, with ΔΦ01 = 0.2π rad and ΔΨ01 = 0, in Fig. 3 the second nonlocal contribution was considered as m2 = 4 with ΔΦ02 of: (a) 0.4π rad, (b) −0.4π rad where ΔΨ02 takes the following values: 0, ± 0.3 rad and ± 0.6 rad. When the refractive contributions of the two nonlocalities have the same sign the amplitude of the z-scan curves decrease as ΔΨ02 increase. When the two refractive phase changes have opposite signs two peaks and two valleys appear in the trace, the relative amplitude between them depends on ΔΨ02 values. This behavior is very similar to that reported in [12] for a thin media with simultaneous third- and fifth-order nonlinearity. The open aperture z-scan curves are in this case only functions of ΔΨ02: for positive values a valley is obtained and for negative ones a peak.
Fig. 3 Numerical z-scan curves for m1 = 2, ΔΦ01 = 0.2π rad, ΔΨ01 = 0 and m2 = 4, for close aperture with a) ΔΦ02 = 0.4π rad, b) ΔΦ02 = − 0.4π rad and c) open aperture, for ΔΨ02 of: −0.6 rad (blue), −0.3 rad (cyan), 0 (black), 0.3 rad (green) and 0.6 rad (red).
In Fig. 4 z-scan curves are plotted for the same values of on-axis nonlinear refractive and absortive phase changes than in the previous case with m2 = 1. When the two nonlocalities have the same sign in the refractive contributions the qualitative behavior are the same, see [Fig. 3(a)] and [Fig. 4(a)]. However, when the refractive contributions have opposite signs the z-scan curves are dominated by the second nonlocality. The amplitudes are reduced as ΔΨ02 increase. In the open aperture z-scan curves the amplitude of the peak or valley are broader and larger than in the previous case.
Fig. 4 Numerical z-scan curves for m1 = 2, ΔΦ01 = 0.2π rad, ΔΨ01 = 0 and m2 = 1, for close aperture with a) ΔΦ02 = 0.4π rad, b) ΔΦ02 = − 0.4π rad and c) open aperture, for ΔΨ02 of: −0.6 rad (blue), −0.3 rad (cyan), 0 (black), 0.3 rad (green) and 0.6 rad (red).
In order to demonstrate the influence of the two nonlocalities of the media with both refractive and absorptive contributions we present z-scan curves considering that m1 parameter is 2 (local case) and m2 parameter takes values of 1, 2, and 4 (see Fig. 5). In [Fig. 5(a)] for close aperture the following values were used: ΔΦ01 = 4π rad, ΔΨ01 = 0.2 rad, ΔΦ02 = −2π rad and ΔΨ02 = 0.2 rad. In [Fig. 5(b)] for close aperture the following values were used: ΔΦ01 = 0.4π rad, ΔΨ01 = 0.2 rad, ΔΦ02 = −0.2π rad and ΔΨ01 = 0.2 rad. These values were used in order to see the behavior of the z-scan curves for large and small refractive contributions. Positive refractive nonlinearity dominates the appearance of the curves because a large positive value was chosen. Due to the magnitude of the absorptive contributions remained constant the open aperture curves only depends on the nonlocality, see [Fig. 5(c)].
Fig. 5 Numerical z-scan curves for m1 = 2 and m2 of: 1 (black), 2 (blue) and 4 (red), nonlinear absorption phase changes of: ΔΨ01 = ΔΨ02 = 0.2 rad. For close aperture a) large nonlinear phase changes with ΔΦ01 = 4 π rad and ΔΦ02 = − 2 π rad, b) small nonlinear phase changes with ΔΦ01 = 0.4 π rad and ΔΦ02 = − 0.2 π rad and (c) open aperture.
From the previous analysis we can say that in some cases the consideration of a second nonlocal contribution in a thin media can give rise to small changes in the z-scan curve compared with that obtained with only one contribution. This is the case when both contributions have the same sign in the nonlinear refractivity, or when one of the nonlinear refractions is larger. Dramatic differences can be observed when the nonlinear refractive contributions have opposite signs or when one of the nonlocalities presents an m parameter smaller than 2.
4. Experimental results
More than one nonlocal contribution in the nonlinear response of a media is expected when it is illuminated with the adequate intensity by a period of time larger than the typical life-times of the physical mechanisms involved. That is the case for long duration laser pulses or continuous wave (cw) illumination. Z-scan experimental results were obtained for a 10 µm sample of hydrogenated amorphous silicon (a:Si-H) under cw illumination of a Helium Neon laser at 633 nm. The laser beam was focused by a convergent lens of 3.5 cm of focal length getting a beam waist of ~17 μm. The sample was scanned along the focus by a computer-controlled servo motor. The transmitted light was measured by a photodetector 1 m away from the lens with and without on axis small aperture. The experimental results for close and open aperture z-scan technique to an incident power of: 2, 4, 6, and 12 mW are shown in Fig. 6, a valley is mainly observed in all cases. The experimental results for close aperture at 6 mW were chosen as reference for the fitting, obtaining a nonlocality of m1 = 1.2. Then the rest of results were calculated numerically for ΔΦ01 = (0.044) P0 and ΔΨ01 = (0.063)P0, where P0 is the incident power in mW. The numerical results are plotted as continuous lines in Fig. 6. We can see that the lines do not follow with a good correspondence the experimental results; in fact a remarkable difference was obtained for the results at 12 mW.
Fig. 6 Experimental ( + ) and numerical (lines) z-scan curves for (a) close and (b) open aperture at incident powers of: 2 (black), 4 (blue), 6 (red), and 12 (green) mW.
From the previous analysis it is clear that considering only one contribution in the response it is not possible to reproduce the experimental results. Then a second contribution was added in the response with m2 = 1.6. This nonlocality participate only with nonlinear absorption coefficient for the first three powers, ΔΨ02 = −0.06 rad, −0.04 rad and 0.04 rad for 2, 4 and 6 mW respectively. For 12 mW it was necessary to include refraction of ΔΦ02 = 0.7π rad and absorption of ΔΨ02 = 0.2 rad. In Fig. 7 are plotted the curves calculated numerically together with the experimental results. In this case we can see that there is a good correspondence between the experimental and numerical results.
Fig. 7 Experimental ( + ) and numerical (lines) z-scan curves for (a) closed and (b) open aperture for incident power of: 2 (black), 4 (blue), 6 (red), and 12 (green) mW.
5. Conclusions
An expression for the output field of a thin nonlinear material, illuminated by a Gaussian beam, which exhibits more than one nonlocal response, was proposed. The expression considers that the nonlocal contributions are independent. The far field intensity distribution is obtained from the numerically calculated Fourier transform of the output field. Z-scan curves for open and close aperture by a thin nonlocal media with two simultaneous nonlocalities, with refractive and absorptive contributions, were numerically studied. The results demonstrated that the shape of the curve depends mainly on the magnitude of the nonlocality and the sign of the nonlinear phase changes involved in the nonlinear response. Unexceptional changes in the z-scan curve, compared with that for one nonlocality, are obtained when both nonlocal contributions present the same signs, however remarkable differences can be obtained when they present opposite signs. Experimental results obtained for a sample of hydrogenated amorphous silicon under cw laser illumination at different incident powers were fitted considering one and two nonlocal nonlinear contributions. A second nonlocality was necessary to include for obtaining a good fit with the experimental results. The proposed model can be extended to include three or more contributions.
Acknowledgments
This work was partially supported by PRODEP, SEP, México. B. A. Martínez Irivas acknowledges grant receipt 209459 of CONACYT, México.
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