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In this work we present numerical results of the far field intensity distributions obtained for a Gaussian beam after crossing a thin nonlinear nonlocal material that exhibit nonlinear refraction and absorption. The distributions are obtained for different positions along the Z axis and different signs of the nonlinear absorption. The results demonstrate that the far field intensity patterns obtained for strong nonlocal media are more affected by the presence of the nonlinear absorption than weak nonlocal media.
© 2015 Optical Society of America
1. Introduction
A large number of spatial effects can be observed when an intense light beam interacts with a nonlinear media. In particular, propagation of a Gaussian beam in a material with an intensity dependent refractive index can produce a far field intensity pattern of concentric rings. This phenomenon is known as spatial self phase modulation, and has been studied for many research groups [1–16]. The first observation of this effect was made by Callen et al [2], in a thermally self-defocusing media. Dabby et al [3], developed a quantitative study of this effect. In order to explain the phenomenon, Santamato and Shen [9], proposed that the far field pattern was composed by two sets of concentric rings, one of these is the result of the interference between self-phase modulation and wavefront curvature associated with a focused beam and the other one is due to the nonlinearity. In [10], based on the far field distributions, a simple method was proposed to determine the sign of the nonlinear refractive index of the sample under illumination. In [11, 12], was presented a qualitative model to explain the spatial profiles produced by an organic nonlinear absorber with a thermal response. In [13], the influence of the wavefront curvature in the far field patterns was numerically studied using the Fresnel-Kirchhoff diffraction formula. In [14], experimental results were presented that confirmed the results presented in [13]. In [15] a z-scan theory based on the solution of the nonlinear paraxial wave equation valid for local response and large nonlinearities including nonlinear refraction and absorption was introduced. Garcia et al [16, 17], presented a model where the nonlocality of the nonlinear response (depicted with an m parameter) exhibited by the media was taken into account. A value of m equal to 2 represents the local case, that is when the profile widths of the nonlinear phase change and the input intensity are the same. A value of m less than 2 represents a nonlinear phase change profile broader than the input intensity profile and a value of m larger than 2 represents a nonlinear phase change profile narrower than the input intensity profile. This nonlocal model has been used to demonstrate that the z-scan curves for local and nonlocal responses are different. Initially the nonlocal model considered solely a refractive nonlinearity, recently in [18], the nonlocal model was extended to consider materials with absorptive nonlinearity. In fact, analytical expressions were obtained to calculate the normalized transmittance in a z-scan experiment for any magnitude of the on-axis nonlinear phase change. These studies demonstrate that the nonlinear absorption modifies the amplitude, width and position of the peak and valley in a z-scan curve. However, a study of how the far field intensity patterns are affected when the nonlocal media exhibits nonlinear absorption has not been presented.
In this work, considering a Gaussian beam that illuminates a thin sample of nonlocal nonlinear media, that exhibit both refractive and absorptive nonlinear index, the far field intensity patterns are numerically calculated for different positions of the sample with respect to the beam waist with a positive and negative absorption coefficient. A nonlocality higher, lower and equal to 2 is also considered. The results will demonstrate that it is important to consider both refractive and absorptive nonlinear effects in the sample, to calculate the far field intensity patterns. In the next section we present the main aspects of the nonlocal model and how the absorptive nonlinearity is considered, then the numerical results under different positions and magnitudes of the nonlocality are presented. Finally the main conclusions are given.
2. Theoretical model
Consider a nonlocal nonlinear material of length L, with a nonlinear refractive coefficient γ and nonlinear absorption coefficient β given by:
where n0 is the linear refractive index, α0 is the linear absorption coefficient and I is the incident intensity. This media is illuminated with a Gaussian beam of wavelength λ and beam waist w0 that propagates along the z-axis, with field amplitude E (r, z). The output field considering that the sample is thin (L<< z0) can be calculated, according to [18], by:where , , ,, 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 andNote that Eq. (3) is reduced to the well accepted expression for the output field in a local media when m = 2. For m values different to 2, Eq. (3) is mainly affected by the function Gloc to the power m/2.
In this work we present numerical results of the intensity patterns at far field obtained from the Fourier transform (FFT), of the output field, given by Eq. (3), for different values of the m parameter and different z positions, close to the beam waist, of the nonlocal media characterized by a given value of ΔΦ0 and ΔΨ0.
3. Numerical results
When the absolute value of the position z of the thin sample was larger than 2z0 the differences in the far field intensity pattern were very small compared with that obtained without nonlinear absorption. For this reason, in our study, we limited the position of the sample to be z = -z0, 0 and z0. The refractive nonlinearity was taken as positive with an on-axis phase change at the waist of ΔΦ0 = 4π. The linear absorption coefficient was fixed at α0 = 1m−1. The absorptive nonlinearity was taken both positive and negative with a magnitude of ΔΨ0 = 0.5 rad. The nonlocal parameter m takes the values 1, 2 and 4. The far field intensity profiles are plotted using ρ as the transversal coordinate that is given by ρ = Dθ where D is the distance to the observation plane and θ is the diffraction angle.
First we present the far field intensity patterns when the media is set before the waist at z = -z0 (convergent beam) and the sample does not present nonlinear absorption for all the values of the nonlocal parameter m, see Fig. 1. We can see that the patterns present the following characteristics: a ring surrounding a central bright spot where the spatial extension of the pattern reduces as m increases. When the nonlinear absorption was considered, see Fig. 2, the changes in the pattern depend on the nonlocality. The far field pattern is more sensitive to the presence of nonlinear absorption for a nonlocal media when the m parameter was smaller than 2. Small changes were produced in the intensity pattern for m larger than 2. For a positive nonlinear absorption the spatial extension of the pattern is reduced, and for a negative the opposite occurs. In order to see the intensity profile changes for nonlinear absorption curves, these have been normalized with the maximum intensity value of the corresponding no absorption curve.
Fig. 1 Far field intensity cross sections obtained for a sample set at z = - z0 with ΔΦ0 = 4π rad, ΔΨ0 = 0 and m of: a) 1, b) 2 and c) 4.
Fig. 2 Far field intensity profiles obtained for a sample set at z = - z0 with ΔΦ0 = 4π rad, ΔΨ0 = 0.5 (red), ΔΨ0 = −0.5 (blue) and m of: a) 1, b) 2 and c) 4. As reference the intensity profiles for ΔΨ0 = 0 are plotted in black.
When the sample was set at z = 0, and does not present nonlinear absorption, the far field intensity patterns for different values of the nonlocal parameter m are shown in Fig. 3. In this case the far field pattern presents two rings, as was to be expected due to the magnitude of ΔΦ0. The spatial extension of the pattern increases as the m parameter. For m = 1 the outer ring presents the most intense region of the pattern while for m = 2 and m = 4 the central lobe is the most intense. The spatial extension of the pattern is reduced when the sample presents a positive nonlinear absorption and the opposite occurs for a negative one, see Fig. 4, as in the previous position. The number of rings of the pattern is reduced for a positive nonlinear absorption when m is equal or less than 2.
Fig. 3 Far field intensity cross sections obtained for a sample set at z = 0 with ΔΦ0 = 4π rad, ΔΨ0 = 0, and m of: a) 1, b) 2 and c) 4.
Fig. 4 Far field intensity profiles obtained for a sample set at z = 0 with ΔΦ0 = 4π rad, ΔΨ0 = 0.5 (red), ΔΨ0 = −0.5 (blue) and m of: a) 1, b) 2 and c) 4. As reference the intensity profiles for ΔΨ0 = 0 are plotted in black.
After the waist of the Gaussian beam (divergent beam), when the thin nonlinear media without nonlinear absorption was set at z = z0, the far field intensity patterns present the following characteristics: a central bright spot surrounded by one ring. The spatial extension of the pattern is reduced as m increases, see Fig. 5. For a positive nonlinear absorption the spatial extension of the pattern is reduced and for a negative the opposite occurs, see Fig. 6.
Fig. 5 Far field intensity cross sections obtained for a sample set at z = z0 with ΔΦ0 = 4 π rad, ΔΨ0 = 0, and m of: a) 1, b) 2 y c) 4.
Fig. 6 Far field intensity profiles obtained for a sample set at z = z0 with ΔΦ0 = 4π rad, ΔΨ0 = 0.5 (red), ΔΨ0 = −0.5 (blue) and m of: a) 1, b) 2 and c) 4. As reference the intensity profiles for ΔΨ0 = 0 in black.
In order to analyze the evolution of the far field intensity pattern when the nonlocal media is moved around the focal region, we show the cross sections for a media with m = 1 and ΔΦ0 = 4π rad that exhibit a negative absorption coefficient, Fig. 7, and a positive one, Fig. 8, for different sample positions between −3z0 and 2z0. We can observe that, for the same nonlocal parameter, the changes in the far field pattern with negative nonlinear absorption are bigger than that of positive nonlinear absorption.
Fig. 7 Far field intensity cross sections for a nonlocal media, m = 1, with ΔΦ0 = 4π rad and ΔΨ0 = - 0.5 rad, for sample positions of: a) −3z0, b) −2z0, c) -z0, d) 0, e) z0 and f) 2z0.
Fig. 8 Far field intensity cross sections for a nonlocal media, m = 1, with ΔΦ0 = 4π rad and ΔΨ0 = 0.5 rad, for sample positions of: a) −3z0, b) −2z0, c) -z0, d) 0, e) z0 and f) 2z0.
The local case, m = 2, with ΔΦ0 = 4π rad and ΔΨ0 = - 0.5 rad presented the far field cross sections shown in Fig. 9, for the same positions than the previous Figs. This local case was studied in ref [15], where the z-scan curve was plotted for a thin material without nonlinear absorption. The influence of the nonlinear absorption in the far field patterns for this locality were less remarkable than in the previous case.
Fig. 9 Far field intensity cross sections for the local case, m = 2, with ΔΦ0 = 4π rad and ΔΨ0 = - 0.5 rad. Sample positions of: a) −3z0, b) −2z0, c) -z0, d) 0, e) z0 and f) 2z0.
The nonlocal case, m = 4, gave the far field intensity cross sections shown in Fig. 10, with ΔΦ0 = 4π rad and ΔΨ0 = - 0.5 rad, for the same sample positions of the previous Figs. The influence of the nonlinear absorption, for this case, produce slight changes in the far field pattern that are practically undetectable.
Fig. 10 Far field intensity cross sections with ΔΦ0 = 4π rad, ΔΨ0 = - 0.5 rad, m = 4 for sample positions of: a) −3z0, b) −2z0, c) -z0, d) 0, e) z0 and f) 2z0.
From the previous analysis is important to remark that the major effects in the far field patterns due to the presence of nonlinear absorption in a thin nonlocal media are observables for sample positions close to the beam waist (−2z0< z< 2z0) of the incident Gaussian beam. Far field patterns of thin nonlinear media with weak nonlocalities (described with values of m larger than 2) are less sensitive to the presence of nonlinear absorption. Far field patterns for local media and nonlocal materials, with m<2, exhibit clear changes with the presence of nonlinear absorption. For materials that present a ΔΦ0<π, the far field patterns are not remarkable affected by the presence of the nonlinear absorption. In this case is better to study the changes introduced by the presence of nonlinear absorption by detecting the on-axis [18], or the radius [19], of the beam.
4. Conclusions
Far field intensity distributions due to thin nonlocal media with simultaneous nonlinear refractive and absorptive contributions were numerically analyzed. The spatial extension of the pattern is affected by the presence of the nonlinear absorption being broaden for the negative and narrowed for the positive one. Depending on the ratio between ΔΦ0 and ΔΨ0 the number of rings at far field can be reduced by the presence of positive nonlinear absorption or increased by the presence of negative nonlinear absorption. Far field patterns due to a nonlocal media with m parameter around or smaller than 2 exhibit more changes by the presence of nonlinear absorption than that with m parameter larger than 2.
Acknowledgements
This work was partially supported by PRODEP, SEP, México. B. A. Martínez Irivas acknowledges grant receipt 209459 of CONACYT, México.
References and links
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