1. Introduction

Nonlinear optical effects are the result of interaction of electromagnetic radiation with matter at very high intensities, causing a change in their optical properties. A large number of materials with nonlinear response have been investigated such as gold nanoparticles [1], silver nanoparticles [2], hybrid materials [3,4], liquid crystals, organic dyes [5,6], etc., determining whether its answer is local or nonlocal. Nonlocal response occurs when nonlinear mechanism involve transport processes such as heat conduction in materials with thermal nonlinearities, drift or diffusion of electric charges in photorefractive media by long-range forces or interaction of several bodies. In particular, organic medium with nonlinear optical response have gained great interest due to low cost, compatibility and high nonlinear response. The origin of this nonlinearity can be attributed to the molecular reorientation induced by light, electronic excitation or thermal effects. There are several techniques for the measurement of optical properties [7]. In particular, due to easy implementation, the Z-scan technique originally proposed by Sheik-Bahae et al. [8–10] has been used to obtain the sign and magnitude of change in nonlinear refractive index and nonlinear absorption. In this technique, the transmittance is measured to far field along optical axis as function of the sample position z with respect to the minimum radius of Gaussian beam (closed Z-scan). When the medium has a refractive nonlinear response, in the transmittance curves is generated a peak-valley or valley-peak for a negative or positive nonlinear response respectively. Nevertheless, for nonlocal nonlinear media [1, 2, 5, 11–13], Bahae model cannot be applied; since it is only valid for local media with phase changes less than π [11]. In particular, in this paper we present a study of Hibiscus Sabdariffa due to high nonlinearity. Previous investigations have reported the magnitude of nonlinear refraction index $n_2$ for phase changes less than π/4 [14]. However, we observed a phase change major than π/4 using closed Z-scan technique and generating self-diffraction rings (self-phase modulation), which indicate a phase change bigger than 2π. Since we observed in the experiments nonlinear absorption, we required a theoretical model that allows us to perform adjustments to the experimental curves with large phase changes and nonlocal response in the simultaneous presence of both nonlinear refraction and nonlinear absorption. Hence, we used output field of a medium with nonlinear absorption and refraction given by [9], where we introduced nonlocal response using the same proposal given by [15, 16]. The experimental curves are in agreement with the theoretical results, indicating that the medium has a high nonlinear nonlocal response.

2. Mathematical model

Our calculations are based on the model reported by García-Ramírez et. al. [15, 16], which calculates the far-field intensity distribution of a Gaussian beam propagating in a nonlocal nonlinear thin medium. They consider a Gaussian beam with amplitude of the form,

Nonetheless, there are materials with nonlocal response and nonlinear absorption. In this work, Hibiscus Sabdariffa was characterized and nonlinear absorption was observed. The electric field was analyzed at the output in Eq. (7) with the model of [9, 10]. The locality or nonlocality is represented by m parameter, in the exponent value of the squared electric field module.

Figure 1 shows theoretical curves of Z-scan with nonlinear refraction and absorption, for three different media: two with nonlocal response and one with local response, considering a phase change due to nonlinear absorption ($\Delta {\Psi }_0=0.5$) and two phase change$\Delta {\Phi }_0$. The first condition happens when$\Delta {\Phi }_0=-\pi$, (R = 6.28), the curves have a moderately suppressed peak and enhanced valley in the three media (see Fig. 1(a)). For a local medium the distance of peak (p) to reference value (r) denoted by $\Delta T_{p-r}$ is smaller than the distance of reference value of valley (v) denoted by $\Delta T_{r-v}$, i.e., $\Delta T_{r-v}>\Delta T_{p-r}$which indicates the presence of nonlinear absorption [17]. According to García-Ramírez et. al. [15, 16], m = 1 represents a strong nonlocal nonlinearity; m = 2 local nonlinearity, and m = 4 weak nonlocal nonlinearity. The second condition occurs when $\Delta {\Phi }_0=-7.5\pi$(R = 47.123), then$\left|\Delta {\Phi }_0\right|>>\Delta {\Psi }_0$, as it is shown in Fig. 1(b). Although nonlinear absorption is present in the medium, the decrease of the maximum peak is not a must. Thus, when $\Delta T_{p-r}>\Delta T_{r-v}$ then the predominant phenomenon is the nonlinear refraction. When $\Delta T_{r-v}>\Delta T_{p-r}$then the predominant phenomenon is the nonlinear absorption. This occurs in the three media despite the peak-valley distance ($\Delta T_{p-v}$) is different for each medium.

 

Fig. 1 Z-Scan curves with $\Delta {\Psi }_0=0.5$ and (a) $\Delta {\Phi }_0=-\pi$ and R = 6.28, (b) $\Delta {\Phi }_0=-7.5\pi$and R = 47.123.

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Figure 2 shows the theoretical curves for a local medium (m = 2) and for a strong nonlocal medium (m = 1) with $\Delta {\Psi }_0=0.5$ at different values of $\Delta {\Phi }_0$. Three cases were observed for the two media. For a local medium [see Fig. 2(a)]; first, with phase changes of $\Delta {\Phi }_0=-\pi /2$ and $\Delta {\Phi }_0=-2\pi$, nonlinear absorption predominates. Second, when $\Delta {\Phi }_0=-3\pi$, a balance between nonlinear absorption and refraction exists ($\Delta T_{r-v}\approx \Delta T_{p-r}$). Third, when $\Delta {\Phi }_0=-6\pi$, nonlinear refraction predominates. For a nonlocal medium (see Fig. 2(b)); first, when $\Delta {\Phi }_0=-\pi /2$, nonlinear absorption predominates. Second, when $\Delta {\Phi }_0=-2\pi$, a balance between nonlinear absorption and refraction exists (ΔT_r-v » ΔT_p-r). Third, with $\Delta {\Phi }_0=-3\pi$, $\Delta {\Phi }_0=-6\pi$, nonlinear refraction predominates.

 

Fig. 2 Z-Scan curves with $\Delta {\Psi }_0=0.5$ and $\Delta {\Phi }_0$ different for (a) m = 2, (b) m = 1.

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Hence, we can say that the transmittance at the output of sample depends on the medium and the parameters $\Delta {\Psi }_0$ and $\Delta {\Phi }_0$. Figure 3 shows the transmittance as function of $\left|\Delta {\Phi }_0\right|$ y $\Delta {\Psi }_0$ for a local medium (m = 2) and for a nonlocal highly medium (m = 1) in z = 0. If absorption and refraction are small, we have a high transmittance because the optical response is nearly linear. If the absorption is small and refraction increases, the transmittance decreases. Due to large phase change, the valley (minimum) is broadening [see Figs. 1(b)-2(a)] in $z=0$. Another interesting case is observed when the refraction is small and absorption increases; then, transmittance decreases. If the refraction and absorption increases, the transmittance tends to increase to a saturation point. This behavior can be used to optical limiting. For nonlocal media we observed the same behavior (see Fig. 3(b)), even though the transmittance value decays faster than a local medium (see Fig. 1(a) with m = 1 and m = 2 in $z=0$).

 

Fig. 3 Dependence of transmittance as function of phase change due to nonlinear refraction |ΔΦ₀|, and phase change due to nonlinear absorption ΔΨ₀ for a medium (a) local (m = 2), (b) nonlocal (m = 1).

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3. Experimental setup

Figure 4 shows absorption spectrum of Hibiscus Sabdariffa. For wavelengths in the visible region, maximum absorption occurs at 540 nm. In our experiments we used a cw Ar laser source at 514 nm of wavelength of variable power and with a spot of minimum radius of${\omega }_{0L}=1.1mm$.

 

Fig. 4 Optical absorption spectrum of hibiscus sabdariffa.

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Hibiscus Sabdariffa is a flower which contains different organic compounds [18]. The extracts were obtained by following a method reported in [19] and filtered to eliminate residual solids. The nonlinear optical properties of hibiscus sabdariffa were studied with Z-scan technique. The experimental setup is showed in Fig. 5. The sample (S) is moved around of the minimum radius of the Gaussian beam (${\omega }_0=22\mu m$) obtained with a lens of focal length 7.5 cm. The transmittance at the output of the sample is captured with a photodetector on the optical axis (opening closed) to the far field.

 

Fig. 5 Z-scan setup with Ar cw laser, lens (L) and sample (S).

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Figure 6 shows the Z-scan curve with 1 mW of power. Transmittance value far from the minimum radius is below unity (approximately 0.5), indicating that the light transmitted in the linear regime is approximately 50% of the incident light; this is due to the strong linear absorption of the sample. For this laser power nonlinear absorption is not observed, since the incident intensity is not enough to excite the medium. We show numerical fit using the Eq. (7) with nonlocality m = 0.4 corresponding to a strong nonlocal response, with maximum phase change of $\Delta {\Phi }_0=-0.16\pi$ and $\Delta {\Psi }_0=1x{10}^{-9}$(negligible). This fit agrees with the results obtained in Eq. (3) due to this equation describes the refractive response of the sample without nonlinear absorption. We obtained a value of $n_2=-2.1x{10}^{-11}{m}^2/W$.

 

Fig. 6 Z-scan curve with closed opening with P = 1 mW. Numerical comparison is made with $\Delta {\Phi }_0=-0.16\pi$ and m = 0.4.

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However, when we increased the power of the laser beam incident in the medium (experimental measurements were made for six different powers), the presence of nonlinear absorption was observed when we performed the experiment with open aperture Z-scan and $\Delta {\Psi }_0$ was calculated for each case, using Eq. (7) with$\Delta {\Phi }_0=0$. Subsequently, we used again the output field given by Eq. (7), for the curves obtained experimentally with closed aperture Z-scan, due to the presence of both nonlinear refraction and nonlinear absorption. We can observe in Fig. 7, the experimental curves of Z-scan have good fit to the theoretical curves with m = 0.4, indicating that the medium has a high nonlocal response. The experimental results are consistent with theoretical results.

 

Fig. 7 Z-scan curves with power of: (a) 4 mW;(b) 7 mW;(c) 8 mW;(d) 9 mW;(e) 10 mW; y (f) 20 mW.

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In this curves we observed atypical and asymmetrical configurations. These indicate the nonlinear refraction in the sample is related to a strong nonlocal nonlinear process of thermal origin. With this methodology we can determine the magnitude of the nonlinear refraction index of the medium at different power of Hibiscus Sabdariffa. We found that the order is 10⁻¹⁰ m²/W.

Figure 8 shows the increment of the magnitude of $n_2$regarding the induced power in the medium. Afterwards, the increment is slow and it shows a tendency to a saturation point.

 

Fig. 8 Dependence of n₂ as function of the power.

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It is well known that the nonlinear refraction index induced by thermal effects is obtained in a theoretical way: studying thermal effects of laser. The relation between the nonlinear refractive index (n₂) and the thermal-optic coefficient is given by [11,13] $n_2=\alpha {\omega }_0^2(dn/\kappa dT)$; where dn/dT is the refractive index ratio regarding temperature T, $\kappa$ is thermal conductivity, α is the linear absorption and ${\omega }_0$ is the minimum radius of the Gaussian beam. If we consider that the energy absorbed by the sample produces its own heating, the value of $dn/\kappa dT$ was calculated and it was obtained 0.0064 m/W.

4. Self-phase modulation

Changes in nonlinear refractive index due to intensity produce a distortion of wave-front and alter the transverse phase of the beam, generating spatial self-phase modulation when $\Delta {\phi }_0>2\pi$. This is seen as a diffraction pattern, which depends on the intensity for a local medium, where the number of rings is related to the maximum phase shift [20]. Still, it does not apply for nonlocal media [15]. For the detection of the intensity distribution to far field, we used the array shown in Fig. 5, where instead of a photodetector, we placed a CCD camera. The images are recorded and their transversal sections are shown in Fig. 9 for different powers. The images are generated when the sample is placed in the negative curvature radius of the beam. We can observe that when the power increases, the number of the rings increases too; nevertheless, it is not proportional to the power. When the characterization of the medium was performed by Z-scan, high nonlinear response was obtained. Therefore, a greater number of diffraction rings would be expected. However, we noted a few rings. For example, when we induced a power of 4mw to medium, we observed only a spot with a concentric tenuous ring, which does not match with the Z-scan curve obtained in Fig. 7(a). These results also show nonlinear refraction and absorption. The rings of SPM are weakened due to absorption in the medium with respect to a purely refractive medium. These results indicate that nonlinear absorption and nonlocal response are present in the medium. In Fig. 9(f) we can see the diffraction pattern becomes asymmetric due to the convection effect generated in the medium by a thermal effect [21]. It is important to note that the SPM effect is evident in Z-scan curve [Fig. 7(f)], where we observed peaks in the experimental and theoretical curve. These peaks correspond to the phase change of the dependent beam of Z position.

 

Fig. 9 Far field diffraction patterns of the laser beam propagating in the sample with laser power of: (a) 4mw, (b) 7mw, (c) 8 mw, (d) 9 mw, (e) 10 mw, (f) 20 mw.

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4. Conclusions

We have found out that Hibiscus Sabdariffa shows a high nonlocal response of 10⁻¹⁰ m²/W order and strong nonlocal response given by m = 0.4. This nonlocal response agrees with the thermal effect, which is the nonlinear origin and it is numerical considered as a nonlocal process. Our numerical analysis present three interesting cases in Z-scan curves: a) absorption predominates over refraction, b) refraction predominates over absorption, and c) absorption and refraction are balanced. In this analysis, we demonstrate that there is not necessarily a decrease in the peak and a widening valley when absorption and refraction are simultaneously present. Another evidence of nonlinear absorption is shown in SPM. The generated number of concentric rings is smaller than those obtained in purely refractive local media. The Hibiscus nonlinearity is an indication of a promising material for nonlinear applications at low power.