Non-linear Optical Response as a Food Authentication: Investigation of Non-linear Optical Properties of Edible Oils by Spatial Self-Phase Modulation (SSPM) Method

The non-linear optical responses of cherry seed oil, avocado seed oil, and sesame oil were investigated by observing spatial self-phase modulation (SSPM) in oil samples using a continuous wave laser beam. The non-linear refraction coefficients and third-order non-linear susceptibility χ(3) of the edible oils were estimated to be 10−6 cm2/W and 10−5 esu, respectively, based on the intensity-dependent number of symmetric diffraction rings observed. Additionally, we examined the potential of the spatial self-phase modulation technique to determine the authenticity of three adulterated oils based on their non-linear optical properties. For this purpose, five different adulterated samples were prepared by diluting the samples with sunflower oil, a cheaper commercial oil, at various dilution ratios. The measured non-linear optical parameters of the adulterated samples revealed a strong correlation between the non-linear optical properties and the level of adulteration in the samples. These results suggest that the SSPM technique could be considered for estimating the degree of adulteration in samples based on their non-linear optical responses.


Introduction
The invention of the laser by Theodore H. Maiman in 1960(Maiman 1960) enabled scientists to observe various previously unknown phenomena due to the availability of highdensity electromagnetic energy sources. In 1961, Franken and his colleagues (Franken et al. 1961) were the first to notice the second-harmonic generation (SHG), providing a foundation for non-linear optics. Non-linear optics studies the optical properties of materials when exposed to highenergy electromagnetic fields. When a sample possesses non-linear optical characteristics and is excited with a higher power density, it can exhibit changes in optical properties such as refractive index, polarization, and variations in absorption coefficients. Non-linear optics describes and studies of all these phenomena (Boyd 2020).
The development of science and technology in recent decades has led to the implementation of new electronic and optoelectronic devices. Researchers are focused on exploring materials with high non-linear optical responses due to their wide range of applications in optical and photonic devices, including optical limiting (Marbello et al. 2020), optical switching (Henari and Cassidy 2012), and optical communication (Li et al. 2020b). Furthermore, studying non-linear optical properties helps ensure sample authenticity and purity (Ribeiro et al. 2019) and enables sample differentiation (de Souza et al. 2020). Several methods are currently available to determine the non-linear characteristics of materials, including elliptical rotation, measurement of beam distortion, interference of two or four waves, z-scan, and spatial self-phase modulation (SSPM) (Li 2017). By employing these techniques, researchers can evaluate the non-linear behavior of various materials, some of which are more experimentally complex than others.
When an intense beam of light passes through a nonlinear medium, the optical Kerr effect causes a change in the medium's refractive index (Boyd 2020). This intensitydependent refractive index variation leads to a phase shift in the light, resulting in a phenomenon known as spatial self-phase modulation. This modulation is characterized by the appearance of ring-shaped diffraction patterns. The origin of these rings can be attributed to the additional phase shift experienced by the Gaussian profile of the light beam as the medium is heated (Yang et al. 2005). The diffraction patterns become noticeable when the absorption coefficient of the medium is sufficiently high. Conversely, when the absorption coefficient is low for a particular light source, the medium only absorbs a small amount of power, which causes minimal heat generation and a negligible phase shift, making the rings invisible.
In recent years, scientists have demonstrated that vegetable oils such as olive oil (Marbello et al. 2020;Mousavi et al. 2019), palm oil (Zamiri et al. 2012), castor oil (Alencar et al. 2006), and other edible oils (Marbello et al. 2019) exhibit highly non-linear responses. Most vegetable oils are edible, particularly those derived from fruit seeds and other parts (Gunstone 2011). Organic oils and fats consist of combinations of triglycerides in liquid or solid phases depending on temperature, along with minor molecules such as free fatty acids, hydrocarbons, vitamins, phospholipids, waxes, pigments, and sterols (Alfred et al. 2002). Edible vegetable oils play a vital role in daily diets and are extensively used in domestic and commercial cooking (Gouilleux et al. 2018). Due to this, there is a significant risk of fraud associated with the production of vegetable oils. One common method of adulteration involves substituting original oils with cheaper alternatives (Moore et al. 2012). To authenticate various vegetable oils, a variety of methods have been suggested, including gas chromatography-mass spectrometry (Mota et al. 2021), Fourier transform infrared (FTIR) spectroscopy (Sota-Uba et al. 2021), and UV-visible spectroscopy (Ok 2017). Consequently, it is crucial to investigate and verify the authenticity of edible vegetable oils.
On the other hand, studies show that the year of crop and geographical origin (i.e., soil, climate, and geographical coordinates) directly affect the minor constituents of vegetable oils, which in turn affect the quality and characteristics of vegetable oil products. For instance, geographical origin (including edaphoclimatic conditions) and production season have significant effects on the fatty acid composition of olive oil (Lechhab et al. 2022a, b). Therefore, the study of minor components of vegetable oils according to different conditions of the year of production and geographical origin can be utilized for the purpose of quality assessment and authenticity, just as studies of lipid composition, especially fatty acid profiles, can be relevant to distinguish oil samples from similar samples obtained from different companies (Arena et al. 2022).
To investigate the non-linear optical response of avocado seed oil, cherry seed oil, and sesame oil, this study measured two optical characteristics: the non-linear refractive index n 2 and the third-order non-linear electric susceptibility χ (3) of these vegetable oils at a specific wavelength of 405nm. The analysis was performed using the well-known SSPM technique, a straightforward and accurate method for investigating the non-linear responses of samples (Jia et al. 2019;Shan et al. 2019). Additionally, the study examined the authenticity of the vegetable oils based on their nonlinear optical properties using the SSPM technique. The non-linear response of vegetable oils was also utilized for sample authentication. This technique offers a cheap, rapid, and simple method for detecting food adulteration, providing valuable results for determining the authenticity of vegetable oils.

Sample Preparation
Edible oils are extracted from various plant and animal sources, which have various metabolic, physical, and chemical properties. Different oil extraction processes are available depending on the source of the oil. In extracting vegetable oil, triglycerides must be separated from oil-containing kernels, seeds, or pulps. Vegetable oil can be produced using various mechanical, chemical, and biological techniques (Hasenhuettl 2016;Nde and Foncha 2020;Qin and Zhong 2016). Mechanical oil extraction extracts vegetable oil from oil seeds using a screw or press (Cakaloglu et al. 2018;Kazempour-Samak et al. 2021). Three types of vegetable oils (sesame oil, avocado seed oil, and cherry seed oil) used in this work were obtained mechanically using the mechanical cold-pressing technique. Oil extraction is performed only under pressure in this technique, and unlike all existing methods, no heat or solvent is employed. All samples are prepared by separating the seeds from external materials; a suitable pressing machine extracts the oils after the seeds have been cold-pressed. The extracted oils were centrifuged and filtered to remove unwanted contaminants. In addition, to evaluate the samples' authenticity and adulteration based on their non-linear optical response, each oil sample is diluted with sunflower oil at five different concentration ratios: 1:0, 2:1, 1:1, 1:2, and 0:1. The ratio 1:0 indicates that the oil is undiluted. Furthermore, the ratios 2:1, 1:1, and 1:2 imply the addition of 33.33%, 50%, and 66.66% of sunflower oil, respectively. The 0:1 ratio refers to the experimental sample containing 100% sunflower oil. It should be noted that the used sunflower oil was purchased from the local market.

SSPM Method
In the optical system, the non-linearity of the medium can be observed when the intense laser beam is transmitted through the non-linear medium, and the non-linearity is demonstrated in the polarization of the material, which the polarization intensity P(t) can express by (Li 2017;Liao et al. 2020): where ε 0 is the vacuum electric permittivity,E(t), χ (1) , χ (2) , and χ (3) are the optical field intensity, linear susceptibility, second-order and third-order susceptibilities, respectively. In the Eq. (1), the first term represents the linear optical effect, the second term refers to the second-order non-linear optical effect, and the third term describes the third-order non-linear optical effect. According to the optical Kerr effect (Sheik-Bahae and Hasselbeck 2009), when the light beam with a Gaussian intensity profile and fundamental mode (TEM 00 ) is transmitted through the non-linear medium length L along the z-axis, the refractive index n will be changed, and this change with the square of the applied light intensity is proportional. With the isotropic medium, the total refractive index n can be expressed as follows: where n 0 is the linear refractive index, and Δn = n 2 I is the change of the refractive index, n 2 is the non-linear refractive index, and I = 2P/πω(z) 2 is the intensity of the laser beam, P is the power of the laser, and (z) = 0 √ 1 + z∕z 0 2 is the beam radius at the propagation length z, ω 0 is beam waist radius, and z 0 = 2 0 ∕ is the Rayleigh length, where λ is the wavelength of the laser beam. The light electric field distribution at the incident plane of the medium (r, z) can be written as follows (Deng et al. 2005): Moreover, the electric field distribution on the exit plane (r, z + L) can be written as follows: (1) where E(0, z) is the electric field of the incident plane center of the medium, r is the radial coordinate, ω(z) is the beam radius at the medium incident plane, k = 2π/λ is the wave vector, R(z) = z( 1 + (z 0 /z) 2 ) is the radius of curvature of the wave front in the corresponding position, α is the linear absorption coefficient, and ϕ(r) is the total phase shift which involves of the additional transient phase shift produced by the transition of the beam through the medium (non-linear phase shift Δϕ NL (r)) and the Gaussian phase shift determined by the radius of curvature (change in linear phase Δϕ L (r)), which expressed as follows (Li et al. 2020a): is the effective length of the optical propagation, L 1 and L 2 are the distance between the front and back surfaces of the cuvette and the light focus position, respectively, as shown in Fig. 1. Also, I(r, z) = I 0 1 + z 2 ∕z 2 0 −1 exp −2r 2 ∕ 2 0 is the intensity distribution of the laser beam and I 0 is the intensity of the laser beam at the center of the Gaussian profile. Now, by using I(r, z) in Eq. (7), we obtain: SSPM diffraction patterns are observed when the phase difference between any two points r 1 and r 2 in the radial direction of the Gaussian beam establishes the relation Δϕ(r 1 ) − Δϕ(r 2 ) = mπ (m is an integer number). The bright and dark rings occur when m is even and odd, respectively.
(4) The phase difference between the center of the Gaussian beam intensity (r 1 = 0) and the infinity (r 2 = ∞) with the N rings, satisfied (Li et al. 2020a): From Eq. (8), Δϕ(∞) = 0 and Δϕ(0) = 2πn 0 n 2 I 0 L eff /λ, Then It is easy to determine the non-linear refractive index n 2 of materials using Eq. (10). The third-order non-linear susceptibility χ (3) (in the Gaussian unit) is defined by (Wu et al. 2016):   filter, the lens (f = 100 mm), and then the sample, which was filled in a 10-mm-thick quartz cuvette. The distance between the focal point of the lens and the sample L 1 was adjusted to 5mm in our experiments. L 2 value was the quartz sample cell thickness plus L 1 (i.e., L 2 = 15 mm). The 1/e 2 intensity radius at the center of the cuvette is measured to be ω 0 = 50 μm, yielding to z 0 = 19.4 mm and L eff = 7.87 mm. The irradiation-induced phase shift in oil samples generated the SSPM phenomenon and resulted to observe diffraction ring as it passed through the samples. The diffraction ring patterns were captured by a CCD camera (Camera EOS Kiss X50) on a black screen 260cm behind the sample quartz cuvette. Figure 2a shows the UV-Vis spectra (PHYSTEC-UVS-2500) of cherry seed oil, avocado seed oil, and sesame oil. These spectra were used to investigate samples' absorption at a given laser wavelength. It is found that all selected samples exhibit absorption at a wavelength of 405nm. It is clear that the cherry seed oil sample, has the highest absorption at 405nm with the absorption coefficient α = 1.26 cm −1 . Meanwhile, avocado seed oil and sesame oil exhibit a lower absorption coefficient value, with α = 0.62 cm −1 and 0.33 cm −1 , respectively. In order to authenticate the various vegetable oils based on their non-linear optical properties, diluted oil samples with cheaper oil (sunflower oil) in five dilution rates of 1:0, 2:1, 1:1, 1:2, and 0:1 were prepared. Figure 2b shows UV-Vis spectrum of sunflower oil. Figure 3a, b, and c show UV-Vis spectra of diluted avocado seed oil, cherry seed oil, and sesame oil at different dilution grades, respectively.

SSPM Results
When the Gaussian laser beam passes through the oil, the temperature of the sample rises due to the medium's light absorption, particularly at the focal point of the laser light.
Increasing the medium's temperature causes a change in the non-linear refractive index. A sample with a larger linear absorption coefficient for a particular wavelength absorbs more light at a specific wavelength, causing a greater thermal effect on the medium. As a result, the non-linear refractive index changes more significantly, indicating that the sample exhibits highly non-linear optical behavior. In our experiment, the first ring was formed when the power of the laser beam reached the threshold power P th . It is found that the required threshold power (or certain power) for observing the first (or a certain number) ring(s) for each sample is distinct, implying that each sample exhibits specific nonlinear behavior. Table 1 shows the required P th for each vegetable oils. The results indicate that the required threshold power P th for cherry seed oil to reveal the first diffraction ring is lower than that of the other two oil samples. This indicates that cheery seed oil has more non-linear optical properties than avocado seed oil, whereas avocado seed oil has more non-linear optical behavior properties than sesame   (Zamiri et al. 2012) oil. These results are compatible with the samples' ability to absorb light at a specific wavelength. The SSPM induced diffraction ring pattern appears as the laser power gradually increases. Increasing the power causing new rings to emerge from the center of the diffraction pattern and increases the number and radius of the rings. Figure 4a, b, and c shows the CCD-captured diffraction ring patterns for cherry seed oil, avocado seed oil, and sesame oil with different incident beam powers at 405-nm wavelengths. The variations in the number of rings generated by changing the laser beam's incident intensity are shown in Fig. 5. Clearly, the relationship between the number of rings and the intensity is linear with a slope of dN/dI. The properties of non-linear optical properties of samples become more evident when the slope of the fitted line dN/ dI is increased. Using n 0 = 1.47 (Xu and Li 2021), and a slope of dN/dI = 0.096 cm 2 /W, dN/dI = 0.175 cm 2 /W, and dN/dI = 0.034 cm 2 /W, respectively, we derived non-linear refractive indexes of n 2 = 1.68 × 10 −6 cm 2 /W for avocado seed oil, n 2 = 3.06 × 10 −6 cm 2 /W for cherry seed oil, and n 2 = 6 × 10 −7 cm 2 /W for sesame oil. According to Eq. (11), the relation between χ (3) and n 2 is proportional. Then, we estimated that for cherry seed oil, avocado seed oil, and sesame oil the value of non-linear susceptibility χ (3) are equal to 16.8 × 10 −5 esu, 9.18 × 10 −5 esu, and 3.28 × 10 −5 esu, respectively. Table 1 lists all the parameters that were measured in this study. The results reveal that the three vegetable oils: cherry seed oil, avocado seed oil, and sesame oil have considerable non-linear optical responses. Estimated values for both parametersn 2 and χ (3) for these oils are compared to reported values for some oils in Table 2. The measured values for n 2 and χ (3) are found to be in the orders of (10 −6 cm 2 /W) and (10 −5 esu), respectively, which are comparable with the values of thermally induced non-linear optical parameters expected for non-linear materials (Boyd 2020).

Authentication of Vegetable Oils Based on SSPM Method
As mentioned above, to evaluate the authenticity of various vegetable oils based on their non-linear optical properties, we produced adulterated oil samples by diluting the original oils with a cheaper oil (sunflower oil) at five different dilution rates: 1:0, 2:1, 1:1, 1:2, and 0:1. The diffraction pattern of a pure sample of sunflower oil is shown in Fig. 4d. At For studying the non-linear optical behavior of all diluted samples (as adulterated oils) at various dilution rates, the diffraction ring patterns were recorded for the adulterated oil samples using the same experimental setup. Fig. 6 displays the laser beam's diffraction patterns after passing through samples of dilution rates at maximum laser power. Diffraction rings appear more regularly in original samples (oils) than in diluted oils, and the number of rings decreases as the concentration of original oils decreases. We found that adding sunflower oil to initial oils results in a reduction in the density of the number of absorbing species and, thus, a decrease in the amount of absorbed energy by the samples (the UV-visible absorption spectra of oils at all concentrations are shown in Fig. 3). As a result of such a reduction in light absorption in diluted oils, the thermal effect in the samples is reduced, and the phase difference also decreases. Hence, dilution-induced changes in the diffraction ring patterns result in a decrease in the number of rings for a given power and an increase in the threshold power, as shown in Fig. 7a. Also, Fig. 7b illustrates the impact of mixing pure oils with sunflower oil in different proportions on the alteration of the n 2 value Figure 8 shows the required laser intensity to observe a certain number of rings, as well as the linear fit of the data. We see that the changes are linear and that the slope of the line decreases as the portion of original oil decreases (the adulteration ratio increases), thereby reducing the non-linear behavior of the samples. Calculated values of the non-linear optical index n 2 and non-linear optical susceptibility χ (3) listed in Table 3 indicate that undiluted oils have greater non-linear optical properties than adulterated oils. In other words, the values closely correlate to samples' adulteration levels.

Conclusion
Spatial self-phase modulation (SSPM) is a non-linear optical effect that can be used to investigate the non-linear optical properties of materials. In this work, for the first time, we investigated the non-linear optical response of three vegetable oils (avocado seed oil, cherry seed oil, and sesame oil) by using the SSPM technique with a continuous laser beam. The results show that the vegetable oil samples have a noticeable non-linear optical response. In addition, studying the non-linear behavior of vegetable oils allows us to evaluate diluted vegetable oils' adulteration by adding cheaper oil. It is suggested that the investigation of non-linear optical properties of vegetable oils using the SSPM technique could be used to authenticate and determine adulteration for these materials as an economical, fast, cheap, accurate, and portable technique. Also, the findings of this study suggest that vegetable oils might have some application, especially in optical switching and photonic devices, based on their non-linear optical characteristics.
Author Contribution All authors contributed to the concept and design of the study. Data preparation, collection, and analysis were performed by all authors. All authors read and approved the final manuscript.
Data Availability All data generated or analyzed during this study are included in this published article