## 1.1 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 (Nde and Foncha 2020; Qin and Zhong 2016; Hasenhuettl 2000). 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 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.

## 1.2 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):

$$P(t)={\varepsilon _0}\left\{ {{\chi ^{(1)}}E(t)+{\chi ^{(2)}}E{{(t)}^2}+{\chi ^{(3)}}E{{(t)}^3}+...} \right\}$$

1

Where\({\varepsilon _0}\)is the vacuum electric permittivity,\(E(t)\),\({\chi ^{(1)}}\),\({\chi ^{(2)}}\), and \({\chi ^{(3)}}\)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 2000), when the light beam with a gaussian intensity profile and fundamental mode (TEM00) is transmitted through the non-linear medium length along the z-axis, the refractive index will be changed, and this change with the square of the applied light intensity is proportional. With the isotropic medium, the total refractive index can be expressed as follows:

Where \({n_0}\) is the linear refractive index, and \(\Delta n={n_2}I\) is the change of the refractive index, \({n_2}\) is the non-linear refractive index, and \(I={{2P} \mathord{\left/ {\vphantom {{2P} {\pi \omega {{(z)}^2}}}} \right. \kern-0pt} {\pi \omega {{(z)}^2}}}\) is the intensity of the laser beam, is the power of the laser, and \(\omega (z)={\omega _0}\sqrt {1+{{\left( {{z \mathord{\left/ {\vphantom {z {{z_0}}}} \right. \kern-0pt} {{z_0}}}} \right)}^2}}\) is the beam radius at the propagation length, \({\omega _0}\) is beam waist radius, and \({z_0}={{\pi \omega _{0}^{2}} \mathord{\left/ {\vphantom {{\pi \omega _{0}^{2}} \lambda }} \right. \kern-0pt} \lambda }\) is the Rayleigh length, where \(\lambda\) is the wavelength of the laser beam. The light electric field distribution at the incident plane of the medium \(\left( {r,z} \right)\) can be written as (Deng et al. 2005):

$$E(r,z)=E(0,z)\exp \left( { - \frac{{{r^2}}}{{\omega {{(z)}^2}}}} \right)\exp \left( { - i\frac{{k{n_0}{r^2}}}{{2R(z)}}} \right)$$

3

Moreover, the electric field distribution on the exit plane \(\left( {r,z+L} \right)\) can be written as:

$$E(r,z+L)=E(0,z)\exp \left( { - \frac{{\alpha L}}{2}} \right)\exp \left( { - \frac{{{r^2}}}{{\omega {{(z)}^2}}}} \right)\exp \left( { - i\phi \left( r \right)} \right)$$

4

where \(E\left( {0,z} \right)\) is the electric field of the incident plane center of the medium, is the radial coordinate, \(\omega (z)\)is the beam radius at the medium incident plane, \(k={{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }\) is the wave vector, \(R(z)=z\left( {\,1+{{\left( {{{{z_0}} \mathord{\left/ {\vphantom {{{z_0}} z}} \right. \kern-0pt} z}} \right)}^2}} \right)\) is the radius of curvature of the wave front in the corresponding position, \(\alpha\) is the linear absorption coefficient, and \(\phi \left( r \right)\) 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\(\Delta {\phi _{NL}}\left( r \right)\)) and the Gaussian phase shift determined by the radius of curvature (change in linear phase\(\Delta {\phi _L}\left( r \right)\)), which expressed as follows (Li et al. 2020):

$$\phi \left( r \right)=\Delta {\phi _L}\left( r \right)+\Delta {\phi _{NL}}\left( r \right)$$

5

where

$$\Delta {\phi _L}\left( r \right)=\frac{{k{n_0}{r^2}}}{{2R\left( z \right)}}$$

6

$$\Delta {\phi _{NL}}\left( r \right)=\left( {\frac{{2\pi {n_0}}}{\lambda }} \right)\int\limits_{0}^{{{L_{eff}}}} {{n_2}I\left( {r,z} \right)} dz$$

7

Where \({L_{eff}}={z_0}\arctan \left. {\left( {{z \mathord{\left/ {\vphantom {z {{z_0}}}} \right. \kern-0pt} {{z_0}}}} \right)} \right|_{{{L_1}}}^{{{L_2}}}\) 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 \(I\left( {r,z} \right)={I_0}{\left( {1+\left( {{{{z^2}} \mathord{\left/ {\vphantom {{{z^2}} {z_{0}^{2}}}} \right. \kern-0pt} {z_{0}^{2}}}} \right)} \right)^{ - 1}}\exp \left( {{{ - 2{r^2}} \mathord{\left/ {\vphantom {{ - 2{r^2}} {\omega _{0}^{2}}}} \right. \kern-0pt} {\omega _{0}^{2}}}} \right)\) 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 \(I\left( {r,z} \right)\) in Eq. (7), we obtain:

$$\Delta {\phi _{NL}}\left( r \right)=\frac{{2\pi {n_0}{n_2}{I_0}{L_{eff}}}}{\lambda }\exp \left( { - \frac{{2{r^2}}}{{\omega _{0}^{2}}}} \right)$$

8

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 \(\Delta \phi \left( {{r_1}} \right) - \Delta \phi \left( {{r_2}} \right)=m\pi\) ( is an integer number). The bright and dark rings occur when is even and odd, respectively. The phase difference between the center of the Gaussian beam intensity (\({r_1}=0\)) and the infinity (\({r_2}=\infty\)) with the rings, satisfied (Durbin et al. 1981):

$$\Delta \phi \left( 0 \right) - \Delta \phi \left( \infty \right)=2N\pi$$

9

From Eq. (8), \(\Delta \phi \left( \infty \right)=0\) and \(\Delta \phi \left( 0 \right)={{2\pi {n_0}{n_2}{I_0}{L_{eff}}} \mathord{\left/ {\vphantom {{2\pi {n_0}{n_2}{I_0}{L_{eff}}} \lambda }} \right. \kern-0pt} \lambda }\), Then

$${n_2}=\frac{\lambda }{{2{n_0}{L_{eff}}}}\frac{N}{I}$$

10

It is easy to determine the non-linear refractive index \({n_2}\) of materials using Eq. (10). The third-order non-linear susceptibility\({\chi ^{\left( 3 \right)}}\)(in the Gaussian unit) is defined by (Wu et al. 2016):

$${\chi ^{\left( 3 \right)}}=\frac{{c\lambda {n_0}}}{{2.4 \times {{10}^4}{\pi ^2}{L_{eff}}}}\frac{{dN}}{{dI}}=\frac{{n_{0}^{2}c}}{{12{\pi ^2}}}{10^{ - 7}}{n_2}$$

11

Figure 1 is a schematic diagram of the SSPM experimental setup. The blue continuous-wave (CW) laser with a wavelength of \(405\,nm\) was utilized as incident light to study the SSPM effect in oil samples. The laser beam passed through the spatial 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 \(5\,mm\) in our experiments. \({L_2}\)value was the quartz sample cell thickness plus\({L_1}\)(i.e.,\({L_2}=15\,mm\)). The\({1 \mathord{\left/ {\vphantom {1 {{e^2}}}} \right. \kern-0pt} {{e^2}}}\)intensity radius at the center of the cuvette is measured to be \({\omega _0}=50\,\mu m\), 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\(260\,cm\)behind the sample quartz cuvette.