In the past several years, due to the wide applications in atomic optics and modern optics, laser beams with various intensity profiles, such as hollow laser beams^[1], Bell–Gauss beams^[2], vortex beams^[3], semi-Gaussian laser beams (SGBs)^[4], and so on, have been developed and studied widely. A laser beam with a special intensity distribution can be obtained by changing the internal structure of the laser or assisting with some external devices, such as a lens, a grating, a prism, and other optical elements^[5–7]. A SGB with no diffraction fringes was produced experimentally in 2011^[8]. A SGB is a kind of laser beam with asymmetric transverse intensity profiles^[8]. The transverse intensity distribution of the SGB is semi-Gaussian in one direction and Gaussian in the orthogonal direction. SGBs have significant potential applications in the fields of modern optics and atom-molecular physics^[9,10]. In practical applications of laser beams, it is necessary and important to study the propagation and transformation of laser beams in different transmission media because of their different characteristics^[11,12]. The propagation properties of a neat nonideal SGB generated experimentally in free-space has been studied in Ref. [13]. In this work, we primarily study the special propagation properties of a neat nonideal SGB through a ZnSe nonlinear crystal in theory and briefly discuss its applications in modern optics.

In our work, a ZnSe crystal is employed as the nonlinear medium. The proposed schematic setup to study the propagation properties of a neat nonideal SGB through a ZnSe crystal is shown in Fig. 1. In Fig. 1, the coordinate in the plane of ZnSe nonlinear crystal is set as (x₀, y₀, 0), and the coordinate in the plane of CCD is set as (x, y, z). A neat SGB is propagating along the z direction, and then incident upon a thin nonlinear dielectric material, i.e., a ZnSe crystal. At a propagation position z after the ZnSe crystal, we find that the SGB has a direction of the shift with an angle θ, which is defined as the deviation angle. This phenomenon is called the self-bending effect, which can be applied extensively to optical switching, bistability and limiting, fast spatial scanning, radiation protection, measurement of the refractive index, and so on^[14,15].

The 1D light field complex amplitude Eideal(x0) of an ideal SGB when it is propagating along the z direction is given by^[4]Eideal(x0)={Emexp[−(x0/w)2]x0≥00x0<0,(1)where (x0) is the coordinate in the plane of ZnSe nonlinear crystal, Em is the light field complex amplitude, and w is the waist radius of the SGB and defined as a width at 1/e2 of the maximum intensity of the semi-Gaussian part of the SGB. Equation (1) is the light field expression of an ideal SGB, but an ideal SGB cannot be obtained experimentally. A SGB generated in experimentally always has a certain ascent border width wb which defined as the width at 1/e2 of the maximum intensity of the sharp border in the SGB. When wb<0.1w, the laser beam has obvious asymmetry. We regard a laser beam which satisfies this condition as a nonideal SGB with high quality, and its theoretical complex amplitude E0(x0) of the incident nonideal SGB can be described by^[4,8]E0(x0)={Emexp[−(x0/w)2]x0≥0Emexp[−(x0/wb)2]x0<0.(2)When this nonideal SGB is incident into a nonlinear medium material, its propagation direction will be affected because of material nonlinearity^[14]. The refractive index n of the nonlinear medium material can be described by^[15]n=n0+n2|E|2,(3)where n0 and n2 are the linear and nonlinear refractive indexes of the nonlinear medium material, respectively. So the refractive index is affected by the characteristics of the medium and the incident light intensity.

If the thickness of the nonlinear medium material is very small, the absorption of the incident light can be neglected. The light field complex amplitude at the end of material can be given by^[7]E1(x0)=E0(x0)exp[iφ0(x0)]⁢exp{−i[n0+n2|E0(x0)|2]kL},(4)where φ0 is the vector angle of initial SGB, k is the wave number of the incident SGB, and L is the thickness of the nonlinear medium material in the optical transmission direction. The thickness L needs to meet the following condition^[7]L∈(D,RNL),(5)where D and R_NL are the diffraction length and the self-focusing length of the incident laser beams, respectively.

According to the Huygens–Fresnel diffraction theory in free-space^[16], the light field complex amplitude at the propagation distance z can be written as $$\begin{gathered} E(x)=eikziλz∫−∞∞eik(x−x0)22zE1(x0)dx0 \\ =Emeikz+iφ0(x0)−in0kLiλz⁢(∫−∞0eik(x−x0)22z−(x0wb)2−in2kLEm2e−2(x0wb)2dx0+∫0∞eik(x−x0)22z−(x0w)2−in2kLEm2e−2(x0w)2dx0). \end{gathered}$$(6)Hence the intensity I(x) of the light field becomes $$\begin{gathered} I(x)=|E(x)|2=Em21λ2z2|∫−∞0eik(x−x0)22z−(x0wb)2−in2kLEm2e−2(x0wb)2dx0+∫0∞eik(x−x0)22z−(x0w)2−in2kLEm2e−2(x0w)2dx0|2 \\ =I01λ2z2|∫−∞0eik(x−x0)22z−(x0wb)2−in2kLI0e−2(x0wb)2dx0+∫0∞eik(x−x0)22z−(x0w)2−in2kLI0e−2(x0w)2dx0|2, \end{gathered}$$(7)where I0=|Em|2 is the intensity amplitude of the input beams.

Based on the aforementioned theoretical analyses, the intensity distributions of the output light beam at a propagation distance z after the nonlinear ZnSe crystal can be calculated by using Eq. (7). In the numerical calculation, the parameters of the incident nonideal SGB are w=300μm, wb=17.1μm, I0=1.1×109W/m2, and wavelength λ=532nm^[8]. These data are the experimental results.

In order to study the propagation properties of the SGB after traveling through nonlinear media, the parameters of the nonlinear medium material ZnSe are taken as L=500μm, n0=2.8, and n2=6.1×10−13m2/W^[7]. The propagation properties of the SGB propagating through the nonlinear medium material ZnSe under this scheme are shown in Fig. 2. Figure 2(a) shows that the distribution of the light field becomes broader and the maximum of the light intensity shifts to the x direction when the propagation distance becomes farther. However, the ratio of the position of the maximum of the light intensity and the propagation distance xmax/z does not change; in other words the deviation angle θ remains the same. Because of the conservation of light energy, when the distribution of the optical field becomes broader, the maximum of the light becomes smaller, which is expressed distinctly in Fig. 2(b). The distribution of the intensity detected in far-field tends to flat which is hard to analyze the experimental results, so the position of the detector should be selected by the appropriate experimental conditions in experiments.

From the aforementioned calculations, the propagation distance z does not affect the deviation angle θ. How to change the deviation angle and the influences of the parameters of the medium material and SGB are the next questions to be studied. The following propagation distance z is taken to be z=1m.

Figure 3 illustrates the intensity distributions of the SGB after traveling through three media with different refraction indices. The parameters of the nonlinear medium material ZnSe are taken to be L=500μm and n0=2.8^[14]. From Fig. 3, we can see that the SGB becomes an approximate symmetric Gaussian beam in the far field after traveling through a linear medium, however, the SGB becomes an asymmetrical beam, whose intensity distribution has a maximum and a secondary maximum because of diffraction, after traveling through the nonlinear medium. Also it has an obvious shift in the x direction (x>0 for n2=6.1×10−13m2/W^[7], and x<0 for n2=−6.1×10−13m2/W), and the deviation angles are ∼1.01mrad (n2=6.1×10−13m2/W) and ∼0.75mrad (n2=−6.1×10−13m2/W), respectively. The results show that the sign of the nonlinear refractive index affects the direction of the shift of the SGB and the positive deviation angle is larger than the negative deviation angle. In this way, we can estimate the sign of the nonlinear refractive index with the direction of the shift using this propagation property.

Figure 4(a) shows that the self-bending of the incident SGB is related to the nonlinear refractive index. A larger nonlinear refractive index corresponds to a larger shift. Furthermore, we also study the dependences of the deviation angle on the nonlinear refractive index. The slope depends linearly on the nonlinear refractive index [shown in Fig. 4(b)], so this propagation property can be used to measure the nonlinear refractive index of the nonlinear medium material.

According to Eq. (7), the intensity of the output light field is related to the intensity of the input beams, and the calculated results when z=1m are shown in Fig. 5. The results show that the distribution of the light field becomes broader and the maximum of the light intensity shifts to the x direction with a larger distance when the intensity of the input beams increase. It is because the larger light intensity, the larger the induced nonlinearity of the nonlinear medium material and consequently a larger divergence angle of the SGB. Also, the deviation angle is proportional to the intensity of the input beams, which is shown in Fig. 5(b).

We also studied the intensity distributions of SGBs propagating through several ZnSe crystals with various thicknesses when z=1m, which are shown in Fig. 6. We can see that the deviation angle increases as the thicknesses of the crystal increases. At the same time there is a maximum and a secondary maximum of the intensity distributions. When the thickness of the crystal increases, the maximum decreases and the secondary maximum increases. The secondary maximum shifts to the right. It can be explained as follows. When the thickness of the nonlinear ZnSe crystal is larger, i.e., the interaction length of the SGB and the material is larger, the propagation properties of the SGB will be affected more, so the divergence distance will be larger.

Finally, we studied the influence of the border width wb of the SGB. From Eqs. (1) and (2), we can see that a nonideal SGB is equal to the consist of a positive axis ideal SGB with waist radius w and a negative axis ideal SGB with waist radius wb. The intensity distribution of a nonideal SGB after traveling through a nonlinear medium is the superposition of two ideal SGBs’ optical field complex amplitude distributions after the nonlinear medium at the far-field. The impact of the border width of a nonideal SGB on the deviation angle is mainly due to the perturbation of the negative secondary maxima of the SGB on the ideal part of the complex amplitude. The results of the calculation are shown in Fig. 7 with w=1.07mm and z=10m. Figure 7 shows that with increasing wb, the deflection of the propagating laser beams decreases, i.e., the deviation angle becomes smaller. Figure 7 also shows that the left half axle distribution of the light field becomes narrower and the maximum of the light intensity becomes stronger when wb becomes larger. When wb=w, the distribution and maximum of the light intensity in both sides of the axis are the same. In other words, a Gaussian laser beam after traveling through the crystal ZnSe becomes a hollow beam, with the same result as that of Ref. [7]. When wb increases, the asymmetry of the nonideal SGB decreases, so the deflection of the SGB becomes smaller.

In conclusion, we study the propagation properties of a nonideal SGB with a small wb generated experimentally through a nonlinear medium material. When a nonideal SGB propagates through a nonlinear medium material, its propagation direction will change and shift away from the initial propagating direction. The calculated results show that with increasing nonlinear refractive indices and thickness of the nonlinear medium material, the incident light intensity and deviation angle become larger; however, the trends of the maximum light intensity are not the same with increasing wb, i.e., the deviation angle becomes smaller. Therefore, we can choose the appropriate parameters according to the specific needs of the experiment. Furthermore, these propagation properties of a nonideal SGB can be used to measure the nonlinear refractive indices of a nonlinear medium and crystal thickness, which can also be used in the context of spatial scanning of laser beams.