1. Introduction

Spatial self-phase modulation (SSPM) is one of the all-optical modulation phenomena caused by the third-order nonlinear refractive effect of materials [1,2]. In the past few decades, researchers have observed SSPM phenomena in various types of materials, from 0D quantum dots [3], 1D nanotubes [4], 2D layered materials [5], to 3D bulk materials [6]. Recently, the rapid development of 2D layered materials has injected new vitality into the SSPM research. On the one hand, the SSPM has been widely used to characterize the third-order nonlinear susceptibility of various 2D layered materials (e.g., graphene [5], boron [7], and black phosphorus (BP) [8]). On the other hand, based on 2D layered materials, SSPM has developed many passive nonlinear photonic devices, such as all-optical switches [9], all-optical information converters [10], all-optical logic gates [7], and all-optical diodes [11]. Up to now, most studies on SSPM use Gaussian beams as excitation light sources, while there are relatively few reports on SSPM excited by other beams (e.g., vortex beams [12] and vector fields [13]).

In the past twenty years, various types of vector fields have been designed and generated by manipulating the state of polarization (SoP) of the light field, including linearly polarized vector fields [14], hybridly polarized vector fields [15], Poincaré beams [16], and arbitrary complex vector fields [17]. Researchers have extensively studied the focusing, propagation, and technical applications of different types of vector fields. Meanwhile, the interaction of vector fields with nonlinear optical media has been also investigated. Under the excitation of vector fields, novel third-order nonlinear optical effects have been reported, such as vectorial self-diffraction [13], controllable nonlinear optical propagation [18], designable optical field collapse [19], etc.

Among various types of vector fields, linearly polarized vector fields (e.g., radially polarized beam and azimuthally polarized beam) are the most widely studied. As an important parameter for describing the linearly polarized vector field, the topological index is usually an integer. We call this vector field as an integer-order linearly polarized vector field (ILPVF). In recent years, linearly polarized vector fields with fractional-order topological indices, known as fractional-order linearly polarized vector fields (FLPVFs), have been reported to have many unique photophysical properties, such as the evolution of polarization singular lines [20], the spatial separation of angular momenta during focusing [21], and the controllable focal field intensity distribution [22]. In the field of nonlinear optics, the interaction of FLPVFs with media is certainly different from that of ILPVFs, although there are few reports on the related nonlinear optical phenomena [23]. With respect to SSPM, theoretical and experimental investigations have been conducted on the self-diffraction phenomenon excited by ILPVFs [13,24], although SSPM phenomena excited by FLPVFs is still seldom.

In this work, we theoretically investigate both the intensity and SoP distributions of FLPVFs passing through isotropic nonlinear Kerr media. Unlike weakly focused ILPVFs with localized linear SoP and doughnut-shaped intensity distributions, weakly focused FLPVFs have hybrid SoP and asymmetric intensity distributions. Different from SSPM pattern of ILPVFs having localized linear SoP and concentric multi-ring structure, SSPM patterns of FLPVFs exhibit hybrid SoP and symmetry-breaking self-diffraction intensity structure. All theoretical results are experimentally demonstrated.

2. Theory

The FLPVF in the cylindrical coordinate system at the initial plane can be expressed as [14,21]

In our vector field generation system (see Fig. 1), a linearly polarized Gaussian beam is modulated by a vortex half-wave plate (VHP) to generate a FLPVF. In this case, the amplitude distribution of the generated FLPVF can be approximately described by

 

Fig. 1. Experimental setup of SSPM excited by FLPVFs. HWP: half-wave plate; L1-L3: lens; VHP: vortex half-wave plate.

Download Full Size | PDF

The FLPVF can be expressed as a modal superposition of all ILPVFs. Accordingly, Eq. (1) can be rewritten as [21]

Adopting a similar procedure as reported in Refs. [21,25], a theoretical analysis of SSPM excited by a FLPVF is described as follows. Using the Rayleigh-Sommerfeld vectorial diffraction formula under the paraxial approximation [26], we obtain the analytical expression for the electric field of the weakly focused FLPVF along the + z direction with the coordinate origin at the lens’ geometrical focus as

The weakly focused FLPVF evolves into a hybridly polarized vector field at the focal region, as we will demonstrate in Sec. 4. The complex electric field of the focused FLPVF can be transformed into a linear superposition of left-hand (LH) and right-hand (RH) circular components as

Considering that the focused FLPVF passes through an optically thin sample with isotropic refractive nonlinearity, one can obtain the electric field at the exit plane of the sample as

Here, n ₀ is the linear refraction index of the sample, $A = 6Re [\chi _{xxyy}^{(3)}]$ and $B = 6Re [\chi _{xyyx}^{(3)}]$ are two independent tensor components of the third-order nonlinear susceptibility in an isotropic sample, and e = (|E ₊|-|E _-|)/(|E ₊|+|E _-|) is the ellipticity of the localized electric field. For quantitative comparison, the peak nonlinear refractive phase shift of linearly polarized light (e = 0) at the focal plane is defined as $\Delta {\Phi _0} = kn_2^{\textrm{lin}}E_{00}^2L$, where $n_2^{\textrm{lin}} = (2\pi /{n_0})(A + B/2)$.

Based on the Rayleigh-Sommerfeld vectorial diffraction formula under the paraxial approximation [26], the complex electric field on the far-field observational plane can be obtained as

3. Experiment

The above-mentioned theory describes a novel SSPM phenomenon excited by a FLPVF. In what follows, we experimentally demonstrate this SSPM phenomenon by implementing a focused FLPVF through BP nanosheets. The experimental setup is illustrated in Fig. 1. The light source used in our experiments is a continuous-wave laser with adjustable power (YBGL1064-4W). It produces a linearly polarized Gaussian beam with a wavelength of λ=1064 nm. The Gaussian beam emitted by the laser is incident onto a half-wave plate (HWP). Then the laser beam is expanded and collimated through the beam expansion system composed of a concave lens (L1, f = -50 mm) and a convex lens (L2, f = 150 mm). Afterwards, the linearly polarized beam passes through a VHP (VR-1064-SP), where the direction of linear polarization is parallel to the fast axis of the VHP, generating a linearly polarized vector field with a Gaussian profile [27]. Note that the HWP is used to adjust the linear polarization direction of the laser beam, so that it matches the fast axis of the VHP. The purpose of using the beam expansion system is to fully illuminate the laser beam onto the VHP, thereby improving the efficiency and quality of generated vector fields. In experiments, using VHPs with different topological indices, we generate FLPVFs with a waist radius of ω=2.14 ± 0.07 mm.

The generated FLPVF is weakly focused by a convex lens (L3) with a focal length of f = 400 mm, producing the beam waist of ω ₀= 63 ± 2 µm at the focal plane. The intensity distributions of the focused FLPVFs with different values of α are captured by a charge-coupled device (CCD) camera located at the focus. To visually analyze the SoP characteristics of the focal field, a polarizer is placed before the camera to record the intensity distribution after passing through the polarizer with different orientation angles.

In SSPM experiments, the nonlinear optical sample is BP nanosheets dispersed in agarose solid. BP nanosheets is chosen because this sample exhibits superior SSPM phenomena under the excitation of Gaussian laser beams [8]. The detailed morphology and structure characterizations of BP nanosheets can be found in Ref. 8. Firstly, BP nanosheets dispersed in the solution of 1-methyl-2-pyrrolidone with the mass concentration of ∼0.025 mg/mL were contained in a 10 mm-thick quartz cell. Then, according the reported preparation process [28], BP nanosheets dispersed in agarose solid are prepared. The reason for using the solid sample is to avoid distortion of the SSPM pattern caused by gravity-promoted thermal convection in the liquids. It is noteworthy that as a carrier for excitation of SSPM, BP nanosheets only provide Kerr-like optical nonlinearity in our experiment. Of course, other 2D materials can also be chosen to study SSPM of FLPVFs. For simplicity, as shown in Fig. 1, the sample is placed at the focus of the convex lens (L3), where the center of the sample coincides with the focal plane. The CCD camera is placed at d = 100 mm behind the focus to record the far-field self-diffraction intensity pattern. In this case, moving the CCD only slightly changes the relative size of the intensity pattern.

4. Results and discussions

Firstly, let’s investigate the intensity patterns and SoP characteristics of FLPVFs with different values of α. Figures 2(a) and 2(b) show the theoretically simulated and experimentally measured intensity patterns of FLPVFs with five typical topological indices α, respectively. Evidently, the theoretically simulated intensity patterns of FLPVFs are consistent with the experimentally measured results. For the sake of comparison, the results of the ILPVF (α=1) are also presented in Fig. 2. Unlike the central polarization singular point of the ILPVF, the FLPVF has a radial polarization singular line [21]. As shown in Fig. 2, the intensity patterns of all generated FLPVFs are nearly indistinguishable for different values of α. After passing through a linear polarizer, the intensity pattern exhibits multi sector extinction due to the asymmetric SoP distribution of FLPVFs (see the first column in Fig. 2(a)) [14,21]. It should be noted that there are some differences between theoretical and experimental results, which is attributed to the following two reasons: (i) theoretically, the polarization singular lines are infinitely fine, while experimentally, the FLPVF generated by the interaction of light with microstructures of a VHP has a certain width of polarization singular line; and (ii) Fig. 2(a) shows the simulations by Eqs. (3) and (4), while the measured intensity patterns shown in Fig. 2(b) are the intensity of FLPVFs generated by VHPs propagating at ∼1 cm to the camera in free space. During the propagation of FLPVF in free space, both its intensity and SoP undergo evolution, resulting in the measured intensity pattern with stripes near the polarization singular line [20].

 

Fig. 2. (a) Theoretically simulated and (b) experimentally measured intensity patterns of FLPVFs without and with a linear polarizer. The first column in (a) also illustrates the schematics of the SoP distributions (Green line: linear polarization). The first rows in (a) and (b) show the transmission direction of the linear polarizer. The size of all these patterns is 2.4ω×2.4ω.

Download Full Size | PDF

Figure 3(a) shows the simulated intensity patterns of the FLPVFs with different topological indices α at the focal plane (z = 0). The corresponding measured results are shown in Fig. 3(b). Obviously, the numerical simulations and experimental results are in good agreement, indicating that our theoretical description is correct. By comparing Figs. 2 and 3, it can be observed that although the total intensity distribution of the incident light field is similar, its focal field intensity distribution is significantly different (e.g., an elliptical focal spot for α=0.5 and two semi symmetric focal spots for α=1.5). More interestingly, unlike the localized linear polarization distribution of the incident light field, the SoP of the focused FLPVF is more abundant at the focal region. As shown in the first column of Fig. 3(a), there is a hybrid SoP distribution of linear polarization, LH and RH elliptical polarizations on the focal plane. The novel distributions of intensity and SoP are derived from the fact that the focal field of FLPVF is a modal superposition of ILPVFs [21]. In a word, fractional topological indices provide an additional degree of freedom for manipulating the intensity and SoP distributions of vector fields. Intriguingly, this hybridly polarized focal field will lead to novel photophysical properties during propagation in free space and exhibit unique SSPM phenomena after passing through nonlinear optical Kerr media, as we will demonstrate in Figs. 4 and 5.

 

Fig. 3. (a) Theoretically simulated and (b) experimentally measured intensity patterns of the focused FLPVFs without and with a linear polarizer at the focal plane. The first column in (a) also illustrates the schematics of the SoP distributions (Green/blue/black: linear/LH/RH polarizations). The first rows in (a) and (b) show the transmission direction of the linear polarizer. The size of all these patterns is 4ω ₀× 4ω ₀.

Download Full Size | PDF

 

Fig. 4. (a) Theoretically simulated and (b) experimentally measured far-field intensity patterns of the focused FLPVFs propagating in free space without and with a linear polarizer. The first column in (a) also illustrates the schematics of the SoP distributions (Green line: linear polarization). The first rows in (a) and (b) show the transmission direction of the linear polarizer. The size of all these patterns is 1.5 mm × 1.5 mm.

Download Full Size | PDF

 

Fig. 5. (a) Theoretically simulated and (b) experimentally measured far-field intensity patterns of the FLPVFs passing through a nonlinear Kerr sample (ΔΦ₀= 2π) without and with a linear polarizer. The first column in (a) also illustrates the schematics of the SoP distributions (Green/blue/black: linear/LH/RH polarizations). The first rows in (a) and (b) show the transmission direction of the linear polarizer. The size of all these patterns is 4.2 mm × 4.2 mm.

Download Full Size | PDF

Using Eq. (5) with z = 100 mm, we simulate the SoP and intensity distributions of focused FLPVFs propagating to the far-field observational plane in free space. Figures 4(a) and 4(b) show the theoretical simulations and corresponding measurement results, respectively. Obviously, there is a good agreement between the theoretical and experimental results. As shown in Fig. 4, when the weakly focused ILPVF (i.e., α=1) propagates to the far-field observational plane, its intensity distribution evolves from a Gaussian-type structure to a doughnut-type structure, but its SoP distribution maintains the initial radial polarization at any propagation position. Interestingly, when weakly focused FLPVFs propagate into the far field, their intensity distributions develop from a Gaussian-type structure at incident plane to a symmetry-breaking structure (e.g., edge breaking for α=0.5). Meanwhile, their SoP distributions evolve from the localized linear polarization at the incident plane of the lens (see Fig. 2), the hybrid polarization at the focal plane (see Fig. 3), and the localized linear polarization at the far-field observational plane (see Fig. 4). It is noteworthy that this localized linear polarization distributions of FLPVFs at the far-field observational plane are very different from those of incident field. This is because both the intensity and SoP distributions of the FLPVFs undergo evolution during the focusing and propagation in free space.

To study the influence of nonlinear Kerr medium on the intensity and SoP of FLPVFs, experimental parameters are taken for numerical simulations as λ=1064 nm, ω=2.14 mm, f = 400 mm, z = 0, d = 100 mm, B/A = 0.5, and ΔΦ₀= 2π. Note that the value of B/A strongly depends on the origin of optical nonlinearity. In our experimental conditions, the SSPM phenomenon of BP nanosheets at the wavelength of 1064 nm arises from the coexistence of optical Kerr nonlinearity (B/A = 1) and thermally induced nonlinearity (B/A = 0) [8]. Under the relatively low intensity, the resulting nonlinearity can be safely regarded as Kerr-like optical nonlinearity with B/A = 0.5. In the experiments, the power of the generated FLPVFs is adjusted to ensure that the peak nonlinear phase shift excited by BP nanosheets ΔΦ₀ is approximately equal to 2π. Figures 5(a) and 5(b) show the numerically simulated and experimentally measured intensity patterns of the FLPVFs through nonlinear Kerr media, respectively. Apparently, numerical simulations are basically consistent with experimental measurements. It is shown that the far-field self-diffraction intensity patterns of FLPVFs exhibit a symmetry-breaking multi-ring structure and hybrid SoP distribution. Comparatively, SSPM patterns of ILPVF (i.e., α=1) shown in Fig. 5 have the localized linear SoP and concentric multi-ring structure [13]. The observed SSPM phenomena of FLPVFs can be understood as follows. The focal field with hybrid polarization and asymmetric intensity distribution produces the spatially asymmetric nonlinear refractive phase shift with the aid of isotropic nonlinear optical effects. This phase shift modulates the propagation of FLPVFs themselves, leading to far-field symmetry-breaking multi-ring intensity patterns. At the same time, due to the difference in nonlinear phase shift caused by the interaction of linearly and circularly polarized lights with isotropic optical nonlinearity in the hybridly polarized field, as described by Eq. (12), the far field of FLPVFs exhibits a hybrid SoP distribution.

5. Conclusion

In summary, we have theoretically and experimentally investigated the weakly focusing and propagation of FLPVFs in free space. It is shown that the ILPVF maintains its initial SoP distribution during weakly focusing and propagation in free space, and its intensity distribution exhibits axial symmetry. Interestingly, the SoP of a FLPVF evolves into a hybrid polarization distribution during focusing and propagation, and its intensity becomes an asymmetric intensity distribution. More importantly, we have studied the SSPM phenomena of FLPVFs passing through an isotropic nonlinear Kerr media. It is shown that SSPM patterns of ILPVFs have localized linear SoP and concentric multi-ring structure. Intriguingly, SSPM patterns of FLPVFs exhibit hybrid SoP and symmetry broken self-diffraction intensity structure. In short, compared with ILPVFs, FLPVFs have one more degree of control freedom, which brings unique photophysical properties and excites novel vectorial SSPM phenomena. The presented work provides a nonlinear optics approach for manipulating both the SoP and intensity distributions of the light field. This type of symmetric breaking light field with hybrid polarization distribution has potential applications in optical micromanipulation, light-matter interaction, passive nonlinear photonic devices, etc.