1. Introduction

Numerous applications are based on the inverse energy flux phenomena [1]. It can be used as an analog to “tractor beams” [2] in the area of optical trapping and manipulation, and in the detection of “invisibility cloaks” [3]. Particularly, inverse energy flux can be observed when an azimuthally polarized beam with vortical phase is sharply focused with a high numerical aperture lens. In this case, the energy flux is directed toward the source near the optical axis of the focal spot and away from the source on the periphery.

Previously it was shown [4] that inverse energy flux could be obtained by the sharp focusing of higher order radially polarized laser beams. Furthermore, the integrated inverse energy flow increases with increase in the topological order of incident radially polarized laser beams. Hence, it is relevant to generate cylindrical vector beams of higher orders.

Moreover, cylindrical vector beams are of interest in areas of optics such as data multiplexing in optical communications [5] and amplitude polarization modulation of the focal distribution [6], which is important for optical trapping, microscopy, laser ablation, and imaging of exoplanets [7].

Presently, there are a few methods for the creation of cylindrical vector beams. One of the main approaches is the polarization transformation of an incident beam using liquid crystal polarization modulators [8], subwavelength gratings [9,10], vector beams superposition [11], nanostructured fused-silica q-plates [12,13], and metal subwavelength gratings [14].

Each method has its own advantages and disadvantages. Metal subwavelength gratings work as a rule in a reflecting mode and are less chemically resistant to an aggressive medium. For the infrared band, only subwavelength gratings are transparent because of limited transparency windows of fused silica. The main disadvantage of subwavelength gratings is the nonuniformity of the Fresnel reflection coefficient that occurs because of the nonuniformity of a crystal’s refractive indices. This disadvantage can be avoided by combining subwavelength gratings with orthogonal ridges in adjacent Fresnel zones [15,16].

Previous works [9,10] show that cylindrical vector beams can be created effectively with inhomogeneous subwavelength gratings. Particularly, A. Niv at al [9]. proposed sectorial subwavelength gratings for radially and azimuthally polarized beams of arbitrary orders. In [10], the authors showed the possibility of creating radially and azimuthally polarized beams of the first order. In another study [15], it was shown that the focusing and polarization phase modulation of an initial beam can be accomplished by subwavelength gratings.

In this study, we propose metasurfaces for the simultaneous creation and focusing of high-order radially polarized beams for an effective formation of inverse energy flux near the optical axis. In contrast to sectorial subwavelength gratings [9,15], we propose and apply the continuous shape of ridges that provide a more homogeneous transformed polarization. To create continuous subwavelength relief, we derive new phase functions equations. The additional advantage of created metasurfaces is its varying period that depends on the coordinates. It increases the efficiency of the polarization transform.

2. Basic parameters of a subwavelength grating as a half-wave plate

Subwavelength diffraction grating is a binary profile etched on a flat substrate (Fig. 1(a)). Thicknesses of the ridges and grooves are $d_1$ and $d_2$, respectively. Refractive indices of the substrate material and surrounding medium are $n_1$ and $n_2$, respectively.

 

Fig. 1 Subwavelength grating as an inhomogeneous half-wave plate; (a) general view of the subwavelength grating; (b) mutual arrangement of the following vectors and angles between them: grating vector $\overrightarrow{K}$, electric vector ${\overrightarrow{E}}_{in}$ of the incident beam, and electric vector ${\overrightarrow{E}}_{out}$ of the output beam. The order of polarization (m) is 5 and polar angle (φ) is 10°.

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If the period of the refractive grating is less than the wavelength of the incident beam, the structure can be considered as homogeneous and anisotropic. Thus, it is possible to average the refractive index of the structure for the chosen direction and obtain effective refractive indices as follows:

Assuming that the refractive index of the substrate is n₁ = 4.206 + 0.42174j for a 633 nm wavelength, and ordinary and extraordinary indices for the subwavelength grating in the air (refractive index n₂ = 1) for a fill factor Q = 0.5 are $n_{eff}^{TE}=3.057$ and $n_{eff}^{TM}=1.3759$, respectively. Thus, the difference between the ordinary and extraordinary refractive indices for the subwavelength grating is considerably greater than that of natural crystals (less than 0.1).

V. Kotlyar et al. [18] provided an adjustment of the subwavelength grating parameter, which works as a half-wave plate and provides an effective polarization transform.

3. Phase function of polarization grating

In this section, we derive the formula for the phase function of the subwavelength grating that transforms the incident linearly polarized beam into a radially polarized beam with an arbitrary order m. In general, the function of the relief height of a subwavelength grating is

Subwavelength grating vector $\overrightarrow{K}$ is the gradient of the grating phase function $\overrightarrow{K}=\nabla f\left(r\right)$. $\overrightarrow{K}$ defines the direction of a crystal’s fast axis at each point of the coordinate system. The vector $\overrightarrow{K}$ is collinear to the substrate plane and perpendicular to the optical axis z (Fig. 1(b)).

We assume that the electric vector of the incident linearly polarized beam is oriented along the x-axis (Fig. 1(b)). The beam propagates in the direction perpendicular to the x-y plane as shown in Fig. 1(b). To create a radially polarized beam of order m with the electric vector ${\overrightarrow{E}}_{out}(r,\phi )$ (Eq. (3)), it is necessary to rotate the electric vector ${\overrightarrow{E}}_{in}$ of the incident linearly polarized beam by an angle $m\phi$ at each point $(r,\phi )$ of the transverse cross-section. To provide this rotation, the angle between the vectors $\overrightarrow{K}$ and ${\overrightarrow{E}}_{in}$ must be equal to mφ/2. The expressions for the electric vector ${\overrightarrow{E}}_{out}$ of the required output beam and for the grating vector $\overrightarrow{K}$ are

The mentioned vectors and angles between them (Eq. (3)) are shown in Fig. 1(b). To illustrate mutual orientation of the vectors and the axes, the order of the radially polarized beam (m) is chosen to be 5, and the polar angle (φ) is 10°.

To define the shape of the grating, it is sufficient to determine the phase function $f\left(x,y\right)$. To create continuous subwavelength relief, we must make the phase functions satisfying the equation $\nabla f\left(x,y\right)=\overrightarrow{K}$.

By considering the gradient of the phase function in polar coordinates and equating it to zero, it is easy to obtain solutions for two cases: when the grating period depends only on the azimuthal angle φ, and only on the radius r. For the first case, the period $d\left(\phi \right)$ and phase function $f\left(r,\phi \right)$ are

Similarly, assuming the case when the grating period does not depend on the azimuthal angle φ, it is possible to derive the equations for the period $d\left(r\right)$ and phase function $f\left(r,\phi \right)$:

In Eqs. (4) and (5), d₀ is the constant that determines the rate of the period’s variation with the coordinates. Equipotential curves of the phase function serve both as the slow axis and edge of the subwavelength grating’s ridge because the edge of the ridge and the slow axis must be perpendicular to the fast axis at each point.

Thus, it can be concluded from Eqs. (4) and (5) that for the period that depends only on the azimuthal angle φ and only on the radius r, the slow axis of the crystal has the following view of r_angle and r_rad:

It should be noticed that the equation of curves for the ridges of the grating with a period that depends only on the azimuthal angle r_angle and that which depends only on the radius r_rad are the same except for the power of C.

Similarly, we can find equations for the fast axis of the crystal. They have the same view as in Eq. (6).

Based on Eqs. (6) and (7), the fast and slow axes of the crystal, represented in orange and black, respectively, are plotted in Fig. 2. There are crystal axes (or edges of ridges) for the different orders of radial polarization m and for different period’s dependencies on coordinates.

 

Fig. 2 Curves of fast (orange) and slow (black) axes of the subwavelength grating that transforms linearly polarized beams into radially polarized beams with order m. In the first row, the period of curves repetition depends only on the azimuthal angle φ. In the second row, the period of curves repetition depends only on the radius r.

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In Fig. 2, we can see that the fast and slow axes of the crystal are perpendicular to each other at each point in the plane of the element. Note that the shape of the slow curve axis defines the shape of the subwavelength grating’s ridges, i.e., two slow axes form two edges of one ridge.

4. Metasurfaces for the generation of inverse energy flux

In this study, we propose an optical element that combines two functions: polarization and phase transformation of radiation. The focusing phase corresponds to a binary Fresnel lens with an annular radially symmetric zone structure. Therefore, we use only those formulas that are achieved for a period that depends only on the radius, so that the period of the grating in one ring of the lens does not change much.

The mechanism of binary phasing of a metasurface is based on the placement of gratings with perpendicular ridges in adjacent rings of the metasurface [15] as shown in Figs. 3(a) and 3(d). The simulation was performed at a wavelength of 633 nm. The numerical aperture of the metasurface was chosen to be 0.99 to achieve sufficient values of the inverse flux [4].

 

Fig. 3 The working of metasurface of (a) second and (d) third order; (b, e) longitudinal and (c, f) transverse cross-sections of power flux longitudinal component distributions S_z; graphs of S_z versus the y-axis of the focal spot are shown in the insets on the right.

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In this study, we propose to change the constant C by increasing the radius of the annular zone of the metasurface and by ensuring that the grating period is approximately constant in all zones and approximately equal to 120 nm. The height of the grating relief is chosen to be 120 nm according to a study [18] in which the authors have determined an optimal height of the subwavelength grating that corresponds to a half-wave plate. The technology of manufacturing such structures is described by S. Stafeev et al. [19].

For the numerical simulation of laser radiation diffraction, the finite element method is used, which is implemented in the Comsol Multiphysics software. Figure 3 shows the results of the simulation of the linearly polarized Gaussian beam focusing using the proposed second and third order metasurface. Mutual arrangement of the elements and an initial beam can be understood with the help of axes that are shown in Fig. 3. The elements lie in the xy-plane. Gaussian beam illuminates the elements along the z-axis which is also an optical axis. Figure 3 shows the distributions of the Poynting vector longitudinal component S_z.

From Fig. 3, it can be concluded that after focusing with the proposed metasurface in the created focal distribution, the energy flux near the optical axis is pointed toward the source of the incident beam. Such an energy flux is a well-known inverse or reverse energy flux [1,4]. When the second order metasurface focuses the incident linearly polarized beam, an inverse energy flow distribution with a circular shape is formed, and the third order metasurface forms an annular shape of the inverse energy flux distribution. The second and third order metasurfaces provide an integral value of the inverse energy flux in the focal spot equal to 2.3 (Fig. 3(c)) and 6.6 a.u. (Fig. 3(f)), respectively. Thus, we can conclude that a higher order metasurface provides a greater inverse energy flux [4].

5. Conclusion

This study proposes a family of optical elements for the formation of an inverse energy flux in a focal region. These elements take the form of subwavelength gratings with curved ridges. New expressions were obtained for the phase functions of the polarizing subwavelength gratings, which convert a linearly polarized beam into a radially polarized beam of arbitrary order. The curves of the slow and fast axes of the proposed photonic-crystal subwavelength gratings were plotted.

Combining two functions of polarization and phase transformations in one element we decrease the number of elements in an arrangement. The advantage of the element is continuous relief that provides more efficient polarization transformation especially in adjacent thin annular zones of high numerical aperture focusing phase.

Moreover, we carried out numerical simulations that showed the formation of the inverse energy flux in the focal region. Integrated inverse energy flux that achieved with the third-order metasurface is 2.8 times higher than the integrated inverse energy flux achieved with the second-order metasurface. This fact reveals that a higher order metasurfaces provide a greater overall inverse energy flux. Also, the shape of the inverse flux area achieved with the third-order metasurface is annular and has bigger force area.