Polarization conversion when focusing cylindrically polarized vortex beams

Currently, cylindrical beams with radial or azimuthal polarization are being used successfully for the optical manipulation of micro- and nano-particles as well as in microscopy, lithography, nonlinear optics, materials processing, and telecommunication applications. The creation of these laser beams is carried out using segmented polarizing plates, subwavelength gratings, interference, or light modulators. Here, we demonstrate the conversion of cylindrically polarized laser beams from a radial to an azimuthal polarization, or vice versa, by introducing a higher-order vortex phase singularity. To simultaneously generate several vortex phase singularities of different orders, we utilized a multi-order diffractive optical element. Both the theoretical and the experimental results regarding the radiation transmitted through the diffractive optical element show that increasing the order of the phase singularity leads to more efficient conversation of the polarization from radial to azimuthal. This demonstrates a close connection between the polarization and phase states of electromagnetic beams, which has important implications in many optical experiments.

In some applications, such as STED (Stimulated Emission Depletion Microscopy) methods 20 , it is important to use a specific combination of laser beam polarization and spatial properties. In other applications, a desired state of polarization during the propagation of the beam must be retained, for example, to increase network throughput by using fibre modes that carry orbital angular momentum 27 . Polarization distribution control of the laser radiation enables some unique methods, like the selective excitation of an anisotropic molecule, focusing on a size smaller than the diffraction limit, and the fabrication of periodic nanostructures with femtosecond laser light 22,23,28 .
The conversion of polarization type can take place when beams with a phase singularity are tightly focused [29][30][31] . The transfer of angular momentum from the spin degree of freedom (which is related to the state of polarization) to the orbital (which is relevant to the phase distribution) degree of freedom can also occur in anisotropic media [32][33][34][35] . The interaction of the polarization singularities and phase singularities is used to detect the polarization state of the laser beam 36,37 . Note that a complete differentiation of polarization types is only possible when the light is sharply focused 37 .
In this paper, we demonstrate the conversion of polarization type in a cylindrically polarized laser beam by introducing a higher-order vortex phase singularity. Earlier, a polarization distribution change for a vortex radially polarized beam in the propagation was shown 12,38,39 . However, the effects of a higher-order vortex phase in a cylindrical vector beam have not been previously studied. Here, the phenomenon of an orthogonal transformation in polarization is described empirically for the first time. The effect of the conversion of an azimuthally polarized beam to a radially polarized beam, and vice versa, is apparent even with weak focusing (in the paraxial case). A theoretical model for this phenomenon in focused cylindrical vector beams using the Debye approximation is also presented. In addition, a numerical study was performed, and the experimental results fully confirmed the theoretical predictions.

Results
Theoretical analysis of polarization conversion. The sharp focusing of beams cannot be handled by the paraxial approximation and is usually solved by the Debye method 40 . The vector generalization of Debye theory is able to explain the behaviour of the polarization and intensity distributions of the electromagnetic field in the focal region. According to this method, the field distribution in the focal region is formed by wave rays that converge inside the cone bounded by the aperture of the optical system. Using the Debye approximation 41,42 for tight focusing, we obtain the following expression for the transverse components in the case of focusing an azimuthally polarized beam (as in this case, with no longitudinal component), having an m-th order vortex phase: We can define the boundary of the region where the radial polarization dominates over the azimuthal with the following radius: where j v,1 is the first zero of the v-th Bessel function of the first kind. When |m| ≥ 2 at the central region of the focal plane, a radial, not azimuthal, polarization will be created. As the order of the vortex phase increases, this region will increase in size. Note also that the region of conversion shrinks as the numerical aperture of the focusing system increases (θ c → 90°). That is, in the paraxial case with a weak focus, this effect will be more significant.
Let us now consider the following incident optical beam: where σ is the angular width of the waist of a Gaussian beam, and sin α is the numerical aperture of the focusing device.
Using certain approximations, including θ ≈ θ α T( ) cos cos 2 , we can transform the integral in Eq. (1) for a beam described by Eq. (5) into a form with tabular integrals 43 and calculate it analytically. Using the notation x = kρ sin σ, after mathematical transformations we obtain: m az m m , In this analysis, we do not take into account factors that do not depend on x and m because the module for both components is the same. The radial component is always positive for m > 0, and the azimuthal one changes sign once in this area. Because, for a constant argument, the function I m (·) decreases with increasing order, then we can show that in the region with positive values of , the radial component is greater than the azimuthal component. Moreover holds over a wider range. As proof, in Eq. (7), there is a difference of functions ⋅ ν I ( ) with orders that differ by one. The recurrence relation for orders that differ by two is known: We use the monotonic continuity of the modified Bessel functions to write the following as an approximation: By substituting Eq. (8) into Eq. (7), we obtain: The ratio of the intensities, similar to Eq. (2), will be the following: I  I   I  I  I   I  I  I  I   I  I  I  I   ( , , ) ( Because, as mentioned, the function I v (·) decreases with increasing order when the argument is held constant, we obtain η > → 1 m az rad . When focusing a radially polarized field possessing an m-th order vortex phase, we obtain the following equation(in the paraxial case, we can consider only the transverse components): In the paraxial case, we can consider only the transverse components. In this way, we obtain a situation opposite to the one that was discussed above. That is, the radial polarization is transformed into an azimuthal polarization. This effect gets stronger as the order of the optical vortex increases:  Figure 2 shows the intensity distribution in the focal plane for the focused radially polarized beam described by Eq. (11) for = m 0, 4. As observed in the absence of a vortex phase (m = 0), an initial radially polarized beam preserves the polarization in the focal plane. If we add a vortex phase of the first order, of either sign, in the focused radially polarized beam, a bright spot with circular polarization will be formed in the focal plane. If we add a vortex phase of a higher order, the conversion from radial polarization to azimuthal is observed. Figure 3 shows the focal distribution of the azimuthal and radial components of the electric field to quantitatively estimate the degree of conversion. We calculated the coefficient η →

Experimental Results
To investigate the polarization conversion experimentally, we utilized the experimental optical setup shown in Fig. 4. The output beam from a solid-state laser (λ = 532 nm) first passed through a pinhole PH (100-μm aperture). Then, a polarizer P was used to obtain linearly polarized light with a predetermined polarization direction. A diaphragm D was used to separate the central spot of the Airy disk resulting from the wave diffraction of the pinhole. The S-plate 44 , oriented in the direction of polarization of the incident laser beam, converted the initially linearly polarized beam into a radially polarized beam. The resulting radially polarized laser beam illuminated the amplitude diffractive optical element (DOE), forming a superposition of eight vortex beams with orders ±1, ±2, ±3, and ±4 in different diffraction orders. The lens L (f = 150 mm) focused the laser beam on the camera sensor.
The inset in Fig. 4 shows the amplitude transmission function for the diffractive optical element. An amplitude mask is obtained by encoding 45 the transmission function of the multi-order vortex DOE: where n is an index of diffractive order, m n is the topological charge of the vortex beam, and (u n , v n ) are the carrier spatial frequencies.
The operating principle this DOE is shown in Fig. 5. The effect is the introduction of optical vortices of different orders in a radially polarized Gaussian beam, which initially does not have a vortex phase. Using a multi-order  . Experimental optical setup. The solid-state laser has an output wavelength λ = 532 nm, PH is a pinhole (100-μm aperture), P is a polarizer, D is a diaphragm, S is an S-waveplate (radial polarization converter), DOE is an amplitude diffractive optical element forming a superposition of eight vortex beams with orders of ±1, ±2, ±3, and ±4, L is a lens with a focal length f = 150 mm, A is an analyser, and CMOS is a CMOS-video camera (LOMO TC-1000, 3664×2740 pixel resolution). DOE, multiple optical vortices of different orders are formed, which allows the simultaneous measurement of the degree of polarization conversion at each spot.
The diffraction pattern formed by the amplitude diffractive optical element is shown in Fig. 6. Figure 6a shows the simulated intensity distribution generated by the DOE. Because we utilized an amplitude DOE, the intensity of the zero diffraction order is too high in comparison with the intensity from the other diffraction orders. To clearly see the non-zero diffraction orders, we removed the zero diffraction order from the simulated picture. Figure 6b shows the intensity distributions obtained without the analyser. These experimentally obtained distributions show simultaneous interaction with eight optical vortices of different orders. Each of them is formed with a separate diffraction order at the corresponding location in the focal plane. In the paraxial case considered here, the displacement of the diffraction orders from the centre of the focal plane has no effect on polarization conversion. To confirm the order of the vortices, we recorded the interference pattern of the vortices with the Gaussian beam, resulting in characteristic fork fringes, as shown in Fig. 6c. The interference patterns confirm the order of the vortices to be m = ±1, ±2, ±3, and ±4, respectively.
Intensity distributions obtained with different orientations of the analyser are shown in Fig. 7. The analyser rotation angles are equal to ±45 and 90 degrees. The analyser orientation in these figures is represented by white arrows. In addition, the zero diffraction order, in which a laser beam has a topological charge of m = 0, gives an indication of the orientation of the analyser. As was predicted theoretically, in the case of optical vortices with m = ±2, ±3, and ±4 the conversion of radially polarized light into azimuthally polarized light is observed. For optical vortices with m = ±1, no conversion is observed. Figure 8 shows the intensity distributions formed in the far-field region using a diffractive optical element forming two vortex beams with topological charge m = ±10. It can be clearly observed that in this case, we obtain a laser beam with a nearly perfect azimuthal polarization, while the radial component decreases significantly. Thus, the experimental results are in good agreement with the simulation results presented above.

Discussion
We conducted a theoretical analysis of the effect of sharply focusing a cylindrically polarized beam in the presence of an optical element with a vortex phase. Analytical expressions for the field in the focal region for radially and azimuthally polarized beams were obtained. In this way, we demonstrated the conversion of polarization state in  cylindrically polarized laser beams by introducing a higher-order vortex phase singularity. In addition, a numerical study was performed. The experimental results are in good agreement with the simulation.
Our theoretical and experimental results show that increasing the order of the phase singularity leads to increased conversation of the radially polarized laser beam into an azimuthally polarized one. Our results demonstrate the close connection between of the polarization and phase states of electromagnetic beams. Specific combinations of the polarization and spatial properties of the laser beam are important for certain applications, such as telecommunication and materials processing. Bozinovic et al. 27 presented multiplexing techniques that use the wavelength, amplitude, phase, and polarization of light to encode information. Taking into account the results presented here, it is necessary to use combinations of orthogonal polarization states and orbital angular momentum carefully. On the other hand, the effect of a phase vortex on the cylindrical polarization shown in our work will allow for a better understanding of the processes occurring during the interaction of laser radiation with matter, as in ablation 23 .

Methods
Model of focusing cylindrically polarized beams. In the Debye approximation of tight focusing, the cylindrical components of the electric field of a monochromatic electromagnetic wave are described by the following expression 31 : z r 0 0 2 where (ρ, ϕ, z) are the cylindrical coordinates in the focal region, (θ, φ) are the spherical angular coordinates of the focusing system's output pupil, α is the maximum value of the azimuthal angle related to the system's numerical aperture, B(θ, φ) is the transmission function, T(θ) is the pupil's apodization function (equal to θ cos for aplanatic systems), = π λ k 2 is the wavenumber, λ is the wavelength, f is the focal length, c r (φ), c φ (φ) are the polarization coefficients of the incident radiation.   (14) can be simplified as follows: