Spatial Dynamics of a Radially Polarized Terahertz Laser Beam with a Phase Singularity

Phase is an important characteristic of electromagnetic waves. It is well known that a beam with a helical wave front characterized by a phase of <inline-formula><tex-math id="M1">\begin{document}$\exp({\rm{i}}l\theta )$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20181677_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20181677_M1.png"/></alternatives></inline-formula> (which depends on azimuthal angle <inline-formula><tex-math id="M2">\begin{document}$\theta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20181677_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20181677_M2.png"/></alternatives></inline-formula> and topological charge <i>l</i>), has a momentum component along the azimuthal direction, resulting in an orbital angular momentum of per photon along the beam axis. Owing to its fascinating properties, the beam has received a great deal of attention and has provided novel applications in manipulation of particles or atoms, optical communication, optical data storage. In order to meet the needs of various applications, techniques for efficiently generating optical beams carrying orbital angular momentum are always required. Current schemes for generating the beams carrying orbital angular momentum include computer-generated holograms, spiral phase plates, spatial light modulators, and silicon integrated optical vortex emitters. Among the usual methods to produce helical beams, the traditional spiral phase plate is an optical device that utilizes the progressive increasing of height of a dielectric material along an azimuthal direction to produce a vortex beam for beam phase modulation with a high conversion efficiency. However, it is difficult to regulate the topological charge <i>l</i> of the outgoing beam through the superposition of the phase plates due to the special geometric feature. In this paper, the flat spiral phase plate is designed by compressing the height of traditional spiral phase plate, and inducing the refractive index to increase in the azimuthal direction based on coordinate transformation. By means of theoretical analysis and numerical simulation, it is found that the flat spiral phase plate can produce high quality vortex beams just as the traditional spiral phase plate can do. Particularly, the height of the flat spiral phase plate and the topological charge <i>l</i> carried by the vortex beams can be arbitrarily adjusted according to the refractive index selection of the dielectric material. In order to meet the needs of practical applications, the vortex beams with different topological charges can be obtained by stacking multiple layers of flat spiral phase plates. The flat spiral phase plate has broad potential applications in the fields of optical transmission and optical communication.

We investigate a multispiral phase plate (MSPP) with multiple centers of phase singularity arbitrarily located in the MSPP plane. Equations to describe the topological charge of an optical vortex in the initial plane immediately behind the MSPP and orbital angular momentum (OAM) normalized relative to the beam power are derived. The topological charge in the initial plane is found as a sum of the topological charges of all singularities if their centers are located inside a finite-radius circular aperture. If the phase singularity centers are partially located on the boundary of a circular diaphragm limiting the MSPP, the total topological charge is found as the sum of all singularities divided by 2. Total OAM that the vortex carries depends on the location of the singularity centers: the farther from the center of the plate the singularity center is located, the smaller is its contribution to the OAM. If all singularity centers are located on the boundary of the diaphragm limiting MSPP, then the OAM of the vortex beam equals zero, although in this case the topological charge of the beam is nonzero.

Spiral phase plate (SPP) is the widely used method in the generation of vortex beam (VB) with fixed topological charges (TCs) for specific wavelength. Although VB with large TCs can be directly generated by using the SPP with high vortex order. The fabrication of high-quality SPPs with high vortex orders usually requires complex manufacturing process and high machining accuracy. An alternative method to generate VBs with large TCs is cascaded multiple SPPs with low order. In this study, we numerically calculate the transmitted light field of cascaded multiple SPPs according to the Huygens–Fresnel diffraction integral, and perform the experimental verifications. Based on cascading 6 SPPs (3 SPPs with TCs of 2, and 3 SPPs with TCs 4, respectively), an VB with TCs as high as 18 is generated. Furthermore, The TCs of the generated VB are detected by coaxial and off-axis interfering with fundamental Gaussian beam or its conjugate beam, respectively. The generated fork and spiral patterns allow us to distinguish the value and sign of TCs carried by the VB. The experimental results coincide well with the theoretical simulations. The fork pattern shows better resolution than the spiral one, and the petal pattern with small spiral allows us to distinguish large TCs with a higher resolution.

Subject and Purpose. Currently, numerous ideas and different methods have been in growth for generating vortex beams — areas of the circular motion of the electromagnetic wave energy flow around the so-called phase singularity points caused by a violation of the wave front topological structure. The purpose of this work is to obtain analytical expressions describing the nonparaxial diffraction of wave modes of the waveguide resonator of a terahertz laser during the wave mode interaction with a spiral phase plate. The resulting vortex beams are examined for their physical features in free space propagation. Methods and Methodology. The Rayleigh-Sommerfeld vector theory is adopted to consider the propagation of vortex laser beams generated by wave modes of the quasi-optical waveguide cavity when interacting with a spiral phase plate in different diffraction zones. Results. For the first time, analytical expressions have been obtained to describe the nonparaxial diffraction of wave modes of the waveguide resonator of a terahertz laser, when resonator modes interact with a spiral phase plate at different topological charges, n. The physical features of the resulting vortex beams were studied in their free space propagation. It has been shown that a spiral phase plate modifies the structure of the linearly polarized EH₁₁ mode so that the original (n=0) intensity profile with the maximum energy at the center turns at n=1 and 2 into a ring-like donut shape with an energy hole in the center. The azimuthally polarized TE₀₁ mode has originally (n=0) a ring-shaped intensity. At n=1, this configuration changes to have the maximum intensity in the center. At n=2, it becomes annular again. In the process, the spherical phase front of the beam of the linearly polarized EH₁₁ mode becomes spiral and have one singularity point on the axis, whereas the phase structure of the azimuthally polarized TE₀₁ mode gains a region with two phase singularity points off the axis. Conclusions. The results of the study can effectively facilitate information transfer in high-speed THz communication systems. They can provide a real platform to perform tasks related to tomography, exploring properties of materials, detecting astrophysical sources, which makes them very promising in modern technologies.

Analytical expressions for the nonparaxial mode diffraction of a terahertz laser metal waveguide resonator are obtained. The study assumes interaction between the modes and a spiral phase plate, considering different topological charges (n). Also, using numerical modeling, the physical features of the emerging vortex beams as they propagate in free space are studied. The Rayleigh-Sommerfeld vector theory is employed to investigate the propagation of vortex laser beams in the Fresnel zone, excited by the modes of a metal waveguide quasi-optical resonator upon incidence on a spiral phase plate. In free space, the spiral phase plate for exciting TE11 mode from the profile with the intensity maximum in the center (n = 0) forms an asymmetric ring one with two maxima (n = 1, 2). For the exciting TE01 mode, the initial ring (n = 0) structure of the field intensity is transformed into a structure with a maximum radiation intensity in the center (n = 1), and later again into a ring (n = 2). The phase front of the beam for the Ey component of the linearly polarized along the y axis TE11 mode changes from spherical to spiral with one on-axis singularity point. In the phase profile of the transverse components of the azimuthally polarized TE01 mode, a region with two and three off-axis phase singularity points appears.

Using simplified Richards-Wolf formulas we show that laser light with azimuthal polarization and singular phase can produce a smaller focal spot than that from a laser beam with radial polarization, other conditions remaining the same. It is numerically shown that when focusing an azimuthally polarized laser beam with phase singularity using a zone plate a 1.3 times smaller focal spot can be attained than when an aplanatic lens is used. A spiral phase plate can be replaced with a phase step with a π-phase shift. In this case the subwavelength focal spot from a laser beam with azimuthal polarization, which is formed near the zone plate surface, loses circular symmetry, while becoming smaller and acquiring an elliptical form with radiuses of 0.273λ and 0.314λ (NA = 1).

In the paper, we present a spiral phase plate (SPP) filter in Fourier domain for detecting the intensity or phase object’s edge information. The proposed edge detection scheme takes advantage of SPP filter and Sobel operator. In contrast to the spiral phase contrast (SPC) imaging adopting common SPP with topological charge l= 1, our scheme offers a superposed-SPP filter with several different topological charge as a substitute for Sobel edge detection operator. The numerical simulations results show that it is feasible to obtain the edge information with the proposed scheme, and the edge performance is enhanced by the proposed scheme in comparison to common SPP filter. With the increase of n, the superposition number of SPPs, the proposed scheme has a better edge information. Since the superposed SPPs can be designed before the experiment, the proposed scheme can archive a clear edge at real time, without the convolutional operations.

In this paper, we studied the properties of speckle, which is produced by passing the radially polarized vortex beams through the scattering medium. We first of all study the generation of the radially polarized vortex beams. The experimental results show that the generated light beams possess the orbital angular momentum (OAM) and the polarization is of radial direction. The radially polarized vortex beams with topological charge zero, 2, or 4, and 6 are obtained by the radial polarization convertor and the spiral phase plate. Then we investigate the speckle properties of these light beams passing through the scattering medium with different sizes of the scattering media particles. The speckle properties is numerically simulated, the related parameters of speckle also are measured, for example, the size of the speckle. Both theoretical and experimental results show that, the size of the speckle formed by the radially polarized beam is smaller than the size of the speckle formed by the non-radially polarized beam, the size of the speckle decreases with the increasing topological charge of radially polarized vortex beams, and the decreasing size of the scattering medium particle. At the same time, we also study the generation of partially coherent vortex light beams, and investigate the speckle properties produced by passing partially coherent vortex light beams through the scattering medium. It is shown that the lower coherence of the light beams will result in the contrast of the speckle. Both theoretical and experimental results of this paper may find applications in laser display et al‥

Method for exploring the topological charge and shape of an optical vortex generated by a spiral phase plate

In this work, the interaction of a supersonic beam of toluene diluted in He with a resonant oscillating RF field is investigated both experimental and theoretically. It is shown how the resonant field induces a peak structure in the transverse beam profile which can be explained by the onset of molecular interferences. Specifically, the interaction of a toluene beam of 0.12 eV of translational energy with a resonant RF field of 1.12 kV/m amplitude, and -610 kV/m(2) of gradient at the horizontal plane, during 160 micros produces a series of maxima in the transverse beam profile. The observed structure was satisfactorily reproduced by a quantum interference model based on the interaction of two coherent superpositions induced by the resonant RF field. It appears the present experimental technique could be useful to investigate the spectroscopy and dynamical behavior of coherent beams of polar molecules.

Generating optical vortex with large topological charges by spiral phase plates in cascaded and double-pass configuration

The generation and application of the vortex beams are part of the hot topics in the optical field. In this paper, the phase structure of the four-step phase plates, analyzed by Fourier series expansion method, is composed of a series of spiral phase plates. When the phase plate is directly irradiated by linearly polarized light, multi-order diffraction waves with different topological charge numbers are generated. Unlike vortex waves, the intensity distribution of the multi-order diffraction has a deviation from the axial symmetry due to the interference with each other. On this basis, a new scheme is proposed to generate vortex beams by the four-step phase plates. With the help of Mach-Zehnder interferometer, the diffraction waves generated by two pieces of the four-step phase plates overlap each other. By adjusting the phase difference of the Mach-Zehnder interferometer, some orders of diffraction waves generate destructive interference while the others generate constructive interference. Thus the linear polarized light can be converted into vortex beams. The diffraction intensity and angular momentum distributions of the four-step phase plates with different cycle numbers are numerically simulated and compared with the spiral phase plates, we can provethat the vortex beams can be obtained by simple four-step phase plates which are the same as those obtained by spiral phase plates. In addition, the four-step phase plates with a small cycle number can generate vortex beams with a large topological charge number and the fabrication difficulty of the phase plates is reduced.

We deduce and study an analytical expression for Fresnel diffraction of a plane wave by a spiral phase plate (SPP) that imparts an arbitrary-order phase singularity on the light field. Estimates for the optical vortex radius that depends on the singularity's integer order n (also termed topological charge, or order of the dislocation) have been derived. The near-zero vortex intensity is shown to be proportional to rho2n, where p is the radial coordinate. Also, an analytical expression for Fresnel diffraction of the Gaussian beam by a SPP with nth-order singularity is analyzed. The far-field intensity distribution is derived. The radius of maximal intensity is shown to depend on the singularity number. The behavior of the Gaussian beam intensity after a SPP with second-order singularity (n = 2) is studied in more detail. The parameters of the light beams generated numerically with the Fresnel transform and via analytical formulas are in good agreement. In addition, the light fields with first- and second-order singularities were generated by a 32-level SPP fabricated on the resist by use of the electron-beam lithography technique.

An analytical expression for the spatial spectrum of the conic wave diffracted by a spiral phase plate (SPP) with arbitrary integer singularity of order n is obtained. Conic wave diffraction by the SPP is equivalent to plane-wave diffraction by a helical axicon. A comparison of the conic wave and Gaussian beam diffraction on a SPP is made. It is shown that in both cases a light ring is formed, with the intensity function growing in proportion to rho(2n) at small values of radial variable rho and decreasing as n(2)rho(-4) at large rho. By use of direct e-beam writing on the resist, a 32 level SPP of the 2nd order and diameter 5 mm is manufactured. By use of this SPP, a He-Ne laser beam is transformed into a beam with phase singularity and ringlike intensity distribution. A four-order binary diffractive optical element (DOE) with its transmittance proportional to a linear superposition of four angular harmonics is also manufactured. With this DOE, simultaneous optical trapping of several polystyrene beads of diameter 5 microm is performed.

The focusing properties of the hyperbolic-cosine-Gaussian beam induced by spiral phase plates are investigated numerically. The phase plate may alter the wavefront phase of the hyperbolic-cosine-Gaussian beam by topological charge which results in spiral optical vortex. Results show that the topological charge of the spiral phase plate influences focal intensity distribution considerably, and some novel focal patterns, focal split appear in focal evolution with different topological charge. The focal evolution differs for different parameters in cosh parts of the hyperbolic-cosine-Gaussian beam. The optical gradient force is also illustrated to show promising application of the hyperbolic-cosine-Gaussian beam containing the spiral optical vortex in optical tweezers array.