Abstract

Linearly and circularly polarized terahertz (THz) vortex beams are generated by adopting a THz quarter wave plate and spiral phase plates with topological charges 1 and 2. Taking advantage of a THz digital holographic imaging system, longitudinal components of THz vortices with different polarizations and topological charges are coherently measured and systemically analyzed in a focusing condition. The application potential of circularly polarized THz vortex beams in microscopy is experimentally demonstrated and the transformation between the spin angular momentums and orbital angular momentums of THz waves is also checked. Modified Richards-Wolf vector diffraction integration equations are applied to successfully simulate experimental phenomena.

© 2016 Optical Society of America

1. Introduction

As an important class of special beams, optical vortices have received more and more attentions in recent years. Owing to their unique properties, including a toroidal amplitude, a phase singularity and quantization orbital angular momentums (OAM), optical vortex beams have immense application potentials in optical trapping [1, 2 ], communication systems [3, 4 ], astronomical observations [5–7 ], and so on. Until now, extensive research has been carried out on the generation and focusing properties of optical vortices from visible light to radio-frequency. J. E. Curtis et al. experimentally and theoretically investigated the relationship between the structure of optical vortices and angular momentum flux [8]. B. S. Chen et al. analyzed the coherence and polarization properties of focused circularly polarized vortex beams based on the vector Debye theory [9]. Z. Zhang et al. proposed a three-dimensional focus shaping technique by combining partially coherent circularly polarized vortex beams and a binary diffractive optical element [10]. P. Schemmel et al. and J. W. He et al. separately adopted a dielectric spiral phase plate (SPP) and a meta-surface device based on a V-shape antenna structure to generate optical vortices in the THz frequency range [11–13 ]. However, as a crucial component, the longitudinal fields of vortex beams around a focal point in an optical focusing system are only less theoretically discussed due to the high detection difficulty [14–16 ]. Fortunately, the coherent measurement of longitudinal components of diffractive electro-magnetic fields in the THz waveband has been realized in our previous work [17].

In this paper, the complex field distributions of the longitudinal components for focused THz vortex beams with linear and circular polarizations are comprehensively characterized by using a THz digital holographic imaging system. The amplitude images, phase patterns and diffraction processes of longitudinal fields with different polarizations and topological charges are precisely observed by implementing a Z-scan measurement. The application possibility of the class of beams in high resolution imaging is also experimentally verified. These experimental procedures are accurately simulated by utilizing modified Richards-Wolf vector diffraction algorithms. The work prompts the research developments of optical vortices in the THz frequency range.

2. Experiment

To generate a THz vortex beam, a dielectric SPP is chosen as a wave front modulator, which is the simplest method to control OAM modes in the THz frequency range [11]. The thickness of the SPP azimuthally varies around the plate surface for imparting a vortex phase shift to an incident plane wave. The step height of the SPP is determined by [12, 18 ]

h=lλΔn,
where l is the change in topological charge induced by the SPP, λ is the wavelength of the incident wave, and Δn is the change of the refractive index between the SPP and surrounding medium. Because the experiment is performed in the free space, the refractive index of the surrounding medium is considered as 1. A polylactic acid medium is picked as the material of the SPP and its refractive index is firstly measured by using a conventional THz time-domain spectroscopy (TDS) system. As shown in Fig. 1(a) , the polylactic acid medium shows a small dispersion feature and its refractive index slowly decreases from 1.65 to 1.62 over the frequency range 0.4-1.2 THz. In the experiment, 484 μm (corresponding to 0.62 THz) is chosen as the central wavelength of the SPP, because the intensity of the spectral component is higher than others in our system. The refractive index of the SPP is 1.63 at 0.62 THz, so Δn is equal to 0.63. Two SPPs with continuous surfaces are designed, which can separately induce topological charge changes with l = 1 and 2. The two SPPs are fabricated by using a commercial three-dimensional printing technique and are polished to reduce their roughness as far as possible. It should be noted that the SPP with l = 2 is designed in a split configuration for reducing its total thickness. The SPP composes of two semicircular phase plates with l = 1. The step heights of both SPPs are 768 μm and their diameters are 20 mm.

 figure: Fig. 1

Fig. 1 (a) Refractive index spectrum of a polylactic acid medium. (b) Photos of spiral phase plates with l = 1 (left) and 2 (right). (c) THz digital holographic imaging system.

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A THz digital holographic imaging system shown in Fig. 1(c) is utilized to characterize complex field distributions of converging THz vortex beams. The light source is a Spectra-Physics femtosecond laser amplifier with an 800 nm central wavelength, a 50 fs pulse duration, a 900 mW average power and a 1 kHz repetition ratio. The laser pulse is split into pump and probe beams for exciting and detecting THz radiations. The pump beam with an 890 mW average power is expanded by a concave lens (L1) with a 25 mm focal length. A <110> ZnTe crystal with a 3 mm thickness is mounted close to L1 and is illuminated by the pump beam. A linearly polarized THz beam is excited due to the optical rectification effect. A parabolic mirror (PM) with a 75 mm focal length is used to collimate the THz beam. The THz beam with a 21 mm diameter successively emerges through the designed SPP and a high resistivity silicon lens with a 25 mm focal length to form a converging THz beam with a vortex wave front. The out-going THz beam is detected by another ZnTe crystal. To measure different THz polarization components, a half wave plate (HWP) and a polarizer (P) are employed to adjust the probe polarization in the path of the probe beam [19]. The probe beam is reflected onto the detection crystal by a 50/50 non-polarization beam splitter (BS). In the detection crystal, the probe polarization is modulated by the THz field to carry the two dimensional THz information due to a linear electro-optic effect. The reflected probe beam is sent into the imaging module of the system, which consists of a quarter wave plate (QWP), a Wollaston prism (PBS), two lenses (L2 and L3), and a CCD camera with a 4 Hz frame rate for measuring the variation in the probe polarization. A mechanical chopper is inserted into the path of the pump beam and is synchronously controlled with the CCD camera. The imaging module can capture the image of the probe beam and extract the two-dimensional THz information by using the dynamic subtraction and balanced electro-optic detection techniques [20, 21 ]. In this system, the size of the imaging region is 8 mm × 8 mm and one pixel size is 32 μm × 32 μm. Varying the optical path difference between the THz and probe beams sequentially, a series of THz images in the time domain are acquired. To enhance the signal-to-noise ratio, 100 frames are averaged at each scan point. Operating the Fourier transformation to THz temporal images, the amplitude and phase information of each spectral component are extracted.

To observe the transverse and longitudinal components of the converging THz field, both of the crystalline orientation of the detection crystal and the polarization of the probe beam need to be carefully selected. A <110> ZnTe crystal is chosen to measure the transverse THz components. Adjusting the probe polarization, the horizontal and vertical components (Ex and Ey) can be separately obtained when the angle between the probe polarization and the <001> axis of the detection crystal is 0° or 45° [19]. A <100> ZnTe crystal is picked to measure the longitudinal component (Ez). To optimize the detection efficiency, the angle between the probe polarization and the <010> axis of the ZnTe is tuned to 45° [22]. Besides, it should still be pointed out that the <110> and <100> ZnTe crystals with identical thickness have the same detection efficiencies [23]. In the experiment, thicknesses of both ZnTe crystals are measured as 0.995 mm by using a micrometer and their thickness tolerances are lower than 0.002 mm. Therefore, it can be considered that both ZnTe crystals have the same measurement sensitivities to THz fields.

To record the diffraction features of focused THz vortices with linear and circular polarizations, a quartz THz quarter wave plate (TQWP, TYDEX Company, Russia) with a 400 μm central wavelength and a 200 μm bandwidth is applied to translate the THz polarization, as shown in Fig. 1(c). In the experiment, the SPP and the silicon lens are together mounted on a one dimensional translation stage and a Z-scan measurement is implemented to record the evolution process of the THz vortices. The focal point is considered as the origin. The scan range is from −12 mm to 12 mm and the scan resolution is 1 mm.

3. Results and discussions

3.1 Linear polarization

First of all, vector field distributions of linearly polarized THz vortex beams are measured and presented. The initial polarization direction of the incident THz field is along the x axis. Because the THz beam is non-tightly focused in the experiment, the THz polarization is not obviously influenced after passing through the focal spot [24]. Therefore, only the Ex and Ez components are recorded. Figures 2(a) and 2(b) show the amplitude and wrapped phase distributions of Ex with l = 1 for the 0.62 THz radiation on the focal plane. It can be seen that the amplitude exhibits a typical doughnut shape with a 0.45 mm radius due to the central phase singularity. Herein, the radius of the light ring is defined as the distance between the position of the amplitude maximum and its central point [25]. The phase of Ex shows an expected vortex pattern, which monotonically increases along the azimuthal direction. These features of the THz complex field are very similar to those of a Laguerre-Gaussian (LG) mode Epl with l = 1 and p = 0, where p is the radial index [26]. It should be explained that the color of a pixel is set as gray on the phase pattern to eliminate the uncertain phase noise when its amplitude is less than 10% of the maximum value on the corresponding amplitude image. The amplitude and wrapped phase maps of Ex with l = 2 at 0.62 THz on the focal plane are presented in Figs. 2(c) and 2(d), respectively. Its amplitude manifests a larger annular shape with a 0.74 mm radius due to a higher topological number. The phase variation shows two periodic 2π shifts around the optical axis. These phenomena demonstrate that SPPs can effectively convert the mode of an optical field to THz vortices with low orders [11, 12 ]. In addition, it should be pointed out that the radius of a light ring increases with a proportion of l/2 as the value of l gets bigger [25]. Therefore, the expected ratio between the radiuses for l = 1 and 2 is 1.41. However, the measurement ratio is 1.64 which is slightly deviated from the expected value. Actually, SPPs are not perfected mode converters which induce some weak LG modes with high orders to result in the ratio difference [11].

 figure: Fig. 2

Fig. 2 (a) Amplitude and (b) wrapped phase distributions of the Ex component at 0.62 THz for a linearly polarized THz vortex beam with l = 1 on the focal plane. Corresponding (c) amplitude and (d) phase images of Ex with l = 2.

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Replacing the detection crystal and operating the Z-scan measurement, the evolution of the longitudinal component Ez in the focusing process is reconstructed. Figure 3(a) shows the amplitude image of Ez at 0.62 THz for the linearly polarized THz vortex beam with l = 1 on the focal plane, which manifests an elliptical main spot on the optical axis and two off-axis lobes. Herein, the full width at half maximum (FWHM) is used as the diameter of a light spot. The lengths of the main spot along the x and y axes are 0.64 mm and 1.2 mm, respectively. The maximum values of two lobes separately appear at x = −0.91 mm and 0.89 mm. In addition, the amplitude maximum value of Ez is approximately l0% of Ex. Figure 3(b) shows the cross-sectional amplitude distribution of Ez on the x-z plane (y = 0 mm). The focal depth of the main spot is about 10 mm and those of the lobes are 15 mm, respectively. To understand the formation of the amplitude distribution, the phase pattern of Ez on the focal plane is presented in Fig. 3(c). It can be seen that the phase is a 180-degree rotationally symmetric modality, which nearly exhibits a flat plane on the region of the main spot and appears two screw points at x = −0.49 mm and 0.47 mm. Therefore, a constructive interference is produced on the optical axis to generate the main spot. Meanwhile, two screw points give rise to the destructive interferences at two sides of the main spot to cause two lobes. Figure 3(d) exhibits the longitudinal phase distribution of Ez on the x-z plane. The two phase singularities always exist in the focal depth range. In addition, the phase image shows that Ez undergoes a 1.2π phase variation on the optical axis as the THz field propagates through the focal spot, which is ascribed to the influence of a Gouy phase shift. It is well known that the Ez component forms a destructive interference on the central interface (x = 0 mm) to result in a similar dipole pattern when an x-linearly polarized plane wave is focused [17, 22 ]. Here, the introduction of the spiral phase offsets the π phase jump on the central interface and induces the special complex field distribution of Ez.

 figure: Fig. 3

Fig. 3 (a) Transverse and (b) longitudinal amplitude distributions of Ez at 0.62 THz for a linearly polarized THz vortex beam with l = 1 on the focal plane and cross-sectional plane (y = 0 mm), respectively. Corresponding (c) transverse and (d) longitudinal wrapped phase patterns. (e), (f), (g) and (h) are simulated amplitude and phase maps of Ez with l = 1 on the focal plane and cross-sectional plane by using the Richards-Wolf vector diffraction equation. (i) Normalized amplitude profile curves extracted from (a) and (e) on the line of y = 0 mm. The red solid line is the experimental result and the blue square dot line is the simulation one.

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To ensure the accuracy of the measurement results, the experimental procedure is simulated by using a modified Richards-Wolf vector integration algorithm [27, 28 ]. The Ez component for a converging THz vortex beam with the linear polarization can be expressed as [28]

Ez=il+1kf0αexp(ilφ)Bl(t,φ)sin2θcosθexp(ikzcosθ)dθ,
where k is the wave number in the vacuum, f is the focal length of the silicon lens, (ρ,φ,z) is the cylindrical coordinate on an observation plane, θ is the angle between the optical axis and the THz beam, α is the maximum focusing angle of the THz beam and is equal to 22° in this work. In addition, the function Bl(t,φ) can be written as
Bl(t,φ)=i2[exp(iφ)Jl+1(t)exp(iφ)Jl1(t)],
where Jl+1(t) and Jl1(t) are Bessel functions of the first kind of different orders and t=kρsinθ. Figures 3(e)-3(h) show the simulated amplitude and phase images of Ez with l = 1 at 0.62 THz on the focal plane and x-z plane. The simulation results are in good agreement with the experimental ones, which present two screw points at x = ± 0.49 mm. To quantitatively analyze the simulation results, the normalized Ez amplitude profile curves on the line of y = 0 mm are extracted from Figs. 3(a) and 3(e) and are compared in Fig. 3(i). Their maximum divergence is approximately 10%. These differences between the experimental and simulation results are mainly due to the machining imperfection of the SPP and the non-coaxial dislocation between the SPP and silicon lens.

Utilizing the same measurement method, the diffraction features of Ez at 0.62 THz for the converging linearly polarized THz vortex beam with l = 2 are also observed. Figures 4(a) and 4(c) show the amplitude and wrapped phase images of Ez on the focal plane, which give a more complicated complex field distribution. The amplitude possesses a nearly circular contour profile with a 1.35 mm radius and three minimum values at x = 0 mm, −0.70 mm, and 0.72 mm along the line of y = 0 mm. The phase presents a bilaterally anti-symmetric modality, which manifests three screw points on the positions of three amplitude minima. Figures 4(b) and 4(d) show the longitudinal amplitude and wrapped phase maps of Ez on the x-z plane. It can be seen that the amplitude is divided into four lobes by the three minima. The focal depths of the two lobes around the optical axis are about 10 mm and those of the outer two lobes are 15 mm, which are similar to the case of Ez with l = 1. The unsymmetric pattern of the amplitude may be ascribed to the non-coaxial dislocation of the SPP and the inhomogeneous distribution of the initial incident THz wave. In the longitudinal phase maps, the three singularities are always maintained before and after the focal spot to lead to the characteristics of Ez. The phase evolutions at two sides of the optical axis (x = ± 0.25 mm) are also observed and both of them present a 2.1π Gouy phase shift. The amplitude and phase images of Ez on the transverse and longitudinal cross-sectional planes are also simulated by using Eq. (2), as shown in Figs. 4(e)-4(h). The simulation results accord with the experimental phenomena very well. In the simulated phase pattern, the three screw points appear at x = 0 mm and ± 0.72 mm. The normalized Ez amplitude profile curves along the y = 0 mm direction are also extracted from Figs. 4(a) and 4(e) and are given in Fig. 4(i). Although the maximum divergence between the experimental and simulation results is about 30%, the main experimental features are duplicated very well to further check the exactness of the measurement.

 figure: Fig. 4

Fig. 4 (a) Amplitude and (c) wrapped phase distributions of Ez at 0.62 THz for a linearly polarized THz vortex beam with l = 2 on the focal plane. Corresponding (b) amplitude and (d) wrapped phase maps of Ez on the x-z plane. (e), (f), (g) and (h) are transverse and longitudinal patterns of the amplitude and wrapped phase of Ez with l = 2 calculated by the modified Richards-Wolf equation. (i) Normalized experimental and simulation amplitude profile curves extracted from (a) and (e) along the y = 0 mm direction.

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3.2 Circular polarization

Compared with linearly polarized optical vortices, optical vortex beams with a circular polarization have more distinctive features and have received much attention [9, 10, 29 ]. Q. W. Zhan has theoretically pointed out that analogous focusing properties of a radially polarized light can be achieved by properly combining handedness of the circular polarization and charges of the phase singularity [29]. In this work, this point has been comprehensively experimentally demonstrated. Right circularly polarized (RCP) THz beams with spiral wave fronts of l = 1 and 2 are generated by using the TQWP and SPPs. Figures 5(a) and 5(b) present the amplitude and wrapped phase images of the Ex component at 0.62 THz for a focused RCP THz vortex beam with l = 1 on the focal plane. The annular amplitude pattern and spiral phase variation are the same as the experimental results shown in Figs. 2(a) and 2(b). The amplitude and phase distributions of the Ey component are given in Figs. 5(c) and 5(d), which exhibit identical features as those of Ex. The amplitude of Ey is approximately 0.8 times of the amplitude of Ex. Actually, it should be noted that the RCP THz beam generated by the TQWP is a quasi-circularly polarized light at 0.62 THz. The properties of the TQWP have been evaluated by using a THz-TDS system. It can achieve a 79% amplitude ratio between Ey and Ex and induce a 0.54π phase retardation between the two polarization components at 0.62 THz. The amplitude and phase patterns of Ex and Ey with l = 2 are shown in Figs. 5(e)-5(h), respectively. Their characteristics are consistent with phenomena exhibited in Figs. 2(c) and 2(d), including larger-size light ring and double 2π vortex phase shift.

 figure: Fig. 5

Fig. 5 (a) Amplitude and (b) wrapped phase images of the Ex component at 0.62 THz for a right circularly polarized (RCP) THz vortex beam with l = 1 on the focal plane. Corresponding (c) amplitude and (d) phase patterns of the Ey component. (e), (f), (g), and (h) present the amplitude and wrapped phase distributions of Ex and Ey at 0.62 THz for a RCP THz beam with a vortex wave front of l = 2 on the focal plane.

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The most special focusing property of a circularly polarized vortex beam is that its longitudinal component Ez is comparable to that of a radially polarized beam in an identical focusing condition. Utilizing the THz imaging system, the diffraction processes of Ez with a circular polarization and different topological charges are coherently recorded. Figures 6(a) and 6(b) present the amplitude distributions of Ez at 0.62 THz for the converging RCP THz vortex beam with l = 1 on the focal plane and x-z plane. It can be seen that the amplitude of Ez presents a main focal point and an annular intensity maximum. The diameter of the main point and its focal depth are 0.80 mm and 10 mm, respectively. The annular intensity maximum always accompanies the main spot in the focusing process and its radius is about 0.91 mm on the focal plane. The corresponding phase of Ez presents an axially symmetrically annular distribution and has not any phase jump and screw point on the optical axis, as shown in Figs. 6(c) and 6(d). These features are very analogous to those of a radially polarized beam [30, 31 ]. Besides, the measurement Gouy phase shift exhibits a phase variation of about 1.1π along the optical axis, which is basically consistent with a recent report [32]. To explain the phenomenon, a RCP optical field can be expressed as ERCP=P(ρ)exp(jφ)(eρjeφ)/2 [29], where eρ and eφ are the unit vectors on radial and azimuthal directions, P(ρ) is the axially symmetric amplitude factor and can be set as 1 for simplicity. After passing through a SPP with l, the transmitted RCP vortex field can be written as E=exp[j(l1)φ](eρjeφ)/2. When l is equal to 1, the vortex phase item exp[j(l1)φ] is eliminated and the optical field can be seen as a linear superposition of radially and azimuthally polarized beams. In an optical focusing system, the distribution of Ez is dominated by the radially polarized beam, because the azimuthally polarized beam cannot create a longitudinal component [30].

 figure: Fig. 6

Fig. 6 (a) and (b) show the amplitude distributions of Ez at 0.62 THz for a converging right circularly polarized (RCP) THz vortex beam with l = 1 on the focal plane and x-z plane. (c) and (d) show the corresponding transverse and longitudinal wrapped phase patterns. (e), (f), (g), and (f) give simulated amplitude and phase images of Ez at 0.62 THz for a focused radially polarized beam with l = 1 on the transverse and longitudinal cross-sectional planes. (i) Comparison of normalized experimental and simulation Ez amplitude distribution plots obtained from (a) and (e) on the line of y = 0 mm.

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To further check this point, the Ez component of a focused radially polarized beam is simulated by using a modified Richards-Wolf equation [27, 28 ]. The complex field of Ez can be written as

Ez=ilkf0αexp[i(l1)φ]Jl1(t)sin2θcosθexp(ikzcosθ)dθ.
Figures 6(e)-6(h) depict the simulated amplitude and wrapped phase images of Ez with l = 1 on the focal plane and x-z plane. The normalized Ez amplitude distribution plots are obtained from Figs. 6(a) and 6(e) on the line of y = 0 mm, as shown in Fig. 6(i). Their maximum divergence is low than 15%. The simulation results conform to the experimental ones, which adequately demonstrate the previous theoretical prediction.

When a RCP THz beam passes through the SPP with l = 2, the transmitted THz field can be expressed as E=exp(jφ)(eρjeφ)/2, which carries a vortex phase item. In this case, its longitudinal component is consistent with that of a radially polarized vortex beam in an identical focusing system. Figures 7(a) and 7(c) exhibit the amplitude and wrapped phase distributions of Ez at 0.62 THz for a RCP THz vortex beam with l = 2 on the focal plane. The Ez component possesses a doughnut-like amplitude with a 0.45 mm radius and an annular lobe with a 1.1 mm radius. Its phase presents a 2π spiral variation. The cross-sectional amplitude and phase distributions of Ez on the x-z plane are shown in Figs. 7(b) and 7(d), which also exhibit diffraction features of a typical optical vortex. The Gouy phase shifts around the optical axis (x = ± 0.25 mm) are measured as 2.2π. These phenomena are in good agreement with measurement results of a radially polarized vortex beam with a topological charge 1 in our previous work [17]. Using Eq. (4), transverse and longitudinal complex field distributions of Ez at 0.62 THz for a radially polarized vortex beam with a topological charge 1 (namely l = 2) are simulated and shown in Figs. 7(e)-7(h). The comparison of the normalized experimental and simulation Ez amplitude distribution plots along the y = 0 mm direction is also given in Fig. 7(i). Despite a 35% maximum divergence due to experimental errors, the simulation result is in accord with main experimental characteristics, including the central doughnut-like amplitude and annular lobe.

 figure: Fig. 7

Fig. 7 (a) and (b) are the transverse and longitudinal amplitude patterns of Ez at 0.62 THz for a RCP THz vortex beam with l = 2. (c) and (d) are the corresponding wrapped phase maps of Ez on the transverse and longitudinal cross-sectional planes. (e), (f), (g), and (h) present simulated amplitude and phase distributions of Ez at 0.62 THz for a focused radially polarized vortex beam with a topological charge 1 on the focal plane and x-z plane, respectively. (i) Comparison of normalized experimental and simulation Ez amplitude distribution plots obtained from (a) and (e) along the y = 0 mm direction.

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It is well known that the Ez component of a converging radially polarized beam has a smaller size than the Ex component of a focused linearly polarized beam in a same focusing condition [31, 33 ]. It is the main reason why a kind of optical beams can be applied in high resolution microscopy [34, 35 ] and particle acceleration [36, 37 ]. Recently, E. A. Nanni et al. have experimentally achieved linear electron acceleration in a metallic waveguide by using the longitudinal component of a radially polarized THz beam [38]. Herein, the focal spot sizes of Ez components of circularly polarized THz vortex beams are also analyzed. The experimental data in Figs. 6(i) and 7(i) are selected and shown in Fig. 8 . For comparison, the TQWP and SPPs are removed from the THz imaging system and the Ex component of a focused x-linearly polarized THz beam is measured on the focal plane. The amplitude profile curve of Ex with a 1.0 mm diameter is also plotted in Fig. 8. It is apparent that Ez of the RCP THz vortex beam with l = 1 has a sharper profile than Ex of the linearly polarized THz beam even in the case of non-tightly focusing, as shown in Fig. 8(a). Meanwhile, Ez of the RCP THz vortex beam with l = 2 has a steep dark region with a 0.27 mm diameter on the central part of the light ring, as shown in Fig. 8(b). These phenomena indicate that the circularly polarized vortex beam can be also employed in high precision confocal imaging [34], subtraction imaging [39], and other optics fields.

 figure: Fig. 8

Fig. 8 Normalized Ez amplitude profile curves of the RCP THz vortex beams with l = 1 (a) and 2 (b). These lines are obtained from Figs. 6(i) and 7(i) . The blue dashed line is the amplitude profile of Ex for a converging linearly polarized THz beam at 0.62 THz on the focal plane.

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Since a circularly polarized optical beam carries both spin angular momentum (SAM) and OAM, the interconversion process between the SAM and OAM can be modulated to derive different longitudinal fields by varying handedness of the circular polarization [9, 29 ]. When a left circularly polarized (LCP) THz beam is generated by adjusting the TQWP and a spiral phase variation is loaded on the THz wave front by SPPs, the transmitted THz field can be expressed as E=exp[j(l+1)φ](eρ+jeφ)/2 and its Ez component exhibits a vortex distribution with a higher order in an optical focusing system. Figures 9(a)-9(d) present the amplitude and wrapped phase images of Ez components of focused LCP THz vortex beams with l = 1 and 2 at 0.62 THz on the focal plane, which manifest pronounced features of optical vortices with topological charges 2 and 3. Figures 9(e)-9(h) give the corresponding simulated amplitude and phase patterns of radially polarized vortex beams with charges 2 and 3, which conform to the measurement results very well. Above experimental and theoretical results depict that formations of various Ez components can be actualized by adjusting the transformation from the SAM to OAM in each THz photon.

 figure: Fig. 9

Fig. 9 (a) and (b) are the amplitude and wrapped phase images of Ez at 0.62 THz for a converging left circularly polarized (LCP) THz vortex beam with l = 1 on the focal plane. (c) and (d) are the corresponding amplitude and phase maps of Ez with l = 2. (e)-(h) present the simulated complex field distributions of Ez components of radially polarized vortex beams with charges 2 and 3.

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4. Conclusion

In conclusion, the longitudinal components Ez of converging linearly and circularly polarized THz vortex beams with topological charges 1 and 2 are coherently observed and analyzed by utilizing the SPPs, TQWP and THz digital holographic imaging system. The Ez components of linearly polarized THz vortex beams with l = 1 and 2 exhibit a 180-degree rotationally symmetric distribution with two phase singularities and a bilaterally anti-symmetric distribution with three phase screw points, respectively. The Ez components of RCP THz vortex beams with l = 1 and 2 separately present characteristics of radially polarized vortex beams with topological charges 0 and 1. The Ez complex field patterns of LCP THz vortex beams with l = 1 and 2 are also measured to show the coupling from the SAMs to OAMs in THz photons. Using modified Richards-Wolf vector diffraction integration equations, the measurement results are totally theoretically duplicated. We believe that the work is valuable for advancements of THz applications in high-resolution microscopy. Furthermore, these experimental rules and theoretical analyses can be directly generalized in the infrared, visible and other spectral ranges.

Acknowledgments

This work was supported by the 973 Program of China (No.2013CBA01702), the National Natural Science Foundation of China (Nos. 11474206, 91233202, 11374216, and 11404224), the Program for New Century Excellent Talents in University (NCET-12-0607), the Scientific Research Project of Beijing Education Commission (KM201310028005), the Specialized Research Fund for the Doctoral Program of Higher Education (20121108120009), the Scientific Research Base Development Program of the Beijing Municipal Commission of Education and the Beijing youth top-notch talent training plan (CITTCD201504080).

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10. Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015). [CrossRef]  

11. P. Schemmel, G. Pisano, and B. Maffei, “Modular spiral phase plate design for orbital angular momentum generation at millimetre wavelengths,” Opt. Express 22(12), 14712–14726 (2014). [CrossRef]   [PubMed]  

12. P. Schemmel, S. Maccalli, G. Pisano, B. Maffei, and M. W. R. Ng, “Three-dimensional measurements of a millimeter wave orbital angular momentum vortex,” Opt. Lett. 39(3), 626–629 (2014). [CrossRef]   [PubMed]  

13. J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013). [CrossRef]   [PubMed]  

14. B. Chen, J. Pu, and O. Korotkova, “Focusing of a femtosecond vortex light pulse through a high numerical aperture objective,” Opt. Express 18(10), 10822–10827 (2010). [CrossRef]   [PubMed]  

15. B. H. Jia, X. S. Gan, and M. Gu, “Direct observation of a pure focused evanescent field of a high numerical aperture objective lens by scanning near-field optical microscopy,” Appl. Phys. Lett. 86(13), 131110 (2005). [CrossRef]  

16. S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013). [CrossRef]  

17. X. Wang, S. Wang, Z. Xie, W. Sun, S. Feng, Y. Cui, J. Ye, and Y. Zhang, “Full vector measurements of converging terahertz beams with linear, circular, and cylindrical vortex polarization,” Opt. Express 22(20), 24622–24634 (2014). [CrossRef]   [PubMed]  

18. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127(4-6), 183–188 (1996). [CrossRef]  

19. X. Wang, Y. Cui, W. Sun, J. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010). [CrossRef]   [PubMed]  

20. Z. Jiang, X. G. Xu, and X.-C. Zhang, “Improvement of terahertz imaging with a dynamic subtraction technique,” Appl. Opt. 39(17), 2982–2987 (2000). [CrossRef]   [PubMed]  

21. X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010). [CrossRef]   [PubMed]  

22. S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012). [CrossRef]  

23. A. Nahata and W. Zhu, “Electric field vector characterization of terahertz surface plasmons,” Opt. Express 15(9), 5616–5624 (2007). [CrossRef]   [PubMed]  

24. J. W. M. Chon, X. S. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002). [CrossRef]  

25. M. J. Padgett and L. Allen, “The poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995). [CrossRef]  

26. A. E. Siegman, Lasers (University Science Books, 1986).

27. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]  

28. S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011). [CrossRef]  

29. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006). [CrossRef]   [PubMed]  

30. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]   [PubMed]  

31. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]   [PubMed]  

32. K. J. Kaltenecker, J. C. Konig-Otto, M. Mittendorff, S. Winnerl, H. Schneider, M. Helm, H. Helm, M. Walther, and B. M. Fischer, “Gouy phase shift of a tightly focused, radially polarized beam,” Optica 3(1), 35–41 (2016). [CrossRef]  

33. H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]  

34. N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001). [CrossRef]   [PubMed]  

35. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000). [CrossRef]   [PubMed]  

36. R. D. Romea and W. D. Kimura, “Modeling of inverse C-caronerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990). [CrossRef]   [PubMed]  

37. J. Rosenzweig, A. Murokh, and C. Pellegrini, “A proposed dielectric-loaded resonant laser accelerator,” Phys. Rev. Lett. 74(13), 2467–2470 (1995). [CrossRef]   [PubMed]  

38. E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015). [CrossRef]   [PubMed]  

39. N. Tian, L. Fu, and M. Gu, “Resolution and contrast enhancement of subtractive second harmonic generation microscopy with a circularly polarized vortex beam,” Sci. Rep. 5, 13580 (2015). [CrossRef]   [PubMed]  

References

  • View by:

  1. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
    [Crossref] [PubMed]
  2. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
    [Crossref] [PubMed]
  3. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  4. F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
    [Crossref]
  5. F. Tamburini, B. Thide, G. M. Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011).
    [Crossref]
  6. B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
    [Crossref] [PubMed]
  7. M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597(2), 1266–1270 (2003).
    [Crossref]
  8. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
    [Crossref] [PubMed]
  9. B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26(4), 862–869 (2009).
    [Crossref] [PubMed]
  10. Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
    [Crossref]
  11. P. Schemmel, G. Pisano, and B. Maffei, “Modular spiral phase plate design for orbital angular momentum generation at millimetre wavelengths,” Opt. Express 22(12), 14712–14726 (2014).
    [Crossref] [PubMed]
  12. P. Schemmel, S. Maccalli, G. Pisano, B. Maffei, and M. W. R. Ng, “Three-dimensional measurements of a millimeter wave orbital angular momentum vortex,” Opt. Lett. 39(3), 626–629 (2014).
    [Crossref] [PubMed]
  13. J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013).
    [Crossref] [PubMed]
  14. B. Chen, J. Pu, and O. Korotkova, “Focusing of a femtosecond vortex light pulse through a high numerical aperture objective,” Opt. Express 18(10), 10822–10827 (2010).
    [Crossref] [PubMed]
  15. B. H. Jia, X. S. Gan, and M. Gu, “Direct observation of a pure focused evanescent field of a high numerical aperture objective lens by scanning near-field optical microscopy,” Appl. Phys. Lett. 86(13), 131110 (2005).
    [Crossref]
  16. S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
    [Crossref]
  17. X. Wang, S. Wang, Z. Xie, W. Sun, S. Feng, Y. Cui, J. Ye, and Y. Zhang, “Full vector measurements of converging terahertz beams with linear, circular, and cylindrical vortex polarization,” Opt. Express 22(20), 24622–24634 (2014).
    [Crossref] [PubMed]
  18. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127(4-6), 183–188 (1996).
    [Crossref]
  19. X. Wang, Y. Cui, W. Sun, J. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010).
    [Crossref] [PubMed]
  20. Z. Jiang, X. G. Xu, and X.-C. Zhang, “Improvement of terahertz imaging with a dynamic subtraction technique,” Appl. Opt. 39(17), 2982–2987 (2000).
    [Crossref] [PubMed]
  21. X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
    [Crossref] [PubMed]
  22. S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012).
    [Crossref]
  23. A. Nahata and W. Zhu, “Electric field vector characterization of terahertz surface plasmons,” Opt. Express 15(9), 5616–5624 (2007).
    [Crossref] [PubMed]
  24. J. W. M. Chon, X. S. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002).
    [Crossref]
  25. M. J. Padgett and L. Allen, “The poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
    [Crossref]
  26. A. E. Siegman, Lasers (University Science Books, 1986).
  27. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [Crossref]
  28. S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
    [Crossref]
  29. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
    [Crossref] [PubMed]
  30. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [Crossref] [PubMed]
  31. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  32. K. J. Kaltenecker, J. C. Konig-Otto, M. Mittendorff, S. Winnerl, H. Schneider, M. Helm, H. Helm, M. Walther, and B. M. Fischer, “Gouy phase shift of a tightly focused, radially polarized beam,” Optica 3(1), 35–41 (2016).
    [Crossref]
  33. H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
    [Crossref]
  34. N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
    [Crossref] [PubMed]
  35. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
    [Crossref] [PubMed]
  36. R. D. Romea and W. D. Kimura, “Modeling of inverse C-caronerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
    [Crossref] [PubMed]
  37. J. Rosenzweig, A. Murokh, and C. Pellegrini, “A proposed dielectric-loaded resonant laser accelerator,” Phys. Rev. Lett. 74(13), 2467–2470 (1995).
    [Crossref] [PubMed]
  38. E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
    [Crossref] [PubMed]
  39. N. Tian, L. Fu, and M. Gu, “Resolution and contrast enhancement of subtractive second harmonic generation microscopy with a circularly polarized vortex beam,” Sci. Rep. 5, 13580 (2015).
    [Crossref] [PubMed]

2016 (1)

2015 (3)

E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
[Crossref] [PubMed]

N. Tian, L. Fu, and M. Gu, “Resolution and contrast enhancement of subtractive second harmonic generation microscopy with a circularly polarized vortex beam,” Sci. Rep. 5, 13580 (2015).
[Crossref] [PubMed]

Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

2014 (3)

2013 (2)

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013).
[Crossref] [PubMed]

2012 (3)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012).
[Crossref]

2011 (2)

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
[Crossref]

F. Tamburini, B. Thide, G. M. Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011).
[Crossref]

2010 (3)

2009 (1)

2008 (1)

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

2007 (2)

A. Nahata and W. Zhu, “Electric field vector characterization of terahertz surface plasmons,” Opt. Express 15(9), 5616–5624 (2007).
[Crossref] [PubMed]

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

B. H. Jia, X. S. Gan, and M. Gu, “Direct observation of a pure focused evanescent field of a high numerical aperture objective lens by scanning near-field optical microscopy,” Appl. Phys. Lett. 86(13), 131110 (2005).
[Crossref]

2003 (3)

M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597(2), 1266–1270 (2003).
[Crossref]

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

2002 (1)

J. W. M. Chon, X. S. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002).
[Crossref]

2001 (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
[Crossref] [PubMed]

2000 (3)

1997 (1)

1996 (1)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

1995 (3)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

M. J. Padgett and L. Allen, “The poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
[Crossref]

J. Rosenzweig, A. Murokh, and C. Pellegrini, “A proposed dielectric-loaded resonant laser accelerator,” Phys. Rev. Lett. 74(13), 2467–2470 (1995).
[Crossref] [PubMed]

1990 (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse C-caronerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
[Crossref] [PubMed]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Ahmed, N.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Alferov, S. V.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Allen, L.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

M. J. Padgett and L. Allen, “The poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
[Crossref]

Anzolin, G.

F. Tamburini, B. Thide, G. M. Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011).
[Crossref]

Bergman, J.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
[Crossref] [PubMed]

Bianchini, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Brown, T.

Carozzi, T. D.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
[Crossref] [PubMed]

Chen, B.

Chon, J. W. M.

J. W. M. Chon, X. S. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002).
[Crossref]

Chong, C. T.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Cui, Y.

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[Crossref] [PubMed]

Dholakia, K.

Dolinar, S.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Fallahi, A.

E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
[Crossref] [PubMed]

Fan, H.

Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Fazal, I. M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Feng, S.

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N. Tian, L. Fu, and M. Gu, “Resolution and contrast enhancement of subtractive second harmonic generation microscopy with a circularly polarized vortex beam,” Sci. Rep. 5, 13580 (2015).
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B. H. Jia, X. S. Gan, and M. Gu, “Direct observation of a pure focused evanescent field of a high numerical aperture objective lens by scanning near-field optical microscopy,” Appl. Phys. Lett. 86(13), 131110 (2005).
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B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
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H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
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N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
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Helm, M.

K. J. Kaltenecker, J. C. Konig-Otto, M. Mittendorff, S. Winnerl, H. Schneider, M. Helm, H. Helm, M. Walther, and B. M. Fischer, “Gouy phase shift of a tightly focused, radially polarized beam,” Optica 3(1), 35–41 (2016).
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E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
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Huang, H.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
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E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
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S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012).
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B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
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B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
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B. H. Jia, X. S. Gan, and M. Gu, “Direct observation of a pure focused evanescent field of a high numerical aperture objective lens by scanning near-field optical microscopy,” Appl. Phys. Lett. 86(13), 131110 (2005).
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Kaltenecker, K. J.

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S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
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S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
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S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
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R. D. Romea and W. D. Kimura, “Modeling of inverse C-caronerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
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S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
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H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
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Maffei, B.

Mari, E.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
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E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
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K. J. Kaltenecker, J. C. Konig-Otto, M. Mittendorff, S. Winnerl, H. Schneider, M. Helm, H. Helm, M. Walther, and B. M. Fischer, “Gouy phase shift of a tightly focused, radially polarized beam,” Optica 3(1), 35–41 (2016).
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S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012).
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E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
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Nanni, E. A.

E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
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Novotny, L.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
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M. J. Padgett and L. Allen, “The poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
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B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
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J. Rosenzweig, A. Murokh, and C. Pellegrini, “A proposed dielectric-loaded resonant laser accelerator,” Phys. Rev. Lett. 74(13), 2467–2470 (1995).
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Pu, J.

Qu, J.

Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

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R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

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E. A. Nanni, W. R. Huang, K. H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
[Crossref] [PubMed]

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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
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G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127(4-6), 183–188 (1996).
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F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

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R. D. Romea and W. D. Kimura, “Modeling of inverse C-caronerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
[Crossref] [PubMed]

Rosenzweig, J.

J. Rosenzweig, A. Murokh, and C. Pellegrini, “A proposed dielectric-loaded resonant laser accelerator,” Phys. Rev. Lett. 74(13), 2467–2470 (1995).
[Crossref] [PubMed]

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H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

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S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

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Schneider, H.

K. J. Kaltenecker, J. C. Konig-Otto, M. Mittendorff, S. Winnerl, H. Schneider, M. Helm, H. Helm, M. Walther, and B. M. Fischer, “Gouy phase shift of a tightly focused, radially polarized beam,” Optica 3(1), 35–41 (2016).
[Crossref]

S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012).
[Crossref]

Schönle, A.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
[Crossref] [PubMed]

Sheppard, C.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Shi, L. P.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Simpson, N. B.

Sjöholm, J.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
[Crossref] [PubMed]

Smith, G. M.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

Sponselli, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Sun, W.

Sun, W. F.

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

Tamburini, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
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F. Tamburini, B. Thide, G. M. Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011).
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F. Tamburini, B. Thide, G. M. Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011).
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Then, H.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
[Crossref] [PubMed]

Thide, B.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

F. Tamburini, B. Thide, G. M. Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011).
[Crossref]

Thidé, B.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007).
[Crossref] [PubMed]

Tian, N.

N. Tian, L. Fu, and M. Gu, “Resolution and contrast enhancement of subtractive second harmonic generation microscopy with a circularly polarized vortex beam,” Sci. Rep. 5, 13580 (2015).
[Crossref] [PubMed]

Tur, M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Turnbull, G. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

Turnnen, J.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turnnen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Volotovsky, S. G.

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
[Crossref]

Walther, M.

Wang, H. F.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Wang, J.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wang, S.

Wang, X.

Wang, X. K.

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

Willner, A. E.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

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K. J. Kaltenecker, J. C. Konig-Otto, M. Mittendorff, S. Winnerl, H. Schneider, M. Helm, H. Helm, M. Walther, and B. M. Fischer, “Gouy phase shift of a tightly focused, radially polarized beam,” Optica 3(1), 35–41 (2016).
[Crossref]

S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Xie, Z.

Xu, H. F.

Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Xu, X. G.

Yan, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ye, J.

Ye, J. S.

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

Youngworth, K.

Yue, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Zhan, Q.

Zhang, X.-C.

Zhang, Y.

Zhang, Z.

Z. Zhang, H. Fan, H. F. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
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B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26(4), 862–869 (2009).
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Figures (9)

Fig. 1
Fig. 1 (a) Refractive index spectrum of a polylactic acid medium. (b) Photos of spiral phase plates with l = 1 (left) and 2 (right). (c) THz digital holographic imaging system.
Fig. 2
Fig. 2 (a) Amplitude and (b) wrapped phase distributions of the Ex component at 0.62 THz for a linearly polarized THz vortex beam with l = 1 on the focal plane. Corresponding (c) amplitude and (d) phase images of Ex with l = 2.
Fig. 3
Fig. 3 (a) Transverse and (b) longitudinal amplitude distributions of Ez at 0.62 THz for a linearly polarized THz vortex beam with l = 1 on the focal plane and cross-sectional plane (y = 0 mm), respectively. Corresponding (c) transverse and (d) longitudinal wrapped phase patterns. (e), (f), (g) and (h) are simulated amplitude and phase maps of Ez with l = 1 on the focal plane and cross-sectional plane by using the Richards-Wolf vector diffraction equation. (i) Normalized amplitude profile curves extracted from (a) and (e) on the line of y = 0 mm. The red solid line is the experimental result and the blue square dot line is the simulation one.
Fig. 4
Fig. 4 (a) Amplitude and (c) wrapped phase distributions of Ez at 0.62 THz for a linearly polarized THz vortex beam with l = 2 on the focal plane. Corresponding (b) amplitude and (d) wrapped phase maps of Ez on the x-z plane. (e), (f), (g) and (h) are transverse and longitudinal patterns of the amplitude and wrapped phase of Ez with l = 2 calculated by the modified Richards-Wolf equation. (i) Normalized experimental and simulation amplitude profile curves extracted from (a) and (e) along the y = 0 mm direction.
Fig. 5
Fig. 5 (a) Amplitude and (b) wrapped phase images of the Ex component at 0.62 THz for a right circularly polarized (RCP) THz vortex beam with l = 1 on the focal plane. Corresponding (c) amplitude and (d) phase patterns of the Ey component. (e), (f), (g), and (h) present the amplitude and wrapped phase distributions of Ex and Ey at 0.62 THz for a RCP THz beam with a vortex wave front of l = 2 on the focal plane.
Fig. 6
Fig. 6 (a) and (b) show the amplitude distributions of Ez at 0.62 THz for a converging right circularly polarized (RCP) THz vortex beam with l = 1 on the focal plane and x-z plane. (c) and (d) show the corresponding transverse and longitudinal wrapped phase patterns. (e), (f), (g), and (f) give simulated amplitude and phase images of Ez at 0.62 THz for a focused radially polarized beam with l = 1 on the transverse and longitudinal cross-sectional planes. (i) Comparison of normalized experimental and simulation Ez amplitude distribution plots obtained from (a) and (e) on the line of y = 0 mm.
Fig. 7
Fig. 7 (a) and (b) are the transverse and longitudinal amplitude patterns of Ez at 0.62 THz for a RCP THz vortex beam with l = 2. (c) and (d) are the corresponding wrapped phase maps of Ez on the transverse and longitudinal cross-sectional planes. (e), (f), (g), and (h) present simulated amplitude and phase distributions of Ez at 0.62 THz for a focused radially polarized vortex beam with a topological charge 1 on the focal plane and x-z plane, respectively. (i) Comparison of normalized experimental and simulation Ez amplitude distribution plots obtained from (a) and (e) along the y = 0 mm direction.
Fig. 8
Fig. 8 Normalized Ez amplitude profile curves of the RCP THz vortex beams with l = 1 (a) and 2 (b). These lines are obtained from Figs. 6(i) and 7(i) . The blue dashed line is the amplitude profile of Ex for a converging linearly polarized THz beam at 0.62 THz on the focal plane.
Fig. 9
Fig. 9 (a) and (b) are the amplitude and wrapped phase images of Ez at 0.62 THz for a converging left circularly polarized (LCP) THz vortex beam with l = 1 on the focal plane. (c) and (d) are the corresponding amplitude and phase maps of Ez with l = 2. (e)-(h) present the simulated complex field distributions of Ez components of radially polarized vortex beams with charges 2 and 3.

Equations (4)

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h = l λ Δ n ,
E z = i l + 1 k f 0 α exp ( i l φ ) B l ( t , φ ) sin 2 θ cos θ e x p ( i k z cos θ ) d θ ,
B l ( t , φ ) = i 2 [ exp ( i φ ) J l + 1 ( t ) exp ( i φ ) J l 1 ( t ) ] ,
E z = i l k f 0 α exp [ i ( l 1 ) φ ] J l 1 ( t ) sin 2 θ cos θ e x p ( i k z cos θ ) d θ .

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