Vectorial diffraction properties of THz vortex Bessel beams

Zhen Wu; Xinke Wang; Wenfeng Sun; Shengfei Feng; Peng Han; Jiasheng Ye; Yue Yu; Yan Zhang

doi:10.1364/OE.26.001506

1. Introduction

The terahertz (THz) wave is belonged to the far-infrared electromagnetic radiation, which attracts attentions of more and more researchers over the past decades due to its distinctive features, including the low photon energy, strong responses to water molecules, high transmissions to non-polar materials, and so on [1]. With the constant advancement of the THz technology, THz sensing and imaging techniques have shown the powerful application abilities in many independent fields, such as non-invasive flaw detection [2], biological imaging [3], chemical identification [4], optical communications [5], and so on. Currently, investigations of special THz beams open up new avenues for the further development of the THz technology, because the special diffraction properties of these THz beams can be utilized to improve the performances of current THz systems [6, 7]. As an important kind of special beams, a Bessel beam has been widely concerned by the mass due to its non-diffractive and self-healing features since it was firstly proposed by Durnin et al. in 1987 [8]. Recently, the generation and application of the Bessel beam have been introduced into the THz frequency range. In 2009, Shaukat et al. used polytetrafluoroethene axicons and a quantum cascade laser to achieve the generation of the THz Bessel beams [9]. In 2012, Bitman et al. utilized a broadband THz Bessel-like beam to enhance the imaging depth of a THz raster-scan imaging system [10]. In 2015, Monnai et al. proposed a THz plasmonic Bessel beamformer based on a periodically corrugated metal surface and achieved the formation of the THz Bessel beam [11].

If a spiral phase modulation is loaded onto a Bessel beam, a vortex Bessel beam is formed, which possesses a phase singularity as well as a quantization orbital angular momentum (OAM) and simultaneously maintains the non-diffractive feature. Currently, the application values of the vortex Bessel beam have been paid attentions by numerous researchers in the visible light range. Arnold et al. investigated the nonlinear propagation phenomena of intense Bessel-Gauss vortices in transparent mediums on the experiment and theory [12]. Cheng et al. utilized the Rytov theory to establish the transmission model for vortex Bessel-Gaussian beams in weak anisotropic turbulence and pointed out its value in quantum optical communications [13]. Yang et al. theoretically computed the radiation pressure force exerted on complex shaped homogenous particles by high-order vortex Bessel beams [14]. Aleksanyan et al. reported the optical spin-orbit engineering of a vortex Bessel beam by using a homogeneous birefringent axicon and theoretically predicted the existence of various kinds of spatially modulated free-space light fields [15]. Actually, studies of the THz vortex beams are always a research hot spot in the THz field. Researchers tried all kinds of methods to generate THz vortex beams, such as medium spiral phaseplate [16], metasurface [17], mode conversion of a THz radially polarized beam [18], and so on. Recently, some investigations about the vortex Bessel beams in the THz frequency range have been also reported [19–21]. However, the vectorial diffraction properties of the THz vortex Bessel beams are rarely concerned in pervious works.

In this work, THz vortex Bessel beams with linear and circular polarizations are generated by employing a quartz THz quarter wave plate (TQWP), a spiral phase plate (SPP), and Teflon axicons with different opening angles. By utilizing a THz focal-plane imaging system, the transverse (E_x, E_y) and longitudinal (E_z) electric field components of the THz vortex Bessel beams are coherently characterized and analyzed. The vectorial Rayleigh diffraction integral is used to simulate the experimental phenomena. By varying the opening angle of the axicon, the divergences between these generated THz vortex Bessel beams are observed and compared in detail, including the light spot size, the diffraction-free range, and the phase evolution. This work comprehensively exhibits the vectorial diffraction properties of a THz vortex Bessel beam.

2. Experiment

To characterize the vectorial diffraction properties of a THz vortex Bessel beam, a THz focal-plane imaging system is used to coherently measure each polarization component of the THz field. The experimental setup is shown in Fig. 1(a). The light source is a Spectra-Physics femtosecond laser amplifier with an 800 nm central wavelength, a 50 fs pulse duration, a 1 kHz repetition ratio, and an 800 mW average power. The output laser is split into two parts to generate and detect the THz radiation, including the pump and probe beams. The pump beam with a 790 mW average power is firstly expanded by a concave lens (L1) with a 50 mm focal length and then irradiates on a <110> ZnTe crystal with a 3 mm thickness. The x-linearly polarized THz wave is excited by the optical rectification [22] and is collimated by a metallic parabolic mirror (PM) with a 100 mm focal length. The diameter of the THz beam approaches approximately 14 mm. The THz beam successively passes through a SPP and a Teflon axicon to form a THz vortex Bessel beam with a topological charge of 1. The out-going THz beam is incident into a sensor crystal for the coherent measurement. On the path of the probe beam, the laser with a 10 mW average power passes through a half wave plate (HWP) and a polarizer (P) in sequence for varying the probe polarization. Then, the probe beam is reflected onto the sensor crystal by a 50/50 non-polarizing beam splitter. The THz field induces the birefringence variation of the sensor crystal due to the Pockels effect and further modulates the polarization of the probe beam. The probe beam with the THz information is captured by an imaging module, including a quarter wave plate (QWP), a Wollaston prism (PBS), two lenses (L2 and L3), and a CCD camera with a 4 Hz frame rate. By adopting a balanced electro-optic detection method [23], a two-dimensional THz complex field is extracted. To remove the background intensity of the probe beam, a mechanical chopper is inserted into the path of pump beam and a dynamics subtraction technique is implemented [24]. Due to the limited size of the sensor crystal, the effective imaging region is about 5.2 mm × 5.2 mm and the size of a pixel is 32 μm × 32 μm. By adjusting the optical path difference between the THz and probe beams, THz images at different time delays are captured. At each scan point, 25 frames are averaged to improve the signal-to-noise ratio and the whole time window is 17 ps. To extract the amplitude and phase images of each spectral component, the Fourier transformation is operated on the THz temporal signal of each pixel.

Fig. 1 Experimental setup and samples. (a) THz focal-plane imaging system. (b) Photograph of a Teflon spiral phase plate. (c) Photographs of Teflon axicons with opening angles of 30°, 25°, 20°, 15°.

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To comprehensively characterize the vectorial properties of a THz vortex Bessel beam, the sensor crystals with different crystalline orientations and the polarization of the probe beam are appropriately picked out for separately measuring the transverse and longitudinal polarization components of the THz field. To obtain the transverse components, a <110> ZnTe with a 1 mm thickness is chosen as the sensor crystal. Moreover, the angle between the polarization direction of the probe beam and the <001> axis of the crystal is adjusted as 0° or 45° for measuring the horizontal (E_x) or vertical (E_y) component [25]. To obtain the longitudinal component (E_z), a <100> ZnTe with a 1 mm thickness is selected as the sensor crystal. The angle between the probe polarization and the <010> axis of the crystal is set to 45° to maximize the detection efficiency [26]. In addition, it should be pointed out that both <110> and <100> ZnTe crystals with an identical thickness have the same detection sensitivities to the THz field [27].

To generate THz vortex Bessel beams with different non-diffractive zones, the Teflon SPP with a topological charge of 1 and Teflon axicons with opening angles of 30°, 25°, 20°, 15° are selected as the wave front modulators, as shown in Figs. 1(b) and 1(c). The thickness of the SPP linearly increases in a counter-clockwise sense to impart a vortex phase modulation to the incident THz wave. The step height of the SPP is determined by [16]

h = \frac{l λ}{Δ n},

where

l

is the change in the topological charge induced by the SPP,

λ

is the wavelength of the incident THz wave, and

Δ n

is the change in the refractive index of the Teflon material with respect to the surrounding air. In the experiment, 600 μm is chosen as the central wavelength and the topological charge of the SPP is fixed as 1. Because the refractive index of the Teflon material is approximately 1.426 in the 0.4-0.8 THz frequency range,

Δ n

is equal to 0.426. According to Eq. (1), the step height of the SPP is calculated as 1.41 mm. In addition, the SPP also possesses a 1.0 mm substrate thickness for fabrication convenience. The aperture diameters of both the SPP and axicons are 19 mm. To observe the discrepancies between the linearly and circularly polarized THz vortex Bessel beams, a quartz TQWP (TYDEX Company, Russia) is used to adjust the polarization of the THz beam. The TQWP possesses a 500 μm central wavelength and a 200 μm bandwidth. To characterize the non-diffractive feature and the spiral wave front of the THz vortex Bessel beam, the TQWP, the SPP, and an axicon are together mounted on a linear motorized translation stage and a Z-scan measurement is implemented for recording the evolution of the THz beam. It should be also noted that the contact plane between the tip of the axicon and the sensor crystal is considered as the original point. In the measurement, the total optical path of the THz beam is not changed, so the influence induced by the linear phase shift of the THz wave is effectively avoided.

3. Results and discussions

3.1 Linear polarization

First of all, an x-linearly polarized THz vortex Bessel beam with a topological charge of 1 is generated by using the SPP and the axicon with a 30° opening angle. To characterize the diffraction properties of the THz beam, the Z-scan measurement with a 30 mm scan range and a 2 mm scan resolution is implemented. The 0.5 THz spectral component is extracted by using the Fourier transformation. Figure 2(a) presents the normalized longitudinal amplitude profile of the E_x component on the x-z plane (y = 0 mm) and the transverse amplitude cross-sections at z = 4 mm, 8 mm, 12 mm, 16 mm, 20 mm, respectively. The positions of these transverse cross-sections are marked by the white dashed lines on the longitudinal amplitude pattern. Obviously, the E_x amplitude shows a main doughnut shape with a 0.57 mm radius and an annular side-lobe with a 1.76 mm radius due to the central phase singularity and the conic phase modulation. Herein, the distance between the amplitude maximum and the central zero-amplitude point is considered as the radius of the THz light ring. The non-diffractive feature of the THz vortex Bessel beam is attributed to the interference effect of the THz beams refracted by the axicon. Comparing these transverse amplitude cross-sections at different propagation distances, the size of the central THz light ring almost keeps invariant. The diffraction-free range Z_max is defined as the full widths at half maximum (FWHM) of the THz amplitude along the z axis. From the longitudinal amplitude profile, Z_max is evaluated as 24 mm. According to the typical formula of the diffraction-free range [28], Z_max can be expressed as

Z_{\max} = \frac{ω_{0}}{\tan [arc \sin (n \sin α) - α]},

where n and α are the refractive index and the opening angle of the Teflon axicon, ω₀ is the radius of the incident THz beam. In the experiment, these parameters are n = 1.426, α = 30°, ω₀ = 7 mm and the calculated Z_max is equal to 25.3 mm, which is basically consistent with the experimental result.

Fig. 2 Characterization of the E_x component at 0.5 THz for a linearly polarized THz vortex Bessel beam generated by using the SPP and the axicon with the opening angle of 30°. (a) shows the normalized longitudinal E_x amplitude profile and transverse amplitude cross-sections at z = 4 mm, 8 mm, 12 mm, 16 mm, 20 mm, respectively. The positions of these transverse cross-sections are marked by the white dashed lines on the longitudinal amplitude pattern. (b) presents the wrapped E_x phase distributions on the longitudinal and transverse cross-sections. (c) and (d) exhibit the corresponding simulated amplitude and phase patterns by using the vectorial Rayleigh diffraction integral. (e) gives the experimental and simulated amplitude profile curves, which are extracted on the line of y = 0 mm at z = 12 mm from (a) and (c). (f) presents the corresponding experimental and simulated phase profile curves.

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To further understand the propagation properties of the THz vortex Bessel beam, the wrapped longitudinal phase profile and transverse phase cross-sections with different propagation distances are extracted for the E_x component, as shown in Fig. 2(b). It can be seen that the E_x phase presents spiral modalities on the regions of the main light ring and the annular side-lobe. During the propagation process, the two vortex phase patterns always rotate in a counter-clockwise sense and their twist directions before and after the focal spot (z = Z_max/2) appear reversed. According to our previous report [17], the rotation of the vortex phase pattern is caused by the Gouy phase shift of the THz converging beam and the variation of the twist direction is ascribed to the phase modulation of the axicon, which are similar to focusing properties of a THz vortex Gaussian beam. The contour profile of the central vortex phase pattern almost keeps a fixed size in the range from z = 0 mm to 24 mm because of the non-diffractive feature of the THz vortex Bessel beam. In addition, a phase jump of π always exists on the interface between the main light ring and the annular side-lobe. Actually, it can be seen that the central phase singularity and the phase jump of π give rise to the peculiar amplitude distribution of the THz vortex Bessel beam.

To confirm the measurement accuracy, the amplitude and phase distributions of E_x are simulated by utilizing the vectorial Rayleigh diffraction integral [29, 30]. The E_x component can be expressed as

\begin{array}{l} E_{x} (ρ, φ, z) = \frac{- j z π \exp (j k r)}{λ r^{2}} \int_{0}^{L} d ρ_{0} \exp (μ) ρ_{0} \\ \times {\begin{cases} [j (t_{p} \cos θ + t_{s}) \exp (j φ) + \frac{j}{2} (t_{p} \cos θ - t_{s}) \exp (- j φ)] J_{1} (η) \\ - \frac{j}{2} (t_{p} \cos θ - t_{s}) \exp (3 j φ) J_{3} (η) \end{cases}}, \end{array}

where k is the wave number in vacuum,

(ρ, φ, z)

is the cylindrical coordinate on the observation plane, t_p and t_s are the Fresnel transmission coefficients through the slope of the axicon for the x- and y-polarized components, L is the radius of the axicon. θ is the deflection angle versus to the optical axis for the transmitted THz beam refracted by the axicon, which can be written as

θ = arc \sin (n \sin α) - α

. In addition,

r = \sqrt{ρ^{2} + z^{2}}

,

ρ = \sqrt{x^{2} + y^{2}}

,

ρ_{0} = \sqrt{x_{0}^{2} + y_{0}^{2}}

, where

(x_{0}, y_{0})

and

(x, y)

are the Cartesian coordinates on the incident and observation planes.

J_{1} (η)

and

J_{3} (η)

are the Bessel functions of the first kind with the orders 1 and 3.

η

and

μ

can be separately written as

η = - \frac{k ρ ρ_{0}}{r},

μ = [- \frac{ρ_{0}^{2}}{ω_{0}^{2}} + j k \frac{ρ_{0}^{2}}{2 r} - j k ρ_{0} (n - 1) \tan α] .

The simulated amplitude and phase distributions on the corresponding longitudinal and transverse cross-sections are exhibited in Figs. 2(c) and 2(d), which are consistent with the experimental phenomena well. To quantitatively check the simulation results, the E_x amplitude profile curves on the line of y = 0 mm at z = 12 mm are extracted from Figs. 2(a) and 2(c) and are compared in Fig. 2(e). The maximum divergence between the experimental and simulation results does not exceed 17%. The corresponding experimental and simulated phase profile curves are also presented in Fig. 2(f), which are in agreement with each other except for some abrupt phase jumps on a few pixels.

To comprehensively understand the vectorial properties of the THz vortex Bessel beam with the linear polarization, the amplitude and phase distributions of the E_z component at 0.5 THz are obtained by using the same measurement method. Figure 3(a) present the longitudinal E_z amplitude profile on the x-z plane (y = 0 mm) and the transverse amplitude cross-sections at z = 4 mm, 8 mm, 12 mm, 16 mm, 20 mm, respectively. The amplitude maximum of E_z is approximately 20% of E_x. The E_z amplitude presents a central elliptical main spot and bilateral symmetric off-axis crescent side-lobes. The FWHMs of the main spot along the x and y directions are 0.7 mm and 1.3 mm, respectively. On the left and right sides of the main spot, two side-lobes are located at x = ± 1.1 mm. From the longitudinal amplitude pattern, it can be seen that the size of the main spot and the positions of the side-lobes almost keep constant in the range from z = 0 mm to 24 mm, because the E_z component also possesses the non-diffractive feature. The corresponding wrapped longitudinal and transverse phase cross-sections are exhibited in Fig. 3(b). The E_z phase manifests a flat plane on the region of the main spot and two screw points at x = ± 0.6 mm. In the diffraction-free range, the central flat region maintains in a fixed scale and two phase screw points always occur on the two sides of the flat region. Clearly, the central main spot is formed by the constructive interference on the optical axis and the appearance of off-axis side-lobes is caused by the destructive interferences induced by the phase screw points. As a whole, the phase pattern shows a 180-degree rotationally symmetric modality, which is analogous to the E_z phase of a THz vortex Gaussian beam [31]. As well-known, E_z of a linearly polarized THz converging beam manifests a central zero-amplitude zone, which is caused by the destructive interference on the optical axis [32]. For the THz vortex Bessel beam, the introduction of the spiral phase modulation eliminates the destructive interference and leads to the distinctive amplitude distribution of the E_z component. The field distributions of E_z are also simulated by using the vectorial Rayleigh diffraction integral [29, 30]. The E_z component can be written as

\begin{array}{l} E_{z} (ρ, φ, z) = \frac{j π \exp (j k r)}{λ r^{2}} \int_{0}^{L} d ρ_{0} \exp (μ) ρ_{0} \\ \times [\begin{array}{l} - \frac{1}{2} (t_{p} \cos θ + t_{s}) ρ_{0} J_{0} (η) + j (t_{p} \cos θ + t_{s}) ρ \cos φ \exp (j φ) J_{1} (η) \\ + \frac{1}{2} (t_{p} \cos θ + t_{s}) ρ_{0} \exp (2 j φ) J_{2} (η) \end{array}], \end{array}

where

J_{2} (η)

is the Bessel function of the first kind with the order 2. Figures 3(c) and 3(d) give the simulated amplitude and phase images on the corresponding longitudinal and transverse cross-sections, which reproduce the experimental phenomena well. To further compare the experimental and simulation results, the E_z amplitude and phase profile curves on the line of y = 0 mm at z = 12 mm are extracted from Figs. 3(a)-3(d) and are presented together in Figs. 3(e) and 3(f), which are coincident with each other well. On the amplitude profile curves, the maximum divergence between the experimental and simulation results is less than 16%.

Fig. 3 Characterization of the E_z component at 0.5 THz for a linearly polarized THz vortex Bessel beam. (a) shows the E_z amplitude distributions on the longitudinal profile and transverse cross-sections at z = 4 mm, 8 mm, 12 mm, 16 mm, 20 mm. (b) presents the E_z phase patterns on the longitudinal and transverse cross-sections. (c) and (d) give the corresponding simulated amplitude and phase distributions of the E_z component. (e) and (f) exhibit the amplitude as well as phase profile curves extracted from the experimental and simulation results on the line of y = 0 mm at z = 12 mm.

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To analyze discrepancies between these generated THz vortex Bessel beams under different focusing conditions, the axicons with α = 30°, 25°, 20°, 15° are measured with a 80 mm scan range and a 2 mm scan resolution. Their longitudinal E_x amplitude profiles at 0.5 THz on the x-z plane (y = 0 mm) are extracted and compared in Fig. 4(a). Obviously, with decreasing the opening angle of the axicon, the converging process of the THz vortex Bessel beam is smoother and the intensity of the E_x component gradually attenuates. To quantitatively analyze the influence of the opening angle to the THz vortex Bessel beam, the normalized transverse amplitude profile curves with α = 30°, 25°, 20°, 15° are extracted at z = 12 mm, 16 mm, 22 mm, 30 mm, respectively. Their positions are marked by the white dashed lines in the corresponding longitudinal amplitude patterns. The experimental results show that the radiuses of these THz light rings are 0.57 mm, 0.78 mm, 0.88 mm, 1.19 mm, respectively. To compare the non-diffractive features of these THz vortex Bessel beams, their normalized longitudinal amplitude profile curves along the z direction are extracted from their corresponding maximal amplitude positions and are plotted together in Fig. 4(a). The experimental results exhibit that the diffraction-free ranges with α = 30°, 25°, 20°, 15° are 24 mm, 32 mm, 42 mm, 59 mm, respectively. There are some oscillations on these curves, which are mainly ascribed to the machining errors of these axicons [33]. The diffraction-free ranges of these THz vortex Bessel beams can be calculated by Eq. (2) and the calculated results are Z_max = 25.3 mm, 32.8 mm, 43.3 mm, 60.0 mm with α = 30°, 25°, 20°, 15°, which basically accord with the experimental phenomena. To further compare the diffraction properties of these THz vortex Bessel beams with different α, their longitudinal E_x phase profiles are extracted and exhibited, as shown in Fig. 4(b). With reducing the value of α, the phase pattern of E_x presents slower evolution tendencies along the longitudinal and transverse propagation directions. The transverse phase profile curves with α = 30°, 25°, 20°, 15° are extracted at z = 12 mm, 16 mm, 22 mm, 30 mm and are compared in Fig. 4(b). It can be seen that the phase curve with a smaller α possesses larger flat regions on the two sides of the central phase singularity. On the corresponding amplitude maximal positions, the longitudinal phase evolution curves of these THz vortex Bessel beams along the z direction are extracted and together presented in Fig. 4(b). For clarity, the phase curves are unwrapped and their original values at z = 0 mm are set as 0. Evidently, the phase curve with a smaller α exhibits a less Gouy phase shift, because the transverse spatial confinement of the THz beam is weakened [34]. Total phase shifts with α = 30°, 25°, 20°, 15° along the z direction are 4.6π, 3.5π, 2.5π, 1.6π, respectively. In addition, the Gouy phase shifts of these THz vortex Bessel beams are almost linear in their diffraction-free ranges, which are similar with that of a THz zero-order Bessel beam [35]. Actually, it can be concluded that the axicons with different α mainly influence the phase modulation to the incident THz beams and further vary the amplitude distributions of these generated THz vortex Bessel beams.

Fig. 4 Comparison of the E_x components for the linearly polarized THz vortex Bessel beams generated by utilizing the SPP and the Teflon axicons with the opening angles of α = 30°, 25°, 20°, 15°. (a) shows the normalized longitudinal amplitude distributions of E_x with α = 30°, 25°, 20°, 15° at 0.5 THz on the x-z plane. The normalized transverse amplitude profile curves with α = 30°, 25°, 20°, 15° are extracted at z = 12 mm, 16 mm, 22 mm, 30 mm, respectively. The white dashed lines indicate the positions of these curves. The longitudinal amplitude profile curves along the z direction are also extracted from the corresponding maximal amplitude positions and are plotted together. (b) gives the wrapped longitudinal phase patterns with different αon the x-z plane. The corresponding transverse phase profile curves with different αare extracted and compared. The unwrapped longitudinal phase evolution curves along the z direction are also extracted and together plotted.

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To observe the influence of the opening angle α to vectorial properties of the THz vortex Bessel beam, propagation characteristics of the E_z components with different α are also analyzed. Figure 5(a) gives the longitudinal E_z amplitude profiles with α = 30°, 25°, 20°, 15° on the x-z plane (y = 0 mm) for the 0.5 THz spectral component. With decreasing α, the transverse wave vector components induced by the axicon progressively diminish so that the intensity of E_z gradually weakens. The transverse and longitudinal scales of the E_z light spot are pronouncedly magnified. The normalized transverse amplitude profile curves with α = 30°, 25°, 20°, 15° are extracted at z = 12 mm, 16 mm, 22 mm, 30 mm and are compared in Fig. 5(a). The FWHMs of these main spots with α = 30°, 25°, 20°, 15° are 0.7 mm, 1.0 mm, 1.3 mm, 1.6 mm along the x direction and the positions of the corresponding off-axis side-lobes are x = ± 1.1 mm, ± 1.3 mm, ± 1.6 mm, ± 2.0 mm, respectively. Figure 5(a) also exhibits the normalized longitudinal amplitude profile curves with different α along the optical axis, which manifests the diffraction-free ranges of the E_z components with α = 30°, 25°, 20°, 15° are 24 mm, 32 mm, 43 mm, 59 mm, respectively. The longitudinal phase patterns of E_z with different α on the x-z plane are also compared, as shown in Fig. 5(b). The experimental results show that the phase patterns of these E_z components have similar distribution characteristics, including the central flat region and bilateral phase screw points. The transverse phase profile curves with α = 30°, 25°, 20°, 15° are obtained at z = 12 mm, 16 mm, 22 mm, 30 mm and are compared in Fig. 5(b). It can be seen that the reduction of α enlarges the scale of the central flat region and alters the positions of the phase screw points. Besides, the unwrapped longitudinal phase evolution curves with different α along the optical axis are extracted and together plotted in Fig. 5(b). These curves show that the Gouy phase shifts of E_z with α = 30°, 25°, 20°, 15° are 5.2π, 4.0π, 2.8π, 1.8π, respectively. Taken altogether, the variation of the opening angle α has analogous effects to the E_x and E_z components. When an axicon with a smaller α is adopted, all of polarization components possess larger main light spots, longer diffraction-free ranges, and slower phase evolution tendencies for the formed THz vortex Bessel beam.

Fig. 5 Comparison of the E_z components for the linearly polarized THz vortex Bessel beams with α = 30°, 25°, 20°, 15°. (a) gives the longitudinal E_z amplitude profiles with α = 30°, 25°, 20°, 15° at 0.5 THz on the x-z plane. The normalized transverse amplitude profile curves with α = 30°, 25°, 20°, 15° are extracted and exhibited at z = 12 mm, 16 mm, 22 mm, 30 mm. The normalized longitudinal amplitude profile curves with different αalong the optical axis are obtained and compared. (b) shows the wrapped longitudinal E_z phase patterns with different α. The corresponding transverse phase profile curves with different α are extracted. The unwrapped longitudinal phase evolution curves with different αare obtained and compared.

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3.2 Circular polarization

When the incident THz beam is converted into a circular polarization, the generated THz vortex Bessel beam will carry a spin angular momentum (SAM). Then, a mutual coupling effect between the SAM and the orbital angular momentum (OAM) induced by the SPP is formed to cause some distinctive properties of the THz beam. In this work, a right circularly polarized THz vortex Bessel beam is generated by using the TQWP, the SPP, and the axicon with α = 30°. Its diffraction characteristics are observed in detail by using the same measurement method. Firstly, the amplitude and phase distributions of E_x and E_y at 0.5 THz are separately measured and presented on the position of the THz focal spot (z = 12 mm), as shown in Fig. 6. The E_x and E_y components show almost the same distribution characteristics, including the main light rings, the annular amplitude side-lobes, as well as the spiral phase patterns. Their focal spot sizes are coincident with the case with the linear polarization. In Figs. 6(a) and 6(c), it should be noted that the inhomogeneous distributions on the E_x and E_y light rings as well as the amplitude divergence of 20% between them are ascribed to the imperfect of the TQWP. In addition, it is easy to find that a phase difference of π/2 occurs between the E_x and E_y components, as shown in Figs. 6(b) and 6(d).

Fig. 6 Distributions of the E_x and E_y components for the right circularly polarized THz vortex Bessel beam generated by using the TQWP, the SPP, and the axicon with α = 30°. (a) and (b) present the normalized amplitude and wrapped phase images of E_x at 0.5 THz on the position of the THz focal spot. (c) and (d) show the amplitude and phase distributions of E_y.

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The E_z component manifests the peculiar converging property of the THz vortex Bessel beam with the circular polarization. By using the Z-scan measurement, the diffraction process of the E_z component at 0.5 THz is recorded with a 2 mm scan resolution and a 30 mm scan range. Figure 7(a) presents the longitudinal E_z amplitude profile (y = 0 mm) and transverse amplitude cross-sections at z = 4 mm, 8 mm, 12 mm, 16 mm, 20 mm. The E_z amplitude possesses a circular main spot with a 0.7 mm FWHM and ring-shaped side-lobes. From the longitudinal amplitude pattern, it can be seen that the size of E_z is non sensitive to the propagation distance and its diffraction-free range is evaluated as 24 mm. In addition, the non-uniform distributions on the main spot and side-lobes are caused by the imperfect circular polarization of the THz beam. The longitudinal E_z phase profile and transverse phase cross-sections with different propagation distances are also extracted and exhibited in Fig. 7(b). The experimental results show the E_z phase is composed of several concentric circular regions and a phase jump of π occurs on the interface between neighbor circles. The longitudinal phase pattern is almost identical to the case with the linear polarization. Along the optical axis, the Gouy phase shift of E_z approaches to 3.2π in the range from z = 0 mm to 30 mm. As well-known, if the SAM and OAM of the THz beam are carefully adjusted, the phase vortex mode excited by the circular polarization can be effectively counteracted by the spiral phase modulation of the SPP for the E_z component. Therefore, E_z of a converging circularly polarized THz vortex beam can be the same as the longitudinal electric field of a focused radially polarized THz beam [36]. Herein, the experimental results show that the E_z component of a circularly polarized THz vortex Bessel beam can achieve an analogous effect as a focused radially polarized THz beam and can maintain the non-diffractive feature at the same time. The property is very valuable for improving optical read-write techniques and confocal imaging systems. To further demonstrate this point, the E_z component of the THz Bessel beam with a radial polarization is simulated by using the vectorial Rayleigh diffraction integral [30, 37]. The E_z component can be expressed as

E_{z} (ρ, φ, z) = \frac{j k \exp (j k r) t_{p} \cos θ}{r^{2}} \int_{0}^{L} d ρ_{0} \exp (μ) ρ_{0} [- ρ_{0} J_{0} (η) + j ρ J_{1} (η)] .

Figures 7(c) and 7(d) present the simulated amplitude and phase patterns on the corresponding longitudinal and transverse cross-sections, which are in agreement with the experimental phenomena well.

Fig. 7 Distributions of the E_z component for the right circularly polarized THz vortex Bessel beam. (a) shows the longitudinal E_z amplitude profile and transverse amplitude cross-sections at z = 4 mm, 8 mm, 12 mm, 16 mm, 20 mm, respectively. (b) gives the corresponding wrapped E_z phase patterns on the longitudinal and transverse cross-sections. (c) and (d) present the simulated amplitude and phase images on the x-z and x-y planes by using the vectorial Rayleigh diffraction integral for a radially polarized THz Bessel beam.

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Interestingly, the coupling effect between the SAM and OAM can be conveniently modulated for the circularly polarized THz vortex Bessel beam. When the incident THz beam is converted into the left circular polarization by adjusting the TQWP, the topological charge of the phase vortex mode is doubled for the E_z component. Figures 8(a) and 8(b) show the amplitude and phase distributions of E_z at 0.5 THz on the position of the THz focal spot. The amplitude shows a main light ring with a 1.0 mm radius and a ring-shaped side-lobe with a 2.2 mm radius. The phase pattern exhibits two periodic 2π shifts around the optical axis on the regions of the main light ring and the side-lobe. The E_z component of the THz Bessel beam with a radial polarization and a topological charge of 2 can be written as [30, 37]

\begin{array}{l} E_{z} (ρ, φ, z) = \frac{j k \exp (j k r) t_{p} \cos θ}{2 r^{2}} \int_{0}^{L} d ρ_{0} \exp (μ) ρ_{0} \\ \times [j ρ J_{1} (η) + 2 ρ_{0} J_{2} (η) - j ρ J_{3} (η)] \exp (2 j φ) . \end{array}

Figures 8(c) and 8(d) give the simulated amplitude and phase distributions by using Eq. (8). It can be seen that the experimental phenomena confirms the theoretical simulation with a good degree of concordance.

Fig. 8 Distributions of the E_z component on the position of the focal spot for the left circularly polarized THz vortex Bessel beam. (a) and (b) give the experimental normalized amplitude and wrapped phase patterns. (c) and (d) show the simulated amplitude and phase patterns for a radially polarized THz Bessel beam with a topological charge of 2.

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Finally, the influence of the opening angle α to the E_z component is also checked for the circularly polarized THz vortex Bessel beam. By utilizing the TQWP, the SPP, and the axicons with α = 30°, 25°, 20°, 15°, the right circularly polarized THz vortex Bessel beams with different parameters are generated. The corresponding amplitude as well as phase distributions are separately exhibited on the positions of their focal spots, as shown in Fig. 9. With decreasing α, the intensity of E_z gradually attenuates and the size of the main spot progressively magnifies. The FWHMs of these main spots with α = 30°, 25°, 20°, 15° are approximately 0.7 mm, 1.0 mm, 1.3 mm, 1.6 mm, respectively. In addition, the phase pattern with a smaller α shows a central circular flat region with a larger scale because the converging process of the THz beam is smoother. Evidently, the influence of α to E_z is the same as the case with the linear polarization.

Fig. 9 Comparison of the E_z components for the circularly polarized THz vortex Bessel beams with α = 30°, 25°, 20°, 15°. (a)-(d) give the E_z amplitude distributions with α = 30°, 25°, 20°, 15° at 0.5 THz on the positions of their focal spots. (e)-(h) show the corresponding phase patterns.

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4. Conclusion

In conclusion, the THz vortex Bessel beams with linear and circular polarizations are generated by employing the TQWP, the SPP, and the Teflcon axicons with different opening angles. The vectorial diffraction characteristics of the THz vortex Bessel beams are systematically measured by utilizing the THz focal-plane imaging system, including the transverse (E_x, E_y) and longitudinal (E_z) electric fields. For the THz vortex Bessel beam with a linear polarization, the E_x component presents the ring-shaped amplitude as well as the spiral phase pattern and shows the same diffraction-free range as a THz zero-order Bessel beam. Simultaneously, the E_z component with a linear polarization shows a central elliptical main spot, off-axis crescent side-lobes, and two phase screw points. The peculiar amplitude and phase distributions of the THz vortex Bessel beam are precisely simulated by using the vectorial Rayleigh diffraction integral. Varying the opening angle of the axicon, the non-diffractive features of E_x and E_z are detailedly compared and analyzed for these generated THz vortex Bessel beams. The experimental results show that E_x and E_z with a smaller opening angle have larger main spot sizes, longer diffraction-free ranges, and slower phase evolution tendencies. For the THz vortex Bessel beam with a circular polarization, the E_z component can exhibit a circular focal spot or a higher order vortex modality in its non-diffractive zone by carefully adjusting the coupling effect between the SAM and OAM of the THz beam. In addition, the E_z component with a circular polarization also presents the attenuation of the intensity and the enlargement of the light spot size with reducing the opening angle of the axicon. In this work, the full view of vectorial diffraction properties of a THz vortex Bessel beam is accurately characterized and elaborately presented. We firmly believe that the work is valuable for the future applications of a THz vortex Bessel beam in THz imaging, THz communications, and micro-particle manipulation.

Funding

Program 973 of China (No. 2013CBA01702); National Natural Science Foundation of China (No. 11474206, 11404224, 11774243, 11374216, and 11774246); National Key R&D Program of China (2017YFF0209704); Youth Innovative Research Team of Capital Normal University; General program of science and technology development project of Beijing Municipal Education Commission under Grant No. KM201510028004; Beijing youth top-notch talent training plan (CIT&TCD201504080); Beijing Nova Program Grant No. Z161100004916100 and Scientific Research Base Development Program of the Beijing Municipal Commission of Education.

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Vectorial diffraction properties of THz vortex Bessel beams

Abstract

1. Introduction

2. Experiment

3. Results and discussions

3.1 Linear polarization

3.2 Circular polarization

4. Conclusion

Funding

References and links

Cited By

Figures (9)

Equations (8)

Optics Express