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Abstract
An effective experiment scheme is proposed to generate the terahertz (THz) perfect optical vortex (POV) beams by diffractive elements at the frequency of 0.1THz. Two diffractive elements are designed and fabricated by 3D-printing to form the generation system. The ring radius of the generated beams is independent of the topological charge and positive linear relationship with the radial wave vector. By controlling the radial wave vector, the ring radius can be freely adjusted. The experiment results are shown to corroborate the numerical simulation ones. Such generated beams hold promise for developing the novel THz fiber communication systems.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, optical vortex beams carrying orbital angular momentum (OAM) have attracted much research interest owing to their helical phase front structure [1–3]. Optical vortex beams are widely applied in many fields, such as optical communication [4], optical tweezer [5], particle manipulation [6] and quantum information [7]. Multiple OAM states of optical vortex beams can be utilized to transmit multiple data streams since the different OAM modes are mutually orthogonal [4,8]. Therefore, optical vortex beams, providing an extra degree of freedom, have the potential for improving the capacity of optical communication systems.
The spectrum range of THz waves is sandwiched between the microwave and the far-infrared. With the increasing demand for communication capacity, THz waves become a research hotspot because it provides sufficient bandwidth for high-speed wireless transmission. THz wireless technology is promising for meeting the wireless transmission of uncompressed high-definition videos at live broadcast sites [9,10]. THz vortex beams, combining the broadband and one extra degree of freedom, will further extend the capacity of wireless communication systems. In 2014, we have demonstrated several basic functionalities for THz vortex beams communications [11,12]. However, the “donut-like” ring radius of the generated THz vortex beams increases with the increase of the topological charge. This property makes it difficult to couple multiple OAM beams into a fixed radius fiber in the THz communication systems [13,14].
The same difficulty also occurred in the conventional optical vortex domain [15]. To overcome this difficulty, Ostrovsky et al. in 2013 proposed the concept of a perfect optical vortex (POV) beam and demonstrated that the ring radius of generated beams does not depend on its topological charge [16]. Many advanced applications of POV beams have emerged in the optical domain, such as microparticle trapping [17], optical communication [18], vector vortex beam [19,20] and plasmonic metasurface [21], according to its compelling property. Typically, the spatial light modulator (SLM) can upload the required amplitude and phase patterns to generate the POV beams in the optical domain. However, the practical devices are absent in the THz domain, which makes the generation of THz POV beams a challenge. Fortunately, Fedotowsky et al. in 1974 [22] proposed the concept of an optimal phase filter. Kotlyar et al. in 2016 [23] generated a POV beam by using an optimal phase element consisting of a pure phase-modulated SLM [24]. The optimal phase element has been demonstrated to be the best suited optical element for generating a POV beam, comparing with the amplitude-phase element and spiral axicon. There are many stationary methods to imitate the pure phase-modulated SLM, such as metasurface and binary zone structure, in the THz domain [25–27]. Recently, 3D printing technology has been widely applied to fabricate the complex diffractive elements [28–30].
In this paper, we extend the optimal phase element to generate THz POV beams. Two diffractive elements, namely, the optimal phase element and the Fourier transform (FT) lens, are designed and fabricated by 3D-printing to form the generation system. The “perfect vortex” property of the generated beams is demonstrated and the ring radius is controllable. Propagation dynamic of the THz beam passing through an FT lens is investigated. Coaxial interferometry is employed to verify the vortex nature of the generated beams. Mode purity of the generated beams is calculated. Numerical simulation results have been given, which are in good agreement with the experiment ones. Such THz POV beams are promising to improve the development of THz fiber communication systems.
2. Design and fabrication
Owing to the work devoted to the generation of POV beams [23], the complex amplitude of a POV beam could be generated by using three different optical elements, namely, an amplitude-phase element, an optimal phase element, and a spiral axicon. The generated POV beam has shown to provide the highest intensity and the weakest relationship with topological charge on the ring when using the optimal phase element. Thus, the optimal phase element is the best suited optical element to generate POV beams. When the collimated Gaussian beam passes through an optimal phase element and an FT lens, a POV beam can be generated in the back focal plane of the FT lens. Hence, the generation systems of POV beams contain two elements, a pure phase-modulated SLM and an FT lens, in the optical domain. In this study, the optimal phase element and FT lens are designed and fabricated by 3D-printing, which can form an optimal phase mask and perform the FT, respectively.
The transmission function of the optimal phase element can be presented in the following form,
where $(r,\varphi )$ are the polar coordinates in the beam cross section, $\textrm{sgn}(x) = 1$ at $x\;>\;0$ and $\textrm{sgn}(x) =- 1$ at $x\;<\;0$, ${J_l}$ is the $l\textrm{th}$ order Bessel function of the first kind, $\alpha $ is the radial wave vector. The phase profile of the optimal phase element is shown as follows,
Fig. 1. Phase profiles (top row), height profiles (second row), and optical pictures (bottom row) of the (a) optimal phase elements and (b) FT lens. The gray range of phase profiles is $[0,2\pi ]$. The diameter and maximum height of diffractive elements are 101.6mm and 6.997mm, respectively.
3. Experimental setup and results
Schematic diagram of the experimental setup is shown in Fig. 2. THz source is an IMPATT diode (Terasense Group Inc.) to radiate the continuous THz radiation with an output power of 80mW at a frequency of 0.1THz. Through a conical horn antenna (WR-8), the THz radiation is emitted into the free space. A high-density polyethylene lens, Lens1, is employed to collimate the THz beam. The inset of Fig. 2 displays the normalized intensity distribution of the collimated THz beam in the backplane of Lens1, which has a quasi-Gaussian intensity distribution with a diameter of beam $2{w_0} = 65.5\textrm{mm}$. The collimated beam is then divided into a transmission beam and a reflection beam by a silicon wafer beam splitter BS1. The transmission beam is directed onto an optimal phase element to impose the phase modulation. An FT lens is placed behind the optimal phase element with a distance of focal length ($f = 125\textrm{mm}$). A THz POV beam will be generated in the back focal plane of the FT lens. The reflection beam, as a reference beam, is reflected by two sliver mirrors M1 and M2, then it is combined with a THz POV beam by a silicon wafer beam splitter BS2. Interfering a THz POV beam with a reference beam can create an interference pattern. THz receiver is a Schottky diode (Virginia Diode Inc.) with an identical conical horn antenna to receive the THz radiation. The emitted THz beam is modulated by a mechanical chopper. A lock-in amplifier (SR830, Stanford Research Systems) connects the receiver and the chopper, which can amplify the dynamic range and improve the signal-to-noise ratio (SNR). The THz signal is detected by the receiver fixed at a three-axis translation stage. Pixel size and detection area are $200 \times 200$ and $200\textrm{mm} \times 200\textrm{mm}$ in $xy$-plane, respectively. In $xz$-plane or $yz$-plane, the pixel size is $200 \times 300$ and the corresponding detection area is $200\textrm{mm} \times 300\textrm{mm}$.
Fig. 2. Schematic diagram of the experimental setup for generating the THz POV beams. Lens1: high-density polyethylene lens. BS1 and BS2: silicon wafer beam splitters. M1 and M2: sliver mirrors. OPE: optimal phase element with topological charge $l = 2$ and radial wave vector $\alpha = 1.5{\alpha _0}$. The focal length of FT lens is $f = 125\textrm{mm}$. The inset: the normalized intensity distribution of the collimated THz beam.
To verify the generation of THz POV beams, the normalized intensity distributions are recorded in the back focal plane of the FT lens ($f = 125\textrm{mm}$). The experiment intensity distributions are shown in Fig. 3(a). As evident from the intensity distributions of the generated beams with topological charge ranging from $l =- 2$ to $l = 2$, the ring radius closely remains constant. A quantitative analysis is then performed to evaluate the relationship between the ring radius and the topological charge. Affected by the scattering effect of diffractive elements, the ring radius of the generated beams has a slight difference along different directions in the experiments. The variation of the ring radius with topological charge is drawn. Along $x$-axis and $y$-axis directions, the red and blue lines are shown in Fig. 3(c). As evident from two lines, the ring radius of the generated beams remains constant, with a slight fluctuation, when the topological charge changes from $l =- 2$ to $l = 2$. These results confirm that the ring radius does not depend on the topological charge. Along with the experimental parameters, the normalized intensity distributions of the generated beams are obtained by the numerical simulations based on the angular spectrum. The simulation intensity distributions are shown in Fig. 3(b). The black line of Fig. 3(c) represents the simulation variation of the ring radius with topological charge. The experiment ring radius is excellently consistent with the simulation ones. Because of the scattering effect of diffractive elements and the aberration of experiment system, the experiment intensity distributions appear a slightly broader ring and an uneven ring compared with the simulation ones. Moreover, a bright spot located at the center of the ring is inevitable because a part of the diffused beam is focused by the FT lens.
Fig. 3. (a) Experiments and (b) simulations obtained the normalized intensity distributions of the generated THz POV beams with topological charge ranging from $l =- 2$ to $l = 2$. The radial wave vector is $\alpha = 1.5{\alpha _0}$. (c) Variation of the ring radius with topological charge ranging from $l =- 2$ to $l = 2$. The red line and blue line represent experimental results along $x$-axis and $y$-axis directions, respectively. The black line represents corresponding simulation result.
To explore the relationship between the ring radius and the radial wave vector for the generated THz POV beams, the normalized intensity distributions are recorded in the back focal plane of the FT lens ($f = 125\textrm{mm}$). The experiment intensity distributions are shown in Fig. 4(a). As evident from the intensity distributions of the generated beams with radial wave vector ranging from $\alpha = {\alpha _0}$ to $\alpha = 2.5{\alpha _0}$, the ring radius increases with the increase of its radial wave vector. A quantitative analysis is performed to draw the variation of the ring radius with radial wave vector. Along $x$-axis and $y$-axis directions, the red and blue lines are shown in Fig. 4(c). As evident from two lines, the ring radius of the generated beams linearly increases, with a slight deviation, when the radial wave vector changes from $\alpha = {\alpha _0}$ to $\alpha = 2.5{\alpha _0}$. Along with the experiment parameters, the normalized intensity distributions of the generated beams are numerically simulated with the results shown in Fig. 4(b). The black line of Fig. 4(c) represents the simulation variation of the ring radius with radial wave vector. The ring radius is excellently consistent between the experiments and the simulations.
Fig. 4. The (a) experiments and (b) simulations obtained the normalized intensity distributions of the generated THz POV beams with radial wave vector ranging from $\alpha = {\alpha _0}$ to $\alpha = 2.5{\alpha _0}$. The topological charge is $l = 2$. (c) Variation of the ring radius with radial wave vector ranging from $\alpha = {\alpha _0}$ to $\alpha = 2.5{\alpha _0}$. The red line and blue line represent experimental results along $x$-axis and $y$-axis directions, respectively. The black line represents corresponding simulation result.
In order to investigate the propagation dynamics of the THz beam passing through an FT lens ($f = 125\textrm{mm}$), the normalized intensity distributions of the beam with topological charge ranging from $l = 0$ to $l = 2$ are recorded in $xz$-plane along the central axis of the beam ($y = 0\textrm{mm}$). The experiment intensity distributions are shown in the first row of Fig. 5(a). The detected range is set from $z = 25\textrm{mm}$ to $z = 325\textrm{mm}$ along the propagation direction. The reason for the setting is that the receiver may hit the FT lens when the both are too close. The THz beam passing through the FT lens clearly presents a ring intensity distribution near the back focal plane ($z = 125\textrm{mm}$) in which a THz POV beam can be generated. The generated POV beam then degenerates into a Bessel-Gaussian beam with corresponding vortex order when the beam propagates away from the back focal plane. As the experiment intensity distributions shown in the second row of Fig. 5(a), the normalized intensity distributions of the beam with topological charge ranging from $l = 0$ to $l = 2$ are recorded in $xy$-plane at the position of $z = 300\textrm{mm}$. The intensity distributions of these beams are evidently similar to the ones of the Bessel-Gaussian beams with corresponding vortex order. The similarity further confirms that the POV beam will degenerate into a Bessel-Gaussian beam with corresponding vortex order when the beam propagates away from the back focal plane. The variation of the ring radius with propagation distance is drawn in the range of $z = 100\textrm{mm}$ to $z = 150\textrm{mm}$. The topological charge ranges from $l = 0$ to $l = 2$. The corresponding curves are shown in the end row of Fig. 5(a). These curves reveal that the ring radius of these beams closely remains constant over a long propagation distance near the back focal plane. Hence, the generation systems proposed in this investigation have excellent tolerance. Along with the experimental parameters, the numerical simulation results are shown in Fig. 5(b), in close agreement with the experimental ones. Comparing the experiments and the simulations, a slightly broad beam, an imperfect main lobe, and a slightly flat curve successively appear in all rows of Fig. 5(a), which can be introduced by the scattering effect of diffractive elements and the aberration of experiment system.
Fig. 5. (a) Experiments and (b) simulations obtained the propagation dynamics of the THz beam passing through the FT lens. The range of topological charge is from $l = 0$ to $l = 2$ and radial wave vector is $\alpha = 1.5{\alpha _0}$. The first row represents the normalized intensity distributions in $xz$-plane at $y = 0\textrm{mm}$. The second row represents the normalized intensity distributions in $xy$-plane at $z = 300\textrm{mm}$. The end row represents the variation of the ring radius with propagation distance in the range of $z = 100\textrm{mm}$ to $z = 150\textrm{mm}$.
There are many methods to discriminate the topological charge of vortex beams, such as triangular aperture diffraction [31], double-slit interferometry [32], tilted interferometry [33] and coordinate transformation [34]. Coaxial interferometry [35], which is a simple and effective method, is chosen to verify the vortex nature of the generated THz POV beams. The brief experimental setup of the coaxial interferometry is shown in Fig. 2. By interfering a generated beam and a reference beam, an interference pattern is recorded in the back focal plane of the FT lens ($f = 125\textrm{mm}$). As shown in Fig. 6(a), the experiment interference patterns with topological charge ranging from $l =- 2$ to $l = 2$ are the spiral fringe patterns that can discriminate the topological charge of the generated THz POV beams. The number of the fringes represents the modulus of topological charge. The spiral direction represents the sign of topological charge. Along with the experiment parameters, the simulation interference patterns are shown in Fig. 6(b), which are in a close agreement with the experiment ones. The slight mismatch appears in the center of the interference patterns between the simulations and the experiments. The reason for the mismatch is that the 3D printing material has some absorption. When the beam passes through two diffractive elements with absorption, the intensity of the generated beam becomes weaker than the reference beam. According to the simulation, we found that a bright spot will appear in the center of the interference pattern when interfering a weaker generated beam and a reference beam.
Fig. 6. (a) Experiments and (b) simulations obtained the interference patterns of topological charges ranging from $l =- 2$ to $l = 2$ in $xy$-plane. The radial wave vector is $\alpha = 1.5{\alpha _0}$.
To characterize the quality of the generated THz POV beams, the simulation phase patterns are acquired along with the parameters of Fig. 3. Phase patterns of the generated THz POV beams with topological charge ranging from $l =- 2$ to $l = 2$ are shown in Fig. 7(a). Complex amplitude distributions of the generated beams can be obtained by combing the intensity patterns of Fig. 3(a) and the phase patterns of Fig. 7(a). Spiral spectra of the complex amplitude distributions are then calculated [36]. The calculation results are shown in Fig. 7(b). The longitudinal coordinate represents the weight of all spiral mode. The mode purity of input OAM or topological charge is equal to its weight in the spiral spectra. As evident from the figure, the mode purity of all input OAM or topological charge close to 1. These results indicate that the generated THz POV beams have high-quality mode purity.
Fig. 7. (a) Phase patterns and (b) spiral spectra of the generated THz POV beams with topological charge ranging from $l =- 2$ to $l = 2$.
4. Conclusion
In summary, we have demonstrated the effectiveness of the experiment scheme for generating the THz POV beams at 0.1THz. Two kinds of diffractive elements contained in the generation system are designed and fabricated by 3D-printing. The ring radius of these generated beams is independent of the topological charge and linearly increases with the increase of the radial wave vector. Therefore, the required THz POV beams can be generated by controlling the magnitude of the radial wave vector. Next, the ring radius of the generated beams remains unchanged over a long propagation distance ($z = 100\textrm{mm}$ to $z = 150\textrm{mm}$) near the focal plane, which indicates the generation systems have excellent tolerance. Finally, the vortex nature of the generated beams is verified by the coaxial interferometry. The mode purity of all input OAM or topological charge close to 1 based on the simulation calculation. The experiment results are in excellent accord with the simulation ones. These generated beams have potential applications for coupling multiple OAM modes into a fixed radius fiber, which will further enlarge the capacity of THz fiber communication systems.
Funding
National Natural Science Foundation of China (11574105).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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