We present an efficient method to discriminate orbital angular momentum (OAM) of the terahertz (THz) vortex beam using a diffractive mode transformer. The mode transformer performs a log-polar coordinate transformation of the input THz vortex beam, which consists of two 3D-printed diffractive elements. A following lens separates each transformed OAM mode to a different lateral position in its focal plane. This method enables a simultaneous measurement over multiple OAM modes of the THz vortex beam. We experimentally demonstrate the measurement of seven individual OAM modes and two multiplexed OAM modes, which is in good agreement with simulations.
© 2016 Optical Society of America
The orbital angular momentum (OAM) has been recognized as a fundamental property of electromagnetic (EM) waves, attracting many research interests in optical fields [1–3]. It is not until 1992 that Allen et al. experimentally demonstrated Laguerre-Gaussian (LG) beams carry well-defined OAM modes . LG beams with helical phase structure are always related to azimuthally dependent phase eilφ, in which φ is the transverse azimuthal coordinate and l is the OAM quantum number. The fact that l can take any integer value gives an unbounded state space, which can be potentially utilized to encode information [5–7]. OAM modes with different l values are mutually orthogonal to each other, allowing them to be used as different carriers for multiplexing and transmitting multiple data streams along the same spatial axis. These features of OAM are very promising to improve the performance of communication scenarios including free-space optical communications, fibre-optic communications, and RF communications.
Indeed, OAM has seen exciting progress with an achievement of transmission capacity up to 1 Pb/s in the optical domain [8–10]. These experiments unequivocally demonstrate the potentials of using OAM multiplexing in both free-space and special vortex fibre . Although most studies of the OAM are associated with the optical domain, it is also of great interest to employ such advanced multiplexing approaches at millimeter and THz band [12–14]. THz beams can be a suitable candidate for information carrier of the future wireless communications due to its high frequency. THz vortex beam would further enhance the future wireless communications with an additional degree of freedom. However, given the different frequency range from optical domain, THz OAM-based communication systems would require different technologies. Efficiently and unambiguously discriminating the OAM modes is one key part among all these required technologies.
Diffractive grating and hologram containing an l-fold dislocation can be used to detect single corresponding OAM mode . A series of projection measurement are capable of detecting the OAM content of light , but projection measurement for N modes requires at least N photons. A more sophisticated hologram design is able to separate several OAM modes using a single phase-screen . Similar techniques using spiral phase plates (SPPs) and q-plates also demonstrated the ability of detecting OAM modes [18, 19]. Nevertheless, none of them has proven to be efficient enough for separating a large number of possible OAM modes at the single-photon level. A method for separating OAM modes at single-photon level has been demonstrated by Leach et al. , but it is a great challenge for building multiply cascaded Mach-Zehnder interferometers. Overall, above-mentioned methods are unfeasible or inefficient when scale for applications in THz OAM-based communication systems.
Recently, it has been demonstrated that two custom refractive elements in conjunction with a lens can spatially separate different OAM modes [21–26]. The first element, the reformatter, transforms azimuthal position of the input beam into a transverse position of the output beam. A second element, the corrector, compensates the residual phase aberration of the output beam arising from variations in the optical path length. Thus, optical waves with helical wavefronts are transformed into tilted plane waves which are separated by a single lens in its focal plane. This method is substantially more efficient than previous methods for discriminating OAM modes, which has potential application for THz OAM-based communication system. Yan Yan et al. have performed a 28 GHz OAM-based communication system by taking advantage of this method . However, such refractive elements are also unfeasible to discriminate OAM modes when scale into the THz regime. In order to discriminate OAM modes of the THz vortex beam with this method, two aspects should be noted: (1) the THz regime lacks practical devices to introduce complex phase patterns ; (2) presently available continuous-wave (CW) THz sources provides only tens mW output power , while THz wave considerably attenuates when travelling through atmosphere and penetrating common materials .
In this paper, we extend the log-polar coordinate transformation method to discriminate seven individual OAM modes and two multiplexed OAM modes. Two diffractive elements are designed and printed to transform OAM modes into transverse momentum modes. Specifically, we propose a novel method to design the corrector which shows a good performance in cooperation with the reformatter.
2. Design and fabrication
Various approaches have been proposed to generate optical beams carrying OAM , but very few of them are practical and efficient in the THz regime. In our previous work [30, 31], a 3D-printed SPP is used to generate a THz vortex beam. A typical SPP has a azimuthal height profile of lλφ/[2π(n-1)], where n is the refractive index of the medium, and λ is the wavelength of the THz beam, thus a spiral phase structure could be imprinted onto the THz beam . The 3D printing technology is a flexible way to fabricate complex element in THz range. In this paper, we use SPPs to produce THz vortex beams experimentally, and extend the 3D printing technique to fabricate the two elements of the OAM mode transformer. In the design process of the corrector, the two elements of the transformation system can be treated as phase-only transmittance functions.
Owing to the efforts devoted in the coordinate transformation method [21–26], the phase profile of the reformatter can be given by
To compensate the residual phase aberration in the corrector plane, the required phase correction is calculated by the stationary phase approximation [25, 26]. Alternatively, it can be obtained via the angular spectrum method. For such a transformation system, a fundamental-mode Gaussian beam with a planar phase front acts as the input. Then the electric field distribution of the Gaussian beam passing through the reformatter is derived as
Due to the lack of practical devices, introducing dynamic phase pattern is difficult to realize in the THz regime. However, imprinting static phase pattern onto the THz beam is available with suitable methods. When passing through a material with thickness of h, the phase shift of the light can be calculated as ΔΦ = 2π (n-1) h/λ, where n is the refractive index of the material. Hence, the refractive OAM mode transformer requires height profiles of
From Eq. (5), one can deduce that the maximum height of the reformatter and the corrector using in the THz regime is about 25λ and 200λ, respectively. In consideration of the size and material absorption, such refractive elements are unpractical in the THz regime. Similar to the Fresnel lens, the phases of the Eq. (1) and Eq. (4) can be wrapped modulo 2π. Hence, the height of the diffractive OAM mode transformer is acquired by the refractive ones modulo λ/(n-1). In addition, a base of height h0 = 2 mm is added to the diffractive OAM mode transformer. The two diffractive elements actually requires height profiles of
After passing through the mode transformer, the THz vortex beam is focused by a single lens of focal length f. The focal spot is produced at an l-dependent horizontal position given by
Figure 1(a) and 1(c) show the CAD models of the diffractive OAM mode transformer derived from Eq. (6). The two diffractive elements were fabricated via an Objet 3D printer, with parameters d = 76.2 × 0.8 mm and L = 50 mm. The 3D printer had a printing resolution of 600 dpi (42 μm) in the xy-plane and 900 dpi (28 μm) along the z-axis. The 3D printing material is a rigid opaque material (VeroWhitePlus), whose optical properties were characterized with a Zomega-Z3 THz time-domain spectrometer (THz-TDS). At the frequency of 0.3 THz, the refractive index and the absorption coefficient of the printing material are about 1.655 and 1.5 cm−1, respectively. The fabricated diffractive elements are depicted in Fig. 1(b) and Fig. 1(d).
3. Experimental setup and results
A diagram of the experimental setup is shown in Fig. 2. In our experiment, the THz transmitter is a Gunn-diode (Spacek Lab Inc.) driven multiplier-chain (Virginia Diode Inc.) delivering CW radiation at a frequency of 0.3 THz with an output power of 0.3mW. The THz radiation is emitted into the free space through a WR-3.4 diagonal horn antenna and perpendicular to the underlying optical table. A HDPE lens, Lens1, is used to collimate the THz beam. The collimated THz beam is then directed onto a SPP with l = 3 for generating the desired THz vortex beam. The OAM mode transformer maps the input THz vortex beam with azimuthal phase gradient into a plane wave with a transverse phase gradient. Subsequent focusing, Lens2, produces a lateral spot in its focal plane. A Schottky-diode (Virginia Diode Inc.) in combination with an identical diagonal horn antenna is used as the THz receiver. The incident THz beam is modulated by an optical chopper in front of the transmitter, and a lock-in amplifier (SR830, Stanford Research Systems) is utilized for detecting the induced photocurrent in the receiver. The THz receiver is mounted on a three-axis stage for detection at various positions with a detection volume of 90 × 90 × 300 mm. All the detected images are obtained in the xy-plane, with 181 × 181 pixels in a square area of 90 × 90 mm.
Figure 2(a) illustrates the normalized intensity profile of the collimated THz radiation, showing a Gaussian-shaped intensity profile with a diameter 2ω0 = 16 mm (measured at FWHM). The Gaussian beam was then directed onto a SPP, and Fig. 2(b) clearly depicts the ring-shaped normalized intensity profile of the generated THz vortex beam with l = + 3. To observe the log-polar coordinate transformation caused by the reformatter, six images of the transforming beam in the xy-plane with 10 mm interval between adjacent images, are captured in Fig. 2(c). The result coincides with the theoretical prediction, and a ring-shaped intensity structure is gradually transformed into a straight intensity line. The corrector is placed a distance L = 50 mm behind the reformatter to remove the residual phase aberration. The transformed beam was focused by Lens2 with a focal length f = 100 mm, producing an elongated lateral spot shown in Fig. 2(d) in its focal plane.
We experimentally investigated the performance of the discriminating system through a serial detection of individual OAM modes of the THz vortex beam. Figure 3(a) shows the normalized intensity profiles of the input OAM modes generated by SPPs with l ranging from −3 to + 3. The experimental results and simulated results of the focal spot are depicted in Fig. 3(b) and Fig. 3(c), respectively. As can be seen, the focal spot moves in the direction orthogonal to the direction in which the spot is elongated. The simulated results are calculated by scalar diffraction theory, and one can see that the experimentally detected spots are slightly broader than the simulated ones, which is due to aberration introduced by the experimental system. The positions of the spots produced by incoming OAM modes ranging from l = −3 to l = + 3 are defined as the maximum intensity positions, as depicted in Fig. 4(a). The experimental spot positions (blue dots) agree well with the predictions (black curve), calculated from Eq. (7). The error bar with a value of ± 0.5 mm refers to the scanning resolution in the y direction.
To directly measure the OAM content of the THz vortex beam, we define seven, equally sized, rectangular regions in the detector plane, all centered around one of the expected spot position for the OAM modes between −3 to + 3. By sending a known OAM mode through the system and measuring the total normalized intensity in the seven regions, we can calculate the OAM content matrix for the detection of seven individual OAM modes. Figure 4(b) and 4(c) show the OAM content matrix for detecting seven pure individual OAM modes ranging from −3 to + 3 in experiment and simulation, respectively. For a single input mode, the majority energy of the detected spot in the rectangular region corresponds to the input OAM mode. From the distribution of the spot in these rectangular regions, it is clearly possible to unambiguously determine single input OAM mode of the THz vortex beam. As described before, the experimental spot is slightly broader than the simulated ones, the crosstalk between the neighboring detected spots becomes larger and hence the off-diagonal elements in Fig. 4(b).
Based on the discrimination results of individual OAM modes in Fig. 4(b), one can increase the separation between OAM modes to minimize the crosstalk. In other words, using every other mode, ∆l = 2, as the input, the average crosstalk value of ± 1 and ± 3 channels can be about −6 dB. The crosstalk value is a bit high when we consider it in a communication system, which could be due to the setup misalignment. When applied in the practical communication systems, the transformation system should be mounted in a cage system with fine position and rotation controls. Moreover, the increase of the channel separation would further decrease the crosstalk value. We believe that this technique is promising to demultiplex multiple OAM channels in a practical communication system by choosing proper channel separation.
Our system also allows us to discriminate a superposition of OAM modes. With our previous work in multicasting OAM modes , two multiplexed modes each comprising two individual modes were generated, as shown in Fig. 5(a), and directed through the OAM mode transformer. As predicted, each multiplexed mode gives two separated spots in the detector plane, accompanied with the simulated results. There is a very good agreement between the experimental and simulated positions, illustrating that the OAM mode transformer is capable of simultaneously measuring the OAM content of the THz vortex beam.
In this work, we have demonstrated a method for efficiently and unambiguously discriminating OAM modes of the THz vortex beam. The log-polar coordinate transformation is implemented with two 3D-printed diffractive elements, and input OAM modes are transformed into a set of tilted plane waves which are then separated by a single lens. Seven individual OAM modes and two multiplexed OAM modes are generated and separated in the detector plane, which are in good agreement with the simulations. Specifically, we have proposed an alternative method for designing the corrector, showing a good performance in combination with the reformatter. In principle, this method can be scaled to arbitrary frequency range in which the scalar diffraction theory is applicable. The OAM mode transformer could be used to efficiently (de)multiplex OAM channels of a THz wireless communication system.
This research is supported by the Wuhan applied basic research project (No. 20140101010009), the National Natural Science Foundation of China (Nos. 11574105,61475054, 61405063), the Fundamental Research Funds for the Central Universities (Nos. 2014ZZGH021, 2014QN023), Technology Innovation Foundation From Innovation Institute of Huazhong University of Science and Technology (Nos. CXY13Q015, CX14-070).
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