1. Introduction

Recently, nondiffraction beams including Bessel beams and Airy beams have attracted many research interests due to their intriguing properties [1–4]. Bessel beams, perhaps the best known diffraction-free beam, have been studied since Durnin et al. firstly proposed and realized in 1987 [1]. However, it is not until 2007 that the finite-energy (power) optical Airy beams were experimentally demonstrated [4]. In 1979, Berry and Balazs firstly observed a nonspreading Airy wave function which is a solution of the quantum mechanical Schrodinger equation [5]. Considering the mathematical correspondence between paraxial equation of diffraction and the Schrödinger equation, Siviloglou et al. demonstrated the accelerating optical Airy beams in 2007 [4, 6].

The accelerating Airy beams possess the well-known ability of maintaining diffraction-free while propagating along a curved parabolic trajectory in the free space [7]. Additionally, the Airy beams exhibits a self-healing feature [8]. Based on these unique features, the Airy beams can possibly have promising applications over numerous challenging fields, such as optical routing [9], particle manipulation [10], light bullets [11], microscopy [12] and plasma physics [13]. Although most studies of the accelerating Airy beams are associated with the optical domain, there are a lot of ingenious implementations of the Airy beams in other fields, such as acoustics [14] and electron-beam [15] fields. Here we show great interest to demonstrate such Airy beams in the THz domain.

In general, the accelerating optical Airy beams are based on dynamic phase modulation devices, mostly are spatial light modulators (SLM), with which it can be very convenient to introduce the required complex phase pattern [16–19]. However, the input photons are substantially lost due to the diffraction efficiency of the SLMs. Thus, Yuan et al. proposed a high efficiency method to generate optical Airy beam using a microfabricated cubic phase plate [20]. Unfortunately, THz domain lacks practical devices to introduce complex phase patterns, which makes the implementation of the accelerating THz Airy beam a challenge [21, 22]. Based on our previous work in the manipulation of the THz beams [23–25], we believe the 3D printing technology could be an efficient and low-cost candidate to introduce the required complex phase patterns in the generation of accelerating THz Airy beams.

In this work, we generate the accelerating THz Airy beam with two 3D-printed diffractive elements, which cannot only imprint the desired complex phase pattern but also perform the required FT. The generated accelerating THz Airy beam propagates along a curved trajectory and obtains a diffraction-free distance of more than 100 mm. In addition, we experimentally investigate the self-healing property of the generated accelerating THz Airy beam. Numerical simulations are also performed and agree well with the experimental results.

2. Concepts and fabrication

Since the ideal Airy beam consumes an infinite space and energy, it is necessary to introduce a truncation function to make it realizable in the experimental environment. Thanks to the great efforts devoted in the research of the finite-energy Airy beams [4,6], an exponential truncation function can be introduced to truncate such an ideal Airy beam. As a result, the initial condition of the truncated Airy beam becomes ϕ(0, s) = Ai(s)exp(as), where a is the truncation factor and s is a dimensionless coordinate. By performing the FT of such an initial condition, one can obtain its Fourier spectrum in the normalized k-space:

From the deduction above, the usual generation system of the accelerating optical Airy beam contains two elements, mostly are a SLM and a FT lens. The SLM is modulated to provide a cubic phase term b(x³ + y³), where b is the scaling factor of the cubic phase term. The usual system possesses a system length of 2f, where f is the focal length of the FT lens. In consideration of the fact that THz wave may dramatically attenuate in the free space propagation, we integrated the FT lens into the input plane and output plane of the usual generation system. As a result, a compact system consuming only the half space of the usual one is developed while avoiding the raise of the element number. The compact generation system is composed of two elements, Element1 (E1) and Element2 (E2), whose phase profiles are as follows,

As we have mentioned the lack of practical devices to manipulate THz beam, introducing dynamic phase pattern is difficult to realize in the THz domain. However, imprinting static phase pattern onto the THz beam is available with 3D printing technology [25], which is due to the equivalency between phase shift and optical path. Specifically, when passing through a uniform material with thickness of h, the phase shift of the light can be calculated as ΔΨ = 2π(n-1)h/λ, where n is the refractive index of the material. Furthermore, to maximally diminish the material absorption [26], the refractive elements can be wrapped modulo λ/(n-1). Hence, the wrapped elements, namely diffractive elements of such a compact generation system requires height profiles of

Figures 1(a) and 1(c) show the CAD models of E1 and E2 derived from Eq. (3). The two elements were fabricated via an Objet 3D printer, with a scaling factor b = 0.009 and f = 50 mm. The 3D printer has a printing resolution of 600 dpi (42 μm) in the xy-plane and 900 dpi (28 μm) along the z-axis. A rigid opaque material (VeroWhitePlus) is chosen as the 3D printing material, 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 3D-printed diffractive elements of E1 and E2 are depicted in Figs. 1(b) and 1(d).

 

Fig. 1 Height profiles (a, c) and photos (b, d) of the E1 (top) and the E2 (bottom). The diameter of them is 76.2 mm, focal length f = 50 mm and b = 0.009.

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We note that, even though at first sight, the E1 appears to look like a normal cubic phase plate. Indeed, E1 contains not only the required cubic phase term but also a lens term, resulting in a slight height profile change compared with the normal cubic phase plate. As mentioned above, E1 imprints the cubic phase into the incident Gaussian beam while introducing an additional phase aberration to perform the desired FT. When the FT of the imprinted beam fulfills at the plane of E2, the transformed beam still involves a residual phase aberration. Thus, E2 is implemented to correct the residual phase aberration. With such a compact system, the generation of the finite-energy accelerating THz Airy beam would occur right behind E2.

3. Experimental setup and results

A schematic 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 continuous wave radiation at a frequency of 0.3 THz with an output power of 0.3 mW. 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 high-density polyethylene (HDPE) lens is used to collimate the THz beam. A Schottky-diode (Virginia Diode Inc.) in combination with an identical diagonal horn antenna serves as the THz receiver. The transmitted THz beam is modulated by an optical chopper in front of the transmitter, and a lock-in amplifier (SR830, Stanford Research Systems) is utilized to detect the induced photocurrent in the receiver. The THz receiver is mounted on a three-axis translation stage for detection at various positions with a maximum detection volume of 90 × 90 × 300 mm.

 

Fig. 2 Schematic diagram of the experimental setup for generating accelerating THz Airy beam. Transmitter: Gunn diode coupled with a diagonal horn antenna. Receiver: Schottky diode coupled with a diagonal horn antenna. HDPE Lens: High-density polyethylene (HDPE) lens. Element1 (E1) and Element2 (E2): 3D printed diffractive elements. The normalized intensity profile of the collimated Gaussian beam at the front plane of E1 is depicted as the inset.

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The inset of Fig. 2 illustrates the normalized intensity profile of the incident THz beam, showing a Gaussian-shaped intensity profile with a diameter 2ω₀ = 17 mm (measured at full width at half maximum, FWHM). The Gaussian beam is then directed onto the compact generation system, including E1 and E2, as shown in Fig. 3. All the captured images obtained in the xy-plane have a pixel size of 181 × 181 and a square detecting area of 90 × 90 mm. The pixel size and detecting area of the images in the xz-plane are 181 × 301 and 90 × 300 mm, respectively.

 

Fig. 3 Experimental (left) and simulation (right) results of the finite-energy accelerating THz Airy beam are illustrated. The top row indicates the xz normalized intensity profiles of the Airy beam. (b)-(e) The xy normalized intensity profiles of the experimentally generated Airy beam measured at the dash line positions of the upper image. (g)-(j) The corresponding simulation results at the same detection planes.

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In order to investigate the generated accelerating THz Airy beam, we demonstrate, both experimentally and numerically, the intensity profiles of the beam in the xz-plane and xy-plane. As depicted in Fig. 3, the top row illustrates the propagation dynamics of the generated Airy beam in the xz-plane. In accordance with theoretical prediction, the observed Airy beam shows the diffraction-free ability with a nondiffraction distance of more than 100 mm. Moreover, the main lobe of the observed beam accelerates along x axis, resulting in a curved trajectory. Since the Airy beam can be considered as a caustic of ray emerging sideways from points that are located far away from the main lobe in the E1 plane [27, 28], this could be able to simply explain these properties above. However, the generated Airy beam did not show the diffraction-free and accelerating abilities as well as the simulated ones, which could be due to the system misalignment and the attenuation of the THz beam in the free space. On the other hand, we consecutively investigated the xy intensity profiles of the generated Airy beam in the propagation direction. We firstly assumed the back plane of E2 as the origin plane, namely z = 0 mm plane, along the propagation direction. Then four xy normalized intensity profiles of the generated Airy beam with propagation distances of z = 0 mm, z = 50 mm, z = 100 mm, z = 150 mm are captured and depicted in Figs. 3(b)-3(e). Clearly, the generated beam shows an accelerating Airy beam behavior and its main lobe keeps the shape with a distance of more than 100 mm. Although the generated Airy beam divergents at z = 150 mm, as shown in Fig. 3(e), it still retains the typical intensity contour.

In addition to the experimental demonstration, a beam propagation method based on the angular spectrum is employed to numerically investigate the propagation dynamics of the Airy beam. The corresponding numerical simulations are demonstrated in the right part of Fig. 3. Both the xz-plane and xy-plane intensity profiles of the Airy beam are calculated with the angular spectrum formulas. It is obvious that the numerical simulation results agree well with the experimental results.

Beside the two intriguing properties we mentioned above, the accelerating Airy beam can exhibit a self-healing property. To investigate such a property, we exploited a metal segment (shown as the inset of the Fig. 4(a)) to partially block the main lobe and the inner lobes of the generated Airy beam at the plane of z = 0 mm. Figure 4(a) shows the propagation dynamics of the partially blocked beam in xz-plane. Figures 4(b)-4(d) corresponds to the xy normalized intensity profiles measured at the dash line positions of the upper image, namely at z = 0 mm, z = 50 mm, z = 100 mm. Obviously, the intensity breach around the dash line (a) in the upper image indicates that the beam was partially blocked. Referring to the image in Fig. 3(b), one can easily find out that the main lobe of the generated Airy beam in Fig. 4(b) was distorted by the insertion of a metal segment. However, in the comparison of the Fig. 3(c) and Fig. 4(c), one can find out that the main lobe of the partially blocked Airy beam almost reconstructs to its shape. Meanwhile, the blocked beam retains the nondiffraction and accelerating properties during propagation. Such self-healing property can be interpreted via a simple geometrical optics model [29,30], in which one can infer that the self-healed main lobe is generated by the uncovered beams in Fig. 4.

 

Fig. 4 The experimental (left) and numerical (right) investigation of the self-healing property of the generated accelerating THz Airy beam. The upper row displays the xz normalized intensity profile of the partially blocked beam. The inset of the Fig. 4(a) illustrates the block situation in detail. (b)-(d) The experimental xy normalized intensity profiles of the beam at z = 0 mm, z = 50 mm, z = 100 mm. (f)-(h) The corresponding simulated results.

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Corresponding numerical simulations are also demonstrated in the right part of Fig. 4 and show a good agreement with the experimental results. The simulations are still based on the angular spectrum method, in which E1, E2 and the metal obstacle are treated as transmittance functions with zero thickness. As we notice that the experimentally generated beam behaves a significant distortion behind the plane of z = 100 mm, this could be caused by the system misalignment and the accuracy in the placement of the metal segment.

4. Conclusion

In this paper, we have demonstrated the accelerating THz Airy beam at the frequency of 0.3 THz. Two 3D-printed diffractive elements are exploited to introduce the desired complex phase pattern and perform the FT. In accordance with the theoretical prediction, the generated Airy beam propagates along a curved trajectory and obtains a diffraction-free distance of more than 100 mm. Moreover, the Airy beam can self-reconstruct during propagation even though the insertion of significant distortion by a metal obstacle. Numerical simulations are also performed and show a good agreement with the experimental results. We believe that the accelerating THz Airy beam may pave the way to improve THz imaging systems and build robust THz communication links.