1 Introduction

Light diffraction can cause problems in many applications, for example, optical transmission of information and photolithography [1, 2]. As a result, non-spreading or diffraction-free waveforms are of interest in a number of fields and have been investigated theoretically and experimentally [3].

One typical example of a diffraction-free optical wave is the Bessel beam, which inspired the discovery of many other interesting diffraction-free beams [4, 5]. The Airy beam is an asymmetric example of a non-spreading wave that was first theorized in 1979 as having unique features, such as self-acceleration along the propagation path without any external potential [6]. In 2007, an Airy beam was successfully demonstrated in the laboratory by modulating a Gaussian beam with a cubic phase, using a computer-controlled liquid crystal based spatial light modulator (LC-SLM) [7]; this approach attracted significant attention [8–11]. Many modulation methods have since been developed to reconstruct an Airy beam from a Gaussian beam, including a continuous transparent phase mask [12] and asymmetric non-linear photonic crystals [13]. In general, it is not easy to fabricate a mask with continuous phase modulation due to the high cost and complicated fabrication process [14].

In this paper, we report a new method based on a digital micromirror device (DMD) to generate an Airy beam. Compared to an LC-SLM, a DMD provides an alternative way to generate special beams having certain advantages. It has been shown that a DMD can project up to seven times more object light than an LC-SLM [15]. Furthermore, the fill factor of a DMD is around 90 %, whereas that of an LC-SLM is less than 70 %, which means that pixels in a DMD are packed closer and could lead to higher resolution [16]. In addition, a DMD provides a wider dynamic range of grayscale [17]. Since a DMD is programmable, holograms could be changed quickly and easily, which is valuable for tunable optical beam shaping [18]. Because of these advantages, a DMD is an ideal holographic media and spatial light modulator and has been used to replace the LC-SLM to generate Laguerre–Gaussian beams [19] and to replace an axicon lens to generate Bessel beams [20].

The normalized paraxial equation of diffraction is shown in the following:

$$i\frac{\partial \phi }{\partial \xi } + \frac{1}{2}\frac{{\partial^{2} \phi }}{{\partial^{2} s}} = 0$$
(1)

where s = y/y 0 is a dimensionless transverse coordinate, y 0 is an arbitrary transverse scale, ξ = z/ky 20 is a normalized distance in the direction of propagation, k = 2πn/λ 0 is the wavenumber [21]. Finite power Airy beams can be obtained by introducing an exponential decaying function:

$$\phi \left( {s,\xi = 0} \right) = Ai(s){ \exp }(\alpha s)$$
(2)

where α is the decay factor to limit the infinite Airy tail and enable the physical realization of such beams. In this paper, the value of α is 0.024.

The angular Fourier spectrum of the finite power Airy beam is shown in the following equation:

$$\varPhi_{0} \left( k \right) = \exp \left( { - \alpha k^{2} } \right)\exp \left( {\frac{i}{3}\left( {k^{3} - 3\alpha^{2} k - i\alpha^{3} } \right)} \right)$$
(3)

where k represents vector in the normalized k-space [21]. It shows that an Airy beam can be generated by modulating a Gaussian beam with a cubic phase.

2 Experimental setup

The experimental setup is illustrated in Fig. 1, where DMD (Texas Instruments) is used to display computer-generated holograms. The inserts of Fig. 1 are illustrations of holograms with slight distortions to show the fine difference. These holograms are interference patterns of finite power Airy beams with unitary plane waves in one-dimensional (1D) and 2D space. A 1-mW, 635-nm diode laser (CPS180, Thorlabs) was collimated and used as a monochromatic light source. Focus lenses L1 and L2 (focal length of 5 cm and 20 cm, respectively) worked as beam expanders that expanded the diameter of the laser beam from 0.4 cm to 1.6 cm. The expanded laser beam was modulated by a DMD loaded with the appropriate hologram and reflected to a converging lens (focal length of 20 cm) to perform Fourier transforms. Finally, the generated Airy beam was recorded with a CMOS camera (DCC1645C, Thorlabs) placed in the vicinity of the rear focal plane of the converging lens.

Fig. 1
figure 1

Illustration of the experimental setup used to reconstruct an Airy beam by a digital micromirror device. The insert shows computer-generated holograms for reconstructing a 1D and b 2D Airy beams

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3 Results and discussion

In order to evaluate the diffraction-free property of the generated Airy beam, we defined the camera position as Z = 0 cm when the camera was located at the focal plane of the converging lens, i.e., 20 cm from the converging lens, and Z = 5 cm when the camera was 25 cm from the converging lens. In Fig. 2, we can see that the width of the Gaussian beam increased significantly from Z = 0 to Z = 5 cm (Fig. 2a, b), whereas the width of the generated Airy beam remained constant (Fig. 2c, d), i.e., non-diffraction. The center of the generated Airy beam shifted significantly (Fig. 2c, d), which represents another special property of an Airy beam—self-acceleration property.

Fig. 2
figure 2

Intensity profiles of a Gaussian beam at a Z = 0 cm and b Z = 5 cm. Intensity profiles of an Airy beam at c Z = 0 cm and d Z = 5 cm

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More details of the self-acceleration property can be seen in Fig. 3, which shows the generated Airy beam profiles recorded at propagation distances of Z = 0, 3, and 5 cm (Fig. 3a, c, e, respectively). Theoretical calculations of Airy beam profiles in the same situations were also conducted (Fig. 3b, d, f, respectively). The experimental results agree well with the theoretical calculations in terms of both beam profiles and deflections [22]. For example, the experimental results showed that deflection of the generated Airy beam was about 0.2 mm from Z = 0 to Z = 5 cm, which is the same as the theoretical calculation.

Fig. 3
figure 3

Intensity profiles of the generated Airy beam at a 0 cm, c 3 cm, and e 5 cm. The center of the main lobe shifts to the right with increasing propagation distance, while the distance of two adjacent lobes remains constant. Theoretical calculation results at the same propagation distance are shown in b, d, and f, respectively

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Figure 4 shows deflections of the generated Airy beam versus propagation distance. Theoretical prediction of deflection y(Z) is calculated using the following equation [7]:

$$y\left( Z \right) = \frac{{Z^{2} }}{{4k^{2} y_{0}^{3} }}$$
(4)
Fig. 4
figure 4

Deflection of the generated Airy beam as a function of propagation distance. Solid dots represent experimental results, while the solid line is the theoretical prediction

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The experimental data in general agree well with the theoretical predictions, while agreement in the near field is not as close as in the far field. This can be attributed to small deflections in the near field compared to the system resolution of the experimental setup.

A 2D Airy beam was also reconstructed using a 2D hologram (insert of Fig. 1). Experimental beam profiles at propagation distances of Z = 0, 2, and 4 cm were plotted (Fig. 5a, c, e) along with theoretical profiles (Fig. 5b, d, f) [22]. Both experimental results and theoretical calculations showed that the 2D Airy beam traveled along a 45˚ axis in the x–y coordinate because the launch angles in the x–z and y–z coordinates were both zero [8, 20].

Fig. 5
figure 5

Images of the generated 2D Airy beam at different propagation distances. a, c, and e are the experimental results at distance Z = 0, 2, and 4 cm, respectively; b, d, and f are the corresponding theoretical results

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Beam power was measured using a power meter (FieldMax II, Coherent). This indicated around 29 μw for the incident Gaussian beam, 2.55 μw for the main lobe of the generated 1D Airy beam, and 2.58 μw for the main lobe of the generated 2D Airy beam.

4 Conclusions

We successfully demonstrated that both 1D and 2D Airy beams can be generated using a DMD and holographic technique. Characteristics of Airy beams, such as the non-spreading and self-acceleration properties, were experimentally verified. Experimental results showed good agreement with theoretical calculations. This approach can also be used to generate other special beams.