Generation of rotationally symmetric power-exponent-phase vortex beams based on digital micromirror devices

Mei Zhang; Jiantai Dou; Jiaqing Xu; Bo Li; Youyou Hu

doi:10.1364/OE.500141

1. Introduction

Structured light refer to beams with specific spatial structure or phase distribution, which can be achieved by introducing special amplitude, phase and polarization structures in the beam cross section [1,2], such as vortex beams [3], cylindrical vector beams [4], Higher-order Poincaré beams [5,6], and Hybrid-order Poincaré beams [7]. Among them, vortex beams have been widely studied because of their helical phase structure and the characteristics of carrying orbital angular momentum (OAM). As a new type of vortex beams, power-exponent-phase vortex beams (PEPVBs) are the non-canonical vortex with power-exponent-phase wavefront [8,9], which are mainly divided into the asymmetric power-exponent-phase vortex beams (APEPVBs) with phase term of exp[i2lπ(φ/2π)ⁿ] [10,11], and rotationally symmetric power-exponent-phase vortex beams (RSPEPVBs) with phase structure of exp{i2π[rem(lφ,2π)/2π]ⁿ} [12,13]. Where the integers l and n are the topological charges (TCs) and power index, respectively [14]. Additionally, in the case of l = 1, the RSPEPVBs reduce to APEPVBs, and both of them degenerate into canonical vortex beam when n = 1. Thus, the RSPEPVBs are more representative and have attracted widespread attentions in past years [15–17].

Due to the special phase structures, the intensity distribution of RSPEPVBs is in the form of rotationally symmetric fan when leaving the source plane, of which the petal number is equal to TCs [12]. Thus, the RSPEPVBs exhibit unique tight-focusing properties [15] and can be utilized in optical tweezers for optical trapping of multiple particles [16]. In addition, the RSPEPVBs show good stability in oceanic turbulence and exhibit distinctive properties [17]. Thus, the RSPEPVBs have great potential in the applications of optical communication, particle capture and so on, while the generation of RSPEPVBs is the prerequisite for the applications.

Currently, spatial light modulator (SLM), digital micromirror device (DMD), vortex phase wave plate (VWP) and spiral phase plates (SPPs) have been widely applied for the generation of vortex beams, which can also be used to generate RSPEPVBs. Among them, SPPs is one of the easiest and controllable methods to generate vortex beams, linearly changing the phase shift of the incident beam to obtain the vortex phase structure [18,19]. Similarly, if the spiral structure of SPP is set to the phase form of RSPEPVBs, the phase conversion can be achieved to generate RSPEPVBs. However, each SPP can only form a specific light field without universality. Besides, the SLMs are the flexible options for the generation of vortex beams by loading the phase mask on the computer and modulating the phase [20,21]. Similarly, SLMs can be implemented for the controlled generation of RSPEPVBs [13]. However, the higher price and lower refresh rate may limit their applications [22]. In addition, DMD, as an emerging light manipulation device, can achieve the amplitude modulation by loading a binary amplitude grating [23,24]. Compared to SLMs, DMDs have the disadvantages of low diffraction efficiency and 24° incidence angle limitation, but their refresh rate are as high as 10k-30kHz and the spectral range reach 355-2000nm, which are better than the several hundred Hz and 450-1700nm of SLMs. Currently, DMDs have been employed to obtain vortex beams with faster switching speed, higher refresh rate, wider coverage of spectral area and non-polarization selectivity [25–27]. Thus, DMD would be an advantageous choice for generating RSPEPVBs. However, to the best of our knowledge, there are no reports on the generation of RSPEPVBs based on DMDs.

In this paper, a new method for the generation of RSPEPVBs based on DMDs was proposed and demonstrated. From binary holograms theory and Lee method, a theoretical model was established to simulate the binary amplitude holograms and verify that the RSPEPVBs can be adjusted flexibly and conveniently by loading different binary holograms on the computer. Then, an experimental system was built for the generation of RSPEPVBs by applying DMDs. The experimental results agreed well with the simulation, and confirmed that the ±1^st-order diffraction beams correspond to the RSPEPVBs with positive and negative TCs, respectively, which is the first time to report the RSPEPVBs with converse TCs. The research will contribute to the development of novel structured light generated by DMD.

2. Theory and method

Different from the APEPVBs, the remainder function of RSPEPVBs can recover the angular periodicity of intensity distribution, resulting in the unique distributions of intensity and phase [11,13]. In this paper, the Laguerre-Gaussian RSPEPVBs were selected as the special case, and the expression of electric field in source plane (z = 0) is [15,17]

(1)$$E(r,\varphi ,0) = {A_{_0}}{\left( {\frac{{\sqrt 2 r}}{w}} \right)^l}\textrm{exp} \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\textrm{exp} \left\{ {i2\pi \mathop {\left[ {\frac{{\textrm{rem} (l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}.$$

where r and φ are the polar coordinates, and w is the waist width. l and n represent the TC and power order, respectively. rem(x,y) means the remainder function. For the ease of calculation, the amplitude A₀ is set to 1.

The phase and intensity distributions on the source plane and far field of RSPEPVBs with different TCs are shown in Fig. 1. It can be found that the RSPEPVBs possess non-uniformly distributed phase, of which there is a phase singularity at the center. Thus, the intensity distribution of RSPEPVBs are ring-shaped on the source plane, and the circle radius is proportional to the values of |l|. However, when leave the source plane (z > 0), the RSPEPVBs evolve into “multi-petaled” shape with rotational symmetry, of which the rotation directions with positive and negative TCs are mirrored to each other. While, as the phase gradient increases with n, the beam intensity becomes more concentrated with larger n. The simulation results are consistent with the theory.

Fig. 1. The phase and intensity distributions of RSPEPVBs on the source plane (z = 0 m) and far field (z = 1 m) with (a1)-(a3) l = −3, n = 2, (b1)-(b3) l = 3, n = 2, (c1)-(c3) l = 4, n = 2 and (d1)-(d3) l = 4, n = 4.

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Commonly, pure amplitude modulation can be considered as both amplitude and phase modulations. As an amplitude-typed spatial light modulator, DMD can be utilized for the generation of vortex beams by modifying the phase profile and reshaping the wavefront amplitudes [28], which is also called binary amplitude holography method. The binary holograms are created accurately to encode intricate signals, and loaded onto the DMD to control the flip of micromirror [29]. Presently, Lee method [30], superpixel method [31] and so on have been applied for the computer holographic coding. Among them, as the most commonly binary holographic methods, Lee method and superpixel method have their own advantages. The superpixel method outperforms the Lee method in terms of efficiency and fidelity, while the Lee holographic method has more advantages in spatial resolution and encoding speed compared to the superpixel method. However, previously, many works based on DMD to generate vortex beams relied on superpixel algorithms, such as LG₁₀ OAM mode [31] and dual color Airy beams [32]. In the paper,the Lee method was selected to create the binary holograms for the generation of RSPEPVBs.

The first-order diffraction with required intensity and phase information is selected as the target light field and the form is U(x,y)=A(x,y)exp[iϕ(x,y)] [33], where target amplitude A(x,y) is a positive normalized function and the target phase ϕ(x,y) takes values from -π to +π. Then, a periodic binary amplitude grating is applied to encode the binary holograms, which can be expressed by a Fourier series [34],

(2)$$f(x,y) = \sum\limits_n {\frac{{\sin (\pi nq)}}{{\pi n}}} \textrm{exp} [in(2\pi ({u_0}x + {\nu _0}y) + 2\pi p)].$$

where (u₀,ν₀) represents the spatial frequency of periodic grating. When using monochromatic plane waves to illuminate the binary grating, the first-order diffraction can be expressed as,

(3)$${f_1}(x,y) \propto \frac{{\sin ({\pi q} )}}{\pi }\textrm{exp} ({i2\pi p} ).$$

Obviously, q and p are respectively related the amplitude and phase, which means to feasibly control the amplitude and phase by creating a transmittance function to encodes amplitude and phase as follows [34],

(4)$$T(x,y) = \frac{1}{2} + \frac{1}{2}{\mathop{\rm sgn}} \{{\cos [{2\pi ({u_0}x + {\nu_0}y) - 2\pi p(x,y)} ]- \cos [{\pi q(x,y)} ]} \},$$

Among them, sgn(x) is the sign function, the expression of the target amplitude q(x,y) and the target phase p(x,y) can be written as,

(5)$$\begin{aligned} q(x,y) &= \frac{1}{\pi }\arcsin [{A(x,y)/{A_{\max }}} ]\\ &= \frac{1}{\pi }\arcsin \left[ {\frac{{\left|{{A_{_0}}{{\left( {\frac{{\sqrt 2 r}}{w}} \right)}^l}\textrm{exp} \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\textrm{exp} \left\{ {i2\pi \mathop {\left[ {\frac{{rem(l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}} \right|}}{{{{\left|{{A_{_0}}{{\left( {\frac{{\sqrt 2 r}}{w}} \right)}^l}\textrm{exp} \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\textrm{exp} \left\{ {i2\pi \mathop {\left[ {\frac{{rem(l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}} \right|}_{\max }}}}} \right], \end{aligned}$$

(6)$$\begin{aligned} p(x,y) &= \frac{1}{{2\pi }}\phi (x,y)\\ &= \frac{1}{{2\pi }}angle\left[ {{A_{_0}}{{\left( {\frac{{\sqrt 2 r}}{w}} \right)}^l}\textrm{exp} \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\textrm{exp} \left\{ {i2\pi \mathop {\left[ {\frac{{rem(l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}} \right]. \end{aligned}$$

where A_max is its maximum value, while | | stands for taking the amplitude, and angle stands for taking the phase angle.

For the fork grating, the diffraction angle and diffraction intensity of the first-order diffraction are controlled by the spacing and duty cycle of grating. Meanwhile, the transverse phase of the grating limits the phase of diffracted beam, and modulate the intensity and phase [35]. According to Eqs. (4–6), binary holograms of beams carrying different information can be calculated.

Then, the software of MATLAB(R2021b) was used to add the phase and amplitude information of RSPEPVBs to the linear grating and generate a fork diffraction grating [36]. As illustrated in Fig. 2, the binary holograms for the generation of RSPEPVBs with different TCs were obtained, where the wavelength is set as 632.8 nm and power order n is equal to 2, which are similar to a bifurcated grating with ring restriction. The reflectance of the black and white parts represents 0 and 1, respectively. As we can see, the number of bifurcations is exactly equal to the corresponding TC of l.

Fig. 2. Two-dimensional amplitude hologram of Lee method for the generation of RSPEPVBs with (a) l = 3, (b) l = 4, (c) l = 5 and (d) l = 6.

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3. Experiments and results

In this section, an experimental setup for the generation of RSPEPVBs based on DMD was established and shown in Fig. 3. The Gaussian beam passes through the beam expansion collimation system, consisted of two lenses with focal length of L1 = 25.4 mm and L2 = 100 mm, to turn into the collimated plane wave and cover the DMD window. Then, it is divided into two beams by the beam splitter (BS1), of which one is incident on the DMD surface at 24^° through the reflector mirror (M1), and the other one is reflected by the reflector mirror (M2) into the beam combiner (BS2) as reference beam. The beam, after DMD modulation, combines with the reference beam in BS2 to form interference fringes.

Fig. 3. Experimental setup for the generation of RSPEPVBs based on DMD. A: attenuating plate; L1 and L2: beam expanding collimation system; BS1 and BS2: beam splitter; M1 and M2: mirror; DMD: digital micromirror device; L3 and L4: 4f system; S: shelter; P: pinhole; CBP: camera beam profiler.

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When the reference light is blocked by shelter (S), the experimental setup can be used to generate RSPEPVBs by loading the above-mentioned binary amplitude grating on the DMD. After emitted perpendicularly to the DMD, the beam passes through a 4f system, composed of two lenses with focal length of L3 = 100 mm and L4 = 100 mm, L3 and L4 are staggered rather than coaxial. The beam is transformed into the Fourier spectrum by Fourier transform through L3. It can be found that the +1^st-order and −1^st-order diffraction occupy the largest energy in all diffraction order. Therefore, the pinhole of adjustable size is configured in the rear focal plane of L3 to select the +1^st-order and −1^st-order diffraction. Then, the spectrum is converted to the real space through the inverse Fourier transformation of L4 [37] and the RSPEPVBs is obtained.

In the experiments, a He-Ne laser (DH-HN250) with wavelength of 632.8 nm is utilized as the light source. So, the visible DMD (FLDISCOVERY F6500 Type-A), with operating wavelength of 400nm-700 nm, and resolution of 1920*1080, was applied, of which the micromirror size is 7.56µm × 7.56µm and the diagonal dimension is 0.65 inches. Besides, the camera beam profiler (CBP, THORLABS BC106N-VIS/M) is utilized to collect the intensity images of mode and interference fringes. Attenuation film (A) is added to the optical path to prevent overexposure.

Then, the experiments were carried out and the results were shown in Figs. 4 and 5. The intensity patterns of RSPEPVBs with TC of 3 to 6 were illustrated in Fig. 4(a2)-(d2), which are consistent with the results of simulation, as shown in Fig. 4(a1)-(d1). As we can see, the +1^st-order diffraction intensity of RSPEPVBs produced by DMD with different TCs was “petal” shaped, and the number of “petals” was equal to the number of TCs, as well as the bifurcations of the two-dimensional hologram in Fig. 2. To further verify the phase structure of the beams, the interference experiments between the plane wave and the generated RSPEPVBs were carried out and the results were given in Fig. 4(a4)-(d4), which agreed well with the simulation, as illustrated Fig. 4(a3)-(d3). It can be found that there were several bifurcations in the interferogram, of which the number was equal to the value of TCs. In addition, the location of the bifurcation was the same as that of the singularity and there was only one bifurcation at each location, which means the TC of each singularity was equal to l [38,39]. These characteristics verified that the RSPEPVBs generated by DMD possessed high quality.

Fig. 4. Intensity distribution and interference results with plane wave of +1^st-order RSPEPVBs generated by DMD. (a1)-(d1) and (a3)-(d3) simulation results, (a2)-(d2) and (a4)-(d4) experimental results.

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Besides, we also studied the properties of the −1^st-order diffracted RSPEPVBs, as illustrated in Fig. 5. As we can see from Fig. 5(a2-d2), the shape of the −1^st-order diffracted RSPEPVBs was similar to that of the +1^st-order, of which the intensity was slightly weaker. However, they had opposite direction of rotation and the shape of the −1^st-order RSPEPVBs was same as the simulation of negative TC, as shown in Fig. 5(a1-d1). Therefore, we thought the TC of −1^st-order diffracted RSPEPVBs was negative and carried out an interference simulation and experiment to verified the judgement, as illustrated in Fig. 5(a3-d3) and (a4-d4). As we can see, there were also interference fringes at the locations of singularities, and the number of individual bifurcations was equal to the number of “petals”. However, the direction of each bifurcation was opposite to that of +1^st-order and the properties agreed with the simulation results when TCs was negative. This property was identical to the vortex beam [3]. Therefore, the TCs of the −1^st-order diffracted beams were negative, which was the first time to report the RSPEPVBs with negative TC.

Fig. 5. Intensity distribution and interference results with plane wave of −1^st-order RSPEPVBs generated by DMD. (a1)-(d1) and (a3)-(d3) simulation results, (a2)-(d2) and (a4)-(d4) experimental results.

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To investigate the stability of the RSPEPVBs generated by DMD, the intensity profiles of the beam were collected at different distances, as shown in Fig. 6. The results showed that, as the distance increased, the beam profile of the RSPEPVBs with l=±3 exhibited a divergent trend, but the overall shape remained stable, which verified that the RSPEPVBs generated by DMD were stable.

Fig. 6. Intensity distribution of the ±1^st-order RSPEPVBs with (a) l = 3 and (b) l = −3 at different distances.

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Diffraction efficiency is one of the important indexes to evaluate the refraction and diffractive hybrid optical systems, containing imaging diffractive optical elements, and theory indicates that the maximum diffraction efficiency of binary amplitude holography based DMD is 10.1% [40,41]. In the experiments, the diffraction efficiencies of the total and the ±1^st-order diffracted RSPEPVBs were measured by using the optical power meter (PM, THORLABS PM400). The results showed that the total diffraction efficiency was up to 7.18%, which was slightly lower than the theoretical maximum value, while the diffraction efficiency of the ±1^st-order diffracted RSPEPVBs was about 1.73%, which was slightly higher than that of the generation of vortex beams by DMD (1.5%) [34].

4. Conclusions

In summary, a new method for the generation of RSPEPVBs based DMD was proposed and demonstrated. Based on the theory of binary amplitude holography and Lee method, the computation model of binary holograms for the generation of RSPEPVBs was derived, which was successfully utilized to obtain the two-dimensional amplitude hologram. Then, the experimental setup for the generation of RSPEPVBs based on DMD was established and the phase structure of the RSPEPVBs was verified by the Mach-Zehnder interferometer. The experimental results showed that the RSPEPVBs can be generated based on DMD with high beam quality and stability, and the ±1^st-order diffracted beams were respectively corresponding to RSPEPVBs with contrary TCs, which was the first time to report the RSPEPVBs with negative TC. Furthermore, we also measured the diffraction efficiency of RSPEPVBs generated by DMD with overall diffraction efficiency of 7.18%, and ±1^st-order diffraction efficiency of 1.73%, which conforms to the theory. The method can be applied for the generation of RSPEPVBs with different parameters and quickly achieve mode switching by loading different binary amplitude holograms, which provides a new choice for the generation of new structure beams based on DMD.

Funding

National Natural Science Foundation of China (62205133); Jiangsu Provincial Key Research and Development Program (BE2022143); Wenzhou Major Scientific and Technological Innovation Project (ZG2022008); Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX23_2123).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Generation of rotationally symmetric power-exponent-phase vortex beams based on digital micromirror devices

Abstract

1. Introduction

2. Theory and method

3. Experiments and results

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (6)

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