Shaping complex fields with a digital micromirror device (DMD) has attracted much attention recently due to its potential application in optical communication and microscopy. In this paper, we present an optimized Lee method to achieve dynamic shaping of orbital-angular-momentum (OAM) beams using a binary DMD. An error diffusion algorithm is introduced to enhance the accuracy for binary-amplitude hologram design, making it possible to achieve high fidelity wavefront shaping while retaining a high resolution. As a proof of concept, we apply this method to create different classes of OAM beams. The numerical simulations verify that a fidelity of F > 0.985 can be achieved for all the test OAM fields with fully independent phase and amplitude modulation. Moreover, we experimentally demonstrate the dynamic shaping of different OAM beams including pure modes and mixed modes with a switching rate of up to 17.8 kHz. On this basis, accurate information encoding into the generated multiplexed OAM beams is accomplished, which provides access to high speed classical and quantum communications that employ spatial mode encoding.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The orbital angular momentum (OAM) of light has become the focus of intensive research because OAM offers an additional degree of freedom, which could greatly benefit applications ranging from optical trapping [1, 2], microscopy , and optical communications [4, 5] to quantum information science . Light beams carrying OAM or OAM beams have been found to possess a spiral phase structure characterized by , where is the azimuthal angle and the integer, , known as the topological charge . The OAM carried by each photon within these beams is ( is Planck’s constant divided by). Unlike spin angular momentum, for which only two states are possible, the OAM states of a photon can take any integer value. Notably, the topological charges of OAM provide intrinsically orthogonal channels, enabling to drastically increase the information channels and capacity of optical communication by OAM multiplexing . Moreover, such orthogonal OAM modes can be used for a physical realization of a high-dimenssional quantum state in quantum information processing .
Note that there exist other orthogonal spatial mode bases that can also be used for mode division multiplexing, e.g. Hermite–Gaussian modes  and vector modes . Distinguished from these types of modal groups, OAM has circular symmetry, making OAM beams conveniently matched to many optical components and subsystems for ease of implementation . Specially, OAM beams can be considered Laguerre-Gaussian (LG) modes with two indices (i.e. an azimuthal index and a radial index ), and OAM can exist for LG modes with different radial orders. If only is used for multiple beams of different values, then this can be considered a subset of the fuller two-dimensional set of LG modes . However, as OAM can exist for as well, then the use of different values of and different values can form a fuller set of modes and consequently a higher system capacity [11, 13]. Recently, both the radial and azimuthal degrees of freedom of the LG modes have been exploited to achieve high capacity communications [13, 14].
To take full advantage of light’s OAM relies on the generation and manipulation of OAM-carrying light beams. Typically, pure OAM modes are produced through tailoring the spatial phase of Gaussian beams via spiral phase plates , spatial light modulators (SLMs) , Q-plates , or metasurface [18, 19]. While, the generation of complex OAM beams such as multiplexed OAM beams, mixed LG beams or Bessel beams regularly requires full control of both the phase and amplitude of the light. Even though phase-only SLMs can be used to shape the complex fields by phase grating design [20, 21], their limited refresh rate (~100 Hz), polarization and wavelength dependency sometimes might become an issue. Specially, these techniques do not allow fast switching among different OAM states, which would be necessary for multiplexing. A digital micro-mirror device (DMD) offers a promising candidate due to their cheaper price, polarization and wavelength insensitivity and high refresh rate up to 32 kHz .
The DMD is actually a rectangular mirror array where each mirror has only two states, ‘ON’ or ‘OFF’, to selectively reflect incident light to the rear optics. In order to modulate the complex field, a binary amplitude hologram is required to encode the desired field. The Lee method is a well-known way to obtain the binary hologram , and has been adopted for high-speed beam shaping using a DMD [24, 25]. For the Lee method, the hologram is generated by directly converting a grayscale hologram to a binary one based on a fixed threshold. However, this deterministic binarization method will inevitably introduce errors to encoding and deteriorate its performance in complex field modulation. Recently, efforts have been made to address this issue. For example, Lee holography with pixel dithering has been used to improve the performance, but the obtained errors still remains at the 5% level for low resolution fields . Besides, a super-pixel encoding scheme, which combines square regions of nearby micromirrors into superpixels, has been proposed to achieve high-fidelity beam shaping, but at the expense of sacrificing the pixel resolution .
Here, we propose an optimized Lee method to achieve high-fidelity wavefront shaping with a binary DMD, while retaining its high pixel resolution. Instead of using the deterministic binarization method, an error diffusion algorithm is introduced to optimize the performance of Lee holography for complex OAM beam shaping. We show that the sharp artificial boundaries arising from the deterministic binarization can be eliminated by applying the error diffusion algorithm. As a proof of concept, we utilize this method to generate different OAM beams including LG beams and Bessel beams, and all the findings are verified by the simulations and experiments. The optimized Lee holography achieves a fidelity of F > 0.985 for all the test OAM states with fully independent phase and amplitude modulation. In addition, we experimentally demonstrate the dynamic shaping of different OAM beams including pure modes and mixed modes with a binary DMD. Further, switching among the generated OAM modes at rates of up to 17.8 kHz is achieved with our DMD-based scheme, and the achieved refresh rate is much higher than those reported in the literatures [24, 28]. On the basis, accurate information encoding into the multiplexed OAM beams is demonstrated, enabling access to both classical and quantum communications where spatial mode encoding is needed.
2. Theoretical considerations
The DMD is a binary-amplitude SLM. Since we intend to fully modulate the phase and amplitude of the light, a binary hologram is required to reconstruct the complex field
Mathematically, the periodic binary grating can be written as a Fourier series [24, 29],
If we allow both and to be functions of positions, the results are still accurate under the condition that the functions and vary much slower than the grating period. Thus, we can obtain the desired complex field by setting the two functions as 
2.1 Lee method
Now one can generate any complex field only if a binary computer-generated-hologram that represents the amplitude grating indicated by Eq. (2) is calculated first. The Lee method is a well-known way to create the binary hologram, and its normalized transmittance is given by Eqs. (4)-(6), the binary mask can be calculated and loaded onto the DMD to reconstruct the target field.
In the Lee method, a bias function is added to introduce amplitude fluctuation to the sinusoidal and then hard-clipped with zero threshold. However, this fixed threshold method introduces strong artifacts especially at boundaries within the hologram during the binarization process. For example, a helical phase front with a topological charge of can be created by using a fork-like pattern as shown in Fig. 1(d). Utilizing the Lee method, a binary amplitude hologram is calculated as presented in Fig. 1(e). It is obvious that the Lee hologram exhibits sharp artificial boundaries compared with the grayscale pattern. The pixelation of these boundaries leads to strong artifacts at isolated spacing frequencies, which is verified by its Fourier spectrum [Fig. 1(h)]. Besides, we can also observe multiple azimuthal discontinuities in the hologram structure. These errors inevitably deteriorate the accuracy in complex field modulation.
2.2 Optimized Lee method via error diffusion algorithm
In order to achieve accurate complex field modulation with an amplitude-only hologram, we actually need to create an analog amplitude pattern with the form of 
A lot of error diffusion algorithms are available, which are originally developed for digital image processing. It should be noted that the gray values of the pattern are normalized in [0, 1] before processed. As illustrated in Fig. 1(a), the blue pixel indicates the pixel currently being processed, and the white neighboring ones are to be processed. The diffused errors from the blue pixel to the white pixels are determined by the given weights and , and all the weights sum up to unity. The processing direction is from left to right and from top to bottom. Figure 1(c) sketches the specific algorithm execution process . Error diffusion begins with comparing the value of the blue pixel with threshold that is chosen to be 0.5 in most cases. If the pixel value is greater than the threshold, we make it equal to 1, otherwise make it 0. Then, the diffused errors of neighbors are calculated by multiplying the difference between original value and binary value by the given error weight filter . Finally, the neighboring pixels get updated values by adding their original values and the diffused errors together. Each time the error is distributed to the neighbors before those neighboring pixels are binarized. Hence, if a particular pixel has been rounded up to 1 during the binarization, it becomes more likely for its neighbors to be rounded down to 0, such that on average, the overall error is close to zero. In this work, the error diffusion algorithm we adopt is the “minimized average error” algorithm proposed by J. F. Jarvis et al. . As illustrated in Fig. 1(b), twelve neighbor pixels are chosen to be processed each time, and their error weights are shown in the blanks.
As an example, the gray scale fork-like pattern is binarized with the proposed method, and the calculated pattern is shown in Fig. 1(f). By contrast, the sharp artificial boundaries arising from the deterministic binarization can be eliminated by applying the error diffusion algorithm. Further, we analyze the Fourier spectra of the three holograms and present their profiles in Figs. 1(g)-(i). It is clear that the undesired diffraction order beams as well as the binarization noise can be significantly suppressed by error diffusion, as illustrated in the insets. So, the Fourier spectra obtained by the proposed method is much closer to that of the gray-scale hologram, indicating that the optimized method provides a more accurate way for binary hologram design and further spatial mode encoding.
3. Numerical simulation of OAM beam modulation
Light beams carrying OAM play an important role in a variety of applications, such as optical trapping, microscopy, and classical or quantum communications. In these applications, dynamic shaping and manipulation of OAM field is essential, which would be necessary for multiplexing. The binary DMD offers a convenient way to realize it benefiting from its high switching rate. To accurately encode the complex field, we use the optimized Lee method to calculate the binary holograms for OAM beam shaping, including the LG modes, Bessel modes and the angular (ANG) modes. ANG modes are well-known set of mixed OAM modes which can be expressed mathematically as 
We numerically study the performance of the proposed method in OAM mode generation by comparison with the traditional Lee method. According to the desired amplitude and phase profiles of LG modes [Figs. 2(b)-(c)], gray-value hologram was calculated with Eq. (7) as shown in Fig. 2(a), and binary patterns were calculated based on the two methods and are presented in Figs. 2(d) and 2(g). Periodic artifacts along rings and sharp artificial boundaries can be observed in the holograms calculated by the Lee method, while the proposed method yields more smooth masks. From the binary patterns we can calculate the resulting fields by first applying a fast Fourier transform, then a multiplication with a circular mask for the spatial filter and finally a second fast Fourier transform . The intensity and phase profiles of the calculated fields are shown in Figs. 2(e), 2(h), 2(f), and 2(i), respectively. Obviously, the proposed method succeeds to eliminate the artifacts and errors caused by the Lee method, greatly improving the quality of beam shaping. In addition, we apply the method to create the Bessel modes, and the numerical results are presented in Fig. 3. It is found that high-quality Bessel modes can also be obtained by applying the error diffusion algorithm.
In order to quantify the modulation accuracy of the proposed method, we further calculate the fidelity of the shaped light field, where is the target field and is the field obtained with our method. Table 1 lists the calculated fidelities of all test OAM modes including LG modes, Bessel modes and ANG modes with different parameters for two methods, respectively. The fidelity of our method is found to be higher with fully independent amplitude and phase. For all the test OAM fields, the calculated fidelity can achieve at least F > 0.985. In spite of the high fidelity, it varies a little bit with the beam parameters for both methods. These parameters determine the complexity of beam structure that affects the encoding accuracy, because the phase and amplitude distribution need to meet the condition that they vary much slower than the grating period [Eq. (4)]. For example, the fidelity decreases with increasing for LG modes because the beam structure tends to be complicated. Overall, when compared to the Lee method, the optimized method reduces the errors and specially eliminates the azimuthal artifacts.
4. Experimental verification
As a proof of concept, we demonstrate generation of various OAM beams with binary masks projected onto a DMD. For dynamic shaping of OAM beams, a state-of-the-art DMD (ALP 4395, Vialux) with an electronic board was used to send and display the binary masks at a high speed up to 17.8 kHz. The experimental arrangement is sketched in Fig. 4(a). A laser (633 nm wavelength, EW-250B, Eachwave Scientific Instrument) with a TEM00 mode was used as a light source. The laser beam was expanded by lenses L1 and L2 and then steered to fully illuminate the surface of the DMD with an incident angle of 24°. The employed DMD has a full HD resolution of 1920 × 1080 pixels. The reflected light from the DMD passed through a 4-f configuration (L3 = 300mm, L4 = 100mm) and was imaged by a CCD camera (PL-D752MU, PixeLINK). To select the first-order diffraction beam, a pinhole placed at the Fourier plane was adopted. At the imaging plane, the desired complex fields were produced. Then the CCD camera could directly record the intensity profiles of the produced field. For phase measurement, a Mach-Zehnder interferometer was constructed to obtain the interferograms from which the phase profile can be extracted. Furthermore, switching among different OAM modes was detected in real time by a photodiode (PDB420A, Thorlabs), and a data acquisition system (NI PXIe5122, National Instruments) was used to acquire the switching waveform. Our setup enables rapid switching among the generated OAM modes at rate of up to 17.8 kHz, but the imaging speed is limited by the maximum frame rate of the camera (180 Hz).
4.1 Shaping of pure OAM modes
First, the pure vortex modes were created with binary holograms using a DMD. Figures 4(c) and 4(d) show the intensity patterns of vortex OAM mode of generated with two methods along with the desired phase distribution [Fig. 4(b)]. From the intensity pattern obtained with Lee method [Fig. 4(c)], we can observe some azimuthal dark lines, while they do not appear in the intensity pattern generated by our method. This example demonstrates that both methods enable to coherently control the phase of a beam. But the error diffusion method can eliminate the periodic artifacts caused by the deterministic binarization method, enabling a high-fidelity beam shaping.
Next, different LG and Bessel modes were also generated with both methods. It was implemented by displaying the corresponding binary masks onto the DMD. We can easily switch among these modes through changing the binary masks. We demonstrate the measured results for some of these modes in Fig. 5. The cross-sectional images of LG beams and Bessel beams obtained with Lee method are shown in Figs. 5(a1)-(b1) and 5(c1)-(d1), while the intensity profiles of those modes obtained with our method are presented in Figs. 5(a2)-(d2). The amplitude profiles of LG or Bessel beam present rings’ shape with a node at the beam center, where a phase singularity exits, associated to azimuthally uniform OAM density. This can be verified by analyzing the phase patterns that were measured and presented in Figs. 5(a3)-(d3) and Figs. 5(a4)-(d4). Obviously, the optimized method produces more accurate amplitude and phase distributions. Additionally, we obtained the experimental fidelity for the tested mode by comparing the measured field with the ideal field distribution. Taking the Bessel beam with as an example, the measured fidelity for our method is 0.94, while the Lee method achieves only 0.89 in the experiment. By comparison, it is verified that the optimized method yields more smooth profiles and higher fidelity, indicating its good performance in wavefront shaping. Note that, we measured the efficiency of generating OAM modes with the DMD, and both methods achieve a comparable light efficiency of around 5%.
4.2 Dynamic shaping of Mixed OAM beams
Apart from the pure modes, the mixed OAM beams have recently drawn much attentions due to their important application in optical communication and quantum key distribution (QKD) systems. Specially, the ANG modes are widely used in OAM-based QKD systems. These mode are mutually unbiased with respect to the OAM basis, so theoretically they consist of the superposition on infinitely many OAM modes. Here, we demonstrated the experimental generation of such mixed OAM modes using the proposed method. Actually, they are under-sampled approximations of ANG modes. The intensity profiles for ANG modes constructed with superposition of vortex OAM modes from the set are presented in Figs. 6(a)-(e). A rotational fan-shaped structure can be observed in the sequence, which is controlled by the angular parameter. These discrete and mutually unbiased modes are necessary for achieving security in an OAM-based QKD system.
Using a DMD for generating OAM modes enables us to switch among different modes at very high speeds. In Fig. 6(f), we show the waveform (blue) of the dynamic switching among the generated ANG modes together with the DMD trigger waveform (red). Further, switching between the OAM modes and non-load mode (green waveform) is also demonstrated. With our current setup, switching rate of 17.8 kHz (corresponding to the period of 56us) can be achieved, which will benefit dynamic application situations. Note that the actual imaging speed is much slower than the refresh rate of DMD because of the limited frame rate (180 Hz) of the CCD camera used. Apart from the OAM modes we focus on here, the proposed method can be further used for shaping other kinds of structured light beams. The dynamic control of both phase and amplitude of a laser beam is an enabling technology for classical communication and QKD systems that employ spatial mode encoding.
5. Encoding information into mixed OAM beams
The multiplexing of the orthogonal OAM charges allows parallel information to be encoded into a single mixed OAM beam [4, 5, 35]. It is based on the creation of vortex beams representing a controllable superposition of OAM charges, whose weight coefficients serve as carriers of information . For example, a letter can be represented using eight binary numbers (i.e.: 0 or 1) in ASCII protocol, so a multiplexed vortex beam with eight different OAM charges allows to encode any letter in the alphabet . Based on such a principle, we can encode a desired word by constructing the vortex beam with a designed superposition of OAM charges. Figure 7(a) shows the multiplexed OAM fields encoding the word “ALP” (the type of our DMD), and the corresponding binary codes are presented in Fig. 7(c).
In practice, the information encoding is achieved by directly generating the multiplexed OAM beams, requiring independent modulation of amplitude and phase. This procedure can be accomplished using a high-speed DMD that will benefit the fast data transmission. Experimentally, we generated the desired mixed OAM fields using our technique. The measured optical fields including amplitude and phase are shown in Fig. 7(b). To test the implementation of the information encoding, the different topological charges of the vortices are used as ‘markers’ enabling OAM decomposition and information decoding . The inner product algorithm offers a good method to demultiplex the information from the superimposed beams . The algorithm can be expressed as
According to the algorithm, we can calculate the coefficients with respect to each OAM basis and compare them with the desired ones. The calculated signals obtained experimentally are shown in Fig. 7(d) where the measured values are labelled on the top of the bars, matching well with the designed codes in Fig. 7(c). Therefore, the optical fields encoding the information “ALP” have been accurately created. Notably, our technique enables fast switching among the generated fields, underlying high-speed information encoding and transmission for optical communications.
In conclusion, we have proposed an optimized Lee method to achieve high-fidelity and dynamic shaping of OAM beams using a binary DMD. An error diffusion algorithm is introduced to eliminate the sharp artificial boundaries of binary holograms induced by the traditional Lee method, enabling enhanced accuracy for field encoding. We applied the proposed method to create OAM modes including both pure and mixed states. Numerical simulations and experiments verify its good performance in fidelity when dealing with complex wavefront modulation. Notably, our DMD-based beam shaping scheme enables dynamic switching among the generated OAM modes at a speed of up to 17.86 kHz, which is much faster than the speed that can be achievable by the liquid crystal SLMs. Furthermore, accurate information encoding into the generated multiplexed OAM beams is demonstrated in the experiments, which has potential applications in classical communication systems and QKD systems. Apart from the OAM modes we focus on here, the proposed method can be further used for shaping other kinds of structured light beams that will bring new opportunities for optical microscopy [38, 39], metrology [40–42], and probing and sensing .
National Natural Science Foundation of China (NSFC) (Grant Nos. 61535011, 11704369).
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