Since Allen et al. demonstrated 30 years ago that beams with helical wavefronts carry orbital angular momentum (OAM), the OAM of beams has attracted extensive attention and stimulated lots of applications in both classical and quantum physics. Akin to an optical frequency comb, a beam can carry multiple various OAM components simultaneously. A series of discrete, equally spaced, and equally weighted OAM modes comprise an OAM comb. Inspired by the spatially extended laser lattice, we demonstrate both theoretically and experimentally an approach to producing OAM combs through azimuthal binary phases. Our study shows that transition points in the azimuth determine the OAM distributions of diffracted beams. Multiple azimuthal transition points lead to a wide OAM spectrum. Moreover, an OAM comb with any mode spacing is achievable through reasonably setting the position and number of azimuthal transition points. The experimental results fit well with theory. This work presents a simple approach that opens new prospects for OAM spectrum manipulation and paves the way for many applications including OAM-based high-security encryption and optical data transmission, and other advanced applications.
- Advanced Photonics Nexus
- Vol. 1, Issue 1, 016003 (2022)
Abstract
Video Introduction to the Article
1 Introduction
Similar to macroscopic objects, particles such as photons,1 electrons,2 and neutrons3 can also carry angular momentum. The angular momentum of a photon is of two types, with the spin angular momentum (SAM) corresponding to circular polarization4 and the orbital angular momentum (OAM) determining the helical wavefronts of a beam.5 Actually, OAM is an inherent feature of a beam whose complex amplitude comprises a helical term , where is the OAM eigenvalue and known as the topological charge associated with an OAM value ( is the reduced Planck’s constant) per photon, and denotes the azimuthal angle.6,7 One or more phase singularities determined by multiple OAM components are present for an OAM beam, thus leading to a doughnut-shaped transverse intensity.8 OAM offers an additional degree of freedom and leads to new possibilities for photonics research. Over the past 30 years, beams carrying OAM have found a wide variety of applications, including large-capacity communications,9
Previous research has provided a variety of schemes to generate OAM beams, such as direct output from a laser24
In this study, we overcome the aforementioned limitations by demonstrating a specially designed azimuthal binary phase-only grating for the selective manipulation of OAM combs. Several transition points were embedded in a spiral phase along the azimuth to tailor the OAM spectrum of the diffractive beams. The number and position of the azimuthal transition points were optimized to produce arbitrary OAM combs, where the optimized results were fixed and could be employed for various scenarios at any time. A proof-of-principle experiment was performed to demonstrate the performance of the proposed azimuthal binary-phase gratings. These favorable results were in good agreement with our simulations. This study offers a real-time, flexible, simple, and accurate method for generating beams with complicated OAM spectra, which paves the way for classical/quantum optical communications, holographic encryption and decryption, and other advanced applications in the future.
2 Results and Discussion
2.1 Azimuthal Phase Modulation: From Continuation to Binarization
Previous studies have illustrated that a continuous azimuthal phase, namely a spiral phase plate (SPP), can bring a pure single OAM mode for the incident beams.29 In other words, such spiral phase mod with nodal lines can transform a Gaussian beam into an OAM beam with topological charge . As displayed in Fig. 1(a), if we binarize the continuous azimuthal phase as 0 and with the binarization threshold :
Figure 1.Binarizing continuous azimuthal phases into
2.2 Azimuthal Binary Phase with Multiple Transition Points
Next, we discuss what happens if multiple azimuthal transition points are brought in. The phase distribution of a binary phase-only grating with () various azimuthal transition points within one grating period reads
Figure 2.The evolution of the obtained OAM spectra with respect to azimuthal transition points. Four transition points as
2.3 OAM Comb Manipulation
To produce the desired OAM combs, a feasible way is to find a proper set of transition points. From the scalar diffraction theory, the far-field diffraction of an incident plane wave can be calculated approximatively by the Fourier transform of the transmittance function of azimuthal binary phases. Thus, from Eq. (2) reads , where denotes Fourier transformation. On the basis of helical harmonic , the diffraction field can be decomposed into the summation of infinite helical term with various complex weights :42
The complex weights in Eq. (3) read
Actually, corresponds to the intensity of OAM channel . By now, a mind map of OAM comb manipulation emerges clearly, as the azimuthal transition points lead to azimuthal nodal lines determine the diffraction fields, and such fields further determine the intensity of each OAM channel . In other words, the final OAM spectrum is a function of azimuthal transition points as indeed (Supplementary Note 2). To find a proper set {} for the desired OAM comb, several evaluation parameters are necessary. Here two parameters are introduced, one of which is the efficiency defined as , where and denote the set of topological charges of desired OAM comb and the integer set separately. The other is the uniformity as ), where , min{} and max{} denote taking the minimum and maximum, respectively. Note that must be because a single phase-only grating cannot produce coaxial multiplexed OAM modes with desired OAM distributions, and there must be irrelevant and undesired OAM modes.38 {} calculation is available through iteration embedded numerical simulation. In each iteration, one must evaluate the value of and to decide how to adjust the transition points set {} until an ideal scenario where the finally calculated transition points contribute to highest efficiency and uniformity . Usually such process is complex and time-consuming, which incommodes the practical application of OAM comb. Zhou and Liu43 have already demonstrated numerical solutions for binary Dammann gratings, which can expand the incident beam along the direction to achieve a one-dimensional (1D) equal-intensity laser array with both high efficiency and high uniformity. In Ref. 43, multiple lateral () transition points are introduced within one grating period , resulting in multiple presented diffraction orders (-expanded modes) featured as , where corresponds to the ’th diffraction order. Here, we move from “-space” to “,” where the azimuthal () transition points in one azimuthal period () lead to OAM superposed modes (-expanded modes) featured as . The resulting fields in the above two “spaces” have identical forms. Their only difference is that one expands the incident beam along the coordinate while the other along the coordinate. Such phenomena imply that the numerical solutions given in Ref. 43 are also effective to be employed in the azimuth to produce various OAM superposed modes with equal intensities.
We map the numerical solutions presented in Ref. 43 from “-space” to “-space” (Supplementary Note 3) to generate azimuthal binary phases, calculate the far-field diffractions through the scalar diffraction theory for incident Gaussian beams, and then analyze the OAM spectra (Supplementary Note 4). As displayed in Fig. 3(a), an azimuthal binary phase can produce nine equal-intensity OAM modes with topological charges ranging from to . It has six azimuthal transition points as 0, 0.4190, 0.8087, 1.7963, 2.8693, and 3.7127. The efficiency and the uniformity of the obtained OAM comb are calculated as 78.32% and 99.61%, which are high enough for OAM comb manipulation. In addition, we also attempt to generate OAM combs with other OAM mode intervals . Such manipulation is based on the scaling property of the Fourier transform, where if , then with the scale factor. So, here scaling the azimuthal phase contributes to , where is the azimuthal angle in the far field. As an example, we scale the azimuthal angle in Fig. 3(a) with scale factors and , and the final obtained OAM combs are given in Figs. 3(b) and 3(c). Note that the OAM comb is present through normalized intensity, where the calculated or experimentally measured intensity of each OAM channel is normalized with respect to the maximum intensity among all the present OAM channels. In both of the two scenarios, there are also nine OAM modes with equal intensities. However, their topological charge distributions are totally different; one is to with , the other is to with . Such phenomena illustrate that it is practicable to produce OAM combs with various OAM mode intervals through azimuth scaling and provide a more flexible scheme of OAM comb manipulation. Additionally, OAM range is limited to to here. When extending such a present range, in Figs. 3(b) and 3(c) one can see undesired side OAM modes distributions similar to Fig. 3(a), but the only difference is the space between adjacent OAM channels.
Figure 3.OAM comb manipulation with various OAM mode intervals. From left to right are azimuthal transition points, the corresponding binary phases, far-field diffractions, and OAM combs, respectively. All the OAM combs in (a), (b), and (c) consist of nine OAM components, but their OAM mode intervals are different as 1, 2, and 3, respectively. The interval difference results by scaling the azimuthal phase
Proof-of-principle experiments are also carried out to show the practical operability (Supplementary Notes 5–7). In the experiment, we encode the azimuthal binary phase onto a liquid-crystal spatial light modulator (SLM) to accomplish the phase-only modulation. The OAM spectra are analyzed with the back-converted method as a series of spiral phases (), also known as anti-SPPs in some literature, are encoded simultaneously on the SLM and then the intensity of center bright spot is calculated to represent the relative intensity of the OAM channel , . We first generate an OAM comb consisting of 64 OAM states ranging from to with the OAM mode interval . The experimentally captured intensity patterns and OAM spectrum and its corresponding simulation results are given in Fig. 4(a). The uniformity of the experimental result is evaluated as 92.04%, which is lower than that of the simulation at 98.16%. In addition, the similarity, which implies the consistency between the experimental and theoretical results suggested by the work44,45 is evaluated as 96.49%. We also generate an OAM comb consisting of 32 OAM states ranging from to with the OAM mode interval , the results of which are shown in Fig. 4(b). In this case, the uniformities of the experimental and simulated results are 90.95% and 98.86%, respectively, and the similarity is analyzed as 93.36%. The numbers of azimuthal transition points {} for the above two cases are 70 and 68 separately (Supplementary Note 6). The measured OAM spectra in the experiment show little difference compared with simulation, where the intensities emerge in some irrelevant OAM channels and some power is likely to leak from desired to undesired OAM components. In fact, such “leakage” is meaningless and makes no physical sense because inevitable interchannel crosstalk is introduced during the data-processing of the back-conversion-based OAM spectrum measurement.
Figure 4.Experimentally generated OAM combs. (a) Experimental results of an OAM comb consisting of 64 OAM states ranging from
3 Conclusions
We showed that binarizing a spiral phase into binarization results in multiple OAM components that are symmetric with respect to the fundamental mode () of equal intensity. In this case, there are two azimuthal transition points constituting the two nodal lines that divide phases 0 and . When more transition points are introduced, the resulting OAM spectrum becomes more complex. Nevertheless, the OAM distribution can be tailored by adjusting the number and value of azimuthal transition points to accomplish arbitrary OAM comb manipulations. Finding a proper set of azimuthal transition points is crucial for OAM comb manipulation; however, it is complicated because of repetitive iterations. Therefore, based on previous reports on the numerical solutions of lateral transition points to construct a 1D beam lattice that expands the incident beam along the axis, we build a mapping between the axis in Cartesian coordinates and the axis in polar coordinates, thus transforming the optimal solution of transition points along the axis into that of the azimuth along the axis to accomplish a fast binary phase calculation. In the simulation, we successfully produce various OAM combs following the above hypothesis. We also show how to adjust the mode interval between adjacent OAM modes inside the OAM comb by scaling the azimuthal coordinates. In the proof-of-principle experiment, OAM combs consisting of up to 64 OAM components with the largest absolute value of OAM state 63 are generated, whose uniformity reaches 92.04% and fits well with the theory. We also attempt to adjust the OAM mode interval in practice and produce an OAM comb comprising 32 OAM components whose OAM states range from −62 to +62 with a mode interval of 4. The uniformity is evaluated to be 90.95%. The favorable experimental results illustrate that our proposed method exhibits good performance for tailoring OAM combs in practice.
In addition, the OAM states can be shifted by illuminating the binary phase-only grating with a higher-order optical vortex. Alternatively, this may be done by integrating the ’th order spiral phase with the proposed binary azimuthal phase to shift the OAM components by , as in Ref. 46. In this manner, OAM combs whose components are asymmetric with respect to the fundamental mode () can also be achieved.
The ability to generate a beam with a specifically tailored OAM spectrum is crucial for OAM applications; e.g, the unique key of holographic encryption and decryption.36,37 In addition, the proposed approach provides a convenient method for the practical generation of high-dimensional OAM combs, namely, an available high-dimensional Hilbert space; thus, it can find potential in high-dimensional photon entanglement and will inspire applications in quantum key distribution and quantum teleportation.47 Iteration methods are typically employed to optimize phase-only gratings to produce a multiplexed OAM beam. However, these complicated processes are time-consuming and may be inconvenient for practical applications. Herein, we demonstrate a simple approach in which azimuthal binary phases are introduced to manipulate arbitrary OAM combs with equal mode intensities. Furthermore, the azimuthal transition points can be easily calculated by mapping the numerical solutions of the binary lateral phases. In summary, our proposal opens new avenues for OAM comb manipulation and lays the foundation for applications in many OAM-based systems.
Shiyao Fu is currently an associate professor at Beijing Institute of Technology. He received his PhD in electronic science and technology from Beijing Institute of Technology in 2019. His current research interests cover complex structured fields manipulation, novel solid-state lasers, and their applications in critical engineering. He has published one book and more than 40 peer-reviewed papers. He has 18 authorized patents. He was selected for the national postdoctoral program for Innovative Talents. He won the Wang Da-Heng Optics Prize, and his doctoral dissertation was recognized by the Chinese Institute of Electronics. He is a senior member of the Chinese Optical Society.
Chunqing Gao is a professor in the School of Optics and Photonics, Beijing Institute of Technology, China since 2001. He received his BS degree and MS degree in optics from Beijing Institute of Technology, and received his PhD in physics from Technical University Berlin, Germany. The current research in his group mainly focuses on novel laser systems, laser beam manipulation, and related applications. He has published three books and more than 260 scientific papers in journals and conferences. He has 30 authorized patents in the fields of optics and lasers.
Biographies of the other authors are not available.
References
[1] G. Molina-Terriza, J. P. Torres, L. Torner. Twisted photons. Nat. Phys., 3, 305-310(2007).
[3] C. W. Clark et al. Controlling neutron orbital angular momentum. Nature, 525, 504-506(2015).
[17] M. Padgett, R. Bowman. Tweezers with a twist. Nat. Photonics, 5, 343-348(2011).
[24] A. Forbes. Structured light from lasers. Laser Photonics Rev., 13, 1900140(2019).
[25] S. Ngcobo et al. A digital laser for on-demand laser modes. Nat. Commun., 4, 2289(2013).
[51] J. Jahns et al. Dammann gratings for laser beam shaping. Opt. Eng., 28, 281267(1989).
Set citation alerts for the article
Please enter your email address