Beams with fast and continuously-tunable orbital angular momentum (OAM) have potential applications in classical and quantum optical communications, sensing, and in the study of beam propagation through turbulence. An acousto-optical deflector (AOD) is a sophisticated, well-studied device that continuously and rapidly tunes the deflection angle of an output beam. The log-polar HOBBIT setup can generate beams with OAM by wrapping elliptically shaped Gaussian beams with linear phase tilt to a ring. By combining the linear tilted output from the AOD with the OAM generation capabilities of the HOBBIT system, the generated OAM modes become continuously tunable at high speeds measured on the order of 400 kHz.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
As interest in the exploration of orbital angular momentum (OAM) properties grows, fast switching between different OAM modes is a key technique needed to thoroughly explore applications. A common approach in both classical and quantum OAM communications uses different OAM modes as symbols or bits. The capability of switching or hopping between OAM modes can increase data rate dramatically [1–5]. Studies of OAM beams in turbulent environments suggests that different OAM modes have different propagation performances through turbulence [6,7]. Fast switching between OAM modes will benefit such studies by enabling the exploration of a wide range of OAM modes. Other sensing related applications that could benefit from rapidly-tunable OAM include beam steering through scattered media , particle manipulation using three dimensional beams , object rotation detection [10,11], temperature sensing , and motion detection .
So far, one of the most popular techniques for mode switching uses spatial light modulators (SLM), a device that has a very limited switching speed. Digital micro-mirror devices (DMD) can boost switching speeds up to tens of kHz , which is comparable with the switching speed of a direct OAM mode emitter . The DMD micro-mirror pitch limits the spatial resolution, while the mode emitter is only capable of tuning integer OAM modes. Fractional OAM modes, also referred to as non-integer , continuous , successive , and rational  modes are another interesting aspect of study. This is primarily because it is almost impossible to generate an entirely pure integer OAM state. Secondly, it has been analytically deduced that fractional OAM Bessel beams could form an infinite number of orthogonal subsets of OAM modes , which can further benefit classical and quantum optical communication. Thirdly, the fractional OAM Bessel beams preserve the non-diffracting properties that integer OAM beams possess . This property is key for beam propagation applications, including propagation through turbulence and turbid environments. Moreover, it has been found that a group of fractional OAM modes generated by a synthesis of Laguerre-Gaussian (LG) modes have good structural stability on propagation to the far field . These LG mode-based fractional OAM states can be used in both classical and quantum communication.
The proposed fast tunable OAM generation technique utilizes an optical geometric transformation [20–23] known as the log-polar transform [24–33]. Refractive log-polar elements were first explored as an efficient OAM mode sorter in 2010 . In 2013, Mirhosseini used a fan-out diffractive beam-copying method to increase the log-polar mode sorter separation efficiency up to 95% . In 2013, Mhlanga successfully sorted more than forty HeNe Bessel beam OAM modes. In 2015, the same elements were successfully used to demultiplex OAM modes with higher mode selectivity and better efficiency than that of cascaded beam splitters . In 2015, Morgan designed and fabricated a diffractive version of the log-polar elements for OAM mode (de)multiplexing . In 2016, Srimathi used the same log-polar elements for an underwater communication link . In 2017, Ruffato made a compact demultiplexing version of the log-polar elements operating at 632.8 nm [30,31]. In 2017, Lightman 3D printed a log-polar mode sorter, which working for a broad-spectrum of light . In 2018, Ruffato redesigned the log-polar diffractive elements and explored the non-paraxial regime property, which is a good example of diffractive device miniaturization .
The acousto-optical deflector (AOD) is a reasonably fast modulation device that is commonly used for stable phase modulation and beam shaping . Bessel beams have been generated using an AOD array  and a cylindrical axisymmetric AOD . In this work, a novel technique for OAM switching and tuning using an AOD in conjunction with a log-polar coordinate transformation system is demonstrated. The maximum mode switching speed for the experimental setup is measured on the order of 400 kHz, which is determined by the acoustic velocity of the crystal as well as the beam diameter. For a different AOD and a reduced beam size, this speed has the potential of reaching tens of MHz with sub-microsecond response time — far higher than the kHz level switching methods mentioned above. Typically, AODs have a very high damage threshold and are widely used in high power laser systems for beam deflecting and laser pulse generation. The integration of an AOD also opens up high power and direct energy related applications for our HOBBIT system.
The two log-polar coordinate transform optics work as a pair to perform the optical transformation of wrapping a linear-distributed beam into an annular-distribution. The mapping process involves two customized diffractive phase optics: the wrapper which performs the line to ring transformation, while the phase-corrector corrects the phase distortion introduced by the wrapper upon propagation. Interestingly, if a length-wise linear phase is applied along the linear-distributed beam, this will be transformed as well, producing a spiral phase which has been encoded onto the wrapped-ring. Therefore, the far field of this ring shape beam carries OAM.
As the basis of this work, the optical setup used to generate OAM modes is shown in Fig. 1(a). In this technique, a Gaussian input beam is passed through an AOD. When a voltage signal with the central frequency of the AOD is applied, the 1st order deflection of the Gaussian input is at the Bragg condition with the Gaussian beam propagating along the optical axis. In this orientation, the Gaussian beam has a flat wavefront and the designed system will generate an OAM mode of charge equal to zero. When the frequency of the acoustic wave deviates from this center frequency, the beam is instead deflected by some additional angle along the horizontal direction as shown in Fig. 1(a). The deviation away from the Bragg condition results in the 1st order deflection with a tilted phase relative to the axis of propagation. The output of the AOD is then passed through a 4-f embedded line generator, used as a dual-axis manipulator. The output of the 4-f embedded line generator is an elliptical beam with an elongated length and a suppressed height with a phase tilt along the horizontal direction specific to the applied acoustic frequency. This elliptical beam then propagates through the log-polar HOBBIT optics that wrap the ellipse into an asymmetric annular-distribution. Overall, this results in an elliptical Gaussian beam with linear phase being wrapped into an asymmetric ring with azimuthal OAM phase, which is the angular spectrum of the asymmetric Bessel-Gauss beams. The seminal concept of asymmetric Bessel beams and its physically realistic version, asymmetric Bessel-Gauss beams, have been proposed and studied by Kotlyar in 2014 [38,39]. In 2018, a general case of asymmetric Mathieu beams has been analytically derived and experimentally generated by Barcelo-Chong, who showed the recovery of asymmetric Bessel modes when the ellipticity parameter approaches zero . Bessel-Gauss beams are well known for their nondiffracting behavior, but this property relies on the size of the Gaussian envelope. A larger Gaussian envelope produces more ringing, which in turn produces the longer Rayleigh range. In this case, a small Gaussian envelop is used, which will reduce the nondiffracting behavior. In addition, the azimuthal distribution of the angular spectrum of the asymmetric Bessel-Gauss beams is different from Kotlyar’s papers [38,39], which will be discussed in detail later in this paper.
The input to the AOD has a Gaussian distribution with the diameter of the beam defined as 2w0 as shown in Fig. 1(b). The momentum vector of incident photon is , that of the diffracted photon is , and that of the phonon is . According to the principle of momentum conservation, the momentum vector of the diffracted photon should be equal to the sum of the momentum vectors of the incident photon and of the acoustic phonon, , shown in Fig. 1(c). The notation and is used for the diffracted photon’s momentum vector when the far-field beam has charge 0 and m, respectively. The OAM mode index m = l + α is a continuous charge number in which l is the integer part and α is the fractional part, which is defined as a positive real number 0≤α<1. The Bragg angle is greatly exaggerated in Fig. 1(a) for visual clarity, and a general Bragg angle equation can be represented asFig. 1. According to the paraxial approximation, this angle deviation corresponds to charge m and can be represented byEqs. (2)–(4) results in an expression for charge m as a function of the frequency change from the Bragg condition given by
As shown in Fig. 1, the 1st order deflected beam exiting the AOD is a Gaussian distribution, which can be expressed as
The elliptical Gaussian beam is then incident on the log-polar optics which have been well studied [24,25,41,42]. The HOBBIT mapping process uses two customized log-polar optics: the wrapper that maps the elliptical Gaussian beam to an asymmetric ring profile, and the phase-corrector that corrects the phase distortion introduced by the wrapper. Since the elliptical line has a horizontal Gaussian distribution, the HOBBIT system wraps it into an asymmetric ring with a ring radius, ρ0, defined from the origin to peak intensity location and width, 2wring, as shown in Fig. 1(c). Given the log-polar mapping equation of , the near-field output from the proposed HOBBIT system is given byEq. (8) can then be derived asEq. (9) is the combination of a group of Bessel-Gaussian (BG) beams carrying OAM. Intuitively, it is a weighted linear combination of every possible integer OAM phase carrying nth-order Bessel function of the 1st kind modulated by the same Gaussian envelop. The parameter Bn is the weighting or selection factor, which distributes the power within the central 2 to 3 modes and decays rapidly as m approaches positive and negative infinity. When α = 0, then m = l, meaning an integer charge will be select as n = l, and Bl is the maximum value. As α increases, the central weighting factor Bn’s maximum value will move from n = l to n = l + 1. This means fractional-charged OAM-carrying BG beams are a linear combination of integer BG beams. Considering the α = 0 case, the Bn parameter has the property ofEq. (9) can be rewritten asEq. (9), the fractional-charged OAM beams are essentially the combination of integer-charged OAM beams. For each of these integer components, the parameter Bn works as a window to distribute the power between the integer charge OAMs and decays rapidly as parameter n approaches positive and negative infinity.
As can be seen in Eq. (11), a change in β only affects the weighting factor Bn. Conceptually, when β is very small, very little power will be contained at the edges of the active zone on the log polar elements. When this whole area is wrapped, there will be a highly asymmetric ring. As β approaches 1, the distribution about the wrapped ring becomes more azimuthally Gaussian. In fact, as β increases beyond 1, the distribution about the wrapped ring becomes more azimuthally uniform and the weighting factors Bl ± 1 decrease, but more of the power will be clipped by the log-polar optic aperture. This results in a lower power efficiency of the system but higher modal symmetry. Equation (9) not only describes the distribution of integer charge numbers, but also fractional charge numbers. Figure 2 shows the analytic intensity and phase profiles using simulation parameters λ = 532 nm, β = 0.66, wring = 329 µm, ρ0 = 850 µm, using 5 central terms, and for the focal length of Fourier lens F = 400 mm. Due to the small radius of the Gaussian envelope, only one faint ring of the 0th order Bessel-Gaussian beam appears in the simulations and experimental results.
The log-polar coordinate transform theory assumes that the input is a rectangular shaped beam [20–23]. This notion in fact reduces the translation efficiency of such systems due to the fact that a Fourier transform of a rectangular function contains high spatial frequency components. On the other hand, the Fourier transform of a Gaussian shape produces another Gaussian distribution. In the HOBBIT system presented above, an elliptical Gaussian beam is easily generated from a Gaussian input. This has the added benefit of a higher power efficiency compared to that of a rectangular beam input.
3. Diffractive phase-only optics and experiment setup
3.1 Diffractive phase-only optics
The diffractive log-polar HOBBIT elements are fabricated using a photolithographic method in our cleanroom facility. As shown in Fig. 3(a), 6 row × 6 column devices has been fabricated on a single wafer. The optics are optimized for the wavelength of 532 nm, and have a pixel size of 2 μm × 2 μm and 24 = 16 phase levels. The design parameter a is 1.8/π mm and b is 2 mm. The microscope profiles of a wrapper and phase-corrector are shown in Fig. 3(b) and (c). Scanning-electron microscopy (SEM) images of the fabricated optics are shown in Fig. 3(d) and (e) with magnification of 130 × . The theoretical diffraction efficiency of a 4-layer lithographic process diffractive phase element is about 98%. After applying a 99.9% transmission anti-reflection (AR) coating on each surface of the HOBBIT optics, the mean transmission efficiency of both the wrapper and phase corrector combined has been measured to be 91% with 0.5% standard deviation from charge −10 to 10.
3.2 Experiment setup and results
The AOD couples up to 70% of the optical energy into its 1st diffraction order. This deflection angle is continuously tunable by adjusting the frequency of the acoustic signal. As mentioned above, our experimental setup applies a 4-f system to image the AOD output deflection angle into the line shape beam’s linear phase and another 4-f system to elongate the circular Gaussian beam into an elliptical Gaussian beam. The elliptical Gaussian beam is incident upon the wrapper and then is mapped into an azimuthally asymmetric ring shaped beam during propagation to the phase corrector. After phase correction at the second optical element, the ring-shaped beam carrying OAM phase will form a BG beam in the far-field. A diagram of the experimental setup is shown in Fig. 4.
The deflected beam was generated using a Gooch & Housego AODF 4120-3. This AOD is constructed using a tellurium dioxide (TeO2) crystal, with a Bragg angle of 2.9°, computed by Eq. (1), as shown in Fig. 5. The acoustic velocity is 0.65 mm/µs, typical for the shear mode of a TeO2 crystal. An input beam with a diameter of approximately 1.5 mm can be deflected at a rate of approximately 434.8 kHz, corresponding to a measured switching speed of 2.3 µs. Higher switching speeds are achievable in other materials such as quartz and fused silica. The acoustic velocity of such devices can be an order of magnitude above the shear-mode TeO2 devices. By decreasing the beam size through a crystal and with a faster acoustic velocity, switching speeds could be further increased into the tens of megahertz. A picture of the compact experimental setup is shown in Fig. 5. The transmission efficiency of each surfaces of the 3 optics in the line generator is 99%, and the total transmission efficiency of log-polar OAM generator is 91%. Taking into account the 70% AOD’s 1st order diffractive efficiency (DE), the total system efficiency is approximately 60%. Given a 30 mW input power, the output BG beam is approximately 18 mW. The current setup is restricted to only one linear polarization due to the requirements of the AOD. However, there are other types of AODs that are polarization insensitive. In this case, different polarization states could be used in the HOBBIT system since the log-polar optics are polarization insensitive.
The focal lengths L1 and L2 are F1 = 50 mm and F2 = 100 mm respectively, parameter a = 1.8/π mm, and the frequency index corresponding to Δm = 1 interval is Δf1 = 0.36 MHz. A series of rings with different OAM phases are output from the log-polar HOBBIT optics. The far-field of this group of ring shape OAM phase carrying beams are BG beams . The generated BG beams are experimental generated, imaged and simulated using Eq. (9) as shown in Fig. 6. The experimental results have good agreement with the simulation results. A comparison of the radius of the dark vortex to the corresponding charge numbers as well as driving signal frequencies is shown in Fig. 7 for both the experimental and simulated beam profiles. This radius was measured by finding the inner radial location of the half-maximum amplitude. The simulation is an approximation of an infinite series. The slight dips in the curve are a result of truncating the infinite series to obtain this approximation. The DE of the m = −5 beam is 8.8% lower than the DE of the m = 5 beam because the deviation away from the Bragg condition that has the highest DE.
The deflection angle of the 1st order AOD output is continuously tunable, therefore the OAM phase is continuously tunable as well. The intensity distributions of the fractional OAM modes spanning from charge −1.2 to + 1.2 in steps of 0.6 are shown in Fig. 8. The charge numbers are verified by the single stationary cylindrical lens method [43–45] as −1.21 ± 0.03, −0.63 ± 0.03, −0.01 ± 0.08, 0.64 ± 0.02 and 1.21 ± 0.02. Two independent methods were used to verify the OAM charge numbers. First, the dark center of the beam intensity was measured and second, a single stationary cylindrical lens method was used. Both independent methods agree and are consistent with the designed OAM charge numbers. See Visualization 1 for a video of the demonstrating scan for charges −3 to + 3 in steps of 0.2.
In conclusion, we have proposed a method of cascading an AOD with the HOBBIT log-polar transform optical system to rapidly and continuously tune the output OAM mode of a BG beam. This means the HOBBIT system has the capability of generating tunable fractional OAM modes. The OAM mode is controlled through the AOD driving frequency, which controls the amount of linear tilt to be wrapped into a ring through the log-polar transformation. The tuning speeds of AODs are limited by the velocity of the acoustic wave through the crystal, which has the potential to well exceed conventional mode manipulators such as DMDs, SLMs, and single electrically contacted thermos-optical controlled vortex emitters. The scalar form of the far-field HOBBIT has been analytically derived, resulting in a group of asymmetric fractional BG beams. See Visualization 1 for a video demonstrating a scan for charges −3 to + 3 in steps of 0.2.
This technique provides a fast and continuous OAM carrying BG beam tuning solution. This HOBBIT system may benefit a multitude of areas not limited to communication from classical to quantum applications, particle optical manipulation, beam shaping, laser beam machining, microscopy, microlithography, direct energy, filamentation, as well as sensing through turbulence in air and underwater environments. OAM is rapidly gaining interest in all of these areas and the tunable capabilities of this system has the potential to open up the in-depth study of these modes under various conditions, including environments that change slowly such as turbulence. Also, the AOD can support a superposition of driving frequencies that result in multiple OAM modes being generated simultaneously. Because of this, future work will consist of exploring coherent combinations of OAM modes from this AOD based HOBBIT system.
The full derivation of the far field asymmetric Bessel Gaussian beam is presented in this appendix. The near-field output from the proposed HOBBIT system represented by Eq. (8) can be rewritten as separable functions with respect to only ρ or ϕ terms,
As shown in Eq. (15), there are two Fourier transforms that need to be solved. It is well known that the Fourier transform of a Gaussian expression is a Gaussian as well. The other Fourier transform term is more interesting and complicated. Starting with the definition of the polar coordinate Fourier transform,
Eq. (15) can be written as,Eq. (17) can be re-written as,46]Eq. (15) reduces to the final analytic terms as given by Eq. (9), in which the Bn term is the weighting term and shifts power between different Bessel terms. Considering the α = 0 case, given n = m + k = l + k, k = 0, 1, 2…Eq. (9) will reduce into Eq. (11).
Office of Naval Research (ONR) (N00014-16-1-3090, N00014-17-1-2779).
1. M. Chagnon, M. Osman, Q. Zhuge, X. Xu, and D. V. Plant, “Analysis and experimental demonstration of novel 8PolSK-QPSK modulation at 5 bits/symbol for passive mitigation of nonlinear impairments,” Opt. Express 21(25), 30204–30220 (2013). [CrossRef] [PubMed]
2. A. J. Willner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, “Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40(24), 5810–5813 (2015). [CrossRef] [PubMed]
4. T. Lei, S. Gao, Z. Li, Y. Yuan, Y. Li, M. Zhang, G. N. Liu, X. Xu, J. Tian, and X. Yuan, “Fast-switchable OAM-based high capacity density optical router,” IEEE Photonics J. 9(1), 1 (2017). [CrossRef]
6. M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012). [CrossRef] [PubMed]
7. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014). [CrossRef]
8. C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8(12), 931–936 (2014). [CrossRef] [PubMed]
9. H. Wang, F. Yuan, S. Chang, P. Sun, S. Zhang, H. Li, Z. Zheng, S. Liu, T. Xie, and C. Wang, “Arbitrary manipulation of micro-particles in three dimensions by steering of multiple orbital angular momentum modes,” Proc. SPIE 10347, 35 (2017). [CrossRef]
11. M. P. J. Lavery, S. M. Barnett, F. C. Speirits, and M. J. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica 1(1), 1–4 (2014). [CrossRef]
12. Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098–5101 (2014). [CrossRef] [PubMed]
14. M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5(1), 4856 (2014). [CrossRef] [PubMed]
15. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004). [CrossRef]
16. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. 10(1), 015009 (2008). [CrossRef]
17. K. Huang, H. Liu, S. Restuccia, M. Q. Mehmood, S. Mei, D. Giovannini, A. Danner, M. J. Padgett, J. Teng, and C. Qiu, “Spiniform phase-encoded metagratings entangling arbitrary rational-order orbital angular momentum,” Light Sci. Appl. 7(3), 17156 (2018). [CrossRef]
19. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008). [CrossRef] [PubMed]
20. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64(8), 1092–1099 (1974). [CrossRef]
21. O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10(2), 164–168 (1974). [CrossRef]
23. M. A. Stuff and J. N. Cederquist, “Coordinate transformations realizable with multiple holographic optical elements,” J. Opt. Soc. Am. 7(6), 977–981 (1990). [CrossRef]
24. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010). [CrossRef] [PubMed]
25. H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015). [CrossRef] [PubMed]
26. T. Mhlanga, A. Dudley, A. Mcdonald, F. S. Roux, M. Lavery, M. Padgett, and A. Forbes, “Efficient sorting of Bessel beams,” Proc. SPIE 8637, 86371C (2013). [CrossRef]
28. I. R. Srimathi, K. Miller, W. Li, K. S. Morgan, J. Baghdady, and E. G. Johnson, “Diffractive orbital angular momentum demultiplexing elements for underwater optical communications,” in Frontiers in Optics (Optical Society of America, 2016), p. FTh4E.2.
29. W. Li, K. Miller, I. R. Srimathi, Y. Li, K. S. Morgan, and E. G. Johnson, “Efficient 1550 nm diffractive log-polar element based orbital angular momentum mode-division multiplexing,” in Frontiers in Optics (Optical Society of America, 2017), p. JTu2A.37.
31. G. Ruffato, M. Massari, G. Parisi, and F. Romanato, “Test of mode-division multiplexing and demultiplexing in free-space with diffractive transformation optics,” Opt. Express 25(7), 7859–7868 (2017). [CrossRef] [PubMed]
32. S. Lightman, G. Hurvitz, R. Gvishi, and A. Arie, “Miniature wide-spectrum mode sorter for vortex beams produced by 3D laser printing,” Optica 4(6), 605–610 (2017). [CrossRef]
33. K. S. Morgan, I. S. Raghu, and E. G. Johnson, “Design and fabrication of diffractive optics for orbital angular momentum space division multiplexing,” Proc. SPIE 9374, 93740Y (2015). [CrossRef]
34. G. Ruffato, M. Girardi, M. Massari, F. Romanato, E. Mafakheri, and P. Capaldo, “Compact diffractive optics for high-resolution sorting of orbital angular momentum beams,” Proc. SPIE 10744, 18 (2018). [CrossRef]
35. W. Akemann, J.-F. Léger, C. Ventalon, B. Mathieu, S. Dieudonné, and L. Bourdieu, “Fast spatial beam shaping by acousto-optic diffraction for 3D non-linear microscopy,” Opt. Express 23(22), 28191–28205 (2015). [CrossRef] [PubMed]
36. A. Grinenko, M. P. MacDonald, C. R. P. Courtney, P. D. Wilcox, C. E. M. Demore, S. Cochran, and B. W. Drinkwater, “Tunable beam shaping with a phased array acousto-optic modulator,” Opt. Express 23(1), 26–32 (2015). [CrossRef] [PubMed]
40. A. B.-C. Arturo Barcelo-Chong, B. E.-P. Brian Estrada-Portillo, A. C.-B. Arturo Canales-Benavides, and S. L.-A. Servando Lopez-Aguayo, “Asymmetric Mathieu beams,” Chin. Opt. Lett. 16(12), 122601 (2018). [CrossRef]
41. G. Ruffato, M. Massari, and F. Romanato, “Diffractive optics for combined spatial- and mode- division demultiplexing of optical vortices: design, fabrication and optical characterization,” Sci. Rep. 6(1), 24760 (2016). [CrossRef] [PubMed]
43. S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016). [CrossRef] [PubMed]
45. J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007). [CrossRef]
46. I. S. G. and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2007).