I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. DETERMINING THE OAM S...III. THE EFFECT OF WAVE F...IV. ANGULAR MISALIGNMENT ...V. EXPERIMENTAL EXAMPLE O...VI. CONCLUSION AND DISCUS...SUPPLEMENTARY MATERIALREFERENCESIn 1992, Allen et al. reported in their seminal paper that optical vortices with an azimuthally varying phase exp(iℓφ) carry a well-defined amount of orbital angular momentum (OAM) ℓℏ per photon.11. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). https://doi.org/10.1103/physreva.45.8185 Here, φ is the azimuthal angle of the polar coordinates (r, φ, z), ℓ is an integer number known as the topological charge, which is associated with the number of times the phase wraps around the optical axis, and ℏ is the reduced Planck constant. This discovery ignited a new era of fundamental and applied research, at both the classical and the quantum levels (see, for example, Ref. 22. M. J. Padgett, “Orbital angular momentum 25 years on (invited),” Opt. Express 25, 11265–11274 (2017). https://doi.org/10.1364/oe.25.011265 and references therein). Applications have been proposed in optical micro- and nano-manipulation, optical communications, high-resolution microscopy, and optical metrology, among many others.3–113. C. Rosales-Guzmán, N. Hermosa, A. Belmonte, and J. P. Torres, “Experimental detection of transverse particle movement with structured light,” Sci. Rep. 3, 2815 (2013). https://doi.org/10.1038/srep028154. A. Belmonte, C. Rosales-Guzmán, and J. P. Torres, “Measurement of flow vorticity with helical beams of light,” Optica 2, 1002–1005 (2015). https://doi.org/10.1364/optica.2.0010025. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013). https://doi.org/10.1126/science.12399366. Y. Yang, Y. Ren, M. Chen, Y. Arita, and C. Rosales-Guzmán, “Optical trapping with structured light: A review,” Adv. Photonics 3, 034001 (2021). https://doi.org/10.1117/1.ap.3.3.0340017. E. Otte and C. Denz, “Optical trapping gets structure: Structured light for advanced optical manipulation,” Appl. Phys. Rev. 7, 041308 (2020). https://doi.org/10.1063/5.00132768. S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007). https://doi.org/10.1126/science.11373959. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66 (2015). https://doi.org/10.1364/aop.7.00006610. A. E. Willner, Z. Zhao, C. Liu, R. Zhang, H. Song, K. Pang, K. Manukyan, H. Song, X. Su, G. Xie, Y. Ren, Y. Yan, M. Tur, A. F. Molisch, R. W. Boyd, H. Zhou, N. Hu, A. Minoofar, and H. Huang, “Perspectives on advances in high-capacity, free-space communications using multiplexing of orbital-angular-momentum beams,” APL Photonics 6, 030901 (2021). https://doi.org/10.1063/5.003123011. D. McArthur, A. M. Yao, and F. Papoff, “Scattering of light with angular momentum from an array of particles,” Phys. Rev. Res. 2, 013100 (2020). https://doi.org/10.1103/physrevresearch.2.013100 In some of these applications, the OAM of light serves as a path-breaking information carrier. For instance, when a light field interacts with chiral structures of scattering nanoparticles,1111. D. McArthur, A. M. Yao, and F. Papoff, “Scattering of light with angular momentum from an array of particles,” Phys. Rev. Res. 2, 013100 (2020). https://doi.org/10.1103/physrevresearch.2.013100 the OAM of light changes, now carrying information about the chiral nanostructure itself. By OAM analysis, this information can be extracted and nanoscale properties can be unveiled.Notably, in most applications, an accurate determination of the OAM spectrum, i.e., unraveling the content of OAM by its topological charge ℓ, is crucial, leading to several proposals on how to measure it. Since the early days, many techniques have relied on interferometric approaches, in some of which the unknown field interferes with a plane wave12,1312. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). https://doi.org/10.1103/physreva.56.406413. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28, 2285–2287 (2003). https://doi.org/10.1364/ol.28.002285 or an inverted copy of itself.14–2014. O. Bryngdahl, “Radial- and circular-fringe interferograms,” J. Opt. Soc. Am. 63, 1098–1104 (1973). https://doi.org/10.1364/josa.63.00109815. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002). https://doi.org/10.1103/physrevlett.88.25790116. H. Wei and X. Xue, “Simplified measurement of the orbital angular momentum of single photons,” Opt. Commun. 223, 117–122 (2003). https://doi.org/10.1016/s0030-4018(03)01619-517. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004). https://doi.org/10.1103/PhysRevLett.92.01360118. S. Li, P. Zhao, X. Feng, K. Cui, F. Liu, W. Zhang, and Y. Huang, “Measuring the orbital angular momentum spectrum with a single point detector,” Opt. Lett. 43, 4607–4610 (2018). https://doi.org/10.1364/ol.43.00460719. G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017). https://doi.org/10.1038/s41467-017-01215-x20. R. Aboushelbaya, K. Glize, A. F. Savin, M. Mayr, B. Spiers, R. Wang, N. Bourgeois, C. Spindloe, R. Bingham, and P. A. Norreys, “Measuring the orbital angular momentum of high-power laser pulses,” Phys. Plasmas 27, 053107 (2020). https://doi.org/10.1063/5.0005140 Thereby, the discrete topological charge is extracted from the interference pattern, for example, by counting the number of arms in the spiraling intensity strips pattern. Other techniques employ diffraction phenomena, e.g., by propagating the OAM-carrying beams through an annular aperture2121. C.-S. Guo, L.-L. Lu, and H.-T. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34, 3686–3688 (2009). https://doi.org/10.1364/ol.34.003686 or an axicon.2222. Y. Han and G. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett. 36, 2017–2019 (2011). https://doi.org/10.1364/ol.36.002017 Since the number of bright rings in the far- or near-field intensity profile, respectively, is equal to the discrete topological charge of the beam, the OAM spectrum can be identified. In a variant of these approaches, a triangular aperture can be implemented.2323. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010). https://doi.org/10.1103/PhysRevLett.105.053904 In this case, the bright points at any side of the triangular diffraction pattern are directly related to its topological charge—an approach that has been generalized to also determine the radial index in Laguerre–Gaussian modes.2424. M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012). https://doi.org/10.1063/1.4728111 Other approaches include phase-shifting digital holography,2525. C.-S. Guo, X. Cheng, X.-Y. Ren, J.-P. Ding, and H.-T. Wang, “Optical vortex phase-shifting digital holography,” Opt. Express 12, 5166–5171 (2004). https://doi.org/10.1364/opex.12.005166 the weak measurement principle,2626. J. Zhu, P. Zhang, Q. Li, F. Wang, C. Wang, Y. Zhou, J. Wang, H. Gao, L. C. Kwek, and F. Li, “Measuring the topological charge of orbital angular momentum beams by utilizing weak measurement principle,” Sci. Rep. 9, 7993 (2019). https://doi.org/10.1038/s41598-019-44465-z and the rotational Doppler effect,2727. H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light: Sci. Appl. 6, e16251 (2017). https://doi.org/10.1038/lsa.2016.251 to name a few. More recently, the use of customized refractive optical elements has led to the development of a device termed the mode sorter, which allows direct measurement of discrete topological charges. Such a device relies on the conversion of OAM modes into modes with a transverse phase gradient. Depending on their topological charge, the modes are focused to different lateral positions.2828. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012). https://doi.org/10.1364/oe.20.002110 Subsequently, a generalized digital version of this mode sorter was proposed, to not only measure the topological charge but also the radial index of Laguerre–Gaussian modes, termed the Laguerre–Gaussian mode sorter.2929. N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019). https://doi.org/10.1038/s41467-019-09840-4Most of the above-mentioned techniques only allow for extracting discrete ℓ values and/or can be successfully applied only to ideal, i.e., undisturbed vortex beams or beams in static experimental configurations. In contrast, an accurate determination of the continuous OAM spectrum under nonideal conditions represents an open challenge, which includes light propagating through a faulty, e.g., astigmatic optical system, under atmospheric turbulence, or when experimental settings change dynamically. This can be the case, for example, in fiber or free-space optical communication with structured light.9,10,309. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66 (2015). https://doi.org/10.1364/aop.7.00006610. A. E. Willner, Z. Zhao, C. Liu, R. Zhang, H. Song, K. Pang, K. Manukyan, H. Song, X. Su, G. Xie, Y. Ren, Y. Yan, M. Tur, A. F. Molisch, R. W. Boyd, H. Zhou, N. Hu, A. Minoofar, and H. Huang, “Perspectives on advances in high-capacity, free-space communications using multiplexing of orbital-angular-momentum beams,” APL Photonics 6, 030901 (2021). https://doi.org/10.1063/5.003123030. J. Wang, S. Chen, and J. Liu, “Orbital angular momentum communications based on standard multi-mode fiber (invited paper),” APL Photonics 6, 060804 (2021). https://doi.org/10.1063/5.0049022 Here, one of the “killing” factors is the resulting modal-crosstalk, which leads to a broadening of the OAM spectrum. This crosstalk ultimately reduces the transmission capacity of the optical link.3131. A. Trichili, M. A. Cox, B. S. Ooi, and M.-S. Alouini, “Roadmap to free space optics,” J. Opt. Soc. Am. B 37, A184–A201 (2020). https://doi.org/10.1364/josab.399168 One of the main effects caused by the example of turbulence is the transverse shift of transmitted modes (also known as tip or tilt). Nonetheless, for an optical vortex beam to be accurately detected, its phase singularity (point of undefined phase) has to be centered with respect to the OAM decoding system, e.g., the transverse phase vortex encoded in a decoding hologram.3232. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016). https://doi.org/10.1364/aop.8.000200 Any misalignment will result in a crosstalk: As OAM beams represent a complete mode set, a misaligned optical beam can be represented as a complex linear combination of different OAM modes.3333. M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, “Analysis of orbital angular momentum of a misaligned optical beam,” New J. Phys. 7, 46 (2005). https://doi.org/10.1088/1367-2630/7/1/046 As a consequence, the OAM spectrum significantly and unintentionally broadens. Therefore, many approaches have been proposed to overcome such issues.34–3834. M. A. Cox, L. Maqondo, R. Kara, G. Milione, L. Cheng, and A. Forbes, “The resilience of Hermite– and Laguerre–Gaussian modes in turbulence,” J. Lightwave Technol. 37, 3911–3917 (2019). https://doi.org/10.1109/jlt.2019.290563035. Y.-D. Liu, C. Gao, X. Qi, and H. Weber, “Orbital angular momentum (OAM) spectrum correction in free space optical communication,” Opt. Express 16, 7091–7101 (2008). https://doi.org/10.1364/oe.16.00709136. J. Lin, X.-C. Yuan, M. Chen, and J. C. Dainty, “Application of orbital angular momentum to simultaneous determination of tilt and lateral displacement of a misaligned laser beam,” J. Opt. Soc. Am. A 27, 2337–2343 (2010). https://doi.org/10.1364/josaa.27.00233737. S. Lohani and R. T. Glasser, “Turbulence correction with artificial neural networks,” Opt. Lett. 43, 2611–2614 (2018). https://doi.org/10.1364/ol.43.00261138. Q. Zhao, S. Hao, Y. Wang, L. Wang, X. Wan, and C. Xu, “Mode detection of misaligned orbital angular momentum beams based on convolutional neural network,” Appl. Opt. 57, 10152–10158 (2018). https://doi.org/10.1364/ao.57.010152 For example in Ref. 3434. M. A. Cox, L. Maqondo, R. Kara, G. Milione, L. Cheng, and A. Forbes, “The resilience of Hermite– and Laguerre–Gaussian modes in turbulence,” J. Lightwave Technol. 37, 3911–3917 (2019). https://doi.org/10.1109/jlt.2019.2905630, the authors proposed to use optical modes with less sensitivity for transverse shifts, such as Hermite–Gaussian modes. Alternative approaches suggest to identify the tilt angle and correct for this aberration either by recovering the phase distribution of the tilted OAM mode or by the use of machine learning techniques.37–3937. S. Lohani and R. T. Glasser, “Turbulence correction with artificial neural networks,” Opt. Lett. 43, 2611–2614 (2018). https://doi.org/10.1364/ol.43.00261138. Q. Zhao, S. Hao, Y. Wang, L. Wang, X. Wan, and C. Xu, “Mode detection of misaligned orbital angular momentum beams based on convolutional neural network,” Appl. Opt. 57, 10152–10158 (2018). https://doi.org/10.1364/ao.57.01015239. P. Zhao, S. Li, Y. Wang, X. Feng, C. Kaiyu, L. Fang, W. Zhang, and Y. Huang, “Identifying the tilt angle and correcting the orbital angular momentum spectrum dispersion of misaligned light beam,” Sci. Rep. 7, 7873 (2017). https://doi.org/10.1038/s41598-017-07734-3

Nevertheless, the accurate alignment of a physical electric field of interest to its likewise physical, analog optical decoding system remains a major challenge, especially when disturbances or dynamic changes are involved. The implementation of digitally controllable elements as spatial light modulators (SLMs) or digital mirror devices (DMDs) eases this process; however, they cannot fully solve the problem. They allow for digital, computer-generated decoding holograms, which can be adapted dynamically and enable including, for instance, correcting functions for some disturbances. However, accurate hologram position and correcting functions have to be determined beforehand, but they might change dynamically. Hence, to enable accurate determination of the OAM spectrum of light under nonideal, maybe dynamically varying conditions, we propose to go all-digital.

In this work, we present an all-digital technique aimed at extracting the continuous OAM spectrum of a disturbed light field, with specific applications to those that have suffered from small perturbations due to, for instance, lens aberrations or angular misalignment. We demonstrate this approach through numerical simulations and a proof-of-concept experiment, using a glass ball lens as the perturbing medium. We transform the physical light field of interest into a digital one by established digital holographic phase metrology4040. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978). https://doi.org/10.1364/ol.3.000027 (DHPM)—the interference of the target optical field with a reference field—and, subsequently, digitally process the recorded interference pattern. This single-shot measurement facilitates the accurate determination of the continuous OAM spectrum by implementing adaptable correction patterns, paving the way to OAM spectrum analysis in dynamic, high-speed systems. For this purpose, we implement tools of singular optics for precise identification of the transverse decoding position in light. Further, we highlight the effect of typical errors occurring in physical decoding systems on the OAM spectrum and present approaches for their correction by our all-digital technique. The proposed all-digital, single-shot approach is particularly beneficial for high-speed investigations, less sensitive to errors such as imperfections of optical systems, and enables the accurate determination of continuous OAM spectra with high precision due to optimized and automated determination of the decoding position and correction of dynamically changing disturbances.

II. DETERMINING THE OAM SPECTRUM OF LIGHT

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. DETERMINING THE OAM S... <<III. THE EFFECT OF WAVE F...IV. ANGULAR MISALIGNMENT ...V. EXPERIMENTAL EXAMPLE O...VI. CONCLUSION AND DISCUS...SUPPLEMENTARY MATERIALREFERENCES

A. Standard holographic OAM decoding

To holographically decode the OAM spectrum of a scalar light field E = E0 · exp(iϕ), a combination of a phase-only spatial light modulator (SLM) and a Fourier lens (LF) is typically applied,3232. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016). https://doi.org/10.1364/aop.8.000200 as illustrated in Fig. 1(a). Thereby, the field E is multiplied by the SLM phase function in the SLM plane (=real space) and the lens creates the far field of the result on a detector, e.g., a camera. This standard decoding approach relies on the inner product of E and each of the OAM-carrying modes uℓ = exp(iℓφ) (φ ∈ [0, 2π]) in the subspace of interest ℓ = [ℓmin, ℓmax]. This technique is based on modal decomposition (for more details, see Ref. 3232. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016). https://doi.org/10.1364/aop.8.000200), where we assume that each optical field can be expressed as a linear combination of basis functions from an orthogonal set of spatial modes,3232. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016). https://doi.org/10.1364/aop.8.000200u(r⃗)=∑ℓ=1∞cℓuℓ(r⃗),(1)with spatial coordinate r⃗ and complex coefficient cℓ = ρℓ exp(iϕℓ), its amplitude ρℓ and intermodal phase Δϕℓ = ϕℓ − ϕ0. By displaying uℓ* as a phase-only function (hologram), that is, arg(uℓ*)=−ℓφ on the SLM, the product ρℓ2=|⟨uℓ|E⟩|2 is shaped. In this case, the on-axis intensity in the Fourier plane of the SLM, i.e., on the detector, is proportional to the power content of the OAM mode of topological charge ℓ. By measuring the power content per ℓ of interest, displaying the according function arg(uℓ*) on the SLM, the OAM spectrum of E is determined.Note that one can choose to display multiple decoding holograms at a time by, e.g., angular/spatial multiplexing41–4341. E. Otte, K. Tekce, and C. Denz, “Spatial multiplexing for tailored fully-structured light,” J. Opt. 20, 105606 (2018). https://doi.org/10.1088/2040-8986/aadef342. C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19, 113501 (2017). https://doi.org/10.1088/2040-8986/aa8b8e43. S. Li and J. Wang, “Simultaneous demultiplexing and steering of multiple orbital angular momentum modes,” Sci. Rep. 5, 15406 (2015). https://doi.org/10.1038/srep15406 in order to speed up the measuring process. In this case, each hologram uℓ* is assigned to a selected spatial carrier frequency by adding a blazed grating such that ρℓ2 is found at different transverse positions in the Fourier space. Thereby, the system has to be calibrated with respect to the diffraction efficiency per grating. Further, the number of simultaneously displayed holograms is limited by, for instance, the resolution of the SLM.We illustrate this standard approach (without multiplexing) in Fig. 1 (simulation) for decoding the OAM spectrum of a helical Laguerre–Gaussian (LG) light field described by a complex amplitude,44–4644. A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).45. G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961). https://doi.org/10.1002/j.1538-7305.1961.tb01626.x46. B. E. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Hoboken, 2019).LGp,ℓ(r,φ,z)=Ap,ℓ(r,z)⋅eikr22R(z)⋅eiϕp,ℓG(z)⋅eiℓφ,(2)Ap,ℓ(r,z)=2p!π(|ℓ|+p)!⋅1w(z)⋅e−r2w2(z)×r2w(z)|ℓ|⋅Lp|ℓ|2r2w2(z),(3)ϕp,ℓG(z)=(2p+|ℓ|+1)ϕ0,0G(z),(4)where, k is the wave number, R(z) the wave front curvature, w(z) the beam radius, w0 = w(0) the beam waist, p∈N0 the radial mode index, Lnℓ(⋅) the eponymous Laguerre polynomial,4747. A. Schutza, “Transmission of quantum information via Laguerre-Gaussian modes,” McNair Scholars J. 14(1), 8 (2010). ϕn,ℓG the Gouy phase shift of LG light fields, and ϕ0,0G the Gouy phase shift of fundamental Gaussian light field. The phase vortex structure exp(iℓφ) is related to a twisted wave front upon propagation and causes the formation of a point of undefined phase, i.e., a phase singularity on the propagation axis of LG fields. The optical decoding by the system of SLM, Fourier lens, and detector is simulated by the two-dimensional (2D) Fourier transform (F) of the product of the electric field and the phase function of the SLM hologram, i.e., F[E⋅exp(−iℓφ)].Note that ℓ is also considered as the singularity index of the embedded phase singularity (ℓsing=∮dφ/2π∈Z) and, for those, represents a conserved quantity.48,4948. M. R. Dennis, K. O’Holleran, and M. J. Padgett, in Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), Chap. 5, pp. 293–363.49. M. S. Soskin and M. V. Vasnetsov, in Singular Optics (Elsevier, 2001), Chap. 4, pp. 219–277. However, ℓ (integer and fractional) is also established as a measure for OAM of vortex beams and is used as such in the context of this work.The exemplary scalar light field has a topological charge ℓ = 1, radial index p = 0, and beam waist w0 = 0.25 mm. Its ring-shaped intensity and vortex phase are presented in Figs. 1(b) and 1(c), respectively. When this light field passes the analysis system of SLM and Fourier lens, in Fourier space, we observe intensity structures as illustrated in Fig. 1(e) for integer ℓ = [−2, 2]. Considering the respective on-axis intensities as the power content per OAM mode, we can determine the OAM content or spectrum, as presented in Fig. 1(d). Note that, here and in the following, we also consider fractional OAM values with ℓ = [−3, 3], ∈R, resulting in continuous OAM spectra, which are of specific interest for applications in optical sensing based on OAM,3,43. C. Rosales-Guzmán, N. Hermosa, A. Belmonte, and J. P. Torres, “Experimental detection of transverse particle movement with structured light,” Sci. Rep. 3, 2815 (2013). https://doi.org/10.1038/srep028154. A. Belmonte, C. Rosales-Guzmán, and J. P. Torres, “Measurement of flow vorticity with helical beams of light,” Optica 2, 1002–1005 (2015). https://doi.org/10.1364/optica.2.001002 in particular in nanoscale systems1111. D. McArthur, A. M. Yao, and F. Papoff, “Scattering of light with angular momentum from an array of particles,” Phys. Rev. Res. 2, 013100 (2020). https://doi.org/10.1103/physrevresearch.2.013100 and/or if minimal changes in OAM content might be of interest. As expected, a clear, absolute maximum is observed for ℓ = 1, reflecting the chosen topological charge of the exemplary LG field. While, as expected, for integer values with ℓ ≠ 1, minima (=0) are found, OAM of fractional ℓ values reveals nonzero contributions to the total spectrum. This is due to the fact that modes of fractional ℓ can again be represented by the complete mode set of integer OAM fields and, therefore, also contain contributions of OAM modes of charge ℓ = 1.Even though this measurement procedure is well established, it still presents some challenges to the general user. One of these challenges is the appropriate transverse positioning of the decoding hologram in relation to the optical axis of the light field or, more precisely, to the center of the embedded optical vortex, which represents the carrier of OAM information. In Fig. 2, we illustrate the effect of misaligning hologram and optical vortex when analyzing the OAM spectrum of light. Again, we use the LG0,1 field as illustrative example. We assume a transverse mismatch between the LG field and the hologram by changing the x-distance between their centers. Note that the position of the hologram is shifted by Δx in the x-direction, while the light field stays centered at (x, y) = (0, 0), assuming that, in the corresponding experiment, the field and Fourier lens are well-adjusted in the optical system. Figure 2(a) reveals an increasing shift (in negative ℓ direction) in the OAM spectrum for increasing Δx (orange, green), with its distribution deviating from the ideal curve (blue). The respective change of the intensity distribution in Fourier space ρl2 (on the detector) is exemplified for Δx = 0.2 mm in Fig. 2(b). For a mismatch of the holograms and the light field center, the main contribution in OAM spectrum is no longer located at ℓ = 1, but it is shifted to fractional values ℓℓ = 2). Note that this shifted spectrum represents the correct OAM for the chosen point of reference, i.e., the central position of the shifted hologram. However, we are interested in the overall OAM carried by the beam in its propagation direction, i.e., the point of reference has to be located on the optical axis of the beam.Deviations from the accurate OAM spectrum, appearing for an incorrect point of reference, may become critical and cause incorrect results when implementing optical spectrum analysis in applications, such as information transfer and encoding50,5150. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). https://doi.org/10.1364/opex.12.00544851. M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015). https://doi.org/10.1088/1367-2630/17/3/033033 or the study of chiral media/structures.1111. D. McArthur, A. M. Yao, and F. Papoff, “Scattering of light with angular momentum from an array of particles,” Phys. Rev. Res. 2, 013100 (2020). https://doi.org/10.1103/physrevresearch.2.013100 Optical systems can generally be optimized such that the light field and the hologram center are matched as closely as possible. However, an exact match is hard or even impossible when the light field is moving in the transverse plane, e.g., as a result of propagation through a refracting, maybe, dynamic medium. The identification of the phase vortex center of the field is further complicated by the fact that the center of OAM-carrying light fields is, in general, dark with low to zero intensity. Tackling these issues, in the following, we propose an alternative all-digital approach to optimize holographic OAM spectrum analysis.

B. Single-shot, all-digital OAM spectrum analysis

The proposed all-digital OAM spectrum analysis is sketched in Fig. 3 (simulations), exemplified for E = LG0,1. For this approach, we first apply the established digital holographic phase metrology (DHPM; for details, see Refs. 4040. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978). https://doi.org/10.1364/ol.3.000027, 5252. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). https://doi.org/10.1364/josa.72.000156, and 5353. A. Zannotti, “Realization and exploration of structured light and photonic structures,” in Caustic Light in Nonlinear Photonic Media (Springer, Cham, 2020), Chap. 2, pp. 31–49.) to extract the amplitude and the phase of the light field E of interest and, simultaneously, digitalize it. Thereby, E (real space) is interfered off-axis with a plane wave of the same polarization. In the experiment, this plane wave is approximated by an expanded, collimated Gaussian laser beam. The formed transverse interference pattern [Fig. 3(a)] is recorded on a detector (camera) in real space—this pattern is all what is needed to subsequently extract the OAM spectrum digitally. In the next step, the interference pattern is Fourier transformed (F), giving a complex 2D field [intensity in Fig. 3(b)]. In this field, we identify and crop [see blue box in (b)] the contribution of E, located at nonzero spatial frequencies in k-space (Fourier space) with the location depending on the interference angle. The inverse Fourier transform (F−1) of the cropped complex field gives the measured electric field E with amplitude E0 and phase ϕ [Fig. 3(c)]. Note that measured phase values are only to be considered at positions where E is not approximately zero. Otherwise, random phase values (speckles) contribute to the later OAM spectrum. To do so in experiment, we derive a binary 2D mask from the measured amplitude, deleting all outer electric field information (i.e., the dark center of a donut beam will be kept) where the intensity |E0(x, y)|2 is smaller than 3% of its maximum.In order to extract the OAM spectrum of the light field, we perform the multiplication of E = E0 · exp(iϕ) with the hologram functions uℓ* per ℓ [Fig. 3(e)], Fourier transform the product [Fig. 3(f)], and determine the on-axis intensity in Fourier space [white cross in (f)] being proportional to the OAM mode contribution [resulting spectrum in Fig. 3(g)]. To avoid deviations due to misalignment of E and the