arxiv: 2605.06566 · v1 · submitted 2026-05-07 · ⚛️ physics.optics · quant-ph

Recognition: unknown

A Unified SU(2) Framework for Vector Beam Transformations and Complex Beam Shaping

Gayathri G T , Gururaj Kadiri

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:10 UTC · model grok-4.3

classification ⚛️ physics.optics quant-ph

keywords vector beamsSU(2) operationsdoubly inhomogeneous waveplatesstructured lightcomplex beam shapingspin-orbit couplingbirefringent elementspolarization transformations

0 comments

The pith

A single doubly inhomogeneous waveplate can implement any prescribed vector beam transformation exactly, including global phase, when a specific condition is met.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a constructive framework that treats transformations between structured light fields as spatially varying SU(2) operations on polarization. This leads to an explicit method for designing doubly inhomogeneous waveplates, called d-plates, that realize the desired mapping. The authors identify the precise condition under which one such element suffices for an exact implementation, including the overall phase factor, and show how to build it from a finite sequence of simpler singly inhomogeneous plates in configurations such as QHQ. The same operations also correspond to quantum channels acting on orbital angular momentum, with polarization acting as an ancilla. This unifies vector beam transformations, spin-orbit dynamics, and complex beam shaping inside one parameter-synthesis procedure.

Core claim

Transformations between vector beams are realized as spatially varying SU(2) operations; when the target mapping satisfies a stated algebraic condition, a single doubly inhomogeneous waveplate implements it exactly, including global phase, and can be constructed from a finite cascade of singly inhomogeneous plates.

What carries the argument

The doubly inhomogeneous waveplate (d-plate), which realizes a prescribed spatially varying SU(2) operation on the polarization degree of freedom and thereby maps one structured light field onto another.

If this is right

A broad class of structured-light problems, including vector-beam transformations, spin-orbital dynamics, and complex beam shaping, can be solved inside one SU(2) synthesis procedure.
The identical SU(2) operations implement quantum channels on the orbital-angular-momentum degree of freedom with polarization as a physical ancilla.
Explicit recipes exist for realizing any qualifying d-plate as a finite sequence of singly inhomogeneous plates, including the QHQ configuration.
The framework supplies a systematic route for designing next-generation photonic elements for structured light and spin-orbit information processing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction to sequences of singly inhomogeneous plates may lower fabrication complexity for certain target transformations.
The ancilla interpretation suggests direct translation of the same hardware into simple quantum-channel simulators on OAM states.
Relaxing the ideal-model assumption to include small fabrication errors would yield quantitative bounds on output fidelity that could be tested experimentally.

Load-bearing premise

Birefringent elements can be fabricated to produce the exact spatially varying SU(2) operations required, including global phase, without losses or deviations from the ideal model.

What would settle it

Fabricate the d-plate prescribed by the construction for a chosen vector-beam pair, then measure the output field’s amplitude, polarization, and phase distribution and compare it directly to the target field including overall phase.

Figures

Figures reproduced from arXiv: 2605.06566 by Gayathri G T, Gururaj Kadiri.

**Figure 1.** Figure 1: FIG. 1: Transformation corresponding to the quantum ˆ view at source ↗

**Figure 2.** Figure 2: FIG. 2: (a), (b) Transformations corresponding to view at source ↗

**Figure 3.** Figure 3: FIG. 3: The d-plate and waveplate parameters corre view at source ↗

**Figure 4.** Figure 4: FIG. 4: (a-e) The d-plates required for generating the five angle states |2, j⟩, for j = −2, . . . , 2 respectively, where the orange line indicates retardance Γ and the blue line indicates fast-axis orientation α. The vector beams emerging at the exit plane of these plates and its transverse plane intensity profiles, for horizontal input is represented in Appendix E. E. Quantum Maps The action of a d-pla… view at source ↗

**Figure 5.** Figure 5: FIG. 5: (a-e) The d-plates required for generating the five view at source ↗

**Figure 7.** Figure 7: FIG. 7: Evolution of the beam with horizontal polar view at source ↗

**Figure 6.** Figure 6: FIG. 6: Evolution of the beam with vertical polarization view at source ↗

**Figure 9.** Figure 9: FIG. 9: Evolution of the state view at source ↗

**Figure 11.** Figure 11: FIG. 11: (a-e) The five angle states view at source ↗

**Figure 10.** Figure 10: FIG. 10: Evolution of an incident light beam through view at source ↗

**Figure 13.** Figure 13: FIG. 13: (a-e) The five OAM states view at source ↗

read the original abstract

We present a constructive framework for designing transformations between structured light fields using birefringent optical elements, formulated in terms of SU(2) operations on polarization. Within this framework, transformations between vector beams are treated as spatially varying SU(2) operations, leading to a direct procedure for designing doubly inhomogeneous waveplates (d-plates) that implement the desired mapping. We identify a condition under which a single element implements a prescribed transformation exactly, including the global phase, and provide an explicit prescription for constructing the corresponding doubly inhomogeneous waveplate (d-plate) when this condition is satisfied, along with its realization using a finite sequence of singly inhomogeneous plates, including a QHQ configuration. Within this formulation, a broad class of problems in structured light can be treated within a single framework, including vector beam transformations, spin-orbital dynamics, and complex beam shaping. Crucially, the same SU(2) operations directly realize quantum channels on the orbital angular momentum degree of freedom, with polarization serving as a physical ancilla. These results establish a unified and explicitly constructive route to complex beam shaping and vector beam transformations based on SU(2) parameter synthesis, and provide a systematic foundation for designing next-generation photonic elements for structured light and spin-orbit information processing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit algebraic recipe for d-plates that realize exact vector-beam maps including global phase, plus a QHQ decomposition into ordinary plates.

read the letter

The main point is a constructive SU(2) procedure that turns a target transformation on vector beams into a single doubly inhomogeneous waveplate when a stated condition holds, and supplies a finite decomposition into singly inhomogeneous plates via QHQ. They also map the same operations onto quantum channels acting on OAM with polarization as ancilla. That unification of beam shaping, spin-orbit coupling, and channel synthesis under one parameter-free construction is the concrete advance. It moves past abstract group-theoretic descriptions by giving a direct build recipe that includes the global phase, which earlier work often left aside. The algebraic route appears self-contained once the condition is met, with no fitted parameters or circular definitions. On the practical side, the framework assumes ideal birefringent elements with exact spatial variation and no losses or fabrication errors. That idealization is standard but limits immediate applicability; real devices will need tolerance analysis the paper does not supply. The single-element condition is probably not universal, so the decomposition route is the more general contribution. The abstract and stress-test note suggest the derivations close without hidden propagation assumptions, but the full steps for condition identification and QHQ verification would need checking. This is useful for researchers who design structured-light components or spin-orbit photonic devices and want a systematic starting point rather than case-by-case intuition. It is not yet a complete experimental toolkit, but the formal construction is sharp enough to merit referee time. I would send it for review.

Referee Report

0 major / 2 minor

Summary. The manuscript presents a constructive SU(2) framework for designing transformations between structured light fields using birefringent optical elements. Transformations between vector beams are formulated as spatially varying SU(2) operations on polarization, yielding a direct procedure for constructing doubly inhomogeneous waveplates (d-plates). A condition is identified under which a single d-plate implements a prescribed transformation exactly (including global phase), together with an explicit prescription for the d-plate and its realization via a finite sequence of singly inhomogeneous plates (including a QHQ configuration). The same operations are shown to realize quantum channels on the orbital angular momentum degree of freedom with polarization as ancilla, unifying vector beam transformations, spin-orbital dynamics, and complex beam shaping.

Significance. If the algebraic constructions and condition hold, the work supplies a parameter-free, explicitly constructive route to photonic-element design for structured light and spin-orbit information processing. The self-contained SU(2) parameterization, the exact single-element realization under the stated condition, the finite decomposition into singly inhomogeneous plates, and the direct mapping to quantum channels on OAM constitute clear strengths that could streamline both theoretical design and experimental realization in optics.

minor comments (2)

The abstract is dense and would benefit from a single concrete example (e.g., a specific vector-beam map and the corresponding d-plate parameters) to illustrate the condition and QHQ sequence before the general claims.
Notation for the spatially varying SU(2) operators and the global-phase inclusion should be introduced with a short table or explicit definition early in the main text to aid readers unfamiliar with the inhomogeneous-waveplate literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive review of our manuscript. We appreciate the recognition of the SU(2) framework's strengths in providing an explicitly constructive route to vector beam transformations and its unification with quantum channel realizations. The recommendation for minor revision is noted; we will prepare a revised version accordingly.

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraic and self-contained

full rationale

The paper's central contribution is an algebraic parameterization of spatially varying SU(2) operations on polarization to construct d-plates and QHQ decompositions for vector beam maps. This is a direct constructive procedure from standard SU(2) group elements, with an explicit condition for single-element exact realization including global phase. No steps reduce by definition to the target result, no fitted inputs are relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems are required for the formal claims. The framework remains parameter-free once the condition is met and is internally self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard mathematical properties of the SU(2) group for polarization and the physical assumption that birefringent elements can be spatially engineered to match the required operations.

axioms (1)

domain assumption Polarization transformations induced by birefringent elements are accurately represented as SU(2) group operations
This is invoked throughout the framework for treating transformations as spatially varying SU(2) operations.

invented entities (1)

doubly inhomogeneous waveplate (d-plate) no independent evidence
purpose: To implement a prescribed vector beam transformation exactly as a single spatially varying SU(2) element
Introduced as the physical device realizing the framework when the condition is satisfied; no independent evidence provided.

pith-pipeline@v0.9.0 · 5522 in / 1495 out tokens · 80822 ms · 2026-05-08T06:10:27.934789+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

63 extracted references · 3 canonical work pages · 1 internal anchor

[1]

A Unified SU(2) Framework for Vector Beam Transformations and Complex Beam Shaping

and optical imaging for biomedical applications [9]. Vector beams are also structured light beams with spa- tially varying states of polarization (SoP) across their transverse plane [10]. In case of higher dimensions, a beam that is inhomogeneous in its Orbital Angular Mo- mentum (OAM) degree of freedom is also a vector beam. OAM plays a major role in the...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Two waveplates Any arbitrary SU(2) transformation can be realized using two waveplates ˆWΓ1(α1) and ˆWΓ2(α2) as: ˆSΓ(k) = ˆWΓ2(α2) ˆWΓ1(α1) (22) For a given ˆSΓ(k) there are infinite pairs of ˆWΓ1(α1) and ˆWΓ2(α2) that satisfy Eq. . (22). So, the parameters of the two waveplates, if one of them is set at half-wave retardance is calculated to be: Γ1 =π, co...
[3]

In this case, the three waveplates ˆWΓ1(α1), ˆWΓ2(α2) and ˆWΓ3(α3) need not be arbitrary

Three waveplates In general, a set of three waveplates is sufficient to real- ize any arbitrary SU(2) transformation. In this case, the three waveplates ˆWΓ1(α1), ˆWΓ2(α2) and ˆWΓ3(α3) need not be arbitrary. They can be set as two quarter-wave plates and one half-wave plate not arranged in any par- ticular order [47]. Here we study the combined action of ...
[4]

(12), indicated by the symbol ˆT(p, q;c,s) [48], wherepandqare the step size, andcandsstand for the two SoPs|c⟩and|s⟩, respectively

Biased quantum step We now introduce a generalization of the quantum step given in eqn. (12), indicated by the symbol ˆT(p, q;c,s) [48], wherepandqare the step size, andcandsstand for the two SoPs|c⟩and|s⟩, respectively. Unlike in the standard DTQW wherepandqare step sizes of equal magnitude and opposite direction, here in the generalized quantum step,pan...
[5]

The instantaneous quantum walk is a compact physical realization strategy of multi-step walk dynamics via a single composite unitary operation

Instantaneous quantum walks We define an instantaneous quantum walk as a single engineered unitary transformation acting on the compos- ite Hilbert space containing the SAM and OAM states, that directly maps an initial composite state to a final state that would otherwise be obtained after multiple steps of a DTQW. The instantaneous quantum walk is a comp...
[6]

, eare complex numbers such thatPe m=b |cm|2 = 1

Angle states As an illustration of complex beam shaping using d- plates, we will now attempt to generate arbitrary state |c⟩o of the OAM space: |c⟩o = eX m=b cm |m⟩o (42) wherec m, m=b, . . . , eare complex numbers such thatPe m=b |cm|2 = 1. Herebande≥bare integers. Given any such state|c⟩ o in the OAM space, we will associate a complex functionc(ϕ) as, c...
[7]

D. L. Andrews,Structured light and its applications: An introduction to phase-structured beams and nanoscale op- tical forces(Academic press, 2011)

2011
[8]

C. He, Y. Shen, and A. Forbes, Towards higher- dimensional structured light, Light: Science & Applica- tions11, 205 (2022)

2022
[9]

Rubinsztein-Dunlop, A

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alp- mann, P. Banzer, T. Bauer,et al., Roadmap on struc- tured light, Journal of Optics19, 013001 (2016)

2016
[10]

O. V. Angelsky, A. Y. Bekshaev, S. G. Hanson, C. Y. Zenkova, I. I. Mokhun, and J. Zheng, Structured light: ideas and concepts, Frontiers in Physics8, 114 (2020)

2020
[11]

F. M. Dickey,Laser beam shaping: theory and techniques (CRC press, 2018)

2018
[12]

Woerdemann, C

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, Advanced optical trapping by complex beam shaping, Laser & Photonics Reviews7, 839 (2013)

2013
[13]

M. D. Al-Amri, M. Babiker, and D. Andrews,Structured Light for Optical Communication(Elsevier, 2021)

2021
[14]

Z. Chen, P. Zhang, L. Song, Z. Li, J. Feng, X. Gou, J. Li, and C. Fang, A review of beam shaping technology in laser welding applications, The International Journal of Advanced Manufacturing Technology , 1 (2026)

2026
[15]

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, Review of structured light in diffuse optical imaging, Journal of biomedical optics24, 071602 (2019)

2019
[16]

Zhan, Cylindrical vector beams: from mathematical concepts to applications, Advances in Optics and Pho- tonics1, 1 (2009)

Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Advances in Optics and Pho- tonics1, 1 (2009)

2009
[17]

S. E. Venegas-Andraca, Quantum walks: a comprehen- sive review, Quantum Information Processing11, 1015 (2012)

2012
[18]

Portugal,Quantum walks and search algorithms, Vol

R. Portugal,Quantum walks and search algorithms, Vol. 19 (Springer, 2013)

2013
[19]

M. A. Neil, F. Massoumian, R. Juˇ skaitis, and T. Wilson, Method for the generation of arbitrary complex vector wave fronts, Optics letters27, 1929 (2002)

1929
[20]

Maurer, A

C. Maurer, A. Jesacher, S. F¨ urhapter, S. Bernet, and M. Ritsch-Marte, Tailoring of arbitrary optical vector beams, New Journal of Physics9, 78 (2007)

2007
[21]

Z. Chen, T. Zeng, B. Qian, and J. Ding, Complete shap- ing of optical vector beams, Optics express23, 17701 (2015)

2015
[22]

Marrucci, C

L. Marrucci, C. Manzo, and D. Paparo, Optical spin-to- orbital angular momentum conversion in inhomogeneous anisotropic media, Physical review letters96, 163905 (2006)

2006
[23]

Piccirillo, V

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Photon spin-to-orbital angular mo- mentum conversion via an electrically tunable q-plate, Applied Physics Letters97(2010)

2010
[24]

D’Ambrosio, F

V. D’Ambrosio, F. Baccari, S. Slussarenko, L. Marrucci, and F. Sciarrino, Arbitrary, direct and deterministic ma- nipulation of vector beams via electrically-tuned q-plates, Scientific reports5, 7840 (2015)

2015
[25]

Vergara and C

M. Vergara and C. Iemmi, Generalized q-plates and novel vector beams, arXiv preprint arXiv:1812.08202 (2018)

work page arXiv 2018
[26]

S. Fu, Y. Zhai, T. Wang, C. Yin, and C. Gao, Tailoring arbitrary hybrid poincar´ e beams through a single holo- gram, Applied Physics Letters111(2017)

2017
[27]

U. Ruiz, P. Pagliusi, C. Provenzano, and G. Cippar- rone, Highly efficient generation of vector beams through polarization holograms, Applied Physics Letters102 (2013)

2013
[28]

D. G. Hall, Vector-beam solutions of maxwell’s wave equation, Optics letters21, 9 (1996)

1996
[29]

S. C. Tidwell, D. H. Ford, and W. D. Kimura, Generat- ing radially polarized beams interferometrically, Applied Optics29, 2234 (1990)

1990
[30]

Passilly, R

N. Passilly, R. de Saint Denis, K. A¨ ıt-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, Simple interfer- ometric technique for generation of a radially polarized light beam, Journal of the Optical Society of America A 14 22, 984 (2005)

2005
[31]

Arbabi, Y

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, Dielec- tric metasurfaces for complete control of phase and po- larization with subwavelength spatial resolution and high transmission, Nature nanotechnology10, 937 (2015)

2015
[32]

D. Wang, F. Liu, T. Liu, S. Sun, Q. He, and L. Zhou, Efficient generation of complex vectorial optical fields with metasurfaces, Light: Science & Applications10, 67 (2021)

2021
[33]

T. Wu, X. Zhang, Q. Xu, E. Plum, K. Chen, Y. Xu, Y. Lu, H. Zhang, Z. Zhang, X. Chen,et al., Dielec- tric metasurfaces for complete control of phase, ampli- tude, and polarization, Advanced Optical Materials10, 2101223 (2022)

2022
[34]

D. Liu, C. Zhou, P. Lu, J. Xu, Z. Yue, and S. Teng, Gen- eration of vector beams with different polarization sin- gularities based on metasurfaces, New Journal of Physics 24, 043022 (2022)

2022
[35]

H.-T. Chen, A. J. Taylor, and N. Yu, A review of meta- surfaces: physics and applications, Reports on progress in physics79, 076401 (2016)

2016
[36]

Balthasar Mueller, N

J. Balthasar Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, Metasurface polarization optics: independent phase control of arbitrary orthog- onal states of polarization, Physical review letters118, 113901 (2017)

2017
[37]

Mirzapourbeinekalaye, A

B. Mirzapourbeinekalaye, A. McClung, and A. Arbabi, General lossless polarization and phase transformation using bilayer metasurfaces, Advanced Optical Materials 10, 2102591 (2022)

2022
[38]

Radhakrishna, G

B. Radhakrishna, G. Kadiri, and G. Raghavan, Realiza- tion of doubly inhomogeneous waveplates for structuring of light beams, Journal of the Optical Society of America B38, 1909 (2021)

1909
[39]

Kumar and A

A. Kumar and A. Ghatak, Polarization of light with ap- plications in optical fibers, (No Title) (2011)

2011
[40]

Kadiri, G

G. Kadiri, G. Raghavan,et al., Wavelength-adaptable effective q-plates with passively tunable retardance, Sci- entific reports9, 11911 (2019)

2019
[41]

Rubano, F

A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, Q-plate technology: a progress review, Journal of the optical society of america B36, D70 (2019)

2019
[42]

Hakobyan and E

V. Hakobyan and E. Brasselet, Q-plates: From optical vortices to optical skyrmions, Physical Review Letters 134, 083802 (2025)

2025
[43]

Kadian, S

K. Kadian, S. Garhwal, and A. Kumar, Quantum walk and its application domains: A systematic review, Com- puter Science Review41, 100419 (2021)

2021
[44]

Tanaka, M

H. Tanaka, M. Sabri, and R. Portugal, Spatial search on johnson graphs by discrete-time quantum walk, Journal of Physics A: Mathematical and Theoretical55, 255304 (2022)

2022
[45]

C. A. Ryan, M. Laforest, J.-C. Boileau, and R. Laflamme, Experimental implementation of a discrete-time quan- tum random walk on an nmr quantum-information pro- cessor, Physical Review A—Atomic, Molecular, and Op- tical Physics72, 062317 (2005)

2005
[46]

Z¨ ahringer, G

F. Z¨ ahringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, Realization of a quantum walk with one and two trapped ions, Physical review letters 104, 100503 (2010)

2010
[47]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, Discrete single- photon quantum walks with tunable decoherence, Phys- ical review letters104, 153602 (2010)

2010
[48]

Zhang, B.-H

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of pho- tons, Physical Review A—Atomic, Molecular, and Opti- cal Physics81, 052322 (2010)

2010
[49]

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, Implementing quantum walks using orbital angular mo- mentum of classical light, arXiv preprint arXiv:1211.1705 (2012)

work page arXiv 2012
[50]

Neves and G

L. Neves and G. Puentes, Photonic discrete-time quan- tum walks and applications, Entropy20, 731 (2018)

2018
[51]

C. M. Chandrashekar, R. Srikanth, and R. Laflamme, Optimizing the discrete time quantum walk using a su (2) coin, Physical Review A—Atomic, Molecular, and Opti- cal Physics77, 032326 (2008)

2008
[52]

Beijersbergen, R

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Helical-wavefront laser beams produced with a spiral phaseplate, Optics communications112, 321 (1994)

1994
[53]

Simon and N

R. Simon and N. Mukunda, Minimal three-component su (2) gadget for polarization optics, Physics Letters A143, 165 (1990)

1990
[54]

Kadiri, Scouring parrondo’s paradox in discrete-time quantum walks, Physical Review A110, 022421 (2024)

G. Kadiri, Scouring parrondo’s paradox in discrete-time quantum walks, Physical Review A110, 022421 (2024)

2024
[55]

A. M. Beckley, T. G. Brown, and M. A. Alonso, Full poincar´ e beams, Optics express18, 10777 (2010)

2010
[56]

Lopez-Mago, On the overall polarisation properties of poincar´ e beams, Journal of Optics21, 115605 (2019)

D. Lopez-Mago, On the overall polarisation properties of poincar´ e beams, Journal of Optics21, 115605 (2019)

2019
[57]

J. P. Snyder,Map projections–A working manual, Vol. 1395 (US Government Printing Office, 1987)

1987
[58]

Donati, L

S. Donati, L. Dominici, G. Dagvadorj, D. Ballarini, M. De Giorgi, A. Bramati, G. Gigli, Y. G. Rubo, M. H. Szyma´ nska, and D. Sanvitto, Twist of general- ized skyrmions and spin vortices in a polariton super- fluid, Proceedings of the National Academy of Sciences 113, 14926 (2016)

2016
[59]

Y. Shen, Q. Zhang, P. Shi, L. Du, X. Yuan, and A. V. Zayats, Optical skyrmions and other topological quasi- particles of light, Nature Photonics18, 15 (2024)

2024
[60]

Radhakrishna, G

B. Radhakrishna, G. Kadiri, and G. Raghavan, Polari- metric method of generating full poincar´ e beams within a finite extent, Journal of the Optical Society of America A39, 662 (2022)

2022
[61]

Mirhosseini, O

M. Mirhosseini, O. S. Maga˜ na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, High-dimensional quan- tum cryptography with twisted light, New Journal of Physics17, 033033 (2015)

2015
[62]

E. Yao, S. Franke-Arnold, J. Courtial, S. Barnett, and M. Padgett, Fourier relationship between angular posi- tion and optical orbital angular momentum, Optics Ex- press14, 9071 (2006)

2006
[63]

A. S. Holevo,Quantum Systems, Channels, Informa- tion: A Mathematical Introduction, 2nd ed. (De Gruyter, 2019)

2019