Coherent ray-wave structured light based on (helical) Ince-Gaussian modes
LLetter Optics Letters 1
Coherent ray-wave structured light based on (helical)Ince-Gaussian modes Z HAOYANG W ANG , Y
IJIE S HEN , Q
IANG L IU , AND X ING F U Key Laboratory of Photonic Control Technology (Tsinghua University), Ministry of Education, Beijing 100084, China State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Tsinghua University, Beijing 100084, China Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, UK [email protected] [email protected] February 5, 2021 The topological evolution of classic eigenmodes in-cluding Hermite-Laguerre-Gaussian and (helical) Ince-Gaussian modes is exploited to construct coherent statemodes, which unifies the representations of traveling-wave (TW) and standing-wave (SW) ray-wave struc-tured light for the first time and realizes the TW-SWunified ray-wave geometric beam with topology of ray-trajectories splitting effect, breaking the boundary ofTW and SW structured light. We experimentally gen-erate these new modes with high purity and dynamiccontrol by digital holography method, revealing poten-tial applications in optical manipulation and communi-cation. © 2021 Optical Society of America http://dx.doi.org/10.1364/ao.XX.XXXXXX
Structured light, with the ability to arbitrarily tailor its am-plitude, phase and polarization [1], has attracted great attentiondue to its wide applications such as optical tweezers, quantumentanglement, and communications [2–5]. Multifarious beammodes as eigensolutions of paraxial wave equation (PWE) wereidentified in structured light family, such as Hermite-Gaussian(HG) mode, Laguerre-Gaussian (LG) mode, Ince-Gaussian (IG)mode and helical-Ince-Gaussian (HIG) mode [6], as well asvarious superposed state of eigenmodes including mixing HGmode [7], vortex lattice [8], and SU(2) geometric beam [9, 10].The topological evolution and unified representation for priorlight modes rised enthusiasm. For example, the generalizedHermite-Laguerre-Gaussian (HLG) modes unify LG and HGmodes [11, 12]. IG modes interpret the transition from LG to HGmodes based on the topology of elliptical coordinate [6]. A sin-gularity hybrid evolution model, as a further generalized family,accommodates the HLG and HIG modes together [13]. Recently,an SU(2) Poincaré sphere model was proposed to universallyreveal the topology of orbital angular momentum (OAM) eigen-modes and coherent state complex modes [14]. Nevertheless,unified representation for more generalized structured modesstill requires further exploration.As another important category of structured light, thestanding-wave (SW) modes, the superposed state of two TWmodes propagating at opposite directions, has distinct charac- �� yxz yxz yxz Fig. 1.
A SW ray-wave mode is composed by two TW ray-wave modes in opposite directions. z ranges from 0 to 2 z R .The lower-right inserts show corresponding phases at focus.teristics and intriguing patterns individually. For instances, thecomplex SW mode composed by vector vortex TW modes wasgenerated to simulate entangled beating and applied in opticalmachining [4]. This concept can also be introduced in exoticray-wave structured light, as shown in Figs. 1, and the TW andSW ray-wave geometric beams unveil the ray-like structures oflight for propagating in freespace [15, 16] and oscillating in res-onator [9, 17], respectively. However, the topological connectionof TW and SW ray-wave modes has never been studied in eithertheory or experiment. Notably, such ray-wave structure has re-cently extened the frontier of modern physics such as nondiffrac-tion effect [18], topological phase [19], and quantum-classicalinformatics [20]. Therefore, breaking the TW-SW boundary inray-wave structured light is significant for unveiling more gen-eral topological evolution and potential applications.In this Letter, a unified representation for topological evolu-tion of TW and SW structured light is proposed and we constructthe novel geometric beams with splitting ray-wave structure andmixing ray-wave structure that have not been observed before.We explore the exotic ray-wave structure in generalized geomet-ric beams by constructing the cluster of classical trajectories. Thesimulated and experimental results intuitively demonstrate thetopological evolution of our generalized structured light.The PWE has various analytical forms in different coordi-nates. In cylindrical coordinate ( r , θ , z ) , the eigenmodes are TWLG modes and SW LG modes [21]. The TW LG modes con-tain spiral phase factor e i (cid:96) θ , noted as LG p , (cid:96) , l where p , (cid:96) , l areradial, azimuthal, and longitudinal indices ( p = min ( n , m ) and a r X i v : . [ phy s i c s . op ti c s ] F e b etter Optics Letters 2 (cid:96) = m − n ). The SW LG modes include even and odd LG modesnoted as LG e p , (cid:96) , l and LG o p , (cid:96) , l , respectively. LG e p , (cid:96) , l contains factorcos ( (cid:96) θ ) instead of e i (cid:96) θ while the LG o p , (cid:96) , l contains factor sin ( (cid:96) θ ) .A SW LG mode is equivalent to the superposition of two TWLG modes with opposite directions. In the elliptic coordinate ( ξ , η , z ) ( ξ ∈ [ ∞ ) , η ∈ [
0, 2 π ) ), which is related to the Carte-sian coordinate by elliptic eccentricity (cid:101) , the eigenmodes aregiven by IG modes as even IG mode IG e u , v , l ( x , y , z | (cid:101) ) and oddIG mode IG o u , v , l ( x , y , z | (cid:101) ) , where 0 ≤ v ≤ u for even IG mode,1 ≤ v ≤ u for odd IG mode, and ( − ) u − v = ( u , v ) are ofthe same parity [6]. Akin to TW and SW LG modes, a mixingmode, HIG mode, can be composed by odd and even IG modesas HIG ± u , v , l = ( IG e u , v , l ± iIG o u , v , l ) [6]. In Cartesian coordinate ( x , y , z ) , the eigensolution of PWE is HG n , m , l mode [22]. In ad-dition, the mixing linear superposition of eigenmodes is alsothe solution of PWE. More exact topological evolution betweenthese various eigenmodes is still needed to research. Hereinafter,we will study the topological evolution between these classicaleigenmodes for a generalized eigenmode family.An elliptic coordinate can topologically evolve into Carte-sian coordinate for (cid:101) → ∞ , and to cylindrical coordinate for (cid:101) →
0, respectively [6, 13]. Thus IG mode can be consideredas the transitional state between TW HG mode and SW LGmode noted as ψ IG n , m , l ( x , y , z | (cid:101) ) by changing the parameter (cid:101) , asshown in Figs. 2 a1-a4 and b1-b4. Besides, HG modes can beconverted to TW LG modes via HLG modes [11, 12] noted as ψ HLG n , m , l ( x , y , z | α ) , as shown in Figs. 2 a4-a7 and b4-b7. Further-more, the TW LG modes can be further extended to mixingHG mode by exploiting HIG mode noted as ψ HIG n , m , l ( x , y , z | (cid:101) ) , asshown in Figs. 2 a7-a10 and b7-b10. The generalized eigenmodefamily (cid:110) ψ IG n , m , l , ψ HLG n , m , l , ψ HIG n , m , l (cid:111) can be exploited to construct com-plex coherent state light, further enriching the structured family. To unify TW and SW ray-wave structured light.
In abovesections, we present the generalized eigenmode family to unifythe topological evolution of TW and SW eigenmodes, whichallows us to explore more complex coherent structured modeas on-demand superposed spatial wavepacket of eigenmodes.Hereinafter, we will demonstrate a family of TW-SW unified ray-wave structured geometric beams with striking ray-trajectorysplitting and merging properties that have not been observedbefore. We exploit the formation of SU(2) coherent state to con-struct such modes [9, 14, 23]: Φ n , m , l ( x , y , z ) = N /2 N ∑ K = NK e i K φ Ψ eigen n + pK , m + qK , l − sK ( x , y , z ) , (1) where φ is coherent phase, ( p , q , s ) are three integers related tofrequency coupling among transverse and longitudinal modes Q = p + q , s = − P , ( P , Q ) are a pair of coprime inte-gers for fulfilling frequency-degenerate condition [9], Ψ eigen n , m , l ∈ (cid:110) ψ IG n , m , l , ψ HLG n , m , l , ψ HIG n , m , l (cid:111) . The corresponding topological evolutionof spatial wavepacket for ( n , m ) = (
10, 0 ) and ( p , q ) = ( Q , 0 ) is shown in Figs. 2 d1-d10 and e1-e10. The SW and TW geo-metric beams with oscillating and propagating ray paths werestudied in [15–17]. But we create new modes as the transitionalstate between them, as shown in Figs. 2 c1-c4 and d1-d4, rayorbits of a planar TW geometric beam gradually split into twofold and distribute into oscillating trajectory of the circular SWgeometric beam, which unify the topological evolution of TW and SW geometric beams. Besides, the generalized geometricbeam also includes the evolution from a planar trajectory beaminto the circular trajectory beam carrying OAM (Figs. 2 c4-c7).Furthermore, we could utilize HIG modes to extend more ex-otic transformation from the OAM state into a mixing planartrajectory beam (Figs. 2 c7-c10), not proposed before either.We also explore the higher-order formation of such coherentstate mode, as the results for ( n , m ) = (
10, 3 ) and ( p , q ) = ( Q , 0 ) demonstrated in Figs. 2 f1-f10, g1-g10 and h1-h10, where thelight on each ray state changes into higher-order HLG modeformation, namely the multi-axis vortex beam [24]. Here wemarkedly generalize such multi-vortex geometric beam that theTW multi-HG beam can topologically evolve into SW multi-LGbeam (Figs. 2 g1-g4) and can also evolve into TW multi-LG beam(Figs. 2 g4-g7) and further into multi-mixing HG beam (Figs. 2g7-g10). The newly proposed ray-wave geometric beams (inred and blue boxes of Figs. 2) largely enrich the structured lightfamily and inspire the tailoring of more exotic structured light. To explore ray-wave structure with splitting ray orbits.
Due to the quantum-classical correspondence nature of coherentstate, the coherent state geometric beam shows intriguing ray-wave duality [9], i.e. the wave pattern of which is localized on acluster of classical ray trajectories. For TW modes, the cluster ofclassical trajectories (cid:110) x b s , y b s , z | α (cid:111) ± is coupled with spatial wavepacket of Eq. (1) for Ψ eigen n , m , l = ψ HLG n , m , l ( x , y , z | α ) (geometric beamsbased on HLG modes) [25], where α ranges from − π /4 to π /4, ± represents two opposite directions, ( x b s , y b s , z ) represents a clas-sical trajectory labelled s where spatial wave packets are locatedon, and (cid:110) x b s , y b s , z | α (cid:111) ± a collection of such classical trajectories,where s =
0, 1, · · · , Q − α ranging from 0to π /4 for TW modes, corresponding to the mode evolutionfrom planar to circular classical orbits, as shown in Figs. 2 c4-c7and f4-f7. Furthermore, we can construct the cluster of classicaltrajectories for generalized geometric beams.SW modes can be decomposed into two TW modes, whichreveals that the cluster of classical trajectories of SW geometricbeams (shown in Figs. 2 c1 and f1) can be interpreted as a super-position of two clusters of classical trajectories of TW modes inopposite directions as: (cid:110) x b s , y b s , z (cid:111) SW = (cid:110) x b s , y b s , z | α = π /4 (cid:111) + (cid:110) x b s , y b s , z | α = − π /4 (cid:111) . (2) Since SW modes correspond to the case (cid:101) =
0, the (cid:110) x b s , y b s , z (cid:111) SW can be noted as a limiting case (cid:110) x b s , y b s , z | (cid:101) = (cid:111) IG , where (cid:110) x b s , y b s , z | (cid:101) (cid:111) IG is the cluster of classical trajectories coupled withspatial wave packet of Eq. (1) for Ψ eigen n , m , l = ψ IG n , m , l ( x , y , z | (cid:101) ) (geo-metric beams based on IG modes), revealing the exotic splittingorbits in evolution of SW-TW modes with (cid:101) increasing from 0 to ∞ (see details in Supplement), as shown in Figs. 2 c1-c4, f1-f4.Besides, the mixing HG modes (HIG modes with (cid:101) → ∞ )are essentially equal to a superposition of two HG modes withdifferent indices, as shown in Figs. 2 c8-c10, f8-f10, where thecluster of classical trajectories coupled with spatial wave packetof Eq. (1) for Ψ eigen n , m , l = ψ HIG n , m , l ( x , y , z | (cid:101) ) (geometric beams basedon HIG modes), revealing exotic mixing orbits with (cid:101) increasingfrom 0 to ∞ . The exotic TW-SW-unified ray-wave structures re-veal the generalized ray-wave duality in generalized geometric etter Optics Letters 3 yxz yxz yxz yxz yxz E i g e n m od es G e o m e t r i c b ea m s H i gh e r - o r d e r g e o m e t r i c b ea m s α = � /16 α = � /16 α = � /16 α = � /8 α = � /8 α = � /4 α = � /8 α = � /4 α = � /4 є =0 є =1 є =30 є =45 є =30 є =45 є =4 є → ∞ є →∞ є →∞ є →∞ є →∞ є →∞ є =0 є =0 α =0 α =0 α =0 є =1.5 є =0 є =4 є =45 є =0 є =30 є =0 є =45 є =30 (a1) (c2)(d2)(f2) (f7) (f10)(f9)(f8)(f6)(g6) (g7) (g9)(g8) (g10)(f5)(g5)(f4)(g4)(f3)(g3)(g2)(c1)(d1)(g1)(f1) (c4)(d4) (c6)(d6)(c3)(d3) (c5)(d5) (d7) (d8) (d9)(c7) (c8) (c10)(c9) (d10)(a7) (a10)(a9)(a8)(a6)(a5)(a4)(a3)(a2)(b1) (b3) (e4)(e3)(b2)(e2)(e1) (h2)(h1) (h7) (h9)(h8) (h10)(h6)(h5)(h4)(h3) (b7) (b10)(b9)(b8)(e7) (e10)(e9)(e8)(b6)(b5) (e6)(b4) (e5) � - � � - � - �� Fig. 2.
The topological evolutions of TW-SW unified eigenmodes (a, b), geometric beams ( n , m ) = (
10, 0 ) (c, d, e), and higher-ordergeometric beams ( n , m ) = (
10, 3 ) (f, g, h). Panels (a1-a10), (d1-d10), and (g1-g10) show the corresponding three-dimensional spatialwave packets. Panels (c1-c10) and (f1-f10) show the corresponding classical trajectories for geometric beams and higher-ordergeoemtric beams, respectively. z ranges from 0 to 2 z R . Panels (b1-b10), (e1-e10), and (h1-h10) show the corresponding transverseintensity and phase distributions at z = − π to π forphase.) etter Optics Letters 4 a1 b1 c1 a2b2 c2 a3 b3 c3 a4 b4 c4 a5b5c5 a6b6c6 a7b7c7 a8b8c8 a9b9c9 a10b10c10 ㎜ ㎜ ㎜ ㎜ Fig. 3.
Experimental results of the topological evolutions of TW-SW unified eigenmodes (a), geometric beams (b), and higher-ordergeometric beams (c). See Visualization 1, Visualization 2, Visualization 3 for dynamic movies of (a1-a10), (b1-b10), and (c1-c10).beam, providing a deeper physical insight of quantum-classicalcorrespondence (ray-wave duality).
Experimental realization.
We experimentally generate thesecomplex modes with high-purity based on the classic digitalholography method by a digital micromirror device [26, 27]. Ex-perimental results of TW-SW unified structured light are shownin Figs. 3, where rows from top to bottom are the patterns of TW-SW unified eigenmodes, multi-path geometric beams and multi-HLG higher-order geometric beams, respectively, recorded at z = Discussion.
Our model of TW-SW-unified structured lightis largely extendable. For instance, it can also be applied tomore complex SU(2) coherent state corresponding to general-ized ray-wave Lissajous and trochoidal wavepacket [28, 29].We can also study its general structure in astigmatic and vec-torial optical fields. In addition, other kinds of coherent super-posed formations are also expected to explore, such as SU(1,1)coherent state [30], and hybrid coherent state [31]. The TW-SWunification also act as a new mechanism to extend topologicalstructure, so as to enable novel applications. The ray-trajectory-splitting topology provides new degrees of freedom to createmulti-partite classical entangled state [20], which can be em-ployed in high-speed optical encryption and communication [5].The multi-singularity and complex OAM evolution of the newstructured light is also in need of advanced optical tweezers andtrapping [3].In summary, we propose a new theory to unify the TW andSW formations of structured light. It generalizes the familyof ray-wave geometric modes based on TW-SW-unified eigen-modes (IG, HLG, and HIG modes), extending the new topolog-ical ray-wave structures as thier complex coherent states. Thegeneralized theoretical framework has strong extensibility andapplicability to construct more complex modes and to studyOAM with multi-singularities, which inspires the explorationof more topological properties of novel structured modes withtheir advanced applications.
Funding.
The National Key Research and Development Pro-gram of China (2017YFB1104500); National Natural ScienceFoundation of China (61975087); Beijing Young Talents SupportProject (2017000020124G044).
Disclosures.
The authors declare no conflicts of interest.
REFERENCES
1. A. Forbes, Opt. Photonics News , 24 (2020).2. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan,Light. Sci. & Appl. , 90 (2019).3. E. Otte and C. Denz, Appl. Phys. Rev. , 041308 (2020).4. E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes,Light. Sci. & Appl. , 18009 (2018).5. B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad,and A. Forbes, Nat. Phys. , 397 (2017).6. M. A. Bandres and J. C. Gutiérrez-Vega, JOSA A , 873 (2004).7. V. Kotlyar and A. Kovalev, JOSA A , 274 (2014).8. Y. Shen, Z. Wan, X. Fu, Q. Liu, and M. Gong, JOSA B , 2940 (2018).9. Y.-F. Chen, C. Jiang, Y.-P. Lan, and K.-F. Huang, Phys. Rev. A ,053807 (2004).10. Y. Shen, X. Yang, X. Fu, and M. Gong, Appl. Opt. , 9543 (2018).11. T. Alieva and M. J. Bastiaans, Opt. Lett. , 1461 (2005).12. E. Abramochkin and T. Alieva, Opt. Lett. , 4032 (2017).13. Y. Shen, Y. Meng, X. Fu, and M. Gong, JOSA A , 578 (2019).14. Y. Shen, Z. Wang, X. Fu, D. Naidoo, and A. Forbes, Phys. Rev. A ,031501 (2020).15. C.-H. Chen and C.-F. Chiu, Opt. Express , 12692 (2007).16. Y. Shen, X. Fu, and M. Gong, Opt. Express , 25545 (2018).17. T.-H. Lu and C. He, Opt. Express , 20876 (2015).18. A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, Nat. Commun. (2020).19. T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. Dennis, A. Vamivakas,and M. Alonso, Phys. review letters , 233602 (2018).20. Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes,Light. Sci. & Appl. (in press) (2021).21. M. W. Beijersbergen, L. Allen, H. Van der Veen, and J. Woerdman, Opt.Commun. , 123 (1993).22. A. E. Siegman, Lasers (University Science, Mill Valley, CA, 1986).23. V. Bužek and T. Quang, JOSA B , 2447 (1989).24. P. Tuan, Y. Hsieh, Y. Lai, K.-F. Huang, and Y.-F. Chen, Opt. Express ,20481 (2018).25. Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K.-F. Huang, Opt. Lett. , 2649 (2019).26. Y.-X. Ren, R.-D. Lu, and L. Gong, Annalen Der Physik , 447 (2015).27. Z. Wan, Z. Wang, X. Yang, Y. Shen, and X. Fu, Opt. Express (2020).28. Y.-F. Chen, Y. Lin, K.-F. Huang, and T.-H. Lu, Phys. Rev. A , 043801(2010).29. Z. Wan, Z. Wang, X. Yang, Y. Shen, and X. Fu, Opt. Express , 31043(2020).30. K. Wodkiewicz and J. Eberly, JOSA B , 458 (1985).31. Y. Shen, X. Yang, D. Naidoo, X. Fu, and A. Forbes, Optica7