Spatiotemporal Vortex Pulses: Angular Momenta and Spin-Orbit Interaction
SSpatiotemporal Vortex Pulses: Angular Momenta and Spin-Orbit Interaction
Konstantin Y. Bliokh
Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
Recently, spatiotemporal optical vortex pulses carrying a purely transverse intrinsic orbital angu-lar momentum were generated experimentally [
Optica , 1547 (2019); Nat. Photon. , 350 (2020)].However, an accurate theoretical analysis of such states and their angular-momentum properties re-mains elusive. Here we provide such analysis, including scalar and vector spatiotemporal Bessel-typesolutions as well as descrption of their propagational, polarization, and angular-momentum proper-ties. Most importantly, we calculate both local densities and integral values of the spin and orbitalangular momenta, and predict observable spin-orbit interaction phenomena related to the couplingbetween the trasnverse spin and orbital angular momentum. Our analysis is readily extended tospatiotemporal vortex pulses of other natures (e.g., acoustic). Introduction.—
Vortex beams carrying intrinsic orbitalangular momentum (OAM) are widely studied and ex-ploited in modern optics [1–7], acoustics [8–13], and elec-tron microscopy [14–20]. They have found numerous ap-plications in a variety of classical and quantum systems.Such beams are monochromatic, and their intrinsic OAMis produced by a screw phase dislocation (vortex) alignedwith the beam axis [21, 22]. Thus, this OAM is longi-tudinal , i.e., aligned with the mean momentum of thebeam.Recently, spatiotemporal analogues of vortex beams, spatiotemporal vortex pulses (STVPs) , were predictedtheoretically [23, 24] and generated experimentally in op-tics [25–28]. Such states are essentially polychromatic,and they carry intrinsic OAM transverse (or, generally,tilted) with respect to the propagation direction of thepulse. This OAM is produced by an edge (or mixed edge-screw) phase dislocation [21, 22]. It is anticipated thatthe STVPs and transverse intrinsic OAM can consider-ably extend functionality and applications of wave vor-tices.Despite the very recent progress in the generation ofSTVPs [26–28], they still lack an accorate theoretical de-scription, including consistent analysis of their angular-momentum and polarization properties. Indeed, despitenumerous mentionings of the OAM in Refs. [26–28], itsvalue has not been obtained there. Instead, the topolog-ical number of the phase dislocation, ℓ , was used, whichgenerically does not coincide with the normalized OAMvalue [2, 29]. Furthermore, accurate calculations of theOAM are impossible without a full vector description andseparation of the spin and orbital parts of the total an-gular momentum [7, 30–32].In this paper, we fill this gap by constructing sim-ple Bessel-type solutions for STVPs. We describe theirpropagational dynamics including ‘temporal diffraction’[24, 26], examine polarization properties, and calculateboth local densities and integral values of the OAM andspin angular momentum (SAM) of such pulses. We showthat the in-plane linear polarization inevitably producesa longitudinal field component and a nonzero transverseSAM density. This produces the spin-orbit interaction ef- fects, known for monochromatic beams [30, 33], such asobservable polarization-dependent intensity distributionsof STVPs. Importantly, an integral value of the trans-verse SAM vanishes, while the integral OAM is quantizedas (cid:126) ℓ per photon only for circularly-symmetric pulseswith equal width and length. For elliptical STVPs withdifferent width and length, which were used in experi-ments [26–28], the OAM value is larger than (cid:126) ℓ per pho-ton.Thus, our work provides the self-consistent full-vectordescription of optical STVPs. It also allows straightfor-ward extension to analogous acoustic pulses. Scalar Bessel-type solutions.—
We first consider scalarwaves and simplest analytical vortex-beam solutions:Bessel beams [30, 34–38]. Monochromatic Bessel beamspropagating along the 𝑧 -axis can be constructed as asuperposition of plane waves with the same frequency 𝜔 = 𝜔 , wavevectors k distributed within a cone of polarangle 𝜃 = 𝜃 , and with an azimuthal phase difference ℓ𝜑 ( 𝜑 is the azimuthal angle in k -space) corresponding to avortex of integer order ℓ , Fig. 1(a). In other words, thewavevectors form a circle in the 𝑘 𝑧 = 𝑘 ‖ plane, with thecenter at (0 , , 𝑘 ‖ ) and radius 𝑘 ⊥ , where 𝑘 ‖ = 𝑘 cos 𝜃 , 𝑘 ⊥ = 𝑘 sin 𝜃 , 𝑘 = 𝜔 /𝑐 , and 𝑐 is the speed of light. Inreal space, this superposition results in the scalar wave-function 𝜓 ( r , 𝑡 ) ∝ 𝐽 ℓ ( 𝑘 ⊥ 𝑟 ) exp( 𝑖𝑘 ‖ 𝑧 + 𝑖ℓ𝜙 − 𝑖𝜔 𝑡 ), where( 𝑟, 𝜙 ) are the polar coordinates in the ( 𝑥, 𝑦 )-plane and 𝐽 𝑛 is the Bessel function of the first kind. The transverseintensity and phase distributions of such Bessel beam areshown in Fig. 1(a).To construct a Bessel-type STVP with a purely trans-verse intrinsic OAM, we use a superposition of planewaves with wavevectors distribute over a circle in the 𝑘 𝑦 = 0 plane with the center at (0 , , 𝑘 ) and radius Δ 𝑘 ,Fig. 1(b). Using the azimuthal angle ˜ 𝜑 with respect tothe center of this circle, we introduce the azimuthal phasedifference ℓ ˜ 𝜑 and write the real-space wavefunction as aFourier-type integral: 𝜓 ( r , 𝑡 ) ∝ ∫︁ 𝜋 𝑒 𝑖 [ 𝑘 +Δ 𝑘 cos ˜ 𝜑 𝑧 +Δ 𝑘 sin ˜ 𝜑 𝑥 + 𝑖ℓ ˜ 𝜑 − 𝜔 ( ˜ 𝜑 ) 𝑡 ] 𝑑 ˜ 𝜑 , (1) a r X i v : . [ phy s i c s . op ti c s ] F e b FIG. 1. The plane-wave spectra (left) and phase-intensity dis-tributions of real-space wavefunctions 𝜓 ( r , 𝑡 ) (right) for (a)the monochromatic Bessel beam with ℓ = 2 and (b) spa-tiotemporal Bessel pulse with ℓ = 2, Eqs. (1) and (2). Inreal-space distribtuions, the brightness is proportional to theintensity | 𝜓 | , while the color indicates the phase Arg( 𝜓 ). where 𝜔 ( ˜ 𝜑 ) = 𝑐 √︁ 𝑘 + Δ 𝑘 + 2 𝑘 Δ 𝑘 cos ˜ 𝜑 . ParameterΔ 𝑘 determines the degree of paraxiality and monochro-maticity of the Bessel STVP. The maximum polar angleof the wavevectors in its spectrum is sin 𝜃 = Δ 𝑘/𝑘 ,Fig. 1(b). For Δ 𝑘 ≪ 𝑘 , 𝜃 ≪
1, the pulse can be consid-ered as near-paraxial and quasi-monochromatic. Belowwe will use this approximation keeping terms linear in 𝜃 ,which describe some post-paraxial phenomena.In this approximation, 𝜔 ( ˜ 𝜑 ) ≃ 𝑐 ( 𝑘 +Δ 𝑘 cos ˜ 𝜑 ), and theintegral (1) results in the analytical Bessel-pulse solution: 𝜓 ( r , 𝑡 ) ∝ 𝐽 ℓ (˜ 𝜌 ) exp( 𝑖𝑘 𝜁 + 𝑖ℓ ˜ 𝜙 ) . (2)Here, 𝜁 = 𝑧 − 𝑐𝑡 = ˜ 𝑟 cos ˜ 𝜙 , ˜ 𝜌 = Δ 𝑘 ˜ 𝑟 , and (˜ 𝑟, ˜ 𝜙 ) are thepolar coordinates in the ( 𝜁, 𝑥 ) plane. The intensity 𝐼 = | 𝜓 | and phase Arg( 𝜓 ) distributions for the Bessel STVP(2) are shown in Fig. 1(b). It has typical Bessel-beamintensity profile 𝐼 ∝ | 𝐽 ℓ (˜ 𝜌 ) | in the ( 𝜁, 𝑥 ) plane, containsan edge phase dislocation of order ℓ , and propagates withthe speed of light along the 𝑧 -axis.Importantly, the nondiffracting solution (2) is a resultof linear expansion of 𝜔 ( ˜ 𝜑 ) with respect to Δ 𝑘 . Theexact solution (1) evolves in time as shown in Fig. 2.Namely, the ℓ th order phase dislocation in the pulse cen-ter splits into a raw of | ℓ | first-order dislocations orienteddiagonally in the ( 𝜁, 𝑥 ) plane. This ‘ temporal diffraction ’was predicted in Ref. [24] and observed in [26]. Akin tothe Rayleigh range characterizing spatial diffraction, atypical scale of the temporal diffraction is given by the FIG. 2. Temporal diffraction of the spatiotemporal Besselpulse (1) with ℓ = 3 and Δ 𝑘/𝑘 = 0 .
3. The characteristictime scale 𝑐𝑡 𝑅 = 𝑘 / Δ 𝑘 is analogous to the Rayleigh rangeof spatially diffracting beams. ‘ temporal Rayleigh range ’ 𝑐𝑡 𝑅 = 𝑘 / Δ 𝑘 , Fig. 2. Notably,nondiffracting Bessel-like STVPs can be constructed us-ing the wavevectors distributed along an ellipse in k spacewhich could be Lorentz-transformed to a monochromaticcircle [24]. Vector solutions and spin-orbit effects.—
We now ex-amine vector Bessel STVPs. For simplicity, we considerthe electric field E of transverse electromagnetic waves;similar arguments could be applied to the magnetic fieldand other types of vector waves. Due to the transversal-ity condition, the electric field of each plane wave in thepulse spectrum must be orthogonal to its wavevector k .This determines two basic polarizations in the problem:(i) out-of-plane, E is directed along the 𝑦 -axis, Fig. 3(a),and (ii) in-plane, E lies in the ( 𝑧, 𝑥 ) plane, Fig. 3(b).For the out-of-plane polarization, the field has only onecomponent, and the problem reduces to the scalar case: 𝐸 𝑦 ( r , 𝑡 ) ∝ 𝜓 ( r , 𝑡 ).For the in-plane polarization, the situation is less triv-ial. Each plane wave in the pulse spectrum has twoelectric-field components, 𝐸 𝑥 and 𝐸 𝑧 , Fig 3(b). Theamplitudes and phases of these components depend onthe wavevector k , which signals the spin-orbit interac-tion [30, 33]. To describe the spin-orbit effects in thelinear approximation in Δ 𝑘 , we write the electric fieldcomnponents for each plane wave as 𝐸 𝑥 = 𝐸 cos 𝜃 ≃ 𝐸 and 𝐸 𝑧 = − 𝐸 sin 𝜃 ≃ − 𝐸 (Δ 𝑘/𝑘 ) sin ˜ 𝜙 , where 𝜃 ≤ 𝜃 is the polar angle of a given wavevector. Since for theinetgrand of Eq. (1), 𝑖 Δ 𝑘 sin ˜ 𝜙 = 𝜕/𝜕𝑥 , one can writethe real-space transverse ( 𝑥 ) and longitudinal ( 𝑧 ) fieldcomponents as 𝐸 𝑥 ( r , 𝑡 ) ≃ 𝜓 ( r , 𝑡 ) , 𝐸 𝑧 ( r , 𝑡 ) ≃ 𝑖𝑘 𝜕𝜓 ( r , 𝑡 ) 𝜕𝑥 . (3)Substituting Eq. (2) into Eq. (3), we obtain the longi-tudinal field 𝐸 𝑧 ∝ 𝑖 Δ 𝑘 𝑒 𝑖𝑘 𝜁 [︁ 𝑒 𝑖 ( ℓ −
1) ˜ 𝜙 𝐽 ℓ − (˜ 𝜌 ) + 𝑒 𝑖 ( ℓ +1) ˜ 𝜙 𝐽 ℓ +1 (˜ 𝜌 ) ]︁ . (4)From Eqs. (2)–(4), the total field intensity, 𝐼 = | 𝐸 𝑥 | + FIG. 3. Electric fields of plane waves in the spectra of Bessel-type vector STVPs (left) and the corresponding real-space in-tensity distributions | E ( r , 𝑡 ) | (right) for (a) the out-of-planepolarization and (b) the in-plane polarization. The parame-ters are: ℓ = 1 and Δ 𝑘 = 0 . | 𝐸 𝑧 | , is given by 𝐼 ∝ 𝐽 ℓ + Δ 𝑘 𝑘 [︀ 𝐽 ℓ − + 𝐽 ℓ +1 + 2 cos(2 ˜ 𝜙 ) 𝐽 ℓ − 𝐽 ℓ +1 ]︀ . (5)Here and hereafter, for brevity, we omit the Bessel-functions argument ˜ 𝜌 .The intensity distribution of the Bessel STVP, Eq. (5),resembles intensity distributions of vector Bessel beamsin optics [30], acoustics [39], and quantum mechanics[40]. In particular, the presence of the Bessel functionsof orders ℓ ± ∼ 𝜃 in the center of the in-plane-polarized STVPs with | ℓ | = 1, Fig. 3. For monochro-matic vortex beams, this phenomenon has been observedin optical experiments [33, 41, 42]. The main difference isthat in the case of monochromatic vortex beams this phe-nomenon occurs for circular polarizations, correspondingto the longitudinal SAM of the beam, while for STVPsit takes place for the in-plane linear polarization. As weshow below, this polarization generates a nonzero trans-verse SAM density directed along the 𝑦 -axis, i.e., alongwith the intrinsic OAM of the pulse. Spin and orbital angular momenta.—
Analysis of thespin and orbital angular momenta of polychromaticSTVPs is a challenging problem because most theoreti-cal methods are developed for monochromatic fields. In-deed, standard formulas for the SAM and OAM densitiesimply averaging over the cycle of periodic temporal os-cillations [7], and they become ill-defined in generic poly-
FIG. 4. Spatial distributions of (a) the normalized trans-verse spin angular momentum density 𝜔 𝑆 𝑦 /𝐼 and (b) theorbital angular momentum density (with the subtracted prop-agational term) 𝜔 𝐿 𝑦 + 𝑘 𝑥𝐼 in the in-plane-polarized BesselSTVP, Eqs. (8) and (5). The parameters are: ℓ = 2 andΔ 𝑘/𝑘 = 0 . chromatic fields [32]. Nonetheless, here we can employthe quasi-monochromaticity of pulses with Δ 𝑘 ≪ 𝑘 andseparate fast temporal oscillations with the central fre-quency 𝜔 = 𝑘 𝑐 and slow temporal evolution with theinverse temporal scale Δ 𝑘 𝑐 . This results in the SAM andOAM densities given by the standard monochromatic for-mulas involving canonical SAM and OAM operators andtime-dependent ‘wavefunction’ E ( r , 𝑡 ) 𝑒 𝑖𝜔 𝑡 / √ 𝜔 [7]: 𝑆 𝑦 = 𝜔 − Im ( E * × E ) 𝑦 , 𝐿 𝑦 = 𝜔 − Im (︂ E * · 𝜕𝜕 ˜ 𝜙 E )︂ . (6)Although the angle ˜ 𝜙 in the ( 𝜁, 𝑥 ) plane involves time,in Eq. (6) we used the fact that at 𝑡 = 0 it becomesthe desired azimuthal angle in the ( 𝑧, 𝑥 ) plane, whereasin the diffractionless approximation the SAM and OAMdistributions are invariantly translated together with thepulse along the 𝑧 -axis.For the out-of-plane polarization (the scalar case),Eqs. (6) yield 𝑆 𝑦 = 0 , 𝐿 𝑦 = 𝜔 − ( ℓ − 𝑘 𝑥 ) 𝐼 . (7)where 𝐼 ∝ 𝐽 | ℓ | (˜ 𝜌 ). The OAM density (7) contains thestandard vortex-related term proportional to ℓ as wellas the 𝑘 𝑥 -dependent term caused by the 𝑧 -propagationof the pulse. After integration of the OAM density 𝐿 𝑦 over the ( 𝑧, 𝑥 ) plane, the 𝑘 𝑥 -term vanishes, while thevortex-induced term yields the integral value of (cid:126) ℓ perphoton [1–7]: 𝜔 ⟨ 𝐿 𝑦 ⟩ / ⟨ 𝐼 ⟩ = ℓ . Here ⟨ ... ⟩ = ∫︀ ... 𝑑𝑧 𝑑𝑥 = ∫︀ ... 𝑑𝜁 𝑑𝑥 and we imply quasi-monochromatic quantiza-tion of photon’s energy, (cid:126) 𝜔 .For the in-plane polarized pulse (3)–(5), Eqs. (6) yield 𝑆 𝑦 ∝ 𝜔 − Δ 𝑘 𝑥 ℓ 𝑘 (︀ 𝐽 ℓ +1 − 𝐽 ℓ − )︀ ,𝐿 𝑦 = 𝜔 − ( ℓ − 𝑘 𝑥 ) 𝐼 + ℓ 𝑘 𝑥 𝑆 𝑦 . (8) FIG. 5. The plane-wave spectrum (left) and the phase-intensity distribution of the real-space wavefunction 𝜓 ( r , 𝑡 )(right) for an elliptical Bessel STVP with ℓ = 2 and ratio ofprincipal axes 𝛾 = 2. The normalized integral OAM of thispulse is 𝜔 ⟨ 𝐿 𝑦 ⟩ / ⟨ 𝐼 ⟩ = 2 .
5, Eq. (10).
These distributions are depicted in Fig. 4. The nonzerospin density 𝑆 𝑦 is a manifestation of the transverse spin phenomenon, which recently attracted great attention[7, 33, 43–45]. Here, the transverse spin arises fromthe interference of plane waves with different directions,phases, and in-plane linear polarizations, Fig. 3(b). Thenormalized SAM density 𝜔 𝑆 𝑦 /𝐼 reaches the minimumand maximum values of − ⟨ 𝑆 𝑦 ⟩ = 0.The OAM density (8) has a form similar to Eq. (7)with an additional spin-related term; this is also a singna-ture of the spin-orbit interaction [30, 33]. Nonethe-less, in contrast to the analogous spin-dependent OAMparts in monochroimatic beams, the integral value ofthis term vanishes. (This follows from the relation ℓ ∫︀ ∞ (︀ 𝐽 ℓ +1 − 𝐽 ℓ − )︀ ˜ 𝜌 𝑑 ˜ 𝜌 ∝ ℓ ∫︀ ∞ 𝐽 ℓ 𝑑𝐽 ℓ /𝑑 ˜ 𝜌 𝑑 ˜ 𝜌 = 0.) Thus,akin to the scalar case, the integral OAM value corre-sponds to (cid:126) ℓ per photon.We conclude that in spite of local spin-orbit interactioneffects, the integral SAM and OAM of STVPs are ratherrobust: ⟨ 𝑆 𝑦 ⟩ = 0 , 𝜔 ⟨ 𝐿 𝑦 ⟩⟨ 𝐼 ⟩ = ℓ . (9)Importantly, the above calculations are made forSTVPs with circularlly symmetric intensity profiles inthe ( 𝜁, 𝑥 ) plane. However, in most cases these profilesare elliptical with some ratio of principal axes 𝛾 in the( 𝜁, 𝑥 ) plane, as shown in Fig. 5. Such an elliptical STVPis described by the substitution 𝑥 → 𝛾𝑥 in the scalarwavefucntion (1) and (2) or, equivalently, by the sub-stitution 𝑘 𝑥 → 𝛾 − 𝑘 𝑥 in the beam spectrum. For thefield of the form ∝ exp( 𝑖ℓ ˜ 𝜙 ) and the OAM operator^ 𝐿 𝑦 = − 𝑖𝜕/𝜕 ˜ 𝜙 = 𝑖 ( 𝑥𝜕/𝜕𝜁 − 𝜁𝜕/𝜕𝑥 ), this results in an additional factor in the intrinsic OAM value [24, 46]: 𝜔 ⟨ 𝐿 𝑦 ⟩⟨ 𝐼 ⟩ = 𝛾 + 𝛾 − ℓ . (10)This factor is significant: for example, the experiments[26, 27] generated STVPs with 𝛾 ≃ . 𝛾 + 𝛾 − ) / ≃ . .
7. Figure 5 shows an example of theSTVPs with ℓ = 2, 𝛾 = 2, and 𝜔 ⟨ 𝐿 𝑦 ⟩ / ⟨ 𝐼 ⟩ = 2 . Conclusions.—
We have examined spatiotemporal vor-tex pulses with purely transverse intrinsic orbital angu-lar momentum. We provided analytical Bessel-type solu-tions, both scalar and vector, and described their propa-gation, polarization, and angular-momentum properties.Most importantly, we provided accurate calculations ofthe spin and orbital angular momenta of STVPs anddescribed observable spin-orbit interaction phenomena.Notably, the polarization and spin-orbit effects mani-fest themselves locally via obsrevable intensity and spin-density distributions, while the integral values of the spinand orbital angular momenta of STVPs are rather robust.At the same time, the integral OAM value is significantlyaffected by the elliptical shape of STVPs with differentwidth and length, which are typically generated in ex-periments [26, 27].The results of our work provide a theoretical platformfor investigations of novel spatiotemporal vortex statesand call for experimental measurements of the predictedpolarization and angular-momentum phenomena. Im-portantly, these results are applicable to waves of dif-ferent natures. For example, STVPs can be generatedin sound waves in fluids or gases. In doing so, one canuse the sclalar approach (1)–(2) for the pressure wave-field 𝑃 ( r , 𝑡 ) or the vector approach similar to Eqs. (3)–(5) for the velocity wavefield V ( r , 𝑡 ). Since sound wavesare longitudinal , i.e., V ‖ k for each plane wave in thespectrum, the velocity field has the 𝑧 and 𝑥 componentsgenerating the transverse spin and related spin-orbit phe-nomena [39, 47]. [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw,and J. P. Woerdman, “Orbital angular momentum oflight and the transformation of Laguerre-Gaussian lasermodes,” Phys. Rev. A , 8185–8189 (1992).[2] L. Allen, S. M. Barnett, and M. J. Padgett, eds., OpticalAngular Momentum (Taylor and Francis, 2003).[3] A. Bekshaev, M. Soskin, and M. Vasnetsov,
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