Knotted Polarizations and Spin in 3D Polychromatic Waves
Danica Sugic, Mark R. Dennis, Franco Nori, Konstantin Y. Bliokh
KKnotted Polarizations and Spin in 3D Polychromatic Waves
Danica Sugic,
1, 2
Mark R. Dennis,
2, 3
Franco Nori,
1, 4 and Konstantin Y. Bliokh Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom EPSRC Doctoral Training Centre for Topological Design,University of Birmingham, Birmingham B15 2TT, United Kingdom Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
We consider complex 3D polarizations in the interference of several vector wave fields with differentcommensurable frequencies and polarizations. We show that resulting polarizations can form knots ,and interfering three waves is sufficient to generate a variety of Lissajous, torus, and other knot types.We describe spin angular momentum, generalized Stokes parameters and degree of polarization forsuch knotted polarizations, which can be regarded as partially-polarized. Our results are generic forany vector wave fields, including, e.g., optical and acoustic waves. As a directly-observable example,we consider knotted trajectories of water particles in the interference of surface water (gravity)waves with three different frequencies.
Introduction.—
Polarization is a fundamental propertyof vector waves of different nature. It is thoroughlystudied in optics and electromagnetism [1], but can beequally applied to any vector wavefields, e.g., elasticand acoustic waves [2–4]. Polarization can be associ-ated with the curve traced by the field vector F ( r , t ) in a given point r . For a monochromatic 3D wave field F ( r , t ) = Re (cid:2) F ( r ) e − iωt (cid:3) , this curve is generically an el-lipse [1].Rotation of the field vector at a given point r canalso be associated with an intrinsic angular momentum(AM), i.e., spin [3–9]. This is one of fundamental dynami-cal properties of vector waves, including electromagnetic,elastic, and acoustic ones. In the most general case, a 2Dpolarization state is described by the four Stokes param-eters (one of which is the normal spin component) [1, 10],while 3D polarized fields require nine generalized Stokesparameters [11–14] (three of which are responsible for thespin components [13, 15]).When the wave contains multiple frequencies, the mo-tion of the field vector becomes more complicated and inthe limit of an irregular chaotic-like motion implies total depolarization of the wave. However, when a polychro-matic field contains only several frequency componentswith well-defined polarizations, the field vector motion iscomplicated but still regular. This regime is only barelystudied; the only properly-described case involves 2Dfields with two commensurable frequencies, which gen-erate closed Lissajous-like polarization curves [16–19].In this work, we show that interfering three or morewaves in 3D with commensurable frequencies producesclosed polarization ‘trajectories’ which are generally knotted [20]. Knotted structures in wave fields have beenintensively studied recently in various contexts [21], suchas knotted electromagnetic field lines [22–25], knots ofwave singularities [26–30], and 2D Lissajous-like polar-izations with an extra synthetic dimension [19]. How-ever, 3D ‘knotted polarizations’ considered here is a novelphysical entity, to the best of our knowledge. We will de- scribe several classes of knots, which are naturally gen-erated in polychromatic fields. We will also analyze thespin angular momentum and generalized Stokes param-eters produced by such knotted polarizations. Our ap-proach is general and can be applied to optical, acoustic,and other vector wave fields. In particular, we show anexample of knotted polarizations in interference of sur-face water (gravity) waves. There, such polarizations cor-respond to real-space trajectories of water molecules andcan be directly observed experimentally [31]. Knotted polarizations.—
We consider an interference of3D vector fields with multiple commensurable frequencies ω n , n = 1 , ..., N , so that F ( r , t ) = (cid:80) n Re (cid:2) F n ( r ) e − iω n t (cid:3) .The field vector in a given point r traces a closed 3Dcurve with the temporal period T = 2 π/ Ω , where Ω isthe lowest common multiple of { ω n } . Such polarizationcurves can be topologically nontrivial and form knots [20],see Fig. 1. Note that the polarization curve depends onlyon frequencies and elliptical polarizations of the interfer-ing waves F n in the given point r . From now on we fixthis point and consider only temporal dependencies ofthe fields.Because of time-harmonic character of the interfer-ing waves, the Cartesian field components F ( t ) =[ F x ( t ) , F y ( t ) , F z ( t )] are sums of cos( ω n t ) and sin( ω n t ) terms with different amplitudes. Knotted curves de-scribed by such harmonic terms are known as Fourieror harmonic knots [32, 33]. These knots are markedwith three integer indices ( i, j, k ) ≥ (1 , , indicatingthe numbers of frequencies in the three Cartesian fieldcomponents. Remarkably, any type of knot can be con-structed as a Fourier- (1 , , k ) knot with some k [34].In the simplest case, when each Cartesian componenthas only one frequency, such Fourier- (1 , , knots arecalled Lissajous knots [37–39], with F ( t ) given by: [ A cos( ω t + φ ) , A cos( ω t + φ ) , A cos( ω t + φ )] , (1)where the frequencies ω n , n = 1 , , , are proportional a r X i v : . [ phy s i c s . op ti c s ] J u l FIG. 1. Examples of knotted polarizations F ( t ) in the interference of three waves with different frequencies and polarizations(shown to the right from the knots). (a) The Lissajous knot (1) with A = A = A , ω /ω = 5 / , ω /ω = 3 / , φ − φ = 0 . ,and φ − φ = 1 . (the or three-twist knot [35]). (b) The torus knot (2) with p = 2 , q = 3 (trefoil knot). (c) Figure-eightknot (3). The spins of the interfering waves and resulted knotted polarizations are shown in purple, whereas the 3D degree ofpolarization for the knotted states (a)–(c) are P = 0 , P = 1 / , and P (cid:39) . , respectively [36]. to three coprime integers, whereas A n and φ n are ampli-tudes and phases. In physical terms, the Lissajous-knotpolarizations are produced by a superposition of three lin-ear polarizations oriented along the three Cartesian axes,as shown in Fig. 1(a) for the example of or ‘three-twist’knot [20, 21, 35, 38].One of the most important classes of knots are torusknots T p,q , which lie on a torus surface and are charac-terized by a pair of coprime integers ( p, q ) [20, 21]. No-tably, every type of torus knots can be represented by aFourier- (1 , , knot with F ( t ) given by [40] A [cos( ω t ) , cos( ω t + φ ) , − sin( ω t ) + cos( ω t + φ )] ,ω ω = qp , ω ω = q − pp , φ = π p , φ = π p − π q . (2)Polarization torus knots are produced by a superpositionof one circular and two linear polarizations , as shown inFig. 1(b) for the example of trefoil knot T , [20, 21].Note that waves with frequencies satisfying Eq. (2) canbe generated in nonlinear wave-mixing processes [19, 41].The Lissajous and torus knots do not exhaust all pos-sible Fourier knots. For example, the figure-eight knot can be generated by the field F ( t ) as follows [33]: A [cos( ωt ) + cos(3 ωt ) , . ωt ) − sin(6 ωt ) , . ωt ) + sin(3 ωt )] . (3)This is a superposition of two elliptical and one linearpolarizations , Fig. 1(c). Spin in optical and acoustic fields.—
The normalizedperiod-averaged spin AM density in a monochromaticvector wavefield can be characterized by the expression S = Im ( F ∗ × F ) / | F | , | S | ≤ . In optical fields, F = E is the electric field [6, 10, 42], while in sound wavefieldsin fluids or gases, F = V is the velocity field [3, 4, 9].Note that this simplified approach ignores the presenceof other fields (magnetic field in optics and pressure fieldin acoustics [4, 5, 7, 9, 43–45]), but this omission is jus-tified in many practical problems, where experimental measurements and phenomena are sensitive only to theelectric and velocity fields.The above expression for the normalized spin origi-nates from the time-averaged non-normalized expression (cid:104) G × F (cid:105) = Im ( F ∗ × F ) / (2 ω ) and the density of “fieldquanta” given by the energy density divided by frequency, (cid:104) F · F (cid:105) /ω = | F | / (2 ω ) , where G is the vector-potential,such that F = ∂ t G . In optics, A = − G is the mag-netic vector-potential in the Coulomb gauge [45], whilein acoustics R = G is the displacement field [9], so that G × F = R × V is the natural mechanical AM form.In the polychromatic field F consideredabove, the vector-potential equals G ( r , t ) = (cid:80) n ω − n Im (cid:2) F n ( r ) e − iω n t (cid:3) . Substituting it into thespin form (cid:104) G × F (cid:105) with time-averaging over the period T yields the normalized spin density in a polychromaticfield: S = (cid:80) n ω − n Im ( F ∗ n × F n ) (cid:80) n ω − n | F n | . (4)This equation shows that spin of a polychromatic fieldrepresents a properly weighted sum of the spins S n ofthe interfering monochromatic components. In particu-lar case of 2D bi-chromatic optical fields, consisting ofcircularly-polarized waves, the general expression (4) co-incides with the one used in Refs. [16, 18]. Obviously, thespin (4) vanishes for interfering linearly-polarized fields,such as the Lissajous knots, Fig. 1(a). For other knot-ted polarizations it is generically nonzero (see [36] andFigs. 1(b,c) for the spin of the torus and figure-eightknotted polarizations, Eqs. (2) and (3)), but restrictedby | S | ≤ .Remarkably, the above spin AM in polychromaticfields with complex 3D polarization curves allows verysimple mechanical analogy. Let us consider a mechanicalpoint particle of unit mass moving in real space along theclosed trajectory r ( t ) = G ( t ) . Then, the period-averagedmechanical AM of this particle, (cid:104) r × ∂ t r (cid:105) equals the spin (cid:104) G × F (cid:105) , whereas its averaged kinetic energy, (cid:104)| ∂ t r | / (cid:105) ishalf of the field energy (cid:104) F · F (cid:105) This hints that the curvetraced by the vector-potential G could be more funda-mental than the one traced by the field F . Of course, allprevious considerations about knotted polarizations canbe equally applied to the vector-potential polarization.Note that for monochromatic fields, the polarizationellipses of the field F and its vector-potential G coincidewith each other (up to a constant factor). In contrast,complex polarization curves of polychromatic fields andtheir vector-potentials generally differ considerably. Thisis because single-harmonic elliptical motion is invariantwith respect to the time-derivative operation, while com-plex polychromatic motion is not. Generalized Stokes parameters and depolarization.—
The above consideration of the spin AM suggests thatany quadratic forms of fields could be considered in asimilar manner, such that interference terms between dif-ferent frequencies are averaged out and the form rep-resents a weighted sum of contributions from each fre-quency component. For example, the canonical momen-tum of a polychromatic field is calculated similarly tothe spin but with the substitution of the quadratic form
Im ( F ∗ n × F n ) → Im [ F ∗ n · ( ∇ ) F n ] [3, 4, 7, 9, 43, 45].Calculation of optical or acoustic radiation forces andtorques also involves similar quadratic forms and allowssimilar approach [46, 47].Here we consider important quadratic forms used forthe description of 3D partially polarized fields, namely,the generalized Stokes parameters Λ l , l = 0 , , ..., [11–14]. For monochromatic fields, these parameters ap-pear from the Hermitian × coherence matrix Γ ij = (cid:104) F ∗ i F j (cid:105) , i, j = x, y, z , and its decomposition Γ ij = (cid:80) l =0 Λ l { λ l } ij via the Gell-Mann matrices λ l [11–14].Here Λ = Tr(ˆΓ) is associated with the field intensity,parameters Λ , Λ , Λ , Λ , and Λ are related to thereal part of ˆΓ , while the three parameters Λ , Λ , and Λ related to the imaginary part of ˆΓ are proportional tothe Cartesian components of the spin density S [13, 15].For polychromatic fields considered here, we introducethe natural analogues of the coherence matrix and thegeneralized Stokes parameters as: Γ ij = (cid:88) n F ∗ ni F nj = 13 (cid:88) l =0 Λ l { λ l } ij , (5)such that the polarization parameters are sums of thecorresponding parameters for the interfering monochro-matic waves: Λ l = (cid:80) n Λ ( n ) l [36]. Note the the vec-tor − Λ , Λ , − Λ ) / (3Λ ) represents an analogue of thenormalized spin (4) but without ω − n weighting factors[13, 15].Among the different definitions of the degree of po-larization for 3D fields [12, 48–50], a natural choice is P = (cid:113)(cid:80) l =1 Λ l (cid:14) √ ∈ [0 , . An elliptically polarized monochromatic light, i.e., each of the interfering compo-nents in our knotted fields, Eqs. (1)–(3) and Fig. 1, is fullypolarized: P ( n ) = 1 [36]. However, calculating the polar-ization parameters (5) for polychromatic knotted fields,we find that its degree of polarization diminishes: P < .This means that polychromatic knotted fields can be re-garded as partially depolarized . In particular, for anyLissajous-knotted field (1) with A = A = A , we findthat P = 0 , i.e., it is totally unpolarized [36]. This is nat-ural, because such fields consists of three independentoscillations with equal amplitudes along the three axes.In turn, for any torus-knotted field (2), the degree of po-larization is P = 1 / , while for the figure-eight-knottedpolarization (3) we find P (cid:39) . [36]. In these cases, thepresence of partial polarization is related to the presenceof nonzero spin (4) in such knotted fields, Figs. 1(b,c)[15]; total depolarization implies zero spin.Note, however, that the spin, polarization parameters,and degree of polarization are not directly related to topological properties of knotted polarizations. These arerather geometrical (or dynamical) properties of the fieldcurve F ( t ) . For example, polarization parameters of theLissajous knots (1) strongly depend on the amplitudes A n (scaling factors along the Cartesian axes) [36], whilethe knot topology is obviously independent of these. Knotted trajectories in water waves.—
Remarkably,complex polarizations of polychromatic waves is a di-rectly observable phenomenon. While measuring time-dependent electric field with optical frequencies is prac-tically impossible, the sound-wave polarization is re-lated to the velocity field V ( r , t ) or the displacement‘vector-potential’ R ( r , t ) [9]. Here, the time-dependentfield R ( r , t ) describes the real-space displacement of themedium particles, so that its polarization curve is the real-space trajectory of the particle. Although it is a chal-lenge to observe the displacement of air or water parti-cles at typical sound frequencies, this can be easily donefor surface water waves with much lower frequencies anddirectly-observable motion of water particles. Indeed, re-cent experiment [31] observed a 2D Lissajous-like motionin the interference of water waves with different frequen-cies. Here we show that a similar experiment can bedesigned to observe 3D knotted trajectories of water par-ticles when taking into account their horizontal and ver-tical motions and interfering three waves with differentfrequencies.Using equations of hydrodynamics, the equation ofmotion for the 3D displacement field R ( r ⊥ , t ) =[ X ( r ⊥ , t ) , Y ( r ⊥ , t ) , Z ( r ⊥ , t )] on the water surface z = 0 for deep-water (gravity) waves can be written as [36] ∂ t Z = g ∇ ⊥ · R ⊥ , where r ⊥ = ( x, y ) , ∇ ⊥ = ( ∂ x , ∂ y ) , R ⊥ = ( X , Y ) , and g is the gravitational accelera-tion. For a monochromatic plane-wave field R ( r ⊥ , t ) =Re (cid:2) R e − iωt + i k · r ⊥ (cid:3) with k = ( k x , k y ) being the wavevec-tor, this yields − ω Z = ig k · R ⊥ , where ω = gk isthe dispersion relation [51]. These equations describe FIG. 2. Interference of three surface water waves, one propagating in x and two standing along y , Eq. (6), with differentfrequencies and phase corresponding to Fig. 1(b). The instantaneous surface shapes (greyscale) and 3D water-particles trajec-tories (colored) are shown for the interference field (a) (see also its animated version [36]) and each of the interfering waves(b). Some of these trajectories correspond to trefoil torus knots, Fig. 1(b), while other are ‘unknots’. The ( x, y ) -regions withknotted and unknotted trajectories are shown in (c). the well-known fact: in a plane gravity wave, the wa-ter particles move along circular trajectories lying in theplane determined by the wavevector and normal to thesurface [52]. In other words, a single deep-water wavehas a purely circular polarization . Then, interferingoppositely-propagating waves with the same frequencyyields a superposition of opposite circular polarization, i.e., a linear polarization with the position-dependent ori-entation. Thus, propagating and standing gravity wavesprovide circular and linear polarizations necessary for thetorus knots (2), Fig. 1(b).Explicitly, we consider the interference of an x -propagating plane wave and two standing waves alongthe y -axis with three different frequencies and phases butequal amplitudes. This results in the displacement field R ( r ⊥ , t ) ∝ cos ( ω t − k x )0 − sin ( ω t − k x ) + ω t + φ ) cos ( k y )cos ( ω t + φ ) sin ( k y ) + − cos ( ω t + φ ) sin ( k y )cos ( ω t + φ ) cos ( k y ) , (6)where k , , = ω , , /g . Choosing the frequencies andphases satisfying Eqs. (2), we find that the real-spacetrajectory of the water motion at r = , R ( , t ) is exactlythe torus knot (2), Fig. 1(b).Figure 2 shows the water surface shape and water-particles trajectories R ( r ⊥ , t ) for the whole interferencefield (6) with p = 2 , q = 3 corresponding to the tre-foil knot in Fig. 1(b), and also for each of the interferingwaves. Note that different points of the water surface ( x, y ) correspond to different mutual phases of the in-terfering waves. Therefore, water motions in differentpoints have different 3D trajectories. In our case, these trajectories represent trefoil knots and ‘unknots’ [20, 21].The ( x, y ) -regions with knotted and unknotted trajec-tories are shown in Fig. 2(c). Numerical calculations[53] show that the fraction of knotted trajectories hereis about . Many of the polarization trajectories arenearly self-intersecting, which can make the precise knottopology hard to resolve. Such self-intersections occurat every transition here between unknot and trefoil knot,but the arcs pass through each other (i.e. not reconnecting like vortices [54, 55]). More complicated superpositionsgive rise to larger areas of knotting, with transitions be-tween multiple knot types. Conclusions.—
We have studied polychromatic 3Dvector waves with closed (periodic) field trajectories. Inparticular, we have found that interference of three ormore vector waves with different frequencies can gener-ate a variety of knotted
3D polarizations, including Lis-sajous, torus, and other knots. We have introduced natu-ral formalism for the spin angular momentum (includinga simple mechanical analogy), generalized Stokes param-eters, and other quadratic forms for polychromatic yetperiodic 3D vector waves. This revealed the presence ofnonzero spin and partial depolarization in generic knot-ted polarizations. Finally, using the generic character ofour consideration, valid for any 3D vector wave fields, wehave provided an example, where knotted polarizationsappear as directly-observable trajectories of water parti-cles in the interference of surface-water (gravity) waves.Thus, our work considerably extends earlier studies ofcomplex polarizations in polychromatic fields, previouslyrestricted to 2D bichromatic Lissajous-like polarizations[16–19], as well as theory and potential applications ofthe spin AM, so far mostly restricted to monochromaticwaves [3–7]. Furthermore, these results provide a newnatural application of the knot theory to wave physics,different from the previously-studied knots of field lines[22–25] or singularities [26–30]. One can expect that non-trivial topological and dynamical features of such knot-ted polarization states will find interesting applicationsin complex wave systems.This work was partially supported by NTT Research,Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (viathe CREST Grant No. JPMJCR1676), Japan Society forthe Promotion of Science (JSPS) (via the KAKENHIGrant No. JP20H00134, and the grant JSPS-RFBRGrant No. JPJSBP120194828), the Grant No. FQXi-IAF19-06 from the Foundational Questions InstituteFund (FQXi), a donor advised fund of the Silicon ValleyCommunity Foundation, and the EPSRC Centre for Doc-toral Training in Topological Design (EP/S02297X/1). [1] R. M. A. Azzam and N. M. Bashara,
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