Global aspects of polarization optics and coset space geometry Arvind ∗ Department of Physical Sciences, Indian Institute of Science Education & Research (IISER) Mohali, Sector 81 SAS Nagar, Manauli PO 140306 Punjab India. S. Chaturvedi † Department of Physics, Indian Institute of Science Education & Research (IISER) Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066 India 7 N. Mukunda 1 ‡ INSA C V Raman Research Professor, Indian Academy of Sciences, 0 C V Raman Avenue, Sadashivanagar, Bangalore 560080 India 2 n Weusegrouptheoreticideasandcosetspacemethodstodealwithproblemsinpolarizationoptics a of a global nature. These include the possibility of a globally smooth phase convention for electric J fields for all points on the Poincaré sphere, and a similar possibility of real or complex bases of 4 transverse electric vectors for all possible propagation directions. It is shown that these methods 2 helpinunderstandingsomeknownresultsinaneffectivemanner,andinansweringnewquestionsas well. WefindthatapartfromthegroupsSU(2)andSO(3)whichoccurnaturallyintheseproblems, ] thegroup SU(3) also plays an important role. h p - s I. INTRODUCTION combed’. WewillseethatrelatingthisquestiontoSO(3) s leadstoanimmediateandilluminatingunderstandingof a l It is well known that the Poincarésphere [1] represen- thisresult. Howeverithasbeenpointedoutrecentlythat c . tation of the pure polarization states of a plane electro- ifonegoestothecomplexdomainandallowselectricfield s magneticwave(withfixedfrequencyandpropagationdi- vectors in all possible (pure) polarization states, the ob- c i rection)isintimatelyrelatedtothepropertiesofthetwo- struction disappears [5]. There exist bases of complex ys dimensional unitary unimodular group SU(2) [2]. Sim- electric fields which are globally smooth over the sphere h ilarly if one considers choices of transverse real electric of directions. Here too it turns out that the use of the p fieldvectorscorrespondingtoallpossiblepropagationdi- group SU(2), and to a limited extent of SU(3) as well, [ rectionsinspace,the realrotationgroupSO(3)becomes clarifies the situation greatly [6]. We develop in full de- relevant. In both cases, questions of a global nature can tail a simple andattractive grouptheory basedchoice of 1 v be analysed particularly effectively if they are cast into acomplexgloballysmoothbasisoverthesphereofdirec- 2 a group theoretical form. tions,bycombiningtheanswerstothefirsttwoquestions 0 Inthispapersuchanapproachisusedtoanalysethree in such a way that the obstructions seen in them com- 5 such questions. The first is the search for a globally pensate, or annihilate, one another. We also describe an 7 smoothphaseconventionfortransverseelectricfieldvec- importantpropertyofthesetofallsuchgloballysmooth 0 tors corresponding to all points on the Poincaré sphere choices. . 1 [3]. This question is shown to have a simple and elegant 0 The material of this paper is organized as follows. In resolutionwhenexaminedinthe frameworkofthe group 7 SectionIIweaddressthefirstofthethreequestionsmen- SU(2). The second and third questions are concerned 1 tioned above and recast it in the language of group the- with choices of globally smooth basis states for trans- : v verse electric field vectors as the propagation direction ory. In particular, we show that the problem of finding i a globally smooth phase convention over the Poincaré X varies overall points on the sphere of directions in phys- sphere is entirely equivalent to the problem of choosing ical space. The second question deals with the case of r a real fields, while the third question asks whether such a representativesof the cosetspace SU(2)/U(1)in a man- nerthatissmoothoverallpointsonthePoincarésphere. smooth basis is possible for complex electric fields. Here it is a well knownresultthat if we limit ourselves to real This reformulation of the originalquestion not only per- mits us to furnish a simple proof of the well known im- fields (corresponding to linear polarizations), such bases do not exist because of an obstruction [4]. possibility of a globally smooth phase convention over TherelevantmathematicalresultisthatthesphereS2 the Poincaré sphere but, on the positive side, enables is not parallelizable, more informally that it ‘cannot be us to provide a parameterization that works at all but one point (the South pole) of the Poincaré sphere. In SectionIII,restrictingourselvesto realelectricfields, we turntothequestionoffindingarealorthonormalbasisin ∗ arvind@iisermohali.ac.in the two dimensional tangent plane at each point on the † subhash@iiserbhopal.ac.in sphere of directions in a smooth manner. Here, in the ‡ nmukunda@gmail.com same spirit as in Section II, we show that this exercise 2 reducestofindingasmoothchoiceofrepresentatives,this with E(nˆ) varying smoothly with respect to nˆ for all nˆ? time for the cosetspace SO(3)/SO(2). Again, as before, We can recastthis question in terms of the groupSU(2) besides providing a simple proof of the well known non as follows. There is a simple one-to-one correspondence existence of such bases, the use of group theoretic lan- betweenSU(2)elements andnormalizedtwo-component guage facilitates construction of bases with the desired complex column vectors (like E above): propertiesatallbutonepointonthesphereofdirections. u SU(2) Next in Section III we reexamine this question allowing ∈ ⇔ the electric field to be complex. In other words we ask α β u= − ∗ , α2+ β 2 =1 thequestionwhetherornotitispossibletocomeupwith (cid:18) β α∗ (cid:19) | | | | ⇔ asmoothchoiceoforthonormalbasesinthecomplexified α tangent spaces at each point on the sphere of directions. ξξξ = C2, ξξξ†ξξξ =1; (cid:18) β (cid:19)∈ The complexification brings the group SU(3) into play in a natural way and we show that by a judicious use u(ξξξ)=(ξξξ, iτ3ξξξ∗). (5) − of the constructs developed earlier, one is not only able This kind of relationship is special to SU(2) [7, 8], and to answer the question at hand in the affirmative but is it shows immediately that, sinceξξξ S3, as a manifold also able to give an elegant and essentially unique con- ∈ struction of the desired bases. Section IV contains our SU(2) S3. (6) concluding remarks and further outlook. ∼ Moreover, u(ξξξ) obeys a ‘covariance condition’ II. GLOBALLY SMOOTH PHASE CONVENTION OVER THE POINCARE´ Su(ξξξ)=u(Sξξξ), any S ∈SU(2), (7) SPHERE– THE OBSTRUCTION which is very useful. With u andξξξ related as in Eqn. (5), we find that Consider plane electromagnetic waves (with fixed fre- quency ω and propagating along the positive z-axis) in uτ u 1 =nˆ τττ, 1 − · various states of pure polarization. Denote the two- nˆ =ξξξ†τττξξξ =(α2 β 2,2 (α∗β),2 (α∗β)); componentcomplexelectricfieldvectorinthetransverse | | −| | ℜ ℑ x y plane, say at z =0, by u′ =ueiφτ1 ⇔ξξξ′ =eiφξξξ. (8) − The set of elements eiφτ1,0 φ < 2π SU(2) E= Ex = E1 C2. (1) defines a U(1) subgrou{p, leadin≤g to the }cos⊂et space (cid:18) Ey (cid:19) (cid:18)E2 (cid:19)∈ SU(2)/U(1) which can be identified with S2 . We see pol The intensity in suitable units is I = E E, and for sim- thatnˆ inEqn.(8)isdeterminedbyξξξ inexactlythesame † way as Eqn. (2) determines the polarization state of E. plicity it will hereafter be set equal to unity. The polar- In the presentcase we view Eqn.(8) as defining the pro- ization state is represented by a point on the Poincaré sphere S2pol, given in terms of E by jection map π :SU(2)→SU(2)/U(1)=S2pol: nˆ =E†τττE∈S2pol. (2) u∈SU(2)→π(u)=nˆ =ξξξ†τττξξξ ∈S2pol, Here we follow the polarization optics conventions for τττ π(ueiφτ1)=π(u). (9) matrices: Conversely, each nˆ S2 determines a corresponding ∈ pol 1 0 0 1 0 i coset in SU(2): τ1 =(cid:18)0 1(cid:19),τ2 =(cid:18)1 0(cid:19),τ3 =(cid:18) i −0 (cid:19) (3) − nˆ S2 ∈ pol ⇔ which are related by a cyclic permutation to the Pauli (nˆ)= u SU(2)π(u)=nˆ =π 1(nˆ) SU(2)(.10) − matrices σσσ normally used in quantum mechanics: τ = C { ∈ | } ⊂ 1 σ3,τ2 =σ1,τ3 =σ2. The three-vector nˆ is real, has unit The searchfor a smooth E(nˆ) for all nˆ ∈S2pol obeying length, and is unchanged by an overall phase transfor- Eqn. (4) is the same as the search for a smoothξξξ(nˆ) for mation E E′ = eiφE. Points on the equator n3 = 0 all nˆ such that (E real ap→art from an overall phase) correspond to linear polarizations. The two poles (0,0, 1) represent cir- ξξξ(nˆ) τττ ξξξ(nˆ)=nˆ. (11) ± † 1 cular polarizations, RCP or LCP, for E = 1 √2(cid:18) i(cid:19) Since by Eqn.(5) u andξξξ determine one another,this in ± apart, again, from overall phases. A physically interest- turn is the same as the search for a smooth coset repre- ing question is this: for each nˆ S2 , is it possible to sentative u(nˆ) (nˆ) for all nˆ. ∈ pol ∈C choose E(nˆ) C2 such that Howeversuchgloballysmoothcosetrepresentativesare ∈ not possible. Suppose one such, u (nˆ) say, did exist. It 0 E(nˆ) τττE(nˆ)=nˆ, (4) would then permit the expression of any u SU(2) in a † ∈ 3 globally smooth way as a product of a coset representa- complex three-component electric field E (omitting the tive and a subgroup element: plane wave factor e iω(t kˆ.x/c)) obeys the transversality − − u SU(2), π(u)=nˆ S2 : condition ∈ ∈ pol u=u (nˆ)eiφτ1, 0 φ<2π. (12) kˆ E=0, (17) 0 ≤ · This would imply that as a manifold SU(2) S3 so it is essentially two-dimensional. S2 U(1) = S2 S1, which is easily seen to ≃be fals≃e. To begin with, consider real fields. Then the vector E The×reforeno glob×allysmoothξ(nˆ)obeying Eqn.(11), or lies in the two-dimensional plane in physical space tan- E(nˆ) obeying Eqn. (4), can be found. gent to S2 at kˆ: dir While it is the case that no smooth coset representa- ativceosue0t(nˆre)pforersaelnltnât∈iveS2pfoolrisalalvnaîleaxbclee,pwt,efcoarnedxaefimnpeles,ucaht E real, kˆ·E=0⇔E∈TkˆS2dir ⊂R3. (18) the South pole (0,0, 1) of S2 . Here is a simple con- Wenowaskwhetheritispossibletochooseanorthonor- struction, with nˆ para−metrisedpobly spherical polar angles mal basis of real vectors E(a)(kˆ) for each kˆ obeying θ,φ: kˆ E(a)(kˆ)=0, E(a)(kˆ) E(b)(kˆ)=δ , nˆ =(sinθcosφ,sinθsinφ,cosθ) S2 , · · ab ∈ pol E(1)(kˆ) E(2)(kˆ)=kˆ, a,b=1,2, (19) 0 θ <π,0 φ<2π; ∧ ≤ ≤ u (nˆ)=(ξξξ (nˆ), iτ ξξξ (nˆ) ), inagloballysmoothway. Wehaveincludedherethecon- 0 0 − 3 0 ∗ ditionthat(E(1)(kˆ),E(2)(kˆ),kˆ)formarighthandedsys- cosθ/2 ξξξ (nˆ)=S , tem;theyobviouslyformanorthonormalsysteminthree 0 0(cid:18)eiφsinθ/2(cid:19) dimensions. (Actually,itsufficestobeabletochooseone real vector E(1)(kˆ) for each kˆ obeying 1 1 1+i 1+i S0 = 2(1+iτ1+iτ2+iτ3)= 2(cid:18) 1+i 1 i(cid:19)∈SU(2) kˆ E(1)(kˆ)=0, E(1)(kˆ) E(1)(kˆ)=1. (20) − − · · (13) where the matrix S connects theτττ andσσσ matrices: Then if we define 0 τj =S0σjS0−1,j =1,2,3. (14) E(2)(kˆ)=kˆ ∧ E(1)(kˆ), (21) One can check that Eqns.(8) are obeyed for all nˆ except all of Eqns. (19) are obeyed). If such a choice were the Southpole. Atthe Northpole, wedo find thatthese possible, for each kˆ S2 we could define an element ∈ dir expressions are all well-defined: R(kˆ) SO(3),theproperrealorthogonalrotationgroup ∈ in three dimensions, carrying e to kˆ: eiπ/4 1 3 θ 0: ξξξ (θ,φ) , → 0 → √2 (cid:18) i (cid:19) kˆ S2 R(kˆ) SO(3): ∈ dir → ∈ u0(θ,φ)→S0. (15) R(kˆ)(e1,e2,e3)=(E(1)(kˆ),E(2)(kˆ),kˆ). (22) Ontheotherhand,attheSouthpolethelimitingexpres- Here e ,j = 1,2,3 are the unit vectors along the three j sions are φ dependent, hence undefined: Cartesian coordinate axes in physical space. Therefore ei(π/4+φ) 1 the columns of the matrix R(kˆ) are: θ π : ξξξ (θ,φ) , → 0 → √2 (cid:18)−i(cid:19) Rj1(kˆ)=E(j1)(kˆ), Rj2(kˆ)=E(j2)(kˆ), Rj3(kˆ)=kˆj, u (θ,φ) iS τ eiφτ1. (16) 0 0 3 j =1,2,3. (23) →− Nevertheless, as we shall see in the next Section, this The elements of SO(3) leavinge invariantare rotations 3 construction is useful in another context. in the xy plane, forming an SO(2) subgroup of SO(3): SO(2)= III. BASES OF ELECTRIC FIELD VECTORS cosφ sinφ 0 FOR ALL PROPAGATION DIRECTIONS − R3(φ)=sinφ cosφ 0 0 φ<2π SO(3),  0 0 1 | ≤ ⊂ Weturnnexttotheconsiderationofsomeglobalprob-    R (φ)e =e .  (24) lems connected with the collection of all plane waves 3 3 3 (withfixedfrequencyω)withallpossiblepropagationdi- SimilartothesituationwithSU(2),heretoowehavethe rections. We denotethe sphereofpropagationdirections coset space identification in physical space by S2 , to be distinguished from the dir Poincaré sphere S2 . For any unit vector kˆ S2 , the SO(3)/SO(2) S2 . (25) pol ∈ dir ≃ dir 4 ThereforeifachoiceofR(kˆ)smoothoverallkˆ S2 did This implies ∈ dir exist,itwouldbeacosetrepresentativeandwecoulduse ε =det =1, (31) it to express every R SO(3) smoothly as a product, jklAk1Al2Aj3 A ∈ that is, SU(3). Then we find that in the struc- R SO(3), Re =kˆ S2 : A ∈ ∈ 3 ∈ dir ture (29) for we must have u SU(2). This will A ∈ R=R(kˆ)R (φ), 0 φ<2π, (26) hereafter be assumed. Such matrices form a subset, 3 ≤ A not a subgroup, in SU(3). The breakup (29) of into A similar to Eqn. (12). However this would mean that two factors is however not unique since there are shared SO(3) has the globalstructure of S2 SO(2). This con- elements: flicts with the fact that it is S3/ , w×here is the identification of antipodal points on∼S3. Ther∼efore globally e iφτ3 0 smooth choices of R(kˆ) SO(3) obeying Eqn. (23) are R3(φ)= − 0 . (32) ∈ 0 0 1 not possible.   As mentioned earlier, this is a well-known result [9]. Let us parametrise kˆ S2 in the same way as nˆ S2 Another even more elementary analytic proof is given ∈ dir ∈ pol in Eqn. (13): later. Now we extend this analysis by considering complex kˆ =(sinθcosφ,sinθsinφ,cosθ) S2 , electric field vectors. This means that at each kˆ 0 θ π,0 φ<2π. ∈ dir (33) S2 , we go from the real two-dimensional tangent plan∈e ≤ ≤ ≤ TdiSr2 R3 to its complexification which is no longer MomentarilyavoidingtheSouthpoleθ =π,letusdefine cokˆntdaiirne⊂d in R3: R0(θ,φ) SO(3) as ∈ R (θ,φ)=R (φ)R (θ)R (φ) 1, 0 3 2 3 − cosθ 0 sinθ kˆ S2 : T S2 (T S2 )c = ∈ dir kˆ dir → kˆ dir R2(θ)= 0 1 0  SO(3), E= complex three-dimensional vector kˆ E=0 . sinθ 0 cosθ ∈ { | · }  −  (27) R (θ,φ)e =kˆ. (34) 0 3 At the North pole R (θ,φ) is obviously well-defined: It now turns out that it is possible to find (in infinitely 0 R (0,φ) = I. However as we approach the South pole many ways) orthonormal bases for these complexified 0 we find a multivaluedness: tangent spaces, which are globally smooth with respect to kˆ [4]. The complexificationin Eqn.(27) suggeststhat cos2φ sin2φ 0 − − it is useful to extend the groups SU(2)and SO(3) so far θ π :R0(θ,φ)  sin2φ cos2φ 0  → → − usedtoSU(3),thegroupofcomplexunitaryunimodular 0 0 1  −  matrices in three dimensions. This contains both SU(2) 0 and SO(3) as subgroups. To begin with, let us write τ e2iφτ3 A = − 1 0 . (35) for a general matrix in the unitary group U(3). If we 0 0 1 demand that the third column j3 be the components   A of a chosen kˆ S2 , unitarity of guarantees that the Let us then write ∈ dir A first two columns of form an orthonormal basis for 0 (TkˆS2dir)c: A (θ,φ)=R0(θ,φ) u(θ,φ) 0  SU(3), A ∈ 0 0 1 =E(a), =kˆ :   Aja j Aj3 j u(θ,φ) SU(2), (θ,φ)e =kˆ. (36) =1 E(a)∗E(b) =δ , kˆ E(a) =0, a,b=1,2. ∈ A 3 A†A 3×3 ⇔ j j ab j j Theconditionsfor (θ,φ)tobewell-definedandglobally (28) smoothoverS2 arAe,apartfromsmoothdependenceson dir θ and φ: We then find easily that has a rather simple form: A 0<θ <π : (θ,φ+2π)= (θ,φ); A A 0 u 0 A=R 0 , R∈SO(3),u∈U(2), θ 0 : (θ,φ) u(0,φ) 0  0 0 1 → A → 0 0 1     Rj3 =kˆj. (29) =φ independent; − 0 As in Eq. (19) let us now add the righthandedness con- τ e2iφτ3u(π,φ) θ π : (θ,φ) − 1 0  dition in the sense → A → 0 0 1   ε = =kˆ . (30) =φ independent. (37) jkl k1 l2 j3 j A A A − 5 These translate into conditions on u(θ,φ): simplifications using the covariance condition (7) we ar- rive at the expressions: 0<θ <π : u(θ,φ+2π)=u(θ,φ); θ 0 : u(0,φ)=u SU(2); u0(θ,φ)=S0e−iφ2τ1e−iθ2τ3eiφ2τ1; 0 θ →π : u(π,φ)=e−2∈iφτ3uπ,uπ SU(2). (38) u(θ,φ)=e−iφτ3eiθ2τ1eiφτ3; → ∈ 0 u(θ,φ) We will develop an expressionfor u(θ,φ) using the re- 0(θ,φ)=R0(θ,φ) 0 . (44) sults of Section II, but let us now show by a simple ar- A 0 0 1 gument the existence of an obstruction in the real do-   main. Such a possibility would correspond to u(θ,φ) = At the poles, (θ,φ) smoothly approaches the unam- 0 e iα(θ,φ)τ3 SO(2) SU(2). Then the conditions (38) biguous limitinAg values − ∈ ⊂ requirethatα(θ,φ)obey,apartfromsmoothnessinθand φ: 0(0,φ)=1, 0(π,φ)=diag( i, i, 1). (45) A A − − − 0<θ <π :α(θ,φ+2π) α(θ,φ)=2nπ, Thus according to Eqn. (28), at the poles we have the − orthonormal bases for the complexified tangent spaces n integer independent of θ; given by θ 0:α(0,φ)=a; → θ π :α(π,φ)=2φ+b; a,b constants. (39) E(a)(0,φ)=ea,E(a)(π,φ)= iea, a=1,2. (46) → − The θ-independence of n follows from smoothness (con- All of these correspond to linear polarizations. On the tinuity) in θ. However these conditions on α(θ,φ) are otherhand,atpointsalongtheequatorθ =π/2,wehave mutually inconsistent: (S =sinφ,C =cosφ,0 φ<2π): ≤ S2 SC C α(0,φ)=a n=0; π − α(π,φ)=2φ⇒+b⇒n=2. (40) R0(cid:16)2,φ(cid:17)= −SCC CS2 S0 ;  − −  This is thereforeasimple analytic proofofthe nonparal- π 1 1+i(C2 S2) 2iSC lelizability of TS2dir, as mentioned earlier. u(cid:16)2,φ(cid:17)= √2(cid:18)2iSC − 1−i(C2−S2) (cid:19); We nowreturntothe problemofconstructinga group S2 SC eiπ/4C elementu(θ,φ)∈SU(2)obeyingtheconditions(38). For 0 π,φ =e−iπ/4 SC −C2 eiπ/4S . (47) this we exploit the construction of u0(θ,φ) in Eqn. (13), A (cid:16)2 (cid:17) −iC iS 0 with behaviours as θ 0,π as given in Eqns. (15,16).  − −  → Apart from a scale change in φ, these are qualitatively Thecomplexorthonormalbasisvectorsforthecomplexi- similar to the desired behaviours of u(θ,φ). If we note fiedtangentspacetoS2 attheequatorialpoint(π/2,φ) dir that are thus τ3eiφτ1 =e−iφτ1τ3, E(1)(π/2,φ)=e−iπ/4(S2,−SC,−iC), e−iφτ1 =e−iπ4τ2e−iφτ3eiπ4τ2, (41) E(2)(π/2,φ)=e−iπ/4( SC,C2, iS), − − S =sinφ,C =cosφ. (48) we can see that a possible solution to our problem is Each of these corresponds to a pure polarization u(θ,φ)=eiπ4τ2S0−1u0(θ,2φ)e−iπ4τ2. (42) state that changes smoothly as φ varies. For φ = 0,π/2,π,3π/2 we have linear polarizations; for φ = As longas θ remains in the range0 θ π, for 0<θ < π/4,3π/4,5π/4,7π/4 we have circular polarizations (of π the 2π periodicity of u(θ,φ) in φ≤is s≤atisfied. At the oppositesensesfor E(1) andE(2)); andforallotherφ we poles we find: have elliptic polarizations. Since we have a continuously changingpropagationvectorkˆ =(cosφ,sinφ,0),itisnot u(θ,φ) θ→0 1, i.e.,u =1; meaningful to display all these features on any common 0 −−−→ or fixed Poincaré sphere. θ π u(θ,φ) → e 2iφτ3 τ τ , i.e.,u =iτ . (43) In a similar manner, the behaviours of − 2 3 π 1 −−−→ · (θ,φ),E(a)(θ,φ) at other points on S2 can be Thus u(θ,φ) is indeed a smooth function of (θ,φ) S2 eAx0amined, but we forego the details. dir ∈ dir atallpointsexcepttheSouthpole,withthedesiredmul- Fromthe manner in which the SU(2) elements u(θ,φ) tivaluedness at that pole. The scale change φ 2φ in- and the SU(3) elements (θ,φ) have been constructed, 0 → A volved in Eqn. (42) does not cause any difficulties. We it would seem that the solution given aboveto the prob- now use Eqn. (42) in Eqn. (36) to get a globally smooth lem of defining globally smooth polarization bases over SU(3) element (θ,φ) obeying Eqns. (28). After some all of S2 is in some sense both minimal and natural. A0 dir 6 The most general globally smooth (θ,φ) is clearly of except at South pole, which is used later for a dif- A the form ferent problem. [B] As to the second problem, here again we have 0 A(θ,φ)=A0(θ,φ) u′(θ,φ) 0 , (49) tohferewaellltaknngoewnnt rveescutlotrs: insopgolsosbiballelyosvmerootthhecehnotiicree 0 0 1   sphere of directions. Here we give a simple proof using SO(3) coset space, as well as a simple ana- whereu′(θ,φ)isanygloballysmoothmapS2 SU(2) lyticproof. Admittedly,thenegativeresultremains S3. Any number of examples of such u′(θ,φ→) are easil≃y unchanged but we go a step further in that we ex- constructed; for instance plicitlyconstructanalmostgloballysmoothSO(3) coset representative, problem only at South pole, u(θ,φ)=exp(isinθ a(θ,cosφ,sinφ) τττ) (50) for use in the next problem. ′ · [C] IfinBaboveonegoesfromrealtocomplextangent for anyreal a(θ,cosφ,sinφ) smoothinθ andpolynomial vectors, it was shown in [4] that globally smooth in cosφ and sinφ is acceptable. bases do exist. We give an explicit construction It is a result of homotopy theory that any two glob- of such a basis by combining the almost globally ally smooth choices of u′(θ,φ) can be continuously de- smooth expressions in A and B , using the proper- formed into one another; the reason is that the homo- ties of SU(3), to get a truly globally smooth solu- topy group π2(S3) is trivial [9]. Thus we see that while tion for problem C. The South pole problems in A thereareinfinitelymanychoicesofgloballysmoothcom- andB canbe made to annihilate one another! The plex orthonormal bases E(a)(kˆ) for electric field vec- noveltyisinuseofSU(3),andthelastresultisthat { } tors for plane waves for all kˆ S2 , any two choices can any two global choices can be smoothly deformed be smoothly deformed into on∈e adnirother. In this precise into one another; in that sense our group theory sense, the solution given above using (θ,φ) is essen- based solution is essentially unique. 0 A tiallyunique;anyothersolutionisinthesamehomotopy The fact that the three-dimensional groups SU(2) and classasthisone,indeedthereisonlyonehomotopyclass. SO(3) play important roles in these problems comes as no surprise. On the other hand, the use of the eight- dimensionalgroupSU(3), in a limited way,in answering the third question is particularly interesting. It is likely IV. CONCLUDING REMARKS that such approaches will be found useful in other problems in the polarization optics context as well. We have looked at three physically important and in- It is likely that such approaches will be found useful teresting problems in classical polarization optics, all of in other problems in polarization optics such as in the whichhaveaglobalcharacter. Ourprincipalresults,can Poincare’ sphere based descriptions and its generaliza- be summarised as follows: tions to situations whichinvolveorbitalangularmomen- tum and higher order effects [10–13]. [A] The first problem regarding the Poincaré sphere wastreated in [3]. It wasshownthere that there is ACKNOWLEDGMENTS nogloballysmoothwayofdefiningelectricfieldvec- torscoveringtheentirePoincarésphere. Ofcourse, we cannot change this result, but we give a much Arvind acknowledges funding from DST India under simpler proof than that in [3], using SU(2) coset Grant No. EMR/2014/000297. NM thanks the Indian spaceproperties. Wealsoconstructanalmostglob- NationalScienceAcademyforenablingthisworkthrough ally smooth coset representative, good everywhere the INSA C V Raman Research Professorship. [1] H. Poincare, Theorie Mathematique de la Lumiere [6] N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, (Gauthiers-Villars, Paris, France, 1892). Phys. Rev.A 65, 121021 (2002). [2] M.BornandE.Wolf,PrinciplesofOptics,7thEd.(Cam- [7] W. Chinn and N. Steenrod, First Concepts of Topology bridge UniversityPress, Cambridge, England, 1999). (MathematicalAssociationofAmerica,Washington,DC, [3] R.Nityananda,Pramana J. Phys. 12, 257 (1979). 1966). [4] R. Nityananda and S. Sridhar, [8] M. Eisenberg and R. Guy, Amer. Math. Monthly 86, AnnalsPhys. 341, 117 (2014). 571574 (1979). [5] N. Mukunda, S. Chaturvedi, and R. Simon, [9] M.Nakahara,Geometry andTopologyandPhysics (IOP, J. Opt.Soc. Amer. A: Opt.Image Sci. Vision 31, 1141 (2014). Bristol and Philadelphia, 2003). 7 [10] M. J. Padgett and J. Courtial, Optics Lett. 24, 430 [12] G. Milione, H. Sztul, D. Nolan, and R. Alfano, (1999). Phys. Rev.Lett. 107, 053601 (2011). [11] G. S. Agarwal, J. Opt. Soc. Amer. A: Opt. Image Sci. [13] M. R. Dennis and M. Alonso, Phil. T. Roy. Soc. A 375, Vision 16, 2914 (1999). 20150441 (2017).

Global aspects of polarization optics and coset space geometry

Similar Global aspects of polarization optics and coset space geometry