Properties of Spin and Orbital Angular Momenta of Light
aa r X i v : . [ qu a n t - ph ] J un Properties of Spin and Orbital Angular Momenta of Light
Arvind ∗ Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali,Sector 81 SAS Nagar, Manauli PO 140306, Punjab, India
S. Chaturvedi † Department of Physics, Indian Institute of Science Education and Research (IISER) Bhopal,Bhopal Bypass Road, Bhauri, Bhopal 462066, India
N. Mukunda ‡ Adjunct Professor, Department of Physics, Indian Institute of Science Education and Research (IISER) Bhopal,Bhopal Bypass Road, Bhauri, Bhopal 462066, India
This paper analyses the algebraic and physical properties of the spin and orbital angular momentaof light in the quantum mechanical framework. The consequences of the fact that these are notangular momenta in the quantum mechanical sense are worked out in mathematical detail. It turnsout that the spin part of the angular momentum has continous eigen values. Particular attention isgiven to the paraxial limit, and to the definition of Laguerre Gaussian modes for photons as wellas classical light fields taking full account of the polarization degree of freedom.
I. INTRODUCTION
There has been great interest for some time now in theangular momentum properties of the Maxwell field [1], inparticular its proposed separation into what have beencalled spin and orbital angular momentum of light [2].In addition to many theoretical investigations [3–21] ex-tensive experimental work [22–28] has also been devotedto understand these concepts.In an earlier work [29] a unified framework for study-ing these novel properties of light, in both classical andquantum domains, has been presented. In particular, thefact that the spin and orbital parts of the total angularmomentum are not truly quantum mechanical angularmomenta at all has been emphasized.The aim of the present paper is to carry this studyfurther and in particular to analyse in full detail thequantum mechanical properties of the spin angular mo-mentum of light at the one photon level. It is seen thatthe eigenvalues and eigenvectors of the spin angular mo-mentum are very different from those of any true an-gular momentum as understood in quantum mechanics.The essential roles of polarization and transversality oflight are brought out, and new vectorial Laguerre–Gaussfields including polarization in the paraxial regime areconstructed.The contents of this paper are organised as follows.Section II reviews the formulation of the free Maxwellequations in a particularly economical form using thecomplex transverse analytic signal vector potential. Theseven basic constants of motion following from Poincar´einvariance are expressed in terms of analytic signal vector ∗ [email protected] † [email protected] ‡ [email protected] potential and electric field. The spin and orbital angu-lar momenta, SAM and OAM, which are also real con-stants of motion, are identified. The description of a gen-eral solution of the Maxwell equations using a complextransverse vector function on wave vector space, and anatural Lorentz invariant Hilbert space made up of suchfunctions, is outlined. Canonical quantisation is recalled,and the operator forms of the seven hermitian constantsof motion, as well as of the SAM and OAM, are listed.A convenient description of the set of all single photonstates in terms of the classical Hilbert space is mentioned.The rest of this paper deals essentially with one photonstates. In Section III some of the properties of the SAMand OAM operators are worked out. The connections tothe helicity operator, the component of the total angu-lar momentum in the momentum direction, are obtainedand its properties are described. Helicity is a well de-fined concept in terms of the generators of the Poincar´egroup. The fact that the SAM components are commu-tative, and that along with the total angular momen-tum they generate a Euclidean group, is brought out.The helicity operator is seen to be invariant under thisEuclidean group. For later comparison, the discrete setof complete orthonormal eigenfunctions of total angularmomentum are recorded. Section IV solves completelythe problem of eigenvalues and eigenfunctions for theSAM along with helicity. It is emphasised that these areideal non normalisable eigenfunctions, as the eigenvaluesof the SAM components are continuous. The contrastwith the total angular momentum eigenfunctions is ex-plicitly seen. To emphasize this aspect, the properties ofSAM in a normalised simultaneous eigenfunction of thehelicity and the third component of total angular momen-tum are worked out. It is shown that such an eigenfunc-tion can never be an eigenfunction of the third componentof SAM as well; the SAM components have a nontrivialvariance matrix in such a state. Section V is devoted toan analysis of the paraxial regime. It is recalled that itis appropriate to perform canonical quantisation beforeconsidering the paraxial limit. The approximate natureof this limit, and the correspondingly approximate con-sequence of transversality, are both clearly brought out.These considerations, combined with the paraxial limitof the general simultaneous eigen functions of helicityand third component of total angular momentum, leadto the development of Laguerre Gauss mode functionsfor the vector Maxwell field. The helicity eigenvalue ofplus or minus ~ appears as a third label added to thetwo that enumerate the modes in the scalar optical case.Section VI is devoted to Concluding Remarks. II. CONSTANTS OF MOTION ANDQUANTIZATION OF THE FREE MAXWELLFIELD
We begin with the classical free Maxwell equationswritten in terms of the complex positive frequency an-alytic signal vector potential A (+) ( x ) where x ≡ ( x , t ).The basic (first order) equation of motion (EOM) is i ∂∂t A (+) ( x , t ) = (ˆ ω A (+) )( x , t ) , ˆ ω = c ( − ∇ ) / . (2.1)This is consistent with the transversality constraint ∇ · A (+) ( x , t ) = 0 . (2.2)The initial data is specified by A (+) ( x , E (+) ( x ) = ic (ˆ ω A (+) )( x ) , B (+) ( x ) = ∇ ∧ A (+) ( x ) . (2.3)They are also transverse and obey first order EOM sim-ilar to A (+) in (2.1). For convenience we will use both A (+) and E (+) in various important expressions.From the relativistic invariance of the Maxwell equa-tions we obtain seven constants of motion (COM) whichhave no explicit time dependence – momentum P , energy P , and total angular momentum J (all real): P j = 12 πc Z d x E (+) ( x ) ∗ · ∂ j A (+) ( x ) ,P = 12 π Z d x E (+) ( x ) ∗ · ∂ A (+) ( x ) ,J j = 12 πc Z d x E (+) m ( x ) ∗ ( δ mn ( x ∧ ∇ ) j + ǫ jmn ) A (+) n ( x ) . (2.4)Here ( ∂ ≡ ∂∂x , x = − x = − ct ). The two terms in thetotal angular momentum J are identified as the orbitalangular momentum (OAM) and spin angular momentum (SAM) respectively of the free field, and both are realCOM’s: L j = 12 πc Z d x E (+) m ( x ) ∗ ( x ∧ ∇ ) j A (+) m ( x ) ,S j = 12 πc Z d x ǫ jmn E (+) m ( x ) ∗ A (+) n ( x ) . (2.5)These will be studied in detail in the sequel.The general solution of (2.1) and (2.2) can be writtenin terms of a complex transverse function v ( k ) of the realwave vector k ∈ R : A (+) ( x , t )= c π Z d k √ ω e ik · x v ( k ) , E (+) ( x , t )= i π Z d k √ ωe ik · x v ( k ) , k · v ( k ) = 0 , ω = ck = c | k | , k · x = k · x − ωt. (2.6)Thus the most general free Maxwell field is given equallywell by A (+) ( x ) or v ( k ). The seven COM’s (2.4) can beexpressed in terms of v ( k ) : P j = Z d k k j v ( k ) ∗ · v ( k ) , P = Z d k ω v ( k ) ∗ · v ( k ) J j = Z d k v m ( k ) ∗ ( − iδ mn ( k ∧ ˜ ∇ ) j − iǫ jmn ) v n ( k ) , ˜ ∂ j = ∂∂k j . (2.7)The OAM and SAM are L j = − i Z d k v m ( k ) ∗ ( k ∧ ˜ ∇ ) j v m ( k ) ,S j = − i Z d k v m ( k ) ∗ ǫ jmn v n ( k ) . (2.8)At the classical level we define a Hilbert space M byusing a metric in the space of amplitudes v ( k ): M = (cid:8) v ( k ) (cid:12)(cid:12) k · v ( k ) = 0 , || v || = Z d k v ( k ) ∗ · v ( k ) < ∞ (cid:9) . (2.9)The norm || v || is Lorentz invariant. The space M willplay an important role after quantization to which wenow turn.The process of canonical quantization involves replac-ing the classical amplitudes v ( k ) , v ( k ) ∗ by vectorial op-erators √ ~ ˆ a ( k ) , √ ~ ˆ a ( k ) † obeying the canonical commu-tation relations (CCR) on a suitable Hilbert space H :[ˆ a j ( k ) , ˆ a l ( k ′ ) † ] = (cid:18) δ jl − k j k l | k | (cid:19) δ (3) ( k − k ′ ) , [ ˆa , ˆa ] = [ ˆa † , ˆa † ] = 0 , k · ˆ a ( k ) = k · ˆ a ( k ) † = 0 . (2.10)The field operators areˆ A (+) ( x ) = c π √ ~ Z d k √ ω e ik · x ˆ a ( k ) , ˆ E (+) ( x ) = i π √ ~ Z d k √ ωe ik · x ˆ a ( k ) . (2.11)The operator forms of the classical COM’s are the her-mitian operatorsˆ P = Z d k ~ ω ˆ a ( k ) † · ˆ a ( k );ˆ P j = Z d k ~ k j ˆ a ( k ) † · ˆ a ( k );ˆ J j = − i ~ Z d k ˆ a m ( k ) † ( δ mn ( k ∧ ˜ ∇ ) j + ǫ jmn ) ˆ a n ( k ); ( a )ˆ L j = − i ~ Z d k ˆ a m ( k ) † ( k ∧ ˜ ∇ ) j ˆ a m ( k ) , ˆ S j = − i ~ Z d k ˆ a m ( k ) † ǫ jmn ˆ a n ( k ) . ( b )(2.12)The commutation relations among the former are deter-mined by the Poincar´e group structure:[ ˆ P µ , ˆ P ν ] = 0;[ ˆ J j , ˆ P ] = 0; [ ˆ J j , ˆ P l ] = i ~ ǫ jln ˆ P n ;[ ˆ J j , ˆ J l ] = i ~ ǫ jln ˆ J n . (2.13)We will examine the important operator properties of theOAM and SAM, ˆ L j and ˆ S j , in the next Section.The Hilbert space H on which the CCR’s (2.10) are re-alized irreducibly is the direct sum of subspaces H n , n =0 , , , · · · , made up of states with definite total photonnumber n . Thus H is the one dimensional subspace ofno photon states ( multiples of the vacuum state | i ); H is the subspace of single photon states; and so on. Theimportance of the classical Hilbert state M , Eq. (2.9),is that there is a one to one correspondence M ↔ H ,given by the following structure: v ( k ) ∈ M , | v i = ˆ a ( v ) † | i ∈ H , ˆ a ( v ) = 1 √ ~ Z d k v ( k ) ∗ · ˆ a ( k ) , ˆ a ( v ) † = 1 √ ~ Z d k v ( k ) · ˆ a ( k ) † ;[ˆ a ( v ) , ˆ a ( v ′ ) † ] = ( v , v ′ ) ~ I ;ˆ a j ( k ) | v i = 1 √ ~ v j ( k ) | i . (2.14)The inner products among one photon states in H areessentially the classical inner products in M : h v ′ | v i = ( v ′ , v ) / ~ (2.15) III. OPERATOR PROPERTIES OF TOTAL,ORBITAL AND SPIN ANGULAR MOMENTUMOF PHOTONS
We now take up a detailed analysis of the operatorsˆ L , ˆ S representing the OAM and SAM of the quantizedMaxwell field respectively. For our purposes it suffices to restrict these (and other) operators to one-photon statesin H . Their actions on a one-photon wavefunction v ( k )can be expressed in a succinct manner. For ˆ P µ and ˆ J wehave:( ˆ P | v i ) j ( k ) = ~ ωv j ( k ) , ( ˆ P l | v i ) j ( k ) = ~ k l v j ( k ) , ( ˆ J l | v i ) j ( k ) = − i ~ (cid:16) ( k ∧ ˜ ∇ ) l v j ( k ) + ǫ ljn v n ( k ) (cid:17) . (3.1)For ˆ L and ˆ S we find:( ˆ L l | v i ) j ( k ) = − i ~ (cid:18) ( k ∧ ˜ ∇ ) l v j ( k ) + k j | k | ( k ∧ v ( k )) l (cid:19) , ( ˆ S l | v i ) j ( k ) = i ~ k l | k | ( k ∧ v ( k )) j . (3.2)Two operator relations follow easily :ˆ P · ˆ L = 0 , ˆ P ∧ ˆ S = 0 . (3.3)The helicity operator ˆ W is defined in terms of Poincar´egroup generators as ˆ W = ˆ P · ˆ J p ˆ P · ˆ P . (3.4)With (3.3) this simplifies toˆ W = ˆ P · ˆ S p ˆ P · ˆ P . (3.5)We next easily find some operator product relations:ˆ J · ˆ S = ˆ S · ˆ S = ˆ W = ~ . (3.6)Therefore we also have ˆ L · ˆ S = 0 . (3.7)Turning to commutators, while Eqs. (2.13) are part ofthe Poincar´e Lie algebra, we now find these additionalones: [ ˆ J l , ˆ W ] = 0;[ ˆ J l , ˆ L m or ˆ S m ] = i ~ ǫ lmn ( ˆ L n or ˆ S n );[ ˆ S l , ˆ P m or ˆ S m or ˆ W ] = 0 , [ ˆ L l , ˆ W ] = 0 . (3.8)As expected, ˆ W is a rotational scalar while ˆ L and ˆ S arevectors. The six hermitian operators ˆ J and ˆ S , all havingthe dimensions of action, realise the Lie algebra of a Eu-clidean group ˜ E (3). This is distinct from the Euclideansubgroup E (3) of the Poincar´e group, generated by ˆ J andˆ P .The result for ˆ W in Eq. (3.6) seems counterintuitive,since all ˆ P j and ˆ S j commute pairwise and all have con-tinuous eigenvalues. The reason of course is the resultˆ P ∧ ˆ S = 0.The operators ˆ J constitute a quantum mechanical an-gular momentum. Thus the eigenvalues of ˆ J · ˆ J and ˆ J are l ( l + 1) ~ and ~ m respectively, for l = 1 , , · · · , and m = l, l − , · · · , − l for photons. As is well known, theirsimultaneous eigenfunctions form a complete orthonor-mal basis for transverse vector functions of the unit wavevector ˆ k ∈ S [30, 31] : { ˆ J , ˆ J } Y ( a ) lm (ˆ k ) = { ~ l ( l + 1) , m ~ } Y ( a ) lm (ˆ k ) , a = 1 , Y (1) lm (ˆ k ) = 1 p l ( l + 1) ( − i k ∧ ˜ ∇ ) Y lm (ˆ k ) , Y (2) lm (ˆ k ) = ˆ k ∧ Y (1) lm (ˆ k ); Z S d Ω(ˆ k ) Y ( a ′ ) l ′ m ′ (ˆ k ) ∗ · Y ( a ) lm (ˆ k ) = δ a ′ ,a δ l ′ ,l δ m ′ ,m ; X a =1 ∞ X l =1 l X m = − l Y ( a ) lm,j (ˆ k ) Y ( a ) lm,j ′ (ˆ k ′ ) ∗ = δ (2) (ˆ k , ˆ k ′ )( δ jj ′ − k j k j ′ | k | ) . (3.9)Here Y lm (ˆ k ) are the usual spherical harmonics and δ (2) (ˆ k , ˆ k ′ ) is the two dimensional surface Dirac deltafunction over S .As we will see, since the ˆ S are not an angular momen-tum, their eigenvalues and eigenvectors have very differ-ent characters. IV. SPIN AND HELICITY EIGENFUNCTIONS,VARIANCE MATRIX FOR SPIN
Now we consider the eigenvalues and eigenvectors ofthe SAM ˆ S . Since the four operators ˆ S j , ˆ W commutepairwise, they can all be simultaneously diagonalized. Asthe ˆ S j transform as a three dimensional vector underspatial rotations, we see from Eqs. (3.6) that the possibleeigenvalues for ˆ S and ˆ W have the formsˆ S → ~ s , ˆ W → ~ w, s ∈ S , w = ± . (4.1)It follows that while ˆ W possesses normalizable eigenvec-tors, for eigenvectors of ˆ S we must use delta functionnormalization on S (cf Eq. (3.9)).Based on the actions given in Eqs. (3.1),(3.2), we caneasily construct the corresponding (ideal) eigenvectors in H . To handle ˆ W , we need to choose, for each ˆ k ∈ S ,a pair of transverse mutually orthogonal circular polar-ization vectors ǫ ( ± ) (ˆ k ). In terms of the spherical polarangles θ, ϕ of ˆ k ∈ S , their definitions and importantproperties are as follows ( with C for cos and S for sin): ǫ (+) (ˆ k ) = e iϕ √ CθCϕ − iSϕ, CθSϕ + iCϕ, − Sθ ) , ǫ ( − ) (ˆ k ) = i ǫ (+) (ˆ k ) ∗ = i e − iϕ √ CθCϕ + iSϕ, CθSϕ − iCϕ, − Sθ );ˆ k · ǫ ( a ) (ˆ k ) = 0 , a = ± ; ǫ ( a ) (ˆ k ) ∗ · ǫ ( b ) (ˆ k ) = δ a,b ;ˆ k ∧ ǫ ( a ) (ˆ k ) = − ia ǫ ( a ) (ˆ k ); ǫ (+) (ˆ k ) ∧ ǫ ( − ) (ˆ k ) = ˆ k . (4.2) As is well known, transverse circular polarization vectorsdefined smoothly all over S do not exist [32–34]. Theabove choices are well defined at θ = 0 but multivaluedat θ = π . Their behaviours under parity are useful, andread: ǫ ( a ) ( − ˆ k ) = iae iaϕ ǫ ( − a ) (ˆ k ) , a = ± , (4.3)so ˆ k ∧ ǫ ( a ) ( − ˆ k ) = ia ǫ ( a ) ( − ˆ k ) . (4.4)After some straightforward analysis, the (ideal) simulta-neous eigenvectors of ˆ S j , ˆ W can be found upto arbitrary‘radial’ functions: s ∈ S , w = ± −→ | s , w i ∈ H :ˆ S | s , w i = ~ s | s , w i , ˆ W | s , w i = ~ w | s , w i ;( | s , w i ) j ( k ) = a ( k, s , w ) δ (2) (ˆ k , w s ) ǫ (+) j ( s ) , any a ( k, s , w ) . (4.5)The inner products have the form expected from or-thonormality: h s ′ , w ′ | s , w i = Z d k ( | s ′ , w ′ i ) j ( k ) ∗ ( | s , w i ) j ( k )= δ w,w ′ δ (2) ( s ′ , s ) Z ∞ k dka ′ ( k, s , w ) ∗ a ( k, s , w ) . (4.6)As for the completeness property, we omit the factor a ( k, s , w ) in Eq. (4.5) and find for the angular part: X w = ± Z S d Ω( s ) (cid:16) δ (2) (ˆ k , w s ) ǫ (+) j ( s ) (cid:17) (cid:16) δ (2) ( ˆ k ′ , w s ) ǫ (+) j ′ ( s ) (cid:17) ∗ = δ (2) (ˆ k , k ′ ) (cid:18) δ jj ′ − k j k j ′ | k | (cid:19) (4.7)This is to be compared to the last line in (3.9) : whilethe right hand sides are the same, the left hand sides havevery different structures, due to the differences betweenˆ J and ˆ S .The fact that ˆ S has continuous eigenvalues (not atall like a quantum mechanical angular momentum),hence no normalisable eigenvectors, has important conse-quences. We illustrate this by examining the propertiesof ˆ S in a normalized simultaneous eigenvector of ˆ J andˆ W . This has the general form :ˆ J → ~ m, ˆ W → ~ w : v m,w ( k ) = a ( k, m, w, θ ) e i ( m − w ) ϕ ǫ ( w ) (ˆ k ); h v m,w | v m,w i = 2 π Z ∞ k dk Z π sin θdθ | a ( k, m, w, θ ) | = 1 . (4.8)Here a ( k, m, w, θ ) is arbitrary. Let us now define an as-sociated probability distribution p ( x ) over [ − ,
1] in thepolar angle θ with x = cos θ , as follows: p ( x ) = 2 π Z ∞ k dk | a ( k, m, w, θ ) | ≥ , Z − dx p ( x ) = 1 . (4.9)The normalisation condition (4.8) implies that p ( x ) is notof delta function type, so it describes a non trivial spreadand variance in x . Then using Eqs. (3.2) and (4.8) wefind the expectation values of the SAM: h v m,w | ˆ S l | v m,w i = ~ Z ∞ k dk Z S d Ω( k ) | a ( k, m, w, θ ) | ˆ k l = ~ h x i δ l, , h f ( x ) i = Z − dxf ( x ) p ( x ) . (4.10)Going a step further, we can obtain the expectation val-ues of quadratics in the ‘spin’ components as a 3 × h v m,w | ˆ S l ˆ S n | v m,w i = ~ (cid:18)Z ∞ k dk Z S d Ω( k ) | a ( k, m, w, θ ) | ˆ k l ˆ k n (cid:19) = ~ diag (cid:18) h (1 − x ) i , h (1 − x ) i , h x i (cid:19) (4.11)Therefore the SAM variance matrix in the normalizedstate | v m,w i is, using (4.10), V = ~ diag (cid:18) h (1 − x ) i , h (1 − x ) i , h (∆ x ) i (cid:19) , (∆ x ) = h x i − h x i . (4.12)From the statements made above regarding the natureof the probability distribution p ( x ), it is clear that thespread (∆ x ) in ˆ S is strictly positive, (∆ x ) >
0. Soin any normalized state | v m,w i with well defined ˆ J andˆ W , there is always a spread in the values of the compo-nents of ˆ S . In particular even though both ˆ J and ˆ W commute with ˆ S , the normalised eigenvector | v m,w i ofˆ J and ˆ W can never be a simultaneous eigenvector of ˆ S as well, whatever be the choice of a ( k, m, w, θ ). By thesame token, the state | v m,w i can never be an eigenvec-tor of the third component ˆ L of OAM, for any choice of a ( k, m, w, θ ). V. PARAXIAL REGIME AND VECTORLAGUERRE-GAUSS MODES
In the previous Sections we have discussed on the onehand the exact simultaneous eigenfunctions of the to-tal squared angular momentum ˆ J and its componentˆ J , and on the other hand those of the three compo-nents of the SAM ˆ S and the helicity ˆ W . These are collected together in Eqs. (3.9) and Eqs. (4.5), (4.6) re-spectively. In both cases, only angular and polarizationdependences are involved. In the general ˆ S , ˆ W eigen-function in Eq. (4.5) for example, an arbitrary, ‘radial’function a ( k, s, w ) appears. Similarly in the general si-multaneous eigenvector of ˆ J , ˆ W in Eq. (4.8) an arbitraryfunction a ( k, m, w, θ ) is present.Now we turn to the physically very important paraxialregime. As argued in earlier work [29], it is reasonableto consider the paraxial limit after canonical quantiza-tion has been completed and the photon picture of lighthas been obtained. Thus once Eqs. (2.10) and their con-sequences and interpretation are in hand, in the subse-quent analysis based on Eqs. (2.14) we limit the choicesof v ( k ) ∈ M to those having the paraxial property. Thatis, the paraxial approximation is made on the choice of v ( k ) within ˆ a ( v ) and ˆ a ( v ) † , not in the canonical quan-tization rule v ( k ) → √ ~ ˆ a ( k ) , v ( k ) ∗ → √ ~ ˆ a ( k ) † in anysense. ’Paraxial photons’ are to be understood in thisway.The paraxial region in wave vector space is defined (in an approximate way) as consisting of those k vectorswhose transverse components k ⊥ are much smaller thantheir (positive) longitudinal components : | k ⊥ | << k, k ≃ k − k ⊥ / k. (5.1)A photon wave function v ( k ) is paraxial if it is negligibleoutside the paraxial region: v ( k ) ≃ k paraxial . (5.2)In that case, transversality determines v ( k ) in terms of v ⊥ ( k ): v ( k ⊥ , k ) ≃ − (cid:18) k ⊥ k (cid:19) k ⊥ · v ⊥ ( k ⊥ , k ) k . (5.3)The longitudinal component is one order of magnitudesmaller than the transverse components.One way in which the paraxial property for v ( k ) can beachieved is if each component v j ( k ) is a common trans-verse Gaussian factor times a polynomial in k ⊥ . Thisrequires that there be a transverse width w and someminimum wave vector magnitude k min > v ( k ⊥ , k ) = (cid:18) a ⊥ ( k ⊥ , k ) c ( k ⊥ , k ) (cid:19) e − w k ⊥ / ,w >> λ max = 2 π/k min ,c ( k ⊥ , k ) ≃ − (cid:18) k ⊥ k (cid:19) k ⊥ · a ⊥ ( k ⊥ , k ) k , (5.4)with a ⊥ and c polynomial in k ⊥ .We can now connect with the exact ˆ J - ˆ W eigenfunc-tions in Eq. (4.8), and their paraxial limits, in this way.For given eigenvalues, ~ m , ~ w of ˆ J , ˆ W the eigenfunctionin Eq. (4.8) contains the arbitrary function a ( k, m, w, θ )as a factor. To make this eigenfunction paraxial meansto impose suitable conditions on this free function. Theparaxial ( small θ ) limits of ǫ ( ± ) (ˆ k ) are : ǫ (+) (ˆ k ) ≃ √ i − θe iϕ ; ǫ ( − ) (ˆ k ) ≃ √ i − iθe − iϕ . (5.5)In scalar paraxial optics the important family ofLaguerre–Gaussian (LG) mode functions have the gen-eral structure of (5.4)–polynomials times a Gaussian fac-tor in transverse variables. These are defined using cylin-drical coordinates, so we have the connection : k = k (sin θ cos ϕ, sin θ sin ϕ, cos θ ) = ( ρ cos ϕ, ρ sin ϕ, k ) : ρ = k sin θ, k = k cos θ, k ⊥ = ρ , k = ρ + k . (5.6)For small θ , we have ρ ≃ kθ , k ≃ k − ρ / k . TheLG mode functions are labelled by two integers: p =0 , , , · · · , m = 0 , ± , ± · · · ; and they are φ m,p ( k ⊥ ) = w √ π s p !( p + | m | )! e imϕ (cid:18) iw ρ √ (cid:19) | m | × L | m | p (cid:18) w ρ (cid:19) e − w ρ / . (5.7)Comparing Eq. (5.4) with Eqs. (4.8),(5.5),(5.7) we areled for each given m to two choices : w = 1 a ( k, m, +1 , θ ) → φ m − ,p ( k ⊥ ) : v m, +1 ,p ( k ⊥ , k ) = 1 √ i − θe iϕ φ m − ,p ( k ⊥ ); ( a ) w = − a ( k, m, − , θ ) → φ m +1 ,p ( k ⊥ ) : v m, − ,p ( k ⊥ , k ) = 1 √ i − iθe − iϕ φ m +1 ,p ( k ⊥ ) . ( b )(5.8)To leading paraxial order, these are the complete–transverse vector LG mode fields. We stress that theseare eigenfunctions of the total angular momentum com-ponent ˆ J and helicity ˆ W with respective eigenvalues ~ m, ± ~ . In addition to the labels m, p in Eq. (5.7) inthe scalar case, now the third helicity label w = ± alsoappears. VI. CONCLUDING REMARKS
We have presented a careful analysis of the propertiesof the so-called spin and orbital angular momenta of light, in the quantum domain, as they apply to single photonstates. It has been known for some time that these op-erators, which are hermitian constants of motion, do nothave the spectral properties expected of an angular mo-mentum in the sense of quantum mechanics. Thus thephoton spin is not such an angular momentum. Its com-ponents do not have discrete quantised eigenvalues. It isa result of transversality of the Maxwell field that thereis no position operator for the photon, therefore no wayof separating the total angular momentum into well de-fined and independent spin and orbital parts. The terms’spin’ and ’ orbital’ angular momenta of light are thusmisnomers which however cannot now be corrected.We show by explicit construction that there exist ideal(non normalisable) eigenvectors for all three spin com-ponents simultaneously. One can, of course, constructnormalisable wave packets out of these eigenvectors, in-volving small patches over the sphere S . At the classicallevel it is an interesting challenge to produce wave fieldscorresponding to such solutions of the Maxwell equations.The helicity and the three spin components do possess si-multaneous ideal eigenvectors, with their eigenvalues be-ing chosen independently. However a normalised eigen-vector of a component of the total angular momentumand helicity can never be an eigenvector of that compo-nent of the spin as well.We recall that a noteworthy feature of the formalismdeveloped in [29] and briefly recapitulated here is the oneto one correspondence between classical radiation fieldconfigurations and the quantum description thereof atthe single photon level. This leads one to expect thatsome of the results arising from the peculiar features ofthe ‘spin’ and ‘orbital’ angular momentum operators atthe one photon level, as discussed here ought to havemeasurable signatures at the classical level as well. ‘Finally we draw attention to the paraxial vectorialLaguerre-Gauss fields which are a physically relevant andnontrivial generalisation of the enormously useful scalarparaxial mode fields of the same name. It is an experi-mental challenge to create such fields, and to bring outtheir characteristic signatures. VII. ACKNOWLEDGEMENTS
NM thanks the Indian National Science Academy forthe INSA Distinguished Professorship, during the tenureof which this work was initiated. Arvind acknowledgesthe financial support from DST/ICPS/QuST/Theme-1/2019/General Project number
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