aa r X i v : . [ phy s i c s . op ti c s ] D ec Classical Light Beams and Geometric Phases
N. Mukunda, ∗ S. Chaturvedi, † and R. Simon ‡ Optics & Quantum Information Group, The Institute of Mathematical Sciences,C.I.T Campus, Tharamani, Chennai 600 113, India. School of Physics, University of Hyderabad, Hyderabad 500 046, India
We present a study of geometric phases in classical wave and polarisation optics using the basicmathematical framework of quantum mechanics. Important physical situations taken from scalarwave optics, pure polarisation optics, and the behaviour of polarisation in the eikonal or ray limitof Maxwell’s equations in a transparent medium are considered. The case of a beam of light whosepropagation direction and polarisation state are both subject to change is dealt with, attentionbeing paid to the validity of Maxwell’s equations at all stages. Global topological aspects of thespace of all propagation directions are discussed using elementary group theoretical ideas, and theeffects on geometric phases are elucidated.
OCIS codes : (030.1640) Coherence; (350.5500) Propagation; (260.5430) Polarization
PACS numbers:
I. INTRODUCTION
The quantum mechanical geometric phase was discovered by Berry in 1983 – 84 [1]. The context was unitaryevolution governed by the Schr¨odinger equation in the adiabatic approximation, i.e., with a hermitian Hamiltonianpossessing a ‘gentle’ time-dependence. Assuming that as an operator the Hamiltonian is cyclic, i.e., it returns to itsoriginal form after a certain interval of time (during which there are no level crossings), the approximate solutions tothe Schr¨odinger equation are also cyclic. The geometric phase is then seen explicitly in these solutions at the end ofthe cycle.Immediately after Berry’s discovery, it was pointed out by Barry Simon [2] that the geometric phase expresses thenon-integrability, or anholonomy, of a natural ‘rule of parallel transport’ (a connection) in a principal fibre bundle, withstructure group U (1), which occurs in the framework of quantum mechanics. This was therefore a characterisation ofthis phase in the language of differential geometry.The ensuing years saw two streams of work relating to the geometric phase. One consisted of extensions of Berry’soriginal work, in the sense of relaxing the conditions under which the phase is definable. The other consisted ofinteresting earlier results which could be reinterpreted as instances of this phase, and so as precursors to it. Werecall three significant efforts of the first kind. Aharonov and Anandan [3] showed that the adiabatic condition is notnecessary—given a cyclic solution to the Schr¨odinger equation involving any (time-dependent) Hamiltonian, one canreconstruct a corresponding geometric phase. This was followed by the work of Samuel and Bhandari [4], in whichthe cyclic condition on a solution was also dispensed with. Given a solution of the Schr¨odinger equation involvingany (time-dependent) Hamiltonian, over any stretch of time, one can in a simple way extend it to a closed or cyclicsolution, then use the Aharonov-Anandan method to identify a geometric phase. Both these extensions of Berry’soriginal framework used quantum mechanical notions, specifically the Schr¨odinger equation. The third step in thedirection of increasing generality was taken by Mukunda and Simon [5, 6] : the geometric phase is entirely kinematicalin content, not requiring a Hamiltonian operator and associated Schr¨odinger equation. It is determined once one isgiven a (sufficiently smooth) curve of unit vectors in any complex Hilbert space, without reference to any specificallyquantum mechanical notions. (The relevant expressions and definitions are recalled below).Turning to the efforts of the second kind, within quantum mechanics we may cite the Bohm-Aharonov effect (alreadydealt with by Berry in his original work), and the clarification of the connection between Bargmann invariants andgeometric phases [7]. Beyond these, it is interesting that many instances of the geometric phase have been identifiedwithin classical (wave) optics—the Gouy phase from 1890 [8, 9]; the work by Rytov, and Vladimirskii [11, 12] in 1938and 1941 on the behavior of light polarisation in the short wavelength limit of wave optics; and Pancharatnam’s ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] studies in 1956 [13] on light polarisation and an associated phase. These classical optics examples will be reviewedbriefly in the next Section. It will be seen that even in these situations the basic idea of a complex linear vector spacecarrying a hermitian inner product, usually regarded as characteristic of quantum mechanics, is essential to be able toidentify ‘classical geometric phases’. In this way, the deep link between Bargmann invariants and geometric phases,and the connection to Berry’s original discovery, are always kept in evidence.With this background, we now describe briefly the kinematic approach to the geometric phase. Let H be a complexHilbert space of any dimension, with vectors ψ, φ, · · · and inner product ( ψ, φ ) [14]. In quantum mechanics | ( ψ, φ ) | is related to a probability; in classical wave optics k ψ k = ( ψ, ψ ) generally stands for light intensity which can, butneed not be, normalised. We next denote by B the unit sphere in H : B = { ψ ∈ H | k ψ k = ( ψ, ψ ) = 1 } ⊂ H . (1.1)The group U (1) of complex phase factors acts on B (also on H ) in a natural way: ψ ∈ B → ψ ′ = e iα ψ ∈ B , ≤ α < π . (1.2)The quotient B /U (1), i.e., collections or equivalence classes of vectors { e iα ψ, ψ fixed , ≤ α < π } differing only byphases, forms the ‘ray space’ R : R = B /U (1) = { ρ ( ψ ) = ψψ † | ψ ∈ B} . (1.3)Whereas B is a subset of H , the ray space R is not : the B − R relationship is that there is a projection map π fromthe former to the latter: π : B → R : ψ ∈ B → π ( ψ ) = ρ ( ψ ) ∈ R . (1.4)Referring to an earlier comment, B is a U (1) principal fibre bundle over the base R . In the quantum mechanicscontext, points in R correspond one-to-one to physical pure states.In this framework, given any continuous piecewise once differentiable parametrised curve C in B , C = { ψ ( s ) ∈ B | s ≤ s ≤ s } ⊂ B , (1.5)with image C in R , C = π [ C ]= { ρ ( s ) = ρ ( ψ ( s )) = ψ ( s ) ψ ( s ) † | s ≤ s ≤ s } ⊂ R , (1.6)the geometric phase ϕ g [ C ] is defined: ϕ g [ C ] = ϕ tot [ C ] − ϕ dyn [ C ] ,ϕ tot [ C ] = arg( ψ ( s ) , ψ ( s )) ,ϕ dyn [ C ] = Im Z s s ds (cid:18) ψ ( s ) , dψ ( s ) ds (cid:19) . (1.7)As indicated, ϕ g [ C ] is a functional of C ⊂ R , while ϕ tot and ϕ dyn are both functionals of C ⊂ B .An important consequence of this definition is a result involving the so-called Bargmann invariants [5]. The simplestsuch invariant involves three pairwise nonorthogonal vectors ψ , ψ , ψ ∈ B and is the expression∆ ( ψ , ψ , ψ ) = ( ψ , ψ )( ψ , ψ )( ψ , ψ ) , (1.8)which is in general complex. The fact that ∆ ( e iα ψ , e iα ψ , e iα ψ ) = ∆ ( ψ , ψ , ψ ) for all real α , α , α showsthat ∆ lives in R rather than in B . Indeed Bargmann, during the course of his famous proof of Wigner’s theorem,introduced ∆ simply to point out this gauge-invariance property and to indicate that it could be used to distinguishbetween unitaries and antiunitaries : while ∆ is invariant under unitaries, its argument changes signature underantiunitaries. It is in [5, 6] that this object introduced by Bargmann almost in passing was elevated to become thebasis of a complete kinematic theory of geometric phase.To relate arg(∆ ( ψ , ψ , ψ )) to a geometric phase, it is necessary to connect the ‘vertices’ ψ , ψ , ψ pairwise insome way, so as to construct a closed continuous piecewise once-differentiable loop in B reminiscent of cyclic quantumevolution. This can be done using the idea of geodesics in R . Given two nonorthogonal vectors ψ, φ ∈ B , and assumingfor definiteness that ( ψ, φ ) is real positive, the (shorter) geodesic in R connecting ρ ( ψ ) to ρ ( φ ) is the image under π of the curve C = { ψ ( s ) } ⊂ B described as follows:( ψ, φ ) = cos θ , < θ < π/ ψ ( s ) = ψ cos s + φ ⊥ sin s sin θ , ≤ s ≤ θ,φ ⊥ = φ − ψ cos θ . (1.9)Along this C one has ( ψ ( s ′ ) , ψ ( s )) = cos( s ′ − s ) , ≤ s ′ , s ≤ θ . (1.10)Then the connection between Bargmann invariants and geometric phases is:arg (∆ ( ψ , ψ , ψ )) = − ϕ g [ C ] ,C = triangle in R with vertices ρ ( ψ ) , ρ ( ψ ) , ρ ( ψ )and connecting geodesics as sides. (1.11)A very far-reaching generalisation of this relation, when dim H ≥
3, has been developed more recently [15–18].It is worth emphasizing that the framework described above, based on the triplet of spaces H , B , R , supports thegeometric phase concept in a simple and direct way. Though suggested by the formal (complex linear space) structureof quantum mechanics, it can be used in other situations as well, such as classical wave optics. We adopt this viewpointin trying to define geometric phases in various physical, particularly classical optical, situations.Before we outline the organization of the material of this paper, it may be useful to add an extended remark by wayof pointing to the precise context of this work. There has been considerable interest in recent times to understand theinterplay between the spatial degree of freedom (coherence) and polarisation degree of electromagnetic beams [see,for instance, [19] and references therein]. It is equally important to understand the behaviour of this interplay asthe Maxwell beam passes through an optical system. Indeed, it turns out that the defining properties of the age oldMueller matrix cannot be correctly enumerated without consideration of this interplay or entanglement [20, 21].A lens of focal length f relates the output field amplitude ψ out ( x , x ) (just after the lens plane) to the input ψ in ( x , x ) (just before the lens plane), ( x , x ) being Cartesian variables in the transverse plane, through ψ out ( x , x ) = exp (cid:18) − i x + x λf (cid:19) ψ in ( x , x ) . (1.12)But when it comes to vector waves, it is clear that the same transformation applied to every (Cartesian) componentof the electric field vector E ( x , x ) will not map solutions of Maxwell equations at the input plane to solutions atthe output, for such a democratic action on the electric field components does not respect the transversality condition ∇ · E = 0. Since this condition is a constraint connecting the spatial degrees of freedom to the polarization degree,it would be respected only if the spatial modulation exp (cid:16) − i x + x λf (cid:17) is accompanied by ‘appropriate’ local rotations of the electric field components (local polarization) [22].Let us arrange the components of the electric and magnetic field amplitude vectors E ( x , x ), B ( x , x ) in atransverse plane z = constant into a six-component electromagnetic vector Λ ( x , x ) = (cid:18) E ( x , x ) B ( x , x ) (cid:19) . (1.13)The approach of [22] rooted at the very Poincar´e symmetry of the Maxwell system of equations led to this fundamentalresult: if T ( x , x ) is the amplitude transmittance function of an optical system in scalar Fourier optics [ a lens, forinstance, has T ( x , x ) = exp (cid:16) − i x + x λf (cid:17) ], then the action T : Λ in ( x , x ) → Λ out ( x , x ) = T ( Q , Q ) Λ in ( x , x ) ,Q = x × + ¯ λG , Q = x × + ¯ λG , (1.14)where G , G are a pair of 6 × numerical matrices arising from the structure of the Poincar´e group [22], does take solutions Λ in ( x , x ) of Maxwell’s equations to solutions Λ out ( x , x ). That is, the matrices G , G effect on thecomponents of Λ ( x , x ) the correct local rotations alluded to above [22]. This result readily leads to Fourier opticsfor Maxwell beams [23] and to electromagnetic Gaussian beams [24], resulting in a straight forward description of notonly the longitudinal component but also the cross-polarisation component [25]. It is in respect of this result thatthe late Henri Bacry anticipated: “it is highly probable that a rigorously gauge theory will be developed in a nearfuture”, the local rotations referred to above constituting “an SO (3) gauge group” [26].The work presented here is only the first step of an ambitious programme constituting our attempt towards a possiblerealization of this anticipation. While the earlier formulation of Fourier optics for Maxwell beams [23] concentrated onparaxial propagation about a fixed direction , the present work aims at laying a global and structurally robust skeletonin the space of directions , handling satisfactorily the well known topological obstructions. In the sequel, we plan toadapt suitably the methods of [23] in local patches of the space of directions, and then ‘stitch’ together the patchesin a smooth manner to arrive at the general case.The contents of this paper are organised as follows. Section II gives brief accounts of three applications of thegeometric phase concept to classical optical situations : the Gouy phase in scalar paraxial wave optics; the Pancharat-nam study of phases in pure polarization optics with fixed propagation direction; and the behaviour of polarisationin the eikonal or ray limit of Maxwell’s equations in a transparent medium with given refractive index function. ThePancharatnam case uses the Poincar´e sphere S of polarisation states, for a fixed direction of propagation, while theray case uses the sphere of propagation directions S . In all these cases, the use of the basic quantum mechanicalframework is highlighted. Section III builds on the last example of Section II in two ways—the generalisation fromthe unphysical case of a single ray to a physical beam of finite cross-sectional area made up of a narrow bundle ofnearly parallel rays; and the inclusion of polarization gadgets in the path of the beam. Once again the quantummechanical framework proves adequate, and now both spheres S , S come into the picture. Section IV takesup certain global features of the sphere of directions S , and builds on a recent suggestion [27] that passage to thecomplex extension of the tangent planes to S removes an obstruction which exists in the real domain. (The workof [27] was motivated in part by earlier works of [28–30].) Using elementary group theoretical arguments, based onthe groups SO (3) and SU (2), a particularly simple global basis of complex orthonormal vector fields tangent to S is constructed. Section V uses the constructions of Section IV to study again the beams of Section III and theirgeometric phases : in appropriate situations, the complete geometric phase separates into a contribution from S and another from S . The final Section VI contains some concluding remarks, while the Appendix compares thepresent framework for handling geometric phases with that proposed in [27]. II. EXAMPLES OF CLASSICAL OPTICAL GEOMETRIC PHASES
In this Section we review three situations in classical optics displaying geometric phases, presenting only the essentialdetails. The first concerns scalar wave optics, the other two include polarisation. Quantum mechanical notation isused when convenient [31].
A. The case of the Gouy phase
We deal with the scalar optical wave field in free space, with fixed (angular) frequency ω , wave number k = ω/c and wavelength λ = 2 π/k . In the paraxial approximation to the Helmholtz equation, with the positive z -axis as thepropagation direction, we obtain the paraxial wave equation in two transverse dimensions : i ¯ λ ∂∂z ψ ( x, y ; z ) = − ¯ λ (cid:18) ∂ ∂x + ∂ ∂y (cid:19) ψ ( x, y ; z ) , (2.1)where ¯ λ = λ/ (2 π ) = k − , reminiscent of ~ . (The exponential factor e i ( kz − ωt ) has been omitted in ψ ). This isformally similar to the Schr¨odinger equation in quantum mechanics for a free nonrelativistic particle of unit mass intwo dimensions, with ¯ λ in place of ~ , and with the longitudinal variable z playing the role of ‘time’. The “Hamiltonianoperator” H for Eq. (2.1) is H = 12 (cid:0) p x + p y (cid:1) , p x = − i ¯ λ ∂∂x , p y = − i ¯ λ ∂∂y . (2.2)If we restrict to one transverse dimension, we have the simpler paraxial wave equation i ¯ λ ∂∂z ψ ( x ; z ) = Hψ ( x ; z ) , H = 12 p x = − ¯ λ ∂ ∂x . (2.3)Our focus is on ‘centred’ Gaussian solutions to this paraxial wave equation, and their phases. Based on grouptheoretical considerations, it is convenient to parametrise normalised centred Gaussians by a complex variable q withnegative imaginary part, i.e., lying in the lower half complex plane:Im q < ψ ( x ; q ) = (cid:18) − Im qπ ¯ λ | q | (cid:19) / exp (cid:18) i x λq (cid:19) , Z ∞−∞ dx | ψ ( x ; q ) | = 1 . (2.4)Then the centred Gaussian solution to Eq. (2.3), with width w in the ‘waist’ plane z = 0, is: ψ ( x ; 0) = ψ (cid:18) x ; − i w λ (cid:19) = (cid:18) πw (cid:19) / exp (cid:18) − x w (cid:19) −→ ψ ( x ; z ) = e iϕ G ( z ) ψ ( x ; q ( z )) ; ϕ G ( z ) = −
12 tan − (cid:18) zz R (cid:19) , q ( z ) = z − iz R ,z R = Rayleigh range = w / λ = πw /λ. (2.5)Here, ϕ G ( z ) is the evolving Gouy phase. It is the argument of ψ (0; z ) (on-axis phase at the plane z = constant) and‘jumps’ by − π/ − nπ/ n transverse dimensions) across the waist plane : ϕ G ( z ) = arg ψ (0; z ) ,ϕ G ( ∞ ) − ϕ G ( −∞ ) = − π/ . (2.6)That the parameter q lives in the lower half plane is a consequence of our taking monochromatic time-dependencein the usual form exp( − iωt ). Had it been taken in the form exp( iωt ), as some authors do, then q would live in theupper half plane. The evolution of Gaussian beams through first order systems described by abcd-matrix is governedby the well known Kogelnik abcd-law [32], q in → q out = a q in + bc q in + d , (2.7)of which the particular case q ( z ) → q ( z ) = q ( z ) + ( z − z ), corresponding to free propagation from z to z [ i.e.,( a, b, c, d ) = (1 , z − z , ,
1) ], is already quoted in Eq. (2.5). It may be noted in passing that the abcd-law has beengeneralized to partially coherent Gaussian beams, the so-called Gaussian Schell-model beams, in [33] and to arbitrarybeams in [34].Our aim now is to show that ϕ G ( z ) is essentially a geometric phase. For this we need the extension of the relation(1.11) to the four-vertex Bargmann invariant, and then specialize it in a particular way. For the moment we usequantum mechanical notation, with ψ denoting a Hilbert space vector. The generalisation of the connection (1.11) is[ ψ , · · · , ψ are unit vectors ]: ∆ ( ψ , ψ , ψ , ψ ) = ( ψ , ψ )( ψ , ψ )( ψ , ψ )( ψ , ψ ) , arg ∆ ( ψ , ψ , ψ , ψ ) = − ϕ g [ C ] ,C = quadrilateral in R with vertices ρ ( ψ ) , · · · , ρ ( ψ )and geodesics connecting ρ ( ψ ) to ρ ( ψ ) , · · · , ρ ( ψ ) to ρ ( ψ ) as sides. (2.8)Now to the specialisation of this relation. Let s be an evolution parameter, and H a ‘Hamiltonian operator’ inde-pendent of s ; and let ψ ( s ) obey the ‘Schr¨odinger equation’ i dds ψ ( s ) = H ψ ( s ) , (2.9)so that ψ ( s ) = e − i ( s − s ) H ψ ( s ) . (2.10)In the relation (2.8) we now choose ψ to be a convenient ‘reference vector’ ψ R , which allows the measurement of thephase ϕ ( s ) of ψ ( s ) with respect to it in the Pancharatnam sense (i.e., through an inner-product) : ϕ ( s ) = arg( ψ R , ψ ( s )) . (2.11)Further, we choose ψ = ψ ( s ) , ψ = a ‘zero energy’ vector ψ E obeying H ψ E = 0, and ψ = ψ ( s ). Then, usingalso Eq. (2.10), the connection (2.8) becomes: ϕ ( s ) − ϕ ( s ) = ϕ g [ quadrilateral in R with vertices ρ ( ψ R ) , ρ ( ψ ( s )) , ρ ( ψ E ) , ρ ( ψ ( s )) , and geodesic sides ] . (2.12)To apply Eq. (2.12) to the present case, we set s = ¯ λ − z and H = p x , so (2.9) becomes (2.3); and associate‘wave functions’ ψ R ( x ) , ψ ( x ; z ) , ψ E ( x ) with the Hilbert space vectors ψ R , ψ ( s ) , ψ E respectively. To arrange ϕ ( s ) inEq. (2.11) to be the Gouy phase ϕ G ( z ), recalling the ‘on-axis’ identification in Eq. (2.6), the wave function ψ R ( x ) mustbecome essentially δ ( x ). Next, to obey the condition H ψ E = 0 the wave function ψ E ( x ) must become x -independent,i.e., a plane wave with wave vector strictly along the z-axis (recall that we have dropped, following (2.1), a factorexp[ i ( kz − ωt )]). With these clues we take ψ R ( x ) and ψ E ( x ) to be particular limiting forms of ψ ( x ; q ) (and as ourinterest is in phases alone we disregard real factors which diverge or vanish in the limits): ψ R ( z ) = lim q =0 ,q → − ψ ( x ; q + iq )= lim q → − − π ¯ λq ) / exp (cid:18) x λq (cid:19) ∼ δ ( x ) ; ψ E ( x ) = lim q → ,q →−∞ ψ ( x ; q + iq )= lim q →−∞ − π ¯ λq ) / exp (cid:18) x λq (cid:19) ∼ constant in x . (2.13)Then indeed, with s = z/ ¯ λ , ϕ ( s ) = arg( ψ R , ψ ( s )) = arg (cid:18)Z ∞−∞ dx ψ R ( x ) ∗ ψ ( x ; z ) (cid:19) = arg ψ (0; z ) = ϕ G ( z ) , (2.14)so Eq. (2.12) becomes: ϕ G ( z ) − ϕ G ( z )= ϕ g [ quadrilateral in R , vertices ρ ( ψ R ( x ) ∼ δ ( x )) ,ρ ( ψ ( x ; z )) , ρ ( ψ E ( x ) ∼ constant) , ρ ( ψ ( x ; z ))and geodesic sides ] . (2.15)This already shows that (differences of) Gouy phases are certain geometric phases. However, for improved under-standing, we can analyse the right hand side further as follows.The argument of ϕ g on the right hand side in the result (2.15) is a quadrilateral in the ‘ray space’ R , with geodesicsides, as it should be. The geodesics needed here are to be constructed in the manner of Eq. (1.9) at the vectorspace or wave amplitude level, followed by projection π to R . A quite subtle analysis [35] (here omitted) shows thatin the present instance (and some others of interest) we can use ‘geodesics’ drawn within the manifold of centredGaussian amplitudes (which may be called ‘constrained geodesics’), and the basic connection (1.11), (2.8) betweenBargmann invariants and geometric phases continues to be valid. Next, as the definition (1.7) of geometric phasesshows, for practical calculations one can choose any convenient ‘lift’ C of C at the level of Hilbert space vectors orwave amplitudes, obeying C = π [ C ]. In particular if C is chosen to be a closed loop (the quadrilateral C in R is ofcourse closed) the piece ϕ tot [ C ] in Eq. (1.7) vanishes and we are left with ϕ g [ C ] = − ϕ dyn [ C ]. Beyond this, one canuse the phase freedom at each point along C to assume, in the present case, that C is a closed loop within the spaceof centred Gaussian wave functions ψ ( x ; q ). It can then be pictured or drawn as a closed curve in the lower half ofthe complex q plane. One must only ensure that the ‘vertices’ are chosen properly, so as to project onto the verticesspecified in R in (2.15), and the connecting curves represent ‘constrained geodesics’ properly. When all this is done,the result is as shown in Fig. 1. z Θ Θ Α Α Α Α z - iz R O O Ψ z - iz R Ψ - iz R q Ψ ~ Ψ R q − planeq q ® − ¥ Ψ ~ Ψ E Ψ ~ Ψ E E E
FIG. 1: Illustrating the hyperbolic geometry of the lower half complex q -plane underlying Gaussian beams and the abcd-law.Free propagation corresponds to the horizontal line passing through q = − iz R . The two circular geodesics are centred at O , O . For the geodesic quandrilateral ψ → ψ → ψ → ψ → ψ the angles at ψ , ψ vanish, while the angles at ψ , ψ become π/ z = − z R and z = z R respectively. Thus 50% of the total Gouy phase ‘jump’ occurs within a propagationdistance 2 z R around the waist, z R decreasing quadratically with decreasing waist size w . The arcs connecting q = 0 to q = − iz R + z (i.e., ψ to ψ ) and q = − iz R + z to q = 0 (i.e., ψ to ψ ) are bothcircular, with centres on the q axis. The straight lines connecting q = − iz R + z to q = − i ∞ (i.e., ψ to ψ ) and q = − i ∞ to q = − iz R + z (i.e., ψ to ψ ) are both vertical, parallel to the q axis. All of them taken in sequence‘represent’ the closed C : C ∼ q = 0 → q = − iz R + z → q = − i ∞→ q = − iz R + z → q = 0 (2.16)and Eq. (2.15) takes the more explicit form ϕ G ( z ) − ϕ G ( z ) = − ϕ dyn [ C ]= − Im I C (cid:26) ( ψ ( x ; q ) , ∂∂q ψ ( x ; q ) ) dq + ( ψ ( x ; q ) , ∂∂q ψ ( x ; q ) ) dq (cid:27) . (2.17)The integration in the q half plane is along the curve (2.16), while the inner products in Hilbert space H = L ( R )(integrations with respect to x ) are left implicit. With some effort one can confirm that the integral over C on theright in Eq. (2.17) indeed reproduces the difference between Gouy phases on the left, as determined by Eq. (2.5).Alternatively, the line integral in Eq. (2.17) equals [10] the negative of one-fourth of the (hyberbolic) area of theenclosed quadrilateral, the abcd-law being a signature of the natural Lobachevskian hyperbolic geometry with metric dℓ = ( dq + dq ) /q , (2.18)underlying the manifold of Gaussian states, the lower half q -plane. The area itself is given by the ‘hyperbolic deficiency’which, for a (geodesic) quadrilateral, equals 2 π minus sum of the interior angles.The interior angle vanishes at R as well as at E . For the other two angles α , α we see from Fig. 1 that α j =2( π/ − θ j ) and tan θ j = z j /z R . Thus the deficiency equals 2( θ + θ ), leading to a geometric phase of − ( θ + θ ) / ϕ G ( z ) − ϕ G ( z ) = −
12 (arctan( z /z R ) − arctan( z /z R )) , (2.19)well known in the context of laser beams, now as a geometric phase.We appreciate that this demonstration of the link between Gouy and geometric phases is fully within the H −B − R framework of quantum mechanics used in the quantum kinematic approach [5, 6] to geometric phase, brieflyrecapitulated in Section 1.
B. The Pancharatnam case
Now we include the polarization degree of freedom, and to begin with consider the extreme case when it is theonly variable. With given frequency ω and wave number k = ω/c , we fix also the direction of propagation to be thepositive z -axis, and consider plane waves in various states of pure polarization. The analysis again falls perfectly intothe quantum mechanical H − B − R scheme. Dropping the standard factor e i ( kz − ωt ) , at each z the electric field isa complex two-component vector in the transverse x-y plane, E = (cid:0) E E (cid:1) . The Hilbert space H for this case is then H = C of dimension two.As is well known, the spaces B and R are S and S respectively, unit spheres in real four and three-dimensionalEuclidean spaces, the latter being the Poincar´e sphere of pure polarization states : H = C → B = S π −→ R = B /U (1) = S . (2.20)Given E at some z , the corresponding pure polarization state is represented by a point ˆ n ∈ S computed as follows: E → ˆ n = ( E † E ) − E † τ E ∈ S , τ = ( τ , τ , τ ) = ( σ , σ , σ ) , (2.21)where the σ ’s are the standard quantum mechanical Pauli matrices. Under free propagation governed by the freeMaxwell equations, the amplitude E and the polarization state ˆ n are both constant: ˆ n is stationary on S .More generally, we imagine the plane wave passing through transparent linear intensity preserving polarizationgadgets which act on E and alter the polarization state ˆ n . These are placed at various locations (lumped) or overvarious stretches (distributed) along the z -axis, separated from one another by intervals of free propagation. Theeffect of such gadgets on E is again governed by Maxwell’s equations for propagation of the field through suitabletransparent material media. As our interest is only in the behaviour of the polarization state ˆ n , the intensity beingheld constant, we can represent each polarization gadget by a corresponding element of the two-dimensional unitaryunimodular group SU (2) [36] —the additional U (1) phase in the full unitary group U (2) is not relevant for this purpose.With this physical picture in place, let us write E ( z ) and ˆ n ( z ) for the field and the polarization state at position z along the propagation axis: E ( z ) ∈ C → ˆ n ( z ) = ( E ( z ) † E ( z )) − E ( z ) † τ E ( z ) ∈ S . (2.22)Then E ( z ) evolves according to the Schr¨odinger-like equation i d E ( z ) dz = H ( z ) E ( z ) , H ( z ) = 12 τ · a ( z ) , (2.23)where a ( z ) is a real three-dimensional vector and H ( z ) is the ‘Hamiltonian’. Correspondingly for ˆ n ( z ) we have d ˆ n ( z ) dz = a ( z ) ∧ ˆ n ( z ) . (2.24)Thus while E ( z ) undergoes a gradually unfolding SU (2) transformation, ˆ n ( z ) experiences a gradual rotation belongingto SO (3) [37]. (Free propagation stretches correspond to H ( z ) = 0, and hence to a ( z ) = 0; for lumped elements likea quarter or half wave plate a ( z ) is a Dirac delta function). Over a finite stretch z to z , we have: E ( z ) = U ( z , z ) E ( z ) , U ( z , z ) ∈ SU (2) ;ˆ n ( z ) = R ( z , z ) ˆ n ( z ) , R ( z , z ) ∈ SO (3) , (2.25)with U ( z , z ) determining R ( z , z ) through the well known SU (2) → SO (3) homomorphism [38].If H ( z ) and a ( z ) are constant from z to z , say H and a respectively, we have U ( z , z ) = e − i ( z − z ) H , R ( z , z ) = R (ˆ a , ( z − z ) | a | ) , ˆ a = a / | a | , (2.26)where R (ˆ a , α ) is the right handed rotation about axis ˆ a by amount α [37]. Then over such a stretch E ( z ) † H E ( z ) = 12 E ( z ) † E ( z ) a · ˆ n ( z ) = constant , (2.27)and as E ( z ) † E ( z ) = constant as well, we see that ˆ n ( z ) moves on a latitude circle in a plane perpendicular to a . Incase a · ˆ n ( z ) = 0, ˆ n ( z ) moves on the great circle perpendicular to ˆ a , the equator with respect to ˆ a ; and as then E ( z ) † H E ( z ) = 0, such stretches contribute zero dynamical phases .A cyclic evolution in this Pancharatnam situation carries the electric field over some curve C ⊂ B = S (assumingfor simplicity E ( z ) † E ( z ) = 1), say from E (1) at z to E (2) = e iθ E (1) at z . Then ˆ n ( z ) describes a closed loop C pol ⊂ S . By Eq. (1.7), the associated geometric phase can be readily computed, and it turns out to be very simplyrelated to the geometry of S : ϕ g [ C pol ] = ϕ tot [ C ] − ϕ dyn [ C ]= arg E (1) † E (2) − Im Z z z dz E ( z ) † d E ( z ) dz = θ + Z z z dz E ( z ) † H ( z ) E ( z )= 12 Ω[ C pol ] , (2.28)where Ω[ C pol ] is the solid angle (with sign) subtended by C pol at the origin of S .In the original Pancharatnam analysis, C pol is a spherical triangle on S with sides being great circle arcs ( i.e.,geodesics ) [39], leading as mentioned above to ϕ dyn [ C ] = 0 if piecewise constant ‘Hamiltonians’ are used. And ϕ g [ C pol ]reduces to the negative of the phase of a three-vertex Bargmann invariant, a special simple instance of Eq. (1.11): iffields E (1) , E (2) , E (3) lead via Eq. (2.21) to the vertices ˆ n , ˆ n , ˆ n of C pol then C pol = spherical triangle on S : ϕ g [ C pol ] = 12 Ω[ C pol ]= − arg( E (1) † E (2) E (2) † E (3) E (3) † E (1) ) . (2.29) C. Polarisation in the eikonal limit
The third situation we consider from the geometric phase perspective is one that has been studied for a long time onaccount of its obvious physical relevance. It is the short wave length—or eikonal or ray—limit of Maxwell’s equations,leading to differential equations for light rays in a given transparent medium, plus the law for evolution of the electricfield along them [40]. We first recall the basic equations resulting from the eikonal limit, then some important previouswork, and finally consider the situation from the geometric phase perspective.In comparison to the previous Pancharatnam case, in the eikonal limit the propagation (ray) direction is allowedto vary while, in a sense to be clarified later, the polarisation state stays constant. We consider Maxwell’s equationsfor propagating electric (and magnetic) fields in a transparent nonconducting non-magnetic material medium char-acterised by a time-independent isotropic refractive index function n ( x ). To leading order, the eikonal limit gives asystem of second order ordinary differential equations whose solutions are rays in the medium: dds (cid:18) n ( x ) d x ds (cid:19) = ▽▽▽ n ( x ) . (2.30)Each solution x ( s ) (for given initial conditions) determines a ray Γ, a curve in physical three-dimensional Euclideanspace. Here s is arc length measured along Γ from some starting point on Γ. We hereafter work with some definite
Γ.As a space curve, Γ is characterized by the following vectors and scalars defined pointwise along it, the dot denotingderivative with respect to s : v ( s ) = ˙ x ( s ) = unit tangent; n ( s ) = ˙ v ( s ) / | ˙ v ( s ) | = unit principal normal; b ( s ) = v ( s ) ∧ n ( s ) = unit binormal; κ ( s ) = | ˙ v ( s ) | = curvature ,τ ( s ) = b ( s ) · ˙ n ( s ) = torsion . (2.31)0At each x ( s ) ∈ Γ, ( v ( s ) , n ( s ) , b ( s )) is a right handed orthonormal triad, unique at generic points with nonzerocurvature; it is locally determined by ˙ x ( s ) and ¨ x ( s ). Formally these vectors obey the ‘equations of motion’˙ v = κ b ∧ v = ( v ∧ ˙ v ) ∧ v , ˙ n = ( κ b + τ v ) ∧ n , ˙ b = τ v ∧ b . (2.32)The first equation (which is actually trivial) means that v obeys the minimal Fermi-Walker transport law [41], while n and b do not do so.Next we consider the evolution of the electric field E ( x ( s )) ≡ E ( s ) along Γ. This comes from the next to leadingorder terms in the eikonal limit of Maxwell’s equations, and when expressed in terms of the normalised electric field Ψ ( s ) we have again the Fermi-Walker transport law along with the transversality condition: Ψ ( s ) = E ( s ) / q E ( s ) † E ( s ) : ˙ Ψ ( s ) = κ ( s ) b ( s ) ∧ Ψ ( s ) , v ( s ) · Ψ ( s ) = 0 . (2.33)That both v ( s ) and Ψ ( s ) obey the Fermi-Walker law is consistent with the need to maintain the transversalitycondition v ( s ) · Ψ ( s ) = 0 along Γ.At each x ( s ) ∈ Γ, n ( s ) and b ( s ) span the transverse plane perpendicular to v ( s ) there. If we introduce anotherorthonormal basis in this plane, e a ( s ), a = 1 ,
2, obeying the Fermi-Walker transport law like Ψ ( s ), we have theevolution equations e a ( s ) · e b ( s ) = δ ab , e ( s ) ∧ e ( s ) = v ( s ) , e a ( s ) · v ( s ) = 0 ;˙ e a ( s ) = κ ( s ) b ( s ) ∧ e a ( s ) , a = 1 , . (2.34)As initial condition we take e ( s ) = n ( s ) , e ( s ) = b ( s ) (2.35)at some s = s . Then the pair ( e , e ) rotates steadily with respect to the pair ( n , b ) at a rate given by the torsion: dds (cid:18) e a ( s ) · n ( s ) e a ( s ) · b ( s ) (cid:19) = (cid:18) τ ( s ) − τ ( s ) 0 (cid:19) (cid:18) e a ( s ) · n ( s ) e a ( s ) · b ( s ) (cid:19) ,a = 1 , (cid:18) e ( s ) e ( s ) (cid:19) = (cid:18) cos χ ( s ) − sin χ ( s )sin χ ( s ) cos χ ( s ) (cid:19) (cid:18) n ( s ) b ( s ) (cid:19) ,χ ( s ) = Z ss ds ′ τ ( s ′ ) . (2.36)Now e a ( s ) · Ψ ( s ) are constants along Γ: Ψ ( s ) = z a e a ( s ) , z a = e a ( s ) · Ψ ( s ) = constant , z † z = ( z ∗ z ∗ ) (cid:18) z z (cid:19) = 1 . (2.37)All the three-dimensional vectors x , v , n , b , E , Ψ , e a have corresponding components with respect to some fixedglobal Cartesian frame in space. The representation (2.37) identifies Ψ ( s ) at each x ( s ) ∈ Γ with a ‘vector’ z in thetwo-dimensional complex linear space C . Using this we can represent the polarization state at x ( s ) ∈ Γ by a pointˆ n ( z ) on the Poincar´e sphere S : Ψ ( s ) → ˆ n ( z ) = z † τ z ∈ S . (2.38)As long as no polarization gadgets are placed anywhere on Γ, the z a are constants, so the polarization state representedby ˆ n ( z ) ∈ S is also constant: only the propagation direction v ( s ) varies. This is to be compared with thePancharatnam situation : under free propagation, both propagation direction k and polarization state ˆ n ∈ S areconstant. If polarization gadgets are placed along the axis, k (by definition) stays constant, while ˆ n ( z ) moves on S .1In the present context we can say the cyclic case occurs when the choice of a ‘later’ point x ( s ) ∈ Γ is such that v ( s ), n ( s ), b ( s ) are the same as v ( s ), n ( s ), b ( s ) respectively at the initial point x ( s ) ∈ Γ. This happens ifcyclic case : ˙ x ( s ) = ˙ x ( s ) , ¨ x ( s ) = ¨ x ( s ) . (2.39)The behaviours of input linear and circular polarizations are then particularly simple. The linear case correspondsto real z a ; then Ψ ( s ) is a vector in space with real Cartesian components all along Γ. From Eqs. (2.35,2.36) we canrelate Ψ ( s ) to Ψ ( s ) as follows:Linear polarisation: : Ψ ( s ) = cos θ e ( s ) + sin θ e ( s ) −→ Ψ ( s ) = cos θ e ( s ) + sin θ e ( s )= cos( θ − χ ( s )) e ( s ) + sin( θ − χ ( s )) e ( s ) ,χ ( s ) = Z s s ds τ ( s ) . (2.40)The two transverse planes at x ( s ), x ( s ) on Γ are parallel to one another, and Ψ ( s ) is obtained from Ψ ( s ) by aright handed rotation by angle χ ( s ) about v ( s ). A detailed calculation shows that χ ( s ) has a geometrical meaning.Over the range s ≤ s ≤ s , the unit tangent v ( s ) to Γ describes a closed loop C dir on the sphere of directions S : C dir = { v ( s ) ∈ S | s ≤ s ≤ s } ⊂ S , v ( s ) = v ( s ) . (2.41)Then we have the result that the integrated torsion is (the negative of) the solid angle subtended by C dir at the centreof S : χ ( s ) = Z s s ds τ ( s ) = − Ω[ C dir ] . (2.42)In the cases of circular polarisations, we get phase shifts rather than a rotation in space. In these cases, Ψ ( s ) is acomplex three-vector at all x ( s ) on Γ :RCP / LCP : Ψ ( s ) = 1 √ e ( s ) ± i e ( s )) −→ Ψ ( s ) = 1 √ e ( s ) ± i e ( s ))= 1 √ e ± iχ ( s ) ( n ( s ) ± i b ( s )) , Ψ ( s ) = e ± iχ ( s ) Ψ ( s ) = e ∓ i Ω[ C dir ] Ψ ( s ) . (2.43)These results on the behaviours of polarization in the ray limit of Maxwell’s equations were obtained very earlyby Rytov and by Vladimirskii [11, 12]. In particular, Rytov showed that the phase difference between RCP and LCPevolves at a rate proportional to the torsion; while Vladimirskii showed that the spatial rotation experienced in thecyclic case for linear polarization is essentially by the solid angle Ω[ C dir ].To cast the above discussion into the geometric phase format of Section I, it is useful to write the evolution equation(2.33) for the (normalised) electric field in a Schr¨odinger-like form with a suitable hermitian Hamiltonian operator.We view Ψ ( s ) (referred to axes fixed in space) as a (normalised) element of H = C , which is the Hilbert space in thepresent context, and find: i dds Ψ ( s ) = H ( s ) Ψ ( s ) ,H ( s ) = iκ ( s ) (cid:0) n ( s ) v ( s ) T − v ( s ) n ( s ) T (cid:1) . (2.44)Thus H ( s ) is a pure imaginary antisymmetric 3 × v ( s ) T Ψ ( s ) = 0 is to beadded as a constraint consistent with the evolution. The definition (1.7) allows us to define a geometric phase forany s and s , and we find that due to transversality the dynamical phase always vanishes . Bringing in the spaces2 B ≃ S and R = CP , the complex two-dimensional projective space appropriate to H = C , we have: C = { Ψ ( s ) ∈ C | s ≤ s ≤ s } ⊂ B ,π [ C ] = C ⊂ R : ϕ g [ C ] = ϕ tot [ C ] − ϕ dyn [ C ] ,ϕ tot [ C ] = arg( Ψ ( s ) † Ψ ( s )) ,ϕ dyn [ C ] = Im Z s s ds Ψ ( s ) † d Ψ ( s ) ds = Im (cid:18) − i Z s s ds Ψ ( s ) † H ( s ) Ψ ( s ) (cid:19) = 0 , i.e. , ϕ g [ C ] = ϕ tot [ C ] . (2.45)Here we recognize that C cannot be drawn freely in B because of the transversality condition, so in this way it isconstrained by Γ.To illustrate the above, let us quote some particular cases in a table, Eq. (2.46) :Choices of s , s Polarisation Behaviour of Ψ ( s ) ϕ g [ C ]Free Linear Real 0 or π Γ cyclic, Eq. (2.39) RCP/LCP Ψ ( s ) = e ∓ i Ω[ C dir ] Ψ ( s ) ∓ Ω[ C dir ] (2.46)The distinction between C ⊂ R and C dir ⊂ S should be kept in mind. III. COMBINED PATH AND POLARISATION GEOMETRIC PHASES
The brief reviews presented in the previous Section show that the Pancharatnam situation and the ray optic limit aremutually complementary. In the former only the polarization state changes, while in the latter only the propagationdirection changes. Now we try to cover the (important) middle ground between them. We endeavour to build up aphysical picture, based ultimately on Maxwell’s equations, with the motivation to arrive at geometric phases in theframework of Section I.As in the eikonal limit, we consider light traveling through a transparent non-magnetic stationary medium withrefractive index n ( x ). We recall that the concept of a single ray is not physically meaningful and cannot be realised.The eikonal limit of Maxwell’s equations leads at first to a first order partial differential equation in three-dimensionalspace for the eikonal, a function S ( x ). A particular eikonal S ( x ) leads to a continuous family or succession of wavefronts over each of which S ( x ) is constant, and which taken together cover some region of physical space. Rays arethen lines drawn in this region, orthogonal at each point to the wave front passing through that point. These rays aresolutions to Eq. (2.30). Thus one eikonal S ( x ) determines a corresponding succession of wave fronts and in turn onefamily F S of rays. There is only one wavefront, and only one ray belonging to F S , through each point in the relevantregion.It is in this sense that single rays are not directly physically realisable. The best that we can do is to consider anarrow or well collimated ( i.e., nearly parallel ) bundle of nearby rays with some nonzero cross-sectional area whichmay vary along the bundle [ Consequently the wavefronts along the bundle, correspondingly limited in their spatialextent, are nearly planar ]. Calling this a beam, at each ‘point’ along it we have some finite spatially limited wavefront.In this picture we have in mind some Γ obeying Eq. (2.30) acting as the ‘backbone’ of the beam. At each location x ( s ) ∈ Γ, we have a propagation direction v ( s ), a spatially limited ‘plane wave’ perpendicular to v ( s ), and a transverseelectric field E ( x ( s )) ≡ E ( s ). Thus we arrive at a physical picture of a continuous succession of limited plane waveseach at a spatial location x ( s ), with propagation direction v ( s ) and in some polarization state. Now we can go astep further and allow the wavelength to be finite, as long as it is much smaller than all other physically relevantdimensions, including the linear dimensions of the limited plane wave elements.In this way we motivate the passage from a physically unrealisable ray to a realisable beam by a process of‘thickening’ of the former. In the sequel, the spatial locations x ( s ) of successive plane wave elements of the beam willsometimes be omitted. The parameter s continues to be distance measured along the beam from some initial point,increasing at each location in the direction of v ( s ).For a beam propagating ‘freely’ in the medium in this way, the evolution equation for Ψ ( s ) is Eq. (2.33). This, aswe have seen, is a consequence of Maxwell’s equations in the medium, and can be put into the Schr¨odinger-like form3(2.44) with a hermitian Hamiltonian operator. The solution Eq. (2.37) with constant z implies a constant polarizationstate ˆ n ( z ) ∈ S given in Eq. (2.38).We can now go another step further and imagine placing various polarisation gadgets over (short) stretches of thebeam, equivalently of Γ, where z varies as function of s , governed by a ‘polarisation Hamiltonian’ as in Eq. (2.23).Thus we arrive at new evolution equations for Ψ ( s ) based on the following ingredients: Ψ ( s ) = z a ( s ) e a ( s ) : i dds e a ( s ) = H (dir) ( s ) e a ( s ) ,H (dir) ( s ) = iκ ( s ) (cid:0) n ( s ) v ( s ) T − v ( s ) n ( s ) T (cid:1) ; i dds z ( s ) = H (pol) ( s ) z ( s ) ,H (pol) ( s ) = 12 τ · a ( s ) , a ( s ) real . (3.1)We have now written H (dir) ( s ) for the ‘direction’ part of the Hamiltonian, appearing in Eq. (2.44); it is completelydetermined by the local geometrical properties of Γ. The other contribution to the evolution of Ψ ( s ) is from the‘polarization’ part of the Hamiltonian, as in Eq. (2.23), written now as H (pol) ( s ). This controls the evolution of thelocal two-component transverse description of Ψ ( s ) resolved along e a ( s ). The complete evolution equation for Ψ ( s )is easily found to be Schr¨odinger-like, with a Hamiltonian which is a (complex) hermitian 3 × i dds Ψ ( s ) = (cid:16) H (dir) ( s ) + H ′ (pol) ( s ) (cid:17) Ψ ( s ) ,H (dir) ( s ) = iκ ( s ) (cid:0) n ( s ) v ( s ) T − v ( s ) n ( s ) T (cid:1) ,H ′ (pol) ( s ) = 12 a j ( s )( τ j ) ab e a ( s ) e b ( s ) T . (3.2)The implied evolution equation for ˆ n ( z ) ∈ S is as in Eq. (2.24):ˆ n ( z ( s )) ≡ ˆ n ( s ) : d ˆ n ( s ) ds = a ( s ) ∧ ˆ n ( s ) . (3.3)Over portions of the beam free of polarization gadgets, where a ( s ) = 0, the local properties of Γ determine thepropagation, and the polarization state is constant. Passage through gadgets leads to changing z ( s ) and ˆ n ( s ). Bothkinds of changes in Ψ ( s ) are ultimately traced back to Maxwells’ equations; and the complete evolution equation (3 . v ( s ) T Ψ ( s ) = 0. In all of this, the separation of effects due to change in beamdirection and those due to presence of polarisation gadgets, is essentially unambiguous.Let us now bring in geometric phase considerations. As in the ray case in Section II(C), we are able to use thebasic quantum mechanical H − B − R framework with H = C , B = S , and R = CP which is of real dimensionfour. For the calculation of dynamical phases we need the resultIm (cid:18) Ψ ( s ) , d Ψ ( s ) ds (cid:19) = Im ( − i ( Ψ ( s ) , ( H (dir) ( s ) + H ′ (pol) ( s )) Ψ ( s ) )) = − a ( s ) · ˆ n ( s ) , (3.4)so there is a contribution only from the presence of polarisation gadgets [This was to be expected since we havearranged the ‘evolution in direction’ to be of vanishing dynamical phase]. For general s and s with initial and finalspatial positions x ( s ), x ( s ) on the beam we define: C = { Ψ ( s ) ∈ H | s ≤ s ≤ s } ⊂ B ,π [ C ] = C ⊂ R . (3.5)(It is implicit that Ψ ( s ) is located in space at x ( s ) and is transverse, so as in Section II it cannot be drawn arbitrarily4in B ). Then we have: ϕ g [ C ] = ϕ tot [ C ] − ϕ dyn [ C ] ,ϕ tot [ C ] = arg( Ψ ( s ) † Ψ ( s )) ,ϕ dyn [ C ] = Im Z s s ds (cid:18) Ψ ( s ) , dds Ψ ( s ) (cid:19) = − Z s s ds a ( s ) · ˆ n ( s ) . (3.6)We illustrate this result in a special situation, where a connection to the results in Section II in the Pancharatnamcase (B) can be made. Let us firstly choose s and s so that this stretch of Γ is ‘cyclic’ in the sense of Eq. (2.39).Then we make an independent additional assumption that the polarisation gadgets placed along the beam between x ( s ) and x ( s ) are such that ( for a particular initial Ψ ( s ) ) z ( s ) turns out to be a phase times z ( s ). This thenmeans that the curve traced by ˆ n ( s ) ∈ S is a closed loop. In all the conditions assumed are:˙ x ( s ) = ˙ x ( s ) , ¨ x ( s ) = ¨ x ( s ) ; z ( s ) = e iθ z ( s ) , ˆ n ( s ) = ˆ n ( s ) ; C pol = { ˆ n ( s ) ∈ S | s ≤ s ≤ s } ⊂ S , closed . (3.7)By Eqs. (2.36,2.42) we relate e a ( s ) to e a ( s ): e ( s ) e ( s ) ! = cos Ω[ C dir ] sin Ω[ C dir ] − sin Ω[ C dir ] cos Ω[ C dir ] ! e ( s ) e ( s ) ! , (3.8)where Ω[ C dir ] is the solid angle subtended at the centre of S by C dir defined in Eq. (2.41). We see that with theconditions (3.7) we deal with two closed loops , C dir ⊂ S and C pol ⊂ S , on the sphere of directions and on thePoincar´e sphere respectively. Now we can calculate the geometric phase for this situation using Eq. (3.6): ϕ g [ C ] = arg z ( s ) † cos Ω[ C dir ] − sin Ω[ C dir ]sin Ω[ C dir ] cos Ω[ C dir ] ! e iθ z ( s ) ! + 12 Z s s ds a ( s ) · ˆ n ( s )= θ + 12 Z s s ds a ( s ) · ˆ n ( s )+ arg ( cos Ω[ C dir ] + 2 i sin Ω[ C dir ] Im z ( s ) z ( s ) ∗ ) . (3.9)On comparing Eqs. (2.27,2.28) of the Pancharatnam situation with Eq. (3.1), we see that the first two terms here addup to Ω[ C pol ]: θ + 12 Z s s ds a ( s ) · ˆ n ( s ) = 12 Ω[ C pol ] . (3.10)If we finally specialize to input circular polarisations, the third term also simplifies:RCP / LCP : z ( s ) = 1 √ , z ( s ) = ± i √ ( cos Ω[ C dir ] + 2 i sin Ω[ C dir ] Im z ( s ) z ( s ) ∗ ) = ∓ Ω[ C dir ] , (3.11)and then the geometric phase becomes, ϕ g [ C ] = 12 Ω[ C pol ] ∓ Ω[ C dir ] . (3.12)We may remark finally that while the geometric phase in the present physical situation is always defined byEq. (3.6), it is only in a quite special situation that we get a simple final expression (3.12), in a way combining thePancharatnam result (2.28) and the pure ray result (2.46). What needs to be stressed however is that the separationof the contributions from the sphere of directions S and from the Poincar´e (polarization) sphere S is essentiallyunambiguous. In the Pancharatnam limit, C dir shrinks to a point and we recover (2.28); while in the pure ray limitwith no polarisation gadgets, it is C pol that shrinks to a point and we get back (2.46) for circular polarisations.5 IV. SOME GLOBAL ASPECTS OF THE SPHERE OF DIRECTIONS
The situation analysed in the previous Section from the geometric phase point of view is that of a (narrow wellcollimated) light beam of fixed frequency travelling in physical space through a given transparent medium, encoun-tering various polarisation gadgets on its way. The path of the beam is based on a ray Γ = { x ( s ) } ⊂ R obey-ing Eq. (2.30) for a given refractive index function n ( x ). From Γ we obtain a particular one-dimensional curve C dir = { v ( s ) = ˙ x ( s ) } ⊂ S , the two-dimensional sphere of directions.The ray Γ also gives a preferred choice of a real orthonormal basis { e a ( s ) } in the transverse plane at each x ( s ) ∈ Γ,perpendicular to v ( s ) there. By resolving the normalised complex transverse electric field Ψ ( x ( s )) with respect tothis basis, we are able to describe it by a normalized complex two-component column vector z ( s ), leading to therepresentation of the polarization state by a point ˆ n ( s ) ∈ S . In particular, real Ψ ( x ( s )) implies real z ( s ) and viceversa, corresponding to linear polarisations.The choice of Γ thus provides both C dir ⊂ S , and { e a ( s ) } . We can regard the latter as a preferred real orthonormalbasis in the real tangent plane T v ( s ) S ≃ R , for each v ( s ) ∈ C dir . As a result, the geometric phase contributionsfrom beam direction and beam polarisation are essentially unambiguously separated.Let us now view the problem from another more global perspective, not immediately related to a ray or to a pictureembedded in physical space . We take the sphere S of plane wave propagation directions as starting point, writing ˆ k for points on it (instead of v ( s ) obtained from Γ as upto now). Each ˆ k is the unit vector in the direction of a wavevector k associated with a possible (spatially limited) propagating plane wave. We now ask if there is a way to choosea real orthonormal basis { e a (ˆ k ) } in the real tangent plane T ˆ k S ≃ R , well defined and varying smoothly with ˆ k forall ˆ k ∈ S .Since this question is posed prior to the possible choice of a ray Γ, even if such { e a (ˆ k ) } exist, it need have nothingto do with the { e a ( s ) } later supplied by a ray Γ at a point on it where v ( s ) = ˆ k . As we have seen, it is { e a ( s ) } whichhas specific advantages from a physical point of view, which may be absent with { e a (ˆ k ) } .It is however a known fact from differential geometry that such choices of { e a (ˆ k ) } for all ˆ k ∈ S do not exist. Thisis expressed by saying that the sphere S is not parallelizable [42] —as a real four-dimensional manifold the tangentbundle T S is not (homeomorphic to) the product S × R . A useful way to display this circumstance, suited forfurther developments, is as follows.In real three-dimensional Euclidean space let us choose a right handed Cartesian system of axes with origin O , andwith ˆ e j , j = 1 , ,
3, the unit vectors along the coordinate axes. Points on the unit sphere S with centre at O willbe written ˆ k = (sin θ cos φ, sin θ sin φ, cos θ ), 0 ≤ θ ≤ π , 0 ≤ φ < π . It is necessary to define two subsets S N , S S of S whose union gives S but which have a nontrivial (indeed, substantial) overlap: S N = { ˆ k ( θ, φ ) ∈ S | ≤ θ < π , ≤ φ < π } ,S S = { ˆ k ( θ, φ ) ∈ S | < θ ≤ π , ≤ φ < π } ,S N ∪ S S = S ,S N ∩ S S = { ˆ k ( θ, φ ) ∈ S | < θ < π , ≤ φ < π } . (4.1)We need the action of proper rotations, elements of the rotation group SO (3), on S . The right handed rotationabout axis ˆ a ∈ S by angle α corresponds to the 3 × R jk (ˆ a , α ) = δ jk cos α + a j a k (1 − cos α ) − ǫ jkl a l sin α , ≤ α ≤ π . (4.2)For any ˆ k ∈ S , there are infinitely many rotations carrying ˆ e to ˆ k . However there is no way to choose one suchrotation for each ˆ k , such that it is globally well-defined and varies smoothly with ˆ k for all ˆ k ∈ S . Over S N , whichis S with just one point (the south pole) removed, a convenient choice does exist : A ′ (ˆ k ) = R (ˆ e , φ ) R (ˆ e , θ ) R (ˆ e , φ ) − = R (ˆ e cos φ − ˆ e sin φ, θ ) , ≤ θ < π, ≤ φ < π ; A ′ (ˆ k )ˆ e = ˆ k . (4.3)This is well defined at θ = 0 but not at θ = π . Acting on ˆ e , ˆ e at the North pole, we get a real orthonormal basis6for the tangent plane T ˆ k S when θ < π : e ′ (ˆ k ) = A ′ (ˆ k )ˆ e = sin φ + cos θ cos φ (cos θ −
1) sin φ cos φ − sin θ cos φ , e ′ (ˆ k ) = A ′ (ˆ k )ˆ e = (cos θ −
1) sin φ cos φ cos φ + cos θ sin φ − sin θ sin φ , ˆ k ∈ S N . (4.4)Over S S a similar choice is: A ′′ (ˆ k ) = R (ˆ e , φ ) R (ˆ e , θ ) R (ˆ e , φ ) , < θ ≤ π , ≤ φ < π : A ′′ (ˆ k )ˆ e = ˆ k . (4.5)Now this is well defined at θ = π but not at θ = 0. Acting on ˆ e , ˆ e at the North pole we get a different realorthonormal basis for T ˆ k S when θ > e ′′ (ˆ k ) = A ′′ (ˆ k )ˆ e = cos θ cos φ − sin φ (1 + cos θ ) sin φ cos φ − sin θ cos φ , e ′′ (ˆ k ) = A ′′ (ˆ k )ˆ e = − (1 + cos θ ) sin φ cos φ cos φ − cos θ sin φ sin θ sin φ , ˆ k ∈ S S . (4.6)In the overlap, which is all of S with just the north and south poles removed, we have connecting or ‘transition’formulae: ˆ k ∈ S N ∩ S S : A ′′ (ˆ k ) = A ′ (ˆ k ) R (ˆ e , φ ) = R (ˆ k , φ ) A ′ (ˆ k ) ; e ′′ a (ˆ k ) = R (ˆ k , φ ) e ′ a (ˆ k ) , a = 1 , . (4.7)There are now two equally good ways to express the nonparallelizable nature of S : (i) it is not possible toextend the definition of A ′ (ˆ k ) (respectively A ′′ (ˆ k )) to cover the South pole θ = π (respectively the North pole θ = 0)possessing smooth behaviour for all ˆ k ∈ S ; (ii) the real orthonormal bases { e ′ a (ˆ k ) } , { e ′′ a (ˆ k ) } for T ˆ k S over S N , S S respectively cannot be modified in any way to yield a real orthonormal basis for T ˆ k S varying smoothly with ˆ k allover S . A more formal statement is this: it is impossible to find two smoothly varying angles χ ′ (ˆ k ) , χ ′′ (ˆ k ) over S N , S S respectively such that the transition group element R (ˆ e , φ ) in Eq. (4.7) can be factorised as R (ˆ e , φ ) = R (ˆ e , χ ′ (ˆ k )) R (ˆ e , χ ′′ (ˆ k )) − , ∀ ˆ k ∈ S N ∩ S S . (4.8)For, if such choices were possible, then A ′ (ˆ k ) R (ˆ e , χ ′ (ˆ k )) = A ′′ (ˆ k ) R (ˆ e , χ ′′ (ˆ k )) would carry ˆ e to ˆ k and be smoothlydefined for all ˆ k ∈ S .Now the ‘topological obstruction’ described above is in the real domain , i.e., viewing each tangent plane T ˆ k S as a real two-dimensional vector space R . It has however been pointed out recently that if one complexifies each T ˆ k S into a complex two-dimensional vector space ( T ˆ k S ) c ≃ C , then the obstruction vanishes [27] : it is possibleto choose orthonormal bases for these complexified tangent spaces in a globally smooth manner. There is naturallyconsiderable freedom in such choices; we describe now a group theory based choice which seems natural and minimalin some sense. This requires the use of the group SU (2) (which is a double cover of SO (3), though this property isnot used in the SU (2) version of Eq. (4.8) established below). What we will show is that the factorisation attemptedin Eq. (4.8) is possible if on the right hand side we allow for elements from SU (2).The defining representation of SU (2) is SU (2) = {U = 2 × | U † U = 11 × , det U = 1 } , (4.9)7group composition being matrix multiplication. The axis-angle description of SU (2) elements is U (ˆ a , α ) = e − i α ˆ a · σ = cos α − i ˆ a · σ sin α , ˆ a ∈ S , ≤ α ≤ π . (4.10)The two-to-one mapping SU (2) → SO (3) respecting the composition laws, the homomorphism, is: U (ˆ a , α ) ∈ SU (2) −→ R (ˆ a , α ) ∈ SO (3) . (4.11)The rotations A ′ (ˆ k ), A ′′ (ˆ k ) defined in Eqs. (4.3, 4.5) are images, in the sense of this mapping, of elements U ′ (ˆ k ), U ′′ (ˆ k )in SU (2) respectively: U ′ (ˆ k ) = e − i φσ e − i θσ e i φσ , ˆ k ∈ S N ; U ′′ (ˆ k ) = e − i φσ e − i θσ e − i φσ , ˆ k ∈ S S . (4.12)Now the overlap transition rule (4.7) involves the subgroup of elements R (ˆ e , φ ) ∈ SO (2) ⊂ SO (3), which happento ‘coincide’ with elements U (ˆ e , φ ) ∈ SU (2) in the following sense: R (ˆ e , φ ) = U (ˆ e , φ ) 000 0 1 , U (ˆ e , φ ) = e − iφσ = cos 2 φ − sin 2 φ sin 2 φ cos 2 φ ! . (4.13)It now turns out that within SU (2) a factorisation of the form (4.8) is possible : U (ˆ e , φ ) = V ′ (ˆ k ) − V ′′ (ˆ k ) , V ′ (ˆ k ) = e − iφσ e − i θ σ e iφσ , ˆ k ∈ S N ; V ′′ (ˆ k ) = e − iφσ e − i θ σ e − iφσ , ˆ k ∈ S S . (4.14)The structures of V ′ (ˆ k ), V ′′ (ˆ k ) are suggested by those of U ′ (ˆ k ), U ′′ (ˆ k ) in Eq. (4.12): in the latter we make thecyclic changes σ → σ → σ → σ , and replace φ by 2 φ . If we use Eq. (4.14) in Eq. (4.13) and then in Eq. (4.7) wesee that A ′′ (ˆ k ) = A ′ (ˆ k ) V ′ (ˆ k ) − V ′′ (ˆ k ) 000 0 1 , i.e., A (ˆ k ) = A ′ (ˆ k ) V ′ (ˆ k ) −
000 0 1 = A ′′ (ˆ k ) V ′′ (ˆ k ) −
000 0 1 (4.15)is a globally well-defined and smoothly varying matrix in SU (3) with the property A (ˆ k )ˆ e = ˆ k , ∀ ˆ k ∈ S . (4.16)Here, the group SU (3) is the three-dimensional extension of SU (2) in Eq. (4.9), and consists of 3 × (not subgroup) of SU (3) carrying ˆ e to ˆ k is easy to characterise : A ∈ SU (3) : A ˆ e = ˆ k ⇔ A = A U
000 0 1 ,A ∈ SO (3) , U ∈ SU (2) , A ˆ e = ˆ k . (4.17)8(This decomposition is however not unique on account of the shared elements (4.13)). And indeed A (ˆ k ) is of thisform, and becomes after simplification: A (ˆ k ) = R (ˆ e , φ ) R (ˆ e , θ ) e i θσ e iφσ
000 0 1 . (4.18)If we act with A (ˆ k ) on ˆ e , ˆ e at the North pole we obtain a globally well defined and smooth complex orthonormalbasis for ( T ˆ k S ) c all over S . To distinguish these vectors from the real ones encountered up to now we write themas g a (ˆ k ), a = 1 ,
2. They are simply related to e ′ a (ˆ k ), e ′′ a (ˆ k ) of Eqs. (4.4, 4.6): g (ˆ k ) = A (ˆ k )ˆ e = cos θ/ e ′ (ˆ k ) + i sin θ/ e ′′ (ˆ k ) , g (ˆ k ) = A (ˆ k )ˆ e = cos θ/ e ′ (ˆ k ) + i sin θ/ e ′′ (ˆ k ) ;ˆ k · g a (ˆ k ) = 0 , g a (ˆ k ) ∗ · g b (ˆ k ) = δ ab . (4.19)While it should be evident a priori that, given the existence of such global complex g a (ˆ k ) all over S , there shouldbe considerable freedom in their choice, the specific form of A (ˆ k ) in Eq. (4.18) suggests that the choice (4.19) isspecially simple. In particular, for the one-parameter family of ˆ k ( θ, φ ) with fixed φ the connection between g a (ˆ k ) and( e ′ a (ˆ k ) , e ′′ a (ˆ k )) is through (portion of) a one-parameter subgroup of SU (2). Hereafter we always use { g a (ˆ k ) } givenabove.Any (complex) three-vector ψ (ˆ k ) orthogonal to ˆ k , thus belonging to ( T ˆ k S ) c , can be expanded as ψ (ˆ k ) = z a g a (ˆ k ) , z a = g a (ˆ k ) ∗ · ψ (ˆ k ) , z = z z ! ∈ C . (4.20)With respect to { g a (ˆ k ) } , and as a convention , we can regard ψ (ˆ k ) at ˆ k and ψ ′ (ˆ k ′ ) at ˆ k ′ as ‘the same’ if g a (ˆ k ) ∗ · ψ (ˆ k ) = g a (ˆ k ′ ) ∗ · ψ ′ (ˆ k ′ ) = z a . (4.21)In this sense we see that the union of ( T ˆ k S ) c over all ˆ k is a Cartesian product, which as discussed earlier is not truefor the usual tangent bundle in the real domain: T S = [ ˆ k ∈ S T ˆ k S S × R , ( T S ) c ≡ [ ˆ k ∈ S ( T ˆ k S ) c ≃ S × C . (4.22)The most obvious use of the above result is in the following context. Suppose ψ (ˆ k ) is a complex vector-valuedtransverse function of ˆ k , which for concreteness we regard as an element of a Hilbert space H as follows: H = { ψ (ˆ k ) ∈ C | ˆ k ∈ S , ˆ k · ψ (ˆ k ) = 0 , k ψ k = Z d Ω(ˆ k ) ψ (ˆ k ) ∗ · ψ (ˆ k ) < ∞ } , (4.23)with d Ω(ˆ k ) = sin θdθdφ the solid angle over S . Then we can expand ψ (ˆ k ) in the basis (4.19) and have: ψ (ˆ k ) = z a (ˆ k ) g a (ˆ k ) , z a (ˆ k ) = g a (ˆ k ) ∗ · ψ (ˆ k ) , k ψ k = Z d Ω(ˆ k ) z (ˆ k ) † z (ˆ k ) . (4.24)This shows that H is the tensor product H = L ( S ) ⊗ H (2) , (4.25)9where L ( S ) is the Hilbert space of (scalar) complex square integrable functions over S , and H (2) is the two-dimensional complex Hilbert space (appropriate for the ‘polarization qubit’).If E (ˆ k ) is a transverse electric field amplitude of a plane wave with propagation direction ˆ k , using the expansion(4.20) we may attempt to represent its polarisation state by a point on the Poincar´e sphere S in the ‘usual’ way:ˆ k · E (ˆ k ) = 0 : E (ˆ k ) = z a g a (ˆ k ) , z a = g a (ˆ k ) ∗ · E (ˆ k ) → ˆ n ( z ) = ( z † z ) − z † τ z ∈ S . (4.26)However this is not in general the ‘usual representation’ of polarisation states in the sense that, for instance, linearpolarisations corresponding to real E (ˆ k ) (upto overall phases) need not imply real z , so ˆ n ( z ) may not lie on theequator of S in the 1-2 plane. Indeed, for ˆ k ∈ S N , from Eqs. (4.7,4.19) we have: e ′ (ˆ k ) e ′ (ˆ k ) ! = U ( θ, φ ) g (ˆ k ) g (ˆ k ) ! , U ( θ, φ ) = cos θ + i sin θ sin 2 φ − i sin θ cos 2 φ − i sin θ cos 2 φ cos θ − i sin θ sin 2 φ ! ;1 √ e ′ (ˆ k ) + i e ′ (ˆ k )) = 1 √ (cid:18) cos θ e iφ sin θ (cid:19) g (ˆ k )+ i √ (cid:18) cos θ − e iφ sin θ (cid:19) g (ˆ k );1 √ e ′ (ˆ k ) − i e ′ (ˆ k )) = 1 √ (cid:18) cos θ − e − iφ sin θ (cid:19) g (ˆ k ) − i √ (cid:18) cos θ e − iφ sin θ (cid:19) g (ˆ k ) . (4.27)Using these in Eq. (4.26) we find that states of RCP and LCP are represented on S by the diametrically oppositepoints ± (sin θ cos 2 φ , sin θ sin 2 φ, cos θ ), not by the usual North and South poles (0 , , ± S in the plane orthogonal to (sin θ cos 2 φ, sin θ sin 2 φ, cos θ ). V. GLOBAL BASES AND GEOMETRIC PHASES
We consider applications of the global results of the previous Section to the calculation of geometric phases. Inthe treatment in Sections II, III the starting point was a ray Γ in a given transparent medium, based on which abeam passing through polarisation gadgets was then considered. From this, a curve C dir ⊂ S was obtained, asin Eq. (2.41). The normalised electric field along the beam was then used to define a curve C ⊂ B in the quantummechanical H − B − R framework, Eqs. (2.45, 3.5), with H = C . At all stages the validity of Maxwell’s equationswas kept in mind.In [27], however, a curve C dir ⊂ S is taken as the starting point for the discussion of geometric phases for beamswith varying direction and polarisation state. From the point of view developed by us, this would mean that inprinciple, for a chosen C dir ⊂ S to be physically realisable, we must imagine a transparent medium with suitablerefractive index function n ( x ), and a ray Γ in this medium, such that a beam traveling along Γ reproduces C dir as wefollow v ( s ) = ˙ x ( s ) along Γ. All this as well as the validity of Maxwell’s equations will be implicitly assumed in whatfollows.In the notation of Section IV, then, we imagine being given a curve C dir = { ˆ k ( s ) ∈ S } ⊂ S , and at each valueof s a normalised transverse electric field Ψ ( s ): Ψ ( s ) ∈ C , Ψ ( s ) ∗ · Ψ ( s ) = 1 , ˆ k ( s ) · Ψ ( s ) = 0 . (5.1)(Though not explicitly stated, the parameter s could be the distance measured from some starting point on a beamin physical space R ). For calculating geometric phases we again use the H − B − R framework with H = C , (theframework used in [27] is different and is briefly recounted in the Appendix), and define C = { Ψ ( s ) ∈ H | Ψ ( s ) † Ψ ( s ) = 1 , ˆ k · Ψ ( s ) = 0 , s ≤ s ≤ s } ⊂ B ,π [ C ] = C ⊂ R . (5.2)0We expand Ψ ( s ) in the complex global basis for ( T ˆ k S ) c described in Eq. (4.19): Ψ ( s ) = z a ( s ) g a (ˆ k ( s )) , z a ( s ) = g a (ˆ k ( s )) ∗ · Ψ ( s ) , z ( s ) = z ( s ) z ( s ) ! , z ( s ) † z ( s ) = 1 . (5.3)To compute the dynamical phase ϕ dyn [ C ] we need as ingredients:˙ Ψ ( s ) = dds Ψ ( s ) = ˙ z a ( s ) g a (ˆ k ( s )) + z a ( s ) ˙ g a (ˆ k ( s )) , ( Ψ ( s ) , ˙ Ψ ( s )) = z ( s ) † ˙ z ( s ) + z a ( s ) ∗ g a (ˆ k ( s )) ∗ · ˙ g b (ˆ k ( s )) z b ( s ) . (5.4)The second term leads us to define a 2 × h ( s ) as h ab ( s ) = h ba ( s ) ∗ = − i g a (ˆ k ( s )) ∗ · ˙ g b (ˆ k ( s )) , (5.5)and then we have: ϕ g [ C ] = ϕ tot [ C ] − ϕ dyn [ C ] ,ϕ tot [ C ] = arg( Ψ ( s ) ∗ · Ψ ( s )) ,ϕ dyn [ C ] = Im Z s s ds ( Ψ ( s ) , ˙ Ψ ( s ))= Im (cid:26)Z s s ds z ( s ) † ˙ z ( s ) + i Z s s ds z ( s ) † h ( s ) z ( s ) (cid:27) . (5.6)With some algebra the elements of h ( s ) can be calculated in terms of θ ( s ), φ ( s ), the spherical polar angles of ˆ k ( s ),and their derivatives ˙ θ ( s ), ˙ φ ( s ): h ( s ) = − h ( s )= − h ˙ θ ( s ) sin 2 φ ( s ) + ˙ φ ( s ) cos 2 φ ( s ) sin 2 θ ( s ) i ,h ( s ) = h ( s ) ∗ = 12 h ˙ θ ( s ) cos 2 φ ( s ) − (sin 2 θ ( s ) sin 2 φ ( s )+ i ( 1 − cos 2 θ ( s ) ) ˙ φ ( s ) i . (5.7)It is interesting that the elements of the matrix h ( s ), which arise from the dependences of g a (ˆ k ) on ˆ k , have ratherelementary forms, which can be ascribed to the group theoretical arguments that led to the construction of { g a (ˆ k ) } .As an illustration, let us consider the case where C dir is a closed loop, i.e., ˆ k ( s ) = ˆ k ( s ). Let us further assumethat Ψ ( s ) differs from Ψ ( s ) just by a phase θ so that C is closed. Since in any case g a (ˆ k )’s are determined by ˆ k ,these assumptions mean that g a (ˆ k ( s )) = g a (ˆ k ( s )) ; Ψ ( s ) = e iθ Ψ ( s ) ⇒ z ( s ) = e iθ z ( s ) , ˆ n ( s ) = ˆ n ( s ) . (5.8)Thus ˆ n ( s ) ≡ ˆ n ( z ( s )) describes a closed loop C pol ⊂ S , and the geometric phase (5.6) becomes: ϕ g [ C ] = θ − Im Z s s ds z ( s ) † ˙ z ( s ) − Z s s ds z ( a ) † h ( s ) z ( s ) . (5.9)Comparing the first two terms with Eq. (2.28) we see that they reproduce exactly Ω[ C pol ], and the net result is ϕ g [ C ] = 12 Ω[ C pol ] − Z s s ds z ( s ) † h ( s ) z ( s ) . (5.10)1Further simplification of the second term seems not possible on general grounds, unless one has some information onthe way Ψ ( s ) varies with s as ˆ k ( s ) traces the loop C dir .The separation of ϕ g [ C ] into the two terms on the right in Eq. (5.10) corresponds to the use of the { g a (ˆ k ) } as abasis for ( T ˆ k S ) c at each ˆ k . A change from { g a (ˆ k ) } to some other globally smooth basis would alter both terms,while preserving the value of ϕ g [ C ]. This could possibly limit the direct physical meaning we may ascribe to, say, Ω[ C pol ] on the right hand side. VI. CONCLUDING REMARKS
We hope to have shown that in all geometric phase considerations in the domain of classical optics, the mathematicalframework of quantum mechanics is adequate and flexible enough to provide a basis for the entire analysis. This isso in scalar wave, pure polarisation, as well as beam propagation problems. We have attempted to provide a clearphysical picture of the situations being considered, fully tracing the phenomena ultimately to Maxwell’s equations inevery case. The relevance of global topological aspects when discussing propagation direction and polarisation statesimultaneously was pointed out in [27]. In our treatment we have addressed these using elementary group theoreticalarguments relevant to the situation — leading, in our view, to particularly simple and elegant results.The approach of this work now needs to be extended to other situations where, in place of a narrow beam endowedwith polarisation properties, an extended polarised wave field in space is contemplated. This and other similarextensions will be taken up elsewhere.
Appendix : Comparison with the approach in [27]
Throughout this paper we have tried to show that the standard
H − B − R structure of quantum mechanics, with B a U (1) principal fibre bundle over base R , can be used under all circumstances to handle geometric phases in classicaloptical situations. In [27] a somewhat different structure has been used. We describe here briefly the connectionbetween the two approaches.For geometric phases associated with light beams we have used the complex three-dimensional Hilbert space H ≃ C with inner product; the unit sphere B ≃ S of real dimension five; and the ray space R ≃ CP of real dimensionfour. Here B is a U (1) principal fibre bundle over base R . We now define characteristic subsets of these spaces asfollows: ˆ k ∈ S : H ˆ k = { E ∈ H | ˆ k · E = 0 } ≃ C ; B ˆ k = B ∩ H ˆ k = { E ∈ H | E † E = 1 , ˆ k · E = 0 } ; R ˆ k = B ˆ k /U (1) = { ρ ( E ) = EE † ∈ R | E ∈ B ˆ k } ; B ˆ k ≃ S , R ˆ k ≃ S , for each ˆ k ∈ S . (A. 1)In more detail in the case of R ˆ k we have: ρ ∈ R ˆ k ⇔ ρ = 3 × ρ † = ρ = ρ ≥ , Tr ρ = 1 , ρ ˆ k = 0 . (A. 2)For two points ˆ k , ˆ k ′ ∈ S , we find easily:ˆ k ∧ ˆ k ′ = 0 : B ˆ k ∩ B ˆ k ′ = { E = q ˆ k ∧ ˆ k ′ | ˆ k ∧ ˆ k ′ | , | q | = 1 } , (A. 3)consisting of essentially real E corresponding to linear polarisations.In contradistinction, the total and base spaces used in [27] are T , L defined as: T = [ ˆ k ∈ S B ˆ k ; (a) L = [ ˆ k ∈ S R ˆ k · (b) (A. 4)2These too are of real dimensions five and four respectively, and T is a U (1) principal fibre bundle over base L .It is easy to see that the second statement in Eq. (4.22) leads to related Cartesian product structures for T and L : T = S × S , L = S × S . (A. 5)In our treatment, as mentioned above, we use uniformly B (the sphere of normalised vectors in C ) rather than T ,and the associated projective space R ≡ CP rather than L . It is important to recognize that T 6 = B , and L 6 = R .Writing p for general points in T : p ∈ T ⇔ p = (ˆ k , E ) , ˆ k ∈ S , E ∈ B ˆ k . (A. 6)Since B ˆ k ⊂ B , the map T → B is well-defined: p = (ˆ k , E ) ∈ T → E ∈ B . (A. 7)However this is a many-to-one map . Given E ∈ B , p = (ˆ k , E ) is not unique as:if E ∧ E ∗ = 0 : ˆ k is fixed upto a sign , resulting in a two-fold ambiguity;if E ∧ E ∗ = 0 : ˆ k is fixed upto an SO (2) rotation , more precisely an O (2) rotation , resulting in a continuous ambiguityinvolving linear polarisation states. (A. 8)Thus, while both T and B are real five-dimensional manifolds, we do not have a one-to-one map between them, sothey are not identical spaces. In a similar way, it can be checked that L and R are nonidentical.In [27], geometric phases are defined for smooth closed curves C ⊂ T , with images C ⊂ L . Such a curve C inparametrised form may be written as C = n p ( s ) = (cid:16) ˆ k ( s ) , E ( s ) (cid:17) ∈ T | s ≤ s ≤ s o ⊂ T , (A. 9)with suitable end point conditions. In our approach, since as seen in Eq. (A. 7) the map T → B is well-defined, wecan pass from C ⊂ T to C ⊂ B in an unambiguous manner : C = n E ( s ) ∈ B ˆ k ( s ) | s ≤ s ≤ s o ⊂ B , (A. 10)and then use Eq. (1.7) to define the geometric phase in the kinematic approach. This is similar to the way in whichin Sections 2 and 3 we take the electric field vector along a ray or a beam and use it to obtain a smooth curve in B for which a geometric phase can be defined using the kinematic approach. The expression for the phase given in [27]is the same as in our treatment, which stays entirely within the standard H − B − R structure of quantum mechanics. [1] M.V. Berry, “Quantal phase factors accompanying adiabatic changes”, Proc. R. Soc. A , 45 – 57 (1984).[2] B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase”, Phys. Rev. Lett. , 2167 – 2170 (1983).[3] Y. Aharonov, J. Anandan, “Phase change during a cyclic quantum evolution”, Phys. Rev. Lett. , 1593 – 1596 (1987).[4] J. Samuel and R. Bhandari. “General setting for Berry’s phase”, Phys. Rev. Lett. , 2339 – 2342 (1988).[5] N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism”, Ann. Phys.(NY) , 205 – 268 (1993).[6] N. Mukunda and R. Simon, Quantum kinematic approach to the geometric phase. II. The case of unitary group represen-tations Ann. Phys.(NY) , 269 – 340 (1993).[7] V. Bargmann, “Note on Wigner’s theorem on symmetry operations”, Jour. Math. Phys. , 862 – 868 (1964).[8] L.G. Gouy, “Sur une propri´et´e nouvelle des ondes lumineuses”, C.R. Acad. Sci. Paris , 1251 – 1253 (1890).[9] L.G. Gouy, “Sur la propagation anomale des ondes”, Ann. Chim. Phys. Ser. 6 , 145 (1891).[10] R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect”, Phys. Rev. Lett. , 880 – 883(1993).[11] S.M. Rytov, “Transition from wave to geometrical optics”, Dokl. Akad. Nauk. USSR , 238 – 242 (1938). [12] V.V. Vladimirskii, “The rotation of polarization plane for curved light ray”, Dokl. Akad. Nauk. USSR , 222 – 227 (1941).[13] S. Pancharatnam, “Generalized theory of interference, and its applications”, Proc. Ind. Acad. Sci. A , 247 – 262 (1956).[14] Inner products in Hilbert spaces will be generally written as ( ψ, φ ) rather than as h ψ | φ i in Dirac notation. For two-component complex column vectors z , z ′ we sometimes write z ′ † z in place of ( z ′ , z ). For three-component complexcolumn vectors, described by Cartesian components in physical space for instance, we write ( ψ , φ ) or ψ † φ or ψ ∗ · φ asconvenient.[15] E.M. Rabei, N. Mukunda, and R. Simon, “Bargmann invariants and geometric phases: A generalized connection”, Phys.Rev. A , 3397 – 3409 (1999).[16] N. Mukunda, S. Chaturvedi, and R. Simon, “Bargmann invariants and off-diagonal geometric phases for multilevel quantumsystems: A unitary-group approach”, Phys. Rev. A , 012102 (2001).[17] N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, anda theory of the geometric phase”, Phys. Rev. A , 042114 (2003).[18] S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase burves andmanifolds in geometric phase theory”, Jour. Math. Phys. , 062106 (2013).[19] M. Santarsiero, J.C.G. de Sande, G. Piquero, and F. Gori, “Coherence–polarization properties of fields radiated fromtransversely periodic electromagnetic sources”, J. Opt. , 055701 (2013).[20] B.N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization ofpre-Mueller and Mueller matrices in polarization optics”, Jour. Opt. Soc. Am. A , 188–199 (2010).[21] B.N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R Simon, “Nonquantum entanglement resolvesa basic issue in polarization optics”, Phys. Rev. Lett. , 023901 (2010).[22] N. Mukunda, R. Simon, and E.C.G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vectortheory”, Phys. Rev. A , 2933–2942 (1983).[23] N. Mukunda, R. Simon, and E.C.G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications”, Jour.Opt. Soc. Am. A , 416–426 (1985).[24] R. Simon, E.C.G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams”, Jour. Opt. Soc. Am. A , 536–540 (1986).[25] R. Simon, E.C.G. Sudarshan,and N. Mukunda, “Cross polarization in laser beams”, Appl. Opt. , 1589–1593 (1987).[26] H. Bacry, “Group theory and paraxial optics” in Group Theoretical methods in Physics , Ed. W.W. Zachary (World Sientific,Singapore, 1985), pp. 215–224.[27] R. Nityananda and S. Sridhar, “Light beams with general direction and polarization: global description and geometricphase”, arXiv :1212.0943[28] R. Bhandari, “Geometric phases in an arbitrary evolution of a light beam”, Phys. Lett. A , 24-244 (1989).[29] J. H. Hannay, “The Majorana representation of polarization, and the Bery phase of light”, Jour. Mod. Opt. , 1001-1008(1998).[30] A. V. Tavrov, Y. Miyamoto, T. Kwabata, and M. Takeda, “Generalized algorithm for the unified analysis and simultaneousevaluation of geometrical spinredirection phase and Pancharatnam phase in complex interferometric system”, Jour. Opt.Soc. Am. A , 154-161 (2000).[31] R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters”, Jour. Opt.Soc. Am. A , 2440-2463 (2000).[32] H. Kogelnik, “Imaging of optical modes-resonators with internal lenses”, Bell Syst. Tech. Jour. , 455-494 (1965).[33] R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gener- alized rays in first-order optics: transformation proper- ties ofGaussian Schell-model fields”, Phys. Rev. A , 3273-3279 (1984).[34] R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law”, Opt. Commun. , 322-328 (1988).[35] See, for instance, E.M. Rabei et al. in Ref. [15].[36] The definition of this group is given later, in Eq. (4.9).[37] The group SO (3) is defined in the axis-angle description later in Eq. (4.2).[38] The relationship between SU (2) and SO (3) is described, using axis-angle variables, later in Eq. (4.11) below.[39] It may be useful to recall that for a spherical triangle on S , the corresponding solid angle (subtended at the centre ofthe sphere) is the ‘spherical excess’, i.e., the amount by which the sum of the three internal angles exceeds π . This excessoccurs because S possesses positive curvature.[40] See, for instance, M.Born and E. Wolf, Principles of Optics , 6 th edition, Chapter III (Pergamon Press, NY, 1987).[41] See, for instance, H. Stephani, General Relativity — An introduction to the Theory of Gravitational Field (CambridgeUniversity Press, Cambridge, UK, 1985), p. 45.[42] M. Eisenberg and R. Guy, “A proof of the hairy ball theorem”, Am. Math. Mon.86