Optics Communications 246 (2005) 437–443

www.elsevier.com/locate/optcom

Scalar Wigner function for vectorial fields andspatial-angular Stokes parameters

Alfredo Luis *

Departamento de Optica, Facultad de Ciencias Fısicas, Universidad Complutense, 28040 Madrid, Spain

Received 19 September 2004; accepted 4 November 2004

Abstract

We introduce a scalar Wigner function for classical vectorial light fields embodying full information about polari-

zation. We introduce global and local Stokes parameters with simple and natural transformation laws under ABCD

transformations and polarization changing devices. We apply this formalism to the free propagation of vectorial Gauss-

ian–Schell waves.

� 2004 Elsevier B.V. All rights reserved.

PACS: 42.25.Ja; 42.60.Jf; 42.50.�pKeywords: Polarization; Wigner function; Stokes parameters

1. Introduction

The Wigner function occupies a distinguished

position in many areas of physics, specially in clas-sical and quantum optics. [1–46]. Since the Wigner

formalism accommodates very different situations

under a common language, it favors a fruitful ex-

change of concepts and results between different

areas of physics. In this work, we take advantage

of this to introduce a scalar Wigner function for

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.11.014

* Tel.: +34913945011; fax: +34913944683.

E-mail address: alluis@fis.ucm.es.

classical vectorial light fields inspired by parallel

results in the area of quantum physics.

Most of the research in theWigner formalismhas

been devoted to scalar waves (optical or mechani-cal), while the vectorial case has received a lesser

attention [47–50]. Given the relevance of polariza-

tion in optics it is worth investigating the proper

inclusion of polarization in the Wigner formalism.

Previous approaches have addressed this issue in

classical optics by defining a 2 · 2 complex matrix,

referred to as Wigner matrix [44–46]. In this work,

we propose a simpler approach closer to the spiritof the Wigner formalism by introducing a real, sca-

lar Wigner function that includes full information

ed.

438 A. Luis / Optics Communications 246 (2005) 437–443

about the polarization of vectorial classical light

waves.

This definition is motivated by parallel results

obtained in the quantum domain. Suitable quan-

tum analogs of polarization are the angularmomentum or spin variables. Currently, there are

satisfactory definitions of scalar Wigner functions

for this vectorial variable [23–43]. For definiteness,

we translate to the classical optical domain the

quantum formalism introduced in [38–43], where

the spin states are represented by real functions

on the sphere, that we shall identify as the Poin-

care sphere. Throughout we will focus on mono-chromatic waves of frequency x.

Our first objective is to generalize the scalar

Wigner function for scalar fields W(r, p) by intro-

ducing a scalar function for vectorial fields

W(r, p, X), where X = (h, /) is the usual parametri-

zation of the Poincare sphere in terms of the polar

angle h and the azimuthal angle /. In order to sim-

plify formulae and to gain insight, the points of thePoincare sphere will be also represented by the fol-

lowing three- and four-dimensional real vectors,

X ¼sin h cos/

sin h sin/

cos h

0B@

1CA; ~X ¼

1ffiffiffi3

psin h cos/ffiffiffi

3p

sin h sin/ffiffiffi3

pcos h

0BBB@

1CCCA:

ð1ÞThe real two-dimensional vector r represents spa-tial position in a plane perpendicular to the axis z,

while the real two-dimensional vector p represents

the usual angular parametrization of the direction

of propagation, being the optical counterpart of

the quantum momentum operator. The real four-dimensional vector a will denote the pair (r, p).For simplicity throughout we assume that the com-

ponent of the electric field along axis z can be

neglected (transversality approximation [51]).

We will show thatW(a, X) has good transforma-

tion properties. As a further consequence of this ap-

proach we find that W(a, X) defines local Stokes

parameters depending both on r and p, representingthe polarization state of light rays. We will refer to

them as spatial-angular local Stokes parameters.

Their angular average gives the more familiar spa-

tial local Stokes parameters. Moreover, we will de-

fine global Stokes parameters as the result of spatial

and angular averages. We will show that they re-

main invariant under ABCD transformations.

2. Quantum Wigner function with spin

In quantum mechanics the scalar or spinless

Wigner function can be defined as [1–13],

W ðaÞ ¼ tr½qDðaÞ�

/Z

d2r0hr� r0=2jqjrþ r0=2ieir0 �p=�h; ð2Þ

where q is the densitymatrix, |ræ are the eigenvectorsof the position operator, andD(a) are the operators,

DðaÞ /Z

d2r0jrþ r0=2ihr� r0=2jeir0 �p=�h: ð3Þ

The inclusion of the spin can be carried out natu-

rally by replacing D(a) by the product D(a, X) =D(a) � D(X), where D(X) is the counterpart ofD(a) for spin variables [47–50]. As we have men-

tioned above, currently there are different suitable

solutions for D(X) fulfilling the desirable propertiesassociated to the Wigner formalism. Maybe, the

best suited for our purposes is the SU(2) Wigner

function for an arbitrary angular momentum j

introduced in [38–43],

DðXÞ ¼Xj

m;m0¼�j

Zm;m0 ðXÞjj;mihj;m0j; ð4Þ

where |j, mæ are the eigenvectors of the component

jz of an angular momentum j and

Zm;m0 ðXÞ ¼ffiffiffiffiffiffi4p

p

2jþ 1

X2j‘¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffi2‘þ 1

phj;m; ‘;m0 � mjj;m0i

� Y ‘;m0�mðXÞ; ð5Þ

being Æj1, m1;j2, m2|j, mæ the Clebsch–Gordan co-

efficients and Y‘,m(X) the spherical harmonics.The complete quantum Wigner function including

spin is [47–50]

W ða;XÞ /Z

d2r0Xj

m;m0¼�j

Zm;m0 ðXÞhj;m0j

� hr� r0=2jqjrþ r0=2ijj;mieir0 �p=�h: ð6Þ

A. Luis / Optics Communications 246 (2005) 437–443 439

3. Scalar Wigner function for classical light waves

including polarization

In classical optics, the Wigner function of an

scalar light wave is given by equation (2) afterreplacing the matrix elements of q by the correla-

tion function C(r1, r2) in the form [21,22],

hr1jqjr2i ! Cðr1; r2Þ ¼ Eðr1ÞE�ðr2Þh i; ð7Þwhere E denotes the scalar complex electric field

and the brackets represent ensemble average. The

reduced Planck constant should also be replacedby the inverse of the wavenumber 1/�h ! k.

We define in classical optics an scalar Wigner

function that includes polarization as the optical

counterpart of the quantum Wigner function with

spin W(a, X) for the case j = 1/2. In this analogy,

the two basis vectors |j = 1/2, m = ±1/2æ representtwo orthogonal polarization states (circular dextro

and levo for example). The Wigner function is de-fined by replacing the matrix elements in (6) by

correlation functions in the form,

h1=2;m0jhr1jqjr2ij1=2;mi ! Cm0 ;mðr1; r2Þ¼ hEm0 ðr1ÞE�

mðr2Þi; ð8Þ

where Em are the components of the complex elec-

tric field vector in the corresponding polarizationbasis.

Therefore, our proposal of Wigner function is

W ða;XÞ ¼X1=2

m;m0¼�1=2

Zm;m0 ðXÞZ

d2r0Cm0 ;mðr� r0=2; rþ r0=2Þeikr0 �p; ð9Þ

being [38–43]

Z�1=2;�1=2ðXÞ ¼1

2ð1�

ffiffiffi3

pcos hÞ;

Z�1=2;1=2ðXÞ ¼ Z�1=2;�1=2ðXÞ ¼

ffiffiffi3

p

2sin hei/:

ð10Þ

We can mention that there are other different rep-

resentations of spin that may be used also to de-scribe polarization and even to define a suitable

measure of the degree of polarization [23–37,52–

55].

4. Spatial-angular Stokes parameters

The Wigner function W(a, X) admits a very

simple expression of the form,

W ða;XÞ ¼ 1

2~SðaÞ � ~X

¼ 1

2S0ðaÞ þ

ffiffiffi3

pSðaÞ �X

h i; ð11Þ

where the four-dimensional real vector~SðaÞ ¼ ðS0ðaÞ;SðaÞÞ is defined as

S0ðaÞ ¼ W þ;þðaÞ þ W �;�ðaÞ;SxðaÞ ¼ W þ;�ðaÞ þ W �;þðaÞ;SyðaÞ ¼ i W þ;�ðaÞ � W �;þðaÞ½ �;SzðaÞ ¼ W þ;þðaÞ � W �;�ðaÞ;

ð12Þ

being

W m0 ;mðaÞ ¼Z

d2r0Cm0 ;mðr� r0=2; rþ r0=2Þeikr0 �p;

ð13Þthe elements of the Wigner matrix [44–46], and ±

represent the m = ±1/2 components.

These expressions can be rearranged in the form

~SðaÞ ¼ tr½WðaÞ~r�; ð14Þwhere

WðaÞ ¼W þ;þðaÞ W þ;�ðaÞW �;þðaÞ W �;�ðaÞ

� �; ð15Þ

and ~r ¼ ðr0; rÞ, being r0 the identity, and r the

Pauli matrices. It also holds that

~SðaÞ ¼ 1

2p

ZdX ~XW ða;XÞ: ð16Þ

The expression (11) is the standard form for the

quantum Wigner function of a spin-1/2 particle

with normalized Stokes parameter (or Bloch vec-

tor) S(a)/S0(a). Therefore, it is natural to identify~SðaÞ as Stokes parameters. Since they depend bothon r and p we can refer to them as spatial-angular

local Stokes parameters. Roughly speaking they

represent the polarization state of light ray at point

r propagating in the direction p.From ~SðaÞ we can derive some other local

and global Stokes parameters by averaging over

the corresponding variables. For example the

440 A. Luis / Optics Communications 246 (2005) 437–443

more familiar spatial local Stokes parameters~sðrÞ are

~sðrÞ ¼Z

d2p~SðaÞ: ð17Þ

On the other hand we may define also angular lo-cal Stokes parameters,

~sðpÞ ¼Z

d2r~SðaÞ: ð18Þ

Moreover, we can define global Stokes parameters

in the form,

~s ¼Z

d4a~SðaÞ: ð19Þ

The above expressions for W(a, X) indicate that

the same procedures measuring the Wigner func-

tion for scalar fields W(a) [56–58] would serve also

to determine W(a, X) in practice. This can be

achieved by replacing the single intensity measure-

ments required for the scalar case by the measure-

ment of the four Stokes parameters. This leads tothe determination of the spatial-angular local

Stokes parameters ~SðaÞ, which in turn determine

W(a, X) via Eq. (11).

5. Transformation properties

One of the most interesting properties ofWigner functions is their good behavior under

transformations. In our context it is known that

the Wigner function for scalar fields transforms

very simply under ABCD transformations [22]:

Wout (a) = Win (T�1a) (transport equation),

where

T ¼A B

C D

� �; ð20Þ

being A, B, C, D two-dimensional real matrices

relating the input and output spatial-angular vari-

ables r, p. Wout and Win are the Wigner functions

for the output and input waves, respectively.As far as we consider ABCD transformations

that do not depend on the polarization, the scalar

transport equation remains valid for the vectorial

case since it is valid for all the components of the

Wigner matrix [46],

W outða;XÞ ¼ W inðT�1a;XÞ: ð21ÞEssentially this is due to the fact that we can re-

gard the above transport equation as a property

of the operators D(a), which is naturally preserved

for the tensor product D(a) � D(X).As a direct consequence of this transformation

law we have the following transport equation for

the spatial-angular local Stokes parameters

derived from (16),

~SoutðaÞ ¼ ~SinðT�1aÞ: ð22Þ

Moreover, from (19), (22) and the symplectic

character of T, we get that the global Stokes

parameters are invariant under ABCD trans-formations,

~Sout ¼ ~Sin: ð23Þ

It is worth noting that no simple transformation

law of this kind holds for the spatial Stokes

parameters ~sðrÞ. This point will be illustrated be-

low with a simple example.

Similarly, the definition of the SU(2) Wigner

function embodies good transformation propertiesunder SU(2) transformations [38–43]. In our con-

text, these are produced by linear, energy conserv-

ing input–output transformations of the field

amplitudes, such as the ones produced by beam

splitters and phase plates. Next we derive from

first principles the corresponding transformation

law for W(a, X) including also the effect of

polarizers.For input–output transformations linear on the

field amplitudes and independent of r, p, we have

that the matricesC in (8) andW in (15) transform as

Coutðr1; r2Þ ¼ UCinðr1; r2ÞUy;

WoutðaÞ ¼ UW inðaÞUy;ð24Þ

where U is a constant 2 · 2 matrix relating input

and output amplitudes. Then, we get from (14) that,

~SoutðaÞ ¼ tr WoutðaÞ~r½ � ¼ tr W inðaÞUy~rU� �

¼ M ~SinðaÞ; ð25Þ

where M is the corresponding Mueller matrix

defined by the relations,

A. Luis / Optics Communications 246 (2005) 437–443 441

Uy~riU ¼X

j¼0;x;y;z

Mi;j~rj: ð26Þ

Finally, using the parametrization ~X we get from

(11) that

W outða; ~XÞ ¼ 1

2~SoutðaÞ � ~X ¼ W inða;M t ~XÞ; ð27Þ

where Mt is the transpose of M. This is the equiv-

alent for polarization of the ABCD transforma-

tion law (21). This equivalence is more exactwhen dealing with energy conserving transforma-

tions (such as the ones produced by phase plates)

for which Mt = M�1.

6. Free propagation of vectorial Gaussian–Schell

waves

We illustrate the above formalism by applying it

to the free propagation of vectorial Gaussian–

Schell waves for which the correlation matrix can

be expressed in the form [59]

Cm;m0 ðr1; r2Þ ¼ Im;m0e�lm;m0 ðr21þr22Þ=2e�mm;m0 ðr1�r2Þ2=4; ð28Þ

where l, m are real symmetric matrices and

Im;m0 ¼ I�m0 ;m. In this case the elements of the

Wigner matrix are

W m;m0 ðaÞ ¼ 4pcm;m0 Im;m0e�lm;m0 r2

e�k2cm;m0 p2

: ð29Þ

where

cm;m0 ¼1

lm;m0 þ mm;m0: ð30Þ

The spatial-angular local Stokes parameters are

S0ðaÞ ¼ 4p cþIþe�lþr

2

e�k2cþp2 þ c�I�e

�l�r2

e�k2c�p2

� �;

SxðaÞ þ iSyðaÞ ¼ 8pc0I0e�l0r

2

e�k2c0p2

;

SzðaÞ ¼ 4p cþIþe�lþr

2

e�k2cþp2 � c�I�e

�l�r2

e�k2c�p2

� �;

ð31Þwhere throughout the subscript + represents the

pair (+, +), the subscript � represents the pair

(�, �), while the subscript 0 represents the pair(�, +) and, as before, ± represent the compo-

nents m = ±1/2. The spatial local Stokes parame-

ters are

s0ðrÞ ¼4p2

k2Iþe�lþr

2 þ I�e�l�r2

� �;

sxðrÞ þ isyðrÞ ¼8p2

k2I0e�l0r

2

;

szðrÞ ¼4p2

k2Iþe�lþr

2 � I�e�l�r2

� �:

ð32Þ

The global Stokes parameters (19) are

S0 ¼4p3

k2Iþlþ

þ I�l�

� �;

Sx þ iSy ¼8p3

k2I0l0

;

Sz ¼4p3

k2Iþlþ

� I�l�

� �:

ð33Þ

The free propagation through a distance z in vac-

uum can be represented by an ABCD matrix with

A = D = I, C = 0, and B = zI, where I is the 2 · 2

identity matrix. After (22) the propagation of the

spatial-angular local Stokes parameters is given

by the transport equation,

~Soutðr; pÞ ¼ ~Sinðr� zp; pÞ: ð34ÞOn the other hand, the propagation of the spatial

local Stokes parameters ~sðrÞ can be expressed as in

(32) after the replacements:

lj ! ljðzÞ ¼ljk

2cjljz2 þ k2cj

;

Ij ! I jðzÞ ¼k2cjI j

ljz2 þ k2cj;

ð35Þ

for j = 0, ±.

Finally, after (23) the global Stokes parameters

do not depend on z.

Most of the works analyzing the propagation of

polarization focus on the spatial local Stokesparameters ~sðrÞ [59–61]. However, it can be appre-

ciated in (32) that they satisfy no simple transfor-

mation law. As we have demonstrated above,

this is compatible with very simple and natural

propagation laws for the spatial-angular local

and the global Stokes parameters. It can be seen

in (34) that the free propagation introduces the

angular variability of the polarization state intothe spatial dependence. This produces the varia-

tion of the spatial polarization even if the

442 A. Luis / Optics Communications 246 (2005) 437–443

spatial-angular polarization state is invariant

along every ray.

As some simple examples we have that com-

pletely unpolarized light at z = 0 (i.e., s(r) = 0 for

all r) may acquire partial polarization during prop-agation [60]. This can be explained with the help of

the spatial-angular local Stokes operators since, in

general, we will have S(a) 6¼ 0, and this partial

polarization hidden in the spatial-angular domain

at z = 0 can manifest itself in the spatial domain at

z 6¼ 0 via the transport Eq. (34). Also, uniformly

polarized light at z = 0 (i.e., s(r)/s0(r) = constant)

becomes nonuniformly polarized at z 6¼ 0 [59]since, in general, S(a)/S0(a) need not be constant.

These examples show that the rather complex

evolution of spatial polarization properties can

be explained in terms of the very simple laws

obeyed by the spatial-angular Stokes parameters

introduced in this work.

7. Conclusions

We have introduced a scalar Wigner function

for classical vectorial light fields embodying com-

plete information about polarization. This infor-

mation is expresses in terms of spatial-angular

local Stokes parameters, that correspond to the

polarization state of light rays. We have shownthat they verify simple and natural transformation

laws, in sharp contrast with the standard spatial

Stokes parameters. If we dismiss the spatial-angu-

lar variability we get global Stokes parameters

which remain invariant under ABCD transforma-

tions. We have applied this approach to the free

propagation of vectorial Gaussian–Schell waves

showing that the complex phenomenology of theevolution of spatial polarization is compatible with

the simple transformation properties satisfied by

the spatial-angular polarization variables.

Acknowledgements

This work has been supported by projectFIS2004-01814 of the Spanish Direccion General

de Investigacion del Ministerio de Educacion y

Ciencia.

References

[1] M. Hillery, R.F. O�Connell, M.O. Scully, E.P. Wigner,

Phys. Rep. 106 (1984) 121.

[2] G.S. Agarwal, E. Wolf, Phys. Lett. A 26 (1968) 485.

[3] G.S. Agarwal, E. Wolf, Phys. Rev. Lett. 21 (1968) 180.

[4] G.S. Agarwal, Phys. Rev. 177 (1969) 400.

[5] K.E. Cahill, R.J. Glauber, Phys. Rev. 177 (1969) 1857.

[6] K.E. Cahill, R.J. Glauber, Phys. Rev. 177 (1969) 1882.

[7] G.S. Agarwal, E. Wolf, Phys. Rev. D 2 (1970) 2161.

[8] G.S. Agarwal, E. Wolf, Phys. Rev. D 2 (1970) 2187.

[9] G.S. Agarwal, E. Wolf, Phys. Rev. D 2 (1970) 2206.

[10] J. Perina, Coherence of Light, D. Reidel, Dordrecht, 1985.

[11] B.-G. Englert, J. Phys. A 22 (1989) 625.

[12] H.-W. Lee, Phys. Rep. 259 (1995) 147.

[13] L. Mandel, E. Wolf, Optical Coherence and Quantum

Optics, Cambridge University Press, Cambridge, 1995.

[14] N. Mukunda, Am. J. Phys. 47 (1979) 182.

[15] J.P. Bizarro, Phys. Rev. A 49 (1994) 3255.

[16] P. Kasperkovitz, M. Peev, Ann. Phys. (N.Y.) 230 (1994)

21.

[17] J. Vaccaro, Phys. Rev. A 52 (1995) 3474.

[18] J. Vaccaro, Opt. Commun. 113 (1995) 421.

[19] A. Luis, L.L. Sanchez-Soto, Phys. Lett. A 261 (1999) 12.

[20] A. Luis, Phys. Rev. A 62 (2000) 054302.

[21] M.J. Bastiaans, Opt. Commun. 25 (1978) 26.

[22] D. Dragoman, Prog. Opt. 37 (1997) 1.

[23] D.M. Kaplan, G.C. Summerfield, Phys. Rev. 187 (1969)

639.

[24] M.O. Scully, Phys. Rev. D 28 (1983) 2477.

[25] G. Ramachandran, A.R. Usha Devi, P. Devi, S. Sirsi,

Found. Phys. 26 (1996) 401.

[26] V.P. Karassiov, A.V. Masalov, J. Opt. B: Quantum S. Opt.

4 (2002) S366.

[27] W.K. Wootters, Ann. Phys. (N.Y.) 176 (1987) 1.

[28] O. Cohendet, Ph. Combe, M. Sirugue, M. Sirugue-Collin,

J. Phys. A 21 (1988) 2875.

[29] D. Galetti, A.F.R. Toledo Piza, Physica A 149 (1988)

267.

[30] J.A. Vaccaro, D.T. Pegg, Phys. Rev. A 41 (1990) 5156.

[31] A. Luks, V. Perinova, Quantum Opt. 6 (1994) 125.

[32] U. Leonhardt, Phys. Rev. Lett. 74 (1995) 4101.

[33] U. Leonhardt, Phys. Rev. A 53 (1996) 2998.

[34] T. Opatrny, D.-G. Welsch, B. Buzek, Phys. Rev. A 53

(1996) 3822.

[35] C. Miquel, J.P. Paz, M. Saraceno, Phys. Rev. A 65 (2002)

062309.

[36] J.P. Paz, Phys. Rev. A 65 (2002) 062311.

[37] A. Luis, J. Perina, J. Phys. A 31 (1998) 1423.

[38] G.S. Agarwal, Phys. Rev. A 24 (1981) 2889.

[39] J.P. Dowling, G.S. Agarwal, W.P. Schleich, Phys. Rev. A

49 (1994) 4101.

[40] J.C. Varilly, J.M. Gracia-Bondıa, Ann. Phys. (N.Y.) 190

(1989) 107.

[41] C. Brif, A. Mann, J. Phys. A 31 (1998) L9.

[42] C. Brif, A. Mann, Phys. Rev. A 59 (1999) 971.

[43] C. Brif, A. Mann, J. Opt. B: Quantum S. Opt. 2 (2000) 245.

A. Luis / Optics Communications 246 (2005) 437–443 443

[44] R. Martınez-Herrero, P.M. Mejıas, J.M. Movilla, Opt.

Lett. 22 (1997) 206.

[45] J.M. Movilla, G. Piquero, R. Martınez-Herrero, P.M.

Mejıas, Opt. Commun. 149 (1998) 230.

[46] P.M. Mejıas, R. Martınez-Herrero, G. Piquero, J.M.

Movilla, Prog. Quantum Electron. 26 (2002) 65.

[47] J.M. Gracia-Bondıa, T.W. Marshall, E. Santos, Phys.

Lett. A 183 (1993) 19.

[48] J.M. Gracia-Bondıa, J.C. Varilly, J. Phys. A 21 (1988)

L879.

[49] A. Czirjak, M.G. Benedict, Quantum S. Opt. 8 (1996) 975.

[50] A. Luis, L.L. Sanchez-Soto, Quantum Opt. 5 (1993) 33.

[51] Three-dimensional fields can be described by replacing

throughout j = 1/2 by j = 1.

[52] F.T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, Phys.

Rev. A 6 (1972) 2211.

[53] A. Luis, Phys. Rev. A 66 (2002) 013806.

[54] A. Luis, Opt. Commun. 216 (2003) 165.

[55] A. Luis, Phys. Rev. A 69 (2004) 023803.

[56] X. Liu, K.-H. Brenner, Opt. Commun. 225 (2003) 19.

[57] M.G. Raymer, M. Beck, D. McAlister, Phys. Rev. Lett 72

(1994) 1137.

[58] D.F. McAlister, M. Beck, L. Clarke, A. Mayer, M.G.

Raymer, Opt. Lett. 20 (1995) 1181.

[59] G.P. Agrawal, E. Wolf, J. Opt. Soc. Am. A 17 (2000) 2019.

[60] D.F.V. James, J. Opt. Soc. Am. A 11 (1994) 1641.

[61] F. Gori, M. Santarsierso, G. Piquero, R. Borghi, A.

Mondello, R. Simon, J. Opt. A: Pure Appl. Opt. 3 (2001) 1.