January 15, 2008 / Vol. 33, No. 2 / OPTICS LETTERS 195

On the vectorial fields with position-independentstochastic behavior

Rosario Martínez-Herrero and Pedro M. Mejías*Departamento de Óptica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

*Corresponding author: pmmejias@fis.ucm.es

Received September 7, 2007; revised December 18, 2007; accepted December 19, 2007;posted December 21, 2007 (Doc. ID 87323); published January 14, 2008

Vectorial fields with position-independent stochastic behavior within a certain region are analyzed. Morespecifically, we deal with the transverse components of this class of beamlike fields (the longitudinal com-ponent assumed to be negligible). The general form of the cross-spectral density tensor (CDT) of these fieldsis shown. Attention is also focused on the properties of these kinds of fields. Thus, among other character-istics, it is seen that the CDT of these fields can be written as the sum of two CDTs associated, respectively,to a totally polarized field and to an unpolarized field. It is also shown that, for such fields, a Young’s inter-ference experiment can always be performed whose fringe visibility is optimized. This behavior has analyti-cally been characterized by means of a certain parameter, valid for general beamlike fields. It is shown that,for the fields studied, this parameter reaches its maximum value. © 2008 Optical Society of America

OCIS codes: 030.1640, 260.5430, 260.2110.

In the past few years, considerable effort has beendevoted to investigating the concept of coherence ofrandom electromagnetic fields [1–14]. As is wellknown, in the scalar case complete coherence of alight field at a certain region � means

(i) maximum visibility of the interference fringesgenerated in a Young’s interferometric arrangement;

(ii) factorization of the cross-spectral density func-tion W�r1 ,r2� [15,16],

W�r1,r2� = �V*�r1�V�r2�� = F*�r1�F�r2�, �1�

where V represents a stochastic process associatedwith the scalar field, ri, i=1,2, are position vectors attwo points of a plane transverse to the propagationaxis z, the asterisk denotes complex conjugation, theangle brackets symbolize an ensemble average, andF�r� is a deterministic function (for brevity, the ex-plicit dependence on the light frequency has beenomitted);

(iii) the amplitude V�r� of the field can be writtenwithin � in the form (in the mean square sense),

V�r� = �g�r�, �2�

where � is a position-independent random variable,which takes values over the stochastic ensemble ofrealizations of the field, and g�r� is a deterministicposition-dependent function.

In summary, in the scalar case, properties (i)–(iii)can be considered as equivalent features. This behav-ior clearly differs with regard to the vectorial regime:in such a case, the light field should be described by arandom vector E. For simplicity, from now on we willconsider a beamlike field essentially propagatingalong the z axis (the longitudinal field component, Ez,is assumed to be negligible), whose transverse com-ponents are Es�r� and Ep�r�. In the rest of the Letter,the fields will be characterized by their transversecomponents. Accordingly, instead of function W, thecross-spectral density tensors (CDTs) Wij, i, j=1,2,

should now be used, namely, [15,16],

0146-9592/08/020195-3/$15.00 ©

Wij � W�r1,r2� = �E+�ri�E�rj��, i,j = 1,2, �3�

where E+ denotes the adjoint (transposed conjugate)of the row vector E= �Es ,Ep�. It has been shown [4]that, for vectorial fields, the factorization of Wij [simi-lar to property (ii) for scalar fields] is equivalent tothe equality �STF

2 =tr�W12W21� / tr W11tr W22=1, where�STF

2 is the parameter introduced not long ago by Set-älä et al. [4] (tr denotes the trace). On the other hand,it has also been shown [1] that the fulfillment of theequation ��W�2= �tr W12�2 / tr W11tr W22=1, where ��W�2represents the degree of coherence proposed by Wolf[1], implies maximum visibility in a Young’s interfer-ence experiment [the vectorial analogy to property (i)for scalar fields].

Let us finally consider property (iii) introduced forcoherent scalar fields: position-independent stochas-tic behavior throughout the region �. The aim of thisLetter is to investigate the generalization of thisproperty on vectorial fields and the consequences in-volved.

Let us then define a kind of beamlike field whoseassociated stochastic process takes the form

E�r� = E0f�r�U�r�, �4�

where E0= �� ,�� is a 1�2 vector whose componentstake random values over the stochastic ensemble,f�r� is a deterministic complex function, and U�r� de-notes a 2�2 deterministic unitary matrix. It shouldbe noted that the randomness of the field E repre-sented by Eq. (4) is contained only in E0, which doesnot depend on the location. If we now write M= fU,since U is unitary we have

M�r�M+�r� = M+�r�M�r� = �f�r��2I, �5�

where I represents the 2�2 identity matrix. FromEq. (4), it then follows that the CDT for this class of

fields reads

2008 Optical Society of America

196 OPTICS LETTERS / Vol. 33, No. 2 / January 15, 2008

W�r1,r2� = M+�r1��M�r2�, �6�

where

� = �E0+E0� = �����2� ��*��

���*� ����2�� .

The above equation analytically establishes thecoherence-polarization structure of the fields intro-duced by Eq. (4). The fields described by Eqs. (4) and(6) can be understood in the following way. Let usconsider a beam whose electric field vector at a cer-tain plane is given by E�r�=E0 g�r�, where, as before,E0= �� ,��, and g�r� denotes a deterministic function[for example, g�r� could represent a Gaussian ampli-tude]. Note that the degree of polarization for thisfield is uniform across the plane. Such a field is thenpropagated through a position-dependent determin-istic anisotropic (reversible) optical element (repre-sented by a unitary matrix U) that alters the polar-ization state locally and an amplitude filtercharacterized by a function t�r�. The final outputbeam would belong to the class of fields we are con-sidering in this Letter, with f�r�= t�r� g�r�. Anothersimple example of these types of beams is the fieldswhose CDT is factorizable [i.e., W�r1 ,r2�=F+�r1�F�r2�, F being a row vector]. We will devotethe remainder of this Letter to showing the proper-ties exhibited by the fields defined by Eqs. (4) and (6).

Property (i): The character of the field, expressedby Eq. (4), is not altered after the application of anylocal unitary transformation (i.e., any deterministicnonsingular Jones matrix, which would represent theaction of a reversible polarization device). As a conse-quence of this property, the field can yet be expressedin the form given by Eq. (4) after any local rotation ofthe coordinate axes (as occurs in a change to curvilin-ear coordinates).

Note that a field represented (in the mean squaresense) by the stochastic process,

E�r� = E0H�r�, �7�

where H is a 2�2 deterministic (but not unitary) ma-trix, can be written in the same form after a unitary(reversible) optical transformation. However, thisbeam does not belong to the types of fields given byEq. (4).

Property (ii): It can be shown that the so-called in-trinsic degrees of coherence [5,8,10,11], �S, �I, of thisclass of fields are equal to 1. The converse is not true:it suffices to note that the intrinsic degrees of coher-ence for any field given by Eq. (7) are also equal to 1.

Property (iii): The (local) degree of polarization[17,18], P�r�, for these kinds of fields reads

P2�r� = 1 −4 Det �

�tr ��2, �8�

where Det stands for determinant. Since the ele-ments of � are position independent, we conclude

from Eq. (8) that, for the fields studied in this Letter,

P is constant throughout the region in which Eq. (6)applies. The converse property is not true: thefields whose CDT is of the form W�r1 ,r2�= f�r1 ,r2�U+�r1��U �r2�, where f�r1 ,r2� is a positivedefinite and not factorizable function, would fulfillEq. (8) but, in general, they do not belong to the classof fields described by Eq. (4).

Property (iv): For the fields we are considering�STF

2 = �1+P2� /2. This follows from direct substitutionof Eq. (6) into the definition of �STF

2 and by applyingEqs. (5) and (8) along with the definition of matrix �.Recall that any field with �STF

2 =1 fulfills Eq. (6). Inother words, any factorizable CDT can be written inthe form given by Eq. (6), with Det �=0.

Property (v): It is well known [15,16] that, for ageneral field, W�r ,r� can be expanded in the form W�r ,r�=WTP�r ,r�+WNP�r ,r�, where WTP�r ,r� andWNP�r ,r� refer to a totally polarized field and to anunpolarized field, respectively. This important prop-erty can be generalized for the CDT W�r1 ,r2� of thefields defined by Eqs. (4) and (6). Note first that, since� is a Hermitian positive definite matrix, we canwrite [15,16]

� = ��11 �12

�12* �22

� = �TP + �NP, �9�

where

�TP = �a11 a12

a12* a22

�and �NP=qI, with Det �TP=0 and

a11 =1

2��11 − �22� +

1

2�tr ��2 − 4 Det �, �10a�

a12 = �12, �10b�

a22 =1

2��22 − �11� +

1

2�tr ��2 − 4 Det �, �10c�

q =1

2��11 + �22� −

1

2�tr ��2 − 4 Det �. �10d�

The validity of Eq. (9) can be tested by substituting ofEqs. (10) into the definitions of matrices �TP and �NP.Taking Eqs. (10) into account, the CDT of the fieldscharacterized by Eq. (6) then becomes

W�r1,r2� = M+�r1��TPM�r2� + M+�r1��NPM�r2�.

�11�

The first term of the right-hand side of Eq. (11) rep-resents the CDT of a totally polarized field, whereasthe second term corresponds to an unpolarized field.Note that Eq. (11) for W�r1 ,r2� is no longer valid for a

general field.

January 15, 2008 / Vol. 33, No. 2 / OPTICS LETTERS 197

It should also be noted that the converse propertyis not true. In other words, the possibility of express-ing the CDT of a field as the sum of two terms (cor-responding to the incoherent superposition of a to-tally polarized and an unpolarized source) does notimply that the beam belongs to the types of fieldsgiven by Eq. (4); for example, in those cases in whichthe matrix M appearing in Eq. (11) cannot be writtenas a product fU, where U is again a unitary matrix.

Property (vi): For the fields we are considering, letus assume we perform the local unitary transforma-tions U+�r1� and U+�r2� at two points, r1 and r2, re-spectively, where U+ denotes the adjoint of the matrixU defining the field [cf. Eq. (4)]. These transforma-tions can be implemented, for instance, by means ofanisotropic phase plates. Since U+U= I, we get thatthe field, after the transformations, becomes E��ri�= f�ri� E0, i=1,2. Consequently, the resulting field,EY�R� in a Young’s interference experiment (the pin-holes placed at r1 and r2), would read EY�R�=E��r1�+E��r2�exp�i��, where R represents the superpositionpoint and � denotes the path difference between theinterfering waves. The fringe visibility V around Rbecomes

V =Imax − Imin

Imax + Imin=

2�f�r1���f�r2��

�f�r1�2� + �f�r2��2,

where Imax and Imin denote the maximum and mini-mum values of the irradiance around R. We thus seethat, in a Young’s interferometric arrangement, wewould get the same visibility as that obtained for sca-lar coherent fields with irradiances �f�r1��2 and�f�r2��2. In particular, the fringe visibility equals 1when �f�r1��= �f�r2��.

To go further into this analysis for the fields stud-ied in this Letter, let us now introduce the followingparameter, g12, defined for general beamlike fields attwo points, r1 and r2, in the form

g12 =tr�W12W21� + 2�Det W12�

tr W11 tr W22

. �12�

Based on the results of recent papers [9,12], it is easyto show that g12 exhibits several general properties,namely, (i) g12 remains invariant under local unitarytransformations; (ii) 0�g12�1; (iii) �STF

2 =1 impliesg12=1 (in general, the converse is not true); (iv) g12 isa measurable parameter (following, for example, theprocedure outlined in [2]).

In addition, for a general beamlike beam, g12 pro-vides the maximum visibility one can obtain in aYoung’s interference experiment by means of localunitary transformations. In particular,

Property (vii): For the fields considered in this Let-ter [defined by Eq. (4)], the parameter g12 reaches itsmaximum value (equal to 1).

The interest of parameter g12 arises because it re-veals an intrinsic (but not hidden) meaningful char-

acteristic of any (general) field, namely, the intimatecapability of the field to improve the fringe visibilityin a well-designed Young’s interference arrangement.Consequently, property (vii) indicates the optimiza-tion of this feature for the fields considered in thisLetter.

Let us finally summarize the main results of thisLetter. A class of vectorial fields has been introduced,which exhibit a number of peculiar coherence and po-larization properties. These beams can be understoodas a natural extension to electromagnetic beamlikefields of a certain behavior associated with completecoherence in the scalar case. The characteristics ofthe present class of fields involve optimum values ofthe intrinsic degrees of coherence and of a parameterg12 that provides the maximum attainable visibility(by means of unitary optical transformations) in aYoung’s interference experiment. Since, in addition,for these fields, �STF

2 can be expressed in terms of thedegree of polarization in a direct and simple way, theresults shown here offer, in our opinion, new perspec-tives to physically interpret and connect, in someway, the parameters ��W�2, �STF

2 , �S, and �I.

We thank an anonymous referee for his/her valu-able suggestions. This work has been supported bythe Ministerio de Educación y Ciencia of Spain,project FIS2004-1900, and by CM-UCM, ResearchGroup Program, 910335 (2007).

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