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1902 J. Opt. Soc. Am. A/Vol. 25, No. 8 /August 2008 R. Martínez-Herrero and P. M. Mejías

Equivalence between optimum Young’s fringevisibility and position-independent

stochastic behavior of electromagnetic fields

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 April 9, 2008; revised June 9, 2008; accepted June 9, 2008;posted June 10, 2008 (Doc. ID 94826); published July 7, 2008

Stochastic electromagnetic fields characterized by optimized fringe visibility in a Young interferometric ar-rangement are shown to be those whose random character is position independent. The optimization procedureinvolves local unitary transformations, which can be implemented by using reversible anisotropic polarizationdevices placed at the two pinholes. It is also shown that the local degree of polarization in the optimized in-terferometer is constant across the superposition region and coincides with the degree of polarization at thetwo pinholes. © 2008 Optical Society of America

OCIS codes: 030.1640, 260.5430.

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. INTRODUCTIONn the past few years, partially coherent and partially po-arized fields have receiving increasing attention in theiterature [1–16]. In particular, considerable effort haseen devoted to investigating the concept of coherence ofandom electromagnetic fields. Recently [16], vectorialelds with position-independent stochastic behaviorithin a certain region were analyzed. These fields ex-ibit peculiar coherence and polarization properties [16].n the present Communication we will try to get deepernsight into the features of these beams. More specifically,e will show that the attainable visibility in a Young in-

erference arrangement can be optimized if, and only if,he beam at the pinholes belongs to this class of fields.

On the other hand, the polarization properties of par-ially coherent electromagnetic fields at the observationlane of a Young interferometer have been studied in re-ent works [17–20]. Thus, it is well known that the polar-zation features of the output field of the Young arrange-

ent are, in general, different from those of the field athe pinholes, even when the fields at the two points ex-ibit identical local polarization behavior. In a recent pa-er [21], conditions were reported ensuring that the po-arization properties at the pinholes are the same as thoset the superposition region (see also [15]). In this connec-ion, it will also be shown in the present work that, for theind of fields we consider here, optimized visibility im-lies that the local degree of polarization is constantcross the observation plane of the interferometer and co-ncides with the degree of polarization at the pinholes.

The paper is then arranged as follows. In the followingection the formalism and key definitions are introduced,nd details are given about the characteristics and theignificance of the type of beams studied in the presentork. In Section 3 an equivalence property is derived be-

ween optimum fringe visibility and position-independenttochastic behavior. Section 4 is devoted to the analysis of

1084-7529/08/081902-4/$15.00 © 2

he local degree of polarization across the superpositionegion. Finally, the main conclusions are summarized inection 5.

. FORMALISM AND KEY DEFINITIONSs is well known, second-order stochastic electromagneticelds are described by the spectral density tensors (CDTs)

ˆij , i , j=1,2 associated with the electric field vector E at

he points r1 and r2[22,23]:

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

here E+ denotes the adjoint (transposed conjugate) ofhe row vector E= �Ex ,Ey�, and the angle brackets symbol-ze an ensemble average (for brevity, the explicit depen-ence on the light frequency has been omitted). Since were interested in electromagnetic beamlike fields propa-ating essentially along the z direction, the longitudinaleld component, Ez, is assumed to be negligible, and theDTs are reduced to 2�2 matrices.In the present Communication, attention is focused on

particular type of fields [16] whose associated stochasticrocess takes the following form at a certain region �:

E�r� = E0f�r�U�r�, �2�

here E0= �� ,�� is a 1�2 vector whose components takeandom values over the stochastic ensemble, f�r� is a de-erministic complex function, and U�r� denotes a 2�2 de-erministic unitary matrix. The fields described by Eq. (2)ould be implemented, for example, by considering aeam whose electric field vector is E�r�=E0f�r�, where0= �� ,�� again denotes a stochastic vector and f�r� rep-

esents, for instance, a Gaussian amplitude. This fieldould propagate through a position-dependent aniso-

ropic optical element [characterized by a unitary matrix

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R. Martínez-Herrero and P. M. Mejías Vol. 25, No. 8 /August 2008 /J. Opt. Soc. Am. A 1903

ˆ �r�]. Note that this element should be reversible [other-ise matrix U�r� would not be unitary, in general]. Thenal output field would then belong to the kind of fields

ntroduced by Eq. (2). Another simple example of thislass of beams is the field whose CDT is factorizable, i.e.,ˆ �r1 ,r2�=F+�r1�F�r2�, with F being a row vector. Let uslso remark that a field represented (in the mean squareense) by the stochastic process E�r�=E0H�r�, where H is2�2 deterministic (but not unitary) matrix, does not

elong to the type of beams described by Eq. (2).Since the randomness of E is contained only in E0,

hich does not depend on the location, these fields have aosition-independent stochastic behavior (including aonstant degree of polarization over �). Moreover, theseelds could be understood as the vectorial counterpart ofhe following scalar fields [16]:

V�r� = �g�r�, �3�

here V denotes the field amplitude, � is a position-ndependent random variable, and g�r� is a deterministicunction. Recall that, for scalar beams, complete coher-nce also involves maximum visibility of the fringes in aoung interferometer, along with factorization of theross-spectral density function [22,23] [in other words,hese two properties together with Eq. (3) can be consid-red as equivalent features in the scalar case]. It shoulde remarked that, in the vectorial case, the factorizationroperty is equivalent to the equality �STF

2 =1, where �STF2

s the parameter introduced not long ago by Setälä et al.4]. Note, however, that in general, �STF

2 differs from 1 forhe beams defined by Eq. (2). Accordingly, these fieldsonstitute a wider class of beams than those fulfilling

STF2 =1.

The beams given by Eq. (2) exhibit a peculiar behavior,hich involves certain characteristic properties. Let usriefly mention three of them [16]:

(i) For the fields studied here, the values of the so-alled intrinsic degrees of coherence, �S, �I, [5,8,10,11]re equal to 1. But the converse is not true: A field withS=�I=1 does not satisfy, in general, Eq. (2). In summary,

he beams with optimum �S and �I belong to a wider typef field than those defined by Eq. (2).

(ii) The CDT, W�r1 ,r2�, of the fields defined by Eq. (2)an be written as the sum of two terms, one of them rep-esenting the CDT of a totally polarized field and thether one associated with an unpolarized field. Note thathis possibility is no longer valid for a general field.

(iii) For the class of fields introduced by Eq. (2), theDT reads [16]

W�r1,r2� = M+�r1��M�r2�, �4�

here

� = �E0+E0� = �����2� ��*��

���*� ����2�� , �5�

nd M� fU [see Eq. (2)]. Thus, Eq. (4) provides theoherence-polarization structure of the fields defined by

q. (2). Other general properties of this class of fields areeported in Ref. [16].

. EQUIVALENCE PROPERTYn essential difference should be noted between the vec-

orial and the scalar cases, represented by Eqs. (2) and3), respectively: V�r� is a completely coherent field (in thecalar sense), whereas E�r�, defined by Eq. (2), is not. Ineneral, for these random electromagnetic fields, the vis-bility of the interference fringes in a two-point superpo-ition experiment would not be equal to 1, even thoughhe beam irradiance at the two pinholes is the same. How-ver, by using unitary transformations at such points, theisibility can be controlled and improved. Examples ofhis type of operation include position-dependent retarderhase plates. Let us then call Vmax the maximum fringeisibility one could attain in a Young interferometricrrangement by means of this class of (reversible) opticallements represented by unitary matrices. We will nexthow the following equivalence property.

Given any pair of pinholes at r1, r2��, the fringeisibility can be brought to its maximum value by localeversible transformations ⇔E=E0f�r�U�r�, ∀r��. Inarticular, Vmax=1 when �f�r1� � = �f�r2��.In other words, the attainable visibility can be opti-ized if and only if the beam belongs to the class of fields

escribed by Eq. (2).Proof. As is well known, given the CDT of a general

andom electromagnetic field at two points r1 and r2, theringe visibility in a two-point interferometric device isbtained from the degree of coherence introduced byolf [1]:

��W�2 =�tr W12�2

tr W11 tr W22

, �6�

here tr denotes the trace. Recently it has been shown9,12,16] that the maximum value one can get by usingocal unitary transformations at the pinholes (placed at r1nd r2) is given by

g12 = ���W�2�max =tr�W12W21� + 2�Det W12�

tr W11 tr W22

, �7�

here Det stands for determinant. It should be noted thatpplication of unitary-matrix devices actually changeshe interference experiment (including the fringe visibil-ty). However, the quantity g12 is not modified. Accord-ngly, g12 may be understood as an intrinsic capability ofhe field to improve the fringe visibility in a well-designedoung arrangement. Among other properties of param-ter g12 [9,12,16], it has been shown that, for a generalandom beamlike field, such a parameter satisfies�g12�1.Let us now assume that, for a certain electromagnetic

eld, the fringe visibility in a suitable two-point interfer-nce experiment can reach its optimum value, i.e.,

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1904 J. Opt. Soc. Am. A/Vol. 25, No. 8 /August 2008 R. Martínez-Herrero and P. M. Mejías

g12 = 1, for any r � �. �8�

his means that two unitary matrices exist, U1 and U2,uch that

��W�2 =�tr W12� �2

tr W11� tr W22�= 1, �9�

here

Wij� = Ui+WijUj, i,j = 1,2. �10�

f course, matrices U1 and U2 represent the action ofptical systems placed at the two pinholes. In Eq. (10), Wij

efers to the CDT evaluated at the pinholes, and Wij� is theDT associated with the field after passing through thenitary-matrix optical devices. It follows at once fromq. (9) that

�tr W12� �2 = �tr��E1��+E2���2 = ��E1��2���E2��2� for any r1,r2 � �,

�11�

here

Ei� = EiUi, I = 1,2, �12�

ith Ei , i=1,2being row vectors at the pinholes (prior tohe passage through the optical elements). But Eq. (11)mplies that the stochastic processes E1� and E2� are lin-arly dependent in the mean square sense (see property Cn [8]). Accordingly, we get [24]

E��r1� = �r1,r2�E��r2�, r1,r2 � �. �13�

pplying this equation to three points in the region �, weave

12 � �r1,r2� = �r1,r3��r3,r2� � 1332, �14�

nd setting r1=r2 one obtains

12 = 1 = 1331, �15�

hich implies

ab = ba−1. �16�

onsequently, Eq. (14) can be written in the form

12 =13

23. �17�

ut the left-hand side of this equation cannot depend on

3, and therefore the r3 dependence of the ratio 13

23must

ancel. We then write

12 =g�r1�

f�r2�, �18�

here f and g denote deterministic functions. Afterubstituting Eq. (18) into Eq. (17), we get

12 =g�r1�

f�r2�= 21

−1 =f�r1�

g�r2�, r1,r2 � �, �19�

hich implies f=g in �. Therefore,

E��r1� =f�r1�

f�r2�E��r2� ⇒

E��r1�

f�r1�=

E��r2�

f�r2�= ¯ = E0,

r1,r2 � �, �20�

here E0 is a random vector, and we finally obtain

��r� = f�r�E0 �in the mean square sense�, for any r � �.

�21�

hus, the CDT associated with E� takes the form

W��r1,r2� = f*�r1�f�r2��, �22�

here �= �E0+E0�, and therefore matrix W reads

W�r1,r2� = f*�r1�f�r2�U1+�U2. �23�

omparing this expression with Eq. (4), we conclude thathe stochastic process E�r� associated with a beam fulfill-ng Eq. (8) corresponds to the type of fields defined byq. (2).The converse property is also true: The CDT of any field

iven by Eq. (2) takes the form (4). Substitution of thisxpression into Eq. (7) gives g12=1, and the proof isompleted. Q.E.D.

. LOCAL DEGREE OF POLARIZATIONCROSS THE SUPERPOSITION REGION

et us now analyze the local degree of polarization PS ofhe field at the superposition region �S. In terms of theDT, PS reads [25,26]

PS2�R� = 1 −

4 Det WS�R,R�

tr WS�R,R�2, �24�

here R is the position vector of a point within �S, andˆ

S�R ,R� denotes the CDT at R. We have

WS�R,R� = W��r1,r1� + W��r2,r2� + W��r1,r2�exp�i�

+ W��r2,r1�exp�− i�, �25�

here r1 and r2 are the position vectors of the pinholes,ˆ � is given by Eq. (22), and =k��R−r2 �−�R−r1 � � is theath difference in free space (k being the wavenumber ofhe light). For simplicity, in Eq. (25) we have omitted theactors 1/ �R−r1 � �1/ �R−r2 � �1/D, where D denotes theistance between the plane of the pinholes and thebservation plane. Taking Eq. (22) into account, WS�R ,R�ecomes

WS�R,R� = �f�r1� + f�r2�exp�i��2�, �26�

nd therefore

PS2�R� = 1 −

4 Det �

�tr ��2, �27�

hich is independent of R. In other words, Eq. (27) showshat the degree of polarization is uniform across theuperposition region � .

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R. Martínez-Herrero and P. M. Mejías Vol. 25, No. 8 /August 2008 /J. Opt. Soc. Am. A 1905

Furthermore, this value of PS2 is just the same to that of

he degree of polarization of the incident field at the pin-oles [cf. Eq. (8) of Ref. [16]]. It should be remarked thathese properties of PS apply once the fringe visibility isptimized.

. CONCLUSIONSrom the general property shown in the present work, it

ollows that the possibility of optimizing the fringe visibil-ty in a Young interference arrangement is intimately con-ected with a position-independent stochastic behavior ofhe field. In other words, the beams defined by Eqs. (2)nd (4) are the only fields that attain optimized visibilityy using reversible unitary optical devices at theinholes. Moreover, in the optimized interferometer, thealue of the degree of polarization at the pinholes isreserved across the superposition region.Let us finally remark that parameter g12 fully identifies

hese fields. In fact, a value g12=1 is equivalent to sayinghat the beam under consideration belongs to the set ofelds defined by Eqs. (2) and (4).

CKNOWLEDGMENTS

his work has been supported by the Ministerio deducación y Ciencia of Spain, project FIS2007-63396, andy CM-UCM, Research Group Program 2008, No. 910335.

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4. The simple proof connecting Eqs. (12) and (21) should beacknowledged to an anonymous referee.

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6. P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M.Movilla, “Parametric characterization of the spatialstructure of non-uniformly polarized laser beams,” Prog.

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