June 1, 2007 / Vol. 32, No. 11 / OPTICS LETTERS 1471

Maximum visibility under unitary transformationsin two-pinhole interference for

electromagnetic fields

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

*Corresponding author: pmmejias@fis.ucm.es

Received February 6, 2007; revised March 1, 2007; accepted March 16, 2007;posted March 20, 2007 (Doc. ID 79831); published May 3, 2007

In terms of the spectral density tensors associated with the electric field vector, the maximum visibility onecan obtain in a two-point interference arrangement by using local (i.e., position-dependent) unitary trans-formations applied at such points is determined. It is also shown that the maximum visibility can be ex-pressed in terms of a number of well-known parameters describing the coherence and polarization featuresof the field. © 2007 Optical Society of America

OCIS codes: 030.1640, 260.5430, 260.2110.

The analysis of partially coherent and partially po-larized fields is at present a topic of new interest[1–12]. For second-order stationary electromagneticfields, the behavior of a light field can be described bythe spectral density tensors Wij , i , j=1,2, associatedwith the electric vector E at two points, r1 and r2,namely [13],

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

�1�

where E+ denotes the adjoint (transposed conjugate)of the row vector E, the angle brackets refer to an en-semble average, and � is the spectral frequency ofthe light (for brevity, it will be omitted from now on).Since, in the present paper, we are interested in elec-tromagnetic beamlike fields propagating essentiallyalong the z-direction, the longitudinal components ofthe field vector may be neglected. In this regime, thetensors Wij, i , j=1,2, reduce to 2�2 matrices. As adescription and measure for the degrees of coherenceof a random electromagnetic field, two definitionswere considered not long ago:

��W�2 =�trW12�2

trW11trW22, �2a�

proposed by Wolf [1], and

uSTF2 =

tr�W12W21�

trW11trW22, �2b�

introduced in Ref. [5] (tr denotes the trace). Here at-tention will be focused on the visibility of the inter-ference fringes in a superposition experiment. Thisquantity (fringe visibility) is obtained from ��W�2, andits value can be controlled and modified, among otherdevices, by means of suitable unitary transforma-tions at r1 and r2. Examples of this type of operationinclude a retarder phase plate whose dephasing ac-tion between the transverse field components de-

pends on the point, and also a rotation of the coordi-

0146-9592/07/111471-3/$15.00 ©

nate system, whose rotation angle is a position-dependent function.

The aim of the present work will be to determinethe maximum visibility one can attain in a two-pointinterference experiment by using local (i.e., position-dependent) unitary transformations applied at suchpoints. In this paper we will directly handle the func-tion ��W�2. More specifically, given the cross-spectraldensity tensor of a field at two points, r1 and r2, itcan be shown (see Appendix A) that the maximumvalue ���W�2�max one can obtain by using local unitarytransformations is given by

���W�2�max =�1

2 + �22 + 2��1���2�

trW11trW22, �3�

where �i2, i=1,2, are the eigenvalues (both are non-

negative) of the matrix product W12W12+ , and ��i�, i

=1,2, denote the positive square root of �i.Let us remark that, although the field at the out-

put of the unitary-matrix device has changed with re-spect to its input value, the quantity ���W�2�max wouldrepresent, in a sense, the intimate capability of afield to improve their visibility in a suitably designedinterference experiment.

An interesting consequence can be inferred fromEq. (3). Let us first recall that, since �1

2 and �22 are the

eigenvalues of matrix W12W12+ , the following relations

apply:

tr�W12W12+ � = �1

2 + �22, �4a�

det�W12W12+ � = �1

2�22, �4b�

where det stands for determinant. Taking theseequations into account, the quantity �STF

2 [see Eq.(2b)] can be written in the form

�STF2 =

�12 + �2

2

trW11trW22, �5�

and ���W�2�max becomes (note that �STF2 does not

change under unitary transformations)

2007 Optical Society of America

1472 OPTICS LETTERS / Vol. 32, No. 11 / June 1, 2007

���W�2�max = �STF2 +

�4 det W12W12+

trW11trW22. �6�

To go further into the physical meaning of this equa-tion, let us introduce the normalized cross-spectraldensity matrix [8,11,12], which has been defined as

M�r1,r2� = W11−1/2W12W22

−1/2. �7�

The singular values of matrix M are the so-calledspectral intrinsic degrees of coherence, �I and �S[8,11,12], which satisfy

�det M� = �I�S. �8�

From Eqs. (7) and (8) and the standard definition forthe degree of polarization [14,15], we get

�det W12� = �I�S�det W11 det W22

=1

4�I�S��1 − P1

2��1 − P22��trW11�2�trW22�2,

�9�

where Pi, i=1,2, represent the degree of polarizationat points r1 and r2. Accordingly, we have

�4 det W12W12+

trW11trW22=

1

2�1�S��1 − P1

2��1 − P22�, �10�

and Eq. (6) finally becomes

���W�2�max = �STF2 +

1

2�I�S��1 − P1

2��1 − P22�. �11�

This is the other main result of this paper: the maxi-mum value of the visibility one can attain under localunitary transformations is a function of several pa-rameters, namely, �STF

2 , the intrinsic degrees of co-herence and the degree of polarization at points r1and r2. In addition, Eq. (11) can be understood as alink between the physical meaning of ��W�2 and �STF

2 .As a further consequence, it is clear from Eq. (11)that one can find fields with the same �STF

2 but whosemaximum visibility in a Young’s interference experi-ment differs (for example, depending on the polariza-tion state of the field). It is also concluded from Eq.(11) that the intrinsic degree of coherence are closelyconnected with the optimum visibility, a result re-cently pointed out in Ref. [12].

To illustrate the above conclusions, let us apply Eq.(11) to a simple example. We first consider a fieldwhose matrices Wij, i , j=1,2 are W11=I,

W12 = �1 1

1 0�, W22 = �2 1

1 1� .

For this field we would have �STF2 =1/2, �I=�S=1,

P12=0, P2

2=5/9, and Eq. (11) gives ���W�2�max=5/6. Letus now consider another field, characterized by the

matrices

W11 = I, W12 = 1

�21

11

�2, and W22 =

3

2�2

�23

2 .

In this case, �STF2 =1/2, �I=�S=1, P1

2=0, P22=8/9, and

we get ���W�2�max=2/3. We thus see that the abovefields exhibit identical values for �STF

2 , �I, �S, and P12,

but their attainable maximum visibility is not thesame due to different values of the degree of polariza-tion at point r2.

APPENDIX A

We will next show Eq. (3). From the singular valuedecomposition of matrix W12, we know that there ex-ist two unitary matrices, namely, U and V, such that

W12 = U+DV, �A1�

with

D = ���1� 0

0 ��2�� .

Let us now introduce two unitary matrices

U1 = U+H1, �A2a�

U2 = V+H2, �A2b�

where H1 and H2 are arbitrary unitary matrices. Af-ter application of the unitary transformations U1 andU2 at point r1 and r2, respectively, we get at the out-put

W11� = H1+UW11U

+H1, �A3a�

W22� = H2+VW11V

+H2, �A3b�

W12� = H1+DH2, �A3c�

where the prime refers to the output values. In termsof these quantities, �W

2 reads after the transforma-tions

��W�2 =�tr�H1

+DH2��2

trW11trW22, �A4�

where the numerator of Eq. (A4) can also be writtenin the form

tr�H1+DH2� = tr�H2H1

+�D�D�, �A5�

�D being the matrix

����1� 0

0 ���2�� .

Now, from the application of the Schwartz inequality,

it follows that

June 1, 2007 / Vol. 32, No. 11 / OPTICS LETTERS 1473

�tr�H2H1+�D�D��2 � tr�H2H1

+�D�DH1H2+� trD,

�A6�

and the equality is reached when

H2H1+�D = ��D, �A7�

which implies

�H2H1+ − �I��D = 0, �A8�

where � denotes a complex number and I is the 2�2 identity matrix. It should be noted that, when Eq.(A6) transforms into an equality, ��W�2 becomes maxi-mized, which is the result we are looking for.

Now let us first assume that det�D�0. The fulfil-ment of Eq. (A7) implies

H2H1+ = �I. �A9�

In addition, since Hi, i=1,2, are unitary matrices,Eq. (A9) gives

H1+ = �H2

+, �A10a�

H2 = �H1, �A10b�

which gives ���2=1 (note that, for �=1, we have H1=H2=I). Consequently, the maximum value of ��W�2after local unitary transformations would finally be

���W�2�max =�tr�H1

+DH2��2

trW11trW22=

�trD�2

trW11trW22

=�1

2 + �22 + 2��1���2�

trW11trW22, �A11�

in accordance with Eq. (3).When det�D=0, one of the diagonal elements of

matrix �D vanishes. Then, by using Eq. (A8) and thefact that H2H1

+ is a unitary matrix, we would again

get Eq. (A11).

This work has been supported by the Ministerio deEducación y Ciencia of Spain, project FIS2004-1900,and by the Universidad Complutense-Comunidad deMadrid, within the framework of the Research GroupProgram, No. 910335.

Note added in proof: Equation (6) of the present pa-per can also be obtained by using the alternative pro-cedure recently published by F. Gori et al. [16].

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