1. Introduction

In recent years, considerable effort has been devoted to describing partially-coherent partially-polarized light beams. In particular, the concept of coherence of random electromagnetic fields has receiving increasing attention [1–22]. To characterize the coherence features of a random electromagnetic field by means of scalar quantities, several parameters have been proposed in the literature: In fact, a measure of the degree of coherence closely related with the fringe visibility in a Young interferometer was proposed by Wolf [1]. In addition, factorization of the cross-spectral density tensor (see below) was linked to a unit value of the so-called electromagnetic degree of coherence, defined in Refs. 2 and 4. Alternative parameters such as the intrinsic degrees of coherence were introduced by Réfrégier et al. [6,13] in connection with the attainable visibility in the Young experiment by using non-singular Jones matrices. More recently, maximization of the fringe visibility by means of (reversible) unitary optical devices has been described through a certain parameter [12,15]. Furthermore, for random electromagnetic fields, the polarization invariance under propagation is also a subject of active research [17–19].

In the scalar case, it is well known that complete coherence at a certain region involves factorization of the cross-spectral density function, W(r ₁,r ₂), according with the expression [23,24].

where r _i, i = 1, 2, are position vectors at two points of a plane transverse to the propagation axis z, f(r) is a deterministic function, and the asterisk denotes complex conjugation (for brevity, the explicit dependence on the light frequency has been omitted). This factorization feature is equivalent, for scalar beams, to a position-independent stochastic behaviour, as well as maximum fringe visibility in a Young interferometric arrangement [20,23,24].

In the vectorial regime, random electromagnetic fields are described by the cross-spectral density tensors (CDTs) Ŵ _ij, i,j = 1,2, associated with the electric field E at points r ₁ and r ₂, namely,

where 〈·〉 denotes ensembles averaging, and the dagger symbolizes the adjoint (transposed conjugate) of the row vector E. As is well established in recent papers, factorization properties of the CDT and unit visibility in the Young scheme are not equivalent concepts. In fact, different types of beams have been reported in the literature according with their visibility and factorization features.

In particular, it has recently been introduced the so-called mean-square coherent (MSC) field, defined as light that is able to interfere with Young’s fringes of unit visibility, when its electromagnetic field propagates through appropriate nonsingular deterministic Jones matrices. In other words, the original state of these fields could be suitably modified at the pinholes by means of certain optical elements in order to optimize the visibility. Two basic kinds of such optical devices can be distinguished: The first type causes reversible transformations, and the original state could be restored. The second type generates irreversible transformations, and some information about the field would be lost in an unrecoverable way.

In this paper, attention will be focused on mean-square coherent light and reversible optical devices (anisotropic nonabsorbing elements), which are represented by unitary matrices. More specifically, given the values P ₁ and P ₂, arbitrary but fixed, of the degree of polarization of the field at the pinholes in a Young arrangement, we study in the next section the maximum attainable visibility under unitary transformations when the illuminating beam is mean-square coherent light. This is the aim of the present work. In Section 3, analytical expressions are also provided for the field vector (in the mean-square sense) and for the CDT of this type of MSCs. This behavior defines a subclass of MSCs. A comparative summary of the main characteristics of well-known types of random electromagnetic fields reported in the literature is given in Section 4. On the basis of inclusive relations, a hierarchy between these sets of fields is also established in Section 4.

2. Maximum fringe visibility for MSC fields through reversible optical devices

Let us consider a MSC field in a spatial domain D, illuminating a Young interference arrangement where the two pinholes are placed at points r ₁ and r ₂ inside D. Since light is mean-square coherent, it was recently shown [21] that the intrinsic degrees of coherence (μ_s, μ_I) of the field E(r ₁) and E(r ₂) at the pinholes are equal to one. Taking this into account, the parameter g ₁₂[15], defined for a general field in the form

reduces, for MSC fields, to the expression

where ${\mu }_{\mathrm{STF}}^2=\genfrac{}{}{0.1ex}{}{\mathrm{Tr}\left({\hat{W}}_{12}{\hat{W}}_{12}^{†}\right)}{\mathrm{Tr}\left({\hat{W}}_{11}\right)\mathrm{Tr}\left({\hat{W}}_{22}\right)}$ represents the electromagnetic degree of coherence [4], and P ₁ and P ₂ denote the values of the degree of polarization at the pinholes. As is known [12,15,20,22], g ₁₂ was defined for general beamlike fields, and can also be understood as the maximum fringe visibility one can get in a Young scheme by using local unitary transformations (reversible optical devices) at the pinholes. In other words, g ₁₂ = ∣μ_W∣² _max, where ${\mid {\mu }_W\mid }^2=\genfrac{}{}{0.1ex}{}{{\mid \mathrm{Tr}{\hat{W}}_{12}\mid }^2}{\mathrm{Tr}{\hat{W}}_{11}\mathrm{Tr}{\hat{W}}_{22}}$ is a measure for the degree of coherence of random electromagnetic fields proposed by Wolf [1].

In general, 0 ≤ g ₁₂ ≤ 1, and the maximum value g ₁₂ = 1 is reached by the set of electromagnetic fields whose random character is position-independent [20,22]. It should be remarked that this kind of fields for which g ₁₂ = 1 constitutes a subclass of mean-square coherent light.

In the present work we assume that MSC light impinges on a Young interferometric device, and we consider fixed (but arbitrary) values of P ₁ and P ₂ (with P ₁, P ₂≠0). Our first purpose is to find the maximum attainable visibility at the superposition plane under unitary (reversible) transformations at the pinholes. This is equivalent to determine the highest value of g ₁₂ for MSC fields with degrees of polarization P(r ₁) = P ₁ and P(r ₂) = P ₂.

Note first (see Eq. (3)) that our problem reduces to maximize μ_STF ² for MSC light with fixed degrees of polarization at two points. In other words, we have to get the maximal value of $\genfrac{}{}{0.1ex}{}{\mathrm{Tr}\left({\hat{W}}_{12}{\hat{W}}_{12}^{†}\right)}{\mathrm{Tr}\left({\hat{W}}_{11}\right)\mathrm{Tr}\left({\hat{W}}_{22}\right)}$ in terms of P ₁ and P ₂. After some algebra (see below), we arrive to the final expression

This equation is a general result: In fact, it provides, for any prescribed values of the degree of polarization at the pinholes, the maximum fringe visibility one can obtain if we use MSC light and reversible optical devices in the Young experiment. In particular, [g(r ₁,r ₂)]_max equals one (optimization) when P(r ₁)=P(r ₂), as expected. This happens, as we know, for the electromagnetic fields with position-independent stochastic behaviour analysed in Refs. 20 and 22.

2.1 Proof of Eq. (4)

We consider a general MSC field in a domain D. It has recently been shown [21] that its CDT can be written as follows

with λ ₁, λ ₂ > 0, and where Ψ ₁(r) and Ψ ₂(r) are non-proportional nonzero deterministic row vectors. But it is not difficult to show that this CDT can also be expressed in the alternative factorizable form

where Ĥ is a 2×2 matrix, namely,

with Det Ĥ ≠ 0 and Ψ_i = (Ψ _i1, Ψ _i2), i = 1,2. In terms of its singular value decomposition, matrix Ĥ reads

where Û and V̂ are unitary matrices, and

with α(r) > β(r) taking nonnegative values. Note that, in terms of the diagonal elements, the degree of polarization of the MSC field at each point r is given by

By using Eq. (9), the CDT becomes

Then

where

which implies that Ŝ is unitary. Taking this into account, after some calculations we get

where θ(r ₁, r ₂) is defined through the relations

where S_ij ij = 1,2, denotes the elements of the unitary matrix Ŝ(r ₁,r ₂). Application of Eq. (11) gives

and

Rearranging Eqs. (15), (17) and (18), we obtain

so that

It is important to remark that Eq. (19) provides an analytical expression of μ_STF ² for a general MSC field. Moreover, Eq. (20) gives ∣μ_W∣² _max for this kind of fields. We then conclude that the maximal value of g ₁₂ one can attain (when the illuminating field is MSC light through reversible devices) is given by the relation (in terms of the degrees of polarization P ₁ and P ₂)

in agreement with Eq. (5). Q. E .D.

Note finally that, in Eq. (20), function cos2θ(r ₁,r ₂) does not depend on the local degree of polarization.

3. Analytical structure of the CDT and of the field vector

To complete the description of the special set of MSC fields we are studying, analytical expressions for both the CDT and the field vector will be provided.

Note first that we have just shown that g ₁₂ reaches its maximal value when θ = 0, so that (see Eqs. (16)) matrix Ŝ adopts the diagonal form

We then have that the unitary matrix V̂ can be expressed as follows

where M̂(r) denotes a diagonal matrix whose nonzero elements are complex exponentials, and Q̂ represents a position-independent unitary matrix. As a consequence, the general form of the CDT of the class of MSC fields that maximizes g ₁₂ reads (cf Eq. (12))

where

Note that this CDT is factorizable in the form given by Eq. (7), as it should be expected for general MSC light. In particular, for the special MSC fields we are considering (those satisfy Eq. (5)), we see that the general form (7) reduces to the analytical structure given by Eqs. (24) and (25).

Another consequence of Eq. (24) refers to the field vector E(r), which can be written (in the mean-square sense) in the form

where E ₀ = (u,v) is a 1×2 vector whose components take complex random values fulfilling <∣u∣² >=<∣v∣² > and <u ^* v >= 0. From this equation, we see at once that E(r) can also be expressed as follows

where ε_i, i = 1, 2, are also complex random variables satisfying < ε ₁ ^* ε ₂ >= 0, and

the symbol ∝ denoting proportionality, and U_ij, i,j = 1,2 being the matrix elements. Since Û is unitary, we finally conclude that vectors Ψ̃ ₁ and Ψ̃ ₂ should be orthogonal. This is an important difference with regard to a general MSC field: In such case (see Eq. (6)), vectors Ψ ₁ and Ψ ₂ are non-proportional, non-orthogonal vector fields. Comparison between different classes of random electromagnetic fields will be further discussed in the next section.

4. Comparative summary of random electromagnetic fields

As was pointed out in the Introduction, maximum visibility, factorization of the cross-spectral density function and position-independent stochastic behavior of the field can be understood as equivalent features in the scalar regime. But this is no longer true in the vectorial case. The above properties should then be considered as separate characteristics which describe different types of fields. In this connection, let us finally summarize and compare the main characteristics of several well-known families of random electromagnetic fields, which exhibit peculiar coherence-polarization features in a certain domain D. We focus our attention on certain similarities and differences concerning the attainable visibility, the factorization of the CDT, and the stochastic structure (in the mean-square sense) of the associated vector field.

{T}: Set of fields with μ_STF ² = 1 [2,4]. The CDT of this kind of beams is factorizable in the form Ŵ(r ₁,r ₂) = F ^†(r ₁)F(r ₂), where F is a deterministic row vector. These fields are totally polarized, with position-independent stochastic character and optimum attainable visibility in a Young scheme.

{V}: Set of fields with g ₁₂ = ∣μ_W∣² _max = 1 [12,15,20,22] (maximum attainable visibility under local unitary transformations in a Young interferometer). The CDT of these fields is factorizable in the sense expressed by Eq. (6) of Ref. [20], with position-independent stochastic behaviour. The vector field reads (cf. Eq. (4) of Ref. [20])

where E ₀ = (u,v) is a row vector whose components take complex random values, f is a deterministic complex function, and Û denotes a unitary matrix.

{R} (MSC light): Set of partially polarized fields with μ_s = μ_I = 1 [21]. The CDT of these fields is factorizable as a product of two matrices, as shown in Eq. (7). The stochastic behaviour depends on the position. The vector field reads (in the mean-square sense)

where ε ₁ and ε ₂ are complex random variables, with <ε ₁ ^* ε ₂>= 0, and Ψ ₁ and Ψ ₂ denote deterministic vectors, which, in general, are not orthogonal.

[R ^*} : Set of MSC fields that, for fixed values of the degree of polarization at the pinholes, optimize the Young fringe visibility under unitary (reversible) transformations. The CDT is also factorizable as a matricial product in the special form given by Eq. (24). The vector field reads

where D̂ ₀ is a diagonal matrix and Ψ̃ ₁ and Ψ̃ ₂ are orthogonal vectors. The stochastic behavior also depends on the position.

Associated with all these features, a hierarchy between these sets of electromagnetic fields can finally be established on the basis of inclusive relations, namely,

This layer structure is illustrated in Fig. 1.

 

Fig. 1. Illustrating the relation {T} ⊂ {V} ⊂ {R ^*} ⊂ {R}

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