1. Introduction

The electric field of an electromagnetic radiation at point r and at time t is classically represented by a complex random vector [1] Ẽ (r,t) that will be written Ẽ (r,t) = E(r,t) e ^-iωt and that will be assumed to be two dimensional E(r,t) = [E_X (r,t),E_Y (r,t)]^T where ^T stands for transpose.

It is generally accepted that the coherency properties between points r ₁ and r ₂ and times t ₁ and t ₂ of the complex random vectors E(r₁ ,t ₁) and E(r₂ ,t ₂) can be represented by the mutual coherence matrix [1, 2, 3, 4, 5] Ω(r₁ ,r₂ ,t ₁,t ₂), defined by:

where ^† denotes conjugate transpose and where <.> denotes ensemble averaging which can correspond to different types of physical averaging. The standard coherency matrix [1] corresponds to the case r₁ = r₂ and t ₁ = t ₂. In the following, the second-order statistical characterization of a set of two electric fields (i.e. their mutual coherence matrix and their coherency matrices) will be denoted an optical situation.

This approach is general and used in many domains [2, 3, 4, 5, 6] but does not provide invariant parameters. Defining such invariant parameters for characterizing the coherence of partially polarized light is still an open question [3, 4, 5, 6, 7]. Different approaches have been proposed in [3, 4, 5] but they do not clearly separate partial polarization and partial coherence. For example, in [3], the following spectral decomposition is considered

where the electromagnetic field is assumed wide sense stationary so that one has Ω(r₁ ,r₂ ,t,t + τ) = Ω(r₁ ,r₂ ,τ). In that case, the author [3] introduced the following spectral degree of coherence

where tr[W] denotes the trace of W. From this expression one can define a degree of coherence in the space time domain by

In [7], the non invariance of this definition to orthogonal curvilinear coordinate system transformations is demonstrated. This appears as one of the arguments in favor of the following definition introduced in [4, 5]

which can also be defined in the temporal domain [7]

Since the coherency matrices Ω(r_i ,r_i ,t_i ,t_i ) play a particular role in polarization theory, they will be denoted Γ(r_i ,t_i ) in the following.

In this paper, we propose to determine the appropriate matrix M(r₁ ,r₂ ,t ₁,t ₂) from which one can determine invariant degrees of coherence of the optical state independently of the particular experimental configuration and of the polarization properties. In particular, it will be shown how this matrix M(r₁ ,r₂ ,t ₁,t ₂) can be determined from Ω(r₁ ,r₂ ,t ₁,t ₂) and how the invariant degrees of coherence are simply related to this matrix. Finally, one will discuss how these proposed degrees of coherence can be related to practical experiments and we will analyze some simple examples.

2. Invariant degrees of coherence

In this section we discuss invariance properties which are introduced in order to separate partial coherence and partial polarization. This analysis will allow us to obtain invariant degrees of coherence.

The degree of coherence defined by Eq. (4) is simply related [3] to the visibility of interference fringes between the electric fields E(r₁ ,t ₁) and E(r₂ ,t ₂). However, this degree of coherence is representative of the particular configuration of the experiment and is not an intrinsic property of the electric waves. More precisely, if one introduces birefringent elements in front of E(r₁ , t ₁) and E(r₂ , t ₂) the interference fringe visibility can be modified. From a mathematical point of view, the action of the birefringent elements corresponds to the application of unitary linear transformations U ₁ and U ₂ on E(r₁ ,t ₁) and E(r₂ ,t ₂). In that case, the obtained electric fields become A_U (r₁ ,t ₁) = U ₁ E(r₁ ,t ₁) and A_U (r₂ ,t ₂) = U ₂ E(r₂ ,t ₂).

In order to introduce degrees of coherence independent of the particular experimental configuration, one needs to define parameters which are invariant when the electric fields E(r₁ ,t ₁) and E(r₂ ,t ₂) are modified by the application of unitary transformations. Since 〈A_U (r₂ ,t ₂) ${\mathbf{A}}_{\mathbf{U}}^{†}$(r₁ ,t ₁)) = U ₂ 〈E(r₂ ,t ₂) E ^†(r₁ ,t ₁)〉 ${\mathbf{U}}_1^{†}$, the previous constraint means that the invariant degrees of coherence have to be the same for Ω(r₁ ,r₂ ,t ₁,t ₂) and for U ₂Ω(r₁ ,r₂ ,t ₁,t ₂) ${\mathbf{U}}_1^{†}$ whatever the unitary transformations U ₁ and U ₂.

The main limitation of considering the matrix Ω(r₁ ,r₂ ,t ₁,t ₂) is that this may not provide degrees of coherence which are independent of the polarization properties of each electric field. Indeed, let us consider the simple case where the coherency matrices are equal and diagonal

It can be natural to consider that the degree of coherence has to be equal to one when r₁ = r₂ and when t ₁ = t ₂. This is indeed the case of the degree of coherence of Eq. (4) but not of the one of Eq. (6). In other words, the degrees of coherence defined in Eq. (4) and Eq. (6) have complementary but different properties. It is in fact possible to define degrees of coherence which are invariant to both unitary transformations and modifications of the polarization states.

To be more explicit, one needs to precise the notion of independence with respect to the polarization state. This requirement means that a local modification of the polarization states should not modify the degrees of coherence. Let us assume that electric fields E(r₁ ,t ₁) and E(r₂ ,t ₂) have coherency matrices Γ_e(r₁ ,t ₁) and Γ_e(r₂ ,t ₂) and mutual coherence matrix Ω_e(r₁ ,r₂ ,t ₁,t ₂). A local modification of the polarization states can be represented by the application of linear operators such that the electric fields become A(r₁ ,t ₁) = B ₁ E(r₁ ,t ₁) and A(r₂ ,t ₂) = B ₂ E(r₂ ,t ₂) where B ₁ and B ₂ are non singular Jones matrices. The mutual coherence matrix of the electric fields A(r₁ ,t ₁) and A(r₂ ,t ₂) will be denoted Ω_a(r₁ ,r₂ ,t ₁,t ₂) in the following.

In this paper, we propose to define degrees of coherence which are invariant when the polarization states of the electric fields are modified by the action of components which are represented by non singular Jones matrices. In other words, the invariant degrees of coherence between E(r₁ ,t ₁) and E(r₂ ,t ₂) have to be equal to the invariant degrees of coherence between A(r₁ ,t ₁) and A(r₂ ,t ₂). Let us define the equivalence class 𝓒[E(r₁ ,t ₁),E(r₂ ,t ₂)] such that two optical situations [E(r₁ ,t ₁),E(r₂ ,t ₂)] and [A(r₁ ,t ₁),A(r₂ ,t ₂)], belong to the same equivalence class if there exists non singular matrices B₁ which depends only on (r₁ ,t ₁) and B₂ which depends only on (r₂ ,t ₂) such that A(r₁ ,t ₁) = B ₁ E(r₁ ,t ₁) and A(r₂ ,t ₂) = B ₂ E(r₂ ,t ₂). In that case, the mutual coherence matrices are related by Ω_a(r₁ ,r₂ ,t ₁,t ₂) = B₂ Ω_e(r₁ ,r₂ ,t ₁,t ₂) ${\mathbf{B}}_{\mathbf{1}}^{†}$ and the coherency matrices are related by Γ_a(r_i ,t_i ) = B _i Γ_e(r_i ,t_i ) ${\mathbf{B}}_i^{†}$ . The invariance principle thus implies that the degrees of coherence must be the same for all couples of electric fields which belong to the same equivalence class 𝓒[E(r₁ ,t ₁),E(r₂ ,t ₂)]. Since one characterizes the optical situation by the second order matrices {Ω(r₁ , r₂ ,t ₁, t ₂), Γ(r₁ , t ₁), Γ(r₂ , t ₂)} all optical situation described by {B₂ Ω(r₁ ,r₂ ,t ₁,t ₂) ${\mathbf{B}}_{\mathbf{1}}^{†}$,B ₁ Γ(r₁ ,t ₁) ${\mathbf{B}}_1^{†}$,B ₂ Γ(r₂ ,t ₂) ${\mathbf{B}}_2^{†}$} should have the same degrees of coherence whatever the non singular Jones matrices B₁ , B ₂.

There exists different possibilities to define the invariant degrees of coherence characterizing the equivalence class 𝓒;[E(r₁ ,t ₁),E(r₂ ,t ₂)] and it is thus important to carefully justify a particular choice. Let us first consider the case of partially polarized light (i.e. for which the coherency matrices Γ(r_i ,t_i ) are non singular). A simple approach consists in considering a set of totally depolarized lights as the representative of 𝓒[E(r₁ ,t ₁),E(r₂ ,t ₂)]. Such a representative is obtained by choosing B _i = Γ^-1/2(r_i ,t_i ). In that case, the optical situation which is representative of 𝓒[E(r₁ ,t ₁),E(r₂ ,t ₂)]corresponds to {M(r₁ ,r₂ ,t ₁,t ₂),I_d ,I_d }with

and where I_d is the identity matrix in dimension two. Since M(r₁ ,r₂ ,t ₁,t ₂) will be central in the following developments, it will be called the normalized mutual coherence matrix of the optical situation defined by the mutual coherence matrix Ω(r₁ ,r₂ ,t ₁,t ₂) and coherency matrices Γ(r_i ,t_i ) with = 1,2.

Furthermore, whatever the unitary matrices U₁ , U₂ , totally depolarized lights with normalized mutual coherence matrices M(r₁ ,r₂ ,t ₁,t ₂) and U₂M(r₁ ,r₂ ,t ₁,t ₂)${\mathbf{U}}_{\mathbf{1}}^{†}$ belong to the same equivalence class and must thus have the same invariant degrees of coherence. In particular, it is possible to choose U₂ = N₂ and to U₁ = N₁ so that N₂ M(r₁ ,r₂ ,t ₁,t ₂) ${\mathbf{N}}_{\mathbf{1}}^{†}$ is diagonal. This choice corresponds to the singular value decomposition of the normalized mutual coherence matrix

where D(r₁ ,r₂ ,t ₁,t ₂) is a diagonal matrix whose real positive diagonal values are called the singular values. One can thus choose the matrix D(r₁ ,r₂ ,t ₁,t ₂) as the representative of the equivalence class 𝓒[E(r₁ ,t ₁),E(r₂ ,t ₂)] and thus define the invariant degrees of coherence from D(r₁ ,r₂ ,t ₁,t ₂).

The matrix D(r₁ ,r₂ ,t ₁,t ₂) corresponds to the normalized mutual coherence matrix of totally depolarized light described in the basis of the singular value decomposition of M(r₁ ,r₂ ,t ₁,t ₂). The diagonal elements μ_S (r₁ ,r₂ ,t ₁,t ₂), μ_I (r₁ ,r₂ ,t ₁,t ₂) of D(r₁ ,r₂ ,t ₁,t ₂) are the singular values of M(r₁ ,r₂ ,t ₁,t ₂) and constitute appropriate general invariant degrees of coherence. In order to simplify the analysis, in the following, the singular values of M(r₁ ,r₂ ,t ₁,t ₂) will be simply noted μ_S , μ_I assuming, with no loss of generality, μ_S ≥ μ_I . The choice of μ_S and μ_I as degrees of coherence corresponds to the intuition. Indeed, let us assume that the electric field is the sum of two statistically independent and orthogonal waves of degrees of coherence μ ₁ and μ ₂ (with μ ₁ > μ ₂), one will see in section 4 that the above approach leads to μ_S = μ ₁ and μ_I = μ ₂.

It can be noted that

where D ²(r₁ ,r₂ ,t ₁,t ₂) is a diagonal matrix of diagonal values ${\mu }_S^2$ and ${\mu }_I^2$. The degrees of coherence can thus be easily obtained from the eigenvalues of M(r₁ ,r₂ ,t ₁,t ₂)M ^†(r₁ ,r₂ ,t ₁,t ₂).

We have seen in this section that it is possible to define invariant degrees of coherence for partially polarized light. This case corresponds to non singular coherency matrices Γ(r_i ,t_i ). We propose to discuss some physical interpretation and to analyze the case of perfectly polarized light in the next section.

3. Physical interpretation and particular case of perfectly polarized lights

We propose in this section to analyze the physical interpretation of the normalized mutual coherence matrix and to show that the proposed approach can be generalized to perfectly polarized lights. It will also allow us to discuss the relation of this approach with the condition at order two of complete electromagnetic coherence introduced by Glauber [11].

Any partially polarized light can be decomposed in its eigenvector basis [1]. More precisely, let us consider the electric fields E(r_i ,t_i ) with coherency matrices Γ_e(r_i ,t_i ). There exists unitary matrices U_i such that the fields A(r_i ,t_i ) = U_i E(r_i ,t_i ) have diagonal coherency matrices Λ_a(r_i ,t_i )

where A(r_i ,t_i ) = [A_X (r_i ,t_i ),A_Y (r_i ,t_i )]^T where |a| is the modulus of a. When the field is not perfectly polarized at point r_i and at time t_i , Λ_a(r_i ,t_i ) is not singular and one has

with I_X (r_i ,t_i ) = 〈|A_X (r_i ,t_i )|²〉 and I_y (r_i ,t_i ) = 〈|A_Y (r_i ,t_i )|²〉.

The mutual coherence matrix between E(r₁ ,t ₁) and E(r₂ ,t ₂) is Ω_e(r₁ ,r₂ ,t ₁,t ₂) = 〈E(r₂ ,t ₂)E ^†(r₁ ,t ₁)〉 and one thus has Ω_e(r₁ ,r₂ ,t ₁,t ₂) = U₂ Ω_a(r₁ ,r₂ ,t ₁,t ₂) ${\mathbf{U}}_{\mathbf{1}}^{†}$, where Ω_a(r₁ ,r₂ ,t ₁,t ₂) is the mutual coherence matrix between A(r₁ ,t ₁) and A(r₂ ,t ₂) (i.e. Ω_a(r₁ , r₂ ,t ₁,t ₂) = 〈A(r₂ ,t ₂)A ^†(r₁ ,t ₁)〉). Obviously Γ_e(r_i ,t_i ) = U_i Λ_a(r_i ,t_i )${\mathbf{U}}_{\mathbf{i}}^{†}$ and using these relations and Eq. (8), one easily gets

with

The physical interpretation of M _a(r₁ ,r₂ ,t ₁,t ₂) is thus simple since

and thus

with

where P, Q = X,Y. The normalized mutual coherence matrix M _a(r₁ ,r₂ ,t ₁,t ₂) is thus made of the standard degrees of coherence of the scalar fields [10] A_X (r,t) and A_Y (r,t) which leads to a simple physical interpretation. One can see with Eq. (13) that the normalized mutual coherence matrix M _e(r₁ ,r₂ ,t ₁,t ₂) corresponds to a change of basis. Indeed, M _e(r₁ ,r₂ ,t ₁,t ₂) is written in the basis in which the light is analyzed while M _a(r₁ ,r₂ ,t ₁,t ₂) is expressed in the basis defined by the eigenvectors of the coherency matrices Γ_e(r_i ,t_i ).

Let us now analyze the case of perfectly polarized lights at (r_i ,t_i ) with i = 1,2. Without loss of generality we assume that A(r_i ,t_i ) = [A_X (r_i ,t_i ),0]^T. One thus gets

and Λ_a(r_i ,t_i ) = Ω_a(r_i ,r_i ,t_i ,t_i ). One can introduce the pseudo inverse ${\mathrm{\Lambda }}_a^{-1/2}$(r_i ,t_i ) of ${\mathrm{\Lambda }}_a^{1/2}$(r_i ,t_i )

which, in that case, simply leads to

One thus sees that when the light is perfectly polarized one has D _a(r₁ ,r₂ ,t ₁,t ₂) = z M _a (r₁ , r₂ , t ₁, t ₂) (where D _a (r₁ , r₂ , t ₁, t ₂) is defined with Eq. (9) and where z is a complex number of modulus one). One thus gets μ_S (r₁ ,r₂ ,t ₁,t ₂) = |η_XX (r₁ ,r₂ ,t ₁,t ₂)| which shows that the proposed invariant degrees of coherence are compatible with the standard degree of coherence classically used for perfectly polarized light.

This analysis can be extended without difficulty to cases where the field is perfectly polarized only at one point r₁ and at one time t ₁ (or only at r₂ and t ₂). This result shows that the definition of the normalized mutual coherence matrix given by Eq. (8) can be generalized to cases of singular coherency matrices Γ(r_i ,t_i ) by considering their pseudo inverses which are defined by ${\mathrm{\Gamma }}_e^{-1/2}$(r_i ,t_i )= U_i ${\mathrm{\Lambda }}_a^{-1/2}$(r_i ,t_i )${\mathbf{U}}_{\mathbf{i}}^{†}$.

Let us now analyze the condition at order two of complete electromagnetic coherence introduced by Glauber [11]. This condition requires that the mutual coherence matrix can be factorized. More precisely, an electric field is said fully coherent with the Glauber approach if there exists a vectorial function Ψ(r, t) so that

which implies that the field has to be totally polarized [11]. Let us define the unitary matrix U_i so that the first column corresponds to the direction of Ψ(r_i ,t_i ). One can thus introduce the field A(r_i ,t_i ) = ${\mathbf{U}}_{\mathbf{i}}^{†}$ E(r_i ,t_i ). The mutual coherence matrix of E(r_i ,t_i ) is Ω(r₁ ,r₂ ,t ₁,t ₂) = U₂ Ω_a(r₁ ,r₂ ,t ₁,t ₂)${\mathbf{U}}_{\mathbf{1}}^{†}$ which leads to Ω_a(r₁ ,r₂ ,t ₁,t ₂) = ${\mathbf{U}}_{\mathbf{2}}^{†}$Ω(r₁ ,r₂ ,t ₁,t ₂)U₁ and thus

The coherency matrices Λ_a(r_i ,t_i ) are thus also diagonal as Ω_a{r₁ ,r₂ ,t ₁,t ₂) but with non zero diagonal elements equal to ||Ψ(r_i ,t_i )||². One easily gets that the normalized mutual coherence matrix has the form of Eq. (20) with |η_XX(r₁ , r₂ , t ₁, t ₂)| = 1. The invariant degrees of coherence μ_S (r₁ , r₂ ,t ₁, t ₂) is thus also equal to 1.

The relation between the proposed approach of this paper and the complete electromagnetic coherence introduced by Glauber [11] is thus analogous to the relation between the standard degree of coherence and the factorization condition for the scalar case which corresponds for example to linearly perfectly polarized lights. In summary, the factorization condition implies that the light has to be perfectly polarized with μ_S (r₁ ,r₂ ,t ₁,t ₂) = 1. In particular, these conditions do not correspond to μ_S (r₁ ,r₂ ,t ₁,t ₂) = μ_I (r₁ ,r₂ ,t ₁,t ₂) = 1 independently of the polarization state. We shall discuss an example of such a case at the end of the following section.

4. Relation with standard definition of scalar optical fields

It is interesting to analyze more precisely how the invariant degrees of coherence introduced above are related to the standard degree of coherence in the case of scalar optical fields. Optical fields can be described with scalar numbers when they are linearly polarized in the same direction. We thus analyze precisely the maximal value of the modulus of the standard degree of coherence one can obtain when the electric fields are modified by non depolarizing elements and polarized linearly in the same direction.

 

Fig. 1. Schematic illustration of the experimental set up used for the measurement of the degrees of coherence. U ₁ and U ₂ denote optical modulators with unitary Jones matrices and P ₁ and P ₂ denote polarizers.

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This analysis will also let us show that μ_S and μ_I have a simple physical interpretation. Indeed, one can follow the concepts developed in [9] for polarimetric Synthetic Aperture Radar interferometry. Let us assume that one analyzes the maximal value of the modulus of the standard degree of coherence of scalar optical fields one can obtain when E(r₁ ,t ₁) and E(r₂ ,t ₂) are modified with non depolarizing components and then polarized with perfect polarizers (see figure 1). This analysis is interesting since the relation between the standard degree of coherence of scalar optical fields and the visibility of interference fringes is well established [10].

The modifications of the polarization state with non depolarizing components are represented by unitary matrices U ₁ and U ₂. The optical fields in front of the polarizer become A_U (r₁ ,t ₁) = U ₁ E(r₁ ,t ₁) and A_U (r₂ ,t ₂) = U ₂ E(r₂ ,t ₂). The actions of the polarizers can be represented by the projection of A_U (r₁ ,t ₁) and A_U (r₂ ,t ₂) on the vector e₁ of unitary modulus.

The modulus of the standard degree of coherence can thus be written

This equation can also be written

The modulus of the standard degree of coherence is maximized by finding the unitary matrices U ₁, U ₂ which maximize η. Using Eq. (8) and if one introduces k_i = Γ^1/2(r_i ,t_i )${\mathbf{U}}_i^{†}$ e₁ one gets

Since for all complex numbers α and β different to 0, η is invariant if one changes k₁ into α k₁ and k₂ into β k₂ , the maximization of η is obtained by finding the unitary vectors a₁ and a₂ which optimize

Using Eq. (9), the normalized mutual coherence matrix can be decomposed into singular values

with ||u_i ||² = ||v_i ||² = 1 and ${\mathbf{u}}_{\mathbf{i}}^{†}$ v_i = 0 where i = 1,2. One thus gets

It can be shown (see Appendix A) that the maximal value of η is μ_S and that this value is reached when a₂ = z ₂ u₂ and a₁ = z ₁ u₁ where z ₁ and z ₂ are complex numbers of modulus equal to one. If a₂ is proportional to v₂ and a₁ is proportional to v₁ , one gets η = μ_I . Finally, if a₂ is proportional to u₂ and a₁ is proportional to v₁ , or if a₂ is proportional to v₂ and a₁ is proportional to u₁ , one gets η = 0.

These results show that one finally obtains an eigenvalue decomposition of the coherence properties. Indeed, u_i and v_i are two orthogonal modes of the normalized mutual coherence matrix of E(r_i , t_i ). In other words, if one considers the coherency properties between points r₁ and r₂ and times t ₁ and t ₂ of the electric fields E(r₁ , t ₁) and E(r₂ , t ₂), the corresponding totally depolarized lights A_D (r₁ ,t ₁) = Γ^-1/2(r₁ ,t ₁) E(r₁ ,t ₁) and A_D (r₂ ,t ₂) = Γ^-1/2(r₂ ,t ₂) E(r₂ ,t ₂) can be decomposed in two independent components. For that purpose, let us write A_D (r_i ,t_i ) = a^u (r_i ,t_i )u_i +a_v (r_i ,t_i )v_i with a_u (r_i ,t_i ) = ${\mathbf{u}}_{\mathbf{i}}^{†}$ A_D (r_i ,t_i ) and a_v (r_i ,t_i ) = ${\mathbf{v}}_{\mathbf{i}}^{†}$ A_D (r_i ,t_i ) and with i = 1,2. One thus gets 〈A_D (r₂ ,t ₂)${\mathbf{A}}_{\mathbf{D}}^{†}$(r₁ ,t ₁)〉 = 〈(a_u (r₂ ,t ₂)u₂ + a_v (r₂ ,t ₂)v₂ )(a_u (r₁ ,t ₁)u₁ + a_v (r₁ ,t ₁)v₁ )^†〉. Using Eq. (27) one gets

It is important to note that u_i and v_i are functions of r_i , t_i . In particular if r₁ = r₂ and t ₁ = t ₂, one obviously gets u₁ = u₂ and v₁ = v₂ and u₁ , u₂ can be chosen arbitrarily. Indeed, a totally depolarized light is still totally depolarized if one applies a unitary transformation to the electric field.

Let us briefly analyze the case where the field is the sum of two independent electric fields with orthogonal polarization states, E(r_i ,t_i ) = E_u (r_i ,t_i )u_i + E_v (r_i ,t_i )v_i . Let us also assume that each component satisfies the factorization condition introduced by Glauber [11] for complete electromagnetic coherence at order two (i.e. 〈E_m (r₂ , t ₂)$E_m^{*}$(r₁ , t ₁)〉 = ψ _m (r₂ , t ₂) ${\psi }_m^{*}$ (r₁ , t ₁) with m = u,v). In that case one gets μ_S = μ_I = 1. However 〈E(r₂ ,t ₂)E ^†(r₁ ,t ₁)〉 cannot be factorized in order to satisfy Eq. (21). Indeed, this field does not fulfil the condition at order two of complete electromagnetic coherence introduced by Glauber [11] since it is not perfectly polarized.

In summary, we have shown in this section that μ_S is the maximal value of the modulus of the standard degree of coherence one can obtain when:

The second quantity (i.e. μ_I ) is the standard degree of coherence obtained when the polarization states of each wave are orthogonal, in the basis (u₁ ,v₁ ) and (u₂ ,v₂ ), to those leading to a standard degree of coherence equal to μ_S .

We have thus analyzed in this section how the invariant degrees of coherence defined as the singular values of the normalized mutual coherence matrix M(r₁ ,r₂ ,t ₁,t ₂) are related to the standard degree of coherence in the case of scalar optical fields.

5. Illustration on simple examples

Let us illustrate these results on some simple examples. Let us first assume that t ₁ = t ₂ and that the processes are wide sense stationary so that time dependency can be dropped out. One will also assume that lights are totally depolarized so that A_D (r_i ) = E(r_i ) and Γ_i = I_d . Let us assume that E(r₂ ) = R_δrE(r₁ ) where δr = r₂ - r₁ and R_δr is a rotation matrix. One thus gets

This example corresponds to the propagation of a wave with rotation of the polarization without depolarization and nor loss of coherence on each rotated component. A singular value decomposition of M(r₁ ,r₂ ) is M(r₁ ,r₂ ) = ${\mathbf{N}}_{\mathbf{2}}^{†}$ D N₁ with N₂ = ${\mathbf{R}}_{\mathbf{\delta }\mathbf{r}}^{†}$, N₁ = I_d and D = I_d (which means that μ_S = μ_I = 1).

Let us now assume that

where α > β > 0, 〈|E_X (r₁ )|²〉 = 〈|E_Y (r₁ )|²〉 = 1, 〈E_X (r₁ )$E_Y^{*}$(r₁ )〉 = 0, 〈|δ_X (r₁ )|²〉 = 1 - α ², 〈|δ_Y (r₁ )|²〉 = 1 - β ², 〈δ_X (r₁ )${\delta }_Y^{*}$(r₁ )〉 = 0, and where E(r₁ ) is assumed independent from δ(r₁ ) = [${\delta }_X^{*}$(r₁ ), ${\delta }_Y^{*}$(r₁ )]^T. This example corresponds to the propagation of a wave with rotation of the polarization and different losses of coherence on the two components. Let us introduce

One still has

and thus

with

The singular decomposition of M(r₁ , r₂ ) is now M(r₁ , r₂ ) = ${\mathbf{N}}_{\mathbf{2}}^{†}$ D N₁ with N₂ = ${\mathbf{R}}_{\mathbf{\delta }\mathbf{r}}^{†}$, N₁ = I_d and D defined by Eq. (35) which means that μ_S = α and μ_I = β.

Since

and since Γ_i = I_d , one obtains with Eq. (4)

It is seen that this degree of coherence depends on the deterministic polarization rotation undergone by the wave. On the other hand, with Eq. (6), one obtains

This degree of coherence is invariant to the polarization rotation since, by construction, it is invariant with respect to any unitary transform of the fields. However, one can notice that if α = β = 1, that is, if E(r₂ ) = R_δrE(r₁ ), $\tilde{\mu }$ ²(r₁ ,r₂ ,t ₁,t ₂) is equal to 1/2 although E(r₁ ) and E(r₂ ) are related by a deterministic relation (i.e. without randomness). In other words, if one introduces E′(r₁ ) = R_δrE(r₁ ) the two components, E′_X(r₁ ) and E_X (r₂ ) on one side, and E′_Y(r₁ ) and E_Y (r₂ ) on the other side, correspond in that case to scalar waves perfectly coherent which leads to μ_S = μ_I = 1 while $\tilde{\mu }$ ²(r₁ ,r₂ ,t ₁,t ₂) = 1/2.

6. Conclusions

In summary, the spatio-temporal coherence properties of partially polarized light have been analyzed with an algebraic approach. This approach allows one to separate partial polarization and partial coherence and to introduce useful invariance properties. It has been shown that one thus obtains invariant degrees of coherence that characterize the intrinsic properties of the optical light independently of the particular experimental conditions. The relation of these degrees of coherence with measurable quantities has also been discussed. These results have been illustrated on some simple examples and compared with other recently introduced degrees of coherence [3, 5, 4].

There exists different perspectives to this work. For simplicity reasons, we have considered the case of 2D electric fields. However, it is straightforward to generalize this approach to 3D electric fields. Indeed, the generalization of the normalized mutual coherence matrix defined by Eq. (8) to the 3D case is direct and degrees of coherence can be defined as the square root of the eigenvalues of the matrix of Eq. (10). We have briefly discussed the relations between the proposed approach of this paper and the condition at order two of complete electromagnetic coherence introduced by Glauber [11]. This is a fundamental subject on which we are investigated more precise studies in our group.