## Invariant degrees of coherence of partially polarized light

Optics Express, Vol. 13, Issue 16, pp. 6051-6060 (2005)

http://dx.doi.org/10.1364/OPEX.13.006051

Acrobat PDF (180 KB)

### Abstract

The spatio-temporal properties of partially polarized light are analyzed in order to separate partial polarization and partial coherence. For that purpose we introduce useful invariance properties which allow one to characterize intrinsic properties of the optical light independently of the particular experimental conditions. This approach leads to new degrees of coherence and their relation with measurable quantities is discussed. These results are illustrated on some simple examples.

© 2005 Optical Society of America

## 1. Introduction

**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.

**r**

_{1}and

**r**

_{2}and times

*t*

_{1}and

*t*

_{2}of the complex random vectors

**E**(

**r**

_{1},

*t*

_{1}) and

**E**(

**r**

_{2},

*t*

_{2}) can be represented by the mutual coherence matrix [1, 2, 3

3. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**, 263–267 (2003). [CrossRef]

4. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. **29**, 328–330 (2004). [CrossRef] [PubMed]

5. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A **21**, 2205–2215 (2004). [CrossRef]

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}), defined by:

^{†}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

**r**

_{1}=

**r**

_{2}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*.

3. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**, 263–267 (2003). [CrossRef]

4. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. **29**, 328–330 (2004). [CrossRef] [PubMed]

5. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A **21**, 2205–2215 (2004). [CrossRef]

6. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A **20**, 78–84 (2003). [CrossRef]

3. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**, 263–267 (2003). [CrossRef]

4. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. **29**, 328–330 (2004). [CrossRef] [PubMed]

5. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A **21**, 2205–2215 (2004). [CrossRef]

6. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A **20**, 78–84 (2003). [CrossRef]

7. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express **11**, 1137–1142 (2003). [CrossRef] [PubMed]

**312**, 263–267 (2003). [CrossRef]

**29**, 328–330 (2004). [CrossRef] [PubMed]

**21**, 2205–2215 (2004). [CrossRef]

**312**, 263–267 (2003). [CrossRef]

*tr*[

*W*] denotes the trace of

*W*. From this expression one can define a degree of coherence in the space time domain by

7. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express **11**, 1137–1142 (2003). [CrossRef] [PubMed]

**29**, 328–330 (2004). [CrossRef] [PubMed]

**21**, 2205–2215 (2004). [CrossRef]

7. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express **11**, 1137–1142 (2003). [CrossRef] [PubMed]

**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.

## 2. Invariant degrees of coherence

**E**(

**r**

_{1},

*t*

_{1}) and

**E**(

**r**

_{2},

*t*

_{2}) are modified by the application of unitary transformations. Since 〈

**A**

_{U}(

**r**

_{2},

*t*

_{2})

**r**

_{1},

*t*

_{1})) =

**U**

_{2}〈

**E**(

**r**

_{2},

*t*

_{2})

**E**

^{†}(

**r**

_{1},

*t*

_{1})〉

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) and for

**U**

_{2}Ω(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})

**U**

_{1}and

**U**

_{2}.

**r**

_{1},

**r**

_{2},

*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

**r**

_{1}=

**r**

_{2}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.

**E**(

**r**

_{1},

*t*

_{1}) and

**E**(

**r**

_{2},

*t*

_{2}) have coherency matrices Γ

_{e}(

**r**

_{1},

*t*

_{1}) and Γ

_{e}(

**r**

_{2},

*t*

_{2}) and mutual coherence matrix Ω

_{e}(

**r**

_{1},

**r**

_{2},

*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*(

**r**

_{1},

*t*

_{1}) =

**B**

_{1}

**E**(

**r**

_{1},

*t*

_{1}) and

**A**(

**r**

_{2},

*t*

_{2}) =

**B**

_{2}

**E**(

**r**

_{2},

*t*

_{2}) where

**B**

_{1}and

**B**

_{2}are non singular Jones matrices. The mutual coherence matrix of the electric fields

*A*(

**r**

_{1},

*t*

_{1}) and

**A**(

**r**

_{2},

*t*

_{2}) will be denoted Ω

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) in the following.

*𝓒;*[

**E**(

**r**

_{1},

*t*

_{1}),

**E**(

**r**

_{2},

*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**(

**r**

_{1},

*t*

_{1}),

**E**(

**r**

_{2},

*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**(

**r**

_{1},

*t*

_{1}),

**E**(

**r**

_{2},

*t*

_{2})]corresponds to {

**M**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}),

**I**

_{d},

**I**

_{d}}with

**I**

_{d}is the identity matrix in dimension two. Since

**M**(

**r**

_{1},

**r**

_{2},

*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(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) and coherency matrices Γ(

**r**

_{i},

*t*

_{i}) with = 1,2.

**U**

_{1},

**U**

_{2}, totally depolarized lights with normalized mutual coherence matrices

**M**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) and

**U**

_{2}

**M**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})

**U**

_{2}=

**N**

_{2}and to

**U**

_{1}=

**N**

_{1}so that

**N**

_{2}

**M**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})

*D*(

**r**

_{1},

**r**

_{2},

*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**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) as the representative of the equivalence class

*𝓒*[

**E**(

**r**

_{1},

*t*

_{1}),

**E**(

**r**

_{2},

*t*

_{2})] and thus define the invariant degrees of coherence from

**D**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}).

**D**(

**r**

_{1},

**r**

_{2},

*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**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}). The diagonal elements

*μ*

_{S}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}),

*μ*

_{I}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) of

**D**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) are the singular values of

**M**(

**r**

_{1},

**r**

_{2},

*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**(

**r**

_{1},

**r**

_{2},

*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}.

**D**

^{2}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) is a diagonal matrix of diagonal values

**M**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})

**M**

^{†}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}).

**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

11. R. J. Glauber, “The Quantum Theory of Optical Coherence,” Phys. Rev. **6**, 2529–2539, (1963) [CrossRef]

**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})

**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

*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}〉.

**E**(

**r**

_{1},

*t*

_{1}) and

**E**(

**r**

_{2},

*t*

_{2}) is Ω

_{e}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) = 〈

**E**(

**r**

_{2},

*t*

_{2})

**E**

^{†}(

**r**

_{1},

*t*

_{1})〉 and one thus has Ω

_{e}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) =

**U**

_{2}Ω

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) is the mutual coherence matrix between

**A**(

**r**

_{1},

*t*

_{1}) and

**A**(

**r**

_{2},

*t*

_{2}) (i.e. Ω

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) = 〈

**A**(

**r**

_{2},

*t*

_{2})

**A**

^{†}(

**r**

_{1},

*t*

_{1})〉). Obviously Γ

_{e}(

**r**

_{i},

*t*

_{i}) =

**U**

_{i}Λ

_{a}(

**r**

_{i},

*t*

_{i})

**M**

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) is thus simple since

*P, Q*=

*X,Y*. The normalized mutual coherence matrix

**M**

_{a}(

**r**

_{1},

**r**

_{2},

*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}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) corresponds to a change of basis. Indeed,

**M**

_{e}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) is written in the basis in which the light is analyzed while

**M**

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) is expressed in the basis defined by the eigenvectors of the coherency matrices Γ

_{e}(

**r**

_{i},

*t*

_{i}).

**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

_{a}(

**r**

_{i},

*t*

_{i}) = Ω

_{a}(

**r**

_{i},

**r**

_{i},

*t*

_{i},

*t*

_{i}). One can introduce the pseudo inverse

**r**

_{i},

*t*

_{i}) of

**r**

_{i},

*t*

_{i})

**D**

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) =

*z*

**M**

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) (where

**D**

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) is defined with Eq. (9) and where

*z*is a complex number of modulus one). One thus gets

*μ*

_{S}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) = |

*η*

_{XX}(

**r**

_{1},

**r**

_{2},

*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.

**r**

_{1}and at one time

*t*

_{1}(or only at

**r**

_{2}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

**r**

_{i},

*t*

_{i})=

**U**

_{i}

**r**

_{i},

*t*

_{i})

11. R. J. Glauber, “The Quantum Theory of Optical Coherence,” Phys. Rev. **6**, 2529–2539, (1963) [CrossRef]

**Ψ**(

**r**,

*t*) so that

11. R. J. Glauber, “The Quantum Theory of Optical Coherence,” Phys. Rev. **6**, 2529–2539, (1963) [CrossRef]

**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}) =

**E**(

**r**

_{i},

*t*

_{i}). The mutual coherence matrix of

**E**(

**r**

_{i},

*t*

_{i}) is Ω(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) =

**U**

_{2}Ω

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})

_{a}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) =

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})

**U**

_{1}and thus

_{a}(

**r**

_{i},

*t*

_{i}) are thus also diagonal as Ω

_{a}{

**r**

_{1},

**r**

_{2},

*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}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2})| = 1. The invariant degrees of coherence

*μ*

_{S}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) is thus also equal to 1.

**6**, 2529–2539, (1963) [CrossRef]

*μ*

_{S}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) = 1. In particular, these conditions do not correspond to

*μ*

_{S}(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) =

*μ*

_{I}(

**r**

_{1},

**r**

_{2},

*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

*μ*

_{S}and

*μ*

_{I}have a simple physical interpretation. Indeed, one can follow the concepts developed in [9

9. R. S. Cloude and K. P. Papathanassiou, “Polarimetric SAR interferometry,” IEEE Trans. Geosci. Remote sens. **36**, 1551–1565 (1998). [CrossRef]

**E**(

**r**

_{1},

*t*

_{1}) and

**E**(

**r**

_{2},

*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].

**U**

_{1}and

**U**

_{2}. The optical fields in front of the polarizer become

**A**

_{U}(

**r**

_{1},

*t*

_{1}) =

**U**

_{1}

**E**(

**r**

_{1},

*t*

_{1}) and

**A**

_{U}(

**r**

_{2},

*t*

_{2}) =

**U**

_{2}

**E**(

**r**

_{2},

*t*

_{2}). The actions of the polarizers can be represented by the projection of

**A**

_{U}(

**r**

_{1},

*t*

_{1}) and

**A**

_{U}(

**r**

_{2},

*t*

_{2}) on the vector

**e**

_{1}of unitary modulus.

**U**

_{1},

**U**

_{2}which maximize

*η*. Using Eq. (8) and if one introduces

**k**

_{i}= Γ

^{1/2}(

**r**

_{i},

*t*

_{i})

**e**

_{1}one gets

*α*and

*β*different to 0,

*η*is invariant if one changes

**k**

_{1}into

*α*

**k**

_{1}and

**k**

_{2}into

*β*

**k**

_{2}, the maximization of

*η*is obtained by finding the unitary vectors

**a**

_{1}and

**a**

_{2}which optimize

**u**

_{i}||

^{2}= ||

**v**

_{i}||

^{2}= 1 and

**v**

_{i}= 0 where

*i*= 1,2. One thus gets

*η*is

*μ*

_{S}and that this value is reached when

**a**

_{2}=

*z*

_{2}

**u**

_{2}and

**a**

_{1}=

*z*

_{1}

**u**

_{1}where

*z*

_{1}and

*z*

_{2}are complex numbers of modulus equal to one. If

**a**

_{2}is proportional to

**v**

_{2}and

**a**

_{1}is proportional to

**v**

_{1}, one gets

*η*=

*μ*

_{I}. Finally, if

**a**

_{2}is proportional to

**u**

_{2}and

**a**

_{1}is proportional to

**v**

_{1}, or if

**a**

_{2}is proportional to

**v**

_{2}and

**a**

_{1}is proportional to

**u**

_{1}, one gets

*η*= 0.

**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

**r**

_{1}and

**r**

_{2}and times

*t*

_{1}and

*t*

_{2}of the electric fields

**E**(

**r**

_{1},

*t*

_{1}) and

**E**(

**r**

_{2},

*t*

_{2}), the corresponding totally depolarized lights

**A**

_{D}(

**r**

_{1},

*t*

_{1}) = Γ

^{-1/2}(

**r**

_{1},

*t*

_{1})

**E**(

**r**

_{1},

*t*

_{1}) and

**A**

_{D}(

**r**

_{2},

*t*

_{2}) = Γ

^{-1/2}(

**r**

_{2},

*t*

_{2})

**E**(

**r**

_{2},

*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}) =

**A**

_{D}(

**r**

_{i},

*t*

_{i}) and

*a*

_{v}(

**r**

_{i},

*t*

_{i}) =

**A**

_{D}(

**r**

_{i},

*t*

_{i}) and with

*i*= 1,2. One thus gets 〈

**A**

_{D}(

**r**

_{2},

*t*

_{2})

**r**

_{1},

*t*

_{1})〉 = 〈(

*a*

_{u}(

**r**

_{2},

*t*

_{2})

**u**

_{2}+

*a*

_{v}(

**r**

_{2},

*t*

_{2})

**v**

_{2})(

*a*

_{u}(

**r**

_{1},

*t*

_{1})

**u**

_{1}+

*a*

_{v}(

**r**

_{1},

*t*

_{1})

**v**

_{1})

^{†}〉. Using Eq. (27) one gets

**u**

_{i}and

**v**

_{i}are functions of

**r**

_{i},

*t*

_{i}. In particular if

**r**

_{1}=

**r**

_{2}and

*t*

_{1}=

*t*

_{2}, one obviously gets

**u**

_{1}=

**u**

_{2}and

**v**

_{1}=

**v**

_{2}and

**u**

_{1},

**u**

_{2}can be chosen arbitrarily. Indeed, a totally depolarized light is still totally depolarized if one applies a unitary transformation to the electric field.

**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

**6**, 2529–2539, (1963) [CrossRef]

*E*

_{m}(

**r**

_{2},

*t*

_{2})

**r**

_{1},

*t*

_{1})〉 =

*ψ*

_{m}(

**r**

_{2},

*t*

_{2})

**r**

_{1},

*t*

_{1}) with

*m*=

*u,v*). In that case one gets

*μ*

_{S}=

*μ*

_{I}= 1. However 〈

**E**(

**r**

_{2},

*t*

_{2})

**E**

^{†}(

**r**

_{1},

*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

**6**, 2529–2539, (1963) [CrossRef]

*μ*

_{S}is the maximal value of the modulus of the standard degree of coherence one can obtain when:

- one puts in front of each wave an optical modulator with variable unitary Jones matrix followed by parallel linear polarizers,
- the scalar fields correspond to the complex amplitude of the two obtained perfectly polarized waves.

*μ*

_{I}) is the standard degree of coherence obtained when the polarization states of each wave are orthogonal, in the basis (

**u**

_{1},

**v**

_{1}) and (

**u**

_{2},

**v**

_{2}), to those leading to a standard degree of coherence equal to

*μ*

_{S}.

## 5. Illustration on simple examples

*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**(

**r**

_{2}) =

**R**

_{δr}

**E**(

**r**

_{1}) where

**δr**=

**r**

_{2}-

**r**

_{1}and

**R**

_{δr}is a rotation matrix. One thus gets

**M**(

**r**

_{1},

**r**

_{2}) is

**M**(

**r**

_{1},

**r**

_{2}) =

**D N**

_{1}with

**N**

_{2}=

**N**

_{1}=

**I**

_{d}and

**D**=

**I**

_{d}(which means that

*μ*

_{S}=

*μ*

_{I}= 1).

*α*>

*β*> 0, 〈|

*E*

_{X}(

**r**

_{1})|

^{2}〉 = 〈|

*E*

_{Y}(

**r**

_{1})|

^{2}〉 = 1, 〈

*E*

_{X}(

**r**

_{1})

**r**

_{1})〉 = 0, 〈|

*δ*

_{X}(

**r**

_{1})|

^{2}〉 = 1 -

*α*

^{2}, 〈|

*δ*

_{Y}(

**r**

_{1})|

^{2}〉 = 1 -

*β*

^{2}, 〈

*δ*

_{X}(

**r**

_{1})

**r**

_{1})〉 = 0, and where

**E**(

**r**

_{1}) is assumed independent from

*δ*(

**r**

_{1}) = [

**r**

_{1}),

**r**

_{1})]

^{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

**M**(

**r**

_{1},

**r**

_{2}) is now

**M**(

**r**

_{1},

**r**

_{2}) =

**D N**

_{1}with

**N**

_{2}=

**N**

_{1}=

**I**

_{d}and

**D**defined by Eq. (35) which means that

*μ*

_{S}=

*α*and

*μ*

_{I}=

*β*.

## 6. Conclusions

**312**, 263–267 (2003). [CrossRef]

**21**, 2205–2215 (2004). [CrossRef]

**29**, 328–330 (2004). [CrossRef] [PubMed]

**6**, 2529–2539, (1963) [CrossRef]

## Appendix A

**u**

_{2}

**v**

_{2}

**N**

_{1}. One thus has

*μ*

_{S}-

*μ*

_{I})

**u**

_{2}

**a**

_{1}and

*μ*

_{I}(

**N**

_{2}

**a**

_{2})

^{†}

**N**

_{1}

**a**

_{1}. This modulus is maximal if these two complex numbers have the same phase and have maximal modulus. The modulus of (

**N**

_{2}

**a**

_{2})

^{†}

**N**

_{1}

**a**

_{1}cannot be greater than one since |(

**N**

_{2}

**a**

_{2})

^{†}

**N**

_{1}

**a**

_{1}| ≤ ||(

**N**

_{2}

**a**

_{2})|| ||

**N**

_{1}

**a**

_{1}|| and ||

**N**

_{i}

**a**

_{i}|| = 1. The choice

**a**

_{1}=

*z*

_{1}

**u**

_{1}and

**a**

_{2}=

*z*

_{2}

**u**

_{2}, where

*z*

_{1}and

**z**

_{2}are complex numbers of modulus one, clearly maximizes |

**u**

_{2}

**a**

_{1}|. Let us show that it also maximizes |(

**N**

_{2}

**a**

_{2})

^{†}

**N**

_{1}

**a**

_{1}|. Indeed, one has |

*z*

_{1}(

**N**

_{2}

**u**

_{2})

^{†}

**N**

_{1}

**u**

_{1}| = |

**u**

_{2}

**v**

_{2}

**u**

_{1}| = |

**u**

_{2}

**u**

_{1}| = 1 which is the maximal possible value of |(

**N**

_{2}

**a**

_{2})

^{†}

**N**

_{1}

**a**

_{1}| as shown above. Furthermore, since (

*z*

_{2}

**u**

_{2})

^{†}

**u**

_{2}

*z*

_{1}

**a**

_{1}) =

*z*

_{1}and (

**N**

_{2}

**a**

_{2})

^{†}

**N**

_{1}

**a**

_{1}=

*z*

_{1}, these two complex numbers have the same phase. This last property proves that the choice

**a**

_{1}=

*z*

_{1}

**u**

_{1}and

**a**

_{2}=

*z*

_{2}

**u**

_{2}maximizes

*η*.

## Acknowledgments

## References and links

1. | J. W. Goodman, “Some first-order properties of light waves,” in |

2. | Ph. Réfrégier, |

3. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

4. | T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. |

5. | J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A |

6. | F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A |

7. | J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express |

8. | P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A: Pure Appl. Opt. |

9. | R. S. Cloude and K. P. Papathanassiou, “Polarimetric SAR interferometry,” IEEE Trans. Geosci. Remote sens. |

10. | L. Mandel and E. Wolf, |

11. | R. J. Glauber, “The Quantum Theory of Optical Coherence,” Phys. Rev. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.6600) Coherence and statistical optics : Statistical optics

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 17, 2005

Revised Manuscript: July 21, 2005

Published: August 8, 2005

**Citation**

Philippe Réfrégier and François Goudail, "Invariant degrees of coherence of partially polarized light," Opt. Express **13**, 6051-6060 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-6051

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### References

- J. W. Goodman, �??Some first-order properties of light waves,�?? in Statistical Optics, 116�??156 (John Wiley and Sons, Inc., New York, 1985).
- Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, New-York, 2004).
- E. Wolf, �??Unified theory of coherence and polarization of random electromagnetic beams,�?? Phys. Lett. A 312, 263�??267 (2003). [CrossRef]
- T. Setälä, J. Tervo, and A. T. Friberg, �??Complete electromagnetic coherence in the space-frequency domain,�?? Opt. Lett. 29, 328�??330 (2004). [CrossRef] [PubMed]
- J. Tervo, T. Setälä, and A. T. Friberg, �??Theory of partially coherent electromagnetic fields in the space-frequency domain,�?? J. Opt. Soc. Am. A 21, 2205�??2215 (2004). [CrossRef]
- F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, �??Coherent-mode decomposition of partially polarized, partially coherent sources,�?? J. Opt. Soc. Am. A 20, 78�??84 (2003). [CrossRef]
- J. Tervo, T. Setälä, and A. T. Friberg, �??Degree of coherence of electromagnetic fields,�?? Opt. Express 11, 1137�??1142 (2003). [CrossRef] [PubMed]
- P. Vahimaa and J. Tervo, �??Unified measures for optical fields: degree of polarization and effective degree of coherence,�?? J. Opt. A: Pure Appl. Opt. 6, 41�??44 (2004). [CrossRef]
- R. S. Cloude and K. P. Papathanassiou, �??Polarimetric SAR interferometry,�?? IEEE Trans. Geosci. Remote sens. 36, 1551�??1565 (1998). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 160�??170 (Cambridge University Press, New York, 1995).
- R. J. Glauber, �??The Quantum Theory of Optical Coherence,�?? Phys. Rev. 6, 2529-2539, (1963) [CrossRef]

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