Abstract

Given the values of the degree of polarization of the fields at the pinholes in a Young interferometer, the maximum attainable visibility under unitary transformations is determined when the illuminating beam is mean-square light. Analytical expressions are also obtained for both the field vector (in the mean-square sense) and the cross-spectral density matrix associated with this kind of beams. A comparative summary is also provided of the main characteristics of well-known types of random electromagnetic fields frequently handled in the literature.

©2009 Optical Society of America

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].

Wr1r2=f*(r1)f(r2),

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,

ŴijŴrirj=E(ri)E(rj),i,j=1,2,

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 (μs, μI) 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

g12=gr1r2=Tr(Ŵ12Ŵ12)+2DetŴ12Tr(Ŵ11)Tr(Ŵ22),

reduces, for MSC fields, to the expression

g12=μSTF2+12(1P12)(1P22),

where μSTF2=Tr(Ŵ12Ŵ12)Tr(Ŵ11)Tr(Ŵ22) 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 = ∣μW2 max, where μW2=TrŴ122TrŴ11TrŴ22 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 μSTF 2 for MSC light with fixed degrees of polarization at two points. In other words, we have to get the maximal value of Tr(Ŵ12Ŵ12)Tr(Ŵ11)Tr(Ŵ22) in terms of P 1 and P 2. After some algebra (see below), we arrive to the final expression

[gr1r2]max=12[1+P(r1)P(r2)+(1P2(r1))(1P2(r2))].

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

Ŵr1r2=λ1Ψ1(r1)Ψ1(r2)+λ2Ψ2(r1)Ψ2(r2),

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

Ŵr1r2=Ĥ(r1)Ĥ(r2),

where Ĥ is a 2×2 matrix, namely,

Ĥ(r)=(λ1ψ11(r)λ1Ψ12(r)λ2Ψ21(r)λ2Ψ22(r)),

with Det Ĥ ≠ 0 and Ψi = (Ψ i1, Ψ i2), i = 1,2. In terms of its singular value decomposition, matrix Ĥ reads

Ĥ(r)=V̂(r)D̂(r)Û(r),

where Û and are unitary matrices, and

D̂(r)=(α(r)00β(r)),

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

P(r)=α2(r)β2(r)α2(r)+β2(r).

By using Eq. (9), the CDT becomes

Ŵr1r2=Û(r1)D̂(r1)V̂(r1)V̂(r2)D̂(r2)Û(r2).

Then

Tr[Ŵr1r2Ŵr1r2]=Tr[D̂2(r1)Ŝr1r2D̂2(r2)Ŝr1r2],

where

Ŝr1r2=V̂(r1)V̂(r2),

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

Tr[Ŵr1r2Ŵr1r2]=
=cos2θ[α2(r1)α2(r2)+β2(r1)β2(r2)]+sin2θ[α2(r1)β2(r2)+β2(r1)α2(r2)],

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

S11=cosθr1r2=S22,
S12=sinθr1r2=S21,

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

1+P(r1)P(r2)=2α2(r1)α2(r2)+2β2(r1)β2(r2)TrŴr1r1TrŴr2r2,

and

1P(r1)P(r2)=2α2(r1)β2(r2)+2β2(r1)α2(r2)TrŴr1r1TrŴr2r2.

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

μSTF2=Tr(Ŵ12Ŵ12)Tr(Ŵ11)Tr(Ŵ22)=12[1+P(r1)P(r2)cos2θr1r2],

so that

gr1r2=12[1+P(r1)P(r2)cos2θr1r2+(1P2(r1))(1P2(r2))].

It is important to remark that Eq. (19) provides an analytical expression of μSTF 2 for a general MSC field. Moreover, Eq. (20) gives ∣μW2 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)

[gr1r2]max=12[1+P(r1)P(r2)+(1P2(r1))(1P2(r2))],

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

Ŝr1r2=V̂(r1)V̂(r2)=(expi[φ(r2)φ(r2)]00expi[ϕ(r2)ϕ(r1)]).

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

V̂(r)=M̂(r)Q̂,

where (r) denotes a diagonal matrix whose nonzero elements are complex exponentials, and 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))

Ŵr1r2=Û(r1)D̂0(r1)D̂0(r2)Û(r2),

where

D̂0(r)=(α(r)expiφ(r)00β(r)expiϕ(r)).

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

E(r)=E0D̂0(r)Û(r),

where E 0 = (u,v) is a 1×2 vector whose components take complex random values fulfilling <∣u2 >=<∣v∣2 > and <u * v >= 0. From this equation, we see at once that E(r) can also be expressed as follows

E(r)=ε1Ψ˜1(r)+ε2Ψ˜2(r),

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

Ψ˜1(r)α(r)expiφ(r)(U11(r)U12(r)),
Ψ˜2(r)β(r)expiϕ(r)(U21(r)U22(r)),

the symbol ∝ denoting proportionality, and Uij, 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 μSTF 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 = ∣μW2 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])

E(r)=E0f(r)Û(r),

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 μs = μI = 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)

E(r)=ε1Ψ1(r)+ε2Ψ2(r),

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

E(r)=E0D̂0(r)Û(r)=ε1Ψ˜1(r)+ε2Ψ˜2(r),

where 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,

{T}{V}{R*}{R}.

This layer structure is illustrated in Fig. 1.

 

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

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Acknowledgments

This work has been supported by the Ministerio de Educación y Ciencia of Spain, project FIS2007-63396, and by CM-UCM, Research Group Program 2008, No. 910335.

References and links

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

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

3. M. Mujat and A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. 28, 2153–2155 (2003). [CrossRef]   [PubMed]  

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

5. H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268–274 (2005). [CrossRef]  

6. Ph. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051–6060 (2005). [CrossRef]   [PubMed]  

7. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a young interference pattern,” Opt. Lett. 31, 688–690 (2006). [CrossRef]   [PubMed]  

8. Ph. Réfrégier and A. Roueff, “Coherence polarization filtering and relation with intrinsic degrees of coehrence,” Opt. Lett. 31, 1175–1177 (2006). [CrossRef]   [PubMed]  

9. Ph. Réfrégier and A. Roueff, “Linear relations of partially polarized and coherent electromagnetic fields,” Opt. Lett. 31, 2827–2829 (2006). [CrossRef]   [PubMed]  

10. F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006). [CrossRef]   [PubMed]  

11. Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63–72 (2006). [CrossRef]  

12. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007). [CrossRef]   [PubMed]  

13. Ph. Réfrégier and A. Roueff, “Intrinsic Coherence: A New Concept in Polarization and Coherence Theory,” Opt. Photon. News 18, 30–35 (2007). [CrossRef]  

14. Ph. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007). [CrossRef]   [PubMed]  

15. R. Martínez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007). [CrossRef]   [PubMed]  

16. R. Martínez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504–1506 (2007). [CrossRef]   [PubMed]  

17. R. Martínez-Herrero and P. M. Mejías, “Electromagnetic fields that remain totally polarized under propagation,” Opt. Commun. 279, 20–22 (2007). [CrossRef]  

18. E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007). [CrossRef]   [PubMed]  

19. M. Santarsiero, “Polarization invariance in a Young interferometer,” J. Opt. Soc. Am. A 24, 3493–3499 (2007). [CrossRef]  

20. R. Martínez-Herrero and P. M. Mejías, “On the vectorial fields with position-independent stochastic behavior,” Opt. Lett. 33, 195–197 (2008). [CrossRef]   [PubMed]  

21. Ph. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008). [CrossRef]   [PubMed]  

22. R. Martínez-Herrero and P. M. Mejías, “Equivalence between optimum Young’s fringe visibility and position-independent stochastic behaviour of electromagnetic fields,” J. Opt. Soc. Am. A 25, 1902–1905 (2008). [CrossRef]  

23. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995).

24. J. Perina, Coherence of Light (Van Nostrand Reinhold Co., London, 1971).

References

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  1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [Crossref]
  2. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express 11, 1137–1142 (2003).
    [Crossref] [PubMed]
  3. M. Mujat and A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. 28, 2153–2155 (2003).
    [Crossref] [PubMed]
  4. T. Setälä, J. Tervo, and A. T. Friberg, “Complete coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [Crossref] [PubMed]
  5. H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268–274 (2005).
    [Crossref]
  6. Ph. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051–6060 (2005).
    [Crossref] [PubMed]
  7. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a young interference pattern,” Opt. Lett. 31, 688–690 (2006).
    [Crossref] [PubMed]
  8. Ph. Réfrégier and A. Roueff, “Coherence polarization filtering and relation with intrinsic degrees of coehrence,” Opt. Lett. 31, 1175–1177 (2006).
    [Crossref] [PubMed]
  9. Ph. Réfrégier and A. Roueff, “Linear relations of partially polarized and coherent electromagnetic fields,” Opt. Lett. 31, 2827–2829 (2006).
    [Crossref] [PubMed]
  10. F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
    [Crossref] [PubMed]
  11. Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63–72 (2006).
    [Crossref]
  12. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
    [Crossref] [PubMed]
  13. Ph. Réfrégier and A. Roueff, “Intrinsic Coherence: A New Concept in Polarization and Coherence Theory,” Opt. Photon. News 18, 30–35 (2007).
    [Crossref]
  14. Ph. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007).
    [Crossref] [PubMed]
  15. R. Martínez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007).
    [Crossref] [PubMed]
  16. R. Martínez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504–1506 (2007).
    [Crossref] [PubMed]
  17. R. Martínez-Herrero and P. M. Mejías, “Electromagnetic fields that remain totally polarized under propagation,” Opt. Commun. 279, 20–22 (2007).
    [Crossref]
  18. E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007).
    [Crossref] [PubMed]
  19. M. Santarsiero, “Polarization invariance in a Young interferometer,” J. Opt. Soc. Am. A 24, 3493–3499 (2007).
    [Crossref]
  20. R. Martínez-Herrero and P. M. Mejías, “On the vectorial fields with position-independent stochastic behavior,” Opt. Lett. 33, 195–197 (2008).
    [Crossref] [PubMed]
  21. Ph. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
    [Crossref] [PubMed]
  22. R. Martínez-Herrero and P. M. Mejías, “Equivalence between optimum Young’s fringe visibility and position-independent stochastic behaviour of electromagnetic fields,” J. Opt. Soc. Am. A 25, 1902–1905 (2008).
    [Crossref]
  23. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995).
  24. J. Perina, Coherence of Light (Van Nostrand Reinhold Co., London, 1971).

2008 (3)

2007 (8)

2006 (5)

2005 (2)

H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268–274 (2005).
[Crossref]

Ph. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051–6060 (2005).
[Crossref] [PubMed]

2004 (1)

2003 (3)

Borghi, R.

Dogariu, A.

Friberg, A. T.

Gori, F.

Goudail, F.

Lee, H.

Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63–72 (2006).
[Crossref]

Li, Y.

Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63–72 (2006).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995).

Martínez-Herrero, R.

Mejías, P. M.

Mujat, M.

Perina, J.

J. Perina, Coherence of Light (Van Nostrand Reinhold Co., London, 1971).

Réfrégier, Ph.

Roueff, A.

Roychowdhury, H.

H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268–274 (2005).
[Crossref]

Santarsiero, M.

Setälä, T.

Tervo, J.

Wolf, E.

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[Crossref] [PubMed]

Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63–72 (2006).
[Crossref]

H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268–274 (2005).
[Crossref]

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995).

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

R. Martínez-Herrero and P. M. Mejías, “Electromagnetic fields that remain totally polarized under propagation,” Opt. Commun. 279, 20–22 (2007).
[Crossref]

H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268–274 (2005).
[Crossref]

Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63–72 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (13)

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[Crossref] [PubMed]

Ph. Réfrégier and A. Roueff, “Coherence polarization filtering and relation with intrinsic degrees of coehrence,” Opt. Lett. 31, 1175–1177 (2006).
[Crossref] [PubMed]

Ph. Réfrégier and A. Roueff, “Linear relations of partially polarized and coherent electromagnetic fields,” Opt. Lett. 31, 2827–2829 (2006).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
[Crossref] [PubMed]

M. Mujat and A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. 28, 2153–2155 (2003).
[Crossref] [PubMed]

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

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[Crossref] [PubMed]

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007).
[Crossref] [PubMed]

Ph. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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Opt. Photon. News (1)

Ph. Réfrégier and A. Roueff, “Intrinsic Coherence: A New Concept in Polarization and Coherence Theory,” Opt. Photon. News 18, 30–35 (2007).
[Crossref]

Phys. Lett. A (1)

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

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Figures (1)

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

Equations (35)

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Wr1r2=f*(r1)f(r2),
ŴijŴrirj=E(ri)E(rj),i,j=1,2,
g12=gr1r2=Tr(Ŵ12Ŵ12)+2DetŴ12Tr(Ŵ11)Tr(Ŵ22),
g12=μSTF2+12(1P12)(1P22),
[gr1r2]max=12[1+P(r1)P(r2)+(1P2(r1))(1P2(r2))].
Ŵr1r2=λ1Ψ1(r1)Ψ1(r2)+λ2Ψ2(r1)Ψ2(r2),
Ŵr1r2=Ĥ(r1)Ĥ(r2),
Ĥ(r)=(λ1ψ11(r)λ1Ψ12(r)λ2Ψ21(r)λ2Ψ22(r)),
Ĥ(r)=V̂(r)D̂(r)Û(r),
D̂(r)=(α(r)00β(r)),
P(r)=α2(r)β2(r)α2(r)+β2(r).
Ŵr1r2=Û(r1)D̂(r1)V̂(r1)V̂(r2)D̂(r2)Û(r2).
Tr[Ŵr1r2Ŵr1r2]=Tr[D̂2(r1)Ŝr1r2D̂2(r2)Ŝr1r2],
Ŝr1r2=V̂(r1)V̂(r2),
Tr[Ŵr1r2Ŵr1r2]=
=cos2θ[α2(r1)α2(r2)+β2(r1)β2(r2)]+sin2θ[α2(r1)β2(r2)+β2(r1)α2(r2)],
S11=cosθr1r2=S22,
S12=sinθr1r2=S21,
1+P(r1)P(r2)=2α2(r1)α2(r2)+2β2(r1)β2(r2)TrŴr1r1TrŴr2r2,
1P(r1)P(r2)=2α2(r1)β2(r2)+2β2(r1)α2(r2)TrŴr1r1TrŴr2r2.
μSTF2=Tr(Ŵ12Ŵ12)Tr(Ŵ11)Tr(Ŵ22)=12[1+P(r1)P(r2)cos2θr1r2],
gr1r2=12[1+P(r1)P(r2)cos2θr1r2+(1P2(r1))(1P2(r2))].
[gr1r2]max=12[1+P(r1)P(r2)+(1P2(r1))(1P2(r2))],
Ŝr1r2=V̂(r1)V̂(r2)=(expi[φ(r2)φ(r2)]00expi[ϕ(r2)ϕ(r1)]).
V̂(r)=M̂(r)Q̂ ,
Ŵr1r2=Û(r1)D̂0(r1)D̂0(r2)Û(r2),
D̂0(r)=(α(r)expiφ(r)00β(r)expiϕ(r)).
E(r)=E0D̂0(r)Û(r),
E(r)=ε1Ψ˜1(r)+ε2Ψ˜2(r),
Ψ˜1(r)α(r)expiφ(r)(U11(r)U12(r)),
Ψ˜2(r)β(r)expiϕ(r)(U21(r)U22(r)),
E(r)=E0f(r)Û(r),
E(r)=ε1Ψ1(r)+ε2Ψ2(r),
E(r)=E0D̂0(r)Û(r)=ε1Ψ˜1(r)+ε2Ψ˜2(r),
{T}{V}{R*}{R}.

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