This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 155.33.16.124

This content was downloaded on 02/06/2014 at 21:13

Please note that terms and conditions apply.

Partially coherent electromagnetic beams propagating through double-wedge depolarizers

View the table of contents for this issue, or go to the journal homepage for more

2014 J. Opt. 16 035708

(http://iopscience.iop.org/2040-8986/16/3/035708)

Home Search Collections Journals About Contact us My IOPscience

Journal of Optics

J. Opt. 16 (2014) 035708 (17pp) doi:10.1088/2040-8978/16/3/035708

Partially coherent electromagnetic beamspropagating through double-wedgedepolarizersJ Carlos G de Sande1, Gemma Piquero2, Massimo Santarsiero3 andFranco Gori3

1 Departamento de Circuitos y Sistemas, Universidad Politécnica de Madrid, Campus Sur, E-28031Madrid, Spain2 Departamento de Óptica, Universidad Complutense de Madrid, E-28040 Madrid, Spain3 Dipartimento di Ingegneria, Università Roma Tre, and CNISM, Via V. Volterra 62, I-00146 Rome, Italy

E-mail: jcgsande@ics.upm.es

Received 26 November 2013, revised 11 January 2014Accepted for publication 14 January 2014Published 3 March 2014

AbstractThe irradiance and polarization characteristics of quasi-monochromatic partially coherentelectromagnetic beams are analyzed when they propagate after passing through a deterministiclinear optical element, i.e., an optical element that can be represented by a Jones matrix. Aclass of such optical elements, which includes double-wedge depolarizers and polarizationgratings, is defined and studied in detail. Analytical expressions are obtained for the case ofdouble-wedge depolarizers and examples are given for an incident Gaussian Schell-modelbeam. For such an input beam, the effects on the irradiance and degree of polarization of thefield propagating beyond the optical element are investigated in detail. A rich variety ofbehaviors is obtained by varying the beam size, coherence width and polarization state of theinput field. The results not only provide a mathematical extension of well-known results to thedomain of partial coherence, but they also exemplify mixing between coherence andpolarization, which is, of course, not possible if, for example, fully spatially coherent fields areanalyzed.

Keywords: coherence, polarization, partially coherent electromagnetic sourcesPACS numbers: 42.25.Ja, 42.25.Kb

(Some figures may appear in colour only in the online journal)

1. Introduction

There is presently an increasing interest in optical elementsthat reduce the degree of polarization of monochromatic lightbeams, in order to use them in those applications where a lowdegree of polarization is required for an optimum performanceof an optical system [1–10]. Among these devices we mentionthe double-wedge depolarizer (DWD), which consists of a pairof uniaxial crystal wedges having their optic axes mutually outof line [1, 9]. The depolarization effect of DWDs is based onscrambling the states of polarization in the space domain, sothey are actually pseudodepolarizers. DWDs have been studied

by considering incident monochromatic plane waves, in bothtotally polarized and unpolarized cases, and interesting resultshave been obtained [9]. To our knowledge, other types ofincident beams, such as partially coherent beams, have notbeen considered yet.

A significant class of partially polarized fields is that ofthe so-called purely polarized fields, whose spatial correlationproperties and polarization states can be decoupled [11–13].Fields belonging to this class present a uniform polarizationacross the transverse plane and their polarization characteris-tics are invariant under propagation through any polarization-

2040-8978/14/035708+17$33.00 1 c© 2014 IOP Publishing Ltd Printed in the UK

J. Opt. 16 (2014) 035708 J C G de Sande et al

insensitive optical system and, as a particular case, during freepropagation [13].

The aim of this work is to investigate the effect of a DWDon the propagation of a purely polarized partially coherentbeam. We limit ourselves to the case of quasi-monochromaticradiation, but the extension to the polychromatic case isstraightforward. The results we are going to present canalso be applied to a wider class of deterministic opticalelements, namely, those that split an incoming field into twoor more components, on introducing a different phase for eachcomponent that depends linearly on the transverse position.Besides DWDs [9], this class of optical elements also includespolarization gratings [14, 15] and any other optical element thatproduces a transversely periodic electromagnetic field [16].Nonetheless, the particular case of DWDs is fully developedin our work.

Numerical examples are given for purely polarized inputbeams of the Gaussian Schell-model (GSM) type, for whichboth the irradiance profile and the coherence degree havea Gaussian form [14, 17–22]. For the two extreme casesof totally polarized and completely unpolarized input fields,the irradiance and polarization characteristics are analyzed indetail. The irradiance and the degree of polarization just at theoutput of the DWD and after free propagation are studied asfunctions of the parameters of both the input beam and theoptical device. It is shown that, by varying the input beamcharacteristics, a rich variety of behaviors can be obtainednot only across the transverse beam section but also uponpropagation. Extension of the numerical examples to the caseof polarization gratings is straightforward.

The paper is structured as follows. The mathematicaltools used throughout the paper are introduced in section 2:in particular, the beam coherence-polarization (BCP) matrix,for treating partially coherent electromagnetic beams, andthe Jones formalism, for studying the effects of a DWDon an incident field. In section 3, the BCP of the fieldpropagating after the DWD is derived, while in section 4examples are given concerning the irradiance distribution andthe degree of polarization of purely polarized GSM inputbeams after passing through a DWD. The two limiting casesof totally polarized and completely unpolarized input beamsare considered in detail, for different choices of the beamparameters. The main conclusions of this work are summarizedin section 5. Moreover, in appendix A all the symbols andnotations used throughout the text are summarized, while theanalytical derivations of some of the expressions used in thepaper are reported in four additional appendices.

2. Preliminaries

2.1. The beam coherence-polarization matrix

Within the paraxial approximation, the coherence and polar-ization characteristics of a quasi-monochromatic light beam,propagating along the z direction of a suitable referenceframe, can be described by means of its BCP matrix [23–25],defined as

J (r1, r2, z)=(

Jxx (r1, r2, z) Jxy (r1, r2, z)Jyx (r1, r2, z) Jyy (r1, r2, z)

), (1)

where

J jk (r1, r2, z)= 〈E j (r1, z; t) E∗k (r2, z; t)〉. (2)

Here, E j (r, z; t), with j, k = x, y, are the Cartesian com-ponents of the time-dependent electric field and r is theposition vector in a plane perpendicular to the z direction. Theasterisk denotes conjugation and the angle brackets representtemporal average. In the case of polychromatic fields, thecross-spectral density tensor defined in the spectral domain[21, 26] should be used, but the two definitions are equivalentif quasi-monochromatic sources are considered.

When evaluated at coincident points, r1 = r2 = r, the BCPmatrix reduces to the polarization matrix, i.e.,

P(r, z)= J (r, r, z) , (3)

containing all information about the irradiance and polariza-tion state of the beam at the point (r, z). In particular, the totalirradiance is defined as the trace of P , while the local degreeof polarization (DoP) is evaluated as [21]

p(r)=

√1−

4 Det{P(r)}Tr2{P(r)}

. (4)

Since the matrix P is Hermitian and semipositive definite,the DoP only takes values in the interval [0, 1], with p = 1corresponding to a perfectly polarized field and p = 0 to acompletely unpolarized one.

The polarization properties of the light fields can be alsodescribed through the Stokes vector, which conveys the sameinformation content as the polarization matrix, and whoseelements are related to the elements of the latter by the relation

S(r)=

S0(r)S1(r)S2(r)S3(r)

=Pxx (r)+ Pyy(r)

Pxx (r)− Pyy(r)2 Re{Pyx (r)}2 Im{Pyx (r)}

. (5)

S0(r) gives the irradiance of the field while the three remainingparameters account for its local polarization state. The Stokesparameters are measurable quantities and can be determinedfrom the irradiances at the output of a quarter wave phase plateand a linear polarizer at different angles [27].

When a partially coherent electromagnetic field passesthorough a deterministic linear optical element, characterizedby a complex 2× 2 Jones matrix T (r), its BCP matrix istransformed according to the following rule [24]:

J (r1, r2, 0)= T (r1) J in (r1, r2, 0) T † (r2) , (6)

where J in is the BCP matrix across the input plane of theelement and the dagger denotes Hermitian conjugation. It hasbeen assumed that the optical element has negligible thicknessand its exit plane coincides with the plane z = 0. Equation (6)can be written for each of the matrix elements of the BCPmatrix as

J jk (r1, r2, 0)=y∑

l,m=x

T jl (r1) T ∗km (r2) J inlm (r1, r2, 0) . (7)

2

J. Opt. 16 (2014) 035708 J C G de Sande et al

Figure 1. Scheme of the double-wedge depolarizer.

It must be recalled that, in general, a deterministic linearoptical element not only changes the polarization state of thepolarized part of the input beam, but also changes the DoPof the input beam [28]. Specifically, the DoP at the exit planeis set to unity for polarizers (singular Jones matrices), whileit remains unchanged when the Jones matrix of the opticalelement is proportional to a unitary matrix. Otherwise, theDoP can be increased, decreased or unchanged by opticalelements represented by nonsingular Jones matrices that arenot proportional to a unitary matrix.

In this work, particular attention will be focused on inputbeams belonging to the class of the so-called purely polarizedfields [11–13], i.e., those fields represented by BCP matricesof the form

J in(r1, r2, 0)= Jsc(r1, r2, 0) P in, (8)

where Jsc(r1, r2, 0) is the mutual intensity of a scalar sourceat the plane z = 0 and P in is a normalized polarization matrix,giving account of the polarization state. Fields of this class areuniformly polarized and their polarization characteristics areinvariant under propagation through polarization-insensitiveoptical systems, such as propagation in free space [13]. When abeam of this class impinges onto a deterministic linear opticalelement, the elements of the BCP matrix across the outputplane become

J jk(r1, r2, 0)= Jsc(r1, r2, 0)y∑

l,m=x

T jl (r1) T ∗km (r2) P inlm . (9)

Since, in general, the Jones matrix T is a function of thetransverse coordinate r, the output field is no longer purelypolarized, implying that the polarization properties of thefield after the optical element may vary under free-spacepropagation. Of course, this does not happen in the caseof optical elements that act uniformly across the transversesection of a purely polarized beam.

2.2. Double-wedge depolarizer

A DWD consists of two birefringent wedges with the samewedge angle ϕ, joined in such a way as to form a parallelepiped(see figure 1). The first wedge has its optic axis along they direction. The optic axis of the second wedge is parallel

to the bisector of the x and y axes. When a plane waveimpinges perpendicularly to the input face of a DWD, theordinary and extraordinary components of the field propagatethrough the first wedge and both of them split into twonew ordinary and extraordinary components in the secondone [29, 30]. Two of these components propagate, after theDWD, following the same direction as the incident planewave but with different phases and orthogonal polarizations.The two other components, also having mutually orthogonalpolarization states, diverge, forming a very small angle (ifcompared to ϕ) with respect to the z axis (see figure 1). Wewill neglect the losses due to reflections at the interfaces andthe small differences between the transmission coefficients forordinary and extraordinary waves or between the parallel andperpendicular components at the exit face.

Under the above hypothesis, the behavior of the DWD canbe described by the following Jones matrix [9]:

T D(x)= 12

(1+ e−iδ2(x) e−iδ1(x)

[1− e−iδ2(x)

]1− e−iδ2(x) e−iδ1(x)

[1+ e−iδ2(x)

]) . (10)

In this equation, δ1 and δ2 are the phase differences betweenthe x and y components of the field after propagating throughthe first and second wedge of the DWD, respectively. It canbe noted that T D(x) is a unitary matrix, therefore the DWDonly changes the polarization state in a different way at eachpoint of the beam transverse section, but it does not modify itsirradiance and DoP.

For small wedge angles (ϕ� 1), the above phase differ-ences can be approximated by [9]

δ1(x)'2πλ|1n| (d1+ x tanϕ)= δ10+ γ x, (11)

δ2(x)'2πλ|1n| (d2− x tanϕ)= δ20− γ x, (12)

where d1 and d2 are the mean thicknesses of the first and secondwedge, respectively, γ = (2π |1n| tanϕ) /λ, and 1n = no−

ne, with no and ne being the ordinary and extraordinaryrefractive indexes, respectively. It can be noticed that bothphases are linear functions of the x coordinate. The explicitdependence of the refractive indexes on the wavelength hasbeen omitted. As is evident from equations (10)–(12), thegeometry of the DWD yields a Jones matrix that dependsonly on the x coordinate, so that the same polarization state

3

J. Opt. 16 (2014) 035708 J C G de Sande et al

is expected at all points along lines with constant x across theexit plane.

It can be shown that a DWD converts a uniformly totallypolarized field into a non-uniformly totally polarized beam [9].The local degree of polarization is unity everywhere, acrossthe transverse section, but the state of polarization changesperiodically along the x direction with a spatial periodicitygiven by

Lx =λ

|1n(λ)| tanϕ=

2πγ. (13)

Several experimental methods have been proposed toobtain periodic variation of the state of polarization acrossthe transverse plane [3, 4, 14, 31]. The use of a DWD isa simple way to obtain such a periodic variation withoutmodifying the irradiance profile of the beam. If the DWDis illuminated with a totally linearly polarized plane wave(impinging perpendicularly to the input face of the DWD),a manifestation of the Talbot effect is observed, with self-replication of the polarization states at integer multiples of theTalbot distance zT = 2 L2

x/λ [16].It is interesting to analyze the behavior of a DWD within

the Stokes formalism. Starting from the expression of the Jonesmatrix in equation (10), it is possible to evaluate the pertinentMuller matrix [32, 33], which turns out to be

MD(x)= T D(x)⊗ T D∗(x)=1 0 0 00 cos δ2(x) sin δ1(x) sin δ2(x) − cos δ1(x) sin δ2(x)0 0 cos δ1(x) sin δ1(x)0 sin δ2(x) − sin δ1(x) cos δ2(x) cos δ1(x) cos δ2(x)

,(14)

where ⊗ represents the direct or Kronecker product. It isevident that, on averaging the Muller matrix elements oversufficiently large distances along the x direction, all the termsvanish except MD

11. Thus, in such conditions the behavior ofthe DWD mimics that of an ideal depolarizer [34].

Finally, let us consider the effects of a DWD on incidentpartially coherent electromagnetic light. In such a case, therelationship between the input and the output BCP matricesof the field is described by equation (7). Taking into accountthe explicit form of the Jones matrix in equation (10), theproducts T jl (r1) T ∗km (r2), appearing in equation (7), turn outto be linear combinations of the phase terms introduced bythe DWD, and such phases are, at least approximately, linearfunctions of the x coordinate (see equations (11) and (12)).As will be seen in the next section, this feature makes simplerthe evaluation of the BCP matrix of the field propagated fromthe DWD, once the propagation of the unperturbed input field(i.e., the field we would have obtained if the optical elementwere not present) is known. The explicit expressions for theseproducts are given in appendix B.

3. Free-space propagation after a deterministiclinear optical element

Knowledge of the BCP matrix across the exit plane of theoptical element enables the evaluation of the BCP matrix

across any transverse plane beyond it, such that the polarizationproperties of the propagated field can be determined. Inparticular, the following expression can be used for freepropagation in paraxial conditions [17, 21]:

J jk (r1, r2, z) =∫∫

K ∗z(r1, ρ1

)J jk

(ρ1, ρ2, 0

)× Kz

(r2, ρ2

)d2ρ1 d2ρ2 (15)

where ρ is the position vector across the z = 0 plane and

Kz (r, ρ)=−iλz

exp[

i2π zλ+

iπλz(r− ρ)2

](16)

is the direct propagator in paraxial conditions.On substituting from equation (16) into equation (15) and

using equation (7), the BCP matrix elements of the propagatedfield can be expressed as

J jk (r1, r2, z) =y∑

l,m=x

1λ2z2 exp

[iζ2

(r2

2 − r21

)]×

∫∫T jl

(ρ1)

T ∗km(ρ2)

J inlm(ρ1, ρ2, 0

)× exp

[iζ2

(ρ2

2 − ρ21 − 2r2 · ρ2

+ 2r1 · ρ1

)]d2ρ1 d2ρ2, (17)

where ζ = 2π/(λz) and · represents the scalar product.On the other hand, the free-space propagation of the

unperturbed input field can be evaluated as

J injk (r1, r2, z)=

exp (iζ s · t)λ2z2 H jk (s, t) , (18)

where the auxiliary function H jk (s, t) is introduced as

H jk (s, t) =∫∫

J injk(ρ1, ρ2, 0

)exp

[iζ2

(ρ2

2 − ρ21

)]× exp

[−iζ

(r2 · ρ2− r1 · ρ1

)]d2ρ1 d2ρ2, (19)

with s= (r1+ r2)/2 and t= r2− r1. The explicit dependenceof H jk on z has been omitted for brevity.

In the following, particular attention will be paid todeterministic linear optical systems that can be representedby Jones matrices whose elements are linear combinations ofphase terms, with phases that linearly depend on the positionacross the optical element plane. Such elements can be writtenin the form

T jk(r)=∑

q

aqjk exp

(iγ q

jk · r), (20)

where aqjk are complex coefficients and the components of

the vectors γqjk give the proportionality along the x and y

coordinates for each term. Jones matrices representing a DWD,equation (10), or a polarization grating [14, 15] belong tothis class. For such class of optical elements it is found, in

4

J. Opt. 16 (2014) 035708 J C G de Sande et al

particular, that

T jl(ρ1)

T ∗km(ρ2)=

∑q,q ′

aqjl

(aq ′

km

)∗× exp

[iγ q

jl · ρ1− iγ q ′km · ρ2

]. (21)

Therefore, from equation (17), the BCP elements correspond-ing to the propagated field behind such an optical element canbe expressed as

J jk (r1, r2, z) =1λ2z2 exp

[iζ2

(r2

2 − r21

)]×

y∑l,m=x

j lmjk (r1, r2, z) , (22)

where the functions

j lmjk (r1, r2, z)=

∑q,q ′

aqjl

(aq ′

jl

)∗× Hlm

(s−

γqjl

ζ+

γq ′kmζ, t+

γqjl

2ζ+

γq ′km

2ζ

)(23)

are different linear combinations of the auxiliary functionsHlm (s, t) and different tilted versions of them. The particularset of linear combinations and tilts depends on the specificoptical element considered. In appendix C, the particular caseof a DWD is developed.

4. Gaussian Schell-model beams throughdouble-wedge depolarizers

The process discussed in the previous section will be illustratedhere for the case of a purely polarized GSM input beam[11–13]. This means that the BCP matrix of the input beamtakes the form shown in equation (8), with the scalar mutualintensity given by [17, 35]

Jsc(r1, r2, 0)= I0 exp

[−

r21 + r2

24σ 2 −

(r1− r2)2

2µ2

], (24)

where I0 is a positive quantity having dimensions of anirradiance, σ 2 is the variance of the irradiance profile, andµ expresses the width of the degree of coherence of the beamacross the input plane of the DWD.

For the case of a GSM beam, the explicit form of theauxiliary function H (s, t) given in equation (19) is [14]

H(s, t)= I0

(λzF

)2

exp(−iζ s · t

F2

)exp

(−αs2

−βt2

F2

),

(25)

where the following definitions have been used:

F2= 1+

4αβζ 2 , (26)

α =1

2σ 2 , (27)

β =1

8σ 2 +1

2µ2 . (28)

When the propagation after the DWD is considered,the expressions of the BCP matrix elements are obtainedfrom equation (22), after substituting from equation (25) intoequations (C.1)–(C.12) of appendix C, and from the latterinto equation (23). The irradiance and the DoP are eventuallyevaluated by means of the polarization matrix, i.e., on lettingr1 = r2 = r in the expression of the BCP matrix, and using theexpressions given in section 2.1.

Two particular cases will be considered in the nextsubsections: a totally polarized (TP) and an unpolarized(UP) incident beam. In the first case, a normalized Jonesvector (cos θ, sin θ exp iφ)T (where T stand for transposed)representing an arbitrary state of polarization specified by theangles θ and φ, yields a normalized polarization matrix givenby (see equations (2) and (3))

P in=

12

(1+ cos 2θ eiφ sin 2θe−iφ sin 2θ 1− cos 2θ

), (29)

while in the second one P in is proportional to the 2× 2 unitmatrix.

4.1. Totally and uniformly polarized input beam

In this case, the BCP matrix for this input beam is of theform

J in,TP(r1, r2, 0)= 12 Jsc(r1, r2, 0)

×

(1+ cos 2θ eiφ sin 2θe−iφ sin 2θ 1− cos 2θ

), (30)

so that, on using equation (9), the BCP across the output planeturns out to be

J TP(r1, r2, 0)= 12 Jsc(r1, r2, 0)

×

(jTPxx (x1, x2) jTP

xy (x1, x2)

jTPxy∗(x2, x1) jTP

yy (x1, x2)

), (31)

with the matrix elements given by

jTPxx (x1, x2)= 2T D

xx (x1)T Dxx∗(x2) cos2 θ

+ 2T Dxy(x1)T D

xy∗(x2) sin2 θ

+ [T Dxx (x1)T D

xy∗(x2)eiφ

+ T Dxy(x1)T D

xx∗(x2)e−iφ

] sin 2θ, (32)

jTPyy (x1, x2)= 2T D

yx (x1)T Dyx∗(x2) cos2 θ

+ 2T Dyy(x1)T D

yy∗(x2) sin2 θ

+ [T Dyx (x1)T D

yy∗(x2)eiφ

+ T Dyy(x1)T D

yx∗(x2)e−iφ

] sin 2θ, (33)

jTPxy (x1, x2)= 2T D

xx (x1)T Dyx∗(x2) cos2 θ

+ 2T Dxy(x1)T D

yy∗(x2) sin2 θ

+ [T Dxx (x1)T D

yy∗(x2)eiφ

+ T Dxy(x1)T D

yx∗(x2)e−iφ

] sin 2θ. (34)

5

J. Opt. 16 (2014) 035708 J C G de Sande et al

Figure 2. Normalized irradiance (left) and degree of polarization (right) for a totally linearly polarized, along x (θ = 0), GSM input beamwith σ = 3Lx for (a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

The local polarization properties of the field are obtainedon letting x1 = x2 = x in equations (32), (33), (34), and thenusing equations (31), (4) and (5). As was expected, the DoPis unity everywhere in the transverse plane of the beam,but the state of polarization varies periodically along the xdirection.

As far as the propagated field is concerned, its BCP matrixis calculated by using equation (22) and takes the form

J TP (r1, r2, z)

=I0

2F2 exp

[iζ(F2− 1

)2F2

(r2

2 − r21

)−αs2

y +βt2y

F2

]

×

(hTP

xx (sx , tx , z) hTPxy (sx , tx , z)

hTPxy∗(sx ,−tx , z) hTP

yy (sx , tx , z)

), (35)

where sx and tx (sy and ty) are the x (y) components of thevectors s and t, and the matrix elements hTP

jk are explicitlyevaluated in appendix D. It can be noticed that for the DWDwe are considering (i.e., oriented as in figure 1), the matrixelements hTP

jk do not depend on y, so that the polarizationproperties of the field are constant along any line parallel to they axis, and the only dependence on y is in the Gaussian profile.Hence, in the following, only the dependence of irradiance andDoP on x and z will be considered.

The optical irradiance along x at any plane z = const canbe obtained by evaluating the trace of the polarization matrixand is

I TP(x, z)=I0

2F2 [hTPxx (x, 0, z)+ hTP

yy (x, 0, z)], (36)

while the local degree of polarization, given by equation (4),

turns out to be

pTP(x, z)=

√[hTP

xx (x, 0, z)− hTPyy (x, 0, z)]2+ 4

∣∣∣hTPxy (x, 0, z)

∣∣∣2hTP

xx (x, 0, z)+ hTPyy (x, 0, z)

.

(37)

For the simplest cases of an incident beam linearlypolarized along the x axis (i.e., θ = 0) or the y axis (θ = π/2),the optical irradiance and the DoP take the following analyticalexpressions:

I TPθ=0,π/2(x, z)

=I0

2F2 exp(−αx2

F2

)×

[1+ exp

(−αγ (γ ± 2xζ )

ζ 2 F2

)], (38)

and

pTPθ=0,π/2(x, z)

=

√exp

[(α−4β)γ 2

2ζ 2 F2

]+ sinh2

[αγ (γ±2xζ )

2ζ 2 F2

]2

cosh[αγ (γ±2xζ )

ζ 2 F2

] , (39)

respectively, where the + (−) sign applies for linear polar-ization along the x (y) axis. The above behaviors of theirradiance distribution and the DoP across the plane (x, z)are reported in figure 2 for a beam polarized along x , havingwidth σ = 3Lx and several values of the coherence width:µ = 3Lx (a), µ = Lx (b), µ = Lx/3 (c). Here and in thefollowing we assume a DWD made in quartz, with ϕ = 2◦,and the wavelength is set to 0.633 µm. As a consequence, theperiodicity of the polarization state at the exit of the DWD

6

J. Opt. 16 (2014) 035708 J C G de Sande et al

Figure 3. Normalized irradiance (left) and degree of polarization (right) for a totally linearly polarized, along x (θ = 0), GSM input beamwith σ = Lx/3 for (a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

is Lx ∼ 2 mm (see equation (13)), corresponding to a Talbotdistance zT ∼ 12 m.

The plots in figure 2 show that the irradiance (left) isalmost uniform beyond the DWD, while the DoP (right) isunity in a region near the DWD, but tends to vanish onincreasing the propagation distance. The latter effect is moreevident on decreasing the coherence width of the incomingbeam.

Such behaviors can be understood if reference is made tothe case of an incident plane wave having the same polarizationas the beam we are considering here. The Jones vector of thisplane wave is of the form U in

∝ (1, 0)T so that, using the Jonesmatrix in equation (10), the output field turns out to be of theform

U out∝

(11

)+

(1−1

)e−iδ2(x). (40)

Since, according to our approximations, the phase δ2 is a linearfunction of x (see equation (12)), the two terms in equation (40)can be recognized as being representative of two plane waves:the first one is polarized atπ/4 and propagates along the z axis;the second one, with the same amplitude, has an orthogonalpolarization and propagates along a tilted axis. From the valuesof the parameters used in these plots, the tilt angle turns outto be of the order of 2× 10−4 rad. Across any transverseplane the superposition of the two plane waves produces apolarization state that varies periodically along the x directionwith period Lx . This polarization pattern displaces laterallyduring propagation, Lx being the lateral displacement withina Talbot distance zT [9, 16]. As far as the DoP is concerned,we first note that, when the incoming field is totally polarizedand perfectly coherent, the DoP at the exit of the DWD mustbe unity everywhere because the field is strictly deterministic.But, when the input beam is still totally polarized but partiallycoherent, the randomness of the field at different points may

be transferred, upon propagation after the DWD, to the fieldcomponents at a single point, so that the DoP may be lessthan unity. This is what is expected to happen when a GSMbeam is used as the input beam. In particular, in figure 2(a)the coherence width is quite large, so that the DoP after theDWD is unity almost everywhere, but it decreases with z whenthe coherence width decreases. It should be stressed that, insuch a way, the DWD acts as a true depolarizer, instead ofa pseudodepolarizer (see comments following equation (14)),because a perfectly polarized input beam is converted into apartially polarized output beam without the need of spatiallyaveraging the states of polarization. In this sense, it couldbe considered as the spatial-coherence counterpart of a Lyotdepolarizer [8, 10].

In figure 3 the same parameters as in the previousfigure have been used, except for the beam width, whichhas been significantly reduced (σ = Lx/3). Even in thiscase, the incoming field is split into two beams: one ofthem propagates along the z axis, and the other one alonga different axis, corresponding to the propagation directionsof the plane waves in equation (40). This effect is moreevident with the highest coherence width (figure 3(a)). Forlower values of µ (figures 3(b) and (c)), the two beamsdiverge more quickly and overlap. On the other hand, theDoP is close to unity everywhere when the input beamis almost perfectly coherent, as expected. However, whenthe coherence width decreases, the DoP decreases with thepropagation distance, as for the previous figure, but thishappens only within the superposition region of the twobeams. Out of this region, when the contribution of one ofthe two beams is dominant, the output field is almost perfectlypolarized.

More interesting behaviors can be observed when thepolarization of the incident beam is linear along the bisector

7

J. Opt. 16 (2014) 035708 J C G de Sande et al

Figure 4. Normalized irradiance (left) and degree of polarization (right) for a totally linearly polarized GSM input beam with θ = π/4 andσ = 3Lx for (a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

of the x and y axes (θ = π/4 and φ = 0), although in such acase the analytical expressions of the irradiance and the DoPare significantly more cumbersome. Basically the same resultswould be obtained for a linear polarization with θ =−π/4 orfor a circularly polarized input beam.

The irradiance distribution and the local DoP after theDWD across the plane (x, z) are shown in figures 4–6, fora totally polarized GSM input beam with azimuth θ = π/4,for several combinations of profile and coherence widths.Figure 4(a) corresponds to σ = µ = 3Lx . In such a case,both the beam width and the coherence width are larger thanthe polarization pattern period produced at the exit planeof the DWD. It can be seen that interference patterns areproduced at certain transverse planes, and such patterns arealmost replicated every half of the Talbot distance (zT/2), butwith a Lx/2 lateral displacement between two consecutivereplicas [9, 16]. The presence of such interference patterns,clearly visible at distances zT/4 and 3zT/4 from the DWD,can be justified in the following way.

As we did for the previous example, let us suppose thatthe input field is a plane wave, and that it is polarized along thebisector of the x and y axes. Its Jones vector is U in

∝ (1, 1)T

so that, using equations (10)–(12), it turns out that the Jonesvector at the output plane of the DWD is

U out∝

(11

)(1+ e−iδ1(x)

)+

(1−1

)(1− e−iδ1(x)

)e−iδ2(x)

=

(11

)(1+ e−iδ1(x)

)+

(1−1

)(−e−i(δ10+δ20)+ e−iδ2(x)

). (41)

Since, within our approximations, both the phases δ1 and δ2are linear functions of the x coordinate, this equation showsthat the output field consists of four linearly polarized planewaves [29, 30]. Two of these waves have polarizations parallelto that of the input field (first term of the right hand sideof equation (41)) and propagate along different directions, sothat they interfere to produce a fringe pattern in the irradianceprofile, with period Lx . During propagation, such a patternmoves along an axis whose orientation is determined bythe propagation directions of the two waves. The other twocomponent waves have linear polarizations as well, but with aperpendicular polarization with respect to the first two (secondterm of the right hand side of equation (41)). They produce aninterference pattern that is transversely shifted by Lx/2 withrespect to the first pattern at the output plane and propagatesalong a different direction. As a consequence, no fringes atall are observed across the exit plane of the DWD when thetwo patterns are in phase opposition, but they appear when,after propagation through zT/4, they superimpose in phase.Of course, the effect reproduces periodically along z. Thissimple model can explain the behavior of the irradiance profileshown in figure 4(a). As far as the DoP is concerned, thesame considerations made for the previous example hold.In particular, the field is expected to be almost perfectlypolarized everywhere if the input beam is almost perfectlycoherent. This is what can be seen in figure 4(a): here,µ= 3Lxand the polarization degree is generally very close to unity,except in some regions where it takes on smaller values, andthis generally happens in some transverse positions at planesaround z = zT/4.

Figure 4(b) corresponds to the same value of the beamwidth, σ = 3Lx , but the spatial coherence is reduced (µ= Lx ).It shows that, even in this case, the irradiance interferencepattern appears at z = zT/4 and, with lower visibility, at

8

J. Opt. 16 (2014) 035708 J C G de Sande et al

Figure 5. Normalized irradiance (left) and degree of polarization (right) for a totally linearly polarized GSM input beam with θ = π/4 andσ = Lx for (a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

Figure 6. Normalized irradiance (left) and degree of polarization (right) for a totally linearly polarized GSM input beam with θ = π/4 andσ = Lx/3 for (a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

3zT/4. The degradation of the visibility, when compared tothat in figure 4(a), is due to the reduction of the coherencelength. The DoP is approximately unity behind the DWD but itreduces with the propagation distance. Furthermore, it presentssmoother variations with respect to the previous case.

Finally, in figure 4(c), the coherence width is furtherreduced (µ = Lx/3). The irradiance interference pattern isappreciated only around the plane z = zT/4. For distancesbeyond zT/2, the interference pattern disappears, and only theapproximately transverse Gaussian profile can be appreciated.On the other hand, the DoP is nearly unity in a reduced regionjust behind the DWD and progressively approaches a constantvalue. This limiting value, reached at propagation distanceswell over zT/2, depends on the characteristics of both the

DWD and the input beam. As has been previously commented,a true depolarizing effect is obtained for spatially incoherentinput fields. For the present case, i.e., θ = π/4, the DoP forlarge z turns out to be

pTPθ=π/4 (x, z→∞)=

1√

2exp

[(α− 4β) γ 2

8αβ

]×

{cosh

(−γ 2

α

)− cos (2δ10+ 2φ)

+ 2 exp[(4β −α) γ 2

8αβ

]cos2 (δ10+φ)

}1/2

× cosh−1(γ 2

8β

). (42)

9

J. Opt. 16 (2014) 035708 J C G de Sande et al

Similar results are shown in figure 5, but for a GSMinput beam with a smaller width (σ = Lx ). The same generalbehaviors as for the previous case are observed, except forthe irradiance distribution, which now presents a more evidenttransverse limitation.

When we consider an input beam with a transverse widthsmaller than the polarization periodicity Lx (see figure 6,with σ = Lx/3), the interference pattern in the irradianceprofile can no longer be appreciated. However, a region withhigher irradiance levels is found around zT/4. In particular,figure 6(a), which is pertinent to the case of an almostcoherent incident beam, shows that three different beams aredistinguishable at the output of the device. The central one, theone with the highest irradiance, propagates along the z axis,while the two other beams follow the propagation directions ofthe plane waves in equation (41). When the incident beam is notperfectly coherent (figure 6(b), with µ= Lx , and figure 6(c),withµ= Lx/3), each of the component beams spreads quickerand, in the overlapping region, the DoP is noticeable lower.Such an effect is better appreciated on reducing the coherencewidths (figure 6(c)).

4.2. Unpolarized input beam

The BCP matrix of a purely unpolarized GSM input beam isof the form

J in,UP(r1, r2, 0)= 12 Jsc(r1, r2, 0)

(1 00 1

), (43)

and the corresponding BCP matrix across the output plane ofthe DWD turns out to be

J UP(r1, r2, 0)

=12 Jsc(r1, r2, 0)

(jUPxx (x1, x2) jUP

xy (x1, x2)

jUPyx∗(x2, x1) jUP

yy (x1, x2)

), (44)

where

jUPxx (x1, x2)= T D

xx (x1)T Dxx∗(x2)+ T D

xy(x1)T Dxy∗(x2), (45)

jUPyy (x1, x2)= T D

yx (x1)T Dyx∗(x2)+ T D

yy(x1)T Dyy∗(x2), (46)

jUPxy (x1, x2)= T D

xx (x1)T Dyx∗(x2)+ T D

xy(x1)T Dyy∗(x2). (47)

A straightforward calculation yields to a zero local degree ofpolarization everywhere in the transverse plane just at the exitof the DWD, as is expected for this case. Nonetheless, changesproduced by the DWD in the BCP matrix elements will affectthe subsequent propagation of the beam, as will be clear in thefollowing.

For the present case, equation (22) takes the form

J UP (r1, r2, z)

=I0

2F2 exp

[iζ(F2− 1

)2F2

(r2

2 − r21

)−αs2

y +βt2y

F2

]

×

(hUP

xx (sx , tx , z) hUPxy (sx , tx , z)

hUPxy∗(sx ,−tx , z) hUP

yy (sx , tx , z)

), (48)

where the matrix elements hUPjk are explicitly reported in ap-

pendix E. As for the previous case, the polarization properties

of the field are independent of y and the irradiance presentsonly a Gaussian dependence on y, so that only the dependenceof irradiance and DoP on x and z will be discussed in thefollowing.

By evaluating the trace of the BCP matrix, the followingoptical irradiance is obtained:

I UP(x, z)=I0

2F2 exp(−αx2

F2

)[1+ exp

(−αγ 2

ζ 2 F2

)× cosh

(2αγ xζ F2

)], (49)

where the Gaussian dependence of the optical irradiance alongthe y coordinate has been omitted and, by using equation (4),the DoP can be obtained as

pUP (x, z) ={

2 exp[− (α+ 4β) γ 2

2ζ 2 F2

]×

[cosh

(2αγ xζ F2

)− cos

(γ 2

ζ F2

)]+ exp

(−2αγ 2

ζ 2 F2

)sinh2

(2αγ xζ F2

)}1/2

×

[1+ exp

(−αγ 2

ζ 2 F2

)× cosh

(2αγ xζ F2

)]−1

. (50)

It can be noted that the DoP is zero for z = 0 andfor z→∞, at any transverse coordinate. At the transversecoordinate x = 0, its dependence on the z coordinate takes theform

pUP(x = 0, z)= exp[(α− 4β) γ 2

4ζ 2 F2

] ∣∣∣∣sin(γ 2

2ζ F2

)∣∣∣∣× cosh−1

(αγ 2

2ζ 2 F2

). (51)

The behaviors of the irradiance and the DoP across theplane (x, z) for a totally unpolarized GSM input beam withdifferent coherence lengths are shown in figures 7–9 (for beamwidths σ = 3Lx , Lx , and Lx/3, respectively).

Figure 7(a) shows that, when the beam and coherencewidths are large enough compared to the polarization peri-odicity Lx , almost the same results obtained for an incidentunpolarized plane wave are reproduced [9, 16]: the irradianceis nearly uniform, except for the transverse Gaussian modula-tion, while the DoP is almost constant across any transverseplane, but varies with z, going from zero to nearly unity, andagain to zero around zT/2. This behavior is repeated alongthe z axis in a quasi-periodic way. The DoP does not reachunity around 3zT/4, as is the case for an incident plane wave,and the successive maxima reached by the DoP progressivelydecrease because of the finite value of the coherence area.

The periodic longitudinal modulation of the DoP can beunderstood on considering that a totally unpolarized incidentplane wave can be thought of as the incoherent superposition oftwo plane waves, linearly polarized along x and y, respectively.

10

J. Opt. 16 (2014) 035708 J C G de Sande et al

Figure 7. Normalized irradiance (left) and degree of polarization (right) for a completely unpolarized GSM input beam with σ = 3Lx for(a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

Figure 8. Normalized irradiance (left) and degree of polarization (right) for a completely unpolarized GSM input beam with σ = Lx for(a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

Equation (40) expresses the Jones vector of the field at theexit plane of the DWD when the input field is a plane wavelinearly polarized along x . The corresponding polarizationstate varies periodically along x with period Lx , and thispolarization pattern shifts transversally during propagation.A similar result is obtained when the input field is a planewave linearly polarized along y, but in such a case the state ofpolarization turns out to be orthogonal to the previous one atany x position across the exit plane of the DWD and, moreover,the polarization pattern shifts in the opposite direction. Thesuperposition of the two polarization patterns gives rise to acompletely unpolarized field across the exit plane of the DWDbut, due to their different shift directions, the two patternsturn out to be perfectly coincident at z = zT/4, so that thedegree of polarization reaches unity [9, 16]. On the other

hand, no irradiance interference patterns appear in this case,because the exiting beams with the same polarization stateare mutually completely incoherent, while the exiting beamsthat are partially coherent present orthogonal polarizations.On reducing the coherence area (µ= Lx in figure 7(b)), noremarkable effects appear on the irradiance distribution, butthe behavior of the DoP changes radically, and its secondmaximum at z = 3zT/4 is quite lower than at zT/4. Forµ = Lx/3 (figure 7(c)), the DoP reaches a maximum, fora propagation distance slightly lower than zT/4, and thenprogressively decreases to zero on increasing z.

In figure 8 the beam width has been reduced to σ = Lx ,while the coherence widths are the same as for the previousfigure. In this case, the spread due to diffraction is moreevident, especially for lower values of the coherence width.

11

J. Opt. 16 (2014) 035708 J C G de Sande et al

Figure 9. Normalized irradiance (left) and degree of polarization (right) for a completely unpolarized GSM input beam with σ = Lx/3 for(a) µ= 3Lx (upper row), (b) µ= Lx (middle row), and (c) µ= Lx/3 (lower row).

The behavior of the DoP as a function of z is similar to thatof the previous case in the neighborhood of the z axis. Out ofthis region, the DoP is significantly higher than zero. This canbe understood, once again, on making reference to the schemein figure 1 and equation (41). In the outer regions, in fact, thecomponent beams do not overlap in a significant way and onlyone perfectly polarized beam is dominant, so that the DoP isexpected to approach unity.

Finally, when the beam width is smaller than the polar-ization periodicity Lx (figure 9), the irradiance splits intothree parts, as was the case for a perfectly polarized inputbeam. This is clearly observable for µ = 3Lx (figure 9(a))and µ= Lx (figure 9(b)), while the three component beamssignificantly overlap when the coherence area is the smallest(µ= Lx/3, in figure 9(c)). As far as the DoP is concerned, thesame considerations hold as the ones made with reference tofigure 8.

5. Conclusions

BCP formalism is used to analyze the polarization character-istics and irradiance profile of partially coherent and partiallypolarized beams propagating after they have passed througha non-uniform deterministic linear optical element. It hasbeen shown that non-uniform deterministic linear optical ele-ments transform purely polarized input fields into non-purelypolarized output beams. The expression of the BCP matrixrepresenting such field is developed for the class of opticalelements described by Jones matrices whose elements arelinear combinations of phases that linearly depend on theposition across the optical element plane (the Jones matricesgiven by equation (20)). This class of optical elements includeDWDs and polarization gratings.

The irradiance profile, the state of polarization and theDoP when purely polarized GSM beams impinge on a DWDand propagate after it have been analyzed in detail. Although

the DoP and the irradiance just at the exit plane of the DWDare the same as those of the incoming beam, the irradianceprofile and the DoP, as well as the state of polarization, varywith the propagation distance after the DWD.

When the transverse profile and the coherence area ofthe incident GSM beam are large enough (with respect tothe transverse polarization periodicity induced by the opticalelement), the polarization state and the DoP of the propagatedfield behave almost as expected for incident plane wavespresenting, in particular, the typical periodicity predicted fromthe Talbot effect [9, 16].

More complex behaviors are obtained on reducing thecoherence width and/or the irradiance width of the input beam.For instance, it has been found that during propagation theDoP may tend to a constant value, lower than unity, even inthe case of a totally polarized input beam. This result shows,in particular, that a DWD may act as a true depolarizer forinput fields partially coherent from the spatial point of view.On the other hand, for small values of the input beam width asplitting of the beam into two or three parts (depending on thepolarization state of the input beam) can be appreciated, whosesuperposition fixes the irradiance distribution and polarizationproperties of the resulting field. The propagation directionsand diffraction widening of such beams depend on the DWDparameters and on the coherence width of the input field,respectively.

The present treatment can be easily extended to the caseof polarization gratings [14, 15] and to any spatially periodicdeterministic optical element.

Appendix A

Tables A.1 and A.2 collect all the symbols used throughoutthe text, their meaning and the first equation where they areused.

12

J. Opt. 16 (2014) 035708 J C G de Sande et al

Table A.1. Latin letters symbols and their meaning.

Symbol Meaning First equation

aqjk Complex coefficients (20)

D (superscript) Double-wedge depolarizer (DWD) characteristic (10)di Mean thickness of the i th wedge (i = 1, 2) (11)E j (r, z; t) Electric field component along j ( j = x, y) (2)F2 1+ 4αβ

ζ 2 (25)

H jk (s, t) Auxiliary function (18)h jk(sx , tx , z) Polarization part of the BCP matrix after the DWD (35)I Irradiance (36)I0 Maximum irradiance of a GSM beam (24)in (superscript) Input beam quantity (6)J (r1, r2, z) Beam coherence-polarization (BCP) matrix (1)Jsc(r1, r2, 0) Scalar mutual intensity (8)j lmjk (r1, r2, z) BCP elements’ terms after a system defined by equation (20) (22)

j jk(x1, x2) Polarization part of the BCP matrix at the DWD exit plane (31)Kz(r, ρ) Direct propagator in paraxial conditions (15)Lx Transverse period of the polarization state (13)MD(x) Muller matrix for a DWD (14)P(r, z) Polarization matrix (3)p(r, z) Degree of polarization (DoP) (4)r Position vector in a transverse plane (1)S(r) Stokes vector (5)T (r) Jones matrix of a deterministic linear optical element (6)s (r1+ r2)/2 (18)t r2− r1 (18)TP (superscript) Quantity derived for totally polarized input beams (29)U Jones vector (40)UP (superscript) Quantity derived for completely unpolarized input beams (43)zT Talbot distance

Table A.2. Greek letter symbols and their meaning.

Symbol Meaning First equation

α 12σ 2 (25)

β 18σ 2 +

12µ2 (25)

γ (2π |1n| tanϕ) /λ (11)γ

qjk Proportionality vectors (20)

δi (x) Phase difference in the i th wedge (i = 1, 2) (10)δi0 Mean phase difference in the i th wedge (i = 1, 2) (11)1n Birefringence (11)ζ 2π/(λz) (17)θ, φ Angles defining the polarization state (29)λ Wavelength (11)µ Width of the degree of coherence (24)ρ Position vector across the source plane (15)σ 2 Variance of the irradiance profile (24)ϕ Wedge angle (11)

13

J. Opt. 16 (2014) 035708 J C G de Sande et al

Appendix B

In this appendix, the particular expressions of productsT jl (r1) T ∗km (r2), which are necessary to evaluate equations (7)and (9) for the case of a DWD are developed.

By using equation (10) it is obtained that

T Dxx (x1)T D

xx∗(x2)=

14 [1+ e−iδ2(x1)

+ eiδ2(x2)+ eiδ2(x2)−iδ2(x1)], (B.1)T D

xy(x1)T Dxx∗(x2)=

14 e−iδ1(x1)[1− e−iδ2(x1)

+ eiδ2(x2)− eiδ2(x2)−iδ2(x1)], (B.2)T D

xx (x1)T Dxy∗(x2)=

14 eiδ1(x2)[1+ e−iδ2(x1)

− eiδ2(x2)− eiδ2(x2)−iδ2(x1)], (B.3)T D

xy(x1)T Dxy∗(x2) =

14 eiδ1(x2)−iδ1(x1)[1− e−iδ2(x1)

− eiδ2(x2)+ eiδ2(x2)−iδ2(x1)], (B.4)T D

yx (x1)T Dyx∗(x2)=

14 [1− e−iδ2(x1)− eiδ2(x2)

+ eiδ2(x2)−iδ2(x1)], (B.5)T D

yy(x1)T Dyx∗(x2)=

14 e−iδ1(x1)[1+ e−iδ2(x1)

− eiδ2(x2)− eiδ2(x2)−iδ2(x1)], (B.6)T D

yx (x1)T Dyy∗(x2)=

14 eiδ1(x2)[1− e−iδ2(x1)+ eiδ2(x2)

− eiδ2(x2)−iδ2(x1)], (B.7)T D

yy(x1)T Dyy∗(x2) =

14 eiδ1(x2)−iδ1(x1)[1+ e−iδ2(x1)

+ eiδ2(x2)+ eiδ2(x2)−iδ2(x1)], (B.8)T D

xx (x1)T Dyx∗(x2)=

14 [1+ e−iδ2(x1)− eiδ2(x2)

− eiδ2(x2)−iδ2(x1)], (B.9)T D

xy(x1)T Dyx∗(x2)=

14 e−iδ1(x1)[1− e−iδ2(x1)

− eiδ2(x2)+ eiδ2(x2)−iδ2(x1)], (B.10)T D

xx (x1)T Dyy∗(x2)=

14 eiδ1(x2)[1+ e−iδ2(x1)

+ eiδ2(x2)+ eiδ2(x2)−iδ2(x1)], (B.11)T D

xy(x1)T Dyy∗(x2) =

14 eiδ1(x2)−iδ1(x1)[1− e−iδ2(x1)

+ eiδ2(x2)− eiδ2(x2)−iδ2(x1)]. (B.12)

Due to the property Jyx (r1, r2, 0)= J ∗xy (r2, r1, 0) [24],it is not necessary to calculate the last four terms.

Appendix C

For the case of a DWD, the terms j lmjk (r1, r2, z) given in

equation (23), which should be substituted in equation (22),can be calculated by taking into account the linearity andthe displacement properties of the Fourier transform in equa-tion (17) and using equations (B.1)–(B.12). Their expressionsare

j xxxx (r1, r2, z)

=14

[Hxx (s, t)+ e−iδ20 Hxx

(s+

γux

2ζ, t−

γux

ζ

)+ eiδ20 Hxx

(s+

γux

2ζ, t+

γux

ζ

)+ Hxx

(s+

γux

ζ, t)], (C.1)

j yxxx (r1, r2, z)

=14

[e−iδ10 Hyx

(s−

γux

2ζ, t+

γux

ζ

)− e−i(δ10+δ20)Hyx (s, t)

+ e−i(δ10−δ20)Hyx

(s, t+

2γux

ζ

)− e−iδ10 Hyx

(s+

γux

2ζ, t+

γux

ζ

)], (C.2)

j xyxx (r1, r2, z)=

14

[eiδ10 Hxy

(s−

γux

2ζ, t−

γux

ζ

)+ ei(δ10−δ20)Hxy

(s, t−

2γux

ζ

)− ei(δ10+δ20)Hxy (s, t)

− eiδ10 Hxy

(s+

γux

2ζ, t−

γux

ζ

)], (C.3)

j yyxx (r1, r2, z)=

14

[Hyy

(s−

γux

ζ, t)

− e−iδ20 Hyy

(s−

γux

2ζ, t−

γux

ζ

)− eiδ20 Hyy

(s−

γux

2ζ, t+

γux

ζ

)+ Hyy (s, t)

], (C.4)

j xxyy (r1, r2, z)

=14

[Hxx (s, t)− e−iδ20 Hxx

(s+

γux

2ζ, t−

γux

ζ

)− eiδ20 Hxx

(s+

γux

2ζ, t+

γux

ζ

)+ Hxx

(s+

γux

ζ, t)], (C.5)

j yxyy (r1, r2, z)

=14

[e−iδ10 Hyx

(s−

γux

2ζ, t+

γux

ζ

)+ e−i(δ10+δ20)Hyx (s, t)

− e−i(δ10−δ20)Hyx

(s, t+

2γux

ζ

)− e−iδ10 Hyx

(s+

γux

2ζ, t+

γux

ζ

)], (C.6)

j xyyy (r1, r2, z)=

14

[eiδ10 Hxy

(s−

γux

2ζ, t−

γux

ζ

)− ei(δ10−δ20)Hxy

(s, t−

2γux

ζ

)+ ei(δ10+δ20)Hxy (s, t)

− eiδ10 Hxy

(s+

γux

2ζ, t−

γux

ζ

)], (C.7)

j yyyy (r1, r2, z)=

14

[Hyy

(s−

γux

ζ, t)

+ e−iδ20 Hyy

(s−

γux

2ζ, t−

γux

ζ

)+ eiδ20 Hyy

(s−

γux

2ζ, t+

γux

ζ

)+ Hyy (s, t)

], (C.8)

14

J. Opt. 16 (2014) 035708 J C G de Sande et al

j xxxy (r1, r2, z)

=14

[Hxx (s, t)+ e−iδ20 Hxx

(s+

γux

2ζ, t−

γux

ζ

)− eiδ20 Hxx

(s+

γux

2ζ, t+

γux

ζ

)− Hxx

(s+

γux

ζ, t)], (C.9)

j yxxy (r1, r2, z)

=14

[e−iδ10 Hyx

(s−

γux

2ζ, t+

γux

ζ

)− e−i(δ10+δ20)Hyx (s, t)

− e−i(δ10−δ20)Hyx

(s, t+

2γux

ζ

)+ e−iδ10 Hyx

(s+

γux

2ζ, t+

γux

ζ

)], (C.10)

j xyxy (r1, r2, z)

=14

[eiδ10 Hxy

(s−

γux

2ζ, t−

γux

ζ

)+ ei(δ10−δ20)Hxy

(s, t−

2γux

ζ

)+ ei(δ10+δ20)Hxy (s, t)

+ eiδ10 Hxy

(s+

γux

2ζ, t−

γux

ζ

)], (C.11)

j yyxy (r1, r2, z)

=14

[Hyy

(s−

γux

ζ, t)

− e−iδ20 Hyy

(s−

γux

2ζ, t−

γux

ζ

)+ eiδ20 Hyy

(s−

γux

2ζ, t+

γux

ζ

)− Hyy (s, t)

]. (C.12)

Again, it is not necessary to calculate the terms j lmyx (r1, r2, z)

because Jxy (r1, r2, z)= Jyx∗ (r2, r1, z).

Appendix D

The matrix elements hTPjk (s, t, z) that appear in equation (35)

can be calculated by substitution of equations (B.1)–(B.12)into (32)–(34) and then applying equation (22) to the particularcase of a DWD. They result in

hTPxx (s, t, z)

=12 H(s, t) [1− cos (δ10+ δ20+φ) sin 2θ ]

+12

[e−iδ20 H

(s+

γ

2ζ, t −

γ

ζ

)+ eiδ20 H

(s+

γ

2ζ, t +

γ

ζ

)]cos2 θ

−12

[e−iδ20 H

(s−

γ

2ζ, t −

γ

ζ

)

+ eiδ20 H(

s−γ

2ζ, t +

γ

ζ

)]sin2 θ

+12

H(

s+γ

ζ, t)

cos2 θ +12

H(

s−γ

ζ, t)

sin2 θ

+14

{e−i(δ10+φ)

[H(

s−γ

2ζ, t +

γ

ζ

)− H

(s+

γ

2ζ, t +

γ

ζ

)]+ ei(δ10+φ)

[H(

s−γ

2ζ, t −

γ

ζ

)− H

(s+

γ

2ζ, t −

γ

ζ

)]+ e−i(δ10−δ20+φ)H

(s, t +

2γζ

)+ ei(δ10−δ20+φ)H

(s, t −

2γζ

)}sin 2θ,

(D.1)hTP

xy (s, t, z)

=12 H(s, t) [cos 2θ + i sin (δ10+ δ20+φ) sin 2θ ]

+12

[e−iδ20 H

(s+

γ

2ζ, t −

γ

ζ

)− eiδ20 H

(s+

γ

2ζ, t +

γ

ζ

)]cos2 θ

−12

[e−iδ20 H

(s−

γ

2ζ, t −

γ

ζ

)− eiδ20 H

(s−

γ

2ζ, t +

γ

ζ

)]sin2 θ

−12

H(

s+γ

ζ, t)

cos2 θ +12

H(

s−γ

ζ, t)

sin2 θ

+14

{e−i(δ10+φ)

[H(

s−γ

2ζ, t +

γ

ζ

)+ H

(s+

γ

2ζ, t +

γ

ζ

)]+ ei(δ10+φ)

[H(

s−γ

2ζ, t −

γ

ζ

)+ H

(s+

γ

2ζ, t −

γ

ζ

)]− e−i(δ10−δ20+φ)H

(s, t +

2γζ

)+ ei(δ10−δ20+φ)H

(s, t −

2γζ

)}sin 2θ, (D.2)

hTPyy (s, t, z)

=12 H(s, t) [1+ cos (δ10+ δ20+φ) sin 2θ ]

−12

[e−iδ20 H

(s+

γ

2ζ, t −

γ

ζ

)+ eiδ20 H

(s+

γ

2ζ, t +

γ

ζ

)]cos2 θ

+12

[e−iδ20 H

(s−

γ

2ζ, t −

γ

ζ

)

15

J. Opt. 16 (2014) 035708 J C G de Sande et al

+ eiδ20 H(

s−γ

2ζ, t +

γ

ζ

)]sin2 θ

+12

H(

s+γ

ζ, t)

cos2 θ +12

H(

s−γ

ζ, t)

sin2 θ

+14

{e−i(δ10+φ)

[H(

s−γ

2ζ, t +

γ

ζ

)− H

(s+

γ

2ζ, t +

γ

ζ

)]+ ei(δ10+φ)

[H(

s−γ

2ζ, t −

γ

ζ

)− H

(s+

γ

2ζ, t −

γ

ζ

)]− e−i(δ10−δ20+φ)H

(s, t +

2γζ

)− ei(δ10−δ20+φ)H

(s, t −

2γζ

)}sin 2θ. (D.3)

In these expressions H(s, t) is given by equation (25) forthe particular case of a GSM input beam.

Appendix E

The matrix elements hUPjk (s, t, z) that appear in equation (44)

can be calculated by substitution of equations (B.1)–(B.12)into equations (45)–(47) and then applying equation (22)to the particular case of a DWD. They can be writtenas

hUPxx (s, t, z)

=12

H(s, t)+14

H(

s+γ

ζ, t)+

14

H(

s−γ

ζ, t)

+14

[e−iδ20 H

(s+

γ

2ζ, t −

γ

ζ

)+ eiδ20 H

(s+

γ

2ζ, t +

γ

ζ

)]−

14

[e−iδ20 H

(s−

γ

2ζ, t −

γ

ζ

)+ eiδ20 H

(s−

γ

2ζ, t +

γ

ζ

)], (E.1)

hUPxy (s, t, z)

=14

[e−iδ20 H

(s+

γ

2ζ, t −

γ

ζ

)− eiδ20 H

(s+

γ

2ζ, t +

γ

ζ

)]−

14

[e−iδ20 H

(s−

γ

2ζ, t −

γ

ζ

)− eiδ20 H

(s−

γ

2ζ, t +

γ

ζ

)]−

14

H(

s+γ

ζ, t)+

14

H(

s−γ

ζ, t), (E.2)

hUPyy (s, t, z)

=12

H(s, t)+14

H(

s+γ

ζ, t)+

14

H(

s−γ

ζ, t)

−14

[e−iδ20 H

(s+

γ

2ζ, t −

γ

ζ

)+ eiδ20 H

(s+

γ

2ζ, t +

γ

ζ

)]+

14

[e−iδ20 H

(s−

γ

2ζ, t −

γ

ζ

)+ eiδ20 H

(s−

γ

2ζ, t +

γ

ζ

)]. (E.3)

In these expressions H(s, t) is given by equation (25) forthe particular case of a GSM input beam.

References

[1] McGuire J P and Chipman R A 1990 Analysis of spatialpseudodepolarizers in imaging systems Opt. Eng.29 1478–84

[2] El Sherif M, Khalil M S, Khodeir S and Nagib N 1996 Simpledepolarizers for spectrophotometric measurements ofanisotropic samples Opt. Laser Technol. 28 561–3

[3] Biener G, Niv A, Kleiner V and Hasman E 2003Computer-generated infrared depolarizer usingspace-variant subwavelength dielectric gratings Opt. Lett.28 1400–2

[4] Honma M and Nose T 2004 Liquid-crystal depolarizerconsisting of randomly aligned hybrid orientation domainsAppl. Opt. 43 4667–71

[5] Vartiainen I, Tervo J and Kuittinen M 2009 Depolarization ofquasi-monochromatic light by thin resonant gratings Opt.Lett. 34 1648–50

[6] Bagan V A, Davydov B L and Samartsev I E 2009Characteristics of Cornu depolarisers made from quartz andparatellurite optically active crystals Quantum Electron.39 73–8

[7] Vena C, Versace C, Strangi G and Bartolino R 2009 Lightdepolarization by non-uniform polarization distribution overa beam cross section J. Opt. A: Pure Appl. Opt. 11 125704

[8] de Sande J C G, Piquero G and Teijeiro C 2012 Polarizationchanges at Lyot depolarizer output for different types ofinput beams J. Opt. Soc. Am. A 29 278–84

[9] de Sande J C G, Santarsiero M, Piquero G and Gori F 2012Longitudinal polarization periodicity of unpolarized lightpassing through a double wedge depolarizer Opt. Express20 27348–60

[10] Makowski P L, Szymanski M Z and Domanski A W 2012Lyot depolarizer in terms of the theory ofcoherence—description for light of any spectrum Appl. Opt.51 626–34

[11] Santarsiero M 2007 Polarization invariance in a Younginterferometer J. Opt. Soc. Am. A 24 3493–9

[12] Gori F, Tervo J and Turunen J 2009 Correlation matrices ofcompletely unpolarized beams Opt. Lett. 34 1447–9

[13] Hassinen T, Tervo J and Friberg A T 2013 Purity of partialpolarization in the frequency and time domains Opt. Lett.38 1221–3

[14] Piquero G, Borghi R and Santarsiero M 2001 GaussianSchell-model beams propagating through polarizationgratings J. Opt. Soc. Am. A 18 1399–405

16

J. Opt. 16 (2014) 035708 J C G de Sande et al

[15] Piquero G, Borghi R, Mondello A and Santarsiero M 2001Far field of beams generated by quasi-homogeneous sourcespassing through polarization gratings Opt. Commun.195 339–50

[16] Santarsiero M, de Sande J C G, Piquero G and Gori F 2013Coherence-polarization properties of fields radiated fromtransversely periodic electromagnetic sources J. Opt.15 055701

[17] Mandel L and Wolf E 1995 Optical Coherence and QuantumOptics (Cambridge: Cambridge University Press)

[18] Gori F, Santarsiero M, Piquero G, Borghi R, Mondello A andSimon R 2001 Partially polarized Gaussian Schell-modelbeams J. Opt. A: Pure Appl. Opt. 3 1–9

[19] Piquero G, Gori F, Romanini P, Santarsiero M, Borghi R andMondello A 2002 Synthesis of partially polarized GaussianSchell-model sources Opt. Commun. 208 9–16

[20] Shirai T, Korotkova O and Wolf E 2005 A method ofgenerating electromagnetic Gaussian Schell-model beamsJ. Opt. A: Pure Appl. Opt. 7 232–7

[21] Wolf E 2007 Introduction to the Theory of Coherence andPolarization of Light (Cambridge: Cambridge UniversityPress)

[22] Martınez-Herrero R, Mejıas P M and Piquero G 2009Characterization of Partially Polarized Light Fields(Springer Series in Optical Sciences vol 147) (Berlin:Springer)

[23] Gori F 1998 Matrix treatment for partially polarized, partiallycoherent beams Opt. Lett. 23 241–3

[24] Gori F, Santarsiero M, Vicalvi S, Borghi R and Guattari G1998 Beam coherence polarization matrix J. Eur. Opt. Soc.A: Pure Appl. Opt. 7 941–51

[25] Gori F, Santarsiero M, Borghi R and Guattari G 1999Irradiance of partially polarized beams in a scalar treatmentOpt. Commun. 163 159–63

[26] Wolf E 2003 Unified theory of coherence and polarizationof random electromagnetic beams Phys. Lett. A 312 263–7

[27] Born M and Wolf E 1999 Principles of Optics (Cambridge:Cambridge University Press) 7th expanded

[28] Simon R 1990 Nondepolarizing systems and degree ofpolarization Opt. Commun. 77 349–54

[29] Alekseeva L V, Povkh I V, Stroganov V I, Kidyarov B I andPasko P G 2002 Four-ray splitting in optical crystals J. Opt.Technol. 39 441–3

[30] Kuznetsov V, Faleiev D, Savin E and Lebedev V 2009Crystal-based device for combining light beams Opt. Lett.34 2856–7

[31] Arrizon V, Tepichin E, Ortiz-Gutierrez M and Lohmann A W1996 Fresnel diffraction at 1/4 of the Talbot distance of ananisotropic grating Opt. Commun. 127 171–5

[32] Fujiwara H 2007 Spectroscopic Ellipsometry: Principlesand Applications. Appendix A (Chichester: Wiley) pp 353–5

[33] Simon B N, Simon S, Gori F, Santarsiero M, Borghi R,Mukunda N and Simon R 2010 Nonquantum entanglementresolves a basic issue in polarization optics Phys. Rev. Lett.104 023901

[34] Chipman R A 2005 Depolarization index and the averagedegree of polarization Appl. Opt. 44 2490–5

[35] Gori F 1983 Mode propagation of the field generated byCollett–Wolf Schell-model sources Opt. Commun.46 149–54

17