Phase-space representation and polarization domainsof random electromagnetic fields
Roman Castaneda,1,* Rafael Betancur,1 Jorge Herrera,1 and Juan Carrasquilla2
1Physics School, Universidad Nacional de Colombia Sede Medellín, A.A. 3840, Medellín, Colombia2CNR-INFM—Democritos National Simulation Centre and International School for Advanced Studies (SISSA),
The theory of electromagnetic spatial coherencewavelets [1] was developed according to the second-order coherence theory of electromagnetic fields[2] for analyzing the fundamental relationshipamong the correlations of the field vectors andthe spatial coherence properties, the polarizationstate, and the energy transport of stationary randomelectromagnetic fields [3,4]. It was also used as theframework for the development of a tensor theoryof electromagnetic radiometry [5]. In the following,it is shown that this theory constitutes the phase-space representation of stationary random electro-magnetic fields.The electromagnetic spatial coherence wavelets
at the frequency ν and wavenumber k ¼ 2π=λ, with
λ the wavelength, are defined as a set of 2 × 2 tensorsof the form [1]
WABðrA � rD=2; ξA; νÞ ¼ SABðrA; ξA; νÞ
× exp�−i
kzrD · ξA
�; ð1Þ
where AB denotes the electric (A ¼ B ¼ E), magnetic(A ¼ B ¼ H), and mixed (A ¼ E; B ¼ H or A ¼ H;B ¼ E) tensors and ðξA; ξDÞ, ðrA; rDÞ are the centerand difference coordinates at the aperture plane(AP) and the observation plane (OP), respectively,which are at a distance z from each other. So, thenotation ξA � ξD=2≡ ðξA þ ξD=2; ξA − ξD=2Þ and rA �rD=2≡ ðrA þ rD=2; rA − rD=2Þ denotes the positions ofany pair of points at those planes, as depictedin Fig. 1. These expressions will be used as shortnotation for the argument of quantities that dependon pairs of points, such the above tensors, forinstance.
is called the (electric, magnetic, or mixed) marginalpower spectrum tensor [1]. Its integral is applied toeach tensor element separately (a notation valid forall tensor expressions in this work), and
denotes the cross-spectral density tensors [2] at AP.A and B are 2 × 2 matrices of real-valued elementsAðþÞlm ðνÞ ¼ hjAlðξA þ ξD=2; νÞj2i1=2δlm and Bð−Þ
lm ðνÞ ¼hjBlðξA − ξD=2; νÞj2i1=2δlm, whose indices lm arerelated to the Cartesian components ðx; yÞ of thevectors
and δlm is the Kronecker delta. AðþÞðνÞ and Bð−ÞðνÞdenote the electric or magnetic field vectors at thepoints ξA � ξD=2, respectively, with ðux; uyÞ the uni-tary vectors along the Cartesian xy axes. ηABðξA �ξD=2; νÞ is the 2 × 2 correlation tensor betweenAðþÞðνÞ and Bð−ÞðνÞ, whose elements take the form
ηð�Þlm ðνÞ ¼ hAðþÞ
l ðνÞBð−Þ�m ðνÞi
hjAðþÞl ðνÞj2i1=2hjBð−Þ
m ðνÞj2i1=2;
where the asterisk denotes the complex conju-gate. Furthermore, ηð�Þ
lm ðνÞ ¼ jηð�Þlm ðνÞj exp½iαð�Þ
lm ðνÞ�holds, with 0 ≤ jηð�Þ
lm ðνÞj ≤ 1, jηllðξA; ξA; νÞj ¼ 1, and
αllðξA; ξA; νÞ ¼ 0. It is worth noting that ηABðξA �ξD=2; νÞ ¼ ½ηBAðξA∓ξD=2; νÞ�T� holds, with T denot-ing transpose. Therefore, WABðξA � ξD=2; νÞ ¼½WBAðξA∓ξD=2; νÞ�T� and SABðrA; ξA; νÞ ¼½SBAðrA; ξA; νÞ�T�. As a consequence, the tensorSAAðrA; ξA; νÞ is Hermitian. In other words,tr½SAAðrA; ξA; νÞ�, where the symbol tr denotes thetensor trace, will be real valued but can take onnegative values, according to Eq. (2). However,taking into account that [1]
WAAðrA � rD=2; νÞ ¼�1λz
�2
× exp�ikzrA · rD
�
×ZAP
SAAðrA; ξA; νÞ
× exp�−i
kzrD · ξA
�d2ξA ð4Þ
represents the corresponding cross-spectral densitytensor at OP, it follows that
SAðrA; νÞ ¼ tr½WAAðrA; rA; νÞ�
¼�1λz
�2ZAP
tr½SAAðrA; ξA; νÞ�d2ξA ≥ 0 ð5Þ
for the power spectrum associated with the vectorAðνÞ at OP. Indeed, the evaluation of the cross-spectral density tensor at OP for rD ¼ 0 gives thetensor SAðrA; νÞ ¼ WAAðrA; rA; νÞ, whose diagonal ele-ments SA
llðrA; νÞ ¼ WAAll ðrA; rA; νÞ are the power spec-
tra [2] at OP contributed by each component AlðνÞof the vector AðνÞ. The quantity tr½SAAðrA; ξA; νÞ�resembles a Wigner distribution function (WDF)[6–10], which provides the phase-space represen-tation for AðνÞ, with ξA and krA=z the space andthe phase variables, respectively. More precisely,tr½SEEðrA; ξA; νÞ� and tr½SHHðrA; ξA; νÞ� are the electricand magnetic WDFs that provide the phase-spacerepresentation for stationary random electromag-netic fields, in any state of spatial coherence andpolarization.
2. Polarization Domains
Because of the Maxwellian coupling [11], the electricand the magnetic field vectors of the electromagneticwave are similarly correlated. Therefore, bothtr½SEEðrA; ξA; νÞ� and tr½SHHðrA; ξA; νÞ� describe thesame physical properties of the wave [1]. On accountof this redundancy, it is enough to regard the phase-space representation given by the electric WDF.From Eq. (3) follows
Accordingly, tr½SEEðrA; ξA; νÞ� describes the electricenergy transfer from AP to OP that depends onthe spatial coherence state of the stationary randomelectromagnetic field, which is determined by thediagonal elements of the ηEE tensor. Indeed, substi-tuting Eq. (6) into Eq. (5) gives the average electricenergy density of this field at OP. It is worth notingthat the Fresnel–Arago interference laws requirethat the complex degree of spatial (or spectral) coher-ence [2–4] of the stationary random electromagneticfield be
The off-diagonal elements of the ηEE tensor deter-mine the polarization state of the field through thepolarization parameter, which estimates how muchthe electromagnetic spatial coherence wavelets arepolarized. For defining the polarization parameter,the electric marginal power spectrum tensor isuniquely expressed in terms of a completely unpolar-ized component and a polarized component, i.e.,SEEðrA; ξA; νÞ ¼ SEE
unpolðrA; ξA; νÞ þ SEEpolðrA; ξA; νÞ with
the property that det½SEEpolðrA; ξA; νÞ� ¼ 0. The deter-
mination of SEEpolðrA; ξA; νÞ is rather involved but is
completely developed in [1]. It allows the polariza-tion parameter to be defined as
where the symbol det denotes the tensor determi-nant. This parameter is real valued in the range0 ≤ PðrA; ξA; νÞ ≤ 1, where the extreme values 0 and1 refer to completely polarized and unpolarized con-tributions to the field, respectively, which propagatealong the ξA → rA axis. Because of Eqs. (2)–(4), thepolarization parameter should exhibit a nonlocalcharacter; i.e., it depends on the contributions pro-
vided by pairs of centers of secondary disturbancewithin the surrounding of ξA. This feature constitu-tes the main difference between it and the degree ofpolarization [2], which is defined instantaneously ona single point on a wavefront.
Because of the above analysis, ηEEðξA � ξD=2; νÞ iscalled the “spatial coherence polarization tensor” [1].It can be written as
ηEEðξA � ξD=2; νÞ¼ μðξA � ξD=2; νÞ
�1 0
0 1
�
þ�
0 ηEExy ðξA � ξD=2; νÞηEEyx ðξA � ξD=2; νÞ 0
�; ð8Þ
where the first term determines the spatial co-herence properties of the field while the secondterm determines its polarization state. It is worthnoting that the first term in Eq. (8) results fromthe above-mentioned condition ηEExx ðξA � ξD=2; νÞ ¼ηEEyy ðξA � ξD=2; νÞ ¼ μðξA � ξD=2; νÞ required by theFresnel–Arago interference laws [4]. Equations (1)–(3) and (8) deal with the electric spatial coherencewavelet tensor
WEEðrA � rD=2; ξA; νÞ
¼�W EE
xx ðrA � rD=2; ξA; νÞ 00 W EE
xx ðrA � rD=2; ξA; νÞ�
þ�
0 W EExy ðrA � rD=2; ξA; νÞ
W EEyx ðrA � rD=2; ξA; νÞ 0
�ð9Þ
with
W EElm ðrA � rD=2; ξA; νÞ ¼ SEE
lm ðrA; ξA; νÞ
× exp�−i
kzrD · ξA
�;
SEElm ðrA; ξA; νÞ ¼
ZAP
hjEðþÞl ðνÞj2i1=2ηEElm
× ðξA � ξD=2; νÞhjEð−Þm ðνÞj2i1=2
× exp�ikzðξA − rAÞ · ξD
�d2ξD
the elements of the electric marginal power spectrumtensor. Therefore, the first term of Eq. (9) is a wavelettensor responsible for the transfer of the spatial co-herence properties and power of the stationary ran-dom electromagnetic field from AP to OP, while thesecond term is the wavelet tensor responsible for the
stands for ηEElm ðξA � ξD=2; νÞ ¼ CδðξDÞ, with C a con-stant for making the expression dimensionless,and δðξDÞ the Dirac delta. In this case, the Cartesiancomponents of the electric field vector at AP will bestrongly correlated only at the same point ξA. Thus,the elements of the electric cross-spectral densitytensor take the form
WEElm ðrA � rD=2; vÞ ¼ C
�1λz
�2
× exp�ikzrA⋅rD
�
×ZAP
hjElðξA; vÞj2i1=2
× hjEmðξA; vÞj2i1=2
× exp�−i
kzrD⋅ξA
�d2ξA:
Its diagonal elements confirm the Van Cittert–Zernike theorem for stationary random electro-magnetic fields, i.e.,
WEEll ðrA � rD=2; νÞ ¼ C
�1λz
�2
× exp�ikzrA · rD
�
×ZAP
hjElðξA; νÞj2i × exp�−i
kzrD · ξA
�d2ξA;
which predicts a gain in spatial coherence of fullyspatially incoherent fields at AP, because of theirpropagation to OP; i.e., the support of the complexdegree of spatial coherence (the region in which ittakes nonnegligible values) grows along the fieldpropagation.
Furthermore, PðrA; ξA; νÞ ¼ 1 stands becausedet½WEEðrA; ξA; νÞ� ¼ 0 holds in this case [1]. Thuscompletely polarized wavelets propagate alongeach ξA → rA axis. Consequently, the supportsof the off-diagonal elements of the ηEE tensorgrow, too; i.e., they are arbitrary narrow at AP be-cause WEE
lm ðξA � ξD=2; νÞ ¼ ChjElðξA þ ξD=2; νÞj2i1=2 ×hjEmðξA − ξD=2; νÞj2i1=2δðξDÞ stands for l ≠ m, butbecome extended at OP because
WEElm ðrA � rD=2; νÞ ¼ C
�1λz
�2exp
�ikzrA · rD
�
×ZAP
hjElðξA; νÞj2i1=2
× hjEmðξA; νÞj2i1=2
× exp�−i
kzrD · ξA
�d2ξA:
Because of the definition of the polarization para-meter, it is expectable that the electromagnetic fieldwill exhibit definite polarization states within suchsupports, although the polarization states in two dif-ferent supports differ, too. In other words, the sup-ports of the off-diagonal elements of the ηEE tensorconform to polarization domains [1], which grow withthe field propagation if the field is locally completelypolarized at AP. The above analysis leads to the con-clusion that the electric cross-spectral density tensor
WEEðrA � rD=2; νÞ ¼ C
�1λz
�2exp
�ikzrA · rD
�
×ZAP
EðξA; νÞ�1 1
1 1
�EðξA; νÞ
× exp�−i
kzrD · ξA
�d2ξA ð10Þ
represents the gain in both spatial coherence and po-larization of a fully spatially incoherent and locallycompletely polarized stationary random electromag-netic field through its propagation, in terms of thegrowth of both the support of its complex degreeof spatial coherence and the size of its polariza-tion domains. Equation (10) can be regarded as a
generalization of the Van Cittert–Zernike theoremthat includes the polarization state of the field.So,
WEEðrA � rD=2; ξA; νÞ ¼ CEðξA; νÞ
�1 1
1 1
�EðξA; νÞ
× exp�−i
kzrD · ξA
�
is the electric spatial coherence wavelet tensorof such a field, and then tr½SEEðrA; ξA; νÞ� ¼C½hjExðξA; νÞj2i þ hjEyðξA; νÞj2i� provides its phase-space representation in terms of an isotropic emis-sion of the electric average energy density by eachcenter of secondary disturbance at AP.
On the other hand, because SABðrA; ξA; νÞ þSBAðrA; ξA; νÞ ¼ 2Re½SABðrA; ξA; νÞ� holds for A ≠ B,with Re denoting the real part, it is enough to con-sider the SABðrA; ξA; νÞ≡ SEHðrA; ξA; νÞ mixed tensor,for mutually orthogonal E and H field vectors. Thusif the matrix
EðξA � ξD=2; νÞ ¼� hjEð�Þ
x ðνÞj2i1=2 00 hjEð�Þ
y ðνÞj2i1=2�
represents the electric field vector at the pointsξA � ξD=2, then the matrix
HðξA � ξD=2; νÞ ¼�
0 −hjHð�Þx ðνÞj2i1=2
hjHð�Þy ðνÞj2i1=2 0
�
will represent the magnetic field vector orthogonal tothe electric field vector at the same points. So, takinginto account the definition of ηEHlm ðξA � ξD=2; νÞ, thetrace of the mixed marginal power spectrum tensorwill be given by
tr½SEHðrA; ξA; νÞ� ¼ZAP
½hE�xðξA þ ξD=2; νÞHy
× ðξA − ξD=2; νÞi− hE�
yðξA þ ξD=2; νÞHx
× ðξA − ξD=2; νÞi�
× exp�ikzðξA − rAÞ · ξD
�d2ξD: ð11Þ
It is apparent that
Re½hE�xðξA þ ξD=2; νÞHyðξA − ξD=2; νÞi
− hE�yðξA þ ξD=2; νÞHxðξA − ξD=2; νÞi�
¼ RejhE�ðξA þ ξD=2; vÞ ×HðξA − ξD=2; vÞij ð12Þholds, in such a way that
Reftr½SEHðrA; ξA; νÞ�g ¼ZAP
RejhE�ðξA þ ξD=2; vÞ
×HðξA − ξD=2; vÞij
× exp�ikzðξA − rAÞ · ξD
�d2ξD;
ð13Þwhere RehE�ðξA þ ξD=2; vÞ ×HðξA − ξD=2; vÞi ¼ ð2π=cÞhSðξA; ξD; vÞi, with c the speed of light in vacuum,is a vector whose magnitude exhibits energetic unitsand whose direction is orthogonal to AP, mainly inthe Fraunhofer domain fexp½iðk=zÞξA · ξD� ≈ 1g.Equation (13) reduces to the Poynting vector [11]at ξA for ξD ¼ 0, i.e., hSðξA; 0; vÞi ¼ ðc=2πÞRe ×hE�ðξA; vÞ ×HðξA; vÞi. By introducing the dimension-less function 1 ¼ KδðξDÞ þ ½1 − KδðξDÞ� in the inte-grand of Eq. (13), with K a constant that ensuresthe dimensionless character of the function andδðξDÞ the Dirac delta function, Eq. (13)becomes
Reftr½SEHðrA; ξA; νÞ�g ¼ 2πc
�K jhSðξA; 0; vÞij
þZAPξD≠0
jhSðξA; ξD; vÞij
× exp�ikzðξA − rAÞ ·ξD
�d2ξD
�:
ð14Þ
The first term of Eq. (14) is proportional to theaverage electromagnetic energy density given bythe magnitude of the Poynting vector at thepoint ξA onAP. The second term provides modulatingelectromagnetic energies, given by the magnitude ofthe vector hSðξA; ξD; vÞi for ξD ≠ 0, which can take onpositive and negative values because of the harmonicfactor in the integrand. Such positive and negativeenergies increase and decrease the value of the firstterm, depending on the argument of the harmonicfactor, in such a way that Reftr½SEHðrA; ξA; νÞ�g cantake on negative values, too.
A similar way to obtain Eq. (5) leads to SEHðrA; νÞ ¼ð1=λzÞ2 RAP trReftr½SEHðrA; ξA; νÞ�gd2ξA andSEHðξA; νÞ ¼ ð1=λzÞ2 ROP Reftr½SEHðrA; ξA; νÞ�gd2rA forthe average electromagnetic energies at OP andAP, respectively. Thus, the conservation law of thetotal electromagnetic energy that flows betweenAP and OP can be expressed as
holds, according to Poynting’s theorem [11]. Further-more, Eq. (14) yields
ZAP
ZOP
Reftr½SEHðrA; ξA; νÞ�gd2rAd2ξA
¼ 2πKc
ZAP
d2rA
ZAP
jhSðξA; 0; vÞijd2ξA
þ 2πc
ZAP
ξD ≠ 0
jhSðξA; ξD; vÞijZOP
× exp�ikzðξA − rAÞ · ξD
�d2rAd2ξDd2ξA: ð16Þ
Consequently, Eqs. (15) and (16) lead to K ¼ ðλzÞ2=RAP d
2rA and the symmetry conditionROP exp½iðk=zÞ ×
ðξA − rAÞ · ξD�d2rA ¼ 0. This means that the positiveand negative modulating electromagnetic energiesdo not contribute to the total electromagnetic energy,but redistribute the electromagnetic energy emittedby AP onto OP. The condition SEHðrA; νÞ ≥ 0 impliesthat
����ZAP
ZAP
ξD ≠ 0
jhSðξA; ξD; vÞij
× exp�ikzðξA − rAÞ · ξD
�d2ξAd2ξD
����≤ K
ZAP
jhSðξA; 0; vÞijd2ξA;
so that 0 ≤ SEHðrA; νÞ ≤ 2KRAP jhSðξA; 0; vÞijd2ξA
holds. According to these results, hSðξA; ξD; vÞi forξD ≠ 0 can be thought of as parallel and antiparallelvectors to the Poynting vector at the position ξA,which add to this Poynting vector, increasing or de-creasing its magnitude, respectively. This behavior isclosely related to the positive and negative values ofthe modulating energies.The above analysis leads to the conclusion that
Reftr½SEHðrA; ξA; νÞ�g provides a phase-space repre-sentation (with the same space and phase variablesas before) of stationary random electromagneticfields, based on the correlation of the mutually ortho-gonal components of the electric and magnetic fieldvectors at pairs of points ξA � ξD=2 on AP. It revealsthe dependence of the flux of the average electromag-netic energy density on the correlation between mu-tually orthogonal components of the electric andmagnetic field vectors at those pairs of points. Thisprediction is more general than the local descriptiongiven by the Poynting theorem [11], and it is in ac-cord with the proper behavior of the wave natureof light that allows one to describe the flux of the
average electromagnetic energy density in presenceof interference [5].
Furthermore, because of the Maxwellian couplingbetween the electric and magnetic field vectors ofelectromagnetic waves in free space, the conditionE ¼ cμ0H holds [11], with μ0 the magnetic permeabil-ity of vacuum. E and H are the amplitudes of thefields at the same point in space–time. This relation-ship also holds for the Cartesian components of thefields ElðξA − ξD=2; νÞ ¼ cμ0HlðξA − ξD=2; νÞ. Conse-quently, ηEHlm ðξA � ξD=2; νÞ ¼ ηEElm ðξA � ξD=2; νÞ holds,too. Thus, due to the Maxwellian coupling, the mixedcorrelation tensor of a stationary random electro-magnetic field at pairs of points on AP equals thespatial coherence-polarization tensor at the samepair of points.
4. Young Experiment with Electromagnetic Waves
According to the phase-space representations pre-viously discussed, the fundamental sources ofstationary random electromagnetic fields in anystate of spatial coherence and polarization are thepairs of centers of secondary disturbance at AP, in-cluding the individual centers as pairs with null se-paration vector. Consequently, the Young experiment[12] becomes the fundamental experience for study-ing such phase-space representations.
A Young interferometer is a simple device, consist-ing of a fixed pinhole pair opened on an opaquescreen and a sensor placed at a distance z fromthe pinhole mask, in the Fraunhofer domain. Thesensor should record the interference pattern pro-duced by the mask, whose openings are small enoughfor neglecting diffraction effects [12]. The basic setupis shown in Fig. 2.
The electric spatial coherence wavelet tensor forthe Young interferometer will have four contribu-tions, i.e., a contribution from each individual pin-hole and two contributions due to the pinhole pair,
Fig. 2. Young interferometer for the analysis of the phase-spacerepresentation of stationary random electromagnetic fields.
The first two terms of Eq. (17) stand for ξD ¼ 0 andξA ¼ �ξA � �ξD=2, while the last two stand for ξA ¼ �ξAand ξD ¼ ��ξD, as indicated by the prefixes�. Accord-ingly, the phase-space representation for the electricfield vector of a uniform stationary random electro-magnetic field that propagates from the pinholemask to the sensor will be provided by
The condition jηEExx ðξA � ξD=2; νÞj ¼ jηEEyy ðξA �ξD=2; νÞj ¼ jμðξA � ξD=2; νÞj was taken into account.So, the electric WDF for the Young interferometerbecomes
Equation (20) reveals the existence of Sudarshan’s“tamasic” rays (from Sanskrit: tamas is “darkness”)[13]. Indeed, the two first terms of Eq. (20) are re-sponsible for the transfer of the electric average en-ergy density emerging from the pinholes to any pointrA on the sensor. The last two terms do not transferenergy but modulate the energetic contributionsfrom the pinholes by means of the cosine factor. Thisterm joins the middle point between the pinholes(which cannot emit energy) with the point rA.Furthermore, the amplitude and phase of the modu-lation depends on the correlation of themutually par-allel components of the electric field vectors at theopenings, specified by the complex degree of spatialcoherence centered at the middle point between thepinholes.
The power spectrum recorded by the sensor is ob-tained by replacing this phase-space representationin Eq. (5), i.e.,
SEðrA; vÞ ¼ SE0 ðvÞ
�1þ V cos
�kzrA · �ξD
− αð�ξA � ξD=2; vÞ��
; ð21Þ
with SE0 ðνÞ ¼ ðC=λzÞ2½hjEðþÞ
x ðνÞj2i þ hjEðþÞy ðνÞj2i þ
hjEð−Þx ðνÞj2i þ hjEð−Þ
y ðνÞj2i� the total power emergingfrom the pinhole mask and
the visibility [12] of the interference (modulation)fringes of the Young’s pattern, which are orthogo-nal to the separation vector �ξD and have a pe-riod of λz=j�ξDj. As expected, 0 ≤ V ≤ 1, andV ¼ jμð�ξA � ξD=2; vÞj stands when hjEðþÞ
l ðνÞj2i1=2 ¼hjEð−Þ
m ðνÞj2i1=2.Now, let us regard the mixed quantities WEHðrA �
rD=2; ξA; νÞ and tr½SEHðrA; ξA; νÞ�. Taking into accountthe properties of ηEHlm ðξA � ξD=2; νÞ, the terms oftr½SEHðrA; ξA; νÞ� take the form
Because of the Maxwellian coupling commentedon above, it is presumable that jhE�ðξA þ �ξD=2; vÞ ×HðξA − �ξD=2; vÞij ¼ jhE�ðξA − �ξD=2; vÞ ×HðξA þ �ξD=2;vÞij holds. Thus, the equations above yield
The first two terms of Eq. (24) correspond to theflux of the average electromagnetic energy from eachpinhole of the Young interferometer. Each contribu-tion is determined by the magnitude of the averagePoynting vector at the corresponding opening. Thethird term confirms Sudarshan’s hypothesis of tama-sic rays. This term joins themiddle point between the
openings to the point rA at the sensor plane and de-pends on the magnitude of the vector hSðξA; ξD; vÞi ¼ðc=2πÞRehE�ðξA þ ξD=2; vÞ ×HðξA − ξD=2; vÞi that isorthogonal to the mask plane at �ξA. Therefore, thisvector cannot transfer electromagnetic energy inspite of its units, but it introduces cosinelike modu-lations onto the electromagnetic energy that flowsfrom the openings, due to the correlation betweenthe mutually orthogonal components of the electricand the magnetic field vectors at the openings. Suchmodulation exhibits a fringe structure that shouldcoincide with the fringes of the electric power spec-trum (interference pattern) given by Eq. (21). Indeed,similar to as in Eq. (5), the electromagnetic powerspectrum at the sensor plane will be
ð25ÞThis distribution exhibits cosinelike fringes of periodλz=j�ξDj, orthogonal to the separation vector �ξD of thepinholes. To specify the visibility and the initialphase of the pattern, the amplitude and phase of theoff-diagonal elements of the electromagnetic correla-tion tensor ηEHlm ðξA � ξD=2; νÞ should be considered. Itis worth noting that Eqs. (21) and (22) reduce to theprediction of the Poynting theorem if the electric andmagnetic field vectors are completely uncorrelated.Therefore, Eq. (24) constitutes the (electromagnetic)phase-space representation of stationary randomelectromagnetic fields in a Young interferometer.
5. Numerical Results
The phase-space representation of a uniform station-ary random electromagnetic field in the Young inter-ferometer (Fig. 2) can be numerically analyzed bysimulating the interferometer operation with a basison the simulation variables in Fig. 3, i.e.,
• �ξD, pinhole separation vector;• �ξA, position of the middle point between the
pinholes;• Equal and constant amplitudes of the electric
field vectors Eð�ξA � �ξD=2Þ at the openings.• ϑ�, random angles of orientation of the electric
field vectors with respect to the line joining the open-ings. Two kinds of random distributions were as-sumed for these angles, i.e., a uniform distributionover the interval ½−π; π� and a Gaussian distribution,with variable standard deviation σ around fixedmean angles �ϑ�.
With these variables, the coherence-polarization ten-sor is expressed as
ηEEðþ;−; νÞ ¼24 hcosϑþ cosϑ−i
hcos2ϑþi1=2hcos2ϑ−i1=2hcosϑþ sinϑ−i
hcos2ϑþi1=2hsin2ϑ−i1=2hsinϑþ cosϑ−i
hsin2ϑþi1=2hcos2ϑ−i1=2hsinϑþ sinϑ−i
hsin2ϑþi1=2hsin2ϑ−i1=2
35:
ð26ÞResults of the numerical simulations for somecases and the corresponding polarization parametersare shown in Table 1. Simulated Young’s interfer-ence patterns were determined from ensembles of30,000 realizations in each case. A realization ofthe Young’s experiment will be specified by a choiceof the orientation angles of the electric field vec-tors at the openings, on account of the given randomdistributions. Specifically, the spatial coherence-polarization tensors, the power spectrum of the pat-tern, and its visibility were numerically calculatedfor the following cases:
1. Statistically independent angles, uniformlydistributed over the interval ½−π; π�.2. Random angles uniformly distributed over the
interval ½−π; π�, under the condition ϑþ ¼ ϑ−.3. Statistically independent angles that fluctuate
following a Gaussian distribution with mean an-gles �ϑþ ¼ �ϑ− ¼ π=4.4. Random angles that fluctuate following a
Gaussian distribution with mean angles �ϑþ ¼ �ϑ− ¼π=4, under the condition ϑþ ¼ ϑ−.5. Statistically independent angles that fluctuate
following a Gaussian distribution with mean angles�ϑþ ¼ π=4 and �ϑ− ¼ 3π=4.6. Random angles that fluctuate following a
Gaussian distribution with mean angles �ϑþ ¼ π=4and �ϑ− ¼ 3π=4, under the condition ϑþ ¼ ϑ−.
The matrix elements of Eq. (26) were calculated bytaking the random orientation angles as ϑ� ¼�ϑ� þΔϑ�, withΔϑ� the random fluctuations aroundthis mean value. As expected, 0 ≤ jηEElm ðþ;−; νÞj ≤ 1 [1]stands in all cases, with ηEExy ðþ;−; νÞ ¼ ηEEyx ðþ;−; νÞand ηEExx ðþ;−; νÞ ¼ ηEEyy ðþ;−; νÞ for high spatial coher-ence. Relative small differences between the diago-nal elements were obtained in cases of low spatialcoherence, due to the fluctuations of the calculationsfor small values. So, the magnitude of such elementscan be regarded as essentially equal, too, as stated bythe first Fresnel–Arago law [4]. Some of them areminus signed, which is due to the orientation ofthe involved electric field vectors in the Cartesianquadrants.
For statistically independent ϑ�, uniformly distri-buted over ½−π; π� (row 1 in Table 1), the random elec-tromagnetic field will be spatially incoherent (verylow visibility of fringes) and unpolarized (very lowpolarization parameter across the pattern), whichcoincides with the experimental results for the Youngexperiment with spatially incoherent natural light. Ifthe randomness of the orientation angles is main-tained but the condition ϑþ ¼ ϑ− is introduced (row2 in Table 1), the diagonal elements of the ηEE tensorbecome equal to one; i.e., the electromagnetic fieldremains unpolarized but becomes spatially coherent.In this case a Young fringe pattern with high visibi-lity will be obtained at OP, like those experimentallyobserved by using a nonpolarized laser beam [14].
However, the orientation angles of the electric fieldvectors at the openings follow Gaussian statistics inmany cases of interest, so that the fields are partiallycorrelated to some extent. If the average orientationof the field vectors is the same (row 3 in Table 1), theelectromagnetic field at OP will be polarized (polar-ization parameter equal to 1 across the pattern) andproduce a Young fringe pattern with high visibilitythere for very narrow fluctuations of the orientationangles (very small standard deviation), even if theangles are statistically independent. But the electro-magnetic field losses both spatial coherence and po-larization when the angular fluctuations increase. Afurther feature appears: the polarization parameterbecomes nonuniform across the pattern at OP, deter-mining the polarization domains.
By introducing the conditionΔϑþ ¼ Δϑ− instead ofthe statistical independence of the angles in theabove Gaussian behavior (row 4 in Table 1), the elec-tromagnetic field at OP becomes highly polarized(polarization parameter equal to 1 across the pat-tern) and spatially coherent for very narrow fluctua-tions of the orientation angles but differs from thecase of statistically independent angles in that thepolarization parameter is nonuniform across the pat-tern in this case; i.e., there should be polarizationdomains, which is not the case for statistically inde-pendent fluctuations. The field will remain spatiallycoherent but loses polarization if the angular fluctua-tions increase.
If the average orientations of the field vectorsat the openings are mutually orthogonal and thefluctuations of the orientation angles follow Gaus-sian statistics (rows 5 and 6 in Table 1), the patternat OP will have very low visibility for both fluc-tuation types, i.e., under statistical independenceand under the condition Δϑþ ¼ Δϑ−, which confirmsthe conformity to the second Fresnel–Arago law [4].However, by very narrow fluctuations the field will bepolarized and exhibit polarization domains at OP;the corresponding Cartesian components of thefield vectors will be correlated, but they cannot pro-duce a fringe pattern because the visibility takesthe form V ¼ j cosð�ϑþ − �ϑ−Þj [3]. This correlationdiminishes when the fluctuations increase, butthe field remains polarized if Δϑþ ¼ Δϑ− stands,while it loses polarization if the orientation anglesfluctuate independently.
6. Conclusions
Electromagnetic spatial coherence wavelets allowthe phase-space representation of stationary randomelectromagnetic fields to be developed. In this con-text, the propagation of the power and the statesof spatial coherence and polarization of the fieldare described as resulting from the properties ofthe correlations between the components of thefield vectors at pairs of points in space. The theorypredicts that the polarization at the observationplane can exhibit a structure named “polarization do-mains,” i.e., delimited regions where the field has aspecific polarization state. In addition, the phase-space representation provides a generalization ofthe Poynting theorem. Theoretical predictions wereexamined by numerical simulations of the Youngexperiment with electromagnetic waves.
Table 1. (Continued)
Polarization Case ηEEðþ;−; vÞCalculatedVisibility Polarization Parameter
Roman Castaneda is obliged to Fernando Medina(Universidad de Antioquia) for his helpful commentsrelating the discussion presented in this paper.This research was supported by the Dirección deInvestigación (DIME) of the Universidad Nacionalde Colombia Sede Medellín.
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