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1894 J. Opt. Soc. Am. A/Vol. 25, No. 8 /August 2008 Castaneda et al.

Holographic features of spatial coherence wavelets

Roman Castaneda,1,* Rafael Betancur,1,2 and Diego Hincapie1,3

1Physics School, Universidad Nacional de Colombia Sede Medellín, A.A. 3840, Medellín, Colombia2E-mail: rabetanc@gmail.com

3E-mail: dahinca@unalmed.edu.co*Corresponding author: rcastane@unalmed.edu.co

Received April 4, 2008; accepted May 10, 2008;posted May 20, 2008 (Doc. ID 94563); published July 7, 2008

The behavior of the marginal power spectrum as a two-channel-multiplexed hologram is analyzed. Its “nega-tive energies” make it quite different from the conventional holograms, i.e., it is not recordable in general andthe objects to be reconstructed (the cross-spectral densities at both the aperture and the observation planes)are virtual. The holographic reconstruction results from the superposition of the spatial coherence waveletsthat carry the marginal power spectrum. These features make the marginal power spectrum a powerful tool foranalysis and synthesis of optical fields, for instance, in optical information processing (signal encryption) andbeam shaping for microlithography. © 2008 Optical Society of America

OCIS codes: 030.1640, 090.1995.

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. INTRODUCTIONhe holographic principle was originally realized by Ga-or [1] in terms of a two-step procedure, i.e., (1) the re-ording of the interference pattern produced by the super-osition of an object wave and a reference wave onto ahysical support named the hologram and (2) the recon-truction of the object wave, in amplitude and phase, byiffracting the reference wave through the hologram afterlacing it at the same position where it was recorded.win images of three-dimensional objects, in addition tohe called zeroth-order of diffraction, are obtained by thisechnique. They appear superposed to each other by Ga-or’s procedure, called on-axis holography, but can beeparated by an alternative procedure first introduced byeith and Upatnieks [2], i.e., the off-axis holography.Digital procedures were later involved in different

tages of the holographic arrangements. So, computerenerated holograms (CGH) [3] replaced the optically re-orded holograms, providing optical images of digitallyonceived objects. In the properly named digital hologra-hy (DH), CCD or complementary metal oxide semicon-uctor sensors (CMOS) record the optical hologram,hich is digitized and reconstructed by using frame grab-ers and computer algorithms [4,5]. Digital processingignificantly improved the versatility of holography byliminating both the zeroth-order of diffraction and one ofhe twin images [4–7], by minimizing the speckle noise8], and by improving the image quality [4–9], amongther methods.

There are many applications of optical holography,GH, and DH. Remarkable among them are the analysisf mechanical behavior of materials and components5,10], the synthesis of diffractive optical elements [11],nd the development of different modalities of digital ho-ography microscopy (DHM) [12]. On the other hand, theelationship between holography and the optical informa-ion processing was analyzed in the framework of thehase-space representation of optical signals [13,14]. In

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his context, the holographic principle and its realizationn specific schemes were fairly described by using thehase-space diagram [14]. With the basis of such a de-cription, the holographic properties of the spatial coher-nce wavelets can be determined, as presented in this pa-er.The spatial coherence wavelets provide the phase-space

epresentation for optical fields in any state of spatial co-erence and polarization [15–17] and for the radiometryf both scalar and electromagnetic optical sources [18,19].oncerning the holographic features of the spatial coher-nce wavelets the scheme of the Fourier DH was appliedor recovering the complex degree of spatial coherence ofchell model beams (laser beams for instance) from theecording of its spot [20,21], and for recovering the map oflasses of radiator pairs [22] from the recording of the in-ensity distribution of the spatial coherence moiré of theeld [22], i.e., the beam spot and the intensity distribu-ion of the spatial coherence moiré should be the Fourierolograms of the complex degree of spatial coherence andhe map of classes of radiator pairs of the field, respec-ively. Such results make the analysis of the holographicroperties of the spatial coherence wavelets an unavoid-ble subject.

. SPATIAL COHERENCE WAVELETSet us consider the far field (Fresnel–Fraunhofer domain

23]) arrangement in Fig. 1, and the center and differenceoordinates [15,16] ��A ,�D� at the aperture plane (AP) andrA ,rD� at the observation plane (OP), for specifying theosition of pairs of points on those planes, i.e., �A±�D /2nd rA±rD /2, respectively. So, the (scalar) spatial coher-nce wavelets at the frequency � that propagate from theP to the OP, placed at a distance z to each other, are de-ned as [15,16]

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Castaneda et al. Vol. 25, No. 8 /August 2008 /J. Opt. Soc. Am. A 1895

W�rA + rD/2,rA − rD/2,�A;�� = S�rA,�A;��exp�− ik

zrD · �A� ,

�1�

here k=2� /� with � being the wavelength, and

S�rA,�A;�� = W�rA,rA,�A;��

=�AP

���A + �D/2,�A − �D/2;��

��S��A + �D/2;��t��A + �D/2�

��S��A − �D/2,��t*��A − �D/2�

�exp�ik

z�A · �D�exp�− i

k

z�D · rA�d2�D,

�2�

s called the marginal power spectrum, with ���A�D /2 ,�A− �D /2 ;�� = ����A+�D /2 ,�A − �D /2 ;��� exp �i� ��A

�D /2 ,�A−�D /2 ;�� being the spectral degree of coherence d

bfi

3ST

rcp[sti

r complex degree of spatial (spectral) coherence at the fre-uency � [24]. S�� ,���0 is the spectral density or powerpectrum of the field across the aperture, t����t����exp�i���� is the aperture transmission, and thesterisk denotes complex conjugate. It is worth notinghat ���A+�D /2 ,�A−�D /2 ;��=�*��A−�D /2 ,�A+�D /2 ;��,��A ,�A ;��=1, and ���A ,�A ;��=0 hold. Both thepatial coherence wavelets and their marginal powerpectra exhibit power units and the marginal power spec-rum is real definite, i.e., S�rA ,�A ;��=S*�rA ,�A ;��.onsequently, the spatial coherence wavelet should beermitian, i.e., W�rA+rD /2 ,rA−rD /2 ,�A ;��=W*�rArD /2 ,rA+rD /2 ,�A ;��.Taking into account that

��A + �D/2,�A − �D/2;�� = ���A + �D/2,�A − �D/2;��

��S��A + �D/2;��

�t��A + �D/2��S��A − �D/2;��

�t*��A − �D/2�,

enotes the cross-spectral density [24] at the AP, so that

W�rA + rD/2,rA − rD/2;�� = � 1

�z�2

exp�ik

zrA · rD��

AP�

AP

W��A + �D/2,�A − �D/2;��exp�ik

z�A · �D�

�exp− ik

z�rA · �D + rD · �A��d2�Ad2�D

ives the cross-spectral density at the OP [24], Eqs. (1) and (2) lead to

W�rA + rD/2,rA − rD/2;�� = � 1

�z�2

exp�ik

zrA · rD��

AP

W�rA + rD/2,rA − rD/2,�A;��d2�A

= � 1

�z�2

exp�ik

zrA · rD��

AP

S�rA,�A;��exp�− ik

zrD · �A�d2�A. �3�

urthermore, W��A ,�A ;��=S��A ;���t��A��2�0 and�rA ,rA ;��=S�rA ;���0 denote the power spectra emerg-

ng from AP and recorded by a detector at the OP, respec-ively, so that

S�rA;�� = � 1

�z�2�

AP

S�rA,�A;��d2�A. �4�

onsequently, the conservation law of the total energy ofhe optical field can be expressed in terms of the propaga-ion invariant �AP�OPS�rA ,�A ;��d2�Ad2rA as [15,16]

�OP

S�rA;��d2rA =�AP

S��A;���t��A��2d2�A

= � 1

�z�2�

OP�

AP

S�rA,�A;��d2�Ad2rA.

�5�

t means that only individual centers of secondary distur-

ance at AP contribute to the total energy of the opticaleld.

. PHASE-SPACE REPRESENTATION OFCALAR OPTICAL FIELDShe function

��A ± �D/2;�� = �S��A ± �D/2;��t��A ± �D/2�

�exp�ik

2z��A ± �D/2�2�

epresents the field amplitude that emerges from eachenter of secondary disturbance at �A±�D /2 on AP andropagates to the OP in the Fresnel–Fraunhofer domain15,16]. After inserting it in Eq. (2) the marginal powerpectrum takes the form of the Wigner distribution func-ion (WDF) for spatially partially coherent fields [25–27],.e.,

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w(t�mc�sstggcaplow. The conservation law of the total energy of the optical

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1896 J. Opt. Soc. Am. A/Vol. 25, No. 8 /August 2008 Castaneda et al.

S�rA,�A;�� =�AP

���A + �D/2,�A − �D/2;����A + �D/2;��

�*��A − �D/2;��exp�− ik

z�D · rA�d2�D, �6�

hich provides a phase-space representation of the opti-al field in any state of spatial coherence, with rA /�z ashe phase coordinate and �A as the spatial coordinate25–27], i.e., its graph constitutes the phase-space dia-ram of the optical field. It is useful to introduce the di-ensionless function 1=C���D�+ �1−C���D�, with C be-

ng a constant with area units, into the integral of Eq. (2)nd to evaluate this integral by applying the swept pro-edure described in [15,16] to express this equation as

S�rA,�A;�� = CS��A;���t��A��2 + 2� AP�D�0

����A + �D/2,�A − �D/2;����S��A + �D/2;���t��A + �D/2��

��S��A − �D/2,���t��A − �D/2��cosk

z�rA − �A� · �D − ���A + �D/2,�A − �D/2;�� − ��d2�D, �7�

ith �=��A+�D /2�−��A−�D /2�. The first term of Eq.7) is proportional to the power spectrum contributed byhe individual center of secondary disturbance placed atA, while the second term determines the cosinelikeodulations due to the correlation between the pairs of

enters of secondary disturbance with separation vectorsD around the position �A. Such modulations are re-tricted by the size of the support of the complex degree ofpatial coherence centred at �A, but it can take on nega-ive values along specific axes �A→rA, with magnitudesreater than the value of the first term, so that the mar-inal power spectrum becomes negative, i.e., the spatialoherence wavelet carries negative energies along suchxes [17–19]. Such negative energies are crucial for ex-laining interference in this framework, as discussed be-

eld [Eq. (5)] imposes C= ��z�2 /�OPd2rA and

�OP

cosk

z�rA − �A� · �D

− ���A + �D/2,�A − �D/2;�� − ��d2rA = 0, �8�

result that confirms the statement above that only indi-idual centers of secondary disturbance at the AP contrib-te to the total energy of the optical field, i.e., the modu-

ation term in Eq. (7) does not contribute but onlyedistributes the energy: it does not constitute a radiantux of the field, as occurs with the first term of Eq. (7).

ndeed, Eqs. (4) and (7) yield

S�rA;�� = C� 1

�z�2�

AP

S��A;���t��A��2d2�A + 2� 1

�z�2�

AP� AP

�D�0

����A + �D/2,�A − �D/2;����S��A + �D/2;��

��t��A + �D/2���S��A − �D/2,���t��A − �D/2��cosk

z�rA − �A� · �D − ���A + �D/2,�A − �D/2;�� − ��d2�Dd2�A. �9�

quation (9) points out that the power spectrum at theP results from the superposition of Young-like interfer-nce patterns provided by the complete set of pairs of cen-ers of secondary disturbance that fill the aperture on the

P. Thus, the cosinelike modulations of S�rA ,�A ;�� pro-uce the interference patterns of S�rA ;��, redistributinghe total energy of the field in such a way that 0S�r ;�� 2C�1/�z�2� S�� ;���t�� ��2d2� . The values

ig. 1. Illustrating the center and difference coordinates for de-oting pairs of points at both the AP and the OP planes.

A AP A A A

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Castaneda et al. Vol. 25, No. 8 /August 2008 /J. Opt. Soc. Am. A 1897

f S�rA ;�� determined by this condition result from theddition of the positive and the negative energies pro-ided by S�rA ,�A ;�� to the first term of Eq. (9), accordingo condition (8). It is the meaning of interference in thehase-space representation provided by the spatial coher-nce wavelets. It also means that the modulation term of�rA ,�A ;�� should be related to the tamasic rays (fromanskrit: tamas=darkness) first introduced by Sudars-an [28] for explaining the destructive interference. In-eed, the value of S�rA ,�A ;�� at each point �rA ,�A� of thehase-space diagram can be thought of as the energyropagated along the path (or the ray) �A→rA. In thisense, the phase-space diagram constitutes the map ofays for the optical field on propagation from the AP to theP. The rays corresponding to the first term of Eq. (7) willropagate the radiant energy of the field and are calledarrier rays, while those corresponding to the second termill only propagate the modulating energy; therefore theyre dark or tamasic and are called modulating rays. Ac-ording to Eq. (7) both carrier and modulating rays can lielong the same path �A→rA. Furthermore, Eq. (9) implieshat specific power distributions can be reached at the OPy properly manipulating the modulating rays, whichhould be very useful in beam shaping [29], for instance.

It is worth noting that the redistribution mechanismxplicitly depends on the spatial coherence properties

ig. 2. Phase-space diagrams (graphs of the marginal power splluminated by a plane wave with a Gaussian degree of spatial card deviation of the degree of coherence is significantly greateriation is 0.5a (partially coherent illumination); and c, the standaiagrams for Fresnel diffraction (the slit subtends two Fresnel zollumination; e, partially coherent illumination; and f, nearly inc

f the optical field. Indeed, fully spatially incoherentelds do not have it. In this case, ���A+�D /2 ,A−�D /2 ;��=0 for �D�0, so that Eqs. (7) and (9) reduceo S�rA ,�A ;��=CS��A ;���t��A��2�0 and S�rA ;��C�1/�z�2�APS��A ;���t��A��2d2�A, respectively.On the other hand, from Eqs. (7) and (8) follows:

S��A;���t��A��2 = � 1

�z�2�

OP

S�rA,�A;��d2rA. �10�

o illustrate the analysis above, Fig. 2 shows the phase-pace diagrams for the Fresnel–Fraunhofer diffraction ofpatially partially coherent light through a slit of widtha, which is uniformly illuminated by a plane wave with aaussian degree of spatial coherence. It is a one-imensional case where the spatial coordinate �A deter-ines positions within the slit, with �A=0 being theiddle point of the slit, and the phase coordinate �AxA /�z, with xA being the spatial coordinate at the OP, de-

ermines the spatial frequencies of the diffracted fieldhose bandwidth is ��A.Fresnel diffraction diagrams are sheared with respect

o the Fraunhofer diffraction diagrams, i.e., the values ofhe marginal power spectrum for each �A are linearlyhifted parallel to the �A axis, according to the Fresnel-hase [15,16] difference �k /z��A�D in the argument of the

for diffraction through a slit of width 2a. The slit is uniformlyce. Upper row: diagrams for Fraunhofer diffraction. a, the stan-the aperture width (coherent illumination); b, the standard de-iation is about 0.2a (nearly incoherent illumination). Lower row:nder the same coherence conditions as in upper row. d, coherentt illumination.

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1898 J. Opt. Soc. Am. A/Vol. 25, No. 8 /August 2008 Castaneda et al.

osine function (Figs. 2(d)–2(f)). A bigger Fresnel number23] yields a greater shear. S��A ,�A ;���0 holds for theright fringes and S��A ,�A ;���0 holds for the dark ones.he fringe contrast decreases when the optical field be-omes more incoherent (Fig. 2). However, its decrement isot uniform: it is stronger in a region around the middleoint of the slit than in the surrounding region of the slitdge. It is due to the distortion of the support of the de-ree of spatial coherence caused by the slit edge [21].

. HOLOGRAPHY AND THE PHASE-SPACEIAGRAM

n a recent inspiring work Testorf and Lohmann [14] dis-ussed the phase-space representation of the holographicrinciple and analyzed the phase-space diagram of differ-nt holographic schemes. Their assumptions for single-ideband holography (extending them to optical fields inny state of spatial coherence) are fulfilled by the mar-inal power spectrum and so their phase-space diagramsre similar. Indeed, the diagrams in the upper rows ofigs. 2 and 3(a) show a central bar and upper and loweridebands symmetric with respect to the bar. The signifi-ant values of the bar are much larger than the values ofhe sidebands and, according to condition (8), the meanalue of the sidebands along the phase coordinate shoulde equal to null. Furthermore, the bar profile for rA /� z0 corresponds to the autocorrelation of the spatially par-

ially coherent field at the AP, i.e.,

S��A;�� =�AP

���A + �D/2,�A − �D/2;���S��A + �D/2;��

�t��A + �D/2��S��A − �D/2,��t*��A − �D/2�d2�D,

�11�hich becomes

S��A;�� =�AP

�S��A + �D/2;��t��A + �D/2�

��S��A − �D/2,��t*��A − �D/2�d2�D,

S�� ;�� = CS�� ;���t�� ��2,

A A A

or fully spatially coherent and incoherent optical fields,espectively. Thus, it is reasonable to assume that the barrofile for rA /� z=0 is the autocorrelation of the referenceave and the sidebands contain the autocorrelation of thebject wave and the twin images [14]. The very sense ofhis analogy is reached after the reconstruction of the ob-ect wave by using the marginal power spectrum as a Fou-ier hologram. As described by Eq. (3) the reconstructedbject wavefront is a complex function of second order,.e., the cross-spectral density at the OP. Consequently,he power spectrum given by Eq. (4) should be the zerothrder of the reconstruction and the twin images resultrom the hermiticity of the cross-spectral density, i.e.,

�rA+rD /2 ,rA−rD /2 ;��=W*�rA−rD /2 ,rA+rD /2 ;��.herefore, such holographic nonoverlapping imageshould be W�rA+rD /2 ,rA−rD /2 ;�� onto two consecutiveuadrants on the OP and W�rA−rD /2 ,rA+rD /2 ;�� on theemaining two quadrants. It is worth noting that theross-spectral density at the OP should be real-valued forhe phase-space diagrams in the upper rows of Figs. 2 and(a) (Fraunhofer diffraction) because S�rA ,�A ;��S�rA ,−�A ;�� holds. It is not the case for the diagrams in

he lower rows of Figs. 2 and 3(b) (Fresnel diffraction), byhich the marginal power spectra are no longer symmet-

ic on account of the quadratic Fresnel phase. The shearf the diagram implies a rotation of the white bar with re-pect to the spatial coordinate axis that depends on theerm �k /z��A ·�D in the argument of the cosine function ofq. (7). It results from the difference between the Fresnelhases �k /2z���A±�D /2�2 associated with the pairs of cen-ers of secondary disturbance at the AP and causes thehear of the whole diagram. Accordingly, the autocorrela-ion of the reference wave is not of null spatial frequency�A=0 for all the positions �A) as by Fraunhofer diffrac-ion, but its spatial frequency linearly increase with theosition �A. However, both the profile of the middle linend the area of the white bar should be the same as byraunhofer diffraction.Although the diagram symmetry with respect to the

xes �A=0 and �A=0 is lost on account of the shear, theidebands remain symmetric with respect to the diagramrigin, in such a way that condition (8) is fulfilled. Conse-uently, the statement that they contain information on a

ig. 3. Morphological features of the phase-space diagrams for a spatially partially coherent optical field by a, the Fraunhofer diffrac-ion and b, the Fresnel diffraction.

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Castaneda et al. Vol. 25, No. 8 /August 2008 /J. Opt. Soc. Am. A 1899

irtual object that causes the redistribution of the energyn the power spectrum at the OP remains valid [Fig.(b)]. Such information is mainly encoded in the fringetructure of the sidebands, whose contrast depends on thepatial coherence state of the optical field at the AP, in theame way as for Fraunhofer diffraction (Fig. 2, lowerow). However, for great enough apertures in comparisono the size of the complex degree of spatial coherence, thehapes of the power spectra at the OP by both Fresnel andraunhofer diffraction tends to be coincident [21], as il-

ustrated in Fig. 4(c) for diffraction through a slit.The profiles of the phase-space diagrams for Fraun-

ofer diffraction in Fig. 2, along the spatial (aperture) co-rdinate, represent the hologram transmissions for theeconstruction of the cross-spectral density at the OP forpecific positions xA=�z�A. For Fig. 2(c) the reconstruc-ion realizes the Van Cittert–Zernike theorem [23,24] be-ause it corresponds to a spatial incoherent source at theP. Accordingly, the spatial coherence wavelet defined inq. (1) is the propagator of the hologram given by the

ig. 4. Power spectrum at OP for the phase-space diagrams: a,n the upper row and b, in the lower row of Fig. 2. The verticalxis is in arbitrary units. The energy redistribution is apparent,aking into account that the diffracted wave is uniform. c, Com-arison between the shapes of the power spectra by both Fraun-ofer and Fresnel diffraction for great enough slit width in com-arison to the size of the support of the complex degree of spatialoherence.

arginal power spectrum along each ray �A→rA, i.e., it isreconstruction element of the cross-spectral density at

he OP, as described by Eq. (3).The analysis above means that the individual center of

econdary disturbance at each position �A contributes tohe reference wave while the pairs of centers with sepa-ation vectors �D, around �A, contribute to the objectave. It means that the reference wave is conformed by

he power emerging from the AP and its redistribution athe OP constitutes the zeroth-order of reconstruction,hich is optically recordable. Furthermore, the objectave should be virtual, in the sense that it cannot propa-ate optical energy and that the marginal power spec-rum should take on negative values. Therefore, it is notptically recordable but numerically determinable. Never-heless, it has significant effects on the second-orderavefront reconstructed by Eq. (3), i.e., it determines the

tate of spatial coherence and redistributes the powerpectrum of the optical field there, depending on the cor-elation properties of the field at the AP. Figure 4 illus-rates the power spectrum redistribution of the uniformlane wave diffracted by a slit. On the other hand, Eq. (2)ields

W��A + �D/2,�A − �D/2;�� = � 1

�z�2

exp�− ik

z�A · �D�

��OP

S�rA,�A;��

�exp�ik

z�D · rA�d2rA, �12�

ith

W��A + �D/2,�A − �D/2;�� = ���A + �D/2,�A − �D/2;��

��S��A + �D/2;��t��A + �D/2�

��S��A − �D/2;��t*��A − �D/2�,

eing the cross-spectral density at AP. So, in a similaray as above, Eq. (12) describes the holographic recon-

truction of the cross-spectral density at the AP, taking�rA ,�A ;�� as its (Fourier) hologram. Its zeroth-order is

he power spectrum at the AP, i.e.,

W��A,�A;�� = S��A;���t��A��2 = � 1

�z�2�

OP

S�rA,�A;��d2rA,

nd its nonoverlapping twin images will be determined byhe hermiticity of the cross-spectral density at the AP, i.e.,hey are W��A+�D /2 ,�A−�D /2 ;�� regarding two consecu-ive quadrants of the AP and W��A−�D /2 ,�A+�D /2 ;�� onhe remaining two quadrants. They have a phase differ-nce of � to each other.

The profiles of the phase-space diagrams for Fraun-ofer diffraction in Fig. 2 (upper row) along the phase co-rdinate represent the hologram transmissions for the re-onstruction of the cross-spectral density at the AP forpecific positions �A within the aperture. It should be real-alued because S�rA ,�A ;��=S�−rA ,�A ;�� holds. But, forhe diagrams in the lower row (Fresnel diffraction), thearginal power spectra are no longer symmetric on ac-

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1900 J. Opt. Soc. Am. A/Vol. 25, No. 8 /August 2008 Castaneda et al.

ount of the quadratic Fresnel phase and hence the cross-pectral density should be complex-valued.

The bandwidth ��A contains the spatial frequencyodes that determine the correlation properties (spatial

oherence state) of the optical field at the OP and itsower spectrum distribution. The contributing modes aressociated to the region of the AP in which the sidebandsxhibit contrasted fringes, which is determined by thepatial coherence state of the optical field at the AP. Thepper rows of Figs. 2 and 3(a) illustrate this feature. Onlyy fully spatially coherent illumination does the whole ap-rture participate in determining the spatial coherencetate of the field at the OP and in redistributing the powerpectrum [Fig. 4(a)]. By spatially partially coherent illu-ination the modes are associated to two regions, sym-etrically placed with respect to the phase-coordinate

xis and limited by the aperture edge [Fig. 3(a)]. The ex-ent of each region ��A is determined by the size of theomplex degree of spatial coherence, i.e., the more inco-erent the field the narrower the regions. Indeed, by in-oherent illuminations such regions constitute the vicini-ies of the aperture edge (Figs. 2(c) and 2(f)).

The region of extent 2�a−��A� is symmetrically distrib-ted with respect to the phase-coordinate axis [Fig. 3(a)]nd is free from fringes. Only the segment of the whitear in this region contributes to both the determination ofhe spatial coherence state of the field at the OP and theistribution of its power spectrum there. Consequently,hrough ideal spatially incoherent fields at the AP onlyhe autocorrelation of the reference wave contributes tohe spatial coherence state of the field at the OP, realizinghe Van Cittert–Zernike theorem. However, ideal spa-ially incoherent illumination is not physically realizable30]. For physical fields, the area of the white bar of thehase-space diagram remains the same for any spatial co-erence state (Fig. 2). Therefore, it should be regarded asfurther distinction of the optical field.

. CONCLUSIONSt was shown that the marginal power spectrum can beegarded as a multiplexed single-sideband hologram con-aining complete information in amplitude and phase ofhe cross-spectral densities at both the aperture and thebservation planes encoded on different channels. Indeed,hey can be reconstructed separately. Morphologically, thehase-space diagram provided by the marginal powerpectrum consists in a central bar representing the auto-orrelation of the reference wave and symmetric side-ands containing the information of the autocorrelation ofhe object wave and the twin images. However, it differsrom the conventional holograms that are intensity distri-utions recorded onto a physical support to be used asnumerical or optical) gratings for reconstructing a com-lex amplitude distribution. The marginal power spec-rum is not optically recordable in general, because it canontain negative energies. However, because of the ac-omplishments of the conservation law of the total energy,t is not an obstacle for the holographic reconstruction atach channel, but it means that the reconstructed objecthould be virtual in the sense that it cannot propagate op-ical energy.

Because of the physical features of the reconstructedbject waves, the marginal power spectrum should be re-arded as a second-order hologram, i.e., first-order (con-entional) holograms reconstruct complex amplitudes of aave field, but second-order holograms reconstruct the

ross-spectral density. The former are positive–definitetructures both optically and numerically realizable, buthe last can take on positive and negative values, so thathey can only be realized numerically.

Due to the holographic properties above, the marginalower spectrum becomes a powerful tool for the analysisnd synthesis of optical fields. In this context, some appli-ations have already been reported, for example, the re-onstruction of the complex degree of spatial coherencerom the spot of Schell model beams and the reconstruc-ion and filtering of classes of radiator pairs. The mostromising potentiality is related to the numerical designf marginal power spectra, an actual research subject.eferences and sideband structures can be determined foreconstructing optical fields with specific coherence prop-rties and power distributions. Applications of this tech-ique include new optical information processing (signalncryption, for instance) and beam shaping for microli-hography among others.

CKNOWLEDGMENTShe authors sincerely thank Jorge Herrera and Mariosuga for their inspiring comments.

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