Spatial coherence wavelets and phase-spacerepresentation of diffraction
Román Castañeda1,* and Juan Carrasquilla2
1Physics School, Universidad Nacional de Colombia Sede Medellín, A.A. 3840 Medellín, Colombia2CNR-INFM-Democritos National Simulation Center and International School for Advanced Studies (SISSA),
The phase-space representation based on Wignerdistribution functions [1] has been used for describ-ing the relationship between the spatial coherenceproperties and the physical features of optical fields[2–5] and radiant sources [6–9]. More recently, scalarand electromagnetic spatial coherence wavelets wereintroduced as primary carriers of correlation proper-ties, polarization states, and power of the field [10–12]. They are Wigner distribution functions, whosephase-space representation gives more insight onthe predictions of the second-order theory of spatialcoherence [13] on optical diffraction. Their structureis determined by the classes of source pairs [14] atthe aperture, and their superposition produces thespatial coherence moiré [15].Classically, diffraction is described as the influence
of the aperture edge onto the behavior of the opticalfield [16]. However, in the framework of the phase-space representation provided by the spatial coher-ence wavelets, diffraction appears as an effect of
the support of the complex degree of spatial coherence[13], which behaves as the effective diffracting aper-ture. The edges of physical apertures enhance the dif-fraction in that they distort the supports in theirvicinity. A further classical prediction is the uncer-tainty relationship between the aperture size andthe extent of the power distributionat the observationplane [16]. Because the support of the complex degreeof spatial coherence is the effective diffracting aper-ture, the uncertainty relationship takes place be-tween the support size and the power distributionat the observation plane. It allows defining the spotof the optical field, which is the Fourier hologram ofthe complex degree of spatial coherence. Its experi-mental recording evidences the physical existenceof the spatial coherence wavelets and has been usedfor determining the complex degree of spatial coher-ence of laser beams [17].
These new concepts provided by the phase-spacerepresentation of diffraction are theoretically estab-lished and both numerically and experimentallyexamined in the following. Section 2 shows the funda-mentals of spatial coherence wavelets, and Section 3concern the phase-space representation of diffraction.
Let us consider the far field (Fresnel–Fraunhofer do-main [16]) arrangement in Fig. 1, and the center anddifference coordinates ðξA; ξDÞ at aperture plane (AP)and ðrA; rDÞ at the observation plane (OP), for specify-ing the position of pairs of points on those planes,i.e., ξA � ξD=2 and rA � rD=2, respectively. The spatialcoherence wavelets at the frequency are defined as[10,11]
WðrA þ rD=2; rA − rD=2; ξA;ωÞ
¼ SðrA; ξA;ωÞ exp�−i
kzrD⋅ξA
�; ð1Þ
where k ¼ 2π=λ, λ is the wavelength, z is the distancebetween AP and OP, and
is the wavefront of the wavelet, called the marginalpower spectrum. The integrand factors in Eq. (2)are the spectral degree of coherence or complexdegreeof spatial (spectral) coherence at the frequency ω [13]μðξA þ ξD=2; ξA − ξD=2;ωÞ ¼ jμðξA þ ξD=2; ξA − ξD=2;ωÞj× exp½iαðξA þ ξD=2; ξA − ξD=2;ωÞ�, the spectral den-sity or power spectrum [13] of the illuminating fieldacross the aperture Sðξ;ωÞ, and the aperture trans-mission tðξÞ ¼ jtðξÞj expðiϕðξÞÞ. The asterisk denotescomplex conjugate. It is worth noticing that μðξAþξD=2; ξA − ξD=2; ωÞ ¼ μ�ðξA − ξD=2; ξA þ ξD=2; ωÞ,μðξA; ξAωÞ ¼ 1, and αðξA; ξA;ωÞ ¼ 0 hold. Because ofthat, WðrA þ rD=2; rA − rD=2; ξA;ωÞ ¼ W�ðrA − rD=2;
rA þ rD=2; ξA;ωÞ and SðrA; ξA;ωÞ ¼ S � ðrA; ξA;ωÞstand too. Furthermore, the spatial coherence wave-lets exhibits units of power density (average energy).
Thecomplexdegreeof spatialcoherence isameasureof the correlation between the spectral amplitudes offrequency ω at two points in the space, with indepen-dence from the used coordinate system. However, itsmeaning slightly differs from the conventional whenusing center and difference coordinates. In the conven-tional description, pairs of centers of secondary distur-bance at arbitrary coordinates ðξ1; ξ2Þ are considered,insuchawaythatthecorrelationpatchisdefinedastheregion inwhichanyarbitrarypair of centerswill be sig-nificantly correlated. In addition, contributions of allindividual centers within the patch to the power spec-trum should be considered. In contrast, the center anddifference coordinates specify surroundingswithin theaperture, centered at each point ξA, whose pair of cen-ters are determined by separation vectors ξD with themiddle point at ξA. The size of a given surroundingwillbe denoted as jξDjMAX, so that the correlation betweenany pair of centers at distances jξDj > jξDjMAX can beneglected. Such a surrounding is called the supportof the complex degree of spatial coherence at the givenξA. It has the following properties:
• Each support contains only one pair of centersof secondary disturbance with the specific separationvector ξD.
• Each specific support provides a spatial coher-ence wavelet onto specific surroundings at OP.
• The cross-spectral density over a specific sur-rounding at OP results from the noninterferingsuperposition of the spatial coherence wavelets pro-vided by all the supports that fill the aperture.
• The power spectrum at the center of a specificsurrounding at OP will result from the superpositionof the marginal power spectra provided by all thesupports that fill the aperture.
By denoting the cross-spectral density and thepower spectrum of the optical field [13] emergingfrom AP by WðξA þ ξD=2; ξA − ξD=2;ωÞ and SðξA;ωÞjtðξAÞj2 ¼ WðξA; ξA;ωÞ, respectively, and the samequantities at OP by WðrA þ rD=2; rA − rD=2;ωÞ andSðrA;ωÞ ¼ WðrA; rA;ωÞ, it was proved that [10,11]
WðrA þ rD=2; rA − rD=2;ωÞ
¼�1λz
�2× exp
�ikzrA⋅rD
�ZAP
WðrA þ rD=2; rA
− rD=2; ξA;ωÞd2ξA; ð3Þ
SðrA;ωÞ ¼�1λz
�2ZAP
SðrA; ξA;ωÞd2ξA ð4Þ
hold, i.e., thespatial coherencewaveletsaretheprimaryvehicles for the transfer of both the correlation proper-ties and the power spectrum of the optical field, fromAP to the OP in free space. From Eq. (4) follows
Fig. 1. Illustration of the center and difference coordinates fordenoting pairs of points at both the AP plane and the OP plane.
for the total energy per unit time at OP. Taking into ac-count that
RAP SðξA;ωÞjtðξAÞj2d2ξA is the total energy
per unit time emerging from AP, the conservation lawof the total energy per unit time that flows from AP toOP will be
ZOP
SðrA;ωÞd2rA ¼ZAP
SðξA;ωÞjtðξAÞj2d2ξA
¼�1λz
�2ZOP
ZAP
SðrA;ξA;ωÞd2ξAd2rA:
ð5Þ
From Eq. (5) it follows that
�1λz
�2ZOP
SðrA; ξA;ωÞd2rA ¼ SðξA;ωÞjtðξAÞj2 ð6Þ
holds. It is worth noticing that the complex degree ofspatial coherence for fully spatially incoherent fieldsat AP is deltalike. Consequently, the spatial coherencewavelet provided by the deltalike support, centered ata givenpoint ξA onAP, takes the formWðrA þ rD=2; rA−rD=2; ξA; ωÞ ¼ CSðξA; ωÞjtðξAÞj2 × expð−iðk=zÞrD ⋅ ξAÞ,withC a constant to be determined by condition (5), i.e.,C ¼ ðλzÞ2= ROP d2rA, with
ROP d
2rA the area illuminatedby the optical field onto OP. Therefore, Eq. (3) gives
WðrA þ rD=2; rA − rD=2;ωÞ ¼ C
�1λz
�2
× exp�ikzrA⋅rD
�ZAP
SðξA;ωÞjtðξAÞj2
× exp�−i
kzrD⋅ξA
�d2ξA; ð7Þ
which represents the Van Cittert–Zernike theorem[13,16] in the framework of the theory of the spatialcoherence wavelets.
3. Phase-Space Representation of Diffraction
A. Wigner Optics
The complex field amplitude that emerges from eachcenter of secondary disturbance at ξA � ξD=2 on APand propagates to OP in the Fresnel–Fraunhoferdomain (i.e., in the paraxial approach) is proportionalto ΨðξA � ξD=2;ωÞ ¼
× expðiðk=2zÞjξA � ξD=2j2Þ, where the Fresnel(squared) phase factor [10,11] results from the distri-bution of Fresnel zones inscribed in the aperture anddetermined from the intersection of OP with the op-tical axis (i.e., rA ¼ 0 in Fig. 1). Replacing it inEq. (2) yields
SðrA; ξA;ωÞ ¼ZAP
μðξA þ ξD=2; ξA − ξD=2;ωÞΨðξA
þ ξD=2;ωÞΨ�ðξA − ξD=2;ωÞ exp�−i
kzξD⋅rA
�d2ξD:
ð8Þ
So, SðrA; ξA;ωÞ has the mathematical form of theWigner distribution function (WDF) for spatiallypartially coherent optical fields [1–7],which is the cor-nerstone of theWigner optics, andprovides thephase-space representation [3–5] for the diffraction of scalaroptical fields in any state of spatial coherence, withkrA=z as the phase coordinate and ξA as the space co-ordinate. Indeed, its realness and the achievement ofEqs. (4) and (6) are canonical properties of the opticalWDF. It can take on negative values as anyWDF, too.Furthermore, the phase carrier of the spatial coher-ence wavelet in Eq. (1) accounts for the paraxial dis-tribution over the surrounding centered at rA.Therefore, this phase-space representation allowsmapping the optical field at AP onto OP, for any stateof spatial coherence.
Figure 2 shows the phase-space representation forFraunhofer diffraction by a slit of width 2a, uni-formly illuminated with light in different states ofspatial coherence. Axes x0A and xA correspond tothe Cartesian components of ξA and rA, respectively,parallel to the slit width. Schell-model field [13] illu-mination is assumed. The graphs representSðxA; x0A;ωÞ only for x0A ≥ 0 because it is symmetricalwith respect to the origin of the phase space. Bright-est fringes correspond to positive contributions,while darkest fringes correspond to the negative con-tributions to the total power spectrum. Three situa-tions can be appreciated. The first, under fullyspatially coherent illumination (Fig. 2(a)), exhibitsfringes along lines of x0A constant over the whole mar-ginal power spectrum. This gives rise to positive andnegative contributions carried by every wavelet con-necting the planes AP and OP, whose superpositionat OP generates the well known diffraction patternsof fully spatially coherent light. In the second, underpartially coherent illumination (Fig. 2(b)), two re-gions can be identified, i.e., the region for 0 ≤ x0A ≤
b from which only positive values are contributedto the power spectrum, by supports within the cen-tral region of the AP. The second region of this graph,b < x0A ≤ a, exhibits positive and negative values(fringes). By fully spatially incoherent illumination(Fig. 2(c)), b → a stands, and the size of the supportof the complex degree of spatial coherence is signifi-cantly smaller than the aperture size. Only waveletsconnecting points close to the aperture edge providenegative contributions to any point at OP.
B. Effective Diffraction Aperture
It is useful to introduce the dimensionless function1 ¼ CδðξDÞ þ ½1 − CδðξDÞ� in the integrand of
with α12 ¼ αðξA þ ξD=2; ξA − ξD=2;ωÞ and ϕ12 ¼ ϕðξAþξD=2Þ − ϕðξA − ξD=2Þ. The cosine function in the in-tegrand of the second term results by separately add-ing the exponentials in Eq. (2), corresponding to thetwo freedom degrees in orientation of each separa-tion vector ξD ≠ 0. Thus, the marginal power spec-trum is expressed in terms of the powercontribution provided by the center of secondary dis-turbance placed at the position ξA (the first term),and the cosinelike modulations provided by the pairsof centers of secondary disturbance with separationvectors ξD within the surrounding centered at ξA.The second term in Eq. (9) can take on negative va-lues withmagnitude bigger than the value of the firstterm [8], i.e., the marginal power spectrum providedby a specific support within the aperture can be ne-gative onto a specific point at OP. Replacing Eq. (8) inthe conservation law of the total energy (5) yields
ZOP
cos�kzðξA − rAÞ · ξD þ α12 þ ϕ12
�d2rA ¼ 0;
ð10Þ
which means that the second term of Eq. (9) does notcontributetothetotalenergyoftheoptical fieldbutonlyredistributes its power spectrum over OP. Neverthe-less, Eq. (4) implies that the superposition of the mar-ginalpower spectraprovidedbyall the supportswithinthe aperture onto any point rA at OP must be positivedefinite. Furthermore, because of conditions (5) and(10), if the second term of Eq. (9) is negative for a givenpoint rA, there should be a different point r0A atOP ontowhich it becomes positive with the same magnitude.This redistribution mechanism is determined by
the shape of the complex degree of spatial coherenceon the support centered at each point ξA within theaperture. Furthermore, the effective integration re-gion of the second term is the smallest betweenthe support of the complex degree of spatial coher-ence and the aperture. So, this integral nullifiesfor fully incoherent illumination, but it runs overthe whole aperture for fully coherent illumination.For partially coherent illumination, the integrationarea is mainly determined by the support of the com-plex degree of spatial coherence centered at ξA.After replacing Eq. (9) in Eq. (4), the power
spectrum at OP will be expressed as SðrA;ωÞ ¼
SindðrA;ωÞ þ SpairsðrA;ωÞ, whose first term is pro-vided by all the individual centers of secondary dis-turbance within the aperture, and the second by the
Fig. 2. (Color online) Marginal power spectrum inFraunhofer do-mainproducedbya slit, of semiwidtha, underuniform illuminationwith a Schell model field (a) fully spatially coherent (b ¼ 0). (b) spa-tially partially coherent, and (c) spatially incoherent (b → a).
pairs of centers within the surroundings centered ateach point ξA. The power spectrum reduces to thefirst term only if the diffracted field is fully spatiallyincoherent at AP, and its distribution corresponds tothe prediction of the geometrical optics. Diffractioneffects predicted by the physical optics will be asso-ciated to the second term of the power spectrum, i.e.,SpairsðrA;ωÞ. For this reason and the analysis above,the support of the complex degree of spatial coher-ence was called the effective diffraction aperture[10,11] Furthermore, conditions (5) and (10) leadto the conclusion that the total energy per unit timeis given by
ROP SindðrA;ωÞd2rA, i.e., only the individual
centers of secondary disturbance contribute to thepower spectrum at OP, while SpairsðrA;ωÞprovides the modulations observed in the diffractionpatterns by taking on positive and negative values,under the condition
ROP SpairsðrA;ωÞd2rA ¼ 0, so that
0 ≤ SðrA;ωÞ ≤ 2SindðrA;ωÞ stands.C. Spatial Coherence Moiré and Map of Classes ofSource Pairs
It is known that the superposition of fringe patternsgives amoiré [18]. Accordingly, the cross-spectral den-sity atOP can be thought as a spatial coherencemoirébecause it results from the noninterfering superposi-tion of spatial coherence wavelets, each one providinga “fringe pattern” given by Eq. (8) [10,11], as obtainedfrom Eqs. (1), (3), and (9). The term
of such superposition in the Fraunhofer domain (i.e.,negligible argument kξA⋅ξD=z of the cosine function[10,11]) gathersall thepairsofcentersofsecondarydis-turbance with a given separation vector ξD ≠ 0 acrosstheaperture;eachpairbelongingtoadifferentsupport.The integrand represents “fringe patterns” at OP,which are orthogonal to the direction of ξD and havethe sameperiodRA ¼ λz=jξDj. So, the set ofpairs of cen-ters of secondary disturbance that provides them is aclass of source pairs for the spatial coherence moiré[14,15].Thesupport of the complexdegreeof spatial co-herence centered at a specific ξA contains one sourcepair of all classes [15]. Its size and shape bound theset of contributing pairs and themagnitude of the com-plex degree of spatial coherence determines theirweights.The map (or set) of classes of source pairs can be re-
trievedbyFourier transformingthepowerspectrumas-sociated to the spatial coherence moiré in theFraunhofer domain, experimentally recorded by a de-tector placed at OP, i.e., ~SðξD;ωÞ ¼
ROP SðrA;
ωÞ exp�iðk=zÞrA⋅ξD�d2rA. Taking into account that the
power spectrum is a real-valued function, its Fourierspectrum will be hermitic, i.e., ~SðξD;ωÞ ¼ ~S�ð−ξD;ωÞ,so that j ~SðξD;ωÞj ¼
��~Sð−ξD;ωÞj and ϑðξD;ωÞ ¼−ϑð−ξD;ωÞ, with ϑðξD;ωÞ denoting its phase. Becauseof this redundancy, the complete map of classes willbe contained in only two consecutive quadrants of theξD domain. Equations (4) and (9) lead to
for the map of classes of source pairs. j ~SðξD;ωÞj repre-sents theweightof the classof sourcepairswithsepara-tion vector ξD in producing the spatial coherencemoiré,andthephaseϑðξD;ωÞdetermineshowthecontributionof that class should be added to the moiré. The deltapeak at the origin of the map ðξD ¼ 0Þ is contributedby the individual centers of secondary disturbance inthe aperture. Its amplitude equals the power densityof the optical field at OP.
The procedure described by Eq. (11) can be bothoptically or numerically performed. In the first case,the power spectrum at OP can be photographicallyrecorded, and the picture can be used as a gratingfor diffracting a fully spatially coherent and uniformplane wave of null phase in the Fraunhofer domain.This diffraction pattern should be the map of classesof source pairs. In the second case, the power spec-trum is recorded by a CCD sensor and digitized forapplying a numerical FFT algorithm. The resultshould be the map. Both procedures resemble therecording and reconstruction of a Fourier hologram,whose twin images are ~SðξD;ωÞ and ~Sð−ξD;ωÞ. Inother words, the recorded power spectrum at OPis the Fourier hologram of the map of classes ofsource pairs of the optical field. Taking into accountEq. (4), the map of classes of source pairs can beexpressed as ~SðξD;ωÞ ¼ ð1=λzÞ2 RAP ~SðξA; ξD;ωÞd2ξA,with ~SðξA; ξD;ωÞ ¼
termination of the map of classes of source pairs[15]. A microscope lens L1 (10 × =0:25) focuses aTEM00 He–Ne laser beam (λ ¼ 632:8nm and Gaus-sian degree of spatial coherence), providing a point-like source at a variable distance d from a rotatingdiffuser D. Shifting L1 along the optical axischanges the spatial coherence of the beam. The dif-fuser is fixed at the focal distance f 2ð40 cmÞ of thelens L2, which collimates the divergent cone of lightin order to illuminate a pinhole mask M with a qua-si-uniform plane wave. A Fourier lens L3 is placedin front of the pinhole mask at its focal distancef 3ð45 cmÞ to obtain the Fraunhofer pattern producedby the mask, at its rear focal plane. A CCD sensorrecords the power spectrum pattern, and a conven-tional FFT algorithm is applied for numerically cal-culating its Fourier transform. So, AP and OPcorrespond to the mask plane M and the CCD-sen-sor plane, respectively.Figure 4 shows the recorded power spectrum (in-
terference patterns on the first and the third rows,from top to bottom) and the moduli of the correspond-ing maps of classes of source pairs (images of pointsin the second and fourth rows). The support diameterof the Gaussian degree of spatial coherence at themask plane was determined for each value of the dis-tance d, by applying the method reported in Ref. [17](values on the upper row of Fig. 4). Two different pin-hole masks (appearing in the first column on the left)were arranged at themask plane in the experimentalsetup of Fig. 3.The maps in the fourth column are the same for
both masks and correspond to the first term ofEq. (12) only, because the second term nullifiesdue to the fully spatially incoherence of the illumi-nation. This peak also appears as the brightest peakin all the maps because it denotes the total powerprovided by each mask. The points around it specifythe classes of source pairs selected (or filtered) bythe corresponding mask, depending on the spatialcoherence state of the illumination. Maps in the sec-ond column contain all the possible classes becauseof the fully spatial coherence of the illumination.Their brightness corresponds to the number of pairsof the class. The number of classes diminishes in themaps on the third column in comparison to themaps on the second column, and their brightnessalso changes, due to the partially spatial coherenceof the illumination, which is revealed by the low vis-
ibility of the fringes of the corresponding interfer-ence patterns.
D. Effect of the Aperture Edge
The deviation of the light paths on account of theedges of an obstacle is called diffraction [16]. How-ever, the phase-space representation of the opticalfield leads to some precision about this classical no-tion by showing the close relationship between dif-fraction and state of spatial coherence of the field.Actually, the power spectrum at OP, obtained fromEqs. (4) and (9),
is the intensity distribution of the diffraction patternproduced by the aperture placed at AP, whose fringeappearance results from the cosinelike modulationsin the second term. Consequently,
• SðrA;ωÞ ¼ Cð1=λzÞ2 RAP SðξA;ωÞjtðξAÞj2d2ξAholds for fully (ideal) spatially incoherent fields, i.e.,such fields are not diffracted.
• For fully (ideal) spatially coherent fields, theaperture shape determines the integration regionfor the variable ξD, as stated by the classical thoughtof diffraction.
• For spatially partially coherent fields, the inte-gration region will depend on the position of the sup-port center within the aperture, given by ξA.Specifically, for the inner region 0 ≤ jξAj ≤ b, with b ¼a − jξDjMAX=2, the support shape determines the inte-gration region, but for the crownb < jξAj ≤ a, theaper-ture edge distorts the support shape, as illustrated inFig. 5 for the particular case of circular shaped aper-ture and support. The subregion of the distorted sup-port that fulfills the symmetry requirements of thecenter and difference coordinates will be the integra-tion region. Such support distortion is the actual con-tribution of the edge to the diffraction of spatiallypartially coherent optical fields. Consequently, thecorrelation between pairs of centers of secondary dis-turbance in the crown increases and approaches to 1at the edge vicinity, and the pair density of the classeswith small separation vectors (and therefore theirweight) increases in the crown.
Although aperture and/or support shapes can bemore complex in practical situations than in theconceptual sketch of Fig. 5, the following analysis
Fig. 3. Experimental setup for determining the map of classes ofsource pairs.
Fig. 4. Experimental results obtained by using the setup in Fig. 3. Two different pinhole masks (left column) and three different supportsizes for the Gaussian degree of spatial coherence were used. Images in the second and fourth rows are the correspondingmaps of classes ofsource pairs to the interference patterns directly over each map.
with the upper integration limit κ ¼ jξDjMAX forSIðrA; ξA;ωÞ, and κ ¼ κðξA; ξDÞ, determined by the con-tour of the distorted support centered at ξA, forSCðrA; ξA;ωÞ. It is apparent that the whole aperturebehaves as a crown with b ¼ 0 under fully spatiallycoherent illumination, so that Eq. (14) reduces toSðrA;ωÞ ¼ ð1=λzÞ2 R0≤jξAj≤a SCðrA; ξA;ωÞd2ξA. On thecontrary, for a ≫ jξDjMAX, the contribution of thecrown becomes negligible, i.e., b → a, and Eq. (14) re-duces to SðrA;ωÞ ¼ ð1=λzÞ2 R0≤jξAj≤a SIðrA; ξA;ωÞd2ξA.For aperture radius bigger than a certain value ofa, this approach should hold under a negligible error.In these cases, the optical field propagates similarly
as in free space. However, cosinelike modulationscould appear at OP because of the complex degreeof spatial coherence in the second term ofSIðrA; ξA;ωÞ. Therefore, diffraction is due to thespatial coherence properties of the optical field, butit is affected by the aperture edge because of the sup-port distortion [10,11]. In addition, the concepts ofspatial frequency and spatial bandwidth should beassociated, more properly, with the effective diffract-ing aperture, so that ηcutoff ¼ kjξDjMAX=2z is the spa-tial cutoff frequency of the diffracted field infree space.
Thus, the crown area-to-aperture area ratio be-comes a practical descriptor of the diffraction of op-tical fields in any state of spatial coherence. For anaperture and a support of radii a and jξDjMAX=2, re-spectively, it takes the form FðEÞ ¼ πða2
− b2Þ=πa2 ¼ 1 − ð1 −
ffiffiffiffiE
p Þ2, where E ¼ jξDj2MAX=4a2 is the
support area-to-aperture area ratio. It is assumedthat E ¼ 1 stands for πjξDj2MAX=4 ≥ πa2, so that 0 ≤
E ≤ 1 and 0 ≤ FðEÞ ≤ 1 hold. Negligible diffraction ef-fects, i.e., FðEÞ → 0 and E → 0, are obtained for fullyspatially incoherent optical fields, while FðEÞ → 1and E → 1 stand for fully spatially coherent opticalfields.
The above description was numerically analyzedand experimentally validated [19]. Both the margin-al power spectra and the power spectra at OP werecalculated, for both Fresnel and Fraunhofer diffrac-tion of light with Gaussian degree of spatial coher-ence and variable variance, through a uniformlyilluminated slit of width 2a. For the experiment,a circular aperture of transmission equal to 1 wasattached at the M plane of the setup in Fig. 3.The same diffracting geometry was used for bothFraunhoffer and Fresnel diffraction, by adjustingthe aperture diameter and the distance betweenthe aperture and the observation planes. In bothcases, the degree of spatial coherence was changedby placing the microscope lens at different distancesd in front of the rotating diffuser, and the supportsize was determined by the method in Ref. [17].So, the parameter E was established for each re-corded power spectrum. Figures 6 and 7 show boththe numerical and the experimental results.Although diffraction by slits differs from diffractionby circular apertures, it is worth noticing that theprofiles change in the same way if the diffractionis performed under similar conditions. It leads tothe conclusion that the theoretical predictions, de-scribed by the numerical results, are experimentallyfulfilled.
E. Spot of the Optical Field
E ≪ 1 stands by making the aperture great enough,because the inner region grows more rapidly thanthe crown as the aperture grows, i.e., Acrown ≈
πjξDmaxja for a ≫ ξDmax, but AI ¼ πa2 so thatlima→∞Acrown=AI ¼ 0 in the conceptual sketchwith circular aperture and support, for instance.
Fig. 5. Conceptual illustration of the support distortion by theaperture edge. The simple case of a circular shaped apertureand support is assumed.
Therefore, SðrA;ωÞ ¼ ð1=λzÞ2 R0≤jξAj≤a SIðrA; ξA;ωÞd2ξAstands and takes the form
SðrA;ωÞ ¼�1λz
�2SoðωÞ
Z0<jξDj≤jξDjMAX
Z0≤jξAj≤a
μðξA
þ ξD=2; ξA − ξD=2;ωÞd2ξA exp�−i
kzrA⋅ξD
�d2ξD ð15Þ
by assuming tðξA � ξD=2Þ ¼ 1 and SðξA � ξD=2;ωÞ ¼SoðωÞ. It only depends on the spatial coherence prop-erties of the optical field across the aperture, repre-sented by the average complex degree of spatialcoherence, i.e., �μðξD;ωÞ ¼ ð1=AIÞ
R0≤jξAj≤a μðξAþ
ξD=2; ξA − ξD=2;ωÞd2ξA, with AI ¼R0≤jξAj≤ad
2ξA. Thisquantity will be recovered by Fourier transformingEq. (15):
�μðξD;ωÞ ¼1
AISoðωÞZOP
SðrA;ωÞ exp�ikzrA⋅ξ0D
�d2rA:
ð16Þ
The procedure described by Eq. (16) can be opticallyor numerically performed, in a similar fashion as byEq. (11), which means that SðrA;ωÞ can be regardedas the Fourier hologram of �μðξD;ωÞ with twin images�μðξD;ωÞ and �μð−ξD;ωÞ. Furthermore, Eq. (11) can be
interpreted as a map of classes of source pairs, whichcontains only a member of each class. The amplitudeand phase for a specific separation vector ξD deter-mine the weight of the modulation provided by thepair with this separation vector onto the optical fieldand how this modulation is applied, respectively. Inaddition, μðξA þ ξD=2; ξA − ξD=2;ωÞ for a specific posi-tion ξA within the aperture should be a subset ofsource pairs of �μðξD;ωÞ, i.e., the submap of sourcepairs within the support centered at ξA.
The power spectrum SðrA;ωÞ in Eqs. (15) and (16)is called the spot of the optical field [17,19]. It is theFourier hologram of the average complex degree ofspatial coherence, which is the basic map of classesof source pairs of the optical field. Accordingly, thereis an uncertainty relationship between the spot ex-tent at OP and the support size of the average com-plex degree of spatial coherence at AP. This is quitedifferent from the well known uncertainty relation-ship between the extent of the central maximum ofthe Fraunhofer diffraction pattern and the aperturesize [16], which is only valid when the aperture isuniformly illuminated by a fully spatially coherentfield. The independence of the spot extent from theaperture size and its only dependence on the supportsize is a criterion for experimental spot identificationand confirms the statement that the support is theeffective diffracting aperture.
Fig. 6. (Color online) Phase-space representation of Fraunhofer diffraction. Theoretical results were numerically calculated for diffrac-tion through a slit of width 2a. Experimental results were obtained by using the setup in Fig. 3 after attaching a circular aperture at themask plane M.
If all the submaps are the same irrespective of theposition ξA of the support centers, �μðξD;ωÞ ¼ μðξD;ωÞholds, i.e., the complex degree of spatial coherencewill be space invariant. Optical fields that satisfy thiscondition are called Schell-model fields [20]. Theirspots take the form SðrA;ωÞ ¼
�1=λz
�2 AISðrA;ωÞ, so
that SðrA;ωÞ ≥ 0. Thus, the recording of a Schell-model field spot proves the existence of the spatialcoherence wavelets. Indeed, Schell-model fieldshave a characteristic spatial coherence wavelet[19] of the form WðrA þ rD=2; rA − rD=2; ξA;ωÞ ¼SðrA;ωÞ expð−iðk=zÞrD⋅ξAÞ. Furthermore, the Van
Cittert–Zernike theorem in Eq. (7) points out thatany uniform primary (fully spatially incoherent)source provides a Schell-model field at a plane inthe Fraunhofer domain.
The above description was experimentally proved[19] by using the setup in Fig. 3 after attaching a cir-cular variable aperture at the planeM and assumingthat the laser beam behaves as a Gaussian Schell-model beam. Figure 8(a) shows the profile of the Airypattern obtained by small enough apertures ðE ¼ 1Þ,in spite of the rotating diffuser. The lower the para-meter E, the bigger the aperture size in comparison
Fig. 7. (Color online) Phase-space representation of Fresnel diffraction, assuming two Fresnel zones inscribed within the diffractingaperture. Theoretical results were numerically calculated for diffraction through a slit of width 2a. Experimental results were obtainedby using the setup in Fig. 3 after attaching a circular aperture at the mask plane M.
Fig. 8. Transversal profiles of the power spectrum recorded by the CCD sensor of the experimental setup in Fig. 3, for different sizes of theaperture stop and values of the parameter E.
to the support of the complex degree of spatial coher-ence. So, the shape of the power spectrum closely ap-proaches to a practically unchangedGaussian profile,as appears in Figs. 8(b) and 8(c), independently fromthe aperture size and the values of the parameter E.Only the brightness increases as the aperture sizegrows because more power flows through the experi-mental setup. It is not apparent in such figuresbecause the profiles were normalized for the presen-tation. Such invariantprofiles belong to the spot of theillumination and constitute the experimental evi-dence of the characteristic spatial coherence waveletof the field at the aperture plane of the setup and thevalidity of the uncertainty relationship between thespot and the support sizes.The spot invariance can be straightforwardly ap-
preciated in the experimental graphs of power spec-trum area versus aperture diameter and versuslnðEÞ shown in Figs. 9. They were determined bya Gaussian degree of spatial coherence with sup-port diameter of 1:5mm, measured by the methodin Ref. [17]. The spot will be recorded by the CCDsensor for E ≤ 0:15ðlnE ≤ −1:9Þ (Fig. 9(b)), i.e., theinfluence of the crown on diffraction can be ne-glected if the support size does not exceed 15%of the aperture area. Under this criterion, the spotappears when the aperture diameter becomes2a≍2:58R0
D, which yields 3:87mm for the per-formed experiment, which is apparent in Fig. 9(a). For E > 0:15 the influence of the crown on dif-fraction should be considered.
4. Summary and Conclusions
Wigner optics includes the phase-space representa-tion of Fresnel–Fraunhofer diffraction, given bythe spatial coherent wavelets. It differs from the con-ventional picture in showing that diffraction mainlyresults from the spatial coherence properties of theoptical field, i.e., the support of the complex degreeof spatial coherence behaves as the effective diffrac-tion aperture. The edges of physical apertures influ-
ence diffraction in that they distort the supports intheir vicinity. This was proved by recording the fieldspot, a characteristic power spectrum distribution atthe observation plane that is the Fourier hologram ofthe average complex degree of spatial coherence atthe aperture plane. The spot of a Schell-model fieldis proportional to the marginal power spectrum car-ried by the characteristic spatial coherence waveletof the field.
The spatial coherence state determines the mar-ginal power spectrum of the field. It is a WDF withpower units, whose negative values are crucial fordescribing interference and diffraction. Actually,each support within the aperture plane provides amarginal power spectrum, which is carried to the ob-servation plane by a spatial coherence wavelet. Aspatial coherence moiré results from the superposi-tion of the fringe structures of the marginal powerspectra. It was also shown that the power spectrumat the observation plane is the Fourier hologram ofthe map of classes of source pairs of the optical field,which determine the fringe structure of the marginalpower spectrum and therefore the interference anddiffraction behavior of light, on the basis of a com-plete description of the spatial coherence propertiesof the optical field.
This investigation was supported by DIME (Uni-versidad Nacional de Colombia Sede Medellín).The authors sincerely thank Jorge Herrera andMario Usuga for their inspiring comments.
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