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Coherence singularities in the field generated by partially coherent sources

T. van Dijk,1 H. F. Schouten,1and T. D. Visser1,2

1

Department of Physics and Astronomy, Free University, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands 2

Department of Electrical Engineering, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands

共Received 13 November 2008; published 5 March 2009兲

We analyze the coherence singularities that occur in the far field that is generated by a broad class of partially coherent sources. It is shown that for rotationally symmetric planar quasihomogeneous sources the coherence singularities form a two-dimensional surface in a reduced three-dimensional space. We illustrate our results by studying the topology of the coherence singularity of a partially coherent vortex beam. We find that the geometry of the phase singularity can be associated with conic sections such as ellipses, lines, and hyperbolas.

DOI:10.1103/PhysRevA.79.033805 PACS number共s兲: 42.25.Kb, 42.25.Fx, 42.60.Jf

I. INTRODUCTION

There is a growing interest in the structure of wave fields in the vicinity of points where certain field parameters are undefined or “singular.” This has given rise to the new sub-discipline of singular optics 关1,2兴. In the past few years,

many different types of singular behavior have been identi-fied. For example, phase singularities occur at positions where the field amplitude vanishes and hence the phase is undefined 关3兴. Polarization singularities arise at locations

where the field is circularly or linearly polarized 关4–10兴.

There either the orientation angle of the polarization ellipse or its handedness is undefined. Also the Poynting vector can exhibit singular behavior at points where its modulus is zero, and hence its orientation is undefined关11兴. Topological

reac-tions of such singularities were studied in关12,13兴.

Optical coherence theory 关14兴 deals with the statistical

properties of light fields. In this theory, correlation functions play a central role关15兴. A form of singular behavior that is

slightly more abstract than those mentioned above occurs in two-point correlation functions. At pairs of points at which the field 共at a particular frequency兲 is completely uncorre-lated, the phase of the correlation function is singular 关16–21兴. When the field at two such points is combined in an

interference experiment, no fringes are produced. These co-herence singularities are points in six-dimensional space. Their relationship to other types of singularities has only recently been clarified 关22–26兴.

Thus far, only one study has been devoted to the descrip-tion of the multidimensional structure of a specific coherence singularity, namely, that of a vortex beam propagating through turbulence 关27兴. In the present article, we analyze

the more general case of coherence singularities in the far zone of the field generated by a broad class of partially co-herent sources. These quasihomogeneous sources are often encountered in practice. We analyze the generic structure of the coherence singularities and also discuss the practical case of a rotationally symmetric source. We illustrate our results by applying them to a recently described new type of “dark core” or vortex beam. For this beam all different cross sec-tions of the singularity are shown to be conic secsec-tions in a suitable coordinate system.

II. PARTIALLY COHERENT SOURCES

Consider a partially coherent planar secondary source, situated in the plane z = 0, that emits radiation into the half space z⬎0 共see Fig.1兲. In the space-frequency domain, the

source is characterized by its cross-spectral density function 关14兴

W共0兲共␳1,␳2,␻兲 = 具U共0兲ⴱ共␳1,␻兲U共0兲共␳2,␻兲典. 共1兲 Here U共0兲共␳,␻兲 represents the source field at frequency␻at position ␳=共x,y兲, the asterisk indicates complex conjuga-tion, and the angled brackets denote an ensemble average. The spectral degree of coherence is the normalized form of the cross-spectral density, i.e.,

␮共0兲 1,␳2,␻兲 = W共0兲共␳1,␳2,␻兲

S共0兲共␳1,␻兲S共0兲共␳2,␻兲 , 共2兲 with S共0兲共␳,␻兲 = W共0兲共␳,␳,␻兲 共3兲 the spectral density 共or “intensity at frequency ␻”兲 of the source. For Schell-model sources 关14兴 the spectral degree of

coherence only depends on position through the difference ␳1−␳2, i.e.,

␮共0兲

1,␳2,␻兲 =␮共0兲共␳1−␳2,␻兲. 共4兲

The field in an arbitrary transverse plane z⬎0 is given by the expression

U共␳,z,␻兲 =

共z=0兲

U共0兲共␳

,␻兲G共␳−␳

,z,兲d2␳

, 共5兲 where G共␳−␳

, z ,␻兲 is an appropriate free-space Green’s function 关14兴, Sec. 5.2. On substituting from Eq. 共5兲 in Eq.

ρ O z = 0 z s1 s2 x y ρ θ

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共1兲, while interchanging the order of integration and

en-semble averaging, it follows that the cross-spectral density at two arbitrary points共␳1, z1兲 and 共␳2, z2兲 satisfies the equation

W共␳1,z1,␳2,z2,␻兲 =

冕冕

共z=0兲

W共0兲共␳1

,␳2

,␻兲Gⴱ共␳1−␳1

,z1,␻兲

⫻ G共␳2−␳2

,z2,␻兲d2␳1

d2␳2

. 共6兲

The spectral density and the spectral degree of coherence at arbitrary points are given by formulas that are quite similar to Eqs.共2兲 and 共3兲, viz.,

␮共␳1,z1,␳2,z2,␻兲 = W共␳1,z1,␳2,z2,␻兲

S共␳1,z1,␻兲S共␳2,z2,␻兲 共7兲 and S共␳,z,兲 = W共,z,,z,␻兲. 共8兲 Coherence singularities are phase singularities of the spectral degree of coherence. They occur at pairs of points at which the field at frequency ␻is completely uncorrelated, i.e.,

␮共␳1,z1,␳2,z2,␻兲 = 0. 共9兲

III. QUASIHOMOGENEOUS SOURCES

An important subclass of Schell-model sources is formed by so-called quasihomogeneous sources 关14兴. For such

sources the spectral density S共0兲共␳,␻兲 varies much more slowly with ␳ than the spectral degree of coherence ␮共0兲

,兲 varies with

. This behavior, that often occurs in

practice, is sketched in Fig.2.

For quasihomogeneous sources the field in the source plane and the field in the far zone are related by two reci-procity relations, namely,

S共⬁兲共s,␻兲 = 共2␲k兲2˜S共0兲共0,␻兲␮˜共0兲共ks,␻兲cos2␪/r2, 共10兲 ␮共⬁兲共r 1s1,r2s2,␻兲 = S ˜共0兲关k共s 2⬜− s1⬜兲,␻兴 S ˜共0兲共0, exp关ik共r2− r1兲兴, 共11兲 with the two-dimensional spatial Fourier transforms given by the expressions S ˜共0兲共f,兲 = 1 共2␲兲2

S共0兲共␳,␻兲e −if·d2, 共12兲˜共0兲共f,␻兲 = 1 共2␲兲2

␮共0兲共␳,␻兲e−if·␳d2␳. 共13兲

In these formulas k =/c is the wave number associated with frequency␻, c being the speed of light in vacuum, sis the projection of the unit direction vector s onto the xy plane, and␪ is the angle that the s direction makes with the z axis 共see Fig. 1兲. The superscript 共⬁兲 indicates points in the far

zone. Equation 共10兲 states that the far-field spectral density

of a planar secondary quasihomogeneous source is propor-tional to the Fourier transform of its spectral degree of co-herence. Equation 共11兲 expresses that the far-field spectral

degree of coherence of such a source is, apart from a geo-metrical factor, given by the Fourier transform of its spectral density.

Even though Eq.共11兲 is quite general, it allows us to draw

several conclusions. First, the far-field spectral degree of co-herence depends on the spectral density of the source, but is independent of its spectral degree of coherence. Second, the dependence of the far-field spectral degree of coherence on the two distances r1 and r2 enters only through the factor

exp关ik共r2− r1兲兴. This means that coherence singularities

oc-cur along certain pairs of observation directions s1

=共sin␪1cos␾1, sin␪1sin␾1, cos␪1兲 and s2 =共sin␪2cos␾2, sin␪2sin␾2, cos␪2兲 for which the prefactor

in Eq. 共11兲 vanishes. Since S˜共0兲共0,␻兲 is both finite and real,

this yields the two constraints Re兵S˜共0兲关k共s

2⬜− s1⬜兲,␻兴其 = 0, 共14兲

Im兵S˜共0兲关k共s2⬜− s1⬜兲,␻兴其 = 0, 共15兲 where Re and Im denote the real and imaginary parts, respec-tively. These two conditions imply that generically 共i.e., when they are independent and commensurate兲, the coher-ence singularities form a two-dimensional surface in the four-dimensional共␪1,␾1,␪2,␾2兲 space.

Let us next consider the specialized case of a source whose spectral density is mirror symmetric with respect to both the x and the y axes. In that case the factor S˜共0兲关k共s2⬜ − s1⬜兴,␻兲 that appears in Eq. 共11兲 is real valued for all values

of its spatial argument关28兴 and hence condition 共15兲 is lifted.

This means that the coherence singularity is a three-dimensional volume in共␪1,␾1,␪2,␾2兲 space. Furthermore, if the spectral density of the source is rotationally symmetric, the spectral degree of coherence in the far zone depends on the observation angles ␾1 and␾2 only through their

differ-ence ␾2−␾1. We therefore conclude that for planar second-ary rotationally symmetric quasihomogeneous sources the coherence singularities are two-dimensional surfaces in the reduced共␪1,␪2,␾2−␾1兲 space. An example of such a source

is examined in Sec.IV.

IV. PARTIALLY COHERENT LAGUERRE-GAUSS SOURCE

We illustrate our results thus far with the analysis of a partially coherent source that generates a Laguerre-Gauss beam 关29兴. For this source we have

S(0)( , )ρ ω ρ 0 ρ' |µ ρ' ω(0)( , )| 0 1

FIG. 2. 共Color online兲 Illustrating the concept of

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S共0兲共␳,␻兲 = A2␳2exp共−␳2/2␴S

2兲, 共16兲

␮共0兲

2−␳1,␻兲 = exp关− 共␳2−␳1兲2/2␴␮2兴, 共17兲

with A a real number, ␳=兩␳兩, and ␴S and ␴␮ the effective widths of the spectral density and of the spectral degree of coherence, respectively. If␴Ⰶ␴Sthe source is quasihomo-geneous. Since S ˜共0兲共f,兲 = 共2 − f2 S 2 S 4 A2exp共− f2␴S 2/2兲/2 , 共18兲 application of the reciprocity relation共11兲 yields

␮共⬁兲共r 1s1,r2s2,␻兲 = 关1 − k2共s2⬜− s1⬜兲2␴S2/2兴 ⫻exp关− k2共s 2⬜− s1⬜兲2␴S2/2兴 ⫻exp关ik共r2− r1兲兴. 共19兲 Because

共s2⬜− s1⬜兲2= sin2␪1+ sin2␪2− 2 sin␪1sin␪2cos共␾1−␾2兲,

共20兲 it follows from Eq. 共19兲 that coherence singularities occur

for those values of␪1,␾1,␪2, and␾2 for which

sin2␪1+ sin2␪2− 2 sin␪1sin␪2cos共␾1−␾2兲 = 2/k2␴S

2

. 共21兲 As remarked at the end of Sec. III, the dependence of the spectral degree of coherence on the two angles␾1and␾2is through their difference␾1−␾2. From now on we set,

with-out loss of generality, ␾2= 0.

An example of the topology of the coherence singularity is shown in Fig.3, from which it can be seen that it forms a

saddlelike surface. Let us consider the ␪1=␪2 cross section. For small values of these two angles, there is no value of␾1

that corresponds with a point on the surface, i.e., there exist no pairs of points at which the field is completely uncorre-lated. When the angles are gradually increased to a critical value ␪1=␪2=␪ca coherence singularity occurs at ␾1= 180°

关29兴. For larger values 共␪1=␪2⬎␪c兲 a value of ␾1⬍180°

corresponds to a point on the surface. Since Eq.共21兲 shows a

dependence of the singularity on cos␾1, this means that the

initial singularity has unfolded into two pairs of singularities: one for ␾1and one for −␾1. It is noted that in this example

k2

S

2= 10 for illustrative purposes. In Fig.4the more realistic

value of 1000 was used. It can be seen that the topological features of the coherence singularity remain unchanged.

V. CONIC SECTIONS

In order to study the coherence singularity depicted in Fig.3 in more detail, it is instructive to rewrite Eq. 共21兲 in

the form x2+ y2+ 2xyz + G = 0, 共22兲 where x = sin1, 共23兲 y = sin2, 共24兲 z = − cos␾1, 共25兲 G = − 2/k2␴S 2 . 共26兲

Although Eq.共22兲 is not a quadratic surface in 共x,y,z兲 space,

θ

1[deg]

θ

2 [deg]

φ

1

[deg]

FIG. 3. 共Color online兲 A two-dimensional coherence singularity in共␪1,␪2,␾1兲 space. In this example k2

S 2= 10. θ2[deg] θ1 [deg] φ1 [deg]

FIG. 4. 共Color online兲 A two-dimensional coherence singularity in共␪1,␪2,␾1兲 space. In this example k2␴S

2

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both the horizontal and the vertical cross sections of the co-herence singularity are quadratic curves. Horizontal cross sections共i.e., fixing the value of z and hence of␾1兲 are conic

sections in共x,y兲 space 关30兴; since x and y are limited to the

interval关0,1兴, only parts of these conic sections are realized. More specifically, if z = −1 the cross section takes the form of two parallel lines. On increasing z it becomes an ellipse共with a circle as a special case when z = 0兲, and finally, for z=1, it becomes two parallel lines again共only one of which lies in the physical domain of x and y兲. Various cross sections of the coherence singularity are shown in Fig.5for selected values of z. Because of the interchangeable roles of x and y in Eq. 共22兲, the cross sections are symmetric about the line x=y.

According to Eq.共22兲 vertical cross sections of the

coher-ence singularities共e.g., fixing the value of y and hence of2兲 are conic sections in 共x,z兲 space; since 0ⱕxⱕ1 and −1 ⱕzⱕ1, only parts of these conic sections are realized. More specifically, if y = 0 the cross section has the form of two parallel lines共only one of which lies in the physical domain

of x and z兲. On increasing y it becomes a branch of a bola, two intersecting lines, and again a branch of a hyper-bola. This is illustrated in Fig. 6. This concludes our identi-fication of various cross section of the coherence singularity with a variety of conic curves.

VI. CONCLUSIONS

We have analyzed the topology of coherence singularities that occur in the far field generated by quasihomogeneous sources. As an specific example we examined the coherence singularity of a partially coherent vortex beam. Its cross sec-tions were found to be different kinds of conic secsec-tions in a modified coordinate system.

ACKNOWLEDGMENTS

This research is supported by The Netherlands Organisa-tion for Scientific Research 共NWO兲, by the Foundation for Fundamental Research on Matter 共FOM兲, and by the Dutch Technology Foundation共STW兲. 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 (a) (b) (c) (d) (e) z = −1 z = −0.5 z = 0 z = 0.5 z = 1 x x x x x y y y y y

FIG. 5. 共Color online兲 Cross sections of the coherence singular-ity in the x , y plane for selected values of z, viz., 共a兲 z=−1, 共b兲 z = −0.5,共c兲 z=0, 共d兲 z=0.5, and 共e兲 z=1. -1.0 -0.5 0 0.5 1.0 0 0.5 1.0 y = 0 y = 0.4 y = 0.445 y = 0.45 y = 0.5 y = 1 x x x x x x (a) (b) (c) (d) (e) (f ) z z z z z z -1.0 -0.5 0 0.5 1.0 -1.0 -0.5 0 0.5 1.0 -1.0 -0.5 0 0.5 1.0 -1.0 -0.5 0 0.5 1.0 -1.0 -0.5 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0

FIG. 6. 共Color online兲 Cross sections of the coherence singular-ity in the x , z plane for selected values of y, viz., 共a兲 y=0, 共b兲 y = 0.4,共c兲 y=0.445, 共d兲 y=0.45, 共e兲y=0.5, and 共f兲 y=1.

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