## Abstract

Points within a fully coherent complex scalar optical field, where the amplitude is identically zero but the optical phase has a jump discontinuity, have been widely investigated by the singular-optics community. More recent researches have extended the domain of singular optics to include partially coherent fields. For example, in coherence vortices the phase of the two-point spectral degree of coherence of a partially coherent field exhibits vortex structure around a point where the magnitude of the spectral degree of coherence vanishes. We show that the spectral degree of coherence of Mie scattered partially coherent statistically stationary electromagnetic fields exhibits a rich set of coherence vortices in both the internal and external fields. Specifically, we look at Mie scattering of a stationary beam from a dielectric sphere and study the formation of coherence vortices and their evolution with both the properties of the scattering sphere, and of the incident partially coherent beam.

©2010 Optical Society of America

## 1. Introduction

In optics, the concept of *coherence* is a measure of the ability of an electromagnetic field to interfere with itself. It is normally defined using the correlation properties between field components and provides a precise mathematical basis to quantify the physically observable quantities of an electromagnetic field [1]. A widely used quantity is the spectral degree of coherence, which quantifies the correlation between two points in the wave field [2]. Its magnitude lies between zero (complete incoherence) and unity (complete coherence). Being a complex valued quantity, the degree of coherence possesses many characteristics analogous to the complex field of an optical wave. One such characteristic is the existence of phase singularities known as coherence vortices [3,4]. At these points the phase is discontinuous [5].

About two centuries ago, William Whewell considered the phase singularities in tide-waves [5]. He noted that rotary systems of tidal waves sometimes have a zero tide-height point at which all cotidal lines meet and a tidal vortex is created. Nye and Berry [6] built upon this work and laid the foundation for the field of singular coherent optics in 1974. Since then, singular optics has progressed rapidly and emerged as an important field of research [7], finding applications in areas as diverse as astronomy [8, 9], imaging [10] and interferometry [11]. At a point where an optical vortex exists, the field amplitude is identically zero but its phase integrated around a closed path enclosing that point yields an integer multiple of 2*π* [12]. This integer multiple signifies the strength (topological charge) of the vortex.

Following the advancements in characterizing phase singularities in coherent optical fields, singular optics has now been extended to partially coherent fields [3,4,13–17]. For example, the singular optics of two-point correlation functions such as the spectral degree of coherence of an optical field have been investigated [3,4,18]. Unlike in a fully coherent field where amplitudes can assume identically zero values [7], partially coherent fields typically have non zero spectral densities. This is because complete destructive interference can occur at certain fixed points in fully coherent fields whereas such sustained destructive interference is often not possible in partially coherent fields. Remarkably, however, two-point correlation functions such as the cross-spectral density of partially coherent fields can have zeros representing pairs of spatial points where the associated stochastic process is completely uncorrelated [4].

As an example of this, Shouten *et al*. [3] discovered that the phase of the spectral degree of coherence has a discontinuity of *π* radians across correlation singularities. Gbur and Visser [4,14] showed that the topological charge of a singularity is a conserved quantity. They showed that correlation singularities can only be created and annihilated such that the total topological charge remains conserved. Gbur and Visser [14] further showed that there exists a relation between optical vortices and coherence vortices. One striking observation was that optical vortices tend to evolve into coherence vortices when the spatial coherence of the optical field is decreased. Such results have been demonstrated theoretically [19] and experimentally [18] for certain systems, and generalized to the case of linear imaging systems [14].

Although extensive work has been done on various scenarios that exhibit optical/correlation phase singularities in fully coherent and partially coherent fields (see for example [2, 7, 20]), no such work exists for scattered partially coherent fields. In this study, we show that Mie scattered, partially coherent statistically stationary light exhibits correlation singularities and coherence vortices. We examine the behavior of these coherence vortices when the coherence of the incident field is varied from fully coherent to partially coherent. Emphasis is given to the creation and annihilation of coherence vortices in the field inside the scatterer and the scattered field relative to scattering system parameters.

This paper is organized as follows: We begin by introducing background theory in Section 2. Basic formalism governing Mie scattering and different measures used for quantifying both coherence and coherence vortices are introduced and briefly described in this section. Thereafter, the details of the scattering setup that we use in our study are described in Section 3. For various particle sizes and coherence states, simulations were carried out and their results are presented in Section 4 along with their physical interpretation. In Section 5, a brief discussion is presented, followed by the conclusion in Section 6.

## 2. Theoretical background

Here we outline the theoretical background to our study on coherence vortices in Mie scattering of partially coherent statistically stationary light. Section 2.1 treats Mie scattering of coherent electromagnetic plane waves by dielectric spheres, Section 2.2 outlines the space-frequency formulation of coherence theory for statistically stationary electromagnetic fields, and Section 2.3 outlines the theory of coherence vortices in stationary electromagnetic fields, from a topological perspective.

#### 2.1. Mie scattering of coherent electromagnetic plane waves

The exact solution to the problem of scattering of a coherent electromagnetic plane wave from a dielectric sphere, now known as Mie Scattering, was first developed by Gustav Mie in 1908 [21]. Since then, Mie theory has been used for analyzing scattering and absorption of light by spheres both theoretically and experimentally [21–23]. We briefly review this theory here.

Consider a dielectric sphere of radius *R* placed at the origin of a rectangular coordinate system as shown in Fig. 1. Suppose an *x*-polarized time-harmonic monochromatic plane wave, propagating in positive *z* direction, to be scattered from this sphere. The incident electromagnetic field can be written as

where *k* is the propagation constant of the surrounding medium, *E*(*t*) is the time-varying amplitude of the incoming plane wave, *ω* is the angular frequency of the incident plane wave and **e**̂* _{x}* is the unit vector along the

*x*-direction.

Once the incoming wave interacts with the dielectric sphere, the corresponding internal field, **E**
* _{int}*, and the scattered electric field,

**E**

*, can be written using Mie theory as*

_{sca}where *a _{n}*,

*b*,

_{n}*c*and

_{n}*d*denote the scattering and absorption coefficients. The functions

_{n}**M**

^{1}

^{o1n},

**N**

_{e1n}

^{1},

**N**

_{e1n}

^{3}and

**M**

^{3}

_{o1n}are vector spherical harmonics as given in [23]. See reference [23] for details on calculation of the above coefficients and functions.

#### 2.2. Coherence theory for statistically stationary electromagnetic fields

The coherence theory of statistically stationary electromagnetic fields [1, 2, 24] is a well-established research area, and here we briefly remind the reader of some key features in this theory. Let {**E**(**r**, *ω*)} denote an ensemble of monochromatic realizations, all of the same angular frequency *ω*, associated with a stationary electric field, in the space-frequency domain
(**r**,*ω*). We can then define the cross spectral density matrix [1, 2, 24] in terms of the electric field components as

where *i*, *j* = *x*,*y*, *z*. The asterisk denotes complex conjugation and angular brackets denote averaging over the ensemble of monochromatic realizations. Note that, under the assumption of ergodicity, ensemble averages are equal to a time average over any single realization of the process. The spectral degree of coherence for this stationary vector process is given by

where Tr[*W _{ij}*] denotes the trace of the cross spectral density matrix and

*S*(

**r**,

*ω*) =

*W*(

**r**,

**r**,

*ω*) is the spectral density of the field at point

**r**and frequency

*ω*.

#### 2.3. Coherence vortices in stationary electromagnetic fields

At pairs of spatial points (**r**
_{1},**r**
_{2}) where the scattering potential is continuous, the spectral degree of coherence *μ*(**r**
_{1},**r**
_{2};*ω*) will be a continuous and single-valued complex function of **r**
_{1},**r**
_{2}, and *ω*.

Suppose for the moment that the variables **r**
_{2} and *ω* are fixed to have the values **r**
^{′}
_{2} and *ω*
^{′}, respectively [4]. With reference to Fig. 2a, consider a directed simple smooth closed curve Γ, over each point of which the following conditions hold: (i) *μ*(**r**
_{1}, **r**
^{′}
_{2}; *ω*
^{′}) is nonzero for all **r**
_{1} ∈ Γ; (ii) the scattering potential is continuous both at **r**
^{′}
_{2}, and for all **r**
_{1} ∈ Γ. These assumptions ensure that arg[*μ*(**r**
_{1},**r**
_{2}
^{′}/;*ω*
^{′})] is defined and continuous for all pairs of spatial points {**r**
_{1} ∈ Γ, **r**
^{′}
_{2}}. Contours Γ obeying these conditions will be termed ‘permissible’ contours.

The single-valuedness of *μ* implies that (*cf*. [25]):

where *m* is an integer. Next, infinitesimally deform the permissible contour Γ to a new permissible contour Γ + *δ*Γ. Since arg[*μ*(**r**
_{1},**r**
_{2}
^{′};*ω*
^{′})] is continuous both for all pairs of points {**r**
_{1} ∈ **r**
^{′}
_{2}} and for all pairs of points {**r**
_{1} ∈ Γ + *δ*Γ, **r**
_{2}
^{′}}, and *δ*Γ is an infinitesimal deformation, it follows that the “circulation” *C*(Γ + *δ*Γ) can differ at most infinitesimally from *C*(Γ). Since Eq. (5) states that *C* is quantized, we have the immediate corollary that [12]

Any finite deformation of Γ can be built up as a sequence of infinitesimal deformations. Thus all oriented permissible simple closed curves Γ, which satisfy conditions (i) and (ii), may be partitioned into a series of equivalence classes labeled by the “topological charge” *m* [4].

Next, assume that a given permissible contour has a non-zero topological charge. Let *σ* be a two-dimensional continuous surface which has Γ = *∂σ* as a boundary. Then there exists at least one pair of points {**r**
_{1} ∈ *σ*, **r**
_{2}
^{′}} at which *μ*(**r**
_{1} ∈*σ*, **r**
_{2}
^{′}; *ω*
^{′}) vanishes. The proof is by *reductio ad absurdum*: Suppose *μ*(**r**
_{1} ∈ *σ*, **r**
_{2}
^{′}; *ω*
^{′}) to be non-vanishing for all pairs of points {**r**
_{1} ∈ *σ*, **r**
_{2}
^{′}}. The curve Γ can therefore be continuously contracted to a point **r**
_{1}
^{″} lying on *σ*, via a family of intermediate simple closed curves all of which lie entirely in *σ*. Once Γ has thereby been continuously deformed to a point, the circulation about the said point must vanish. This is a logical contradiction, since, by the argument given earlier, the circulation must be unchanged from its original value of 2*πm* ≠ 0, under the said continuous contraction of Γ to a point in *σ*. Hence the initial premise—namely that *μ*(**r**
_{1} ∈ *σ*,**r**
_{2}
^{′};*ω*
^{′}) is non-vanishing for all pairs of points {**r**
_{1} ∈ *σ*, **r**
_{2}
^{′}}—is incorrect. Hence our desired result: there exists at least one pair of points {**r**
_{1} = ∈ *σ*,**r**
_{2}
^{′}} at which *μ* (**r**
_{1} ∈ *σ*, **r**
_{2}
^{′}; *ω*
^{′}) vanishes. The locus of the set of points *P* will form continuous nodal lines *N* in volumes of space where the scattering potential is continuous; however, these nodal lines may terminate at spatial points where the scattering potential is discontinuous, and/or extend to infinity. The associated coherence vortices, at points *A* and *B* in Fig. 2a, will have opposite sense. Note that these nodal lines will be stable with respect to continuous perturbations of the scattering potential. Nodal planes are also possible, but these are non-generic in the sense that they are unstable with respect to perturbation.

Zero fringe visibility would be observed in a Young-type interference experiment in which the pinholes were located at **r**
_{1} = *P* ∈ *N* and **r**
_{2}≡*E* [3], as in Fig. 2b. Nonzero visibility will in general result if the pinhole *P* is moved to lie on a small simple closed curve Γ encircling the nodal line *N*. As **r**
_{1} traverses one cycle of Γ, the interference fringes over a screen *S* due to pinhole pairs at **r**
_{2} = *E* and **r**
_{1} = *X*
_{1},*X*
_{2}, ⋯ ,*X*
_{7} will shift by *m* periods; the case *m* = 1 is illustrated in Fig. 2c. Alternatively, if **r**
_{2} is fixed at *E* while **r**
_{1} traverses the sequence *X*
_{1} → *P*→*X*
_{4}, there will be a contrast reversal in the fringes: maxima (minima) for {**r**
_{1} =*X*
_{1},**r**
_{2}=*E*} become minima (maxima) for {**r**
_{1} = *X*
_{4}, **r**
_{2} = *E*}, with zero fringe visibility at the ‘crossover point’ {**r**
_{1} = *P*, **r**
_{2} = *E*} (*cf*. [3]).

The nodal lines *N* constitute the cores of the coherence vortices associated with *μ*(**r**
_{1}, **r**
^{′}
_{2}, *ω*
^{′}). The variables **r**
^{′}
_{2}, *ω*
^{′}—hitherto considered fixed—may now be treated as control variables (*cf*.[20]), the variation of which continuously deforms the nodal lines *N* associated with the coherence vortices in the space of independent variables **r**
_{1}, **r**
_{2}, *ω*. Thus the nodal lines *N*, in the three-dimensional space sketched in Fig. 2a (corresponding to fixed **r**
_{2} = **r**
_{2}
^{′}, *ω* = *ω*
^{′}), trace out ‘tubes’ or ‘membranes’ in the higher-dimensional space where **r**
_{2}, *ω* are not fixed. These membranes will be branched if the nodal lines themselves are branched, as will be the case in topological reactions where higher-order coherence vortices split, or when coherence vortices fuse [4,17].

## 3. Model for Mie scattering of partially-coherent stationary laser fields

Here we describe our model for the partially coherent stationary electromagnetic field incident upon a dielectric sphere. At frequency *ω*, under the space–frequency description of partially coherent electromagnetic fields, the incident field is described by the following ensemble of monochromatic realizations:

Here, *k* = *ω*/*c* is the free-space wavenumber, *c* is the speed of light in vacuum, and **s**̂_{0} represents the direction of propagation which is perpendicular to the polarization **e**̂_{S0⊥}. In our model, we take **e**̂_{s0⊥} ·**e**̂_{y′} = 0, where **e**̂_{y′} is the unit vector along the *y*
^{′}-direction, and *y*
^{′} is as defined in Fig. reffig:rotatedaxes using a convenient rotation matrix that is specified once and for all. The vector **s**̂_{0} is taken to be uniformly randomly distributed in a cone of solid angle ΔΩ, with vertex on the optic axis *z*, as illustrated in Fig. 3a.

This model approximates the beam-like output from a realistic continuous-wave laser, for which a given realization in the space-time domain can be denoted by:

Here, *A*(**r**,*t*) is a complex envelope that varies slowly both in time and space, **s**
$\widehat{\u0303}$
_{0}(*t*) denotes the temporally varying propagation direction on account of the *pointing stability* of the laser [26–28] and **e**
$\widehat{\u0303}$
_{S0⊥} (*t*) denotes the temporally fluctuating polarization. This directional instability of the laser source leads it to be a partially coherent light source. The range of solid angles subtended by **s**
$\widehat{\u0303}$
_{0}(*t*) may be taken as an estimate for ΔΩ in our model. A typical value for ΔΩ is a few pico steradians (psr).

Figure 3b shows two distinct realizations from the ensemble of incident fields, which point along two different directions inside the cone angle ΔΩ. For each such realization, Eq. (6) will correspond to an incident *z*
^{′}-directed plane wave which can be scattered from the dielectric sphere, using Eqs (1) and (2) from coherent Mie theory together with a rotation matrix which takes the *z* axis onto **s**̂_{0}. Once we have determined a suitably large number of realizations of the electric field at a given angular frequency via this procedure, the two-point cross spectral density function and the spectral densities can be directly estimated by averaging over the ensemble in Eq. (6). Consequently the spectral degree of coherence of the fields can be found using Eq. (4). As the ensemble average of statistically stationary ergodic processes is equal to the time average, the parameters calculated in this work can be readily measured using suitable optical detectors with finite response times.

## 4. Results

Here we present computational results obtained by implementing the theoretical model described above, and describing the coherence-vortex structures that are thereby obtained. The wavelength of the stationary laser beam model used for the numerical simulation is *λ*
_{0} = 837 nm. The scattering particle is treated as a homogeneous isotropic non-magnetic sphere with refractive index *n*̃ = 2 - 0.2*i*, which is placed in vacuum. The field is incident on the particle from the negative *z* axis, as shown in Fig. 4a and 4b. MATLAB^{®}(R2007b) codes written by Mätzler [29] were incorporated into our computational Mie scattering calculations. For the purpose of coherence calculation according to Eq. (4), the frequency *ω* was taken as the frequency of the stationary laser beam and the reference point **r**
_{2} was taken to be fixed to the point where the surface of the sphere meets the negative *z* axis. All ensemble averages utilized 500 random realizations of Eq. (6), as all our computational results were seen to have converged for this number of realizations. Section 4.1 considers coherence vortices in the interior of the dielectric sphere, with Section 4.2 considering coherence vortices external to the sphere.

#### 4.1. Coherence vortices in internal field of dielectric sphere

Here we study coherence vortices associated with the spectral degree of coherence inside the sphere. We consider the *xz* and *yz* planes internal to the sphere, as illustrated in Figs. 4a and 4b. Using the model described in Section 3, for a pointing stability of ΔΩ = *π* psr, we obtain the spectral densities shown in Figs. 4c (*xz* plane) and 4d (*yz* plane). The spectral density is concentrated in a smaller ‘hotspot’ area while the majority of the volume of the sphere is illuminated with a comparatively lower spectral intensity. Figures 4e and 4f show the magnitude of the spectral degree of coherence inside the particle. From these figures it is evident that inside the scatterer, the magnitude of the spectral degree of coherence varies between the expected limits of zero and unity. Furthermore, along the axis in the propagation direction we can see that the disturbance is more coherent compared to the other points. In the context of the present analysis, we are particularly interested in the zeros of *μ*, which correspond to pairs of spatial points {**r**
_{1}, **r**
_{2}} for which the electric field is completely uncorrelated at angular frequency *ω*.

When we examine the phase plot of the spectral degree of coherence, it is evident that the phase varies from zero to 2*π* within the scattering volume (see Figs. 4g and 4h). Within the particle we can see branch points where the phase of the spectral degree of coherence demonstrates vortex structure, marked by black dots. These screw-type topological defects, in the phase of the spectral degree of coherence, constitute coherence vortices in the internal coherence field of the dielectric spheres (*cf*. Eq. (5)). Note that the location of the coherence vortices will in general not coincide with the location of zeros of the spectral intensity, as can be seen by comparison of Fig. 4g to Fig. 4c, and comparison of Fig. 4h to Fig. 4d. Note also that all *xz* and *yz* plots are symmetric about the propagation direction *z*.

With a view to examining topological reactions [4,17] of coherence vortices within the dielectric sphere, Fig. 5 plots the phase of the spectral degree of coherence when the cone angle (ΔΩ) is increased, *i.e*. as the pointing stability of the laser is degraded. The left (right) panel of this figure shows the phase of *μ* in the *xz* (*yz*) plane for cone angles of 0 sr, 0.234 msr and 0.96 msr.

No coherence vortices are present inside the region marked by a rectangle in Fig. 5a (*xz*, 0 sr cone angle); upon increasing the cone angle to 0.234 msr, a coherence vortex-antivortex pair has been nucleated as shown in Fig. 5c. This pair annihilates when the cone angle is increased to 0.96 msr (see Fig. 5e). Similar topological reactions occur in the *yz* plane also: the coherence vortex-antivortex pair in Fig. 5b (0 sr cone angle) is annihilated when the cone angle is increased to 0.234 msr (see Fig. 5d) and it reappears when the cone angle is 0.96 msr (see Fig. 5f).

This process of creation and subsequent annihilation, of a coherence vortex-antivortex pair, may be pictured from a topological perspective by replacing the *z* axis of Fig. 2a by ΔΩ, again keeping **r**
_{2} and *ω* fixed, and constraining **r**
_{1} to lie in the *xz* or *yz* plane ∏. The topological arguments of Section 2.3, based solely on the continuity and single-valuedness of the spectral degree of coherence as a function of **r**1, **r**
_{2}, and *ω*, remain applicable. Consider the plane ∏ to sweep through the closed nodal line *N* from left to right, as ΔΩ (rather than *z*) is varied through the range of angles in our simulation. A coherence vortex-antivortex pair will nucleate when ∏ touches the left side of *N* at a single tangent point; the pair will then move apart and coalesce; they annihilate at the value of ΔΩ for which ∏ touches the right side of *N* at a single point.

In addition to coherence vortices in the spectral degree of coherence, the ensemble averaged Poynting vector may also display vortical behavior. Poynting-vector vortices and coherence vortices will in general not coincide, as illustrated in Fig. 6a. Here, the *xy* projection of the ensemble-averaged Poynting vector is displayed as a vector field, with the background color-map denoting phase of the spectral degree of coherence. The vortex structure of the ensemble-averaged Poynting vector does not coincide with the coherence vortex. A converse situation, where a coherence vortex is present but there is no accompanying Poynting-vector vortex, is shown in Fig. 6b. With reference to Fig. 6c, we note that the Poynting-vector field may possess topological defects which are non-vortical—as an example of this, Fig. 6c shows a saddle-type defect which is not accompanied by a topological defect in the phase of the spectral degree of coherence. Lastly, Fig. 6d shows a Poynting vortex that is in the vicinity of, but not coincident with, a coherence vortex.

The ensemble averaged Poynting vector, defined in the space both inside and outside the sphere, will in general be expected to display a rich series of topological defects associated with continuous 3-D vector fields [30, 31]. There is an evident analogy between the singularities of the ensemble averaged Poynting vector considered in this paper, and the physics of polarization singularities in fully coherent fields. Accordingly, a study of the singularities associated with the ensemble averaged Poynting vector, for partially coherent Mie scattering, would form a very interesting investigation that is beyond the scope of this paper.

#### 4.2. Coherence vortices in external field of dielectric sphere

We now consider coherence vortices in the scattered field **E**
* _{sca}* external to the volume of the dielectric sphere (see Eq. (2)). Consider the simulations in Fig. 7. This is a sequences of plots of the phase of the spectral degree of coherence for the scattered field, with

**r**

_{2}fixed and with

**r**

_{1}allowed to range over the sequences of four parallel planes

*z*=

*R*,1.1

*R*,1.2

*R*,1.3

*R*downstream of the sphere with radius

*R*.

Figure 7a, in the tangent plane *z* = *R* downstream of the sphere, shows two coherence vortex-antivortex pairs. Each of these pairs spread apart in propagating to *z* = 1.1*R*, as shown in Fig. 7b. The pairs then approach one another as in propagating to the plane *z* = 1.2*R*; see Fig. 7c. Finally, both pairs have mutually annihilated upon reaching the plane *z* = 1.3*R*, as shown in Fig. 7d.

This coalescence of coherence vortex–antivortex pairs can be visualized via Fig. 2a, by identifying ∏ with the tangent plane *z* = *R* downstream of the sphere. The coherence vortex-antivortex pair at *A* and *B* is seen to expand apart, contract and finally mutually annihilate, as *z* is increased. Thus the annihilation events in Fig. 7 represent a sweeping through the nodal ‘hairpin’ structure indicated by the curved nodal line downstream of the plane ∏ in Fig. 2a. The annihilation occurs when ∏ is at a tangent to the nodal line *N*, with all but one of the points in *N* lying upstream of ∏. An interesting corollary is that this point of coalescence is not well defined, since a tilting of the arbitrary *z* axis (*i.e*., an alternative foliation of the space with a sequence of parallel places) will alter the ‘observed’ point of annihilation. This serves to emphasize that the nodal lines (more generally, nodal hypersurfaces) associated with the coherence vortices are more fundamental than two-dimensional representations of the coherence vortices under a given foliation.

We conclude this section by remarking that, while topological reactions such as coherence vortex–antivortex annihilation can occur in three-dimensional physical space (*e.g*. in Fig. 7), such reactions may also occur in the control space which coordinatizes the spectral degree of coherence. An example of such topological reactions of coherence vortices, associated with non-spatial control variables, was seen in Fig. 5.

## 5. Discussion

It would be very interesting to extend the results of this investigation, from Mie scattering of partially coherent statistically stationary continuous-wave-laser illumination, to partially coherent pulsed-laser illumination. Such pulsed illumination is intrinsically non-stationary, implying a loss of ergodicity meaning that ensemble averages can be no longer be identified with the time averages associated with any real experiment. A way forward invokes the assumption of a pulsed source that is cyclostationary, *i.e*. a process which is periodically stationary [32–36]. If such a process may be taken as cycloergodic, and one works with the generalized two-frequency cross-spectral density *W _{ij}*(

**r**

_{1},

**r**

_{2};

*ω*

_{1},

*ω*

_{2}), then the findings of the present paper might be generalized to the case of such a sub-class of non-stationary processes. This will be the subject of future work.

One further extension would be to delve more deeply into the non-vortical singularities associated with the ensemble-averaged Poynting-vector field. It would be interesting to explore any connection between such singularities and coherence vortices. Moreover, the Poynting-vector singularities in partially coherent Mie scattering would also be of significant interest, independent of coherence singularities in the phase of the spectral degree of coherence.

As a final point, which might make an interesting topic for future research, it would be interesting to further investigate the topological reactions of coherence vortices, upon varying “control” parameters such as *ω* or **r**
_{2} associated with our model. Topological reactions of the Poynting-vector singularities could be studied in a similar manner.

## 6. Conclusion

We have developed a model for calculating the two-point spectral degree of coherence for stationary electromagnetic fields scattered from a dielectric sphere. This models a statistically-stationary stochastic process corresponding to radiation from a continuous wave laser source with a specified degree of pointing stability. Mie’s exact theory for scattering of a linearly polarized coherent plane wave from a dielectric sphere was generalized to our scattering scenario via the space–frequency formulation of partially coherent statistically-stationary electromagnetic fields. Detailed numerical calculations carried out using our model demonstrated the existence of coherence vortices in the spectral degree of coherence, both inside and outside the dielectric sphere. The locations of these coherence-vortex cores did not in general coincide with the zeros of the ensemble-averaged spectral density, as expected. The creation and subsequent annihilation of coherence vortex–antivortex pairs was observed. These topological coherence-vortex reactions were studied, both in the physical space inside and outside the sphere, and in the parameter space occupied by the spectral degree of coherence. We also modeled the formation of topological defects such as vortices and saddle points in the ensemble-averaged Poynting vector field. In general, these Poynting-vector singularities do not coincide with the coherence-vortex cores. Since the underlying Mie scattering theory is dependent only on the ratio of the incident wavelength to the radius of the sphere, the results presented in this paper can theoretically be applied to much larger spheres excited using longer-wavelength radiation, or, conversely, to smaller spheres excited by shorter wavelengths.

## Acknowledgements

We would like thank the anonymous reviewers for their insightful comments, which led to a considerably improved manuscript.

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