1. Introduction

Optical coherence is a measure of the correlations that exist between different points in optical fields. It also expresses the ability of an electromagnetic field to form interference fringes [1,2]. Several parameters have been defined that give precise mathematical measures characterizing the degree of coherence in partially coherent electromagnetic fields. The spectral degree of coherence (μ) is one such parameter that is used extensively in optical coherence theory, which quantifies the correlation between any two space-frequency points in a partially coherent wave field [3]. μ is a complex quantity whose magnitude lies between zero and unity, where the lower limit corresponds to completely incoherent fields and the upper limit corresponds to completely coherent fields [2, 3]. The phase of the spectral degree of coherence sometimes exhibits wavefront dislocations such as edge, screw or mixed type edge-screw dislocations [4, 5]. Screw type dislocations in the phase of correlation functions are also known as coherence vortices [6]. At a coherence vortex, the phase of μ exhibits a helicoidal or vortical structure about a point where its magnitude vanishes [4, 5]. When traversing a closed path around the screw type singular point associated with a coherence vortex, the net phase change is 2qπ [4,5] where the integer q is the strength or the topological charge associated with the singularity.

Field singularities have been observed and identified in various wave fields [7, 8]. In 1832, Hamilton identified a new type of singularity in optical fields known as polarization singularities. In 1833 William Whewell studied the phase singularities in tide-waves and observed that rotary systems of tidal waves sometimes have a zero tide-height point at which all cotidal lines meet, generating a tidal vortex [9]. Extending Young’s work on rainbow theory, in 1833, Airy identified and explained the rainbow as an example of an optical caustic singularity. Inspired by these findings, later in the 20th century, Nye and Berry [4] carried out pioneering research on singularities in optical fields and laid the foundation for a new field of science called singular optics. Subsequently, many others also contributed to the advancement of the singular-optics field [5, 7, 8] which opened up many promising applications in various areas, such as “optical spanners” for the rotation of microscopic particles [10] or for imprinting a vortex state into Bose-Einstein condensates [11]. Note that the caustic singularities of geometric optics are complementary to the vortex singularities of scalar wave optics, as (i) the former are infinities whereas the latter are zeros; and (ii) they are not simultaneously observable, since they arise at mutually exclusive length scales [12–14]. Regarding the caustic infinities, the “no singularity principle” [15] implies that these infinities should be treated as singularities of the theory rather than of nature, which can be rendered finite via passage to a higher theory such as classical wave optics [14, 15]. The zeros associated with vortical singularities of the wave theory also herald a breakdown (in this case of classical wave theory), since the quantum fluctuations of light will become important in the vicinity of such classical zeros [14, 16, 17]. However, this last statement does not in general apply to coherence vortices, since the zeros associated with these correlation singularities are not necessarily associated with field zeros.

Although the majority of researchers studied singularities in deterministic fields in the early days, at the start of the present century, the coherence function of partially coherent fields attracted their attention [6, 18–27]. Although we easily observe that field intensities vanish at singular points in coherent fields, spectral densities typically cannot vanish in partially coherent fields [8] because complete destructive interference is not likely to occur in these fields. However, researchers have observed singular points in correlation metrics such as the spectral degree of coherence of partially coherent fields. At these singular points, the fields at corresponding pairs of space-frequency points are completely uncorrelated [6]. An interesting feature of these correlation singularities is that the magnitude of the correlation metric vanishes at a pair of points while the phase becomes undefined. Furthermore it has been shown that the strength or the topological charge associated with the correlation singularity is a conserved quantity [18].

We previously observed that when certain partially coherent statistically stationary fields are Mie scattered from dielectric spheres, the scattered field as well as the field inside the scatterer possesses coherence vortices [26]. Furthermore, the generation and annihilation of coherence vortices in the scattered field was also demonstrated in that work (see also [24]). Later, we studied the correlation properties of partially coherent light scattered from a system of spheres and saw that a lattice of coherence vortices is generated in the scattered field [27]. The evolution of coherence vortices in the scattered field was also illustrated using a nodal-line plot.

In the majority of these studies, emphasis was given to correlation singularities arising from interfering, diffracting and scattering of plane waves. But in practise such ideal fields cannot be realized, hence optical beams (including focused fields) are of more interest. In this context, Fischer and Visser [28] investigated the coherence properties of spatially partially coherent focused light fields and demonstrated conditions under which the phase of the spectral degree of coherence exhibits singularities.

Although focused fields can be easily expressed as a superposition of plane waves via the angular-spectrum formalism, difficulties arise when studying scattering of those fields due to the intensive numerical computations required. For this reason, a more direct approach, known as the generalized Lorenz-Mie theory (GLMT), was developed by Gouesbet and collaborators [29–34] to model the scattering off spherical obstacles of focused beams, with an emphasis on Gaussian beams. This approach has been shown to lead to great computational savings compared to the plane-wave superposition approach [35, 36]. Other authors have proposed techniques that can be considered as extensions of this theory for other beam or particle shapes [37,38]. An alternative approach was proposed recently by Cui et al. [39], who presented a numerical method based on surface integral equations (SIE) to model the scattering of an arbitrarily incident focused Gaussian beam from an arbitrarily shaped homogeneous dielectric particle.

In 2008, Moore and Alonso [40] gave formulas for describing the scattering of a type of incident field referred to as complex focus (CF) field, which includes paraxial and nonparaxial versions of Gaussian beams, donut beams with radial and azimuthal polarizations, as well as any linear combination of these. This model can be regarded as a special case of the GLMT, but where the expansion coefficients are given by a simple closed-form expression, so no numerical integration is needed. In this work, we extend the work in [40] by studying the coherence properties of the scattered partially coherent CF fields. We demonstrate that Mie scattering of partially coherent CF fields can give rise to coherence vortices, and discuss the evolution of these correlation singularities when the incident field properties are varied.

This paper is organized as follows: In Section 2 we present some background theory related to our study. This includes a concise introduction to Mie scattering of CF fields, an outline of the relevant coherence theory concepts for partially coherent electromagnetic fields, and a brief summary of phase singularities in the phase of correlation functions. Then in Section 3, the mathematical model used for our numerical simulations is described in detail, together with the parameters used in our simulation based on this model. Next, in Section 4 coherence vortices arising from the Mie scattering of a focused partially coherent beam and the corresponding nodal line structures of the scattered field are presented together with a brief discussion. The polarization variation of the scattered field is also studied briefly in Section 4, followed by the conclusion in Section 5.

2. Theoretical background

Here we briefly introduce the main theoretical concepts that underpin our study. Section 2.1 summarizes theoretical details on Mie scattering of CF fields, followed by Section 2.2 which discusses the relevant fundamental concepts of coherence theory as applied to partially coherent electromagnetic fields. Finally, in Section 2.3 coherence vortices are reviewed.

2.1. Mie scattering of CF fields

Berry [41] and Sheppard and Saghafi [42] showed that a coherent spherical wave focused at a complex point generates a scalar field that tends toward a paraxial Gaussian beam, without encountering a singularity in the focal plane as was demonstrated by Cullen and Yu [43]. The complex field at any point r due to a focused spherical wave “converging” to the complex focal point at ρ₀ = r₀ + i q is given by [40]

Nonparaxial extensions of Gaussian electromagnetic fields can also be realized by displacing an electric or magnetic dipole to a complex point ρ₀ [44]. Furthermore, these fields can be expanded in terms of electromagnetic multipole fields $\left[{\mathbf{\Lambda }}_{\mathit{lm}}^{(\text{I},\text{II})}\hspace{0.17em}(\mathbf{r})\right]$ as [40]

When complex fields ${\mathbf{E}}_{(CF)}^{(\text{E})}(\mathbf{r};{\mathit{\rho }}_0,\mathbf{p})$ and ${\mathbf{E}}_{(CF)}^{(\text{B})}(\mathbf{r};{\mathit{\rho }}_0,\mathbf{p})$ are incident on a dielectric scatterer, the corresponding Mie scattered fields can be obtained respectively as

2.2. Coherence properties of partially coherent electromagnetic fields

Optical coherence theory describes the statistical properties of stationary light fields [2, 3]. Some key concepts of coherence theory used in our work are briefly described here.

Denote the ensemble of monochromatic realizations of a statistically stationary electric field by {E(r, ω)}, in the space-frequency domain (r, ω) [47]. The cross spectral density matrix is [2, 3]

Several measures of the correlation between electric fields at any two space-frequency points (r₁, ω) and (r₂, ω) in the field have been proposed. One of them is given by the complex spectral degree of coherence (μ) as [48]

Let us consider two situations that highlight the main differences between these two measures. The first corresponds to the limit as the evaluation points coincide, i.e. r₁ = r₂ = r. In this case

Alternative definitions have also been given for the degree of polarization of nonparaxial fields. Let us start reviewing the case for paraxial fields, where the degree of polarization (P) presented by Wolf in 1959 [3, 52] is given in the three equivalent forms

Extending Wolf’s treatment of polarization, Ellis et al. [53] presented a formula for the degree of polarization of any random, statistically stationary electromagnetic field beyond the paraxial approximation as

It should be noted that of all the metrics discussed so far, only μ [see Eq. (3)] is a complex quantity. Hence neither μ_ξ nor P or P_ξ admit phase dislocations such as vortices or domain walls.

2.3. Phase singularities in correlation functions and coherence vortices

Following the seminal work by Nye and Berry [4], singular optics has emerged and developed as a new field of optics. Although many studies were undertaken on singularities in deterministic optical fields in the early days, singularities in the coherence functions of partially coherent fields attracted attention of researchers later [6, 19–21, 26–28, 55, 56].

Singularities in coherent optical fields occur when the field intensities at some field points become zero, whereas in partially coherent fields such zero-intensity points are not generally realizable. However, it is possible for correlation functions such as the spectral degree of coherence to have zero magnitude at certain pairs of points. Around these points, the phase of the correlation function demonstrates vortical behavior, known as coherence vortices [6]. When traversing a smooth positively oriented simple closed path around the correlation singularity, at each point of which the magnitude of μ is strictly positive, the net phase change is given by

For the case of electromagnetic fields, the degree of coherence μ can and will often present zeros. This is easily seen from Eq. (5): μ vanishes if the real and imaginary parts of Tr[W(r₁, r₂, ω)] vanish, i.e. only two constraints are required. Therefore, if we fix, say, r₂, then μ is expected to vanish along certain curves (i.e. one-dimensional manifolds) in the three-dimensional space spanned by r₁. On the other hand, for μ_ξ to vanish, all nine complex elements of W(r₁, r₂, ω) have to vanish, as can be seen from Eq. (6). This amounts to eighteen constraints that must be satisfied simultaneously (eight in the paraxial regime) for this measure of correlation to vanish. Therefore, for a generic random field, μ_ξ is very unlikely to present true zeros for this degree of coherence within regions where the intensity is not negligible. In the unlikely case that these zeros exist or if the field is specifically designed to contain them, they are unstable to perturbations. The concept of a coherence vortex is therefore not appropriate for this measure.

3. Mathematical model

In this work we study the coherence and polarization variation of Mie-scattered complex-focused beams. Scattering of z-directed, azimuthally and radially polarized partially coherent CF fields is explored while changing the focus of the incident field. The scattering particle is placed at the origin of the Cartesian coordinate system as shown in Fig. 1. As we are extending the work in [40], we use the same values for the scattering sphere radius (kR = 5) and the refractive index of the dielectric sphere (n = 2).

 

Fig. 1 Schematic illustration of the scattering sphere centered at the origin of the coordinate system. The incident CF beam propagates in the positive z-direction.

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In order to model the partial coherence properties, the location of the focus of the incident field and its direction are varied by multiplying ρ₀ by a rotation matrix that performs rotations with respect to the origin by random angles uniformly distributed over a solid angle ΔΩ = 0.125 psr (pico steradians), analogous to the method utilized in Marasinghe et al. [26]. Note that, when the focus point ρ₀ is complex, the waist of the resulting focused beam will be centered on the real part of ρ₀. For all ensemble average calculations, 250 random realizations are considered so that all results converge acceptably.

4. Results and discussion

Here we present the results obtained by numerically simulating the Mie scattering of partially coherent CF fields from a dielectric sphere. For coherence calculations according to Eqs. (3) and (4), the reference point r₂ was arbitrarily taken to be fixed at (1, 1, $\sqrt{2}$) in the Cartesian coordinate system, corresponding to a point in the interior of the sphere. Following Ref. [40], summations in Eq. (1) and Eq. (2) were calculated using terms from l = 0 to l = 12 while m goes from −l to l. In all figures, we report distances relative to the value R/5 = k⁻¹ where R is the scatterer radius of the configuration studied. Scattered fields are observed on xy planes at different z values as illustrated in Fig. 2. Figure 3 shows the variation of the spectral degree of coherence μ of the scattered field (both magnitude and phase) in the XY plane tangential to the scattering sphere [see Fig. 3(a)] when the complex focus of the incident field is varied for radially polarized fields.

In Fig. 3(b) a coherence vortex-antivortex pair ( ${\text{A}}_{\text{P}}^{\text{r}}$ and ${\text{B}}_{\text{P}}^{\text{r}}$) can be clearly seen, when the central complex focus is ρ₀ = (2, 0, 4i). The magnitude of μ at these points goes to zero as shown in Fig. 3(c) ( ${\text{A}}_{\text{M}}^{\text{r}}$ and ${\text{B}}_{\text{M}}^{\text{r}}$), attesting the fact that at correlation singularities |μ| vanishes [4, 5]. When the focus of the CF field is changed to (5, 0, 4i), it can be clearly seen that the location of the new coherence vortex-antivortex pair [see ${\text{C}}_{\text{P}}^{\text{r}}$ and ${\text{D}}_{\text{P}}^{\text{r}}$ in Fig. 3(d)] is different from that of Fig. 3(b). As ρ₀ is changed to (8, 0, 4i), the variation of the spectral degree of coherence of the scattered field further changes, generating several coherence vortex-antivortex pairs as depicted in Fig. 3(f) and 3(g).

 

Fig. 2 Observation plane (XY) of the spectral degree of coherence.

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Fig. 3 Variation of the spectral degree of coherence (μ) of the scattered field in XY plane (z = 5), when the incident beam is radially polarized. As described in Section 2.1, the incident beam is focused at a complex point ρ₀ = r + iq, corresponding to the waist of the resulting focused beam being centered at r. (a) Observation plane XY (z = R). (b) Arg(μ) when ρ₀ = (2, 0, 4i). (c) Abs(μ) when ρ₀ = (2, 0, 4i). (d) Arg(μ) when ρ₀ = (5, 0, 4i). (e) Abs(μ) when ρ₀ = (5, 0, 4i). (f) Arg(μ) when ρ₀ = (8, 0, 4i). (g) Abs(μ) when ρ ₀ = (8, 0, 4i).

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In all phase plots [Figs. 3(b), 3(d) and 3(f)] the phase values are folded to the period between 0 and 2π, while in the magnitude plots [Figs. 3(c), 3(e) and 3(g)] |μ| varies between 0 and 1. Furthermore, in the magnitude plots we can see ‘hot-spot’ areas where |μ| attains values close to unity. At these hot-spots, the fields exhibit near-maximal correlations while fields are completely incoherent at the phase singularities. Furthermore, we can observe that coherence variation of the scattered field exhibits an approximately symmetric behavior about the y = 0 axis in all these figures, despite the fact that r₂ is not contained within the axis of symmetry.

Similarly, when the incident CF field is azimuthally polarized, we can again observe coherence vortex-antivortex pairs in the scattered field, as shown in Fig. 4. Here also, when the average location of the complex focus (ρ₀) is changed, the pattern of the coherence variation in the scattered field varies. Comparing Figs. 4(b), 4(c) and Figs. 4(d), 4(e) it is clear that when ρ ₀ = (5, 0, 4i), the coherence variation is nearly symmetric about the y = 0 axis, whereas the coherence variation is nearly symmetric about the x = 0 axis when the focus is at ρ ₀ = (0, 5, 4i).

 

Fig. 4 Variation of the spectral degree of coherence (μ) of the scattered field in XY plane (z = 5), when the azimuthally polarized beam is focused at different points ρ₀. (a) Observation plane XY (z = R). (b) Arg(μ) when ρ₀ = (5, 0, 4i). (c) Abs(μ) when ρ₀ = (5, 0, 4i). (d) Arg(μ) when ρ₀ = (0, 5, 4i). (e) Abs(μ) when ρ₀ = (0, 5, 4i). (f) Arg(μ) when ρ₀ = (6, 0, 4i). (g) Abs(μ) when ρ₀ = (6, 0, 4i). (h) Arg(μ) when ρ₀ = (8, 0, 4i). (i) Abs(μ) when ρ₀ = (8, 0, 4i).

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Furthermore, we can see in the magnitude plots [Figs. 4(c), 4(g) and 4(i)] that the field becomes more coherent along the (approximate) axis of symmetry. Here also, |μ| vanishes in the magnitude plots at the points corresponding to coherence vortex-antivortices in phase plots [see ${\text{A}}_{\text{P}}^{\text{a}}$, ${\text{B}}_{\text{P}}^{\text{a}}$, ${\text{C}}_{\text{P}}^{\text{a}}$, ${\text{D}}_{\text{P}}^{\text{a}}$, ... etc. in Figs. 4(b), 4(d), 4(f) and 4(h) and ${\text{A}}_{\text{M}}^{\text{a}}$, ${\text{B}}_{\text{M}}^{\text{a}}$, ${\text{C}}_{\text{M}}^{\text{a}}$, ${\text{D}}_{\text{M}}^{\text{a}}$, ... etc. in Figs. 4(c), 4(e), 4(g) and 4(i)].

Figure 5 presents the variation of the spectral degree of coherence, calculated using the definitions presented by Mandel and Wolf (i.e. μ) [48] and Tervo and co-workers (i.e. μ_ξ) [49–51]. As stated in Section 2.2, μ is a complex quantity which contains phase information whereas μ_ξ is a purely real quantity. Hence we can observe phase dislocations only in μ defined in Eq. (3). Accordingly, in Fig. 5(a) we can clearly see a pair of phase dislocation points where Arg(μ) exhibits vortical behavior. Coherence vortices in Fig. 5(a) correspond to zero magnitude points in Fig. 5(b). For comparison, Fig. 5(c) plots μ_ξ. It can be seen that zeros of μ [see Fig. 5(b)] do not necessarily coincide with minima of μ_ξ [see Fig. 5(c)]. These zeros in |μ| are topologically protected while zeros in μ_ξ, in the unlikely case that they existed, are not stable under small perturbations.

 

Fig. 5 Variation of spectral degree of coherence of the scattered field in XY plane (z = 5), when the azimuthally polarized beam is focused beam at (3, 0, 4i). Panels (a) and (b) respectively show the phase and the magnitude of the complex measure μ in Eq. (3) [48], with panel (c) giving the real measure μ_ξ in Eq. (4) [49–51].

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When comparing the magnitude plots of μ and μ_ξ it can be seen that the upper and lower limits of the plots are different from each other. However, the variation of the degree of coherence calculated in both methods exhibit a symmetry about the y = 0 axis. According to Mandel and Wolf’s definition [48], the scattered field is almost completely correlated along the y = 0 axis whereas the field in the same area is only partially correlated according to the definition by Tervo and co-workers. This suggests that each Cartesian component of the field along the y = 0 line is highly correlated with the corresponding field component at r₂, but that the different Cartesian components have a smaller correlation between them.

Figure 6 shows the variation of μ for the scattered field, as the field propagates towards the positive z direction. Here the phase and magnitude of μ are plotted at several XY planes at different z values. From these figures we can clearly see that the coherence vortex-antivortex pair marked as A and B are annihilated as the field propagates along the positive z direction.

 

Fig. 6 Variation of μ of propagating scattered field in different XY planes, when the azimuthally polarized incident beam is focused at (5, 0, 4i). (a) Observation plane XY (z > R). (b) Arg(μ) in z = 1.02kR plane. (c) Abs(μ) in z = 1.02kR plane. (d) Arg(μ) in z = 1.1kR plane. (e) Abs(μ) in z = 1.1kR plane. (f) Arg(μ) in z = 1.5kR plane. (g) Abs(μ) in z = 1.5kR plane.

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However the vortex-antivortex pair marked as C and D travel away from each other. When the vortex-antivortex pair [A and B in Fig. 6(b)] is present, the magnitude of μ vanishes at the corresponding pair of points in Fig. 6(c). However after they annihilate, we can see that the magnitude of μ is no longer zero at those points [see Fig. 6(g)].

The phenomenon of coherence vortex-antivortex annihilation can be more clearly seen by observing the three-dimensional nodal line structure corresponding to the evolution of these correlation singularities, as the field propagates for fixed r₂ and variable r₁. Figure 7 clearly depicts the evolution of the coherence vortices in a three-dimensional rectangular volume.

 

Fig. 7 Nodal lines of the coherence vortices in μ for the scattered field. Phase in upper and lower planes is indicated by contour lines colored according to the bar on right.

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In the base plane of Fig. 7 we can clearly see two sets of coherence vortex-antivortex pairs. As the scattered field propagates in the positive z direction, coherence vortex-antivortex pair A and B is annihilated forming a hair-pin structure. This is an example of a topological reaction of a correlation vortex, a phenomenon studied in a different context by Gu and Gbur [24]. However, coherence vortex-antivortex pair C and D moves away from each other as the field propagates. Nodal lines corresponding to this vortex-antivortex pair exhibit oscillatory behavior similar to the observations in our previous work [27].

Figure 8 shows the degree of polarization of the scattered field observed in the same plane as shown in Fig. 3(a). Here, the left panel shows the P calculated utilizing Ellis et al.’s [53] definition while the right panel shows the P_ξ calculated using Setälä and co-workers’ definition [49, 54].

 

Fig. 8 Degree of polarization of the scattered field in the XY plane (z = 5) when the azimuthally polarized incident field is focused at (5, 0, 4i). (a) P calculated using the Eq. (8) [53]. (b) P_ξ calculated using Eq. (9) [49, 54].

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It can be seen that according to both definitions, the polarization of the scattered field varies strongly as a function of position. Furthermore, the maximum and minimum values of the degree of polarization in the panels are slightly different from each other. In both panels, the variation of the polarization exhibits symmetrical behavior about the y = 0 axis. Both plots have similar areas of hot-spots where the field is highly polarized. Although Fig. 8(a) exhibits minimum polarization along the y = 0 axis, Fig. 8(b) does not exhibit similar characteristics. This is because, along this line, one of the eigenvalues of the polarization matrix is much smaller than the other two, which is the situation in which both definitions differ the most. In summary, according to both definitions the scattered field exhibits highly polarized, partially polarized and highly unpolarized states in the observation plane, although the incident field is completely polarized.

5. Conclusion

In this paper, Mie scattering of coherent CF beams was generalized to investigate the behavior of Mie scattered partially coherent CF fields. Detailed numerical calculations were performed in the space-frequency domain. It was shown that when a partially coherent CF field (both azimuthally and radially polarized) is scattered from a dielectric sphere, the scattered field consists of correlation singularities where coherence vortices are present. When the location of the complex focus is changed in the incident field, the resulting scattered field’s spectral degree of coherence also varies. The correlation singularities also vary accordingly.

Furthermore, it was observed that as the scattered field propagates, some coherence vortex-antivortex pairs are annihilated while some pairs travel apart from each other. Nodal lines corresponding to annihilating coherence vortex-antivortex pairs form a hair-pin structure. Finally we noted that, although the incident CF fields are completely polarized, the degree of polarization of the scattered field varies in a highly structured manner, as a function of position.

When different definitions were utilized to calculate the degree of coherence and the degree of polarization of the scattered field, we obtained different results although there were some similarities as well. One significant distinction is that μ presented by Mandel and Wolf [48] is a complex quantity which contains phase information whereas μ_ξ is a real quantity only [49–51], which typically does not present localized zeros. Hence phase dislocations such as coherence vortices were observed only in μ. It would be interesting to see whether μ_ξ presented by Tervo and others can also be complexified, so that it will also contain interesting phase data similar to μ.

Thus we conclude that although the incident CF field is partially coherent and fully polarized, after Mie scattered by a dielectric sphere, the coherence and polarization properties of the scattered field varies greatly between the lower and higher extremes. i.e. the scattered field at different space-frequency points exhibits highly coherent, completely incoherent and partially coherent characteristics as well as fully polarized, unpolarized and partially polarized characteristics.