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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2858–2875
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Coherence vortices in Mie scattered nonparaxial partially coherent beams

Madara L. Marasinghe, Malin Premaratne, David M. Paganin, and Miguel A. Alonso  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 2858-2875 (2012)
http://dx.doi.org/10.1364/OE.20.002858


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Abstract

We have previously demonstrated that Mie scattering of partially coherent plane waves can create coherence vortices, namely screw-type dislocations in the phase of the spectral degree of coherence. However, plane waves are an idealization and in practice, optical beams are often much closer to reality. Thus, in this paper, we consider coherence vortices created by Mie scattering of partially coherent focused beams. We demonstrate that Mie scattering of partially coherent complex focused beams can give rise to coherence vortices. As the scattered fields propagate coherence vortex-antivortex pairs are annihilated thus creating hair-pin structures in the coherence-vortex nodal lines. The evolution of correlation singularities in the scattered field with the variation of the complex focus point of the incident beam is also discussed. The variation of the degree of polarization of the scattered field is also studied.

© 2012 OSA

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

1. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938). [CrossRef]

,2

2. M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge University Press, Cambridge, 1999).

]. 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

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

]. 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

2. M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge University Press, Cambridge, 1999).

, 3

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

]. The phase of the spectral degree of coherence sometimes exhibits wavefront dislocations such as edge, screw or mixed type edge-screw dislocations [4

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

, 5

5. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]

]. Screw type dislocations in the phase of correlation functions are also known as coherence vortices [6

6. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003). [CrossRef]

]. At a coherence vortex, the phase of μ exhibits a helicoidal or vortical structure about a point where its magnitude vanishes [4

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

, 5

5. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]

]. When traversing a closed path around the screw type singular point associated with a coherence vortex, the net phase change is 2qπ [4

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

,5

5. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]

] 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

7. M. Berry, “Making waves in physics,” Nature (London) 403, 21 (2000). [CrossRef]

, 8

8. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

]. 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

9. W. Whewell, “Essay towards a first approximation to a map of cotidal lines,” Phil. Trans. R. Soc. Lond. 123, 147–236 (1833). [CrossRef]

]. 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

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

] 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

5. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]

, 7

7. M. Berry, “Making waves in physics,” Nature (London) 403, 21 (2000). [CrossRef]

, 8

8. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

] which opened up many promising applications in various areas, such as “optical spanners” for the rotation of microscopic particles [10

10. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature (London) 394, 348–350 (1998). [CrossRef]

] or for imprinting a vortex state into Bose-Einstein condensates [11

11. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006). [CrossRef] [PubMed]

]. 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

12. M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980). [CrossRef]

14

14. M. V. Berry, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices...,” Proc SPIE 3487, 1–5 (1998). [CrossRef]

]. Regarding the caustic infinities, the “no singularity principle” [15

15. G. Gouesbet, “Hypotheses on the a priori rational necessity of quantum mechanics,” Principia, an international journal of epistemology 14, 393–404 (2010).

] 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

14. M. V. Berry, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices...,” Proc SPIE 3487, 1–5 (1998). [CrossRef]

, 15

15. G. Gouesbet, “Hypotheses on the a priori rational necessity of quantum mechanics,” Principia, an international journal of epistemology 14, 393–404 (2010).

]. 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

14. M. V. Berry, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices...,” Proc SPIE 3487, 1–5 (1998). [CrossRef]

, 16

16. M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A: Pure Appl. Opt. 6, S178–S180 (2004). [CrossRef]

, 17

17. S. M. Barnett, “On the quantum core of an optical vortex,” J. Mod. Opt. 55, 2279–2292 (2008). [CrossRef]

]. 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

6. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003). [CrossRef]

, 18

18. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003). [CrossRef] [PubMed]

27

27. M. L. Marasinghe, D. M. Paganin, and M. Premaratne, “Coherence-vortex lattice formed via Mie scattering of partially coherent light by several dielectric nanospheres,” Opt. Lett. 36, 936–938 (2011). [CrossRef] [PubMed]

]. Although we easily observe that field intensities vanish at singular points in coherent fields, spectral densities typically cannot vanish in partially coherent fields [8

8. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

] 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

6. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003). [CrossRef]

]. 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

18. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003). [CrossRef] [PubMed]

].

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

26. M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010). [CrossRef] [PubMed]

]. Furthermore, the generation and annihilation of coherence vortices in the scattered field was also demonstrated in that work (see also [24

24. Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009). [CrossRef]

]). 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

27. M. L. Marasinghe, D. M. Paganin, and M. Premaratne, “Coherence-vortex lattice formed via Mie scattering of partially coherent light by several dielectric nanospheres,” Opt. Lett. 36, 936–938 (2011). [CrossRef] [PubMed]

]. 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

28. D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004). [CrossRef]

] 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

29. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982). [CrossRef]

34

34. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories, (Springer, Berlin, 2011). [CrossRef]

] 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

35. J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–571 (1995). [CrossRef] [PubMed]

, 36

36. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories, (Springer, Berlin, 2011) 48–50. [CrossRef]

]. Other authors have proposed techniques that can be considered as extensions of this theory for other beam or particle shapes [37

37. R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case I: symmetrically polarized beams,” J. Mod. Opt. 52, 2067–2092 (2005). [CrossRef]

,38

38. A. S. van de Nes and P. Torok, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13360–13374 (2007). [CrossRef] [PubMed]

]. An alternative approach was proposed recently by Cui et al. [39

39. Z. Cui, Y. Han, and H. Zhang, “Scattering of an arbitrarily incident focused Gaussian beam by arbitrarily shaped dielectric particles,” J. Opt. Soc. Am. B 28, 2625–2632 (2011). [CrossRef]

], 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

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

] 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

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

] 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.

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

41. M.V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A: Math. Gen. 27, L391–L398 (1994). [CrossRef]

] and Sheppard and Saghafi [42

42. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). [CrossRef]

] 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

43. F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. Lond. A 366, 155–171 (1979). [CrossRef]

]. 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

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

]
UCF(r;ρ0)=4πU0sin{k[(rρ0)(rρ0)]1/2}k[(rρ0)(rρ0)]1/2,
where i is the imaginary unit, U0 is a constant and k is the free space propagation constant. For k|q| ≫ 1 these complex focus (CF) fields approach paraxial Gaussian beams that propagate in the q direction with focus at r0. Based on this observation, for k|q| ≳ 3, these CF fields can be regarded as nonparaxial extensions of Gaussian beams.

Nonparaxial extensions of Gaussian electromagnetic fields can also be realized by displacing an electric or magnetic dipole to a complex point ρ0 [44

44. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999). [CrossRef]

]. Furthermore, these fields can be expanded in terms of electromagnetic multipole fields [Λlm(I,II)(r)] as [40

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

]
E(CF)(E)(r;ρ0,p)=U0l=1m=ll[γlm(I)(ρ0,p)Λlm(I)(r)+γlm(II)(ρ0,p)Λlm(II)(r)],
(1a)
E(CF)(B)(r;ρ0,p)=U0l=1m=ll[γlm(II)(ρ0,p)Λlm(I)(r)γlm(I)(ρ0,p)Λlm(II)(r)],
(1b)
where p is the unit vector in the direction of the electric or magnetic dipole moment and E(CF)(E)(r;ρ0,p), E(CF)(B)(r;ρ0,p) are the CF fields generated by displacing an electric and magnetic dipole to a complex point ρ0, respectively. The beam shape coefficients γlm(I,II)(ρ0,p) are defined as [40

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

]
γlm(I,II)(ρ0,p)=p[Λlm(I,II)(r)]*.
Sheppard and Saghafi [44

44. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999). [CrossRef]

,45

45. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999). [CrossRef]

] further showed that differently polarized fields can be obtained by using various combinations of E(CF)(E)(r;ρ0,p) and E(CF)(B)(r;ρ0,p). By aligning the electric and magnetic dipole moments (p) along the z-direction and focusing the complex field at ρ0 = iq, radially and azimuthally polarized fields can be realized respectively as E(CF)(E)(r;iqz^,z^) and E(CF)(B)(r;iqz^,z^), where is the unit vector along the z-direction.

When complex fields E(CF)(E)(r;ρ0,p) and E(CF)(B)(r;ρ0,p) are incident on a dielectric scatterer, the corresponding Mie scattered fields can be obtained respectively as
E(sc)(E)(r;ρ0,p)=U0l=1m=ll[alγlm(I)(ρ0,p)Πlm(I)(r)+blγlm(II)(ρ0,p)Πlm(II)(r)],
(2a)
E(sc)(B)(r;ρ0,p)=U0l=1m=ll[alγlm(II)(ρ0,p)Πlm(I)(r)+blγlm(I)(ρ0,p)Πlm(II)(r)],
(2b)
where Eq. (17) of [40

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

] explicitly gives the scattering coefficients al and bl, which can be derived by applying standard boundary conditions at the scattering dielectric interface [46

46. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York, 1969).

].

2.2. Coherence properties of partially coherent electromagnetic fields

Optical coherence theory describes the statistical properties of stationary light fields [2

2. M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge University Press, Cambridge, 1999).

, 3

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

]. 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

47. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982). [CrossRef]

]. The cross spectral density matrix is [2

2. M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge University Press, Cambridge, 1999).

, 3

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

]
Wij(r1,r2,ω)=Ei*(r1,ω)Ej(r2,ω),
where i, j = x, y, z. The asterisk denotes complex conjugation and angular brackets denote averaging over the ensemble of monochromatic realizations. If these stationary fields are ergodic, then the ensemble averages will be equal to the corresponding time averages.

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
μ(r,r,ω)=1,μξ(r,r,ω)=Tr[J2(r,ω)]S(r,ω)1,
(7)
where J(r, ω) = W(r, r, ω) is the polarization matrix. Note that Tr[J(r, ω)] = S(r, ω). Equation (7) states that μξ does not necessarily go to unity as the two points of observation approach each other. This is because, even at the same point, different Cartesian components might not be fully correlated. The second situation corresponds to the case of fields that are statistically fully correlated, i.e. for which the cross-spectral density has only one coherent mode. In this case,
Wij(r1,r2,ω)=Ei*(r1,ω)Ej(r2,ω)
, and the two measures give
μ(r1,r2,ω)=E*(r1,ω)E(r2,ω)|E(r1,ω)||E(r2,ω)|1,μξ(r1,r2,ω)=1.
That is, for μ to be unity, the field at the two observation points must not only be statistically fully correlated, but it must also have the same polarization. In general, μ = 1 states that each Cartesian component of the field at r1 is fully correlated with the corresponding component at r2 and that the state of polarization at the two points is similar, while μ = 0 can result either from statistical independence between similar components [so that every term in the sum within the right-hand side of Eq. (5) vanishes independently] or from orthogonality of the polarizations at the two points (so that the zero results from the cancellation between these terms). On the other hand, μξ = 1 states that each component of the field at r1 is fully correlated statistically with each component at r2, even if the state of polarization at the two points is different, while μξ = 0 states that there is no statistical correlation between any component at one point and any component at the other point.

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

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

, 52

52. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento 13, 1165–1181 (1959). [CrossRef]

] is given in the three equivalent forms
P=14DetJ[Tr(J)]2=2[Tr(J2)[Tr(J)]212]=λ1λ2λ1+λ2,
where J is the 2 × 2 submatrix of J containing the x and y components, and Det denotes the determinant. In the second form, λ1 and λ2 are the two eigenvalues of J, with λ1λ2 ≥ 0. The degree of polarization is a real scalar quantity, constrained to lie between zero and unity, with lower and upper limits representing fully unpolarized and fully polarized fields, respectively [3

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

]. Being a real quantity, the degree of polarization does not exhibit phase dislocations such as vortices or domain walls.

Extending Wolf’s treatment of polarization, Ellis et al. [53

53. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005). [CrossRef]

] presented a formula for the degree of polarization of any random, statistically stationary electromagnetic field beyond the paraxial approximation as
P=λ1λ2λ1+λ2+λ3,
(8)
where λ1λ2λ3 ≥ 0 are the eigenvalues of J. On the other hand, Setälä et al. proposed an alternative degree of polarization (Pξ) [54

54. T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002). [CrossRef] [PubMed]

] as
Pξ=32[Tr(J2)[Tr(J)]213]=λ12+λ22+λ32λ1λ2λ1λ3λ2λ3λ1+λ2+λ3.
(9)
The two degrees of polarization P and Pξ have similar behavior when λ3λ2. This includes significantly polarized fields. i.e. when one of the eigenvalues is much larger than the other two. Note that Eq. (7) can then be rewritten as
μξ(r,r,ω)=2Pξ2+13.
That is, μξ at the same point takes its minimum value of 1/3 when the field is fully unpolarized, and is unity only for a fully polarized field.

It should be noted that of all the metrics discussed so far, only μ [see Eq. (3)] is a complex quantity. Hence neither μξ nor P or Pξ 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

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

], 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

6. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003). [CrossRef]

, 19

19. I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004). [CrossRef]

21

21. G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6, S239–S242 (2004). [CrossRef]

, 26

26. M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010). [CrossRef] [PubMed]

28

28. D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004). [CrossRef]

, 55

55. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: birth and evolution of phase singularities in the spatial coherence function,” in Fringe 2005, W. Osten, ed. (Springer BerlinHeidelberg, 2006), 46–53. [CrossRef]

, 56

56. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006). [CrossRef] [PubMed]

].

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

6. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003). [CrossRef]

]. 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
2πq=Γr1arg[μ(r1,r2;ω)]dt1,
where the integer q is the topological charge associated with the correlation singularity [4

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

, 25

25. I. Maleev, “Partial coherence and optical vortices,” Ph.D. dissertation, Worcester Polytechnic Institute, 2004.

, 27

27. M. L. Marasinghe, D. M. Paganin, and M. Premaratne, “Coherence-vortex lattice formed via Mie scattering of partially coherent light by several dielectric nanospheres,” Opt. Lett. 36, 936–938 (2011). [CrossRef] [PubMed]

], r2 = r2 and ω = ω′ are considered fixed, ∇r1 is the gradient operator with respect to r1 and t1 is the unit tangent vector at each point r1 on Γ.

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 μξ 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, μξ 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

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

], 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.

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

26. M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010). [CrossRef] [PubMed]

]. 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, 2) in the Cartesian coordinate system, corresponding to a point in the interior of the sphere. Following Ref. [40

40. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008). [CrossRef] [PubMed]

], 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 ( APr and BPr) 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) ( AMr and BMr), attesting the fact that at correlation singularities |μ| vanishes [4

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

, 5

5. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]

]. 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 CPr and DPr 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.
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).

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).

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 APa, BPa, CPa, DPa, ... etc. in Figs. 4(b), 4(d), 4(f) and 4(h) and AMa, BMa, CMa, DMa, ... 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

48. L. Mandel and E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976). [CrossRef]

] and Tervo and co-workers (i.e. μξ) [49

49. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003). [CrossRef] [PubMed]

51

51. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004). [CrossRef]

]. As stated in Section 2.2, μ is a complex quantity which contains phase information whereas μξ 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 μξ. It can be seen that zeros of μ [see Fig. 5(b)] do not necessarily coincide with minima of μξ [see Fig. 5(c)]. These zeros in |μ| are topologically protected while zeros in μξ, 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 μξ in Eq. (4) [4951].

When comparing the magnitude plots of μ and μξ 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

48. L. Mandel and E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976). [CrossRef]

], 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.

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.

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

53. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005). [CrossRef]

] definition while the right panel shows the Pξ calculated using Setälä and co-workers’ definition [49

49. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003). [CrossRef] [PubMed]

, 54

54. T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002). [CrossRef] [PubMed]

].

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) Pξ calculated using Eq. (9) [49, 54].

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

48. L. Mandel and E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976). [CrossRef]

] is a complex quantity which contains phase information whereas μξ is a real quantity only [49

49. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003). [CrossRef] [PubMed]

51

51. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004). [CrossRef]

], 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 μξ 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.

Acknowledgments

Madara L. Marasinghe acknowledges support from both the Monash Research Graduate School and the Faculty of Engineering, Monash University.

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27.

M. L. Marasinghe, D. M. Paganin, and M. Premaratne, “Coherence-vortex lattice formed via Mie scattering of partially coherent light by several dielectric nanospheres,” Opt. Lett. 36, 936–938 (2011). [CrossRef] [PubMed]

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J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005). [CrossRef]

54.

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002). [CrossRef] [PubMed]

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OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(290.4020) Scattering : Mie theory
(290.5850) Scattering : Scattering, particles
(050.4865) Diffraction and gratings : Optical vortices
(290.5855) Scattering : Scattering, polarization
(260.6042) Physical optics : Singular optics

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: December 14, 2011
Revised Manuscript: January 9, 2012
Manuscript Accepted: January 11, 2012
Published: January 23, 2012

Citation
Madara L. Marasinghe, Malin Premaratne, David M. Paganin, and Miguel A. Alonso, "Coherence vortices in Mie scattered nonparaxial partially coherent beams," Opt. Express 20, 2858-2875 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2858


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References

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