## Coherence vortices in Mie scattered nonparaxial partially coherent beams |

Optics Express, Vol. 20, Issue 3, pp. 2858-2875 (2012)

http://dx.doi.org/10.1364/OE.20.002858

Acrobat PDF (2571 KB)

### 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

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

*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

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

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

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

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

*q*

*π*[4

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

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

*q*is the strength or the topological charge associated with the singularity.

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

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

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

**336**, 165–190 (1974). [CrossRef]

**119**, 604–612 (1995). [CrossRef]

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

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

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]

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]

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

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

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

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

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

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]

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

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]

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]

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]

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]

## 2. Theoretical background

### 2.1. Mie scattering of CF fields

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

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]

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]

**r**due to a focused spherical wave “converging” to the complex focal point at

**ρ**

_{0}=

**r**

_{0}+

*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]

*i*is the imaginary unit,

*U*

_{0}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

**r**

_{0}. Based on this observation, for

*k*|

**q**| ≳ 3, these CF fields can be regarded as nonparaxial extensions of Gaussian beams.

**ρ**

_{0}[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]

**16**, 5926–5933 (2008). [CrossRef] [PubMed]

**p**is the unit vector in the direction of the electric or magnetic dipole moment and

**ρ**

_{0}, respectively. The beam shape coefficients

**16**, 5926–5933 (2008). [CrossRef] [PubMed]

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. 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]

**p**) along the

*z*-direction and focusing the complex field at

**ρ**

_{0}=

*iq*

**ẑ**, radially and azimuthally polarized fields can be realized respectively as

**ẑ**is the unit vector along the

*z*-direction.

**16**, 5926–5933 (2008). [CrossRef] [PubMed]

*a*

*and*

_{l}*b*

*, which can be derived by applying standard boundary conditions at the scattering dielectric interface [46].*

_{l}### 2.2. Coherence properties of partially coherent electromagnetic fields

**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]

*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.

*P*) presented by Wolf in 1959 [3, 52

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

**J**

_{⊥}is the 2 × 2 submatrix of

**J**containing the

*x*and

*y*components, and Det denotes the determinant. In the second form,

*λ*

_{1}and

*λ*

_{2}are the two eigenvalues of

**J**

_{⊥}, with

*λ*

_{1}≥

*λ*

_{2}≥ 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]. Being a real quantity, the degree of polarization does not exhibit phase dislocations such as vortices or domain walls.

*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]

*λ*

_{1}≥

*λ*

_{2}≥

*λ*

_{3}≥ 0 are the eigenvalues of

**J**. On the other hand, Setälä et al. proposed an alternative degree of polarization (

*P*

*) [54*

_{ξ}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]

*P*and

*P*

*have similar behavior when*

_{ξ}*λ*

_{3}≈

*λ*

_{2}. 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 That is,

*μ*

*at the same point takes its minimum value of*

_{ξ}*μ*[see Eq. (3)] is a complex quantity. Hence neither

*μ*

*nor*

_{ξ}*P*or

*P*

*admit phase dislocations such as vortices or domain walls.*

_{ξ}### 2.3. Phase singularities in correlation functions and coherence vortices

**336**, 165–190 (1974). [CrossRef]

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

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. 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. 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. 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. 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. 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]

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

*μ*is strictly positive, the net phase change is given by where the integer

*q*is the topological charge associated with the correlation singularity [4

**336**, 165–190 (1974). [CrossRef]

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]

**r**

_{2}=

**r**′

_{2}and

*ω*=

*ω*′ are considered fixed, ∇

_{r}_{1}is the gradient operator with respect to

**r**

_{1}and

**t**

_{1}is the unit tangent vector at each point

**r**

_{1}on Γ.

*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**(

**r**

_{1},

**r**

_{2},

*ω*)] vanish,

*i.e.*only two constraints are required. Therefore, if we fix, say,

**r**

_{2}, then

*μ*is expected to vanish along certain curves (

*i.e.*one-dimensional manifolds) in the three-dimensional space spanned by

**r**

_{1}. On the other hand, for

*μ*

*to vanish, all nine complex elements of*

_{ξ}**W**(

**r**

_{1},

**r**

_{2},

*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*

*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.*

_{ξ}## 3. Mathematical model

*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

**16**, 5926–5933 (2008). [CrossRef] [PubMed]

*kR*= 5) and the refractive index of the dielectric sphere (

*n*= 2).

**ρ**

_{0}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]

**ρ**

_{0}is complex, the waist of the resulting focused beam will be centered on the real part of

**ρ**

_{0}. For all ensemble average calculations, 250 random realizations are considered so that all results converge acceptably.

## 4. Results and discussion

**r**

_{2}was arbitrarily taken to be fixed at (1, 1,

**16**, 5926–5933 (2008). [CrossRef] [PubMed]

*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.

**ρ**

_{0}= (2, 0, 4

*i*). The magnitude of

*μ*at these points goes to zero as shown in Fig. 3(c) (

*μ*| vanishes [4

**336**, 165–190 (1974). [CrossRef]

**119**, 604–612 (1995). [CrossRef]

*i*), it can be clearly seen that the location of the new coherence vortex-antivortex pair [see

**ρ**

_{0}is changed to (8, 0, 4

*i*), 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).

*π*, 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

**r**

_{2}is not contained within the axis of symmetry.

**ρ**

_{0}) 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, 4

*i*), 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, 4

*i*).

*μ*| vanishes in the magnitude plots at the points corresponding to coherence vortex-antivortices in phase plots [see

*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]

*i.e.*

*μ*

*) [49*

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

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]

*μ*is a complex quantity which contains phase information whereas

*μ*

*is a purely real quantity. Hence we can observe phase dislocations only in*

_{ξ}*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*

*. It can be seen that zeros of*

_{ξ}*μ*[see Fig. 5(b)] do not necessarily coincide with minima of

*μ*

*[see Fig. 5(c)]. These zeros in |*

_{ξ}*μ*| are topologically protected while zeros in

*μ*

*, in the unlikely case that they existed, are not stable under small perturbations.*

_{ξ}*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.

*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)].

**r**

_{2}and variable

**r**

_{1}. Figure 7 clearly depicts the evolution of the coherence vortices in a three-dimensional rectangular volume.

*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

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

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]

*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]

*P*

*calculated using Setälä and co-workers’ definition [49*

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

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]

*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

*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]

*μ*

*is a real quantity only [49*

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

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]

*μ*. It would be interesting to see whether

*μ*

*presented by Tervo and others can also be complexified, so that it will also contain interesting phase data similar to*

_{ξ}*μ*.

*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

## References and links

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30. | G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) |

31. | G. Gouesbet, G. Gréhan, and B. Maheu, “Computations of the g |

32. | G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A |

33. | G. Gouesbet, “T-matrix formulation and generalized Lorenz-Mie theories in spherical coordinates,” Opt. Commun. |

34. | G. Gouesbet and G. Gréhan, |

35. | J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. |

36. | G. Gouesbet and G. Gréhan, |

37. | R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case I: symmetrically polarized beams,” J. Mod. Opt. |

38. | A. S. van de Nes and P. Torok, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express |

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 |

40. | N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express |

41. | M.V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A: Math. Gen. |

42. | C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A |

43. | F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. Lond. A |

44. | C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A |

45. | C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. |

46. | M. Kerker, |

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. |

48. | L. Mandel and E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. |

49. | J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express |

50. | T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. |

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 |

52. | E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento |

53. | J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. |

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. |

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 |

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. |

**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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