## Argand-plane vorticity singularities in complex scalar optical fields: An experimental study using optical speckle |

Optics Express, Vol. 22, Issue 6, pp. 6495-6510 (2014)

http://dx.doi.org/10.1364/OE.22.006495

Acrobat PDF (4304 KB)

### Abstract

The Cornu spiral is, in essence, the image resulting from an Argand-plane map associated with monochromatic complex scalar plane waves diffracting from an infinite edge. Argand-plane maps can be useful in the analysis of more general optical fields. We experimentally study particular features of Argand-plane mappings known as “vorticity singularities” that are associated with mapping continuous single-valued complex scalar speckle fields to the Argand plane. Vorticity singularities possess a hierarchy of Argand-plane catastrophes including the fold, cusp and elliptic umbilic. We also confirm their connection to vortices in two-dimensional complex scalar waves. The study of vorticity singularities may also have implications for higher-dimensional fields such as coherence functions and multi-component fields such as vector and spinor fields.

© 2014 Optical Society of America

## 1. Introduction

1. M. Born and E. Wolf, *Principles of Optics*, 7 (Cambridge University, 1999). [CrossRef]

*𝒞*(

*s*) and

*𝒮*(

*s*) denote the Fresnel integrals,

*b*is a constant and

*s*is a variable and both depend on the position of the point of observation. Adopting the perspective of Keller’s geometrical theory of diffraction [2

2. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. **52**, 116–130 (1962). [CrossRef] [PubMed]

*et al.*[3

3. K. S. Morgan, K. K. W. Siu, and D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express **18**, 9865–9878 (2010). [CrossRef] [PubMed]

3. K. S. Morgan, K. K. W. Siu, and D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express **18**, 9865–9878 (2010). [CrossRef] [PubMed]

*et al.*also found that the magnitudes of the oscillation of the Cornu spiral and the hypocycloid decrease with propagation distance, and wrote down approximate analytical expressions for these fields using the geometric theory of diffraction.

*et al.*[4

4. F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, and D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. **285**, 4141–4151 (2012). [CrossRef]

*et al.*[4

4. F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, and D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. **285**, 4141–4151 (2012). [CrossRef]

5. K. O’Holleran, M. R. Dennis, F. Flossman, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. **100**, 053902 (2008). [CrossRef]

*et al.*[6

6. K. O’Holleran, F. Flossman, M. R. Dennis, and M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. **11**, 094020 (2009). [CrossRef]

7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. **11**, 094001 (2009). [CrossRef]

7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. **11**, 094001 (2009). [CrossRef]

## 2. Theory of Argand-plane vorticity singularities

*x*,

*y*) induces a mapping

*ℳ*: ℝ

^{2}→ ℂ from two-dimensional real space to the Argand plane, given by Here, Ψ

*and Ψ*

_{R}*denote the real and imaginary parts of Ψ(*

_{I}*x*,

*y*), which is the boundary value of the spatial part of a forward-propagating monochromatic scalar three-dimensional wave-field over a given planar surface [8

8. H. S. Green and E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A **66**, 1129–1137 (1953). [CrossRef]

*x*,

*y*) that is associated with a single Cartesian co-ordinate (

*x*,

*y*) in real space to a point on the Argand plane. The set of all such image points for a given Ψ(

*x*,

*y*) may be viewed as a two-dimensional generalization of the Cornu spiral. This generalization forms the core subject of this paper.

*x*,

*y*) is only defined up to a global phase factor exp(

*iϕ*

_{0}), where

*ψ*

_{0}is any real number. This freedom, which for time-independent field equations typically arises from the invariance of the said equations with respect to the origin of time, implies that the Argand plane image corresponding to Eq. 4 may only be meaningfully defined modulo an arbitrary rigid rotation about the Argand plane origin. Global phase factors may also be introduced into a two-dimensional complex wavefunction, by, for example. passing the wave through a thin sheet of non-absorbing glass. Such global phase factors effect a rigid Argand-plane rotation, which amounts to a re-coordinatization of the Argand plane, they do not alter the conclusions drawn below.

*ℳ*corresponding to a specified small patch of (

*x*,

*y*) space, we can perform a jet [9

9. I. Kolar, J. Slovak, and P. W. Michor, *Natural Operations in Differential Geometry* (Springer, 1993). [CrossRef]

*x*,

*y*) and producing a low-order Taylor polynomial at every point of its domain. Truncating at second order about a given fixed point (

*x*,

_{p}*y*), we have where

_{p}*A*≡

*A*(

*x*,

_{p}*y*),

_{p}*B*≡

*B*(

*x*,

_{p}*y*),...,

_{p}*F*≡

*F*(

*x*,

_{p}*y*) ∈ ℂ. The Jacobian determinant (“Jacobian”) of the mapping

_{p}*ℳ*to the Argand plane associated with Ψ(

*x*,

*y*) is given by which is quadratic in

*x*and

*y*for the Taylor expansion to second order, and, when set to zero, provides the location of singularities in the Argand plane for an arbitrary second-order wave-function. Therefore for a patch of space centered about (

*x*,

_{p}*y*) that is sufficiently small for Eq. (5) to be a good approximation, the locus of points for which

_{p}*J*= 0 corresponds to a conic section in (

*x*,

*y*) space, that is, a parabola, hyperbola, ellipse or straight line. The Argand-plane image of this locus of points will then correspond to the associated Argand-plane singularity. These conic sections will evolve into more general curves if the Taylor expansion in Eq. (5) is taken to higher than second order, although we note that the method of analysis presented here is readily generalized to such a case.

*ℳ*at a point (

*x*,

*y*) provides important information about the transformation of Ψ(

*x*,

*y*) under the mapping from an infinitesimal two-dimensional patch enclosing this point. The absolute value of the Jacobian

*J*at some point (

*x*,

_{p}*y*) gives the factor by which Ψ expands or contract infinitesimal patches at

_{p}*p*upon being mapped from real space to the Argand plane, while the sign of

*J*indicates whether the patch has been flipped (

*J*< 0) or not (

*J*> 0). A value of

*J*= 0 indicates that a patch of space in the

*xy*plane has collapsed into a single point and an Argand-plane singularity has formed for

*ℳ*(Ψ(

*x*,

*y*)).

*x*,

*y*,

*z*) can be expressed as [10

10. M. V. Berry and M. R. Dennis, “Topological events on wave dislocation lines: Birth and death of loops, and reconnection,” J. Phys. A Math. Theor. **40**, 65–74 (2007). [CrossRef]

**j**= ImΨ

^{*}∇Ψ is the current up to a multiplicative constant which is set to unity here. The vorticity gives the amount of local rotation in the field. The

*z*-component of the local vorticity, represents the local current rotation at (

*x*,

*y*) and is equivalent to the Jacobian, as seen from Eq. 6. When the local current rotation of the field changes from clockwise to anti-clockwise, that is, where Ω

*= 0, a singularity will be induced by*

_{z}*ℳ*. This fact arises from the continuity of the vorticity, i.e. if Ω

*changes sign as one traverses a given path then the vorticity must vanish for at least one point along the path. As with the Jacobian, points where the vorticity are equal to zero will map to Argand-plane singularities under*

_{z}*ℳ*.

**L**=

**r**×

**j**. The longitudinal orbital angular momentum density is given by

*x*,

*y*). Such phase vortices are characterized by the vanishing of the real and imaginary part of the wave function at the core of the vortex with a change of phase by an integer multiple of 2

*π*around it. These structures yield interesting features in the Argand plane. For detailed descriptions of optical vortices, see e.g. the seminal works of [11

11. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. **53**, 293–363 (2009). [CrossRef]

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

*x*

_{0},

*y*

_{0}) can be locally given by where an infinitesimal patch at (

*x*

_{0},

*y*

_{0}) maps under

*ℳ*to cover the Argand-plane origin, at which point Re(Ψ) = Im(Ψ) = 0. Ψ

_{+}denotes a vortex (anti-clockwise phase winding) and Ψ

_{−}an anti vortex (clockwise phase winding). In Ψ

_{+}, the infinitesimal patch in

*xy*space maps directly onto the Argand-plane origin; in Ψ

_{−}it must be “flipped” in order to cover the Argand-plane origin with the correct orientation. Thus, when there are two vortices of opposite helicity within a simply-connected region in the field of Ψ(

*x*,

*y*), the patches of space will map to the Argand plane origin with one patch being flipped relative to the other, implying the presence of an Argand singularity such as a fold that is induced under the mapping

*ℳ*of the region. The continuity of Ψ(

*x*,

*y*) yields a many-to-one mapping where the fold occurs, thus implying the presence of a singularity. This is an example wherein a function can be transformed under

*ℳ*to bring about

*Argand-plane singularities*, that, in this case, are intimately connected to the presence of vortices of opposite helicity in Ψ(

*x*,

*y*) (see [4

4. F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, and D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. **285**, 4141–4151 (2012). [CrossRef]

*ℳ*vanishes on the unit circle: To determine the Argand-plane image of the unit circle, under the Argand-plane map

*ℳ*which is induced by the wavefunction given in Eq. 11, let us parameterize the unit circle via

*θ*∈ [0, 2

*π*), as

*ψ*, current vorticity |Ω

*| and orbital angular momentum |*

_{z}*L*| associated with Eqn. 11 are shown in Fig. 3. A plot of Ψ

_{z}*against Ψ*

_{R}*as defined by Eq. 14 is shown in Fig. 3(e) and takes the form of the cross-section of an elliptic umbilic catastrophe [13]. The zeros of |Ω*

_{I}*| and |*

_{z}*L*| have no direct association with one another, as predicted by Berry [7

_{z}7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. **11**, 094001 (2009). [CrossRef]

*x*,

*y*) is defined. Assume also that the field does not vanish at any point on the boundary, which in turn implies that the phase

*ϕ*(

*x*,

*y*) = ArgΨ(

*x*,

*y*) is defined at each point of the boundary. Assume that any phase vortices within the boundary will have topological charges that have a magnitude of unity, which is typically a good assumption for generic fields on account of the instability of higher-order vortices. Knowledge of the phase over the boundary will then allow one to determine the net topological charge of the field, namely

*Q*=

*N*(+) −

*N*(−), where

*N*(+) and

*N*(−) are the number of positive and negative unit charges, respectively. Also, as pointed out in Ref. [4

**285**, 4141–4151 (2012). [CrossRef]

*N*=

*N*(+) +

*N*(−) of vortices. Thus knowledge of both

*Q*and

*N*allows

*N*(+) = (

*N*+

*Q*)/2 and

*N*(−) = (

*N*−

*Q*)/2 to be independently measured without needing to located each vortex individually or needing to determine the topological charge of each such vortex.

## 3. Experimental method

*et al.*[4

**285**, 4141–4151 (2012). [CrossRef]

*f*= 18.4 mm aspheric lens, and a 20

*μ*m pinhole located at the focus, followed by an iris downstream set to pass only the zero-order central maxima. The filtered beam was collimated by an

*f*= 100 mm plano-convex lens to give a planar, near-Gaussian beam of approximately 6 mm diameter. The collimated beam passed through a polarising beamsplitter cube to ensure that the beam used in the experiment had pure vertical polarisation. The neutral density filter attenuated the beam to eliminate saturation of the camera.

*f*= 300 mm) located one focal length away from both the iris and the camera ensured that the Fourier transform of the speckle pattern was recorded on the camera. This corresponded to the far-field (Fraunhofer) diffraction of the speckle field present at the iris.

*λ*/4 and a

*λ*/2 waveplate before being recombined with the random phase field on the camera. Each beam traversed an equal length to the camera and the optical configuration was that of a Mach-Zehnder interferometer. A neutral density filter (OD 1) attenuated the beam intensity to match the intensity of the beam containing the optical vortices. Rotating the waveplates to change the alignment of the optical axis of a waveplate from the slow axis to the fast axis with respect to the beam polarisation introduced the characteristic phase shift retardance of the wave-plate without introducing additional phase variations that would be associated with removing the waveplate. Using a combination of waveplate retardances, a 90° optical phase shift was introduced into the reference beam between each sequentially recorded interferogram. These interferograms corresponded to the reference beam being retarded by 0°, 90°, 180° and 270°. The phase

*ϕ*(

*x*,

*y*) was calculated using where

*I*is the intensity of the interferogram recorded for phase retardances

_{i}*δ*= 0,

_{i}*π*/2,

*π*, 3

*π*/2, where

*i*= 1, 2, 3, 4 [14].

## 4. Results

7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. **11**, 094001 (2009). [CrossRef]

7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. **11**, 094001 (2009). [CrossRef]

## 5. Analysis

*J*| <

*ε*= 0.5 × 10

^{−7}. As explained in Sec. 2, the regions where

*J*= 0 possess a twofold meaning: That a patch of space has induced a many-to-one mapping under

*ℳ*, forming an Argand-plane singularity and that a region of zero vorticity exists there. We previously predicted that a region of the wave field containing vortices of opposite helicity must fold at least once under

*ℳ*so that each vortex covers the Argand-plane origin, inducing a fold singularity. The red loop in Fig. 9(a) contains a vortex and an anti-vortex separated by a single “zero Jacobian line”. Therefore we expected that the patch of space will fold once, inducing a fold singularity under

*ℳ*. Indeed, in Fig. 9(c), the Argand image of the area enclosed by the red loop, confirms this. The green loop encloses two vortices of the same helicity. This would imply that the patch of space that contains these does not have to fold under

*ℳ*as both vortices must map to the Argand plane origin in the same orientation. Indeed, the corresponding Argand image, shown in Fig. 9(d) does not contain a fold. Rather the patch of space covers the origin for the first vortex and then

*loops around*so that the other vortex covers the origin. Hence we have two patches covering the origin, corresponding to two vortices, however, no singularity has been induced as the direction of the phase winding did not change over the area enclosed by the arrow loop. Finally, the blue loop in Fig. 9(a) also contains two vortices of the same helicity. Here, though, there are two Jacobian lines separating the two vortices. According to Rothschild

*et al.*[4

**285**, 4141–4151 (2012). [CrossRef]

*ℳ*. Figure 9(e) indeed contains two singularities – a fold and a cusp – corresponding to instances where the patch of space folded and then twisted around again, mapping both vortices over the Argand plane origin at the same orientation. Thus the cusp singularity is the result of a “twist” in the Argand plane origin, induced by a change in the direction of the phase winding.

*ℳ*in a complex scalar function.

*μ*m from its initial position. The zeros of the angular momentum of this data are shown in Fig. 10(b) and display no direct association with the zeros of the vorticity. This is all consistent with our simulation of the Jacobian ellipse and elliptic umbilic catastrophe in Fig. 3.

*N*(+) of vortices and

*N*(−) of anti-vortices in a simply-connected two-dimensional region, given (i) the knowledge of the phase at each point of the boundary of the region, and (ii) the associated Argand-plane map, under the assumption that (iii) all vortices have topological charges of magnitude unity. By counting the number of black-white and white-black phase wraps along the boundaries of the red loop in Fig. 9(a) one can determine that the net topological charge of the phase map within the loop is

*Q*= 0, while zooming in on the Argand-plane origin of the field enclosed in the red loop, shown in Fig. 9(c), shows that there are

*N*= 2 sheets covering the Argand plane origin. Since

*Q*=

*N*(+) −

*N*(−) = 0 and

*N*=

*N*(+) +

*N*(−) = 2, we calculate that there are

*N*(+) = (

*N*+

*Q*)/2 = 1 vortex and

*N*(−) = (

*N*−

*Q*)/2 = 1 anti-vortex, which is consistent with the number of black dots (vortices) and the number of white dots (anti-vortices) enclosed in the red loop in Fig. 9(a). The same method can be shown to derive the number of vortices in the blue and green loops in Fig. 9(a). This example shows that the number of vortices, together with the number of anti-vortices, may be determined without needing to locate each individual vortex and anti-vortex.

## 6. Discussion

*J*= 0, as done for the elliptic umbilic in Sec. 2. Other vortical fields such as the focal volumes of aberrated lenses [15

15. L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy **88**, 85–97 (2001). [CrossRef] [PubMed]

17. M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. **49**, 4335–4348 (2004). [CrossRef] [PubMed]

18. E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, and V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A **79**, 043618 (2009). [CrossRef]

20. C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, and W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. **87**, 210402 (2001). [CrossRef]

*ℳ*could help clarify the connection between phase vortices and vorticity singularities. Berry [7

7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. **11**, 094001 (2009). [CrossRef]

21. I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. **159**, 99–117 (1999). [CrossRef]

*ℳ*as an interesting insight into the so-called “vortex-vorticity” duality. Studying the “unfolding” of vorticity singularities with certain parameter changes such as propagation distance or aperture size may also provides insights into the meaning of vorticity singularities.

*s*, will be described by 2

*s*+ 1 complex wave-functions. The mapping of spinor fields to the Bloch, or Poincaré, sphere may possess vorticity singularities. As a generalisation of the two-dimensional complex scalar function, we could look at coherence functions – which are two-point correlation functions and can exist in seven dimensions. In this case, “coherence vortices” become manifest (corresponsing to a pair of points in three–dimensional space, together with a time lag or angular frequency [22

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

24. P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. **42**, 99–104 (2010). [CrossRef]

25. W. Wang and M. Takeda, “Coherence current, coherence vortex and the conservation law of coherence,” Phys. Rev. Lett. **69**, 223904 (2006). [CrossRef]

## 7. Conclusion

7. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. **11**, 094001 (2009). [CrossRef]

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. |

3. | K. S. Morgan, K. K. W. Siu, and D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express |

4. | F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, and D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. |

5. | K. O’Holleran, M. R. Dennis, F. Flossman, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. |

6. | K. O’Holleran, F. Flossman, M. R. Dennis, and M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. |

7. | M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. |

8. | H. S. Green and E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A |

9. | I. Kolar, J. Slovak, and P. W. Michor, |

10. | M. V. Berry and M. R. Dennis, “Topological events on wave dislocation lines: Birth and death of loops, and reconnection,” J. Phys. A Math. Theor. |

11. | M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. |

12. | H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence function in Young’s interference pattern,” Opt. Lett. |

13. | J. F. Nye, |

14. | H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in |

15. | L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. M. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy |

16. | L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. M. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E |

17. | M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung issue,” Phys. Med. Biol. |

18. | E. A. L. Henn, J. A. Seman, E. R. F. Ramos, M. Caracanhas, P. Castilho, E. P. Olimpio, G. Roati, D. V. Magalhaes, K. M. F. Magalhaes, and V. S. Bagnato, “Observation of vortex formation in an oscillation trapped Bose–Einstein condensate,” Phys. Rev. A |

19. | M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. |

20. | C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, and W. Ketterle, “Vortex nucleation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. |

21. | I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. |

22. | G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. |

23. | 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. |

24. | P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. |

25. | W. Wang and M. Takeda, “Coherence current, coherence vortex and the conservation law of coherence,” Phys. Rev. Lett. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(350.6980) Other areas of optics : Transforms

(110.3175) Imaging systems : Interferometric imaging

(110.4153) Imaging systems : Motion estimation and optical flow

(050.4865) Diffraction and gratings : Optical vortices

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 3, 2014

Revised Manuscript: March 2, 2014

Manuscript Accepted: March 3, 2014

Published: March 12, 2014

**Citation**

Freda Rothschild, Alexis I. Bishop, Marcus J. Kitchen, and David M. Paganin, "Argand-plane vorticity singularities in complex scalar optical fields: An experimental study using optical speckle," Opt. Express **22**, 6495-6510 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6495

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### References

- M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University, 1999). [CrossRef]
- J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962). [CrossRef] [PubMed]
- K. S. Morgan, K. K. W. Siu, D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express 18, 9865–9878 (2010). [CrossRef] [PubMed]
- F. Rothschild, M. J. Kitchen, H. M. L. Faulkner, D. M. Paganin, “Duality between phase vortices and Argand-plane caustics,” Opt. Commun. 285, 4141–4151 (2012). [CrossRef]
- K. O’Holleran, M. R. Dennis, F. Flossman, M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008). [CrossRef]
- K. O’Holleran, F. Flossman, M. R. Dennis, M. J. Padgett, “Methodology for imaging the 3D structure of singularities in scalar and vector optical fields,” J. Opt. A Pure Appl. Opt. 11, 094020 (2009). [CrossRef]
- M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009). [CrossRef]
- H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953). [CrossRef]
- I. Kolar, J. Slovak, P. W. Michor, Natural Operations in Differential Geometry (Springer, 1993). [CrossRef]
- M. V. Berry, M. R. Dennis, “Topological events on wave dislocation lines: Birth and death of loops, and reconnection,” J. Phys. A Math. Theor. 40, 65–74 (2007). [CrossRef]
- M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009). [CrossRef]
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