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
The Cornu spiral is, in essence, the mapping to the Argand plane of the optical field over a plane perpendicular to the optical axis, which results when monochromatic complex scalar plane waves are normally incident upon an infinite edge. This is illustrated in Fig. 1(a), which plots the Argand-plane image of the complex optical field1, sec. 8.7.3] 2], (i) the bottom-left lobe of the spiral in Fig. 1(a) corresponds to the phasor associated with cylindrical waves scattered from the diffracting edge into the region of geometrical shadow; while (ii) the top right lobe results from the coherent superposition of the incident plane wave phasor with the scattered cylindrical edge wave.
The Cornu spiral can be generalized. Morgan et al.  found that in the geometrical shadow of an infinitely long uniform dielectric cylinder normally illuminated by monochromatic scalar plane waves, the Cornu spiral turns into a hypocycloid (see Fig. 1(b) ) as a result of the distorted plane wave (having travelled through the cylinder) superposing with the wave diffracting from the edge of the cylinder. Morgan 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.
In Rothschild et al. , we considered the role of the Argand plane as a means to add insight into vortex behaviour in a complex scalar two-dimensional wavefield. We simulated a vortex-laden speckle field by spatially filtering two-dimensional complex white noise. Note that, in passing from complex fields that vary in only one of the two transverse dimensions to those that vary in both transverse dimensions, the associated Argand-plane map evolves from a line trace (generalized Cornu spiral) into a fully two-dimensional mapping. The appearance of this mapping is typically reminiscent of the caustic-network pattern seen at the bottom of a swimming pool on a bright day. In particular, for such maps we looked at the connection between Argand-plane singularities and quantized phase vortices in the physical field. We described the role of vortices and vorticity in the appearance of singularities of the mapping ℳ of the complex wavefunction to the Argand plane – namely, the cusp and the fold vorticity singularities. Note that we previously referred to these singularities as “Argand-plane caustics” . However, following advice from Sir M.V. Berry in a private communication to the corresponding author, we herein use the term “vorticity singularities” in recognition of the fact that Argand-plane singularities are not associated with gradient maps and do not correspond to caustics, or a focussing of rays in real space.
In the present paper we experimentally verify the numerical predictions made in Rothschild et al.  by calculating Argand mappings of coherent two-dimensional complex scalar wave-functions associated with optical speckle fields. We generate such a speckle field by experimentally recreating the process used for simulations in Rothschild et al. . Additionally, we use the Taylor series expansion of the complex wavefunction to present a local means to locate Argand-plane singularities of any order. Vortices in speckle fields have been studied previously (see e.g. ) and similar experimental and data smoothing techniques to those adopted here have been used by O’Holleran et al. . However, the present work considers the detailed structure of vortices and relates it to the seminal work on “Optical currents” by Berry . We also provide an experimental verification of aspects of the work laid down by Berry .
We close this introduction with an outline of the remainder of the paper. Section 2 details the theory of Argand-plane vorticity singularities. Section 3 provides a description of the experimental set-up and execution whereby an optical phase-stepping interferometer is used to measure Argand-plane maps of coherent vortical speckle. Section 4 reports on the results of the experiment and the methods used to analyse the data. Using the results, we confirm our predictions regarding Argand-plane singularities, shown in Sec. 5. We provide a discussion in Sec. 6, and conclude with Sec. 7.
2. Theory of Argand-plane vorticity singularities
An arbitrary two-dimensional differentiable continuous single-valued complex function Ψ(x, y) induces a mapping ℳ : ℝ2 → ℂ from two-dimensional real space to the Argand plane, given by8]. In this paper, coherent scalar fields are assumed and harmonic time dependence is suppressed.
Figure 2 shows the Argand mapping of a complex number Ψ(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.
Before proceeding, we recall that Ψ(x, y) is only defined up to a global phase factor exp(iϕ0), where ψ0 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.
To locate the singularities induced by the mapping ℳ corresponding to a specified small patch of (x, y) space, we can perform a jet  operation, taking the complex wavefunction Ψ(x, y) and producing a low-order Taylor polynomial at every point of its domain. Truncating at second order about a given fixed point (xp, yp), we haveEq. (5) to be a good approximation, the locus of points for which 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.
The Jacobian determinant of 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 (xp, yp) gives the factor by which Ψ expands or contract infinitesimal patches at 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)).
We can assign a physical meaning to the Jacobian. The vorticity Ω of a three-dimensional complex scalar field Ψ(x, y, z) can be expressed as Eq. 6. When the local current rotation of the field changes from clockwise to anti-clockwise, that is, where Ωz = 0, a singularity will be induced by ℳ. This fact arises from the continuity of the vorticity, i.e. if Ωz 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 ℳ.
The vorticity Ω should not be confused with the local orbital angular momentum density L = r×j. The longitudinal orbital angular momentum density is given by
Note the distinction between the local current rotation referred to above, and the quantized phase vortices associated with screw-type singularities which are a regular feature of complex scalar functions such as Ψ(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 2π 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, 12]. Modulo a continuous deformation, a phase vortex at (x0, y0) can be locally given by4] for more detail).
Returning to the main thread of the argument, we can exemplify Eqn. 6 by taking the special case of a second-order complex polynomial, given byEq. 11, let us parameterize the unit circle via θ ∈ [0, 2π), as Eqn. 11 are shown in Fig. 3. A plot of ΨR against ΨI as defined by Eq. 14 is shown in Fig. 3(e) and takes the form of the cross-section of an elliptic umbilic catastrophe . The zeros of |Ωz| and |Lz| have no direct association with one another, as predicted by Berry .
In the following sections, we will show that Argand mappings containing vorticity singularities can be observed experimentally using an optical speckle field.
We close this theory section with an idea raised by one of the anonymous referees of this paper. Consider a simply-connected two-dimensional region, over which the continuous complex single-valued scalar field Ψ(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. , the number of Argand-plane sheets covering the Argand origin (cf. Fig. 8) is equal to the total number 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
In Rothschild et al. , we computationally generated vortices using the spatial filtering of two-dimensional complex white noise. The Fourier transform of resulting wavefield was truncated using a low-pass filter, resulting in a speckle field once an inverse two-dimensional Fourier transform was applied. The cutoff of the low-pass filter was used to control the size of the speckle.
This process can be achieved in an experiment using a coherent laser, a ground glass screen, a circular iris and a lens, among other apparatus. A schematic of the experiment is shown in Fig. 4.
The output of a linearly-polarised Helium-Neon laser (Thorlabs, 5 mW) was passed through a neutral density filter, and was then spatially filtered by an f = 18.4 mm aspheric lens, and a 20 μm 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.
A random phase field containing optical vortices was created by passing the expanded planar laser beam through a random phase plate, composed of a fine ground glass plate from a film-camera viewing screen. An iris (aperture set to approximately 0.8 mm) immediately after the screen limited the aperture for the system and determined the characteristic size of the speckle interference pattern that propagated towards the CCD camera (Prosilica GE1650).
A plano-convex lens (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.
In order to plot the data in the Argand plane, it was necessary to recover the phase information. This was done using phase stepping interferometry . The algorithm we used requires that four separate interferograms are recorded. As shown in Fig. 4, a reference beam was split from the laser beam before it encountered the glass screen, which was then passed through a 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 using14].
The results of the experiment for the reconstructed intensity and phase are shown in Fig. 5. The region of interest (ROI) is outlined by a white broken rectangle and enlarged for both intensity and phase in Fig. 5. This selection lies near the local minimum of the gentle background curvature in the phase, on the order of 10 wavelength per millimetre. Additionally, the size of the ROI is small enough to allow us to clearly see features in the Argand plane. We note the presence of interference fringes in several areas of the ROI, evident as ripples in the phase and intensity, and are due to unwanted scattering by optical elements.
The Argand-plane mapping of the data in Fig. 5 is shown in Fig. 6. It is difficult to see any of the behaviour described in Sec. 2, in particular that of vorticity singularities, due to an overlying high frequency ripple. These Argand-plane ripples are due to the high frequency interference fringes noted in Fig. 5.
To clean up the Argand image of the data and reveal the underlying vorticity singularities, a Gaussian (Fourier) filter was applied at the level of the complex field. This strategy was necessary because one can not directly smooth the branch cuts in the multi-valued phase map. The process is shown in Fig. 7. Looking at the power spectrum in Fig. 7(a), there is a diagonal streaking effect at medium to high spatial frequencies. This is unrelated to the vertical and horizontal cross-shaped streaking, which is due to truncating to a rectangular region of interest. In order to achieve smoother results, we can apodise the power spectrum by a Fourier-space Gaussian filter (Fig. 7(b)) that is small enough to exclude any unwanted features.
Upon smoothing the data, a cubic interpolation was performed to increase the density of points in the ROI by a multiple of 15. Due to the smoothing and interpolation, the interference fringes that were visible in Fig. 5 are not present and the sampling has become finer. The final result is shown in Fig. 8.
Figures 8(c) and 8(d) show the orbital angular momentum density and vorticity calculated corresponding to the indicated region of data, respectively. The position of the vortices in these images is consistent with the work of Berry in his seminal Optical Currents work , in which he stated that the location of vortex cores has no special relationship to the vorticity of a wave-field. This is indeed the case in Fig. 8(c). Figure 8(d) shows that each of the vortex cores coincide with a zero in the orbital angular momentum, consistent with the prediction by Berry .
Vorticity singularities were distinguished in the Argand plane and linked to particular behaviours in the wave field. The Jacobian determinant was calculated to obtain a visual indication of the phase winding behavior. Figure 9(a) shows the phase of the first frame of data. The black lines are the regions where the magnitude of the Jacobian is sufficiently close to zero, |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. , this means the enclosed space must induce two singularities under 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.
It is easy to see from these examples that, with an understanding of both the vortex distribution and phase winding behaviour of the wave field, the presence of vorticity singularities is a natural result of inducing a mapping ℳ in a complex scalar function.
There are other, higher order, vorticity singularities that can be observed in the Argand plane. Figure 10(c) shows an image of a vorticity singularity that resembles the cross section of the elliptic umbilic catastrophe. This was induced by the mapping of a “Jacobian ellipse”, shown in Fig. 10(a). This feature was spotted in data that was obtained by transversely shifting the ground glass screen (see Fig. 4) 10 μm 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.
We close this Analysis by giving an experimental demonstration of the method outlined in the final paragraph of Sec. 2, which gives a means to determine the total number 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.
Our study of the Argand plane provides a useful complementary method for analysing optical fields. In fact, this device could be used for the study of any complex scalar function, not just optical. Already, the discovery of vorticity singularities that are induced by such a mapping to the Argand plane, and our study of them, has provided us with additional insight into the behaviour of our wavefield. For example, we note the connection between the orbital angular momentum density and vorticity and the significance of vortex locations.
The role of the Argand plane in the analysis of optical fields is an interesting one, and we intend to continue our investigations into its usefulness. An immediate extension of this work is to continue to catalogue vorticity singularities, associating each one with a solution for J = 0, as done for the elliptic umbilic in Sec. 2. Other vortical fields such as the focal volumes of aberrated lenses [15–17] and turbulent Bose-Einstein condensates (see [18–20], for instance) and their resulting Argand images under ℳ could help clarify the connection between phase vortices and vorticity singularities. Berry  established that vorticity in an optical field is not associated with the location of vortex cores. It is interesting that we have shown that vorticity singularities are a manifestation of vorticity and are strongly influenced by the location of vortex cores. Further study, such as the extensions described above, could add insight into the work already laid down by Berry.
Further, one could study critical point explosions (see ) of high order vortices, and the evolution of its image under 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.
The investigation into two-dimensional Argand maps is a generalisation of the Cornu spiral mentioned in Sec. 1. We can further generalise Argand maps by looking at vector fields and other multicomponent fields. For example, arbitrary spin matter waves, with spin s, will be described by 2s + 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 [22–24]). The circulating coherence current (see ) associated with a coherence vortex is analogous to the current in a complex scalar wavefield. This begs the question: Can the two-point correlation function associated with a partially coherent field littered with coherence vortices induce a mapping to the Argand plane to reveal coherence-current vorticity singularities? If so, what can we learn about the coherence current, and, finally, what is the physical meaning and practical utility of such a construction?
We have presented the Argand plane as an interesting tool for the analysis of optical fields. We have verified the existence of vorticity singularities via an experiment involving the spatial filtering of two-dimensional white noise. We have observed the fold, cusp and elliptic umbilic singularities, and showed them to be consistent with theory. The locations of the zeros of the vorticity and orbital angular momentum density of the field were determined and we observed that their association with the location of vortex cores is consistent with the work of Berry . Finally, we have used the second-order Taylor series expansion of the complex wavefunction to locate vorticity singularities induced by a mapping to the Argand plane. Possible extensions of this work could involve critical point explosions, vector and other multi–component fields and coherence functions.
FR thanks Professor Sir Michael Berry for his interest in the concepts discussed in this paper and for interesting discussions. FR also acknowledges funding from the Monash University Faculty of Science Dean’s Scholarship scheme. DMP and MJK acknowledge financial support from the Australian Research Council. MJK is an Australian Research Fellow. All authors acknowledge the very useful comments of the referees of this paper, in particular for putting forward the method given in the final paragraph of both Sec. 2 and Sec. 5 of the manuscript.
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