Generalised Cornu spirals: an experimental study using hard x-rays

Freda Werdiger; Marcus J. Kitchen; David M. Paganin

doi:10.1364/OE.24.010620

1. Introduction

The Cornu spiral is, in essence, an Argand-plane representation of the Fresnel diffraction of light that is normally incident upon an infinite opaque edge. Historically, it was used as a graphical aid for the calculation of the Fresnel integrals associated with this diffraction. Figure 1(a) shows the Argand-plane plot of the complex optical field which results from a plane wave diffracting from a semi–transparent thin aluminium screen. Adopting the perspective of Keller’s geometrical theory of diffraction [1], (i) the bottom right 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 left lobe results from the coherent superposition of the incident plane wave phasor with the scattered cylindrical edge wave.

Fig. 1 (a) A Cornu spiral resulting from the Argand-plane mapping of a monochromatic plane wave diffracted by a partially absorbing aluminium half-plane screen (the dotted line is a unit circle that represents the unscattered plane wave), and (b) a hypocycloid resulting from a monochromatic plane wave diffracted by a partially absorbing cylinder, described by Morgan et al. [2]. The diffracted light that emerges outside the geometric shadow of the cylinder is indicated in blue.

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Morgan et al. [2] described such a mapping of diffraction from a cylindrical edge. This is demonstrated in Fig. 1(b), which shows the simulated Argand mapping of a monochromatic plane wave diffracted by a partially absorbing cylinder. The resulting wavefield takes the form of a hypocycloid when mapped to the Argand plane. There, the lobe on the bottom left (blue), representing the outside of the cylinder where the diffraction pattern lies against the unscattered plane wave, is identical to that of a screen. Crossing the edge of the cylinder, however, the changing thickness of the cylinder causes the Argand-plane trace to evolve into a hypocycloid (black): As the thickness of the cylinder increases, the phase accumulated by passage of the light ray through the cylinder increases in magnitude, causing the one-dimensional trace to rotate in the Argand plane. At the same time, the intensity decreases as more of the incident light is absorbed, causing the Argand trace to spiral in towards the origin. Finally, the light that is scattered from the edge of the cylinder interferes with the light that is incident upon the cylinder, resulting in oscillations in phase and intensity that decrease in amplitude as the distance from the edge increases.

Both the Cornu spiral and the Cornu-come-hypocycloid in Figs. 1(a) and 1(b) are one-dimensional traces embedded in two-dimensional Argand space; while the thickness varies across the cylinder, it retains the one-dimensional nature of the straight edge when projected onto a two-dimensional surface. In order to find a complex function that leaves a two-dimensional image of x-y space in the Argand plane, we need to add a higher degree of spatial complexity to the object of interest. A sphere, when projected onto a surface, has a thickness that varies in both transverse dimensions. However, if the sphere is lying against a homogeneous background, such as the normally incident monochromatic plane wave, that distinction does not translate to a two-dimensional image in the Argand plane. By placing a sphere inside a cylinder, however, we can achieve the varying background necessary to evolve the Argand-plane trace into the sort of two-dimensional Argand-plane image seen in our previous works [3, 4].

Here, we experimentally reconstruct the Cornu spiral and two of its generalisations through the imaging of three objects with increasing complexity: the straight edge, cylinder and sphere-in-cylinder.

Section 2 will provide some background theory, namely Argand-plane mapping theory, including a description of vorticity singularities (Sec. 2.1), and the Geometrical Theory of Diffraction (GTD) (Sec. 2.2). We can use GTD to write down equations for the field downstream of each of the three objects to be imaged and to understand and predict the morphology of the Argand-plane mapping of each field. Section 3 includes the experimental results obtained using hard x-rays. Methods used for capturing images and processing the data are described in Sec. 3.1. Section 3.2 reports on the results of using a piece of Kapton tape to observe x-ray diffraction from a straight edge and subsequent Argand-plane mapping of the reconstructed field. Sections 3.3 and 3.4 do the same for an aluminium cylinder and a spherical bubble trapped in an agar-filled perspex cylinder. Section 4 discusses our results, including possible applications and plans for future work. We conclude with Sec. 5.

2. Theory

Here, we briefly review some relevant background theory, with the reader being referred to Refs [3] and [4] for further detail.

2.1. Argand plane mappings

For an arbitrary two-dimensional differentiable single-valued continuous complex function Ψ(x,y), a mapping $ℳ$ to the Argand plane is given by

ℳ (Ψ (x, y)) \to (Ψ_{R}, Ψ_{I}),

where Ψ_R and Ψ_I are the respective real and imaginary parts of Ψ(x,y), which is here taken to be the boundary value of the spatial part of a forward-propagating three-dimensional monochromatic scalar wavefunction [5], evaluated over a planar surface.

For a complex wavefunction that lies in x-y space, there is in general a loss of information that occurs under a mapping $ℳ$ to the Argand plane, rendering the mapping essentially non-invertible due to the mapping being many-to-one. In such cases, it is possible for a singularity to form under a mapping to the Argand plane.

In order to locate singularities induced by a many-to-one mapping $ℳ$ to the Argand plane associated with Ψ(x,y), we can define the Jacobian determinant (“Jacobian”) of $ℳ$ as [4]

J (x, y) = | \begin{array}{l} \partial_{x} Ψ_{R} & \partial_{y} Ψ_{R} \\ \partial_{x} Ψ_{I} & \partial_{y} Ψ_{I} \end{array} | .

The Jacobian of $ℳ$ provides valuable information about the transformation of Ψ(x,y). The absolute value of J at a point P = (x_p,y_p) in x-y space gives the factor by which infinitesimal patches at p expand or contract under the transformation from real space to Argand space. The sign of J indicates whether a patch has been inverted (J < 0) or not (J > 0). A value of J = 0 indicates that an infinitesimal patch of space in the xy-plane has collapsed onto a single point under $ℳ$ and a singularity of the Argand plane has formed for $ℳ$ (Ψ(x,y)). Setting J(x,y) = 0 provides the location of all singularities of the mapping. The set $P$ of all (x,y) points for which J(x,y) = 0 maps to a hierarchy of singularities in Argand space, such as the fold, which is induced by, as the name suggests, a fold in the local image of x-y space along a single line of zero vorticity.

One can assign a physical meaning to the “zero lines” of the Jacobian and subsequent singularities in the Argand plane. The vorticity Ω of a three-dimensional complex scalar field Ψ(x,y,z) can be expressed as [6, 7]

Ω = \nabla \times j = Im (\nabla Ψ^{*} \times \nabla Ψ) = \nabla Ψ_{R} \times \nabla Ψ_{I},

where ∇ is the gradient operator and j= ImΨ^*∇Ψ is the current up to a multiplicative constant, which is set to unity here. The vorticity quantifies the local rotation in the field. The z-component of the local vorticity,

Ω_{z} = \frac{\partial Ψ_{R}}{\partial x} \frac{\partial Ψ_{I}}{\partial y} - \frac{\partial Ψ_{I}}{\partial x} \frac{\partial Ψ_{R}}{\partial y},

represents the local current rotation at (x,y) and is equivalent to the Jacobian, as seen from Eq. 2. When the local current rotation changes from clockwise to anti-clockwise, implying that when Ω_z = 0, a singularity will be induced by

ℳ

, due to the assumed continuity of the vorticity. Stated alternatively, the Jacobian zero lines are present where the vorticity has vanished.

2.2. Geometrical theory of diffraction

In observing the diffraction of x-rays from the hierarchy of objects defined here, it is necessary to define a theory that describes a diffracted light field. A consequence of the finite wavelength of light, diffraction is not a property described by Geometrical Optics (GO), which uses geometrical rays to calculate light fields accounting for the incidence, refraction and reflection of electromagnetic waves. The Geometrical Theory of Diffraction (GTD), however, adds to this a description of rays that are diffracted around edges and smooth objects. Introduced by Joseph B. Keller in 1953, this theory augmented GO by using geometric rays to describe wave diffraction [1]. GTD, like GO, assumes light travels in rays but introduces diffracted rays, which are produced by incident rays hitting or grazing the edges, corners or vertices of boundaries. Some of these enter the shadow region and others go into the illuminated regions, contributing to the light there. Diffracted rays can be calculated using a modified Fermat’s principle, assigning a phase to each ray and letting the total field at a given point be the sum of all complex rays passing through it.

We can use the geometrical theory of diffraction to describe the field downstream of a semi-opaque phase-amplitude screen illuminated by a monochromatic plane wave. The total field at a point (x,y) will be the sum of all rays passing through that point, each of which can be placed into one of three categories: Those which have passed through the screen, having picked up a subsequent phase change equal to −kδT, where k is the wave number, 1−δ the refractive index of the screen material and T the projected thickness of the screen; those which have passed through uninterrupted, having incurred no phase change; finally, those which have scattered off the boundary of the screen, emanating in the form of a locally cylindrical wavefront, given that the source of the boundary wave is a line.

Thus we can state that for the field Ψ(x,y) over an image plane lying perpendicular to the axis of propagation which is at a distance z from the exit surface of a semi-opaque screen of thickness T which has been illuminated by a rigidly-translating monochromatic plane wave with wavenumber k,

Ψ (x, y) = e^{i k z} e^{- μ T / 2} e^{- i k δ T} + A \frac{e^{i k R}}{\sqrt{R}},

where μ is the attenuation coefficient of the material, R is the distance from the point (x,y) in the image plane and the edge of the screen, and

A

is the amplitude of the scattered wave. If the thickness of the screen varies in one transverse dimension, say the x- direction, we can write

R = \sqrt{x^{2} + z^{2}}

. The two lobes of the Cornu spiral in Fig. 1(a) are due to the phasor addition of the cylindrical wave with the wavefront passing uninterrupted on the outside of the screen and with the wavefront transmitted through the screen.

In the case of a cylinder with radius R_c, whose axis is perpendicular to the optic axis, we can adjust Eq. (5) to account for two separate line sources for a cylindrical boundary wave, one on either side of the cylinder, and a thickness $T (x) = 2 Re \sqrt{R_{c}^{2} - x^{2}}$ that varies across the object. Thus,

Ψ (x, y) = e^{i k z} e^{- μ T (x) / 2} e^{- i k δ T (x)} + A \frac{e^{i k R_{1}}}{\sqrt{R_{1}}} + ℬ \frac{e^{i k R_{2}}}{\sqrt{R_{2}}},

where

R_{1} = \sqrt{{(x + R_{c})}^{2} + z^{2}}

,

R_{2} = \sqrt{{(x - R_{c})}^{2} + z^{2}}

. As x moves closer to the centre of the cylinder, T increases. Referring back to Fig. 1(b), the black lobe of the Cornu spiral unravels in the geometric shadow of the cylinder, due to the change in thickness. The oscillations in phase and intensity are brought about via the addition of phasors, accounting for the cyclodic motion of the Argand-plane trace.

Continuing with the hierarchy of objects investigated for diffraction in this paper, we can adjust Eqs. (5) and (6) to account for a sphere lying within the cylinder. In the event of a sphere normally illuminated by planar complex scalar electromagnetic waves, the source of the diffracted wave would be a ring following the outermost edge around the sphere. Theoretically, a ring source would result in a toroidal wavefront accounting for the diffraction from the edge of the sphere. For a point P lying in the x-y plane at some distance z from the sphere, the diffracted rays that pass through that point are a result of the locally toroidal wavefront emanating from the closest point on the circle. Figure 2.2 demonstrates this point.

Fig. 2 For a sphere lying in the object plane, the source of the diffracted rays is a ring around the outermost edge (red). To find the contribution of the diffracted rays through point P lying at a distance of z = Δ from the object plane, we find the shortest distance from P in the image plane to the ring source in the object plane, defined by R₃. The diffracted rays through P are a result of a toroidal wavefront emanating from the point where R₃ meets the ring.

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Therefore we find that for a sphere of radius R_s lying within in a cylinder that is illuminated by a monochromatic plane wave, the field through a point P(x,y) lying in the image plane at some distance z from the object is given by

Ψ (x, y) = e^{i k z} e^{- μ T (x, y) / 2} e^{- i k δ T (x, y)} + A \frac{e^{i k R_{1}}}{\sqrt{R_{1}}} + ℬ \frac{e^{i k R_{2}}}{\sqrt{R_{2}}} + ℰ \frac{e^{i k {\tilde{R}}_{3}}}{\sqrt{{\tilde{R}}_{3}}},

where

{\tilde{R}}_{3}

is the optical path length of the ray which possesses a geometrical path length given by

R_{3} = \sqrt{{(\sqrt{x^{2} + y^{2}} - R_{s})}^{2} + z^{2}}

. If the sphere is a bubble in the cylinder, the entire object can be a single material with a single refractive index 1−δ and attenuation coefficient μ, with projected thickness

T (x, y) = 2 \sqrt{R_{c}^{2} - x^{2}} - 2 \sqrt{R_{s}^{2} - x^{2} - y^{2}}

.

3. Experimental realisation using hard x-rays

Here we present our experimental results, obtained using hard x-rays. Our reason for choosing this form of radiation is a long-term goal, which lies beyond the scope of the present paper, namely the application of Argand-plane analyses to improve methods for the speckle imaging of lung tissue using coherent x-ray images [8].

3.1. Experimental and analytical methods

The results in this section were obtained via x-ray experiments conducted using a 215m-long beamline (BL20B2) at the SPring-8 synchrotron radiation facility in Hyogo, Japan. The length of the beam allowed for a large field and spatially coherent beam [9]. At the end of the beamline, the distance from source to detector can be adjusted to allow for propagation-based phase contrast imaging (PBI) [9], which is used here to reconstruct the various objects that are featured in the following sections. A schematic of the experimental set-up is shown in Fig. 3. The technique used to obtain images used in this paper is PBI. Unlike other methods that adopt phase contrast, such as x-ray interferometry [10], analyser-based phase-contrast imaging [11, 12], or x-ray diffraction grating methods [13], PBI does not require the use of any additional optical elements between the sample and detector. The act of free-space propagation enables the phase shifts that the sample imposes upon the incident radiation to become visible.

Fig. 3 The general set up of the experiments used to obtain results shown in Secs 3.2 – 3.4. The object to be imaged lies downstream of the x-ray source. The partially coherent light is then allowed to propagate further in order to observe edge enhancement via interference.

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Pertinent to the results presented in this section is the phase retrieval algorithm developed by Paganin et al. [14], which requires a single PBI image per projection. In applying this algorithm, we assume that the sample in question is comprised of a single homogeneous material imaged under paraxial coherent x-ray radiation.

An optical system, such as the system used in the methods described above, can also add specific contrast modes, such as Zernike phase contrast [15]. These techniques have widespread use in light microscopy, particularly for biological specimens, due to their usefulness in defining fine structure in the images. The introduction of certain appropriate optical elements in microscopy serves as a means to generate the desired mode of contrast; this is hard-wired into the experimental set-up. In holography, however the desired contrast mode can be generated numerically using virtual optics. The appropriate transfer function is used to simulate the analog of any optical element, given the boundary value of a forward propagating complex scalar electromagnetic wave over a given plane perpendicular to the optic axis. Thus, the computer becomes a part of the imaging system. This method was termed omni-microscopy and demonstrated using hard x-rays by Paganin et al. [16]. These methods are exploited in the following sub-sections to achieved the desired contrast. The complete wavefield information, obtained using experimental images, is subject to virtual optics, virtually propagated through space to obtain an appropriate amount of phase contrast. In particular, the wavefield derived in Sec. 3.4 is propagated beyond the limits of the particular experimental hardware involved in obtaining the original images.

3.2. Straight edge

In order to reconstruct a Cornu spiral, it was necessary to use an object with a straight edge and illuminate it with electromagnetic radiation to observe diffraction from the edge. A piece of Kapton (polyimide) tape was used for this purpose. The experiment was performed using 30 keV x-rays (μ = 35.05 m⁻¹, δ = 3.38 × 10⁻⁷ [17]) to illuminate the sample. Figure 4(a), together with the profile plot in Fig. 4(b), shows the PBI image of the tape that was taken at a distance of 2.0 m from the sample with a 4000 × 2672 pixel Hamamatsu CCD camera (C9300-124) with a 16.2 μm pixel size. Single image phase retrieval [14] was used to construct an image of the tape at the exit surface, seen in Fig. 4(c). The noise is significant in the profile plot in Fig. 4(d). The resolution of the image was increased by a factor of two using cubic interpolation before using the angular spectrum formulation to forward propagate [18], employing virtual optics to observe a diffraction pattern. Figure 4(e) shows a forward propagation to 2.0 m. The profile plots of the original and simulated images at a distance of 2.0 m from the contact surface, Figs 4(b) and 4(f), respectively, closely resemble one another. The central fringe is brighter in Fig. 4(f), presumably a result of the increased resolution.

Fig. 4 (a) A 1.10 cm × 0.24 cm image 2.0 m from a sample containing a piece of Kapton (polyimide) tape, taken using 24 keV x-rays. A profile plot of the raw image is shown in (b), wherein lies a single bright phase contrast fringe pair. Note as well the highly transmissive quality of the tape; (c) An image of the projected thickness of the tape, with a profile plot shown in (d), highlighting the low signal-to-noise ratio; (e) For comparison, a simulated image of the tape at 2.0 m is shown, along with a profile plot (f) showing the previously noted single bright fringe. The fringe in (f) appears brighter than the one in (d) as the pixel size was halved for the purpose of the simulation.

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Figure 5(a) shows the intensity of the field forward-propagated with virtual optics to eight metres, where there are several fringes. This field is mapped to the Argand plane in Fig. 5(b), where the noise obscures the Cornu spiral, which has emerged due to the boundary wave. Integrating the image along 341 columns to produce a field that varies in only one spatial direction, a clean image of the Cornu spiral can be seen in Fig. 5(c). Cropping the image close to the edge of the tape, a clean Argand trace can be obtained.

Fig. 5 (a) Intensity of the field forward propagated to 8.0 m, where several fringes are visible; (b) Argand mapping of the field propagated to 8.0 m, where the spiral is obscured by noise; (c) Argand mapping of the field after being summed and averaged along the vertical axis, showing a clean Cornu spiral.

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Figure 6 shows the evolution of the trace with distance through 0.0 m, 2.0 m and 6.0 m. Diffraction fringes multiply upon virtual-optics propagation; oscillations in phase and intensity manifest at each end of the Cornu spiral. The characteristic bright central fringe becomes prominent in Fig. 6(b), at 2.0 m from the contact surface. Propagating further to 6.0 m, there are several fringes which are apparent in oscillations seen in Fig. 6(c). The arm that extends outside of the unit circle (dashed line) represents the outside of the Kapton tape which is occupied by the uninterrupted plane wave. Due to the low absorption coefficient and thickness (see Fig. 4(d)) of the material, only a very small amount of radiation is absorbed by the tape; both the uninterrupted and transmitted waves are close to unity and the Cornu spiral clings to the unit circle in each image.

Fig. 6 The evolution of the Cornu spiral with propagation distance: (a) 0.0 m, (b) 2.0 m and (c) 6.0 m. The spiral clings to the unit circle (dashed line) in each instance due to the low absorption of the Kapton tape. The field in (a) traces out an arc due to its smooth and continuous nature, rather than two discrete points representing either side of the tape.

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3.3. Cylindrical edge

Further to the goal of generalising the Cornu spiral, we sought to observe its evolution upon the straight edge being replaced with a cylindrical one. Under these conditions, one side of the object would experience a varying phase and intensity, due to the changing thickness of the cylindrical object, which would deform the corresponding arm of the Cornu spiral. Figure 7(a) shows an image of a 3 mm aluminium cylinder taken with 24 keV x-rays using the Hamamatsu ORCA Flash C11440-52U fibre optic detector with a pixel size of 6.5 μm. The sample-to-detector distance is 35 cm. At 24 keV, aluminium has an absorption coefficient of μ = 464.48 m⁻¹ and refractive index decrement δ = 9.398 × 10⁻⁷ [17]. Observing a profile plot of the raw image, Fig. 7(b), a single fringe is seen on each side of the cylinder. Note as well the effects of the point spread function (PSF) in the field as it approaches the edges of the cylinder. Using single-image phase retrieval, the projected thickness of the cylinder was reconstructed (Fig. 7(c)). Looking at the profile plot in Fig. 7(d), the fringes have been suppressed, as would be expected at the contact surface. The reconstructed image shows a cylinder diameter of approximately 3.25 mm. For comparison, the field at the exit surface was interpolated by a factor of four and then forward-propagated to 35 cm using the angular spectrum formula [18]. The intensity at 35cm is shown in Fig. 7(e), and a profile plot in Fig. 7(f). The noise has been suppressed and there is a single sharp peak at each boundary. The peak in Fig. 7(f) is higher than that in Fig. 7(b) due to the process of cubic interpolation whereby the number of pixels across the peak has been inflated by a factor of four. Note as well that there is no smearing of the point spread function.

Fig. 7 (a) A 9.1 mm × 1.6 mm image taken 35 cm downstream of an aluminium rod illuminated by 24 keV x-rays with a plot profile (b) showing a single bright fringe at the edge of the highly absorbent cylinder; (c) a reconstruction of the projected thickness of the rod and profile in (d); (e) forward propagation to 35 cm for comparison (the intensity is shown) and profile in (f) showing a brighter central fringe due to the interpolation as part of the simulation.

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Figure 8 shows the Argand-plane mapping of the propagated field at various distances; 0.35 m, 1.0 m and 4.0 m. Diffraction fringes multiply as propagation distance increases, as with the Kapton tape in Sec. 3.2. Oscillations in phase and intensity on the outside of the cylinder manifest in the same way as with the straight edge of Fig. 6, spiraling in towards the value of the uninterrupted plane wave. Crossing the edge, the thickness of the sample increases, resulting in changes in phase and intensity, due to the subsequent refraction and absorption of the incident radiation. This causes the trace to travel around the Argand plane, instead of being fixed around a constant background. This ultimately results in the hypocycloid that manifests in Fig. 8(c), at a distance of 4.0 m from the contact surface. At this distance, there is significant diffraction and the resulting phase and intensity oscillations stand out against a rapidly varying background.

Fig. 8 Evolution of the Argand-plane mapping of the aluminium cylinder with propagation distance: (a) 35 cm, (b) 1.0 m and (c) 4.0 m. The spiral in each image represents the outside of the cylinder, where the intensity and phase oscillate against the unscattered plane wave. Within the cylinder, the varying phase causes the trace to move around the Argand plane as the thickness increases and the intensity falls to a minimum. The hypocycloid, as described in Fig. 1(b), is noticable in (c).

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In order to observe the evolution of the hypocycloid under circumstances where there is significantly more diffraction, the energy of the x-rays used in the angular spectrum formula was reduced. For Fig. 9(a), the energy has been reduced to 18 keV (μ = 1132.0 m⁻¹,δ = 1.67 × 10⁻⁶ [17]) and the wavefield allowed to propagate to 4.0 m before being mapped to the Argand plane. Drawing comparison with Fig. 8(c), which is also at 4.0 m from the contact surface, there are considerably more revolutions in the lobe at the bottom-left of Fig. 9(a) than in the corresponding lobe for Fig. 8(c), signifying more diffraction. Additionally, the hypocycloid noted in Fig. 8(c) is more dramatic in Fig. 9(a) due to the increased absorption at the energy. Decreasing the energy to 16 keV (μ = 1622.98 m⁻¹, δ = 2.12 × 10⁻⁶ [17]) heightens this effect. Reducing again to 15 keV (μ = 1974.14 m⁻¹, δ = 2.41 × 10⁻⁶ [17]) and propagating further to 6.0 m from the contact surface, increasing the diffraction from the edge of the cylinder, results in the retrograde motion of the Argand-plane trace described by Morgan et al. [2], seen in Fig. 9(c) [17].

Fig. 9 The projected thickness is forward propagated using x-rays with reduced energy. (a) 18 keV x-rays propagated to 4.0 m; (b) 16 keV x-rays at 4.0 m; (c) 15 keV x-rays at 6.0 m. The retrograde motion of the Argand trace can be seen in (c). The lower energy rays have a higher absorption coefficient and diffract more heavily.

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3.4. Sphere-in-cylinder

In order to observe the evolution of the Argand-plane mapping from a one-dimensional trace to the fully-realized two-dimensional Argand mapping, we imaged a sphere embedded in a cylinder. A perspex cylinder was filled with a solution of 2% agar and a spherical air bubble was trapped in the solution as it set. The sample was then illuminated by 24 keV x-rays and a PBI image was captured using the ORCA Flash detector, as used to obtain the images in Sec. 3.3. Placing the detector at a distance of 2.0 m from the sample, the image in Fig. 10(a) was obtained. A profile plot in Fig. 10(a) shows some contrast at the boundaries between air, perspex and agar. As the aim was to obtain the Argand mapping of a sphere against a cylindrical background, the properties of the agar solution (μ =46.95 m⁻¹, δ = 4.00 × 10⁻⁷ [17]) were used for single-image phase retrieval. The resulting projected thickness is shown in Fig. 10(c), and corresponding profile plot taken across the inside of cylinder in Fig. 10(d), where the fringes have been suppressed. The pixel size was reduced by a factor of four using cubic interpolation.

Fig. 10 (a) A 1 cm × 1 cm image taken 2.0 m from a bubble trapped in a perspex cylinder filled with agar illuminated by 24 keV x-rays. A plot profile (b) taken along the red line in (a) shows the three distinct regions: air, perspex and agar. The reconstruction uses μ and δ for the latter; (c) The projected thickness is shown, with plot profile (d) taken along the red line in (c), within the boundary of the cylinder. Noise and fringes are suppressed.

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The field was numerically forward-propagated to 50 m to observe stark contrast between the bubble and its surroundings. The intensity at this distance is shown in Fig. 11(a). Figure 11(b) shows the profile plot across the mid-section of the bubble, showing some fringes at the boundary of the bubble. The Argand mapping of the bubble is shown in Fig. 11(c), where two small but distinct spiral-like structures are seen, separated due to the varying cylindrical background as well as the noise. Figure 11(d) shows a close-up image of the lower spiral-structure. This structure is the Argand mapping of a field normally incident on a sphere that is lying within a cylinder. It is effectively the Argand mapping resulting from light incident upon a series of infinitesimally flat cylinders, with varying radii, lying against a varying background that together map to the Argand plane in the form of this two-dimensional structure consisting of a continuous infinity of Cornu spirals.

Fig. 11 (a) Image of the cylinder propagated to a distance of 50 m; (b) profile across the mid-section of the bubble, which has a strong signal at this distance; (c) Argand mapping of the bubble; (d) Close-up image of (c), with behaviour seen in fully-realized two-dimensional vorticity singularities, such as the fold singularity indicated by α; (e) With energy halved to 12 keV, the x-rays diffract more strongly and are absorbed heavily, resulting in some hypocylodic behaviour seen on the left side of the image.

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In order to observe some hypocyclodic behaviour, the energy of the x-rays was halved to 12 keV for the angular spectrum formula. The result in Fig. 11(e) shows some such behaviour due to more extreme diffraction that is expected at this lower energy level. There is also greater absorption, with this spiral having a magnitude of approximately one tenth of that in Fig. 11(c).

It is with Figs. 11(c), 11(d) and 11(e) that we finally obtain Argand-plane singularities associated with configuration-space vorticity zeros. These structures, reminiscent of the caustics of geometric optics, are exemplified by the fold-type singularity marked α in Fig. 11(d). As previously explained, all such Argand-plane singularities correspond to the image of points (x,y) with both zero vorticity and zero Jacobian.

4. Discussion

Our previous works [3, 4] focused on vortical behaviour in continuous two-dimensional complex scalar wavefunctions and noted the existence of singularities induced by the associated mapping $ℳ$ to the Argand plane – coined “vorticity singularities” as they correspond to lines of zero vorticity in the xy-plane. We have so far used singularities induced by $ℳ$ to make general and particular observations about the behaviour of screw-type phase defects in complex scalar wavefunctions – such as the behaviour of the wavefield in and around vortex pairs.

At the same time, the field of “singularimetry” (a termed coined recently by Dennis and Götte [19, 20]), is emerging as an important result of modern singular optics. The locality and stability of zeroes of an optical field makes them useful in measuring beam shifts to subwavelength-accuracy. More recent work by Petersen et al. [21] demonstrates the use of vortex lattices to determine small phase shifts imparted on the field by various spherical and cylindrical specimens. One also has recent experimental work on optical currents, such as that of Angelsky et al. [22] and Pavlov et al. [23].

Our work in this paper shifts the focus away from exclusively vortical fields, namely those with screw-type phase defects. The Cornu spiral, the Argand-plane mapping of the diffraction of a field by a straight edge, is a convenient platform for the study of Argand-plane maps, having already been somewhat generalized by Morgan et al. [2]. Mirroring the work of Petersen et al. [21], we have looked beyond the straight edge to explore phase shifts imposed on a field by cylindrical and spherical objects, which manifest as generalized Cornu spirals in the Argand plane. The evolution of the Cornu spiral from a one-dimensional trace (albeit embedded in a two-dimensional plane) for a straight edge to the fully-realized two-dimensional Argand-plane map is in our view an interesting tool that is worthy of further investigation in the future.

5. Conclusion

We have presented a study of generalised Cornu spirals, which are a series of Argand-plane mappings of monochromatic wavefields that are incident upon objects with varying degrees of complexity. Here we have focused on a hierarchy of three objects: a straight edge, a cylinder and a sphere embedded within a cylinder. Each object was imaged using propagation-based phase contrast imaging with hard x-rays. After phase retrieval, generalised Cornu spirals were then constructed for each object.

Acknowledgments

FR acknowledges the assistance of Genevieve A. Buckley, Katie L. Lee and Rhiannon P. Murrie in obtaining the images used here. FR also thanks Dr. Kaye S. Morgan for useful discussions. MJK acknowledges financial support from the Australian Research Council. MJK is an Australian Research Fellow (DP110101941).

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Generalised Cornu spirals: an experimental study using hard x-rays

Abstract

1. Introduction

2. Theory

2.1. Argand plane mappings

2.2. Geometrical theory of diffraction

3. Experimental realisation using hard x-rays

3.1. Experimental and analytical methods

3.2. Straight edge

3.3. Cylindrical edge

3.4. Sphere-in-cylinder

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (7)

Optics Express