1. Introduction

The ubiquity of wavefront dislocations, including point singularities in the transverse phase commonly referred to as screw-type dislocations or optical vortices, is, following the analysis of Nye and Berry [1], now generally recognized [2,3]. Likewise, optical vortices have exercised the imagination of the optics community to a considerable degree over the past decade [4]. Potential applications include wave-guides for atom optics [5], optical computing [6], optical communication [7], optical tweezers, optical spanners [8], high-resolution imaging [9], and probing scattering media [10]. Optical vortices also exhibit the ability to regenerate their initial phase distribution after propagation through an obstacle [11,12] which offers promise in micromanipulation and in lithography. Several studies [13,14] have also been undertaken because of insights that may be gained into vortex systems in media other than light such as fluids, superfluids and superconductors [15]. Indeed, one of the earliest descriptions of a vortex system appears in a study on amphidronic tidal points [16]. Optical vortices will be generated when coherent light is scattered by a rough medium [10], hence their ubiquity. While, for the purposes of investigation, controlled production via laser cavity modes [15], cylindrical lens mode conversion of laser modes [17], computer generated holograms - including spiral zone plates [18] and forked diffraction gratings [5] - and phase plates, both spiral [19,20] and non-spiral [21] have all been demonstrated.

A recent publication made the perhaps surprising demonstration that production and detection of optical vortices was possible at x-ray wavelengths [22]. This opens up a whole new parameter space of possible application for optical vortices. Possible x-ray applications include using the dark vortex core to mask a bright source, as in an astronomical x-ray source [23], or in a synchrotron-based small angle scattering experiment or, as a component in an x-ray phase-shifting mask, where the central zero in the vortex intensity pattern defines a small dot in the negative photoresist that creates a contact that can be used to connect layers [24]. Another idea might be to take advantage of elemental characteristic absorption energies to allow the selective transfer of orbital angular momentum thus allowing the manipulation of components of a structure that contain elements matching the tuned beam energy - for instance microfabricated objects carrying certain metal compounds may be moved while leaving their surroundings unaffected.

The stability of optical vortices on propagation makes them particularly problematic in the context of non-interferometric phase measurements as these fail where there are zeros in the intensity or where there is rotational symmetry in the phase [25,26]. It appears that singularities also play a significant role in certain methods to recover correlations in the wavefield. Techniques that use only measurements of intensity will fail when the intensity distribution is unchanged after propagation through a rotationally symmetric optical system, which is the case for a vortex beam [27]. Both these types of measurements are of particular interest in the x-ray regime [28] and in the context of synchrotron experiments where experimental difficulties are greatly alleviated by minimal instrumentation.

The propagation of a vortex beam has been considered analytically elsewhere [5,29–35]. However, while some of the results of these analyses scale readily to x-ray wavelengths, there are some features, particularly in the context of a synchrotron experiment, which are not dealt with. For instance, all analyses that consider propagation of an optical vortex implanted in a Gaussian beam take the vortex as implanted at the beam waist, allowing the initial phase in the beam to be zero. A feature of the beam from a third-generation synchrotron source is high coherence and low divergence. We wish to retain this feature in a vortex experiment and, if we model the flux distribution as approximately Gaussian, then the beam waist occurs at the source position, which is at the electrons in the insertion device, and is therefore inaccessible. In practice the vortex is implanted by insertion of a phase plate some tens of metres [22] away from the source, consequently the phase terms in the beam at the phase plate must be retained. We consider this case in Section 2 below. In addition, there has been little analysis of the interaction of a propagated vortex beam with an object. The case of a Gaussian beam vortex generated by a diffraction grating, imaged by a lens and diffracted by a half plane or slit has been presented [36]. Also vortex-edge interactions where both the vortex and the edge dislocation are implanted simultaneously in the beam have been examined [37,38]. In Section 3 we analyse the Fresnel diffraction of a propagated x-ray vortex beam diffracted from a wire and compare with numerical and experimental results. In Section 4 we use the results of the previous sections to follow the evolution in the phase structure of a vortex beam as a function of energy.

2. Vortex beam propagation

Based on typical experimentally observed distributions (for instance as seen by the authors in the work reported in [2225,28]), we model the synchrotron source as a Gaussian beam with constant phase at the source and amplitude given by:

where r and θ are cylindrical coordinates and ω₀ is the 1/e half-waist at z=0. In the following derivations we explicitly retain the physical parameters so that their effect may be readily discerned. Typically, for a synchrotron ω₀ is of the order of 10 microns in the vertical direction and order 100’s of microns in the horizontal direction. The Fresnel condition is satisfied if:

where r and r′ are the detector and source positions respectively. For the synchrotron experiment reported in [22] where r is of the order of millimeters, r′ of the order of 10’s or 100’s of microns, the wavelength was 0.13 nm and z was approximately 40 m, this condition is satisfied. The Fresnel diffraction integral of Eq. (1) evaluated at z=z ₁ gives [39]:

where: ρ and ϕ are the cylindrical coordinates in the z ₁ plane, k=2π/λ is the wavenumber;

Our technique for generating a vortex beam is to insert a phase plate that, ideally, is non-absorbing and imparts a helical phase to the beam. It should be noted that material properties at x-ray wavelengths are such that the real part of the refractive index, 1-δ, is less than unity, with δ<10^-3. Consequently, for a phase plate that consists of a spiral ramp cut into a substrate, the phase of the beam will be advanced through the thick parts of the ramp compared to the thin parts. This is opposite to the case in visible wavelength optics and results in a vortex with the opposite helicity to that which might be expected. After insertion of the phase plate so that the centre of the plate is coaxial with the beam the amplitude is:

where m is an integer referred to as the charge of the vortex:

We can then calculate the Fresnel propagation of this vortex beam by evaluating the Fresnel integral at some detector plane Z=Z′-z ₁, where Z′ is the distance of the detector plane from the synchrotron source:

where R and Θ are the cylindrical coordinates in the Z plane. Recalling the definition of a Bessel function of the first kind of order m:

and inserting Eq. (4a) into Eq. (5) we can write:

This can be evaluated [40] to give:

where: I_n is a Modified Bessel function of the first kind of order n;

Typically, at visible wavelengths, Eq. (1) is used as the input to Eq. (5). However, as discussed above, in the synchrotron context we use Eq. (3). The retention of the phase terms in the input beam before implanting the vortex via a phase plate greatly complicates Eq. (8) compared to the corresponding result for visible wavelength.

The detected intensity is given by:

A plot of Eq. (9) is shown in Fig. 1 for the synchrotron parameters described above and with Z=5.8 m, thus satisfying Eq. (2) above. We assume ω₀ is the same both vertically and horizontally. We also calculate the detector plane intensity using a standard numerical propagator to calculate the detector plane complex amplitude:

Here F and F^-1 are the Fourier transform of the argument, k_x and k_y are the Fourier transform spatial frequencies, a is the complex amplitude at coordinates x,y,z, and a(x,y,0) is given by Eq. (3a). The resultant intensity plotted as for Fig. 1 is indistinguishable from Fig. 1.

 

Fig. 1. Vortex detector plane intensity from Eq. (9). ω₀=239.5 µm, z₁=41.4 m, Z=5.8 m, λ=0.13 nm.

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3. Vortex ‘interferograms’ produced by division of wavefront

In practice the existence of a vortex is not demonstrated merely by producing the type of intensity distribution shown in Fig. 1. Ideally some demonstration of the phase distribution should be made. This can be done interferometrically. In the visible regime division of amplitude interferometric techniques are commonly used, however at x-ray wavelengths this type of experiment is not straightforward. Accordingly, in the x-ray context, we use a division of wavefront technique such as the introduction of a wire to act as a secondary source. The wire is placed close to the phase plate and the Fresnel integral can be calculated for the resultant field. The result in [36] for diffraction of a vortex beam through an aperture with appropriate modification for the limits of the integral could be used to derive a complicated expression for the diffracted field. However, the parameters imposed by the typical synchrotron experiment allow us to analyse the experiment as if it were the interference of the vortex beam in Eq. (8) and a cylindrical wave generated at the wire which has a phase gradient along it due to the incident vortex phase. Strictly, the wave generated by the wire will not be a cylindrical wave due to the curvature in the beam and the vortex phase imparted by the phase plate, however, the approximation is shown to be valid under numerical simulation for the synchrotron parameters used above. The resulting intensity is:

where A_v is the real amplitude of the vortex beam given by Eq. (8), A_cyl is the real amplitude of the cylindrical wave and there is a modulation to the intensity distribution given by the cosine of the interference term:

where x_offs and z_offs are the offset coordinates of the wire with respect to the phase plate and where the first atan expression evaluates the phase of its complex argument. If we approximate the vortex beam as a plane wave with a vortex phase then Eq. (12), after switching to Cartesian coordinates, becomes:

Accordingly, we see that the analytic form for the interferogram for x-rays at a synchrotron becomes similar to the form used in calculating a hologram for visible wavelength vortices [3,5].

 

Fig. 2. Interferograms of vortex phase structure produced analytically (a and b), numerically (c) and experimentally (d). From left to right, (a) Eq. (12), (b) Eq. (13), (c) Intensity distribution based on Eq. (10) method, (d) experimental result.

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Figure 2(a) shows the modulation of the intensity distribution produced by the cosine of θ_interf given by Eq. (12), while Fig. 2(b) shows the result for Eq. (13). Figure 2(c) shows the result for a numerical simulation based on Eq. (10). Figure 2d shows an experimental result as reported in [22]. In all cases it can be seen that the phase discontinuity introduces a fork in one of the fringes. The main difference between the first two and the last two images is that the intensity distribution due to the amplitude terms in Eq. (11) are not included in the first two images and that the finite extent of the wire, absorption through the wire and the effects of partial coherence are included in the latter pair.

4. Vortex and an edge discontinuity: non-integer charge

Consider now what happens if the spiral phase plate is not matched to the energy of the incident x-ray beam so that the phase step across the step on the ramp is not an integer multiple of 2π. In general the ‘charge’ of the spiral will be some non-integer, ν. In the derivation above Eq. (6) is modified by replacing m with ν and the Bessel function of the first kind of order m becomes a sum of an Anger and a Weber function of the first kind of order ν. Note that the function is not replaced by a Bessel function of order ν as this has a quite different integral representation [40]. We could proceed via the modified form of Eq. (7), however the success of Eq. (13) in predicting the qualitative form of the downstream ‘interferogram’ suggests that replacing m with ν in that equation will suffice. The approach here becomes similar to that taken elsewhere in calculating the hologram required to generate a vortex [3,5,37].

 

Fig. 3. Interferograms of vortex phase structure for non-integer charge and a rotated phaseplate. From left to right; (a) Modified form of Eq. (13) with ν=0.5, (b) Eq. (14) with ν=0.5 and α=π/4, and (c) Eq. (14) with ν=0.5 and α=π/2.

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Figure 3(a) shows a plot of the cosine of the modified form of θ_interf given by Eq. (13) with ν=0.5. The phase indicated is due to an edge discontinuity terminating at the position of the point discontinuity. It is also straightforward to examine the effect of rotating the phaseplate about the beam propagation axis by rotating the vortex phase terms in the modified form of Eq. (13) to give:

where α is the rotation angle of the phaseplate. Figure 3(b) shows a plot of the cosine of θ_interf given by Eq. (14) with ν=0.5 and α=π/4, while in Fig. 3(c) α=π/2.

 

Fig. 4. (1.2 MB) Movie of evolution in the interferogram as the energy varies from 4.5 keV (charge=2.01) to 9 keV (charge=1).

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In the synchrotron context it is relatively straightforward to continually vary the incident energy over a broad range. The charge of the phaseplate will vary as a function of the refractive index decrement times the energy. Away from absorption edges the refractive index decrement is roughly inversely proportional to the square of the energy, so the charge will be roughly inversely proportional with energy in this regime. Figure 4 is a link to a movie that tracks the evolution in the ‘interferogram’ produced by Eq. (14), with α=0, as a function of equally spaced energy steps.

5. Conclusions

We have developed a useful range of results dealing with vortex phenomena at x-ray energies and for experimental parameters appropriate to a synchrotron source. Analytic expressions that differ from those obtained with visible wavelength assumptions are obtained and care has been taken to retain physical parameters explicitly. It is shown that the analytic forms can usefully predict intensity and phase distributions obtained numerically and experimentally. Finally, we have generated a prediction of vortex phase structure as a function of energy for a given phaseplate. We will compare this with experimental results in future work.