1. Introduction

Young’s interference experiment [1–3] is one of the fundamental experiments in Physics that proved the wave nature of light. By replacing light by atoms, electrons, neutrons and molecules [4–11] it has been shown that similar interference patterns can be obtained thereby establishing duality between particle and wave natures. In the Young’s double slit experiment, coherent states of light is wavefront split and a spatial overlap of electromagnetic fields from these slits is observed on a screen that depicts the independent evolution of these fields. The superposition results in interference fringes which can be understood only by wave theory. Interference fringes are there, only as long as, through which slit the light has passed through, is unknown. Any attempt to find which way light has passed through destroys the interference pattern. The choice of exhibiting one of the two natures namely the particle and wave natures is complementarity.

Vortices are ubiquitous and are being studied in various branches of science and engineering [12–14]. Spiral galaxies, tornados, cyclones, vortices in water and in tea cups, wing-tip vortices of a moving aircraft, Abrikosov lattices, quantum Hall-effect, chaos, Bose Einstein condensates, superfluidity are some of the examples where vortices are present. The circulatory motion of matter or any other physical parameter about the vortex core is being studied for its positive and negative impacts. An optical phase singularity [15–17], also called as a vortex is a space curve of amplitude null, around which the optical currents curl and is observed as a point phase defect in any two dimensional observation plane. The core of a singularity is a zero of the field where phase is indeterminate. The wavefronts are helical ramps winding about the vortex core. The interest in optical vortices has grown leaps and bounds recently, as their presence, usefulness and their properties have evoked more curiosity in the research community. Efforts to generate, detect, examine and use or avoid them are growing day by day.

Diffraction experiments involving vortices are strikingly interesting. The diffraction patterns obtained by non-singular beam illumination of apertures are completely different to their singular beam counterparts. The single slit, double slit and triangular aperture diffraction patterns [18–22] reveal the topological charge of the incident singular wave. Here we report the anomalous phenomenon that the fringe spacing does not depend on pin-hole separation in the Young’s experiment with waves near zero of an optical phase singularity. But for larger pinhole separation such an anomalous phenomenon is not discernible and the usual Young’s fringes appear. The anomalous phenomenon on fringe spacing can be understood if one considers the presence of an additional radial component to the phase gradient field near the core. Many diffraction experiments reported so far have missed this aspect as the phase gradient in a vortex beam is normally considered to have components that are only azimuthal and longitudinal. This work reveals the vortex core structure and is the first experimental evidence to the existence of a radial component of this phase gradient.

2. Young’s experiment

Superposition of Huygen’s secondary waves from two pinholes S₁ and S₂ illuminated by a coherent monochromatic plane wave results in the interference pattern, in which the formation of bright fringes are governed by the phase difference condition $k\left(l_1-l_2\right)=2q\pi$ where $k$ is the propagation constant, q is an integer and $l_1,l_2$ are respectively the distances S₁P and S₂P in Fig. 1. The Fraunhofer diffraction pattern [23,24] and the aperture transmittance function are connected by a Fourier transform – an operation which is shift invariant. Shift invariance means that the diffracted field amplitudes from the two individual slits overlap exactly in the Fraunhofer diffraction plane. Such an overlap of wavefunctions is partial when the pattern is observed in the near field – nevertheless the arguments presented here are applicable where interference fringes are seen.

 

Fig. 1 Schematic of the Young’s Experiment

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Under plane wave illumination at normal incidence, the phase distribution across the aperture plane is constant and the propagation vector points in the direction of the optical axis of the system. But when the two pinholes are illuminated by a singular beam, the propagation vector is no longer unidirectional and is now a function of spatial coordinates. At the two pin-hole locations the local propagation vectors point in two different directions and are represented here by vectors ${\overrightarrow{k}}_1$ and ${\overrightarrow{k}}_2$ respectively.

Consider a scalar optical field $\psi \left(\overrightarrow{r}\right)=A\left(\overrightarrow{r}\right)\exp \left\{i\chi \left(\overrightarrow{r}\right)\right\}$ which has a phase singularity at $\overrightarrow{r}=0$. The optical field in the immediate neighbourhood of the core of the vortex is characterized by high phase gradients. Phase gradient which is normal to phase contour surface $\chi \left(\overrightarrow{r}\right)=\text{constant}$ is the propagation vector $\overrightarrow{k}=\nabla \chi$. The propagation vector (in cylindrical coordinate system) for a singular beam of charge $m$ is given by$\overrightarrow{k}=k_r\widehat{r}+k_{\varphi }\widehat{\varphi }+k_z\widehat{z}$. The origin of the coordinate system is fixed at the vortex core in the aperture plane and $\widehat{z}$ coincides with the optical axis of the experimental setup. The azimuthal component is $k_{\varphi }=m/r$ and the longitudinal component is $k_z=\sqrt{k^2-k_r^2-k_{\varphi }^2}$. The presence of azimuthal components has been experimentally studied by many research groups [25–29]. Since the azimuthal component varies as $\left(m/r\right)$, in the regions very close to the vortex core, the phase gradient will be very high. In this region waves can vary arbitrarily faster than the wavelength [30–32] and can have an evanescent wave component [33]. The phase gradient $\left(\nabla \chi \right)$ in this superosicllatory region, with faster than Fourier behaviour, must have a radial component in addition to the pure circulating phase gradients associated with a vortex. Here we provide the first experimental evidence to the presence of a radial component.

As the direction of the propagation vector is not constant over the wavefront, the phase distribution and the angle of incidence at the two apertures are not same. Assuming that the two pinholes are small, the phase at the two pin-hole locations are ${\chi }_1$ and ${\chi }_2$ respectively. There is also a relative tilt between ${\overrightarrow{k}}_1$ and ${\overrightarrow{k}}_2$vectors. These relative tilt components lead to the change in the number of Young’s fringes.

The Poynting vector (time-averaged energy flow) in optical fields is given by$\stackrel{\rightharpoonup }{j}\left(r\right)=\mathrm{Im}\psi *\left(r\right)\nabla \psi \left(r\right)=I\nabla \chi$, where I is the intensity. Also known as optical current, it flows in a direction normal to the wavefronts, which is the direction of the propagation vector. Berry [34,35] has suggested that the geometry of these lines are such that they can spiral into or out of the core of a vortex, if the scalar field $\psi \left(\overrightarrow{r}\right)$ is expanded about its zero, up to third order in the distance r from the vortex.

Berry [35] has shown that optical beams whose amplitude distribution $A(r)$ is a function of the form $r^n$, where n is an integer, have radial phase gradient. In a typical example, consider a LG beam [36] given by

3. Experimental results and discussion

In our experiment (Fig. 1), we have used two small circular apertures each of radius $a=0.3mm$. The four pin hole separations that were tried are $d=0.9mm,1.2mm,1.5mm$and 2.4mm. We actually tried to have the pin-hole separation to be as small as possible, but as the distance between the pin-holes is reduced, the intensity of light diffracted also becomes very low and that puts limitation on the detection. We have used Imagingsource DFK51AU02 Color camera and its gain has to be adjusted to capture low intensity levels. Thus the intensity is presented in arbitrary units. A spatially filtered collimated light from a He-Ne laser source ($\lambda =632.8nm$) is infested with a phase singularity by a spiral phase plate (RPC Photonics). This singular beam is incident on the two pin-hole arrangement and the diffraction pattern is observed by using a camera. Use of two long slits was avoided, as there would be considerable phase variation across each of the slits. Hence the radius of each of the circular apertures is kept small and at the same time they allow enough light to pass through to carry out the experiment. The Young’s fringes are hence modulated by diffraction envelope which is given by the Airy function $\left(2J_1(\nu )/\nu \right)$, where $J_1\left(\nu \right)$is the Bessel function of first kind, $\nu =\left(2\pi /\lambda \right)a\sin \theta$ and $\theta$ is the diffraction angle.

In the Young’s experiment performed in our lab, we have varied m, the magnitude of the topological charge of the vortex from 1 to 4. The two pin-holes have been positioned equidistant from the core such that the pin-holes and the vortex core are collinear in the aperture plane. Results obtained by varying their separation are presented in rows whereas columns present the results when the magnitude of the topological charge is varied. The first column shows the Young’s fringes obtained under non-singular beam $\left(m=0\right)$ illumination. From the results shown in Fig. 2, one can observe that there is an increase in the number of maxima and minima as the topological charge is increased even though the pin-hole separation distance is kept constant. A comparison with the fringes obtained with non-singular beam illumination shows this increase in the number of fringes. This can be accounted only by the presence of a radial component of phase gradient near the core of the vortex. This result points to the fact that as the charge of vortex is increased the stronger phase gradient at the pin-hole location with concomitant increase in the radial component. Similarly the variation in the radial component of phase gradient as we go closer to the core can also be noticed.

 

Fig. 2 Experimentally observed Young’s fringe patterns

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Since the experiment involves low intensity levels of the vortex core neighbourhood, we also computed the Fraunhofer diffraction pattern (Fig. 3). Even if at the aperture plane the vortex phase distribution is considered to have only azimuthal phase gradient, diffraction due to propagation plays a greater role in the propagation vector spiraling inwards. The first three rows of Figs. 2 and 3 present the interference patterns for the increasing pin-hole separation. The last row depicts the case where the pin-holes are well separated so that they are not in the regions of high vortex phase gradient [22].

 

Fig. 3 Simulated Young’s fringe patterns

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To put things in a simpler perspective, non-singular plane wave illumination corresponds to the case where the propagation vectors at the two pinhole locations do not have a relative tilt between them where as, in the case of singular beam the two propagation vectors have a relative tilt between them and this extra tilt is responsible for the extra fringes in the diffraction pattern. As the charge of the vortex or the distance between the two pinholes is changed the tilt between the propagation vectors at the pinhole locations also changes and hence the number of fringes.

4. Conclusion

An interesting observation in the formation of Young’s fringes has been seen and reported in this work. The number of fringes has shown to defy the general relation between the fringe spacing and pinhole separation. This has been attributed to the presence of the radial component of the phase gradient of singular beams. The transverse component of the phase gradient, spirals in and out of the vortex core. This behaviour is seen very near to the phase singularity where faster than Fourier phenomenon is observed. Outside this superoscillatory region, it circulates about the singularity. This radial component is responsible for the observed change in the Young’s fringe pattern and has been experimentally verified. This aspect has not been seen and reported so far. This work is the first experimental evidence of this phenomenon.