Phase vortices from a Young’s three-pinhole interferometer

An analysis is presented of the phase vortices generated in the far field by an arbitrary arrangement of three monochromatic point sources of complex spherical waves. In contrast with the case of three interfering plane waves, in which an infinitely extended vortex lattice is generated, the spherical sources generate a finite number of phase vortices. Analytical expressions for the vortex core locations are developed and shown to have a convenient representation in a discrete parameter space. Our far-field analysis may be mapped onto the case of a coherently illuminated Young’s interferometer, in which the screen is punctured by three rather than two pinholes.

Figure 1

(Color online) In this schematic of a Young’s three-pinhole interferometer, incident illumination (a) impinges upon a punctured screen (b), producing three divergent waves (c), which interfere, giving rise to a network of nodal lines (d). These are observed as phase vortices over a plane (e).

Figure 2

(Color online) Interference of three plane waves giving rise to an infinite, regular vortex lattice [see Eq. (2)]. Here, the wave vectors are oriented symmetrically with respect to the $z$ axis, with ${\mathbf{k}}_1=(\sqrt{3},-1,10)$, ${\mathbf{k}}_2=(-\sqrt{3},-1,10)$, and ${\mathbf{k}}_3=(0,2,10)$. The real (a) and imaginary (b) parts, amplitude (c), and phase (d) of the wave function $\Psi (x,y,z=0)$ are shown. In (b), zero contours of the real (dashed lines) and imaginary (solid lines) parts are overlaid. In (c) is overlaid the vector field of the probability current density, which is seen to rotate counterclockwise around vortices and clockwise around antivortices. In (d), vortices are visible as the ends of branch cuts indicated by dark to light $(-\pi \to \pi )$ steps. Values in (a)–(c) are represented by linear levels from dark to light (minimum to maximum).

Figure 3

Coordinate system. The point sources $r_1$, $r_2$, and $r_3$ lie in the plane $z=0$. The position vector $\mathbf{r}$ is to a point with cylindrical-polar coordinates ($r_{\perp }$, $\theta$, $z_0$).

Figure 4

Phasor diagram from which vortex solution conditions are established [27]. Three phasors ${\mathbf{p}}_1$, ${\mathbf{p}}_2$, and ${\mathbf{p}}_3$—which, respectively, correspond to the three terms on the right-hand side of Eq. (4), at a given point $\mathbf{r}$ on a nodal line—form a closed triangle corresponding to a resultant zero amplitude. Two angles $\gamma$ and $\eta$ arise in the construction, as indicated. (a) In the case of arbitrary amplitudes for ${\mathbf{p}}_1$, ${\mathbf{p}}_2$, and ${\mathbf{p}}_3$, the tip of ${\mathbf{p}}_2$ is constrained to lie on circle $A$ of radius $\mid {\mathbf{p}}_2\mid$, with the tail of ${\mathbf{p}}_3$ being constrained to lie on circle $B$ of radius $\mid {\mathbf{p}}_3\mid$. (b) For equal amplitudes $\mid {\mathbf{p}}_1\mid =\mid {\mathbf{p}}_2\mid =\mid {\mathbf{p}}_3\mid$ an equilateral triangle is formed. The dashed construction represents another of the six equivalent alternatives formed by permutating the phasor order.

Figure 5

A parameter-space ellipse, in the $(m,n)$ plane, sets an upper limit on the number of vortices that are created by the interfering radiation from three point sources. (a) The bounding ellipse has center $(m_0,n_0)$, makes an angle $\phi$ to the positive $m$ axis, and has semiaxis lengths $a^{\prime }$ and $b^{\prime }$. (b) The enclosed $(m,n)$ pairs lie on a discrete square lattice of unit spacing. Each of these interior points corresponds to two vortices.

Figure 6

(Color online) (a), (c) Amplitude $\mid \Psi \mid$ and (b), (d) associated phase $\chi$ from Eq. (4) at $z_0=25{\lambda }_0$ with different source arrangements, shown pictographically alongside the (a) and (c) labels. Small circles overlaid on numerical simulations show the vortex locations as approximated by Eqs. (13, 17). Values in (a) and (c) are represented by linear levels from dark to light (minimum to maximum). The phase ranges from $-\pi$ (dark) to $\pi$ (light). All sources are in phase (i.e., ${\varphi }_2={\varphi }_3=0$), ${\lambda }_0=\pi$, and the source arrangements are (a), (b) $r_2=r_3=3{\lambda }_0$, ${\theta }_3=0°$ and (c), (d) $r_2=r_3=3{\lambda }_0$, ${\theta }_3=60°$. Note that a spherical background has been subtracted from all phase maps, as described in the main text.

Figure 7

(Color online) Phase $\chi$ from Eq. (4) and vortex-core locations approximated by Eqs. (13, 17), with variation in source arrangement. The representation and parameters are as described in Fig. 6. The scale bar in (c) applies to all images and corresponds to $z_0=25{\lambda }_0$, except for (d) which corresponds to $z_0=250{\lambda }_0$. Small arrows in (b), (c) point to the exact vortex locations, to indicate their deviation from the far-field predictions. ${\lambda }_0=\pi$ and source arrangements are (a) $r_2=r_3=3{\lambda }_0$, ${\theta }_3=10°$, (b) $r_2=r_3=3{\lambda }_0$, ${\theta }_3=90°$, (c) $r_2=r_3=3{\lambda }_0$, ${\theta }_3=175°$, (d) $r_2=r_3=3{\lambda }_0$, ${\theta }_3=175°$, $z_0=250{\lambda }_0$, (e) $r_2=r_3=3{\lambda }_0$, ${\theta }_3=180°$, and (f) $r_2=3{\lambda }_0$, $r_3=1.5{\lambda }_0$, ${\theta }_3=60°$. A spherical background has been subtracted from all phase maps (see main text).

Figure 8

Parameter-space ellipses corresponding to Figs. 6, 7. Enclosed lattice points define vortex locations.

Figure 9

(Color online) Vortices move with varying source phase along trajectories and lie at the intersections of two independent sets of hyperbolas. Two independent phases give rise to two sets of hyperbolas indexed by $m$ (solid lines) and $n$ (dashed lines), respectively. The hyperbolas are shown overlaying the phase $\chi$ from Eq. (6). Note that the result of this equation is $\ge 0$ for arbitrary $(r_{\perp },\theta )$. The parameters are the same as Fig. 7f.

Figure 10

(Color online) Numerical simulations of nodal line structures developed by three sources arranged symmetrically at the corners of an equilateral triangle of side length $r_2=r_3=3{\lambda }_0$ [cf. Figs. 6c, 6d]. The region shown is $z=0-5{\lambda }_0$ for a source wavelength of ${\lambda }_0=\pi$. (a) Three spherical wave sources. (b) Three pinholes.