Source: https://www.osapublishing.org/oe/abstract.cfm?uri=oe-14-7-3039
Timestamp: 2019-04-22 01:08:37+00:00

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When three or more plane waves overlap in space, complete destructive interference occurs on nodal lines, also called phase singularities or optical vortices. For super positions of three plane waves, the vortices are straight, parallel lines. For four plane waves the vortices form an array of closed or open loops. For five or more plane waves the loops are irregular. We illustrate these patterns numerically and experimentally and explain the three-, four- and five-wave topologies with a phasor argument.
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Fig. 1. Optical vortex line in an interference field. Left: periodic cell containing a vortex loop (dotted lines are parts of the loop outside the primitive cell). The intensity on the front and back planes is also represented. Right: phase cross section of the front/back face of the cell. The phase circulates in opposite directions at different points on the vortex line.
Fig. 2. Calculated vortex structure for three, four and five equal-amplitude plane waves (a, b, c) and their respective k-space configuration (d, e, f). Each is calculated on a 256×256×256 Talbot cell.
Fig. 3. The calculated vortex structures for four plane wave superpositions with amplitudes: a), a 1 + a 4 = a 2 + a 3 (b), a 1+a 4<a 2+a 3 (c), a 1+a 4 >a 2+a 3. The choice of k-vectors is the same as Fig. 2 (b), given in Fig. 2 (e). Attached multimedia shows rotation of cells around the optical axes (a, b, c) (1.4Mb, 1.3Mb, 1.1Mb).
Fig. 4. Experimentally measured vortex structures from the superposition of 4 plane waves with amplitudes giving rise to a) twisted vortex lines and b) vortex loops. The choice of k-vectors differs slightly from that in Fig. 3(b) and Fig. 3(c). Attached multimedia shows rotation of each cell around the optical axis (1.9Mb, 1.8Mb).
Fig. 5. Left: Vortex line in (x, y, z). Right: phasor diagrams at the points highlighted on a vortex a) loop and b) a line. For clarity, the orientation of the largest magnitude phasor has been fixed. The red arrows show the angular range of the smallest phasor, with respect to the largest. Note that in (a) this is restricted to less than 2π. Attached multimedia shows phasor configuration changing as the vortex paths are followed (2.3Mb, 2.1Mb).
Fig. 6. Examples of different vortex topologies that can be obtained from five waves of the same magnitude but different relative phase. Fig. 6 (a) is the same Talbot cell as Fig. 2(c). Figs. 6(b) and 6(c) are Talbot cells from the same wave magnitudes as (a) but with one wave advanced in phase by π/2 and π/3. Figs. 6(d),6(e), and 6(f) are their unwrapped counterparts.

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