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Break in Math by Peter Wolff

By Peter Wolff

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The zero surfaces of Young’s twopinhole experiment do not satisfy this condition, while the vortices of the three-pinhole experiment do. In general, we will find that optical vortices are generic features of wavefields, regardless of how they are generated; our next step is to look more closely at their structure. 2 TYPICAL FORMS OF WAVE DISLOCATIONS We have seen through various examples that phase singularities are typically lines in three-dimensional space, but have not yet explained why; in fact, this is quite easy to do in a qualitative manner.

16) This is a typical example of what we will call a screw dislocation. ) We may also write this in terms of cylindrical coordinates as U(ρ, φ, z) = αρeiφ eikz . 7 Illustration of a surface of constant phase for a screw dislocation. where the pitch p of the helix is given by p = 2π/k = λ. 7, from which the origin of “screw” in “screw dislocation” should be obvious. ) This is precisely the sort of vortex singularity that appears in our Laguerre–Gauss beams of order m = 1, as discussed in Chapter 2.

7, we consider the propagation of an n = 0, m = 2 field through 3 km of atmospheric turbulence. We now see that there are evidently four intersections of the Re(u) = 0 and Im(u) = 0, implying four vortices—what has happened? 7 The intensity and zero crossings of an m = 2 vortex after propagating through 3 km of atmospheric turbulence. 6. White circles have been placed over the intersections that indicate vortex locations. 22 Singular Optics all the vortices visible in the region, they would still add up to two: evidently a pair of vortices of opposite handedness was created as the field penetrated further into the turbulence.

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