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New PDF release: Companion to the Weekly Problem Papers

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By John Milne

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1 Screw Dislocation Let us begin by imagining that the phase singularity line extends along the z-axis, passing through the origin x = y = z = 0. In the immediate neighborhood of the axis, the field is almost zero, and because the zero line is parallel to the axis, there will be very little change in the field behavior in the z-direction. We then have, to a good approximation, ∂u ≈ 0. 12) This means that, in the immediate neighborhood of the origin, the slow component of the wavefield satisfies the equation ∂ 2u ∂ 2u + 2 = 0.

What do we expect to be the behavior of the phase fronts? Let us take the simplest case, a = −1. 7. Working in an (x , y, z )coordinate system, we write (x + iy) = ρ eiφ , which implies that the field may be written as U(x , y, z ) = ρeiφ eik(z cos θ+x sin θ) . 34) The surface of constant phase will satisfy z = − tan θx − φ/ cos θ. 9. 9 Illustration of (a) a mixed edge-screw dislocation for θ = π/8 and (b) a mixed edge-screw dislocation for θ = 12π/25. helix becomes infinite and we end up with a pure edge dislocation.

In keeping with our earlier discussion, we expect derivatives that act in the direction of this line to be negligible. 26) z = −x sin θ + z cos θ. 27) We have to be careful here, because the use of the paraxial equation automatically treats the derivatives along x and z on a different footing. 22, and rederive our paraxial expression in these terms. 28) which in terms of our new coordinates becomes ∂2 ∂2 ∂2 U(r) = 0. 29) Generic Properties of Phase Singularities 45 Using U(r) = u(r)eikz , we may write a reduced equation, 2ik sin θ ∂ ∂ ∂2 ∂2 ∂2 u(r) = 0.

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Companion to the Weekly Problem Papers by John Milne

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