By Thomas Henry Eagles
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Extra info for Constructive geometry of plane curves. With numerous examples.
21) for ψ = E is exact. §5. Optical conformal mapping In this Section we develop the concept of transformation media for the physically simplest case: isotropic materials in 2D. Such optical media perform conformal mappings of the plane. We also introduce some aspects of complex analysis, because analytic functions turn out to generate conformal maps. 21) in two dimensions. They may be light waves in planar integrated optics or light rays propagating in 34 2 Fermat’s principle a plane in 3D, for example in spherically symmetric refractive–index proﬁles.
24) for Δ in the last step. We diﬀerentiate ct with respect to b and get from Eqs. 24) √ r+ r+ n2 r2 − b2 ∂r b dr ∂Δ ∂t = +Δ+b − c 2 2 ∂b r ∂b r− r ∂b ρ −b r− ∂Δ = b . 23) the angle diﬀerence Δ between two turning points is constant. 28) ∂b the time delay is uniform. If we employ proﬁles with closed loops in optical conformal mapping the time delay in virtual space is the same as the time delay in physical space, because only space is transformed, not physical time. ) As we discussed in §5, such closed loops in virtual space are needed in invisibility devices.
13) w(ξ± ) = w± = ei(φ0 ±2γ) , dw dξ = ∓2 eiφ0 . 23: Eaton lens. 8) for r ≤ 1. The rays leave in the same direction they came from; they are retroreﬂected. The Eaton lens makes a perfect cat’s eye. As |w± |2 = 1, we see that the half ellipses indeed begin and end at the unit circle. 14) shows that dw/dξ = |dz/dw|2 dw/dξ, so the Newtonian velocity dw/dξ (with transformed parameter) points in the direction of dw/dξ and hence in the φ0 direction for ξ− , the direction of the incident rays. The half ellipses thus match the incoming rays.
Constructive geometry of plane curves. With numerous examples. by Thomas Henry Eagles