By Ulf Leonhardt

Appropriate for complex undergraduate and graduate scholars of engineering, physics, and arithmetic and clinical researchers of every kind, this is often the 1st authoritative textual content on invisibility and the technology at the back of it. It introduces the mathematical foundations of differential geometry and demonstrates sensible functions from basic relativity to electric and optical engineering. greater than a hundred full-color illustrations, plus routines with ideas.

Content:

• entrance subject

• desk of Contents

1. Prologue

2. Fermat's precept

three. Differential Geometry

four. Maxwell's Equations

five. Geometries and Media

• Appendix

• Bibliography

Index

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**Example text**

Solving for z we obtain the inverse M¨obius transformation: dw − b z= . 18) can be expressed by the following chain of ele- §5 Optical conformal mapping 39 mentary transformations w = z4 = z3 + a , c z3 = bc − ad z2 , c2 z2 = 1 , z1 z1 = z + d . 7 Verify by direct calculation that w = z4 (z3 (z2 (z1 (z)))). 20) describe geometric operations: the M¨obius transformation consists of a spatial shift, followed by an inversion, then a complex multiplication—geometrically, a rotation and re–scaling, ﬁnished by a further shift.

15). But in virtual space the line is straight. Curved objects thus may appear ﬂat in optical conformal mapping (as if the fugu of Fig. 20 turns into a ﬂatﬁsh). Flat objects may be hidden by camouﬂage, assuming the colour and texture of the ﬂoor (as ﬂatﬁsh are masters of). Such partial cloaking is less demanding to implement∗ and may be equally eﬀective (Li and Pendry [2008]). §6. Transmutation We have seen how optical conformal mapping transforms a refractive–index proﬁle in physical space into a new proﬁle in virtual space.

Integration gives ln(∂y n) = ln n + const and so leads to the diﬀerential equation ∂y n = an with the solution n = n0 exp(ay) where both a and n0 are real constants. Neither the interface in refraction (Fig. 1) nor the index proﬁle of the mirage shown in Fig. 1 are of this type: such optical illusions are created by curved geometries. 4 Find all transformation media in 2D with radially symmetric refractive–index proﬁles n(r). Why is the Luneburg lens not a transformation medium? 13) as a diﬀerential equation in r.