Isotropic Transformation Optics and Approximate Cloaking Allan Greenleaf joint with Yaroslav Kurylev Matti Lassas Gunther Uhlmann CLK08 @ CSCAMM September 24, 2008 Challenges of cloaking and other transformation optics (TO) designs: Challenges of cloaking and other transformation optics (TO) designs: Material parameters are • Anisotropic Challenges of cloaking and other transformation optics (TO) designs: Material parameters are • Anisotropic • Singular: −→ ∞ At least one eigenvalue 0 or at some points • Material parameters transform as tensors: (cid:2) Conductivity σ, permittivity (cid:3), permeability μ, effective mass m, ... • Material parameters transform as tensors: (cid:2) Conductivity σ, permittivity (cid:3), permeability μ, effective mass m, ... • −→ F : Ω Ω a smooth transformation: σ(x) pushes forward to a new conductivity, σ˜ = F σ, ∗ (cid:3) n 1 ∂Fj ∂Fk jk pq (F σ) (y) = σ ∗ det[∂Fj ] ∂xp ∂xq ∂xk p,q=1 − with the RHS evaluated at x = F 1(y) • F is a diffeomorphism, then ∇ · ∇ ⇐⇒ ∇ · ∇ (σ˜ )u˜ = 0 (σ )u = 0, where u(x) = u˜(F(x)). • F is a diffeomorphism, then ∇ · ∇ ⇐⇒ ∇ · ∇ (σ˜ )u˜ = 0 (σ )u = 0, where u(x) = u˜(F(x)). • For many TO designs, F is singular • F is a diffeomorphism, then ∇ · ∇ ⇐⇒ ∇ · ∇ (σ˜ )u˜ = 0 (σ )u = 0, where u(x) = u˜(F(x)). • For many TO designs, F is singular • ⇒ ∃ Removable singularity theory can = one-to-one correspondence { ∇ · ∇ } ↔ { ∇ · ∇ } Solutions of (σ˜ u˜) = 0 Solutions of (σ u) = 0 Electromagnetic wormholes [GKLU,2007] • Invisible tunnels (optical cables, waveguides)