Isotropic Transformation Optics Approximate Cloaking Allan

50 Pages·2008·0.42 MB·English

by  Allan Greenleaf

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)