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FDTD Simulations of Metamaterials inTransformation Optics
Presentation by Reece Boston
March 7, 2016
What is Transformation Optics?
I Transformation Optics (TO) is the theoretical prediction ofmaterial parameters, ε̂, µ̂ for a medium that effects any desiredtransformation in the paths of light rays, by association betweenMaxwell’s equations in a material with Maxwell’s Equations in acurved space.
I Theoretical underpinnings in differential geometry, with tie-insto GR.
I Materials predicted are, in general, inhomogenous andbi-anisotropic.
~D = ε̂ ~E + γ̂1~H
~B = µ̂ ~H + γ̂2~E
I In most simple applications, γ̂1 = γ̂2 = 0.
Why Does Anyone Care About Transformation Optics?
I Using this procedure, we have near-perfectcontrol over the movement of light.
I Most famously used to design Pendry’sCloak of Invisibility. Not magic, but realscience! −→ $$$ from military.
I Useful in antenna design, focusingincoming radio waves more efficiently.
I Design of optical devices in systems whereGR effects are relevant, such as satellites inorbit.
I Studying cosmological models in thelaboratory.
I Creating anything our imagination desires!I’ll show you!
Example: Beam TurnerI Start in flat empty optical space, (x ′, y ′).I Perform transformation to curved physical space, (x , y):
x ′ −→ x = x ′ cos
(πy ′
2R2
)y ′ −→ y = x ′ sin
(πy ′
2R2
)
I This transformation will turn straight lines in to either rays orcircular arcs
Example: Beam Turner
I Require distances to be conserved between two spaces:
ds ′2 = dx ′2 + dy ′2 = gijdxidx j
I We take Jacobian matrix of transformation, invert to finddx ′ = dx cos θ + dy sin θ
dy ′ =2R2
πr(−dx sin θ + dy cos θ)
I Leads to gij =
(cos2 θ + A2 sin2 θ (1− A2) sin θ cos θ
(1− A2) sin θ cos θ sin2 θ + A2 cos2 θ
)where A = 2R2/πr .
I This gij is metric of curved physical space.I Light rays “see” optical space, we “see” curved space.
Interlude
I So far, we have the “Transformation” part of “TransformationOptics”.
I Transformation step generates curved space metrics, gij . Usefulfor clearly specifying how rays should be distorted. Not strictlynecessary.
I Now that we have a curved space, we’d like to examineMaxwell’s Equations in curved space, to get the “Optics” part.
I This requires a slight detour as we discuss spatial derivatives incurved space.
I We shall now learn everything we need about DifferentialGeometry in four short slides.
Curved Space Derivatives
I The Physics Major’s Dream:
∇·~F (r , θ, φ) =
(∂
∂r,∂
∂θ,∂
∂φ
)·(Fr ,Fθ,Fφ) =
∂Fr∂r
+∂Fθ∂θ
+∂Fφ∂φ
I Wouldn’t it be great! But why isn’t it?(Using ∇ =
(∂∂r ,
1r∂∂θ ,
1r sin θ
∂∂φ
)isn’t any better. )
I Vector operators∇,∇·, and∇× are formally defined in terms ofvolume and area elements:
T (∇) = limV→0
1
V
{
∂V
T (n̂)dS
where T is a linear expression, e.g. T (~a) = ~a · ~F
Curved Space Derivatives, cont.I In spherical coordinates, volume and area elements are
dV = r2 sin θ dr dθdφ,
dSr = r2 sin θ dθdφ, dSθ = r sin θ dr dφ, dSφ = r drdθI Taking sides of a cuboid,
T (∇) =1
r2 sin θdrdθdφ
([T (r2r̂)+ − T (r2r̂)−] sin θdθdφ
+[T (sin θθ̂)+ − T (sin θθ̂)−]rdrdφ
+[T (φ̂)+ − T (φ̂)−]rdrdθ)
=1
r2 sin θ
(∂T (r2 sin θr̂)
∂r+∂T (r sin θθ̂)
∂θ+∂T (r φ̂)
∂φ
)
=1
r2
∂T (r2r̂)
∂r+
1
r sin θ
∂T (sin θθ̂)
∂θ+
1
r sin θ
∂T (φ̂)
∂φI This is correct answer in spherical coordinates for any linear
expression T .
But Wait, There’s More!I Switch from orthonormal basis to coordinate basis,
r̂ = ~er , θ̂ = r~eθ, φ̂ = r sin θ~eφ.
I Then gij =
1 0 00 r2 00 0 r2 sin2 θ
, and√
det gij = r2 sin θ.
I Then,
T (∇) =1
r2 sin θ
(∂T (r2 sin θr̂)
∂r+∂T (r sin θθ̂)
∂θ+∂T (r φ̂)
∂φ
)
=1
r2 sin θ
(∂T (r2 sin θ~er )
∂r+∂T (r2 sin θ~eθ)
∂θ+∂T (r2 sin θ~eφ)
∂φ
)
=1√
det gij
∂T (√
det gij~ea)
∂xa
I This is general expression in any coordinate system, in anycurved space, for any linear expression T .
Divergence and Curl in Curvilinear Coordinates
I Maxwell’s Equations require divergence and curl.I Divergence is easy: T (~a) = ~a · ~F
∇ · ~F =1√g
∂(√g~ea · ~F )
∂xa=
1√g
∂(√gF a)
∂xa
I Curl is slightly more difficult: T (~a) = ~a× ~F
∇× ~F =1√g
∂(√g~ea × ~F )
∂xa=
1√g
∂(√g~ea)
∂xa× ~F + ~ea × ∂ ~F
∂xa
= ~ea × ~eb ∂Fb∂xa
= εabc~ec∂Fb∂xa
I Now let’s go back to Beam Turner and put these in Maxwell’sEquations.
Back to the Beam Turner
I We found gij =
(cos2 θ + A2 sin2 θ (1− A2) sin θ cos θ
(1− A2) sin θ cos θ sin2 θ + A2 cos2 θ
)as
metric tensor of curved space.I Maxwell’s Equations in curved space (without sources, c = 1))
are
1√g
∂
∂x i(√gE i ) = 0
1√g
∂
∂x i(√gB i ) = 0
1√g
[ijk]∂Ek
∂x j= −∂B
i
∂t
1√g
[ijk]∂Bk
∂x j=∂E i
∂t,
I Rearranging slightly, using ~H = ~B in free space
∂
∂x i(√gg ilEl) = 0,
∂
∂x i(√gg ilHl) = 0
[ijk]∂Ek
∂x j= −
∂(√gg ilHl)
∂t[ijk]
∂Hk
∂x j=∂(√gg ilEl)
∂t,
Beam Turner, cont.I Maxwell Equation in our curved space (c = 1):
∂
∂x i(√gg ijEj) = 0,
∂
∂x i(√gg ijHj) = 0
[ijk]∂Ek
∂x j= −
∂(√gg ijHj)
∂t[ijk]
∂Hk
∂x j=∂(√gg ijEj)
∂t,
I Maxwell Equations in a medium:
∂
∂x iD i =
∂
∂x i(εijEj) = 0,
∂
∂x iB i =
∂
∂x i(µijHj) = 0
[ijk]∂Ek
∂x j= −∂B
i
∂t= −
∂(µijHj)
∂t[ijk]
∂Hk
∂x j=∂D i
∂t=∂(εijEj)
∂t,
I Now we transform again, form curved space to flat space, bymeans of a medium, called “Transformation Medium”:
εjk = µjk =√gg jk =
(A cos2 θ + 1
A sin2 θ (A− 1A) sin θ cos θ
(A− 1A) sin θ cos θ 1
A cos2 θ + A sin2 θ
).
I Light can’t tell the difference between gik and εjk , µjk .
Beam Turner Wrap UpI To track what we did:
1. We started in a flat optical space, where light moves alongstraight lines.
2. We performed a transformation to a curved space to cause lightrays to bend.
3. Then we undid the curvature in the space with an equivalentmedium.
I This medium we found to be anisotropic and inhomogenous.I This is the general procedure for any desired spatial curvature.
Now What?
I We want to verify theoretical result before wasting time andmoney building it.
I Simulate light impinging upon a material with ε, µ as givenabove.
I In E&M simulations, two main methods are:I Finite Element Method (FEM).I Finite Difference Time Domain (FDTD).
I FEM discretizes functional solution space, approximatessolution as sum of basis functions. Solves for steady-state(infinite time) solution.
I FDTD discretizes spatial and temporal grid, find field values atgrid points. Marches forward in the time domain.
I In our calculations, we used a 2-dimensional FDTD calculationfor Transverse Magnetic case.
Simulation of Beam Turner
Quater-ring shaped device with
εjk = µjk =
(2R2πr cos2 θ + πr
2R2sin2 θ ( 2R2
πr −πr
2R2) sin θ cos θ
( 2R2πr −
πr2R2
) sin θ cos θ πr2R2
cos2 θ + 2R2πr sin2 θ
).
The Update ProcedureI Consider the Maxwell-Ampere Equation for ~D ,
∂ ~D
∂t= ∇× ~H.
I Discretize time in to timestep ∆t, space by ∆x . Then
~Dn+1/2 − ~Dn−1/2
∆t=
1
∆x∇̃ × ~Hn
(∇̃×~F )z = Fx(i , j+1, k)−Fx(i , j , k)−Fy (i+1, j , k)+Fy (i , j , k).I Rearranging
~Dn+1/2 = ~Dn−1/2 +∆t
∆x∇̃ × ~Hn
I Likewise, for Maxwell-Faraday Equation,
~Bn+1 = ~Bn − ∆t
∆x∇̃ × ~En+1/2
I In between these two, we perform
~En+1/2 =1
εoε−1 ~Dn+1/2, ~Hn+1 =
1
µoµ−1 ~Bn+1
The Yee CellI ~E and ~B fields are staggered in time and space.I This process tends to even out errors due to grid approximation.I In 1D:
Taken from Understanding the Finite Difference Time-Domain Method, John Schneider, www.eecs.wsu.edu/~schneidj/ufdtd
The Yee CellI ~E and ~B fields are staggered in time and space.I This process tends to even out errors due to grid approximation.I In 2D:
Taken from Understanding the Finite Difference Time-Domain Method, John Schneider, www.eecs.wsu.edu/~schneidj/ufdtd
The Yee Cell
I ~E and ~B fields are staggered in time and space.I This process tends to even out errors due to grid approximation.I In 3D:
The FDTD Method in Summary
I Divide space and time according to Yee cell.
I Specify ε, µ over entire spatial domain.
I Introduce source field by altering ~E value at some point(s).
I Propagate source field through space and time by leap-frogalgorithm:
1. ~Dn+1/2 = ~Dn−1/2 + ∆t∆x ∇̃ × ~Hn
2. ~E n+1/2 = 1εoε−1 ~Dn+1/2.
3. ~Bn+1 = ~Bn − ∆t∆x ∇̃ × ~E n+1/2
4. ~Hn+1 = 1µoµ−1 ~Bn+1
I Continue iterating in time, as long as you wish.
Example: Cloak of InvisibilityI Long a staple of fantasy and science fiction:
How do we make it a reality?I Transform the single point of the origin in to a circle of radius R1
r ′ → r = R1 + r ′R2 − R1
R2, θ′ → θ = θ′, z ′ → z = z ′
Example: Cloak of InvisibilityI Long a staple of fantasy and science fiction:
How do we make it a reality?I Transform the single point of the origin in to a circle of radius R1
r ′ → r = R1 + r ′R2 − R1
R2, θ′ → θ = θ′, z ′ → z = z ′
Example: Cloak of InvisibilityI Long a staple of fantasy and science fiction:
How do we make it a reality?I Transform the single point of the origin in to a circle of radius R1
r ′ → r = R1 + r ′R2 − R1
R2, θ′ → θ = θ′, z ′ → z = z ′
I This leads to
gij =
(R2
R2 − R1
)2
cos2 θ + α2 sin2 θ (1− α2) sin θ cos θ 0(1− α2) sin θ cos θ sin2 θ + α2 cos2 θ 0
0 0(R2−R1R2
)2
where α = r−R1
r .I As above, an equivalent medium is given by εij = µij =
√gg ij ,
εij = µij =
α cos2 θ + 1α sin2 θ (α− 1
α) sin θ cos θ 0(α− 1
α) sin θ cos θ 1α cos2 θ + α sin2 θ 0
0 0(
R2R2−R1
)2α
.
Simulation of Cloak of Invisibility
Optical ‘Bag of Holding’
I Transform spatial distances inside a cylinder so that the inside isbigger than the outside.
Optical ‘Bag of Holding’
I Transform spatial distances inside a volume so that the inside isbigger than the outside.
I This means ds2 = B2(dx2 + dy2 + dz2) for some scale factor B .
I gij =
B2 0 00 B2 00 0 B2
⇒ εij =√gg ij =
B 0 00 B 00 0 B
I This is similar normal dielectric, ε = B , like glass or plastic!
I Inside dielectric, phase velocity v < c .
I Same speed in optical space −→ slower speed in physical space.
I We also have εij = µij : this is non-scattering condition. Trulynon-glare glasses.
Ilustratration of Velocity Distotion
Example: Schwarzschild Black HoleI Don’t need transformation; we already know the metric.
ds2 = −(
1− R∗r
)dt2+
(1− R∗
r
)−1
dr2+r2dθ2+r2 sin2 θdφ2.
I More complicated, as now we have spacetime curvature.I Equations for this first discovered by Plebanski, listed without
derivation:
εij = µij = −√|g |
g00g ij
(γT1 )ij = (γ2)ij = −[ijk]g0j
g00,
where D i = εijEj + γ ij1 Hj , B i = µijHj + γ ij2 Ej .I Using this, we find, for 2D case:
εij = µij =r
r − R∗
1− R∗r
x2
r2 −R∗r
xyr2 0
−R∗r
xyr2 1− R∗
ry2
r2 00 0 1
.
I Let’s watch it go!
Illustration of Gravitational Lensing
Further WorkI Simulations analyzing metamaterial periodic elements for actual
construction.
I Reduced cloaks for broadband cloaking.I Perfect black body layer, based off black hole metric. Makes
perfect ‘one-way mirror’, possibly for solar panels.I Anti-telephonic device that I have affectionately named
“Galadriel’s Mirror.”I Possibilities are limited only by our imagination.
We are the Masters of Time and Space!
Light bends to our whim and caprice!
Questions?