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?