This presentation discusses using transformation optics and finite-difference time-domain (FDTD) simulations to design metamaterials that manipulate light in desired ways. It begins with an overview of transformation optics and how material parameters can be derived to effect a spatial transformation on light rays. An example of a "beam turner" is presented, along with the calculations to determine the required inhomogeneous, bi-anisotropic material properties. The presentation then discusses using FDTD simulations to model light propagation through these designed materials by discretizing Maxwell's equations in space and time. Examples shown include simulations of the beam turner and cloaking devices.

1. FDTD Simulations of Metamaterials in
Transformation Optics
Presentation by Reece Boston
March 7, 2016

2. What is Transformation Optics?
Transformation Optics (TO) is the theoretical prediction of
material parameters, ˆ, ˆµ for a medium that effects any desired
transformation in the paths of light rays, by association between
Maxwell’s equations in a material with Maxwell’s Equations in a
curved space.
Theoretical underpinnings in differential geometry, with tie-ins
to GR.
Materials predicted are, in general, inhomogenous and
bi-anisotropic.
D = ˆE + ˆγ1H
B = ˆµH + ˆγ2E
In most simple applications, ˆγ1 = ˆγ2 = 0.

3. Why Does Anyone Care About Transformation Optics?
Using this procedure, we have near-perfect
control over the movement of light.
Most famously used to design Pendry’s
Cloak of Invisibility. Not magic, but real
science! −→ $$$ from military.
Useful in antenna design, focusing
incoming radio waves more efficiently.
Design of optical devices in systems where
GR effects are relevant, such as satellites in
orbit.
Studying cosmological models in the
laboratory.
Creating anything our imagination desires!
I’ll show you!

4. Example: Beam Turner
Start in flat empty optical space, (x , y ).
Perform transformation to curved physical space, (x, y):
x −→ x = x cos
πy
2R2
y −→ y = x sin
πy
2R2
This transformation will turn straight lines in to either rays or
circular arcs

5. Example: Beam Turner
Require distances to be conserved between two spaces:
ds 2
= dx 2
+ dy 2
= gij dxi
dxj
We take Jacobian matrix of transformation, invert to find
dx = dx cos θ + dy sin θ
dy =
2R2
πr
(−dx sin θ + dy cos θ)
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.
This gij is metric of curved physical space.
Light rays “see” optical space, we “see” curved space.

6. Interlude
So far, we have the “Transformation” part of “Transformation
Optics”.
Transformation step generates curved space metrics, gij . Useful
for clearly specifying how rays should be distorted. Not strictly
necessary.
Now that we have a curved space, we’d like to examine
Maxwell’s Equations in curved space, to get the “Optics” part.
This requires a slight detour as we discuss spatial derivatives in
curved space.
We shall now learn everything we need about Differential
Geometry in four short slides.

7. Curved Space Derivatives
The Physics Major’s Dream:
·F(r, θ, φ) =
∂
∂r
,
∂
∂θ
,
∂
∂φ
·(Fr , Fθ, Fφ) =
∂Fr
∂r
+
∂Fθ
∂θ
+
∂Fφ
∂φ
Wouldn’t it be great! But why isn’t it?
(Using = ∂
∂r , 1
r
∂
∂θ , 1
r sin θ
∂
∂φ isn’t any better. )
Vector operators , ·, and × are formally defined in terms of
volume and area elements:
T( ) = lim
V →0
1
V
∂V
T(ˆn)dS
where T is a linear expression, e.g. T(a) = a · F

8. Curved Space Derivatives, cont.
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 drdθ
Taking sides of a cuboid,
T( ) =
1
r2 sin θdrdθdφ
[T(r2
ˆr)+ − T(r2
ˆr)−] sin θdθdφ
+[T(sin θˆθ)+ − T(sin θˆθ)−]rdrdφ
+[T(ˆφ)+ − T(ˆφ)−]rdrdθ
=
1
r2 sin θ
∂T(r2 sin θˆr)
∂r
+
∂T(r sin θˆθ)
∂θ
+
∂T(r ˆφ)
∂φ
=
1
r2
∂T(r2ˆr)
∂r
+
1
r sin θ
∂T(sin θˆθ)
∂θ
+
1
r sin θ
∂T(ˆφ)
∂φ
This is correct answer in spherical coordinates for any linear
expression T.

9. But Wait, There’s More!
Switch from orthonormal basis to coordinate basis,
ˆr = er
, ˆθ = reθ
, ˆφ = r sin θeφ
.
Then gij =


1 0 0
0 r2 0
0 0 r2 sin2
θ

, and det gij = r2 sin θ.
Then,
T( ) =
1
r2 sin θ
∂T(r2 sin θˆr)
∂r
+
∂T(r sin θˆθ)
∂θ
+
∂T(r ˆφ)
∂φ
=
1
r2 sin θ
∂T(r2 sin θer )
∂r
+
∂T(r2 sin θeθ)
∂θ
+
∂T(r2 sin θeφ)
∂φ
=
1
det gij
∂T( det gij ea)
∂xa
This is general expression in any coordinate system, in any
curved space, for any linear expression T.

10. Divergence and Curl in Curvilinear Coordinates
Maxwell’s Equations require divergence and curl.
Divergence is easy: T(a) = a · F
· F =
1
√
g
∂(
√
gea · F)
∂xa
=
1
√
g
∂(
√
gFa)
∂xa
Curl is slightly more difficult: T(a) = a × F
× F =
1
√
g
∂(
√
gea × F)
∂xa
=
1
√
g
∂(
√
gea)
∂xa
× F + ea
×
∂F
∂xa
= ea
× eb ∂Fb
∂xa
= εabc
ec
∂Fb
∂xa
Now let’s go back to Beam Turner and put these in Maxwell’s
Equations.

11. Back to the Beam Turner
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.
Maxwell’s Equations in curved space (without sources, c = 1))
are
1
√
g
∂
∂xi
(
√
gEi
) = 0
1
√
g
∂
∂xi
(
√
gBi
) = 0
1
√
g
[ijk]
∂Ek
∂xj
= −
∂Bi
∂t
1
√
g
[ijk]
∂Bk
∂xj
=
∂Ei
∂t
,
Rearranging slightly, using H = B in free space
∂
∂xi
(
√
ggil
El ) = 0,
∂
∂xi
(
√
ggil
Hl ) = 0
[ijk]
∂Ek
∂xj
= −
∂(
√
ggil Hl )
∂t
[ijk]
∂Hk
∂xj
=
∂(
√
ggil El )
∂t
,

12. Beam Turner, cont.
Maxwell Equation in our curved space (c = 1):
∂
∂xi
(
√
ggij
Ej ) = 0,
∂
∂xi
(
√
ggij
Hj ) = 0
[ijk]
∂Ek
∂xj
= −
∂(
√
ggij Hj )
∂t
[ijk]
∂Hk
∂xj
=
∂(
√
ggij Ej )
∂t
,
Maxwell Equations in a medium:
∂
∂xi
Di
=
∂
∂xi
( ij
Ej ) = 0,
∂
∂xi
Bi
=
∂
∂xi
(µij
Hj ) = 0
[ijk]
∂Ek
∂xj
= −
∂Bi
∂t
= −
∂(µij Hj )
∂t
[ijk]
∂Hk
∂xj
=
∂Di
∂t
=
∂( ij Ej )
∂t
,
Now we transform again, form curved space to flat space, by
means of a medium, called “Transformation Medium”:
jk
= µjk
=
√
ggjk
=
A cos2 θ + 1
A sin2
θ (A − 1
A ) sin θ cos θ
(A − 1
A ) sin θ cos θ 1
A cos2 θ + A sin2
θ
.
Light can’t tell the difference between gik and jk, µjk.

13. Beam Turner Wrap Up
To track what we did:
1. We started in a flat optical space, where light moves along
straight lines.
2. We performed a transformation to a curved space to cause light
rays to bend.
3. Then we undid the curvature in the space with an equivalent
medium.
This medium we found to be anisotropic and inhomogenous.
This is the general procedure for any desired spatial curvature.

14. Now What?
We want to verify theoretical result before wasting time and
money building it.
Simulate light impinging upon a material with , µ as given
above.
In E&M simulations, two main methods are:
Finite Element Method (FEM).
Finite Difference Time Domain (FDTD).
FEM discretizes functional solution space, approximates
solution as sum of basis functions. Solves for steady-state
(infinite time) solution.
FDTD discretizes spatial and temporal grid, find field values at
grid points. Marches forward in the time domain.
In our calculations, we used a 2-dimensional FDTD calculation
for Transverse Magnetic case.

15. Simulation of Beam Turner
Quater-ring shaped device with
jk
= µjk
=
2R2
πr cos2 θ + πr
2R2
sin2
θ (2R2
πr − πr
2R2
) sin θ cos θ
(2R2
πr − πr
2R2
) sin θ cos θ πr
2R2
cos2 θ + 2R2
πr sin2
θ
.

16. The Update Procedure
Consider the Maxwell-Ampere Equation for D,
∂D
∂t
= × H.
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).
Rearranging
Dn+1/2
= Dn−1/2
+
∆t
∆x
˜ × Hn
Likewise, for Maxwell-Faraday Equation,
Bn+1
= Bn
−
∆t
∆x
˜ × En+1/2
In between these two, we perform
En+1/2
=
1
o
−1
Dn+1/2
, Hn+1
=
1
µo
µ−1
Bn+1

17. The Yee Cell
E and B fields are staggered in time and space.
This process tends to even out errors due to grid approximation.
In 1D:
Taken from Understanding the Finite Difference Time-Domain Method, John Schneider, www.eecs.wsu.edu/~schneidj/ufdtd

18. The Yee Cell
E and B fields are staggered in time and space.
This process tends to even out errors due to grid approximation.
In 2D:
Taken from Understanding the Finite Difference Time-Domain Method, John Schneider, www.eecs.wsu.edu/~schneidj/ufdtd

19. The Yee Cell
E and B fields are staggered in time and space.
This process tends to even out errors due to grid approximation.
In 3D:

20. The FDTD Method in Summary
Divide space and time according to Yee cell.
Specify , µ over entire spatial domain.
Introduce source field by altering E value at some point(s).
Propagate source field through space and time by leap-frog
algorithm:
1. Dn+1/2
= Dn−1/2
+ ∆t
∆x
˜ × Hn
2. En+1/2
= 1
o
−1
Dn+1/2
.
3. Bn+1
= Bn
− ∆t
∆x
˜ × En+1/2
4. Hn+1
= 1
µo
µ−1
Bn+1
Continue iterating in time, as long as you wish.

21. Example: Cloak of Invisibility
Long a staple of fantasy and science fiction:
How do we make it a reality?
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

22. Example: Cloak of Invisibility
Long a staple of fantasy and science fiction:
How do we make it a reality?
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

23. Example: Cloak of Invisibility
Long a staple of fantasy and science fiction:
How do we make it a reality?
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
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−R1
R2
2



where α = r−R1
r .
As above, an equivalent medium is given by ij = µij =
√
ggij ,
ij
= µij
=



α cos2 θ + 1
α sin2
θ (α − 1
α ) sin θ cos θ 0
(α − 1
α) sin θ cos θ 1
α cos2 θ + α sin2
θ 0
0 0 R2
R2−R1
2
α


 .

24. Simulation of Cloak of Invisibility

25. Optical ‘Bag of Holding’
Transform spatial distances inside a cylinder so that the inside is
bigger than the outside.

26. Optical ‘Bag of Holding’
Transform spatial distances inside a volume so that the inside is
bigger than the outside.
This means ds2 = B2(dx2 + dy2 + dz2) for some scale factor B.
gij =


B2 0 0
0 B2 0
0 0 B2

 ⇒ ij =
√
ggij =


B 0 0
0 B 0
0 0 B


This is similar normal dielectric, = B, like glass or plastic!
Inside dielectric, phase velocity v < c.
Same speed in optical space −→ slower speed in physical space.
We also have ij = µij : this is non-scattering condition. Truly
non-glare glasses.

27. Ilustratration of Velocity Distotion

28. Example: Schwarzschild Black Hole
Don’t need transformation; we already know the metric.
ds2
= − 1 −
R∗
r
dt2
+ 1 −
R∗
r
−1
dr2
+r2
dθ2
+r2
sin2
θdφ2
.
More complicated, as now we have spacetime curvature.
Equations for this first discovered by Plebanski, listed without
derivation:
ij
= µij
= −
|g|
g00
gij
(γT
1 )ij
= (γ2)ij
= −[ijk]
g0j
g00
,
where Di = ij Ej + γij
1 Hj , Bi = µij Hj + γij
2 Ej .
Using this, we find, for 2D case:
ij
= µij
=
r
r − R∗



1 − R∗
r
x2
r2 −R∗
r
xy
r2 0
−R∗
r
xy
r2 1 − R∗
r
y2
r2 0
0 0 1


 .

29. Illustration of Gravitational Lensing

30. Further Work
Simulations analyzing metamaterial periodic elements for actual
construction.
Reduced cloaks for broadband cloaking.
Perfect black body layer, based off black hole metric. Makes
perfect ‘one-way mirror’, possibly for solar panels.
Anti-telephonic device that I have affectionately named
“Galadriel’s Mirror.”
Possibilities are limited only by our imagination.

31. We are the Masters of Time and Space!
Light bends to our whim and caprice!
Questions?