Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Casimir Energies for Surface Relief Gratings
C-Method in Casimir Calculations
Jef Wagner
Roya Zandi
Department of Physics and Astronomy
University of California, Riverside
Wagner et al March
C-Method
Quantum Vacuum Meeting,
17forthCasimir
2012, Texas A&M
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Outline
1
2
3
4
5
Scattering
Basics
Periodic Systems
C Method
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
Numerical Results
Algorithm
Test System
Results
Perturbative Expansion
Standard Perturbation Theory
The Rayleigh Hypothesis
Corrections to the Casimir Energy
Wagner et al
C-Method for Casimir
Conclusion
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Basics
Periodic Systems
Given the wave equation
− ∇2 + ζ 2 + V 1 + V 2 f = 0
With two disjoint potentials
sup V i = Ωi
T i = V i (1 + G0 V i )−1
The Casimir interaction energy can be written
Z
1
E=
dζ Tr ln 1 − T 1 G0 T 2 G0
4π
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Basics
Periodic Systems
The trace is over the spatial coordinates
Z
1
E=
dζ Tr ln 1 − T 1 G0 T 2 G0
4π
The spatial integrals can be done explicitly by expanding the
Green’s function
X
out 0
G(x, x 0 ) =
φin
x ∈ Ω1 , x 0 ∈ Ω2
α (x)φα (x )
α
and then defining translation and scattering matrices
Z
in
X
Tαβ = dxφin
β (x)T φα (x)
φout
Uαβ φin
α (x1 ) =
β (x2 )
β
The trace in the new form is over separation constants
Z
1
E=
dζ Tr ln 1 − T1 UT2 U
4π
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Basics
Periodic Systems
The trace is over the spatial coordinates
Z
1
E=
dζ Tr ln 1 − T 1 G0 T 2 G0
4π
The spatial integrals can be done explicitly by expanding the
Green’s function
X
out 0
G(x, x 0 ) =
φin
x ∈ Ω1 , x 0 ∈ Ω2
α (x)φα (x )
α
and then defining translation and scattering matrices
X
X
φin
Tαβ φout
φout
Uαβ φin
α (x) +
α (x1 ) =
β (x) β (x2 )
β
β
The trace in the new form is over separation constants
Z
1
E=
dζ Tr ln 1 − T1 UT2 U
4π
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C-Method for Casimir
x∈δΩ
=0
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Basics
Periodic Systems
For a 1-D periodic system the
free wave equation becomes
− ∂x2 − ∂z2 + κ2 f (x, z) = 0
The basis functions are plane waves
q
(±)
φm (x, z) = exp ıKm x ± κ2 + K2m z
where the wave vector has been replaced with a Bloch wave vector
k⊥ → Km ,
Km = k⊥ + Gm ,
Wagner et al
Gm =
C-Method for Casimir
2πm
Lx
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Basics
Periodic Systems
The field can be written in a
Rayleigh expansion
X
(−)
(+)
Rmm0 φm0
f = φm +
m0
The scattering matrix is exponentially suppressed for large m
q
U = exp − κ2 + K2m d
The Casimir interaction energy between two periodic structures is
E
1
= 2
Ly
8π
Z
∞
Z
π/Lx
κdκ
0
dkx ln det 1 − R1 UR2 U
−π/Lx
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
The C method
Numerical method for calculating Rayleigh coefficients
Established method from E&M grating theory
Specialized for surface relief gratings
Begins with a change in variables
{u, v , w } = {x, y , z − h(x)}
The partial derivative is
∂x2 → ∂u − (∂u h)∂w
Wagner et al
2
.
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Expand the height profile and
field in a Fourier and Block series
X
h(u) =
e ıGm u hm
m
f (u, w ) =
X
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
Define the following vectors and
matrices
f(w ) m = fm (w )
(K)m,m0 = δm,m0 Km .
e ıKm u fm (w )
(Gh)m,m0 = G(m−m0 ) h(m−m0 ) .
Separate the Fourier modes - The wave equations becomes a
system of ODEs
(K − Gh∂w )2 − I∂w2 + Iκ2 · f(w ) = 0
m
Proceeding in the standard method
f(w ) = Ve λw
yields an quadratic eigenvalue problem
λ2q A2 · Vq + λq A1 · Vq + A0 Vq = 0.
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
A quick note about the quadratic eigenvalue problem
For an N × N matrix there will be 2N eigenvalues
A general solution to the wave equation can be written
X
X
f (u, w ) =
e ıKm u
cq Vq m e λq w
m
q
The eigenvalues will separate into two sets
{λ+ | All λq such that <(λq ) > 0}
{λ− | All λq such that <(λq ) < 0}
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
Assume Dirichlet boundary conditions on the field
ftot (u, 0) = 0
The field is written in terms on an incident wave and a reflected
field
(+)
ftot (u, w ) = φm (u, w ) + frefl (u, w )
The incident wave and be rewritten in the {u, w } coordinates
(±)
φm (u, w ) = e ıKm u±λm (w +h(u))
X
e
m0 (±) ±λ
e ıKm0 u Lm
=
e mw
e
m0
where
m0 (±)
Lm
Z
=
due −ıGm0 −m u±λ̃m h(u)
Wagner et al
and λ̃m =
C-Method for Casimir
q
κ2 + K2m
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
Assume Dirichlet boundary conditions on the field
ftot (u, 0) = 0
The field is written in terms on an incident wave and a reflected
field
(+)
ftot (u, w ) = φm (u, w ) + frefl (u, w )
The reflected wave is written with eigenvalue from {λ− }
X
X
frefl (u, w ) =
e ıKm0 u
cmq Vq m0 e λq w ,
m0
Wagner et al
q∈{λ− }
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
Assume Dirichlet boundary conditions on the field
ftot (u, 0) = 0
The field is written in terms on an incident wave and a reflected
field
(+)
ftot (u, w ) = φm (u, w ) + frefl (u, w )
The reflected wave is written with eigenvalue from {λ− }
X
X
frefl (u, w ) =
e ıKm0 u
cmq Vq m0 e λq w ,
m0
q∈{λ− }
The boundary condition yields a linear system of equation for cmq
X
m0 (+)
cmq Vq m0 = −Lm
q∈{λ− }
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
The Rayleigh coefficients can by comparing the Rayleigh expansion
X
(−)
Rmm00 φm0 (u, w )
frefl (u, w ) =
m00
With the eigenvector expansion
X
X
e ıKm0 u
frefl (u, w ) =
cmq Vq m0 e λq w ,
m0
q∈{λ− }
For all q where we can make the identification
q
m0 (−)
λq ≈ − κ2 + K2m00 and Vq m0 ∝ Lm00
The Rayleigh coefficients are
Rmm00 = cmq
Wagner et al
Vm
m
m(−)
Lm
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Forming the Eigenvalue Problem
Apply Boundary Conditions
Identify Rayleigh Coefficients
The Rayleigh coefficients can by comparing the Rayleigh expansion
X
X
e
m0 (−)
Rmm00 Lm00 e −λm00 w
e ıKm0 u
frefl (u, w ) =
m0
m00
With the eigenvector expansion
X
X
e ıKm0 u
cmq Vq m0 e λq w ,
frefl (u, w ) =
m0
q∈{λ− }
For all q where we can make the identification
q
m0 (−)
λq ≈ − κ2 + K2m00 and Vq m0 ∝ Lm00
The Rayleigh coefficients are
Rmm00 = cmq
Wagner et al
Vm
m
m(−)
Lm
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
Function RayleighCoefficient( κ, kx , h(x), N)
Form N × N matrices for eigenvalue problem
Solve eigenvalue problem
Use eigenvectors to solve boundary conditions for cmq
Find all indices q where eigenvalues match expected
Return matched indices {q}, and Rayleigh Coefficients
Function LogDet( κ, kx , h(x), d, N)
({q},R) = RaylieghCoefficient
Use {q} to calculates U
Form N = I − RU
Take ln det N
Function Ecas( h(x), d, N )
Numerically Integrate LogDet over κ and kx
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
The test system is a sinusoidal grating and a flat plate
Three parameters:
Two dimensionless paramters:
Amplitude a
a/Lx
Average separation d
d/Lx
Wavelength Lx
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
The energy should converge exponentially as the number of Fourier
modes kept is increased
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
There are two analytic approximations to compare to
The Proximity Force Approximation
EPFA
π2
2d 2 + a2
=−
.
Ly Lx
1440 2(d 2 − a2 )5/2
Wagner et al
Emig’s perturbative approximation
EEmig
π2 1
a2
=−
− 5 GTM
3
Ly Lx
1440 d
d
C-Method for Casimir
d
Lx
.
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
The following plots follow the red paths through parameters space
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
d/Lx = 0.1
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
d/Lx = 0.5
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Algorithm
Test System
Results
d/Lx = 2.0
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Standard Perturbation Theory
The Rayleigh Hypothesis
Corrections to the Casimir Energy
For small amplitudes (h(x) small) it is possible to solve the
eigenvalue problem perturbatively.
λ2q (I − B2 )Vq + λq B1 Vq − A0 Vq = 0
The matrices are
O(h2 )
B2 = Gh · Gh
B1 = (K · Gh + Gh · K)
O(h)
2
A0 = (Iκ + K · K)
O(1)
Following standard perturbation theory
X
X
λ=
λ(i) and V =
V(i)
i
where the superscript
i
(i)
denotes the order of the expression.
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Standard Perturbation Theory
The Rayleigh Hypothesis
Corrections to the Casimir Energy
Perturbative Expansion
Rayleigh Expansion
q
(0)
λq = − κ2 + K2q
q
em = − κ2 + K2
−λ
m
(1)
(2)
λq
λq = Kq G0 h0
X
(0)
= λq Kq
|hm−q |2 Gm−q
m0 (±)
Lm
=
X
i
m
(0) Vq m
(1) Vq
(2) Vq m
m
m0 (±)(i)
Lm
m0 (±)(i)
= δqm
Lm
(0)
(∓λm )i
=
Z
i!
(0)
= λq hm−q
m0 (±)(0)
(0)
(λq )2 X
=
hm−m0 hm0 −q −
2
0
m
(0)
Gm+q−2m0
(λq )2 X
hm−m0 hm0 −q
2
Gm−q
0
Lm
= δmm0
m (±)(1)
= ∓λm hm0 −m
Lm
m0 (±)(2)
Lm
0
=
(0)
(0)
(λm )2 X
hm0 −m00 hm00 −m
2
00
m
m
Wagner et al
du e −ıGm0 −m u hi (u)
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Standard Perturbation Theory
The Rayleigh Hypothesis
Corrections to the Casimir Energy
In the large N limit the perturbative solution (through second
order) matches the Rayleigh expansion.
We can now proceed using only the Rayleigh expansion
X
m0 (−)
m0 (+)
Rmm00 Lm00 = −Lm
m00
This is equivalent to the Rayleigh hypothesis? The first few
reflection coefficients are
(0)
Rmm0 = −δmm0
(1)
(0)
Rmm0 = 2λm hm0 −m
X (0)
(2)
(0)
Rmm0 = 2λm
λm00 hm0 −m00 hm00 −m
m00
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Standard Perturbation Theory
The Rayleigh Hypothesis
Corrections to the Casimir Energy
The zeroth order term gives the Casimir energy for flat plates
π2 1
E (0)
=−
Ly Lx
1440 d 3
The first correction only depends on the average h0
E (1)
π 2 h0
=−
Ly Lx
480 d 4
The second term depends explicitly on radio d/Lx
E (2)
π 2 X |hm |2
=−
J(4πmd/Lx )
Ly Lx
240 m d 5
This is NOT the same expression from Emig (and Prachi).
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Standard Perturbation Theory
The Rayleigh Hypothesis
Corrections to the Casimir Energy
The J function is explicitly given by
Z ∞ 2 −z Z 1 √ 2
z e
15
z + A2 + 2zAx
√
dx
J(A) = 4 dz
4π 0 1 − e −z −1 1 − e − z 2 +A2 +2zAx
J(0) = 1
J(x) ∼ x/4
For Large x
Wagner et al
C-Method for Casimir
Scattering
C Method
Numerical Results
Perturbative Expansion
Conclusion
Conclusions
Scattering method allows us to leverage existing techniques
(such as the C method) for Casimir calculations
I get converged results for a wide range of parameters
Perturbatively the C method is equivalent the Rayleigh
hypothesis
I do not agree with Emig’s approximation either numerically
or perturbatively
I would like to thank Kim, Prachi, Elom, and Nima for letting me
discuss this work when it was first starting. I would also like to
thank Steve for inviting me and hosting this conference.
Wagner et al
C-Method for Casimir