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Casimir forces in the time domain: I. Theory

Alejandro W. Rodriguez,1Alexander P. McCauley,1John D. Joannopoulos,1and Steven G. Johnson2

1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139

2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139

We introduce a method to compute Casimir forces in arbitrary geometries and for arbitrary

materials based on the finite-difference time-domain (FDTD) scheme. The method involves the

time-evolution of electric and magnetic fields in response to a set of current sources, in a modified

medium with frequency-independent conductivity. The advantage of this approach is that it allows

one to exploit existing FDTD software, without modification, to compute Casimir forces. In this

manuscript, part I, we focus on the derivation, implementation choices, and essential properties of

the time-domain algorithm, both considered analytically and illustrated in the simplest parallel-plate

geometry. Part II presents results for more complex two- and three-dimensional geometries.

I.INTRODUCTION

In recent years, Casimir forces arising from quantum

vacuum fluctuations of the electromagnetic field [1, 2, 3]

have become the focus of intense theoretical and ex-

perimental effort [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,

15, 16, 17, 18, 19, 20, 21]. This effect has been veri-

fied via many experiments [22, 23, 24, 25], most com-

monly in simple, one-dimensional geometries involving

parallel plates or approximations thereof, with some ex-

ceptions [26]. A particular topic of interest is the ge-

ometry and material dependence of the force, a sub-

ject that has only recently begun to be addressed in ex-

periments [26] and by promising new theoretical meth-

ods [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

For example, recent works have shown that it is possi-

ble to find unusual effects arising from many-body in-

teractions or from systems exhibiting strongly coupled

material and geometric dispersion [39, 40, 41, 42, 43].

These numerical studies have been mainly focused in two-

dimensional [13, 44, 45, 46] or simple three-dimensional

constant-cross-section geometries [33, 40, 47] for which

numerical calculations are tractable.

In this manuscript, we present a simple and general

method to compute Casimir forces in arbitrary geome-

tries and for arbitrary materials that is based on a

finite-difference time-domain (FDTD) scheme in which

Maxwell’s equations are evolved in time [48]. A time-

domain approach offers a number of advantages over

previous methods. First, and foremost, it enables re-

searchers to exploit powerful free and commercial FDTD

software with no modification. The generality of many

available FDTD solvers provides yet another means to

explore the material and geometry dependence of the

force, including calculations involving anisotropic di-

electrics [49] and/or three-dimensional problems. Sec-

ond, this formulation also offers a fundamentally different

viewpoint on Casimir phenomena, and thus new oppor-

tunities for the theoretical and numerical understanding

of the force in complex geometries.

Our time-domain method is based on a standard for-

mulation in which the Casimir force is expressed as

a contour integral of the frequency-domain stress ten-

sor [2]. Like most other methods for Casimir calculations,

the stress tensor method typically involves evaluation at

imaginary frequencies, which we show to be unsuitable

for FDTD. We overcome this difficulty by exploiting a

recently-developed exact equivalence between the system

for which we wish to compute the Casimir force and a

transformed problem in which all material properties are

modified to include dissipation [50]. To illustrate this

approach, we consider a simple choice of contour, corre-

sponding to a conductive medium, that leads to a simple

and efficient time-domain implementation. Finally, using

a free, widely-available FDTD code [51], we compute the

force between two vacuum-separated perfectly-metallic

plates, a geometry that is amenable to analytical calcu-

lations and which we use to analyze various important

features of our method. An illustration of the power and

flexibility of this method will be provided in a subse-

quent article [52], currently in preparation, in which we

will demonstrate computations of the force in a number

of non-trivial (dispersive, three-dimensional) geometries

as well as further refinements to the method.

II. METHOD

In what follows, we derive a numerical method to

compute the Casimir force on a body using the FDTD

method. The basic steps involved in computing the force

are:

(1) Map the problem exactly onto a new problem with

dissipation given by a frequency-independent con-

ductivity σ.

(2) Measure the electric E and magnetic H fields in

response to current pulses placed separately at each

point along a surface enclosing the body of interest.

(3) Integrate these fields in space over the enclosing

surface and then integrate this result, multiplied by

a known function g(−t), over time t, via Eq. (29).

The result of this process is the exact Casimir force

(in the limit of sufficient computational resolution), ex-

pressed via Eq. (29) and requiring only the time-evolution

of Eqs. (15–16).

arXiv:0904.0267v2 [quant-ph] 21 Apr 2009

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In this section, we describe the mathematical develop-

ment of our time-domain computational method, start-

ing from a standard formulation in which the Casimir

force is expressed as a contour integral of the frequency-

domain stress tensor.We consider the frequency do-

main for derivation purposes only, since the final tech-

nique outlined above resides entirely in the time domain.

In this framework, computing the Casimir force involves

the repeated evaluation of the photon Green’s function

Gij over a surface S surrounding the object of interest.

Our goal is then to compute Gijvia the FDTD method.

The straightforward way to achieve this involves com-

puting the Fourier transform of the electric field in re-

sponse to a short pulse. However, in most methods a

crucial step for evaluating the resulting frequency inte-

gral is the passage to imaginary frequencies, correspond-

ing to imaginary time. We show that, in the FDTD

this, gives rise to exponentially growing solutions and is

therefore unsuitable. Instead, we describe an alternative

formulation of the problem that exploits a recently pro-

posed equivalence in which contour deformations in the

complex frequency-domain ω(ξ) correspond to introduc-

ing an effective dispersive, dissipative medium at a real

“frequency” ξ. From this perspective, it becomes simple

to modify the FDTD Maxwell’s equations for the pur-

pose of obtaining well-behaved stress tensor frequency

integrands. We illustrate our approach by considering

a contour corresponding to a medium with frequency-

independent conductivity σ. This contour has the ad-

vantage of being easily implemented in the FDTD, and

in fact is already incorporated in most FDTD solvers.

Finally, we show that it is possible to abandon the fre-

quency domain entirely in favor of evaluating the force

integral directly in the time domain, which offers several

conceptual and numerical advantages.

A. Stress Tensor Formulation

The Casimir force on a body can be expressed [2] as an

integral over any closed surface S (enclosing the body) of

the mean electromagnetic stress tensor ?Tij(r,ω)?. Here

r denotes spatial position and ω frequency. In particular,

the force in the ith direction is given by:

?∞

S

Fi=

0

dω

?

?

j

?Tij(r,ω)?dSj,

(1)

The stress tensor is expressed in terms of correlation

functions of the the field operators ?Ei(r,ω)Ej(r?,ω)?

and ?Hi(r,ω)Hj(r?,ω)?:

?Tij(r,ω)? =

µ(r,ω)

?

?

?Hi(r)Hj(r)?ω−1

?Ei(r)Ej(r)?ω−1

2δij

?

?

k

?Hk(r)Hk(r)?ω

?

?

+ ε(r,ω)

2δij

k

?Ek(r)Ek(r)?ω

,

(2)

where both the electric and magnetic field correlation

functions can be written as derivatives of a vector poten-

tial operator AE(r,ω):

Ei(r,ω) = −iωAE

µHi(r,ω) = (∇×)ijAE

i(r,ω)

j(r,ω)

(3)

(4)

We explicitly place a superscript on the vector poten-

tial in order to refer to our choice of gauge [Eqs. (3–

4)], in which E is obtained as a time-derivative of A.

The fluctuation-dissipation theorem relates the corre-

lation function of AEto the photon Green’s function

GE

ij(ω;r,r?):

?AE

i(r,ω)AE

j(r?,ω)? = −?

πImGE

ij(ω,r,r?),

(5)

where GE

electric dipole current J along the ˆ ejdirection:

?

Given GE

Eq. (5) to express the field correlation functions at points

r and r?in terms of the photon Green’s function:

ijis the vector potential AE

i in response to an

∇ ×

1

µ(r,ω)∇ × − ω2ε(r,ω)

?

GE

j(ω;r,r?) = δ(r−r?)ˆ ej,

(6)

ij, one can use Eqs. (3–4) in conjunction with

?Ei(r,ω)Ej(r?,ω)? =?

?Hi(r,ω)Hj(r?,ω)? = −?

πω2ImGE

ij(ω;r,r?) (7)

π(∇×)il(∇?×)jmImGE

lm(r,r?,ω),

(8)

In order to find the force via Eq. (1), we must first

compute GE

integration S, and for every ω [2].

be solved numerically in a number of ways, such as by

a finite-difference discretization [30]: this involves dis-

cretizing space and solving the resulting matrix eigen-

value equation using standard numerical linear algebra

techniques [53, 54]. We note that finite spatial discretiza-

tion automatically regularizes the singularity in GE

r = r?, making GE

ij(r,r?= r,ω) at every r on the surface of

Equation (6) can

ijat

ijfinite everywhere [30].

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B. Complex Frequency Domain

The present form of Eq. (6) is of limited computational

utility because it gives rise to an oscillatory integrand

with non-negligible contributions at all frequencies, mak-

ing numerical integration difficult [30]. However, the in-

tegral over ω can be re-expressed as the imaginary part

of a contour integral of an analytic function by commut-

ing the ω integration with the Im operator in Eqs. (7–8).

Physical causality implies that there can be no poles in

the integrand in the upper complex plane.

gral, considered as a complex contour integral, is then

invariant if the contour of integration is deformed above

the real frequency axis and into the first quadrant of the

complex frequency plane, via some mapping ω → ω(ξ).

This allows us to add a positive imaginary component

to the frequency, which causes the force integrand to de-

cay rapidly with increasing ξ [50]. In particular, upon

deformation, Eq. (6) is mapped to:

?

and Eqs. (7–8) are mapped to:

The inte-

∇ ×

1

µ(r,ω)∇ × − ω2(ξ)ε(r,ω)

?

GE

j(ξ;r,r?) = δ(r−r?)ˆ ej,

(9)

?Ei(r,ω)Ej(r?,ω)? =?

?Hi(r,ω)Hj(r?,ω)? = −?

πω2GE

ij(ω;r,r?)(10)

π(∇×)il(∇?×)jmGE

lm(r,r?,ω),

(11)

Equation (1) becomes:

Fi= Im

?∞

0

dξdω

dξ

?

surface

?

j

?Tij(r,ω)?dSj,

(12)

[Note that a finite spatial grid (as used in the present

approach) requires no further regularization of the inte-

grand, and the finite value of all quantities means there

is no difficulty in commuting the Im operator with the

integration.]

We can choose from a general class of contours, pro-

vided that they satisfy ω(0) = 0 and remain above

the real ξ axis. The standard contour ω(ξ) = iξ is a

Wick rotation, which is known to yield a force integrand

that is smooth and exponentially decaying in ξ [2]. In

general, the most suitable contour will depend on the

numerical method being employed.

guarantees a strictly positive-definite and real-symmetric

Green’s function, making Eq. (6) solvable by the most ef-

ficient numerical techniques (e.g. the conjugate-gradient

method) [54]. One can also solve Eq. (6) for arbitrary

ω(ξ) [50], but this will generally involve the use of di-

rect solvers or more complicated iterative techniques [53].

However, the class of contours amenable to an efficient

time-domain solution is more restricted. For instance, a

Wick rotation turns out to be unstable in the time do-

main because it implies the presence of gain [50].

A Wick rotation

C.Time Domain Approach

It is possible to solve Eq. (6) in the time domain

by evolving Maxwell’s equations in response to a delta-

function current impulse J(r,t) = δ(r − r?)δ(t − t?)ˆ ejin

the direction of ˆ ej. GE

from the Fourier transform of the resulting E field. How-

ever, obtaining a smooth and decaying force integrand

requires expressing the mapping ω → ω(ξ) in the time-

domain equations of motion. A simple way to see the

effect of this mapping is to notice that Eq. (9) can be

viewed as the Green’s function at real “frequency” ξ and

complex dielectric [50]:

ijcan then be directly computed

εc(r,ξ) =ω2(ξ)

ξ2

ε(r)(13)

where for simplicity we have taken µ and ε to be

frequency-independent. We assume this to be the case

for the remainder of the manuscript. At this point, it is

important to emphasize that the original physical system

ε at a frequency ω is the one in which Casimir forces and

fluctuations appear; the dissipative system εc at a fre-

quency ξ is merely an artificial technique introduced to

compute the Green’s function.

Integrating along a frequency contour ω(ξ) is therefore

equivalent to making the medium dispersive in the form

of Eq. (13). Consequently, the time domain equations

of motion under this mapping correspond to evolution

of the fields in an effective dispersive medium given by

εc(r,ξ).

To be suitable for FDTD, this medium should have

three properties: it must respect causality, it cannot sup-

port gain (which leads to exponential blowup in time-

domain), and it should be easy to implement. A Wick

rotation is very easy to implement in the time-domain: it

corresponds to setting εc= −ε. However, a negative ep-

silon represents gain (the refractive index is ±√ε, where

one of the signs corresponds to an exponentially grow-

ing solution). We are therefore forced to consider a more

general, frequency-dependent εc.

Implementing arbitrary dispersion in FDTD generally

requires the introduction of auxiliary fields or higher or-

der time-derivative terms into Maxwell’s equations, and

can in general become computationally expensive [48].

The precise implementation will depend strongly on the

choice of contour ω(ξ). However, almost any dispersion

will suit our needs, as long as it is causal and dissipative

(excluding gain). A simple choice is an εc(r,ξ) corre-

sponding to a medium with frequency-independent con-

ductivity σ:

εc(r,ξ) = ε(r)

?

1 +iσ

ξ

?

(14)

This has three main advantages: first, it is imple-

mented in many FDTD solvers currently in use; second,

it is numerically stable; and third, it can be efficiently im-

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plemented without an auxiliary differential equation [48].

In this case, the equations of motion in the time domain

are given by:

∂µH

∂t

∂εE

∂t

= −∇ × E

(15)

= ∇ × H − σεE − J

(16)

Writing the conductivity term as σε is slightly nonstan-

dard, but is convenient here for numerical reasons. In

conjunction with Eqs. (3–4), and a Fourier transform in

ξ, this yields a photon Green’s function given by:

?

This corresponds to picking a frequency contour of the

form:

?

∇ ×

1

µ(r)∇ × − ξ2ε(r)

?

1 +iσ

ξ

??

Gj(ξ;r,r?) = δ(r−r?)ˆ ej,

(17)

ω(ξ) ≡ ξ

1 +iσ

ξ,

(18)

Note that, in the time domain, the frequency of the fields

is ξ, and not ω, i.e. their time dependence is e−iξt. The

only role of the conductivity σ here is to introduce an

imaginary component to Eq. (17) in correspondence with

a complex-frequency mapping. It also explicitly appears

in the final expression for the force, Eq. (12), as a multi-

plicative (Jacobian) factor.

The standard FDTD method involves a discretized

form of Eqs. (15–16), from which one obtains E and

B, not GE

However, in the frequency domain, the

photon Green’s function, being the solution to Eq. (6),

solves exactly the same equations as those satisfied by the

electric field E, except for a simple multiplicative factor

in Eq. (3). Specifically, GE

ij.

ijis given in terms of E by:

GE

ij(ξ;r,r?) = −Ei,j(r,ξ)

iξJ(ξ),

(19)

where Ei,j(r,ξ) denotes the field in the ith direction due

to a dipole current source J(r,t) = J(t)δ(r−r?)ˆ ejplaced

at r?with time-dependence J(t), e.g. J(t) = δ(t).

In principle, we can now compute the electric- and

magnetic-field correlation functions by using Eqs. (10–

11), with ω(ξ) given by Eq. (18), and by setting r = r?in

Eq. (11). Since we assume a discrete spatial grid, no sin-

gularities arise for r = r?, and in fact any r-independent

contribution is canceled upon integration over S. This is

straightforward for Eq. (7), since the E-field correlation

function only involves a simple multiplication by ω2(ξ).

However, the H-field correlation function, Eq. (8), in-

volves derivatives in space. Although it is possible to

compute these derivatives numerically as finite differ-

ences, it is conceptually much simpler to pick a differ-

ent vector potential, analogous to Eqs. (3–4), in which H

is the time-derivative of a vector potential AH. As dis-

cussed in the Appendix, this choice of vector potential

implies a frequency-independent magnetic conductivity,

and a magnetic, instead of electric, current. The result-

ing time-domain equations of motion are:

∂µH

∂t

∂εE

∂t

= −∇ × E + σµH − J

(20)

= ∇ × H

(21)

In this gauge, the new photon Green’s function GH

?AH

current source J are related by:

ij=

i(r,ξ)AH

j(r?,ξ)? and the field H in response to the

GH

ij(ξ;r,r?) = −Hi,j(r,ξ)

iξJ(ξ)

,

(22)

where the magnetic-field correlation function:

?Hi(r,ξ)Hj(r?,ξ)? =?

πω2(ξ)GH

ij(ξ;r,r?),

(23)

is now defined as a frequency multiple of GH

by a spatial derivative of GE

This approach to computing the magnetic correlation

function has the advantage of treating the electric and

magnetic fields on the same footing, and also allows us

to examine only the field response at the location of the

current source. The removal of spatial derivatives also

greatly simplifies the incorporation of discretization into

our equations (see Appendix for further discussion). The

use of magnetic currents and conductivities, while un-

physical, are easily implemented numerically. Alterna-

tively, one could simply interchange ε and µ, E and H,

and run the simulation entirely as in Eqs. (15–16).

The full force integral is then expressed in the sym-

metric form:

?∞

where

?

?

S

j

ijrather than

ij.

Fi= Im?

π

−∞

dξ g(ξ)?ΓE

i(ξ) + ΓH

i(ξ)?,

(24)

ΓE

i(ξ) ≡

?

S

?

?

j

ε(r)

Ei,j(r) −1

?

2δij

?

?

k

Ek,k(r)

?

dSj

(25)

ΓH

i(ξ) ≡

1

µ(r)

Hi,j(r) −1

2δij

k

Hk,k(r)

?

dSj

(26)

represent the surface-integrated field responses in the fre-

quency domain, with Ei,j(r) ≡ Ei,j(r;ξ). For notational

simplicity, we have also defined:

g(ξ) ≡

ω2

iξJ(ξ)

dω

dξΘ(ξ)(27)

Here, the path of integration has been extended to the

entire real ξ-axis with the use of the unit-step function

Θ(ξ) for later convenience.

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The product of the fields with g(ξ) naturally decom-

poses the problem into two parts: computation of the

surface integral of the field correlations Γ, and of the

function g(ξ). The Γicontain all the structural informa-

tion, and are straightforward to compute as the output

of any available FDTD solver with no modification to the

code. This output is then combined with g(ξ), which is

easily computed analytically, and integrated in Eq. (24)

to obtain the Casimir force. As discussed in Sec. IVA,

the effect of spatial and temporal discretization enters ex-

plicitly only as a slight modification to g(ξ) in Eq. (24),

leaving the basic conclusions unchanged.

D. Evaluation in the Time Domain

It is straightforward to evaluate Eq. (24) in the fre-

quency domain via a dipole current J(t) = δ(t), which

yields a constant-amplitude current J(ξ) = 1. Using the

frequency-independent conductivity contour Eq. (18),

corresponding to Eqs. (15–16), we find the following ex-

plicit form for g(ξ):

?

One important feature of Eq. (28) is that g(ξ) →?iσ3/ξ

ΓE(ξ) and ΓH(ξ) are continuous at ξ = 0 (in general they

will not be zero), this singularity is integrable. However,

it is cumbersome to integrate in the frequency domain,

as it requires careful consideration of the time window

for calculation of the field Fourier transforms to ensure

accurate integration over the singularity.

As a simple alternative, we use the convolution theo-

rem to re-express the frequency (ξ) integral of the prod-

uct of g(ξ) and ΓE(ξ) arising in Eq. (24) as an integral

over time t of their Fourier transforms g(−t) and ΓE(t).

Technically, the Fourier transform of g(ξ) does not exist

because g(ξ) ∼ ξ for large ξ. However, the integral is

regularized below using the time discretization, just as

the Green’s function above was regularized by the spa-

tial discretization. (As a convenient abuse of notation, ξ

arguments will always denote functions in the frequency

domain, and t arguments their Fourier transforms in the

time domain.)

Takingadvantage of

(ΓE(t), ΓH(t) = 0 for t < 0) yields the following

expression for the force expressed purely in the time

domain:

?∞

The advantage of evaluating the force integral in the

time domain is that, due to the finite conductivity and

lack of sources for t > 0, Γ(t) will rapidly decay in time.

As will be shown in the next section, g(−t) also decays

g(ξ) = −iξ

1 +iσ

ξ

?

1 + iσ/2ξ

?1 + iσ/ξΘ(ξ) (28)

becomes singular in the limit as ξ → 0. Assuming that

the causalityconditions

Fi= Im?

π

0

dt g(−t)?ΓE

i(t) + ΓH

i(t)?

(29)

with time. Hence, although dissipation was originally

introduced to permit a natural high-frequency cutoff to

our computations, it also allows for a natural time cutoff

T. We pick T such that, for times t > T, knowledge

of the fields will not change the force result in Eq. (29)

beyond a predetermined error threshold. This approach

is very general as it requires no precise knowledge of how

the fields decay with time.

E.Properties of g(−t)

Given g(ξ), the desired function g(−t) is a Fourier

transform. However, the discretization of time in FDTD

implies that the frequency domain becomes periodic and

that g(t) = g(n∆t) are actually Fourier series coefficients,

given by:

g(n∆t) =

?2π/∆t

0

dξ gd(ξ)e−iξn∆t,

(30)

where gd(ξ) is the discretized form of Eq. (27) and is

given in the Appendix by Eq. (38). These Fourier series

coefficients are computed by a sequence of numeric in-

tegrals that can be evaluated in a variety of ways. It is

important to evaluate them accurately in order to resolve

the effect of the ξ = 0 singularity. For example, one could

use a Clenshaw-Curtis scheme developed specifically for

Fourier integrals [55], or simply a trapezoidal rule with a

large number of points that can be evaluated relatively

quickly by an FFT (e.g. for this particular g(ξ), 107

points is sufficient).

Since it is possible to employ strictly-real current

sources in FDTD, giving rise to real Γ, and since we

are only interested in analyzing the influence of g(t) on

Eq. (29), it suffices to look at Img(−t). Furthermore, g(t)

will exhibit rapid oscillations at the Nyquist frequency

due to the delta-function current, and therefore it is more

convenient to look at its absolute value. Figure 1, below,

plots the envelope of |Img(−t)| as a function of t, where

again, g(t) is the Fourier transform of Eq. (27).

As anticipated in the previous section, g(t) decays in

time. Interestingly, it exhibits a transition from ∼ t−1

decay at σ = 0 to ∼ t−1/2decay for large σ. The slower

decay at long times for larger σ arises from a transition

in the behavior of Eq. (28) from the singularity at ξ = 0.

III.PROPERTIES OF THE METHOD

In this section we discuss the practical implementation

of the time-domain algorithm (using a freely-available

time domain solver [51] that required no modification).

We analyze its properties applied to the simplest parallel-

plate geometry [Fig. 2], which illustrate the essential fea-

tures in the simplest possible context. In particular, we

analyze important computational properties such as the

convergence rate and the impact of different conductiv-