Non-monotonic Casimir interaction: The role of amplifying dielectrics

1.6 MB·

by  Morteza Soltani

Non-monotonic Casimir interaction: The role of amplifying dielectrics
Morteza Soltani,1 Jalal Sarabadani,2,∗ and S. Peyman Zakeri1
1Department of Physics, University of Isfahan, Isfahan 81746, Iran
2Department of Applied Physics, and COMP Center of Excellence,
Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, Finland
The normal and the lateral Casimir interactions between corrugated ideal metallic plates in the
presence of an amplifying or an absorptive dielectric slab has been studied by the path-integral
6
quantization technique. The effect of the amplifying slab, which is located between corrugated
1
conductors, is to increase the normal and lateral Casimir interactions, while the presence of the
0
absorptive slab diminishes the interactions. These effects are more pronounced if the thickness of
2
theslab increases, and also if theslab comes closer toone oftheboundingconductors. When both

n bounding ideal conductors are flat, the normal Casimir force is non-monotonic in the presence of
a the amplifying slab and the system has a stable mechanical equilibrium state, while the force is
J attractive and is weakened by intervening the absorptive dielectric slab in the cavity. By replacing

5 oneoftheflatconductorsbyaflatidealpermeableplatetheforcebecomesnon-monotonicandthe
2 system has an unstable mechanical equilibrium state in the presence of either an amplifying or an
absorptive slab. When the left side plate is conductor and the right one is permeable, the force is

] non-monotonic in the presence of a double-layer DA (dissipative-amplifying) dielectric slab with a
l
l stablemechanicalequilibriumstate,whileitispurelyrepulsiveinthepresenceofadouble-layerAD
a
(amplifying-dissipative) dielectric slab.
h
-
s PACSnumbers: 42.50.Lc,03.70.+k,42.50.-p
e
m
. I. INTRODUCTION lateral direction occur when l = (n+ 1)λ and l = nλ,
t 2
a respectively [10], whereas for the system composed of a
m roughidealconductorandaninfinitely permeablecorru-
Aninterestingmacroscopicresultofconfinementofthe
- quantum electromagnetic (EM) field between two ideal gated plate, with the same H and λ, the stable and un-
d parallel flat conductors is the Casimir effect [1]. Com- stable equilibrium states for the position of the plates in
on paring the energy of the system in the presence and in thelateraldirectionoccurwhenl =nλandl =(n+21)λ,
c the absence of bounding ideal metallic plates leads to a respectively [12].
[ finite attractive potential energy. The derivation of this Moreover, the Casimir force also strongly depends on

finite energy with respect to the distance between two
1
plates, H, is the Casimir force, F . The Casimir force
v C
2 perunitarea,S,isthenreadas FSC =−2~4c0πH24,where~is 1 m 2
4 the Planck constantmultiplied by 1 , and c is the speed
8 2π H
of light in the vacuum [2]. The magnitude of this force
6
0 issignificantwhenH isless thanamicron[3]. Therefore a H b H
1 2
. the Casimir effect must be taken into account in design-
1
ing micro- and nano-devices [4–7]. l
0
In addition to the normal Casimir force, corrugated
6 x
conductors can experience lateral Casimir force due to 1
1
: the translationalsymmetry breaking which has been ob- h (x) x x3=z h (x)
v 1 2 2
served experimentally in 2002 [8]. The lateral Casimir
i z= 0
X effectbetweentworoughconductorswithsinusoidalcor-
r rugation has been studied theoretically in the context of FIG. 1. (Color online) The schematic figure of a dielectric
a path-integral formalism [9–11]. This force is very sensi- slabwithdielectricfunctionεandthicknessbimmersedinthe
tive to the characteristics of the rough surfaces and also quantum vacuum and enclosed by two rough metallic plates.
sensitive to the medium in between them. For a system ThemeandistancebetweentwoplatesisH. Thecorrugation
composedoftworoughidealconductorsimmersedinthe amplitude and wavelength for both plates are a and λ, re-
quantum vacuum, at fixed mean separation distance be- spectively. Themeandistancebetweentheleftmetallicplate
and the left side of the dielectric slab is H , while the mean
tween two conductors, H, with λ as the wavelength of 1
distance between the right metallic plate and the right side
corrugationsonbothplates,thestableandunstableequi-
of the dielectric slab is H . The lateral mismatch between
librium states for the position of the rough plates in the 2
metallic plates is l. The corrugations on the left and on the
right conductorsaregiven bytwo heightfunctions h (x)and
1
h (x), respectively. The regions labeled by 1 and 2 are filled
2
bythequantumvacuumwhilethedielectricslabislabeledby
∗ jalal.sarabadani@aalto.fi m.
2
the characteristics of the medium between the bound- II. FORMALISM
ing plates. In a series of papers, it has been shown that
thepresenceofabsorptivedielectricmediumbetweentwo A. Quantization of the EM field
bounding conductors can weaken the normal and lateral
Casimir interactions [13–16]. The reason is that the ab-
In this section we investigate the Casimir interaction
sorptive dielectrics can dissipate the EM energy [17–19].
between two ideal corrugated conductive plates in the
In the other hand, it has been shown that if the energy
presence of either a dissipative or an amplifying slab as
is pumped into the medium artificially, for example by
depictedinFig.1. X =(x ,x)isthecoordinateofapoint
lasers, the EM energy can be amplified in some regions 0
in four-dimensional time-space with the temporal coor-
of frequency [20–24]. These kinds of media are called
dinate x , and the spatial coordinates x = (x ,x ,x ),
amplifying media, where the EM energy can be ampli- 0 1 2 3
x = (x ,x ). z component of X , which is in the direc-
fied. Althoughthephysicalpropertiesofthesemediaare 1 2 1
tionofx ,ontheleftconductorisz =−H −b/2+h (x),
different from those of dissipative dielectrics, the same 3 1 1
while z component of X on the right conductor is
formal methods can be used to quantize the EM field 2
z =H +b/2+h (x), where h (x) andh (x) arethe de-
in the presence of amplifying dielectrics as of dissipative 2 2 1 2
formation fields on the left and on the right ideal plates,
dielectrics [20, 21, 24].
respectively. The average distance between two conduc-
torsisH,the corrugationamplitude forbothconductors
To study an amplifying medium, one can introduce
is a, the wave length of the corrugation in x direction
a susceptibility which must satisfy the Kramers-Kronig 1
for both conductors is λ, l is the lateral shift between
relations. Similar to the dissipative (or absorptive) di-
surfaces in x direction, H is the mean distance of the
electrics,thedielectricfunctionofanamplifyingmedium, 1 1
left side of the slab and the left conductor, H is the
that is ε (ω), should have an imaginary part. But 2
amp
mean distance of the right side of the slab and the right
for dissipative dielectrics, the imaginary part of the di-
conductor, and b is the thickness of the slab.
electric function is positive, i.e. ε (ω) > 0, whereas
I,disp
for amplifying media, the imaginarypartis negative,i.e. To study the Casimir interaction between two corru-
ε (ω) < 0 [21]. The same is true for the imaginary gated ideal metallic plates in the presence of an inter-
I,amp
parts of the permeability of the dissipative and ampli- veningdissipativeoramplifying slab(seeFig.1), wefirst
fying media [23]. One can quantize the EM field in the use the path-integral method to quantize the EM field.
presence of the amplifying medium by adding a noise As we are only considering a uni-axial cavity, therefore
term into Maxwell equations [21, 24, 25], or using the theEMfieldcanbedecomposedintothetransversemag-
canonical approach [21]. For quantization of the EM netic(TM)andtransverseelectric(TE)waveswhichsat-
field in the presence of an amplifying medium, we use isfyDirichlet(D)andNeumann(N)boundaryconditions
the path-integralformalism[13]. Itshouldbe notedthat (BCs),respectivelyonbothplates[10,11,26]. TheDBC
althoughthemethodsforfieldquantizationwhichweuse should be satisfied by TM waves, that is φ(Xa) = 0, on
here for the amplifying and dissipative systems are the both plates, while TE waves satisfy the N BC, that is
same, however the characteristics of these two kinds of ∂nφ(Xa) = 0, where a = 1,2 determines the left and
systems are completely different. For example, vacuum rightplates,respectively. Consequently,after decompos-
fluctuationsinthepresenceofdissipativeandamplifying ing the EM field into TM and TE waves we quantize a
mediabehaveindifferentways. Moreover,anamplifying massless scalar Klein-Gordon field, φ, in the presence of
medium can be used to compensate the effect of dissipa- a dielectric slab between two conducting plates. To this
tion [21]. end, weuse the same methodasofRefs.[13–16]. The di-
electric medium can be a dissipative dielectric slab with
Inthepresentpaper,westudyandcomparethenormal a positive imaginary part of its dielectric function, that
and the lateral Casimir interactions between two rough is ε (ω) > 0, or it can be an amplifying slab with a
I,disp
ideal conductors in the presence of an intervening dissi- negativeimaginarypartofits dielectric functioninsome
pativeoramplifying slab,oradouble-layerwithmixture frequency ranges, that is ε (ω) < 0. The dielectric
I,amp
of them. This paper is organized as follows: In Sec. II medium can be a layereddielectrics with mixture of dis-
wegenerallyintroducethepath-integralformalismtoob- sipative andamplifying layers. Here, for the sakeofsim-
tain the Casimir interaction energy for the system com- plicity and brevity we just present the formalism for a
posedofadielectricslabbetweentwoidealroughmetallic slab with only one dielectric layer, that can be either an
plates. Section III is devotedto study the Casimir inter- absorptive or an amplifying, in between the bounding
actioninasystemcomposedofeitheranabsorptiveoran conductors. To extend the path-integral formalism to
amplifying slab or a double-layer dielectric slab between study the Casimir effect in the presence of an absorptive
two flat bounding ideal plates. In Sec. IV we calculate [13] or an amplifying medium [27], one must write the
and compare the normal and the lateral Casimir inter- Lagrangian of the system with appropriate terms that
actions due to the corrugation on the bounding plates leadsto the correctequationsofmotion. To this end, we
in the presence of either an absorptive or an amplifying consider the following Lagrangiandensity
slab. Finally we wrap up the paper with a conclusion in
Sec. V. L=L +L +L , (1)
sys mat int
3
where L = 1∂αφ(X)∂ φ(X) is the Lagrangian den- thatarecoupledto the free fields Y andφ, respectively.
sys 2 α ω
sity of Klein-Gordon field, the summation over α = Then by integrating over φ and Y the generating
m
x ,x ,x and x is assumed, and as explained in the function can be written as
0 1 2 3
above, X = (x ,x) is a coordinates of a point in four-
0
dimensional time-space. L = ∞dω sgn[ε (ω)](1Y˙2− Z[Jφ,JP]=exp(cid:26)ıZ dXZ dX′(cid:20)JφGφ,φ(X −X′)Jφ
mat I 2 ω
R0
1ω2Y2) is Lagrangian density of a matter field, where +J G (X −X′)J +J G (X −X′)J , (6)
2 ω φ P,φ P P P,P P(cid:21)(cid:27)
the matter is modeled by a continuum oscillator’s field
Y [28],ε (ω)istheimaginarypartofthedielectricfunc- wheretheFouriertransformationoftheGreen’sfunction,
ω I
tion, and Lint = φP˙ is the interaction Lagrangian den- Gφ,φ, is
sitybetweenmatterandscalarfields. P = dων(ω)Yω is sgn[ε (q′)]ν2(q′) −1
the polarizationofthe medium, andν(ω) iRs the coupling G (q ,q)= q2−q2 1+ dq′ I 0 0 ,
φ,φ 0 (cid:20) 0(cid:18) Z 0 q′2−q2+ı0+ (cid:19)(cid:21)
function between the scalar and the matter fields [13]. 0 0
(7)
Using the Lagrangian in Eq.(1), the conjugate of the
and for the other Green’s functions can be calculated as
canonical momentum can be found. By employing the
equal time commutation relations, one can quantize the G (q ,q)=ıq dq′ sgn[εI(q0′)]ν2(q0′)G (q ,q),
fieldandobtaintheHamiltonianofthesystem,andshow φ,P 0 0Z 0 q′2−q2+ı0+ φ,φ 0
0 0
that the scalar field φ satisfies the following equation sgn[ε (q′)]ν2(q′) 2
G (q ,q)= dq′ I 0 0 G (q ,q)
(∇2−ε(ω)ω2)φ+(ω)=j+(r,ω). (2) P,P 0 (cid:20)Z 0 q′2−q2+ı0+ (cid:21) φ,φ 0
N 0 0
ν2(q )
The positive frequency part of the current density oper- + 0 +q2, (8)
ator, j+(r,ω), is q0 0
N
where ~q = (q ,q), q = (q ,q ) and q is the tempo-
0 1 2 0
j+(r,ω)=Θ[ε (ω)] 2ωε0|εI(ω)|aˆ (r,ω) ral Fourier component. The above Green’s functions
N I r π N are fully in agreement with those in Refs.[30, 31]. Us-
+Θ[−ε (ω)] 2ωε0|εI(ω)|aˆ† (r,ω), (3) ing the definition χ(q0) = dq0′sgnq[′ε2I−(qq0′2)+]νı20(+q0′), it can
I r π N be shown that G is the GRreen’s 0func0tion of Eq.(2)
φ,φ
whereΘ(...)istheHeavisidestepfunction,aˆisabosonic and together with using the residues theorem, one can
field with bosonic commutation relation, and aˆ† is the show that Imχ(q0) = πsgn[εI(q0)]ν2(q0)/(2q0). Combin-
complex transpose of aˆ. Then, the Hamiltonian can be ingImχ(q0)withthedefinitionsofGφ,P andGP,P,yields
written as P+ = ıq0χφ+ +PN+, where ıq0PN+ = jN+. Here, we have
assumedthatthematterishomogeneous. Asithasbeen
H = dω d3rsin[ε (ω)]~ωaˆ† (r,ω)aˆ (r,ω), (4) shown in Ref.[13], among the Green’s functions Gφ,φ,
Z Z I N N G and G , only G has contribution to the par-
φ,P P,P φ,φ
tition function and consequently to the Casimir interac-
where εI(ω)=1+ dω′sgnω[ε2I−(ωω′′)2]+|νı20(+ω′)| leads to ν2(ω)= tion. Therefore, hereafter we drop the subscript φ from
sgn[εI(ω)]Im[εI(ω)]R. The main difference between the Gφ,φ. Moreover,for both amplifying and dissipative me-
calculations for the amplifying medium and dissipative dia, G is the Green’s function of Eq.(2), therefore,
φ,φ
one is the presence of Θ[εI(ω)] and sgn[εI(ω)] in the the Casimir interaction between ideal conductors in the
above equations that have been discussed extensively in presenceofadissipativeoranamplifyingslab(seeFig.1)
[21, 22, 24, 25]. It should be mentioned that the Hamil- has the same mathematical form. The only difference is
tonianin Eq.(4)is consistentwith the result ofRef.[24]. in the sign of ε (ω).
I
Toquantizethefield,wecalculatethegeneratingfunc- TM and TE waves can be treated as two scalar fields
tionfortheinteractingsystem[29]. Thegeneratingfunc- separately which satisfy the following differential equa-
tion is defined as tion ∇2−q2[1+χ(q ,x)] φ = 0. For TE modes, φ
0 0
Z[Jφ,JP]=Z0−1eıRdYLint[δJφδ(Y),δJωδ(Y)]Z0[Jφ,Jω] aconndti∂n(cid:8)zuφo,usanodnftohreTleMftwanavde(cid:9)sa,lsεo(qo0n)φthaendrig∂hztφssuhrofaucledsboef
∞ 1 δ ∂ δ n the slab, and in addition, the Green’s function, G, must
=Z0−1 n!(cid:20)ıZ dY δJ (Y)∂y δJ (Y)(cid:21) satisfy the same BCs. Moreover, D and N BCs should
nX=0 φ 0 P besatisfiedbyTMandTEwavesonthesurfacesofboth
×Z0[Jφ,Jω], (5) conductors, respectively.
where Z = D[φ]exp ı dXL is the gen-
0 sys
erating functionR for free(cid:0) Rspace, (cid:1)Z0[Jφ,Jω] = B. Casimir interaction
D[φ] D[Y ]exp ı dX[L +L + J φ +
ω sys mat φ
R Qω (cid:8) R
dωJ Y ]} is the generating function for noninteract- ToobtaintheCasimirinteraction,oneshouldcalculate
ω ω
Ring part of the system, and Jω and Jφ are source fields the partition function of the system from the generating
4
function. To this end, Wick rotation, x → ıτ, on the where the subscript, γ = TM or TE, stands for Dirich-
0
time axis must be applied. After applying the above let or Neumann BCs, respectively. It should be men-
BCs on all fields, expressing D and N BCs in terms of tioned that the first order term of the logarithm of
path-integral over the axillary fields [11, 13, 14], and in- the partitionfunction vanishes because we assumedthat
tegrating over all Gaussian fields, the partition function h (x)d3x = 0, where i = 1 and 2. The explicit forms
i
can be cast into Rof K− and K+ for the geometry depicted in Fig.1 are
γ γ
presentedin AppendicesA and B, andthe explicit forms
1
lnZTM(TE) =−2ln detΓTM(TE) , (9) of Kγ0 for different geometries can be seen in Sec.III and
(cid:2) (cid:3) Appendices C and D. Using Eq.(11), one can obtain the
where ΓTM(TE) is a second rank matrix with rele- Casimir energy per unit area, S, as
′
vant elements of [Γ ] = ∂ ∂ G(x−y,z (x),z (y)),
TE αβ n n′ α β ~c
[ΓTM]αβ =G(x−y,zα(x),zβ(y)), where α,β =1 and 2, E(H)=− lnZγ(H)−lnZγ(H →∞) , (14)
SL
andz1(x)=−H1−2b+h1(x)andz2(x)=H2+2b+h2(x). Xγ (cid:2) (cid:3)
The Green’s functions in the Fourier space can be read
where L is the overall Euclidean length in the temporal
as
direction [10]. Then, Eq.(14) is employed to study the
b b
G ~q,−H − ,H + = CasimirinteractionindifferentgeometriesinSecs.IIIand
TM(TE) 1 2 2 2 IV.
(cid:0) (cid:1)
b b
G ~q,H + ,−H − =
TM(TE) 2 2 1 2
(cid:0) (cid:1)
2εmQme−Q1(H1+H2−b) III. FLAT BOUNDARIES
= ,
(εmQ1+Qm)2 −(∆T1mM(TE))2+e2Qmb In this section, we study the normal Casimir interac-
h i
b b tionbetweeneithertwoflatidealconductors(ε→∞),or
G ~q,(−1)j(H + ),(−1)j(H + ) =
TM(TE) j 2 j 2 betweenaflatidealconductorandaflatidealpermeable
(cid:0) (cid:1)
1 ∆TM(TE)(−1+e2Qmb)eQ1(b−2Hj) plate(µ→∞),inthepresenceofaflatdielectricoraflat
= − 1m , (10) double-layer dielectric slab in between the ideal plates.
2Q1 2Q1 −(∆T1mM(TE))2+e2Q2b Subsection IIIA is devoted to investigate the Casimir
h i force between two ideal conductors in the presence of a
where j = 1,2, ∆TM = Qm−ε(ı˙q0)Q1, ∆TE = Qm−Q1, dissipative or an amplifying slab (see Fig.2a), or in the
1m Qm+ε(ı˙q0)Q1 1m Qm+Q1
Q ≡ı˙ ε (ıq )q2+q2+q2 withζ =1,m,2corresponds presence of a double-layer dielectric slab with one layer
ζ ζ 0 0 1 2
of dissipative and one layer of amplifying dielectrics (see
toregiopnsz <−b/2,|z|<b/2,andz >b/2,respectively.
Fig.2b),whileinsubsectionIIIBweconsidertheCasimir
The dielectric function of the slab is ε (ıq ) = ε(ıq )
m 0 0
forcebetweenanidealconductorandanidealpermeable
while for ζ = 1,2, which corresponds to two vacuum
plateinthepresenceofeitheranabsorptiveoranampli-
space regions between slab and rough plates, ε (ıq ) =
1 0
fying or a double-layer dielectric slab.
ε (ıq ) = 1. For a general case where the dielectric slab
2 0
To this end, we model the susceptibility of the ampli-
has magnetic properties in addition to the electric ones,
fying slab by Lorentz model with gain and loss which
the definition of ∆ must be adjusted accordingly. The
is one of the best choices to model the amplifying me-
general form of ∆ is presented in AppendixC.
dia [21, 24, 27]. Linear gain occurs when the medium is
Then,afterperformingtheperturbativeexpansionup-
pumped below the lasing threshold [32]. Therefore, we
tothesecondorderwithrespecttothedeformationfields,
consider the following model for an amplifying medium
h (x) and h (x), the partition function is written as
1 2 ω2
ε (ω) = 1 − p , where ω = 0.75ω , γ =
ln Z =ln Z +ln Z , (11) amp ω2−ω2−ıγω p 0
γ γ,0 γ,2 0
10−3ω and ω = 103Hz [21]. It can be shown that for
0 0
where the zeroth order of the partition function in the
amplifying media, the imaginary part of the dielectric
presence of the flat bounding conductors is
function is negative, i.e. ε (ω) < 0. After applying
I,amp
ln Z =−S d3~q lnK0(~q), (12) Wick rotation, ω → ıq0, the above dielectric function is
γ,0 Z (2π)3 γ rewritten as
and the second order is ωp2
ε (ıq )=1− . (15)
amp 0 ω2+q2+γq
ln Z = d3xd3y× 0 0 0
γ,2
Z
Thedielectricfunctionoftheabsorptiveslabisalsomod-
1 eledhere byLorentz-oscillatormodel thatafter applying
× − K−(x−y)[h (x)h (y)+h (x)h (y)]
(cid:26) 2 γ 1 2 2 1 Wick rotation can be read as
−1K+(x−y){[h (x)−h (y)]2+[h (x)−h (y)]2} , ωp2
4 γ 1 1 2 2 (cid:27) εdisp(ıq0)=1+ ω2+q2+γq . (16)
0 0 0
(13)
5
at fixed b can be obtained as
FCC ∂ECC
m =− m . (19)
S ∂H (cid:12)
2 (cid:12)H1,b
(cid:12)
FIG. 2. (Color online) (a) Schematic picture of a dielectric (cid:12)
slabinbetweentwoboundingidealconductors(ǫ→∞). The Similar procedure is performed to calculate the Casimir
distance between the left conductor from the left side of the
forceontherightconductorinthepresenceofthedouble-
slab is H , while the distance between the right conductor
1 layer dielectrics (see Fig.2b) as
from the right side of the slab is H , and the slab thickness
2
isb. (b)Schematicpictureofadouble-layerdielectricslab in
between two bounding ideal conductors. The left dielectric
FCC ∂ECC
layer thickness is b, while the right dielectric layer thickness mn =− mn . (20)
isa. Theregionslabeledby1and2arefilledbythequantum S ∂H2 (cid:12)(cid:12)H1,a,b
vacuum. (cid:12)
(cid:12)
To compare the difference between FCC, the Casimir
A. Conductor-Conductor D
force in the presence of the dissipative slab, and FCC,
A
the Casimir force in the presence of an amplifying slab,
ToobtaintheCasimirinteractionbetweentwoflatcon- in Fig.3a the force per unit area has been plotted as a
ductorswesetthedeformationfieldsforbothconductors function of the distance between the bounding conduc-
to h1(x) = h2(x) = 0. Then the Casimir interaction en- tors, H =H1+H2+b, with H1 =H2, for two values of
ergyperunitareaforthesystemcomposedofadielectric thedissipative(blacksolidandblackdashedcurves)and
slab in between two flat conductors, depicted in Fig.2a amplifying (red dashed-dotted and red dashed-dotted-
reads as dotted curves) slab thicknesses b = 100, and 125nm. As
it is clear, the force in the presence of the dissipative
ECC d3~q BTM
m =~c ln 1+ m +[(TM)→(TE)],(17) dielectric slab is attractive for the whole distance range
S Z (2π)3 (cid:20) DmTM(cid:21) between the bounding conductors, while in the presence
ofthe amplifying slab,for smalldistancesthe forceis re-
where the index m=D, A stands for a dissipative or an
pulsive and as the distance increases the force decreases
amplifying slab,respectively,andDmTM(TE) andBmTM(TE) andbeyondacertaindistance its signchangesanditbe-
are functions of H1, H2, b, ε1, ε2, εm (permittivities comes attractive. Therefore, there is an equilibrium dis-
of different layers), µ1, µ2, µm (permeabilities of differ- tance, Hequil, where the value of the force is zero. This
ent layers), ~q, and c. The explicit forms of DTM(TE) equilibrium distance is a function of the amplifying slab
m
TM(TE) thickness,itsdielectricfunctionandthedistancebetween
and B are presented in Appendix C. According
m
to Eq.(12), the kernel at the zeroth order with respect theboundingconductors,H. Byincreasingthethickness
to the deformation fields, where h (x) = h (x) = 0, is of the amplifying slab, the location of the equilibrium
1 2
KT0M(TE) =1+ DBmTmTMM((TTEE)). pthoeinltogis-lsohgifptelodttooft−heFmClaCrgaesradifsutnanctcieosn. oTfhHe ,inwsehticshhoawps-
By using either path-integral formalism [13–15, 29] or proximately reveals thaSt when H >b/8 the force scales
2
transfer matrix method [33–35] the Casimir interaction asH−4. InFig.3b,theCasimirforceperunitarea, FmCCn,
energyperunitareaforthesystemcomposedofadouble- S
inthepresenceofadouble-layerslabbetweentwobound-
layer dielectric slab in between two flat conductors, de-
ing conductors depicted in Fig.2b, has been plotted as a
picted in Fig.2b can be written as
function of H, with H = H , for two values of the dis-
1 2
ECC d3~q BTM sipative and amplifying slabs thicknesses a,b = 40 and
mn =~c ln 1+ mn +[(TM)→(TE)],(18) 50nm. Here,m,n=DandAindicatethedissipativeand
S Z (2π)3 (cid:20) DTM(cid:21)
mn amplifying dielectrics, respectively. This force is attrac-
tive when both layersare dissipative (DD). When one of
wherethe indicesm,n=DandAstandforadissipative
the layeris dissipative andthe other is amplifying (DA),
and an amplifying slab, respectively, and DTM(TE) and
mn atsmalldistancestheforceisrepulsive. ByincreasingH,
BmTMn(TE) are functions of a and µn in addition to H1, the force becomes attractive and it gets a minimum and
H2, b, ε1, ε2, εm, εn, µ1, µ2, µm, µn, ~q and c. The then it grows. For DA slab the system has a mechanical
explicit forms of DmTMn(TE) and BmTMn(TE) are written in equilibriumstatewiththeequilibriumdistanceHDA,equil.
AppendixC. When both layers of the slab are amplifying (AA) the
Toshowthe effectofthepresenceofthedielectricslab equilibrium point is shifted to the larger distances and
in between two conductors on the Casimir interaction in H > H . The inset represents the normal-
AA,equil DA,equil
thesystemdepictedinFig.2a,theCasimirforceperunit ized force, FCC/|F |, as a function of H. Again when
mn C
areathattherightconductorexperiencesatfixedH and approximately H >(a+b)/8, the force scales as H−4.
1 2
6
FIG. 3. (a) The force per unit area, FmCC, as a function of the distance between the bounding conductors, H =H +H +b,
S 1 2
with H = H , in the presence of a single dielectric slab between the bounding conductors, for two values of the dissipative
1 2
(blacksolidandblackdashedcurves)andamplifying(reddashed-dottedandreddashed-dotted-dottedcurves)slabthicknesses
b = 100 and 125nm. The inset shows the log-log plot of −FmCC as a function of H. (b) The force per unit area, FmCCn, as a
S S
functionofthedistancebetweentheboundingconductors,H =H +H +a+b,withH =H ,inthepresenceofadoublelayer
1 2 1 2
dielectric slab between the bounding conductors, for two values of the dissipative and amplifying dielectric layers thicknesses
a=b=40 and 50nm. The inset presents the normalized force FCC/|F | as a function of H. Here, m,n=D and A indicate
mn C
thedissipative and amplifying dielectrics, respectively.
interaction energy per unit area for a system composed
of a single dielectric slab in between an ideal conductor
and an ideal permeable plate (see Fig.4a) reads as
ECP d3~q ITM
m =~c ln 1+ m +[(TM)→(TE)],(21)
S Z (2π)3 (cid:20) JTM(cid:21)
FIG. 4. (Color online) (a) Schematic picture of a dielectric m
slabinbetweentwoboundingplates,oneisflatidealconduc-
and for a system composed of a double-layer dielectric
tor (ǫ → ∞) while the other is a flat ideal permeable plate
slab in between an ideal conductor and an ideal perme-
(µ → ∞). The distance between the ideal conductor from
able plate (see Fig.4b) as
theleft sideoftheslab isH ,whilethedistancebetween the
1
permeable platefrom theright sideof theslab is H and the
2 ECP d3~q ITM
slab thickness is b. (b) Schematic picture of a double-layer mn =~c ln 1+ mn +[(TM)→(TE)],(22)
dielectric slab in between an ideal conductor and an ideal S Z (2π)3 (cid:20) JTM(cid:21)
mn
permeableplate. Theleftdielectriclayerthicknessisb,while
the right dielectric layer thickness is a. The regions labeled whereagaintheindicesm,n=DandAstandforadissi-
by 1 and 2 are filled by thequantumvacuum. pativeandanamplifyingslab,respectively,ITM(TE) and
m
JTM(TE) are functions of H , H , b, ε , ε , ε , µ , µ ,
m 1 2 1 2 m 1 2
µ , ~q and c, and ITM(TE) and JTM(TE) are functions of
m mn mn
B. Conductor-Permeable the above parameters in addition to a and µ . The ex-
n
TM(TE) TM(TE) TM(TE) TM(TE)
plicitformsofI ,J ,I andJ
m m mn mn
Toobservehowmuchthecharacteristicsofthebound- are presented in AppendixD.
ingplates canaffectthe Casimirinteraction,weconsider Tocomparetheeffectsofadissipativeoranamplifying
asystemcomposedofasingledielectricslab,oradouble- slabbetweentwoidealflatplates,oneconductiveandthe
layer dielectric slab in between a flat ideal conductor in other permeable, on the Casimir interaction, the force
the left side and a flat ideal permeable plate in the right per unit area that the permeable plate on the right side
sideofthesystem,depictedinFig.4aandb,respectively. experiencescanbecalculatedatthefixedH andatfixed
1
As in1974Boyershowed[36], aflatconductorandaflat b as
permeable plate immersed in the quantum vacuum and
face each other at a distance H in the absence of di- FmCP =−∂EmCP . (23)
electric slab, experience a repulsive force F = −7F S ∂H (cid:12)
B 8 C 2 (cid:12)H1,b
[12, 37, 38]. Here, our aim is to investigate the effect of (cid:12)
(cid:12)
the dissipative and amplifying slabs on the Boyer repul- Similar procedure can be performed to calculate the
sive force, F . Using either path-integral technique [13– Casimir force on the right side permeable plate in the
B
15, 29] or transfer matrix formalism [33–35] the Casimir presence of the double-layer dielectric slab in between,
7
FIG. 5. (a) The force per unit area, FmCP, as a function of the distance between the bounding conductor and the permeable
S
plate, H =H +H +b, with H =H , in the presence of a single dielectric slab between the bounding ideal plates, for two
1 2 1 2
valuesofthedissipative(blacksolidandblackdashedcurves)andamplifying(reddashed-dottedandreddashed-dotted-dotted
curves) slab thicknesses b = 100 and 125nm. The inset shows the log-log plot of FmCP as a function of H. (b) The force per
S
unit area, FmCPn, as a function of the distance between the bounding ideal plates, H =H +H +a+b, with H =H , in the
S 1 2 1 2
presence of a double layer dielectric slab between the bounding ideal plates, for two values of the dissipative and amplifying
slab thicknesses a=b=40 and 50nm. The inset presentsthe normalized force, FCP/F , where F =−7F , as a function of
mn B B 8 C
H. Here, m,n=D and A indicate thedissipative and amplifying dielectrics, respectively.
for the fixed values of H ,b and a, as been plotted as a function of the distance between the
1
bounding plates, H = H +H +a+b, for DA and AD
1 2
FmCnP =−∂EmCPn . (24) geometries, for two values of the dissipative and ampli-
S ∂H (cid:12) fying slabs thicknesses a = b = 40 and 50nm. The force
2 (cid:12)H1,a,b
(cid:12) is purely repulsive for AD geometry, while for DA, the
(cid:12)
In Fig.5a, the force per unit area, FmCP, has been plot- cross overfrom attractive to repulsive force occurs when
ted as a function of the distance betwSeen the bounding H increases. By increasing the thickness of the double-
conductor and the permeable plate, H = H +H +b, layer slab for DA geometry, the location of the equilib-
1 2
with H = H , in the presence of a single dielectric riumpointisshiftedtothe largerdistances. Itshouldbe
1 2
slab, for two values of the dissipative (black solid and mentioned that in both Figs.5a and b, the equilibrium
black dashed curves) and amplifying (red dashed-dotted states for the position of the right ideal plate are unsta-
and red dashed-dotted-dotted curves) slabs thicknesses ble. The inset presents the normalized force, FmCnP/FB,
b=100, and 125nm. The inset shows the log-log plot of as a function of H.
FmCP asafunctionofH. Theoverallbehavioroftheforces
S
duetothepresenceofthe dissipativeslabandduetothe
presenceof the amplifying slabaresimilarto eachother. IV. CORRUGATED BOUNDARIES:
Atverysmalldistances the bounding plates attracteach CONDUCTOR-CONDUCTOR
other. Byincreasingthedistance,theforceincreasesand
it gets its zero value at the distance H , where
D(A),equil A. Normal Interaction Between a Flat and a
H <H .
D,equil A,equil Corrugated Conductors
To show the effect of the asymmetry in the sys-
tem due to the presence of conducting and permeable
To investigate the effect of corrugation on the nor-
plates, on the Casimir force in the presence of the
mal Casimir interaction between two conductors in the
double-layer dielectric slab, the slab is located at the
presence of a dissipative or an amplifying slab, we set
middle of the cavity by setting H = H , and we
1 2
the deformation fields on the conductors h (x) = 0 and
choose b = a. The Casimir force is then calculated for 1
h (x) = acos[2πx /λ] (see Fig.1). Then the Casimir
the geometries i) conductor-vacuum-dissipative dielec- 2 1
energy due to the corrugation on one of the conductors,
tric layer-amplifying dielectric layer-vacuum-permeable
labeled by 2, is obtained as
plate (DA), ii) conductor-vacuum-amplifying dielectric
layer-dissipativedielectric layer-vacuum-permeableplate
~c
(AD). Quite interestingly, the force is different for the
E = lnZ (H)−lnZ (H →∞) | ,
geometries i and ii. To present this interesting phe- CF AL γ,2 γ,2 h1(x)=0
Xγ (cid:2) (cid:3)
nomenon, in Fig.5b, the force per unit area, FmCPn, has (25)
S
8
FIG. 6. (a) The normalized Casimir energy, E /EPFA, as a function of H = H +H +b with H = H and λ = 1µm in
CF CF 1 2 1 2
the presence of a dissipative (D, black curves) or an amplifying (A, red curves) dielectric slab for various values of the slab
thicknessesb=0,100,150and200nm. Theblacksolidcurveshowsthenormalized energyintheabsenceofdielectricslab. (b)
The normalized Casimir energy, ECF , as a function of H +b/2 which is the distance between theleft conductor and the
ECF(H/2) 1
middleoftheabsorptive(blackcurves)andamplifyingslabs(redcurves),forfixedvaluesofλ=H =1µmanddifferentvalues
of b thesame as those of panel (a). Here, E (H/2) is the valueof E when the centerof theslab is located at the distance
CF CF
H/2 from each conductors. i.e. H =H .
1 2
where index CF stands for corrugated-flat, and sum is is EPFA =−a2∂FC.
CF 4 ∂H
overγ =TM and TE. This energycanthen be castinto Tostudytheeffectofthelocationofthedielectricslab
on the Casimir interactionbetween the bounding plates,
ECF =−π242a02H~5c KT+Mreg(Hλ)+KT+Ereg(Hλ) , (26) ibneeFnigs.h6obw,nthaesnaorfmunaclitzieodnCofasHimi+r ebn/e2rgwyh,iEchCEFi(CsHFt/h2)e,dhiass-
(cid:2) (cid:3) 1
tance between the left conductor and the middle of the
where according to Eq.(25), the regular part of the TM
dissipativeslab(blackcurves)orthemiddleoftheampli-
and TE kernels is defined as K+reg = K+ −
TM(TE) TM(TE) fying slab (red curves), for fixed values of λ=H =1µm
Lim K+ . In Eq.(26), K+ (q) is the Fourier and different values of b which are the same as those of
tHr1a→ns∞formTMat(iToEn)of K+ (x) aTtM~q(T=E)(0,2π,0). The ex- panel(a). Here, ECF(H/2) is the value of ECF when the
TM(TE) λ center of the slab is located at the distance H/2 from
plicit forms of K+ (H) and K+ (H) are presented in
TM λ TE λ eachconductors. As it can be seen, when the amplifying
AppendicesA and B. In Fig.6a the normalized contri-
slab is closer to the rough conductor in the right side,
bution of the corrugation to the Casimir energy, EEPCFFA, this effect is more pronounced.
CF
has been plotted as a function of H =H +H +b with
1 2
H = H and λ = 1µm in the presence of a dissipative
1 2
(D,blackcurves)oranamplifying(A,redcurves)dielec- B. Lateral Force Between Two Corrugated
tricslabwiththicknessesb=0,100,150and200nm. The Conductors
black solidcurve shows the normalizedenergy in the ab-
sence of dielectric slab. Quite interestingly, the presence
Twocorrugatedconductors,immersedinthequantum
of an amplifying slab enhances the Casimir interaction
vacuum, can experience the lateral Casimir force due to
due to the roughness while the interaction is weakened
the translational symmetry breaking [10, 11, 16] in ad-
bythepresenceoftheabsorptiveslab. Thiseffectismore
dition to the normal Casimir force. In this section, we
pronouncedby increasingthe slabthickness. The dielec-
compare the effect of the presence of the absorptive or
tric functions of the amplifying and dissipative slabs are
amplifying dielectric slabs on the lateral Casimir force
the same as of Eqs.(15) and (16), respectively.
between the bounding corrugated conductors depicted
The superscript PFA in EPFA stands for proximity
CF in Fig. 1. The lateral Casimir force is calculated by
force approximation. PFA is used when a, which is the F = −∂E, where E is the Casimir energy in Eq.(14).
corrugation amplitude, is much smaller than the other l ∂l
By setting the corrugation fields on the rough conduc-
length scales in the system such as H and λ. In the
tors depicted in Fig. 1 as h (x) = acos[2πx /λ] and
1 1
proximity force approximation, the surface elements of
h (x) = acos[2π(x +l)/λ], the lateral Casimir force is
2 1
a curved surface around each point are simply replaced
then obtained as
by surface elements parallel to the plane of (x ,x ) at
1 2
the same point [39, 40]. The Casimir energy in a system 2π~ca2 2πl H H
composed of a corrugated and a flat conductors in PFA Fl = λH5 sin( λ )[KT−M(λ)+KT−E(λ)], (27)
9
FIG. 7. (a) The normalized amplitude of the lateral Casimir force, |F|/|FPFA|, between rough metallic plates (depicted in
l l
Fig.1) as a function of themean distance between corrugated conductors, H =H +H +b, for fixed corrugation wavelength
1 2
λ=1µm, and various values of dissipative slab thicknesses b=0,50,100 and 150nm (black curves, from top to bottom) and
amplifying slab thicknesses (red curves, from bottom to top) the same as of the dissipative one. Here, the center of the slab
is fixedat themiddledistance between boundingideal conductors. (b) Thenormalized amplitudeof thelateral Casimir force,
|F|/|F(H/2)|, as a function of H +b/2 which is the distance between the left conductor and the middle of the dissipative
l l 1
(black curves) or amplifying slab (red curves), for fixed values of λ = H = 1µm and different values of the dielectric slab
thicknessesb=0,20,50and100nm. Here,|F(H/2)|istheamplitudeofthelateralforce whenthecenteroftheslab islocated
l
at thedistance H/2 from each conductors.
where K− (q) and K− (q) are the Fourier transforma- anddifferentvaluesofb=0,20,50and100nm. |F (H/2)|
TM TE l
tions of K− (x) and K− (x) at ~q = (0,2π,0), respec- is the amplitude of the lateral force when the center of
TM TE λ
tively, and their explicit forms can be seen in Appen- theslabisfixedatthemiddleofthecavity. Thedielectric
dicesA and B. functions ofthe amplifying andabsorptiveslabsarecho-
sen the same as in Eqs.(15) and (16), respectively. The
In Fig. 7(a), we have plotted the normalized ampli-
tude of the lateral Casimir force, |F |/|FPFA|, between sinusoidalbehavioroftheforcedoesnotalterbychanging
l l
H , while its amplitude alters. Moreover,if the amplify-
tworoughmetallic plates depictedinFig.1 asa function 1
ingslabcomesclosertoeachconductor,theamplitudeof
of H (the mean separation distance between corrugated
thelateralCasimirforceincreases. Incontrasttotheam-
conductors) for fixed corrugation wavelength, λ = 1µm
plifying slab, the amplitude of the lateral Casimir force
and various values of the absorptive (black curves) am-
decreaseswhenthedissipativedielectricscomescloserto
plifying (red curves) slabs thicknesses b = 0,50,100 and
oneoftheboundingconductors. Byincreasingthethick-
150nm. Here, the center of the slab is fixed at the mid-
dle distance between plates. |FPFA| is the magnitude of ness of the slab, this effect is more pronounced.
l
the lateral Casimir force for the same geometry in PFA
[39,40],whichis|FPFA|= πa2∂FC. Comparingthevalue
l λ ∂H
ofthe normalizedlateralCasimirforceinthe presenceof V. CONCLUSION
the dissipative and amplifying slabs yields that at small
distances, the lateral Casimir force in the presence of an Using either path-integral formalism or transfer ma-
amplifyingslabisstrongerthanthatofabsorptivedielec- trix method, we investigated the normal Casimir inter-
tric slab with the same thickness, whereas by increasing action between flat ideal plates in the presence of a dis-
the distance, in both cases, the values of the normalized sipative or an amplifying slab, or a double-layer dielec-
forceapproachtothevaluesoftheforceintheabsenceof tricslab. Thenormalforcebetweenflatidealconductors
thedielectricslab. Moreover,byincreasingthethickness is non-monotonic due to the presence of the amplifying
of an amplifying slab, the lateral Casimir force increases slabandthe systemhasa stableequilibrium state,while
whereas for the absorptive one decreases. the force is attractive and is weakened by intervening
Wehavealsoconsideredtheeffectofthelocationofthe the absorptive dielectric slab in the cavity. By replac-
center of the dissipative or amplifying slabs, H +b/2, ing the right flat conductor by an ideal permeable plate,
1
on the lateral Casimir force. To this end, in Fig. 7b, the overall behaviors of the Casimir force in the pres-
the normalized amplitude of the lateral Casimir force, ence of the absorptive slab and in the presence of the
|F |/|F(H/2)|, hasbeen shownasafunction ofH +b/2 amplifying slab are the same, and for both cases, the
l l 1
which is the mean distance between the left conductor force is non-monotonic and the system has an unstable
andthemiddleofthedissipative(blackcurves)orampli- equilibrium state. Quite interestingly, the Casimir force
fying (red curves)slab, for fixed values of λ=H =1µm is non-monotonic in the presence of a double-layer di-
10
electric slab in the DA geometry while it is purely re- the behavior of the Casimir interactions can be under-
pulsive for the AD geometry. Then employing the path- stood if one looks at the characteristics of the vacuum
integral technique, we calculated the correction to the fluctuations of the EM field near the slab. The dissipa-
normalCasimirinteractionduetothecorrugationonone tivemediumdecreasesthevacuumfluctuationsoftheEM
ofthe bounding conductors,and alsothe lateralCasimir fieldwhereastheamplifyingmediumincreasesthesefluc-
force has been obtained due to the roughness on both tuations [24, 41]. The method that we employed in this
bounding conductors in the presence of the absorptive paper can be used to study the dynamic Casimir effect
or amplifying slab. While the presence of the amplify- [11,43]andtheCasimirinteractionatfinitetemperature
ing slab enhances both normal and lateral Casimir in- [35,44]inthepresenceofabsorptiveoramplifyingslabs.
teractions between the bounding conductors due to the
roughness, the presence of the absorptive slab weakens
both normal and lateral interactions compared to those ACKNOWLEDGMENTS
of a cavity contains only vacuum between the bounding
conductors. We also showed that both normal and lat-
J.S. acknowledges support from the Academy of Fin-
eral Casimir interactions depend on the distance of the
land through its Centers of Excellence Program (2012-
slab from the bounding conductors. By approaching the
2017) under Project No. 915804.
amplifying slab to one of the conductors, both normal
andlateralCasimirinteractionsenhancecomparedtothe
Casimirinteractionsin a cavitywhichcontainsonly vac-
uum between the bounding conductors. If instead, the Appendix A: Fourier transformation of KT+M(−)(x)
dissipative dielectric slab is brought closer to one of the
boundingconductorsbothnormalandlateralCasimirin- The Fourier transformations of the kernels for TM
teractionsareweakened. The reasonforthe differencein waves at (0,2π,0) are
λ
2π d3~q 2π b b 2π
K+ (0, ,0)= +F (~q,H ,H )Q2G ~q+ ˆi,H + ,H + +F (~q,H ,H )F (~q+ ˆi,H ,H )
TM λ Z (2π)3(cid:20) 1 1 1 1 TM λ 2 2 2 2 1 1 1 5 λ 2 2
(cid:0) (cid:1)
2π
+F (~q,H ,H )F (~q+ ˆi,H ,H ) ,
3 1 1 3 2 2
λ (cid:21)
2π d3~q 2π b b 2π
K− (0, ,0)= +F (~q,H ,H )Q2G (~q+ ˆi,H + ,H + )+F (~q,H ,H )F (~q+ ˆi,H ,H )
TM λ Z (2π)3(cid:20) 2 1 1 1 TM λ 2 2 2 2 4 1 1 6 λ 2 2
2π
+F (~q,H ,H )F (~q+ ˆi,H ,H ) , (A1)
2 1 1 2 2 2
λ (cid:21)
where
G ~q,(−1)i(H + b),(−1)i(H + b)
F (~q,H ,H )= TM i 2 i 2 ,
1 i i (cid:0) N(~q,H ,H ) (cid:1)
1 2
G ~q,(−1)i(H + b),(−1)i(H + b) b b
F (~q,H ,H )= TM i 2 i 2 Q G ~q,−H − ,H + ,
2 i i (cid:0) N(~q,H ,H ) (cid:1) 1 TM 1 2 2 2
1 2 (cid:0) (cid:1)
F (~q,H ,H )= GTM(~q,−H1− 2b,H2+ 2b Q G ~q,−H − b,H + b ,
3 1 2 N(~q,H ,H ) (cid:1) 1 TM 1 2 2 2
1 2 (cid:0) (cid:1)
G ~q,−H − b,H + b
F (~q,H ,H )= TM 1 2 2 2 ,
4 1 2 (cid:0) N(~q,H ,H ) (cid:1)
1 2
G ~q,(−1)i(H + b,(−1)i(H + b) b b
F (~q,H ,H )= TM i 2 i 2 Q2G ~q,−H − ,H + ,
5 i i (cid:0) N(~q,H ,H ) (cid:1) 1 TM 1 2 2 2
1 2 (cid:0) (cid:1)
F (~q,H ,H )= GTM ~q,−H1− 2b,H2+ 2b Q2G ~q,−H − b,H + b , (A2)
6 1 2 (cid:0) N(~q,H ,H ) (cid:1) 1 TM 1 2 2 2
1 2 (cid:0) (cid:1)
and
b b b b b b
N(~q,H ,H )=G ~q,−H − ,−H − G ~q,H + ,H + −G2 ~q,−H − ,H + . (A3)
1 2 TM 1 2 1 2 TM 2 2 2 2 TM 1 2 2 2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Appendix B: Fourier transformation of K−(+)(x)
TE
The Fourier transformations of the kernels for TE
waves at (0,2π,0) are
λ