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January26,2010 1:13 WSPC-ProceedingsTrimSize:9inx6in qfext09 1 0 1 0 2 n a J ELECTROMAGNETIC NON-CONTACT GEARS: PRELUDE 2 2 PrachiParashar∗ andKimballA.Milton† ] Oklahoma Centerfor HighEnergy Physics and Homer L. Dodge Department of r e Physics and Astronomy, Universityof Oklahoma, Norman, OK 73019, USA h E-mail: ∗[email protected], †[email protected] t o . InésCavero-Pelaéz t a Theoretical Physics Department, Zaragoza University,Zaragoza 50009, Spain m E-mail: [email protected] - d K.V.Shajesh n o Saint Edward’s School, Vero Beach, FL 32963, USA c E-mail: [email protected] [ 1 We calculate the lateral Lifshitz force between corrugated dielectric slabs of v finite thickness. Taking the thickness of the plates to infinity leads us to the 5 lateral Lifshitz force between corrugated dielectric surfaces of infinite extent. 0 Takingthedielectricconstanttoinfinityleadsustotheconductorlimitwhich 1 hasbeenevaluatedearlierintheliterature. 4 . 1 1. Introduction 0 0 In past decade significant attention has been given to evaluation of the 1 lateralforcebetweencorrugatedsurfaces(forexampleseeRef.1–5andref- : v erences there-in). In an earlier work we calculated the contribution of the i X next-to-leading order to the lateral Casimir force between two corrugated r semi-transparent δ-function plates interacting with a scalar field,4 and the a leading order contribution for the case of two concentric semi-transparent corrugatedcylinders5 using the multiple scattering formalism (see Ref. 6,7 andreferencesthere-in).Weobservedthatincludingthenext-to-leadingor- dercontributionsignificantlyreducedthedeviationfromtheexactresultin the caseofweakcoupling.Comparisonwithexperiments requiresthe anal- ogouscalculationfortheelectromagneticcase.Herewepresentpreliminary results of our ongoing work on the evaluation of the lateral Lifshitz force between two corrugated dielectric (non-magnetic) slabs of finite thickness January26,2010 1:13 WSPC-ProceedingsTrimSize:9inx6in qfext09 2 interacting through the electromagnetic field (see Fig. 1). From the generalresult it is easy to take various limiting cases. Taking the thickness of the dielectric slabs to infinity leads us to the lateral Lif- shitz force between dielectric slabs of infinite extent. The lateral Casimir force between corrugated conductors was evaluated by Emig et al.1 In our situation this is achieved by taking the dieletric constants ε → ∞. Our i results agree with the results in Emig et al.1 Taking the thin-plate approx- imation based on the plasma model we have calculated the lateral force between corrugated plasma sheets. Our goal is to extend these results to next-to-leading order. Most of these will appear in a forthcoming paper. 2. Interaction energy We consider two dielectric slabs of infinite extent in x-y plane, which have corrugationsiny-direction,asdescribedinFig.1.Wedescribethedielectric slabs by the potentials V (z,y)=(ε −1) [θ(z−a −h (y))−θ(z−b −h (y))], (1) i i i i i i where i = 1,2, designates the individual dielectric slabs. θ(z) is the Heav- iside theta function defined to equal 1 for z > 0, and 0 when z < 0. h (y) i describes the corrugationson the surface of the slabs. We define the thickness of the individual slabs as d = b −a , such that a = a −b > 0 i i i 2 1 represents the distance between the slabs. The permittivities of the slabs are represented by ε . i y0 ε1 ε2 d1 d2 d= 2π k0 a h1 h2 Fig.1. Paralleldielectricslabswithsinusoidalcorrugations. Using the multiple scattering formalism for the case of the electromag- January26,2010 1:13 WSPC-ProceedingsTrimSize:9inx6in qfext09 3 neticfield8,9basedonSchwinger’sGreen’sdyadicformalism10andfollowing the formalism described in Gears-I4 we can obtain the contribution to the interactionenergybetweenthetwoslabsinleadingorderinthecorrugation amplitudes to be i dω E(2) = Tr Γ(0)∆V(1)·Γ(0)∆V(1) , (2) 12 2 2π 1 2 Z h i where ∆V(1) are the leading order contributions in the potentials due to i the presence of corrugations.In particular, we have ∆V(1)(z,y)=−h (y)(ε −1)[δ(z−a )−δ(z−b )]. (3) i i i i i NotethatV(0)describesthepotentialforthecasewhenthecorrugationsare i absentandrepresentthebackgroundintheformalism.Γ(0) =Γ(0)(x,x′;ω) is the Green’s dyadic in the presence of background potential V(0) and i satisfies 1 − ∇×∇× +1+V(0)+V(0) ·Γ(0) =−1. (4) ω2 1 2 (cid:20) (cid:21) The corresponding reduced Green’s dyadic γ(0)(z,z′;k ,k ,ω) is defined x y by Fourier transforming in the transverse variables as Γ(0)(x,x′;ω)= dkxdky eikx(x−x′)eiky(y−y′)γ(0)(z,z′;k ,k ,ω). (5) x y 2π 2π Z Usingthefactthatoursystemistranslationallyinvariantinthex-direction, we can write E(2) ∞ dk ∞ dk′ 12 = y y ˜h (k −k′)h˜ (k′ −k )L(2)(k ,k′), (6) L 2π 2π 1 y y 2 y y y y x Z−∞ Z−∞ where L is the length in the x-direction and h˜ (k ) are the Fourier x i y transforms of the functions h (y) describing the corrugations. The kernel i L(2)(k ,k′) is given by y y 1 dζ dk L(2)(k ,k′)=− x I(2)(k ,ζ,k ,k′), (7) y y 2 2π 2π x y y Z Z where I(2)(k ,ζ,k ,k′)= (ε −1)(ε −1) x y y a b × γ(0)(a ,a ;k ,k ,ω)·γ(0)(a ,a ;k ,k′,ω) 2 1 x y 1 2 x y h−γ(0)(b ,a ;k ,k ,ω)·γ(0)(a ,b ;k ,k′,ω) 2 1 x y 1 2 x y −γ(0)(a ,b ;k ,k ,ω)·γ(0)(b ,a ;k ,k′,ω) 2 1 x y 1 2 x y +γ(0)(b ,b ;k ,k ,ω)·γ(0)(b ,b ;k ,k′,ω) , (8) 2 1 x y 1 2 x y i January26,2010 1:13 WSPC-ProceedingsTrimSize:9inx6in qfext09 4 where the reduced Green’s dyadics are evaluated after solving Eq.(4). We note that γ(0)(z,z′;k ,k ,ω) = γ(0)†(z′,z;k ,k ,ω). Our task reduces to x y x y evaluating the reduced Green’s dyadic in the presence of the background. The details of this evaluation will be described in the forthcoming paper. 2.1. Evaluation of the reduced Green’s dyadic The Green’s dyadic satisfies Eq.(4) whose solution can be determined by following the procedure decribed in Schwinger et al.10 The expression for the reduced Green’s dyadic 1 ∂ 1 ∂ gH 0 ik 1 ∂ gH ε(z)∂zε(z′)∂z′ ε(z′)ε(z)∂z γ(0)(z,z′;k,0,ζ)= 0 −ζ2gE 0  (9)  − ik 1 ∂ gH 0 k2 gH   ε(z)ε(z′)∂z′ ε(z)ε(z′)    is given in terms of the electric and magnetic Green’s functions∗ gE(z,z′) and gH(z,z′), which satisfy the following differential equations: ∂2 − −k2−ζ2ε(z) gE(z,z′)=δ(z−z′), (10) ∂z2 (cid:20) (cid:21) ∂ 1 ∂ k2 − − −ζ2 gH(z,z′)=δ(z−z′). (11) ∂zε(z)∂z ε(z) (cid:20) (cid:21) Wehaveusedthedefinitionsk2 =k2+k2 andε(z)=1+V(0)(z)+V(0)(z). x y 1 2 ThereducedGreen’sdyadicforarbitraryk isgeneratedbytherotation y γ(0)(z,z′;k ,k ,ζ)=R·γ(0)(z,z′;k,0,ζ)·RT, (12) x y where k −k 0 1 x y R= k k 0 . (13) y x k   0 0 k   We have dropped delta functions in Eq.(9) because they are evaluated at different points and thus do not contribute. We shall not present explicit solutions to the electric and magnetic Green’s functions here which will be presented in our forthcoming paper. ∗HereweusethenotationinSchwinger etal10 whichwasreversedinmanyofMilton’s publications,forexampleinMilton’sbook.11 January26,2010 1:13 WSPC-ProceedingsTrimSize:9inx6in qfext09 5 2.2. Interaction energy for corrugated dielectric slabs Usingthesolutionstothe electricandmagneticGreen’sfunctioninEq.(9) we can evaluate I(2)(k ,ζ,k ,k′) in Eq.(8) as x y y 1 1 1 1 1 1 − M(−α ,−α′)M(−α ,−α′)(k2+k k′)2ζ4 k2k′22κ2κ′"∆∆′ 1 1 2 2 x y y 1 1 + M(−α ,α¯′)M(−α ,α¯′)k2(k −k′)2ζ2κ′2 ∆∆¯′ 1 1 2 2 x y y 1 1 + M(α¯ ,−α′)M(α¯ ,−α′)k2(k −k′)2ζ2κ2 ∆¯ ∆′ 1 1 2 2 x y y 1 1 1 + M(α¯ ,α¯′)(k2+k k′)κκ′+M(−α¯ ,−α¯′)k2k′2 ∆¯ ∆¯′ 1 1 x y y 1 1 ε (cid:26) 1(cid:27) 1 × M(α¯ ,α¯′)(k2+k k′)κκ′+M(−α¯ ,−α¯′)k2k′2 , (14) 2 2 x y y 2 2 ε (cid:26) 2(cid:27)# where ∆= (1−α2e−2κ1d1)(1−α2e−2κ2d2)eκa 1 2 (cid:2) −α1α2(1−e−2κ1d1)(1−e−2κ2d2)e−κa , (15) M(αi,α′i)= (εi−1) (1−α2i)e−κidi(1−α′i2)e−κ′idi (cid:3) −(1(cid:2)+αi)(1−αie−2κidi)(1+α′i)(1−α′ie−2κ′idi) ,(16) where κ2i = k2+ζ2εi, κ¯i = κi/εi, and αi = (κi −κ)/(κi+κ). Quan(cid:3)tities with primes are obtained by replacing k →k′ everywhere,and quantities y y with bars are obtained by replacing κ with κ¯ except in the exponentials. i i 2.3. Conductor limit In the conductor limit (ε →∞) the above expression takes the form i I(2) (κ,κ′,k −k′)=− κ κ′ 1+ {κ2+κ′2−(ky −ky′)2}2 . ε→∞ y y sinhκasinhκ′a" 4κ2κ′2 # (17) For the case of sinusoidal corrugations described by h (y) = h sin[k (y+ 1 1 0 y )] and h (y)=h sin[k y] the lateral force can be evaluated to be 0 2 2 0 h h F(2) =2k a sin(k y ) F(0) 1 2A(1,1) (k a), (18) ε→∞ 0 0 0 Cas a a ε→∞ 0 (cid:12) (cid:12) where (cid:12) (cid:12) (cid:12) (cid:12) 15 ∞ ∞ s s 1 (s2+s2 −t2)2 A(1,1) (t )= dt s¯ds¯ + + + 0 , ε→∞ 0 π4 sinhssinhs 2 8s2s2 Z−∞ Z0 + (cid:20) + (cid:21) (19) January26,2010 1:13 WSPC-ProceedingsTrimSize:9inx6in qfext09 6 where s2 = s¯2 +t2 and s2 = s¯2 + (t +t )2. The first term in Eq.(19) + 0 corresponds to the Dirichlet scalar case,4 which here corresponds to the E mode (referredto in Ref. 1 as TM mode). We note that A(1,1) (0)=1. See ε→∞ Fig. 2 for the plot of A(1,1) (k a) versus k a. We observe that only in the ε→∞ 0 0 PFA limit is the electromagnetic contribution twice that of the Dirichlet case, and in general the electromagnetic case is less than twice that of the Dirichlet case. A(ε1→,1∞) (k0a) 1.0 0.8 0.6 0.4 0.2 2 4 6 8 10 k a 0 Fig.2. PlotofAε(1→,1∞) (k0a)versusk0a.Thedottedcurverepresents2timestheDirichlet case. Since the above expression involves a convolution of two functions we can evaluate one of the integrals to get 15 ∞ sin(2t u/π) sinh2u 7 A(1,1) (t )= du 0 −sinh2u ε→∞ 0 4 Z0 (2t0u/π) "cosh6u(cid:18)2 (cid:19) 1 2t 2 sinh2u 1 2t 4 sinh2u 0 0 − + , (20) 2(cid:18) π (cid:19) cosh4u 16(cid:18) π (cid:19) cosh2u# which reproduces the result in Emig et al1 apart from an overall factor of 2,whichpresumablyisatranscriptionerror.EventhoughEq.(20)involves onlyasingleintegralitturns outthatthe doubleintegralrepresentationin Eq. (19) is more useful for numerical evaluation because of the oscillatory nature of the function sinx/x in the former. January26,2010 1:13 WSPC-ProceedingsTrimSize:9inx6in qfext09 7 3. Conclusion We have evaluated leading order contribution to the lateral Lifshitz force between two corrugated dielectric slabs. Taking the dielectric constants of thetwobodiestoinfinitygivesthelateralCasimirforcebetweencorrugated conductors. We shall extend these results to next-to-leading order contribution for a better comparison with experiments in future publication as well as include various other limiting cases, which can be readily obtained from Eq. (14). Acknowledgements WethanktheUSDepartmentofEnergyforpartialsupportofthiswork.We extend our appreciation to Jef Wagner, Elom Abalo and Nima Pourtolami for useful comments throughout the work. References 1. T.Emig,A.Hanke,R.GolestanianandM.Kardar,Phys.Rev.A67,022114 (2003). 2. F.Chen,U.Mohideen,G.L.KlimchitskayaandV.M.Mostepanenko,Phys. Rev.A 66, 032113 (2002). 3. A.LambrechtandV.N.Marachevsky,Phys.Rev.Lett.101,160403 (2008). 4. I. Cavero-Peláez, K. A. Milton, P. Parashar and K. V. Shajesh, Phys. Rev. D 78, 065018 (2008) [arXiv:0805.2776 [hep-th]]. 5. I. Cavero-Peláez, K. A. Milton, P. Parashar and K. V. Shajesh, Phys. Rev. D 78, 065019 (2008) [arXiv:0805.2777 [hep-th]]. 6. K.A.Milton andJ.Wagner,J.Phys.A41,155402 (2008) [arXiv:0712.3811 [hep-th]]. 7. K.A.Milton,J.Phys.Conf.Ser.161,012001 (2009) [arXiv:0809.2564 [hep- th]]. 8. K. A. Milton, P. Parashar and J. Wagner, Phys. Rev. Lett. 101, 160402 (2008) [arXiv:0806.2880 [hep-th]]. 9. K. A. Milton, P. Parashar and J. Wagner, in The Casimir Effect and Cos- mology, ed. S. D. Odintsov, E. Elizalde, and O. B. Gorbunova, in honor of Iver Brevik (Tomsk State Pedagogical University), pp. 107-116 (2009) [arXiv:0811.0128 [math-ph]]. 10. J. S. Schwinger, L. L. . DeRaad and K. A. Milton, Annals Phys. 115, 1 (1979). 11. K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, Singapore, 2001).