Casimir-Lifshitz Interaction between Dielectric Heterostructures

The interaction between arbitrary dielectric heterostructures is studied within the framework of a recently developed dielectric contrast perturbation theory. It is shown that periodically patterned dielectric or metallic structures lead to oscillatory lateral Casimir-Lifshitz forces, as well as modulations in the normal force as they are displaced with respect to one another. The strength of these oscillatory contributions increases with decreasing gap size and increasing contrast in the dielectric properties of the materials used in the heterostructures.


I. INTRODUCTION
In light of the ongoing miniaturization of mechanical devices and the recent developments in Casimir-Lifshitz interactions [1,2,3,4,5], there has been some recent interest in the effect of these interactions between the components of small mechanical devices [6]. Since these interaction are particularly strong at small distances, it will be interesting to know how they can be utilized for designing novel mechanical systems that could work without physical contact and could potentially help solve the wear problem [7].
In the past few years there have been a surge of interest in developing techniques that can be used to study the Casimir-Lifshitz interaction in non-ideal geometries, including geometry perturbation theories [8,9,10], semiclassical [11] and classical ray optics [12] approximations, multiple scattering and multipole expansions [13,14,15,16,17], world-line method [18] and exact numerical diagonalization methods [19,20], as well as the numerical Green's function calculation method [21]. These methods have been used in studying the Casimir force in a variety of different geometries, which have improved significantly our understanding of the nontrivial geometry dependence of this effect.
The effect of non-ideal geometry has been shown to lead to a number interesting effects. For example, it has been suggested that corrugated surfaces opposite one another can experience an oscillatory lateral Casimir force [8], which was subsequently observed experimentally [4]. A recent experiment probing the normal Casimir force between a smooth surface and surface with tall rectangular corrugations also revealed further evidence on the nonadditive nature of the Casimir force [5]. Here, we study the Casimir-Lifshitz interaction between arbitrary dielectric heterostructures within the framework of a recently developed formalism [23,27]. We derive a closed form expression for the Casimir-Lifshitz energy between two dielectric heterostructures (such as the example depicted in Fig. 1) up to the second order in the perturbation theory and show that a coherent coupling between the different modes of the spectrum of the dielectric pattern takes place across the gap. As a special example, we consider unidirectional periodic heterostructures (see Fig. 1) and calculate the lateral and normal Casimir-Lifshitz force between them within the same order in the perturbation theory. We find that coupling between modes with identical wavevectors of the pattern structures between the different objects can lead to modulations in the normal force and can give rise to oscillatory later forces, reminiscent of the lateral Casimir force that appears due to coupling between geometrical features such as corrugations [4,8].
This paper is organized as follows. Section II sketches the dielectric contrast perturbation theory, and Sec. III elaborates on how it can be used for dielectric heterostructures giving closed form expressions for the second order term in the perturbation theory. Section IV gives the results for the lateral and normal Casimir-Lifshitz force for a number of choices of materials, and Sec. V contains some discussions and concluding remarks.

II. THEORETICAL FORMULATION
To calculate the Casimir-Lifshitz interaction we need to quantize the electromagnetic field in a background that includes the dielectric or metallic objects that modify the quantum fluctuations of the field. Describing a general assortment of dielectric and metallic objects in space via a frequency dependent dielectric profile ǫ(ω, r), we can write a general expression for the Casimir-Lifshitz energy as [27] FIG. 1: Schematic representation of two identical semi-infinite and periodic objects made of intercalated layers of high and low dielectric functions, occupying the fractions of f and 1 − f , respectively. Here H is the separation between them, a is a dimensionless lateral displacement, and λ is the wavelength of the periodic structure. where We can consider the dielectric function as ǫ(iζ, r) = 1 + δǫ(iζ, r), and expand Eq. (1) in powers of the dielectric contrast. A similar approach has been the subject of a few recent studies [22,23,24,25,26]. The expansion leads to the decomposition of K ij into a diagonal part K 0,ij , corresponding to the empty space, and a perturbation part δK ij , namely where and δK ij (ζ; q, q ′ ) = ζ 2 c 2 δ ij δǫ(iζ, q + q ′ ).
This yields an expansion where The first term is the vacuum energy in the absence of the objects, and the terms in the series take account of their effect in a perturbative scheme. The n-th order term in Eq. (6) takes on the explicit form δǫ(iζ, −q (1) +q (2) ) · · · δǫ(iζ, −q (n) +q (1) ), which involves the Fourier transform of the dielectric contrast profile. Going to real space, we can rewrite the energy of the system as [27] We now use this formulation to study the Casimir-Lifshitz interaction between structures with inhomogeneous or patterned dielectric properties.

III. DIELECTRIC HETEROSTRUCTURES
Let us now consider a configuration similar to the one depicted in Fig. 1, namely two dielectric heterostructures that are placed parallel to each other at a separation H. Using the definition r = (x, z), the dielectric profile can be written as using the labels u and d for the "up" and "down" bodies respectively. To keep the calculations tractable, we now focus on the second order term in the series expansion in Eq. (9). For such two semi-infinite bodies, the second order interaction term between the bodies can be written as for any lateral dielectric function profile, where and We now focus on the specific example of unidirectional periodic structures as depicted in Fig. 1, which is made of subsequent layers of materials with relatively high and low dielectric functions. We can use the periodic properties of the dielectrics and write them in Fourier series expansion. As Fig. 1 shows, we can define the dielectric profile of the d-object as where s is an integer number. We define the Fourier series as where for m = 0, and We can find the corresponding expansion for the u-object by changing x → x + aλ.
Using the Fourier series expansion, one can find the Casimir-Lifshitz energy between two dielectric heterostructures as depicted in Fig. 1 [up to second order in the Clausius-Mossotti expansion of Eq. (9)] as where the prime on the summation sign indicates that the m = 0 term is counted with half the weight, and the pp index means the energy calculated for the plate-plate geometry. This result shows that similar to the case of two corrugated surfaces, two patterned dielectric heterostructures also couple to each other at the leading order when the two wavelengths of the modulations are equal [8]. Moreover, higher harmonics contribute to the Casimir-Lifshitz energy with exponentially decaying contributions, such that at large separations only the fundamental mode (lowest harmonic) will survive [19].

IV. THE NORMAL AND LATERAL FORCES
We now use Eq. (18) to calculate the normal and lateral forces between different types of dielectric and metallic heterostructures. We look at three different types of materials as examples, namely, gold, silicon, and air/vacuum, and consider layered materials made of gold-silicon, silicon-air, and gold-air. We describe the dielectric function of gold using a plasma model, namely, where ω p is the plasma frequency, which is given as ω p (Au) = 1.37 × 10 16 rad/s [28]. For silicon we use the Drude-Lorentz form ǫ(iζ) = 1 + ω 2 ω 0 (Si) = 6.6 × 10 15 rad/s [28]. Finally, for air/vacuum we use ǫ(iζ) = 1. Due to difficulties in keeping the surfaces of the objects parallel to each other, most experiments are performed in plate-sphere geometry. To perform the calculation of the forces for the plate-sphere configuration, we can use the Derjaguin Approximation [29], where we replace one of the semi-infinite objects with a planar surface with a sphere with radius R. The approximation is valid provided that the radius of sphere is much larger than the distance between the dielectric heterostructures, namely, R ≫ H. Using this approximation we can find the normal force between a semi-infinite dielectric heterostructure and a sphere of the same material composition as [29] F nor ps = 2πR Using this result, we can find the Casimir-Lifshitz energy for plate-sphere configuration as which we can now use to calculate the lateral Casimir force as Substituting Eqs. (19) and (20) into Eq. (21), it reads The above equations are the basis of the results that will be presented below. and gold-air are considered each at three different gap sizes of H = 100 nm, H = 300 nm, and H = 600 nm. The normal forces are normalized using the normal force F 0 ps that corresponds to the Casimir-Lifshitz force calculated within the same scheme but with laterally averaged dielectric profile, which corresponds to the m = 0 term in the expansion in Eq. (18). The normal force is found to oscillate as a function of the lateral displacement, having the maximum value when the regions of high dielectric constant from both sides are exactly opposite one another, and the minimum value when in the staggered configuration where regions of higher dielectric constant face regions of lower dielectric constant. The amplitude of the oscillations increases by decreasing the gap size, and the effect is progressively stronger when the contrast between the dielectric properties of the two regions is more pronounced, with a maximum relative change of 0.7 % for gold-silicon, 7 % for silicon-air, and 65 % for gold-air, at the closest separation of H = 100 nm.
In Figs. 2d-f the normal Casimir-Lifshitz forces between the same types of structures as above are presented, for the asymmetric case of f = 0.2. One can see two noticeable differences with the symmetric case. First, the oscillations are now asymmetric, as enforced by the asymmetry of the dielectric profile, although the asymmetry weakens as the gaps size increases and eventually disappears-i.e. the oscillations become symmetric and harmonic-at sufficiently large separations. This is consistent with the picture that different harmonics of the dielectric contrast profile in Eq. (18) couple with each other via an exponential terms that decays with the corresponding wavelengths of each harmonic and as a result any asymmetry caused by higher harmonics will die out at large gap sizes. The second new feature is the significant enhancement of the amplitude of the oscillatory behavior as a function of the lateral displacement. While it is still the case that this amplitude increases with increasing contrast between the dielectric properties of the two materials used in the layered structure, the maximum relative change is 0.4 % for gold-silicon, 6 % for silicon-air, and 200 % for gold-air, at the closest separation of H = 100 nm.
The lateral Casimir-Lifshitz forces for the same layered structures as above are shown in Figs. 3a-c for the symmetric case with f = 0.5. In this case, we have assumed R = 180 µm and λ = 1 µm. Similar to the previous study, three different compositions of gold-silicon, silicon-air, and gold-air are considered each at three different gap sizes of H = 100 nm, H = 200 nm, and H = 400 nm. The lateral force is found to oscillate as a function of the lateral displacement, reminiscent of the lateral Casimir force that is induced by geometrical corrugations [4,8]. The shape of the oscillatory function approaches a sinusoidal behavior as the gap size increases, consistent with the fact that higher harmonics do not contribute to the force in that limit as also seen in geometrical lateral Casimir effect [19]. The amplitude of the oscillations increases by decreasing the gap size as well as the contrast between the dielectric properties of the two regions. Numerically, we find an amplitude of 0.5 pN for gold-silicon, 8 pN for silicon-air, and 12 pN for gold-air, at the closest separation of H = 100 nm. as the gap size is increased and the shape of the profile approaches that of a sinusoidal function (single harmonic).
We also see comparatively more significant enhancement of the amplitude of the oscillatory behavior as a function of the lateral displacement. The amplitude of the oscillations is found as 0.3 pN for gold-silicon, 5 pN for silicon-air, and 7 pN for gold-air, at the closest separation of H = 100 nm.

V. DISCUSSION
In this paper, we have proposed a mechanism by which it is possible to create a lateral Casimir-Lifshitz force as well as controlled modulations in the normal Casimir-Lifshitz force without geometrical corrugations. A coupling similar to what exists in the case of corrugated surfaces gives rise to these oscillatory forces, namely identical modes of the dielectric patterns couple across the gap to generate a macroscopic coherence in the fluctuations. The generic features of these oscillatory forces are very similar to those of the forces caused by corrugations; the effect is stronger and involves more harmonics at closer separations, while it weakens and only involves the lowest mode of the pattern in the dielectric contrast at larger separations.
While the difference in the dielectric properties of the materials controls the general strength of the above results, comparison between Fig. 2 and 3 shows that the modulations in the normal force are more strongly affected by the contrast in the dielectric properties. The choice of air/vacuum as one component also allows us to make predictions about geometrical features with large corrugation amplitudes, which provides an approximation scheme for the nonperturbative geometrical regime.
In the present calculations we have only used the second order terms in the dielectric contrast perturbative series. Higher order terms shown in Eq. (8) will introduce coupling between different modes of the dielectric pattern in a systematic way, as imposed by the overall conservation of the sum of all wavevectors (momenta). While the present is aimed at showing in terms of tractable calculations, one can in principle carry out the calculation of the Casimir-Lifshitz interaction in such dielectric heterostructures using numerical diagonalization methods [20].
Controlled interactions between dielectric heterostructures with smooth outer surfaces could be very useful in practical applications because it will help avoid the complications of bringing surfaces with geometrical protrusions close to each other while avoiding contact between them and controlling their separations. Moreover, it is much easier to pattern dielectric properties of materials in a controlled way than it is to shape them with the high precision that is needed for Casimir effect type experiments.

I. INTRODUCTION
In light of the ongoing miniaturization of mechanical devices and the recent developments in Casimir-Lifshitz interactions [? ? ? ? ? ], there has been some recent interest in the effect of these interactions between the components of small mechanical devices [? ]. Since these interaction are particularly strong at small distances, it will be interesting to know how they can be utilized for designing novel mechanical systems that could work without physical contact and could potentially help solve the wear problem [? ].
In The effect of non-ideal geometry has been shown to lead to a number interesting effects. For example, it has been suggested that corrugated surfaces opposite one another can experience an oscillatory lateral Casimir force [? ], which was subsequently observed experimentally [? ]. A recent experiment probing the normal Casimir force between a smooth surface and surface with tall rectangular corrugations also revealed further evidence on the nonadditive nature of the Casimir force [? ]. Here, we study the Casimir-Lifshitz interaction between arbitrary dielectric heterostructures within the framework of a recently developed formalism [? ? ]. We derive a closed form expression for the Casimir-Lifshitz energy between two dielectric heterostructures (such as the example depicted in Fig. ??) up to the second order in the perturbation theory and show that a coherent coupling between the different modes of the spectrum of the dielectric pattern takes place across the gap. As a special example, we consider unidirectional periodic heterostructures (see Fig. ??) and calculate the lateral and normal Casimir-Lifshitz force between them within the same order in the perturbation theory. We find that coupling between modes with identical wavevectors of the pattern structures between the different objects can lead to modulations in the normal force and can give rise to oscillatory later forces, reminiscent of the lateral Casimir force that appears due to coupling between geometrical features such as corrugations [? ? ].
This paper is organized as follows. Section ?? sketches the dielectric contrast perturbation theory, and Sec. ?? elaborates on how it can be used for dielectric heterostructures giving closed form expressions for the second order term in the perturbation theory. Section ?? gives the results for the lateral and normal Casimir-Lifshitz force for a number of choices of materials, and Sec. ?? contains some discussions and concluding remarks.

II. THEORETICAL FORMULATION
To calculate the Casimir-Lifshitz interaction we need to quantize the electromagnetic field in a background that includes the dielectric or metallic objects that modify the quantum fluctuations of the field. Describing a general assortment of dielectric and metallic objects in space via a frequency dependent dielectric profile ǫ(ω, r), we can write a general expression for the Casimir-Lifshitz energy as [? ] where We can consider the dielectric function as ǫ(iζ, r) = 1 + δǫ(iζ, r), and expand Eq. (??) in powers of the dielectric contrast. A similar approach has been the subject of a few recent studies [? ? ? ? ? ].