Abstract

The Casimir force between electric and magnetic hyperbolic metamaterial slabs is investigated. Due to hyperbolic dispersion, the electromagnetic features of these metamaterials along the optical axis are different from those perpendicular to the optical axis; consequently, these features contribute differently to the Casimir effect. The repulsive Casimir force is formed between electric and magnetic hyperbolic metamaterial slabs; moreover, hyperbolic dispersion can enhance the repulsive effect. However, by utilizing the extremely anisotropic behavior of hyperbolic metamaterials and changing the separation distance between the two slabs, the restoring Casimir force emerges. Additionally, by considering the dispersion of both the permittivity and the permeability of hyperbolic metamaterials, the Casimir force reaches several equilibrium points at different separation distances. Furthermore, the Casimir force at room temperature is discussed. Although the temperature can weaken the effect of the restoring Casimir force, stable equilibria may remain upon choosing suitable filling factors. This work shows that hyperbolic metamaterials have potential applications in micro- and nanoelectromechanical systems, especially for maintaining stability and overcoming adhesion problems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The Casimir effect is a macroscopic quantum effect existing between neutral macroscopic bodies arising from electromagnetic zero-point energy fluctuations in the vacuum due to the existence of the boundaries. In 1948, Casimir theoretically predicted the presence of an attractive force between two neutral, perfectly conducting parallel planes in the vacuum [1]. A few years later, Lifshitz [2] generalized a method to study the attractive force between two semi-infinite dielectric plates containing dispersion and absorption at finite temperatures. A large number of papers have been devoted to obtaining a theoretical understanding of the Casimir effect [3–5] and acquiring experimental measurements of the attractive force [6–8] in recent decades. Generally, the attractive Casimir force between two usual nonmagnetic bodies at small separations is always overwhelming [9]. However, this attractive force may cause irreversible adhesion between neighboring elements in micro- and nanoelectromechanical systems (MEMS and NEMS) [10–12]. Therefore, changing the sign of the Casimir force has become a popular research topic. Repulsive Casimir forces can arise in the systems containing nanowire materials [13,14], or graphene sheets [15,16], or materials with a high magnetic permeability [17,18]. Therefore, repulsive Casimir force systems usually contain magnetically responsive materials, such as saturated ferrite materials [19], or magnetoelectric materials, such as topological insulators [20–23]. In fact, metamaterials [24–29] are extensively used to investigate the repulsive Casimir force; metamaterials are desirable for this purpose because they are artificial, and thus, their electromagnetic properties are controllable in the optical and near-infrared regimes [30]. In addition, the repulsive Casimir force has been observed experimentally between a large plate and a polystyrene sphere coated with a thick gold film, which is separated from the plate by a fluid [31].

Recently, a new kind of artificial material with an unusual permittivity tensor has attracted a great deal of attention [32]. Specifically, the diagonal elements of the permittivity tensor are different and have opposite signs [33]. As a result, the isofrequency contour for TM polarization, which depends on the permittivity, is a hyperboloid. Therefore, this material has been labeled a hyperbolic metamaterial (HMM). Generally, most HMMs are electrically anisotropic; we call this type of HMM an electric hyperbolic metamaterial (ε-HMM). Many potential applications of ε-HMMs, such as sub-diffraction imaging [34,35], negative refraction [36], efficient single-photon sources [37,38], and Casimir interaction torque [39], have been proposed. We previously studied the attractive Casimir force between two ε-HMM slabs, and we demonstrated that hyperbolic dispersion is essential to achieving an enormous increase in the Casimir force [40]. In experiments, ε-HMMs are easily fabricated by metallic nanowire arrays embedded in a dielectric host [35] or by metal-dielectric layered structures [37].

To obtain full flexibility for arbitrarily polarized waves, the magnetic responses of metamaterials that define the dispersion characteristics for TE polarization should be developed to produce magnetic hyperbolic metamaterials (μ-HMMs). Similar to ε-HMMs, μ-HMMs are anisotropic materials with a permeability tensor whose diagonal elements possess opposite signs. Therefore, the resulting isofrequency surface for TE polarization is a hyperboloid. A popular microstructure employed to manipulate the magnetic response is the multilayer fishnet structure, which was predicted theoretically to possess a magnetic hyperbolic dispersion [41]; subsequently, the magnetic hyperbolic dispersion of this structure was confirmed experimentally in a metal-dielectric multilayer fishnet system [42]. Additionally, a theoretical study demonstrated that a wire medium consisting of high-index ε-positive nanowires could also be treated as a μ-HMM at a certain frequency [43]. Therefore, μ-HMMs provide a new platform for modifying the electromagnetic field.

It is well known that the Casimir force can be modified due to the electromagnetic properties of the boundary material [4]. Here, we investigate the formation of the repulsive Casimir effect between HMMs. In particular, we discuss the conditions that are necessary for achieving repulsive and restoring Casimir forces between an ε-HMM slab and a μ-HMM slab in the light of their particular hyperbolic dispersion characteristics. This study indeed is different from our previous research on the influence of hyperbolic dispersion on the attractive Casimir force between two ε-HMM slabs [40]. The remainder of this paper is organized as follows. In Sec. II, we introduce a model considering the Casimir force between two slabs and the electromagnetic responses of HMMs. In Sec. III, we calculate the Casimir force; the repulsive and restoring forces as well as multiple equilibria are obtained, and the influence of the temperature is also considered. In Sec. IV, we draw our conclusions.

2. Model and theoretical framework

The scheme of the system considered here is depicted in Fig. 1. A represents the ε-HMM, while B is the μ-HMM. They are parallel to each other and separated by a distance a. The x-y plane is set to be parallel to the surfaces of the slabs, while the z axis is the out-of-plane optical axis of the anisotropic slabs.

 

Fig. 1 Scheme of the considered system. A and B are an ε-HMM and a μ-HMM, respectively.

Download Full Size | PPT Slide | PDF

2.1 Effective medium theory for hyperbolic metamaterials (HMMs)

HMMs are a particular kind of composite medium. It is important to choose a suitable effective medium theory for HMMs since the calculation of the Casimir force is very sensitive to the choice of which effective medium model is used [44]. In our previous work [40], we demonstrated that the anisotropic Maxwell Garnett mixing formula [45] is suitable for multilayered HMMs when the thicknesses of the layers are much smaller than the operating wavelength. Therefore, based on our research [40] and previous experimental investigations [37,42], we focus here on the case in which both ε-HMM and μ-HMM slabs are constructed from multilayer systems.

The ε-HMM slab A is a multilayer structure with alternating metal and dielectric layers. The metal is nonmagnetic and described by εmA. Similarly, the dielectric layer is described by εdA. Based on the anisotropic Maxwell Garnett mixing formula [45], the permittivity of slab A is a tensor and can be expressed as

εA=(εxxA000εyyA=εxxA000εzzA),
where the elements of the anisotropic permittivity tensor are
εxxA=fAεmA+(1fA)εdA,
εzzA=εmAεdAfAεdA+(1fA)εmA.
Here,fA is the filling factor of the metal in the unit cell of the ε-HMM, i.e., fA=dmA/(dmA+ddA).The metal and the dielectric thicknesses are dmA and ddA. Because both of the ingredient materials are nonmagnetic, the permeability of the ε-HMM is μA=1.

The μ-HMM slab B is a combination of nonmagnetic dielectric and magnetic metamaterials with an effective μeffB. Similar to the expression for the permittivity of the ε-HMM slab, the permeability of the μ-HMM slab is described by the tensor

μB=(μxxB000μyyB=μxxB000μzzB),
where the elements are expressed as
μxxB=fBμeffB+(1fB),
μzzB=μeffBfB+(1fB)μeffB.
Here, fB is the filling factor of the magnetic metamaterial in the unit cell of the μ-HMM, and fB=deffB/(deffB+ddB). deffB and deffB are the thicknesses of the unit cells of the magnetic metamaterial and the nonmagnetic dielectric, respectively. The permittivity of the μ-HMM εB is isotropic for simplicity.

2.2 Casimir force between two parallel slabs

Based on the Maxwell electromagnetic stress tensor method with the properties of macroscopic field operators, the Casimir force at zero temperature can eventually be written as [5]

FC=πRe0dωd2k||(2π)2ω2c2k||2p=TE,TMrpA(ω,k||)rpB(ω,k||)e2iaω2/c2k||21rpA(ω,k||)rpB(ω,k||)e2iaω2/c2k||2,
where k|| is the component of the wave vector parallel to the x-y plane k||=kx2+ky2, and rpA(B) is the reflection coefficient of slab A (B) for p (p=TE,TM)-polarized waves. The integral in Eq. (7) is carried out over all electromagnetic modes. By converting the integral of positive real ω to that of positive imaginary frequencies, i.e., ξ=iω, the Casimir force can be expressed as

FC=2π20dξ0k||dk||ξ2c2+k||2p=TE,TMrpA(iξ,k||)rpB(iξ,k||)e2aξ2/c2+k||21rpA(iξ,k||)rpB(iξ,k||)e2aξ2/c2+k||2.

Since slabs A and B have been considered as bulk anisotropic metamaterials by using the effective medium theory, in this work, we suppose that A and B are both semi-infinite slabs. Then, the reflection coefficients between the vacuum and the ε-HMM (A) described by Eq. (1) can be expressed as [40]

rTEA=μAk0zkAzTEμAk0z+kAzTE,rTMA=εxxAk0zkAzTMεxxAk0z+kAzTM.
Here, k0z is the z component of the wave vector in the vacuum satisfying the electromagnetic dispersion relation k0z2=K2k||2, where K=ω/c is the wavenumber in the vacuum. kAzp (p=TE,TM) is the quantity that is key to electric hyperbolic dispersion, which satisfies the dispersion relation as follows:
k||2+(kAzTE)2=K2εyyAμA,
k||2εzzA+(kAzTM)2εxxA=K2μA.
When εxxAεzzA<0, Eq. (11) describes a hyperboloid consisting of k and kAzTM; this is the reason why uniaxially anisotropic materials with εxxAεzzA<0 were labeled ε-HMMs.

Meanwhile, the reflection coefficients between the vacuum and the μ-HMM (B) described by Eq. (4) are obtained as follows:

rTEB=μxxBk0zkBzTEμxxBk0z+kBzTE,rTMB=εBk0zkBzTMεBk0z+kBzTM.
Here, kBzp is related to magnetic hyperbolic dispersion, which satisfies the dispersion relation
k||2μzzB+(kBzTE)2μxxB=K2εB,
k||2+(kBzTM)2=K2εBμyyB.
Similar to the case for the ε-HMM, Eq. (13) describes a hyperboloid consisting of k and kBzTE when μxxB and μzzB have different signs; this is why uniaxially anisotropic materials with μxxBμzzB<0 were labeled μ-HMMs.

3. Repulsive and restoring Casimir forces

In this section, we discuss the repulsive Casimir force between an ε-HMM slab and a μ-HMM slab. Since the Casimir force is calculated over all frequencies, all ingredient materials should be considered frequency dependent without any loss of generality. These ingredient materials are then reasonably described by the Drude-Lorentz model [24,28,29,46]. For an ε-HMM, the ingredient materials can be described as

εmA=1Ωm2ω2+iγmω,
εdA=1Ωd2ω2ωd2+iγdω.
The ingredient material of a μ-HMM is expressed as
μeffB=1Ωeff2ω2ωeff2+iγeffω.
Here, Ωv, ωv, and γv (v=d,eff) are the plasma frequency, resonance frequency, and damping coefficient, respectively. Ωm and γm are characteristic parameters of the ingredient metal.

3.1 The hyperbolic dispersion regions

The real parts of εxxA, εzzA, μxxB and μzzB are plotted as functions of ω in Fig. 2 for different filling factors; in the corresponding calculations, these characteristic frequencies are chosen as Ωm=ω0, Ωd=0.1ω0, Ωeff=0.3ω0, ωd=ωeff=0.25ω0, γm=0.01ω0 and γd=γeff=0.006ω0. The frequencies are scaled with a reference frequency ω0. The frequencies that satisfy Re[εxxA(ω)]Re[εzzA(ω)]<0 or Re[μxxB(ω)]Re[μzzB(ω)]<0 are called regions of electric or magnetic hyperbolic dispersion, which are illustrated by the shaded areas in Fig. 2. It is clear that an increase in the filling factor can expand the bandwidth of hyperbolic dispersion and enhance the electromagnetic response. In this work, we mainly discuss the influence of this region on the Casimir force.

 

Fig. 2 Real parts of εxxA (solid blue lines) and εzzA (dotted red lines) as functions of ω with different filling factors: (a) fA=0.2, and (b) fA=0.5. Real parts of μxxB (solid blue lines) and μzzB (dotted red lines) with different filling factors: (c) fB=0.2, and (d) fB=0.5. Other parameters are mentioned in the text. Shaded areas indicate regions of hyperbolic dispersion.

Download Full Size | PPT Slide | PDF

3.2 The repulsive Casimir force

To investigate the role of the hyperbolic relation on the Casimir force, we first assume that μA=1 and εB=1.37, indicating that only the electric (magnetic) response is frequency dependent for the ε-HMM (μ-HMM). The relative Casimir force Fr=FC/F0 as a function of the separation a is plotted in Fig. 3(a) for different values of fA and fixed fB=0.5, while Fig. 3(b) shows the results for the opposite case for fixed fA=0.5 but different fB, and the other parameters are the same as in Fig. 2. Here, F0=cπ2/240a4 is the well-known formula for the Casimir force between two perfectly conducting plates with a separation a.

 

Fig. 3 The relative Casimir force Fr=FC/F0 between an ε-HMM and a μ-HMM as a function of the separation a with different filling factors. (a) The filling factor of the ε-HMM fA varies, but the filling factor of the μ-HMM fB is fixed as fB=0.5. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when fB=0.5. (b) The filling factor of the μ-HMM fB varies, but the filling factor of the ε-HMM fA is fixed as fA=0.5. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when fB=0.2. Other parameters are mentioned in the text.

Download Full Size | PPT Slide | PDF

The separation a is scaled by the unit λ0=2πc/ω0. Repulsive Casimir forces, denoted by negative values of Fr, are evident in the results.

It is well known that the repulsive Casimir force can be obtained between an electric plate and a magnetic plate [17,18]. In the present work, slab A is an ε-HMM, for which the electric response dominates across the whole frequency range. From Fig. 3(a), the separation distance of the attraction-repulsion transition decreases with the growth of fA, indicating that the separation region of the repulsive force becomes larger. The force at a small separation is related to the electromagnetic modes of the high frequency region [28], where both slabs are mainly electric. Therefore, the force at a small separation is attractive. Furthermore, when the separation increases, the frequency range of the main contribution shifts to lower frequencies. From the inset of Fig. 3(a), the electromagnetic properties of the μ-HMM are mainly magnetic in these hyperbolic dispersion regions. Consequently, the attractive Casimir forces gradually decrease and finally become repulsive. In addition, as shown in Fig. 2, the increased fA of the ε-HMM leads to a wider hyperbolic dispersion region and a stronger electric response. Thus, the attractive and repulsive forces are both strengthened.

The influence of fB on the force is shown in Fig. 3(b). When fB=0.2, although the μ-HMM has a magnetic hyperbolic response in some frequencies, its electric response is stronger across the whole frequency range, as shown in the inset of Fig. 3(b); thus, the force is attractive. However, the permeability of the μ-HMM is higher than its permittivity εB at lower frequencies for fB=0.5, as shown in the inset of Fig. 3(a). For larger separations, the Casimir forces mainly originate from contributions at lower frequencies, where the A slab (ε-HMM) is mainly electric, and the B slab (μ-HMM) is mainly magnetic. Thus, repulsive forces are found at large separations. This means that the repulsive force will emerge between the ε-HMM and the μ-HMM only if the magnetic response of the μ-HMM is stronger than its electric response.

3.3 The restoring Casimir force and multiequilibrium

From the above discussion, we note that forces can transform from attractive to repulsive with an increase in the separation. However, the forces do not become attractive again with a further increase in the separation. These conditions do not represent a stable equilibrium; with the initial separation possessing Fr=0, they will be either attracted together or repulsed away after a small perturbation. A stable equilibrium can be established when the force becomes attractive with an increasing separation. The Casimir forces at different separations originate from different frequency ranges [28]. Thus, when the electromagnetic response of the μ-HMM changes between electric and magnetic several times across the whole frequency, the polarity of the force converts correspondingly. Consequently, a stable equilibrium may emerge. Therefore, we suppose that the μ-HMM possesses not only the magnetic dispersion described in Eq. (17) but also the electric dispersion whose εB is described by the Drude-Lorentz model as

εB=1ΩB2ω2+iγBω.
The values of the characteristic frequencies are ΩB=γB=0.01ω0.

For the μ-HMM with this frequency-dependent electric dispersion, we plot the Casimir force as a function of the separation in Fig. 4. The restoring forces, defined as the transition from repulsive to attractive with increasing separation, appear in the cases of fB=0.2, 0.5 and 1. As mentioned above, increasing fB can expand the hyperbolic region and enhance its magnetic response. Thus, the repulsion effect is stronger. In the meantime, for low frequencies, the μ-HMM has a strong electric response, as shown by the inset in Fig. 4; therefore, the corresponding repulsive forces at large separations are weakened and finally become attractive.

 

Fig. 4 The relative Casimir force Fr between an ε-HMM and a μ-HMM as a function of the separation a with different fB. The restoring forces are shown in the curves for fB=0.2 (dashed blue line), 0.5 (dotted magenta line), and 1 (dash-dotted red line). The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with fB=0.2.

Download Full Size | PPT Slide | PDF

Because the permittivity of an ε-HMM and the permeability of a μ-HMM are anisotropic, the electromagnetic properties of two slabs are complex across the whole frequency spectrum. Here, we focus on the possibility of multiequilibrium, i.e., the existence of more than one restoring Casimir force with different separations. As an example, the adjusted parameters are chosen as follows: Ωm=0.01ω0, γm=0.001ω0, Ωeff=1.7ω0, ωeff=0.1ω0, and γB=0.0006ω0; the other parameters are the same as those in Fig. 4. The resulting Casimir forces varying with the separation are shown in Fig. 5. It is evident that for fB=0.5, there exist four polarity transitions with an increase in the separation, two of which correspond to stable restoring forces and equilibria, as shown by the squares in Fig. 5. This result can be easily explained as follows. First, along the real frequencies for the μ-HMM, there are two resonances corresponding to μxxB and μzzB, which are the main contributors to the Casimir repulsion at different separations; thus, there are two different separation regions corresponding to the repulsive force. Second, along the imaginary frequencies (see the inset in Fig. 5), the electromagnetic responses are different according to different imaginary frequencies. The attractive forces at very long separations can be explained via the dispersive behavior within the band where εB(iξ)>μxxB(iξ),μzzB(iξ). When the frequencies in the band μxxB(iξ)>εB(iξ)>μzzB(iξ) provide the main contribution, these electromagnetic response properties are involved. Thus, the Casimir force converts its polarity from attractive to repulsive and then to attractive again with decreasing separation. Subsequently, due to the frequency band where μxxB(iξ),μzzB(iξ)>εB(iξ), the force becomes repulsive again at relatively short separations. Because the ε-HMM and μ-HMM are both mainly electric across the entire range of frequencies as mentioned above, the force finally becomes attractive at very short separations. Due to the frequent switching between attraction and repulsion, the magnitude of the force is small. However, the corresponding effect may still be enhanced by carefully adjusting the parameters and filling factors.

 

Fig. 5 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation a with different fB. Multiequilibrium appears when fB=0.5 (dash-dotted magenta line). Circles and squares are the transition points of the force polarity. Squares indicate stable equilibria. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with fB=0.5. Other parameters are mentioned in the text.

Download Full Size | PPT Slide | PDF

3.4 The Casimir force at finite temperatures

Here, we consider the Casimir force at finite temperatures. Equation (8) is rewritten in the form of a summation over the Matsubara frequencies ξm=2πmkBT/ as

FC=kBTπm=0(1δm02)0k||dk||ξm2c2+k||2p=TE,TMrpA(iξm,k||)rpB(iξm,k||)e2aξm2/c2+k||21rpA(iξm,k||)rpB(iξm,k||)e2aξm2/c2+k||2,
where kB is the Boltzmann constant. The Casimir forces at a room temperature of T=300K are illustrated in Fig. 6. Compared with the zero-temperature situations illustrated in Fig. 5, it is obvious that transitions of the force polarity are obtained at larger separations. Although a valley is observed for fB=0.5, the force is always attractive. Therefore, room temperature destroys the multiple stable equilibria at the zero-temperature limit. However, by adjusting a suitable filling factor, the multiple stable equilibria emerge again, as shown by the squares in the case of fB=0.9 in Fig. 6. Although the transitions occur at larger separations, the multiple stable equilibria are maintained.

 

Fig. 6 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation a with different fB at room temperature T=300K. Circles and squares are the transition points of the force polarity. Squares indicate a stable equilibrium. The parameters are the same as those in Fig. 5.

Download Full Size | PPT Slide | PDF

3.5 Discussion

It is known that, all materials are mainly electric at high frequencies. Therefore, the Casimir force at a small separation is attractive since it is related to the electromagnetic modes in the high frequency region. In order to get the repulsive Casimir force, one of the slabs must be magnetic at low frequencies, which is guaranteed by Eq. (17) in the present work. Previous works [47,48] analyzed the Casimir interaction between metal-dielectric metamaterials, and concluded that the Casimir force is always attractive in such systems. However, one still may not eliminate the possibility of fabricating HMMs with the magnetic materials. Recently, a theoretical work provides a new approach to realize hyperbolic metamaterial [49]. They demonstrated that a hyperbolic dispersion relation in diamond with NV centers can be engineered and dynamically tuned by applying a magnetic field. Here we suggest that one can utilize ferrimagnets or saturated ferrite materials as an ingredient material to fabricate the required magnetic metamaterial. The effects of HMM model beyond the Drude-Lorentz expression are left as a subject for further work. The present effects can be measurable in the micron range, thus the reference frequency ω0 should be chosen to be about 1014 rad/s, which can be achieved in recent experiments. Take a=λ0=1μm for example, the force is about 107~105 N/m2. Furthermore, our purpose is to achieve multi-stable equilibria with the Casimir force. As we demonstrate above, to realize the multi-equilibrium, the permeability of μ-HMM should be anisotropic, and the dispersion of permittivity must be taken into account. Correspondingly, the electromagnetic response of the μ-HMM changes between electric and magnetic several times across the whole frequency, then the multiequilibrium will emerge. Therefore, the different resonant frequencies for μxxB and μzzB of μ-HMM are essential to the multi-equilibrium. From these standpoints, the present results do not contradict the conclusions of Refs [47,48].

4. Conclusion

In conclusion, we study the Casimir effect between two types of HMMs, i.e., electric and magnetic hyperbolic metamaterials, both of which exhibit multilayer structures. Due to the effective medium theory, a region of hyperbolic dispersion will emerge, thereby offering a different contribution to the Casimir effect. Repulsive Casimir forces emerge at the zero-temperature limit, and they can be enhanced by adjusting the hyperbolic dispersion. Additionally, we find that the restoring Casimir force occurs between two slabs by manipulating the permittivity dispersion of the μ-HMM. Furthermore, multiple stable equilibria can be established due to the extreme anisotropy of the hyperbolic metamaterials. This phenomenon is understood by analyzing the properties of the electromagnetic responses within the corresponding frequency range at certain separations. Finally, the forces between two types of HMMs at room temperature are also calculated. The formation of multiple stable equilibria is maintained, and they appear at larger separations when one chooses suitable parameters. This work could be helpful for overcoming adhesion problems in MEMS and NEMS in the future.

Funding

National Natural Science Foundation of China (NSFC) (11747080, 11874287, 11574229, 11774262, 11574068); Program 973 (2016YFA0302800); the Shanghai Science and Technology Committee (STCSM) (Grants No.18JC1410900); the Shanghai Education Commission Foundation; A grant from the King Abdulaziz City for Science and Technology (KACST); the Doctoral Scientific Research Foundation of Shanghai Ocean University (A2-0203-00-100375).

Acknowledgments

We thank Springer Nature Author Services (https://authorservices.springernature.com) for its linguistic assistance during the preparation of this manuscript.

References

1. H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51(7), 793–795 (1948).

2. E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2(2), 73–83 (1956).

3. M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001). [CrossRef]  

4. M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir effect (OUP Oxford, 2009), Vol. 145.

5. G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009). [CrossRef]  

6. U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 μ m,” Phys. Rev. Lett. 81(21), 4549–4552 (1998). [CrossRef]  

7. G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002). [CrossRef]   [PubMed]  

8. R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003). [CrossRef]   [PubMed]  

9. O. Kenneth and I. Klich, “Opposites attract: a theorem about the Casimir Force,” Phys. Rev. Lett. 97(16), 160401 (2006). [CrossRef]   [PubMed]  

10. F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998). [CrossRef]  

11. E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001). [CrossRef]  

12. A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011). [CrossRef]  

13. S. I. Maslovski and M. G. Silveirinha, “Ultralong-range Casimir-Lifshitz forces mediated by nanowire materials,” Phys. Rev. A 82(2), 022511 (2010). [CrossRef]  

14. S. I. Maslovski and M. G. Silveirinha, “Mimicking Boyer’s Casimir repulsion with a nanowire material,” Phys. Rev. A 83(2), 022508 (2011). [CrossRef]  

15. J. C. Martinez and M. B. A. Jalil, “Casimir force between metal and graphene sheets,” J. Opt. Soc. Am. B 32(1), 157–162 (2015). [CrossRef]  

16. J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018). [CrossRef]  

17. E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002). [CrossRef]   [PubMed]  

18. O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002). [CrossRef]   [PubMed]  

19. R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011). [CrossRef]  

20. A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011). [CrossRef]   [PubMed]  

21. W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013). [CrossRef]  

22. J. C. Martinez and M. B. A. Jalil, “Tuning the Casimir force via modification of interface properties of three-dimensional topological insulators,” J. Appl. Phys. 113(20), 204302 (2013). [CrossRef]  

23. R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016). [CrossRef]  

24. F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008). [CrossRef]   [PubMed]  

25. F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008). [CrossRef]  

26. V. Yannopapas and N. V. Vitanov, “First-principles study of Casimir repulsion in metamaterials,” Phys. Rev. Lett. 103(12), 120401 (2009). [CrossRef]   [PubMed]  

27. R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009). [CrossRef]   [PubMed]  

28. Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010). [CrossRef]  

29. R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013). [CrossRef]  

30. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef]   [PubMed]  

31. J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009). [CrossRef]   [PubMed]  

32. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013). [CrossRef]  

33. P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014). [CrossRef]   [PubMed]  

34. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef]   [PubMed]  

35. M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009). [CrossRef]  

36. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef]   [PubMed]  

37. Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012). [CrossRef]  

38. H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef]   [PubMed]  

39. T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21(12), 14943–14955 (2013). [CrossRef]   [PubMed]  

40. G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017). [CrossRef]  

41. S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20(14), 15100–15105 (2012). [CrossRef]   [PubMed]  

42. S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016). [CrossRef]   [PubMed]  

43. M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016). [CrossRef]  

44. R. Esquivel-Sirvent and G. C. Schatz, “Mixing rules and the Casimir force between composite systems,” Phys. Rev. A 83(4), 042512 (2011). [CrossRef]  

45. V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016). [CrossRef]   [PubMed]  

46. G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009). [CrossRef]  

47. M. G. Silveirinha and S. I. Maslovski, “Physical restrictions on the Casimir interaction of metal-dielectric metamaterials: An effective-medium approach,” Phys. Rev. A 82(5), 052508 (2010). [CrossRef]  

48. M. G. Silveirinha, “Casimir interaction between metal-dielectric metamaterial slabs: Attraction at all macroscopic distances,” Phys. Rev. B Condens. Matter Mater. Phys. 82(8), 085101 (2010). [CrossRef]  

49. Q. Ai, P.-B. Li, W. Qin, C. P. Sun, and F. Nori, “NV-Metamaterial: Tunable Quantum Hyperbolic Metamaterial Using Nitrogen-Vacancy Centers in Diamond,” arXiv:1802.01280 (2018).

References

  • View by:
  • |
  • |
  • |

  1. H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51(7), 793–795 (1948).
  2. E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2(2), 73–83 (1956).
  3. M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
    [Crossref]
  4. M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir effect (OUP Oxford, 2009), Vol. 145.
  5. G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009).
    [Crossref]
  6. U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 μ m,” Phys. Rev. Lett. 81(21), 4549–4552 (1998).
    [Crossref]
  7. G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
    [Crossref] [PubMed]
  8. R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
    [Crossref] [PubMed]
  9. O. Kenneth and I. Klich, “Opposites attract: a theorem about the Casimir Force,” Phys. Rev. Lett. 97(16), 160401 (2006).
    [Crossref] [PubMed]
  10. F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
    [Crossref]
  11. E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001).
    [Crossref]
  12. A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
    [Crossref]
  13. S. I. Maslovski and M. G. Silveirinha, “Ultralong-range Casimir-Lifshitz forces mediated by nanowire materials,” Phys. Rev. A 82(2), 022511 (2010).
    [Crossref]
  14. S. I. Maslovski and M. G. Silveirinha, “Mimicking Boyer’s Casimir repulsion with a nanowire material,” Phys. Rev. A 83(2), 022508 (2011).
    [Crossref]
  15. J. C. Martinez and M. B. A. Jalil, “Casimir force between metal and graphene sheets,” J. Opt. Soc. Am. B 32(1), 157–162 (2015).
    [Crossref]
  16. J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
    [Crossref]
  17. E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002).
    [Crossref] [PubMed]
  18. O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
    [Crossref] [PubMed]
  19. R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
    [Crossref]
  20. A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011).
    [Crossref] [PubMed]
  21. W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
    [Crossref]
  22. J. C. Martinez and M. B. A. Jalil, “Tuning the Casimir force via modification of interface properties of three-dimensional topological insulators,” J. Appl. Phys. 113(20), 204302 (2013).
    [Crossref]
  23. R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
    [Crossref]
  24. F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
    [Crossref] [PubMed]
  25. F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008).
    [Crossref]
  26. V. Yannopapas and N. V. Vitanov, “First-principles study of Casimir repulsion in metamaterials,” Phys. Rev. Lett. 103(12), 120401 (2009).
    [Crossref] [PubMed]
  27. R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
    [Crossref] [PubMed]
  28. Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
    [Crossref]
  29. R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
    [Crossref]
  30. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
    [Crossref] [PubMed]
  31. J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
    [Crossref] [PubMed]
  32. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
    [Crossref]
  33. P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
    [Crossref] [PubMed]
  34. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [Crossref] [PubMed]
  35. M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
    [Crossref]
  36. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
    [Crossref] [PubMed]
  37. Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
    [Crossref]
  38. H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
    [Crossref] [PubMed]
  39. T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21(12), 14943–14955 (2013).
    [Crossref] [PubMed]
  40. G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
    [Crossref]
  41. S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20(14), 15100–15105 (2012).
    [Crossref] [PubMed]
  42. S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
    [Crossref] [PubMed]
  43. M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016).
    [Crossref]
  44. R. Esquivel-Sirvent and G. C. Schatz, “Mixing rules and the Casimir force between composite systems,” Phys. Rev. A 83(4), 042512 (2011).
    [Crossref]
  45. V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016).
    [Crossref] [PubMed]
  46. G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
    [Crossref]
  47. M. G. Silveirinha and S. I. Maslovski, “Physical restrictions on the Casimir interaction of metal-dielectric metamaterials: An effective-medium approach,” Phys. Rev. A 82(5), 052508 (2010).
    [Crossref]
  48. M. G. Silveirinha, “Casimir interaction between metal-dielectric metamaterial slabs: Attraction at all macroscopic distances,” Phys. Rev. B Condens. Matter Mater. Phys. 82(8), 085101 (2010).
    [Crossref]
  49. Q. Ai, P.-B. Li, W. Qin, C. P. Sun, and F. Nori, “NV-Metamaterial: Tunable Quantum Hyperbolic Metamaterial Using Nitrogen-Vacancy Centers in Diamond,” arXiv:1802.01280 (2018).

2018 (1)

J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
[Crossref]

2017 (1)

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

2016 (4)

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016).
[Crossref]

V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016).
[Crossref] [PubMed]

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

2015 (1)

2014 (1)

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
[Crossref] [PubMed]

2013 (5)

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

J. C. Martinez and M. B. A. Jalil, “Tuning the Casimir force via modification of interface properties of three-dimensional topological insulators,” J. Appl. Phys. 113(20), 204302 (2013).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21(12), 14943–14955 (2013).
[Crossref] [PubMed]

2012 (3)

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20(14), 15100–15105 (2012).
[Crossref] [PubMed]

2011 (5)

R. Esquivel-Sirvent and G. C. Schatz, “Mixing rules and the Casimir force between composite systems,” Phys. Rev. A 83(4), 042512 (2011).
[Crossref]

R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
[Crossref]

A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011).
[Crossref] [PubMed]

S. I. Maslovski and M. G. Silveirinha, “Mimicking Boyer’s Casimir repulsion with a nanowire material,” Phys. Rev. A 83(2), 022508 (2011).
[Crossref]

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

2010 (4)

S. I. Maslovski and M. G. Silveirinha, “Ultralong-range Casimir-Lifshitz forces mediated by nanowire materials,” Phys. Rev. A 82(2), 022511 (2010).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

M. G. Silveirinha and S. I. Maslovski, “Physical restrictions on the Casimir interaction of metal-dielectric metamaterials: An effective-medium approach,” Phys. Rev. A 82(5), 052508 (2010).
[Crossref]

M. G. Silveirinha, “Casimir interaction between metal-dielectric metamaterial slabs: Attraction at all macroscopic distances,” Phys. Rev. B Condens. Matter Mater. Phys. 82(8), 085101 (2010).
[Crossref]

2009 (6)

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
[Crossref] [PubMed]

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

V. Yannopapas and N. V. Vitanov, “First-principles study of Casimir repulsion in metamaterials,” Phys. Rev. Lett. 103(12), 120401 (2009).
[Crossref] [PubMed]

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009).
[Crossref]

2008 (3)

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
[Crossref] [PubMed]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008).
[Crossref]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

2007 (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

2006 (2)

2003 (1)

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
[Crossref] [PubMed]

2002 (3)

E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002).
[Crossref] [PubMed]

O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
[Crossref] [PubMed]

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

2001 (2)

M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
[Crossref]

E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001).
[Crossref]

1998 (2)

F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
[Crossref]

U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 μ m,” Phys. Rev. Lett. 81(21), 4549–4552 (1998).
[Crossref]

1956 (1)

E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2(2), 73–83 (1956).

1948 (1)

H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51(7), 793–795 (1948).

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Alekseyev, L. V.

Atkinson, J.

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
[Crossref] [PubMed]

Bandlow, B.

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

Barnakov, Y. A.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Bi, M. H.

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

Bordag, M.

M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
[Crossref]

Bressi, G.

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

Buks, E.

E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002).
[Crossref] [PubMed]

E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001).
[Crossref]

Capasso, F.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
[Crossref] [PubMed]

Carugno, G.

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

Casimir, H. B. G.

H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51(7), 793–795 (1948).

Chen, H.

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

Chen, L.

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

Chen, X.

J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
[Crossref]

Cortijo, A.

A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011).
[Crossref] [PubMed]

Dalvit, D. A. R.

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
[Crossref] [PubMed]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008).
[Crossref]

Decca, R. S.

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
[Crossref] [PubMed]

Economou, E. N.

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

Esquivel-Sirvent, R.

R. Esquivel-Sirvent and G. C. Schatz, “Mixing rules and the Casimir force between composite systems,” Phys. Rev. A 83(4), 042512 (2011).
[Crossref]

Fischbach, E.

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
[Crossref] [PubMed]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Grushin, A. G.

A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011).
[Crossref] [PubMed]

He, P. F.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Hui, P.-C.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

Iorsh, I.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Iwase, E.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

Jacob, Z.

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
[Crossref] [PubMed]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

Jalil, M. B. A.

J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
[Crossref]

J. C. Martinez and M. B. A. Jalil, “Casimir force between metal and graphene sheets,” J. Opt. Soc. Am. B 32(1), 157–162 (2015).
[Crossref]

J. C. Martinez and M. B. A. Jalil, “Tuning the Casimir force via modification of interface properties of three-dimensional topological insulators,” J. Appl. Phys. 113(20), 204302 (2013).
[Crossref]

Johnson, S. G.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

Kenneth, O.

O. Kenneth and I. Klich, “Opposites attract: a theorem about the Casimir Force,” Phys. Rev. Lett. 97(16), 160401 (2006).
[Crossref] [PubMed]

O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
[Crossref] [PubMed]

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Kivshar, Y. S.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20(14), 15100–15105 (2012).
[Crossref] [PubMed]

Klich, I.

O. Kenneth and I. Klich, “Opposites attract: a theorem about the Casimir Force,” Phys. Rev. Lett. 97(16), 160401 (2006).
[Crossref] [PubMed]

O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
[Crossref] [PubMed]

Klimchitskaya, G. L.

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009).
[Crossref]

Koschny, T.

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

Kosulnikov, S. Y.

M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016).
[Crossref]

Krause, D. E.

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
[Crossref] [PubMed]

Kretzschmar, I.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Krishnamoorthy, H. N.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Kruk, S. S.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20(14), 15100–15105 (2012).
[Crossref] [PubMed]

Lan, Y. H.

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

Li, H.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Lifshitz, E. M.

E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2(2), 73–83 (1956).

Loncar, M.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

López, D.

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
[Crossref] [PubMed]

Lubkowski, G.

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

Maclay, G. J.

F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
[Crossref]

Mann, A.

O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
[Crossref] [PubMed]

Markel, V. A.

Martinez, J. C.

J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
[Crossref]

J. C. Martinez and M. B. A. Jalil, “Casimir force between metal and graphene sheets,” J. Opt. Soc. Am. B 32(1), 157–162 (2015).
[Crossref]

J. C. Martinez and M. B. A. Jalil, “Tuning the Casimir force via modification of interface properties of three-dimensional topological insulators,” J. Appl. Phys. 113(20), 204302 (2013).
[Crossref]

Maslovski, S. I.

T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21(12), 14943–14955 (2013).
[Crossref] [PubMed]

S. I. Maslovski and M. G. Silveirinha, “Mimicking Boyer’s Casimir repulsion with a nanowire material,” Phys. Rev. A 83(2), 022508 (2011).
[Crossref]

S. I. Maslovski and M. G. Silveirinha, “Ultralong-range Casimir-Lifshitz forces mediated by nanowire materials,” Phys. Rev. A 82(2), 022511 (2010).
[Crossref]

M. G. Silveirinha and S. I. Maslovski, “Physical restrictions on the Casimir interaction of metal-dielectric metamaterials: An effective-medium approach,” Phys. Rev. A 82(5), 052508 (2010).
[Crossref]

McCauley, A. P.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

Menon, V. M.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Milonni, P. W.

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
[Crossref] [PubMed]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008).
[Crossref]

Minovich, A.

Mirmoosa, M. S.

M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016).
[Crossref]

Mohideen, U.

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009).
[Crossref]

M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
[Crossref]

U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 μ m,” Phys. Rev. Lett. 81(21), 4549–4552 (1998).
[Crossref]

Morgado, T. A.

Mostepanenko, V. M.

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009).
[Crossref]

M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
[Crossref]

Munday, J. N.

J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
[Crossref] [PubMed]

Narimanov, E.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

Narimanov, E. E.

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Neshev, D. N.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20(14), 15100–15105 (2012).
[Crossref] [PubMed]

Nie, W. J.

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

Noginov, M. A.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

O’Brien, K.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

Onofrio, R.

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

Parsegian, V. A.

J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
[Crossref] [PubMed]

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Podolskiy, V. A.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Powell, D. A.

Pshenay-Severin, E.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

Revzen, M.

O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
[Crossref] [PubMed]

Rodriguez, A. W.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

Rosa, F. S. S.

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008).
[Crossref]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
[Crossref] [PubMed]

Roukes, M. L.

E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002).
[Crossref] [PubMed]

E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001).
[Crossref]

Roy, A.

U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 μ m,” Phys. Rev. Lett. 81(21), 4549–4552 (1998).
[Crossref]

Ruoso, G.

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

Schatz, G. C.

R. Esquivel-Sirvent and G. C. Schatz, “Mixing rules and the Casimir force between composite systems,” Phys. Rev. A 83(4), 042512 (2011).
[Crossref]

Schuhmann, R.

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

Serry, F. M.

F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
[Crossref]

Shekhar, P.

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
[Crossref] [PubMed]

Silveirinha, M. G.

T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21(12), 14943–14955 (2013).
[Crossref] [PubMed]

S. I. Maslovski and M. G. Silveirinha, “Mimicking Boyer’s Casimir repulsion with a nanowire material,” Phys. Rev. A 83(2), 022508 (2011).
[Crossref]

S. I. Maslovski and M. G. Silveirinha, “Ultralong-range Casimir-Lifshitz forces mediated by nanowire materials,” Phys. Rev. A 82(2), 022511 (2010).
[Crossref]

M. G. Silveirinha, “Casimir interaction between metal-dielectric metamaterial slabs: Attraction at all macroscopic distances,” Phys. Rev. B Condens. Matter Mater. Phys. 82(8), 085101 (2010).
[Crossref]

M. G. Silveirinha and S. I. Maslovski, “Physical restrictions on the Casimir interaction of metal-dielectric metamaterials: An effective-medium approach,” Phys. Rev. A 82(5), 052508 (2010).
[Crossref]

Simovski, C. R.

M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016).
[Crossref]

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Smolyaninov, I. I.

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

Song, G.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

Soukoulis, C. M.

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

Tumkur, T.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Vitanov, N. V.

V. Yannopapas and N. V. Vitanov, “First-principles study of Casimir repulsion in metamaterials,” Phys. Rev. Lett. 103(12), 120401 (2009).
[Crossref] [PubMed]

Walliser, D.

F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
[Crossref]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Weiland, T.

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

Wong, Z. J.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

Woolf, D.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

Xu, J. P.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

Yang, Y. P.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

Yannopapas, V.

V. Yannopapas and N. V. Vitanov, “First-principles study of Casimir repulsion in metamaterials,” Phys. Rev. Lett. 103(12), 120401 (2009).
[Crossref] [PubMed]

Zeng, R.

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Zhang, X.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Zhao, R.

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

Zhou, J.

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

Zhu, C. J.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

Zhu, G.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Zhu, S. Y.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

Zubairy, M. S.

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

AIP Adv. (1)

J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
[Crossref]

Appl. Phys. Lett. (3)

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

J. Appl. Phys. (2)

J. C. Martinez and M. B. A. Jalil, “Tuning the Casimir force via modification of interface properties of three-dimensional topological insulators,” J. Appl. Phys. 113(20), 204302 (2013).
[Crossref]

F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nano Converg. (1)

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
[Crossref] [PubMed]

Nat. Commun. (1)

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7(1), 11329 (2016).
[Crossref] [PubMed]

Nat. Mater. (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Nature (3)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
[Crossref] [PubMed]

E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002).
[Crossref] [PubMed]

Opt. Express (3)

Phys. Lett. A (1)

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

Phys. Rep. (1)

M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
[Crossref]

Phys. Rev. A (9)

S. I. Maslovski and M. G. Silveirinha, “Ultralong-range Casimir-Lifshitz forces mediated by nanowire materials,” Phys. Rev. A 82(2), 022511 (2010).
[Crossref]

S. I. Maslovski and M. G. Silveirinha, “Mimicking Boyer’s Casimir repulsion with a nanowire material,” Phys. Rev. A 83(2), 022508 (2011).
[Crossref]

R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
[Crossref]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

R. Esquivel-Sirvent and G. C. Schatz, “Mixing rules and the Casimir force between composite systems,” Phys. Rev. A 83(4), 042512 (2011).
[Crossref]

M. G. Silveirinha and S. I. Maslovski, “Physical restrictions on the Casimir interaction of metal-dielectric metamaterials: An effective-medium approach,” Phys. Rev. A 82(5), 052508 (2010).
[Crossref]

Phys. Rev. B (1)

M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016).
[Crossref]

Phys. Rev. B Condens. Matter Mater. Phys. (3)

M. G. Silveirinha, “Casimir interaction between metal-dielectric metamaterial slabs: Attraction at all macroscopic distances,” Phys. Rev. B Condens. Matter Mater. Phys. 82(8), 085101 (2010).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001).
[Crossref]

Phys. Rev. Lett. (9)

O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
[Crossref] [PubMed]

U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 μ m,” Phys. Rev. Lett. 81(21), 4549–4552 (1998).
[Crossref]

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
[Crossref] [PubMed]

O. Kenneth and I. Klich, “Opposites attract: a theorem about the Casimir Force,” Phys. Rev. Lett. 97(16), 160401 (2006).
[Crossref] [PubMed]

V. Yannopapas and N. V. Vitanov, “First-principles study of Casimir repulsion in metamaterials,” Phys. Rev. Lett. 103(12), 120401 (2009).
[Crossref] [PubMed]

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011).
[Crossref] [PubMed]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
[Crossref] [PubMed]

Proc. K. Ned. Akad. Wet. (1)

H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51(7), 793–795 (1948).

Rev. Mod. Phys. (1)

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009).
[Crossref]

Science (1)

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2(2), 73–83 (1956).

Other (2)

M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir effect (OUP Oxford, 2009), Vol. 145.

Q. Ai, P.-B. Li, W. Qin, C. P. Sun, and F. Nori, “NV-Metamaterial: Tunable Quantum Hyperbolic Metamaterial Using Nitrogen-Vacancy Centers in Diamond,” arXiv:1802.01280 (2018).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Scheme of the considered system. A and B are an ε-HMM and a μ-HMM, respectively.
Fig. 2
Fig. 2 Real parts of ε x x A (solid blue lines) and ε z z A (dotted red lines) as functions of ω with different filling factors: (a) f A = 0.2 , and (b) f A = 0.5 . Real parts of μ x x B (solid blue lines) and μ z z B (dotted red lines) with different filling factors: (c) f B = 0.2 , and (d) f B = 0.5 . Other parameters are mentioned in the text. Shaded areas indicate regions of hyperbolic dispersion.
Fig. 3
Fig. 3 The relative Casimir force F r = F C / F 0 between an ε-HMM and a μ-HMM as a function of the separation a with different filling factors. (a) The filling factor of the ε-HMM f A varies, but the filling factor of the μ-HMM f B is fixed as f B = 0.5 . The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when f B = 0.5 . (b) The filling factor of the μ-HMM f B varies, but the filling factor of the ε-HMM f A is fixed as f A = 0.5 . The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when f B = 0.2 . Other parameters are mentioned in the text.
Fig. 4
Fig. 4 The relative Casimir force F r between an ε-HMM and a μ-HMM as a function of the separation a with different f B . The restoring forces are shown in the curves for f B = 0.2 (dashed blue line), 0.5 (dotted magenta line), and 1 (dash-dotted red line). The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with f B = 0.2 .
Fig. 5
Fig. 5 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation a with different f B . Multiequilibrium appears when f B = 0.5 (dash-dotted magenta line). Circles and squares are the transition points of the force polarity. Squares indicate stable equilibria. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with f B = 0.5 . Other parameters are mentioned in the text.
Fig. 6
Fig. 6 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation a with different f B at room temperature T = 300 K . Circles and squares are the transition points of the force polarity. Squares indicate a stable equilibrium. The parameters are the same as those in Fig. 5.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

ε A = ( ε x x A 0 0 0 ε y y A = ε x x A 0 0 0 ε z z A ) ,
ε x x A = f A ε m A + ( 1 f A ) ε d A ,
ε z z A = ε m A ε d A f A ε d A + ( 1 f A ) ε m A .
μ B = ( μ x x B 0 0 0 μ y y B = μ x x B 0 0 0 μ z z B ) ,
μ x x B = f B μ e f f B + ( 1 f B ) ,
μ z z B = μ e f f B f B + ( 1 f B ) μ e f f B .
F C = π Re 0 d ω d 2 k | | ( 2 π ) 2 ω 2 c 2 k | | 2 p = TE,TM r p A ( ω , k | | ) r p B ( ω , k | | ) e 2 i a ω 2 / c 2 k | | 2 1 r p A ( ω , k | | ) r p B ( ω , k | | ) e 2 i a ω 2 / c 2 k | | 2 ,
F C = 2 π 2 0 d ξ 0 k | | d k | | ξ 2 c 2 + k | | 2 p = TE,TM r p A ( i ξ , k | | ) r p B ( i ξ , k | | ) e 2 a ξ 2 / c 2 + k | | 2 1 r p A ( i ξ , k | | ) r p B ( i ξ , k | | ) e 2 a ξ 2 / c 2 + k | | 2 .
r TE A = μ A k 0 z k A z TE μ A k 0 z + k A z TE , r TM A = ε x x A k 0 z k A z TM ε x x A k 0 z + k A z TM .
k | | 2 + ( k A z TE ) 2 = K 2 ε y y A μ A ,
k | | 2 ε z z A + ( k A z TM ) 2 ε x x A = K 2 μ A .
r TE B = μ x x B k 0 z k B z TE μ x x B k 0 z + k B z TE , r TM B = ε B k 0 z k B z TM ε B k 0 z + k B z TM .
k | | 2 μ z z B + ( k B z TE ) 2 μ x x B = K 2 ε B ,
k | | 2 + ( k B z TM ) 2 = K 2 ε B μ y y B .
ε m A = 1 Ω m 2 ω 2 + i γ m ω ,
ε d A = 1 Ω d 2 ω 2 ω d 2 + i γ d ω .
μ e f f B = 1 Ω e f f 2 ω 2 ω e f f 2 + i γ e f f ω .
ε B = 1 Ω B 2 ω 2 + i γ B ω .
F C = k B T π m = 0 ( 1 δ m 0 2 ) 0 k | | d k | | ξ m 2 c 2 + k | | 2 p = TE,TM r p A ( i ξ m , k | | ) r p B ( i ξ m , k | | ) e 2 a ξ m 2 / c 2 + k | | 2 1 r p A ( i ξ m , k | | ) r p B ( i ξ m , k | | ) e 2 a ξ m 2 / c 2 + k | | 2 ,

Metrics