## Abstract

Based on the scattering formalism and transfer matrix method, we calculate the Casimir energy in multilayer system containing general anisotropic media and apply the result to the anisotropic saturated ferrite three-layer structure. We investigate the stable equilibrium resulting from repulsive Casimir force in the three-layer anisotropic ferrite structure, focusing on the control of the equilibrium position by means of the external magnetic field, which might provide possibility for Casimir actuation under external manipulation. Furthermore, we propose a Casimir torque switch where the torque acting on the intermediate layer can be switched on and off by tuning the relative orientation between the external magnetic fields applied on the outer ferrite layers. The relation between the feature of torque-off/torque-on state and the weak/strong anisotropy of the ferrite is studied. These findings suggest potential application of Casimir torque in, e.g., cooling the rotation of a thin slab in micromachining process via external magnetic field.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Zero-point quantum fluctuations of the electromagnetic (EM) field are shown to cause the universal van der Waals forces [1]. Resulting from the EM field confined by the boundaries, such fluctuation-induced forces can be detected macroscopically, which was first postulated by Casimir [2]. He predicted that due to the momentum of virtual photons in vacuum, two neutral ideal conducting plates would experience an attractive interaction in the mesoscopic scales. Lifshitz later generalized the result to include two dielectric half-spaces [3], and since then a large body of work has been devoted to developing methods for evaluating the Casimir force for the cases of various material and geometrical configurations [4,5]. The experiments for Casimir force measurement with good accuracy have been performed with present-day technology [6–10], and also from a technological point of view, the Casimir force is found to become significant in the architecture of micro- and nanodevices, where, e.g., the force can be used to drive the devices that are thus known as the Casimir actuated systems [11–13]. Under practical consideration, the change of the sign of Casimir force that is demonstrated to be determined by the frequency-dependent electromagnetic response of the materials, is extensively investigated in recent years [10,14–16]. The transition from attraction to repulsion with the decreasing separation between objects, which corresponds to the restoring Casimir force near a stable equilibrium position, has been a topic of considerable interest [17–22] and is recently verified experimentally [23]. The Casimir force can be regarded as originating from the dependence of the free energy on the relative distance between objects, and similarly as far as the anisotropic objects are concerned the Casimir energy may vary with the relative orientation between objects, such variation results in a rotational interaction that is commonly referred to as the Casimir torque [24–26], an angular analogue of the Casimir force. Due to the anisotropy of the Casimir energy between plates made of anisotropic materials, the plates tend to rotate with respect to each other to minimize the Casimir energy. Theoretical work has addressed the Casimir torque generated with different anisotropic materials, including birefringent materials [27,28], metamaterials [29,30] and anisotropic topological insulators [31], and experimental setups were proposed to measure the torque [32–35]. There has been significant interest about the Casimir torque due to its potential applications, such as noncontact gears [36,37], torsional Casimir actuation [38,39], and Casimir rotor [40].

The multilayer structure suggests a broad variety of different options in the viewpoint of Casimir effect, and the Casimir force and torque in multilayer structures would also be relevant to the integrated device application. In this paper, we develop the method for calculating the Casimir energy in multilayer structure consisting of magnetodielectric anisotropic material, and then focus on the effects in the three-layer anisotropic saturated ferrite system. Between parallel plates embedded in vacuum, the Casimir force is proven to be always attractive if the plates are composed of nonmagnetic materials [41–44], while the repulsive force is only obtained when the magnetic material is involved, motivating the study of the Casimir effect in the context of magnetic plates [45–48], among which the saturated ferrite material is considered [49]. The saturated ferrite possesses nontrivial magnetic permeability that is tunable by the applied external magnetic field, and moreover, the ferrite material is uniaxial anisotropic with respect to the applied field orientation, which provides the possibility for manipulation of the Casimir effect by means of external magnetic field. Stable equilibrium of the material layer inside the multilayer structure can be formed by entrapping the layer with Casimir repulsion on both sides of it, which, in comparison to the stable equilibrium in the two-layer structure, is much more effective, because either the attraction near equilibrium in the two-layer structure is rather weak or the two-layer structure exhibits a rather finite region of stability. Besides, since the case of the repulsion between two plates may not correspond to the case of the repulsion in the multilayer, it is reasonable that special attention should be given to the formation of the Casimir repulsion and stable equilibrium with an overall consideration of the multilayer structure. In the three-layer structure containing saturated ferrites, we show that the equilibrium position of the intermediate layer can be dynamically adjusted via the magnetic-field-tuned permeability of the ferrites, meaning that the layer may be actuated through Casimir force by manipulating the external magnetic field. The influence of the anisotropy of the saturated ferrite material, i.e., the dependence on the orientation of magnetic field, is also examined.

Furthermore, the Casimir torque in three-layer structure containing saturated ferrites is studied in our work. The Casimir torque has potential application for actuating, or, decelerating the rotation of an object, in the latter of which the torque can be considered as a rotational “friction” that cools the rotation state and would therefore be useful in, e.g., the micromachining processes of the devices. Such rotational friction is an interaction that exists between objects even when they are static with respect to each other, in distinction from the so-called quantum friction. Quantum friction between bodies in relative motion [50–53] is essentially connected to the dynamical Casimir effect [52,54,55], known as the energy radiation emitted by bodies in relative motion [56–59]; while here in this paper the torque under the control of external magnetic field is essentially the consequence of the static Casimir effect. In addition, we note that Casimir torque can be considered to make possible the cooling of a rotational body, where, for instance, the torque prevents the body from rotating due to the quantum friction; and such application indicates a process of suppressing the dynamic Casimir effect via its static counterpart.

We propose a Casimir torque switch in a three-layer anisotropic ferrite structure that allows the torque acting on the intermediate ferrite layer to be switched on and off. When the principal axes of the two outer layers that sandwich the intermediate layer are perpendicular to each other, the Casimir torques acting on the two sides of the intermediate layer exhibits opposite sinusoidal-like dependence on the angle and thus may cancel each other out, which corresponds to an “off” state of the torque on the intermediate layer. The switch between the “on” state and the “off” state can be conveniently controlled by the angular variation of the external magnetic fields applied on the outer saturated ferrite layers. In the “off” state of an ideal torque switch in the three-layer structure, the intermediate layer can be placed statically at any arbitrary angular position, which will also be unaffected by the direction of external magnetic field applied on the intermediate layer, while in the “on” state, the intermediate layer is rotated by the torque whose magnitude and sign can be tuned by the direction of the applied magnetic field on it. Such switch of Casimir torque can be further explored in connection with other possible frictional forces to form a rotor that is switched between rotational motion state and stationary state. Therefore, the three-layer ferrite structure is a convenient geometry to use for a range of investigations and applications such as contact-free normal/rotational actuator, rotational cooling, and switchable rotor.

## 2. General formalism for anisotropic multilayer structure systems

In this section, we present a general formalism for the calculation of the Casimir energy between two parallel multilayer structure systems where the layers can be made of anisotropic materials. From the point of view of the scattering formalism, the Casimir energy is calculated between two scatterers in electromagnetic vacuum, and the scattering formula has been used in problems dealing with Casimir interaction for various material configurations and arbitrarily shaped boundaries [60–62]. Using the scattering formalism, the expression for the Casimir energy density between two multilayer structure consisting of anisotropic materials can be given as follows:

*a*is the distance between the two multilayer structure system, and $\{{{d_{l(r)}}} \}= {d_{l(r)1}},{d_{l(r)2}},{d_{l(r)3}} \ldots$ indicate the thicknesses of each layer in the left (right) multilayer structure, and $\{{{\theta_{l(r)}}} \}= {\theta _{l(r)1}},{\theta _{l(r)2}},{\theta _{l(r)3}} \ldots$ indicate the orientation of the principal axes of each layer in the left (right) multilayer structure. ${{\mathbf R}_l}$ and ${{\mathbf R}_r}$ are the $2 \times 2$ Fresnel reflection matrices of the left and right multilayer structures, respectively, which are the functions of layer thicknesses $\{{{d_{l(r)}}} \}$ and principal-axis orientations $\{{{\theta_{l(r)}}} \}$. For a multilayer composed of general anisotropic media, the reflection matrix is defined as The diagonal terms ${r_1}$ and ${r_4}$ describe the reflection amplitudes of transverse electric (TE) polarized incident wave that is reflected with TE polarization, and of transverse magnetic (TM) polarized incident wave that is reflected with TM polarization, respectively. The off-diagonal terms correspond to the mixing of polarizations, where ${r_2}$ is the TM reflection amplitude of a TE incidence, and ${r_3}$ is the TE reflection amplitude of a TM incidence.

Based on the four-component wave function vector constructed from the *x* and *y* components of the electric and magnetic fields, the transfer matrix corresponding to each anisotropic layer of the multilayer structure can be formed. The wave function vectors of electromagnetic field propagating before and after the multilayer structure is calculated by using the transfer matrices of the multilayer, where the boundary conditions of the electromagnetic field at each interface of the multilayer structure are applied. The elements of the total reflection matrix of the multilayer structure can be eventually obtained via the above transfer matrix method, and the details and the results are given in Sec. A.1 in the Appendix. We next present the derivation of the transfer matrices for the anisotropic layers.

Derivation details and results for uniaxial material layer with out-of-plane anisotropic direction are given in Sec. A.2 in the Appendix. In the cases of the in-plane uniaxial anisotropic material and the biaxial anisotropic material, the transfer matrix is more difficult to deal with due to that it is needed to re-solve the Maxwell equations to find the complicated wave vector of the electromagnetic field in the interior of the anisotropic layer. In the following we derive the transfer matrix for the general biaxial anisotropic material layer; for the uniaxial anisotropic material with in-plane principal axis, the result can also be obtained by setting the *yy* (or *xx*) component equal to *zz* component in the electromagnetic tensors. The coordinate system is chosen so that the principal axes of the biaxial anisotropic material coincide with *x* and *y* axes and the permittivity and permeability tensors are diagonal: $\boldsymbol{ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \varepsilon } } = diag({{\varepsilon_{xx}},{\varepsilon_{yy}},{\varepsilon_{zz}}} )$, $\boldsymbol{ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \mu } } = diag({{\mu_{xx}},{\mu_{yy}},{\mu_{zz}}} )$. In order to calculate the transfer matrix for any possible plane of incidence, we let $y^{\prime}z$ plane be the plane of incidence, between which and *y* axis the angle is $\varphi $. In the $x^{\prime}y^{\prime}z$ coordinate system, we assume that the electric and magnetic field solutions to wave equations are

By using ${e_{x^{\prime}}}(0)$ to express the other amplitude quantities, we have

## 3. Casimir effect in three-layer ferrite structure

#### 3.1 Casimir force in three-layer structure: Stable Casimir equilibrium

As is seen from the previous section, the Casimir energy varies with the distance *a* and the angles $\{{{\theta_{l(r)}}} \}$ of each anisotropic layer in the multilayer structure. The distance dependence and the angular dependence of the Casimir energy causes the normal force that acts on the bodies perpendicular to the surface and the torque that leads to rotational motion, respectively. The normal Casimir force can be expressed in terms of energy derivative from Eq. (1) as

*μ*is dependent on the externally applied magnetic field. For the magnetic component of the EM wave parallel to the external field, the permeability of the ferrite remains ${\mu _\parallel } = 1$, while in the EM wave mode with magnetic component perpendicular to the external field, a precession of magnetic dipole around the external field is induced, and the permeability is a function of the strength of the external field ${H_{ex}}$ and the mode frequency $\omega$, which can be described as [63]

Here we consider three-layer anisotropic saturated ferrite structure shown in the inset of Fig. 1(a). Repulsive Casimir force can be obtained between two parallel plate separated by a distance of vacuum where one plate is made of saturated ferrite (the other one is mainly electric); however, the case of the repulsion between two plate may not correspond to the case of the repulsion in the three-layer structure. The force cannot be estimated between two adjacent layers while neglecting the third layer, and an overall consideration should be given to the three-layer structure by means of the general method for calculation of the force between multilayer stacks described in the previous section. In what follows, we focus on the net force acting on the intermediate layer and the possible equilibrium position. The force on the intermediate layer C is the resultant value of the force acting on its two sides, and for example, the force on the left side of layer C can be evaluated using Eq. (14) where *a* is substituted by ${a_1}$, and ${{\mathbf R}_l}$ and ${{\mathbf R}_r}$ are the reflection matrix of layer A and reflection matrix of the structure consisting of layers C and B, respectively. In order to obtain the Casimir repulsion, layer A and layer B in the configuration are assumed to be saturate ferrite, and the intermediate layer C is a slab of material with strong electric response, such as dielectric or metal. The value of the dielectric permittivity of the typical ferrite ${\varepsilon _A} = {\varepsilon _B} = 12$ is chosen [64].

We first examine the force on the intermediate layer C when it is made of dielectric material with the permittivity ${\varepsilon _C}$=10 and is nonmagnetic with ${\mu _C}$=1. The net force on the layer C as a function of distance ${a_1}$ for different ${\omega _{ex}}$ values (i.e., different magnitudes of the external magnetic field ${H_{ex}}$), is depicted in Fig. 1(a), where the total separation distance ${a_1} + {a_2}$ is fixed to be $4{\lambda _m}$. It is assumed that ${\omega _{exA}} = {\omega _{exB}}$ and the thicknesses ${d_A} = {d_B} = {d_C} = 1.0{\lambda _m}$, where the frequency scale is chosen as ${\omega _m}$ of the ferrite, and the distances and thicknesses are scaled with the corresponding wavelength ${\lambda _m} = c/{\omega _m}$. The external magnetic field is applied along the z axis, and thus the elements of the anisotropic permeability tensor are ${\mu _z} = {\mu _\parallel } = 1$ and ${\mu _x} = {\mu _y} = {\mu _ \bot }(\omega )$. The positive values of the force in the figure correspond to the attraction to layer A (or the repulsion from layer B), while the negative values of the force correspond to the repulsion from layer A (or the attraction from layer B). It can be seen that when the intermediate layer is off-center positioned, relative weak repulsion from the nearer outer layer may be obtained under certain values of the applied magnetic field; the Casimir interaction of the intermediate layer with the nearer outer layer is predominantly attractive.

We next consider the case where the intermediate layer is made of metal with dispersive permittivity of the Drude model $\varepsilon = 1 - {{\Omega _m^2} \mathord{\left/ {\vphantom {{\Omega _m^2} {{\omega^2}}}} \right.} {{\omega ^2}}}$. The external magnetic field applied on the outer layers of ferrite material is still assumed to be along the *z* direction. The force experienced by the intermediate layer for different values of magnitude of the applied field on the ferrite layer A and B is shown in Fig. 1(b). When the intermediate layer is moved to deviate from the zero-force position, it can be pulled back by a significant Casimir restoring force and remarkable stability is found at the zero-force equilibrium. Furthermore, such stable equilibrium position can be shifted by altering the applied magnetic field on the outer layers. When the magnitudes of the applied field are identical, ${\omega _{exA}} = {\omega _{exB}} = 0.001{\omega _m}$, the intermediate layer is stably balanced at the center of the structure with ${a_1} = {a_2} = 2{\lambda _m}$. If the magnitude of external field applied on layer A is changed so that ${\omega _{exA}} = 0.01{\omega _m}$, the stable equilibrium position will be shifted towards layer A at ${a_1} = 1.75{\lambda _m}$; or if ${\omega _{exB}}$ is changed to $0.01{\omega _m}$ by altering the field magnitude applied on layer B, the equilibrium will be shifted towards layer B at ${a_1} = 2.25{\lambda _m}$. In the case of the intermediate dielectric layer [Fig. 1(a)], the Casimir repulsion is relatively weak since the electric response of the dielectric is not sufficiently strong, therefore remarkable restoring effect near the equilibrium may not be obtained. In comparison, as seen here in Fig. 1(b), the intermediate metal layer experiences significant Casimir repulsion, which ensures that it is easy to shift the layer dynamically between different stable equilibrium positions by changing the applied field magnitude, thus allowing for application such as Casimir actuated systems. In addition, the alternative structure can also be considered where the outer layers are made of dielectric or metal, while the intermediate layer is saturated ferrite under the control of an external magnetic field. For such structure, provided that the two outer layers are made of materials with different electric responses, the movement of stable equilibrium of the intermediate ferrite layer can be obtained by manipulating the field applied on the intermediate layer.

Moreover, we examine the case where the external magnetic field is considered to be applied along *x* axis. The permeability tensor elements of the ferrite are ${\mu _x} = {\mu _\parallel } = 1$ and ${\mu _y} = {\mu _z} = {\mu _ \bot }(\omega )$. For the structure with two ferrite layer sandwiching a metal layer, it can be found from Fig. 2 that there is no Casimir repulsion for whatever value of the applied field. The zero-force position of the intermediate layer corresponds to an unstable equilibrium, i.e., the layer will collapse into the nearby outer layer if it deviates from the equilibrium position. The strong magnetic response of the ferrite under the *x* direction external field results from the nontrivial permeability in the *y* and *z* direction, while for the *z* direction external field the strong magnetic response is shown in the *x* and *y* direction. Therefore, the magnetic response contribution from *y* direction is the same in both cases, which means that the total effect in the cases of *x* and *z* direction external field is mainly determined by the contribution from the magnetic response in *z* and *x* direction, respectively. When the external field is along *z* axis, strong magnetic response in the *x* direction contributes to both TE and TM wave reflections, but under the external field applied along *x* axis, strong magnetic response in the *z* direction does not contribute to the TM wave reflection. That is to say, the contribution from the *x* direction magnetic response to the repulsive Casimir force when the external field is along *z* axis, is more significant than that from the *z* direction magnetic response to the repulsion when the external field is along *x* axis, which leads to the result that, in the latter case, Casimir repulsion may not be obtained and consequently there is no stable equilibrium in three-layer structure.

#### 3.2 Casimir torque in three-layer structure: Casimir torque switch

We proceed to study the Casimir torque in the three-layer structure. One can obtain the torque from angle derivative of the Casimir energy, $\tau ={-} \partial {E_C}/\partial \theta$, where $\theta$ is the alignment angle between the principal axes of the anisotropic layers. In this section we consider the structure consisting of three anisotropic saturated ferrite layers in which a Casimir torque switch can be formed. In such a system, the torque acting on the intermediate layer can be switched on and off by adjusting the alignment angle between principal axes of the outer layers where, when the torque is switched off, the intermediate ferrite layer can be statically placed at any arbitrary angular position uninfluenced by the direction of external magnetic field that is applied on the layer.

The motivation of the torque switch originates from the phenomenon that the dependence of the Casimir torque for two-plate system on the alignment angle $\theta $ between two principal axes obeys a sinusoidal law $\tau \sim \sin (2\theta )$, which is obtained under the approximation of weak anisotropy, or, slight birefringence [27]. When the relative orientation of the principal axes of the two plates goes from parallel to orthogonal ($\theta = 0 \sim \pi /2$), the torque increases from zero to its maximum (at $\theta = \pi /4$) and reduces to zero in a manner opposite to that of the increasing process, and then repeats the trend in an antisymmetric manner when the two principal axes are changed from orthogonal to antiparallel ($\theta = \pi /2 \sim \pi $). Consider such sinusoidal angular dependence in the configuration of three anisotropic layers. Sandwiched between two anisotropic plates whose principal axes are orthogonal to each other, the intermediate plate may experience the respective torques between it and each of the two outer plates that are of the same magnitude but of opposite signs and consequently counteract each other in whatever direction the principal axis of the intermediate plate is, therefore, a “torque-off” state is formed for the intermediate plate. When the principal axes of the two outer layers are rotated so that they are nonorthogonal to each other, the torques on the two sides of the intermediate plate cannot cancel each other in all of the directions of its principal axis, and thus this corresponds to a “torque-on” state (the maximum magnitude of the torque is reached when the principal axes of the two outer plates are parallel aligned). One may find that the Casimir torque switch formed in the three-layer system is different from the “on/off” state of the torque between two plates of birefringent material, because the “off” state of the torque in two-plate system exists only when the principal axes of two plates are parallel aligned and thus the plates are “caught” at the corresponding position where the principal axes are parallel aligned. In the torque-off state here in the three-layer system, the intermediate plate is not subjected to the torque in any principal axis direction, therefore it is a free state for the intermediate plate.

The sinusoidal-law dependence of the Casimir torque between two anisotropic plates with respect to the alignment angle, is obtained based on the approximation of the weak anisotropy of the two plates. However, the setting of the weak anisotropy may be at the expense of weakening the strength of the torque. On the other hand, for the case of strong anisotropy where the electric/magnetic responses in different directions differ greatly, such as the situation in the saturated ferrite structure system examined in this paper, the angular dependence of the Casimir torque may deviate from the sinusoidal law (but in fact, in order to realize the torque switch, it is not necessary that the angular dependence of the torque between adjacent plates conforms to a strict sinusoidal law; in any case where the angular dependence in the angle ranges of $0 \sim \pi /4$ and $\pi /4 \sim \pi /2$ is symmetrical, the torque switch may possibly be obtained in the three-layer system). In addition, the above proposal for the torque switch is from an estimate based on the angular dependence of the torque between two plates, but the actual torque between two adjacent layers in the three-layer system will also include the influence of the presence of the third layer, therefore, the exact case of the torque switch in the three-layer structure needs to be numerically studied.

A three-layer structure consisting of three anisotropic saturated ferrite layers with in-plane applied external magnetic fields is schematically shown in Fig. 3(a). The torque acting on the three layers as a function of the angle $\theta $ is plotted in Fig. 3(b), where the external fields applied on the two outer layers are orthogonal, $\theta ^{\prime} = \pi /2$. The torque on layer A (B) is evaluated using Eq. (1) where *a* is substituted by ${a_1}$ (${a_2}$), and ${{\mathbf R}_l}$ and ${{\mathbf R}_r}$ are the reflection matrix of layer A (B) and reflection matrix of the structure consisting of layers C and B (A), respectively. The torque on the intermediate layer C represented by the solid line is the resultant value of the torque acting on its two sides. The applied fields are considered to be the same in magnitude ${\omega _{exA}} = {\omega _{exB}} = {\omega _{exC}} = {10^{ - 3}}{\omega _m}$, and the thickness of the layers are all $d = 0.5{\lambda _m}$. The torque $\tau $ is plotted in units of ${\tau _0} = \hbar \omega _m^3/64{c^2}{\pi ^3}$. In order that the torque on the two sides of the intermediate layer can be balanced out so as to achieve the torque switch, the separation distances between adjacent layers are also set to the same value ${a_1} = {a_2} = 0.1{\lambda _m}$. It is obviously seen that the angular dependence of the torque between adjacent layers exhibits a certain deviation from the sinusoidal law [see the dashed line and dotted line in Fig. 3(b)], where the angle at which the maximum value of the torque occurs deviates from $\pi /4$, leading to that the torque on both sides of the intermediate layer cannot completely offset each other and thus the result differs from the expected “off” state of a torque switch.

When larger layer separation distances are taken, the angular dependence of the torque between adjacent layers shows greater deviation from the sinusoidal law, as shown in Fig. 4(a) where ${a_1} = {a_2} \equiv a$. Since the Casimir torque is greatly suppressed with the increase of the separation distances, the normalized Casimir torque ${\tau _n}({a,\theta } )\equiv {{\tau ({a,\theta } )} \mathord{\left/ {\vphantom {{\tau ({a,\theta } )} {{{|\tau |}_{\max }}(a )}}} \right.} {{{|\tau |}_{\max }}(a )}}$ is adopted in Fig. 4(a) in order to see clearly the change of the angular law with the varying separation distances, where $\tau ({a,\theta } )$ is the torque on the outer layer at the angle $\theta $ and ${|\tau |_{\max }}(a )$ is the maximum torque on the outer layer for fixed layer separation *a*. In Fig. 4(b), the torque on each layer is plotted when the separation distance ${a_1} = {a_2} = 0.5{\lambda _m}$ is fixed. Although the angular dependence deviates from the sinusoidal law, compared with the result in Fig. 3(b), the more symmetrical angular dependence between the ranges of $0 \sim \pi /4$ and $\pi /4 \sim \pi /2$ results in that the torques on the two sides of the intermediate layer tend to be of the same magnitude and opposite signs, and the net torque is relatively closer to zero. But it also has to be noted that the result is given for the case of larger distance where the Casimir torque between two plates is much weakened compared with that at short distance.

We proceed to examine the influence of the magnitude of the external magnetic field. Figure 5 shows the angular dependence of the Casimir torque on each ferrite layer when the external field applied on the intermediate layer of the three-layer system is strengthened. ${\omega _{exC}}$ is set to $1.0{\omega _m}$, while the values of ${\omega _{exA}}$ and ${\omega _{exB}}$ are unchanged. The separation distances are ${a_1} = {a_2} = 0.1{\lambda _m}$. The permeability perpendicular to the external field of the intermediate ferrite layer is decreased with the increased external field, indicating the weakened anisotropy of the intermediate layer, which is close to the approximation of obtaining the sinusoidal angular dependence. It can be found from the figure that in the case where the two outer layer still exhibit strong anisotropy (to ensure that the torque is not weak), the torque between adjacent layers in the three-layer system may tend to follow the sinusoidal law $\tau \sim \sin (2\theta )$, and then the intermediate layer experiences an almost zero net torque, which is close to the “off” state of an ideal Casimir torque switch.

In order to quantitatively show how close the result is to the torque-off state, we calculate and plot in Fig. 6 the maximum torque on each layer ${|\tau |_{\max }}$ for fixed layer separations ${a_1} = {a_2} = 0.1{\lambda _m}$ and the relative maximum torque which is defined as the ratio of maximum torque on the intermediate layer C to that on the outer layer A(B), ${{|\tau |_{\max }^\textrm{C}} \mathord{\left/ {\vphantom {{|\tau |_{\max }^\textrm{C}} {|\tau |_{\max }^{\textrm{A}(\textrm{B})}}}} \right.} {|\tau |_{\max }^{\textrm{A}(\textrm{B})}}}$. The blue circles represent the case where only ${\omega _{exC}} \equiv {\omega _{ex}}$ is varied while ${\omega _{exA}} = {\omega _{exB}} = {10^{ - 3}}{\omega _m}$ are fixed, and the red squares represent the case where ${\omega _{exA}} = {\omega _{exB}} = {\omega _{exC}} \equiv {\omega _{ex}}$ are varied simultaneously. When the value of ${{{\omega _{ex}}} \mathord{\left/ {\vphantom {{{\omega_{ex}}} {{\omega_m}}}} \right.} {{\omega _m}}}$ decreases from 1, the perpendicular permeability of ferrite increases and the anisotropy is enhanced, and the torque on the outer layer becomes stronger [see Fig. 6(a)]. When ${{{\omega _{ex}}} \mathord{\left/ {\vphantom {{{\omega_{ex}}} {{\omega_m}}}} \right.} {{\omega _m}}}$ is smaller than about 0.1, the maximum torque on the outer layer tends to a saturation limit. The maximum torque on the intermediate layer exhibits similar behavior [see Fig. 6(b)]: the torque is significantly suppressed with the increase of external magnetic field in the range of ${{{\omega _{ex}}} \mathord{\left/ {\vphantom {{{\omega_{ex}}} {{\omega_m}}}} \right.} {{\omega _m}}} = 0.05\sim 1$. The smaller value of the relative maximum torque ${{|\tau |_{\max }^\textrm{C}} \mathord{\left/ {\vphantom {{|\tau |_{\max }^\textrm{C}} {|\tau |_{\max }^{\textrm{A}(\textrm{B})}}}} \right.} {|\tau |_{\max }^{\textrm{A}(\textrm{B})}}}$ indicates that the system becomes close to the ideal torque-off state and may also has a remarkable torque-on state (the maximum torque on the outer layer is strong, and consequently the torque on intermediate layer is significant when the principal axes of the outer layers are nonorthogonal to each other). It is seen from Fig. 6(c) that the relative maximum torque is generally decreased with the weakening of the anisotropy of ferrite (the increase of ${\omega _{ex}}$). In the case with the decreased anisotropy of the three ferrite layers (red squares), both the maximum torque on intermediate layer and the relative maximum torque are smaller than those in the case with only decreased anisotropy of the intermediate layer (blue circles), however, the torque on the outer layer in the former case is also weakened because of weak anisotropy of all the layers.

## 4. Conclusion

In conclusion, we have derived the theoretical expressions for calculating the Casimir torque and force in the multilayer system containing general anisotropic media based on the transfer matrix method. The results were presented for the uniaxial material with out-of-plane and in-plane anisotropic directions as well as for the biaxial anisotropic material. We then applied our general result to the three-layer structure consisting of anisotropic saturated ferrite whose permeability is tuned by an external magnetic field. The formation and control of the stable equilibrium based on the repulsive Casimir forces in the three-layer system was discussed, and the influences of the magnitude and direction of the external magnetic field were studied. Furthermore, we investigated the Casimir torque acting on the layers of the three-layer system. We proposed a Casimir torque switch based on the three-layer ferrite structure, where the torque on the intermediate layer can be switched on and off. It was found that although a near-ideal torque-off state can be formed with weak anisotropic layers, but a system containing strong anisotropic layers can also similarly correspond to a good torque switch with an “off” state that is close to zero-torque and a more remarkable torque-on state. The report in this work could provide some insight into the Casimir effect in three-layer anisotropic system and could be potentially useful in the research fields related to, e.g., the stability, the actuation, and the rotational cooling of macroscopic objects in micro- and nanodevices under the control of externally applied fields in the future.

## A. Appendix

## A.1 Fresnel reflection coefficients

The Fresnel reflection matrix can be derived using the method similar to that presented in [65]. In this Appendix, we give the detailed expression for the reflection matrix elements. The elements of the total reflection matrix of the multilayer structure can be obtained as

*M*of each layer in the structure:

## A.2 Derivation of the transfer matrix of out-of-plane uniaxial anisotropic material layer

In this Appendix, we give the detailed derivation of the transfer matrix of uniaxial material layer with out-of-plane anisotropic direction. The coordinate system is chosen so that the interfaces of the multilayer structure coincide with the *xy* plane. The anisotropic layer occupies the region $z \sim z + \Delta z$, and the electromagnetic tensors are given by

*j*th uniaxial anisotropic layer in the multilayer structure, the four-component wave function vector is written in terms of the electric and magnetic fields in the

*j*th layer as In the out-of-plane uniaxial anisotropic material layer, the electric and magnetic field solutions to the wave equations are assumed to be as follows:

*z*direction and that with perpendicular wave vector along negative

*z*direction, which will be indicated in what follows by superscripts $+$ and $-$, respectively. The above expressions shows the relations for the positive

*z*direction propagation terms, while those for negative

*z*direction propagation terms can be given by

*x*,

*y*,

*z*) inside the layer can be written as

*M*describes the relation between the wave function vectors at (

*x*,

*y*,

*z*) and at (

*x*,

*y*, $z + \Delta z$): By substituting Eq. (32) into Eq. (33), one can obtain the explicit expression for the transfer matrix of out-of-plane uniaxial anisotropic material layer

## Funding

National Natural Science Foundation of China (11574068, 11804219, 11874287, 61901148); Zhejiang Provincial Natural Science Foundation of China (LY14A040008, LQ18F050002).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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