# Influence of retardation and dispersive surfaces on the regimes of the
lateral Casimir-Polder force

Lucas Queiroz<EMAIL_ADDRESS>Instituto Federal de Educação,
Ciência e Tecnologia do Pará, Campus Marabá Rural, 68508-970, Marabá, Pará,
Brazil  Faculdade de Física, Universidade Federal do Pará, 66075-110, Belém,
Pará, Brazil

(August 28, 2024)

###### Abstract

We investigate, by means of the scattering approach, the Casimir-Polder
interaction between a neutral anisotropic polarizable particle and a
corrugated surface made of a realistic material. By focusing on the lateral
force (arising from the presence of corrugation on the surface), we
investigate the conditions for the particle to be attracted to the nearest
corrugation peak, valley, or to an intermediate point between a peak and a
valley, with such behaviors called peak, valley and intermediate regimes,
respectively. Such regimes of the lateral force were recently predicted in the
literature, but in the context of the van der Waals interaction and
considering the surface made of some ideal material (a perfectly conducting or
a nondispersive dielectric). Here, we investigate how the occurrence of the
mentioned regimes is affected by the consideration of realistic dielectric
properties for the surface and also of the retardation in the interaction. In
this context, we show that the consideration of a dispersive surface, when
compared to the mentioned idealized materials, can amplify the occurrence of
the valley and intermediate regimes. Moreover, regarding the consideration of
retardation, we show that it has a small influence on the occurrence of the
valley regime, but, for the intermediate ones, can either amplify or inhibit
them. Such investigation provides a preciser description of the interaction
between an anisotropic particle and a corrugated surface, giving a better
understanding of the nontrivial aspects of the lateral Casimir-Polder force.

##  I Introduction

The Casimir-Polder (CP) interaction between a neutral polarizable particle and
a surface is a quantum phenomenon that arises due to the quantum fluctuations
of the electromagnetic field in vacuum [1, 2]. In recent years, it has been
shown that such interaction could present nontrivial behaviors when
considering anisotropic polarizable particles [3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16]. In Ref. [12], specifically, it was investigated the
behavior of the lateral van der Waals (vdW) force acting on an anisotropic
particle due to the presence of corrugations on the surface, and it was
discussed the existence of regimes of this force. Such regimes are
characterized by the possibility of the particle to be attracted towards the
nearest corrugation peak, valley, or an intermediate point between a peak and
a valley, with such behaviors called peak, valley and intermediate regimes,
respectively [12]. In Ref. [13], it was investigated how these regimes are
affected by the consideration of dielectric medias, but considering only
nondispersive ones, since the focus was on obtaining only a first estimate
about the behavior of the regimes in the presence of dielectrics.

The investigations done in Refs. [12, 13] are very useful to obtain
preliminary insights about the mentioned regimes. On the other hand, to obtain
more precise results which can be compared with experimental data, it is
essential to consider retardation effects and real material properties of the
surface. Thus, in the present paper, we investigate the CP interaction between
an anisotropic polarizable particle and a dielectric corrugated surface made
of realistic materials. Following the discussion made in Ref. [17], we use the
scattering approach, at zero-temperature, to calculate the CP energy for this
system. In this way, we generalize a formula found in Ref. [17] in such a way
that we include the case of an anisotropic electrically polarizable particle,
and study the nontrivial regimes of the lateral force discussed in Refs. [12,
13] in a more general context, showing how they are affected by the
consideration of a surface described by a frequency-dependent electric
permittivity, as well as retardation effects on the interaction.

The paper is organized as follows. In Sec. II, we develop the scattering
approach to describe the interaction between an anisotropic particle and a
corrugated surface, and explore such interaction in the vdW and CP regimes. In
Sec. III, we apply the obtained formulas to the case of a sinusoidal
corrugated surface. In Sec. IV, we discuss some implications of our results.
In Sec. V, we present our final comments.

##  II Interaction energy for an anisotropic polarizable particle and a
corrugated surface

Let us start considering a neutral anisotropic polarizable particle (here we
consider only electrically polarizable ones) in vacuum situated at ${\bf
r}_{0}={\bf r}_{0||}+z_{0}\hat{{\bf z}}$ (with ${\bf r}_{0||}=x_{0}\hat{{\bf
x}}+y_{0}\hat{{\bf y}}$), and interacting with a corrugated surface, as shown
in Fig. 1. The corrugation profile of the surface is described by the function
$h(x,y)$ [$z_{0}>h(x,y)$], which defines a suitable modification of a
reference plane at $z=0$.

Figure 1:  Illustration of an anisotropic polarizable particle (represented by
an ellipsoid), located at ${\bf r}_{0}=x_{0}\hat{{\bf x}}+y_{0}\hat{{\bf
y}}+z_{0}\hat{{\bf z}}$, interacting with a general corrugated surface, whose
corrugation profile is described by $z=h(x,y)$ [$z_{0}>h(x,y)$].

By using the scattering approach, in Ref. [17] the authors derived a zero-
temperature scattering formula to compute the interaction between an isotropic
particle and a corrugated surface. Here, we take as basis such derivation, but
considering an anisotropic particle interacting with the corrugated surface.
This allows us to investigate the nontrivial regimes of the lateral force
discussed in Refs. [12, 13] in a more general context, considering realistic
dispersive materials to describe the surface, as well as arbitrary particle-
surface distances (of course, obeying the restrictions of the discussed case,
as shown later).

As in Ref. [17], it is convenient to derive the scattering formula to the
problem discussed here using the plane-wave basis $\left|{\bf
k},p\right\rangle$, where ${\bf k}$ represents the lateral wave-vector of a
fluctuating electromagnetic field with polarization $p$ (TE for transverse
electric and TM for transverse magnetic). Assuming that $z_{0}$ is much larger
than the dimensions of the particle, the zero-temperature scattering formula
to compute the Casimir-Polder interaction is given by [17]

$\displaystyle U\left({\bf r}_{0}\right)$
$\displaystyle=-\hbar\int_{0}^{\infty}\frac{d\xi}{2\pi}\int\frac{d^{2}{\bf
k}}{\left(2\pi\right)^{2}}\int\frac{d^{2}{\bf
k}^{\prime}}{\left(2\pi\right)^{2}}e^{-\left(\kappa+\kappa^{\prime}\right)z_{0}}$
$\displaystyle\times\sum_{p,p^{\prime}}\left\langle{\bf k},p\right|{\cal
R}_{S}\left|{\bf k}^{\prime},p^{\prime}\right\rangle\left\langle{\bf
k}^{\prime},p^{\prime}\right|{\cal R}_{P}\left|{\bf k},p\right\rangle,$ (1)

where $\kappa=\sqrt{\xi^{2}/c^{2}+\left|{\bf k}\right|^{2}}$ is the imaginary
$z$-component of the wave-vector associated with the imaginary frequency
$\xi$. The factor $\exp[-\left(\kappa+\kappa^{\prime}\right)z_{0}]$ represents
the round-trip propagation of the field between the surface and the particle.
Finally, ${\cal R}_{P}$ and ${\cal R}_{S}$ are the reflection operators for
the particle and the surface, respectively (we remark that these operators are
frequency dependent, but to avoid an overload of the notation we are omitting
it).

###  II.1 The consideration of an anisotropic particle

The reflection operator for an anisotropic particle can be calculated
following the same steps discussed in Ref. [17] in the context of an isotropic
one. The consideration of a ground-state anisotropic particle consists in
describing it as an induced electric dipole with a dipole moment given by

${\bf d}\left(\omega\right)=\boldsymbol{\alpha}\left(\omega\right)\cdot{\bf
E}\left({\bf r}_{0},\omega\right),$ (2)

where $\boldsymbol{\alpha}\left(\omega\right)$ is the dynamic polarizability
tensor of the particle written in terms of real frequencies $\omega$ (later we
consider imaginary frequencies by making $\omega\to i\xi$). We remark that the
polarizability of an anisotropic particle is described by a symmetric tensor
of rank two. For an isotropic particle, this tensor is given by
$\boldsymbol{\alpha}=\alpha{\bf I}$ (with ${\bf I}$ being the identity matrix)
and thus its polarizability can be simply described by the scalar quantity
$\alpha$. Following the steps discussed in Ref. [17], but considering Eq. (2),
one obtains

$\displaystyle\left\langle{\bf k}^{\prime},p^{\prime}\right|{\cal
R}_{P}\left|{\bf k},p\right\rangle$
$\displaystyle=-\frac{\xi^{2}}{2\epsilon_{0}\kappa^{\prime}c^{2}}e^{i\left({\bf
k}-{\bf k}^{\prime}\right)\cdot{\bf
r}_{0||}}e^{-\left(\kappa+\kappa^{\prime}\right)z_{0}}$
$\displaystyle\times\sum_{m,n}\alpha_{mn}\left(i\xi\right)\left[\hat{\boldsymbol{e}}_{p^{\prime}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}\left[\hat{\boldsymbol{e}}_{p}^{+}\left({\bf
k}\right)\right]_{n},$ (3)

where $\alpha_{mn}\left(i\xi\right)$ $(m,n=x,y,z)$ are the components of the
polarizability tensor $\boldsymbol{\alpha}$, and
$\left[\hat{\boldsymbol{e}}_{p^{\prime}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}$ and
$\left[\hat{\boldsymbol{e}}_{p}^{+}\left({\bf k}\right)\right]_{n}$ are the
components of the unit polarization vectors for the outgoing and the incoming
fields, respectively. The $\pm$ notation represents the propagation direction
of the field along the $z$-axis, and thus the superscript $+$ refers to the
incoming field on the particle, and $-$ refers to the outgoing field. For a
field propagating with complete wave-vector ${\bf K}^{\pm}={\bf k}\pm
k_{z}\hat{{\bf z}}$, with
$k_{z}=\text{sgn}(\omega)\sqrt{\omega^{2}/c^{2}-\left|{\bf k}\right|^{2}}$
(when we make $\omega\to i\xi$, we get $k_{z}\to i\kappa$), the unit
polarization vectors corresponding to the TE and TM polarizations are defined
as

$\displaystyle\hat{\boldsymbol{e}}_{\text{TE}}^{+}\left({\bf k}\right)$
$\displaystyle=\hat{\boldsymbol{e}}_{\text{TE}}^{-}\left({\bf
k}\right)=\hat{{\bf z}}\times\hat{{\bf k}},$ (4)
$\displaystyle\hat{\boldsymbol{e}}_{\text{TM}}^{\pm}\left({\bf k}\right)$
$\displaystyle=\hat{\boldsymbol{e}}_{\text{TE}}^{\pm}\left({\bf
k}\right)\times\hat{{\bf K}}^{\pm}.$ (5)

Besides, the product $\left[\hat{\boldsymbol{e}}_{p^{\prime}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}\left[\hat{\boldsymbol{e}}_{p}^{+}\left({\bf
k}\right)\right]_{n}$ can be viewed as components of a $3\times 3$ matrix, so
that we have one matrix for each combination of field polarizations
($p^{\prime}$ and $p$). Thus, using Eqs. (4) and (5), one obtains that

$\displaystyle\left[\hat{\boldsymbol{e}}_{\text{TE}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}\left[\hat{\boldsymbol{e}}_{\text{TE}}^{+}\left({\bf
k}\right)\right]_{n}$ $\displaystyle=\frac{1}{\left|{\bf
k}^{\prime}\right|\left|{\bf
k}\right|}\left(\begin{array}[]{ccc}k_{y}^{\prime}k_{y}&-k_{y}^{\prime}k_{x}&0\\\
-k_{x}^{\prime}k_{y}&k_{x}^{\prime}k_{x}&0\\\ 0&0&0\end{array}\right);$ (9)
$\displaystyle\left[\hat{\boldsymbol{e}}_{\text{TE}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}\left[\hat{\boldsymbol{e}}_{\text{TM}}^{+}\left({\bf
k}\right)\right]_{n}$ $\displaystyle=\frac{1}{\left|{\bf
k}^{\prime}\right|\left|{\bf
k}\right|}\frac{c}{\xi}\left(\begin{array}[]{ccc}-k_{y}^{\prime}\kappa
k_{x}&-k_{y}^{\prime}\kappa k_{y}&-ik_{y}^{\prime}\left|{\bf k}\right|^{2}\\\
k_{x}^{\prime}\kappa k_{x}&k_{x}^{\prime}\kappa
k_{y}&ik_{x}^{\prime}\left|{\bf k}\right|^{2}\\\ 0&0&0\end{array}\right);$
(13) $\displaystyle\left[\hat{\boldsymbol{e}}_{\text{TM}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}\left[\hat{\boldsymbol{e}}_{\text{TE}}^{+}\left({\bf
k}\right)\right]_{n}$ $\displaystyle=\frac{1}{\left|{\bf
k}^{\prime}\right|\left|{\bf
k}\right|}\frac{c}{\xi}\left(\begin{array}[]{ccc}\kappa^{\prime}k_{x}^{\prime}k_{y}&-\kappa^{\prime}k_{x}^{\prime}k_{x}&0\\\
\kappa^{\prime}k_{y}^{\prime}k_{y}&-\kappa^{\prime}k_{y}^{\prime}k_{x}&0\\\
-ik_{y}\left|{\bf k}^{\prime}\right|^{2}&ik_{x}\left|{\bf
k}^{\prime}\right|^{2}&0\end{array}\right);$ (17)
$\displaystyle\left[\hat{\boldsymbol{e}}_{\text{TM}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}\left[\hat{\boldsymbol{e}}_{\text{TM}}^{+}\left({\bf
k}\right)\right]_{n}$ $\displaystyle=\frac{1}{\left|{\bf
k}^{\prime}\right|\left|{\bf
k}\right|}\frac{c^{2}}{\xi^{2}}\left(\begin{array}[]{ccc}-\kappa^{\prime}k_{x}^{\prime}\kappa
k_{x}&-\kappa^{\prime}k_{x}^{\prime}\kappa
k_{y}&-i\kappa^{\prime}k_{x}^{\prime}\left|{\bf k}\right|^{2}\\\
-\kappa^{\prime}k_{y}^{\prime}\kappa
k_{x}&-\kappa^{\prime}k_{y}^{\prime}\kappa
k_{y}&-i\kappa^{\prime}k_{y}^{\prime}\left|{\bf k}\right|^{2}\\\ i\kappa
k_{x}\left|{\bf k}^{\prime}\right|^{2}&i\kappa k_{y}\left|{\bf
k}^{\prime}\right|^{2}&-\left|{\bf k}^{\prime}\right|^{2}\left|{\bf
k}\right|^{2}\end{array}\right).$ (21)

Now that we found the reflection operator for an anisotropic polarizable
particle, the interaction energy between it and a general surface is computed
by substituting Eq. (3) into Eq. (1), but taking $z_{0}=0$ in the former,
because the propagation of the field along the $z$-axis is already taken into
account in Eq. (1) [described by the factor
$\exp[-\left(\kappa+\kappa^{\prime}\right)z_{0}]$]. Thus, when performing
this, one obtains

$\displaystyle U\left({\bf r}_{0}\right)$
$\displaystyle=\frac{\hbar}{\epsilon_{0}c^{2}}\int_{0}^{\infty}\frac{d\xi}{2\pi}\xi^{2}\sum_{m,n}\alpha_{mn}\left(i\xi\right)$
$\displaystyle\times\int\frac{d^{2}{\bf
k}}{\left(2\pi\right)^{2}}\int\frac{d^{2}{\bf
k}^{\prime}}{\left(2\pi\right)^{2}}\frac{e^{-\left(\kappa+\kappa^{\prime}\right)z_{0}}}{2\kappa^{\prime}}e^{i\left({\bf
k}-{\bf k}^{\prime}\right)\cdot{\bf r}_{0||}}$
$\displaystyle\times\sum_{p,p^{\prime}}\left\langle{\bf k},p\right|{\cal
R}_{S}\left|{\bf
k}^{\prime},p^{\prime}\right\rangle\left[\hat{\boldsymbol{e}}_{p^{\prime}}^{-}\left({\bf
k}^{\prime}\right)\right]_{m}\left[\hat{\boldsymbol{e}}_{p}^{+}\left({\bf
k}\right)\right]_{n}.$ (22)

Note that, when we consider an isotropic particle
[$\alpha_{mn}\left(i\xi\right)=\alpha\left(i\xi\right)\delta_{mn}$], we
recover the expression for $U\left({\bf r}_{0}\right)$ obtained in Ref. [17]
[Eq. (11) of this reference].

###  II.2 The consideration of a corrugated surface

As discussed in Ref. [17], the calculation of the reflection operator for a
corrugated surface is a highly nontrivial problem. One way to do this is by
writing this operator as a perturbative expansion in powers of $h(x,y)$. Such
calculation was already done in Refs. [18, 19] considering that the
corrugation amplitude [$\text{max}|h(x,y)|=a$] is the smallest length scale in
the problem. The solution obtained from this calculation was used in Ref. [17]
up to first perturbative order. Following Ref. [17], let us also consider the
reflection operator for a corrugated surface as a perturbative expansion in
powers of $h$. Assuming that $a\ll z_{0}$, we can consider this perturbative
expansion up to the first order, i.e.:

${\cal R}_{S}\approx{\cal R}_{S}^{(0)}+{\cal R}_{S}^{(1)},$ (23)

where ${\cal R}_{S}^{(0)}$ is the unperturbed solution which describes the
specular reflection of the field on a plane surface, whereas ${\cal
R}_{S}^{(1)}$ is the first-order correction to ${\cal R}_{S}^{(0)}$ due to the
surface corrugation. In this context, the Casimir-Polder interaction can be
written as the sum

$U\left({\bf r}_{0}\right)\approx U^{(0)}\left(z_{0}\right)+U^{(1)}\left({\bf
r}_{0}\right).$ (24)

Note that the dependence of $U^{(1)}$ on the variables $x_{0}$ and $y_{0}$
gives rise to a lateral force (a force parallel to the reference plane $z=0$)
that acts on the particle. Since we are interested only in the behavior of
this lateral force, we focus our attention only on $U^{(1)}$, which involves
${\cal R}_{S}^{(1)}$.

The matrix elements of the first-order reflection operator ${\cal
R}_{S}^{(1)}$ can be written as [19, 17]

$\langle{\bf k},p|{\cal R}_{S}^{(1)}|{\bf
k}^{\prime},p^{\prime}\rangle=R_{pp^{\prime}}^{(1)}\left({\bf k},{\bf
k}^{\prime}\right)H\left({\bf k}-{\bf k}^{\prime}\right),$ (25)

where $H\left({\bf k}\right)$ is the Fourier transform of $h(x,y)$, and the
components of the matrix $R_{pp^{\prime}}^{(1)}$ are given by [19, 17]:

$\displaystyle R_{\text{TE},\text{TE}}^{(1)}({\bf k},{\bf k}^{\prime})$
$\displaystyle=2\kappa^{\prime}\frac{\kappa-\kappa_{t}}{\kappa^{\prime}+\kappa_{t}^{\prime}}C,$
(26) $\displaystyle R_{\text{TE},\text{TM}}^{(1)}({\bf k},{\bf k}^{\prime})$
$\displaystyle=2\kappa^{\prime}\frac{c}{\xi}\frac{\kappa_{t}^{\prime}\left(\kappa-\kappa_{t}\right)}{\epsilon\left(i\xi\right)\kappa^{\prime}+\kappa_{t}^{\prime}}S,$
(27) $\displaystyle R_{\text{TM},\text{TE}}^{(1)}({\bf k},{\bf k}^{\prime})$
$\displaystyle=2\kappa^{\prime}\frac{\xi/c}{\left(\kappa^{\prime}+\kappa_{t}^{\prime}\right)}\frac{\left[\epsilon\left(i\xi\right)\kappa-\kappa_{t}\right]\kappa_{t}S}{\frac{\xi^{2}}{c^{2}}-\kappa^{2}\left[\epsilon\left(i\xi\right)+1\right]},$
(28) $\displaystyle R_{\text{TM},\text{TM}}^{(1)}({\bf k},{\bf k}^{\prime})$
$\displaystyle=-2\kappa^{\prime}\frac{\epsilon(i\xi)\kappa-\kappa_{t}}{\epsilon(i\xi)\kappa^{\prime}+\kappa_{t}^{\prime}}\frac{[\epsilon(i\xi)|{\bf
k}||{\bf
k}^{\prime}|+\kappa_{t}\kappa_{t}^{\prime}C]}{\frac{\xi^{2}}{c^{2}}-\kappa^{2}[\epsilon(i\xi)+1]},$
(29)

with $\kappa_{t}=\sqrt{\epsilon\left(i\xi\right)\xi^{2}/c^{2}+\left|{\bf
k}\right|^{2}}$ and

$\displaystyle S$
$\displaystyle=\sin\left(\phi-\phi^{\prime}\right)=\frac{k_{y}k_{x}^{\prime}-k_{x}k_{y}^{\prime}}{\left|{\bf
k}\right|\left|{\bf k}^{\prime}\right|},$ (30) $\displaystyle C$
$\displaystyle=\cos\left(\phi-\phi^{\prime}\right)=\frac{k_{x}k_{x}^{\prime}+k_{y}k_{y}^{\prime}}{\left|{\bf
k}\right|\left|{\bf k}^{\prime}\right|}.$ (31)

Thus, substituting Eq. (25) into Eq. (22), we can write $U^{(1)}$ in a similar
way to Ref. [17], namely

$U^{(1)}\left({\bf r}_{0}\right)=\int\frac{d^{2}{\bf
k}}{\left(2\pi\right)^{2}}e^{i{\bf k}\cdot{\bf r}_{0||}}g\left({\bf
k},z_{0}\right)H\left({\bf k}\right),$ (32)

where $g\left({\bf k},z_{0}\right)$ is the response function given by

$g({\bf
k},z_{0})=\frac{\hbar}{\epsilon_{0}c^{2}}\int_{0}^{\infty}\frac{d\xi}{2\pi}\xi^{2}\sum_{m,n}\alpha_{mn}(i\xi)\int\frac{d^{2}{\bf
k}^{\prime}}{(2\pi)^{2}}a_{{\bf k}^{\prime},{\bf k}^{\prime}-{\bf k}}^{mn},$
(33)

with

$\displaystyle a_{{\bf k}^{\prime},{\bf k}^{\prime\prime}}^{mn}$
$\displaystyle=\frac{e^{-(\kappa^{\prime}+\kappa^{\prime\prime})z_{0}}}{2\kappa^{\prime\prime}}$
$\displaystyle\times\sum_{p^{\prime},p^{\prime\prime}}R_{p^{\prime},p^{\prime\prime}}^{(1)}({\bf
k}^{\prime},{\bf k}^{\prime\prime})[\hat{e}_{p^{\prime\prime}}^{-}({\bf
k}^{\prime\prime})]_{m}[\hat{e}_{p^{\prime}}^{+}({\bf k}^{\prime})]_{n}.$ (34)

We remark that these results are a generalization of those found in Refs. [12,
13] since they take into account the realistic properties for the surface and
retardation effects in the interaction. Besides this, they also generalize
Eqs. (16) and (17) of Ref. [17] by the consideration of an anisotropic
electrically polarizable particle. Using these results, we later investigate
the behavior of $U^{(1)}$ to obtain information about the lateral force acting
on the particle. But before this, let us discuss these results within the vdW
and CP regimes.

###  II.3 Van der Waals regime

The vdW regime is the limiting case where retardation effects can be
neglected, which occurs when $z_{0}\ll\lambda_{P},\lambda_{S}$, with
$\lambda_{P}$ and $\lambda_{S}$ being typical wavelengths that characterize
the optical responses of the particle and the surface, respectively. In this
regime, it is assumed that the particle-surface interaction is instantaneous,
which means that the limit $c\to\infty$ can be considered. Thus, by performing
$c\to\infty$ in Eqs. (32)-(34), one obtains

$U^{(1)}_{\text{vdW}}\left({\bf r}_{0}\right)=\int\frac{d^{2}{\bf
k}}{\left(2\pi\right)^{2}}e^{i{\bf k}\cdot{\bf
r}_{0||}}g_{\text{vdW}}\left({\bf k},z_{0}\right)H\left({\bf k}\right),$ (35)

where

$\displaystyle g_{\text{vdW}}({\bf k},z_{0})$
$\displaystyle=\frac{\hbar}{\epsilon_{0}}\sum_{m,n}\int_{0}^{\infty}\frac{d\xi}{2\pi}\alpha_{mn}(i\xi)\int\frac{d^{2}{\bf
k}^{\prime}}{(2\pi)^{2}}e^{-(|{\bf k}^{\prime}|+|{\bf
k}^{\prime\prime}|)z_{0}}$
$\displaystyle\times\frac{[\epsilon(i\xi)-1]}{[\epsilon(i\xi)+1]^{2}}\left[\epsilon(i\xi)+\frac{k_{x}^{\prime}k_{x}^{\prime\prime}+k_{y}^{\prime}k_{y}^{\prime\prime}}{|{\bf
k}^{\prime}||{\bf k}^{\prime\prime}|}\right]G_{mn},$ (36)

with ${\bf k}^{\prime\prime}={\bf k}^{\prime}-{\bf k}$, and

$G_{mn}=\left(\begin{array}[]{ccc}-k_{x}^{\prime\prime}k_{x}^{\prime}&-k_{x}^{\prime\prime}k_{y}^{\prime}&-ik_{x}^{\prime\prime}|{\bf
k}^{\prime}|\\\
-k_{y}^{\prime\prime}k_{x}^{\prime}&-k_{y}^{\prime\prime}k_{y}^{\prime}&-ik_{y}^{\prime\prime}|{\bf
k}^{\prime}|\\\ ik_{x}^{\prime}|{\bf k}^{\prime\prime}|&ik_{y}^{\prime}|{\bf
k}^{\prime\prime}|&-|{\bf k}^{\prime\prime}||{\bf
k}^{\prime}|\end{array}\right).$ (37)

Such result provides the vdW interaction between an anisotropic polarizable
particle and a corrugated surface made of a real dispersive material.

When considering a nondispersive surface, one has $\epsilon(i\xi)=\epsilon$,
so that the integration over $\xi$ has to be performed only on the
polarizability tensor components $\alpha_{mn}(i\xi)$. One can write these
components as [20]

$\alpha_{mn}\left(i\xi\right)=\frac{2}{\hbar}\sum_{l\neq
k}\frac{\omega_{lk}\langle k|\hat{d}_{m}|l\rangle\langle
l|\hat{d}_{n}|k\rangle}{\omega_{lk}^{2}+\xi^{2}},$ (38)

where $\hat{d}_{i}(i=x,y,z)$ are the components of the dipole moment operator
and $\omega_{lk}$ is the frequency of a typical transition between the states
$l$ and $k$ of the particle. Thus, by performing the $\xi$-integration on
(38), one obtains

$\int_{0}^{\infty}d\xi\alpha_{mn}\left(i\xi\right)=\frac{\pi}{\hbar}\langle\hat{d}_{m}\hat{d}_{n}\rangle.$
(39)

By substituting this result into Eq. (36), and performing the integrals over
${\bf k}^{\prime}$, Eqs. (35)-(37) recover the result for
$U^{(1)}_{\text{vdW}}$ obtained in Ref. [13] [Eq. (50) of this reference with
$\epsilon_{2}=1$]. In addition, when considering a perfectly conducting
surface [$\epsilon(i\xi)\to\infty$], Eqs. (35)-(37) recover the result for
$U^{(1)}_{\text{vdW}}$ obtained in Ref. [12] [Eq. (14) of this reference].

###  II.4 Casimir-Polder regime

The CP regime is the limiting case where retardation effects must be taken
into account, which occurs when $z_{0}\gg\lambda_{P},\lambda_{S}$. In this
regime, $U^{(1)}_{\text{CP}}$ is given by Eqs. (32)-(34) with
$\alpha_{mn}\left(i\xi\right)\to\alpha_{mn}\left(0\right)$ and
$\epsilon\left(i\xi\right)\to\epsilon\left(0\right)$. When considering
metallic surfaces, one can also perform the limit
$\epsilon\left(0\right)\to\infty$ in Eqs. (32)-(34), which acts only on the
components of the matrix $R_{pp^{\prime}}^{(1)}$ [Eqs. (26)-(29)] leading to
[21]

$\displaystyle\lim_{\epsilon\to\infty}R_{\text{TE},\text{TE}}^{(1)}({\bf
k},{\bf k}^{\prime})$ $\displaystyle=-2\kappa^{\prime}C,$ (40)
$\displaystyle\lim_{\epsilon\to\infty}R_{\text{TE},\text{TM}}^{(1)}({\bf
k},{\bf k}^{\prime})$ $\displaystyle=-2\frac{\xi}{c}S,$ (41)
$\displaystyle\lim_{\epsilon\to\infty}R_{\text{TM},\text{TE}}^{(1)}({\bf
k},{\bf k}^{\prime})$
$\displaystyle=-2\frac{\xi}{c}\frac{\kappa^{\prime}}{\kappa}S,$ (42)
$\displaystyle\lim_{\epsilon\to\infty}R_{\text{TM},\text{TM}}^{(1)}({\bf
k},{\bf k}^{\prime})$ $\displaystyle=\frac{2}{\kappa}\left(\left|{\bf
k}\right|\left|{\bf k}^{\prime}\right|+\frac{\xi^{2}}{c^{2}}C\right),$ (43)

with $S$ and $C$ given by Eqs. (30) and (31).

##  III Sinusoidal corrugation

To investigate the behavior of $U^{(1)}$, it is convenient to choose a
corrugation profile for the surface. Let us investigate the case of a
sinusoidal corrugated surface with amplitude $a$ and corrugation period
$\lambda_{c}$, which is described by $h(x,y)=a\cos(k_{c}x)$, where
$k_{c}=2\pi/\lambda_{c}$ and $a\ll z_{0}$. For this case, substituting the
Fourier transform of $h$ into Eq. (32), and performing the k-integration, one
obtains

$U^{(1)}(x_{0},z_{0})=\frac{a}{2}[e^{ik_{c}x_{0}}g(\tilde{{\bf
k}},z_{0})+e^{-ik_{c}x_{0}}g(-\tilde{{\bf k}},z_{0})],$ (44)

where $\tilde{{\bf k}}=k_{c}\hat{{\bf x}}$ and $g$ is given by Eqs. (33) and
(34). By carefully inspecting the ${\bf k}^{\prime}$-integration $\int
d^{2}{\bf k}^{\prime}a_{{\bf k}^{\prime},{\bf k}^{\prime}+\tilde{{\bf
k}}}^{mn}$ within $g(-\tilde{{\bf k}},z_{0})$, we find that

$\displaystyle\int d^{2}{\bf k}^{\prime}$ $\displaystyle a_{{\bf
k}^{\prime},{\bf k}^{\prime}+\tilde{{\bf k}}}^{mn}=$ $\displaystyle\int
d^{2}{\bf k}^{\prime}\left(\begin{array}[]{ccc}a_{{\bf k}^{\prime},{\bf
k}^{\prime}-\tilde{{\bf k}}}^{xx}&0&-a_{{\bf k}^{\prime},{\bf
k}^{\prime}-\tilde{{\bf k}}}^{xz}\\\ 0&a_{{\bf k}^{\prime},{\bf
k}^{\prime}-\tilde{{\bf k}}}^{yy}&0\\\ -a_{{\bf k}^{\prime},{\bf
k}^{\prime}-\tilde{{\bf k}}}^{xz}&0&a_{{\bf k}^{\prime},{\bf
k}^{\prime}-\tilde{{\bf k}}}^{zz}\end{array}\right),$ (48)

so that $U^{(1)}(x_{0},z_{0})$ is written as

$\displaystyle U^{(1)}(x_{0},z_{0})$
$\displaystyle=\frac{a\hbar}{8\epsilon_{0}\pi^{3}}\int_{0}^{\infty}d\xi\int
d^{2}{\bf k}^{\prime}\frac{\xi^{2}}{c^{2}}$
$\displaystyle\times[\sum_{m}\alpha_{mm}(i\xi)a_{{\bf k}^{\prime},{\bf
k}^{\prime}-\tilde{{\bf k}}}^{mm}\cos(k_{c}x_{0})$
$\displaystyle+2i\alpha_{xz}(i\xi)a_{{\bf k}^{\prime},{\bf
k}^{\prime}-\tilde{{\bf k}}}^{xz}\sin(k_{c}x_{0})].$ (49)

By defining

$\displaystyle{\cal V}_{xx}(k_{c},z_{0})$
$\displaystyle=\int_{0}^{\infty}d\xi\int d^{2}{\bf
k}^{\prime}\frac{\xi^{2}}{c^{2}}\alpha_{xx}\left(i\xi\right)a_{{\bf
k}^{\prime},{\bf k}^{\prime}-\tilde{{\bf k}}}^{xx},$ (50) $\displaystyle{\cal
V}_{yy}(k_{c},z_{0})$ $\displaystyle=\int_{0}^{\infty}d\xi\int d^{2}{\bf
k}^{\prime}\frac{\xi^{2}}{c^{2}}\alpha_{yy}\left(i\xi\right)a_{{\bf
k}^{\prime},{\bf k}^{\prime}-\tilde{{\bf k}}}^{yy},$ (51) $\displaystyle{\cal
V}_{zz}(k_{c},z_{0})$ $\displaystyle=\int_{0}^{\infty}d\xi\int d^{2}{\bf
k}^{\prime}\frac{\xi^{2}}{c^{2}}\alpha_{zz}\left(i\xi\right)a_{{\bf
k}^{\prime},{\bf k}^{\prime}-\tilde{{\bf k}}}^{zz},$ (52) $\displaystyle{\cal
V}_{xz}(k_{c},z_{0})$ $\displaystyle=2i\int_{0}^{\infty}d\xi\int d^{2}{\bf
k}^{\prime}\frac{\xi^{2}}{c^{2}}\alpha_{xz}\left(i\xi\right)a_{{\bf
k}^{\prime},{\bf k}^{\prime}-\tilde{{\bf k}}}^{xz},$ (53)

we can write Eq. (49) as

$U^{(1)}(x_{0},z_{0})=\frac{a\hbar}{8\pi^{3}\epsilon_{0}}A(k_{c},z_{0})\cos[k_{c}x_{0}-\delta(k_{c},z_{0})],$
(54)

where

$A(k_{c},z_{0})=\sqrt{{\cal V}_{\text{Sum}}^{2}+{\cal V}_{xz}^{2}},$ (55)

with ${\cal V}_{\text{Sum}}={\cal V}_{xx}+{\cal V}_{yy}+{\cal V}_{zz}$, and
$\delta(k_{c},z_{0})$ is a nontrivial phase function defined by

$\delta(k_{c},z_{0})=\arctan\left(\frac{{\cal V}_{xz}}{{\cal
V}_{\text{Sum}}}\right).$ (56)

From Eq. (54), considering the behavior of $U^{(1)}$ with respect to $x_{0}$,
one can see that the stable equilibrium points of $U^{(1)}$ can be over the
corrugation peaks $(\delta=\pi)$, valleys $(\delta=0)$ or over intermediate
points between a peak and a valley $(\delta\neq 0,\pi)$. Such possibilities
were first investigated in Refs. [12, 13] and were called peak, valley and
intermediate regimes, respectively (see Fig. 2). In both Refs. [12, 13], these
effects were studied within the vdW regime of the interaction, but the
corrugated surface was considered as a perfect conductor in Ref. [12], while
it was considered as a nondispersive surface in Ref. [13]. Thus, Eqs.
(54)-(56) generalize the results obtained in these references by considering
both realistic material properties for the surface and retardation effects in
the interaction, enabling us to investigate their effects on the occurrence of
the regimes of the lateral force.

Figure 2:  Illustration of a neutral anisotropic particle (represented by an
ellipsoid) interacting with a corrugated surface (solid lines). The stable
equilibrium points (indicated by the circles) of $U^{(1)}(x_{0},z_{0})$
(dashed lines), with respect to $x_{0}$, can be over the corrugation peaks
[peak regime, (i)], valleys [valley regime, (ii)], or over intermediate points
between a peak and a valley [intermediate regime, (iii)].

###  III.1 Van der Waals regime of Eq. (54)

For our purposes, it is interesting to calculate Eq. (54) in the vdW regime.
When performing the limit $c\to\infty$ on this equation, one can see that it
acts only on the functions ${\cal V}_{mn}$, enabling us to calculate the
integrals over ${\bf k}^{\prime}$, obtaining:

$\displaystyle{\cal V}_{xx}^{\text{vdW}}$
$\displaystyle=-\frac{3\pi}{64z_{A}^{4}}\int_{0}^{\infty}d\xi\alpha_{xx}\left(i\xi\right)\frac{\left[\epsilon\left(i\xi\right)-1\right]}{\left[\epsilon\left(i\xi\right)+1\right]^{2}}\left[\epsilon\left(i\xi\right){\cal
K}_{xx}^{\text{cond}}\left(k_{c}z_{0}\right)+{\cal
K}_{xx}^{\text{diel}}\left(k_{c}z_{0}\right)\right],$ (57) $\displaystyle{\cal
V}_{yy}^{\text{vdW}}$
$\displaystyle=-\frac{3\pi}{64z_{A}^{4}}\int_{0}^{\infty}d\xi\alpha_{yy}\left(i\xi\right)\frac{\left[\epsilon\left(i\xi\right)-1\right]}{\left[\epsilon\left(i\xi\right)+1\right]^{2}}\left[\epsilon\left(i\xi\right){\cal
K}_{yy}^{\text{cond}}\left(k_{c}z_{0}\right)+{\cal
K}_{yy}^{\text{diel}}\left(k_{c}z_{0}\right)\right],$ (58) $\displaystyle{\cal
V}_{zz}^{\text{vdW}}$
$\displaystyle=-\frac{3\pi}{64z_{A}^{4}}\int_{0}^{\infty}d\xi\alpha_{zz}\left(i\xi\right)\frac{\left[\epsilon\left(i\xi\right)-1\right]}{\left[\epsilon\left(i\xi\right)+1\right]^{2}}\left[\epsilon\left(i\xi\right){\cal
K}_{zz}^{\text{cond}}\left(k_{c}z_{0}\right)+{\cal
K}_{zz}^{\text{diel}}\left(k_{c}z_{0}\right)\right],$ (59) $\displaystyle{\cal
V}_{xz}^{\text{vdW}}$
$\displaystyle=\frac{3\pi}{32z_{A}^{4}}\int_{0}^{\infty}d\xi\alpha_{xz}\left(i\xi\right)\frac{\left[\epsilon\left(i\xi\right)-1\right]}{\left[\epsilon\left(i\xi\right)+1\right]^{2}}\left[\epsilon\left(i\xi\right){\cal
K}_{xz}^{\text{cond}}\left(k_{c}z_{0}\right)+{\cal
K}_{xz}^{\text{diel}}\left(k_{c}z_{0}\right)\right],$ (60)

where the functions ${\cal K}_{mn}^{\text{cond}}$ and ${\cal
K}_{mn}^{\text{diel}}$ are given by [13]:

$\displaystyle\mathcal{K}_{xx}^{\text{cond}}\left(u\right)$
$\displaystyle=u^{3}K_{3}\left(u\right)-u^{4}K_{2}\left(u\right),$
$\displaystyle\mathcal{K}_{xx}^{\text{diel}}\left(u\right)$
$\displaystyle=\left[\frac{56}{3}u^{2}+u^{4}\right]K_{2}\left(u\right)-\frac{11}{3}u^{3}K_{3}\left(u\right),$
(61) $\displaystyle\mathcal{K}_{yy}^{\text{cond}}\left(u\right)$
$\displaystyle=u^{3}K_{3}\left(u\right),$
$\displaystyle\mathcal{K}_{yy}^{\text{diel}}\left(u\right)$
$\displaystyle=8u^{2}K_{2}\left(u\right)-u^{3}K_{3}\left(u\right),$ (62)
$\displaystyle\mathcal{K}_{zz}^{\text{cond}}\left(u\right)$
$\displaystyle=\left[\frac{16}{3}u^{2}+u^{4}\right]K_{2}\left(u\right)+\frac{2}{3}u^{3}K_{3}\left(u\right),$
$\displaystyle\mathcal{K}_{zz}^{\text{diel}}\left(u\right)$
$\displaystyle=2u^{3}K_{3}\left(u\right)-u^{4}K_{2}\left(u\right),$ (63)
$\displaystyle\mathcal{K}_{xz}^{\text{cond}}\left(u\right)$
$\displaystyle=\frac{8}{3}u^{3}K_{2}\left(u\right)-u^{4}K_{3}\left(u\right),$
$\displaystyle\mathcal{K}_{xz}^{\text{diel}}\left(u\right)$
$\displaystyle=u^{4}K_{3}\left(u\right)-\frac{16}{3}u^{3}K_{2}\left(u\right),$
(64)

with $K_{2}$ and $K_{3}$ being modified Bessel functions of the second kind.
Thus, the vdW interaction between an anisotropic polarizable particle and a
dispersive sinusoidal surface is given by Eqs. (54)-(56) with ${\cal
V}_{mn}\to{\cal V}_{mn}^{\text{vdW}}$ given by Eqs. (57)-(64). We remark that
depending on the models for $\alpha_{mn}\left(i\xi\right)$ and
$\epsilon\left(i\xi\right)$ the $\xi$-integrals in Eqs. (57)-(60) can also be
performed. Lastly, Eqs. (57)-(64) could also be obtained from Eqs. (35)-(37)
by considering in them a sinusoidal corrugation and performing the ${\bf
k}^{\prime}$-integration.

###  III.2 Model for $\alpha_{mn}\left(i\xi\right)$

Let us describe, for simplicity (but without loss of generality), the
anisotropic polarizable particle as a prolate spheroidal nanoparticle. This
particle has a rotational symmetry axis (named particle axis), which we
consider initially oriented in $z$-direction, such that its polarizability
tensor is represented by the matrix

$\alpha_{mn}(i\xi)=\left(\begin{array}[]{ccc}\alpha_{\perp}(i\xi)&0&0\\\
0&\alpha_{\perp}(i\xi)&0\\\ 0&0&\alpha_{\parallel}(i\xi)\end{array}\right),$
(65)

with [9]

$\displaystyle\alpha_{\parallel}(i\xi)$
$\displaystyle=\epsilon_{0}V\frac{\epsilon_{P}(i\xi)-1}{1+[\epsilon_{P}(i\xi)-1]d},$
(66) $\displaystyle\alpha_{\perp}(i\xi)$
$\displaystyle=\epsilon_{0}V\frac{\epsilon_{P}(i\xi)-1}{1+\frac{1}{2}[\epsilon_{P}(i\xi)-1](1-d)},$
(67)

which are the polarizabilities in the directions parallel and normal to the
particle axis, respectively. In these equations, $V$ is the spheroid’s volume,
$\epsilon_{P}$ is the dielectric permittivity of the material in which the
nanoparticle is made of, and $d$ is the depolarizing factor of the spheroid
which is written in terms of the particle aspect ratio $r$ (ratio between its
length and width) as [9]

$d=\frac{1}{1-r^{2}}+\frac{r\log\left(\frac{r+\sqrt{r^{2}-1}}{r-\sqrt{r^{2}-1}}\right)}{2\left(r^{2}-1\right)^{3/2}}.$
(68)

For a prolate spheroid, one has $r>1$, so that one can note that the limit
$r\to 1$ corresponds to the case where the particle is a sphere and its
polarizability is isotropic, on the other hand, in the limit $r\to\infty$ the
particle is like a needle and it is highly anisotropic. For a generic
orientation of the particle, the polarizability tensor is represented by a
non-diagonal matrix and we can write its components in terms of the spherical
angles $(\theta,\phi)$ as:

$\displaystyle\alpha_{xx}$
$\displaystyle=\alpha_{\perp}+\left(\alpha_{\parallel}-\alpha_{\perp}\right)\sin^{2}\theta\cos^{2}\phi,$
(69) $\displaystyle\alpha_{yy}$
$\displaystyle=\alpha_{\perp}+\left(\alpha_{\parallel}-\alpha_{\perp}\right)\sin^{2}\theta\sin^{2}\phi,$
(70) $\displaystyle\alpha_{zz}$
$\displaystyle=\alpha_{\perp}+\left(\alpha_{\parallel}-\alpha_{\perp}\right)\cos^{2}\theta,$
(71) $\displaystyle\alpha_{xy}$
$\displaystyle=\frac{\alpha_{\parallel}-\alpha_{\perp}}{2}\sin
2\phi\sin^{2}\theta,$ (72) $\displaystyle\alpha_{xz}$
$\displaystyle=\frac{\alpha_{\parallel}-\alpha_{\perp}}{2}\sin
2\theta\cos\phi,$ (73) $\displaystyle\alpha_{yz}$
$\displaystyle=\frac{\alpha_{\parallel}-\alpha_{\perp}}{2}\sin
2\theta\sin\phi.$ (74)

We remark that the polarizability tensor is a symmetric tensor, thus one has
$\alpha_{yx}=\alpha_{xy}$, $\alpha_{zx}=\alpha_{xz}$ and
$\alpha_{zy}=\alpha_{yz}$.

###  III.3 Models for $\epsilon\left(i\xi\right)$ and
$\epsilon_{P}\left(i\xi\right)$

Let us consider our formulas for the case of a gold nanoparticle interacting
with a corrugated gold surface, whose permittivities are both described by the
plasma model as

$\epsilon\left(i\xi\right)=\epsilon_{P}\left(i\xi\right)=1+\frac{\omega_{p}^{2}}{\xi^{2}},$
(75)

where $\omega_{p}$ is the plasma frequency of the metal. We can also write the
plasma frequency in terms of the plasma wavelength $\lambda_{p}$ as
$\omega_{p}=2\pi c/\lambda_{p}$. For gold, we have $\omega_{p}\approx
1.385\times 10^{16}\text{ rad/s}$, which results in $\lambda_{p}\approx
136\text{ nm}$. Since both the nanoparticle and the surface are considered to
be made of gold, the value for $\lambda_{p}$ is the reference to determine if
we are within the vdW or CP regimes, which are obtained when
$z_{0}\ll\lambda_{p}$ and $z_{0}\gg\lambda_{p}$, respectively. In addition,
for the chosen situation, the $\xi$-integrals in Eqs. (57)-(60) can be
performed, which means that we have an analytic expression for the vdW
interaction between a gold spheroidal nanoparticle and a gold corrugated
surface.

##  IV Discussions

To investigate the behavior of the minimum points of $U^{(1)}$, one has to
study the behavior of the function $\delta$, which depends on the functions
${\cal V}_{mn}$, as one can see in Eq. (56). Let us start making some general
discussions on the behavior of the function $\delta$, which are similar to
those made in Refs. [12, 13]. From this equation, we can see that the
occurrence of intermediate regimes just requires ${\cal V}_{xz}\neq 0$, which
occurs when $\alpha_{xz}\neq 0$. For the considered spheroidal particle, this
happens when it is oriented such that $\sin 2\theta\cos\phi\neq 0$ in Eq.
(73). When this is not the case (for instance, when the particle is oriented
such that its axis coincides with $x$, $y$ or $z$), one has $\alpha_{xz}=0$
(thus, ${\cal V}_{xz}=0$), and we have that the function $\delta$ can only
assume the values $0$ or $\pi$, which correspond to valley and peak regimes,
respectively. In addition, we have one or the other of these regimes depending
on the sign of ${\cal V}_{\text{Sum}}$, where we have valley regime when it is
positive, and peak regime when negative [see Eq. (56)]. Among the functions
${\cal V}_{xx}$, ${\cal V}_{yy}$ and ${\cal V}_{zz}$, the only one that
changes its sign is ${\cal V}_{xx}$, and thus it is responsible for the change
of sign of ${\cal V}_{\text{Sum}}$. As a consequence, considering Eqs.
(50)-(52), the greater $\alpha_{xx}$ is in relation to $\alpha_{yy}$ and
$\alpha_{zz}$, the greater can be the occurrence of valley regime. In other
words, the more anisotropic the particle is in the direction of the
corrugation, the greater can be the occurrence of valley regime. Next, we
investigate how the consideration of realistic material properties for the
surface and of retardation in the interaction affects such occurrence.

Let us start considering the particle oriented such that its axis is parallel
to the $x$-axis. In this situation, $\alpha_{mn}$ is described by Eqs.
(69)-(74) with $\theta=\pi/2$ and $\phi=0$, and, as discussed above, we have
valley regime when ${\cal V}_{\text{Sum}}>0$, or peak regime when ${\cal
V}_{\text{Sum}}<0$. We begin investigating the effects of the consideration of
realistic material properties for the surface on the occurrence of these
regimes in the short distance limit (vdW regime). For this, in Fig. 3, we
consider Eqs. (57)-(59) and we show the behavior of ${\cal V}_{\text{Sum}}$,
in terms of $\lambda_{c}/z_{0}$, for the cases in which the corrugated surface
is considered dispersive, nondispersive [$\epsilon(i\xi)=\text{cte}$] and
perfectly conductive [$\epsilon(i\xi)\to\infty$]. In this figure, it can be
seen that the general behavior of ${\cal V}_{\text{Sum}}$ is similar for all
cases, differing only numerically (this conclusion was first mentioned in Ref.
[12] through rough estimates). In Fig. 3(i), one notes that when the surface
is considered as a perfect conductor one has the greatest magnitudes of ${\cal
V}_{\text{Sum}}$, since its reflectivity is perfect (for the nondispersive
cases, the surface has transparency and the mentioned magnitudes decrease, as
discussed in Ref. [13]). The opposite happens when the surface is dispersive,
since in this case its reflectivity depends on the frequency of the field
fluctuation, decreasing as the frequency increases. Despite this, ${\cal
V}_{\text{Sum}}$ can change its sign, as shown in Fig. 3(ii), which means that
a transition between peak and valley regimes can occur. Moreover, the
consideration of realistic material properties for the surface can amplify the
occurrence of valley regime, since the mentioned change of sign for the
dispersive case occurs at a greater value of $\lambda_{c}/z_{0}$ than the
other cases [as shown, for instance, in Fig. 3(ii)]. In other words, although
the consideration of realistic material properties for the surface weakens the
interaction, it can increase the range of $\lambda_{c}/z_{0}$ values in which
the valley regime occurs, contributing to a greater occurrence of this regime.

Figure 3:  Behavior of ${\cal V}_{\text{Sum}}$ versus $\lambda_{c}/z_{0}$ for
a gold spheroidal nanoparticle interacting with a gold sinusoidal surface
within the vdW regime [Eqs. (57)-(59)]. The particle is oriented such that its
axis is parallel to the $x$-axis, and $\alpha_{mn}$ is described by Eqs.
(69)-(74) with $\theta=\pi/2$ and $\phi=0$. The highlighted interval in panel
(i) is shown in panel (ii) in a zoomed in view. In each panel it is considered
$z_{0}=30\text{ nm}$ and $r=2$, and it is shown the cases where the surface is
considered dispersive (solid line), perfectly conductive (dashed line) and
nondispersive [dotted line for $\epsilon(i8\times 10^{15})$ and dot-dashed for
$\epsilon(i5\times 10^{15})$]. We remark that we have valley regime when
${\cal V}_{\text{Sum}}>0$, and peak regime when ${\cal V}_{\text{Sum}}<0$.

It is also interesting to perform the investigations above in the long
distance case. For this, in Fig. 4, we consider Eqs. (50)-(52) for the
particle oriented with its axis parallel to the $x$-axis, and we show the
behavior of ${\cal V}_{\text{Sum}}$, in terms of $\lambda_{c}/z_{0}$, for the
cases in which the corrugated surface is considered dispersive, nondispersive
and perfectly conductive. Similar to the vdW case, in this figure we have that
the general behavior of ${\cal V}_{\text{Sum}}$ is similar for all cases,
differing only numerically. Moreover, it is known that for long distances from
the surface the smallest frequencies give the main contribution to the
interaction and, thus, $\epsilon(i\xi)$ can be replaced to its zero-frequency
value in Eqs. (50)-(52). Because of this, the magnitudes of ${\cal
V}_{\text{Sum}}$ for a dispersive surface are close to those for a perfect
conductor (thus, are also greater than those for a nondispersive surface), as
expected, since $\epsilon(0)\to\infty$ for a metallic surface. Despite this,
in the long distances case, we also have that the consideration of realistic
material properties for the surface can amplify the occurrence of valley
regime, since ${\cal V}_{\text{Sum}}$ changes its sign at a greater value of
$\lambda_{c}/z_{0}$ for the dispersive case than the other cases. Thus,
although the consideration of realistic material properties for the surface
weakens the interaction, it can increase the range of $\lambda_{c}/z_{0}$
values in which the valley regime occurs for both short and long distances
cases.

Figure 4:  Behavior of ${\cal V}_{\text{Sum}}$ versus $\lambda_{c}/z_{0}$ for
a gold spheroidal nanoparticle interacting with a gold sinusoidal surface
computed using Eqs. (50)-(52). The particle is oriented such that its axis is
parallel to the $x$-axis, and $\alpha_{mn}$ is described by Eqs. (69)-(74)
with $\theta=\pi/2$ and $\phi=0$. The highlighted interval in panel (i) is
shown in panel (ii) in a zoomed in view. In each panel it is considered
$z_{0}=1\,\mu\text{m}$ and $r=2$, and it is shown the cases where the surface
is considered dispersive (solid line), perfectly conductive (dashed line) and
nondispersive [dotted line for $\epsilon(i8\times 10^{15})$ and dot-dashed for
$\epsilon(i5\times 10^{15})$]. We remark that we have valley regime when
${\cal V}_{\text{Sum}}>0$, and peak regime when ${\cal V}_{\text{Sum}}<0$.

The same previous discussions can also be made for the intermediate regimes,
and we obtain the same conclusions. To illustrate this, let us consider the
particle oriented such that $\alpha_{mn}$ is described by Eqs. (69)-(74) with
$\theta=\pi/3$ and $\phi=0$ (its axis is oriented in the $xz$-plane). In this
situation, $\alpha_{xz}\neq 0$, and we have intermediate regimes. In Fig. 5 we
show the behavior of the function $\delta$ [Eq. 56], versus
$\lambda_{c}/z_{0}$, within the vdW regime for the cases in which the surface
is considered dispersive, nondispersive and perfectly conductive. Remembering
that $\delta=0,\pi$ means valley and peak regimes, respectively, it is noted
that when the surface is considered dispersive, the minimum points of
$U^{(1)}$ tend to be further away from the corrugation peaks than in the other
cases. Thus, the consideration of realistic material properties for the
surface also can amplify the occurrence of intermediate regimes. Although we
made this analysis within the vdW regime, we remark that the same conclusions
are obtained for the long distance case.

Figure 5:  Behavior of $\delta$ versus $\lambda_{c}/z_{0}$ for a gold
spheroidal nanoparticle interacting with a gold sinusoidal surface within the
vdW regime [Eqs. (57)-(59)]. The particle is oriented such that $\alpha_{mn}$
is described by Eqs. (69)-(74) with $\theta=\pi/3$ and $\phi=0$ (its axis is
in the $xz$-plane). We consider $z_{0}=30\text{ nm}$ and $r=2$, and it is
shown the cases where the surface is considered dispersive (solid line),
perfectly conductive (dashed line) and nondispersive [dotted line for
$\epsilon(i8\times 10^{15})$ and dot-dashed for $\epsilon(i5\times 10^{15})$].

The investigation of the effect of retardation in the discussed interaction
can be made by considering the particle at a long distance from the surface
and comparing the results obtained from Eqs. (50)-(53) with those obtained
from Eqs. (57)-(60). In this way, let us consider, again, the particle
oriented such that its axis is parallel to the $x$-axis, and $\alpha_{mn}$ is
described by Eqs. (69)-(74) with $\theta=\pi/2$ and $\phi=0$. As previously
discussed, in this situation we have valley regime when ${\cal
V}_{\text{Sum}}>0$, or peak regime when ${\cal V}_{\text{Sum}}<0$. In Fig. 6,
we consider two values for $z_{0}$ outside the vdW regime and we show the
behaviors of ${\cal V}_{\text{Sum}}$ obtained from Eqs. (50)-(52) and from
Eqs. (57)-(59). Note that, as expected, the consideration of retardation
weakens the interaction. On the other hand, it has a small effect on the
occurrence of valley regime, since ${\cal V}_{\text{Sum}}$ changes its sign by
a slightly smaller value of $\lambda_{c}/z_{0}$ in both cases shown in Fig. 6.
This means that, when the goal is to estimate the value of $\lambda_{c}/z_{0}$
in which we have a transition between peak and valley regimes, the formulas
within the vdW regime [i.e., Eqs. (57)-(59)] provide good approximate results.
Otherwise, when the goal is in obtaining precise values of the magnitudes of
the energy, Eqs. (50)-(52) must be used.

Figure 6:  Behavior of ${\cal V}_{\text{Sum}}$ versus $\lambda_{c}/z_{0}$ for
a gold spheroidal nanoparticle interacting with a gold sinusoidal surface. In
both figures, the solid line shows this behavior disregarding retardation
effects, i.e., using Eqs. (57)-(59), while the dashed line shows it
considering these effects by using Eqs. (50)-(52). In addition, we consider a
particle with $r=2$ and oriented such that its axis is parallel to the
$x$-axis, so that $\alpha_{mn}$ is described by Eqs. (69)-(74) with
$\theta=\pi/2$ and $\phi=0$. In (a) we have $z_{0}=0.5\mu\text{m}$, while in
(b) $z_{0}=1.0\mu\text{m}$. We remark that we have valley regime when ${\cal
V}_{\text{Sum}}>0$, and peak regime when ${\cal V}_{\text{Sum}}<0$.

We also investigate the effects of retardation on the occurrence of the
intermediate regimes. This is shown in Fig. 7, where we consider the particle
at a distance from the surface outside the vdW regime and we show the
behaviors of $\delta$ obtained from Eqs. (50)-(53) and from Eqs. (57)-(60). We
also consider the particle oriented such that $\alpha_{mn}$ is described by
Eqs. (69)-(74) with $\theta=\pi/3$ and $\phi=0$ (its axis is oriented in the
$xz$-plane). Remembering that $\delta=0,\pi$ means valley and peak regimes,
respectively, it is noted in Fig. 7 that the consideration of retardation can
amplify the occurrence of the intermediate regimes up to a certain value of
$\lambda_{c}/z_{0}$, above which it begins to inhibit the effect. This is
reinforced in the inset of Fig. 7, where we show the ratio between the results
calculated considering retardation effects and those disregarding them.

Figure 7:  Behavior of $\delta$ versus $\lambda_{c}/z_{0}$ for a gold
spheroidal nanoparticle interacting with a gold sinusoidal surface. The solid
line shows this behavior disregarding retardation effects, i.e., using Eqs.
(57)-(59), while the dashed one shows it considering these effects by using
Eqs. (50)-(52). In addition, we consider the particle at
$z_{0}=1.0\mu\text{m}$, with $r=2$, and oriented such that $\alpha_{mn}$ is
described by Eqs. (69)-(74) with $\theta=\pi/3$ and $\phi=0$ (its axis is in
the $xz$-axis). The inset shows the ratio between the results calculated
considering retardation effects and those disregarding them.

##  V Final remarks

By means of the scattering approach, we obtained the Casimir-Polder
interaction between a neutral anisotropic polarizable particle and a
corrugated surface. Focusing on the lateral force, which arises due to the
presence of corrugation on the surface, we investigated how the occurrence of
the peak, valley and intermediate regimes is affected by the consideration of
realistic dielectric properties for the surface and of retardation on the
interaction. In this context, the discussions made here go beyond those made
in Refs. [12, 13], which considered idealized situations only in the vdW
regime and also only obtained preliminary insights about the behavior of the
mentioned effects.

Our main results are given by Eqs. (32)-(34), whose application to a
sinusoidal corrugation results in Eqs. (54)-(56), with the functions ${\cal
V}_{mn}$ given by Eqs. (50)-(53). These equations, which take into account
realistic properties for the surface and retardation effects in the
interaction, not only generalize those found in Refs. [12, 13], but also
generalize the corresponding ones found in Ref. [17], since we consider an
anisotropic electrically polarizable particle. Moreover, we showed that the
consideration of realistic material properties for the surface, when compared
to idealized materials (perfectly conductive or nondispersive), amplifies the
occurrence of the valley and intermediate regimes. Specifically, it increases
the range of the values of $\lambda_{c}/z_{0}$ in which the valley regime
occurs, or, when we have intermediate regimes, it shifts the minimum points of
$U^{(1)}$ further away from the corrugation peaks (see Figs. 3-5). In
addition, we also showed that the effect of retardation on the occurrence of
the valley regime is small (see Fig. 6), indicating that our vdW formulas
[Eqs. (57)-(60)] provide good approximate results when the interest is in
estimating the value of $\lambda_{c}/z_{0}$ in which we have a transition
between peak and valley regimes. On the occurrence of intermediate regimes,
however, the consideration of retardation can amplify or inhibit it, which
depends on the value of $\lambda_{c}/z_{0}$ (see Fig. 7). The results obtained
here, which consider more realistic aspects for the particle-surface
interaction, provide a preciser description of the interaction between an
anisotropic particle and a corrugated surface, giving a better understanding
of the nontrivial aspects of the lateral Casimir-Polder force.

###### Acknowledgements.

The author expresses sincere gratitude to Danilo T. Alves for the careful
reading of this paper and for the valuable orientation throughout the PhD. The
author also thanks Paulo A. Maia Neto for valuable discussions and comments.
This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

## References

  * Casimir and Polder [1948] H. B. G. Casimir and D. Polder, The Influence of Retardation on the London-van der Waals Forces, Phys. Rev. 73, 360 (1948).
  * Milonni [1994] P. W. Milonni,  _The Quantum Vacuum. An Introduction to Quantum Electrodynamics_ (Academic Press, San Diego, 1994).
  * Levin  _et al._ [2010] M. Levin, A. P. McCauley, A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, Casimir repulsion between metallic objects in vacuum, Phys. Rev. Lett. 105, 090403 (2010).
  * Eberlein and Zietal [2011] C. Eberlein and R. Zietal, Casimir-Polder interaction between a polarizable particle and a plate with a hole, Phys. Rev. A 83, 052514 (2011).
  * Buhmann  _et al._ [2016] S. Y. Buhmann, V. N. Marachevsky, and S. Scheel, Impact of anisotropy on the interaction of an atom with a one-dimensional nano-grating, Int. J. Mod. Phys. A 31, 1641029 (2016).
  * Abrantes  _et al._ [2018] P. P. Abrantes, Y. França, F. S. S. da Rosa, C. Farina, and R. de Melo e Souza, Repulsive van der Waals interaction between a quantum particle and a conducting toroid, Phys. Rev. A 98, 012511 (2018).
  * Venkataram  _et al._ [2020] P. S. Venkataram, S. Molesky, P. Chao, and A. W. Rodriguez, Fundamental limits to attractive and repulsive Casimir-Polder forces, Phys. Rev. A 101, 052115 (2020).
  * Marchetta  _et al._ [2021] J. J. Marchetta, P. Parashar, and K. V. Shajesh, Geometrical dependence in Casimir-Polder repulsion, Phys. Rev. A 104, 032209 (2021).
  * Bimonte  _et al._ [2015] G. Bimonte, T. Emig, and M. Kardar, Casimir-Polder force between anisotropic nanoparticles and gently curved surfaces, Phys. Rev. D 92, 025028 (2015).
  * Gangaraj  _et al._ [2018] S. A. H. Gangaraj, M. G. Silveirinha, G. W. Hanson, M. Antezza, and F. Monticone, Optical torque on a two-level system near a strongly nonreciprocal medium, Phys. Rev. B 98, 125146 (2018).
  * Antezza  _et al._ [2020] M. Antezza, I. Fialkovsky, and N. Khusnutdinov, Casimir-Polder force and torque for anisotropic molecules close to conducting planes and their effect on ${\text{CO}}_{2}$, Phys. Rev. B 102, 195422 (2020).
  * Nogueira  _et al._ [2021] E. C. M. Nogueira, L. Queiroz, and D. T. Alves, Peak, valley, and intermediate regimes in the lateral van der Waals force, Phys. Rev. A 104, 012816 (2021).
  * Queiroz  _et al._ [2021] L. Queiroz, E. C. M. Nogueira, and D. T. Alves, Regimes of the lateral van der Waals force in the presence of dielectrics, Phys. Rev. A 104, 062802 (2021).
  * Nogueira  _et al._ [2022] E. C. M. Nogueira, L. Queiroz, and D. T. Alves, Sign inversion in the lateral van der Waals force, Phys. Rev. A 105, 062816 (2022).
  * Queiroz  _et al._ [2023] L. Queiroz, E. C. M. Nogueira, and D. T. Alves, Sign inversion in the lateral van der Waals force between an anisotropic particle and a plane with a hemispherical protuberance: an exact calculation, J. Phys. A: Math. Theor. 56, 115301 (2023).
  * Alves  _et al._ [2023] D. T. Alves, L. Queiroz, E. C. M. Nogueira, and N. M. R. Peres, Curvature-induced repulsive effect on the lateral Casimir-Polder–van der Waals force, Phys. Rev. A 107, 062821 (2023).
  * Messina  _et al._ [2009] R. Messina, D. A. R. Dalvit, P. A. M. Neto, A. Lambrecht, and S. Reynaud, Dispersive interactions between atoms and nonplanar surfaces, Phys. Rev. A 80, 022119 (2009).
  * Neto  _et al._ [2005a] P. A. M. Neto, A. Lambrecht, and S. Reynaud, Roughness correction to the Casimir force: Beyond the Proximity Force Approximation, Europhys. Lett. (EPL) 69, 924 (2005a).
  * Neto  _et al._ [2005b] P. A. M. Neto, A. Lambrecht, and S. Reynaud, Casimir effect with rough metallic mirrors, Phys. Rev. A 72, 012115 (2005b).
  * Buhmann [2012] S. Y. Buhmann,  _Dispersion Forces II_, Springer Tracts in Modern Physics, Vol. 248 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012).
  * Dalvit  _et al._ [2008] D. A. R. Dalvit, P. A. M. Neto, A. Lambrecht, and S. Reynaud, Lateral Casimir-Polder force with corrugated surfaces, J. Phys. A: Math. Theor. 41, 164028 (2008).