Quasianalytical modal approach for computing Casimir interactions in periodic nanostructures

We present an almost fully analytical technique for computing Casimir interactions between periodic lamellar gratings based on a modal approach. Our method improves on previous work on Casimir modal approaches for nanostructures [Phys. Rev. A 82, 062111 (2010)] by using the exact form of the eigenvectors of such structures, and computing eigenvalues by solving numerically a simple transcendental equation. In some cases eigenvalues can be solved for exactly, such as the zero-frequency limit of gratings modeled by a Drude permittivity. Our technique also allows us to predict analytically the behavior of the Casimir interaction in limiting cases, such as the large-separation asymptotics. The method can be generalized to more complex grating structures and may provide a deeper understanding of the geometry-composition-temperature interplay in Casimir forces between nanostructures.

Figure 1

Schematic representation of scattering of an electromagnetic field on a 1D grating.

Figure 2

Numerical solution of the transcendental equation (34) for the Drude model. Only the lowest eleven eigenvalues are shown. (a) $s=h$ polarization. The dotted line corresponds to the smallest eigenvalue ${\eta }_{\nu =0}^{\left(h\right)}$. Solid lines are the eigenvalues obtained using as seeds the expression (37) [$\nu$ going from 1 (bottom curve) to 4 (top curve)]. Dashed lines are the eigenvalues obtained using as seeds the expression (38) [$\nu$ going from 1 (bottom curve) to 2 (top curve)]. (b) $s=e$ polarization. The dotted line corresponds to the smallest eigenvalue ${\eta }_{\nu =0}^{\left(e\right)}$. Solid and dashed lines are the eigenvalues obtained using as seeds the expressions (41) for the two possible signs [$\nu$ going from 1 (bottom curve) to 4 (top curve)]. Dashed lines are the eigenvalues obtained using as seeds the expression (38) [$\nu$ going from 1 (bottom curve) to 3 (top curve)]. Parameters are $p_1=160$ nm, $p_2=90$ nm, and ${\alpha }_0=0.5\pi /p$. The optical parameters chosen for these plots are ${\omega }_p=8.39$ eV, $\gamma =0.043$ eV; the general structure of the curves remains unchanged for other choices of Drude parameters.

Figure 3

Spatial structure of electromagnetic modes in grating region in units of plasma wavelength ${\lambda }_p=2\pi /{\omega }_p$, for $k_y=0$, ${\alpha }_0=0.2\pi /p$ for second Matsubara frequency $\xi =4\pi k_{B}T/\left(\hslash c\right)$ at $T=300$ K. The curves represent the modes with $\nu =2$, corresponding to the two seeds in Eqs. (37) and (38) and to the two seeds ($\pm$ solutions) in Eq. (41). Our choice $k_y=0$ implies that the $e$ and $h$ polarizations decouple, and that $E_x$ and $H_y$ depend only on the $h$ polarization, while $E_y$ and $H_x$ depend only on the $e$ polarization. For the $h$ polarization two categories of modes exist: the first mainly vibrates within the grooves, and the second mainly vibrates within the teeth (this is particularly clear for $H_y$). In agreement with Maxwell's equations, the component of the electric field along the modulation direction is discontinuous at the groove walls while the remaining ones are all continuous. The numerical values inside the plots indicate the effective refractive index at imaginary frequency, $n_{\mathrm{eff}}\left(i\xi \right)\equiv k_z/\xi$, for the corresponding mode (the value on the left corresponds to the full-line mode, the one on the right to the dashed-line mode). The Drude and grating parameters are the same as in the previous figure.

Figure 4

Density plot of the intensity of the electromagnetic field within the modulated region. Upper box corresponds to $|E_x{|}^2$ and lower box to $|E_y{|}^2$. In each of the plots the horizontal axis is the modulation $x$ direction and the vertical axis is the invariant $z$ direction. The parameters are the same as in the previous figure and, therefore, $|E_x^2|$ depends only on the $h$ polarization [Eqs. (37) and(38)] while $|E_y^2|$ depends only on the $e$ polarization [Eq. (41)]. The first five modes for each polarization and for each set of eigenvalues are represented. There is only one zero mode per polarization (see discussion in the text). For the parameters chosen here the zero mode is almost constant for the $h$ polarization.

Figure 5

Matrix elements of reflection operator corresponding to zeroth-order reflection. The full lines are the result for the grating for the first seven Matsubara frequencies. The dashed lines show the equivalent matrix elements for the flat surface (Fresnel coefficients). (a) Zeroth-order reflection coefficients for the $h$ polarization. The first seven Matsubara frequencies are shown (zeroth to the sixth from the top to the bottom). (b) Zeroth-order reflection coefficients for the $e$ polarization. Once again, the first seven Matsubara frequencies are shown. The zeroth-frequency term is zero, while the other ones (first to the sixth from the bottom to the top) decrease in absolute value. The depth of the grating is $d=216$ nm, and the remaining parameters are the same as in Fig. 2.

Figure 6

Casimir pressure between a metallic grating and a metallic plane, computed using our quasianalytical modal approach. At large separation the pressure tends towards the value $\zeta \left(3\right)k_{B}T/\left(8\pi a^3\right)$ (dashed curve). At short separations the pressure is $\propto a^{-3}$ because of the finite grating conductivity. The prefactor used for the dotted curve is the one for the plane-plane case multiplied by the filling factor $f=p_2/p$ (see definition of $P_{\mathrm{filling}}$ in the text). The Drude parameters are ${\omega }_p=8.39$ eV and $\gamma =0.043$ eV. The geometrical parameters of the grating are width of the grooves $p_1=160$ nm, width of the teeth $p_2=90$ nm, and height $d=216$ nm. Temperature is set to $T=300$ K.

Figure 7

Plane-plane (dashed) and plane-grating (solid) Casimir pressure normalized by $P_{\mathrm{filling}}$ with the respective filling factor ($f=1$ for the plane-plane and $f=90/250$ for the plane-grating). The two vertical lines are located at distances corresponding to the plasma wavelength (${\lambda }_p=2\pi /{\omega }_p$) and half of the thermal wavelength [${\lambda }_T=\hslash c/\left(2k_{B}T\right)$]. The transition from the $a^{-3}$ nonretarded behavior to the $a^{-4}$ retarded behavior for the plane-plane configuration occurs much faster than for the plane-grating configuration. The large-distance $a^{-3}$ thermal regime is, however, not affected by the grating and starts roughly at the same point for both configurations. Parameters are the same as in previous figures.