Calculating Casimir interactions for periodic surface-relief gratings using the $C$ method

We develop a formalism to calculate the fluctuation-induced interactions in periodic systems. The formalism, which combines the scattering theory with the $C$ method borrowed from electromagnetic gratings studies, is suitable and efficient for the calculation of the Casimir forces involving surface-relief gratings. We apply the developed technique to obtain the energy and lateral force for simple one-dimensional sinusoidal gratings. Using this formalism, we derived known asymptotic expressions that were previously obtained through perturbative approximations. At close separation, our numerical results match those obtained by the proximity force approximation and its first correction using the derivative expansion.

Figure 1

Scattering problem for surface-relief grating.

Figure 2

A scatter plot of the eigenvalues $\left\{{\lambda }_{-}\right\}$ (circles) in units of the inverse wavelength $L_x^{-1}$ in the complex plane. The problem corresponds to the sinusoidal grating $h\left(x\right)=0.1sin\left(2\pi x\right)$, $\kappa =1$, and $k_x=1$. The matrix size is $21\times 21$ with the Fourier index $m$ ranging from $-10,10$. The first 15 negative Rayleigh wave numbers (crosses) defined in Eq. (33) are also shown on the plot. The figure shows that 11 of the 21 eigenvalues match well with the Rayleigh wave numbers.

Figure 3

The test system for the energy calculations. The system consists of a flat plate separated from a sinusoidal grating with an average separation $d$. The sinusoidal grating has an amplitude $a$ and a wavelength $L_x$.

Figure 4

The test system for the lateral force calculations. The system consists of two sinusoidal gratings, of equal amplitude $a$ and wavelength $L_x$. The gratings have an average separation $d$, and a lateral displacement between the peaks $b$.

Figure 5

Parameter space plot for the energy system shown in Fig. 3. The two dimensionless quantities are the amplitude of corrugations divided by the wavelength $a/L_x$ and the separation divided by the wavelength $d/L_x$. The lighter (red) shaded region shows where for a fixed matrix size $N$, the $C$ method gives converged results. The (blue) horizontally hashed region shows the region of validity for the PFA and DE. The (green) vertically hashed region shows the region of validity for the perturbative expansion.

Figure 6

Casimir energy normalized to the PFA as defined by Eq. (67) versus $a/d$ for fixed $d/L_x=0.1$ (upper plot) and $d/L_x=0.5$ (lower plot). All parameters are defined in Fig. 3. The solid lines are the analytic formulas for the PFA plus the DE [Eqs. (67) and (68)]. The solid (blue) curve corresponds to the TM mode, the dashed (red) curve corresponds to the TE mode. The triangles and squares correspond to the converged numerical results using the $C$ method for the TM and TE modes, respectively.

Figure 7

Lateral Casimir force per unit area in units of $\hslash c/L_x^4$ vs lateral displacement. The curve is for a mean separation of $d/L_x=0.1$, and an amplitude of $a/d=0.3$. The solid (blue) and dashed (red) lines correspond to the prediction of the PFA+DE for the TM and TE modes, respectively. The triangles and squares correspond to the numerical results for the TM and TE modes, respectively.

Figure 8

Correction to the Casimir energy in units of $\hslash c{d}^5/a^2$ versus $d/L_x$. The solid (blue) and dashed (red) lines are based on the analytical results given in Eq. (59) for the TM and TE modes, respectively. The empty triangles and squares are the numerical calculations using the $C$ method for the TM and TE modes for fixed $a/d=0.1$, respectively. The filled triangles and squares are the numerical calculations using the $C$ method for TM and TE modes for fixed $a/L_x=0.05$, respectively. The vertical dotted line is at $d/L_x=0.05$ and corresponds to the separation at which tips of the corrugations would touch the flat plate for $a/L_x=0.05$.

Figure 9

Lateral Casimir force per unit area in units of $\hslash c/L_x^4$ versus the lateral displacement. The triangles and squares are numerical calculations for TM and TE modes for constant mean separation $d/L_x=0.5$ and amplitude $a/d=0.1$. The solid lines are the second-order perturbative expressions in Eq. (A31) and the dashed line is the fourth-order correction as calculated in Ref. [24] for the TM mode.