Comment on “Electromagnetic force on structured metallic surfaces”

In [Phys. Rev. B 92, 115416 (2015)], Velzen and Webb proposed that by structuring a metal film suspended in free space with a periodic array of nanoscale slots, the steady-state optical pressure on the film resulting from an incident plane wave can be substantially enhanced compared to the standard radiation-pressure limit. This result appears to be in violation of the law of momentum conservation, given that the momentum of free-space electromagnetic plane waves is well established. We recalculated the optical fields and forces for the same configuration as Velzen and Webb and found that, although the calculated optical fields and force densities agree with the published result, the resulting total optical forces were not enhanced and, instead, matched the standard radiation-pressure relation. Our calculations imply that the diverging results arise due to different choices of the region used for integration of the force density.

Figure 1

(a) An Au film in free space with periodic slits on the top surface illuminated from the top. (b) and (c) Two choices of the force-calculation region $\mathrm{\Omega }$ for the periodic structure. (b) Outside choice: $\partial \mathrm{\Omega }$ consists of two segments in free space and two segments that overlap the periodic boundaries. (c) Inside choice: $\partial {\mathrm{\Omega }}^{\prime }\mathrm{s}$ are within the homogeneous object but as close as possible to the boundaries of the object.

Figure 2

E and H magnitudes for (a) and (c) slot width $W=60\mathrm{nm}$ and slot depths $D=55\mathrm{nm}$ and (b) and (d) 75 nm. The simulation mesh size is 0.25 nm.

Figure 3

The $y$-component of the calculated Lorentz (a-b) and EL (c-d) force densities with slot width $W=30$ nm. For (a) and (c), the slot depth is $D=51$ nm; for (b) and (d), the slot width is $D=81$ nm. The power density of the incident light is 1 mW over a circular spot with diameter of 1 $\mu \mathrm{m}$. The mesh size is 0.25 nm. For the Lorentz formulation, we calculated the force density within the object and across the material boundary. For the EL formulation, we only calculated the force density within the object.

Figure 4

Calculated time-averaged optical pressure as a function of slot depth based on (a) the volumetric Lorentz method [Eq. (4) with (2)], (b) the MST method [Eq. (6) with ${\overleftrightarrow{\mathit{T}}}_{\mathrm{L}}$], (c) the EL tensor method [Eq. (6) with ${\overleftrightarrow{\mathit{T}}}_{\mathrm{EL}}$], and (d) Eq. (1). In all cases, $\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{outside}}$. The minus sign means that the direction of the optical force is in the $-y$ direction. The far-field reflectance as a function of slot depth is shown as an inset.

Figure 5

Total time-averaged optical pressure on the structure in Fig. 1 (with $W$ = 30 nm and $D$ = 46 nm), calculated using (a) the volumetric Lorentz method, (b) the volumetric EL method, (c) the MST method, and (d) the EL tensor method with both ${\mathrm{\Omega }}_{\mathrm{outside}}$ and ${\mathrm{\Omega }}_{\mathrm{inside}}$ applied as a function of simulation mesh size. For panels (a), (c), and (d), we show the zoomed-in plots of the results using ${\mathrm{\Omega }}_{\mathrm{outside}}$ to demonstrate convergence.

Figure 6

(a) The finite-width Au structure in free space, illuminated by a plane wave injected at the top boundary of the total-field scattered-field area. The incident field amplitude is the same as that for the infinite-width case. The red box with all boundaries in free space is the force-calculation region. (b) The time-averaged optical pressure as a function of the number of periods using the MST and volumetric Lorentz methods, compared to the infinite-width case.