1. Introduction

Light-induced forces are of great interest not only for research, but also for modern technologies, e.g. in order to develop manipulation techniques for macro-, micro- and nano-objects. In laser tweezers [1–4], these forces are used to displace dielectric particles. As for metal particles, these are moved by optical tweezers if their size is smaller than 100 nm. It is, however, practically impossible to handle metal particles having a size of more than 100 nanometers. In recent works [5, 6], a light-induced force was predicted in a subwavelength slit between two metal planes. In contrast to the pressure of light, the force described is attractive if the magnetic field is parallel to the planes. This is due to the interaction of surface plasmons which are excited at the conducting surfaces. This force could be exploited for metal particle manipulation and is promising for laser microswitches.

For real metals with their complex permittivity ε = ε₁ + iε₂, the light-induced force for a slit with the dimension L in the z-direction (see Fig. 1) is given by

 

Fig. 1 Scheme of the slit geometry. The yellow areas denote the perfect metal.

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The aim of this paper is to improve the theoretical model and to study the width dependence of the entering field. We normalize the incident wave by unity. If the width of slit 2l approaches zero, the field can be found by using the Fresnel formulas. In this limiting case, the amplitude at the metal surface equals 2, since the field is formed by two waves of equal amplitudes, the incident one and the reflected one. If the width is much greater than the wavelength 2l ≫ λ, we pass into geometric optics [9] and the entering amplitude approaches 1, as in free space. For experimental purposes, the intermediate case 2l ∼ λ is of interest. An explicit analytical expression for the estimation of the light-induced attractive force in the limiting case of a narrow slit (a = 2πl/λ ≪ 1) is derived.

This calculation is carried out below. In Section 2, the equations are solved for the slit in the perfect metal. We find the solution in the form of the Fourier series. In Section 3, the set for Fourier coefficients is solved and explicit analytical formulas are derived for a narrow slit. Section 4 describes the finite element method computation for the gold film covering the slit surface. Section 5 summarizes the conclusions.

2. Analytical solution

Following paper [8], we examine the perfect metallic half-space with a slit having the half-width l. Figure 1 shows the cross section of the metal and the slit. The normal incidence of a wave running upwards is indicated by the arrow at the bottom. We choose the y-axis normal to the half-space boundary, the x-axis normal to the slit walls, the z-axis normal to the xy-plane, and the magnetic field vector H⃗ along the z-axis. In this geometry, the Maxwell equations for a monochromatic wave are reduced to the 2D Helmholtz equation for the magnetic field amplitude H_z = H(x, y)e^−iωt:

The boundary conditions for the chosen polarization are reduced to continuity of the tangential magnetic field and continuity of the normal electric displacement vector field. The conditions at the perfect conductor surface are:

The solution at y > 0 can be found as a Fourier series, whereas at y < 0, it is a Fourier integral:

3. Asymptotic formulas

At k₀l ≪ 1, the higher spatial modes are almost negligible: |b_m| ≪ |b₀|, m ≠ 0. Set (7) reduces to one equation, which yields

As the width increases, the next modes emerge. Equation (5) shows that value ${\beta }_m^2$ is positive for mode numbers m ⩽ k₀l/π, and with increasing l, the number of propagating modes increases. The rest of the modes m > k₀l/π has the imaginary propagation constant. They are then evanescent and decay along the y-axis. In particular near the threshold of the first mode 2l = λ, set (7) reduces as follows to a pair of equations as follows:

Figure 2 shows the square |H(0, 0)|² within the zeroth (10) and the first (12) approximations as functions of the dimensionless width a = k₀l. The solid curve and the dots coincide in the subwavelength region; then, the zeroth approximation is valid at a ≪ 1. The first approximation works up to a ≲ π. It is clear that a breaking point (the dip with a jump of the first derivative) is formed in the curve at the first mode threshold, whereas the zeroth approximation is a smooth function. The explicit expression (10) with the matrix element T₀₀ involves integral (8). Since it still works well for the small parameter a, we may replace it by a simpler expansion in the parameter:

The solving of a truncated set confirms the behavior of the curve for a narrow slit. Figure 3(a) shows the absolute value of the field at the coordinate origin as a function of a for m, m′ ⩽ M = 50. At k₀l > 1, the numerical solution has many saw-tooth-like oscillations. Each breaking point corresponds to the birth of a new mode at k₀l = πm, m = 1, 2,... In the phase characteristics in Fig. 3(b), we see similar oscillations in the saw-tooth curve with a slow damping.

 

Fig. 2 Square of the normalized field amplitude |H(0, 0)|² as a function of the dimensionless slit width a = k₀l: (a) calculated by formula (10) (solid curve), by (12) (dots), and by the simple asymptotic expression (13) at small a (crosses).

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Fig. 3 The square of absolute value (a) and phase (b) of the normalized field H(0,0) as a function of the dimensionless slit half-width a = k₀l: the perfect metal at M = 50 (solid line) and JCMsuite simulations for gold (dots).

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4. Numerical modeling

The calculation for real metal by eigen-mode expansion is more complicated since it includes the numerical solution of a transcendental equation for the slit modes. In addition, as shown in paper [10], the zeroth mode vanishes when one takes the damping in real metal into consideration. The damping influences the field, as well as the surface plasmon excited at the edges of the slit [11]. The plasmon excitation is taken into account by the adaptive solver JCMsuite [12]. This program solves the Maxwell equations in a defined calculation area by means of the finite element method. A special feature of the problem is the semi-infinite domain (see Fig. 1). JCMsuite offers, however, the possibility to truncate the computational domain by applying virtual, perfectly matched absorbing layers [13]. With these boundary conditions implemented, the calculation area is free of feedback from the rim. Moreover, the area can be kept sufficiently small.

The thickness of the gold layer covering the silicon substrate is 100 nm in order to practically prevent the penetration of the electromagnetic wave. The gold film at the silicon substrate is a set-up to be employed in the forthcoming experiment. The refractive index of the gold used is n = 0.3226 + 10.6200i at the wavelength λ = 1.55 μm [14]. A number of quality laser sources has been developed for telecommunication carrier wave then this wavelength is chosen. The magnetic field and the associated phase of the wave are calculated numerically as a function of parameter a. The results are shown in Fig. 3. As the plot shows, the result for gold is in good agreement with the eigen-mode calculation for the perfect metal.

5. Conclusions

The field has been studied at the entrance to the slit between two parallel metal planes as a function of the slit width. At zero and infinite slit width, the field amplitude is well known: 2 and 1, respectively, whilst the incident wave is normalized by unity. The problem is to calculate the transient curves between these limiting cases. The entering field is obtained as a series (9); its coefficients are found by the solving of set (7), where the matrix elements are integrals (8). The number of terms of the series which needs to be considered is approximately equal to the number of modes propagating trough the slit.

While the frequency is less than the critical value for the slit considered as a planar waveguide, i.e. in the subwavelength regime, the field variation is monotonic and smooth. In the case of a wider slit, the next modes must be accounted for. The amplitude and the phase characteristics are found. Both curves turn out to be non-monotonic and involve saw-tooth-like oscillations. Each bend of the curves corresponds to a new mode excitation. The finite element method was applied for the slit covered with gold taking into account the Joule energy losses. The computation is in agreement with the perfect metal solution. The simple analytical expression for estimation of the light-induced attractive force in the limiting case of a narrow slit (a = 2πl/λ ≪ 1) has been derived.