# Light induced force

### Project content

In line of a project founded by the German Research Foundation (DFG) experimental work is done to verify a theoretical predicted light induced force. This attractive force is equivalent to negative light pressure and, according to the theory, arises when polarized light falls into a slit between two macroscopic metallic bodies and the slit width is smaller than half the wavelength of the light.
For the verification of this force the nano force facility of the PTB will be used. Beside the experimental verification of the effect, the improvement of the theoretical model that forms the base for the light induced force is also a goal of the project.
As part of the project the following steps are planed:

• Modification of the PTB-nano force facility
• Development of measurement techniques and performance of direct measurements for the proof of the new light induced forces between two parallel metallic bodies with sub wavelength distance in vacuum
• Calculation of the reflection coefficient for the TM and TE mode for two parallel metallic plates with a sub wavelength distance
• Analysis of measurement uncertainty

### Theoretical background

Below a brief summary of the theoretical background of the light induced forces will be given, which can be found in full in the literature [1,2]. The described light induced forces occurs when light falls into a slit between two metallic bodies, whose width 2l is smaller than half the wavelength of the light (2l < λ/2) (fig. 1).
The wave vector k0 of the incident wave is orthogonal to the incident plane. The dimensions of the slit are designated by L (height), d (depth) und 2l (width) (fig. 1). In the theoretical model it is assumed that the slit height (along the y-axis) is much bigger than the slit width and wavelength (L >> l, λ). In this case the lateral forces in y-direction can be neglected.

Figure 1: Propagation of the TM mode at the incidence of a linear polarized electromagnetic wave from below the slit. The wave vector k0 proceeds in z-direction. The magnetic field Hy of the wave is oriented parallel and the electric field Ex perpendicular to the walls of the slit (TM mode).

It can also be assumed, that the wall thickness in x-direction is much greater than the skin depth δs of the light.  For the case of an incident light beam from below, as shown in figure 1, only the transversal magnetic main mode is present in the slit and produces and attractive force between the two metallic plates according to the theoretical calculations [1,2]. For the generation of the force the metallic properties of both surfaces are important, since the dielectric constant of metals can be described by a complex function: εM = ε1 + i ε2. Some metals like gold and silver have a high negative real part ε1 (-ε1 >> 1) and a small imaginary part ε2 ((-ε1) >>ε2) in the visible and infrared (IR) region of the spectrum (compare fig. 2). To simplify the estimation of the force the case of a homogenous beam with constant field amplitude at the intersection will be considered. The effective penetration depth δ of the light beam in the slit is

(1)

Figure 2: The factor λ⋅(-ε1 )/ε2 for gold and silver depending on the wavelength λ ([1,2,3]).

The light induced attractive force FTM (cf. fig. 1) for the case of deep slits (d >> δ) is given by:

(2)

(with the vacuum permeability μ0, half of the magnetic field intensity amplitude of the TM mode in the slit h0 and the wavelength λ of the light).During the theoretical analysis it has to be considered, that a 90 degree rotation of the polarization plane results in a transversal electrical mode (TE) in the slit. This TE mode results in a repelling force FTE, which however is small compared to the attractive force (FTM  >> FTE). An estimation of the magnitude of the light induced forces according to equation (2) with the target parameters of the planed experiment shows, that detection of the attractive force as well as practical applications should be possible.With the proof of the new light induces forces new possibility would arise for the manipulation of metallic macro, micro, and nano objects in vacuum and liquids. Possible areas of application would be e.g. optical switchable structure.

### Literature

1] Nesterov V, Frumin L and Podivilov E 2011 Negative light pressure force between two metal bodies separated by a subwavelength slit, EPL. 94 64002

[2] Nesterov V and Frumin L 2011 Light-induced attractive force between two metal bodies separated by a subwavelength slit, Meas. Sci. Technol. 22 094008 (7pp)

[3] Palik E D 1998 Handbook of Optical Constants of Solids (Academic Press, London)