Diffractive Focusing of Waves in Time and in Space

We study the general wave phenomenon of diffractive focusing from a single slit for two types of waves and demonstrate several properties of this effect. Whereas in the first situation, the envelope of a surface gravity water wave is modulated in time by a rectangular function, leading to temporal focusing, in the second example, surface plasmon polariton waves are focused in space by a thin metal slit to a transverse width narrower than the slit itself. The observed evolution of the phase carrier of the water waves is measured for the first time and reveals a nearly flat phase as well as an 80% increase in the intensity at the focal point. We then utilize this flat phase with plasmonic beams in the spatial domain, and study the case of two successive slits, creating a tighter focusing of the waves by putting the second slit at the focal point of the first slit.

Figure 1

Theoretical prediction and experimental observation of diffractive focusing of water wave packets with width $t_0=4.18\mathrm{s}$ corresponding to six cycles. A schematic illustration is given of the experimental setup (a), theoretical and experimental results representing the intensity evolution $|a{|}^2/|a_0{|}^2$ of the wave packets along the tank [(b) and (d)], and of the envelope phase evolution [(c) and (e)] given by $\cos \psi$. Panels (f) and (g) display the spatial and temporal evolution of intensity and phase along the dashed lines in (d) and (e), respectively.

Figure 2

Theoretical predictions (solid line) and experimental results (dots) of the normalized Gaussian temporal width $\delta W$ with a value of $\kappa t_0=4.7$ for a slit with a width of $t_0=4.18\mathrm{s}$ (a), and the focal length, $x_f$, in its dependence on the pulse width $t_0$ (b).

Figure 3

Diffractive focusing of a plasmonic wave from a $14\times 10\mu \mathrm{m}$ slit: scanning electron microscope image of the sample (a) and a picture taken by the near-field scanning optical microscope (NSOM) camera (b); three-dimensional representation of the NSOM measurements of the topography of the sample (c), and of the intensity distribution of the SPP (d); numerical simulation (e) of diffractive focusing for another slit with dimensions $10\times 10\mu \mathrm{m}$; and the NSOM measurement of the corresponding SPP intensity distribution (f). The white dashed line in (f) represents the slit topography.

Figure 4

Diffractive focusing of a plasmonic wave from two successive slits: the dimensions of the first slit are $14\times 10\mu \mathrm{m}$, and of the second $5\times 4\mu \mathrm{m}$. Picture of the sample taken by the NSOM camera (a), and the simulation result (b) under the assumption of an infinitesimal narrow slit. The NSOM measurement of the SPP intensity distribution (c) combines two separate images that are connected at the location of the second slit. The white dashed line represents the slit topography. Theoretical and experimental results (d) of the normalized Gaussian spatial width $\delta W$ with a value of $\kappa a_1=3.15$ for the area after the first slit ($a_1=14\mu \mathrm{m}$), and a value of $\kappa a_2=4.89$ for the domain after the second slit ($a_2=5\mu \mathrm{m}$). The black dashed lines in (d) represent the location of the second slit. The experimental results were calculated for a reduced area given approximately by the slit size.