Tighter spots of light with superposed orbital-angular-momentum beams

The possibility of focusing light to an ever tighter spot has important implications for many applications and fields of optics research, such as nano-optics and plasmonics, laser-scanning microscopy, optical data storage, and many more. The size of lateral features of the field at the focus depends on several parameters, including the numerical aperture of the focusing system, but also the wavelength and polarization, phase and intensity distribution of the input beam. Here, we study the smallest achievable focal feature sizes of coherent superpositions of two copropagating beams carrying opposite orbital angular momentum. We investigate the feature sizes for this class of beams not only in the scalar limit, but also use a fully vectorial treatment to discuss the case of tight focusing. Both our numerical simulations and our experimental results confirm that lateral feature sizes considerably smaller than those of a tightly focused Gaussian light beam can be observed. These findings may pave the way for improving the resolution of imaging systems or may find applications in nano-optics experiments.

Figure 1

(a) Numerically calculated electric energy density (normalized to the maximum value of ${\left|\mathbf{E}\right|}^2$) and (b) phase distributions of an $x$-polarized petal beam for $\left|l\right|=8, \lambda =535$ nm and $w_0=0.65$ mm.

Figure 2

(a) Focal distributions of the electric energy density of the individual electric field components $|E_x{|}^2, |E_y{|}^2$, and $|E_z{|}^2$ as well as the total electric energy density ${\left|\mathbf{E}\right|}^2$ of a tightly focused petal-like beam with $\left|l\right|=8, \lambda =535$ nm, $w_0=0.65$ mm (Fig. 1). All distributions were numerically calculated using vectorial diffraction theory [29] and normalized to the maximum of the total electric energy density. (b) Schematic of the experimental setup (similar to Ref. [12]). An $x$-polarized Gaussian beam of wavelength 535 nm is converted into a petal beam by means of a liquid-crystal-based spatial light modulator (SLM; phase-only). The beam is focused by a microscope objective with a numerical aperture (NA) of 0.9. A 150-nm-diameter gold bead is raster scanned using a three-dimensional piezo stage through the focal plane of the light beam to determine the focal distribution. The transmitted and forward-scattered light is collected by a high-NA (1.3) oil-immersion lens and measured with a photodiode. (c) An experimental scan showing the total electric energy density distribution of the beam under study as shown in (a).

Figure 3

(a) Dependence of peak-to-valley distances on $\left|l\right|$ retrieved from the analytical paraxial treatment ($d_p^{\text{scalar}}$; green) and from vectorial diffraction theory ($d_p^{\text{vector}}$; red) for $\lambda =535$ nm focused with a microscope objective of $\text{NA}=0.9$. The experimental data (blue) is plotted for the experimentally accessible range of values for $\left|l\right|=0–40$. The inset shows the dependence of the beam radius in the focus on $\left|l\right|$. (b) Distributions of the total electric energy density ${\left|\mathbf{E}\right|}^2$ of tightly focused petal beams for the special cases of $\left|l\right|=0,1,2$.