Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie

An analytical description of arbitrary strongly aberrated axially symmetric focusing is developed. This is done by matching the solution of geometrical optics with a wave pattern which is universal for the underlying ray structure. The corresponding canonical integral is the Bessoid integral, which is a three-dimensional generalization of the Pearcey integral that approximates the field near an arbitrary two-dimensional cusp. We first develop the description for scalar fields and then generalize it to the vector case. As a practical example the formalism is applied to the focusing of light by transparent dielectric spheres with a few wavelengths in diameter. The results demonstrate good agreement with the Mie theory down to Mie parameters of about 30. Compact analytical expressions are derived for the intensity on the axis and the position of the diffraction focus both for the general case and for the focusing by microspheres. The high intensity region is narrower than for an ideal lens of the same aperture at the expense of longitudinal localization and has a polarization dependent fine structure, which can be explained quantitatively. The results are relevant for aerosol and colloid science where natural light focusing occurs and can be used in laser micro- and nano-processing of materials.

Figure 1

Absolute square of the Pearcey integral $I_P$ (top) and the Bessoid integral $I$ (bottom). The latter is proportional to the field of a spherically aberrated wave within small angles approximation.

Figure 2

(a) Three-ray region inside the cuspoid (dashed line). (b) One-ray region outside. The $z$ axis is represented by a dashed-dotted line.

Figure 3

Refraction of a ray—propagating from $Q$ to $P$—by a sphere with radius $a$ and refractive index $n$. The picture is drawn in the meridional plane and all indicated angles are positive.

Figure 4

(a) A ray propagates from $Q$ to $P$ in the meridional plane (plane of incidence). (b) Decomposition of the initial electric field vector with length $E_0$ into its $\pi$- and $\sigma$-component parallel and perpendicular to the meridional plane.

Figure 5

Normalized intensity ${\mid E∕E_0\mid }^2$ in the normalized $x$,$z$ plane (top) and in the $y$,$z$ plane (bottom). Contour shadings go from white (zero) to black $(\approx 700)$. Parameters: refractive index $n=1.5$, dimensionless wave number $ka=100$. The initial electric field vector is ${\mathbf{E}}_0=E_0{\mathbf{e}}_x$. The sphere with radius $a$ is positioned at the origin of the coordinate system. The focus of geometrical optics is located at $z=f=1.5a$, whereas the diffraction focus (the point of maximum intensity) is significantly shifted towards the sphere: $f_d\approx 1.25a$. In dimensional units, for a wavelength of $\lambda =0.248\mu \mathrm{m}$ the sphere radius is $a\approx 4\mu \mathrm{m}$.

Figure 6

Left: Diffraction focus in units of the sphere radius as a function of $n$ and $ka$ (contour lines from top to bottom go from 1.1 to 3.0 in steps of 0.1). Right: Intensity enhancement at $f_d$ (contour lines from bottom left to top right are 20, 50, 100, 200, 500, 1000 and 2000).

Figure 7

${\mid E∕E_0\mid }^2$ on the axis. Dashed lines represent the Mie theory, solid lines are the results of Bessoid matching. The parameters are as in Fig. 5 and the cases (a), (b), (c), and (d) correspond to $q=300$, 100, 30, and 10, respectively. In dimensional units, for $\lambda =0.248\mu \mathrm{m}$ this corresponds to sphere radii of $a\approx 12$, 4, 1.2, and $0.4\mu \mathrm{m}$.

Figure 8

Normalized ${\mid E\mid }^2$, ${\mid H\mid }^2$ and $S_z$ in the normalized $x,y$ plane for $z=1.02a$ calculated with Bessoid matching (left) and with the Mie theory (right). The parameters are the same as in Fig. 5.

Figure 9

Near the axis, the phases of the rays 1 and 3 differ from the phase of the nonparaxial rays np (crossing the axis at the same $z$) by $\pm k\rho \sin \beta$, whereas the phases of ray 2 and the (par)axial ray are the same in first order. The nonparaxial ray and ray 3 cross the axis at an angle $\beta >0$.

Figure 10

Meridional cross section for the determination of the meridional radius of curvature, $R_m$. $M$ is the center of the sphere.

Figure 11

Meridional cross section for the determination of the sagittal radius of curvature, $R_s$.