Fig. 1. (a) Ray tracing for a big particle with radius R≫λ. We introduce the incidence angle φ and the refraction angle θ inside the sphere sinφ = nsinθ. The ray enter into the particle at the point with coordinates y_in= tanφ and x_in= −cosφ. The angles χ and α are given by χ = 2θ − φ and α = 2φ − 2θ. Two close rays y_c and y_cc (corresponding to angles φ and φ + δφ) emerged from the sphere after the second refraction are crossing at the caustic point x_c= x_out + ∂_φsinχ/∂_φtanα. This yields the Eq. (1) for caustic. (b) The shape of the caustic from the Eq. (1) for the sphere with n = 1.5 is shown by dashed black line.

Fig. 2. (a) Distribution of intensity calculated from the Mie theory with n = 1.5 and q = 70. Such distribution is typical for Bessoid matching solution, see e.g., Fig. 5 in ref.²⁴. (b) Intensity distribution according to Bessoid approximation²⁴ (solid blue line) and from the Mie theory (dotted red line).

Fig. 3. Amplitudes |Fl(e)| and |Fl(m)| for ℓ = 30 and n_p= 1.5 versus size parameter q.The first sharp resonance arise at q by the order of l. Insert shows how the position of the first sharp resonance vary with l number.

Fig. 4. (a) Spherical Bessel function ψl(kr) at big index l=1000²⁸. (b) Spacial distribution of the modulus of the Πl(e) function (9) at φ=0 and l=30 for n=1.5 and q=23.855.

Fig. 5. Distribution of electric intensity (E/E₀)² within the yz plane of the particle with refractive index n = 1.515 and size parameter q = 11.

Fig. 6. (a) We introduce the same incidence angle φ and the refraction angle θ as in Fig. 1(a). Here h is the height of truncation normalized to particle radius R. Ray emerges from the sphere after the second refraction with the angle γ follows the Snell’s law sinγ = n sin(φ − θ). The shape of the caustic for the truncated sphere with h = 1 − 1/n and n = 1.5 is shown by dashed black line. The solid green line shows the caustic of the spherical particle with the same refractive index. (b) The same parameters and the exact solution of the Maxwell equation, corresponding to size parameter q = 2πR/λ = 100.

Fig. 7. Distribution of electric E² intensity (picures on the top) and magnetic H² intensity (down pictures) within the cross section of the Janus cylinder with refractive index n = 1.5 (down), n = 1.3 (top), and size parameter q = 5π.

Fig. 8. Maximal field enhancement around the truncated cylindrical versus the depth of truncated element.

Fig. 9. Distribution of the field intensity for a resonant value of truncation (a) and zoom in (b) and further (c).

Fig. 10. Internal and external electric (a) and magnetic (b) intensities versus size parameter for the cylinder with fixed truncation parameter h = 0.02. Size parameter q = 32.5 correspnds to radius of the cylinder 2R≈ 10λ.

Fig. 11. Schematic for a lithographic process with truncated cylinders.Here a thin protected layer between the matrix and photoresist plays an important role of anti-reflective coating, depending on the thickness of the coating.