Tight focusing of electromagnetic fields by large-aperture mirrors

We derive nonparaxial input conditions for simulations of tightly focused electromagnetic fields by means of unidirectional nonparaxial vectorial propagation equations. The derivation is based on the geometrical optics transfer of the incident electric field from significantly curved reflecting surfaces such as parabolic and conical mirrors to the input plane, with consideration of the finite thickness of the focusing element and large convergence angles, making the propagation vectorial and nonparaxial. We have benchmarked numerical solutions of propagation equations initiated with the nonparaxial input conditions against the solutions of Maxwell equations obtained by vectorial diffraction integrals. Both transverse and longitudinal components of the electric field obtained by these methods are in excellent agreement.

Figure 1

(a) The geometry of reflection on an axially symmetric optical element. Solid red lines show the incident and reflected rays at an angle $\alpha$ with the normal $\mathbf{n}$ to the mirror at the reflection point $\{r,z_s\left(r\right)\}$. Green arrows show the optical path if the reflection point on the mirror surface is replaced by an effective emitter in the plane $z=0$. Blue arrows show the optical path if the reflection point is projected on the same plane $z=0$ in the paraxial approximation. (b) Illustration of energy flux conservation in the ray tube (filled); ${\mathbf{S}}_r$ and ${\mathbf{S}}_e$ are the Poynting vectors of reflected and emitted waves, respectively.

Figure 2

Focusing by the parabolic mirror with the focal length $f\approx 3$ mm $\left(f/1.5,\mathrm{NA}=0.33\right)$. (a) Variation of $|E_x{|}^2$ on the beam axis $r=0$ with propagation distance $z$ calculated by the UHPE with paraxial or standard (solid black curve) and nonparaxial input conditions (red curve). Comparison is made with vectorial diffraction integrals (black dots). (b) Phase and (d) amplitude of standard (dashed curve) and nonparaxial (solid curve) input conditions. (c) The log-scaled difference between nonparaxial $\phi$ and standard ${\phi }_p$ phases.

Figure 3

Focusing of the beam, (16), by the parabolic mirror, (17), with the focal length $f\approx 1.2$ mm $\left(f/0.6,\mathrm{NA}=0.64\right)$. The transverse $(X,Y)$ distributions of (a) $|E_x{|}^2$, (b) $|E_y{|}^2$, and (c) $|E_z{|}^2$ in the focus (upper row; in each panel the squared modulus of the field is normalized by its maximum value). Marked panels show the longitudinal distributions with propagation distance $z$ of the quantities $|E_x{|}^2$ (left column), $|E_y{|}^2$ (middle column), and $|E_z{|}^2$ (right column) in the transverse positions indicated by the corresponding markers in the upper row. Simulation results are obtained from the UHPE with nonparaxial input conditions (solid curve) and the VDI (dots).

Figure 4

Same as Fig. 3, for the propagation after reflection from a conical mirror, (18), with angle $\alpha ={20}^{\circ }(f/0.6,\mathrm{NA}=0.64)$. The upper row corresponds to $z=1.26$ mm.

Figure 5

Same as Fig. 3, for the propagation after reflection from the mirror $z_s\propto r^{1.1}(f/0.6,\mathrm{NA}=0.64)$. The upper row corresponds to $z=1.195$ mm.