# Polynomial Apodizers for Centrally Obscured Vortex Coronagraphs

## TL;DR

This paper introduces polynomial apodizations for vortex coronagraphs that compensate for central obscurations, significantly improving throughput and nulling performance for complex telescope pupils.

## Contribution

It presents analytically derived polynomial apodizations for vortex coronagraphs that maintain perfect nulling with central obscurations, enhancing throughput over previous ring-apodized designs.

## Key findings

- Achieves >70% throughput for charge 6 vortex with central obscurations.
- Designs polynomial apodizations for charges 2, 4, and 6.
- Demonstrates potential for complex pupil optimization in exoplanet imaging.

## Abstract

Several coronagraph designs have been proposed over the last two decades to directly image exoplanets. Among these designs, the vector vortex coronagraphs provide theoretically perfect starlight cancellation along with small inner working angles when deployed on telescopes with unobstructed pupils. However, current and planned space missions and ground-based extremely large telescopes present complex pupil geometries, including secondary mirror central obscurations, that prevent vortex coronagraphs from rejecting on-axis sources entirely. Recent solutions combining the vortex phase mask with a ring-apodized pupil have been proposed to circumvent this issue, but provide a limited throughput for vortex charges $>2$. We present a family of pupil plane apodizations that compensate for pupil geometries with circularly symmetric central obstructions caused by on-axis secondary mirrors for charge 2, 4, and 6 vector vortex coronagraphs. These apodizations are derived analytically and allow the vortex coronagraph to retain theoretically perfect nulling in the presence of central obscurations. For a charge 4 vortex, we design polynomial apodization functions assuming a greyscale apodizing filter that represent a substantial gain in throughput over the ring-apodized vortex coronagraph design, while for a charge 6 vortex, we design polynomial apodized vortex coronagraphs that have $\gtrsim 70\%$ total energy throughput for the entire range of central obscuration sizes studied. We propose methods for optimizing apodizations produced with either greyscale apodizing filters or shaped mirrors. We conclude by demonstrating how this design may be combined with apodizations numerically optimized for struts and segment gaps in telescope pupils to design terrestrial exoplanet imagers for complex pupils.