Zernike Mode Sorting with Vortex Phase Filters: Perfect Coronagraphs and Ideal Wavefront Sensors

TL;DR

The paper tackles the challenge of efficient spatial mode sorting for circular apertures by introducing a Zernike-based sorter built from even-charge vortex phase filters (VPFs). It shows that a VPF of charge $l$ maps complex Zernike modes $Z_n^m$ to $Z_n^{m+l}$ when $|m+l|\le n$, ejecting others via destructive interference, enabling lossless, crosstalk-free demultiplexing. A practical two-stage VPF–Mach–Zehnder architecture, plus a recursive, diagonal-by-diagonal sorting strategy, enables complete complex Zernike decomposition, with a prefilter for real-valued Zernikes. The authors outline applications to wavefront sensing and coronagraphy that saturate quantum sensitivity limits, discuss extensions to fourth-order coronagraphy, and propose approaches to handle arbitrary apertures via single-mode conversion and MPLC, highlighting both the potential and practical challenges for broadband operation and precise alignment.

Abstract

Spatial mode sorting has come to prominence as an optical processing modality capable of saturating fundamental limits to numerous sensing tasks including wavefront sensing, coronagraphy, and superresolution imaging. But despite their promising theoretical advantages, contemporary mode sorters often feature large crosstalk, high loss, or sort modes that are poorly adapted to conventional imaging systems (e.g., Hermite- and Laguerre-Gauss). Here, we introduce an alternative architecture that sorts spatial modes natural to circularly symmetric apertures: Zernike polynomials. Using conventional optics hardware and even-order vortex phase plates, we show how to assemble a series of vortex phase filters that can in principle separate the various Zernike polynomials losslessly and without crosstalk. This idea is demonstrated via application to wavefront sensing and coronagraphy, where we propose an optical system that saturates the quantum sensitivity limits to both tasks. We expect our work to prove useful for high-contrast imaging of extrasolar planets, improving both wavefront control and coronagraph performance.

Figures (7)

• Figure 1: (a) First ten real- and (b) complex-valued Zernike polynomials. In both plots value represents mode amplitude and hue the phase. Real Zernikes are ubiquitous in optics, but here complex Zernikes prove more valuable because of their definite OAM, a feature vital for developing a theoretically perfect mode sorter.
• Figure 2: (a) Optical layout of a VPF, shown here as charge 2. Light from the pupil is brought to focus and phase shifted using a vortex phase mask. A second lens inverts the imaging, relaying the field to a conjugate pupil plane. (b) - (e) Select complex Zernike polynomials (labels bottom left) after propagating through a VPF2. Pupil edges are marked by the white lines. Maximum radial distance shown is one diameter. Notice that modes $Z_n^m$ having $m \geq n$ are removed from the pupil while modes with $m < n$ have been simply transformed into $Z_{n}^{m+2}$. This type of nulling was first observed for piston by Foo et al. Foo:05 but manifests for any polynomial shifted to an invalid Zernike index.
• Figure 3: A mnemonic for understanding the action of a VPF on the complex Zernike pyramid, shown here for charge 4. We can imagine a VPF as shifting the entire pyramid left or right (positive or negative charge), moving some modes outside the pyramid boundary (oblique lines). Zernikes that cross these boundaries are ejected from the pupil. All other modes are retained, but transform into new Zernike polynomials $Z_{n'}^{m'} = Z_{n}^{m+l}$, where here $l=4$ but could in general be any even integer.
• Figure 4: Our two-stage approach for separating pyramid edges in the complex Zernike pyramid. We first pass light through a VPF2, separating out the $m=n$ OAM modes from $m < n$. Following the second lens, we intercept the reforming pupil light with a balanced MZI, using phase shifting stops to bind the two different mode groups into orthogonal output ports.
• Figure 5: An optical processor effecting perfect complex Zernike decompositions. We rely on iterated use of the VPF-MZI developed in Fig. \ref{['fig:first-stage']} to sort Zernike pyramid diagonal-by-diagonal, and then one-by-one along each diagonal. Mode appearing immediately right and below a VPF-MZI optic represent the two groups just separated. When only a single mode appears we have perfectly isolated it.