Adding colour to the Zernike wavefront sensor: Advantages of including multi-wavelength measurements for wavefront reconstruction

TL;DR

This work addresses the challenge of achieving nm-level wavefront control for direct imaging of Earth-like planets by augmenting the Zernike wavefront sensor (ZWFS) with multi-wavelength measurements. It introduces a model-based, accelerated gradient-descent reconstruction that fuses data across wavelengths (and polarizations for the vector ZWFS), enabling phase-diverse information and improved robustness. Key findings show that scalar ZWFS gains significant dynamic range from multi-wavelength diversity and benefits from broader photon counts, while two-wavelength phase unwrapping can recover large discontinuities such as petal errors, albeit with increased noise; the vector ZWFS gains are more pronounced in usable bandwidth than in dynamic range. The approach is practical with MKID detectors and existing optical components, and a SRON test bed is being prepared to validate multi-wavelength ZWFS concepts for next-generation extreme adaptive optics systems.

Abstract

To directly image Earth-like planets, contrast levels of 10^-8 - 10^-10 are required. The next generation of instruments will need wavefront control below the nanometer level to achieve these goals. The Zernike wavefront sensor (ZWFS) is a promising candidate thanks to its sensitivity, which reaches the fundamental quantum information limits. However, its highly non-linear response restricts its practical use case. We aim to demonstrate the improvement in robustness of the ZWFS by reconstructing the wavefront based on multi-wavelength measurements facilitated by technologies such as the microwave kinetic inductance detectors (MKIDs). We performed numerical simulations using an accelerated multi-wavelength gradient descent reconstruction algorithm. Three aspects are considered: dynamic range, photon noise sensitivity, and phase unwrapping. We examined both the scalar and vector ZWFS. Firstly, we find that using multiple wavelengths improves the dynamic range of the scalar ZWFS. However, for the vector ZWFS, its already extended range was not further increased. In addition, a multi-wavelength reconstruction allowed us to take advantage of a broader bandpass, which increases the number of available photons, making the reconstruction more robust to photon noise. Finally, multi-wavelength phase unwrapping enabled the measurement of large discontinuities such as petal errors with a trade-off in noise performance.

Figures (8)

• Figure 1: Gradient descent based wavefront reconstruction using multiple wavelength measurements for a scalar and a vector ZWFS. The input wavefront is described by a set of modal coefficients, $\theta$. The current estimate is propagated through a model of the considered ZWFS which produces the expect output intensity images at the considered wavelengths. These are compared to the measured intensity images and the similarity is captured in a cost function $J(\theta)$. The gradients of this cost function with respect to $\theta$ are calculated by a back-propagation process and these are used to update the current estimate by using an optimizer such as the Newton-CG.
• Figure 2: Dynamic range of the scalar and vector ZWFS using the gradient descent based reconstruction. For each configuration, 500 wavefronts are reconstructed consisting of 75 Zernike modes with a power law exponent between -1 and -3 and an RMS up to 250 nm. (a) and (b) Residual RMS for the scalar ZWFS using a monochromatic reconstruction for two different wavelengths (600, 1000 nm). (c) Residual RMS when employing the multi-wavelength reconstructor. (d), (e), and (f) Same results, but for a vector ZWFS.
• Figure 3: Example of the multi-wavelength gradient descent algorithm not finding the correct solution. The scalar ZWFS is used with wavelengths 600 and 1000 nm and the mask consists of a dot with $5\pi/2$ phase shift and 2 $\lambda/D$ dot diameter at 600 nm. (a) Input wavefront of 168 nm RMS. (b) Residual error after reconstruction of 38 nm RMS. (c) Calculated gradients at convergence at each wavelength and the overall gradient obtained by averaging.
• Figure 4: Reconstruction error at different photon levels for the monochromatic scalar ZWFS and the multi-wavelength scalar ZWFS using 600 nm and either 428 or 1000 nm. It is assumed that each wavelength has the same number of photons. 100 photon noise samples are reconstructed at each considered input level. Coloured areas correspond to $\pm1$ standard deviation.
• Figure 5: Reconstruction error at different bandwidth sizes using different reconstruction approaches for the vector ZWFS. The first method (vZWFS) does not separate the wavelengths and reconstructs at the central wavelength using the broadband image. The second method (mw-vZWFS) separates the bandwidth into ten wavelength bins and then uses the multi-wavelength algorithm. Then, 50 photon noise samples are taken at each bandwidth size. Coloured areas correspond to $\pm1$ standard deviation.