tokens: the same linear assumptions previously applied to yield kernel phases, analogous self-calibrating observables also exist for propagation through more complex optical systems such as coronagraphs. However, for the example configuration explored, the rapid onset of a nonlinear response indicates they may be of limited practical applicability to coronagraphic observations except in the extremely high-wave-front-quality regime. Subnanometer wave-front precision may nevertheless be achieved in future large space telescopes with high-order deformable mirrors. 2.5. Zernike Basis Taking the Jacobian of one large pixel grid by another is expensive in terms of memory; it is therefore desirable to bring down the dimensionality of either the input or target space. One way to do this is to use a Zernike basis to represent the wave front (Zernike 1934 ). One advantage of this is that we can examine only up to a certain order in Zernike polynomials if we believe that wave-front errors are negligible below a certain spatial scale. We can then differentiate just with respect to Zernike coefficients, which is much more computationally efficient and allows us to run simulations on grids that are many times larger than otherwise. Similar bases (e.g., the “hexike” basis) can be constructed for arbitrary noncircular telescope apertures. We have rerun the simple diffraction simulation from Section 2.1 with 200 Zernike modes on a 256 × 256 grid, and the coronagraph from Section 2.4 on a 128 × 128 grid using 300 Zernike modes. The calculations now take around 4 s, rather than many minutes, with peak memory use around 1.5 and 3 GB rather than 7 and 9 GB respectively. The Jacobian derived from this basis for simple diffraction is an excellent approximation to the full calculation on a grid. Given that in problems such as HST data analysis we believe that PSF variations are dominated by low-order modes (Lallo et al. 2006 ), this allows us to calculate a larger kernel space with a more accurate pupil model than has previously been possible. While it is a huge advantage in simulation speed and fidelity, this basis does not accurately represent the high-spatial-frequency sine waves corresponding to speckles far from the core of the PSF. Nevertheless, it does an adequate job representing the effects of low-order wave-front error, and this may be sufficient for many purposes, for example in optimization of optical design. 3. Discussion A phase transfer matrix constructed analytically with a discrete pupil model contains no a priori information about the pixel sampling, binning, or windowing. This means that the implicit convolutions in the plane, and therefore the associated correlation between adjacent baselines, are not taken into account. Therefore, the normally orthonormal kernel phases (calculated analytically for an infinite field of view and fine sampling) are no longer fully linearly independent. For example, Martinache et al. ( 2020 ) note that in the analysis of
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