Subaperture translation estimation accuracy in transverse translation diversity phase retrieval

Abstract

For optical metrology by transverse translation diversity phase retrieval (or ptychography), information theoretic limits on the ability to estimate subaperture translation, essential for accurate metrology, are assessed as a function of the optical aberrations of the system being measured. Special attention is given to the case that an unknown linear phase aberration, or equivalent detector or target motion, is present that varies with each point spread function in the measured data.

Ptychography for optical metrology with limited translation knowledge

“…In particular, Step 1 alleviates stagnation induced by the model PSF not having significant intensity overlapping regions of the measured PSF with significant intensity. Also, as noted in [16], the second and third radial degree phase aberrations have a critical role in contributing subaperture translation information to the subaperture translation estimation process. Consequently, we observe that estimating these lowest-order exit pupil phase in Step 2 prior to estimating subaperture translations in Step 3 aids successful retrieval.…”

“…where a 2;k and a 3;k are coefficients of the tip and tilt linear phase terms of the kth PSF, and a j is the jth Zernike polynomial coefficient for all of the PSFs for j ≥ 4. This model allows the linear phase terms to differ between PSFs in a condition we refer to as the unshared linear phase case, which accommodates for a translating detector or a target translating within an isoplanatic patch of the optical system as outlined in [16]. If the linear phase coefficients are known to be identical for all PSFs, the model reduces to the simpler expression,…”

“…For the optical metrology of imaging systems [7,9,10,[14][15][16], the generalized exit pupil function [17], rather than an object, is the unknown complex field of interest. A subaperture or mask is introduced in a plane ideally conjugate to the pupil.…”

“…When probing the central portion of the pupil, there are few sharp-edged amplitude or phase features to act as fiducials, as is the case with coherent diffractive imaging with biological or manmade specimens that have sharp-edged features. In [16], we calculated the Cramer-Rao lower bound (CRLB) for estimating subaperture translation from the measured PSF data in the optical metrology application. Though the CRLB analysis does not indicate how accurate PTE must be for success, it does indicate situations in which there is insufficient information for a unique solution of exit pupil phase and subaperture translation.…”

“…Also, the ideally unresolved point target being imaged moved in an unknown fashion from PSF to PSF for some tests. Since this target motion must be estimated and because the residual aberrations are mostly of only second radial degree, like defocus and astigmatism, translation estimation is challenging consistent with the CRLB theory and simulations in [16]. Consequently, we are concerned that systematic errors in the PTE arising from imprecise modeling of the Lyot stop motion could go uncorrected by the translation estimation steps in conventional TTDPR.…”