A New Method for Wavefront Sensing using Optical Masking Interferometry

TL;DR

This work tackles direct, full-field wavefront sensing by measuring electromagnetic path-length delays across an optical aperture. It proposes optical aperture masking interferometry combined with self-calibration of complex visibilities, yielding a linear relation between element-gain phases and path-length delays via $δL = λ × (φ_G / 360^\circ)$. Experimental tests at 400 nm on the ALBA Xanadu bench achieved nm-scale path-length precision per 1 ms exposure, with wavefront tilt measured to about $0.1''$ accuracy and static non-planar distortions down to a few nanometers. The method is moving-part- and reference-beam-free, can be extended by adding more holes or multi-frequency data to increase dynamic range, and holds promise for real-time adaptive optics and high-precision metrology.

Abstract

Wave front sensing of the surface of equal phase for a propagating electromagnetic wave is a vital technology in fields ranging from real time adaptive optics, to high accuracy metrology, to medical optometry. We have developed a new method of wavefront sensing that makes a direct measurement of the electromagnetic phase distribution, or path-length delay, across an optical wavefront. The method is based on techniques developed in radio astronomical interferometric imaging. The method employs optical interferometry using a 2-D aperture mask, a Fourier transform of the interferogram to derive interferometric visibilities, and self-calibration of the complex visibilities to derive the voltage amplitude and phase gains at each hole in the mask, corresponding to corrections for non-uniform illumination and wavefront distortions across the aperture, respectively. The derived self-calibration gain phases are linearly proportional to the electromagnetic path-length distribution to each hole in the aperture mask, relative to the path-length to the reference hole, and hence represent a wavefront sensor with a precision of a small fraction of a wavelength. The method was tested at $λ=400\,$nm at the Xanadu optical bench at the ALBA synchrotron light source using a rotating mirror to insert tip-tilt changes in the wavefront. We reproduce the wavefront tilts to within $0.1''$ ($5\times 10^{-7}$~radians). We also derive the static metrology though the optical system for non-planar wavefront distortions to $\sim \pm1$~nm repeatability. Lastly, we derive frame-to-frame variations of the wavefront tilt due to vibrations of the optical components which range up to $\sim 0.5"$. These variations are relevant to adaptive optics applications. Based on the measured visibility phase noise after self-calibration, we estimate an rms path-length precision per 1~ms exposure of 0.6 nm.

Figures (9)

• Figure 1: Schematic of the optical system with cartoon of wavefront distortions. Note that multiple mirrors in vacuum and out are not shown, including the rotating mirror (see Figure \ref{['fig:rotmir']}). The rotating mirror can be considered a planar 'distortion' of just the tip-tilt term for the wavefront.
• Figure 2: The self-calibration algorithm. The "Generate Source Model" subroutine can involve Gaussian fitting to the corrected visibilities described in Section \ref{['sec:method']}, or other model fitting or deconvolution algorithms.
• Figure 3: Picture of the optical bench setup at Xanadu, ALBA, showing the location of the rotating mirror. The photons from the synchrotron beam line enter from the right of the picture, as indicated by the yellow lines and arrows. NRA = non-redundant aperture mask.
• Figure 4: Top Left: Photograph of the 7 hole mask illuminated by the synchrotron light beam. Holes are 2 mm in diameter, and they are numbered. Top right: interferogram made using the 7 hole mask. The pixel scale is $0.153"$ per pixel, which, using the 15.03 m distance from the list source to the mask, corresponds to 11.1 $\mu$m in the source plane. Bottom left: Fourier transform of the interferogram showing the visibility amplitudes. Interferometric baselines between holes are numbered below each visibility sample (0-0 corresponds to the autocorrelation = total power; Hermitian conjugate samples are not numbered). The pixel scale is baseline length measured in wavelengths (or inverse radians), with 657 wavelengths per pixel. Bottom right: Same, showing the visibility phases. The color scale is in radians. Note that the sizes of the discrete regions in the $u$--$v$ coordinates is linearly related to the size of the holes, or the inverse size of the Airy diskCarilli2024.
• Figure 5: The visibility phase time series of baselines 1-6 (red) and 2-5 (orange). These are two neighboring and parallel baselines that do not have a hole in common (see Figure \ref{['fig:4frame']}). Left shows the phases for the u,v points before self-calibration. Note that the Y-axis range is much larger for the left-hand figure.