Phase Retrieval using Nonlinear Curvature Sensing within Convergent Beams

Abstract

Path-length diversity methods may be used for adaptive optics (AO) systems to retrieve phase and amplitude information by measuring intensity across multiple planes. Observations that rely on free-space propagation, such as the nonlinear curvature wavefront sensor (WFS), have been shown to offer excellent sensitivity and robustness to scintillation. However, the default design results in a large opto-mechanical footprint due to unavoidable geometric-optics and wave-optics effects. Measurements recorded in a convergent beam would improve instrument compactness, while concentrating light into smaller detector regions of interest, improving signal-to-noise ratio and possibly wavefront reconstruction speed. In this paper, we study path-length diversity wavefront sensing using four planes of contemporaneous intensity measurements made in a convergent beam. We develop a physical optics propagation model and validate the model by performing wavefront reconstructions in both simulations and lab experiments. The manuscripts core contribution is a practical, intensity-domain, Fourier-transform-based recipe to use a conventional multi-plane Gerchberg-Saxton (or comparable) reconstruction pipeline with convergent-beam measurements, enabling a compact optical layout. We find that this approach offers practical benefits over an equivalent free-space wavefront sensor, in particular reducing size, weight, complexity and cost.

Figures (8)

• Figure 1: Cartoon level optical design concepts for capturing the sensing channels of a multi-plane wavefront sensor. Beam-splitting components introduce optical path-length differences between channels (individual derivative beams are not shown). Operating in collimated space requires a large opto-mechanical foot-print (top three designs). Operating in a converging beam improves compactness (bottom design).
• Figure 2: Quadratic kernel phase, $\Phi_{\rm kernel}$ (radians), appearing in the Collins/Fresnel propagation integral for a thin lens plus free-space gap plotted across the entrance pupil for several axial distances. The model assumes a wavelength of $\lambda=633$ nm, beam diameter of $d=2$ mm, and $f=300$ mm focal length.
• Figure 3: Image scaling factor, $S = z_{\rm eff} / z$, evaluated for several different focal lengths. A focusing optic provides access to many multiples of the linear distance, $z$, used for creating optical path-length differences.
• Figure 4: Simulated $f=300$ mm lens images (top row) and rescaled images (bottom row) including atmospheric turbulence. Measured $z$-distances are listed in the top panel while $z_{\rm eff}$ and $S=z_{\rm eff}/z$ are listed in the bottom panel. Color-bars are in units of counts.
• Figure 5: Wavefront reconstruction of a Kolmogorov phase screen representing atmospheric turbulence using simulations.