To investigate on-axis image quality of my Takahashi FSQ-106EDX 530 mm f/5 lens, I implemented the wavefront curvature estimator outlined in Roddier & Roddier (1993, 1991). With this method, the wavefront *W* can be reconstructed from an intra-focal image *I* of a bright star and an inverted (rotated by 180°) extra-focal image *E* of the same star by solving a partial differential equation of the inhomogeneous Poisson form

Δ*W* ≈ *k* (*I* - *E*) / (*I* + *E*),

subject to the Neumann boundary condition

δ*W */ δ*n* = 0,

where *k* is a constant that depends on the degree of defocus and on the focal length of the lens and *n* is a vector normal to the circular boundary of the projection of the pupil on the image plane.

Both sides of the equation are measures of wavefront curvature. The left hand side is the wavefront Laplacian, the departure of the wavefront from its local average. The right hand side is the sensor signal that scales with the difference between images. An aberrated optical zone with positive wavefront Laplacian will cause light to focus at a position short of the nominal focal plane. This gives a higher fluence intra-focal image and a lower fluence extra-focal image in the projection of the zone and a corresponding positive value in the right hand side of the equation.

The equation is solved using Fourier techniques and refined with an iterative algorithm that simulates closed-loop wavefront compensation in adaptive optics. Residual aberrations are compensated by geometrically distorting the images until the noise level is reached. Geometric distortions are determined by direct numerical differentiation of the refined wavefront. Poisson solver self-consistency conditions are satisfied by the method outlined in Ftaclas & Kostinski (2001).

The results in *r*-band, *g*-band, and *b*-band with Astrodon Generation 2 E-Series filters are shown below for the Takahashi lens. Strehl ratios of 0.91, 0.93, and 0.94 are estimated for the three bands, respectively. Since the wavefronts were estimated from images that were affected by seeing, the delivered Strehl in each band from the lens may be larger than estimated.

The intra-focal (left) and inverted extra-focal (right) images obtained in *r*-band are shown below with linear stretch. Each image is the mean of sixteen 5.0 second bias-subtracted and co-centered integrations of Capella. The defocus was ~3.4 mm (40000 digital focuser set points). This gives a ~10.5 mm lower bound on the size of the resolved wavefront corrugation period in *r*-band. Note the outer Fresnel ring is brighter in the extra-focal image than in the intra-focal image. Conversely, the inner Fresnel rings are brighter in the intra-focal image. This is indicative of spherical aberration.

The computed wavefront and the corresponding synthetic interferogram in *r*-band are shown below. The wavefront contours are labeled in nanometers. Piston, tilt, and defocus Zernike aberrations were pre-subtracted from the wavefront. The wavefront RMS is 32.9 nm. This gives a Strehl ratio of 0.91.

A three-dimension plot of the computed *r*-band wavefront surface is shown below with contours in nanometers.

The point spread function of the computed *r*-band wavefront for a monochromatic, midband wavelength of 656 nm is shown below with rational function stretch. Note the lack of separation between the first and second airy rings.

The plot below shows encircled energy in the point spread function of the computed *r*-band wavefront for a monochromatic, midband wavelength of 656 nm as a function of diameter. For comparison, the encircled energy in the point spread function of an aberration-free 530 mm f/5 lens is also shown. 80% of *r*-band energy is contained within a diameter of 10.9 microns, 4.3 arcseconds.

The table below shows coefficients of a least-squares fit to the computed *r*-band wavefront with the first twenty-two Noll indexed Zernike polynomials. The Z11 primary spherical, Z22 secondary spherical, and Z8 primary coma horizontal terms have the largest coefficients. The adjusted R^2 coefficient of determination of the least-squares fit equals 0.979 and the fitting residual RMS error is 4.8 nm. The t-statistic equals the coefficient estimate divided by the coefficient standard error estimate. A term likely has a non-zero coefficient if the absolute value of its t-statistic exceeds three.

The intra-focal (left) and inverted extra-focal (right) images obtained in *g*-band are shown below. Note the outer Fresnel ring is again brighter in the extra-focal image.

The computed wavefront and the corresponding synthetic interferogram in *g*-band are shown below. The wavefront RMS is 23.0 nm, smaller than in *r*-band. This gives a Strehl ratio of 0.93.

A three-dimension plot of the computed *g*-band wavefront surface is shown below.

The point spread function of the computed *g*-band wavefront for a monochromatic, midband wavelength of 546 nm is shown below with rational function stretch. Note that the airy rings are non-uniform.

The plot below shows encircled energy in the point spread function of the computed *g*-band wavefront for a monochromatic, midband wavelength of 546 nm as a function of diameter. 80% of *g*-band energy is contained within a diameter of 8.3 microns, 3.2 arcseconds.

The table below shows coefficients of a least-squares fit to the computed *g*-band wavefront. Primary spherical, secondary spherical, and primary coma vertical terms have the largest coefficients.

The intra-focal (left) and inverted extra-focal (right) images obtained in *b*-band are shown below. Note the outer Fresnel ring is once again brighter in the extra-focal image.

The computed wavefront (left) and the corresponding synthetic interferogram (right) in *b*-band are shown below. The wavefront RMS is 16.6 nm, smaller than in both *r*-band and *g*-band. This gives a Strehl ratio of 0.94.

A three-dimension plot of the computed *b*-band wavefront surface is shown below.

The point spread function of the computed *b*-band wavefront for a monochromatic, midband wavelength of 436 nm is shown below with rational function stretch. Note that the airy rings are again non-uniform.

The plot below shows encircled energy in the point spread function of the computed *b*-band waveband for a monochromatic, midband wavelength of 436 nm as a function of diameter. 80% of *b*-band energy is contained within a diameter of 5.5 microns, 2.1 arcseconds.

The table below shows coefficients of a least-squares fit to the computed *b*-band wavefront. The secondary spherical term has the largest coefficient. Primary coma, primary astigmatism, and primary spherical aberrations are also present.

C. Roddier and F. Roddier, "Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes", *Journal of the Optical Society of America A*, 10(11):2277-2287, November 1, 1993.

F. Roddier and C. Roddier, "Wavefront reconstruction using iterative Fourier transforms", *Applied Optics*, 30(11):1325-1327, April 10, 1991.

C. Ftaclas and A. Kostinski, "Curvature Sensors, Adaptive Optics, and Neumann Boundary Conditions", *Applied Optics*, 40(4):435-438, Feburary 1, 2001.

D. Malacara, M. Servín, and Z. Malacara, *Interferogram Analysis for Optical Testing*, ISBN 1-57444-682-7, CRC Press, Taylor & Francis Group, 2005.

J. Beckers, "Adaptive Optics for Astronomy: Principles, Performance, and Applications", *Annual Review of Astronomy and Astrophysics*, 31:13-62, September 1993.

Keywords:
Strehl,
Zernike,
aberration,
curvature,
encircled energy,
interferogram,
on-axis,
point spread function,
quality,
wavefront

Copyright © 2012-2016 M. Schuster. All Rights Reserved.