As a test of accuracy of my Roddier & Roddier wavefront reconstruction implementation, defocused images were simulated for a 530 mm f/5 lens with known low-order Zernike polynomial aberrations. Images were computed by numerical integration of the approximate Rayleigh-Sommerfeld scalar diffraction model (Gillen and Guha, 2004). The reconstruction technique was applied to these images using the default defocused pupil image diameter estimation heuristics.
The table below shows test results. Each row in the table corresponds to a single simulation with the Zernike polynomial coefficient indicated by the Index column set equal to 20 nm RMS. The Coefficient column equals the recovered Zernike polynomial coefficient RMS and the Wavefront column equals the estimated wavefront RMS. Zernike polynomials with equal radial and absolute azimuthal degree (Z5, Z6, Z9, Z10, Z14, Z15, Z20, and Z21) are underestimated. Primary astigmatism and primary trefoil aberrations have zero Laplacian and are therefore more difficult to retrieve since all of the information comes from boundary conditions. The estimated wavefront RMS of secondary aberrations is larger than the coefficient RMS due to cross talk with the corresponding primary aberration (Z12, Z13, Z16, Z17, Z18, Z19, and Z22).
The mean and standard deviation of coefficient RMS is 19.3 and 2.2 nm, respectively. The mean and standard deviation of estimated wavefront RMS is 20.0 and 2.9 nm, respectively. Hence on these tests wavefront recovery of 20 nm RMS aberrations is accurate to within ~15% at the one standard deviation level. The last row in the table shows a result for an aberration-free simulation. The small 0.3 nm estimated wavefront RMS of the aberration-free test represents systematic error due to aliasing artifacts in the simulated images.
|Index||Degree||Coefficient (nm RMS)||Wavefront (nm RMS)||Aberration|
|Z5||(2, -2)||17.8||17.9||Primary astigmatism oblique|
|Z6||(2, 2)||15.8||15.8||Primary astigmatism vertical|
|Z7||(3, -1)||21.0||21.0||Primary coma vertical|
|Z8||(3, 1)||21.0||21.0||Primary coma horizontal|
|Z9||(3, -3)||16.8||16.8||Primary trefoil vertical|
|Z10||(3, 3)||16.8||16.9||Primary trefoil oblique|
|Z11||(4, 0)||22.1||22.1||Primary spherical|
|Z12||(4, 2)||21.1||24.6||Secondary astigmatism vertical|
|Z13||(4, -2)||21.0||23.5||Secondary astigmatism oblique|
|Z14||(4, 4)||18.4||18.3||Primary quadrafoil vertical|
|Z15||(4, -4)||16.0||16.1||Primary quadrafoil oblique|
|Z16||(5, 1)||20.3||20.8||Secondary coma horizontal|
|Z17||(5, -1)||20.3||20.8||Secondary coma vertical|
|Z18||(5, 3)||21.2||22.7||Secondary trefoil oblique|
|Z19||(5, -3)||21.2||22.7||Secondary trefoil vertical|
|Z20||(5, 5)||17.6||17.6||Primary pentafoil oblique|
|Z21||(5, -5)||17.6||17.6||Primary pentafoil vertical|
|Z22||(6, 0)||21.8||23.5||Secondary spherical|
The intra-focal (left) and inverted extra-focal (right) images for several of the tests are shown below. The first, labeled Z0, are the aberration-free test images. The rest are Z12 secondary astigmatism vertical, Z18 secondary trefoil vertical, and Z22 secondary spherical. Aliasing artifacts are visible in all of the images.
G. Gillen and S. Guha, "Modeling and propagation of near-field diffraction patterns: A more complete approach", American Journal of Physics, 72(9):1195-2001, September 2004.