Wavefront reconstruction accuracy

February 09, 2014

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
- - - 0.3 None

 

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.

 

blog accuracy Z0blog accuracy Z0

 

blog accuracy Z12blog accuracy Z12

 

blog accuracy Z18blog accuracy Z18

 

blog accuracy Z22blog accuracy Z22

 

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.

 


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