Tomographic alignment algorithm for an extremely large three-mirror telescope: invisible modes

Piotr Piatrou; Gary Chanan

doi:10.1364/AO.49.006395

Applied Optics
Vol. 49,
Issue 33,
pp. 6395-6401
(2010)
•https://doi.org/10.1364/AO.49.006395

Tomographic alignment algorithm for an extremely large three-mirror telescope: invisible modes

Piotr Piatrou and Gary Chanan

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Piotr Piatrou and Gary Chanan, "Tomographic alignment algorithm for an extremely large three-mirror telescope: invisible modes," Appl. Opt. 49, 6395-6401 (2010)

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Abstract

We analyze the optical effects due to distortions of a three-mirror telescope that is sufficiently large that all three mirrors must be actively controlled. Numerical experiments on telescopes with both monolithic and segmented primary mirrors reveal the existence of telescope misalignment configurations (modes) that are invisible to a fixed focal station wavefront sensor, even for highly redundant multidirectional tomographic measurement schemes. We describe these modes and give a theoretical explanation for them.

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k	$b_{x}^{l - k} b_{y}^{k}$	$\partial_{x}^{l - k} \partial_{y}^{k}$	$x^{5}$	$x^{4} y$	$x^{3} y^{2}$	$x^{2} y^{3}$	$x y^{4}$	$y^{5}$
0	$b_{x}$	$\partial_{x}$	$5 x^{4}$	$4 x^{3} y$	$3 x^{2} y^{2}$	$2 x y^{3}$	$y^{4}$	0
1	$b_{y}$	$\partial_{y}$	0	$x^{4}$	$2 x^{3} y$	$3 x^{2} y^{2}$	$4 x y^{3}$	$5 y^{4}$

k	$b_{x}^{l - k} b_{y}^{k}$	$\partial_{x}^{l - k} \partial_{y}^{k}$	$x^{5}$	$x^{4} y$	$x^{3} y^{2}$	$x^{2} y^{3}$	$x y^{4}$	$y^{5}$
0	$b_{x}^{2}$	$\partial_{x}^{2}$	$20 x^{3}$	$12 x^{2} y$	$6 x y^{2}$	$2 y^{3}$	0	0
1	$b_{x} b_{y}$	$\partial_{x} \partial_{y}$	0	$4 x^{3}$	$6 x^{2} y$	$6 x y^{2}$	$4 y^{3}$	0
2	$b_{y}^{2}$	$\partial_{y}^{2}$	0	0	$2 x^{3}$	$6 x^{2} y$	$12 x y^{2}$	$20 y^{3}$

k	$b_{x}^{l - k} b_{y}^{k}$	$\partial_{x}^{l - k} \partial_{y}^{k}$	$x^{4}$	$x^{3} y$	$x^{2} y^{2}$	$x y^{3}$	$y^{4}$
0	$b_{x}^{2}$	$\partial_{x}^{2}$	$12 x^{2}$	$6 x y$	$2 y^{2}$	0	0
1	$b_{x} b_{y}$	$\partial_{x} \partial_{y}$	0	$3 x^{2}$	$4 x y$	$3 y^{2}$	0
2	$b_{y}^{2}$	$\partial_{y}^{2}$	0	0	$2 x^{2}$	$6 x y$	$12 y^{2}$

k	$b_{x}^{l - k} b_{y}^{k}$	$\partial_{x}^{l - k} \partial_{y}^{k}$	$x^{5}$	$x^{4} y$	$x^{3} y^{2}$	$x^{2} y^{3}$	$x y^{4}$	$y^{5}$
0	$b_{x}$	$\partial_{x}$	$5 x^{4}$	$4 x^{3} y$	$3 x^{2} y^{2}$	$2 x y^{3}$	$y^{4}$	0
1	$b_{y}$	$\partial_{y}$	0	$x^{4}$	$2 x^{3} y$	$3 x^{2} y^{2}$	$4 x y^{3}$	$5 y^{4}$

k	$b_{x}^{l - k} b_{y}^{k}$	$\partial_{x}^{l - k} \partial_{y}^{k}$	$x^{5}$	$x^{4} y$	$x^{3} y^{2}$	$x^{2} y^{3}$	$x y^{4}$	$y^{5}$
0	$b_{x}^{2}$	$\partial_{x}^{2}$	$20 x^{3}$	$12 x^{2} y$	$6 x y^{2}$	$2 y^{3}$	0	0
1	$b_{x} b_{y}$	$\partial_{x} \partial_{y}$	0	$4 x^{3}$	$6 x^{2} y$	$6 x y^{2}$	$4 y^{3}$	0
2	$b_{y}^{2}$	$\partial_{y}^{2}$	0	0	$2 x^{3}$	$6 x^{2} y$	$12 x y^{2}$	$20 y^{3}$

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Applied Optics