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References

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  1. G.-m. Dai, "Wavefront expansion basis functions and their relationships," J. Opt. Soc. Am. A 23, 1657-1668 (2006).
    [CrossRef]
  2. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]

2006

1976

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Equations (9)

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F u v ( x , y ) = exp [ j 2 π N ( u x + v y ) ] .
T p q ( ρ , θ ) x = q T p 1 q 1 ( ρ , θ ) ,
T p q ( ρ , θ ) y = ( p q ) T p 1 q ( ρ , θ ) .
f ( m , t ) = { 1 ( 2 t ) ! ( m 2 t ) ! if t + t = ( p q ) 2 and m 0 1 ( 2 t + 1 ) ! ( m 2 t 1 ) ! if t + t = ( p q 1 ) 2 and m < 0 0 otherwise } .
Z 3 1 x = x ( 2 8 T 1 1 + 3 8 T 3 1 + 9 8 T 3 3 ) = 2 8 T 1 1 x + 3 8 T 3 1 x + 9 8 T 3 3 x = 2 8 T 0 0 + 3 8 T 2 0 + 9 8 T 2 2 = 2 8 Z 0 0 + 3 8 ( 1 4 Z 0 0 + 1 4 3 Z 2 0 1 2 6 Z 2 2 ) + 9 8 ( 1 4 Z 0 0 + 1 4 3 Z 2 0 + 1 2 6 Z 2 2 ) = 8 Z 0 0 + 2 6 Z 2 0 + 2 3 Z 2 2 ,
cos ϕ = { 1 + [ W ( x , y ) x ] 2 + [ W ( x , y ) y ] 2 } 1 2 ,
W ( x , y ) = p , q a p q T p q ( x , y ) ,
[ W ( x , y ) x ] 2 = p , q p , q a p q a p q q q T p + p 2 q + q 2 ( x , y ) ,
[ W ( x , y ) y ] 2 = p , q p , q a p q a p q ( p q ) ( p q ) T p + p 2 q + q ( x , y ) .

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