Abstract

New algorithms for reconstructing wavefront from slopes data are developed, which exhibit high accuracy over broad spatial-frequency bandwidth. Analyzing wavefront reconstructors in the frequency domain lends new insight into ways to improve frequency response and to understand noise propagation. The mathematical tools required to analyze the frequency domain are first developed for discrete band-limited signals. These tools are shown to improve frequency response in either spatial-or frequency-domain reconstruction algorithms. A new spatial-domain iterative reconstruction algorithm based on the Simpson rule is presented. The local phase estimate is averaged over 8 neighboring points whereas the traditional reconstructors use 4 points. Analytic results and numerical simulations show that the Simpson-rule–based reconstructor provides high accuracy up to 85% of the bandwidth. The previously developed rectangular-geometry band-limited algorithm in frequency domain is adapted to hexagonal geometry, which adds flexibility when applying frequency-domain algorithms. Finally, a generalized analytic expression for error propagation coefficient is found for different reconstructors and compared with numerical simulations.

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References

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2010 (1)

2008 (2)

2006 (1)

2005 (1)

2004 (2)

2002 (3)

1996 (1)

1995 (1)

1991 (1)

1986 (1)

1980 (1)

1978 (1)

1977 (2)

1962 (1)

B. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9, 84–97 (1962).
[CrossRef]

1949 (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. Inst. Radio Eng. 37, 10–21 (1949).

Bahk, S.-W.

Boreman, G. D.

Brase, J. M.

Bromage, J.

Campos, J.

Chung, J. H.

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Modeling and control of a deformable mirror,” J. Dyn. Sys. Meas. Control 124, 297–302 (2002).
[CrossRef]

Dainty, C.

Dainty, J. C.

Dubra, A.

Elster, C.

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function f(x) when only ddxf(x) or d2dx2f(x) is known at discrete measurement points,” in X-Ray Mirrors, Crystals, and Multilayers II, A. K. Freund, A. T. Macrander, T. Ishikawa, and J. L. Wood, eds., Proc. SPIE 4782, 12–20 (2002).

Fess, E.

Freischlad, K.

Fried, D. L.

Gavel, D. T.

Hudgin, R. H.

Irwin, D.

Koliopoulos, C. L.

Kruschwitz, B. E.

Kwiatkowski, J.

Macintosh, B.

Malacara, D.

Marroquin, J. L.

Millecchia, M.

Moore, M.

Moreno, A.

Nicholls, T. W.

Noll, R. J.

Paterson, C.

Phillips, B. L.

B. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9, 84–97 (1962).
[CrossRef]

Poyneer, L. A.

Pruyne, A.

Roddier, C.

Roddier, F.

Rolland, J. P.

Servin, M.

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. Inst. Radio Eng. 37, 10–21 (1949).

Southwell, W. H.

Velinsky, S. A.

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Modeling and control of a deformable mirror,” J. Dyn. Sys. Meas. Control 124, 297–302 (2002).
[CrossRef]

Weingärtner, I.

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function f(x) when only ddxf(x) or d2dx2f(x) is known at discrete measurement points,” in X-Ray Mirrors, Crystals, and Multilayers II, A. K. Freund, A. T. Macrander, T. Ishikawa, and J. L. Wood, eds., Proc. SPIE 4782, 12–20 (2002).

Winters, S. E.

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Modeling and control of a deformable mirror,” J. Dyn. Sys. Meas. Control 124, 297–302 (2002).
[CrossRef]

Yaroslavsky, L. P.

Yzuel, M. J.

Zou, W.

Zuegel, J. D.

Appl. Opt. (3)

J. ACM (1)

B. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9, 84–97 (1962).
[CrossRef]

J. Dyn. Sys. Meas. Control (1)

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Modeling and control of a deformable mirror,” J. Dyn. Sys. Meas. Control 124, 297–302 (2002).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Opt. Express (3)

Opt. Lett. (3)

Proc. Inst. Radio Eng. (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. Inst. Radio Eng. 37, 10–21 (1949).

Other (2)

See http://www.jwst.nasa.gov/ .

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function f(x) when only ddxf(x) or d2dx2f(x) is known at discrete measurement points,” in X-Ray Mirrors, Crystals, and Multilayers II, A. K. Freund, A. T. Macrander, T. Ishikawa, and J. L. Wood, eds., Proc. SPIE 4782, 12–20 (2002).

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Figures (4)

Fig. 1
Fig. 1

Simpson geometry.

Fig. 2
Fig. 2

Frequency response of Simpson reconstructor with λ =0.07489 (a) 3-D view of the frequency response. (b) cross-section along fx axis. Solid line is calculated from the analytic expression and circles are from simiulations

Fig. 3
Fig. 3

(a) Prostrate hexagon array; (b) standing hexagon array.

Fig. 4
Fig. 4

Flowchart of band-limited reconstruction for a hexagonal geometry. is the band-limited filter function [Eq. (53)]. Ω1 and Ω2 are the regions where the slopes data exist. The subscripts ‘m’ in Step 2 and 7 denote the measured slopes. IDFT stands for inverse discrete Fourier transform.

Tables (3)

Tables Icon

Table 1 Summary of Band-Limited Derivative Operators (D x,0)

Tables Icon

Table 2 Summary of the Frequency-Domain Operators for the Associated Finite Difference Schemes

Tables Icon

Table 3 Summary of Noise Propagation

Equations (70)

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φ ( x ) = n = φ ( n Δ x ) sinc [ 2 W ( x n Δ x ) ] ,
d d x sinc ( x ) = π j 1 ( π x ) ,
d φ ( x ) d x = 1 Δ x n = φ ( n Δ x ) π j 1 [ 2 π W ( x n Δ x ) ] .
d φ d x | ( x = m Δ x ) φ x ( m ) = 1 Δ x n = 1 π j 1 ( n π ) { φ [ ( m + n ) Δ x ] φ [ ( m n ) Δ x ] } ,
π j 1 ( n π ) = ( 1 ) n + 1 n , n = 1 , 2 ,
m = 0 N 1 d φ d x | ( x = m Δ x ) = 0 .
φ ˜ x ( k ) = 2 i Δ x [ n = 1 ( 1 ) n + 1 n sin ( 2 π n N k ) ] φ ˜ ( k ) ,
φ ˜ x ( k ) = i 2 π Δ x 𝖲 ( k ) φ ˜ ( k ) ,
𝖲 ( k ) = { for even N { k / N , k = 0 , , N / 2 1 0 , k = N / 2 k / N 1 , k = N / 2 + 1 , , N 1 for odd N { k / N , k = 0 , , ( N 1 ) / 2 k / N 1 , k = ( N + 1 ) / 2 , , N 1 .
d φ d x | ( x = m Δ x + 1 2 Δ x ) φ x , 1 2 ( m ) = 1 Δ x n = 1 π j 1 ( n π 1 2 π ) { φ [ ( m + n ) Δ x ] φ [ ( m n + 1 ) Δ x ] } ,
π j 1 ( n π 1 2 π ) = 4 π ( 1 ) n + 1 ( 2 n 1 ) 2 , n = 1 , 2 ,
φ ˜ x , 1 2 ( k ) = i 2 π Δ x exp ( i π N k ) 𝖳 ( k ) φ ˜ ( k )
𝖳 ( k ) = { k / N , k = 0 , , N 1 2 1 k / N , k = N 1 2 + 1 , , N 1 .
φ ( x = m Δ x + 1 2 Δ x ) φ 1 2 ( m ) = n = 1 sinc ( n 1 2 ) { φ [ ( m + n ) Δ x ] φ [ ( m n + 1 ) Δ x ] } ,
φ ˜ 1 2 ( k ) = exp ( i π N k ) 𝖱 ( k ) φ ˜ ( k )
𝖱 ( k ) = { for even N { 1 , k = 0 , , N / 2 1 0 , k = N / 2 1 , k = N / 2 + 1 , , N 1 for odd N { 1 , k = 0 , , ( N 1 ) / 2 1 , k = ( N + 1 ) / 2 , , N 1 .
φ ˜ x , 1 2 , 1 2 ( p , q ) = i 2 π Δ x exp [ i π N ( p + q ) ] 𝖳 ( q ) 𝖱 ( p ) φ ˜ ( p , q )
exp ( ± π i N k ) 𝖱 ( k ) = exp [ ± π i 𝖲 ( k ) ] .
2 π Δ x 𝖲 ( p ) = { 2 π L [ N 1 2 + p ] , N 1 2 } ,
2 π Δ x 𝖲 ( p ) = { 2 π L ( N 2 + p ) , N 2 } ,
2 π Δ x 𝖲 ( p ) = { 2 π L [ ( N 2 1 ) + p ] , N 2 1 } .
φ ˜ x = i k x ¯ φ ˜ ,
φ ˜ x , 1 2 = i k x ¯ exp ( i Δ x 2 k x ¯ ) φ ˜ ,
φ ˜ 1 2 = exp ( i Δ x 2 k x ¯ ) φ ˜ .
j a ( j ) φ ^ ( i + j ) = k b ( k ) S ( i + k ) ,
φ ^ ( i + 1 ) φ ^ ( i ) = Δ x 2 [ S ( i ) + S ( i + 1 ) ] .
φ ^ ( i + 1 ) φ ^ ( i 1 ) = Δ x 3 [ S ( i 1 ) + 4 S ( i ) + S ( i + 1 ) ] .
ɛ = i , j { 1 2 Δ x [ φ ^ ( i , j + 1 ) φ ^ ( i , j 1 ) ] 1 6 [ S x ( i , j + 1 ) + 4 S x ( i , j ) + S x ( i , j 1 ) ] } 2 + { 1 2 Δ y [ φ ^ ( i + 1 , j ) φ ^ ( i 1 , j ) ] 1 6 [ S y ( i + 1 , j ) + 4 S y ( i , j ) + S y ( i 1 , j ) ] } 2 .
g L [ φ ^ ( i , j ) φ ^ ( i , j 2 ) ] + g R [ φ ^ ( i , j ) φ ^ ( i , j + 2 ) ] + g U ( Δ x Δ y ) 2 [ φ ^ ( i , j ) φ ^ ( i 2 , j ) ] + g D ( Δ x Δ y ) 2 [ φ ^ ( i , j ) φ ^ ( i + 2 , j ) ] = g L [ S x ( i , j 2 ) + 4 S x ( i , j 1 ) + S x ( i , j ) ] Δ x 3 g R [ S x ( i , j ) + 4 S x ( i , j + 1 ) + S x ( i , j + 2 ) ] Δ x 3 + g U ( Δ x Δ y ) 2 [ S y ( i 2 , j ) + 4 S y ( i 1 , j ) + S y ( i , j ) ] Δ y 3 g D ( Δ x Δ y ) 2 [ S y ( i , j ) + 4 S y ( i + 1 , j ) + S y ( i + 2 , j ) ] Δ y 3 Δ S ( i , j )
g L [ φ ^ ( i , j ) φ ^ ( i , j 1 ) ] + g R [ φ ^ ( i , j ) φ ^ ( i , j + 1 ) ] + g U ( Δ x Δ y ) 2 [ φ ^ ( i , j ) φ ^ ( i 1 , j ) ] + g D ( Δ x Δ y ) 2 [ φ ^ ( i , j ) φ ^ ( i + 1 , j ) ] = g L [ S x ( i , j 1 ) + S x ( i , j ) ] Δ x 2 g R [ S x ( i , j ) + S x ( i , j + 1 ) ] Δ x 2 + g U ( Δ x Δ y ) 2 [ S y ( i 1 , j ) + S y ( i , j ) ] Δ y 2 g D ( Δ x Δ y ) 2 [ S y ( i , j ) + S y ( i + 1 , j ) ] Δ y 2
φ ^ ( m + 1 ) ( i , j ) = φ ^ ( m ) ( i , j ) + ω [ φ ^ ( m ) ¯ ( i , j ) φ ^ ( m ) ( i , j ) ] ,
φ ^ ( m ) ¯ ( i , j ) = [ φ ^ 0 ( m ) ¯ ( i , j ) + Δ S ( i , j ) ] / ( g L + g R + g U + g D ) ,
φ ^ 0 ( m ) ¯ ( i , j ) = g L φ ^ ( m ) ( i , j 2 ) + g R φ ^ ( m ) ( i , j + 2 ) + g U φ ^ ( m ) ( i 2 , j ) + g D φ ^ ( m ) ( i + 2 , j ) .
ɛ = 1 N 2 Σ { | D x φ ^ ˜ A x S ˜ x | 2 + | D y φ ^ ˜ A y S ˜ y | 2 } ,
D x = 1 2 Δ x [ exp ( i k x ¯ Δ x ) exp ( i k x ¯ Δ x ) ] D x , Simpson
A x = 1 6 [ exp ( i k x ¯ Δ x ) + 4 + exp ( i k x ¯ Δ x ) ] A x , Simpson .
φ ^ ˜ = ( D x * A x S ˜ x + D y * A y S ˜ y ) | D x | 2 + | D y | 2 .
H = φ ^ ˜ φ ˜ = ( D x * A x D x , 0 + D y * A y D y , 0 ) | D x | 2 + | D y | 2 ,
H Simpson = 1 2 ω x sin ω x ( 2 + cos ω x ) + ω y sin ω y ( 2 + cos ω y ) sin 2 ω x + sin 2 ω y ,
ɛ reg = λ i , j { 1 ( 2 Δ x ) 2 [ φ ^ ( i , j + 1 ) 2 φ ^ ( i , j ) + φ ^ ( i , j 1 ) ] 2 + 1 ( 2 Δ y ) 2 [ φ ^ ( i + 1 , j ) 2 φ ^ ( i , j ) + φ ^ ( i 1 , j ) ] 2 } .
ɛ reg = λ 1 N 2 i , j { | D x , reg φ ^ ˜ | 2 + | D y , reg φ ^ ˜ | 2 } ,
D x , reg = 2 Δ x sin 2 ( k x ¯ Δ x 2 )
H Simpson ( λ ) = 1 3 ω x sin ω x ( 2 + cos ω x ) + ω y sin ω y ( 2 + cos ω y ) sin 2 ω x + sin 2 ω y + 4 λ ( sin 4 1 2 ω x + sin 4 1 2 ω y ) .
( 2 Δ x ) 2 ɛ reg φ ^ ( i , j ) = λ g L [ φ ^ ( i , j ) 2 φ ^ ( i , j 1 ) + φ ^ ( i , j 2 ) ] + λ g R [ φ ^ ( i , j + 2 ) 2 φ ^ ( i , j + 1 ) + φ ^ ( i , j ) ] + λ g U [ φ ^ ( i , j ) 2 φ ^ ( i 1 , j ) + φ ^ ( i 2 , j ) ] + λ g D [ φ ^ ( i + 2 , j ) 2 φ ^ ( i + 1 , j ) + φ ^ ( i , j ) ] 2 λ g L R [ φ ^ ( i , j + 1 ) 2 φ ^ ( i , j ) + φ ^ ( i , j , 1 ) ] 2 λ g U D [ φ ^ ( i + 1 , j ) 2 φ ^ ( i , j ) + φ ^ ( i 1 , j ) ] .
φ ^ ¯ ( i , j ) = [ φ 0 ^ ¯ ( i , j ) + Δ S ( i , j ) + λ Δ φ ^ ( i , j ) ] / [ ( 1 + λ ) ( g L + g R + g U + g D ) + 4 λ ( g L R + g U D ) ] ,
Δ φ ^ g L [ 2 φ ^ ( i , j 1 ) φ ^ ( i , j 2 ) ] + g R [ 2 φ ^ ( i , j + 1 ) φ ^ ( i , j + 2 ) ] + g U [ 2 φ ^ ( i 1 , j ) φ ^ ( i 2 , j ) ] + g D [ 2 φ ^ ( i + 1 , j ) φ ^ ( i + 2 , j ) ] + 2 g L R [ φ ^ ( i , j + 1 ) + φ ^ ( i , j 1 ) ] + 2 g U D [ φ ^ ( i + 1 , j ) + φ ^ ( i 1 , j ) ] .
S ˜ x = [ S ˜ 1 , x + T hex S ˜ 2 , x S ˜ 1 , x T hex S ˜ 2 , x ] [ S ˜ x , I S ˜ x , II ]
T hex ( p , q ) = exp [ π i M p π i 𝖲 ( q ) ] .
S ˜ x = [ S ˜ 1 , x + T hex S ˜ 2 , x S ˜ 1 , x T hex S ˜ 2 , x ] [ S ˜ x , I S ˜ x , II ]
T hex ( p , q ) = exp [ π i 𝖲 ( p ) π i N q ] .
φ 1 ˜ = 1 2 ( φ I ˜ + φ II ˜ ) ,
φ 2 ˜ = 1 2 T hex * ( φ I ˜ φ II ˜ ) .
[ S ˜ x ( p , q ) , S ˜ y ( p , q ) ] = D x * ( p , q ) S ˜ x ( p , q ) + D y * ( p , q ) S ˜ y ( p , q ) | D x ( p , q ) | 2 + | D y ( p , q ) | 2 ,
D x ( p , q ) = 2 π i Δ x 𝖲 ( q ) ,
D y ( p , q ) = 2 π i ( 2 Δ y ) 𝖲 ( p ) ,
D x ( p , q ) = 2 π i ( 2 Δ x ) 𝖲 ( q ) ,
D y ( p , q ) = 2 π i Δ y 𝖲 ( p ) ,
σ φ 2 = 1 N t Σ Δ φ 2 = 1 N t N 2 Σ | Δ φ ˜ | 2 ,
φ ˜ = ( D x * A x S ˜ x + D y * A y S ˜ y ) | D x | 2 + | D y | 2 + λ ( | D x , reg | 2 + | D y , reg | 2 ) ,
| Δ φ ˜ | 2 = ( | D x A x Δ S x | ˜ 2 + | D y A y Δ S y ˜ | 2 ) [ | D x | 2 + | D y | 2 + λ ( | D x , reg | 2 + | D y , reg | 2 ) ] 2 .
η = σ φ 2 h 2 σ S 2 = 1 L 2 k x , k y ( | D x A x | 2 + | D y A y | 2 ) [ | D x | 2 + | D y | 2 + λ ( | D x , reg | 2 + | D y , reg | 2 ) ] 2 ,
S N = ( | D 0 , x | 2 + | D 0 , y | 2 ) ( | D x A x | 2 + | D y A y | 2 ) [ | D x | 2 + | D y | 2 + λ ( | D x , reg | 2 + | D y , reg | 2 ) ] 2 .
| H | 2 S N .
k x , k y | H | 2 ( k x 2 + k y 2 ) k x , k y S N ( k x 2 + k y 2 ) = η
S ˜ x ( p , q ) = n = 0 N 1 m = 0 M 1 S x ( m , n ) exp ( 2 π i M m p 2 π i N n q )
S ˜ x ( p , q ) = n = 0 N 1 m = 0 M 1 S x ( 2 m , n ) exp ( 2 π i M m p 2 π i N n q ) + n = 0 N m = 0 M 1 S x ( 2 m + 1 , n ) exp ( 2 π i M m p 2 π i N n q π i M p ) ,
S ˜ x ( p = p , q ) = S x , 1 ˜ ( p , q ) + exp ( π i M p ) S x , 2 ˜ ( p , q )
S ˜ x ( p = p + M , q ) = S x , 1 ˜ ( p , q ) exp ( π i M p ) S x , 2 ˜ ( p , q )
S x , 1 ˜ = 1 2 ( S x , I ˜ + S x , II ˜ ) ,
S x , 2 ˜ = 1 2 T hex * ( S x , I ˜ S x , II ˜ ) .

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