Abstract

Wavefronts reconstructed from measured gradients are composed of a straightforward integration of the measured data, plus a correction term that disappears when there are no measurement errors. For regions of any shape, this term is a solution of Poisson’s equation with Dirichlet conditions (V=0 on the boundaries). We show that for rectangular regions, the correct solution is not a periodic one, but one expressed with Fourier cosine series. The correct solution has a lower variance than the periodic Fourier transform solution. Similar formulas exist for a circular region with obscuration. We present a near-optimal solution that is much faster than fast-Fourier-transform methods. By use of diagonal multigrid methods, a single iteration brings the correction term to within a standard deviation of 0.08, two iterations, to within 0.0064, etc.

© 2006 Optical Society of America

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References

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  2. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).
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  4. Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215, 285-288 (2003).
    [CrossRef]
  5. Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4657 (2004).
    [CrossRef]
  6. A. Talmi and E. N. Ribak, "Direct demodulation of Hartmann-Shack patterns," J. Opt. Soc. Am. A 21, 632-639 (2004).
    [CrossRef]
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  22. A MATLAB program with these examples can be read at physics.technion.ac.il/~eribak/s2p/.

2004 (3)

2003 (3)

2002 (2)

2001 (1)

A. J. Roberts, "Simple and fast multigrid solution of Poisson's equation using diagonally oriented grids," Aust N. Z. Ind. Appl. Math. J. 43, E1-36 (2001).

2000 (2)

1999 (1)

S. Rios and E. Acosta, "Orthogonal modal reconstruction of a wave front from phase difference measurements," J. Mod. Opt. 46, 931-939 (1999).

1993 (1)

1991 (1)

1986 (1)

1980 (1)

1977 (2)

Acosta, E.

S. Rios and E. Acosta, "Orthogonal modal reconstruction of a wave front from phase difference measurements," J. Mod. Opt. 46, 931-939 (1999).

Carmon, Y.

Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4657 (2004).
[CrossRef]

Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215, 285-288 (2003).
[CrossRef]

Dainty, C.

Dubra, A.

Ellerbroek, B. L.

Ellerbroek, L.

Elster, C.

Freischlad, K. R.

Fried, D. L.

Gavel, D.

Gilles, L.

Hudgin, R. H.

Koliopoulos, C. L.

Macintosh, B.

Paterson, C.

Poyneer, L. A.

Ribak, E. N.

A. Talmi and E. N. Ribak, "Direct demodulation of Hartmann-Shack patterns," J. Opt. Soc. Am. A 21, 632-639 (2004).
[CrossRef]

Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4657 (2004).
[CrossRef]

Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215, 285-288 (2003).
[CrossRef]

Rios, S.

S. Rios and E. Acosta, "Orthogonal modal reconstruction of a wave front from phase difference measurements," J. Mod. Opt. 46, 931-939 (1999).

Roberts, A. J.

A. J. Roberts, "Simple and fast multigrid solution of Poisson's equation using diagonally oriented grids," Aust N. Z. Ind. Appl. Math. J. 43, E1-36 (2001).

Roddier, C.

Roddier, F.

Southwell, W. H.

Talmi, A.

Troy, M.

Tyler, G. A.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

Vogel, C. R.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4657 (2004).
[CrossRef]

Aust N. Z. Ind. Appl. Math. J. (1)

A. J. Roberts, "Simple and fast multigrid solution of Poisson's equation using diagonally oriented grids," Aust N. Z. Ind. Appl. Math. J. 43, E1-36 (2001).

J. Mod. Opt. (1)

S. Rios and E. Acosta, "Orthogonal modal reconstruction of a wave front from phase difference measurements," J. Mod. Opt. 46, 931-939 (1999).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215, 285-288 (2003).
[CrossRef]

Opt. Lett. (2)

Other (4)

D.Malacara, ed., Optical Shop Testing (Wiley, 1978).

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

R.K.Tyson, ed., Adaptive Optics Engineering Handbook (Marcel Decker, 2000).

A MATLAB program with these examples can be read at physics.technion.ac.il/~eribak/s2p/.

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Figures (1)

Fig. 1
Fig. 1

(Left) input, (center) cosine transform errors, (right) Fourier transform errors for examples 5–7 in Table 1: (top) analytic periodic function, (center) random function, (bottom) with 50% noise on the slopes. In this last case we show the cosine output (bottom left) as it changes little compared with the noiseless input above it. The frame sizes were 64 × 128 .

Tables (1)

Tables Icon

Table 1 Errors Produced in Calculation of Phase from Gradients over a Rectangular Region

Equations (63)

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Var ( S a ) = Var ( S b ) + Var ( S a S b ) ,
Φ ( n + 1 , m ) Φ ( n , m ) Δ x Φ ( n + 1 2 , m ) S x ( n + 1 2 , m ) S x ( n ̑ , m ) ,
Φ ( n , m + 1 ) Φ ( n , m ) Δ y Φ ( n , m + 1 2 ) S y ( n , m + 1 2 ) S y ( n , m ̑ ) ,
ϵ 2 = n , m R [ Δ x Φ ( n ̑ , m ) S x ( n ̑ , m ) ] 2 + [ Δ y Φ ( n , m ̑ ) S y ( n , m ̑ ) ] 2 = minimum .
Δ x Φ ( n ̑ , m ) = S x ( n ̑ , m ) + C x ( n ̑ , m ) ;
Δ y Φ ( n , m ̑ ) = S y ( n , m ̑ ) + C y ( n , m ̑ )
Φ ( n , m ) = Φ ( 1 , 1 ) + x = 3 2 n 1 2 Δ x Φ ( x , 1 ) + y = 3 2 m 1 2 Δ y Φ ( n , y ) = Φ ( 1 , 1 ) + x = 3 2 n 1 2 S x ( x , 1 ) + y = 3 2 m 1 2 S y ( n , y ) + x = 3 2 n 1 2 C x ( x , 1 ) + x = 3 2 m 1 2 C y ( n , y ) .
0 = Δ x Δ y Φ ( n ̑ , m ̑ ) Δ y Δ x Φ ( n ̑ , m ̑ ) = Δ y Φ ( n + 1 , m ̑ ) Δ y Φ ( n , m ̑ ) Δ x Φ ( n ̑ , m + 1 ) + Δ x Φ ( n , m ̑ ) ,
Δ x C y ( n ̑ , m ̑ ) Δ y C x ( n ̑ , m ̑ ) = Δ y S x ( n ̑ , m ̑ ) Δ x S y ( n ̑ , m ̑ ) ρ ( n ̑ , m ̑ ) ,
ε 2 = n , m R C x 2 ( n ̑ , m ) + C y 2 ( n , m ̑ ) = minimum .
d C x d y d C y d x = ρ ( n ̑ , m ̑ ) .
C y = d V d x , C x = d V d y ;
d 2 V d x 2 + d 2 V d y 2 = ρ , { x , y } ϶ R .
ε 2 = R [ ( d V d x ) 2 + ( d V d y ) 2 + ( d W d x ) 2 + ( d W d y ) 2 ] d x d y + 2 R ( d V d x d W d y d V d y d W d x ) d x d y .
2 R ( d V d x d W d y d V d y d W d x ) d x d y = 2 R V ( d 2 V d x d y d 2 W d y d x ) d x d y + B V d l W = B V d l W ,
C x ( n ̑ , m ) = Δ y V ( n ̑ , m ) , C y ( n , m ̑ ) = Δ x V ( n , m ̑ ) ,
( Δ x 2 + Δ y 2 ) V ( n ̑ , m ̑ ) = ρ ( n ̑ , m ̑ ) = Δ y S x ( n ̑ , m ̑ ) Δ x S y ( n ̑ , m ̑ ) .
C x ( n ̑ , m ) = Δ y V ( n ̑ , m ) + Δ x W ( n ̑ , m ) ,
C y ( n , m ̑ ) = Δ x V ( n , m ̑ ) + Δ y W ( n , m ̑ ) .
ϵ 2 = n ̑ , m C x ( n ̑ , m ) 2 + n , m ̑ C y ( n , m ̑ ) 2 = n ̑ , m [ Δ y V ( n ̑ , m ) ] 2 + [ Δ x W ( n ̑ , m ) ] 2 + n , m ̑ [ Δ x V ( n , m ̑ ) ] 2 + [ Δ y W ( n , m ̑ ) ] 2 + 2 n , m ̑ Δ x V ( n , m ̑ ) Δ y W ( n , m ̑ ) 2 n ̑ , m Δ y V ( n ̑ , m ) Δ x W ( n ̑ , m ) ,
n = 1 , , N , m = 1 , , M ; n ̑ = 3 2 , 5 2 , , N 1 2 , m ̑ = 3 2 , 5 2 , , M 1 2 .
2 n , m ̑ Δ x V ( n , m ̑ ) Δ y W ( n , m ̑ ) 2 n ̑ , m Δ y V ( n ̑ , m ) Δ x W ( n ̑ , m ) = 2 n ̑ , m ̑ V ( n ̑ , m ̑ ) [ Δ x Δ y W ( n ̑ , m ̑ ) Δ y Δ x W ( n ̑ , m ̑ ) ] + boundary terms .
H [ S + C ] d l = H S d l + H C d l K + H C d l = 0
V ( n ̑ , m ̑ ) = V ( n + 1 2 , m + 1 2 ) = q x = 1 N 1 q y = 1 M 1 V ̃ ( q x , q y ) sin ( n q x π N ) sin ( m q y π M ) ,
ρ ( n ̑ , m ̑ ) = q x = 1 N 1 q y = 1 M 1 ρ ̃ ( q x , q y ) sin ( n q x π N ) sin ( m q y π M ) .
ρ ̃ ( q x , q y ) = 4 N M n = 1 N 1 m = 1 M 1 ρ ( n ̑ , m ̑ ) sin ( n q x π N ) sin ( m q y π M ) ,
V ̃ ( q x , q y ) = 4 N M n = 1 N 1 m = 1 M 1 V ( n ̑ , m ̑ ) sin ( n q x π N ) sin ( q y π M ) .
( Δ x 2 + Δ y 2 ) V ̃ ( q x , q y ) = 4 ( sin 2 Q x 2 + sin 2 Q y 2 ) V ̃ ( q x , q y ) = ρ ̃ ( q x , q y ) .
V ̃ ( q x , q y ) = ρ ̃ ( q x , q y ) T 2 ( q x ) + T 2 ( q y ) , q x , q y > 0 ,
C x ( q x , q y ) = T ( q y ) V ̃ ( q x , q y ) , C ̃ y ( q x , q y ) = T ( q x ) V ̃ ( q x , q y ) .
S x ( n ̑ , m ) = S x ( n + 1 2 , m ) = q x = 1 N 1 q y = 0 M 1 S ̃ x ( q x , q y ) sin ( n Q x ) cos [ ( m 1 2 ) Q y ] ,
Φ ( n , m ) = q x = 0 N 1 q y = 0 M 1 Φ ̃ ( q x , q y ) cos [ ( n 1 2 ) Q x ] cos [ ( m 1 2 ) Q y ] ,
Φ ̃ ( q x , q y ) = T ( q x ) S ̃ x ( q x , q y ) + T ( q y ) S ̃ y ( q x , q y ) T 2 ( q x ) + T 2 ( q y ) , Φ ̃ ( 0 , 0 ) = 0
S ̃ x ( q x , q y ) = 4 2 δ ( q y ) N M n = 1 N 1 m = 1 M S x ( n ̑ , m ) sin ( n Q x ) cos [ ( m 1 2 ) Q y ] ,
S ̃ y ( q x , q y ) = 4 2 δ ( q x ) N M n = 1 N m = 1 M 1 S y ( n , m ̑ ) cos [ ( n 1 2 ) Q x ] sin ( m Q y ) ,
Φ ( n + 1 , m ) Φ ( n , m ) S x ( n + 1 2 , m ) = 1 2 [ U x ( n + 1 , m ) + U x ( n , m ) ] ,
Φ ( n , m + 1 ) Φ ( n , m ) S y ( n , m + 1 2 ) = 1 2 [ U y ( n , m + 1 ) + U y ( n , m ) ] .
S x ( n + 1 2 , m ) 13 24 [ U x ( n + 1 , m ) + U x ( n , m ) ] 1 24 [ U x ( n + 2 , m ) + U x ( n 1 , m ) ] ,
S y ( n , m + 1 2 ) 13 24 [ U y ( n , m + 1 ) + U y ( n , m ) ] 1 24 [ U y ( n , m + 2 ) + U y ( n , m 1 ) ] .
Φ ( x , y ) = q x q y Φ ̃ ( q x , q y ) cos [ ( x 1 2 ) Q x ] cos [ ( y 1 2 ) Q y ] ,
U x ( x , y ) = d Φ d x = q x = 1 N q y = 0 M 1 U ̃ x ( q x , q y ) sin [ ( x 1 2 ) Q x ] cos [ ( y 1 2 ) Q y ] ,
U y ( x , y ) = d Φ d y = q x = 0 N 1 q y = 1 M U ̃ y ( q x , q y ) cos [ ( x 1 2 ) Q x ] sin [ ( y 1 2 ) Q y ] .
Φ ̃ ( q x , q y ) = U ̃ x ( q x , q y ) Q x + U ̃ y ( q x , q y ) Q y Q x 2 + Q y 2 δ ( q x , q y ) ,
S ̃ x ( q x , q y ) = U ̃ x ( q x , q y ) cos ( Q x 2 ) ,
S ̃ y ( q x , q y ) = U ̃ y ( q x , q y ) cos ( Q y 2 ) .
Φ ̃ ( q x , q y ) = 1 4 U ̃ x ( q x , q y ) sin ( Q x ) + U ̃ y ( q x , q y ) sin ( Q y ) sin 2 ( Q x 2 ) + sin 2 ( Q y 2 ) δ ( q x , q y ) .
S ̃ x ( q x , q y ) = U ̃ x ( q x , q y ) [ 13 12 cos ( Q x 2 ) 1 12 cos ( 3 Q x 2 ) ] = U ̃ x ( q x , q y ) cos ( Q x 2 ) P ( Q x ) ,
Φ ̃ ( q x , q y ) = U ̃ x ( q x , q y ) sin ( Q x ) P ( Q x ) + U ̃ y ( q x , q y ) sin ( Q y ) P ( Q y ) 4 [ sin 2 ( Q x 2 ) + sin 2 ( Q y 2 ) ] δ ( q x , q y ) .
r 1 2 = 0.5 N 0.5 sin ( r h π k N ) sin ( r h π j N ) = N 2 δ ( k j ) ,
q = 1 N 1 sin ( π r h q N ) sin ( π t h q N ) = N 2 δ ( t h r h ) .
r 1 2 = 0.5 N 0.5 cos ( r h π k N ) cos ( r h π j N ) = N 2 δ ( k j ) [ 1 + δ ( k ) ] ,
q = 0 N 1 1 1 + δ ( q ) cos ( π r h q N ) cos ( π t h q N ) = N 2 δ ( t h r h ) .
r = 1 N 1 sin ( r π k N ) sin ( r π j N ) = N 2 δ ( k j ) ,
q = 1 N 1 sin ( π r q N ) sin ( π t q N ) = N 2 δ ( t r ) .
k j = r = 1 N 1 cos ( r π k N ) cos ( r π j N ) + 1 2 cos 0 cos 0 + 1 2 cos ( π k ) cos ( π j ) = N 2 δ ( k j ) [ 1 + δ ( k ) + δ ( k N ) ] .
R ̃ Ci ( q ) = r = 1 N R ( r ) cos ( π q r N ) = r = 1 N R ( r ) cos ( 2 π q r 2 N ) = R F ( q ) + R F ( 2 N q ) 2 ,
R F ( q ) = r = 1 N R ( r ) exp ( i 2 π q r 2 N ) + r = N + 1 2 N 0 * exp ( i 2 π q r 2 N ) ,
R ̃ Si ( q ) = r = 1 N R ( r ) sin ( π q r N ) = R F ( q ) R F ( 2 N q ) 2 i ,
R ̃ Ch ( q ) = n = 1 N R ( n ) cos [ π q ( n 1 2 ) N ] = R ̃ Ci ( q ) cos ( π q 2 N ) + R ̃ Si ( q ) sin ( π q 2 N ) ,
R ̃ Sh ( q ) = n = 1 N R ( n ) sin [ π q ( n 1 2 ) N ] = R ̃ Si ( q ) cos ( π q 2 N ) + R ̃ Ci ( q ) sin ( π q 2 N ) .
R Ch ( r ) = 1 N q = 0 N 1 R ̃ h ( q ) cos [ π q ( r 1 2 ) N ] = F ( r ) + F ( 2 N r + 1 ) 2 ,
R Sh ( r ) = 1 N q = 1 N 1 R ̃ h ( q ) sin [ π q ( r 1 2 ) N ] = F ( r ) F ( 2 N r + 1 ) 2 i ,
F ( r ) = 2 2 N q = 0 N 1 R ̃ h ( q ) exp ( i π q 2 N ) exp ( i 2 r π q 2 N ) = 2 F 1 [ R ̃ h ( q ) exp ( i π q 2 N ) ] ,

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