Abstract

Curvature sensors are used to measure wave-front aberrations in a number of different applications ranging from adaptive optics to optical testing. In practice, their performance is limited not only by the quality of the detector used for irradiance measurements but also by the separation between measurement planes used for the calculation of the axial derivative of intensity. This work resolves the problem of determining the separation between intensity measurement planes thus optimizing the variance in experimental measurements. To do this, the variance of the local curvature of the phase will be analyzed as a function of the noise level in the measurements and the separation between planes. Moreover, error bounds will be established for experimental measurements.

© 2003 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. F. Roddier, C. Roddier, N. Roddier, �??Curvature Sensing: a new wavefront sensing method,�?? Proc. Soc. Photo-Opt Instrum. Eng. 976, 203-209 (1988).
  3. C. Roddier, F. Roddier, �??Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,�?? J. Opt. Soc. Am. A 11, 2277-2287 (1993).
    [CrossRef]
  4. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, �??Quantitative optical phase microscopy,�?? Optics Letters, 11, 817-819 (1998).
    [CrossRef]
  5. A. Barty, K. A. Nugent, A. Roberts, D. Paganin, �??Quantitative phase tomography,�?? Opt. Commun. 175, 329-336 (2000).
    [CrossRef]
  6. G. Ganesh Chandan, R. M. Vasu, S. Asokan, �??Tomographic imaging of phase objects in turbid media through quantitative estimate of phase of ballistic light,�?? Opt. Commun. 191, 9-14 (2001).
    [CrossRef]
  7. J. A. Quiroga, J. A. Gómez-Pedrero, J. C. Martínez-Antón, �??Wavefront measurement by solving the irradiance transport equation for multifocal systems,�?? Opt. Eng. 40, 2885-2891 (2001).
    [CrossRef]
  8. M. Toyoda, K. Araki, Y. Suzuki, �??Wave-front tilt sensor with two cuadrant detectors and its application to a laser beam pointing system,�?? Appl. Opt. 41, 2219-2223 (2002).
    [CrossRef]
  9. P.J. Fox, T. R. Mackin, L. D. Turner, I. Colton, K. A. Nugent, R. E. Scholten, �??Noninterferometric phase imaging of a neutral atomic beam,�?? J. Opt. Soc. Am. B 8, 1773-1776 (2002) 1773.
    [CrossRef]
  10. M. R. Teague, �??Deterministic phase retrieval: a Green�??s function solution,�?? J. Opt. Soc. Am. A 1, 1434-1441 (1983).
  11. M. Milman, D. Redding, L. Needels, �??Analysis of curvature sensing for large-aperture adaptive optics systems,�?? J. Opt. Soc. Am. A 13, 1226-1238 (1996).
    [CrossRef]
  12. K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical methods for physics and engineering (University Press, Cambridge, 1997), Chap. 3.
  13. M. Born, E. Wolf, Principles of Optics (University Press, Cambridge, 1980), App. III.
  14. M. C. Roggemann, B. Welsh, Imaging through turbulence (CRC Press, Florida, 1996), Chap. 3.

Appl. Opt. (2)

J. Opt. Soc. Am. A (3)

C. Roddier, F. Roddier, �??Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,�?? J. Opt. Soc. Am. A 11, 2277-2287 (1993).
[CrossRef]

M. R. Teague, �??Deterministic phase retrieval: a Green�??s function solution,�?? J. Opt. Soc. Am. A 1, 1434-1441 (1983).

M. Milman, D. Redding, L. Needels, �??Analysis of curvature sensing for large-aperture adaptive optics systems,�?? J. Opt. Soc. Am. A 13, 1226-1238 (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, �??Quantitative phase tomography,�?? Opt. Commun. 175, 329-336 (2000).
[CrossRef]

G. Ganesh Chandan, R. M. Vasu, S. Asokan, �??Tomographic imaging of phase objects in turbid media through quantitative estimate of phase of ballistic light,�?? Opt. Commun. 191, 9-14 (2001).
[CrossRef]

Opt. Eng. (1)

J. A. Quiroga, J. A. Gómez-Pedrero, J. C. Martínez-Antón, �??Wavefront measurement by solving the irradiance transport equation for multifocal systems,�?? Opt. Eng. 40, 2885-2891 (2001).
[CrossRef]

Optics Letters (1)

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, �??Quantitative optical phase microscopy,�?? Optics Letters, 11, 817-819 (1998).
[CrossRef]

Proc. SPIE (1)

F. Roddier, C. Roddier, N. Roddier, �??Curvature Sensing: a new wavefront sensing method,�?? Proc. Soc. Photo-Opt Instrum. Eng. 976, 203-209 (1988).

Other (3)

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical methods for physics and engineering (University Press, Cambridge, 1997), Chap. 3.

M. Born, E. Wolf, Principles of Optics (University Press, Cambridge, 1980), App. III.

M. C. Roggemann, B. Welsh, Imaging through turbulence (CRC Press, Florida, 1996), Chap. 3.

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Figures (6)

Fig.1.
Fig.1.

Standard deviation of estimated axial derivatives within Ω versus measurement planes position for (a) cylindrical wave fronts, (b) elliptical wave fronts, (c) spherical wave fronts and (d) hyperbolic wave fronts. In all cases σ=0.01.

Fig. 2.
Fig. 2.

Error curves corresponding to four paraxial spherocylindrical phases with 2(|D|+|E|)=1. z opt denotes the position of the plane which minimizes the standard deviation of the estimated intensity derivative for the cylindrical wave front.

Fig. 3.
Fig. 3.

Evolution of the intensity along the z axis. The curves plotted show the behavior for combinations of D and E for which 2(|D|+|E|)=1.

Fig. 4.
Fig. 4.

Upper bound (solid line) and lower bound (dashed line) of error in the estimation of the axial derivative of the intensity for several values of the noise level σ.

Fig. 5.
Fig. 5.

Optimum positioning versus noise level plotted within the operational range of the curvature sensor. The points in blue are evaluated by finding the minimum value of Eq. (18) for intensity normalized to unity. In red the linear curve fit is represented.

Fig. 6.
Fig. 6.

Standard deviation of the estimated derivative (blue line) and upper error bound (red line) and lower error bounds (green line) predicted by the theory.

Equations (25)

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I 0 2 ϕ ( x , y ; z = 0 ) = [ I ( x , y ; z ) z ] z = 0
2 ϕ ( x , y ; z = 0 ) = [ I ( x , y ; z ) z ] z = 0 I 0 , I ( x , y ; z = 0 ) = I 0
i ( x , y ; z ) = I ( x , y ; z ) + n ( x , y ; z )
I ̂ ( x , y ; 0 ) = i ( x , y ; z ) i ( x , y ; z ) 2 z
I ̂ ¯ ( x , y ; 0 ) = I ( x , y ; 0 ) , s 2 [ I ̂ ( x , y ; 0 ) ] = I ̂ ( x , y ; 0 ) I ( x , y ; 0 ) 2 ¯ = min
I ( x , y ; ± z ) = I 0 ± I ( x , y ; 0 ) z + 1 2 I ( x , y ; ξ ± ( x , y ) ) z 2
I ̂ ( x , y ; 0 ) = n ( x , y ; z ) n ( x , y ; z ) 2 z + I ( x , y ; 0 )
+ 1 4 [ I ( x , y ; ξ + ( x , y ) ) I ( x , y ; ξ ( x , y ) ) ] z
s 2 [ I ̂ ( x , y ; 0 ) ] = σ 2 2 z 2 + 1 16 [ I ( x , y ; ξ + ( x , y ) ) I ( x , y ; ξ ( x , y ) ) ] 2 z 2
ϕ loc ( x , y ) = a ( x 0 , y 0 ) + b ( x 0 , y 0 ) ( x x 0 ) + c ( x 0 , y 0 ) ( y y 0 ) + d ( x 0 , y 0 ) ( x x 0 ) 2
+ e ( x 0 , y 0 ) ( y y 0 ) 2 + f ( x 0 , y 0 ) ( x x 0 ) ( y y 0 )
I ( x , y ; z ) = I 0 1 + z 2 ϕ ( x c , y c ) + z 2 H [ ϕ ( x c , y c ) ]
I loc ( x , y ; z ) = I 0 1 + 2 z ( d + e ) + z 2 ( 4 de f 2 )
I loc ( x , y ; ξ + ( x , y ) ) I loc ( x , y ; ξ ( x , y ) ) =
4 z [ I loc ( x , y , z ) I loc ( x , y , z ) 2 z I loc ( x , y ; 0 ) ] =
8 I 0 ( d + e ) z { 4 ( d + e ) 2 [ 2 + ( 4 de f 2 ) z 2 ] ( 4 de f 2 ) 4 ( d + e ) 2 z 2 [ 1 + ( 4 de f 2 ) z 2 ] 2 }
s loc 2 ( x , y ; z ) = s loc 2 ( z ) = σ 2 2 z 2 + 4 I 0 2 ( d + e ) 2 z 4 { 4 ( d + e ) 2 [ 2 + ( 4 de f 2 ) z 2 ] ( 4 de f 2 ) 4 ( d + e ) 2 z 2 [ 1 + ( 4 de f 2 ) z 2 ] 2 } 2
ϕ loc ( x , y ) = A + Bx + Cy + D x 2 + E y 2
s loc 2 ( z ) = σ 2 2 z 2 + 4 I 0 2 ( D + E ) 2 z 4 [ 4 ( D + E ) 2 ( 2 + 4 DE z 2 ) 4 DE 4 ( D + E ) 2 z 2 ( 1 + 4 DE z 2 ) 2 ] 2
σ 2 z opt s [ I ̂ ( x , y ; 0 ) ] V { σ 2 2 ( V z opt ) 2 + I 0 2 [ ( V z opt ) 2 ( V z opt ) 2 1 ] 2 }
s ( D = V 2 , E = 0 ) 2 ( z ) = V 2 { σ 2 2 ( Vz ) 2 + I 0 2 [ ( Vz ) 2 ( Vz ) 2 1 ] 2 }
s ( D = V 2 , E = 0 ) 2 ( h ) = V 2 { σ 2 2 h 2 + I 0 2 [ h 2 h 2 1 ] 2 }
s D = 0 , E = 0 ( h ) = V σ 2 h
h opt = α + β ln σ + γ ( ln σ ) 2
2 ϕ = 0 , ( 2 ϕ ) 2 = 8 3 0.0232 ( r 0 2 R ) 5 6 λ 2 π R 2

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