Abstract

In this paper we determine the optimum propagation distance between measurement planes and the plane of the lens in a wavefront curvature sensor with the diffraction optics approach. From the diffraction viewpoint, the measured wavefront aberration can be decomposed into Fourier harmonics at various frequencies. The curvature signal produced by a single harmonic is analyzed with the wave propagation transfer function approach, which is the frequency analysis of wavefront curvature sensing. The intensity of the curvature signal is a sine function of the product of the propagation distance and the squared frequency. To maximize the curvature signal, the optimum propagation distance is proposed as one quarter of the Talbot length at the critical frequency (average power point at which the power spectrum density is the average power spectrum density). Following the determination of the propagation distance, the intensity of the curvature signal varies sinusoidally with the squared frequencies, vanishing at some higher frequency bands just like a comb filter. To cover these insensitive bands, wavefront curvature sensing with dual propagation distances or with multi-propagation distances is proposed.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

2007

2004

2003

M. Soto, E. Acosta, and S. Ríos, "Performance analysis of curvature sensors: optimum positioning of the measurement planes," Opt. Express 11, 2577-2588 (2003).
[CrossRef] [PubMed]

S. Woods and A. Greenaway, "Wavefront sensing by use of a Green’s function solution to the intensity transport equation," J. Opt. Soc. Am. A. 20, 508-512 (2003).
[CrossRef]

2002

2000

1999

1996

T. Gureyev and K. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A. 13, 1670-1682 (1996).
[CrossRef]

1994

D. Johnston, B. Ellerbroek, and S. Pompeat, "Curvature sensing analysis," SPIE 2201, 528-538 (1994).
[CrossRef]

1992

G. Yang, B. Gu, and B. Dong, "Theory of the amplitude-phase retrieval in an any linear transform system and its applications," SPIE 1767, 457-478 (1992).
[CrossRef]

V. Arrizόn and J. Ojeda-Castaňeda, "Irradiance at Fresnel planes of a phase grating," J. Opt. Soc. Am. A 9, 1801-1806 (1992).
[CrossRef]

1991

J. Graves and D. McKenna, "University of Hawaii adaptive optics system: III. Wavefront curvature sensor," SPIE 1542, 262-272 (1991).
[CrossRef]

1990

1988

1983

1972

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

1960

J. Cowley and A. Moodie, "Fourier images IV: the phase grating," Proc. Phys. Soc. London Sect. B 76, 378-384 (1960).

Acosta, E.

Blain, C.

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

Blanchard, P.

Cortés-Martínez, R.

Cowley, J.

J. Cowley and A. Moodie, "Fourier images IV: the phase grating," Proc. Phys. Soc. London Sect. B 76, 378-384 (1960).

Dong, B.

G. Yang, B. Gu, and B. Dong, "Theory of the amplitude-phase retrieval in an any linear transform system and its applications," SPIE 1767, 457-478 (1992).
[CrossRef]

Ellerbroek, B.

D. Johnston, B. Ellerbroek, and S. Pompeat, "Curvature sensing analysis," SPIE 2201, 528-538 (1994).
[CrossRef]

Fisher, D.

Geng, Y.

Gerchberg, R.

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Graves, J.

J. Graves and D. McKenna, "University of Hawaii adaptive optics system: III. Wavefront curvature sensor," SPIE 1542, 262-272 (1991).
[CrossRef]

Greenaway, A.

S. Woods and A. Greenaway, "Wavefront sensing by use of a Green’s function solution to the intensity transport equation," J. Opt. Soc. Am. A. 20, 508-512 (2003).
[CrossRef]

P. Blanchard and A. Greenaway, "Simultaneous multiplane imaging with a distorted diffraction grating," Appl. Opt. 38, 6692-6699 (1999).
[CrossRef]

Greenaway, A. H.

Gu, B.

G. Yang, B. Gu, and B. Dong, "Theory of the amplitude-phase retrieval in an any linear transform system and its applications," SPIE 1767, 457-478 (1992).
[CrossRef]

Gureyev, T.

T. Gureyev and K. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A. 13, 1670-1682 (1996).
[CrossRef]

Guyon, O.

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

O. Guyon, "High-performance curvature wavefront sensing for extreme AO," SPIE 6691, 66910G (2007).
[CrossRef]

Hand, D. P.

Hattori, M.

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

Hayano, Y.

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

Jiang, Z.

Johnston, D.

D. Johnston, B. Ellerbroek, and S. Pompeat, "Curvature sensing analysis," SPIE 2201, 528-538 (1994).
[CrossRef]

Lambert, R. W.

Lane, R.

McKenna, D.

J. Graves and D. McKenna, "University of Hawaii adaptive optics system: III. Wavefront curvature sensor," SPIE 1542, 262-272 (1991).
[CrossRef]

Moodie, A.

J. Cowley and A. Moodie, "Fourier images IV: the phase grating," Proc. Phys. Soc. London Sect. B 76, 378-384 (1960).

Nugent, K.

T. Gureyev and K. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A. 13, 1670-1682 (1996).
[CrossRef]

Pompeat, S.

D. Johnston, B. Ellerbroek, and S. Pompeat, "Curvature sensing analysis," SPIE 2201, 528-538 (1994).
[CrossRef]

Ríos, S.

Roddier, F.

Saxton, W.

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Shephard, J. D.

Soto, M.

Taghizadeh, M. R.

Takami, H.

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

Teague, M.

van Dam, M.

Waddie, A. J.

Watanabe, M.

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

Woods, S.

S. Woods and A. Greenaway, "Wavefront sensing by use of a Green’s function solution to the intensity transport equation," J. Opt. Soc. Am. A. 20, 508-512 (2003).
[CrossRef]

Woods, S. C.

Xi, F.

Xu, X.

Yang, G.

G. Yang, B. Gu, and B. Dong, "Theory of the amplitude-phase retrieval in an any linear transform system and its applications," SPIE 1767, 457-478 (1992).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

T. Gureyev and K. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A. 13, 1670-1682 (1996).
[CrossRef]

S. Woods and A. Greenaway, "Wavefront sensing by use of a Green’s function solution to the intensity transport equation," J. Opt. Soc. Am. A. 20, 508-512 (2003).
[CrossRef]

Opt. Express

Optik

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

PASP

O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP.  120, 655-664 (2008).
[CrossRef]

Proc. Phys. Soc. London Sect. B

J. Cowley and A. Moodie, "Fourier images IV: the phase grating," Proc. Phys. Soc. London Sect. B 76, 378-384 (1960).

SPIE

D. Johnston, B. Ellerbroek, and S. Pompeat, "Curvature sensing analysis," SPIE 2201, 528-538 (1994).
[CrossRef]

O. Guyon, "High-performance curvature wavefront sensing for extreme AO," SPIE 6691, 66910G (2007).
[CrossRef]

G. Yang, B. Gu, and B. Dong, "Theory of the amplitude-phase retrieval in an any linear transform system and its applications," SPIE 1767, 457-478 (1992).
[CrossRef]

J. Graves and D. McKenna, "University of Hawaii adaptive optics system: III. Wavefront curvature sensor," SPIE 1542, 262-272 (1991).
[CrossRef]

Other

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999), Chap. 5.
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of curvature sensing; two defocus planes M 1 and M 2 correspond to measurement planes P 1 and P 2, respectively.

Fig. 2.
Fig. 2.

Comparison of the coef icients of the first 5 harmonic components.

Fig. 3.
Fig. 3.

Filter model of curvature sensing shown in Eq. (14).

Fig. 4.
Fig. 4.

Covering effect of dual-z or multi-z curvature sensing.

Equations (21)

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I P 1 ( φ ) 1 z k 2 φ
I P 2 ( φ ) 1 + z k 2 φ ,
φ n ( ξ ) = B n sin ( n 2 π f 0 ξ + θ n ) ,
2 B = B ( 2 π f ) 2 sin ( 2 π f ξ θ )
= ( 2 π f ) 2 B .
I P 1 ( B ) 1 + z k ( 2 π f ) 2 B
I P 2 ( B ) 1 z k ( 2 π f ) 2 B .
U 0 ( ξ , f ) = exp [ j B sin ( 2 π f ξ ) ] = q = J q ( B ) exp ( j 2 π q f ξ ) ,
U ( x , z , f ) = q = J q ( B ) exp ( j π λ z q 2 f 2 ) exp ( j 2 π q f x ) ,
I ( x , z , f ) = 1 + 4 J 0 ( B ) J 1 ( B ) sin ( π λ z f 2 ) sin ( 2 π f x ) + else ,
I ( x , ( 2 n 1 ) Z T / 4 , f ) = 1 + sin [ 2 B sin ( 2 π f x ) ]
1 + 2 B sin ( 2 π f x ) i f B 1
I ( x , z , f ) 1 + 2 B sin ( π λ z f 2 ) sin ( 2 π f x ) i f B 1 .
I ( x , z , f ) 1 2 B sin ( π λ z f 2 ) sin ( 2 π f x ) i f B 1 .
S ( x , z , f ) = I ( x , z , f ) I ( x , z , f ) I ( x , z , f ) + I ( x , z , f )
2 B sin ( π λ z f 2 ) sin ( 2 π f x ) .
S ( x , ( 2 n 1 ) Z T / 4 ) ± 2 B sin ( 2 π f x ) .
z opt ( f ) = 1 2 λ f 2 .
S ( x , f ) 2 B sin [ π 2 ( f f k ) 2 ] sin ( 2 π f x ) .
z k 2 = 1 2 λ ( 2 f k ) 2 = 1 2 z k .
z k 3 = 1 4 z k .

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