Abstract

The pyramid wavefront sensor is very similar to the Fourier knife-edge test, but employs dynamic modulation to quantify the phase derivative. For circular modulation, we compare approximate geometrical optics calculations, more exact diffraction calculations, and experimental results. We show that both the sinusoidal and the approximate linear relationship between wavefront derivative and wavefront sensor response can be derived rigorously from diffraction theory. We also show that geometrical, diffraction and experimental results are very similar, and conclude that the approximate geometrical predictions can be used in place of the more complex diffraction results.

© 2006 Optical Society of America

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References

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  1. R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
    [CrossRef]
  2. E. Gaviola, "On the quantitative use of the Foucault knife-edge test," J. Opt. Soc. Am. 26, 163-169 (1936).
    [CrossRef]
  3. E. H. Linfoot, Recent Advances in Optics (Oxford University Press, London, 1955).
  4. R. Ragazzoni and J. Farinato, "Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics," Astron. Astrophys. 350, L23-L26 (1999).
  5. S. Esposito and A. Riccardi, "Pyramid wavefront sensor behaviour in partial correction adaptive optics system," Astron. Astrophys. 369, L9-L12 (2001).
    [CrossRef]
  6. R. Ragazzoni, A. Diolaiti, and E. Vernet, "A pyramid wavefront sensor with no dynamic modulation," Opt. Commun. 208, 51-60 (2002).
    [CrossRef]
  7. J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, and J. Farinato, "Is there need of any modulation in the pyramid wavefront sensor," in Adaptive Optical System Technologies II, P. L.Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 288-298 (2003).
    [CrossRef]
  8. J. B. Costa, "Modulation effect of the atmosphere in a pyramid wavefront sensor," Appl. Opt. 44, 60-66 (2005).
    [PubMed]
  9. I. Iglesias, R. Ragazzoni, Y. Julien, and P. Artal, "Extended source pyramid wavefront sensor for the human eye," Opt. Express 10, 419-428 (2002).
    [PubMed]
  10. S. R. Chamot, C. Dainty, and S. Esposito, "Adaptive optics for ophthalmic applications using a pyramid wavefront sensor," Opt. Express 14, 518-526 (2006).
    [CrossRef] [PubMed]
  11. A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, "Laboratory characterization of a Foucaultlike wavefront sensor for adaptive optics," in Adaptive Optical System Technologies, R. K. Tyson, ed., Proc. SPIE 3353, 941-951 (1998).
    [CrossRef]
  12. O. Feeney, "Theory and Laboratory Characterisation of Novel Wavefront Sensor for Adaptive Optics Systems," PhD thesis, National University of Ireland, Galway (2001).
  13. S. Esposito, O. Feeney, and A. Riccardi, "Laboratory test of a pyramid wavefront sensor," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 416-422 (2000).
    [CrossRef]
  14. R. GaleWilson, "Wavefront-error evaluation by mathematical analysis of experimental Foucault-test data," Appl. Opt. 14, 2286-2297 (1975).
    [CrossRef] [PubMed]
  15. C. Vérinaud, "On the nature of the measurements provided by a pyramid wavefront sensor," Opt. Commun. 233, 27-38 (2004).
    [CrossRef]
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, sixth edition (Academic Press, San Diego, 2000).

2006 (1)

2005 (1)

2004 (1)

C. Vérinaud, "On the nature of the measurements provided by a pyramid wavefront sensor," Opt. Commun. 233, 27-38 (2004).
[CrossRef]

2002 (2)

R. Ragazzoni, A. Diolaiti, and E. Vernet, "A pyramid wavefront sensor with no dynamic modulation," Opt. Commun. 208, 51-60 (2002).
[CrossRef]

I. Iglesias, R. Ragazzoni, Y. Julien, and P. Artal, "Extended source pyramid wavefront sensor for the human eye," Opt. Express 10, 419-428 (2002).
[PubMed]

2001 (1)

S. Esposito and A. Riccardi, "Pyramid wavefront sensor behaviour in partial correction adaptive optics system," Astron. Astrophys. 369, L9-L12 (2001).
[CrossRef]

1999 (1)

R. Ragazzoni and J. Farinato, "Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics," Astron. Astrophys. 350, L23-L26 (1999).

1996 (1)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

1975 (1)

1936 (1)

Artal, P.

Chamot, S. R.

Costa, J. B.

Dainty, C.

Diolaiti, A.

R. Ragazzoni, A. Diolaiti, and E. Vernet, "A pyramid wavefront sensor with no dynamic modulation," Opt. Commun. 208, 51-60 (2002).
[CrossRef]

Esposito, S.

S. R. Chamot, C. Dainty, and S. Esposito, "Adaptive optics for ophthalmic applications using a pyramid wavefront sensor," Opt. Express 14, 518-526 (2006).
[CrossRef] [PubMed]

S. Esposito and A. Riccardi, "Pyramid wavefront sensor behaviour in partial correction adaptive optics system," Astron. Astrophys. 369, L9-L12 (2001).
[CrossRef]

Farinato, J.

R. Ragazzoni and J. Farinato, "Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics," Astron. Astrophys. 350, L23-L26 (1999).

Gaviola, E.

Iglesias, I.

Julien, Y.

Ragazzoni, R.

R. Ragazzoni, A. Diolaiti, and E. Vernet, "A pyramid wavefront sensor with no dynamic modulation," Opt. Commun. 208, 51-60 (2002).
[CrossRef]

I. Iglesias, R. Ragazzoni, Y. Julien, and P. Artal, "Extended source pyramid wavefront sensor for the human eye," Opt. Express 10, 419-428 (2002).
[PubMed]

R. Ragazzoni and J. Farinato, "Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics," Astron. Astrophys. 350, L23-L26 (1999).

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

Riccardi, A.

S. Esposito and A. Riccardi, "Pyramid wavefront sensor behaviour in partial correction adaptive optics system," Astron. Astrophys. 369, L9-L12 (2001).
[CrossRef]

Vérinaud, C.

C. Vérinaud, "On the nature of the measurements provided by a pyramid wavefront sensor," Opt. Commun. 233, 27-38 (2004).
[CrossRef]

Vernet, E.

R. Ragazzoni, A. Diolaiti, and E. Vernet, "A pyramid wavefront sensor with no dynamic modulation," Opt. Commun. 208, 51-60 (2002).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. (2)

R. Ragazzoni and J. Farinato, "Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics," Astron. Astrophys. 350, L23-L26 (1999).

S. Esposito and A. Riccardi, "Pyramid wavefront sensor behaviour in partial correction adaptive optics system," Astron. Astrophys. 369, L9-L12 (2001).
[CrossRef]

J. Mod. Opt. (1)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

C. Vérinaud, "On the nature of the measurements provided by a pyramid wavefront sensor," Opt. Commun. 233, 27-38 (2004).
[CrossRef]

R. Ragazzoni, A. Diolaiti, and E. Vernet, "A pyramid wavefront sensor with no dynamic modulation," Opt. Commun. 208, 51-60 (2002).
[CrossRef]

Opt. Express (2)

Other (6)

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, "Laboratory characterization of a Foucaultlike wavefront sensor for adaptive optics," in Adaptive Optical System Technologies, R. K. Tyson, ed., Proc. SPIE 3353, 941-951 (1998).
[CrossRef]

O. Feeney, "Theory and Laboratory Characterisation of Novel Wavefront Sensor for Adaptive Optics Systems," PhD thesis, National University of Ireland, Galway (2001).

S. Esposito, O. Feeney, and A. Riccardi, "Laboratory test of a pyramid wavefront sensor," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 416-422 (2000).
[CrossRef]

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, and J. Farinato, "Is there need of any modulation in the pyramid wavefront sensor," in Adaptive Optical System Technologies II, P. L.Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 288-298 (2003).
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, sixth edition (Academic Press, San Diego, 2000).

E. H. Linfoot, Recent Advances in Optics (Oxford University Press, London, 1955).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Principle of the pyramid wavefront sensor. Light from the pupil is split at the Fourier plane and re-imaged as four separate pupils (only two are shown in the image). (b) The beam is modulated, i.e., rotated around the prism center.

Fig. 2.
Fig. 2.

Values of 〈I 1-I 2〉 in the aperture, numerically obtained from the integral in Eq. (11). System parameters are λ=632.8nm, ξ 0=1.281mm, f=163mm, and R=1.64mm. The aberration is defocus as given by Eq. (19), where n ranges from -40 to 40 in steps of 5.

Fig. 3.
Fig. 3.

Pyramid response 〈I 1-I 2〉 as a function of the incident-phase derivative, according to the geometrical prediction (solid lines) and diffraction (dots) and assuming a sinusoidal modulation. System parameters are the same as in Fig. 2, except that the modulation amplitude is ξ 0=1.281mm (3V) and ξ 0=0.427mm (1V).

Fig. 4.
Fig. 4.

Same as Fig. 3, except Eq. (10) was used to calculate the diffraction result. For linear modulation the diffraction result and the geometrical approximation (not shown) are indistinguishable when placed on the same graph. The geometrical approximation with circular modulation (solid line) is therefore plotted here to allow easier comparison with Figs. 3 and 6.

Fig. 5.
Fig. 5.

Values of Sx in the aperture, obtained experimentally at 3 V modulation (ξ0=1.281mm). Each curve corresponds to one Badal stage position and consequently to a certain amount of defocus given by Eq. (20). The stage position b = 0, 1, …, 27 mm.

Fig. 6.
Fig. 6.

Pyramid response Sx as a function of the incident-phase derivative, according to the geometrical prediction (solid lines) and experiment (dots). System parameters are the same as in Fig. 2, except that the modulation amplitude is ξ 0=1.281mm (3V) and ξ 0=0.427mm (1V). Values of defocus for the geometrical curves are obtained from Eq. (20).

Equations (27)

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W ( x , y ) x = ξ 0 f sin ( π 2 S x ( x , y ) )
W ( x , y ) y = ξ 0 f sin ( π 2 S y ( x , y ) )
S x ( x , y ) = I a ( x , y ) I b ( x , y ) I c ( x , y ) + I d ( x , y ) I a ( x , y ) + I b ( x , y ) + I c ( x , y ) + I d ( x , y ) ,
S y ( x , y ) = I a ( x , y ) + I b ( x , y ) I c ( x , y ) I d ( x , y ) I a ( x , y ) + I b ( x , y ) + I c ( x , y ) + I d ( x , y ) .
W ( x , y ) x π ξ 0 2 f S x ( x , y ) ,
W ( x , y ) y π ξ 0 2 f S y ( x , y ) ,
W x ξ 0 f ,
U i ( x ) U 0 ( x ) h i ( x ) , i = 1 , 2
I 1 ( x ) I 2 ( x ) A 0 2 R R d x sin [ ϕ ( x ) ϕ ( x ) ] x x ,
I 1 ( x ) I 2 ( x ) A 0 2 R R d x sin [ φ ( x ) φ ( x ) ] x x sin [ a 0 ( x x ) ] a 0 ( x x ) .
I 1 ( x ) I 2 ( x ) A 0 2 R R d x sin [ φ ( x ) φ ( x ) ] x x J 0 [ a 0 ( x x ) ]
lim x x f ( x ) g ( x ) = lim x x f ( x ) g ( x ) .
I 1 I 2 A 0 2 φ x d x J 0 [ a 0 ( x x ) ] ,
I 1 I 2 A 0 2 ξ 0 φ x .
φ x π 2.4048 k ξ 0 f .
I 1 ( x ) I 2 ( x ) A 0 2 d z sin ( cz a 0 ) z J 0 ( z )
0 d x x J 0 ( x ) sin ( β x ) = { π 2 : β > 1 arcsin β : 1 < β < 1 π 2 : β < 1
I 1 ( x ) I 2 ( x ) A 0 2 arcsin ( f ξ 0 k φ x )
φ ( x ) = 2 π n x 2 R 2
C 2 0 = 0.0055 b 2 0.3883 b + 4.2474
sin [ ϕ ( x ) ϕ ( x ) ] = sin [ ϕ mod ( x ) + φ ( x ) ϕ mod ( x ) φ ( x ) ]
= sin [ φ ( x ) φ ( x ) ] cos [ ϕ mod ( x ) ϕ mod ( x ) ]
+ cos [ φ ( x ) φ ( x ) ] sin [ ϕ mod ( x ) ϕ mod ( x ) ]
S x 1 T R R d x sin [ φ ( x ) φ ( x ) ] x x T 2 T 2 d t cos [ a 0 ( x x ) sin ( 2 π T t ) ]
+ 1 T R R d x cos [ φ ( x ) φ ( x ) ] x x T 2 T 2 d t sin [ a 0 ( x x ) sin ( 2 π T t ) ]
J 0 ( x ) = 1 π 0 π d y cos ( x sin y ) ,
S x R R d x sin [ φ ( x ) φ ( x ) ] x x J 0 [ a 0 ( x x ) ]

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