Abstract

A wavefront sensing technique is proposed based on the principle of aberration-mode filtering and detection. The mathematical foundation of the method is provided by a series of orthogonal and binary functions, for the optical aperture, derived from the Walsh series. It is shown that the expansion of a wavefront using these basis functions is explicitly related to the expansion of the optical field itself on the same basis. This permits the determination of the coefficients associated with the binary aberration modes through simple intensity measurements with the help of a phase-only spatial light modulator and a single-mode optical fiber. These coefficients can be independently acquired in sets that characterize wavefronts in various spatial resolutions. A numerical simulation and practical implementation of the technique are also discussed.

© 2009 Optical Society of America

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References

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  1. G. Rousset, “Wavefront sensing,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University, 1999).
  2. Adaptive Optics for Vision Science, J. Porter, H. Queener, J. Lin, K. Thorn, and A. Awwal, eds. (Wiley-Interscience, 2006).
    [Crossref]
  3. J. Liang, B. Grimm, S. Goelz, and J. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shark wave-front sensor,” J. Opt. Soc. Am. A 11, 1949-1957 (1994).
    [Crossref]
  4. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223-1225 (1988).
    [Crossref] [PubMed]
  5. J. Schwider, “Continuous lateral shearing interferometer,” Appl. Opt. 23, 4403-4409 (1984).
    [Crossref] [PubMed]
  6. R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  8. F. Wang, “Binary phase masking for optical interrogation of matters in turbid media,” Opt. Lett. 33, 2587-2589(2008).
    [Crossref] [PubMed]
  9. K. G. Beauchamp, Walsh Functions and Their Applications (Academic Press, 1975).
  10. E. G. Neumann, Single-Mode Fibers: Fundamentals, T. Tamir, ed., Springer Series in Optical Sciences (Springer-Verlag, 1988).

2008 (1)

1994 (1)

1988 (1)

1984 (1)

1979 (1)

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

Beauchamp, K. G.

K. G. Beauchamp, Walsh Functions and Their Applications (Academic Press, 1975).

Bille, J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Chidlaw, R.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

Goelz, S.

Gonsalves, R. A.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

Grimm, B.

Liang, J.

Neumann, E. G.

E. G. Neumann, Single-Mode Fibers: Fundamentals, T. Tamir, ed., Springer Series in Optical Sciences (Springer-Verlag, 1988).

Roddier, F.

Rousset, G.

G. Rousset, “Wavefront sensing,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University, 1999).

Schwider, J.

Wang, F.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Appl. Opt. (2)

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

Opt. Lett. (1)

Proc. SPIE (1)

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

K. G. Beauchamp, Walsh Functions and Their Applications (Academic Press, 1975).

E. G. Neumann, Single-Mode Fibers: Fundamentals, T. Tamir, ed., Springer Series in Optical Sciences (Springer-Verlag, 1988).

G. Rousset, “Wavefront sensing,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University, 1999).

Adaptive Optics for Vision Science, J. Porter, H. Queener, J. Lin, K. Thorn, and A. Awwal, eds. (Wiley-Interscience, 2006).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Mode-field fitted Walsh functions for the polar coordinate system. These functions have only two values, - 1 and 1. The shaded areas are assigned the value of 1; the clear areas are assigned the value of - 1 .

Fig. 2
Fig. 2

Optical arrangement for wavefront sensing using the proposed method.

Fig. 3
Fig. 3

Simulated wavefronts and the resultant intensity distributions in the focal plane. (a) synthesized incoming wavefront; (b) corrected wavefront; (c) reference intensity distribution for plane incoming wave; (d) intensity distribution formed by the synthesized wavefront; and (e) intensity distribution formed by the corrected wavefront.

Equations (12)

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a ρ < b e ρ 2 σ 2 ρ · d ρ d θ ,
A e ρ 2 σ 2 W l ( ρ , θ ) · W l ( ρ , θ ) · ρ d ρ d θ = C δ l , l ,
i | A e ρ 2 σ 2 · E * ( ρ , θ ) · ρ d ρ d θ | 2 ,
E * ( ρ , θ ) = E o * e j l a l W l ,
e j a l W l = cos ( a l W l ) + j sin ( a l W l ) = cos ( a l ) + j sin ( a l ) W l .
E * ( ρ , θ ) = E o * l [ cos ( a l ) + j sin ( a l ) W l ] = E o * l B l ( a 1 , a 2 , a 3 , ) W l .
{ i 0 = D | B 0 ( a 1 , a 2 , a 3 , ) | 2 i 1 = D | B 0 ( a 1 ξ , a 2 , a 3 , ) | 2 i 2 = D | B 0 ( a 1 , a 2 ξ , a 3 , ) | 2 i N = D | B 0 ( a 1 , a 2 , a 3 , , a N ξ ) | 2 .
E * ( ρ , θ ) = E o * e j 3 l = 0 a l W l = E o * l = 0 3 [ cos ( a l ) + j sin ( a l ) W l ] .
E * ( ρ , θ ) = E o * { [ cos ( a 1 ) cos ( a 2 ) cos ( a 3 ) + j sin ( a 1 ) sin ( a 2 ) sin ( a 3 ) ] W 0 , [ cos ( a 1 ) sin ( a 2 ) sin ( a 3 ) j sin ( a 1 ) cos ( a 2 ) cos ( a 3 ) ] W 1 , [ sin ( a 1 ) cos ( a 2 ) sin ( a 3 ) j cos ( a 1 ) sin ( a 2 ) cos ( a 3 ) ] W 2 , [ sin ( a 1 ) sin ( a 2 ) cos ( a 3 ) j cos ( a 1 ) cos ( a 2 ) sin ( a 3 ) ] W 3 } .
i 0 = D { [ cos ( a 1 ) cos ( a 2 ) cos ( a 3 ) ] 2 + [ sin ( a 1 ) sin ( a 2 ) sin ( a 3 ) ] 2 } ; i 1 = D { [ cos ( a 1 ξ ) cos ( a 2 ) cos ( a 3 ) ] 2 + [ sin ( a 1 ξ ) sin ( a 2 ) sin ( a 3 ) ] 2 } ; i 2 = D { [ cos ( a 1 ) cos ( a 2 ξ ) cos ( a 3 ) ] 2 + [ sin ( a 1 ) sin ( a 2 ξ ) sin ( a 3 ) ] 2 } ; i 3 = D { [ cos ( a 1 ) cos ( a 2 ) cos ( a 3 ξ ) ] 2 + [ sin ( a 1 ) sin ( a 2 ) sin ( a 3 ξ ) ] 2 } .
Z 2 0 = 3 ( 2 ρ 2 1 ) ; Z 2 2 = 6 ρ 2 cos ( 2 θ ) ; Z c = Z 2 0 · Z 2 2 .
Φ M ( x , y ) = 0.268 W 4 + 0.598 W 5 0.166 W 9 .

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