Abstract

We present a wave front sensorless adaptive optics scheme for an incoherent imaging system. Aberration correction is performed through the optimisation of an image quality metric based upon the low spatial frequency content of the image. A sequence of images is acquired, each with a different aberration bias applied and the correction aberration is estimated from the information in this image sequence. It is shown, by representing aberrations as an expansion in Lukosz modes, that the effects of different modes can be separated. The optimisation of each mode becomes independent and can be performed as the maximisation of a quadratic function, requiring only three image measurements per mode. This efficient correction scheme is demonstrated experimentally in an incoherent transmission microscope. We show that the sensitivity to different aberration magnitudes can be tuned by changing the range of spatial frequencies used in the metric. We also explain how the optimisation scheme is related to other methods that use image sharpness metrics.

© 2007 Optical Society of America

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References

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2007

2006

2005

M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, "Methods for the characterisation of deformable membrane mirrors," Appl. Opt. 44, 5131-5139 (2005).
[CrossRef] [PubMed]

L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," in Proc. S.P.I.E., ‘Opto-Ireland 2005: Imaging and Vision’  5823, 40-47 (2005).Q2

2003

2002

D. R. Luke, J. V. Burke and R. G. Lyon, "Optical wavefront reconstruction: theory and numerical methods," SIAM Review 44, 169-224 (2002).Q1
[CrossRef]

2000

1997

1996

1987

1982

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Engineering 21, 829-832 (1982).

1977

1974

1963

W. Lukosz, "Der Einfluß der Aberrationen auf die optische Übertragungsfunktion bei kleinen Orts-Frequenzen," Optica Acta 10, 1-19 (1963).
[CrossRef]

Appl. Opt.

Imaging and Vision

L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," in Proc. S.P.I.E., ‘Opto-Ireland 2005: Imaging and Vision’  5823, 40-47 (2005).Q2

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Engineering

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Engineering 21, 829-832 (1982).

Opt. Express

Opt. Lett.

Optica Acta

W. Lukosz, "Der Einfluß der Aberrationen auf die optische Übertragungsfunktion bei kleinen Orts-Frequenzen," Optica Acta 10, 1-19 (1963).
[CrossRef]

SIAM Review

D. R. Luke, J. V. Burke and R. G. Lyon, "Optical wavefront reconstruction: theory and numerical methods," SIAM Review 44, 169-224 (2002).Q1
[CrossRef]

Other

R. K. Tyson, Principles of Adaptive Optics, Academic Press, London, 1991.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford University Press, 1998).

N. Doble, Image Sharpness Metrics and Search Strategies for Indirect Adaptive Optics. PhD thesis, University of Durham, United Kingdom, 2000.

V. N. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics, (SPIE, Bellingham, WA, 2001).

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, (Cambridge University Press, 2nd ed., 1992).

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Figures (7)

Fig. 1.
Fig. 1.

The calculation of the incoherent optical transfer function: (a) the circular pupil P with circumference C; (b) The geometry used in the autocorrelation calculation, showing the pupil overlap A; (c) The resulting aberration-free incoherent optical transfer function.

Fig. 2.
Fig. 2.

(a) Schematic diagram of the experimental apparatus; (b) Raw image of scatterer without aberrations; (c) Spectral density of the scatterer image (log scale) with M 1, M 2 and incoherent cut-off frequencies marked. The horizontal and vertical lines at the edge of this image are FT artefacts arising from the sharp image boundaries.

Fig. 3.
Fig. 3.

Experimental measurement of the optimisation metric: (a) Variation of g with aberration magnitude a = |a| for the different frequency ranges given by the figures in parentheses (M 1,M 2). The solid lines are Lorentzian fits to the mean values; (b) The measured and calculated half width of g(a) for different frequency ranges. The theoretical curve is valid for small frequencies.

Fig. 4.
Fig. 4.

Correction of a single Lukosz aberration mode (astigmatism, i = 5) using the scatterer specimen with M 1 = 0.06 and M 2 = 0.4. The first row shows the raw images and the second row contains the corresponding spectral densities. The third row illustrates schematically the sampling of the Lorentzian curve used in the optimisation calculation. The diagrams correspond to: (a1-a3) initial aberration of magnitude a 5 = -4.9; (b1-b3) additional negative bias -b = -11.5 applied; (c1-c3) additional positive bias b = 11.5 applied; (d1-d3) correction applied.

Fig. 5.
Fig. 5.

Correction of multiple Lukosz aberration modes showing images (upper row) and spectral densities (lower row). The initial aberration was a random combination of eight modes, i = 4 to 11, with overall amplitude a = 11.5. For the correction procedure M 1 = 0.06, M 2 = 0.4, and the bias b = 9.8. The images show: (a) Scatterer, initial aberration; (b) Scatterer, corrected; (c) USAF test chart, initial aberration; (d) USAF test chart, corrected. The values of ϕ rms show the root mean square phase aberration in radians.

Fig. 6.
Fig. 6.

Correction accuracy for the correction of eight Lukosz modes (i = 4 to 11) using different frequency ranges A-F (see Table 1). The upper graphs correspond to bias b = 4.9 whereas in the lower graphs b = 1.6. (a1,a2) show the mean correction error ε; (b1,b2) show the mean Strehl ratio S.

Fig. 7.
Fig. 7.

The effect of additional modes on the correction procedure using M 1 = 0.06, M 2 = 0.4, and b = 4.9.

Tables (2)

Tables Icon

Table 1. Spatial frequency ranges for the results of Fig. 6.

Tables Icon

Table 2. Zernike and Lukosz mode definitions

Equations (46)

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I ( x ) = t ( x ) * h ( x ) .
J ( m ) = H ( m ) T ( m ) ,
S J ( m ) = H ( m ) 2 S T ( m ) ,
H ( m ) = P ( r ) P * ( r ) = P ( r m ) P * ( r ) d A ,
H 0 ( m ) = 1 π ( r m ) ( r ) d A = 2 π [ cos 1 ( m 2 ) m 2 1 ( m 2 ) 2 ] ,
H ( m ) = 1 π ( r m ) ( r ) exp j [ Φ ( r m ) Φ ( r ) ] d A .
H ( m ) 1 π ( r m ) ( r ) exp j [ m Φ + O ( m 2 ) ] d A ,
H ( m ) 1 π ( r m ) ( r ) d A j π ( r m ) ( r ) ( m Φ ) d A 1 2 π ( r m ) ( r ) ( m Φ ) 2 d A + O ( m 2 ) .
H ( m ) H 0 ( m ) j π P ( m Φ ) d A 1 2 π P ( m Φ ) 2 d A + O ( m 2 ) .
H ( m ) 2 H 0 ( m ) 2 1 π P ( m Φ ) 2 d A + [ 1 π P ( m Φ ) d A ] 2 ,
H ( m ) 2 H 0 ( m ) 2 1 π P ( m Φ ) 2 d A .
S J ( m ) [ H 0 ( m ) 2 1 π P ( m Φ ) 2 d A ] S T ( m ) .
g ( M 1 , M 2 ) = ξ = 0 2 π m = M 1 M 2 S J ( m ) m d m d ξ
m = M 1 M 2 { H 0 ( m ) 2 ξ = 0 2 π S T ( m ) d ξ 1 π ξ = 0 2 π S T ( m ) [ P ( m Φ ) 2 d A ] d ξ } m d m .
S T ( m ) = α 0 ( m ) 2 + i = 1 [ α i ( m ) cos ( 2 ) + β i ( m ) sin ( 2 ) ] .
ξ = 0 2 π S T ( m ) d ξ = π α 0 ( m ) .
P ( m Φ ) 2 d A = m 2 2 P Φ 2 [ 1 + cos ( 2 ξ 2 χ ) ] d A ,
ξ = 0 2 π S T ( m ) [ P ( m Φ ) 2 d A ] d ξ = π m 2 2 P Φ 2 [ α 0 ( m ) + α 1 ( m ) cos ( 2 χ ) + β 1 ( m ) sin ( 2 χ ) ] d A .
g ( M 1 , M 2 ) q 0 ( M 1 , M 2 ) q 1 ( M 1 , M 2 ) 1 π P Φ 2 d A ,
q 0 ( M 1 , M 2 ) = π m = M 1 M 2 H 0 ( m ) 2 α 0 ( m ) m d m
q 1 ( M 1 , M 2 ) = π 2 m = M 1 M 2 α 0 ( m ) m 3 d m
I 1 = 1 π P Φ 2 d A .
L n m ( r , θ ) = B n m ( r ) × { cos ( ) m 0 sin ( ) m < 0
B n m ( r ) = { 1 2 n [ R n 0 ( r ) R n 2 0 ( r ) ] n m = 0 1 2 n [ R n m ( r ) R n 2 m ( r ) ] n m 0 1 n R n n ( r ) m = n 0 1 m = n = 0
R n m ( r ) = k = 0 n m 2 ( 1 ) k ( n k ) ! r n 2 k k ! ( n + m 2 k ) ! ( n m 2 k ) ! .
Φ ( r ) = i = 4 N + 3 a i L i ( r , θ ) ,
I 1 = 1 π p Φ 2 d A = i = 4 N + 3 a i 2 .
g ( { a i } ) q 0 q 1 i = 4 N + 3 a i 2 ,
g ( { a i } ) 1 q 2 + q 3 i = 4 N + 3 a i 2 ,
c = B D a ,
G ( { a i } ) = g ( { a i } ) 1 q 2 + q 3 i = 4 N + 3 a i 2 ,
G ( a k ) q 2 + q 3 a k 2 ,
q 2 = q 2 + q 3 i k a i 2
a corr = b ( G + G ) 2 G + 4 G 0 + 2 G ,
ε = a err = a in + a corr .
S = exp ( ϕ rms 2 ) = exp ( D a err 2 ) .
σ = I ( x ) 2 d x d y .
σ = ξ = 0 2 π m = 0 2 S J ( m ) m d m d ξ
I 2 = 1 π P ( m Φ ) d A .
I 2 = 1 π m P ( Φ ) d A = 1 π m C Φ n d c ,
I 2 = 1 π m θ = 0 2 π ϕ ( θ ) cos ( ξ θ ) d θ .
ϕ ( θ ) = μ 0 2 + i = 1 [ μ i cos ( ) + v i sin ( ) ] ,
I 2 = m [ μ 1 cos ( ξ ) + v 1 sin ( ξ ) ] .
z n m = { 1 2 n ( n + 1 ) a n m 1 2 ( n + 1 ) ( n + 2 ) a n + 2 m m n 1 2 n ( n + 1 ) a n n m = n 0 a 0 0 m = n = 0
ϕ rms = Da .
ρ rms = λ 2 πNA a .

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