Abstract

We present a fast and flexible non-interferometric method for the correction of small surface deviations on spatial light modulators, based on the Gerchberg-Saxton algorithm. The surface distortion information is extracted from the shape of a single optical vortex, which is created by the light modulator. The method can be implemented in optical tweezers systems for an optimization of trapping fields, or in an imaging system for an optimization of the point-spread-function of the entire image path.

© 2007 Optical Society of America

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References

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  1. E. R. Dufresne and D. G. Grier, "Optical tweezer arrays and optical substrates created with diffractive optics," Rev. Sci. Instrum. 69, 1974-1977 (1998).
    [CrossRef]
  2. S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral phase contrast imaging in microscopy," Opt. Express 13, 689-694 (2005).
    [CrossRef] [PubMed]
  3. K. D. Wulff, D. G. Cole, R. L. Clark, R. DiLeonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett, "Aberration correction in holographic optical tweezers," Opt. Express 14, 4169-4174 (2006).
    [CrossRef] [PubMed]
  4. R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237246 (1972).
  5. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  6. A. Muller, and A. Buffington, "Real-time correction of atmospherically degraded telescope images through image sharpening," J. Opt. Soc. Am. 64, 1200-1210 (1974).
    [CrossRef]
  7. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, "Synthesis of digital holograms by direct binary search," Appl. Opt. 26, 2788-2798 (1987).
    [CrossRef] [PubMed]
  8. J. R. Fienup, "Phase retrieval using boundary conditions," J. Opt. Soc. Am. A 3, 284-288 (1985).
    [CrossRef]
  9. S. Kirkpatrick, C. D. GelattJr., M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 4598 (1983).
    [CrossRef]
  10. S. W. Hell, and J. Wichmann, "Breaking the diffraction resolution limit by stimulated emission: stimulatedemission- depletion fluorescence microscopy," Opt. Lett. 19, 780-782 (1994).
    [CrossRef] [PubMed]

2006

2005

1998

E. R. Dufresne and D. G. Grier, "Optical tweezer arrays and optical substrates created with diffractive optics," Rev. Sci. Instrum. 69, 1974-1977 (1998).
[CrossRef]

1994

1987

1985

1983

S. Kirkpatrick, C. D. GelattJr., M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 4598 (1983).
[CrossRef]

1982

1974

1972

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237246 (1972).

Allebach, J. P.

Buffington, A.

Clark, R. L.

Cole, D. G.

Cooper, J.

DiLeonardo, R.

Dufresne, E. R.

E. R. Dufresne and D. G. Grier, "Optical tweezer arrays and optical substrates created with diffractive optics," Rev. Sci. Instrum. 69, 1974-1977 (1998).
[CrossRef]

Fienup, J. R.

Gelatt, C. D.

S. Kirkpatrick, C. D. GelattJr., M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 4598 (1983).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237246 (1972).

Gibson, G.

Grier, D. G.

E. R. Dufresne and D. G. Grier, "Optical tweezer arrays and optical substrates created with diffractive optics," Rev. Sci. Instrum. 69, 1974-1977 (1998).
[CrossRef]

Hell, S. W.

Kirkpatrick, S.

S. Kirkpatrick, C. D. GelattJr., M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 4598 (1983).
[CrossRef]

Leach, J.

Muller, A.

Padgett, M. J.

Saxton, W. O.

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237246 (1972).

Seldowitz, M. A.

Sweeney, D. W.

Vecchi, M. P.

S. Kirkpatrick, C. D. GelattJr., M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 4598 (1983).
[CrossRef]

Wichmann, J.

Wulff, K. D.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Optik

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237246 (1972).

Rev. Sci. Instrum.

E. R. Dufresne and D. G. Grier, "Optical tweezer arrays and optical substrates created with diffractive optics," Rev. Sci. Instrum. 69, 1974-1977 (1998).
[CrossRef]

Science

S. Kirkpatrick, C. D. GelattJr., M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 4598 (1983).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Sketch of the experimental setup. The off-axis vortex lens (superposed by a blazed grating) shown in the picture should create a focussed doughnut mode on the CCD. Due to surface deviations of the SLM, however the doughnut appears distorted. A single image of the distorted doughnut enables an iterative “phase retrieval” algorithm to find the corresponding phase function in the SLM plane.

Fig. 2.
Fig. 2.

GS algorithm for finding phase errors. The starting pattern consists of a uniform intensity distribution, limited by a circular aperture that defines the numerical aperture of the imaging system, and a phase vortex (gray scales correspond to phase values). After a few iterative cycles, the error function ε converges. H(x,y) then corresponds to a phase pattern that would produce the actually observed distorted doughnut image, if displayed on a perfect SLM. It represents a perfect vortex superposed by phase errors. By subtracting the starting pattern, the phase errors are extracted.

Fig. 3.
Fig. 3.

Numerical simulation assuming a distorted SLM surface. (A) Phase image of the simulated SLM surface distortion, extending over a range of about 4 rad. (B) shows the intensity of an accordingly disturbed optical vortex in the far field. After 30 iterations, the SLM surface has been reproduced accurately. (C) shows the remaining deviation. Note that compared to (A) the scaling has changed to milliradians. (D) Optical vortex after correction.

Fig. 4.
Fig. 4.

Experimental results, achieved with the setup of Fig. 1: the small images in (A) show doughnut modes prior (first column) and after (second and third column) correction patterns have been added to the hologram function. The corresponding experimentally measured point spread functions are plotted below. In this experiment, the GS algorithm was applied two times: after it converged for the first time, the resulting correction pattern was displayed on the SLM. Then, a second run of the algorithm was performed on the base of the now less distorted doughnut image (second column). Finally, displaying the sum of both correction functions led to a further improved doughnut quality (third column). In the next step an additional distortion in the optical path was introduced by inserting a tilted glass plate in the beam path (see Fig. 1). Also this distortion could be compensated by an additional run of the optimization algorithm (fifth column). The two images in the darker box on the right show the ideal intensity distributions as they were produced by a perfectly flat SLM. The colored patterns in (B) represent the corresponding correction functions.

Fig. 5.
Fig. 5.

Optimization of a simple imaging path. (A) Sketch of the setup. Just the first diffraction order is shown. Lens 2 images the resolution target on the CCD, when the SLM is used as a mirror. (B) Images of the resolution target and a doughnut mode, before (upper image) and after (lower image) SLM correction.

Equations (4)

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A ( k x , k y ) exp [ i Φ ( k x , k y ) ] = F { exp [ i H ( x , y ) ] } .
H ( x , y ) = modulo 2 π [ L arg ( x + i y ) + k x + π λ f ( x 2 + y 2 ) ] ,
A ( x , y ) = F { exp [ i H ( x , y ) + i D ( x , y ) ] } ,
A = M Holo M Array Δ k 2 π Δ ,

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