## Wavefront correction of spatial light modulators using an optical vortex image

Optics Express, Vol. 15, Issue 9, pp. 5801-5808 (2007)

http://dx.doi.org/10.1364/OE.15.005801

Acrobat PDF (1195 KB)

### 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

## 1. Introduction

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. Fürhapter, 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]

*determine*the phase errors just from the distorted shape of a focussed doughnut mode. The basic idea is that we use a phase retrieval algorithm to find the hologram that would produce the observed distorted doughnut if displayed on a perfectly flat SLM and imaged with “perfect” optics. Consequently, the acquired hologram will include the information about imperfections like the surface deviation of our “real” SLM or phase errors of the additional imaging optics. Once the phase errors are known, a corrective phase pattern can be calculated, which can be superposed to each subsequently produced phase pattern to compensate for any phase errors of the SLM or the optical pathway.

*L*= 1 and focusses it on a CCD. The vortex lens is superposed by a blazed grating in order to spatially separate the optical vortex generated in the first diffraction order from other orders. According to surface aberrations of the light modulator, the doughnut appears distorted. A CCD image of this doughnut is taken as input for the Gerchberg-Saxton (GS) algorithm [4] – an iterative phase retrieval algorithm – which is able to find the corresponding phase hologram that would produce the observed intensity pattern, if it were displayed on a non-distorted SLM. The acquired hologram will then consist of an ideal phase vortex pattern overlaid by the phase errors that are retrieved with this method.

*every*error can be completely compensated by a phase-only modulator).

*H*(

*x*,

*y*) of a light field by just knowing the modulus

*A*(

*k*,

_{x}*k*) of its Fourier transform:

_{y}*H*(

*x*,

*y*) corresponds to the hologram function, and

*A*(

*k*,

_{x}*k*) to the amplitude of the observed doughnut mode. The problem is of relevance in many fields of research, for instance electron microscopy or astronomy, which led to the development of various techniques to solve this problem [5

_{y}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]

*A*(

*k*,

_{x}*k*). The deviation can be quantified by the root-mean-square deviation as an error function

_{y}*ε*. Since

*ε*depends on the phase value of every single pixel of

*H*(

*x*,

*y*), it can be understood as a scalar field in a

*M*×

*N*dimensional space where each pixel of the phase pattern

*H*(

*x*,

*y*) corresponds to an independent axis. Phase functions which produce amplitude fields similar to

*A*(

*k*,

_{x}*k*) can correspond to local minima of

_{y}*ε*in this multidimensional space. The GS algorithm, as well as so-called “steepest-descent” methods like for instance the direct binary search [7

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]

*ε*, until it reaches a minimum. Which minimum is finally reached, depends solely on the initial hologram phase, which corresponds to an initial starting position in the space of

*ε*. To handle this problem, one can define detailed boundary conditions [8

8. J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A **3**, 284–288 (1985). [CrossRef]

*ε*.

*ε*-space, which lies close enough to the global minimum of

*ε*for the algorithm to find it.

## 2. Phase retrieval procedure

*x*+

*i y*) with a finite circular aperture in the SLM plane, where arg denotes the complex phase angle (in a range between 0 and 2

*π*) of a complex number. The aperture is necessary, since we want the algorithm to find a phase hologram which produces a doughnut mode of the size we

*observe*. Because the size of the experimentally produced focussed doughnut depends on the numerical aperture of the imaging pathway, a correspondingly matched numerical aperture has to be introduced in the numerical algorithm by limiting the theoretically assumed illumination aperture by a circular mask. Additional information about why such an aperture has to be used and how the right aperture size can be determined will follow later.

*k*,

_{x}*k*) is replaced by the square root of the doughnut intensity image, followed by an inverse Fourier transform. In the SLM plane, the amplitude

_{y}*a*(

*x*,

*y*) is replaced by a constant value (including the aperture), which corresponds to a homogenous illumination of the SLM hologram in the experiment. The cycle is performed until

*ε*converges (i.e., until it decreases less than a certain percentage per iterative step, which is 1% in our case). The resulting pattern

*H*(

*x*,

*y*) corresponds to a pure phase hologram, that would produce the observed distorted doughnut pattern. In order to find the corresponding wavefront distortion

*C*(

*x*,

*y*) introduced by the non-optimal surface of the SLM and the remaining optical components, the phase wavefront of an ideal doughnut mode has to be subtracted from

*H*(

*x*,

*y*), i.e.,

*C*(

*x*,

*y*) =

*H*(

*x*,

*y*) - arg(

*x*+

*i y*). Finally, in order to program the SLM such that it compensates these distortions,

*C*(

*x*,

*y*) has to be subtracted from every hologram function, that is displayed on the SLM.

## 3. Simulation results

## 4. Experiments

*LC-R 720*from

*Holoeye*, with a resolution of 1280×768 pixels at a pixel size of 20 μm) is illuminated by the beam of a helium-neon laser, which was expanded to a diameter much larger than the displayed phase hologram, which has a diameter of

*M*=512 pixels. Thus the light intensity is approximately constant over the whole diffraction pattern. The linear polarization state of the laser is rotated to maximize the diffraction efficiency. The SLM then shows efficient phase modulation (and a small residual polarization modulation).

_{Holo}*L*, superposed by a grating of periodicity 2

*π*/

*k*, and a lens with focal length

*f*.

*λ*represents the wavelength of the used light. If

*f*is chosen to match the distance between SLM and CCD, the modulus of the light field on the CCD is that of the Fourier transform of the hologram on the SLM:

*D*(

*x*,

*y*) denoting the phase distortion function of the SLM.

*L*=1 and

*L*=0, respectively. The doughnut mode seems to be much more distorted than the focussed spot corresponding to the point spread function of the setup.

*M*, e.g.

_{Array}*M*= 1024 pixels. This step gets rid of the noise outside the relevant part of the image, greatly improving the performance of the approach. It is necessary to “mask” the simulation hologram with a circular aperture, as sketched in Fig. 2, in order to ensure that the computer-simulated light field has exactly the same size (in pixels) as the real image on the CCD. The masking thus represents a correction for the different size scaling factors of the software (numerical FFT) and hardware (optical lens) Fourier transform. The optical FT incorporates length scaling factors which the FFT disregards, like wavelength, focal lengths, and the pixel size ratio between SLM and CCD. Because of the aperture masking, the correction pattern (which has the size of the aperture) will always be smaller than

_{Array}*M*. Consequently it is reasonable to choose

_{Array}*M*larger than the hologram size in the first place.

_{Array}*k*(in inverse pixels), and measuring the corresponding spatial shift of the laser spot on the CCD. The desired aperture diameter

*A*(in pixel units) is then determined by

*M*the side length of the data array, and

_{Array}*M*the diameter of the hologram on the SLM, both measured in pixels. Eq. 4 loses its validity if additional apertures narrow the beam diameter. According to Eq. 4, the aperture diameter (and hence the size of the correction pattern) depends linearly on

_{Holo}*M*. Consequently the resolution of the correction pattern increases with the size of

_{Array}*M*, however, at the cost of higher computational effort. If

_{Array}*M*is chosen to be 2

_{Array}*π*∆ℓ/∆

*k*, the correction pattern will have the resolution of the SLM hologram. However, for our experiments, this value is in the range of 5000 pixels, i.e. too large for reasonably fast computation. Thus we choose

*M*= 1024 pixels, and “re-sample” the achieved correction pattern to the size of

_{Array}*M*.

_{Holo}## 5. Discussion

*L*=1 has proven to be much more suitable for this technique than LG modes of significantly higher order, and also more suitable than trying to optimize the point spread function directly. A possible reason for this may lie in the fact that the point spread function has too small an extension to be clearly resolved by the CCD, and intensity deviations due to phase errors have little impact on the scale of the very intense center of the spot. On the other hand, LG modes of higher order show violation of the Nyquist criteria in a larger zone around the hologram center. Generally, it seems plausible that the

*L*=1 doughnut mode hologram is an “optimal” test pattern for obtaining a phase correction function, since it generates an extended doughnut image with an isotropic structure that is created purely by interference and depends critically on the perfect phase relation between all field components in its Fourier plane, i.e., the SLM plane.

*ε*. This is expected because the phase vortex is in this case no longer an adequate initial condition. Using more sophisticated search algorithms like for example

*Simulated Annealing*[9

9. S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, “Optimization by Simulated Annealing,” Science **220**, 4598 (1983). [CrossRef]

*ε*. Alternatively, after each iterative step, the correction function could be directly applied to the SLM. The convergence to wrong solutions can immediately be noticed by an abruptly decreasing doughnut quality.

10. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. **19**, 780–782 (1994). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. |

2. | S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express |

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 |

4. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

5. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

6. | A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. |

7. | M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. |

8. | J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A |

9. | S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, “Optimization by Simulated Annealing,” Science |

10. | S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 20, 2007

Revised Manuscript: April 5, 2007

Manuscript Accepted: April 10, 2007

Published: April 27, 2007

**Virtual Issues**

Vol. 2, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Wavefront correction of spatial light modulators using an optical vortex image," Opt. Express **15**, 5801-5808 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-9-5801

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### References

- 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]
- S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral phase contrast imaging in microscopy," Opt. Express 13, 689-694 (2005). [CrossRef] [PubMed]
- 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]
- 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).
- J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- 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]
- 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]
- J. R. Fienup, "Phase retrieval using boundary conditions," J. Opt. Soc. Am. A 3, 284-288 (1985). [CrossRef]
- S. Kirkpatrick, C. D. GelattJr., M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 4598 (1983). [CrossRef]
- 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]

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