## Characterization of deformable mirrors for spherical aberration correction in optical sectioning microscopy

Optics Express, Vol. 18, Issue 7, pp. 6900-6913 (2010)

http://dx.doi.org/10.1364/OE.18.006900

Acrobat PDF (607 KB)

### Abstract

In this paper we describe the wavefront aberrations that arise when imaging biological specimens using an optical sectioning microscope and generate simulated wavefronts for a planar refractive index mismatch. We then investigate the capability of two deformable mirrors for correcting spherical aberration at different focusing depths for three different microscope objective lenses. Along with measurement and analysis of the mirror influence functions we determine the optimum mirror pupil size and number of spatial modes included in the wavefront expansion and we present measurements of actuator linearity and hysteresis. We find that both mirrors are capable of correcting the wavefront aberration to improve imaging and greatly extend the depth at which diffraction limited imaging is possible.

© 2010 OSA

## 1. Introduction

1. M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. **213**(1), 11–19 (2004). [CrossRef]

3. M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. **228**(1), 97–102 (2007). [CrossRef] [PubMed]

5. P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express **11**(10), 1123–1130 (2003). [CrossRef] [PubMed]

6. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. **206**(1), 65–71 (2002). [CrossRef] [PubMed]

7. M. Schwertner, M. Booth, T. Tanaka, T. Wilson, and S. Kawata, “Spherical aberration correction system using an adaptive optics deformable mirror,” Opt. Commun. **263**(2), 147–151 (2006). [CrossRef]

8. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. **32**(1), 5–7 (2007). [CrossRef]

9. M. J. Booth, M. A. A. Neil, and T. Wilson, “New modal wave-front sensor: application to adaptive confocal fluorescence microscopy and two-photon excitation fluorescence microscopy,” J. Opt. Soc. Am. A **19**(10), 2112–2120 (2002). [CrossRef]

11. M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. **103**(46), 17137–17142 (2006). [CrossRef] [PubMed]

12. S. N. S. Reihani and L. B. Oddershede, “Confocal microscopy of thick specimens,” J. Biomed. Opt. **14**(3), 030513 (2009). [CrossRef] [PubMed]

13. H. Itoh, N. Matsumoto, and T. Inoue, “Spherical aberration correction suitable for a wavefront controller,” Opt. Express **17**(16), 14367–14373 (2009). [CrossRef] [PubMed]

## 2. Wavefront aberrations in optical sectioning microscopy

16. P. Torok, S. J. Hewlett, and P. Varga, “The role of specimen-induced spherical aberration in confocal microscopy,” J. Microsc. **188**(2), 158–172 (1997). [CrossRef]

17. M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. **192**(2), 90–98 (1998). [CrossRef]

13. H. Itoh, N. Matsumoto, and T. Inoue, “Spherical aberration correction suitable for a wavefront controller,” Opt. Express **17**(16), 14367–14373 (2009). [CrossRef] [PubMed]

18. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. **36**(11), 2305–2312 (1997). [CrossRef] [PubMed]

19. P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express **16**(2), 507–523 (2008). [CrossRef] [PubMed]

20. M. Mansuripur, “Effects of High-Numerical-Aperture Focusing on the State of Polarization in Optical and Magnetooptic Data-Storage Systems,” Appl. Opt. **30**(22), 3154–3162 (1991). [CrossRef] [PubMed]

*φ*, is given by Eq. (1) [17

17. M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. **192**(2), 90–98 (1998). [CrossRef]

*d*is the depth of the nominal focal plane (in the absence of a refractive index boundary),

*ρ*is the normalized pupil radius and

*α*and

*β*are the angles of the marginal ray in the first and second media respectively (i.e. NA = n

_{1}sin(

*α*)) with respect to the optic axis.

*Z*with coefficients

_{n}^{0}(ρ)*A*(Eq. (2)).

_{n0}17. M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. **192**(2), 90–98 (1998). [CrossRef]

_{2}

^{0}) from the aberration function. Note that the amplitude of the wavefront aberration is largest close to the edge of the pupil and that the overall wavefront deviation from flatness increases with the NA (from Eq. (1) it is also apparent that the amplitude of the wavefront aberration also increases linearly with focusing depth).

## 3. Characterisation of Deformable Mirrors

### 3.1 Spatial Modes of the Mirrors

*φ*, for a given set of actuator signals,

_{m}*x*, is given by:

_{m}*A*is the influence matrix of the mirror. The actuator signals which give the best fit to a given target wavefront

_{m}*φ*, can then be calculated by premultiplying

_{0}*φ*by the pseudo inverse of the influence matrix which itself can be found from a singular value decomposition of

_{0}*A*.

_{m}*U*matrix comprise a set of N orthogonal spatial modes of the mirror (where N is the number of mirror actuators) and the diagonal elements of

*S*represent the gain of each mode, the columns of

*V*make up an orthogonal set of actuator signals. A smaller gain means that a large actuator signal is required to produce a given amount of that mode. The spatial modes included in the wavefront expansion can be altered by setting the relevant diagonal elements of

*S*to zero or by conditioning the

*U*matrix. To restrict the expansion to the first n (<N) spatial modes, elements of columns n + 1 to N of U are set to zero.

*φ*) can be determined from.

_{0}*f*is a clipping function to account for the limited stroke of the mirror actuators and is defined by

25. S. P. Poland, A. J. Wright, and J. M. Girkin, “Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy,” Appl. Opt. **47**(6), 731–736 (2008). [CrossRef] [PubMed]

^{4}. For each different imaging condition we fix the mirror aperture ratio, at 0.59 for the PDM and 0.87 for the Mirao 52-e, and determine the optimum wavefront correction which each mirror can achieve as a function of depth, optimizing the number of spatial modes used in the expansion at each depth. Fixing the aperture ratio is a practical restriction as it is unlikely to be feasible to arbitrarily change the magnification of the relay optics for different focusing depths. For a Strehl ratio of 0.8 or greater, imaging is generally considered to be diffraction limited.

### 3.2 Linearity and hysteresis of mirror actuators

_{m}in Eq. (6) is independent of x

_{m}and the deformation of the mirror deformation, φ

_{m}, obeys Eq. (11) for all permitted combinations of actuator signals x

_{1}and x

_{2}.

## 4. Conclusions

## Acknowledgements

## References and links

1. | M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. |

2. | M. Schwertner, M. J. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express |

3. | M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. |

4. | S. W. S. Hell, E. H. K., “Lens Aberrations in Confocal Fluorescence Microscopy,” in |

5. | P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express |

6. | L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. |

7. | M. Schwertner, M. Booth, T. Tanaka, T. Wilson, and S. Kawata, “Spherical aberration correction system using an adaptive optics deformable mirror,” Opt. Commun. |

8. | M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. |

9. | M. J. Booth, M. A. A. Neil, and T. Wilson, “New modal wave-front sensor: application to adaptive confocal fluorescence microscopy and two-photon excitation fluorescence microscopy,” J. Opt. Soc. Am. A |

10. | M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A |

11. | M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. |

12. | S. N. S. Reihani and L. B. Oddershede, “Confocal microscopy of thick specimens,” J. Biomed. Opt. |

13. | H. Itoh, N. Matsumoto, and T. Inoue, “Spherical aberration correction suitable for a wavefront controller,” Opt. Express |

14. | N. Devaney, E. Dalimier, T. Farrell, D. Coburn, R. Mackey, D. Mackey, F. Laurent, E. Daly, and C. Dainty, “Correction of ocular and atmospheric wavefronts: a comparison of the performance of various deformable mirrors,” Appl. Opt. |

15. | C. Paterson, I. Munro, and J. C. Dainty, “A low cost adaptive optics system using a membrane mirror,” Opt. Express |

16. | P. Torok, S. J. Hewlett, and P. Varga, “The role of specimen-induced spherical aberration in confocal microscopy,” J. Microsc. |

17. | M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. |

18. | P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. |

19. | P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express |

20. | M. Mansuripur, “Effects of High-Numerical-Aperture Focusing on the State of Polarization in Optical and Magnetooptic Data-Storage Systems,” Appl. Opt. |

21. | Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. |

22. | J. C. Lai, Z. H. Li, C. Y. Wang, and A. Z. He, “Experimental measurement of the refractive index of biological tissues by total internal reflection,” Appl. Opt. |

23. | M. Schwertner, M. J. Booth, and T. Wilson, “Simulation of specimen-induced aberrations for objects with spherical and cylindrical symmetry,” J. Microsc. |

24. | E. J. Fernandez, L. Vabre, B. Hermann, A. Unterhuber, B. Povazay, and W. Drexler, “Adaptive optics with a magnetic deformable mirror: applications in the human eye,” Opt. Express |

25. | S. P. Poland, A. J. Wright, and J. M. Girkin, “Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy,” Appl. Opt. |

**OCIS Codes**

(110.0180) Imaging systems : Microscopy

(220.1000) Optical design and fabrication : Aberration compensation

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: November 30, 2009

Revised Manuscript: March 10, 2010

Manuscript Accepted: March 11, 2010

Published: March 19, 2010

**Virtual Issues**

Vol. 5, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Michael Shaw, Simon Hall, Steven Knox, Richard Stevens, and Carl Paterson, "Characterization of deformable mirrors for spherical aberration correction in optical sectioning microscopy," Opt. Express **18**, 6900-6913 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6900

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### References

- M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004). [CrossRef]
- M. Schwertner, M. J. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express 12(26), 6540–6552 (2004). [CrossRef] [PubMed]
- M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228(1), 97–102 (2007). [CrossRef] [PubMed]
- S. W. S. Hell, E. H. K., “Lens Aberrations in Confocal Fluorescence Microscopy,” in Handbook of Biological Confocal Microscopy, Second ed., J. B. Pawley, ed. (Plenum Press, 1995), pp. 347–354.
- P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express 11(10), 1123–1130 (2003). [CrossRef] [PubMed]
- L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002). [CrossRef] [PubMed]
- M. Schwertner, M. Booth, T. Tanaka, T. Wilson, and S. Kawata, “Spherical aberration correction system using an adaptive optics deformable mirror,” Opt. Commun. 263(2), 147–151 (2006). [CrossRef]
- M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]
- M. J. Booth, M. A. A. Neil, and T. Wilson, “New modal wave-front sensor: application to adaptive confocal fluorescence microscopy and two-photon excitation fluorescence microscopy,” J. Opt. Soc. Am. A 19(10), 2112–2120 (2002). [CrossRef]
- M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A 17(6), 1098–1107 (2000). [CrossRef]
- M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006). [CrossRef] [PubMed]
- S. N. S. Reihani and L. B. Oddershede, “Confocal microscopy of thick specimens,” J. Biomed. Opt. 14(3), 030513 (2009). [CrossRef] [PubMed]
- H. Itoh, N. Matsumoto, and T. Inoue, “Spherical aberration correction suitable for a wavefront controller,” Opt. Express 17(16), 14367–14373 (2009). [CrossRef] [PubMed]
- N. Devaney, E. Dalimier, T. Farrell, D. Coburn, R. Mackey, D. Mackey, F. Laurent, E. Daly, and C. Dainty, “Correction of ocular and atmospheric wavefronts: a comparison of the performance of various deformable mirrors,” Appl. Opt. 47(35), 6550–6562 (2008). [CrossRef] [PubMed]
- C. Paterson, I. Munro, and J. C. Dainty, “A low cost adaptive optics system using a membrane mirror,” Opt. Express 6(9), 175–185 (2000). [CrossRef] [PubMed]
- P. Torok, S. J. Hewlett, and P. Varga, “The role of specimen-induced spherical aberration in confocal microscopy,” J. Microsc. 188(2), 158–172 (1997). [CrossRef]
- M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192(2), 90–98 (1998). [CrossRef]
- P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. 36(11), 2305–2312 (1997). [CrossRef] [PubMed]
- P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express 16(2), 507–523 (2008). [CrossRef] [PubMed]
- M. Mansuripur, “Effects of High-Numerical-Aperture Focusing on the State of Polarization in Optical and Magnetooptic Data-Storage Systems,” Appl. Opt. 30(22), 3154–3162 (1991). [CrossRef] [PubMed]
- Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(371–N), 379 (2006). [CrossRef]
- J. C. Lai, Z. H. Li, C. Y. Wang, and A. Z. He, “Experimental measurement of the refractive index of biological tissues by total internal reflection,” Appl. Opt. 44(10), 1845–1849 (2005). [CrossRef] [PubMed]
- M. Schwertner, M. J. Booth, and T. Wilson, “Simulation of specimen-induced aberrations for objects with spherical and cylindrical symmetry,” J. Microsc. 215(3), 271–280 (2004). [CrossRef] [PubMed]
- E. J. Fernandez, L. Vabre, B. Hermann, A. Unterhuber, B. Povazay, and W. Drexler, “Adaptive optics with a magnetic deformable mirror: applications in the human eye,” Opt. Express 14(20), 8900–8917 (2006). [CrossRef] [PubMed]
- S. P. Poland, A. J. Wright, and J. M. Girkin, “Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy,” Appl. Opt. 47(6), 731–736 (2008). [CrossRef] [PubMed]

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