## Accuracy of correction in modal sensorless adaptive optics |

Optics Express, Vol. 20, Issue 3, pp. 2598-2612 (2012)

http://dx.doi.org/10.1364/OE.20.002598

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### Abstract

We investigate theoretically and experimentally the parameters governing the accuracy of correction in modal sensorless adaptive optics for microscopy. On the example of two-photon fluorescence imaging, we show that using a suitable number of measurements, precise correction can be obtained for up to 2 radians rms aberrations without optimising the aberration modes used for correction. We also investigate the number of photons required for accurate correction when signal acquisition is shot-noise limited. We show that only 10^{4} to 10^{5} photons are required for complete correction so that the correction process can be implemented with limited extra-illumination and associated photoperturbation. Finally, we provide guidelines for implementing an optimal correction algorithm depending on the experimental conditions.

© 2012 OSA

## 1. Introduction

1. M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Nat. Acad. Sci. **99**, 5788–5792 (2002). [CrossRef] [PubMed]

6. R. Aviles-Espinosa, J. Andilla, R. Porcar-Guezenec, O. E. Olarte, M. Nieto, X. Levecq, D. Artigas, and P. Loza-Alvarez, “Measurement and correction of in vivo sample aberrations employing a nonlinear guide-star in two-photon excited fluorescence microscopy,” Biomed. Opt. Express **2**, 3135–3149 (2011). [CrossRef] [PubMed]

## 2. Principle of model-based modal aberration correction

4. D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. **34**, 2495–2497 (2009). [CrossRef] [PubMed]

8. D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express **16**, 9290–9305 (2008). [CrossRef] [PubMed]

**a**is the vector of coefficients of aberration in different modes, and the matrix

**A**describes the influence of each aberration mode on the value of M. The diagonal elements of

**A**are the mode eigenvalues and the non-diagonal elements are the crosstalks between different modes. Under this assumption, the crosstalks are independent of the amount of aberration, and can thus be suppressed using an appropriate linear transformation of the initial mode basis. This transformation can be determined theoretically or experimentally (see [8

8. D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express **16**, 9290–9305 (2008). [CrossRef] [PubMed]

8. D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express **16**, 9290–9305 (2008). [CrossRef] [PubMed]

7. N. Olivier, D. Débarre, and E. Beaurepaire, “Dynamic aberration correction for multiharmonic microscopy”, Opt. Lett. **34**, 3145–3147 (2009). [CrossRef] [PubMed]

4. D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. **34**, 2495–2497 (2009). [CrossRef] [PubMed]

9. D. Débarre, T. Vieille, and E. Beaurepaire, “Simple characterisation of a deformable mirror inside a high numerical aperture microscope using phase diversity,” J. Microsc. **244**, 136–143 (2011). [CrossRef] [PubMed]

10. M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A **17**, 1098–1107 (2000). [CrossRef]

## 3. Influence of residual linear crosstalk

^{2}].

## 4. Influence of the measurement bias in a 3-measurements scheme

7. N. Olivier, D. Débarre, and E. Beaurepaire, “Dynamic aberration correction for multiharmonic microscopy”, Opt. Lett. **34**, 3145–3147 (2009). [CrossRef] [PubMed]

*Drosophila*embryos and fresh mouse brain tissue, that the shape of the curve was well described by the square root of a Lorentzian curve (see e.g. Fig. 3(c)), a shape which we used throughout this paper. We found, however, that using a Gaussian or Lorentzian fit did not significantly affect the accuracy of correction. In any case, the results presented here can be straightforwardly extended to the case where another such simple function is used to fit the variations of M as a function of aberrations.

## 5. Influence of the number of measurements per mode

## 6. Signal level and accuracy of correction

4. D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. **34**, 2495–2497 (2009). [CrossRef] [PubMed]

11. A. J. Wright, S. P. Poland, J. M. Girkin, C. W. Freudiger, C. L. Evans, and X. S. Xie, “Adaptive optics for enhanced signal in CARS microscopy,” Opt. Express **15**, 18209–18219 (2007). [CrossRef] [PubMed]

*i*, the theoretical curve

*f*(

*a*

*) for M perfectly fits the noiseless experimental data, and that the only sources of imprecision in the measurements are the shot noise in the signal and the dark noise from the detectors. If the fit parameters (amplitude, width, centre and if required offset due to the dark noise) are perfectly known, the residual error between the fitting curve and the measurement points,*

_{i}*a*

*the amplitude of aberration in mode*

_{i}*i*and

*j*th measurement in mode

*i*performed with bias

*b*

*), is given by the sum of the square of the noise in each measurement. Assuming that the dark noise also exhibits a Poissonian distribution, the error becomes : where*

_{j}*N*is the total number of photons in the P measurements of one mode

_{tot}*i*, and

*B*is the average value of the dark noise. Let us now assume that there is no noise in any of the P measured values. For each mode

*i*, the total squared difference between these values and the fitting function

*f*is given by : where

*I*

_{0},

*w*,

*B*,

*c*are respectively the estimated amplitude, width, offset and centre of the curve, and

*I*

_{0}

*,*

_{c}*w*,

_{c}*B*,

_{c}*c*are their true values. If the parameters are set to their correct values, this error is zero. If now a small error

_{c}*dc*is introduced on

*c*, the error becomes : where the partial derivatives of

*f*are all taken at

*I*

_{0}

*,*

_{c}*w*,

_{c}*B*,

_{c}*c*and

_{c}*I*

_{0},

*w*,

*B*as a function of the value set for

*c*. When a least square fit is performed,

*ε*is the quantity that is minimised and thus yields the residual error when

*c*is evaluated with an error of

*dc*. It is therefore reasonable to assume that the error

*dc*on the value of

*c*due to the presence of noise is obtained by setting

_{c}*ε*(

*dc*) =

*ε*, so that :

_{noise}*E*determined above to account for imprecise knowledge of the initial aberration :

*F*has been calculated from the measured shape of the curve of M as a function of aberrations and the values of the biases (Fig. 6, inset). Dark noise level

*B*was determined by acquiring an image with equivalent size and pixel dwell time in absence of sample, and

*E*

_{0}was estimated by considering the best accuracy achieved, obtained for

*N*= 6 × 10

_{tot}^{5}: since the initial correction was performed using this value for

*N*and set as zero, and assuming that the initial error and the subsequent measure are independent, the value obtained at this point is

_{tot}*E*

_{0}= 0.0148. The theoretical curve is thus plotted with all parameters fixed and remarkably fits the experimental data.

^{4}. This is illustrated on Fig. 7 on a lily pollen grain imaged with 3-colour 2PEF. Here, both system aberrations and sample-induced distortion (mostly due to the index mismatch between the sample embedding medium and the objective immersion medium) are corrected. The total corrected aberration in (b) is 0.96 rad rms, and correction is performed using 2 iterations of the 5N algorithm. Here the total number of photons used for correction was 3.9 ×10

^{5}, about 9 times the minimum value determined above. This value was obtained at the maximum speed of our scanner and using the minimum power that could be set for the excitation source. As expected, the correction results in a significant improvement in brightness and resolution of the image.

## 7. Discussion

7. N. Olivier, D. Débarre, and E. Beaurepaire, “Dynamic aberration correction for multiharmonic microscopy”, Opt. Lett. **34**, 3145–3147 (2009). [CrossRef] [PubMed]

*F*(see equation 4), on the modulation depth of the metric curve that is probed during these P measurements. As a result we note that using a signal that varies faster with the amount of aberrations such as THG [7

**34**, 3145–3147 (2009). [CrossRef] [PubMed]

*F*, which depends on the shape of the curve of M as a function of aberrations. This shape in turn depends mostly on two factors: first, the structure of the sample determines how strongly the signal is affected by aberrations. For example, in two-photon microscopy, it has been shown analytically and experimentally that the signal from point objects in a 2D image is more affected than that of a bulk sample [4

**34**, 2495–2497 (2009). [CrossRef] [PubMed]

*F*will be greater for the former and smaller for the latter. This leads to the intuitive result that a stronger signal is needed to correct for aberrations when this signal is weakly sensitive to aberrations.

*F*decreases.

*F*(and hence the optimal exposure during correction) can be calibrated before imaging using a similar sample: for example, it can be expected that when imaging at a given depth in a given region of fresh mouse brain tissue, the shape of the curve for M will be more or less identical from one sample to the next. It can thus be measured as a function of, e.g., depth, excitation wavelength (which might affect aberrations in high order modes) or immersion medium so that the optimal exposure is known once and for all for the subsequent studies. In case exposure is less critical, a conservative value of 10–20 times the threshold obtained here should compensate for a possible drop in modulation of the curve for M in most samples. The results presented here can in summary be easily used for optimising the correction process in any kind of sample.

^{4}photons for 11 aberration modes. As a comparison, a 512 × 512 image with a moderate average intensity of 15 photon counts per pixel corresponds to about 4 ×10

^{6}detected photons. Even if the number of measurements is significant, the exposure required for correction is therefore limited and compatible with biological imaging. This number of photons required for correction grows linearly with the number of corrected modes: here we used only 11 low order Zernike modes that could be accurately produced by our deformable mirror, but correction of higher order modes might be beneficial in certain cases. It is beyond the scope of this paper to determine the number of modes that should be used for correction in biological samples, and this question should be addressed in further studies.

*in vivo*imaging is the time required to perform the correction. Thank to the simplicity of our correction algorithm, the processing time is very small compared to the acquisition time of the data. In our setup, this time was in most cases not limited by the signal intensity, but rather by the speed of our deformable mirror and galvanometer mirrors. The minimum time necessary for each measurement of the metric M was around 60 ms, so that the total time for correction ranged from 1.3 s (2N+1 algorithm, 23 measurements) to 3.3 s (5N algorithm, 55 measurements). In order to keep the number of photons to a low level at this acquisition time, the excitation power was decreased by an amount calculated from the initial image intensity. Once correction was achieved, greater sampling and pixel dwell time were used to increase image quality, so that the acquisition time for the corrected image was usually a few seconds. Although the time required for correction in our setup is still significant, the resulting delay is nevertheless compatible with numerous studies of biological samples, e.g. study of slow developmental processes or of the 3D morphology of tissues, or static aberration correction for the study of a rapid process such as neuronal activity.

^{7}photons/s. As a result, the minimal time required for correction could in principle be decreased to 1.4ms for the 2N+1 algorithm or 3.6ms for the 5N algorithm, enabling fast aberration correction during live imaging.

## 8. Conclusion

- Modal sensorless aberration correction can be implemented using any convenient set of N modes, provided that the crosstalk between the modes for the chosen metric is moderate, i.e. the error induced in mode A by aberrations in mode B does not exceed 10–20% of the amplitude of aberration in mode B.
- If the aberration amplitude is small (0–0.5 rad rms), one iteration of the 2N+1algorithm is sufficient to correct for it. If the amplitude is moderate (0.5–1 rad rms) the 5N algorithm yields accurate results. For greater amplitudes, several iterations may be used.
- When P=3 measurements per modes are used, the bias should be set as roughly equal to the width of the M curve. When P is greater than 3, the probed range can be accordingly extended to ensure stability of the algorithm over a greater aberration range.
- If more than 3 free parameters per mode are used to fit the curve of M as a function of aberration amplitude, the number of measurement per mode, P, should be increased accordingly to ensure good correction accuracy.
- Accuracy can be further increased using a priori knowledge of the aberrations, e.g. by changing the order in which the modes are corrected (modes with large amounts of aberration being corrected first).
- Sampling and illumination can be reduced during correction to minimise exposure. A conservative value of 2 −5 ×10
^{4}photons per mode and per algorithm iteration can be used for accurate correction when phototoxicity or photobleaching are not critical. - If exposure is critical, illumination can be optimised using a priori knowledge of the shape of the curve of metric M as a function of aberrations, which can be measured on a similar sample for calibration.
- Correction should be performed on a region of the field of view over which aberrations are roughly homogeneous, else the correction accuracy may be degraded by residual aberration remaining locally.

## Appendix A Zernike modes and numbering scheme used in this work

## Appendix B Experimental removal of residual tip, tilt and defocus from the modes used for correction

*i*modes. This is performed experimentally so as to take into account the exact intensity profile of the excitation beam which strongly influences the shift induced by each mode. Our approach is similar to the method described in [12].

*i*as well as for tip, tilt and defocus. The amount of tip and tilt to add to mode

*i*is then calculated as the opposite of the ratio between the displacement for mode

*i*and that for tip or tilt, respectively. This is also applied to defocus so that subsequent correction for the axial shift does not reintroduce lateral shift.

*i*and defocus in turn. Following the same principle as for the lateral shift, the axial shift for each mode

*i*is then removed by adding the appropriate amount of defocus.

## Acknowledgments

## References and links

1. | M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Nat. Acad. Sci. |

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

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

4. | D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. |

5. | N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods |

6. | R. Aviles-Espinosa, J. Andilla, R. Porcar-Guezenec, O. E. Olarte, M. Nieto, X. Levecq, D. Artigas, and P. Loza-Alvarez, “Measurement and correction of in vivo sample aberrations employing a nonlinear guide-star in two-photon excited fluorescence microscopy,” Biomed. Opt. Express |

7. | N. Olivier, D. Débarre, and E. Beaurepaire, “Dynamic aberration correction for multiharmonic microscopy”, Opt. Lett. |

8. | D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express |

9. | D. Débarre, T. Vieille, and E. Beaurepaire, “Simple characterisation of a deformable mirror inside a high numerical aperture microscope using phase diversity,” J. Microsc. |

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. | A. J. Wright, S. P. Poland, J. M. Girkin, C. W. Freudiger, C. L. Evans, and X. S. Xie, “Adaptive optics for enhanced signal in CARS microscopy,” Opt. Express |

12. | A. Thayil and M. J. Booth, “Self calibration of sensorless adaptive optical microscopes,” J. Eur. Opt. Soc. |

**OCIS Codes**

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(180.6900) Microscopy : Three-dimensional microscopy

(180.4315) Microscopy : Nonlinear microscopy

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: November 23, 2011

Revised Manuscript: January 2, 2012

Manuscript Accepted: January 10, 2012

Published: January 20, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Aurélie Facomprez, Emmanuel Beaurepaire, and Delphine Débarre, "Accuracy of correction in modal sensorless adaptive optics," Opt. Express **20**, 2598-2612 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-3-2598

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### References

- M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Nat. Acad. Sci.99, 5788–5792 (2002). [CrossRef] [PubMed]
- P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express11, 1123–1130 (2003). [CrossRef] [PubMed]
- M. Rueckel, J. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Nat. Acad. Sci.103, 17137–17142 (2006). [CrossRef] [PubMed]
- D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett.34, 2495–2497 (2009). [CrossRef] [PubMed]
- N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods7, 141–147 (2009). [CrossRef] [PubMed]
- R. Aviles-Espinosa, J. Andilla, R. Porcar-Guezenec, O. E. Olarte, M. Nieto, X. Levecq, D. Artigas, and P. Loza-Alvarez, “Measurement and correction of in vivo sample aberrations employing a nonlinear guide-star in two-photon excited fluorescence microscopy,” Biomed. Opt. Express2, 3135–3149 (2011). [CrossRef] [PubMed]
- N. Olivier, D. Débarre, and E. Beaurepaire, “Dynamic aberration correction for multiharmonic microscopy”, Opt. Lett.34, 3145–3147 (2009). [CrossRef] [PubMed]
- D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express16, 9290–9305 (2008). [CrossRef] [PubMed]
- D. Débarre, T. Vieille, and E. Beaurepaire, “Simple characterisation of a deformable mirror inside a high numerical aperture microscope using phase diversity,” J. Microsc.244, 136–143 (2011). [CrossRef] [PubMed]
- M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A17, 1098–1107 (2000). [CrossRef]
- A. J. Wright, S. P. Poland, J. M. Girkin, C. W. Freudiger, C. L. Evans, and X. S. Xie, “Adaptive optics for enhanced signal in CARS microscopy,” Opt. Express15, 18209–18219 (2007). [CrossRef] [PubMed]
- A. Thayil and M. J. Booth, “Self calibration of sensorless adaptive optical microscopes,” J. Eur. Opt. Soc.6, 11045 (2011).

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