## Reduction of spherical-aberration impact in microscopy by wavefront coding

Optics Express, Vol. 17, Issue 16, pp. 13810-13818 (2009)

http://dx.doi.org/10.1364/OE.17.013810

Acrobat PDF (367 KB)

### Abstract

In modern high-NA optical scanning instruments, like scanning microscopes, the refractive-index mismatch between the sample and the immersion medium introduces a significant amount of spherical aberration when imaging deep inside the specimen, spreading out the impulse response. Since such aberration depends on the focalization depth, it is not possible to achieve a static global compensation for the whole 3D sample in scanning microscopy. Therefore a depth-variant impulse response is generated. Consequently, the design of pupil elements that increase the tolerance to this aberration is of great interest. In this paper we report a hybrid technique that provides a focal spot that remains almost invariant in the depth-scanning processing of thick samples. This invariance allows the application of 3D deconvolution techniques to that provide an improved recovery of the specimen structure when imaging thick samples.

© 2009 OSA

## 1. Introduction

1. J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” Prog. Opt. **2**, 131–180 (1963). [CrossRef]

6. S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. **28**(10), 771–773 (2003). [CrossRef] [PubMed]

7. C. J. R. Sheppard and M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. **30**(25), 3563–3568 (1991). [CrossRef] [PubMed]

8. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. **99**(9), 5788–5792 (2002). [CrossRef] [PubMed]

10. C. Mauclair, A. Mermillod-Blondin, N. Huot, E. Audouard, and R. Stoian, “Ultrafast laser writing of homogeneous longitudinal waveguides in glasses using dynamic wavefront correction,” Opt. Express **16**(8), 5481–5492 (2008). [CrossRef] [PubMed]

11. C. Preza and J. A. Conchello, “Depth-variant maximum likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A **21**(9), 1593–1601 (2004). [CrossRef]

12. J. Ojeda-Castañeda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. **27**(12), 2583–2586 (1988). [CrossRef] [PubMed]

13. E. R. Dowski Jr. and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**(11), 1859–1865 (1995). [CrossRef] [PubMed]

14. W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. **41**(29), 6080–6092 (2002). [CrossRef] [PubMed]

## 2. Spherical aberration induced in the axial scanning

*θ*) accounts for the amplitude transmittance at the objective exit pupil, and

*α*is the maximum value for the aperture angle

*θ*. Lateral and axial positions in the focal region are expressed through cylindrical coordinates (

*r*,

*z*) as measured from the Gaussian focus,

*F*. The phase factor

*W*(

*θ*) accounts for potential phase distortions occurred during the focusing.

16. In a real microscopy experiments there is, still, a coverslip between the immersion liquid and the sample medium. Such slab may induce a constant amount of SA that does not depend on the focusing depth. This SA can be compensated statically either with the correction collar, or with a proper index-matching immersion medium. Therefore it is not necessary to consider it in the calculations.

*F*, and the exit pupil. Each plane-wave component of the field emerging from the objective obeys Snell’s law,

*n*

_{1}sin

*θ*=

*n*

_{2}sin

*θ*’, when refracted at the interface. The resulting field is reconstructed as the superposition of refracted plane waves.

*d*

_{0}, the distance between the interface and the focus,

*F*. On the contrary, we have named as the penetration depth,

*d*′

_{0}, the distance between the interface and the place where the wave effectively focuses.

*z*

_{S}towards the sample, the focusing depth is

*d*

_{S}=

*d*

_{0}+

*z*

_{S}. In such case the penetration depth is

*d*′

_{S}=

*d*′

_{0}+

*z*′

_{S}(

*z*′

_{S}being the axial displacement suffered by the effective focus). If we assume that by use of the correction collar [17] the spherical aberration is statically compensated for the focusing depth

*d*

_{0}, the wave distortions

*W*(

*θ*;

*z*

_{S}) induced at a focusing depth

*d*

_{S}can be easily evaluated by [18,19

19. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A **12**(2), 325–332 (1995). [CrossRef]

*W*(

*θ*;

*z*

_{S}) into a power series and making a fourth-order approximation. Besides, we have introduced the well-known defocus and SA coefficients, which are given, respectively, by

*n*

_{1}=1.52), and a specimen whose refractive index is close to

*n*

_{2}=1.33, which is the index of the water solution where the specimen is embedded. We have assumed that the SA has been compensated, with a correction collar, for a penetration depth

*d*′

_{0}=10

*µm*, which corresponds to a focusing depth

*d*

_{0}=11.7

*µm*.

20. V. N. Mahajan, “Gaussian apodization and beam propagation,” Prog. Opt. **49**, 1–95 (2006). [CrossRef]

*w*

_{20}=-

*w*

_{40}, but at the axial position of the PSF main peak. The images are shown in Fig. 3. A significant feature comes out from this figure: there is only a very small degradation of images of 2D samples. This is because, although the 3D PSF is strongly spread, the lateral PSF at the best image plane still has a sharp peak even in case of significant amount of SA.

## 3. Effect of the spherical aberration in 3D scanning microscopy

21. I. Escobar, G. Saavedra, M. Martínez-Corral, and J. Lancis, “Reduction of the spherical aberration effect in high-numerical-aperture optical scanning instruments,” J. Opt. Soc. Am. A **23**(12), 3150–3155 (2006). [CrossRef]

*q*(

*ζ*). The same result can be encountered in the analysis of 1D paraxial focusing systems when considering the case of a cylindrical lens illuminated by a monochromatic plane wave. When the 1D aperture stop (of transmittance

*p*(

*x*

_{0})) is placed, as usually, at the front focal plane of the cylindrical lens of focal length

*f*

_{x}, the intensity distribution on planes parallel to the back focal plane is given by

22. J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. **30**(13), 1647–1649 (2005). [CrossRef] [PubMed]

*w*

^{x}_{20}plays in Eq. (8) a role similar to the one played by coefficient

*w*

_{40}in Eq. (7). Thus, one can conclude that the axial intensity distribution produced by a high-NA scanning optical microscope in which certain amount of SA is induced, is the same as the transverse intensity distribution obtained at a defocused plane

*z*=-2

*λf*

^{2}

_{x}w_{40}/Δ

^{2}in the 1D paraxial focusing experiment. Since very efficient solutions have been given for decreasing the effect of defocusing in the 1D case (i.e., for increasing the

*depth-of-field*of such systems), we can now apply the analogous techniques to efficiently desensitize the axial response of the high-NA objective to the sample-induced SA.

## 4. Application of the wavefront coding technique

*wavefront coding*(WFC). In the WFC technique a pupil mask designed for increasing the depth-of-field is used in combination with a digital image-restoration process. Thus, as a first stage in the WFC method, the defocused patterns are uniformly blurred over a large axial range by effect of the mask. The almost-constant transverse impulse response in this case allows a sharp reconstruction inside this extended range by a single step deconvolution. A successful proposal in this sense is the 1D cubic phase filter proposed by Dowski and Cathey [13

13. E. R. Dowski Jr. and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**(11), 1859–1865 (1995). [CrossRef] [PubMed]

*q*(

*ζ*)=exp-(-

*i*2

*πAζ*

^{3}). Following the suggestions made in [13

13. E. R. Dowski Jr. and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**(11), 1859–1865 (1995). [CrossRef] [PubMed]

*A*much higher than 5. Thus we set

*A*=50.

*d*′

_{o}=10

*µm*. We calculated, as the squared modulus of Eq. (7), the confocal PSFs corresponding to seven equally-spaced scanning steps (see Fig. 4). In the video we compare, for varying penetration depth, the focal spots produced by the circular aperture (CA) and by the cubic filter (CF). We see that the PSFs obtained with the cubic filter are very different from the ones obtained with the circular aperture. In fact they are much more spread. But, in contrast with the circular-aperture case, they do not change significantly as the penetration depth increases. So, we can affirm that the cubic phase filter produces a spread focus that is insensitive to the amount of SA.

*d*′

_{S}and

*d*

_{S}, and show the corresponding value of the SA coefficient. Note that there is not any proportional relation between the penetration depth and the induced spherical aberration [23

23. D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. **13**(9), 2126–2133 (1974). [CrossRef] [PubMed]

d′_{S} (mm) | 10.0 | 12.5 | 15.0 | 17.5 | 20.0 | 22.5 | 25.0 |

d
_{S} (mm) | 13.2 | 16.5 | 19.7 | 22.9 | 26.3 | 29.7 | 33.1 |

w_{40}
| 0.0 | -0.6 | -1.1 | -1.6 | -2.1 | -2.6 | -3.2 |

*z*′

_{S}=7.5

*µm*+

*iδ*, with

*i*=0,…,7 and

*δ*=1.25

*µm*. The plates are shown in Fig. 5.

*z*′

_{S}, is calculated as the incoherent superposition of eight 2D images. Each of these eight images is obtained as the 2D convolution between the corresponding plate and the defocused transverse section of the 3D PSF associated to the selected

*z*′

_{S}.

*z*′

_{S}=10.00

*µm*. In Fig. 6(b) we show five sections of the deconvolved 3D image. These images are, or course much worse than the original. This is because it is in fact nonsense to apply deconvolution to systems whose impulse response varies very fast.

*z*′

_{S}=10.00

*µm*. After performing the 3D deconvolution we have obtained a new 3D image of the synthetic object. In Fig. 6(d) we show five 2D sections of the said object. We conclude from the figure that a wavefront coding technique provides 3D images that are highly immune to the SA induced by axial scanning.

## 5. Conclusions

25. C. J. Cogswell, S. V. King, S. R. Prasanna Pavani, D. B. Conkey, and R. H. Cormack, “Microscope imaging capabilities improve using computational optics,” Proc. Focus on Microscopy 09 (http://focusonmicroscopy.org/2009/PDF/341_Cogswell.pdf)

26. S. V. King, A. R. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. **13**(2), 024020 (2008). [CrossRef] [PubMed]

## Acknowledgements

## References and Links

1. | J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” Prog. Opt. |

2. | R. Barakat and A. Houston, “Transfer function of an annular aperture in the presence of spherical aberration,” J. Opt. Soc. Am. A |

3. | J. P. Mills and B. J. Thompson, “Effect of aberrations and apodization on the performance of coherent optical systems. I. The amplitude impulse response,” J. Opt. Soc. Am. A |

4. | J. Ojeda-Castañeda, P. Andrés, and A. Díaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. |

5. | J. Ojeda-Castañeda, P. Andrés, and A. Díaz, “Strehl ratio with low sensitivity to spherical aberration,” J. Opt. Soc. Am. A |

6. | S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. |

7. | C. J. R. Sheppard and M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. |

8. | M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. |

9. | M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. |

10. | C. Mauclair, A. Mermillod-Blondin, N. Huot, E. Audouard, and R. Stoian, “Ultrafast laser writing of homogeneous longitudinal waveguides in glasses using dynamic wavefront correction,” Opt. Express |

11. | C. Preza and J. A. Conchello, “Depth-variant maximum likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A |

12. | J. Ojeda-Castañeda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. |

13. | E. R. Dowski Jr. and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. |

14. | W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. |

15. | M. Gu, Advanced optical imaging theory, Springer-Verlag (Berlin) 2000. |

16. | In a real microscopy experiments there is, still, a coverslip between the immersion liquid and the sample medium. Such slab may induce a constant amount of SA that does not depend on the focusing depth. This SA can be compensated statically either with the correction collar, or with a proper index-matching immersion medium. Therefore it is not necessary to consider it in the calculations. |

17. | The highly corrected objectives are equipped with a correction collar to allow adjustment of the central lens group position to coincide with fluctuations in cover-glass thickness. |

18. | C. J. R. Sheppard and C. J. Cogswell, “Effects of aberrating layers and the tube length on confocal imaging properties,” Optik (Stuttg.) |

19. | P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A |

20. | V. N. Mahajan, “Gaussian apodization and beam propagation,” Prog. Opt. |

21. | I. Escobar, G. Saavedra, M. Martínez-Corral, and J. Lancis, “Reduction of the spherical aberration effect in high-numerical-aperture optical scanning instruments,” J. Opt. Soc. Am. A |

22. | J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. |

23. | D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. |

24. | M. Born and E. Wolf, Principles of Optics. University Press (Cambrigde) 1999. Ch 13. |

25. | C. J. Cogswell, S. V. King, S. R. Prasanna Pavani, D. B. Conkey, and R. H. Cormack, “Microscope imaging capabilities improve using computational optics,” Proc. Focus on Microscopy 09 (http://focusonmicroscopy.org/2009/PDF/341_Cogswell.pdf) |

26. | S. V. King, A. R. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.6890) Image processing : Three-dimensional image processing

(110.6880) Imaging systems : Three-dimensional image acquisition

(180.6900) Microscopy : Three-dimensional microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: May 4, 2009

Revised Manuscript: June 23, 2009

Manuscript Accepted: June 24, 2009

Published: July 24, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

G. Saavedra, I. Escobar, R. Martínez-Cuenca, E. Sánchez-Ortiga, and M. Martínez-Corral, "Reduction of spherical-aberration impact in microscopy by wavefront coding," Opt. Express **17**, 13810-13818 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-16-13810

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### References

- J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” Prog. Opt. 2, 131–180 (1963). [CrossRef]
- R. Barakat and A. Houston, “Transfer function of an annular aperture in the presence of spherical aberration,” J. Opt. Soc. Am. A 55(5), 538–541 (1965). [CrossRef]
- J. P. Mills and B. J. Thompson, “Effect of aberrations and apodization on the performance of coherent optical systems. I. The amplitude impulse response,” J. Opt. Soc. Am. A 3(5), 694–703 (1986). [CrossRef]
- J. Ojeda-Castañeda, P. Andrés, and A. Díaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. 11(8), 487–489 (1986). [CrossRef] [PubMed]
- J. Ojeda-Castañeda, P. Andrés, and A. Díaz, “Strehl ratio with low sensitivity to spherical aberration,” J. Opt. Soc. Am. A 5(8), 1233–1236 (1988). [CrossRef]
- S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003). [CrossRef] [PubMed]
- C. J. R. Sheppard and M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. 30(25), 3563–3568 (1991). [CrossRef] [PubMed]
- M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002). [CrossRef] [PubMed]
- M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88(3), 031109 (2006). [CrossRef]
- C. Mauclair, A. Mermillod-Blondin, N. Huot, E. Audouard, and R. Stoian, “Ultrafast laser writing of homogeneous longitudinal waveguides in glasses using dynamic wavefront correction,” Opt. Express 16(8), 5481–5492 (2008). [CrossRef] [PubMed]
- C. Preza and J. A. Conchello, “Depth-variant maximum likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004). [CrossRef]
- J. Ojeda-Castañeda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27(12), 2583–2586 (1988). [CrossRef] [PubMed]
- E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1865 (1995). [CrossRef] [PubMed]
- W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. 41(29), 6080–6092 (2002). [CrossRef] [PubMed]
- M. Gu, Advanced Optical Imaging Theory, (Springer-Verlag, Berlin) 2000.
- In a real microscopy experiments there is, still, a coverslip between the immersion liquid and the sample medium. Such slab may induce a constant amount of SA that does not depend on the focusing depth. This SA can be compensated statically either with the correction collar, or with a proper index-matching immersion medium. Therefore it is not necessary to consider it in the calculations.
- The highly corrected objectives are equipped with a correction collar to allow adjustment of the central lens group position to coincide with fluctuations in cover-glass thickness.
- C. J. R. Sheppard and C. J. Cogswell, “Effects of aberrating layers and the tube length on confocal imaging properties,” Optik (Stuttg.) 87, 34–38 (1991).
- P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12(2), 325–332 (1995). [CrossRef]
- V. N. Mahajan, “Gaussian apodization and beam propagation,” Prog. Opt. 49, 1–95 (2006). [CrossRef]
- I. Escobar, G. Saavedra, M. Martínez-Corral, and J. Lancis, “Reduction of the spherical aberration effect in high-numerical-aperture optical scanning instruments,” J. Opt. Soc. Am. A 23(12), 3150–3155 (2006). [CrossRef]
- J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. 30(13), 1647–1649 (2005). [CrossRef] [PubMed]
- D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. 13(9), 2126–2133 (1974). [CrossRef] [PubMed]
- M. Born, and E. Wolf, Principles of Optics. (University Press, Cambrigde) 1999, Ch 13.
- C. J. Cogswell, S. V. King, S. R. Prasanna Pavani, D. B. Conkey, and R. H. Cormack, “Microscope imaging capabilities improve using computational optics,” Proc. Focus on Microscopy 09 ( http://focusonmicroscopy.org/2009/PDF/341_Cogswell.pdf )
- S. V. King, A. R. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13(2), 024020 (2008). [CrossRef] [PubMed]

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