Abstract

In this work, a wavefront encoded (WFE) imaging system built using a squared cubic phase mask, designed to reduce the sensitivity of the imaging system to spherical aberration, is investigated. The proposed system allows the use of a space-invariant image restoration algorithm, which uses a single PSF, to restore intensity distribution in images suffering aberration, such as sample–induced aberration in thick tissue. This provides a computational advantage over depth-variant image restoration algorithms developed previously to address this aberration. Simulated PSFs of the proposed system are shown to change up to 25% compared to the 0 µm depth PSF (quantified by the structural similarity index) over a 100 µm depth range, while the conventional system PSFs change up to 84%. Results from experimental test-sample images show that restoration error is reduced by 29% when the proposed WFE system is used instead of the conventional system over a 30 µm depth range.

© 2016 Optical Society of America

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Experimental validation of a customized phase mask designed to enable efficient computational optical sectioning microscopy through wavefront encoding

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Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy

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Appl. Opt. 54(29) 8587-8595 (2015)

References

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2016 (1)

S. Ghosh and C. Preza, “Three-Dimensional Block-Based Restoration Integrated with Wide-field Fluorescence Microscopy for the Investigation Of Thick Specimens with Spatially Variant Refractive Index,” J. Biomed. Opt. 21(4), 046010 (2016).
[Crossref] [PubMed]

2015 (6)

B. Kim and T. Naemura, “Blind Depth-variant Deconvolution of 3D Data in Wide-field Fluorescence Microscopy,” Sci. Rep. 5, 9894 (2015).
[Crossref] [PubMed]

A. Wong, X. Y. Wang, and M. Gorbet, “Bayesian-based deconvolution fluorescence microscopy using dynamically updated nonstationary expectation estimates,” Sci. Rep. 5, 10849 (2015).
[Crossref] [PubMed]

N. Patwary, S. V. King, and C. Preza, “3D microscope imaging robust to restoration artifacts introduced by optically thick specimens,” Proc. SPIE 9330, 93300O (2015).
[Crossref]

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

N. Patwary and C. Preza, “Image restoration for three-dimensional fluorescence microscopy using an orthonormal basis for efficient representation of depth-variant point-spread functions,” Biomed. Opt. Express 6(10), 3826–3841 (2015).
[Crossref] [PubMed]

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54(29), 8587–8595 (2015).
[Crossref] [PubMed]

2014 (3)

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

N. Patwary, A. Doblas, S. V. King, and C. Preza, “Reducing depth induced spherical aberration in 3D widefield fluorescence microscopy by wavefront coding using the SQUBIC phase mask,” Proc. SPIE 8949, 894911 (2014).
[Crossref]

L. Silvestri, L. Sacconi, and F. S. Pavone, “Correcting spherical aberrations in confocal light sheet microscopy: A theoretical study,” Microsc. Res. Tech. 77(7), 483–491 (2014).
[Crossref] [PubMed]

2012 (2)

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

M. Persson, D. Engström, and M. Goksör, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express 20(20), 22334–22343 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (1)

2009 (1)

2004 (2)

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] [PubMed]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

2003 (1)

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216(1), 55–63 (2003).
[Crossref]

1999 (1)

1998 (1)

1996 (1)

J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, 199–208 (1996).
[Crossref]

1992 (1)

1990 (1)

1971 (1)

I. J. Good, “Non-Parametric Roughness Penalty for Probability Densities,” Nature 229(1), 29–30 (1971).

Agard, D. A.

Arigovindan, M.

Babcock, H.

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Cathey, W. T.

Clouvel, G.

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

Colicchio, B.

Conchello, J.-A.

Dieterlen, A.

Doblas, A.

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54(29), 8587–8595 (2015).
[Crossref] [PubMed]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

N. Patwary, A. Doblas, S. V. King, and C. Preza, “Reducing depth induced spherical aberration in 3D widefield fluorescence microscopy by wavefront coding using the SQUBIC phase mask,” Proc. SPIE 8949, 894911 (2014).
[Crossref]

Dowski, E.

Ducommun, B.

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

Engström, D.

Escande, P.

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

Escobar, I.

Frongia, C.

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

Ghosh, S.

S. Ghosh and C. Preza, “Three-Dimensional Block-Based Restoration Integrated with Wide-field Fluorescence Microscopy for the Investigation Of Thick Specimens with Spatially Variant Refractive Index,” J. Biomed. Opt. 21(4), 046010 (2016).
[Crossref] [PubMed]

Gibson, S. F.

Goksör, M.

Good, I. J.

I. J. Good, “Non-Parametric Roughness Penalty for Probability Densities,” Nature 229(1), 29–30 (1971).

Gorbet, M.

A. Wong, X. Y. Wang, and M. Gorbet, “Bayesian-based deconvolution fluorescence microscopy using dynamically updated nonstationary expectation estimates,” Sci. Rep. 5, 10849 (2015).
[Crossref] [PubMed]

Haeberlé, O.

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216(1), 55–63 (2003).
[Crossref]

Hansen, E. W.

Kim, B.

B. Kim and T. Naemura, “Blind Depth-variant Deconvolution of 3D Data in Wide-field Fluorescence Microscopy,” Sci. Rep. 5, 9894 (2015).
[Crossref] [PubMed]

King, S. V.

N. Patwary, S. V. King, and C. Preza, “3D microscope imaging robust to restoration artifacts introduced by optically thick specimens,” Proc. SPIE 9330, 93300O (2015).
[Crossref]

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54(29), 8587–8595 (2015).
[Crossref] [PubMed]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

N. Patwary, A. Doblas, S. V. King, and C. Preza, “Reducing depth induced spherical aberration in 3D widefield fluorescence microscopy by wavefront coding using the SQUBIC phase mask,” Proc. SPIE 8949, 894911 (2014).
[Crossref]

Lanni, F.

Lorenzo, C.

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

Maalouf, E.

Martínez-Corral, M.

Martínez-Cuenca, R.

Masson, A.

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

McGowan, J.

McNally, J. G.

J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, 199–208 (1996).
[Crossref]

Mukamel, E. A.

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

Naemura, T.

B. Kim and T. Naemura, “Blind Depth-variant Deconvolution of 3D Data in Wide-field Fluorescence Microscopy,” Sci. Rep. 5, 9894 (2015).
[Crossref] [PubMed]

Patwary, N.

N. Patwary and C. Preza, “Image restoration for three-dimensional fluorescence microscopy using an orthonormal basis for efficient representation of depth-variant point-spread functions,” Biomed. Opt. Express 6(10), 3826–3841 (2015).
[Crossref] [PubMed]

N. Patwary, S. V. King, and C. Preza, “3D microscope imaging robust to restoration artifacts introduced by optically thick specimens,” Proc. SPIE 9330, 93300O (2015).
[Crossref]

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54(29), 8587–8595 (2015).
[Crossref] [PubMed]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

N. Patwary, A. Doblas, S. V. King, and C. Preza, “Reducing depth induced spherical aberration in 3D widefield fluorescence microscopy by wavefront coding using the SQUBIC phase mask,” Proc. SPIE 8949, 894911 (2014).
[Crossref]

Pavone, F. S.

L. Silvestri, L. Sacconi, and F. S. Pavone, “Correcting spherical aberrations in confocal light sheet microscopy: A theoretical study,” Microsc. Res. Tech. 77(7), 483–491 (2014).
[Crossref] [PubMed]

Persson, M.

Preza, C.

S. Ghosh and C. Preza, “Three-Dimensional Block-Based Restoration Integrated with Wide-field Fluorescence Microscopy for the Investigation Of Thick Specimens with Spatially Variant Refractive Index,” J. Biomed. Opt. 21(4), 046010 (2016).
[Crossref] [PubMed]

N. Patwary, S. V. King, and C. Preza, “3D microscope imaging robust to restoration artifacts introduced by optically thick specimens,” Proc. SPIE 9330, 93300O (2015).
[Crossref]

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54(29), 8587–8595 (2015).
[Crossref] [PubMed]

N. Patwary and C. Preza, “Image restoration for three-dimensional fluorescence microscopy using an orthonormal basis for efficient representation of depth-variant point-spread functions,” Biomed. Opt. Express 6(10), 3826–3841 (2015).
[Crossref] [PubMed]

N. Patwary, A. Doblas, S. V. King, and C. Preza, “Reducing depth induced spherical aberration in 3D widefield fluorescence microscopy by wavefront coding using the SQUBIC phase mask,” Proc. SPIE 8949, 894911 (2014).
[Crossref]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

S. Yuan and C. Preza, “Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging,” Opt. Express 19(23), 23298–23314 (2011).
[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] [PubMed]

Saavedra, G.

Sacconi, L.

L. Silvestri, L. Sacconi, and F. S. Pavone, “Correcting spherical aberrations in confocal light sheet microscopy: A theoretical study,” Microsc. Res. Tech. 77(7), 483–491 (2014).
[Crossref] [PubMed]

Sánchez-Ortiga, E.

Sedat, J. W.

Shaevitz, J.

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Silvestri, L.

L. Silvestri, L. Sacconi, and F. S. Pavone, “Correcting spherical aberrations in confocal light sheet microscopy: A theoretical study,” Microsc. Res. Tech. 77(7), 483–491 (2014).
[Crossref] [PubMed]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Tucker, S.

Wang, X. Y.

A. Wong, X. Y. Wang, and M. Gorbet, “Bayesian-based deconvolution fluorescence microscopy using dynamically updated nonstationary expectation estimates,” Sci. Rep. 5, 10849 (2015).
[Crossref] [PubMed]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Wong, A.

A. Wong, X. Y. Wang, and M. Gorbet, “Bayesian-based deconvolution fluorescence microscopy using dynamically updated nonstationary expectation estimates,” Sci. Rep. 5, 10849 (2015).
[Crossref] [PubMed]

Yuan, S.

Zhuang, X.

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

Appl. Opt. (2)

Biomed. Opt. Express (1)

Biophys. J. (1)

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

IEEE Trans. Image Process. (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

J. Biomed. Opt. (1)

S. Ghosh and C. Preza, “Three-Dimensional Block-Based Restoration Integrated with Wide-field Fluorescence Microscopy for the Investigation Of Thick Specimens with Spatially Variant Refractive Index,” J. Biomed. Opt. 21(4), 046010 (2016).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (4)

Microsc. Res. Tech. (1)

L. Silvestri, L. Sacconi, and F. S. Pavone, “Correcting spherical aberrations in confocal light sheet microscopy: A theoretical study,” Microsc. Res. Tech. 77(7), 483–491 (2014).
[Crossref] [PubMed]

Nature (1)

I. J. Good, “Non-Parametric Roughness Penalty for Probability Densities,” Nature 229(1), 29–30 (1971).

Opt. Commun. (1)

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216(1), 55–63 (2003).
[Crossref]

Opt. Express (5)

Proc. SPIE (4)

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, 199–208 (1996).
[Crossref]

N. Patwary, S. V. King, and C. Preza, “3D microscope imaging robust to restoration artifacts introduced by optically thick specimens,” Proc. SPIE 9330, 93300O (2015).
[Crossref]

N. Patwary, A. Doblas, S. V. King, and C. Preza, “Reducing depth induced spherical aberration in 3D widefield fluorescence microscopy by wavefront coding using the SQUBIC phase mask,” Proc. SPIE 8949, 894911 (2014).
[Crossref]

Sci. Rep. (3)

A. Masson, P. Escande, C. Frongia, G. Clouvel, B. Ducommun, and C. Lorenzo, “High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM,” Sci. Rep. 5, 16898 (2015).
[Crossref] [PubMed]

B. Kim and T. Naemura, “Blind Depth-variant Deconvolution of 3D Data in Wide-field Fluorescence Microscopy,” Sci. Rep. 5, 9894 (2015).
[Crossref] [PubMed]

A. Wong, X. Y. Wang, and M. Gorbet, “Bayesian-based deconvolution fluorescence microscopy using dynamically updated nonstationary expectation estimates,” Sci. Rep. 5, 10849 (2015).
[Crossref] [PubMed]

Other (3)

J. R. Swedlow, J. W. Sedat, and D. A. Agard, Deconvolution of images and spectra (Academic Press, 1997), Chap. 9.

N. Patwary and C. Preza, “Wavefront encoded computational optical sectioning microscopy reduces depth variability in 3D imaging,” in Imaging and Applied Optics 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper CM2E.4.

Computational Imaging Research Laboratory, “Computational Optical Sectioning Microscopy Open Source (COSMOS) software package,” http://cirl.memphis.edu/COSMOS ].

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Figures (8)

Fig. 1
Fig. 1

SQUBIC phase mask. Unwrapped mask (gray level presentation of the phase values onto the pupil) for: (a) NA = 0.8, A = 88; (b) NA = 1.4, A = 20. (c) Phase depth for low NA and high NA MO; (d) and (e) show the wrapped phase masks corresponding to (a) and (b), respectively.

Fig. 2
Fig. 2

Effect of residual aberration (RA) in the conventional imaging system. (a) Experimental PSF without any SA ( ΔRI=0 ); (b) Theoretical PSF without SA; (c) Theoretical PSF computed considering the residual aberration (RA); Intensity profiles thorough the center of the XZ sectional view images. Lens: 63X/1.4 NA oil immersion; Specimen embedding medium is ProLong® Diamond (RI = 1.47 when cured); Emission wavelength: 515 nm.

Fig. 3
Fig. 3

Depth invariability of the SQUBIC PSF compared to the conventional PSF. Simulated XZ cut view images of the PSFs at depth 0 µm and 100 µm: (a) & (b) conventional, (c) & (d) SQUBIC (A = 20), and (e) & (f) SQUBIC (A = 80). (g) The SSIM between the PSF computed at 0 µm depth and PSFs computed at other depths for the conventional and the SQUBIC imaging system for A = 20, 50, and 80. Lens: 63X/1.4 NA oil-immersion, specimen mounting medium is Prolong Diamond with RI = 1.47 when cured, and the emission wavelength is 515 nm.

Fig. 4
Fig. 4

Impact of system depth sensitivity on restoration of simulated images from SQUBIC-WFE and conventional systems. (a) XZ sectional view of a simulated object with a 3 µm in diameter sphere centered at 30 µm; (b) Correlation coefficients between the simulated true object and the corresponding restored images in the case of conventional and SQUBIC system (beads centered at depths, d= d o ±mΔd where, d 0 =30μm, Δd=2μm, and 5m5 ). All the images are restored using a PSF computed at 30 µm depth for both the systems. Lens: 20X/0.8NA dry lens, specimen embedding medium is water (RI = 1.33), and the emission wavelength is 515 nm.

Fig. 5
Fig. 5

Reduction of image restoration artifacts while imaging deep into the sample using the SQUBIC-WFE imaging system in simulations: XZ cut view images from: (a) an object with five 3 µm in diameter spherical beads in a depth range from 10 µm to 50 µm; (b) Conventional simulated image; (c) SQUBIC-WFE simulated image; Restored images using the PSFs computed at 30 µm depth in the case of (d) conventional imaging, and (e) SQUBIC-WFE imaging. (f) Intensity profiles comparison through the center of the object and the restored images as show in Fig. (a). Lens: 20X/0.8NA air-immersion, specimen embedding medium is water (RI = 1.33), and the emission wavelength is 515 nm. SQUBIC design parameter A = 88.

Fig. 6
Fig. 6

Comparison between the conventional and SQUBIC-WFE intermediate images. The XZ sectional view images of a 6 µm in diameter spherical shell having shell thickness equal to 1 µm in the case of: (a)-(c) Conventional system, and (d)-(f) SQUBIC (A = 20) WFE system. (a) and (d) are experimentally acquired images from depth 3 µm; whereas (b-c) and (e-f) are corresponding simulated images at depths 3 µm and 27 µm, respectively. Lens: 63X/1.4 NA oil immersion; Specimen embedding medium is prolong diamond (RI = 1.47 when cured); Emission wavelength: 515 nm. conventional case the intensity distributions of the images at depths 3 µm and 27 µm are different in simulation; on the other hand, in the SQUBIC-WFE case the intensity distribution of the simulated images not only match with the experiment, but the intensity distribution is also similar at depths 3 µm and 27 µm.

Fig. 7
Fig. 7

Performance comparison between the conventional and the SQUBIC-WFE system in terms of image restoration through simulations. (a) XZ sectional views of the conventional restored images of the spherical shells centered at depths 3 µm, 12 µm and 27 µm an restored using PSFs at 3 µm, 12 µm and 27 µm; (b) Corresponding restored images as (a) in the case of SQUBIC (A = 80) system. The values on the images are the correlation coefficients between the true object and the corresponding restored images. (c) Quantitative comparison in terms of the standard deviation σ(ρ) of the correlation values. Mean value of σ(ρ) is indicated by color coded (corresponding to each case) horizontal lines on the bar plot. (d) Intensity profile comparison between the conventional and the SQUBIC images at 27 µm, restored with the corresponding PSFs at 3 µm. Lens: 63X/1.4 NA oil immersion; RI of the specimen embedding is equal to 1.47; emission wavelength is 515 nm.

Fig. 8
Fig. 8

Performance comparison between the conventional and the SQUBIC-WFE system through the experimental image restoration. (a) Restored images of three different objects located at depths d0 µm, d0 + 12 µm, and d0 + 27 µm using the PSFs at 0 µm (unaberrated PSF), 12 µm and 27 µm depths. (b) Corresponding restored images in the case of SQUBIC-WFE system. The numbers on the images are the correlation coefficients. The corresponding best restored images (from a visual inspection) are used as reference images in each case to compute the correlations. Lens: 63X/1.4 NA oil immersion; Specimen embedding medium is prolong diamond (RI = 1.47 when cured); Emission wavelength: 515 nm; Design parameter of the SQUBIC phase mask A = 20.

Tables (1)

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Table 1 List of Simulations and Experiments Performed in the Investigation Studies

Equations (9)

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φ( f ˜ x , f ˜ y )={ 2πA [ 1( f ˜ x 2 + f ˜ y 2 ) sin 2 (α) 1 1cos(α) + 1 2 ] 3 ; 0 ( f ˜ x 2 + f ˜ y 2 ) 1 0 otherwise
h z i , z o (x,y)= | F 1 { H( f x , f y ) e j(2π/λ)W( f x , f y ; z i , z o ) e jφ( f x , f y ) } | 2 ,
h(x,y,z; z o )= h z, z o (x,y).
g( x i )= O h( x i x o , y i y o , z i ; z o )f( x o ) d x o ,
s k+1 (x,y,z)= s k (x,y,z) H z o [ h(x,y,z; z o ) d(x,y,z) g k (x,y,z) ],
L[ s(x)|g(x) ]+γR[ s(x) ],
d pixles =[ 2(NA)Δx λ ]κ,
Γ(X,Y)= ( 2 μ X μ Y + c 1 )( 2 σ XY + c 2 ) ( μ X 2 + μ Y 2 + c 1 )( σ X 2 + σ Y 2 + c 2 ) ,
ρ(X,Y)= σ XY + c 3 σ X σ Y + c 3 ,

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