Abstract

Wavefront encoding (WFE) with different cubic phase mask designs was investigated in engineering 3D point-spread functions (PSF) to reduce their sensitivity to depth-induced spherical aberration (SA) which affects computational complexity in 3D microscopy imaging. The sensitivity of WFE-PSFs to defocus and to SA was evaluated as a function of phase mask parameters using mean-square-error metrics to facilitate the selection of mask designs for extended-depth-of-field (EDOF) microscopy and for computational optical sectioning microscopy (COSM). Further studies on pupil phase contribution and simulated WFE-microscope images evaluated the engineered PSFs and demonstrated SA insensitivity over sample depths of 30 μm. Despite its low sensitivity to SA, the successful WFE design for COSM maintains a high sensitivity to defocus as it is desired for optical sectioning.

© 2011 OSA

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2010 (3)

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010).
[CrossRef]

G. Carles, A. Carnicer, and S. Bosch, “Phase mask selection in wavefront coding systems: A design approach,” Opt. Lasers Eng. 48(7-8), 779–785 (2010).
[CrossRef]

2009 (1)

2008 (1)

2007 (2)

Z. Kam, P. Kner, D. A. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[CrossRef] [PubMed]

J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A 24(9), 2622–2627 (2007).
[CrossRef] [PubMed]

2005 (1)

J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[CrossRef] [PubMed]

2004 (4)

2003 (2)

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(1-3), 55–63 (2003).
[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]

1999 (2)

1995 (2)

1994 (1)

1992 (1)

1984 (1)

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[CrossRef] [PubMed]

Agard, D. A.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Z. Kam, P. Kner, D. A. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[CrossRef] [PubMed]

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[CrossRef] [PubMed]

Bosch, S.

G. Carles, A. Carnicer, and S. Bosch, “Phase mask selection in wavefront coding systems: A design approach,” Opt. Lasers Eng. 48(7-8), 779–785 (2010).
[CrossRef]

Boulanger, J.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Bressan, D.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Carles, G.

G. Carles, A. Carnicer, and S. Bosch, “Phase mask selection in wavefront coding systems: A design approach,” Opt. Lasers Eng. 48(7-8), 779–785 (2010).
[CrossRef]

Carlton, P. M.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Carnicer, A.

G. Carles, A. Carnicer, and S. Bosch, “Phase mask selection in wavefront coding systems: A design approach,” Opt. Lasers Eng. 48(7-8), 779–785 (2010).
[CrossRef]

Cathey, W. T.

Conchello, J. A.

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

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Conchello, J.-A.

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Dowski, E.

Dowski, E. R.

Escobar, I.

Feng, H. J.

H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010).
[CrossRef]

Fletcher, D. A.

Gibson, S. F.

Gordon-Messer, S.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Gustafsson, M.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Haase, S.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Haber, J. E.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[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(1-3), 55–63 (2003).
[CrossRef]

Harvey, A. R.

Kam, Z.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Z. Kam, P. Kner, D. A. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[CrossRef] [PubMed]

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Kervrann, C.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Kner, P.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Z. Kam, P. Kner, D. A. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[CrossRef] [PubMed]

Lanni, F.

Li, Q.

H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010).
[CrossRef]

Li, Y. C.

H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010).
[CrossRef]

Lichtman, J. W.

J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[CrossRef] [PubMed]

Martínez-Corral, M.

Martinez-Cuenca, R.

Matsuda, A.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas., “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11(3), 1056–1067 (1994).
[CrossRef] [PubMed]

Mezouari, S.

Nemeth, G.

Pauca, V. P.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).
[CrossRef]

Pavani, S. R. P.

Piestun, R.

Plemmons, R. J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).
[CrossRef]

Prasad, S.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).
[CrossRef]

Preza, C.

Saavedra, G.

Salamero, J.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Sanchez-Ortiga, E.

Sedat, J. W.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Z. Kam, P. Kner, D. A. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[CrossRef] [PubMed]

Shaevitz, J. W.

Shao, L.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Sherif, S. S.

Sibarita, J.-B.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Thomas, L. J.

Torgersen, T. C.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).
[CrossRef]

Török, P.

Tucker, S. C.

Uzawa, S.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

van der Gracht, J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).
[CrossRef]

Varga, P.

Winoto, L.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010).
[CrossRef] [PubMed]

Xu, Z. H.

H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010).
[CrossRef]

Zhao, H.

H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010).
[CrossRef]

Annu. Rev. Biophys. Bioeng. (1)

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
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Figures (12)

Fig. 1
Fig. 1

Phase wrapped from –π to π for phase-mask designs computed within a circular pupil aperture. Phase pattern for: (a) CPM (α = 30); (b) SCPM (α = 30, β = α/2 and ω = π/2); (c) GCPM (α = 30 and β = −3α); (d) GCPM selected for EDOF microscopy (α = 150 and β = −3α); and (e) GCPM selected for COSM (α = 50 and β = -α).

Fig. 2
Fig. 2

XY and XZ view of WFE-PSFs for a 60x/1.2NA oil-immersion lens using different types of mask designs with and without SA. PSFs without SA, i.e., zo = 0 μm (top row) computed using a: (a) CCA; (b) CPM (α = 30); (c) GCPM (α = 30 and β = −3α); and (d) SCPM (α = 30, β = α/2 and ω = π/2). PSFs with SA for zo = 20 μm (bottom row) computed with a: (e) clear circular aperture; (f) CPM (α = 30); (g) GCPM (α = 30 and β = −3α); and (h) SCPM (α = 30, β = α/2 and ω = π/2).

Fig. 10
Fig. 10

Simulated unprocessed EDOF images of 3 objects with different amounts of SA, computed using a model for a high-NA WFE-microscope that includes the selected GCPM-EDOF mask design. (a) XZ (top) and XY (bottom) projection views of Object 1. (b) XZ center cut views of the simulated image when the top bead is at depth 0 μm (top), 10 μm (middle) and 20 μm (bottom). (c) XY cut views of simulated images at different distances away from the best focal plane shown by the dotted lines in the XZ image of (b): zi = −5 μm (left); 0 μm (middle); and 5 μm (right).

Fig. 11
Fig. 11

Simulated intermediate images of the 3-bead objects with different amounts of depth-induced SA computed using a model for a high-NA WFE-microscope that includes the selected GCPM-COSM mask design. (a) XZ center cut views of simulated image when the top bead in the object is at depth: 0 μm (top); 10 μm (middle); and 20 μm (bottom). (b) XY cut views of simulated images at different distances away from the best focal plane shown by the dotted lines in the top XZ image of (a): zi = −5 μm (left); 0 μm (middle); and 5 μm (right).

Fig. 3
Fig. 3

NMSE plots for CPM-PSFs for different α values reported in panel (c). NMSE2D values plotted vs. defocus reported as the distance from the designed best focal plane (FP), for PSFs: (a) without SA (zo = 0 μm); and (b) with SA (zo = 20 μm). (c) NMSE3D computed for PSFs with increasing amount of SA, plotted vs. zo. A semi-log scale is used for plotting the NMSE2D to facilitate visualization of the small differences between the curves especially near the minimum values of the curves.

Fig. 4
Fig. 4

NMSE plots for GCPM-PSFs for different α values reported in panel (a). NMSE2D computed for PSFs without SA and plotted vs. defocus: (a) β = -α/3; (b) β = -α; and (c) β =  3α. NMSE3D computed for PSFs with increasing amount of SA and plotted vs. zo: (d) for β = −3α and different values of α; and (e) for α = 150 and different values of β. A semi-log scale is used for plotting the NMSE2D to facilitate visualization of the small differences between the curves.

Fig. 5
Fig. 5

NMSE plots for SCPM-PSFs for different α values when β = α/3 and ω = -π/2. (a) NMSE2D computed for PSFs without SA and plotted vs. defocus using a semi-log scale. (b) NMSE3D computed for PSFs with increasing amount of SA and plotted vs. zo.

Fig. 6
Fig. 6

Comparison of NMSE plots for WFE-PSFs engineered with selected masks designs suitable for EDOF microscopy using a CPM (α = 200), a GCPM (α = 150 and β = −3α) and a SCPM (α = 150, β = α/3 and ω = -π/2). NMSE2D computed from PSFs without SA (a) and with SA when zo = 20 μm (b), plotted using a semi-log scale vs. defocus. (c) NMSE3D computed for PSFs with increasing amount of SA plotted vs. zo.

Fig. 7
Fig. 7

Comparison of NMSE plots for WFE-PSFs engineered with selected masks designs suitable for COSM using a CPM (α = 50), a GCPM (α = 50 and β = -α) and a SCPM (α = 30, β = α/2 and ω = -π/4). NMSE2D computed from PSFs without SA (a) and with SA when zo = 20 μm (b), plotted vs. defocus. (c) NMSE3D computed for PSFs with increasing amount of SA plotted vs. zo. (d) R plotted as a function of α. The straight dashed line marks the CCA R value.

Fig. 8
Fig. 8

Comparison of the GCPM phase to CCA-ATF phase changes due to defocus and SA. Phase profile vs. normalized frequency, f x , for f y =0 of the CCA-ATF phase due to: (a) defocus only, for different defocus values; (b) SA only, for different amounts of SA. (c) Comparison of the phase change due to: SA (zi = 0 μm, zo = 30 μm); defocus (zi = −15 μm, zo = 0 μm), the selected GCPM-EDOF phase mask (α = 150, β = −3α), and the selected GCPM-COSM phase mask (α = 50, β = -α). (d) Derivative with respect to f x of the phase functions shown in (c).

Fig. 9
Fig. 9

XY and XZ cut-view images through the center of 3D WFE-PSFs for a 60x/1.2NA oil-immersion lens computed without SA for zo = 0 μm (top row) and with SA for zo = 10 μm (second row), and for zo = 20 μm (third row) using different types of mask designs: (a) CCA, i.e. a conventional PSF suitable for COSM; (b) GCPM (α = 50, β = -α) suitable for COSM; and (c) GCPM (α = 150, β = −3α) suitable for EDOF microscopy.

Fig. 12
Fig. 12

The effect of α on the CPM-PSF (a & b), on the GCPM-PSF (c & d) and on frequency content (e). PZ cut views (where P is the diagonal line along x = y) of the WFE-PSF using: (a) CPM with α = 30; (b) CPM with α = 150; (c) GCPM with α = 30 and β = −3α and (d) GCPM with α = 150, and β = −3α. We note that the CPM-PSF image in (a) looks qualitatively similar to the image in Fig. 4.8(a) of [21]. (e) Normalized MTF intensity profiles vs. f p (the frequency along the diagonal line f x = f y ). MTFs shown are without SA (zo = 0 μm) and with SA (zo = 20 μm) using different masks: (i) CCA; (ii) GCPM with α = 150 and β = −3α; and (iii) GCPM with α = 30 and β = −3α.

Tables (1)

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Table 1 Equations for Evaluation Metrics

Equations (5)

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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 ,
θ( f x , f y )= 2π λ W( f x , f y ; z i, z o )+ϕ( f x , f y ).
h WFE (x,y, z i , z o )= h z i , z o (x,y),
g( x i )= O h WFE ( x i x o , y i y 0 , z i , z o )s( x o ) d x o ,
R= NMSE ¯ 2D / NMSE ¯ 3D ,

Metrics