Abstract

Numerical refocusing can be seen as a method of compensating the defocus aberration based on deconvolution by inverse filtering [1] in digital holographic microscopy (DHM). It is well-understood in cases when a coherent (ie point and monochromatic) light source such as a collimated laser beam is used [2]. This paper extends the theory to the case of illumination by a quasi-monochromatic extended (spatially incoherent) source. Refocusing methods for spatially incoherent illumination are derived and benefits of this type of illumination are demonstrated. We have proved both theoretically and experimentally that coherent-based refocusing gives incorrect results for extended-source illumination, while results obtained using the newly derived method are correct.

© 2013 Optical Society of America

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References

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  1. M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE 7746774620 (2010).
    [Crossref]
  2. F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38, 7085–7094 (1999).
    [Crossref]
  3. L. Lovicar, J. Komrska, and R. Chmelík, “Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process,” Opt. Express 18, 20585–20594 (2010).
    [Crossref] [PubMed]
  4. R. Barer, “Interference microscopy and mass determination,” Nature (London) 169, 366–367 (1952).
    [Crossref]
  5. Y. Cotte, F. Toy, C. Arfire, S. Kou, D. Boss, I. Bergoënd, and C. Depeursinge, “Realistic 3D coherent transfer function inverse filtering of complex fields,” Biomed. Opt. Express 2, 2216–2230 (2011).
    [Crossref] [PubMed]
  6. F. Dubois, C. Yourassowsky, N. Callens, C. Minetti, and P. Queeckers, “Applications of digital holographic microscopes with partially spatial coherence sources,” JPCS, 139(IOP Publishing, 2008), p. 012027.
  7. T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun 252, 188–201 (2005).
    [Crossref]
  8. T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošt’ák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express 21, 14747–14762 (2013).
    [Crossref] [PubMed]
  9. F. Dubois, M. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. 43, 1131–1139 (2004).
    [Crossref] [PubMed]
  10. F. Dubois, O. Monnom, C. Yourassowsky, and J–C. Legros, “Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies,” Appl. Opt. 41, 2621–2626 (2002).
    [Crossref] [PubMed]
  11. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
    [Crossref] [PubMed]
  12. O. Carriere, J. Hermand, and F. Dubois, “Underwater microorganisms observation with off-axis digital holography microscopy using partially coherent illumination,” in “OCEANS 2011,” (IEEE, 2011), pp. 1–7.
  13. P. Kolman and R. Chmelík, “Coherence-controlled holographic microscope,” Opt. Express 18, 21990–22003 (2010).
    [Crossref] [PubMed]
  14. R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. 53, 2673–2689 (2006).
    [Crossref]
  15. J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996), international editions 1996
  16. Y. Cotte, M. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express 18, 19462–19478 (2010).
    [Crossref] [PubMed]
  17. R. Chmelík and Z. Harna, “Parallel-mode confocal microscope,” Opt. Eng. 38, 1635–1639 (1999).
    [Crossref]
  18. L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE,  7141p. 71411S (2008).
    [Crossref]
  19. Y. Geerts, M. Steyaert, and W. Sansen, Design of multi-bit delta-sigma A/D converters, vol. 686 (Springer, 2002).
  20. F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [Crossref]
  21. E. Olson, “On computing the average orientation of vectors and lines,” in “Robotics and Automation (ICRA)”, Proc. IEEE, pp. 3869–3874, (2011).
  22. C. Agostinelli and U. Lund, R package circular: Circular Statistics (version 0.4-3), CA: Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University, Venice, Italy. UL: Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA (2011).

2013 (1)

2011 (1)

2010 (4)

2008 (1)

L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE,  7141p. 71411S (2008).
[Crossref]

2006 (2)

R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. 53, 2673–2689 (2006).
[Crossref]

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[Crossref] [PubMed]

2005 (1)

T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun 252, 188–201 (2005).
[Crossref]

2004 (1)

2002 (1)

1999 (2)

1978 (1)

F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[Crossref]

1952 (1)

R. Barer, “Interference microscopy and mass determination,” Nature (London) 169, 366–367 (1952).
[Crossref]

Agostinelli, C.

C. Agostinelli and U. Lund, R package circular: Circular Statistics (version 0.4-3), CA: Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University, Venice, Italy. UL: Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA (2011).

Antoš, M.

Arfire, C.

Barer, R.

R. Barer, “Interference microscopy and mass determination,” Nature (London) 169, 366–367 (1952).
[Crossref]

Bergoënd, I.

Boss, D.

Callens, N.

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[Crossref] [PubMed]

F. Dubois, C. Yourassowsky, N. Callens, C. Minetti, and P. Queeckers, “Applications of digital holographic microscopes with partially spatial coherence sources,” JPCS, 139(IOP Publishing, 2008), p. 012027.

Carriere, O.

O. Carriere, J. Hermand, and F. Dubois, “Underwater microorganisms observation with off-axis digital holography microscopy using partially coherent illumination,” in “OCEANS 2011,” (IEEE, 2011), pp. 1–7.

Chmelík, R.

T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošt’ák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express 21, 14747–14762 (2013).
[Crossref] [PubMed]

M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE 7746774620 (2010).
[Crossref]

L. Lovicar, J. Komrska, and R. Chmelík, “Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process,” Opt. Express 18, 20585–20594 (2010).
[Crossref] [PubMed]

P. Kolman and R. Chmelík, “Coherence-controlled holographic microscope,” Opt. Express 18, 21990–22003 (2010).
[Crossref] [PubMed]

L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE,  7141p. 71411S (2008).
[Crossref]

R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. 53, 2673–2689 (2006).
[Crossref]

R. Chmelík and Z. Harna, “Parallel-mode confocal microscope,” Opt. Eng. 38, 1635–1639 (1999).
[Crossref]

Cotte, Y.

Depeursinge, C.

Dostál, Z.

Dubois, F.

Geerts, Y.

Y. Geerts, M. Steyaert, and W. Sansen, Design of multi-bit delta-sigma A/D converters, vol. 686 (Springer, 2002).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996), international editions 1996

Harna, Z.

R. Chmelík and Z. Harna, “Parallel-mode confocal microscope,” Opt. Eng. 38, 1635–1639 (1999).
[Crossref]

Harris, F.

F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[Crossref]

Hermand, J.

O. Carriere, J. Hermand, and F. Dubois, “Underwater microorganisms observation with off-axis digital holography microscopy using partially coherent illumination,” in “OCEANS 2011,” (IEEE, 2011), pp. 1–7.

Istasse, E.

Joannes, L.

Józwicki, R.

T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun 252, 188–201 (2005).
[Crossref]

Kolman, P.

Komrska, J.

Kou, S.

Kozacki, T.

T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun 252, 188–201 (2005).
[Crossref]

Kvasnica, L.

L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE,  7141p. 71411S (2008).
[Crossref]

Legros, J.

Legros, J–C.

Lošt’ák, M.

Lovicar, L.

L. Lovicar, J. Komrska, and R. Chmelík, “Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process,” Opt. Express 18, 20585–20594 (2010).
[Crossref] [PubMed]

L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE,  7141p. 71411S (2008).
[Crossref]

Lund, U.

C. Agostinelli and U. Lund, R package circular: Circular Statistics (version 0.4-3), CA: Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University, Venice, Italy. UL: Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA (2011).

Minetti, C.

F. Dubois, M. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. 43, 1131–1139 (2004).
[Crossref] [PubMed]

F. Dubois, C. Yourassowsky, N. Callens, C. Minetti, and P. Queeckers, “Applications of digital holographic microscopes with partially spatial coherence sources,” JPCS, 139(IOP Publishing, 2008), p. 012027.

Monnom, O.

Novella Requena, M.

Olson, E.

E. Olson, “On computing the average orientation of vectors and lines,” in “Robotics and Automation (ICRA)”, Proc. IEEE, pp. 3869–3874, (2011).

Pavillon, N.

Queeckers, P.

F. Dubois, C. Yourassowsky, N. Callens, C. Minetti, and P. Queeckers, “Applications of digital holographic microscopes with partially spatial coherence sources,” JPCS, 139(IOP Publishing, 2008), p. 012027.

Sansen, W.

Y. Geerts, M. Steyaert, and W. Sansen, Design of multi-bit delta-sigma A/D converters, vol. 686 (Springer, 2002).

Schockaert, C.

Slabý, T.

Steyaert, M.

Y. Geerts, M. Steyaert, and W. Sansen, Design of multi-bit delta-sigma A/D converters, vol. 686 (Springer, 2002).

Toy, F.

Toy, M.

Týc, M.

M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE 7746774620 (2010).
[Crossref]

Yourassowsky, C.

Appl. Opt. (3)

Biomed. Opt. Express (1)

J. Mod. Opt. (1)

R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. 53, 2673–2689 (2006).
[Crossref]

Nature (London) (1)

R. Barer, “Interference microscopy and mass determination,” Nature (London) 169, 366–367 (1952).
[Crossref]

Opt Commun (1)

T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun 252, 188–201 (2005).
[Crossref]

Opt. Eng. (1)

R. Chmelík and Z. Harna, “Parallel-mode confocal microscope,” Opt. Eng. 38, 1635–1639 (1999).
[Crossref]

Opt. Express (5)

Proc. IEEE (1)

F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[Crossref]

Proc. SPIE (2)

L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE,  7141p. 71411S (2008).
[Crossref]

M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE 7746774620 (2010).
[Crossref]

Other (6)

Y. Geerts, M. Steyaert, and W. Sansen, Design of multi-bit delta-sigma A/D converters, vol. 686 (Springer, 2002).

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996), international editions 1996

O. Carriere, J. Hermand, and F. Dubois, “Underwater microorganisms observation with off-axis digital holography microscopy using partially coherent illumination,” in “OCEANS 2011,” (IEEE, 2011), pp. 1–7.

F. Dubois, C. Yourassowsky, N. Callens, C. Minetti, and P. Queeckers, “Applications of digital holographic microscopes with partially spatial coherence sources,” JPCS, 139(IOP Publishing, 2008), p. 012027.

E. Olson, “On computing the average orientation of vectors and lines,” in “Robotics and Automation (ICRA)”, Proc. IEEE, pp. 3869–3874, (2011).

C. Agostinelli and U. Lund, R package circular: Circular Statistics (version 0.4-3), CA: Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University, Venice, Italy. UL: Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA (2011).

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Figures (10)

Fig. 1
Fig. 1 Cross-sections of support of CTFs for an objective with NA = 0.5 in the reflected-light (HR) and transmitted-light mode (HT). Calculated for (a) point-source illumination and (b) extended-source illumination with NAi = NA, BL denotes the longitudinal frequency bandwidth (see section 4.2). CTFs are rotationally symmetric with respect to the Z axis.

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Fig. 2
Fig. 2 Dependence of |Ht(0, Qt; z)| (ie transmission of spatial frequencies) on defocus. Computed for RDHM, NA = NAi = 0.25 (full lines) and NAi = 0.05 (dashed lines). Narrow-band filter with central λ = 550nm, full width in the half maximum (FWHM) = 10 nm.

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Fig. 3
Fig. 3 Phase of the 2D CTF for defocus of 8 μm) for TDHM (a) and RDHM (b). Monochromatic extended-source illumination for selected values of NAi. NA = 0.25, λ = 550nm. Shifted so that argHt = 0 for Qt = 0.

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Fig. 4
Fig. 4 Visualized |wt(qt; 0)| of the field of view recorded by RDHM in the extended-source imaging setup (setup E, see the list in section 4.1). Edges of the field of view were apodized (see section 4.4). The stroke in the upper right corner is a diffraction grating artifact, the white rectangle points out the section of the sample shown further in the paper in more detail.

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Fig. 5
Fig. 5 The unwrapped phase difference Pφ (Qt; 8 μm) between Wt(Qt; 0) and Wt(Qt; 8 μm). Both phase shifts are unwrapped in such way that the zero frequency has phase equal to 0 in both cases. (a) extended-source, setup E and (b) quasi-point-source P.

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Fig. 6
Fig. 6 Comparison of in-focus sample images (see the full sample in Fig. 4): (a) reference intensity image (setup W), (b) RDHM (setup E) filtered extended-source amplitude image | t c * ( q t ; 0 ) |, (c) RDHM (setup E) extended-source amplitude image |wt(qt; 0)| and (d) RDHM (setup P) point-source amplitude image |wt(qt; 0)|. The black circle has radius of 1.22(NA + NAi)/λ, see section 4.3. The colormap is linear and adjusted separately for each image to provide the highest contrast possible.

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Fig. 7
Fig. 7 Comparison of close-ups corresponding to Fig. 6 (center of the upper edge): (a) reference intensity image (setup W), (b) RDHM (setup E) filtered extended-source amplitude image | t c * ( q t ; 0 ) |, (c) RDHM (setup E) extended-source (raw) amplitude image |wt(qt; 0)|. The colormap is common for all three images. The cross-section at the bottom is taken along the path marked by the white bar in individual images above.

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Fig. 8
Fig. 8 Phase difference Pφ′(Qt) corresponding to defocus (a) 8 μm (see also Fig. 5) and (b) −8 μm. Solid lines correspond to experimental data, dashed lines to the theory and the bands show 95% confidence interval for the experimental data. The experimental data has been smoothed out to reduce the quantization error by applying convolution with a Gaussian kernel of σ = 0.05NA/λ.

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Fig. 9
Fig. 9 Refocusing in the setup P (quasi-monochromatic, quasi-point-source) amplitude images: (a) in-focus deconvolved | t c * ( q t ; 0 ) |, (b) defocused by 8 μm |wt(qt; 8 μm)|, (c) phase refocused | t p * ( q t ; 8   μ m ) | using the correct propagator and (d) phase refocused | t f * ( q t ; 8   μ m ) | using the approximative (coherent) propagator. See Fig. 4 for the colorbar. The cross-section on the right is took along the path marked by the white bar in individual images on the left.

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Fig. 10
Fig. 10 Refocusing in the setup E (quasi-monochromatic, extended-source) amplitude images: (a) in-focus deconvolved | t c * ( q t ; 0 ) |, (b) defocused by 8 μm |wt(qt; 8 μm)|, (c) phase refocused | t p * ( q t ; 8   μ m ) | using the correct propagator and (d) phase refocused | t f * ( q t ; 8   μ m ) | using inappropriate (coherent) propagator. See Fig. 4 for the colorbar. The cross-section on the right is took along the path marked by the white bar in individual images on the left.

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Equations (16)

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| Δ z | ≪ λ 1 − NA i 2 NA   NA i ,
O ( Q t ) ∝ S ( c | Δ z | Q t ) ,
W t ( Q t ; z ) = ∬ ℝ 2 w t ( q t ; z ) exp ( − 2 π i q t ⋅ Q t )       d 2 q t ,
w ( q ) = ∭ ℝ 3 H ( Q ) T ( Q ) exp ( 2 π i q ⋅ Q )       d 3 Q ,
W t ( Q t ; z ) = T t ( Q t ) H t ( Q t ; z ) ,
H t ( Q t ; z ) = ∫ ℝ H ( Q ) exp ( 2 π i   z Z )       d Z .
W t ( Q t ; z + Δ z ) = W t ( Q t ; z ) P ( Q t ; Δ z ; z ) ,
P ( Q t ; Δ z ; z ) = H t ( Q t ; z + Δ z ) H t ( Q t ; z ) ,                     for     all     Q t   so   that   H t ( Q t ; z ) ≠ 0.
H ( Q ) = P o ( Q t )     δ [ K + Z + ( K 2 − Q t 2 ) 1 2 ]             for               Q t ≤ NA / λ ,
P ( Q t ; Δ z ) = exp [ 2 π i Δ z ( K 2 − Q t 2 ) 1 2 ]               for       Q t ≤ NA / λ .
T f * ( Q t ; Δ z ) = W t ( Q t ; Δ z ) P ( Q t ; Δ z )
G { f ; c } = exp ( i arg f ) | f | + c 1 + c ,               c ∈ [ 0 , ∞ ) .
P ( Q t ; Δ z ) = H t ( Q t ; Δ z ) G { H t ( Q t ; 0 ) ; c } .
T * ( Q t ; Δ z ; c 1 ) = W t ( Q t ; Δ z ) G { H t ; c 1 max | H t | } ,
N t = 4 NA λ D t ,                           N L = 2 1 − 1 − NA 2 λ D L ,
P φ ( Q t ; Δ z ) = arg W t ( Q t ; Δ z ) W t ( Q t ; 0 ) .

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