Abstract

In a number of previous studies on light focusing, the asymmetric axial intensity distribution with intensity peak shifted away from the paraxial focal plane was demonstrated for lenses working with a low Fresnel number. Here, the axial asymmetry of the three-dimensional point spread function (PSF) and the aberration effects are examined in a magnified phase-shifting holographic imaging achieved by the mismatch of reference and reconstruction waves. In the analysis, an optimal combination of experimental parameters and the range of applicable lateral magnifications are found for which the axial asymmetry of the PSF is not apparent and the aberration effects are acceptable. The focal shift and the axial asymmetry of the PSF and the effects of holographic aberrations are evaluated by approximate quantitative criteria whose validity is verified in exact numerical models and experiments. The optimal design of in-line holographic geometry is demonstrated by reconstructing the three-dimensional PSFs and the image of the resolution target recorded in the experimental setup using a spatial light modulator.

© 2017 Optical Society of America

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References

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2016 (1)

2015 (1)

2012 (4)

2011 (2)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nature Photon. 5, 335–342 (2011).
[Crossref]

X. Yu, J. Hong, Ch. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53(11), 112306 (2011).
[Crossref]

2010 (2)

2007 (1)

2006 (1)

2005 (1)

2004 (2)

2003 (1)

2002 (1)

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9) R85–R101 (2002).
[Crossref]

1997 (1)

1982 (1)

1981 (2)

J. H. Erkkila and M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. A 71(7), 904–905 (1981).
[Crossref]

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[Crossref]

1965 (1)

W. Reinhard, “Magnification and Third-Order Aberrations in Holography,” J. Opt. Soc. Am. A 55(8), 987–992 (1965).
[Crossref]

Booth, M. J.

Bouchal, P.

Bouchal, Z.

Brooker, G.

Cai, L. Z.

Celechovský, R.

Cheng, X.

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nature Photon. 5, 335–342 (2011).
[Crossref]

Colomb, T.

Cuche, E.

DeLuca, J.

DeLuca, K.

Depeursinge, C.

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nature Photon. 5, 335–342 (2011).
[Crossref]

Ding, J.

Dong, Y.

Emery, Y.

Erkkila, J. H.

J. H. Erkkila and M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. A 71(7), 904–905 (1981).
[Crossref]

Ferraro, P.

P. Ferraro, A. Wax, and Z. Zalevsky, Coherent Light Microscopy: Imaging and Quantitative Phase Analysis (Springer, 2011), Chap. 1.

Garcia, J.

García, J.

Garcia-Sucerquia, J.

Goodman, J. W.

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

Grover, G.

Gu, M.

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
[Crossref]

Guo, C.

Hong, J.

X. Yu, J. Hong, Ch. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53(11), 112306 (2011).
[Crossref]

Jackin, B. J.

Jericho, M. H.

Jericho, S. K.

Juptner, W.

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9) R85–R101 (2002).
[Crossref]

Katz, B.

Kelner, R.

Kim, M. K.

X. Yu, J. Hong, Ch. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53(11), 112306 (2011).
[Crossref]

Klages, P.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH Verlag, 2005), Chap. 4.

Kreuzer, H. J.

Lee, H. I. D.

H. I. D. Lee, S. J. Sahl, M. D. Lew, and W. E. Moerner, “The double-helix microscope super-resolves extended biological structures by localizing single blinking molecules in three dimensions with nanoscale precision,” Appl. Phys. Lett. 100, 153701 (2012).
[Crossref] [PubMed]

Lew, M. D.

H. I. D. Lee, S. J. Sahl, M. D. Lew, and W. E. Moerner, “The double-helix microscope super-resolves extended biological structures by localizing single blinking molecules in three dimensions with nanoscale precision,” Appl. Phys. Lett. 100, 153701 (2012).
[Crossref] [PubMed]

Li, Y.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[Crossref]

Liu, Ch.

X. Yu, J. Hong, Ch. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53(11), 112306 (2011).
[Crossref]

Magistretti, P.J.

Mahajan, V. N.

Marquet, P.

Micó, V.

Moerner, W. E.

H. I. D. Lee, S. J. Sahl, M. D. Lew, and W. E. Moerner, “The double-helix microscope super-resolves extended biological structures by localizing single blinking molecules in three dimensions with nanoscale precision,” Appl. Phys. Lett. 100, 153701 (2012).
[Crossref] [PubMed]

Narayanamurthy, C. S.

Piestun, R.

Quirin, S.

Rappaz, B.

Reinhard, W.

W. Reinhard, “Magnification and Third-Order Aberrations in Holography,” J. Opt. Soc. Am. A 55(8), 987–992 (1965).
[Crossref]

Ren, X.

Rogers, M. E.

J. H. Erkkila and M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. A 71(7), 904–905 (1981).
[Crossref]

Rong, Z.

Rosen, J.

Sahl, S. J.

H. I. D. Lee, S. J. Sahl, M. D. Lew, and W. E. Moerner, “The double-helix microscope super-resolves extended biological structures by localizing single blinking molecules in three dimensions with nanoscale precision,” Appl. Phys. Lett. 100, 153701 (2012).
[Crossref] [PubMed]

Schnars, U.

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9) R85–R101 (2002).
[Crossref]

Schwertner, S.

Wang, H.

Wang, Y.

Wax, A.

P. Ferraro, A. Wax, and Z. Zalevsky, Coherent Light Microscopy: Imaging and Quantitative Phase Analysis (Springer, 2011), Chap. 1.

Wilson, T.

Wolf, E.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[Crossref]

Wu, J.

Xu, W.

Yamaguchi, I.

Yatagai, T.

Yu, X.

X. Yu, J. Hong, Ch. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53(11), 112306 (2011).
[Crossref]

Zalevsky, Z.

Zhang, T.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

H. I. D. Lee, S. J. Sahl, M. D. Lew, and W. E. Moerner, “The double-helix microscope super-resolves extended biological structures by localizing single blinking molecules in three dimensions with nanoscale precision,” Appl. Phys. Lett. 100, 153701 (2012).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

W. Reinhard, “Magnification and Third-Order Aberrations in Holography,” J. Opt. Soc. Am. A 55(8), 987–992 (1965).
[Crossref]

J. H. Erkkila and M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. A 71(7), 904–905 (1981).
[Crossref]

Meas. Sci. Technol. (1)

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9) R85–R101 (2002).
[Crossref]

Nature Photon. (1)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nature Photon. 5, 335–342 (2011).
[Crossref]

Opt. Commun. (1)

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[Crossref]

Opt. Eng. (1)

X. Yu, J. Hong, Ch. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53(11), 112306 (2011).
[Crossref]

Opt. Express (4)

Opt. Lett. (8)

Other (4)

P. Ferraro, A. Wax, and Z. Zalevsky, Coherent Light Microscopy: Imaging and Quantitative Phase Analysis (Springer, 2011), Chap. 1.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH Verlag, 2005), Chap. 4.

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

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Optical scheme for evaluation of the 3D PSF in lensless holographic microscopy: (a) in-line recording geometry, (b) reconstruction geometry, (c) asymmetric axial profile of the PSF with geometrical parameters used in the quantitative evaluation of the focal shift and the axial PSF asymmetry.
Fig. 2
Fig. 2 In-line holographic setup used for reconstruction of the 3D PSFs and image of the resolution target in testing an optimal design of geometric parameters: SMF ···single-mode fiber, L1 ··· collimating lens, BS··· beam splitter, SLM··· spatial light modulator creating converging/diverging lenses with varying focal lengths ± fm, L2 ··· imaging lens.
Fig. 3
Fig. 3 Image deterioration due to spherical aberration demonstrated by the color-coded Strehl ratio (image reconstructed by the plane wave and evaluated at the optimal image plane): (a) pinhole radius ρp = 2.75 μm, hologram radius ρh = αλ|zr|/ρp, (b) constant hologram radius ρh = 5.6 mm.
Fig. 4
Fig. 4 Dependence of the indicators of imaging performance on the pinhole to object distance Δ (evaluation performed for zr = −43.5 mm and ρp = 2.75 μm): object space diffraction-limited lateral resolution in multiples of λ, Δr (green line), resolution limit in multiples of λ given by sampling conditions for CCD QImaging Retiga 4000R (thin black line), asymmetry coefficient, Q (red line), Strehl ratio for optimal image plane, K (black line) and coefficient of spherical aberration in multiples of λ, S (blue line). Ranges of parameters: I-axially asymmetric PSF, II-symmetric nearly diffraction-limited PSF (Q ≥ 0.8, K ≥ 0.8), III-strong spherical aberration.
Fig. 5
Fig. 5 Diffraction-limited axially asymmetric PSF reconstructed by the plane wave from three phase-shifted point holograms taken with the geometric parameters belonging to the area I in Fig. 4 (ρh = 6.1 mm, zr = −43.5 mm, Δ = 0.1 mm). The axial and radial PSF profiles obtained from experimental data were transformed into the object space and compared with numerical simulations of the hologram recording and reconstruction.
Fig. 6
Fig. 6 Symmetric low-aberration PSF reconstructed by the plane wave from three phase-shifted point holograms taken with the geometric parameters belonging to the area II in Fig. 4 (ρh = 6.1 mm, zr = −43.5 mm, Δ = 1.5 mm).
Fig. 7
Fig. 7 PSF deteriorated by a strong holographic aberration and reconstructed by the plane wave from three phase-shifted point holograms taken with the geometric parameters belonging to the area III in Fig. 4 (ρh = 6.1 mm, zr = −43.5 mm, Δ = 4.9 mm).
Fig. 8
Fig. 8 Plane wave reconstructions of the resolution target verifying an optimal choice of geometric parameters for hologram recording: (a) aberration-free imaging with reduced diffraction-limited lateral resolution (zr = −46.4 mm, Δ = 0.7 mm), (b) low-aberration imaging with optimal lateral resolution approaching the limit allowed by the sampling conditions (zr = −46.4 mm, Δ = 3 mm), (c) imaging with resolution reduced by holographic aberrations (zr = −46.4 mm, Δ = 10 mm).

Equations (37)

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ψ z i i λ t ( r ) exp ( i k | r i | ) | r i | 2 d r ,
t ( r ) = T ( r ) ψ c | ψ s + ψ r | 2 .
t ( r ) = T ( r ) A s A r A c exp [ i ( Φ s Φ r + Φ c ) ] ,
A j = a j | r 0 j | , Φ j = k ( | r j | | r 0 j | ) , j = s , r , c ,
Φ j = k ( | r | 2 2 z j r r 0 j z j ) , j = s , r , c ,
ψ 1 z i T ( r ) exp ( i k Ω | r | 2 2 ) exp ( i 2 π r R ) d r ,
Ω = 1 z i 1 z s + 1 z r 1 z c ,
X = 1 λ ( x i z i x s z s x c z c ) , Y = 1 λ ( y i z i y s z s y c z c ) .
1 f = 1 z c Δ z r ( z r + Δ ) ,
β = f z r + Δ .
I ( q ) = ( 1 q π n i ) 2 sinc 2 ( q 2 ) ,
q = π n i Δ z i f + Δ z i , n i = ρ h 2 λ f .
Δ z i + = 2 f n i 2 , Δ z i = 2 f n i + 2 .
tan ( q 2 ) = ( q 2 ) ( 1 q π n i ) .
Δ z i max = f q max π n i q max .
Λ i ± = Δ z i ± Δ z i max .
Q i = | Λ i + Λ i | .
Δ r = ρ p ( 1 Δ | z r | ) ,
N i = α 2 λ κ ρ p Δ r ,
Δ | z r | Δ r 2 α p CCD .
W = S ρ N 4 + C ρ N 3 cos φ + A ρ N 2 cos 2 φ + F ρ N 2 + D ρ N cos φ ,
S = ρ h 4 8 ( 1 z i 3 + 1 z r 3 1 z s 3 1 z c 3 ) , C = x s ρ h 3 2 ( β z i 3 1 z s 3 ) , A = x s 2 ρ h 2 2 ( β 2 z i 3 1 z s 3 ) , F = 1 2 A , D = x s 3 ρ h 2 ( β 3 z i 3 1 z s 3 ) .
S = 3 8 ρ h 4 z s z r f .
S = 3 8 ρ h 4 Δ z r 2 ( z r + Δ ) 2 .
K = | 1 π 0 2 π 0 1 exp ( i k W ) ρ N d ρ N d φ | 2 .
K = 1 k 2 σ W 2 ,
W 2 = 1 π 0 2 π 0 1 W 2 ρ N d ρ N d φ , W = 1 π 0 2 π 0 1 W ρ N d ρ N d φ .
K = 1 4 45 k 2 S 2 .
K = 1 1 180 k 2 S 2 .
z r 2 Δ ( 1 + z r Δ ) 2 2 ρ h 4 5 λ .
Δ z r 2 ( 1 + z r Δ ) 2 λ 3 18 ρ p 4 .
t j = exp ( i k r m 2 2 f m + i ϑ j ) + exp ( i k r m 2 2 f m ) , j = 1 , 2 , 3 ,
Φ j = Φ j ( 1 ) Φ j ( 3 ) , j = s , r , c , i ,
Φ j ( 1 ) = q j k | r j | 2 | r 0 j | 2 2 z j , Φ j ( 3 ) = q j k | r j | 4 | r 0 j | 4 8 z j 3 ,
W = 1 k ( Φ i 3 Φ r ( 3 ) + Φ s ( 3 ) + Φ c ( 3 ) ) .
q j k Φ j ( 3 ) = A j 040 ρ 4 + A j 131 x j ρ 3 cos φ + A j 222 x j 2 ρ 2 cos 2 φ + A j 220 x j 2 ρ 2 + A j 311 x j 3 ρ cos φ , j = s , r , c , i ,
A j 040 = 1 8 z j 3 , A j 131 = 4 A j 040 , A j 222 = 4 A j 040 , A j 220 = 2 A j 040 , A j 311 = 4 A j 040 .

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