Abstract

The transport of intensity equation (TIE) relates the variation of intensity of a wave-front along its mean direction of propagation with its phase. In experimental application, characteristic artefacts may affect the retrieved phase. These originate from inadequacies in estimating the axial derivative and the amplification of noise in the inversion of the TIE. To tackle these issues, images recorded at multiple planes of focus can be integrated into a multi-focus TIE (MFTIE) solution. This methodology relies on the efficient sampling of phase information in the spatial-frequency domain, typically by the definition of band pass filters implemented as a progression of sharp spatial frequency cut-offs. We present a convenient MFTIE implementation which avoids the need for recording images at very specific planes of focus and applies overlapping cut-offs, greatly simplifying the experimental application. This new approach additionally also accounts for partial spatial coherence in a flux-preserving framework. Using simulated data as well as experimental data from optical microscopy and electron microscopy we show that the frequency response of this MFTIE algorithm recovers efficiently a wide range of spatial frequencies of the phase that can be further extended by simple iterative refinement.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  20. Code available at: http://github.com/AEljarrat/inline_holo
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    [Crossref]

2016 (2)

A. Parvizi, E. Van den Broek, and C. T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. 2, 3 (2016).
[Crossref]

A. Parvizi, W. Van den Broek, and C. T. Koch, “Gradient flipping algorithm: introducing non-convex constraints in wavefront reconstructions with the transport of intensity equation,” Opt. Express 24, 8344–8359 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (5)

2012 (1)

C. Ophus and T. Ewalds, “Guidelines for quantitative reconstruction of complex exit waves in HRTEM,” Ultramicrosc. 113, 88–95 (2012).
[Crossref]

2008 (2)

C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicrosc. 108, 141–150 (2008).
[Crossref]

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Cryst. A 64, 123–134 (2008).
[Crossref]

2005 (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).

2004 (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III: The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

2003 (2)

2002 (1)

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

1995 (1)

1990 (1)

1983 (1)

1982 (1)

W. O. Saxton and W. Baumeister, “The correlation averaging of a regularly arranged bacterial cell envelope protein,” J. Microsc. 127, 127–138 (1982).
[Crossref] [PubMed]

Agour, M.

M. Agour, “Optimal strategies for wave fields sensing by means of multiple planes phase retrieval,” J. Opt. 17, 85604–85616 (2015).
[Crossref]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).

Asundi, A.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III: The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Baumeister, W.

W. O. Saxton and W. Baumeister, “The correlation averaging of a regularly arranged bacterial cell envelope protein,” J. Microsc. 127, 127–138 (1982).
[Crossref] [PubMed]

Berty, A.

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

Chen, Q.

Claus, R. A.

Conejo-Rodríguez, A.

Ewalds, T.

C. Ophus and T. Ewalds, “Guidelines for quantitative reconstruction of complex exit waves in HRTEM,” Ultramicrosc. 113, 88–95 (2012).
[Crossref]

Falaggis, K.

Fernández-Guasti, M.

Gaylord, T. H.

M. H. Jenkins, J. M. Long, and T. H. Gaylord, “Multifilter phase imaging with partially coherent light,” Appl. Opt. 53, 29–39 (2014).
[Crossref]

Gaylord, T. K.

Granados-Agustín, F.

Gureyev, T. E.

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).

Jenkins, M. H.

M. H. Jenkins and T. K. Gaylord, “Quantitative phase microscopy via optimized inversion of the phase optical transfer function,” Appl. Opt. 54, 8566–8579 (2015).
[Crossref] [PubMed]

M. H. Jenkins, J. M. Long, and T. H. Gaylord, “Multifilter phase imaging with partially coherent light,” Appl. Opt. 53, 29–39 (2014).
[Crossref]

Jiménez, J. L.

Jingshan, Z.

Koch, C. T.

A. Parvizi, W. Van den Broek, and C. T. Koch, “Gradient flipping algorithm: introducing non-convex constraints in wavefront reconstructions with the transport of intensity equation,” Opt. Express 24, 8344–8359 (2016).
[Crossref] [PubMed]

A. Parvizi, E. Van den Broek, and C. T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. 2, 3 (2016).
[Crossref]

C. T. Koch, “Towards full-resolution inline electron holography,” Micron 63, 69–75 (2014).
[Crossref]

C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicrosc. 108, 141–150 (2008).
[Crossref]

Kozacki, T.

Long, J. M.

M. H. Jenkins, J. M. Long, and T. H. Gaylord, “Multifilter phase imaging with partially coherent light,” Appl. Opt. 53, 29–39 (2014).
[Crossref]

Martínez-Carranza, J.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III: The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III: The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[Crossref]

Ophus, C.

C. Ophus and T. Ewalds, “Guidelines for quantitative reconstruction of complex exit waves in HRTEM,” Ultramicrosc. 113, 88–95 (2012).
[Crossref]

Oszlányi, G.

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Cryst. A 64, 123–134 (2008).
[Crossref]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III: The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Parvizi, A.

A. Parvizi, E. Van den Broek, and C. T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. 2, 3 (2016).
[Crossref]

A. Parvizi, W. Van den Broek, and C. T. Koch, “Gradient flipping algorithm: introducing non-convex constraints in wavefront reconstructions with the transport of intensity equation,” Opt. Express 24, 8344–8359 (2016).
[Crossref] [PubMed]

Popescu, G.

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011).

Roberts, A.

Roddier, F.

Saxton, W. O.

W. O. Saxton and W. Baumeister, “The correlation averaging of a regularly arranged bacterial cell envelope protein,” J. Microsc. 127, 127–138 (1982).
[Crossref] [PubMed]

Süto, A.

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Cryst. A 64, 123–134 (2008).
[Crossref]

Teague, M. R.

Tian, L.

Van den Broek, E.

A. Parvizi, E. Van den Broek, and C. T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. 2, 3 (2016).
[Crossref]

Van den Broek, W.

Waller, L.

Zou, C.

Acta Cryst. A (1)

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Cryst. A 64, 123–134 (2008).
[Crossref]

Adv. Struct. Chem. Imag. (1)

A. Parvizi, E. Van den Broek, and C. T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. 2, 3 (2016).
[Crossref]

Appl. Opt. (4)

J. Electron Microsc. (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).

J. Microsc. (3)

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III: The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

W. O. Saxton and W. Baumeister, “The correlation averaging of a regularly arranged bacterial cell envelope protein,” J. Microsc. 127, 127–138 (1982).
[Crossref] [PubMed]

J. Opt. (1)

M. Agour, “Optimal strategies for wave fields sensing by means of multiple planes phase retrieval,” J. Opt. 17, 85604–85616 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

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

Micron (1)

C. T. Koch, “Towards full-resolution inline electron holography,” Micron 63, 69–75 (2014).
[Crossref]

Opt. Commun. (1)

T. E. Gureyev, “Composite techniques for phase retrieval in the fresnel region,” Opt. Commun. 220, 49–58 (2003).
[Crossref]

Opt. Express (4)

Ultramicrosc. (2)

C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicrosc. 108, 141–150 (2008).
[Crossref]

C. Ophus and T. Ewalds, “Guidelines for quantitative reconstruction of complex exit waves in HRTEM,” Ultramicrosc. 113, 88–95 (2012).
[Crossref]

Other (3)

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011).

F. de la Peña and et al., “HyperSpy 1.3”, DOI: , http://hyperspy.org
[Crossref]

Code available at: http://github.com/AEljarrat/inline_holo

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Figures (7)

Fig. 1
Fig. 1 Panels (a) and (b) respectively depict the amplitude and phase of the input wave used for simulations in this paper, while panel (c) is the phase retrieved from three-plane TIE using a noiseless simulated dataset.
Fig. 2
Fig. 2 The images at the left present three-plane TIE phases obtained for each defocus pair in a simulated dataset with added noise (SNR = 20 dB) and, from (a) to (c), defocus values δz = 1, 5 and 10 μm, respectively. Each recovered phase has been compared to the original phase in the simulation using FRC and RMSE, depicted in the panels to the right using blue and red lines, respectively. Light blue lines correspond to negative parts of the FRC, that have been flipped for representation purposes. Dashed red lines indicate the theoretical noise error, RMSE n 2, calculated for this level of noise and these defoci.
Fig. 3
Fig. 3 The images at the left present the phase of the complex waves obtained from reconstructions carried out using, in the first two rows, MFTIE and, in the last row, GPTIE. All the recovered waves were refined using 25 iterations of the Gerchberg-Saxton algorithm. Panel (a) is showing the results obtained using only 3 defocus image pairs at δz = 1, 5 and 10 μm. Panels (b) and (c) use a dataset of 21 planes log-spaced between 1 and 150 μm, with a SNR = 15 dB. Blue and red lines are used to depict FRC and RMSE validations of the recovered phases, with solid and dashed lines indicating if the phase before or after the refinement was used, respectively.
Fig. 4
Fig. 4 In the first column, panels (a) and (d) show the phases obtained from of HeLa cells using MFTIE (top) and GPTIE (bottom) in-line holography algorithms, respectively. The second and third columns show the phase and amplitudes of the complex wave obtained after 25 iterations of GS refinement of the TIE results.
Fig. 5
Fig. 5 Validation tests with the experimental images for the phase reconstructions in Fig. 4. In the top panel, r-value tests for the validation of the MFTIE convergence envelope. Here, blue circles and black crosses indicate the r-value calculated before and after GS refinement, respectively, performed for different values of the convergence angle. In the bottom panel, the χ2-tests, with black circles and red squares representing MFTIE and GPTIE algorithms and dashed or solid lines indicating whether the validation was performed before or after GS refinement.
Fig. 6
Fig. 6 Even-odd FRC validation tests of the MFTIE (black) and GPTIE (red) phase reconstructions in Fig. 4. The top and bottom panels include the validations performed before and after GS refinement, respectively.
Fig. 7
Fig. 7 Phase reconstruction from an experimental TEM dataset of sample with gold NPs (bright) sputtered over lacey carbon featuring a hole in the middle (dark). Panel (a) is showing the MFTIE result, obtained using HNDS down to qm ≃ 2π(150 nm)−1. Panel (b) shows the phase recovered after 20 iterations of GF-refinement using the former result as input. In this refinement, spatial-frequencies below qm ≃ 2π(8 nm)−1 were iteratively flipped to obtain a phase with a smooth gradient over large portions of the measured and pad area. Panel (c) is showing the phase retrieved by non-linear refinement using FRWR, which is able to retrieve the sharp phase excursion between vacuum and sample.

Equations (16)

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Ψ ( x , y , z ) = u z ( r ) e i k c t = I z ( r ) e i Φ z ( r ) e i k c t
k z I = ( I Φ )
( z I ) 0 δ z ( z I ) 0 , δ z = I ( + δ z ) I ( δ z ) 2 δ z
Φ 0 = TIE 1 ( z I ) 0 = k 1 I 0 1 1 ( z I ) 0 = k 2 I 0 1 2 ( z I ) 0
Φ ^ 0 , δ z ( q ) = k ^ δ z 2 q ( I 0 1 1 [ q ^ δ z 2 ( z I ) 0 , δ z ] )
Φ 0 ( r ) = 1 ( δ z Φ ^ 0 , δ z )
^ δ z 2 = H δ z ( q ) q 2 f δ z ( q )
f δ z = δ z H δ z ( q ) S δ z ( q ) q 2
q H 2 ( ϕ ) = 2 k 6 ϕ δ z
χ δ z R S q χ δ z F K + 1 2 δ z k ( χ δ z F K ) 2 + o ( q 2 ) = χ δ z F K + χ q H 2 ( χ ) = 8 χ k 3 δ z
RMSE n 2 ( q L , q H ) = 1 π ( k σ n L I ¯ 0 δ z 2 ) 2 ( q L 2 q H 2 ) = A δ z ( q L 2 q H 2 )
q L 2 = q H 2 ( RMSE n 2 A δ z q H 2 + 1 ) 1 δ z A δ z RMSE n 2
RMSE n 2 ( SNR , Δ z ) = A δ z q m 2
q L 2 ( ρ ) = q H 2 [ ρ A δ z RMSE n 2 ( SNR , Δ z ) + 1 ] 1
r = x , y , z | I EXP I SIM | I EXP
χ 2 = x , y ( I EXP I SIM ) 2 I EXP

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