Abstract

A novel quantitative phase imaging method is shown to estimate phase accurately over a wide range of length scales using Köhler illumination from an extended incoherent source. The method is based on estimating the longitudinal intensity derivative in the transport-of-intensity equation via convolution with multiple Savitzky–Golay differentiation filters and generalizes methods previously developed for coherent imaging to the practical scenario of partially coherent imaging. The resulting noise and resolution performance are evaluated via numerical simulation and demonstrated experimentally using a blazed transmission grating as well as a single-mode fiber as test objects.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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  26. C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
    [CrossRef]

2013

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[CrossRef]

L. Waller, “Phase imaging with partially coherent light,” Proc. SPIE 8589, 85890K (2013).
[CrossRef]

C. Zuo, Q. Chen, Y. J. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differential filter—theory and applications,” Opt. Express 21, 5346–5362 (2013).
[CrossRef]

J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
[CrossRef]

2012

2011

2010

2007

2005

2004

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A 21, 828–831 (2004).
[CrossRef]

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

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]

2002

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

C. Sheppard, “Three-dimensional phase imaging with the intensity transport equation,” Appl. Opt. 41, 5951–5955 (2002).
[CrossRef]

1997

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

1985

1983

1978

1955

G. Nomarski and A. R. Weill, “Application à la métallographie desméthodes interférentielles à deux ondes polarisées,” Rev. Metall. 2, 121–128 (1955).

1942

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent object,” Physica (Amsterdam) 9, 686–698 (1942).
[CrossRef]

Acosta, E.

Allman, B. E.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

Anastasio, M. A.

Asundi, A.

Bachim, B. L.

Barbastathis, G.

Barone-Nugent, E. D.

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

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]

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

Bellair, C. J.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

Bie, R.

Carney, P. S.

Chen, Q.

Curl, C. L.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

Delbridge, L.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

Depeursinge, C.

Ding, H.

Falaggis, K.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[CrossRef]

Fienup, J. R.

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Gaylord, T. K.

Gillette, M. U.

Harris, P. J.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

Hutsel, M. R.

Jenkins, M.

J. Long, M. Jenkins, and T. K. Gaylord, “Comparison of transport-of-intensity derivative methods for optical fibers under partially coherent illumination,” in Frontiers in Optics 2013, OSA Technical Digest (Optical Society of America, 2013), paper FTh1F.3.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Kou, S. S.

Kozacki, T.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[CrossRef]

Kujawinska, M.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[CrossRef]

Long, J.

J. Long, M. Jenkins, and T. K. Gaylord, “Comparison of transport-of-intensity derivative methods for optical fibers under partially coherent illumination,” in Frontiers in Optics 2013, OSA Technical Digest (Optical Society of America, 2013), paper FTh1F.3.

Marquet, P.

Martinez-Carranza, J.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[CrossRef]

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]

Mettler, S. C.

Millet, L.

Mir, M.

Nomarski, G.

G. Nomarski and A. R. Weill, “Application à la métallographie desméthodes interférentielles à deux ondes polarisées,” Rev. Metall. 2, 121–128 (1955).

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]

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

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

Petruccelli, J. C.

Pogany, A.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Popescu, G.

Roberts, A.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

Rogers, J.

Schoonover, R. W.

Sheppard, C.

Sheppard, C. J. R.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Soto, M.

Streibl, N.

Teague, M. R.

Tian, L.

Unarununotai, S.

Waller, L.

Wang, Z.

Weill, A. R.

G. Nomarski and A. R. Weill, “Application à la métallographie desméthodes interférentielles à deux ondes polarisées,” Rev. Metall. 2, 121–128 (1955).

Wilkins, S. W.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Yu, Y. J.

Yuan, X. H.

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent object,” Physica (Amsterdam) 9, 686–698 (1942).
[CrossRef]

Zhang, L.

Zhao, M.

Zuo, C.

Zysk, A. M.

Appl. Opt.

J. Microsc.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. Delbridge, and K. A. Nugent, “Quantitative phase amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef]

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

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Physica (Amsterdam)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent object,” Physica (Amsterdam) 9, 686–698 (1942).
[CrossRef]

Proc. SPIE

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[CrossRef]

L. Waller, “Phase imaging with partially coherent light,” Proc. SPIE 8589, 85890K (2013).
[CrossRef]

Rev. Metall.

G. Nomarski and A. R. Weill, “Application à la métallographie desméthodes interférentielles à deux ondes polarisées,” Rev. Metall. 2, 121–128 (1955).

Rev. Sci. Instrum.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Other

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

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

J. Long, M. Jenkins, and T. K. Gaylord, “Comparison of transport-of-intensity derivative methods for optical fibers under partially coherent illumination,” in Frontiers in Optics 2013, OSA Technical Digest (Optical Society of America, 2013), paper FTh1F.3.

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Figures (10)

Fig. 1.
Fig. 1.

Block diagram representation of the MFPI method.

Fig. 2.
Fig. 2.

Normalized PCTFs (TP(2)Δz/B,p=λρ/NAobj) for orders m=1,7,13, and 19 and coherence parameters (a) S=0.1, (b) S=0.5, and (c) S=0.9. WD and TIE PCTFs are also plotted.

Fig. 3.
Fig. 3.

Normalized cut-off frequencies (ρc=λρc/NAobj) for odd orders m=1 through 25 and coherence parameters S=0,0.1,0.5, and 0.9.

Fig. 4.
Fig. 4.

PTFs for orders m=1,7,13,19 and for coherence parameters (a) S=0.1, (b) S=0.5, and (c) S=0.9, where (p=λρ/NAobj). WD PTFs are also plotted.

Fig. 5.
Fig. 5.

Simulated phase object.

Fig. 6.
Fig. 6.

Recovered phases (colorbar units—radians) for (a) S=0.1 and fixed-order m=1, (b) S=0.1 and fixed-order m=27, (c) S=0.1 and MFPI-PC including orders m=127, (d) S=0.5 and fixed-order m=1, (e) S=0.5 and fixed-order m=27, (f) S=0.5 and MFPI-PC including orders m=127, (g) S=0.9 and fixed-order m=1, (h) S=0.9 and fixed-order m=27, and (i) S=0.9 and MFPI-PC including orders m=127. In all figures the registered trademark symbol is expanded in the top right with its associated location given by the dashed square outlines.

Fig. 7.
Fig. 7.

Recovered phase for S=0.5 and coherent MFPI (MFPI-C) including orders m=127. The registered trademark symbol is expanded in the top right with its associated location given by the dashed square outlines.

Fig. 8.
Fig. 8.

Phase RMSE as a function of normalized noise standard deviation σ (unitless) for (a) S=0.1, (b) S=0.5, and (c) S=0.9 and fixed orders m=1,7,13, and 17 compared with the MFPI-PC result. The result for MFPI-C is also plotted in each case.

Fig. 9.
Fig. 9.

Visible transmission grating phase measurement results. Interpolated surface plots of the measured phase on a 24.5×24.5μm patch for (a) S=0.1 and (b) S=0.5. Measured line profiles for (c) S=0.1 and (d) S=0.5. Ideal (assuming 90° groove angles) and predicted (filtered using the associated WD PTF) line profiles are also plotted for reference.

Fig. 10.
Fig. 10.

Various representations of optical fiber tomography experimental data processed using conventional TIE recovery with defocus distances of (a)–(c) 9 μm and (d)–(f) 0.6 μm as well as the established (g)–(i) MFPI-C method and the proposed (j)–(l) MFPI-PC method. The phase sinogram data are represented in the first column [(a), (d), (g), and (j)]. The logarithm of the phase spectral density is represented in sinogram format in the second column [(b), (e), (h), and (k)]. The resulting tomograms, after applying filtered backprojection using the data in the first column, are represented in the third column [(c), (f), (i), and (l)]. Within the third column, the top-right insets are zoomed-in images of the fiber core (region indicated by the dashed squares), and the bottom-left insets enhance contrast in the cladding using different colorbar limits. Partially coherent (S=0.5) illumination was used to produce the intensity data for the measurement.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

I(r,z)z=(λ2π)·[I(r,z)ϕ(r,z)].
I˜(ρ,η)=Bδ(ρ,η)+P˜(ρ,η)TP(3)(ρ,η)+A˜(ρ,η)TA(3)(ρ,η),
S=NAcond/NAobj.
dI(r,0)dz=i=nnaiI(r,iΔz)Δz.
dI˜(ρ)dz=ϕ˜(ρ)TP(2)(ρ),
TP(2)(ρ)=4πλHSG(ej2πη)TP(3)(ρ,η)dη,
TPW(2)(ρ)=4πλj2πηTP(3)(ρ,η)dη.
dI˜(ρ)dz=ϕ˜(ρ)2πλ¯Bρ2,
TPTIE(2)(ρ)=2πλ¯Bρ2.
ξ=TP(2)(ρc)/TPW(2)(ρc).
PTF(ρ)=TP(2)(ρ)/TPTIE(2)(ρ).

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