Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process

Luděk Lovicar; Jiří Komrska; Radim Chmelík

doi:10.1364/OE.18.020585

Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process

Luděk Lovicar, Jiří Komrska, and Radim Chmelík

Optics Express
Vol. 18,
Issue 20,
pp. 20585-20594
(2010)
•https://doi.org/10.1364/OE.18.020585

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Luděk Lovicar, Jiří Komrska, and Radim Chmelík, "Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process," Opt. Express 18, 20585-20594 (2010)

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Related Topics
Table of Contents Category
- Instrumentation, Measurement, and Metrology
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Interferometry (120.3180)
- Phase measurement (120.5050)
- Surface measurements, figure (120.6650)
- Interference microscopy (180.3170)

About this Article
History
- Original Manuscript: July 22, 2010
- Revised Manuscript: September 4, 2010
- Manuscript Accepted: September 5, 2010
- Published: September 14, 2010

Accessible

Open Access

Abstract

The paper deals with quantitative phase imaging of two-height-level surface reliefs. The imaging is considered to be a linear system and, consequently, the Fourier transform of the image is the product of the Fourier transform of a 2D function characterizing the surface and a specific 2D coherent transfer function. The Fourier transform of functions specifying periodic surface reliefs is factorized into two functions similar to lattice and structure amplitudes in crystal structure analysis. The approach to the imaging process described in the paper enables us to examine the dependence of the phase image on the surface geometry. Theoretical results are verified experimentally by means of a digital holographic microscope.

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References

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|
Year
|
Author
|
Publication

T. Wilson, and C. J. R. Sheppard, Theory and practice of scanning optical microscopy (Academic Press, 1984).
J. T. Sheridan and C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105(5-6), 367–378 (1994).
[Crossref]
J. T. Sheridan and C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10(4), 614–632 (1993).
[Crossref]
M. Gu and C. J. R. Sheppard, “Effects of defocus and primary spherical aberrations on images of a straight edge in confocal microscopy,” Appl. Opt. 33(4), 625–630 (1994).
[Crossref] [PubMed]
J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19(7), 074007 (2008).
[Crossref]
S. D. Nicola, P. Ferraro, A. Finizio, S. Grilli, G. Coppola, M. Iodice, P. D. Natale, and M. Chiarini, “Surface topography of microstructures in lithium niobate by digital holographic microscopy,” Meas. Sci. Technol. 15(5), 961–968 (2004).
[Crossref]
Y. Emery, E. Cuche, F. Marquet, S. Bourquin, P. Marquet, J. Kühn, N. Aspert, M. Botkin, and Ch. Depeursinge, “Digital Holographic Microscopy (DHM): Fast and Robust 3D measurements with interferometric resolution for industrial inspection,” in Fringe 2005 – New Optical Sensors and Measurement Systems (Springer, (2006), pp. 667-671.
Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kühn, M. Botkine, T. Colomb, F. Montfort, F. Charrière, Ch. Depeursinge, P. Debergh, and R. Conde, “Digital Holographic Microscopy (DHM) for metrology and dynamic characterization of MEMS and MOEMS,“ Proc. SPIE 6186, art. no. 61860N (2006).
R. Chmelík, “Three–dimensional scalar imaging in high–aperture low–coherence interference and holographic microscopes,” J. Mod. Opt. 53(18), 2673–2689 (2006).
[Crossref]
R. Chmelik, H. Uhlirova, P. Kolman, and P. Vesely, “Wide Range Coherence Digital Holographic Microscope,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2010), paper DTuA6. http://www.opticsinfobase.org/abstract.cfm?URI=DH-2010-DTuA6
J. Komrska, “ The Fourier transform of lattices,” in Proceedings of the International Summer School 27th May – 5thJune1991, Chlum u Třeboně, Czechoslovakia, L. Eckertová and T. Růžička, eds. (Institute of Physics Publishing, Bristol and Philadephia), pp. 87–113.
R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965), Chap. 6.
M. von Laue, and E. H. Wagner, Röntgenstrahl-Interferenzen (Akademische Verlagsgesellschaft, 1960), Chap. 3.
www.Luxpop.com : Thin film and bulk index of refraction and photonics calculations.
M. Born, and E. Wolf, Principles of Optics. 7th ed. (Cambridge University Press, 2002), Chap. 1.
J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik (Stuttg.) 80, 171–183 (1988).
J. Komrska, “Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures,” J. Opt. Soc. 72(10), 1382–1384 (1982).
[Crossref]
R. Chmelík and Z. Harna, “Parallel-mode confocal microscope,” Opt. Eng. 38, 1635–1639 (1999).
[Crossref]
E. N. Leith and J. Upatnieks, “Holography with Achromatic-Fringe Systems,” J. Opt. Soc. Am. A 57(8), 975–980 (1967).
[Crossref]

2008 (1)

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19(7), 074007 (2008).
[Crossref]

2006 (1)

R. Chmelík, “Three–dimensional scalar imaging in high–aperture low–coherence interference and holographic microscopes,” J. Mod. Opt. 53(18), 2673–2689 (2006).
[Crossref]

2004 (1)

S. D. Nicola, P. Ferraro, A. Finizio, S. Grilli, G. Coppola, M. Iodice, P. D. Natale, and M. Chiarini, “Surface topography of microstructures in lithium niobate by digital holographic microscopy,” Meas. Sci. Technol. 15(5), 961–968 (2004).
[Crossref]

1999 (1)

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

1994 (2)

J. T. Sheridan and C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105(5-6), 367–378 (1994).
[Crossref]

M. Gu and C. J. R. Sheppard, “Effects of defocus and primary spherical aberrations on images of a straight edge in confocal microscopy,” Appl. Opt. 33(4), 625–630 (1994).
[Crossref] [PubMed]

1993 (1)

J. T. Sheridan and C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10(4), 614–632 (1993).
[Crossref]

1988 (1)

J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik (Stuttg.) 80, 171–183 (1988).

1982 (1)

J. Komrska, “Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures,” J. Opt. Soc. 72(10), 1382–1384 (1982).
[Crossref]

1967 (1)

E. N. Leith and J. Upatnieks, “Holography with Achromatic-Fringe Systems,” J. Opt. Soc. Am. A 57(8), 975–980 (1967).
[Crossref]

Charrière, F.

Chiarini, M.

Chmelík, R.

R. Chmelík, “Three–dimensional scalar imaging in high–aperture low–coherence interference and holographic microscopes,” J. Mod. Opt. 53(18), 2673–2689 (2006).
[Crossref]

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

Colomb, T.

Coppola, G.

Cuche, E.

Depeursinge, C.

Emery, Y.

Ferraro, P.

Finizio, A.

Grilli, S.

Gu, M.

M. Gu and C. J. R. Sheppard, “Effects of defocus and primary spherical aberrations on images of a straight edge in confocal microscopy,” Appl. Opt. 33(4), 625–630 (1994).
[Crossref] [PubMed]

Harna, Z.

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

Iodice, M.

Komrska, J.

J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik (Stuttg.) 80, 171–183 (1988).

J. Komrska, “Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures,” J. Opt. Soc. 72(10), 1382–1384 (1982).
[Crossref]

Kühn, J.

Leith, E. N.

E. N. Leith and J. Upatnieks, “Holography with Achromatic-Fringe Systems,” J. Opt. Soc. Am. A 57(8), 975–980 (1967).
[Crossref]

Marquet, P.

Montfort, F.

Natale, P. D.

Nicola, S. D.

Sheppard, C. J. R.

J. T. Sheridan and C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105(5-6), 367–378 (1994).
[Crossref]

M. Gu and C. J. R. Sheppard, “Effects of defocus and primary spherical aberrations on images of a straight edge in confocal microscopy,” Appl. Opt. 33(4), 625–630 (1994).
[Crossref] [PubMed]

J. T. Sheridan and C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10(4), 614–632 (1993).
[Crossref]

Sheridan, J. T.

J. T. Sheridan and C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105(5-6), 367–378 (1994).
[Crossref]

J. T. Sheridan and C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10(4), 614–632 (1993).
[Crossref]

Upatnieks, J.

E. N. Leith and J. Upatnieks, “Holography with Achromatic-Fringe Systems,” J. Opt. Soc. Am. A 57(8), 975–980 (1967).
[Crossref]

Appl. Opt. (1)

M. Gu and C. J. R. Sheppard, “Effects of defocus and primary spherical aberrations on images of a straight edge in confocal microscopy,” Appl. Opt. 33(4), 625–630 (1994).
[Crossref] [PubMed]

J. Mod. Opt. (1)

R. Chmelík, “Three–dimensional scalar imaging in high–aperture low–coherence interference and holographic microscopes,” J. Mod. Opt. 53(18), 2673–2689 (2006).
[Crossref]

J. Opt. Soc. (1)

J. Komrska, “Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures,” J. Opt. Soc. 72(10), 1382–1384 (1982).
[Crossref]

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

E. N. Leith and J. Upatnieks, “Holography with Achromatic-Fringe Systems,” J. Opt. Soc. Am. A 57(8), 975–980 (1967).
[Crossref]

J. T. Sheridan and C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10(4), 614–632 (1993).
[Crossref]

Meas. Sci. Technol. (2)

Opt. Commun. (1)

J. T. Sheridan and C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105(5-6), 367–378 (1994).
[Crossref]

Opt. Eng. (1)

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

Optik (Stuttg.) (1)

J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik (Stuttg.) 80, 171–183 (1988).

Other (9)

T. Wilson, and C. J. R. Sheppard, Theory and practice of scanning optical microscopy (Academic Press, 1984).

Y. Emery, E. Cuche, F. Marquet, S. Bourquin, P. Marquet, J. Kühn, N. Aspert, M. Botkin, and Ch. Depeursinge, “Digital Holographic Microscopy (DHM): Fast and Robust 3D measurements with interferometric resolution for industrial inspection,” in Fringe 2005 – New Optical Sensors and Measurement Systems (Springer, (2006), pp. 667-671.

Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kühn, M. Botkine, T. Colomb, F. Montfort, F. Charrière, Ch. Depeursinge, P. Debergh, and R. Conde, “Digital Holographic Microscopy (DHM) for metrology and dynamic characterization of MEMS and MOEMS,“ Proc. SPIE 6186, art. no. 61860N (2006).

R. Chmelik, H. Uhlirova, P. Kolman, and P. Vesely, “Wide Range Coherence Digital Holographic Microscope,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2010), paper DTuA6. http://www.opticsinfobase.org/abstract.cfm?URI=DH-2010-DTuA6

J. Komrska, “ The Fourier transform of lattices,” in Proceedings of the International Summer School 27th May – 5thJune1991, Chlum u Třeboně, Czechoslovakia, L. Eckertová and T. Růžička, eds. (Institute of Physics Publishing, Bristol and Philadephia), pp. 87–113.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965), Chap. 6.

M. von Laue, and E. H. Wagner, Röntgenstrahl-Interferenzen (Akademische Verlagsgesellschaft, 1960), Chap. 3.

www.Luxpop.com : Thin film and bulk index of refraction and photonics calculations.

M. Born, and E. Wolf, Principles of Optics. 7th ed. (Cambridge University Press, 2002), Chap. 1.

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Fig. 1 Test surface relief: a) AFM image, b) SEM image, theoretical model with protrusions of c) block shape with rounded side edges (MB), d) cylindrical shape (MC).

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Fig. 2 Coherence-controlled holographic microscope. S..source, L1..lens, D..diffraction grating, M..mirror, BS..beam splitter, Obj..objective lens, Sp..specimen, RM..reference mirror, OP..output plane, L2..objective lens, CCD..camera.

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Fig. 3 Detail of the specimen central part a) image-plane hologram and b) reconstructed image phase.

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Fig. 4 Contour plot of a quantitative phase image (a) observed experimentally, (b) computed theoretically with MB model, (c) with MC model. Contour lines span the range (−1.8; 0) radians with the step 0.1 radian. The three results are compared by means of their cross sections d) – f) made in three directions marked in a).

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Tables (2)

Table 1 Correspondence of a Periodic Relief Description with That of a Periodic Crystal Lattice

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Table 2 Modulations of Curves in Fig. 4 and Their Mutual Deviations Converted into OPD

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Equations (18)

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u^{i} (x^{i}, y^{i}, z^{i}) = \underset{\infty}{∭} T (m, n, s) c (m, n, s) \exp [2 π i (m x^{i} + n y^{i} + s z^{i})] d m d n d s,

T (m, n, s) = \underset{\infty}{∭} t (x, y, z) \exp [- 2 π i (m x + n y + s z)] d x d y d z,

t (x, y, z) = δ (z),

u_{0}^{i} (x^{i}, y^{i}, z^{i}) = \int_{- \infty}^{\infty} c (0, 0, s) \exp (2 π i s z^{i}) d s .

u_{0}^{i} (x^{i}, y^{i}, z^{i}) = C \exp [- i k (1 + \cos α) z^{i}] sinc [k (1 - \cos α) z^{i}],

φ (x^{i}, y^{i}, z^{i}) = - k (1 + \cos α) z^{i}

t_{b i n} (x, y) = {\begin{matrix} 0 \dots for regions where z = 0 \\ 1 \dots for regions where z = z_{1} \end{matrix}

t (x, y, z) = t_{b i n} (x, y) r_{1} δ (z - z_{1}) + [1 - t_{b i n} (x, y)] r_{0} δ (z),

T (m, n, s) = T_{b i n} (m, n) [r_{1} \exp (- 2 π i s z_{1}) - r_{0}] + r_{0} δ (m) δ (n),

u^{i} (x^{i}, y^{i}, z^{i}) = u_{0}^{i} (x^{i}, y^{i}, z^{i}) + u_{b i n}^{i} (x^{i}, y^{i}, z^{i}),

u_{b i n}^{i} (x^{i}, y^{i}, z^{i}) = \iint_{\infty} T_{b i n} (m, n) c_{b i n} (m, n; z^{i}, z_{1}) \exp [2 π i (m x^{i} + n y^{i})] d m d n,

c_{b i n} (m, n; z^{i}, z_{1}) = \int_{\infty} c (m, n, s) {r_{1} \exp [2 π i s (z^{i} - z_{1})] - r_{0} \exp (2 π i s z^{i})} d s .

t_{b i n} (x, y) = f_{U} (x, y) * l (x, y),

T_{b i n} (m, n) = F_{U} (m, n) L (m, n),

l (x, y) = \sum_{n_{x} = 1}^{N_{x}} \sum_{n_{y} = 1}^{N_{y}} δ (x - n_{x} a) δ (y - n_{y} a)

L (m, n) = \exp {- π i a [(N_{x} - 1) m + (N_{y} - 1) n} \frac{\sin (π a N_{x} m) \sin (π a N_{y} n)}{\sin (π a m) \sin (π a n)} .

F_{U} (m, n) = π r^{2} \frac{2 J_{1} (2 π r \sqrt{m^{2} + n^{2}})}{2 π r \sqrt{m^{2} + n^{2}}},

\begin{array}{l} F_{U} (m, n) = \\ = \frac{1}{2 π^{2}} {\sin η {\frac{\cos 2 π [m (d + ρ) - n d]}{n (m \sin η + n \cos η)} + \frac{\cos 2 π [m d + n (d + ρ)]}{m (m \cos η - n \sin η)} + \\ + \frac{\cos 2 π [m (d + ρ) + n d]}{n (- m \sin η + n \cos η)} + \frac{\cos 2 π [- m d + n (d + ρ)]}{m (m \cos η + n \sin η)}} + \\ + \sin 2 η \sum_{r = 1}^{l - 1} {\frac{\cos 2 π [m (d + ρ \cos 2 r η) + n (d + ρ \sin 2 r η)]}{[m \sin (2 r - 1) η - n \cos (2 r - 1) η] [m \sin (2 r + 1) η - n \cos (2 r + 1) η]} + \\ + \frac{\cos 2 π [- m (d + ρ \sin 2 r η) + n (d + ρ \cos 2 r η)]}{[m \cos (2 r - 1) η + n \sin (2 r - 1) η] [m \cos (2 r + 1) η + n \sin (2 r + 1) η]}}}, \end{array}

Periodic relief:	lattice function	motif	periodic relief
Crystal lattice:	lattice function	unit cell	crystal lattice
Notation:	l(x, y)	f_u (x, y)	t_bin (x, y) = f_u (x, y) *l(x, y)
Scheme:

Cross-section	Experiment (nm)	MB model (nm)	Difference (MB from experiment)	MC model (nm)	Difference (MC from experiment)	Plane-surface approx. (nm)
d	55 ± 3	60	5 nm 9.0%	58	3 nm 5.4%	199
e	30 ± 3	21	9 nm 30.0%	25	5 nm 16.7%	0
f	73 ± 3	72	1 nm 1.4%	73	0 nm 0%	199

Periodic relief:	lattice function	motif	periodic relief
Crystal lattice:	lattice function	unit cell	crystal lattice
Notation:	l(x, y)	f_u (x, y)	t_bin (x, y) = f_u (x, y) *l(x, y)
Scheme:

Cross-section	Experiment (nm)	MB model (nm)	Difference (MB from experiment)	MC model (nm)	Difference (MC from experiment)	Plane-surface approx. (nm)
d	55 ± 3	60	5 nm 9.0%	58	3 nm 5.4%	199
e	30 ± 3	21	9 nm 30.0%	25	5 nm 16.7%	0
f	73 ± 3	72	1 nm 1.4%	73	0 nm 0%	199

Abstract

References

2008 (1)

2006 (1)

2004 (1)

1999 (1)

1994 (2)

1993 (1)

1988 (1)

1982 (1)

1967 (1)

Charrière, F.

Chiarini, M.

Chmelík, R.

Colomb, T.

Coppola, G.

Cuche, E.

Depeursinge, C.

Emery, Y.

Ferraro, P.

Finizio, A.

Grilli, S.

Gu, M.

Harna, Z.

Iodice, M.

Komrska, J.

Kühn, J.

Leith, E. N.

Marquet, P.

Montfort, F.

Natale, P. D.

Nicola, S. D.

Sheppard, C. J. R.

Sheridan, J. T.

Upatnieks, J.

Appl. Opt. (1)

J. Mod. Opt. (1)

J. Opt. Soc. (1)

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

Meas. Sci. Technol. (2)

Opt. Commun. (1)

Opt. Eng. (1)

Optik (Stuttg.) (1)

Other (9)

Cited By

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Tables (2)

Equations (18)

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