Abstract

The paper reviews a technique for fringe analysis referred to as Fourier fringe analysis (FFA) or the Fourier transform method, with a particular focus on its application to metrology of extreme physical phenomena. Examples include the measurement of extremely small magnetic fields with subfluxon sensitivity by electron wave interferometry, subnanometer wavefront evaluation of projection optics for extreme UV lithography, the detection of sub-Ångstrom distortion of a crystal lattice, and the measurement of ultrashort optical pulses in the femotsecond to attosecond range, which show how the advantages of FFA are exploited in these cutting edge applications.

© 2012 Optical Society of America

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2012 (1)

2011 (1)

M. Takeda, “Measurement of extreme physical phenomena by Fourier fringe analysis: a review,” Proc. SPIE 8011, 80116S (2011).
[CrossRef]

2010 (1)

X. Su, Q. Zhang, “Dynamic 3-D shape measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (2010).
[CrossRef]

2009 (2)

2007 (1)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementation,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

2006 (3)

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

C. Ouchi, S. Kato, M. Hasegawa, T. Hasegawa, H. Yokota, K. Sugisaki, M. Okada, K. Murakami, J. Saito, M. Niibe, M. Takeda, “EUV wavefront metrology at EUVA,” Proc. SPIE 6152, 61522O (2006).
[CrossRef]

T. Rementter, P. Johnsson, J. Mauritsson, K. Varju, Y. Ni, F. L’epine, E. Gustafsson, M. Kling, J. Khan, R. Lo’pez-Martens, K. J. Schafer, M. J. J. Vrakking, A. L’huillier, “Attosecond electron wave packet interferometry,” Nat. Phys. 2, 323–326 (2006).
[CrossRef]

2005 (5)

2004 (2)

X. Su, W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

A. Dubra, C. Paterson, C. Dainty, “Study of the tear topography dynamics using a lateral shearing interferometer,” Opt. Express 12, 6278–6288 (2004).
[CrossRef]

2003 (2)

M. J. Hÿtch, J.-L. Putaux, J.-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03 Å by electron microscopy,” Nature 423, 270–273 (2003).
[CrossRef]

C. Karaalioglu, Y. Skarlatos, “Fourier transform method for measurement of thin film thickness by speckle interferometer,” Opt. Eng. 42, 1694–1698 (2003).
[CrossRef]

2002 (2)

2001 (3)

T. Fricke-Begemann, J. Burke, “Speckle interferometry: three-dimensional deformation field measurement with a single interferogram,” Appl. Opt. 40, 5011–5022 (2001).
[CrossRef]

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, J. P. Gilles, “3D measurement of micro-mechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

2000 (3)

P. P. Naulleau, K. A. Goldberg, J. Bokor, “Extreme ultraviolet carrier-frequency shearing interferometry of a lithographic four-mirror optical system,” J. Vac. Sci. Technol. B 18, 2939–2943 (2000).
[CrossRef]

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

S. Schedin, G. Pedrini, H. J. Tiziani, “Pulsed digital holography for deformation measurements on biological tissues,” Appl. Opt. 39, 2853–2857 (2000).
[CrossRef]

1999 (3)

1998 (1)

1997 (2)

J. B. Liu, P. D. Ronney, “Modified Fourier transform method for interferogram fringe pattern analysis,” Appl. Opt. 36, 6231–6241 (1997).
[CrossRef]

D. Charraut, C. Bainier, D. Courjon, C. Girard, “Near-field phase measurement by Fourier analysis of the fringe pattern,” Pure Appl. Opt. 6, 491–502 (1997).
[CrossRef]

1996 (4)

M. Borghesi, A. Giulietti, D. Giulietti, L. A. Gizzi, A. Macchi, O. Willi, “Characterization of laser plasmas for interaction studies: progress in time-resolved density mapping,” Phys. Rev. E 54, 6769–6773 (1996).
[CrossRef]

M. Takeda, “Recent progress in phase unwrapping techniques,” Proc. SPIE 2782, 334–343 (1996).
[CrossRef]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef]

M. Takeda, J. Suzuki, “Crystallographic heterodyne phase detection for highly sensitive lattice-distortion measurements,” J. Opt. Soc. Am. A 13, 1495–1500 (1996).
[CrossRef]

1995 (1)

M. J. Hÿtch, M. Gandais, “Quantitative criteria for the detection and characterization of nanocrystals from high-resolution electron microscopy images,” Philos. Mag. A 72, 619–634 (1995).
[CrossRef]

1994 (3)

Y. Ohtsuka, K. Oka, “Contour mapping of the spatiotemporal state of polarization of light,” Appl. Opt. 33, 2633–2636 (1994).
[CrossRef]

T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

M. B. Whitworth, J. M. Huntley, “Dynamic stress analysis by high-resolution reflection moiré photography,” Opt. Eng. 33, 924–931 (1994).
[CrossRef]

1993 (1)

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

1992 (2)

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

M. Takeda, M. Kitoh, “Spatiotemporal frequency multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 9, 1607–1613 (1992).
[CrossRef]

1990 (3)

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

M. Takeda, “Spatial-carrier fringe pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
[CrossRef]

1989 (1)

M. Kujawinska, A. Spik, J. Wojciak, “Fringe pattern analysis using Fourier transform techniques,” Proc. SPIE 1121, 130–135 (1989).
[CrossRef]

1988 (3)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

M. Kalal, B. Luther-Davies, K. A. Nugent, “Phase-amplitude imaging: the fully automated analysis of mega-gauss magnetic field measurements in laser-produced plasmas,” J. Appl. Phys. 64, 3845–3850 (1988).
[CrossRef]

R. Snyder, L. Hesselink, “Measurement of mixing fluid flows with optical tomography,” Opt. Lett. 13, 87–89 (1988).
[CrossRef]

1987 (2)

1986 (1)

1985 (2)

B. Doerband, H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors, Appl. Opt. 24, 2604–2611 (1985).
[CrossRef]

M. Takeda, Q. S. Ru, “Computer-based highly sensitive electron-wave interferometry,” Appl. Opt. 24, 3068–3071 (1985).
[CrossRef]

1984 (2)

M. Takeda, S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

1983 (3)

N. Osakabe, K. Yoshida, Y. Horiuchi, T. Matsuda, H. Tanabe, T. Okuwaki, J. Endo, H. Fujiwara, A. Tonomura, “Observation of recorded magnetization pattern by electron holography,” Appl. Phys. Lett. 42, 746–748 (1983).
[CrossRef]

L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22, 1535–1539 (1983).
[CrossRef]

M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[CrossRef]

1982 (1)

1974 (1)

1972 (1)

1968 (1)

1959 (1)

Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Aharonov, Y.

Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Akiba, M.

Anderson, E.

Attwood, D.

Austin, D. R.

Bachor, H.-A.

Bainier, C.

D. Charraut, C. Bainier, D. Courjon, C. Girard, “Near-field phase measurement by Fourier analysis of the fringe pattern,” Pure Appl. Opt. 6, 491–502 (1997).
[CrossRef]

Benedetti, E.

G. Sansone, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S. De Silvestri, M. Nisoli, “Measurement of harmonic phase differences by interference of attosecond light pulses,” Phys. Rev. Lett. 94, 193903 (2005).
[CrossRef]

Bohm, D.

Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Bokor, J.

P. P. Naulleau, K. A. Goldberg, J. Bokor, “Extreme ultraviolet carrier-frequency shearing interferometry of a lithographic four-mirror optical system,” J. Vac. Sci. Technol. B 18, 2939–2943 (2000).
[CrossRef]

Bone, D. J.

Borghesi, M.

M. Borghesi, A. Giulietti, D. Giulietti, L. A. Gizzi, A. Macchi, O. Willi, “Characterization of laser plasmas for interaction studies: progress in time-resolved density mapping,” Phys. Rev. E 54, 6769–6773 (1996).
[CrossRef]

Bosseboeuf, A.

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, J. P. Gilles, “3D measurement of micro-mechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

Brangaccio, D. J.

Brock, N.

Brundavanam, M. M.

Bruning, J. H.

Bryanston-Cross, P. J.

T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Bryngdahl, O.

Burke, J.

Caumes, J.-P.

G. Sansone, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S. De Silvestri, M. Nisoli, “Measurement of harmonic phase differences by interference of attosecond light pulses,” Phys. Rev. Lett. 94, 193903 (2005).
[CrossRef]

Chan, K.-P.

Chang, C.

Charraut, D.

D. Charraut, C. Bainier, D. Courjon, C. Girard, “Near-field phase measurement by Fourier analysis of the fringe pattern,” Pure Appl. Opt. 6, 491–502 (1997).
[CrossRef]

Chen, W.

X. Su, W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

Chong, C.

Courjon, D.

D. Charraut, C. Bainier, D. Courjon, C. Girard, “Near-field phase measurement by Fourier analysis of the fringe pattern,” Pure Appl. Opt. 6, 491–502 (1997).
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C. Ouchi, S. Kato, M. Hasegawa, T. Hasegawa, H. Yokota, K. Sugisaki, M. Okada, K. Murakami, J. Saito, M. Niibe, M. Takeda, “EUV wavefront metrology at EUVA,” Proc. SPIE 6152, 61522O (2006).
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G. Sansone, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S. De Silvestri, M. Nisoli, “Measurement of harmonic phase differences by interference of attosecond light pulses,” Phys. Rev. Lett. 94, 193903 (2005).
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G. Sansone, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S. De Silvestri, M. Nisoli, “Measurement of harmonic phase differences by interference of attosecond light pulses,” Phys. Rev. Lett. 94, 193903 (2005).
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M. Takeda, J. Suzuki, “Sub-ångstrom measurement of crystal lattice distortions by Fourier fringe analysis,” in Frontiers in Information Optics, Topical Meeting of ICO 1994, Technical Digest (Japan Society of Applied Physics, 1994), p. 105.

M. Takeda, J. Suzuki, “Crystallographic heterodyne phase detection technique for highly-sensitive lattice-distortion measurements,” in Signal Recovery and Synthesis, Vol 11 of Technical Digest Series (Optical Society of America, 1995), pp. 23–25.

D. W. Robinson, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).

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Figures (13)

Fig. 1.
Fig. 1.

Principle of generic Fourier transform (FT) method. IFT, inverse FT. (After Takeda [9] with permission from SPIE.)

Fig. 2.
Fig. 2.

Application areas of the Fourier transform method. (After Takeda [9] with permission from SPIE.)

Fig. 3.
Fig. 3.

Schematic of electron holography microscope. (After Tonomura [52] with permission from APS.)

Fig. 4.
Fig. 4.

(a) Electron phase and (b) its contour map obtained by FTM (contours are drawn at intervals of 1 / 18.5 fringes). (c) Fringe contour map obtained by Osakabe et al. [53] using an optical reconstruction with ten-times phase magnification (contours appear at intervals of 1 / 10 fringes). (After Takeda and Ru [55] with permission from OSA.)

Fig. 5.
Fig. 5.

Schematic diagram of multifunctional EUV interferometer. (After Ouchi et al. [59] with permission from SPIE.)

Fig. 6.
Fig. 6.

LSI and digital Talbot interferometer using a crossed grating. (After Ouchi et al. [59] with permission from SPIE.)

Fig. 7.
Fig. 7.

FTM-based wavefront detection for crossed grating lateral shearing interferometer (CGLSI). iFFT, inverse FFT. (After Ouchi et al. [59] with permission from SPIE.)

Fig. 8.
Fig. 8.

(a) Simulated lattice image with an edge dislocation, (b) spatial frequency spectra of the lattice image, and (c) vertical displacement of simulated atoms represented by the phase value. (After Takeda and Suzuki [65] with permission from OSA.)

Fig. 9.
Fig. 9.

FFA of an edge dislocation seen end-on in silicon. (a) High-resolution electron microscope image in [ 1 1 ¯ 0 ] orientation, Burgers vector b = 1 / 2 [ 110 ] (Fourier transform in inset), (b)  ( 111 ) lattice fringes obtained by filtering (magnified for display purposes), (c)  ( 11 1 ¯ ) lattice fringes, (d) phase image of ( 111 ) lattice fringes, and (e) phase image of ( 11 1 ¯ ) lattice fringes. Color range 0 to 2 π rad . (After Hÿtch et al. [67] with permission from Nature.)

Fig. 10.
Fig. 10.

SPIDER setup. GDD, SF10 glass block; PR, periscope for polarization rotation; BS, beam splitters; TS1, translation stage for adjustment of delay τ ; TS2, translation stage for adjustment of temporal overlap of the short pulse pair with the stretched pulse; HA, periscope for height adjustment; FM, focusing mirror (30 cm radius of curvature); SFG, upconversion crystal [30 μm thick type II β -barium borate (BBO)], and OMA, optical multichannel analyzer. The filled shapes represent silver-coated mirrors, and the filled circles and arrows on the beam path display the polarization state of the beam. (After Gallmann et al. [68] with permission from OSA.)

Fig. 11.
Fig. 11.

SPIDER spectral interferogram of a sub-6-fs pulse (dotted curve). Additionally, the spectra of the individual upconverted pulses are shown (solid and dashed–dotted curves). (After Gallmann et al. [68] with permission from OSA.)

Fig. 12.
Fig. 12.

Reconstructed temporal intensity profile (left) and spectral phase (right; dashed curve) [62]. The independently measured power spectrum of the pulse (right; solid curve) has a transform limit of 5.3 fs. The solid curves are referenced to the left and the dashed curves to the right vertical axes. (After Gallmann et al. [68] with permission from OSA.)

Fig. 13.
Fig. 13.

Spatial analogue of SPIDER. (After Takeda [9] with permission from SPIE.)

Equations (8)

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g ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π ( f 0 X x + f 0 Y y ) + ϕ ( x , y ) ] = a ( x , y ) + 1 2 b ( x , y ) exp [ i ϕ ( x , y ) ] exp [ 2 π i ( f 0 X x + f 0 Y y ) ] + 1 2 b ( x , y ) exp [ i ϕ ( x , y ) ] exp [ 2 π i ( f 0 X x + f 0 Y y ) ] ,
G ( f X , f Y ) = A ( f X , f Y ) + C ( f X f X 0 , f Y f Y 0 ) + C * [ ( f X + f X 0 ) , ( f Y + f Y 0 ) ] .
c ( x , y ) = 1 2 b ( x , y ) exp [ i ϕ ( x , y ) ] .
b ( x , y ) = 2 | c ( x , y ) | ,
ϕ ( x , y ) = tan 1 { Im [ c ( x , y ) ] / Re [ c ( x , y ) ] } ,
Δ ϕ = e S B n d S = e C A · d l ,
S ( ω ) = | E ˜ ( ω ) | 2 + | E ˜ ( ω + Ω ) | 2 + 2 | E ˜ ( ω ) E ˜ ( ω + Ω ) | cos [ ϕ ( ω + Ω ) ϕ ( ω ) + τ ω ] ,
S ( ω ) = | E ˜ ( ω Ω / 2 ) | 2 + | E ˜ ( ω + Ω / 2 ) | 2 + 2 | E ˜ ( ω Ω / 2 ) E ˜ ( ω + Ω / 2 ) | cos [ ϕ ( ω + Ω / 2 ) ϕ ( ω Ω / 2 ) + Δ x ω ] | E ˜ ( ω ) | 2 + | E ˜ ( ω + Ω ) | 2 + 2 | E ˜ ( ω ) E ˜ ( ω + Ω ) | cos [ ϕ ( ω + Ω ) ϕ ( ω ) + Δ x ω ] ,

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