Abstract

An amplitude-division two-beam interferometer illuminated by a quasi-monochromatic, spatially incoherent, and periodic source yields multiple localization planes of interference fringes. If a thick transmission sample with a few localized phase disturbances in various layers is placed in the interferometer, the disturbances in a layer can be detected, making its images through the two arms coincide with a chosen localization plane. Different layers can be analyzed by means of shifting the localization plane by a variation of the source period without any other changes in the device. Here we illustrate this method by applying it to a shearing interferometer, a classical Wollaston prism placed between crossed polarizers. Experimental images of different observation planes are obtained, and they are in good agreement with the theoretical expectations.

© 2001 Optical Society of America

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References

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  1. W. H. Steel, Interferometry (Cambridge University, London, 1967).
  2. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  3. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
    [CrossRef]
  4. D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).
  5. B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement,” Meas. Sci. Technol. 10, 33–55 (1999).
    [CrossRef]
  6. J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
    [CrossRef]
  7. J. M. Simon, S. A. Comastri, “Fringe localization depth,” Appl. Opt. 26, 5125–5129 (1987).
    [CrossRef] [PubMed]
  8. J. M. Simon, S. A. Comastri, “Interferometers: equivalent sine condition,” Appl. Opt. 27, 4725–4730 (1988).
    [CrossRef] [PubMed]
  9. S. Cha, C. M. Vest, “Interferometry and reconstruction of strongly refracting asymmetric-refractive-index fields,” Opt. Lett. 4, 311–313 (1979).
    [CrossRef] [PubMed]
  10. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  11. J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
    [CrossRef]
  12. S. A. Comastri, J. M. Simon, “Multilocalization and van Cittert–Zernike theorem. 1. Theory,” J. Opt. Soc. Am. 17, 1265–1276 (2000).
    [CrossRef]
  13. J. M. Simon, S. A. Comastri, “Multilocalization and van Cittert–Zernike theorem. 2. Application to the Wollaston prism,” J. Opt. Soc. Am. 17, 1277–1283 (2000).
    [CrossRef]
  14. J. M. Simon, S. A. Comastri, R. Echarri, “The Mach–Zehnder interferometer: examination of a volume by non-classical localization plane shifting,” Pure Appl. Opt. 3, 242–249 (2001).

2001 (1)

J. M. Simon, S. A. Comastri, R. Echarri, “The Mach–Zehnder interferometer: examination of a volume by non-classical localization plane shifting,” Pure Appl. Opt. 3, 242–249 (2001).

2000 (2)

S. A. Comastri, J. M. Simon, “Multilocalization and van Cittert–Zernike theorem. 1. Theory,” J. Opt. Soc. Am. 17, 1265–1276 (2000).
[CrossRef]

J. M. Simon, S. A. Comastri, “Multilocalization and van Cittert–Zernike theorem. 2. Application to the Wollaston prism,” J. Opt. Soc. Am. 17, 1277–1283 (2000).
[CrossRef]

1999 (1)

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement,” Meas. Sci. Technol. 10, 33–55 (1999).
[CrossRef]

1998 (1)

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

1997 (1)

1988 (1)

1987 (2)

1980 (1)

J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
[CrossRef]

1979 (2)

S. Cha, C. M. Vest, “Interferometry and reconstruction of strongly refracting asymmetric-refractive-index fields,” Opt. Lett. 4, 311–313 (1979).
[CrossRef] [PubMed]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Cha, S.

Comastri, S. A.

J. M. Simon, S. A. Comastri, R. Echarri, “The Mach–Zehnder interferometer: examination of a volume by non-classical localization plane shifting,” Pure Appl. Opt. 3, 242–249 (2001).

J. M. Simon, S. A. Comastri, “Multilocalization and van Cittert–Zernike theorem. 2. Application to the Wollaston prism,” J. Opt. Soc. Am. 17, 1277–1283 (2000).
[CrossRef]

S. A. Comastri, J. M. Simon, “Multilocalization and van Cittert–Zernike theorem. 1. Theory,” J. Opt. Soc. Am. 17, 1265–1276 (2000).
[CrossRef]

J. M. Simon, S. A. Comastri, “Interferometers: equivalent sine condition,” Appl. Opt. 27, 4725–4730 (1988).
[CrossRef] [PubMed]

J. M. Simon, S. A. Comastri, “Fringe localization depth,” Appl. Opt. 26, 5125–5129 (1987).
[CrossRef] [PubMed]

J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
[CrossRef]

Dorrio, B. V.

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement,” Meas. Sci. Technol. 10, 33–55 (1999).
[CrossRef]

Echarri, R.

J. M. Simon, S. A. Comastri, R. Echarri, “The Mach–Zehnder interferometer: examination of a volume by non-classical localization plane shifting,” Pure Appl. Opt. 3, 242–249 (2001).

Echarri, R. M.

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

Eiju, T.

Fernandez, J. L.

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement,” Meas. Sci. Technol. 10, 33–55 (1999).
[CrossRef]

Garea, M. T.

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

Hariharan, P.

Jahns, J.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Lohmann, A. W.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Oreb, B. F.

Phillion, D. W.

Servin, M.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Simon, J. M.

J. M. Simon, S. A. Comastri, R. Echarri, “The Mach–Zehnder interferometer: examination of a volume by non-classical localization plane shifting,” Pure Appl. Opt. 3, 242–249 (2001).

S. A. Comastri, J. M. Simon, “Multilocalization and van Cittert–Zernike theorem. 1. Theory,” J. Opt. Soc. Am. 17, 1265–1276 (2000).
[CrossRef]

J. M. Simon, S. A. Comastri, “Multilocalization and van Cittert–Zernike theorem. 2. Application to the Wollaston prism,” J. Opt. Soc. Am. 17, 1277–1283 (2000).
[CrossRef]

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

J. M. Simon, S. A. Comastri, “Interferometers: equivalent sine condition,” Appl. Opt. 27, 4725–4730 (1988).
[CrossRef] [PubMed]

J. M. Simon, S. A. Comastri, “Fringe localization depth,” Appl. Opt. 26, 5125–5129 (1987).
[CrossRef] [PubMed]

J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
[CrossRef]

Simon, M. C.

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

Steel, W. H.

W. H. Steel, Interferometry (Cambridge University, London, 1967).

Vest, C. M.

Am. J. Phys. (1)

J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
[CrossRef]

Appl. Opt. (4)

J. Mod. Opt. (1)

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

S. A. Comastri, J. M. Simon, “Multilocalization and van Cittert–Zernike theorem. 1. Theory,” J. Opt. Soc. Am. 17, 1265–1276 (2000).
[CrossRef]

J. M. Simon, S. A. Comastri, “Multilocalization and van Cittert–Zernike theorem. 2. Application to the Wollaston prism,” J. Opt. Soc. Am. 17, 1277–1283 (2000).
[CrossRef]

Meas. Sci. Technol. (1)

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement,” Meas. Sci. Technol. 10, 33–55 (1999).
[CrossRef]

Opt. Commun. (1)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (1)

J. M. Simon, S. A. Comastri, R. Echarri, “The Mach–Zehnder interferometer: examination of a volume by non-classical localization plane shifting,” Pure Appl. Opt. 3, 242–249 (2001).

Other (2)

W. H. Steel, Interferometry (Cambridge University, London, 1967).

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

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Figures (9)

Fig. 1
Fig. 1

Shearing interferometer (S.I.): σ, source; (x, y), coordinates at the source plane; z, normal to the source; O, central source point; , sample with a disturbance surrounding one point; σ̃, image of σ through the sample; Õ, effective central source point giving rise to a wave front that has a disturbance around point Q when it immediately leaves ; Õ′ or Õ″ (and Q′ or Q″), images of Õ (and Q) through arms I or II; ′, axis containing Õ′ and Õ″; S˜k and S˜k, images of an arbitrary source point; Ploc,o, point of the classical localization plane Σ0 such that ÕPloc,o = ÕPloc,o; “I o ” and “II o ”, rays ÕPloc,o and ÕPloc,o; ′, axis perpendicular to ′ and containing Ploc,o; Po and P′, points on the observation plane Σ′; ξ′, axis parallel to ′ with origin at Po; B′ (and B″), point where the ray “I o ” (and “II o ”) intersects the ξ′ axis; D′, distance from plane (′, ′) to (ξ′, η′) (′ and η′ not shown for simplicity); so = BB″, shear.

Fig. 2
Fig. 2

Experimental device: (x, y, z), orthogonal coordinate system with z along the optical bench, y parallel to the source slits, and x in the source plane; I.S., illumination optical system consisting of a continuous mercury source, a condenser that is a convergent lens, a mask with slits parallel to y, and a sample with front and back surfaces T 1 and T 2; Õ, image of the central source point O through the sample; Wollaston prism with optical axes E 1 and E 2, angle κ, and between two crossed polarizers 1 and 2; O.S., observation optical system consisting of a relay lens, a filter, and a block composed of a microscope objective and a charged-coupled device; Σ′: arbitrary observation plane; Π, reference plane to measure longitudinal distances; g, distance from Σ′ to the right-hand border of , G Σ′, distance from the right-hand border of to Π.

Fig. 3
Fig. 3

Images of the sample when the source is a slit: (a1) surface T 2, (a2) surface T 1.

Fig. 4
Fig. 4

Images of (a) the calibration grating, (b) the localization plane of order 0.

Fig. 5
Fig. 5

Images of the gratings used: (a) period δx 1, (b) period δx 2.

Fig. 6
Fig. 6

Coherence patterns for (a) grating of period δx 1 (b) grating of period δx 2.

Fig. 7
Fig. 7

Images obtained experimentally with a periodic source: (a1) [and (b1)] correspond to surface T 2 of the sample when the grating of period δx 1 (and δx 2) is used; (a2) [and (b2)] correspond to surface T 1 of the sample when the grating of period δx 1 (and δx 2) is used; (a3) [and (b3)] interferogram at the localization plane of order m = -2 when the grating of period δx 1 (and δx 2) is used.

Fig. 8
Fig. 8

Amplitude-division interferometer: Ploc (and P′), point at the classical localization (and at any observation) plane where ray “I” (and “Ĩ”) intersects “II”; P I,loc and P II,loc (or P I and P II ), images of Ploc (or P′) back through the interferometer. (a) Source space: O, central source point; S d and S u , points on the lower and the upper borders of the source; (x, y, z), orthogonal coordinates (z normal to the source); β II and β I = β II + Δβ, angles from the z axis to rays OP II and OP I ; J I , point on the ray OP II,loc such that δξ = J I P I is parallel to x; n, refractive index. (b) Observation space: O′ and O″, images of O through arms I and II; Sd and Su, images of S d and S u through arm I; (x′, y′, z′), coordinates at the source image through arm I (z′ normal to Sd Su); Z″, distance from Ploc to P′; βII, angle from the z′ axis to the ray OPloc; J′, image of J I ; δξ′ = JP′, relative coordinate; F′ = OPloc; F″ = O″Ploc; Λ′: bifurcation angle (from ray “II” to “I”). n′, refractive index.

Fig. 9
Fig. 9

Interferometer parameters: Õ, central source point; P Io and P IIo , images of the observation point Po back through the two arms; Σ(I) and Σ(II), images of Σ′ back through the two arms; σ̃′ and σ̃″, images of the effective source σ̃ through arms I and II; α o , angle of tilt; Õ P Io - Õ P IIo = h o , shift; s o , shear; τ o = ([Õ Po] II - [Õ Po] I )/c, delay; to, tilt; Õ″Po - Õ′Po = lo, lead.

Tables (1)

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Table 1 Longitudinal Distance and Fringe Spacing

Equations (27)

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K=Z cosΛo/2Z, F˜=O˜Ploc,o=D-KcosΛo/2D-K.
IOP=IIP+IIIP+2IIP×IIIP1/2 cosΨOξ, η.
ΨOξ, η=ΨO-2π/λ¯O˜P-O˜P,
ΨO=θI,II-2π/λ¯O˜O˜-O˜O˜,
D=ξ0j+1-ξ0j=λ¯n|D||to|=λ¯n|Λo|1+KF˜=01+KF˜,
ΨOξ, η=ΨO+2πD ξ-2π/λ¯wξ, η-wξ, η,
ξj-ξ0j=D/λ¯wξj, ηj-wξj, ηj,
ξj-ξ0j=Dλ¯ sowξξ,η,
μI,IIP|Φ=0=1Nsin ζζsinϑNsinϑ, ζ=ba10P2,  ϑ=δxa10P2, a10P=2πq10KF˜+K,
Km=F˜m0q1δx-m0,  q1=H˜H˜.
Ψσ=argμI,IIP+θI,II-2π/λ¯O˜O˜-O˜O˜,
ecal=0.1 mm,  δcal=14.00±0.003 mm, eΣ=ecalδΣ/δcal.
Kexp=g+G0-g+GΣ=G0-GΣ,
G0=46.0±0.5 mm, 0=0.0571±0.0002 mm.
GF=172.5±0.5 mm, F˜=GF-G0=127.0±1.0 mm,
δx1=0.199±0.002 mm, b1=0.086±0.004 mm, δx2=0.257±0.004 mm, b2=0.114±0.004 mm.
Z=PlocP, F=OPloc, F=OPloc.
VOPD=LIP-LIIP-LI,OP+LII,OP=-λ/2πa10Px+a01Py+Φ(x, y, P,
IP=IIP+IIIP+2IIP×IIIP1/2|μI,IIP|cosΨP,
ΨP=argμI,IIP+θI,II-a00P,
μI,IIP|Φ=0=1Nsin ζζsinϑNsinϑ,
ζ=ba10P2=nπbλ¯sin βI-sin βII=πq1b0ZF+Z, ϑ=δxa10P2=nπδxλ¯sin βI-sin βII=πq1δx0ZF+Z,
a10P=2πnλ¯sin βI-sin βII=2πq10ZF+Z, 0=λ¯n|F||d0|=λ¯n1|Λ|, q1=H cos βIIH, q˜=q1δx0,
Zm=Fm0q1δx-m0  with q1=HH.
mF+Zmλ¯n|d0|=01+ZmF=01-m/q˜.
L.D.|m=Zm+1/N-Zm-1/N=2FNq˜1-m/q˜2-1/Nq˜.
n×H˜×soD=n×H˜×soD,

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