Abstract

Phase-shifting interferometry is a preferred technique for high-precision surface form measurements, but the difficulty in handling the intensity distortions from multiple-surface interference has limited the general use of the technique to interferometer cavities producing strict two-beam interference. I show how the capabilities of phase-shifting interferometry can be extended to address this problem using wavelength tuning techniques. The basic theory behind the technique is reviewed and applied specifically to the measurement of parallel plates, where surfaces, optical and physical thickness, and homogeneity are simultaneously obtained. Basic system requirements are derived, common error sources are discussed, and the results of the measurements are compared with theory and alternative measurement methods.

© 2003 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.
  2. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  3. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  4. L. Deck, P. de Groot, “Punctuated quadrature phase-shifting interferometry,” Opt. Lett. 23, 19–21 (1998).
    [CrossRef]
  5. K. Freischlad, “Large flat panel profiler,” in Flatness, Roughness and Discrete Defect Characterization for Computer Disks, Wafers and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 163–171 (1996).
    [CrossRef]
  6. P. de Groot, “Grating interferometer for flatness testing,” Opt. Lett. 21, 228–230 (1996).
    [CrossRef] [PubMed]
  7. C. Ai, “Multimode-laser interferometric apparatus for eliminating background interference fringes from thin-plate measurements,” U.S. patent5,452,088 (19September1995).
  8. P. Dewa, A. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).
  9. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
    [CrossRef]
  10. K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
    [CrossRef] [PubMed]
  11. M. Suematsu, M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique for fringe analysis,” Appl. Opt. 30, 4046–4055 (1991).
    [CrossRef] [PubMed]
  12. R. J. Bell, Introductory Fourier Transform Spectroscopy, 1st ed. (Academic, New York, 1972), Chap. 1.
  13. FTPSI is the topic of U.S. and foreign patents pending assigned to Zygo Corporation.
  14. L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” in Interferometry XI: Applications, W. Osten, ed., Proc. SPIE4778, 218–226 (2002).
  15. L. Deck, “Simultaneous multiple surface measurements using Fourier-transform phase shifting interferometry,” in New Methods for the Acquisition, Processing and Evaluation of Data in Optical Metrology, W. Osten, W. Jüptner, eds., (Elsevier, Paris, 2001), pp. 230–236.
  16. L. Deck, “Multiple surface phase-shifting interferometry,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
    [CrossRef]
  17. C. Ai, J. C. Wyant, “Measurement of the inhomogeneity of a window,” Opt. Eng. 30, 1399–1404 (1991).
    [CrossRef]
  18. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. 12, 354–365 (1995).
    [CrossRef]
  19. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]

2000 (1)

1998 (1)

1996 (1)

1995 (2)

1992 (1)

1991 (2)

1990 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Ai, C.

C. Ai, J. C. Wyant, “Measurement of the inhomogeneity of a window,” Opt. Eng. 30, 1399–1404 (1991).
[CrossRef]

C. Ai, “Multimode-laser interferometric apparatus for eliminating background interference fringes from thin-plate measurements,” U.S. patent5,452,088 (19September1995).

Bell, R. J.

R. J. Bell, Introductory Fourier Transform Spectroscopy, 1st ed. (Academic, New York, 1972), Chap. 1.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

de Groot, P.

Deck, L.

L. Deck, P. de Groot, “Punctuated quadrature phase-shifting interferometry,” Opt. Lett. 23, 19–21 (1998).
[CrossRef]

L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” in Interferometry XI: Applications, W. Osten, ed., Proc. SPIE4778, 218–226 (2002).

L. Deck, “Simultaneous multiple surface measurements using Fourier-transform phase shifting interferometry,” in New Methods for the Acquisition, Processing and Evaluation of Data in Optical Metrology, W. Osten, W. Jüptner, eds., (Elsevier, Paris, 2001), pp. 230–236.

L. Deck, “Multiple surface phase-shifting interferometry,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
[CrossRef]

Dewa, P.

P. Dewa, A. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).

Freischlad, K.

K. Freischlad, “Large flat panel profiler,” in Flatness, Roughness and Discrete Defect Characterization for Computer Disks, Wafers and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 163–171 (1996).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Kulawiec, A.

P. Dewa, A. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).

Larkin, K. G.

Okada, K.

Oreb, B. F.

Ose, T.

Sakuta, H.

Suematsu, M.

Takeda, M.

Tsujiuchi, J.

Wyant, J. C.

C. Ai, J. C. Wyant, “Measurement of the inhomogeneity of a window,” Opt. Eng. 30, 1399–1404 (1991).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. 12, 354–365 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

C. Ai, J. C. Wyant, “Measurement of the inhomogeneity of a window,” Opt. Eng. 30, 1399–1404 (1991).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Other (9)

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

K. Freischlad, “Large flat panel profiler,” in Flatness, Roughness and Discrete Defect Characterization for Computer Disks, Wafers and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 163–171 (1996).
[CrossRef]

C. Ai, “Multimode-laser interferometric apparatus for eliminating background interference fringes from thin-plate measurements,” U.S. patent5,452,088 (19September1995).

P. Dewa, A. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).

R. J. Bell, Introductory Fourier Transform Spectroscopy, 1st ed. (Academic, New York, 1972), Chap. 1.

FTPSI is the topic of U.S. and foreign patents pending assigned to Zygo Corporation.

L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” in Interferometry XI: Applications, W. Osten, ed., Proc. SPIE4778, 218–226 (2002).

L. Deck, “Simultaneous multiple surface measurements using Fourier-transform phase shifting interferometry,” in New Methods for the Acquisition, Processing and Evaluation of Data in Optical Metrology, W. Osten, W. Jüptner, eds., (Elsevier, Paris, 2001), pp. 230–236.

L. Deck, “Multiple surface phase-shifting interferometry,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Four-surface Fizeau cavity geometry used to measure the profiles, optical thickness, homogeneity, and physical thickness of a parallel plate.

Fig. 2
Fig. 2

Elemental two-surface cavity.

Fig. 3
Fig. 3

First-, second-, and third-order spectra for a four-surface cavity geometry with M = 3 and μ = 3 and a Blackman window. Note that the second- and third-order frequencies are well separated from the first-order peaks. The frequency is normalized to the sample rate.

Fig. 4
Fig. 4

rms phase error sensitivity to intensity distortions for a 120-frame acquisition by 90° phase-shift increments and a Hamming window. The frequency is normalized to the sample rate. The sensitivity of the seven-frame PSI algorithm is shown for comparison.

Fig. 5
Fig. 5

rms phase error sensitivity spectrum to intensity distortions of the same amplitude for the 0.225 primary frequency as a function of μ when a Hamming window is used. The vertical dashed lines indicate the proximity of the closest neighboring first-order peak in a four-surface cavity.

Fig. 6
Fig. 6

rms phase error sensitivity spectrum to intensity distortions of the same amplitude for the 0.225 primary frequency as a function of μ when a Blackman window is used. The vertical dashed lines indicate the proximity of the closest neighboring first-order peak in a four-surface cavity.

Fig. 7
Fig. 7

Phase error sensitivity to small-amplitude vibrations with a 64-frame FTPSI acquisition with a Blackman window. The sensitivity of the de Groot seven-frame PSI algorithm is also shown for comparison.

Fig. 8
Fig. 8

Detail of the WM. The HDS is oriented at 45° to the HSPMI polarization axis and the four detectors A, B, C, and D observe interference in phase quadrature. PBS, polarization beam splitter; BS, beam splitter.

Fig. 9
Fig. 9

Apparatus used in the experiments to verify FTPSI.

Fig. 10
Fig. 10

Interferogram of the 8-mm parallel plate in a four-surface cavity.

Fig. 11
Fig. 11

Measured OPL spectrum of optical gap lengths of the four-surface geometry with μ = 3 and M = 3.

Fig. 12
Fig. 12

Surface profile of the first surface (S 2) of the 8-mm parallel plate: (a) with FTPSI and (b) with a seven-frame PSI algorithm and index-matching lacquer to minimize reflections from the opposite surface. The ripple distortion in (b) is 5.5-nm PV. Tilt was removed from both (a) and (b).

Fig. 13
Fig. 13

Surface profile of the second surface (S 3) of the 8-mm parallel plate (a) with FTPSI and (b) with a seven-frame PSI algorithm and index-matching lacquer to minimize reflections from the opposite surface. The ripple distortion in (b) is 5.8-nm PV. Tilt was removed from both (a) and (b).

Fig. 14
Fig. 14

Optical thickness variation of the 8-mm parallel plate with tilt removed.

Fig. 15
Fig. 15

Transmission measurements of the 8-mm parallel plate (a) with FTPSI and (b) with a seven-frame PSI algorithm. The ripple distortion in (b) is approximately 6-nm PV, and tilt was removed in both (a) and (b).

Fig. 16
Fig. 16

Homogeneity variation of the 8-mm parallel plate (a) with FTPSI and (b) with oil-on plates and a seven-frame PSI algorithm. Tilt was removed in both (a) and (b).

Fig. 17
Fig. 17

Homogeneity of the 8-mm parallel plate by use of FTPSI without tilt removed. The overall variation was 183 nm, implying a 3-nm/mm2 linear homogeneity variation along the long axis of the plate.

Fig. 18
Fig. 18

Measurement of a surface under the influence of a sinusoidal vibration with a normalized frequency of 0.21- and 20-nm amplitude. (a) Shows that the cavity interferogram contained approximately six fringes of tilt. (b) The surface profile produced with a seven-frame PSI algorithm and exhibiting ripple with 9–10-nm PV. (c) Produced with a 64-frame FTPSI measurement; no ripple is observed.

Equations (34)

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Φx, y=4πmnx, yGx, yνc+Θm.
Φx, yt=4πmnx, yGx, ycνt1+ηx, y,
fmx, y=Lmx, ycνt,
Lmx, y=2mnx, yGx, y1+ηx, y.
ϕx, y=argAx, y, f,
Ax, y, f=- ax, y, twtexp-i2πftdt,
fW=LWcνt.
f=LLWfW,
Ax, y, L=-ax, y, twtexp-i2πLLWfWtdt.
ϕW=2πfWt,
Ax, y, L=-ax, y, ϕWwϕW×exp-iϕWL/LWdϕW.
Ax, y, L=j=0P-1ajx, ywj exp-iϕWjL/LWΔϕWj,
SL=|AL|2,
ϕx, y=argAx, y, L.
fmin=1+μΔT=1+μfSP,
Γ=c1+μΔν.
G1=Γ, G2=G1M+1,  GN-1=GN-2M+1=ΓM+1N-2.
fS-Mfmaxfmax+1+μfSP,
fmax=Γj=0N-2M+1jΔνcPfS.
P1+μj=0N-1M+1j.
fi=M+1i-1j=0N-1M+1j,
Õx, y=λ4πϕG2x, y,
S˜2x, y=λ4πϕG1x, y,
S˜3x, y=λ4πϕG3x, y-ϕECx, y,
G˜2x, y=λ4πϕECx, y-ϕG3x, y-ϕG1x, y.
ñx, y=Õx, y-n¯G˜2x, yT¯.
hj=Wj cos2πj 1fs fC+Wj sin2πj 1fs fCfor j=0P-1,
Sf=2|HfBf|,
Hf=j=0P-1 hj exp-i2πjf/fsj=0P-1 Wj.
2/Q, 2M+1/Q, 2+M+1/Q, 1+2M+1/Q, |M+1-1|/Q, 2+2M+1/Q,
M+1-1+xQ=1Q  x=1-M=-1,
2Q=M+1+xQ  x=1-M=-1.
1-MM+1+x+1Q=M+1+xQ  x=2M+1=23,
ϕWM=tan-1A-CD-B,

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