Abstract

We propose a computationally efficient theoretical model for low-coherence interferometric profilers that measure surface heights by scanning the optical path difference of the interferometer. The model incorporates both geometric and spectral effects by means of an incoherent superposition of ray bundles through the interferometer spanning a range of wavelengths, incident angles, and pupil plane coordinates. This superposition sum is efficiently performed in the frequency domain, followed by a Fourier transform to generate the desired simulated interference signal. Example applications include white-light interferometry, high-numerical-aperture microscopy with a near-monochromatic light source, and interference microscopy for thickness and topography analysis of thin-film structures and other complex surface features.

© 2004 Optical Society of America

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References

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  1. H. Haneishi, “Signal processing for film thickness measurements by white light interferometry,” Graduate thesis (Department of Communications and Systems Engineering, University of Electro-communications, Chofu, Tokyo, 1984).
  2. P. de Groot, “Method and apparatus for surface topography measurement by spatial-frequency analysis of interferograms,” U.S. patent5,398,113 (14March1995).
  3. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  4. P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” in Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
    [CrossRef]
  5. J. Biegen, “Determination of the phase change on reflection from two-beam interference,” Opt. Lett. 19, 1690–1692 (1994).
    [CrossRef] [PubMed]
  6. M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc SPIE775, 233–247 (1987).
    [CrossRef]
  7. G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef] [PubMed]
  8. C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4733 (1995).
    [CrossRef] [PubMed]
  9. K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).
  10. R. C. Gutierrez, K. V. Shcheglov, T. K. Tang, “Pulsed-source interferometry for characterization of resonant micromachined structures,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head Island, S.C., 8–11 June 1998.
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1987), p. 40.
  12. Ref. 11, p. 62.
  13. P. A. Flournoy, R. W. McClure, G. Wyntjes, “White-light interferometric thickness. gauge,” Appl. Opt. 11, 1907–1915 (1972).
    [CrossRef] [PubMed]
  14. A. Bosseboeuf, S. Petigrand, “Application of microscopic interferometry techniques in the MEMS field,” in Microsystems Engineering: Metrology and Inspection III, C. Soreeki, ed., Proc. SPIE5145, 1–16 (2003).
    [CrossRef]
  15. P. de Groot, X. Colonna de Lega, “Signal modeling for modern interference microscopes,” in Optical Metrology in Production Engineering, W. Osten, M. Takeda, eds., Proc. SPIE5457, 26–34 (2004).
    [CrossRef]
  16. A. Pförtner, J. Schwider, “White-light-interferometry suffering from dispersion deviations,” Annual report (Lehrstuhl für Optik, University of Erlangen-Nürnberg, Berlin, 2000), p. 40.
  17. S. W. Kim, G. H. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8April2003).

1995 (2)

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4733 (1995).
[CrossRef] [PubMed]

K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).

1994 (1)

1992 (1)

1990 (1)

1972 (1)

Biegen, J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1987), p. 40.

Bosseboeuf, A.

A. Bosseboeuf, S. Petigrand, “Application of microscopic interferometry techniques in the MEMS field,” in Microsystems Engineering: Metrology and Inspection III, C. Soreeki, ed., Proc. SPIE5145, 1–16 (2003).
[CrossRef]

Caber, P. J.

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” in Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

Chim, S. S. C.

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc SPIE775, 233–247 (1987).
[CrossRef]

Colonna de Lega, X.

P. de Groot, X. Colonna de Lega, “Signal modeling for modern interference microscopes,” in Optical Metrology in Production Engineering, W. Osten, M. Takeda, eds., Proc. SPIE5457, 26–34 (2004).
[CrossRef]

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc SPIE775, 233–247 (1987).
[CrossRef]

de Groot, P.

P. de Groot, “Method and apparatus for surface topography measurement by spatial-frequency analysis of interferograms,” U.S. patent5,398,113 (14March1995).

P. de Groot, X. Colonna de Lega, “Signal modeling for modern interference microscopes,” in Optical Metrology in Production Engineering, W. Osten, M. Takeda, eds., Proc. SPIE5457, 26–34 (2004).
[CrossRef]

Dresel, T.

Eguchi, T.

K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).

Flournoy, P. A.

Gutierrez, R. C.

R. C. Gutierrez, K. V. Shcheglov, T. K. Tang, “Pulsed-source interferometry for characterization of resonant micromachined structures,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head Island, S.C., 8–11 June 1998.

Hane, K.

K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).

Haneishi, H.

H. Haneishi, “Signal processing for film thickness measurements by white light interferometry,” Graduate thesis (Department of Communications and Systems Engineering, University of Electro-communications, Chofu, Tokyo, 1984).

Häusler, G.

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc SPIE775, 233–247 (1987).
[CrossRef]

Kim, G. H.

S. W. Kim, G. H. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8April2003).

Kim, S. W.

S. W. Kim, G. H. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8April2003).

Kino, G. S.

Larkin, K. G.

Martinek, S. J.

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” in Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc SPIE775, 233–247 (1987).
[CrossRef]

McClure, R. W.

Nakano, K.

K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).

Niemann, R. J.

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” in Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

Okuma, S.

K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).

Petigrand, S.

A. Bosseboeuf, S. Petigrand, “Application of microscopic interferometry techniques in the MEMS field,” in Microsystems Engineering: Metrology and Inspection III, C. Soreeki, ed., Proc. SPIE5145, 1–16 (2003).
[CrossRef]

Pförtner, A.

A. Pförtner, J. Schwider, “White-light-interferometry suffering from dispersion deviations,” Annual report (Lehrstuhl für Optik, University of Erlangen-Nürnberg, Berlin, 2000), p. 40.

Schwider, J.

A. Pförtner, J. Schwider, “White-light-interferometry suffering from dispersion deviations,” Annual report (Lehrstuhl für Optik, University of Erlangen-Nürnberg, Berlin, 2000), p. 40.

Shcheglov, K. V.

R. C. Gutierrez, K. V. Shcheglov, T. K. Tang, “Pulsed-source interferometry for characterization of resonant micromachined structures,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head Island, S.C., 8–11 June 1998.

Sheppard, C. J. R.

Tang, T. K.

R. C. Gutierrez, K. V. Shcheglov, T. K. Tang, “Pulsed-source interferometry for characterization of resonant micromachined structures,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head Island, S.C., 8–11 June 1998.

Venzke, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1987), p. 40.

Wyntjes, G.

Yoshida, H.

K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).

Appl. Opt. (4)

Opt. Lett. (1)

Trans. SICE (1)

K. Nakano, H. Yoshida, K. Hane, S. Okuma, T. Eguchi, “Fringe scannning interferometric imaging of small vibration using pulsed laser diode,” Trans. SICE 31, 454–460 (1995).

Other (11)

R. C. Gutierrez, K. V. Shcheglov, T. K. Tang, “Pulsed-source interferometry for characterization of resonant micromachined structures,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head Island, S.C., 8–11 June 1998.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1987), p. 40.

Ref. 11, p. 62.

P. J. Caber, S. J. Martinek, R. J. Niemann, “A new interferometric profiler for smooth and rough surfaces,” in Laser Dimensional Metrology: Recent Advances for Industrial Application, M. J. Downs, ed., Proc. SPIE2088, 195–203 (1993).
[CrossRef]

H. Haneishi, “Signal processing for film thickness measurements by white light interferometry,” Graduate thesis (Department of Communications and Systems Engineering, University of Electro-communications, Chofu, Tokyo, 1984).

P. de Groot, “Method and apparatus for surface topography measurement by spatial-frequency analysis of interferograms,” U.S. patent5,398,113 (14March1995).

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc SPIE775, 233–247 (1987).
[CrossRef]

A. Bosseboeuf, S. Petigrand, “Application of microscopic interferometry techniques in the MEMS field,” in Microsystems Engineering: Metrology and Inspection III, C. Soreeki, ed., Proc. SPIE5145, 1–16 (2003).
[CrossRef]

P. de Groot, X. Colonna de Lega, “Signal modeling for modern interference microscopes,” in Optical Metrology in Production Engineering, W. Osten, M. Takeda, eds., Proc. SPIE5457, 26–34 (2004).
[CrossRef]

A. Pförtner, J. Schwider, “White-light-interferometry suffering from dispersion deviations,” Annual report (Lehrstuhl für Optik, University of Erlangen-Nürnberg, Berlin, 2000), p. 40.

S. W. Kim, G. H. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8April2003).

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Figures (12)

Fig. 1
Fig. 1

Interference microscope physical model. PZT, piezoelectric transducer.

Fig. 2
Fig. 2

Geometry of a single ray bundle illumination and imaging through an objective obeying the sine condition. The focal length is normalized.

Fig. 3
Fig. 3

Theoretical Fourier magnitude for height-scanning interferometry in the limit of a white-light source and collimated object illumination. The magnitude curve follows that of the emission spectrum of the source.

Fig. 4
Fig. 4

Theoretical Fourier magnitude for height-scanning interferometry in the limit of a spatially incoherent, monochromatic source uniformly filling the pupil plane. The magnitude curve follows the distribution function of the pupil plane illumination, here a top hat, weighted by the spatial frequency, which is proportional to the directional cosine of the object illumination.

Fig. 5
Fig. 5

Height-scanning interference signal simulations for a 500-nm light source: (a) broad 100-nm bandwidth and a more narrowly collimated 0.2-NA illumination; (b) narrow 20-nm Gaussian bandwidth and a wide, uniformly filled 0.6-NA pupil.

Fig. 6
Fig. 6

Height-scanning signal simulations for a 2-μm-thick film of index 2 deposited on a substrate of index 4, viewed with a 500-nm center wavelength: (a) broad 200-nm Gaussian bandwidth, narrow 0.28-NA illumination; (b) narrow 5-nm bandwidth, wide 0.80-NA illumination; (c) both broad 200-nm bandwidth and wide 0.80-NA illumination.

Fig. 7
Fig. 7

Coherence broadening effect of nonlinear chromatic dispersion: (a) no dispersion, 500-nm, 100-nm bandwidth light source; (b) same source characteristics but unbalanced quadratic dispersion ℘ = 0.2 μm2 (∼0.8 fringe over the full spectrum) in the interferometer.

Fig. 8
Fig. 8

Measured spectrum of a white-light LED source, nominally centered at 556 nm with a bandwidth of 62 nm, used for the experimental data acquisition in Fig. 9

Fig. 9
Fig. 9

Comparison of experimental and theoretical interference signals for a solid surface. The sample is a SiC optical flat, the white-light source spectrum is as in Fig. 8, and the Mirau objective has an unobscured NA range from 0.27 to 0.78.

Fig. 10
Fig. 10

Comparison of experimental and theoretical interference signals for a thin-film sample. The sample is 1025 nm of SiO2 (index 1.45 on Si, the 498-nm LED source has an approximately Gaussian spectral bandwidth of 27 nm, and the Mirau objective has an unobscured NA range from 0.27 to 0.78.

Fig. 11
Fig. 11

Comparison of the experimental and theoretical Fourier spectrum magnitudes for the thin-film sample signal shown in Fig. 10.

Fig. 12
Fig. 12

Comparison of the experimental and theoretical Fourier spectrum phases for the thin-film sample signal shown in Fig. 10.

Equations (46)

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gβ, k, ζ=R+Z+2 RZ cos2βkh-ζ+υ-ω,
β=cosψ,
k=2π/λ.
Iζ=001 gβ, k, ζUβVkβdβdk,
cosψsinψdψ=-βdβ.
κˆ=2βk.
qK=001 UβVk×- gβ, k, ζexpiKζdζβdβdk.
2 cosκˆζ+ =expiκˆζ++exp-iκˆζ-,
δK±κˆ=-expK±κˆiζdζ,
-gβ, k, ζexpiKζdζ=δKR+Z+δK-κˆ RZ expiκˆh+iυ-ω+δK+κˆ RZ exp×-iκˆh-iυ-ω.
β=κˆ/2k,
dβ=dκˆ/2k.
qK=002k δKR + ZΓdκˆdk+002k δK-κˆ RZ expiκˆh+iυ-ωΓdκˆdk+002k δK+κˆ RZ×exp-iκˆh-iυ-ωΓdκˆdk,
Γκˆ, k=Uβκˆ, kVkκˆ/4k2.
qK=δK0κ/2R+ZΓdkdκˆ+HKexpiKhK/2RZ expiυ-ωΓ κˆ=+Kdk+H-Kexp-iKh×-K/2 RZ exp-iυ-ωΓ κˆ=-Kdk,
Hu=0for u < 01otherwise.
qK=qsK+qpK,
Iζ=-qKexp-iKζdK.
q0=K0k>K/2R+ZΓ,
qK>0=expiKhk>K/2 RZ expiυ-ωΓ.
Kstep=2πNζstep.
qK=δKR+Z0κˆ/2Γκˆ, kdkdκˆ+HKexpiKh RZ K/2 ΓK, kdk+H-Kexp-iKh RZ -K/2 Γ-K, kdk.
Uβ=δβ-1.
UK, k=δK/2k-1.
δfk=δk-ξ|df/dk|k=ξ,
ΓK, k=K28Vkk2 δk-K/2.
qK=δKR+Z20Vκˆ/2dκˆ+HK RZ expiKh2 VK/2+H-K RZ exp-iKh2 V-K/2.
|qK>0|  Vk,
Vk=δk-k0.
qK=δKR+Z0 κˆUκˆ/2k0dκˆ+HKHk0-K/2expiKh RZ4k02 KUK/2k0+H-KHk0+K/2×exp-iKh RZ4k02 KU-K/2k0.
|qK>0|  βUβ,
Zβ, k=|zβ, k|2,
ωβ, k=argzβ, k.
zβ, k=ϑ+ϑ exp2ikLββn1+ϑϑ exp2ikLββn,
ββ= 1-1-β2/n2 1/2
υk=υ0+ k-k0 2,
BK=sinKζstep/2Kζstep/2,
qK=002k δKR+ZΓdκˆdk+002k δK-κˆ RZ expiκˆh+iυ-ωΓdκˆdk+002k δK+κˆ RZ×exp-iκˆh-iυ-ωΓdκˆdk.
02k δKfκˆ, kdκˆ=δK0 H2k-κˆfκˆ, kdκˆ,
02k δK-κˆfκˆ, kdκˆ=HKfK, kH2k-K,
02k δK+κˆfκˆ, kdκˆ=HKf-K, kH2k+K.
qK=δK00 H2k-κˆR+ZΓdκˆdk+0 HKH2k-K RZ expiκˆh+iυ-ωΓdk+0 HKH2k+K RZ×exp-iκˆh-iυ-ωΓdk.
00 H2k-κˆfκˆ, kdκˆdk=00 H2k-κˆfκˆ, kdkdκˆ,
0 HKfK, kH2k-Kdk=HKK/2 fK, kdk,
0 H-Kf-K, kH2k+Kdk=H-K-k/2 f-K, kdk,
qK=δK0κˆ/2R+ZΓdkdκˆ+HKexpiKhK/2 RZ expi υ-ωΓ κˆ=+Kdk+H-Kexp-iKh×-K/2 RZ exp-i υ-ωΓ κˆ=-Kdk.

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