Abstract

We have introduced the techniques of phase-shifting interferometry into a laser feedback interference microscope based on a helium–neon laser. With moderate feedback, multiple reflections between the sample and the laser are shown to be negligible, and the interferometer responds sinusoidally with a well-characterized fringe modulation. One can obtain higher signal-to-noise ratios by determining the number of additional terms required for modeling the effect of multiple reflections on the phase and visibility measurements in the high-feedback regime. Changes in optical path length are determined with nanometer precision without phase averaging or lock-in detection.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Stephan and D. Hugon, Phys. Rev. Lett. 55, 703 (1985).
    [CrossRef] [PubMed]
  2. A. Bearden, M. P. O’Neill, L. C. Osborne, and T. L. Wong, Opt. Lett. 18, 238 (1993).
    [CrossRef]
  3. Th. H. Peek, P. T. Bolwijn, and C. Th. Alkemade, Am. J. Phys. 35, 820 (1967).
    [CrossRef]
  4. G. A. Acket, D. Lenstra, A. J. Den Boef, and B. H. Verbeek, IEEE J. Quantum Electron. QE-20, 1163 (1984).
    [CrossRef]
  5. J. Mork, B. Tromberg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
    [CrossRef]
  6. R. Juskaitis, T. Wilson, and N. P. Rea, Opt. Commun. 109, 167 (1994).
    [CrossRef]
  7. D. Sarid, V. Weissenberger, D. A. Iams, and J. T. Ingle, IEEE J. Quantum Electron. 25, 1968 (1989).
    [CrossRef]
  8. K. Creath, in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
    [CrossRef]
  9. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
    [CrossRef]
  10. E. B. Hooper and G. Bekefi, J. Appl. Phys. 37, 4083 (1966); erratum,  38, 1998 (1967).
    [CrossRef]
  11. A. Yariv, Quantum Electronics (Wiley, New York, 1989), pp. 192ff.
  12. D. Lenstra, M. van Vaalen, and B. Jaskorzynska, Physica C 125, 255 (1984).
    [CrossRef]
  13. Preliminary calibration of the instrument reveals that the fringe visibility does not exceed 0.5; see B. Ovryn, J. H. Andrews, and S. Eppell, Proc. SPIE 2655, 153 (1996).
    [CrossRef]
  14. J. E. Greivenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992); B. Ovryn and E. M. Haacke, Appl. Opt. 32, 147 (1993).
    [CrossRef] [PubMed]
  15. In the absence of systematic errors, the signal-to-noise ratio of a phase measurement depends on environmental perturbations, errors in the phase shifts, and the error in the intensity measurements and is inversely proportional to the fringe visibility; see J. Schwider, in Progress in Optics XXVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 272–343.
  16. P. Hariharan, B. F. Oreb, and T. Eiju, Appl. Opt. 26, 2504 (1987); P. Hariharan, Appl. Opt. 26, 2506 (1987). The phase, ?=2k?, and the visibility m are determined by measurement of the intensities at each of five phase shifts ? in Eq.??(1).?Representing the measured intensities for the five phase shifts ?=-?, ?=-?/2, ?=0, ?=?/2, and ?=? as I1, I2, I3, I4, and I5, respectively, we calculated the quantities ? and m: ?=a tan?/?, m=3?2+?21/22I1+I2+2I3+I4+I5, where ?=2I2-2I4 and ?=2I3-I1-I5.
    [CrossRef] [PubMed]
  17. The phase error with j=4 in Eq.??(1) is tan?measured=1-b2cos2?+b4cos4?1+b2cos2?+b4cos4?×tan?actual.
  18. Because there is no ambiguity in the direction of the change in OPL, the increased OPL indicates the presence of an etched trough in the silicon.?This finding agrees with a topological scan by use of atomic-force microscopy, which indicates a mean step height of 76.2??nm for the sample shown in Fig.??2.?A rms surface roughness of greater than 10??nm was measured within the narrow trough.?The surface was smooth to less than 1??nm, however, outside the trough.

1996 (1)

Preliminary calibration of the instrument reveals that the fringe visibility does not exceed 0.5; see B. Ovryn, J. H. Andrews, and S. Eppell, Proc. SPIE 2655, 153 (1996).
[CrossRef]

1994 (1)

R. Juskaitis, T. Wilson, and N. P. Rea, Opt. Commun. 109, 167 (1994).
[CrossRef]

1993 (1)

1992 (1)

J. Mork, B. Tromberg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

1989 (1)

D. Sarid, V. Weissenberger, D. A. Iams, and J. T. Ingle, IEEE J. Quantum Electron. 25, 1968 (1989).
[CrossRef]

1987 (1)

1985 (1)

G. Stephan and D. Hugon, Phys. Rev. Lett. 55, 703 (1985).
[CrossRef] [PubMed]

1984 (2)

D. Lenstra, M. van Vaalen, and B. Jaskorzynska, Physica C 125, 255 (1984).
[CrossRef]

G. A. Acket, D. Lenstra, A. J. Den Boef, and B. H. Verbeek, IEEE J. Quantum Electron. QE-20, 1163 (1984).
[CrossRef]

1980 (1)

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

1967 (1)

Th. H. Peek, P. T. Bolwijn, and C. Th. Alkemade, Am. J. Phys. 35, 820 (1967).
[CrossRef]

1966 (1)

E. B. Hooper and G. Bekefi, J. Appl. Phys. 37, 4083 (1966); erratum,  38, 1998 (1967).
[CrossRef]

Acket, G. A.

G. A. Acket, D. Lenstra, A. J. Den Boef, and B. H. Verbeek, IEEE J. Quantum Electron. QE-20, 1163 (1984).
[CrossRef]

Alkemade, C. Th.

Th. H. Peek, P. T. Bolwijn, and C. Th. Alkemade, Am. J. Phys. 35, 820 (1967).
[CrossRef]

Andrews, J. H.

Preliminary calibration of the instrument reveals that the fringe visibility does not exceed 0.5; see B. Ovryn, J. H. Andrews, and S. Eppell, Proc. SPIE 2655, 153 (1996).
[CrossRef]

Bearden, A.

Bekefi, G.

E. B. Hooper and G. Bekefi, J. Appl. Phys. 37, 4083 (1966); erratum,  38, 1998 (1967).
[CrossRef]

Bolwijn, P. T.

Th. H. Peek, P. T. Bolwijn, and C. Th. Alkemade, Am. J. Phys. 35, 820 (1967).
[CrossRef]

Bruning, J. H.

J. E. Greivenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992); B. Ovryn and E. M. Haacke, Appl. Opt. 32, 147 (1993).
[CrossRef] [PubMed]

Creath, K.

K. Creath, in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
[CrossRef]

Den Boef, A. J.

G. A. Acket, D. Lenstra, A. J. Den Boef, and B. H. Verbeek, IEEE J. Quantum Electron. QE-20, 1163 (1984).
[CrossRef]

Eiju, T.

Eppell, S.

Preliminary calibration of the instrument reveals that the fringe visibility does not exceed 0.5; see B. Ovryn, J. H. Andrews, and S. Eppell, Proc. SPIE 2655, 153 (1996).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992); B. Ovryn and E. M. Haacke, Appl. Opt. 32, 147 (1993).
[CrossRef] [PubMed]

Hariharan, P.

Hooper, E. B.

E. B. Hooper and G. Bekefi, J. Appl. Phys. 37, 4083 (1966); erratum,  38, 1998 (1967).
[CrossRef]

Hugon, D.

G. Stephan and D. Hugon, Phys. Rev. Lett. 55, 703 (1985).
[CrossRef] [PubMed]

Iams, D. A.

D. Sarid, V. Weissenberger, D. A. Iams, and J. T. Ingle, IEEE J. Quantum Electron. 25, 1968 (1989).
[CrossRef]

Ingle, J. T.

D. Sarid, V. Weissenberger, D. A. Iams, and J. T. Ingle, IEEE J. Quantum Electron. 25, 1968 (1989).
[CrossRef]

Jaskorzynska, B.

D. Lenstra, M. van Vaalen, and B. Jaskorzynska, Physica C 125, 255 (1984).
[CrossRef]

Juskaitis, R.

R. Juskaitis, T. Wilson, and N. P. Rea, Opt. Commun. 109, 167 (1994).
[CrossRef]

Kobayashi, K.

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

Lang, R.

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

Lenstra, D.

D. Lenstra, M. van Vaalen, and B. Jaskorzynska, Physica C 125, 255 (1984).
[CrossRef]

G. A. Acket, D. Lenstra, A. J. Den Boef, and B. H. Verbeek, IEEE J. Quantum Electron. QE-20, 1163 (1984).
[CrossRef]

Mark, J.

J. Mork, B. Tromberg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Mork, J.

J. Mork, B. Tromberg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

O’Neill, M. P.

Oreb, B. F.

Osborne, L. C.

Ovryn, B.

Preliminary calibration of the instrument reveals that the fringe visibility does not exceed 0.5; see B. Ovryn, J. H. Andrews, and S. Eppell, Proc. SPIE 2655, 153 (1996).
[CrossRef]

Peek, Th. H.

Th. H. Peek, P. T. Bolwijn, and C. Th. Alkemade, Am. J. Phys. 35, 820 (1967).
[CrossRef]

Rea, N. P.

R. Juskaitis, T. Wilson, and N. P. Rea, Opt. Commun. 109, 167 (1994).
[CrossRef]

Sarid, D.

D. Sarid, V. Weissenberger, D. A. Iams, and J. T. Ingle, IEEE J. Quantum Electron. 25, 1968 (1989).
[CrossRef]

Schwider, J.

In the absence of systematic errors, the signal-to-noise ratio of a phase measurement depends on environmental perturbations, errors in the phase shifts, and the error in the intensity measurements and is inversely proportional to the fringe visibility; see J. Schwider, in Progress in Optics XXVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 272–343.

Stephan, G.

G. Stephan and D. Hugon, Phys. Rev. Lett. 55, 703 (1985).
[CrossRef] [PubMed]

Tromberg, B.

J. Mork, B. Tromberg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

van Vaalen, M.

D. Lenstra, M. van Vaalen, and B. Jaskorzynska, Physica C 125, 255 (1984).
[CrossRef]

Verbeek, B. H.

G. A. Acket, D. Lenstra, A. J. Den Boef, and B. H. Verbeek, IEEE J. Quantum Electron. QE-20, 1163 (1984).
[CrossRef]

Weissenberger, V.

D. Sarid, V. Weissenberger, D. A. Iams, and J. T. Ingle, IEEE J. Quantum Electron. 25, 1968 (1989).
[CrossRef]

Wilson, T.

R. Juskaitis, T. Wilson, and N. P. Rea, Opt. Commun. 109, 167 (1994).
[CrossRef]

Wong, T. L.

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1989), pp. 192ff.

Am. J. Phys. (1)

Th. H. Peek, P. T. Bolwijn, and C. Th. Alkemade, Am. J. Phys. 35, 820 (1967).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (4)

G. A. Acket, D. Lenstra, A. J. Den Boef, and B. H. Verbeek, IEEE J. Quantum Electron. QE-20, 1163 (1984).
[CrossRef]

J. Mork, B. Tromberg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

D. Sarid, V. Weissenberger, D. A. Iams, and J. T. Ingle, IEEE J. Quantum Electron. 25, 1968 (1989).
[CrossRef]

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

J. Appl. Phys. (1)

E. B. Hooper and G. Bekefi, J. Appl. Phys. 37, 4083 (1966); erratum,  38, 1998 (1967).
[CrossRef]

Opt. Commun. (1)

R. Juskaitis, T. Wilson, and N. P. Rea, Opt. Commun. 109, 167 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

G. Stephan and D. Hugon, Phys. Rev. Lett. 55, 703 (1985).
[CrossRef] [PubMed]

Physica C (1)

D. Lenstra, M. van Vaalen, and B. Jaskorzynska, Physica C 125, 255 (1984).
[CrossRef]

Proc. SPIE (1)

Preliminary calibration of the instrument reveals that the fringe visibility does not exceed 0.5; see B. Ovryn, J. H. Andrews, and S. Eppell, Proc. SPIE 2655, 153 (1996).
[CrossRef]

Other (6)

J. E. Greivenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992); B. Ovryn and E. M. Haacke, Appl. Opt. 32, 147 (1993).
[CrossRef] [PubMed]

In the absence of systematic errors, the signal-to-noise ratio of a phase measurement depends on environmental perturbations, errors in the phase shifts, and the error in the intensity measurements and is inversely proportional to the fringe visibility; see J. Schwider, in Progress in Optics XXVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 272–343.

The phase error with j=4 in Eq.??(1) is tan?measured=1-b2cos2?+b4cos4?1+b2cos2?+b4cos4?×tan?actual.

Because there is no ambiguity in the direction of the change in OPL, the increased OPL indicates the presence of an etched trough in the silicon.?This finding agrees with a topological scan by use of atomic-force microscopy, which indicates a mean step height of 76.2??nm for the sample shown in Fig.??2.?A rms surface roughness of greater than 10??nm was measured within the narrow trough.?The surface was smooth to less than 1??nm, however, outside the trough.

A. Yariv, Quantum Electronics (Wiley, New York, 1989), pp. 192ff.

K. Creath, in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Visibility and phase measurements when the phase-shifted LFI operates in two regimes: moderate (left) and high (right) feedback. The rms error in the OPL (0.3  nm) was determined from the difference between a linear least-squares fit and a measurement of the λ/2 phase ramp, which was applied to the electro-optic modulator. Superimposed upon the data (right) are fits based on a simultaneous nonlinear least-squares estimate of the visibility and the OPL by use of four additional terms from Eq.  (1).

Fig. 2
Fig. 2

Visibility image obtained when an etched silicon sample was raster scanned in a phase-shifted laser feedback microscope with a 50/0.8N.A. objective.

Fig. 3
Fig. 3

Visibility and OPL obtained from a single line scan across the bottommost 2.5µm pitch bar shown in Fig.  2. The data indicate operation within the moderate feedback regime. Both the visibility and the phase indicate the steps in the silicon. A linear slope of 7.6 nm/µm was subtracted from the OPL data.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

Im,b,δ,Ψ=I01+m cos2kδ+Ψ×j=0-bjcosj2kδ+Ψ,
m=K1-R2R2Reff1-R1+1-R2=γReff,
b=R2Reff.

Metrics