Abstract

Common-path imaging interferometers offer some advantages over other interferometers, such as insensitivity to vibrations and the ability to be attached to any optical system to analyze an imaged wavefront. We introduce the spatial-phase-shift imaging interferometry technique for surface measurements and wavefront analysis in which different parts of the wavefront undergo certain manipulations in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing interferometry methods. We present the mathematical algorithm for reconstructing the wavefront from the interference patterns and detail the optical considerations for implementing the optical system. We implemented the spatial phase shift into a working system and used it to measure a variety of objects. Measurement results and comparison with other measurement methods indicate that this approach improves measurement accuracy with respect to existing quantitative phase-measurement methods.

© 2006 Optical Society of America

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References

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  1. O. Soloviev and G. Vdovin, " Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Exp. 13, 3743- 3753 ( 2005).
    [CrossRef]
  2. S. Cain, " Design of an image projection correlating wavefront sensor for adaptive optics," Opt. Eng. 43, 1670- 1681 ( 2004).
    [CrossRef]
  3. A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, " Self-focusing in a nematic liquid crystal for measurements of wavefront distortions," Opt. Commun. 232, 439- 442 ( 2004).
    [CrossRef]
  4. T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
    [CrossRef]
  5. M. A. Vorontsov, E. W. Justh, and L. A. Beresnev, " Adaptive optics with advanced phase-contrast techniques. I. High-resolution wave-front sensing," J. Opt. Soc. Am. A 18, 1289- 1299 ( 2001).
    [CrossRef]
  6. S. Wolfling, N. Ben-Yosef, and Y. Arieli, " A generalized method for wavefront analysis," Opt. Lett. 29, 462- 464 ( 2004).
    [CrossRef] [PubMed]
  7. S. Wolfling, D. Banitt, N. Ben-Yosef, and Y. Arieli, " Innovative metrology method for the 3-dimensional measurement of MEMS structures," in Reliability, Testing and Characterization of MEMS/MOEMS III, D. E. Tanner and R. Ramesham, eds., Proc. SPIE 5343, 255- 263 ( 2004).
    [CrossRef]
  8. S. Wolfling, E. Lanzmann, M. Israeli, N. Ben-Yosef, and Y. Arieli, " Spatial phase shift interferometry--A wavefront analysis technique for three-dimensional topometry," J. Opt. Soc. Am. A 22, 2498- 2509 ( 2005).
    [CrossRef]
  9. D. W. Phillion, " General methods for generating phase-shifting interferometry algorithms," Appl. Opt. 36, 8098- 8115 ( 1997).
    [CrossRef]
  10. F. Zernike, " Diffraction theory of knife-edge test and its improved form, the phase-contrast," Mon. Not. R. Astron. Soc. 94, 371 ( 1934).
  11. J. A. Ferrari, E. Garbusi, and E. M. Frins, " Modified Michelson interferometer with electro-optic phase control," Opt. Commun. 209, 245- 253 ( 2002).
    [CrossRef]
  12. C. Dunsby, Y. Gu, and P. M. W. French, " Single-shot phase-stepped wide-field coherence-gated imaging," Opt. Exp. 11, 105- 115 ( 2003).
    [CrossRef]
  13. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  14. K. Creath, " Phase measurements interferometry techniques," in Progress in Optics, Vol. XXVI, E.Wolf, ed. (Elsevier Science, 1988), pp. 349- 393.
    [CrossRef]
  15. T. Noda and S. Kawata, " Separation of phase and absorption images in phase-contrast microscopy," J. Opt. Soc. Am. A 9, 924- 931 ( 1992).
    [CrossRef]
  16. R. Liang, J. K. Erwin, and M. Mansuripur, " Variation on Zernike's phase-contrast microscope," Appl. Opt. 39, 2152- 2158 ( 2000).
    [CrossRef]
  17. H. Kadono, M. Ogusu, and S. Toyooka, " Phase shifting common path interferometer using a liquid-crystal phase modulator," Opt. Commun. 110, 391- 400 ( 1994).
    [CrossRef]
  18. C. R. Mercer and K. Creath, " Liquid-crystal point-diffraction interferometer for wave-front measurements," Appl. Opt. 35, 1633- 1642 ( 1996).
    [CrossRef] [PubMed]
  19. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998), Chaps. 4 and 5, pp. 100-278.
  20. A. VanderLugt, Optical Signal Processing (Wiley, 1992), pp. 118- 123.
  21. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
    [CrossRef]

2005

O. Soloviev and G. Vdovin, " Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Exp. 13, 3743- 3753 ( 2005).
[CrossRef]

S. Wolfling, E. Lanzmann, M. Israeli, N. Ben-Yosef, and Y. Arieli, " Spatial phase shift interferometry--A wavefront analysis technique for three-dimensional topometry," J. Opt. Soc. Am. A 22, 2498- 2509 ( 2005).
[CrossRef]

2004

S. Wolfling, N. Ben-Yosef, and Y. Arieli, " A generalized method for wavefront analysis," Opt. Lett. 29, 462- 464 ( 2004).
[CrossRef] [PubMed]

S. Wolfling, D. Banitt, N. Ben-Yosef, and Y. Arieli, " Innovative metrology method for the 3-dimensional measurement of MEMS structures," in Reliability, Testing and Characterization of MEMS/MOEMS III, D. E. Tanner and R. Ramesham, eds., Proc. SPIE 5343, 255- 263 ( 2004).
[CrossRef]

S. Cain, " Design of an image projection correlating wavefront sensor for adaptive optics," Opt. Eng. 43, 1670- 1681 ( 2004).
[CrossRef]

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, " Self-focusing in a nematic liquid crystal for measurements of wavefront distortions," Opt. Commun. 232, 439- 442 ( 2004).
[CrossRef]

T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
[CrossRef]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

2003

C. Dunsby, Y. Gu, and P. M. W. French, " Single-shot phase-stepped wide-field coherence-gated imaging," Opt. Exp. 11, 105- 115 ( 2003).
[CrossRef]

2002

J. A. Ferrari, E. Garbusi, and E. M. Frins, " Modified Michelson interferometer with electro-optic phase control," Opt. Commun. 209, 245- 253 ( 2002).
[CrossRef]

2001

2000

1997

1996

1994

H. Kadono, M. Ogusu, and S. Toyooka, " Phase shifting common path interferometer using a liquid-crystal phase modulator," Opt. Commun. 110, 391- 400 ( 1994).
[CrossRef]

1992

1934

F. Zernike, " Diffraction theory of knife-edge test and its improved form, the phase-contrast," Mon. Not. R. Astron. Soc. 94, 371 ( 1934).

Arieli, Y.

S. Wolfling, E. Lanzmann, M. Israeli, N. Ben-Yosef, and Y. Arieli, " Spatial phase shift interferometry--A wavefront analysis technique for three-dimensional topometry," J. Opt. Soc. Am. A 22, 2498- 2509 ( 2005).
[CrossRef]

S. Wolfling, N. Ben-Yosef, and Y. Arieli, " A generalized method for wavefront analysis," Opt. Lett. 29, 462- 464 ( 2004).
[CrossRef] [PubMed]

S. Wolfling, D. Banitt, N. Ben-Yosef, and Y. Arieli, " Innovative metrology method for the 3-dimensional measurement of MEMS structures," in Reliability, Testing and Characterization of MEMS/MOEMS III, D. E. Tanner and R. Ramesham, eds., Proc. SPIE 5343, 255- 263 ( 2004).
[CrossRef]

Banitt, D.

S. Wolfling, D. Banitt, N. Ben-Yosef, and Y. Arieli, " Innovative metrology method for the 3-dimensional measurement of MEMS structures," in Reliability, Testing and Characterization of MEMS/MOEMS III, D. E. Tanner and R. Ramesham, eds., Proc. SPIE 5343, 255- 263 ( 2004).
[CrossRef]

Ben-Yosef, N.

S. Wolfling, E. Lanzmann, M. Israeli, N. Ben-Yosef, and Y. Arieli, " Spatial phase shift interferometry--A wavefront analysis technique for three-dimensional topometry," J. Opt. Soc. Am. A 22, 2498- 2509 ( 2005).
[CrossRef]

S. Wolfling, N. Ben-Yosef, and Y. Arieli, " A generalized method for wavefront analysis," Opt. Lett. 29, 462- 464 ( 2004).
[CrossRef] [PubMed]

S. Wolfling, D. Banitt, N. Ben-Yosef, and Y. Arieli, " Innovative metrology method for the 3-dimensional measurement of MEMS structures," in Reliability, Testing and Characterization of MEMS/MOEMS III, D. E. Tanner and R. Ramesham, eds., Proc. SPIE 5343, 255- 263 ( 2004).
[CrossRef]

Beresnev, L. A.

Brock, N.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

Buchter, S. C.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, " Self-focusing in a nematic liquid crystal for measurements of wavefront distortions," Opt. Commun. 232, 439- 442 ( 2004).
[CrossRef]

Cain, S.

S. Cain, " Design of an image projection correlating wavefront sensor for adaptive optics," Opt. Eng. 43, 1670- 1681 ( 2004).
[CrossRef]

Creath, K.

C. R. Mercer and K. Creath, " Liquid-crystal point-diffraction interferometer for wave-front measurements," Appl. Opt. 35, 1633- 1642 ( 1996).
[CrossRef] [PubMed]

K. Creath, " Phase measurements interferometry techniques," in Progress in Optics, Vol. XXVI, E.Wolf, ed. (Elsevier Science, 1988), pp. 349- 393.
[CrossRef]

Dunsby, C.

C. Dunsby, Y. Gu, and P. M. W. French, " Single-shot phase-stepped wide-field coherence-gated imaging," Opt. Exp. 11, 105- 115 ( 2003).
[CrossRef]

Erwin, J. K.

Ferrari, J. A.

J. A. Ferrari, E. Garbusi, and E. M. Frins, " Modified Michelson interferometer with electro-optic phase control," Opt. Commun. 209, 245- 253 ( 2002).
[CrossRef]

French, P. M. W.

C. Dunsby, Y. Gu, and P. M. W. French, " Single-shot phase-stepped wide-field coherence-gated imaging," Opt. Exp. 11, 105- 115 ( 2003).
[CrossRef]

Frins, E. M.

J. A. Ferrari, E. Garbusi, and E. M. Frins, " Modified Michelson interferometer with electro-optic phase control," Opt. Commun. 209, 245- 253 ( 2002).
[CrossRef]

Fukuchi, T.

T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
[CrossRef]

Garbusi, E.

J. A. Ferrari, E. Garbusi, and E. M. Frins, " Modified Michelson interferometer with electro-optic phase control," Opt. Commun. 209, 245- 253 ( 2002).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998), Chaps. 4 and 5, pp. 100-278.

Gu, Y.

C. Dunsby, Y. Gu, and P. M. W. French, " Single-shot phase-stepped wide-field coherence-gated imaging," Opt. Exp. 11, 105- 115 ( 2003).
[CrossRef]

Hayes, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

Israeli, M.

Justh, E. W.

Kadono, H.

H. Kadono, M. Ogusu, and S. Toyooka, " Phase shifting common path interferometer using a liquid-crystal phase modulator," Opt. Commun. 110, 391- 400 ( 1994).
[CrossRef]

Kaivola, M.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, " Self-focusing in a nematic liquid crystal for measurements of wavefront distortions," Opt. Commun. 232, 439- 442 ( 2004).
[CrossRef]

Kawata, S.

Lanzmann, E.

Liang, R.

Malacara, D.

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

Malacara, Z.

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

Mansuripur, M.

Mercer, C. R.

Millerd, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

Nayuki, T.

T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
[CrossRef]

Nemoto, K.

T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
[CrossRef]

Noda, T.

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

Ogusu, M.

H. Kadono, M. Ogusu, and S. Toyooka, " Phase shifting common path interferometer using a liquid-crystal phase modulator," Opt. Commun. 110, 391- 400 ( 1994).
[CrossRef]

Phillion, D. W.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998), Chaps. 4 and 5, pp. 100-278.

Servin, M.

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

Shevchenko, A.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, " Self-focusing in a nematic liquid crystal for measurements of wavefront distortions," Opt. Commun. 232, 439- 442 ( 2004).
[CrossRef]

Soloviev, O.

O. Soloviev and G. Vdovin, " Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Exp. 13, 3743- 3753 ( 2005).
[CrossRef]

Tabiryan, N. V.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, " Self-focusing in a nematic liquid crystal for measurements of wavefront distortions," Opt. Commun. 232, 439- 442 ( 2004).
[CrossRef]

Toyooka, S.

H. Kadono, M. Ogusu, and S. Toyooka, " Phase shifting common path interferometer using a liquid-crystal phase modulator," Opt. Commun. 110, 391- 400 ( 1994).
[CrossRef]

Uchino, K.

T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
[CrossRef]

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, 1992), pp. 118- 123.

Vdovin, G.

O. Soloviev and G. Vdovin, " Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Exp. 13, 3743- 3753 ( 2005).
[CrossRef]

Vorontsov, M. A.

Wolfling, S.

S. Wolfling, E. Lanzmann, M. Israeli, N. Ben-Yosef, and Y. Arieli, " Spatial phase shift interferometry--A wavefront analysis technique for three-dimensional topometry," J. Opt. Soc. Am. A 22, 2498- 2509 ( 2005).
[CrossRef]

S. Wolfling, N. Ben-Yosef, and Y. Arieli, " A generalized method for wavefront analysis," Opt. Lett. 29, 462- 464 ( 2004).
[CrossRef] [PubMed]

S. Wolfling, D. Banitt, N. Ben-Yosef, and Y. Arieli, " Innovative metrology method for the 3-dimensional measurement of MEMS structures," in Reliability, Testing and Characterization of MEMS/MOEMS III, D. E. Tanner and R. Ramesham, eds., Proc. SPIE 5343, 255- 263 ( 2004).
[CrossRef]

Wyant, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

Yamaguchi, Y.

T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
[CrossRef]

Zernike, F.

F. Zernike, " Diffraction theory of knife-edge test and its improved form, the phase-contrast," Mon. Not. R. Astron. Soc. 94, 371 ( 1934).

Appl. Opt.

Electr. Eng. Jpn.

T. Fukuchi, Y. Yamaguchi, T. Nayuki, K. Nemoto, and K. Uchino, " Development of a laser wavefront sensor for measurement of discharges in air," Electr. Eng. Jpn. 146, 10- 17 ( 2004).
[CrossRef]

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

F. Zernike, " Diffraction theory of knife-edge test and its improved form, the phase-contrast," Mon. Not. R. Astron. Soc. 94, 371 ( 1934).

Opt. Commun.

J. A. Ferrari, E. Garbusi, and E. M. Frins, " Modified Michelson interferometer with electro-optic phase control," Opt. Commun. 209, 245- 253 ( 2002).
[CrossRef]

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, " Self-focusing in a nematic liquid crystal for measurements of wavefront distortions," Opt. Commun. 232, 439- 442 ( 2004).
[CrossRef]

H. Kadono, M. Ogusu, and S. Toyooka, " Phase shifting common path interferometer using a liquid-crystal phase modulator," Opt. Commun. 110, 391- 400 ( 1994).
[CrossRef]

Opt. Eng.

S. Cain, " Design of an image projection correlating wavefront sensor for adaptive optics," Opt. Eng. 43, 1670- 1681 ( 2004).
[CrossRef]

Opt. Exp.

O. Soloviev and G. Vdovin, " Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Exp. 13, 3743- 3753 ( 2005).
[CrossRef]

C. Dunsby, Y. Gu, and P. M. W. French, " Single-shot phase-stepped wide-field coherence-gated imaging," Opt. Exp. 11, 105- 115 ( 2003).
[CrossRef]

Opt. Lett.

Proc. SPIE

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, " Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304- 314 ( 2004).
[CrossRef]

S. Wolfling, D. Banitt, N. Ben-Yosef, and Y. Arieli, " Innovative metrology method for the 3-dimensional measurement of MEMS structures," in Reliability, Testing and Characterization of MEMS/MOEMS III, D. E. Tanner and R. Ramesham, eds., Proc. SPIE 5343, 255- 263 ( 2004).
[CrossRef]

Other

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

K. Creath, " Phase measurements interferometry techniques," in Progress in Optics, Vol. XXVI, E.Wolf, ed. (Elsevier Science, 1988), pp. 349- 393.
[CrossRef]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998), Chaps. 4 and 5, pp. 100-278.

A. VanderLugt, Optical Signal Processing (Wiley, 1992), pp. 118- 123.

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Figures (9)

Fig. 1
Fig. 1

(Color online) General characteristic optical setup of the SPS interferometry technique.

Fig. 2
Fig. 2

(Color online) Orientation of LC molecules under different applied voltages to the active noncentral area: A vertical polarizer in a direction parallel to the LC's optical axis is placed in the optical path. Below a threshold voltage (Vc ) there is no change in orientation (left); as the voltage increases, a change of orientation occurs (middle); above a certain voltage (Vsat ) all molecules in the active area are aligned perpendicular to their original direction (right).

Fig. 3
Fig. 3

(Color online) Measured retardation (in wavelengths when measuring with the source of a 637.8-nm wavelength) at different applied voltages of the active area (circles) and nonactive area (crosses) of a LC device having two parts.

Fig. 4
Fig. 4

(Color online) Phase manipulation by passive phase plates. The reflected wavefront is amplitude split into three different channels, each including a different passive phase plate (p.p), an imaging lens, and a CCD to collect the intensity of the manipulated wavefront.

Fig. 5
Fig. 5

(Color online) 3D views of various industrial applications as measured by the SPS sensor head: (a) laser texture marks on a hard disk, (b) fiber connector, (c) part of a copper-plated wafer, and (d) a MEMS mirror array. In all the measurements, the lateral axes are presented in micrometers and the Z axis is in nanometers. The various heights are also represented by different gradations of gray.

Fig. 6
Fig. 6

(Color online) SPS system performance at various objectives: (a) accuracy measured against three certified step heights, (b) static repeatability of 30 consecutive measurements, (c) dynamic repeatability of eight measurements in 1-h intervals.

Fig. 7
Fig. 7

(Color online) Topometry of trenches 4 μm wide and 300 nm deep as measured in the same location by an AFM (gray, somewhat jagged curve) and by the SPS system with a 100× objective (dark, more continuous curve). The location along the X axis is in micrometers, and the height is in nanometers.

Fig. 8
Fig. 8

(Color online) Topometry of a 1-μm-wide trench as measured by an AFM and by the 100× objective of the SPS system. The location along the X axis is in micrometers, and the height is in nanometers.

Fig. 9
Fig. 9

(Color online) Height measurement of a certified height standard in the SPS system (gray, relatively smooth curve) compared with a measurement done with a phase-shift interferometer (dark, more jagged curve). The location along the X axis is in micrometers, and the height is in nanometers.

Equations (129)

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

o ( x , y ) = A ( x , y ) exp [ i ϕ ( x , y ) ] ,
ϕ ( x , y ) = 4 π h ( x , y ) λ mod 2 π ,
2 π
2 π
λ / 4
( x , y )
x = u λ L
y = v λ L
H ( x , y ) = k exp ( i θ ) G ( x , y ) + [ 1 G ( x , y ) ] = G ( x , y ) [ k exp ( i θ ) 1 ] + 1 ,
O ( x λ L , y λ L ) = O ( x λ L , y λ L ) H ( x , y ) .
o ( x , y ) = C 1 F 1 [ O ( x λ L , y λ L ) ] = C 2 { [ k exp ( i θ ) 1 ] S ( x , y ) + o ( x , y ) } ,
F 1
C 1
C 2
S ( x , y ) o ( x , y ) g ( x , y )
g ( x , y ) = F 1 [ G ( u λ L , v λ L ) ] .
[ k exp ( i θ ) 1 ]
( θ , k )
I θ ( x , y ) = | [ k exp ( i θ ) 1 ] S ( x , y ) + o ( x , y ) | 2 .
SAN = Δ D λ L ,
o ( x , y ) = 0
I θ ( x , y )
I θ ( x , y )
o ( x , y ) = 0
I θ ( x , y ) = | k exp ( i θ ) 1 | 2 | S ( x , y ) | 2 .
α ( x , y )
I θ ( x , y ) = | [ k exp ( i θ ) 1 ] C ( x , y ) exp [ i ψ ( x , y ) ] + A ( x , y ) | 2 ,
ψ ( x , y )
ψ ( x , y ) ϕ ( x , y ) α ( x , y )
ϕ ( x , y )
ψ ( x , y )
ϕ ( x , y )
A ( x , y )
ϕ ( x , y )
C ( x , y ) = | S ( x , y ) |
β 0 ( x , y ) , β c ( x , y )
β s ( x , y )
I θ j
C ( x , y )
ψ ( x , y )
A ( x , y )
β 0 ,
β c ,
β s
α ( x , y )
α ( x , y )
ϕ ( x , y )
ϕ = ψ + α
I θ ( x , y ) = β 0 ( x , y ) + β c ( x , y ) cos θ + β s ( x , y ) sin θ ,
β 0 ( x , y ) = A 2 ( x , y ) + ( 1 + k 2 ) C 2 ( x , y ) 2 A ( x , y ) C ( x , y ) cos ψ ( x , y ) ,
β c ( x , y ) = 2 k A ( x , y ) C ( x , y ) cos ψ ( x , y ) 2 k C 2 ( x , y ) ,
β s ( x , y ) = 2 k A ( x , y ) C ( x , y ) sin ψ ( x , y ) .
β 0 , β c , β s
I θ 1 , , I θ j , j 3
3 × 3
θ 1 , , θ j
C ( x , y )
X = [ k C ( x , y ) ] 2
X 2 β 0 X + β c     2 + β s     2 4 = 0.
C 2 = | S | 2
C 2
λ / 400
A ( x , y )   and   ψ ( x , y )
A ( x , y ) = [ β 0 ( x , y ) + 1 k β c ( x , y ) + ( 1 k 2 ) C 2 ( x , y ) ] 1 / 2 ,
ψ ( x , y ) = arg [ β c ( x , y ) + 2 k C 2 ( x , y ) + i β s ( x , y ) ] .
α ( x , y )
S = ( Γ S ) g ,
Γ ( x , y ) A ( x , y ) exp [ i ψ ( x , y ) ] C ( x , y ) .
α ( x , y )
g n ( x , y )
S = i + j n s i , j x i y j
1   nm
690 - nm
768 × 1024   pixels
4 .65 - μm
2 π t n e / λ
n e
n e
n θ
n θ
n 0
Δφ
Δφ ( V ) = φ center φ peripherial ( V ) = 2 π λ t ( n e n θ ) ;
n 0 n θ ( V ) n e .
Δ φ = 2.16 π
λ = 637.8   nm
5 - V
Δ n = n e n 0 = 0.184
t ( n e n θ ) = 689   nm
2 π / 10
2 π / 100
5 × / 0.15 NA
10 × / 0.3 NA
20 × / 0.4 NA
100 × / 0.9 NA
24 - nm
3 - nm
30 - μm
60 - μm
100 - nm
250   nm
35   nm
step-height   error = i = 1 25 ( h i b i ) 25 H ,
h i   and   b i
1   nm
0.02   nm
( 1 σ )   of   0.43   nm
( 1 σ )
0.5 %
929.0   nm
Δ j
Δ j = i = 1 30 ( h i b i ) 30 ( h j b j ) ,
h i
b i
( 1 σ )
0.6   Å
0.9   Å
929.0 - nm
( 1 σ )
0.26   nm
0 .33   nm
14   μm
300   nm
2   nm
1 - μm
88 .0   nm
0 .6   nm
1 - nm
0 .1 - nm

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