Abstract

The primary limitation of conventional phase-shifting interferometry (PSI) is its inability to measure surfaces with large aspheric departures. A new method of data analysis, sub-Nyquist interferometry (SNI), is described and demonstrated to overcome this problem. SNI is an extension of PSI, and it preserves the measurement precision that is inherent to PSI. For some types of wavefronts, measurement range improvements of more than 2 orders of magnitude are shown, and these improvements result from the utilization of a priori knowledge about the wavefront. Simple and reasonable assumptions are found to be very powerful for improving the aspheric measurement capability of the interferometer system.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. H. Brunning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978). pp. 409–437.
  2. P. Hariharan, Optical Interferometry (Academic, New York, 1985), pp. 151–165.
  3. J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Electro Optics, 118 (Nov.1985).
  4. J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350 (1984).
    [CrossRef]
  5. J. C. Wyant, “Use of an ac Heterodyne Lateral Shear Interferometer with Real-Time Wavefront Correction Systems,” Appl. Opt. 14, 2622 (1975).
    [CrossRef] [PubMed]
  6. J. L. Seligson, C. A. Callari, J. E. Greivenkamp, J. W. Ward, “Stability of a Lateral-Shearing Heterodyne Twyman-Green Interferometer,” Opt. Eng. 23, 353 (1984).
    [CrossRef]
  7. J. M. Bennett, “Comparison of Techniques for Measuring the Roughness of Optical Surfaces,” Opt. Eng. 24, 380 (1985).
    [CrossRef]
  8. A. Offner, “Null Tests Using Compensators,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 439–458.
  9. J. C. Wyant, “Holographic and Moire Techniques,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 381–408.
  10. B. E. Truax, “Programmable Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 10 (1987).
  11. O. Y. Kwon, J. C. Wyant, C. R. Hayslett, “Rough Surface Interferometry at 10.6 μm,” Appl. Opt. 19, 1862 (1980).
    [CrossRef] [PubMed]
  12. K. N. Prettyjohns, S. L. DeVore, E. L. Dereniak, J. C. Wyant, “Direct Phase Measurement Interferometer Working at 3.8 μm,” Appl. Opt. 24, 2211 (1985).
    [CrossRef] [PubMed]
  13. Y.-Y. Cheng, J. C. Wyant, “Two-Wavelength Phase Shifting Interferometry,” Appl. Opt. 23, 4539 (1984).
    [CrossRef] [PubMed]
  14. K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
    [CrossRef]
  15. See, for example, J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 266–285.
  16. P. Mertz, F. Gray, “A Theory of Scanning and Its Relation to the Characteristics of the Transmitted Signal in Telephotography and Television,” Bell Syst. Tech. J. 13, 464 (1934).
  17. J. P. Rossi, “Sub-Nyquist-Encoded PCM NTSC Color Television,” J. SMPTE 85, 1 (1976).
    [CrossRef]
  18. K. H. Barratt, K. Lucas, “An Introduction to Sub-Nyquist Sampling,” IBA Tech. Rev. (GB) 12, 3 (1979).
  19. K. Creath, J. C. Wyant, “Direct Phase Measurement of Aspheric Surface Contours,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 101 (1986).

1987 (1)

B. E. Truax, “Programmable Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 10 (1987).

1986 (1)

K. Creath, J. C. Wyant, “Direct Phase Measurement of Aspheric Surface Contours,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 101 (1986).

1985 (4)

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

K. N. Prettyjohns, S. L. DeVore, E. L. Dereniak, J. C. Wyant, “Direct Phase Measurement Interferometer Working at 3.8 μm,” Appl. Opt. 24, 2211 (1985).
[CrossRef] [PubMed]

J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Electro Optics, 118 (Nov.1985).

J. M. Bennett, “Comparison of Techniques for Measuring the Roughness of Optical Surfaces,” Opt. Eng. 24, 380 (1985).
[CrossRef]

1984 (3)

J. L. Seligson, C. A. Callari, J. E. Greivenkamp, J. W. Ward, “Stability of a Lateral-Shearing Heterodyne Twyman-Green Interferometer,” Opt. Eng. 23, 353 (1984).
[CrossRef]

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350 (1984).
[CrossRef]

Y.-Y. Cheng, J. C. Wyant, “Two-Wavelength Phase Shifting Interferometry,” Appl. Opt. 23, 4539 (1984).
[CrossRef] [PubMed]

1980 (1)

1979 (1)

K. H. Barratt, K. Lucas, “An Introduction to Sub-Nyquist Sampling,” IBA Tech. Rev. (GB) 12, 3 (1979).

1976 (1)

J. P. Rossi, “Sub-Nyquist-Encoded PCM NTSC Color Television,” J. SMPTE 85, 1 (1976).
[CrossRef]

1975 (1)

1934 (1)

P. Mertz, F. Gray, “A Theory of Scanning and Its Relation to the Characteristics of the Transmitted Signal in Telephotography and Television,” Bell Syst. Tech. J. 13, 464 (1934).

Barratt, K. H.

K. H. Barratt, K. Lucas, “An Introduction to Sub-Nyquist Sampling,” IBA Tech. Rev. (GB) 12, 3 (1979).

Bennett, J. M.

J. M. Bennett, “Comparison of Techniques for Measuring the Roughness of Optical Surfaces,” Opt. Eng. 24, 380 (1985).
[CrossRef]

Brunning, J. H.

J. H. Brunning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978). pp. 409–437.

Callari, C. A.

J. L. Seligson, C. A. Callari, J. E. Greivenkamp, J. W. Ward, “Stability of a Lateral-Shearing Heterodyne Twyman-Green Interferometer,” Opt. Eng. 23, 353 (1984).
[CrossRef]

Cheng, Y.-Y.

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

Y.-Y. Cheng, J. C. Wyant, “Two-Wavelength Phase Shifting Interferometry,” Appl. Opt. 23, 4539 (1984).
[CrossRef] [PubMed]

Creath, K.

K. Creath, J. C. Wyant, “Direct Phase Measurement of Aspheric Surface Contours,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 101 (1986).

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Electro Optics, 118 (Nov.1985).

Dereniak, E. L.

DeVore, S. L.

Gaskill, J. D.

See, for example, J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 266–285.

Gray, F.

P. Mertz, F. Gray, “A Theory of Scanning and Its Relation to the Characteristics of the Transmitted Signal in Telephotography and Television,” Bell Syst. Tech. J. 13, 464 (1934).

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350 (1984).
[CrossRef]

J. L. Seligson, C. A. Callari, J. E. Greivenkamp, J. W. Ward, “Stability of a Lateral-Shearing Heterodyne Twyman-Green Interferometer,” Opt. Eng. 23, 353 (1984).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic, New York, 1985), pp. 151–165.

Hayslett, C. R.

Kwon, O. Y.

Lucas, K.

K. H. Barratt, K. Lucas, “An Introduction to Sub-Nyquist Sampling,” IBA Tech. Rev. (GB) 12, 3 (1979).

Mertz, P.

P. Mertz, F. Gray, “A Theory of Scanning and Its Relation to the Characteristics of the Transmitted Signal in Telephotography and Television,” Bell Syst. Tech. J. 13, 464 (1934).

Offner, A.

A. Offner, “Null Tests Using Compensators,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 439–458.

Prettyjohns, K. N.

Rossi, J. P.

J. P. Rossi, “Sub-Nyquist-Encoded PCM NTSC Color Television,” J. SMPTE 85, 1 (1976).
[CrossRef]

Seligson, J. L.

J. L. Seligson, C. A. Callari, J. E. Greivenkamp, J. W. Ward, “Stability of a Lateral-Shearing Heterodyne Twyman-Green Interferometer,” Opt. Eng. 23, 353 (1984).
[CrossRef]

Truax, B. E.

B. E. Truax, “Programmable Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 10 (1987).

Ward, J. W.

J. L. Seligson, C. A. Callari, J. E. Greivenkamp, J. W. Ward, “Stability of a Lateral-Shearing Heterodyne Twyman-Green Interferometer,” Opt. Eng. 23, 353 (1984).
[CrossRef]

Wyant, J. C.

K. Creath, J. C. Wyant, “Direct Phase Measurement of Aspheric Surface Contours,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 101 (1986).

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Electro Optics, 118 (Nov.1985).

K. N. Prettyjohns, S. L. DeVore, E. L. Dereniak, J. C. Wyant, “Direct Phase Measurement Interferometer Working at 3.8 μm,” Appl. Opt. 24, 2211 (1985).
[CrossRef] [PubMed]

Y.-Y. Cheng, J. C. Wyant, “Two-Wavelength Phase Shifting Interferometry,” Appl. Opt. 23, 4539 (1984).
[CrossRef] [PubMed]

O. Y. Kwon, J. C. Wyant, C. R. Hayslett, “Rough Surface Interferometry at 10.6 μm,” Appl. Opt. 19, 1862 (1980).
[CrossRef] [PubMed]

J. C. Wyant, “Use of an ac Heterodyne Lateral Shear Interferometer with Real-Time Wavefront Correction Systems,” Appl. Opt. 14, 2622 (1975).
[CrossRef] [PubMed]

J. C. Wyant, “Holographic and Moire Techniques,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 381–408.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

P. Mertz, F. Gray, “A Theory of Scanning and Its Relation to the Characteristics of the Transmitted Signal in Telephotography and Television,” Bell Syst. Tech. J. 13, 464 (1934).

IBA Tech. Rev. (GB) (1)

K. H. Barratt, K. Lucas, “An Introduction to Sub-Nyquist Sampling,” IBA Tech. Rev. (GB) 12, 3 (1979).

J. SMPTE (1)

J. P. Rossi, “Sub-Nyquist-Encoded PCM NTSC Color Television,” J. SMPTE 85, 1 (1976).
[CrossRef]

Laser Focus/Electro Optics (1)

J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Electro Optics, 118 (Nov.1985).

Opt. Acta (1)

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

Opt. Eng. (3)

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350 (1984).
[CrossRef]

J. L. Seligson, C. A. Callari, J. E. Greivenkamp, J. W. Ward, “Stability of a Lateral-Shearing Heterodyne Twyman-Green Interferometer,” Opt. Eng. 23, 353 (1984).
[CrossRef]

J. M. Bennett, “Comparison of Techniques for Measuring the Roughness of Optical Surfaces,” Opt. Eng. 24, 380 (1985).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

K. Creath, J. C. Wyant, “Direct Phase Measurement of Aspheric Surface Contours,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 101 (1986).

B. E. Truax, “Programmable Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 10 (1987).

Other (5)

See, for example, J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 266–285.

A. Offner, “Null Tests Using Compensators,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 439–458.

J. C. Wyant, “Holographic and Moire Techniques,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 381–408.

J. H. Brunning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978). pp. 409–437.

P. Hariharan, Optical Interferometry (Academic, New York, 1985), pp. 151–165.

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 (15)

Fig. 1
Fig. 1

Actual wavefront and compressed wavefront as calculated modulo 2π by phase-shifting interferometry.

Fig. 2
Fig. 2

Removal of the 2π phase discontinuities by the PSI algorithm.

Fig. 3
Fig. 3

Sensor geometry.

Fig. 4
Fig. 4

Frequency space representation of the introduction of aliased fringes: (A) bandlimited scene, (B) image spectrum when the Nyquist frequency is greater than the scene bandwidth; (C) image spectrum when the Nyquist frequency is less than the scene bandwidth.

Fig. 5
Fig. 5

Aliasing of fringes at a high spatial frequency to a low spatial frequency.

Fig. 6
Fig. 6

Representation of sub-Nyquist sampling: (A) original scene spectrum; (B) spectrum of the sampled image; (C) portion of the image spectrum occurring in the sensor baseband; (D) reconstructed image formed by moving portions of the spectrum in (C) that occur in locations where it is known that there is no scene content.

Fig. 7
Fig. 7

For a measured fringe frequency ξm, there are several choices for the original fringe frequency.

Fig. 8
Fig. 8

The MTF due to the pixel aperture for a standard sensor and a sparse-array sensor.

Fig. 9
Fig. 9

Graphic representation of sub-Nyquist interferometry: (A) PSI data as calculated modulo 2π; (B) possible solutions for the wavefront at each pixel that are permitted by the data in (A); (C) same points as in (B), showing that the original wavefront goes through one of these points at each pixel; (D) PSI reconstruction through the possible solutions; (E) SNI reconstruction of the same data using derivative continuity.

Fig. 10
Fig. 10

Computer simulations that demonstrate the performance of SNI. A cubic wavefront and a value of G = 0.1 are assumed: (A) original wavefront used for the simulations; (B) PSI reconstruction of this wavefront; (C) SNI reconstruction of this wavefront using derivative continuity.

Fig. 11
Fig. 11

Computer simulation of a reentrant type wavefront measured with SNI.

Fig. 12
Fig. 12

Graphic representation of two-wavelength phase-shifting interferometry.

Fig. 13
Fig. 13

Experimental simulations of sub-Nyquist interferometry: (A) Experimental data from 60 pixels as reconstructed by the conventional PSI technique. This result serves as the standard to judge the SNI results. (B)–(F) The PSI and SNI reconstructions using the data from every nth pixel: (B) every other pixel, G = ½; (C) every fourth pixel, G = ¼; (D) every sixth pixel, G = ⅙; (E) every eighth pixel, G = ⅛; (F) every ninth pixel, G = 1/9.

Fig. 14
Fig. 14

Experimental results showing the measurement range improvements that are possible by implementing SNI on an existing interferometer with a standard sensor.

Fig. 15
Fig. 15

Use of sub-Nyquist interferometry for measuring discontinuous surfaces.

Equations (15)

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

i ( x , y , δ ) = i ( x , y ) + i ( x , y ) cos [ ϕ ( x , y ) δ ] ,
i n ( x , y ) = 1 Δ δ n Δ / 2 δ n + Δ / 2 i ( x , y , δ ) d δ ,
i n ( x , y ) = i ( x , y ) + i ( x , y ) [ sin ( Δ / 2 ) Δ / 2 ] cos [ ϕ ( x , y ) δ n ] .
δ n = 0 , π / 2 , π , 3 π / 2 ,
i 1 ( x , y ) = i ( x , y ) + i ( x , y ) cos [ ϕ ( x , y ) ] , i 2 ( x , y ) = i ( x , y ) + i ( x , y ) sin [ ϕ ( x , y ) ] , i 3 ( x , y ) = i ( x , y ) i ( x , y ) cos [ ϕ ( x , y ) ] , i 4 ( x , y ) = i ( x , y ) i ( x , y ) sin [ ϕ ( x , y ) ] .
ϕ ( x , y ) = tan 1 [ i 2 ( x , y ) i 4 ( x , y ) i 1 ( x , y ) i 3 ( x , y ) ] .
OPD ( x , y ) = ϕ ( x , y ) λ / 2 π ,
i s ( x , y ) = [ i ( x , y ) * * rect ( x / a , y / b ) ] comb ( x / x s , y / y s ) ,
I s ( ξ , η ) = [ I ( ξ , η ) sinc ( a ξ , b η ) ] * * comb ( x s ξ , y s η ) ,
sinc ( a ξ , b η ) = sin ( π a ξ ) π a ξ sin ( π b η ) π b η .
ξ o = | ξ m ± 2 n f N | , n = 0 , 1 , 2 .
G = a / x s ,
ϕ i = ϕ ̂ i ± 2 π n i ,
λ eq = λ 1 λ 2 / | λ 1 λ 2 | ,
h = h 0 + n λ / 2 ,

Metrics