Abstract

A simple method for the measurement of the pixel modulation transfer function (MTF) of sparse-array (extended MTF) sensors has been developed. We use a phase-shifting Twyman–Green interferometer to generate a series of single spatial-frequency fringe patterns incident on the sensor The resulting signal modulation is measured. We achieve self-calibration by restricting the measured spatial frequencies to multiples of the Nyquist frequency. The aliased patterns at these frequencies are unique and easily identifiable. Spatial frequencies of 480 cycles/mm are generated and measured. This frequency value is more than ten times that of the sensor sampling frequency. The expected MTF shape is obtained at multiples of the sampling frequency. At odd multiples of the Nyquist frequency, the MTF’s are affected by the electronic bandwidth and cross talk in the charge-injection device sensor.

© 1994 Optical Society of America

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References

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  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, 464–515 (1934).
  2. J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.
  3. J. E. Greivenkamp, “Color dependent optical prefilter for the suppression of aliasing artifacts,” Appl. Opt. 29, 676–684 (1990).
    [CrossRef] [PubMed]
  4. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
    [CrossRef] [PubMed]
  5. See, for example, J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978) pp. 266–285.
  6. R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist Interferometry: results and implementation issues,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 378–388 (1990).
  7. D. Malacara, “Twyman–Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.
  8. See, for example, R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 10–12.
  9. M. Marchywka, D. G. Socker, “Modulation transfer function measurement technique for small-pixel detectors,” Appl. Opt. 31, 7198–7213 (1992).
    [CrossRef] [PubMed]
  10. G. R. Sims, M. B. Denton, “Spatial pixel crosstalk in a charge-injection device,” Opt. Eng. 26, 999–1007 (1987).
  11. A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry: Applications II, R. J. Pryputniewicz, W. P. Jueptner, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2004, 173–181 (1993).

1992 (1)

1990 (1)

1987 (2)

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
[CrossRef] [PubMed]

G. R. Sims, M. B. Denton, “Spatial pixel crosstalk in a charge-injection device,” Opt. Eng. 26, 999–1007 (1987).

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, 464–515 (1934).

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), pp. 501–598.

Burckhardt, C. B.

See, for example, R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 10–12.

Collier, R. J.

See, for example, R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 10–12.

Denton, M. B.

G. R. Sims, M. B. Denton, “Spatial pixel crosstalk in a charge-injection device,” Opt. Eng. 26, 999–1007 (1987).

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, 464–515 (1934).

Greivenkamp, J. E.

J. E. Greivenkamp, “Color dependent optical prefilter for the suppression of aliasing artifacts,” Appl. Opt. 29, 676–684 (1990).
[CrossRef] [PubMed]

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
[CrossRef] [PubMed]

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist Interferometry: results and implementation issues,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 378–388 (1990).

A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry: Applications II, R. J. Pryputniewicz, W. P. Jueptner, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2004, 173–181 (1993).

Lin, L. H.

See, for example, R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 10–12.

Lowman, A. E.

A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry: Applications II, R. J. Pryputniewicz, W. P. Jueptner, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2004, 173–181 (1993).

Malacara, D.

D. Malacara, “Twyman–Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.

Marchywka, M.

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, 464–515 (1934).

Palum, R. J.

R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist Interferometry: results and implementation issues,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 378–388 (1990).

Sims, G. R.

G. R. Sims, M. B. Denton, “Spatial pixel crosstalk in a charge-injection device,” Opt. Eng. 26, 999–1007 (1987).

Socker, D. G.

Appl. Opt. (3)

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, 464–515 (1934).

Opt. Eng. (1)

G. R. Sims, M. B. Denton, “Spatial pixel crosstalk in a charge-injection device,” Opt. Eng. 26, 999–1007 (1987).

Other (6)

A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry: Applications II, R. J. Pryputniewicz, W. P. Jueptner, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2004, 173–181 (1993).

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

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

R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist Interferometry: results and implementation issues,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 378–388 (1990).

D. Malacara, “Twyman–Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.

See, for example, R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 10–12.

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Figures (9)

Fig. 1
Fig. 1

Sensor geometry.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Theoretical pixel MTF’s for sensors with width-to-pitch ratios G = 1 and G = 1/2.

Fig. 4
Fig. 4

Twyman–Green interferometer with the mirrors positioned as close to the sensor as possible so that beam walk-off is minimized; PZT, piezoelectric transducer.

Fig. 5
Fig. 5

Tilt fringes generated by plane waves of equal but opposite angle of incidence.

Fig. 6
Fig. 6

MTF of a sparse-array sensor that uses only data at multiples of the sampling frequency: (a) horizontal spatial frequencies, (b) vertical spatial frequencies.

Fig. 7
Fig. 7

MTF for the same sparse-array sensor as in Fig. 6, but with data plotted at all multiples of the Nyquist frequency: (a) horizontal spatial frequencies, (b) vertical spatial frequencies.

Fig. 8
Fig. 8

MTF of a second sparse-array sensor with smaller apertures that uses data at multiples of the sampling frequency: (a) horizontal spatial frequencies, (b) vertical spatial frequencies.

Fig. 9
Fig. 9

MTF of a standard CID sensor assuming G = 1: (a) horizontal spatial frequencies, (b) vertical spatial frequencies.

Equations (8)

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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 ,
p = λ 2 sin θ ,
γ = 2 I 1 I 2 I 1 + I 2 ,
γ ( x , y ) = 3 [ 4 ( I 4 - I 2 ) 2 + ( I 1 + I 5 - 2 I 3 ) 2 ] 1 / 2 2 ( I 1 + I 2 + 2 I 3 + I 4 + I 5 ) ,
I i = I 0 { 1 + γ ( x , y ) cos [ ϕ ( x , y ) + δ i ] } ,
δ i = - 180 ° , - 90 ° , 0 ° , 90 ° , 180 ° .

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