Abstract

We establish an encoding figure of merit for a detector-noise limited Fourier transform spectrometer (FTS) and compare it to the comparable figure for a Hadamard transform spectrometer (HTS). If N measurements are made to establish N spectral densities, the mean square errors obtained with the Fourier system are a factor of 2 greater than for the analogous Hadamard system. The limitation of the Fourier system is partly that it does not truly Fourier analyze the radiation. Instead a cosine squared modulation is imposed on the different spectral frequencies. An additional difficulty is that neither the cosine nor the cosine squared functions form an orthonormal set. This makes the Fellgett’s advantage (root-mean-squared figure of merit) for a single detector Michelson interferometer a factor of (N/8)1/2 greater than for a conventional grating instrument—rather than (N/2)1/2 as maintained in standard texts. The theoretical limit, which may not be realizable with practical instruments, would be (N)1/2.

© 1976 Optical Society of America

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References

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  1. T. Hirschfeld, G. Wintjes, Appl. Opt. 12, 2876 (1973); Appl. Opt. 13, 1740 (1974).
    [Crossref] [PubMed]
  2. J. A. Decker, Appl. Opt. 13, 1296 (1974).
    [Crossref] [PubMed]
  3. N. M. Blachman, Proc. IEEE 62, 346 (1974); Proc. IEEE 63, 329 (1975).
    [Crossref]
  4. C. K. Yuen, Proc. IEEE 63, 329 (1975).
    [Crossref]
  5. See, for example, J. E. Stewart, Infrared Spectroscopy (Dekker, New York, 1970), p. 96.
  6. See, for example, P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 454.
  7. N. J. A. Sloane, T. Fine, P. G. Phillips, M. Harwit, Appl. Opt. 8, 2103 (1969).
    [Crossref] [PubMed]
  8. E. D. Nelson, M. L. Fredman, J. Opt. Soc. Am. 60, 1664 (1971).
    [Crossref]
  9. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  10. M. Harwit, P. G. Phillips, L. W. King, D. A. Briotta, Appl. Opt. 13, 2669 (1974).
    [Crossref] [PubMed]
  11. M. H. Tai, M. Harwit, N. J. A. Sloane, Appl. Opt. 14, 2678 (1975).
    [Crossref] [PubMed]

1975 (2)

1974 (3)

1973 (1)

1971 (1)

1969 (1)

Blachman, N. M.

N. M. Blachman, Proc. IEEE 62, 346 (1974); Proc. IEEE 63, 329 (1975).
[Crossref]

Briotta, D. A.

Decker, J. A.

Feshbach, H.

See, for example, P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 454.

Fine, T.

Fredman, M. L.

Harwit, M.

Hirschfeld, T.

King, L. W.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Morse, P. M.

See, for example, P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 454.

Nelson, E. D.

Phillips, P. G.

Sloane, N. J. A.

Stewart, J. E.

See, for example, J. E. Stewart, Infrared Spectroscopy (Dekker, New York, 1970), p. 96.

Tai, M. H.

Wintjes, G.

Yuen, C. K.

C. K. Yuen, Proc. IEEE 63, 329 (1975).
[Crossref]

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

N. M. Blachman, Proc. IEEE 62, 346 (1974); Proc. IEEE 63, 329 (1975).
[Crossref]

C. K. Yuen, Proc. IEEE 63, 329 (1975).
[Crossref]

Other (3)

See, for example, J. E. Stewart, Infrared Spectroscopy (Dekker, New York, 1970), p. 96.

See, for example, P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 454.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

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Equations (9)

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S ( ξ ) = 0 P ( ν ) cos 2 ( 2 π ξ ν ) d ν
= 1 2 P 0 + 1 2 0 P ( ν ) cos ( 4 π ξ ν ) d ν .
P ( ν ) = 16 0 cos ( 4 π ξ ν ) S ( ξ ) d ξ .
P ( ν ) = 16 n = 0 N S ( n Δ ) cos ( 4 π ν n Δ ) Δ .
π ( ν ) = 4 N n = 0 N S ( n Δ ) cos ( 4 π ν n Δ ) .
π = W - 1 S .
= σ 2 Trace [ W - 1 ( W - 1 ) T ] ,
/ ( σ 2 ) 8             ( two beam interferometer FTS )
/ ( σ 2 ) 4             ( single mask HTS ) .

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