Abstract

The general problem of obtaining an acceptable estimate of a spectrum in Fourier spectroscopy is discussed for four specific cases. The two known techniques currently used for the spectral recovery from an asymmetric interferogram are shown to be mathematically equivalent. A discussion of their practical merit is also included.

© 1968 Optical Society of America

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References

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  1. For general reference see G. A. Vanasse and H. Sakai, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1967).
  2. The factor 2 is introduced for convenience.
  3. M. Forman, W. H. Steel, and G. A. Vanasse, J. Opt. Soc. Am. 56, 59 (1966).
    [Crossref]
  4. L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965).
  5. L. Mertz, Infrared Phys. 7, 17 (1967).
    [Crossref]
  6. H. Sakai and G. A. Vanasse, J. Opt. Soc. Am. 56, 131 (1966).
    [Crossref]
  7. W. H. Steel and M. L. Forman, J. Opt. Soc. Am. 56, 982 (1966).
    [Crossref]

1967 (1)

L. Mertz, Infrared Phys. 7, 17 (1967).
[Crossref]

1966 (3)

Forman, M.

Forman, M. L.

Mertz, L.

L. Mertz, Infrared Phys. 7, 17 (1967).
[Crossref]

L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965).

Sakai, H.

H. Sakai and G. A. Vanasse, J. Opt. Soc. Am. 56, 131 (1966).
[Crossref]

For general reference see G. A. Vanasse and H. Sakai, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1967).

Steel, W. H.

Vanasse, G. A.

M. Forman, W. H. Steel, and G. A. Vanasse, J. Opt. Soc. Am. 56, 59 (1966).
[Crossref]

H. Sakai and G. A. Vanasse, J. Opt. Soc. Am. 56, 131 (1966).
[Crossref]

For general reference see G. A. Vanasse and H. Sakai, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1967).

Infrared Phys. (1)

L. Mertz, Infrared Phys. 7, 17 (1967).
[Crossref]

J. Opt. Soc. Am. (3)

Other (3)

L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965).

For general reference see G. A. Vanasse and H. Sakai, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1967).

The factor 2 is introduced for convenience.

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Figures (3)

Fig. 1
Fig. 1

Spectral estimates obtained by the multiplicative method. The phase error φ0(σ) is given by

Fig. 2
Fig. 2

Comparison of spectral estimate obtained by the multiplicative method (solid line) with that obtained by the simple cosine transformation (dotted line).

Fig. 3
Fig. 3

Spectral estimate obtained by the convolution method. The phase error is given by

Equations (62)

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F ( x ) = 2 0 B ( σ ) cos 2 π σ x d σ ,
F ( x ) = 2 0 B ( σ ) cos [ 2 π σ x + φ ( σ ) ] d σ ,
F ( x ) = - B ( σ ) e i φ ( σ ) e i 2 π σ x d σ .
and             B ( σ ) = B ( - σ ) φ ( σ ) = - φ ( - σ ) .
B ( σ ) = | - F ( x ) e - i 2 π σ x d x |
S ˜ ( σ ) = - A ( x ) F ( x ) e - i 2 π σ x d x = Ã ( σ ) * F ˜ ( σ ) ,
à ( σ ) = - A ( x ) e - i 2 π σ x d x
à ( σ ) * F ˜ ( σ ) = - à ( σ ) F ˜ ( σ - σ ) d σ .
S ˜ ( σ ) = S ˜ ( σ ) e i ψ ( σ ) = Ã ( σ ) * F ˜ ( σ ) e i ψ ( σ ) .
B ( σ ) * Ã c ( σ ) = B ( σ ) * - A ( x ) cos 2 π σ x d x .
F ˜ ( σ ) = - F ( x ) e - i 2 π σ x d x = B ( σ ) e i φ ( σ )
à ( σ ) = - A ( x ) e - i 2 π σ x d x = à e ( σ ) e i β ( σ ) .
- ( x ) e - i 2 π σ x d x = B ( σ )
- a ( x ) e - i 2 π σ x d x = Ã e ( σ ) .
φ ( σ ) = 2 π σ + φ 0 ( σ ) .
φ ( σ ) = φ ( σ ) + d φ d σ ( σ - σ ) + 1 2 ! d 2 φ d σ 2 ( σ - σ ) 2 + .
φ ( σ ) = φ ( σ ) + ( d φ / d σ ) ( σ - σ ) .
S ˜ ( σ ) = B ( σ ) e i φ ( σ ) * Ã ( σ ) e i φ ( σ ) { B ( σ ) * Ã ( σ ) - e i 2 π δ σ } ,
2 π δ = d φ / d σ .
S ˜ ( σ ) = exp { i [ φ ( σ ) + α ( σ ) ] } B ( σ ) * Ã ( σ ) e - i 2 π δ σ ,
ψ ( σ ) = φ ( σ ) + α ( σ ) .
α ( σ ) = tan - 1 B ( σ ) * Im [ Ã ( σ ) e - i 2 π δ σ ] B ( σ ) * Re [ Ã ( σ ) e - i 2 π δ σ ] .
à ( σ ) e - i 2 π δ σ = e - i 2 π δ σ - A ( x ) e - i 2 π σ x d x = - A ( x ) exp [ - i 2 π σ ( x + δ ) ] d x - A ( x - δ ) e - i 2 π σ x d x .
B ( σ ) * Ã ( σ ) e - i 2 π δ σ = - ( x ) A ( x - δ ) e - i 2 π σ x d x .
S ˜ ( σ ) = e i φ ( σ ) - ( x ) A ( x - δ ) e - i 2 π σ x d x .
F ( x ) = F ( - x )
B ( σ ) = - F ( x ) e - i 2 π σ x d x = - F ( x ) cos 2 π σ x d x .
à ( σ ) = - A ( x ) e - i 2 π σ x d x = - A ( x ) cos 2 π σ x d x = à e ( σ ) = à c ( σ ) ,
S ˜ ( σ ) = B ( σ ) * Ã c ( σ ) .
S ˜ ( σ ) = B ( σ ) * Ã e ( σ ) e i β ( σ ) ,
à ( σ ) = à e ( σ ) e i β ( σ ) .
R e S ˜ ( σ ) = - A ( x ) F ( x ) cos 2 π σ x d x
B ( σ ) * Ã e ( σ ) cos β ( σ ) = B ( σ ) * - A ( x ) cos 2 π σ x d x
B ( σ ) = - 2 π σ .
x = ,
à ( σ ) = à e ( σ ) e - i 2 π σ .
L ,
à ( σ ) = à e ( σ ) ,
S ˜ ( σ ) = B ( σ ) * Ã e ( σ ) .
à e ( σ ) = - a ( x ) e - i 2 π σ x d x = - a ( x ) cos 2 π σ x d x , S ˜ ( σ ) = B ( σ ) * - a ( x ) cos 2 π σ x d x ,
F ( x ) F ( - x )
S ˜ ( σ ) = Ã ( σ ) * B ( σ ) e i φ ( σ )
S ˜ ( σ ) = e i φ ( σ ) { B ( σ ) * Ã ( σ ) e - i 2 π δ σ } .
à ( σ ) = à e ( σ ) ,
δ L .
B ( σ ) * Ã e ( σ ) e - i 2 π δ σ = B ( σ ) * Ã e ( σ ) = B ( σ ) * - a ( x ) cos 2 π σ x d x .
ψ ( σ ) = φ ( σ ) ,
S ˜ ( σ ) = B ( σ ) * Ã e ( σ ) .
S ˜ ( σ ) = e i φ ( σ ) { B ( σ ) * Ã ( σ ) e - i 2 π δ σ } .
à ( σ ) = à e ( σ ) e i β ( σ ) .
β ( σ ) = 2 π σ ,
B ( σ ) * Ã ( σ ) e - i 2 π δ σ = S ˜ ( σ ) e i α ( σ ) = S ˜ ( σ ) exp { - [ ψ ( σ ) - φ ( σ ) ] } .
δ L ,
S ˜ ( σ ) cos [ ψ ( σ ) - φ ( σ ) ] = B ( σ ) * - A ( x - δ ) cos 2 π σ x d x .
δ L
B ( σ ) * δ L + δ cos 2 π σ x d x .
- F ( x ) A ( x ) cos 2 π σ x d x = cos [ φ ( σ ) + α ( σ ) ] B ( σ ) * Ã ( σ ) e - i 2 π δ σ = cos φ ( σ ) { B ( σ ) * R e [ Ã ( σ ) e - i 2 π δ σ ] } - sin φ ( σ ) { B ( σ ) * I m [ Ã ( σ ) e - i 2 π δ σ ] } ,
S ˜ ( σ ) = Ã ( σ ) * F ˜ ( σ ) = Ã ( σ ) * F ˜ ( σ ) e i ψ ( σ ) .
F ( x ) A ( x ) * { - σ M σ M e - i φ ( σ ) e i 2 π σ x d σ } ,
R e { S ˜ ( σ ) e - i φ ( σ ) } = S ˜ ( σ ) cos { ψ ( σ ) - φ ( σ ) } .
φ 0 ( σ ) = c ( σ / σ M ) 2 ,
φ 0 ( σ ) = c ( σ / σ M ) 2 ,