Abstract

An interpolation formula is derived which gives an apodized spectrum as the convolution of the unapodized spectrum (sampled at suitable points) with the apodized apparatus function. This allows many apodizations to be applied to a single interferogram with the performance of only a single Fourier transformation. A further saving in computation effort is possible if the apodized apparatus function decays rapidly away from its center. Examples of such cases are presented, where the interferogram is weighted by a function which is a cosine series of only a few terms.

© 1964 Optical Society of America

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References

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  1. P. Fellgett, J. Phys. Radium 19, 187 (1958).
    [CrossRef]
  2. J. Strong, J. Opt. Soc. Am. 47, 354 (1957).
    [CrossRef]
  3. P. Jacquinot, Rept. Progr. Phys. 23, 268 (1960).
    [CrossRef]
  4. B. Dossier, Rev. Opt. 33, 57 (1954).
  5. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Company, Inc., New York, 1962), p. 51.
  6. J. Connes, Rev. Opt. 40, 62 (1961).
  7. G. A. Vanasse, J. Opt. Soc. Am. 52, 472 (1962).
    [CrossRef]
  8. H. A. Gebbie, Symposium on Interferometry (Teddington, 1959), p. 427.

1962 (1)

1961 (1)

J. Connes, Rev. Opt. 40, 62 (1961).

1960 (1)

P. Jacquinot, Rept. Progr. Phys. 23, 268 (1960).
[CrossRef]

1958 (1)

P. Fellgett, J. Phys. Radium 19, 187 (1958).
[CrossRef]

1957 (1)

1954 (1)

B. Dossier, Rev. Opt. 33, 57 (1954).

Connes, J.

J. Connes, Rev. Opt. 40, 62 (1961).

Dossier, B.

B. Dossier, Rev. Opt. 33, 57 (1954).

Fellgett, P.

P. Fellgett, J. Phys. Radium 19, 187 (1958).
[CrossRef]

Gebbie, H. A.

H. A. Gebbie, Symposium on Interferometry (Teddington, 1959), p. 427.

Jacquinot, P.

P. Jacquinot, Rept. Progr. Phys. 23, 268 (1960).
[CrossRef]

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Company, Inc., New York, 1962), p. 51.

Strong, J.

Vanasse, G. A.

J. Opt. Soc. Am. (2)

J. Phys. Radium (1)

P. Fellgett, J. Phys. Radium 19, 187 (1958).
[CrossRef]

Rept. Progr. Phys. (1)

P. Jacquinot, Rept. Progr. Phys. 23, 268 (1960).
[CrossRef]

Rev. Opt. (2)

B. Dossier, Rev. Opt. 33, 57 (1954).

J. Connes, Rev. Opt. 40, 62 (1961).

Other (2)

H. A. Gebbie, Symposium on Interferometry (Teddington, 1959), p. 427.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Company, Inc., New York, 1962), p. 51.

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Figures (7)

F. 1
F. 1

Functions used to weight interferograms. l is the path difference between the two interfering beams. L is the maximum path difference.

F. 2
F. 2

Apparatus functions corresponding to the weighting functions shown in Fig. 1, normalized to equal 1 at k = 0. k is the wavenumber. w is the width of the central peak at half-maximum amplitude.

F. 3
F. 3

Logarithmic plot of the absolute values of the apparatus functions shown in Fig. 2. The index ν corresponds to the summation index in Eq. (8). Corresponding curves in Figs. 2 and 3 are drawn with same style.

F. 4
F. 4

Summary plot, showing the logarithms of apodization and decay parameters as functions of half-widths, w is the half-width, h is the amplitude of the first side lobe, ν (s) is the range of the summation index in Eq. (8) for a maximum error equal to s.

F. 5
F. 5

Apparatus functions, A″(k) for Cases Eα, where the weighting function is proportional to 1+(1+α) cosπl/L +α cos2πl/L. The symbols at A″(k) = 0.5 give the half-widths of the apparatus functions shown in Fig. 2. Solid curve, α = 0; dashes, α = 0.06; dash–dot, 0.10; dash–2 dots, 0.14; 2 dashes–dot, α = 0.18.

F. 6
F. 6

Apparatus functions A″(k) for Cases Dα, where the weighting function is proportional to cosπl/2L+α cos3πl/2L. The symbols at A″(k) = 0.5 give the half-widths of the apparatus functions shown in Fig. 2. Solid curve, α = 0; 2 dashes–dot, α = 0.12; dashes,α = 0.16; 2 dots–dash, α = 0.20; dash–dot, α = 0.24.

F. 7
F. 7

Logarithmic plot of the apodization parameter |hm| vs the half-width w of the central peak of the apodized apparatus function, for Cases Dα and Eα. The dashed line corresponds to the similar plot in Fig. 4.

Tables (2)

Tables Icon

Table I Definition of symbols.

Tables Icon

Table II Half-widths (w), amplitudes (h) of largest lobes, and values of the index ν for maximum summation errors of 0.01 and 0.001.

Equations (18)

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f ( k ) = n f ( n / 2 L ) A ( k n / 2 L ) .
p ( x ) * q ( x ) = p ( y ) q ( x y ) d y .
T ( l ) · T ( l ) = T ( l ) ,
A ( k ) * A ( k ) = A ( k ) .
I ( l ) = I ( l ) · T ( l ) · T ( l ) · V ( l ) = I ( l ) · W ( l ) ,
S ( k ) = S ( k ) * A ( k ) = S ( α ) A ( k α ) d α = S ( α ) A ( k α β ) U ( β ) d α d β .
S ( k ) = ν S ( ν / 2 L ) A ( α ν / 2 L ) × A ( k α β ) U ( β ) d α d β .
S ( k ) = ν S ( ν / 2 L ) A ( k ν / 2 L β ) U ( β ) d β = ν S ( ν / 2 L ) A ( k ν / 2 L ) .
V ( l ) m = 0 M B m cos m π l L ,
U ( k ) m = 0 M B m δ ( k m 2 L ) + m = 0 M B m δ ( k + m 2 L ) .
A ( k ) = m = 0 M B m A ( k m 2 L ) + m = 0 M B m A ( k + m 2 L ) .
A ( p L + 1 4 L ) = m = 0 M B m [ sin 2 π ( p + 1 4 m / 2 ) 2 π ( p + 1 4 m / 2 ) + sin 2 π ( p + 1 4 + m / 2 ) 2 π ( p + 1 4 + m / 2 ) ] = m = 0 M B m cos m π π [ 1 p 1 4 p 2 + 1 + 4 m 2 16 p 3 + ] .
m = 0 M B m ( 1 ) m = 0
m = 0 M m 2 B m ( 1 ) m = 0
V ( l ) 1 + ( 1 + α ) cos π l / L + α cos 2 π l / L ,
V ( l ) m = 0 M B m cos m π l 2 L ( m odd ) ,
A ( k ) m = 0 M B m A ( k m 4 L ) + m = 0 M B m A ( k + m 4 L ) ( m odd ) .
V ( l ) cos π l / 2 L + α cos 3 π / 2 L .