Abstract

Apodizing functions are used in Fourier transform spectroscopy (FTS) to reduce the magnitude of the sidelobes in the instrumental line shape (ILS), which are a direct result of the finite maximum optical path difference in the measured interferogram. Three apodizing functions, which are considered optimal in the sense of producing the smallest loss in spectral resolution for a given reduction in the magnitude of the largest sidelobe, find frequent use in FTS [J. Opt. Soc. Am. 66, 259 (1976) ]. We extend this series to include optimal apodizing functions corresponding to increases in the width of the ILS ranging from factors of 1.1 to 2.0 compared with its unapodized value, and we compare the results with other commonly used apodizing functions.

© 2007 Optical Society of America

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References

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  1. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, 1972).
  2. A. S. Filler, "Apodization and interpolation in Fourier-transform spectroscopy," J. Opt. Soc. Am. 54, 762-767 (1964).
    [CrossRef]
  3. R. H. Norton and R. Beer, "New apodizing functions for Fourier spectrometry," J. Opt. Soc. Am. 66, 259-264 (1976).
    [CrossRef]
  4. R. H. Norton and R. Beer, "New apodizing functions for Fourier spectrometry: errata," J. Opt. Soc. Am. 67, 419 (1977).
  5. F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform," Proc. IEEE 66, 51-83 (1978).
    [CrossRef]
  6. The Interactive Data Language (Research Systems Inc, 2007).
  7. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, (Cambridge U. Press, 1992).
  8. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, From the Point of View of Communications Engineering (Dover, 1959).
  9. R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, "Phase correction of emission line Fourier transform spectra," J. Opt. Soc. Am. A 12, 2165-2171 (1995).
    [CrossRef]

1995 (1)

1978 (1)

F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform," Proc. IEEE 66, 51-83 (1978).
[CrossRef]

1977 (1)

1976 (1)

1964 (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform," Proc. IEEE 66, 51-83 (1978).
[CrossRef]

Other (4)

The Interactive Data Language (Research Systems Inc, 2007).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, (Cambridge U. Press, 1992).

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, From the Point of View of Communications Engineering (Dover, 1959).

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, 1972).

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Figures (3)

Fig. 1
Fig. 1

The profile of the extended apodizing functions.

Fig. 2
Fig. 2

The corresponding ILS of the extended apodizing functions in ascending order. The lowest trace shows the sinc function for reference.

Fig. 3
Fig. 3

The loci of the extended apodizing functions on the Filler diagram (open circles). The solid diamonds show the loci of the original Norton–Beer apodizing functions. For comparison the loci of other commonly used apodizing functions are shown; the Bartlett and Hann functions give inferior results, while the Gaussian, Hamming and Blackman–Harris functions are seen to be near optimum. The solid line is the empirical boundary given by Eq. (4).

Tables (5)

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Table 1 Coefficients of the Original Norton–Beer Apodizing Functions

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Table 2 Coefficients, C i , of the Extended Norton–Beer Apodizing Functions

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Table 3 FWHM, Relative Height with Respect to the Peak of the ILS, a and Position in Units of 1 L b of the First Five Minima of the Apodizing Functions Presented in This Paper

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Table 4 FWHM, Relative Height with Respect to the Peak of the ILS, a and Position in Units of 1 L b of the First Five Maxima of the Apodizing Functions Presented in this Paper

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Table 5 FWHM of ILS Relative to the Sinc and the Magnitude of the Largest Secondary Lobe Relative to the Maximum, Expressed as a Percentage, for the Extended Norton–Beer Apodizing Functions

Equations (17)

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I ( δ ) = + B ( σ ) [ 1 + cos ( 2 π σ δ ) ] d σ ,
B ( σ ) = + I ( δ ) cos ( 2 π σ δ ) d δ .
B ( σ ) = L + L I ( δ ) cos ( 2 π σ δ ) d δ ,
Π ( δ ) = 1 , δ L
Π ( δ ) = 0 , δ > L .
I { Π ( δ ) } = L + L cos ( 2 π σ δ ) d δ = 2 L sin ( 2 π σ L ) 2 π σ L = 2 L sinc ( 2 π σ L ) ,
D α ( δ L ) = cos ( π δ 2 L ) + α cos ( 3 π δ 2 L ) , 0 α 1 ,
E α ( δ L ) = 1 + ( 1 + α ) cos ( π δ L ) + α cos ( 2 π δ L ) , 0 α 1 ,
P α , p ( δ L ) = 1 + p + ( 1 + α ) cos ( π δ L ) + α cos ( 2 π δ L ) , 1 α 1 ; 0 p 1 .
NB ( δ L ) = i = 0 n C i ( 1 ( δ L ) 2 ) i , where i = 0 n C i = 1 , n = 0 , 1 , 2 , 3
log h h 0 1.939 1.401 ( W W 0 ) 0.597 ( W W 0 ) 2 ,
A ( δ L ) = exp ( δ L ) 2 , 0 δ L ;
A ( δ L ) = 0.54 + 0.46 cos ( π δ L ) , 0 δ L ;
A ( δ L ) = 0.5 ( 1 + cos ( π δ L ) ) , 0 δ L ;
A ( δ L ) = 0.42323 + 0.49755 cos ( π δ L ) + 0.07922 cos ( 2 π δ L ) , 0 δ L ,
A ( δ L ) = 0.35875 + 0.48829 cos ( π δ L ) + 0.14128 cos ( 2 π δ L ) + 0.01168 cos ( 3 π δ L ) , 0 δ L .
A ( δ L ) = 0.355766 + 0.487395 cos ( π δ L ) + 0.144234 cos ( 2 π δ L ) + 0.012605 cos ( 3 π δ L ) , 0 δ L ,

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