Abstract

A method is described for deconvolving into Voigt profiles a spectrum obtained with a Fabry-Perot spectrometer. This technique removes many of the limitations of some recently discussed schemes for analytically describing such spectroscopic data and reducing it to Voigt profiles. The use of a digital computer is assumed for all computations. The analytical description of the spectrometer includes all the important instrumental parameters as well as four adjustable profile parameters (position, amplitude, Lorentzian width, and Gaussian width) for each line present in the source spectrum. A least-squares adjustment of all nonfixed profile parameters is performed such that the sum of the individually fitted lines reproduces the recorded spectrum. This technique has been successfully applied to the thirteen fine structure components of the 4686-Å line of He ii obtained with a double etalon Fabry-Perot spectrometer.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Hernandez, Appl. Opt. 5, 1745 (1966).
    [CrossRef] [PubMed]
  2. E. A. Ballik, Appl. Opt. 5, 170 (1966).
    [CrossRef] [PubMed]
  3. J. O. Stoner, Opt. Soc. Am. 36, 370 (1966).
    [CrossRef]
  4. G. T. Best, Appl. Opt. 6, 287 (1967).
    [CrossRef] [PubMed]
  5. R. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).
  6. G. Goertzel, in Mathematical Methods for Digital Computers, A. Ralston, H. S. Wilf, Eds. (John Wiley & Sons, Inc., New York, 1962).
  7. J. W. Cooley, J. W. Tukey, Math. Computation 19, 296 (1965).
    [CrossRef]
  8. M. L. Forman, J. Opt. Soc. Am. 56, 978 (1966).
    [CrossRef]
  9. A. Savitzky, M. J. E. Golay, Ann. Chem. 36, 1627 (1964).
    [CrossRef]

1967

1966

1965

J. W. Cooley, J. W. Tukey, Math. Computation 19, 296 (1965).
[CrossRef]

1964

A. Savitzky, M. J. E. Golay, Ann. Chem. 36, 1627 (1964).
[CrossRef]

1953

R. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).

Ballik, E. A.

Best, G. T.

Chabbal, R.

R. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Computation 19, 296 (1965).
[CrossRef]

Forman, M. L.

Goertzel, G.

G. Goertzel, in Mathematical Methods for Digital Computers, A. Ralston, H. S. Wilf, Eds. (John Wiley & Sons, Inc., New York, 1962).

Golay, M. J. E.

A. Savitzky, M. J. E. Golay, Ann. Chem. 36, 1627 (1964).
[CrossRef]

Hernandez, G.

Savitzky, A.

A. Savitzky, M. J. E. Golay, Ann. Chem. 36, 1627 (1964).
[CrossRef]

Stoner, J. O.

J. O. Stoner, Opt. Soc. Am. 36, 370 (1966).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Computation 19, 296 (1965).
[CrossRef]

Ann. Chem.

A. Savitzky, M. J. E. Golay, Ann. Chem. 36, 1627 (1964).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris)

R. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).

Math. Computation

J. W. Cooley, J. W. Tukey, Math. Computation 19, 296 (1965).
[CrossRef]

Opt. Soc. Am.

J. O. Stoner, Opt. Soc. Am. 36, 370 (1966).
[CrossRef]

Other

G. Goertzel, in Mathematical Methods for Digital Computers, A. Ralston, H. S. Wilf, Eds. (John Wiley & Sons, Inc., New York, 1962).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Deconvolution of He ii 4686 Å. Eleven of the thirteen fine structure components are drawn using the least-squares adjusted parameters. Each of these profiles is free of instrumental effects. Also plotted is the sum of the fitted components, each convolved with the instrumental function. This sum is for the most part superimposed on the original spectrum which, although plotted, cannot readily be discerned. All curves were drawn directly from the output of the deconvolution program by an incremental plotter.

Fig. 2
Fig. 2

The instrumental function of a double etalon, Fabry-Perot spectrometer for spacers in a 9 2 ratio. One free spectral range of the multiple etalon is shown which includes the main passband (center) and eight instrumental ghosts.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

Y = I * V = I * L * G ,
I = [ ( A 1 * D 1 ) ( A 2 * D 2 ) ] * F .
A ( σ ) = { 1 + 4 R ( 1 - R ) - 2 sin 2 [ 2 π n t ( σ - σ 0 ) ] } - 1 ,
G ( σ ) = exp [ - 4 ln 2 ( σ - σ 0 ) 2 / Δ σ G 2 ] ,
L ( σ ) = [ 1 + ( σ - σ 0 ) 2 / ( 1 2 Δ σ L ) 2 ] - 1 ,
F ( σ ) = { 1 for σ 0 - 1 2 Δ σ F σ σ 0 + 1 2 Δ σ F . 0 for all other σ .
Y ( σ j ) = { A 0 / 2 + n = 1 n max A n cos ( n ξ j ) } * L * G .
Y ( σ j ) = A 0 / 2 + n = 1 n max A n exp ( - n l ) exp ( - n 2 g 2 / 4 ) cos ( n ξ j ) ,
ξ j = π ( σ j - σ 0 ) / Δ σ , l = π ( Δ σ L ) / ( 2 Δ σ ) , g = π ( Δ σ G ) / [ 2 ( ln 2 ) 1 2 Δ σ ] , and Δ σ = 1 / ( 2 t ) .
Y ( σ j ) = i = 1 i max I i A 0 / 2 + n = 1 n max A n exp ( - n l i ) exp ( - n 2 g i 2 / 4 ) cos ( n ξ j i ) A 0 / 2 + n = 1 n max A n exp ( - n l i ) exp ( - n 2 g i 2 / 4 ) .

Metrics