Abstract

The detection and the correction of instrumental line-shape distortions have been studied. Some common distortion principles have been mathematically examined. A procedure for error correction is presented. In addition to a theoretical study, we also investigated simulated and experimental examples. An application for the procedure is presented.

© 2000 Optical Society of America

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References

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  1. P. Raspollini, P. Ade, B. Carli, M. Ridolfi, “Correction of instrument line-shape distortions in Fourier transform spectroscopy,” Appl. Opt. 37, 3696–3705 (1998).
    [CrossRef]
  2. P. Saarinen, J. Kauppinen, “Spectral line-shape distortions in Michelson interferometers due to off-focus radiation source,” Appl. Opt. 31, 2353–2359 (1992).
    [CrossRef] [PubMed]
  3. J. Kauppinen, P. Saarinen, “Line-shape distortions in misaligned cube corner interferometers,” Appl. Opt. 31, 68–75 (1992).
    [CrossRef]
  4. P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).
  5. J. Kauppinen, T. Kärkkäinen, E. Kyrö, “Correcting errors in the optical path difference in Fourier spectroscopy: a new accurate method,” Appl. Opt. 17, 1587–1594 (1978).
    [CrossRef] [PubMed]
  6. P. Saarinen, J. Kauppinen, “Multicomponent analysis of FT-IR spectra,” Appl. Spectrosc. 45, 953–963 (1991).
    [CrossRef]
  7. J. K. Kauppinen, D. J. Moffat, H. H. Mantsch, D. G. Cameron, “Fourier self-deconvolution: a method for resolving intrinsically overlapped bands,” Appl. Spectrosc. 35, 271–276 (1981).
    [CrossRef]

1998 (1)

P. Raspollini, P. Ade, B. Carli, M. Ridolfi, “Correction of instrument line-shape distortions in Fourier transform spectroscopy,” Appl. Opt. 37, 3696–3705 (1998).
[CrossRef]

1992 (2)

P. Saarinen, J. Kauppinen, “Spectral line-shape distortions in Michelson interferometers due to off-focus radiation source,” Appl. Opt. 31, 2353–2359 (1992).
[CrossRef] [PubMed]

J. Kauppinen, P. Saarinen, “Line-shape distortions in misaligned cube corner interferometers,” Appl. Opt. 31, 68–75 (1992).
[CrossRef]

1991 (1)

1981 (1)

1978 (1)

Ade, P.

P. Raspollini, P. Ade, B. Carli, M. Ridolfi, “Correction of instrument line-shape distortions in Fourier transform spectroscopy,” Appl. Opt. 37, 3696–3705 (1998).
[CrossRef]

Cameron, D. G.

Carli, B.

P. Raspollini, P. Ade, B. Carli, M. Ridolfi, “Correction of instrument line-shape distortions in Fourier transform spectroscopy,” Appl. Opt. 37, 3696–3705 (1998).
[CrossRef]

de Haseth, J. A.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).

Griffiths, P. R.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).

Kärkkäinen, T.

Kauppinen, J.

Kauppinen, J. K.

Kyrö, E.

Mantsch, H. H.

Moffat, D. J.

Raspollini, P.

P. Raspollini, P. Ade, B. Carli, M. Ridolfi, “Correction of instrument line-shape distortions in Fourier transform spectroscopy,” Appl. Opt. 37, 3696–3705 (1998).
[CrossRef]

Ridolfi, M.

P. Raspollini, P. Ade, B. Carli, M. Ridolfi, “Correction of instrument line-shape distortions in Fourier transform spectroscopy,” Appl. Opt. 37, 3696–3705 (1998).
[CrossRef]

Saarinen, P.

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Figures (14)

Fig. 1
Fig. 1

Aperture-broadened line at 4000 cm-1 and the same line with off-axis radiation source distortion.

Fig. 2
Fig. 2

Lines from Fig. 1 after truncation of the signal.

Fig. 3
Fig. 3

Same as in Fig. 1, but the lines are at 2000 cm-1.

Fig. 4
Fig. 4

∊(x)-error curves that correspond to off-axis radiation source distortions at 4000 and at 2000 cm-1. The error curves were truncated at points where the boxcar signal has its first zero. The error curves behave exactly as predicted.

Fig. 5
Fig. 5

Lower frequency ∊(x) curve shrunk by a factor of 2 and scaled by 0.5 is clearly equal to the higher frequency ∊(x) curve.

Fig. 6
Fig. 6

Amplitude curves that correspond to distorted lines at 2000 and at 4000 cm-1. The curves behave according to the equation A a ν r (x) = aA ν R (ax).

Fig. 7
Fig. 7

Amplitude curves corresponding to the undistorted and the off-axis radiation source distorted lines at 4000 cm-1.

Fig. 8
Fig. 8

Ratio of the amplitudes of the distorted and the undistorted signals. These curves demonstrate the predicted behavior of the signals as a function of wave number. The a coefficient is canceled, because there is an a coefficient in aA(ax) in the signal that corresponds to the boxcar lines and in the signal that corresponds to the off-axis distorted line shape. We multiplied the distorted signal by this ratio curve when we performed the correction. If a nondecaying cosine wave were used as a standard signal this multiplying would be a deconvolution.

Fig. 9
Fig. 9

Higher frequency amplitude ratio curve stretched by a factor of 2 equal to a lower frequency amplitude ratio curve.

Fig. 10
Fig. 10

Difference between the undistorted and the distorted lines and the difference between the undistorted and the corrected lines at 4000 cm-1. The difference diminished to a 1/400 part of its original value.

Fig. 11
Fig. 11

Distorted and undistorted (or less distorted) lines from the 3000-cm-1 methane band.

Fig. 12
Fig. 12

Distorted and undistorted (or less distorted) lines from the 1300-cm-1 methane band. The growth of the distortion as a function of wave number is clear.

Fig. 13
Fig. 13

∊(x) curves corresponding to the distortions at 1300 and at 3000 cm-1. The stretching and scaling effect as in Eq. (15) is shown.

Fig. 14
Fig. 14

Demonstration of the correction procedure. The phase correction symmetrizes the asymmetric distortion. The amplitude correction deconvolves the broadened line.

Equations (25)

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Ix=Eν=- Eνexpi2πνxdν,
Eν=-1Ix=- Ixexp-i2πνxdx.
Ix=- Eposνexpi2πνxdν=Axexpiϕx,
EΩν, ν0=ΠΩν, ν0=2π/ν0,ν0-ν0Ω/2πνν00,elsewhere,
EΩν, ν0=Ω sincν0xΩ/2.
Eν=2L sinc2πνL*EΩν, ν0,
Eν=-1Ix=-1Axexpi2πνx,
Eν=-1Ix+x=-1(Ax+xexpi2πνx+x).
x=ϕ2x-ϕ1x2πνR=12πνRarctanIm2Re2-arctanIm1Re1=12πνRarctanIm2Re1 - Im1Re2Re2Re1 + Im2Im1,
Eν=Axexpiψx,
Eν/a=aAaxexpiψax.
Eν, νR=Axexpiψxexpi2πνRx=Axexpi2πνRx+ψx2πνR,
Eν/a, aνR=aAaxexpiψaxexpi2πaνRx=aAaxexpi2πaνRx+ψax2πaνR.
νRx=ψx2πνR,  aνRx=ψax2πaνR.
aνRx=1a νRax,
AaνRx=aAνRax.
x=0x+1x,
νRx=0x+1x, aνRx=0x+1/a1ax,
νRx-aaνRax=0x-a0x/a.
0x-a0x/a=kx-akx/a=0.
Iji=j-dj+d Ii sincπ2j-i-icosπ2j-i-i,
AxAx+x,
Bx=Adistorted0Aundistorted0AundistortedxAdistortedx.
Ijj-dj+d BiIi sincπ2j-i-icosπ2j-i-i,
Eν, s=1,s<R0, ρνR0-s1πarccosρν2+s2-R022ρνs,ρν|R0-s|, r0+s,0,elsewhere

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