Abstract

Line-shape distortions caused by the misalignment of the moving cube mirror in Fourier transform spectrometers have been described. A method of studying and correcting these distortions is presented. By using this method we can estimate the accuracy of the line position, which is especially important in high-resolution Fourier transform spectroscopy. The method is verified in simulations, and in practice it has been used to align the Oulu Fourier transform spectrometer.

© 1992 Optical Society of America

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References

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  1. M. V. R. K. Murty, “Some more aspects of the Michelson interferometer with cube corners,” J. Opt. Soc. Am. 50, 7–10 (1960).
    [CrossRef]
  2. 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]
  3. J. Kauppinen, “Double-beam high resolution Fourier spectrometer for the far infrared,” Appl. Opt. 14, 1987–1992 (1975).
    [CrossRef] [PubMed]
  4. J. Kauppinen, “Working resolution of 0.010 cm−1 between 20 cm−1 and 1200 cm−1 by a Fourier spectrometer,” Appl. Opt. 18, 1788–1796 (1979).
    [CrossRef] [PubMed]
  5. J. Kauppinen, V.-M. Horneman, “Cube corner interferometer with the resolution of about 0.001 cm−1,” in Technical Digest of the Ninth Colloquium on High Resolution Spectroscopy, Riccione, Italy, 16–20 September 1985 (Cooperativa Libraria Universitaria Editrice, Bologna, 1985); J. Kauppinen, V.-M. Horneman, “Large aperture cube corner interferometer with a resolution of 0.001 cm−1,” Appl. Opt. 30, 2575–2578 (1991).
    [CrossRef] [PubMed]

1979

1978

1975

1960

Horneman, V.-M.

J. Kauppinen, V.-M. Horneman, “Cube corner interferometer with the resolution of about 0.001 cm−1,” in Technical Digest of the Ninth Colloquium on High Resolution Spectroscopy, Riccione, Italy, 16–20 September 1985 (Cooperativa Libraria Universitaria Editrice, Bologna, 1985); J. Kauppinen, V.-M. Horneman, “Large aperture cube corner interferometer with a resolution of 0.001 cm−1,” Appl. Opt. 30, 2575–2578 (1991).
[CrossRef] [PubMed]

Kärkkäinen, T.

Kauppinen, J.

J. Kauppinen, “Working resolution of 0.010 cm−1 between 20 cm−1 and 1200 cm−1 by a Fourier spectrometer,” Appl. Opt. 18, 1788–1796 (1979).
[CrossRef] [PubMed]

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]

J. Kauppinen, “Double-beam high resolution Fourier spectrometer for the far infrared,” Appl. Opt. 14, 1987–1992 (1975).
[CrossRef] [PubMed]

J. Kauppinen, V.-M. Horneman, “Cube corner interferometer with the resolution of about 0.001 cm−1,” in Technical Digest of the Ninth Colloquium on High Resolution Spectroscopy, Riccione, Italy, 16–20 September 1985 (Cooperativa Libraria Universitaria Editrice, Bologna, 1985); J. Kauppinen, V.-M. Horneman, “Large aperture cube corner interferometer with a resolution of 0.001 cm−1,” Appl. Opt. 30, 2575–2578 (1991).
[CrossRef] [PubMed]

Kyrö, E.

Murty, M. V. R. K.

Appl. Opt.

J. Opt. Soc. Am.

Other

J. Kauppinen, V.-M. Horneman, “Cube corner interferometer with the resolution of about 0.001 cm−1,” in Technical Digest of the Ninth Colloquium on High Resolution Spectroscopy, Riccione, Italy, 16–20 September 1985 (Cooperativa Libraria Universitaria Editrice, Bologna, 1985); J. Kauppinen, V.-M. Horneman, “Large aperture cube corner interferometer with a resolution of 0.001 cm−1,” Appl. Opt. 30, 2575–2578 (1991).
[CrossRef] [PubMed]

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Figures (13)

Fig. 1
Fig. 1

Misaligned cube-corner interferometer. M1 and M2 are fixed and moving cube corners, respectively. BS is a beam splitter. M′2 is the mirror image of M2 formed by BS. S is the radiation source, and S0 is the mirror image of S formed by BS. S′0 and S″0 are the mirror images of S0 formed by M1 and M′2, respectively. The lateral shift of the moving cube corner M2 is d. The angle between the optical and effective axes is β, and x is the optical path difference in the direction of the effective axis.

Fig. 2
Fig. 2

Solid angle Ω in which the extended source S is seen at point P.

Fig. 3
Fig. 3

(A) The Ω effect, which decreases modulation in the interferogram and broadens the spectral line on the wave-number scale. (B) Situation after the optimum truncation of the interferogram at L0.

Fig. 4
Fig. 4

Probable paths of the moving cube corner M2.

Fig. 5
Fig. 5

Origin of the line-shape distortion caused by the aperture (Ω) in a misaligned interferometer. (A) The angle β between the directions of the optical and effective axes is smaller than θ. The effective axis hits the source S with radius R at point A at a distance s from the center of the source. (B) The distorted part of the line is located on the left-hand side (a lower wave number) between ν0 cos(θ + β) and ν0 cos (θ − β). The columns shown by the line shape correspond to the radiation from the parts shown by the source S, respectively.

Fig. 6
Fig. 6

(A) Distorted line shapes with β = 0.1θ, 0.2θ, 0.3θ, and 0.4θ; (B) Distorted line shapes with β = 0.6θ, θ, 1.05θ, 1.5θ, and 2θ.

Fig. 7
Fig. 7

Difference between the zero crossing points xj and xj0, which gives an error ∊j in the optical path difference; xj and xj0 are the zero crossing points of the interferograms that correspond to an asymmetric distorted line and a monochromatic reference line, respectively.

Fig. 8
Fig. 8

Error function ∊(x) in the case of a perfectly aligned interferometer. The bottom line corresponds to a symmetric line, and the top line corresponds to a line with a linear phase error ∊0.

Fig. 9
Fig. 9

Phase error functions ϕ(x) that correspond to the line shapes shown in Fig. 6 (A) with β = 0, 0.1θ, 0.2θ, 0.3θ, and 0.4θ.

Fig. 10
Fig. 10

Optical layout of the cube corner interferometer with a resolution of 0.001 cm−1: M’s, mirrors; W’s, windows; S, source; B’s, beam splitters; L’s, lenses; G, Golay detector; D, light diode; FP, Fabry–Perot interferometer; MO’s, motors; GB, gear box, SH, shutter.

Fig. 11
Fig. 11

Spectra of D2O and CO with a practical resolution of 0.001 cm−1 (only four scans with a bolometer).

Fig. 12
Fig. 12

Error curve ∊(x) derived from the OCS line at 885.84 cm−1 by using the method derived above (β ≠ 0).

Fig. 13
Fig. 13

Error curve ∊(x) derived from the OCS Une at 536.52 cm−1 after realignment of the interferometer (β ≈ 0).

Equations (16)

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Δ ν = ν 0 Ω 2 π ,
L 0 1 . 21 π ν 0 Ω .
E ( α ) = 1 , β < θ, α θ β , E ( α ) = 1 π arccos ( α 2 + β 2 θ 2 2 αβ ) α ( | θ β | , θ + β ) , E ( α ) = 0 elsewhere ,
m ( 0 ) = 2 J 1 ( q ) q ,
q = 4 π ν 0 d ( 0 ) θ 4 π ν 0 d ( 0 ) ( Ω π ) 1 / 2 ,
d ( 0 ) < 1 8 π ( Δ ν ν 0 ) 1 / 2 ,
β < θ ( S / N ) 1 / 2 ,
x i 0 = j 1 / 2 2 ν R , j = 1 , 2 , 3 , .
j = x j x j 0 = x j j 1 / 2 2 ν R .
ν true = ν R d ( x ) d x ν R ,
ν true x = ν R x R = ν R [ x ( x ) ]
ν true = ν R [ 1 ( x ) x ] .
F 1 [ E ( ν ν R ) ] = Re ( x ) + i Im ( x )
ϕ ( x ) = arctan [ Im ( x ) Re ( x ) ] = 2 π ν R ( x ) .
I ( j Δ x ) = k = j N j + N I k g j k ,
g j = sinc [ π 2 ( j j ) ] cos [ π 2 ( j j ) ] ,

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