Abstract

A new computational method for calculating and correcting the errors of the optical path difference in Fourier spectrometers is presented. This method only requires an one-sided interferogram and a single well-separated line in the spectrum. The method also cancels out the linear phase error. The practical theory of the method is included, and an example of the progress of the method is illustrated by simulations. The method is also verified by several simulations in order to estimate its usefulness and accuracy. An example of the use of this method in practice is also given.

© 1978 Optical Society of America

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References

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  1. M. L. Forman et al., J. Opt. Soc. Am. 56, 59 (1966).
    [CrossRef]
  2. M. Mertz, Infrared Phys. 7, 17 (1967).
    [CrossRef]
  3. H. Sakai et al., J. Opt. Soc. Am. 58, 84 (1968).
    [CrossRef]
  4. R. B. Sanderson, E. E. Bell, Appl. Opt. 12, 266 (1973).
    [CrossRef] [PubMed]
  5. J. Kauppinen, Infrared Phys. 16, 359 (1976).
    [CrossRef]
  6. J. Connes, V. Nozal, J. Phys. Rad. 22, 359 (1961).
    [CrossRef]
  7. V. S. Bukreev, V. A. Vagin, J. Appl. Spectrosc. 23, 1396 (1975).
    [CrossRef]
  8. J. Connes, Rev. Opt. 40, 74 (1961).
  9. J. Kauppinen, Appl. Opt. 14, 1987 (1975).
    [CrossRef] [PubMed]
  10. J. Kauppinen, Acta Univ. Oul. A38 (1975).

1976 (1)

J. Kauppinen, Infrared Phys. 16, 359 (1976).
[CrossRef]

1975 (3)

V. S. Bukreev, V. A. Vagin, J. Appl. Spectrosc. 23, 1396 (1975).
[CrossRef]

J. Kauppinen, Acta Univ. Oul. A38 (1975).

J. Kauppinen, Appl. Opt. 14, 1987 (1975).
[CrossRef] [PubMed]

1973 (1)

1968 (1)

1967 (1)

M. Mertz, Infrared Phys. 7, 17 (1967).
[CrossRef]

1966 (1)

1961 (2)

J. Connes, Rev. Opt. 40, 74 (1961).

J. Connes, V. Nozal, J. Phys. Rad. 22, 359 (1961).
[CrossRef]

Bell, E. E.

Bukreev, V. S.

V. S. Bukreev, V. A. Vagin, J. Appl. Spectrosc. 23, 1396 (1975).
[CrossRef]

Connes, J.

J. Connes, V. Nozal, J. Phys. Rad. 22, 359 (1961).
[CrossRef]

J. Connes, Rev. Opt. 40, 74 (1961).

Forman, M. L.

Kauppinen, J.

J. Kauppinen, Infrared Phys. 16, 359 (1976).
[CrossRef]

J. Kauppinen, Appl. Opt. 14, 1987 (1975).
[CrossRef] [PubMed]

J. Kauppinen, Acta Univ. Oul. A38 (1975).

Mertz, M.

M. Mertz, Infrared Phys. 7, 17 (1967).
[CrossRef]

Nozal, V.

J. Connes, V. Nozal, J. Phys. Rad. 22, 359 (1961).
[CrossRef]

Sakai, H.

Sanderson, R. B.

Vagin, V. A.

V. S. Bukreev, V. A. Vagin, J. Appl. Spectrosc. 23, 1396 (1975).
[CrossRef]

Acta Univ. Oul. (1)

J. Kauppinen, Acta Univ. Oul. A38 (1975).

Appl. Opt. (2)

Infrared Phys. (2)

M. Mertz, Infrared Phys. 7, 17 (1967).
[CrossRef]

J. Kauppinen, Infrared Phys. 16, 359 (1976).
[CrossRef]

J. Appl. Spectrosc. (1)

V. S. Bukreev, V. A. Vagin, J. Appl. Spectrosc. 23, 1396 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. Rad. (1)

J. Connes, V. Nozal, J. Phys. Rad. 22, 359 (1961).
[CrossRef]

Rev. Opt. (1)

J. Connes, Rev. Opt. 40, 74 (1961).

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Figures (8)

Fig. 1
Fig. 1

An undistorted and a distorted line and the corresponding Fourier transforms in the z domain.

Fig. 2
Fig. 2

Graphical illustrations of a shifting of the wavenumber origin.

Fig. 3
Fig. 3

Fourier transforms of the shifted reference lines Rs(ν) and R s ( ν ).

Fig. 4
Fig. 4

Relationship between the actual Δ(z) and Δc(z) used in the convolution.

Fig. 5
Fig. 5

Lines at 300 cm−1 and 500 cm−1 transformed with triangular apodization from correct (upper) and incorrect (lower) sampling.

Fig. 6
Fig. 6

Example of the use of the correction method starting from the distorted line at 500 cm−1 shown in Fig. 5.

Fig. 7
Fig. 7

Simulations of the correction method for a parabolic and a sinusoidal error function.

Fig. 8
Fig. 8

Example of the practical application of the method to correct the distortions in a water vapor spectrum in the far ir.

Equations (37)

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E ( ν ) = S ( ν ) + R ( ν ) = I ( x ) exp ( i 2 π ν x ) d x = 1 [ I ( x ) ] ,
I ( x ) = E ( ν ) exp ( i 2 π ν x ) d ν = [ E ( ν ) ] .
x j = j h + Δ j = ( j + j ) h ,
x = z + Δ ( z ) ,
I ( x ) = E ( ν ) exp { i 2 π ν [ z + Δ ( z ) ] } d ν = z { E ( ν ) exp [ i 2 π ν Δ ( z ) ] } ,
E ( ν ) = S ( ν ) + R ( ν ) = I [ z + Δ ( z ) ] exp ( i 2 π ν z ) d x = z 1 [ I ( x ) ] .
R ( ν ) = exp [ L ( ν ν o ) d ] 1 I σ d / π σ 2 + ( ν ν o ) 2 ,
z [ R ( ν ) ] = δ ( z ) 2 I d exp ( 2 π σ | z | ) cos 2 π ν o z ,
ν = ν o [ 1 + d Δ ( z ) d z ]
z [ R ( ν ) ] = δ ( z ) 2 I d exp ( 2 π σ | z | ) cos 2 π ν z .
ϕ k = 2 π ν o ( x k z k ) = 2 π ν o Δ ( z k ) = 2 π ν o [ 1 + d Δ ( z ) d z z k ] ( z k z k ) ( z k z k ) π z k + 1 z k
Δ ( z k ) = [ 1 + d Δ ( z ) d z z k ] ( z k z k ) .
z [ R s ( ν ) ] = δ ( z ) 2 I d exp ( 2 π σ | z | ) cos 2 π ( ν o ν s ) z .
z k = ( k + ½ ) / 2 ( ν o ν s ) = ( k + ½ ) D h = j h ,
z [ R s ( ν ) ] = δ ( z ) 2 I d exp ( 2 π σ | z | ) cos 2 π { ν o [ 1 + d Δ ( z ) d z ] ν s } z ,
ϕ k = 2 π ν o Δ ( z k ) = ϕ k ( z k z k ) π z k + 1 z k = μ k π z k + 1 z k ,
z k + 1 z k = ( 2 { ν o [ 1 + d Δ ( z ) d z z k ] ν s } ) 1 ,
Δ ( z k ) = k h [ ν o ν s ν o + d Δ ( z ) d z z k ] μ k = ν max μ k D ν o [ 1 + D ν o ν max d Δ ( z ) d z z k ] .
ν max ν o D | d Δ ( z ) d z | max
Δ ( z ) k = k h ν max D ν o μ k .
ν o { 1 + [ d Δ ( z ) d z ] min } ν s to ν o { 1 + [ d Δ ( z ) d z ] max } ν s
E ( ν ) = I ( x ) exp ( i 2 π ν x ) d x
I ( z ) = ν max ν max E ( ν ) exp ( i 2 π ν z ) d ν = ν max ν max I ( x ) exp [ i 2 π ν ( z x ) ] d x d ν = I ( x ) { ν max ν max exp [ i 2 π ν ( z x ) ] d ν } d x = I ( x ) F ( z x ) d x = [ I ( x ) * F ( x ) ] z ,
F ( x ) = ν max ν max exp ( i 2 π ν x ) d ν = 2 ν max sinc ( π ν max x ) cos ) π ν max x ) ,
I ( z ) = I ( x ) δ ( z x ) d x = I ( z ) .
d x = d z + d Δ ( z ) = d z [ 1 + d Δ ( z ) d z ] d z ,
I ( z ) I [ z + Δ ( z ) ] 2 ν max sinc { π ν max [ z z Δ ( z ) ] } cos { π ν max [ z z Δ ( z ) ] } d z .
I j i = j d j + d I i sinc [ π 2 ( j i i ) ] cos [ π 2 ( j i i ) ] ,
I c ( z ) = z { E ( ν ) exp [ i 2 π ν Δ ( z ) ] exp [ i 2 π ν Δ c ( z ) ] } = z { E ( ν ) exp [ i 2 π ν ζ ( z ) ] } .
| d Δ ( z ) d z |
2 π ν max | ζ ( z ) | max 2 π ν max | d Δ ( z ) d z | max D h 2 0.01 π .
D 0.02 / | d Δ ( z ) d z | max .
[ 1 + d Δ ( z ) d z j h ] .
I ( x ) = cos 2 π 100 x + cos 2 π 300 x + cos 2 π 500 x + cos 2 π 700 x ,
x j = j h j = 0 , 1 , 2 , ... , 999 x j = j h + 0.002 ( j 999 ) h j = 1000 , 1001 , ... , 2999 x j = j h + 4 h j = 3000 , 3001 , ... , 3999 x j = ( j + 4 ) h 0.001 ( j 3999 ) h j = 4000 , 4001 , ... , 7999 .
D ν o ν max | d Δ ( z ) d z | max = 0.02 .
x j = j h + ( j / 5000 ) 2 h ( parabolic ) x j = j h + 0.3 sin ( π j / 3000 ) ( sinusoidal ) . }

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