Abstract

The interferogram equations for a Michelson interferometer operated with a mosaic array of detectors are derived. The effects on the instrumental line shape function (ILF) due to an individual detector field subtense and its displacement from the optical axis of the interferometer have been numerically computed from the interferogram equations. Theoretical predictions of the dependence of the ILF on various system parameters are presented as well as comparisons of the theoretical ILF with those measured in the laboratory.

© 1982 Optical Society of America

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References

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  1. E. V. Loewenstein, “Fourier Spectroscopy: An Introduction,” in Proceedings, Aspen International Conference on Fourier Spectroscopy, 1970, G. A. Vanasse, A. T. Stair, J. D. Baker, Eds. AFCRF-71-0019.
  2. R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).
  3. E. R. Peck, J. Opt. Soc. Am. 45, 931 (1955).
    [CrossRef]
  4. M. V. R. K. Murty, J. Opt. Soc. Am. 50, 7 (1960).
    [CrossRef]
  5. J. E. Stewart, J Opt. Soc. Am. 58, 434 (1968).
    [CrossRef]
  6. P. F. Parshin, Opt. Spektrosk. 31, 1017 (1971) [Opt. Spectrosc. 31, 548 (1971)].
  7. M. A. Gershun, N. I. Pivovar, Sov. J. Opt. Technol. 42, 1 (1975).
  8. G. Guelachvili, in Spectrometric Techniques, Vol. 2, G. A. Vanasse, Ed. (Academic, Press, New York, 1981), pp. 38–40.

1980

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

1975

M. A. Gershun, N. I. Pivovar, Sov. J. Opt. Technol. 42, 1 (1975).

1971

P. F. Parshin, Opt. Spektrosk. 31, 1017 (1971) [Opt. Spectrosc. 31, 548 (1971)].

1968

J. E. Stewart, J Opt. Soc. Am. 58, 434 (1968).
[CrossRef]

1960

1955

Andrada, T.

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

Billingsley, F.

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

Cook, F.

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

Gershun, M. A.

M. A. Gershun, N. I. Pivovar, Sov. J. Opt. Technol. 42, 1 (1975).

Grieder, W. F.

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

Guelachvili, G.

G. Guelachvili, in Spectrometric Techniques, Vol. 2, G. A. Vanasse, Ed. (Academic, Press, New York, 1981), pp. 38–40.

Loewenstein, E. V.

E. V. Loewenstein, “Fourier Spectroscopy: An Introduction,” in Proceedings, Aspen International Conference on Fourier Spectroscopy, 1970, G. A. Vanasse, A. T. Stair, J. D. Baker, Eds. AFCRF-71-0019.

Murphy, R. E.

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

Murty, M. V. R. K.

Parshin, P. F.

P. F. Parshin, Opt. Spektrosk. 31, 1017 (1971) [Opt. Spectrosc. 31, 548 (1971)].

Peck, E. R.

Pivovar, N. I.

M. A. Gershun, N. I. Pivovar, Sov. J. Opt. Technol. 42, 1 (1975).

Stewart, J. E.

J. E. Stewart, J Opt. Soc. Am. 58, 434 (1968).
[CrossRef]

Yap, B. K.

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

J Opt. Soc. Am.

J. E. Stewart, J Opt. Soc. Am. 58, 434 (1968).
[CrossRef]

J. Opt. Soc. Am.

Opt. Spektrosk.

P. F. Parshin, Opt. Spektrosk. 31, 1017 (1971) [Opt. Spectrosc. 31, 548 (1971)].

Proc. Soc. Photo-Opt. Instrum. Eng.

R. E. Murphy, T. Andrada, F. Cook, F. Billingsley, W. F. Grieder, B. K. Yap, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 152 (1980).

Sov. J. Opt. Technol.

M. A. Gershun, N. I. Pivovar, Sov. J. Opt. Technol. 42, 1 (1975).

Other

G. Guelachvili, in Spectrometric Techniques, Vol. 2, G. A. Vanasse, Ed. (Academic, Press, New York, 1981), pp. 38–40.

E. V. Loewenstein, “Fourier Spectroscopy: An Introduction,” in Proceedings, Aspen International Conference on Fourier Spectroscopy, 1970, G. A. Vanasse, A. T. Stair, J. D. Baker, Eds. AFCRF-71-0019.

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Figures (18)

Fig. 1
Fig. 1

Central fringe pattern and focal plane configuration.

Fig. 2
Fig. 2

Interferogram moduli for detector 1.

Fig. 3
Fig. 3

Interferogram moduli for detector 2.

Fig. 4
Fig. 4

Interferogram moduli for detector 3.

Fig. 5
Fig. 5

Interferogram moduli for detector 4.

Fig. 6
Fig. 6

Interferogram moduli for circular detector matched to fringe.

Fig. 7
Fig. 7

Interferogram moduli for a small detector.

Fig. 8
Fig. 8

ILF components for detector 4, Xn = 1.0. The ILF is shown as a function of relative wave numbers measured in units of X o - 1.

Fig. 9
Fig. 9

ILF with Xn = 1.0; see caption of Fig. 8.

Fig. 10
Fig. 10

ILF with Xn = 2.0; see caption of Fig. 8.

Fig. 11
Fig. 11

ILF with Xn = 5.0; see caption of Fig. 8.

Fig. 12
Fig. 12

ILF with Xn = 10.0; see caption of Fig. 8.

Fig. 13
Fig. 13

ILF fidelity for various maximum OPDs.

Fig. 14
Fig. 14

ILF spectral resolution for various maximum OPDs.

Fig. 15
Fig. 15

Detector FOV effects on spectral shift.

Fig. 16
Fig. 16

Laboratory instrument field response.

Fig. 17
Fig. 17

Calculated and measured spectral resolution.

Fig. 18
Fig. 18

Calculated and measured spectral shift.

Equations (18)

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I ( σ , x ) = ( A / 2 ) B ( σ ) cos ( 2 π σ x cos θ ) d Q ,
I ( σ , x ) = ( A / 2 ) B ( σ ) - b 1 b 2 - a 1 a 2 cos { 2 π σ x [ 1 - ( a 2 + b 2 ) / 2 ] } d a d b .
I ( σ , x ) = A B ( σ ) 4 σ x [ ( F 1 F 2 - F 3 F 4 ) cos 2 π σ x + ( F 1 F 4 + F 2 F 3 ) sign ( x ) sin 2 π σ x ] ,
F 1 = C ( a 1 2 σ x ) + C ( a 2 2 σ x ) , F 2 = C ( b 1 2 σ x ) + C ( b 2 2 σ x ) , F 3 = S ( a 1 2 σ x ) + S ( a 2 2 σ x ) , F 4 = S ( b 1 2 σ x ) + S ( b 2 2 σ x ) ,
sign ( x ) = { + 1 for positive x , - 1 for negative x , C ( z ) = 0 z cos ( π y 2 / 2 ) d y ,             S ( z ) = 0 z sin ( π y 2 / 2 ) d y
I ( σ , x ) = ( A / 2 ) B ( σ ) D [ H c cos 2 π σ x + H s sign ( x ) sin 2 π σ x ] ,
H c = ( F 1 F 2 - F 3 F 4 ) / ( 2 D σ x ) ,             H s = ( F 1 F 4 + F 2 F 3 ) / ( 2 D σ x ) ,
a o = 1 / ( σ o x o ) 1 / 2 ,
x n = x / x o ,             a n = a / a o , σ n = σ / σ o ,             b n = b / a o , D n = ( a 1 n + a 2 n ) ( b 1 n + b 2 n ) .
I n ( σ n , x n ) = H c cos 2 π σ n x n + H s sign ( x ) sin 2 π σ n x n ,
H c = F 1 F 2 - F 3 F 4 2 ( a 1 n + a 2 n ) ( b 1 n + b 2 n ) σ x x n , H s = F 1 F 4 + F 2 F 3 2 ( a 1 n + a 2 n ) ( b 1 n + b 2 n ) σ x x n ,
F 1 = C [ a 1 n ( 2 σ n x n ) 1 / 2 ] + C [ a 2 n ( 2 σ n x n ) 1 / 2 ] .
I n c ( σ x , x n ) = H c c cos ( π σ n x n / 2 ) + H s c sign ( x ) sin ( π σ n x n / 2 ) ,
H c c = sin ( π x n σ n / 2 ) ( π x n σ n / 2 ) cos ( π x n σ n / 2 ) , H s c = sin ( π x n σ n / 2 ) ( π x n σ n / 2 ) sin ( π x n σ n / 2 ) .
F 1 F 2 = - a 1 n a 2 n - b 1 n b 2 n V ( Q n ) cos ( π σ n x n a n 2 ) cos ( π σ n x n b n 2 ) d Q n , F 3 F 4 = - a 1 n a 2 n - b 1 n b 2 n V ( Q n ) sin ( π σ n x n a n 2 ) sin ( π σ n x n b n 2 ) d Q n , F 1 F 4 = - a 1 n a 2 n - b 1 n b 2 n V ( Q n ) sin ( π σ n x n a n 2 ) cos ( π σ n x n b n 2 ) d Q n , F 2 F 3 = - a 1 n a 2 n - b 1 n b 2 n V ( Q n ) cos ( π σ n x n a n 2 ) sin ( π σ n x n b n 2 ) d Q n ,
Q = ( a 2 + b 2 ) 1 / 2 , Q n = ( a n 2 + b n 2 ) 1 / 2 , L = separation between apertures times Q ,
V ( Q ) = ( π - t + s D 2 2 D 1 2 - 2 L D 1 sin t ) / π ,
t = arccos ( D 2 2 - D 1 2 - 4 L 2 4 D 1 L ) , s = arcsin [ sin ( t ) D 1 / D 2 ] .

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