Abstract

A quantitative investigation of the effects of noise inherent in the signal received by the detector of a Fabry-Perot spectrometer has been carried out in detail. The study also includes the effects of the instrumental profile width and the presence of unwanted continuum radiation in the signal, such as background radiation and detector dark current. The results show the spectrum of the noise in the transform plane to have a quantitative description in terms of the instrumental and line shapes as well as the signal strength, or emission rate, of both the line profile and the unwanted radiation. The magnitude and shape of this noise spectrum not only limits in a known quantitative manner the amount of useful information available from a given measured profile, but it also provides information on the profile itself, thus allowing for more information to be obtained from the measurement than would otherwise be possible. A detailed example is given for the retrieval of winds, temperature, and emission rate from line profile measurements, and it is quantitatively shown how the (noise) limited information available from a measured profile appears as uncertainties of determination. These uncertainties are shown to be dependent on the signal strength, more correctly the noise inherent in this signal, as well as the shapes and relative widths of the line and instrumental profiles, with the temperature uncertainty showing the most sensitivity to these factors. Although in some cases the results are qualitatively obvious, their quantitative nature allows the design of experiments for least uncertainty and/or define the limitations inherent in such experiments.

© 1978 Optical Society of America

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References

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  1. G. Hernandez, Appl. Opt. 5, 1745 (1966).
    [CrossRef] [PubMed]
  2. G. Hernandez, Appl. Opt. 9, 1591 (1970).
    [CrossRef] [PubMed]
  3. G. Hernandez, Appl. Opt. 13, 2654 (1974).
    [CrossRef] [PubMed]
  4. J. Connes, Rev. Opt. 40, 231 (1961).
  5. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323ff.
  6. J. O. Stoner, J. Opt. Soc. Am. 56, 370 (1966).
    [CrossRef]
  7. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 108ff.
  8. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 189ff.
  9. G. Hernandez, O. A. Mills, Appl. Opt. 12, 126 (1973).
    [CrossRef] [PubMed]
  10. J. Gagné, J. Saint-Dizier, M. Picard, Appl. Opt. 13, 581 (1974).
    [CrossRef] [PubMed]
  11. P. B. Hays, R. G. Roble, Appl. Opt. 10, 193 (1971).
    [CrossRef] [PubMed]

1974 (2)

1973 (1)

1971 (1)

1970 (1)

1966 (2)

1961 (1)

J. Connes, Rev. Opt. 40, 231 (1961).

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 108ff.

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323ff.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 189ff.

Connes, J.

J. Connes, Rev. Opt. 40, 231 (1961).

Gagné, J.

Hays, P. B.

Hernandez, G.

Mills, O. A.

Picard, M.

Roble, R. G.

Saint-Dizier, J.

Stoner, J. O.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323ff.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

Rev. Opt. (1)

J. Connes, Rev. Opt. 40, 231 (1961).

Other (3)

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323ff.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 108ff.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 189ff.

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Equations (40)

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g ( x ) = T 1 [ 1 + 2 k = 1 a k cos ( 2 π k x T 1 ) ] .
f ( x ) = B + Q g ( x ) .
Q = Q p g ( O ) 1 ; g ( O ) = T 1 ( 1 + 2 k = 1 a k ) .
b n = 2 T 1 Q a n = 2 [ T g ( O ) ] 1 Q p a n ,
b o = 2 T 1 Q ( 1 + B T Q 1 ) = 2 T 1 [ B T + Q p g ( O ) 1 ] .
σ b n 2 = x = 1 T σ f ( x ) 2 ( b n f ( x ) ) 2 .
σ b n 2 = 2 T 2 Q [ B T Q 1 + ( 1 + a 2 n ) ] = 2 T 2 Q ( B T Q 1 + 1 ) + T 1 ( 2 T 1 Q a 2 n )
= 2 T 2 [ B T + Q p g ( O ) 1 ( 1 + a 2 n ) ] .
lim n σ b n 2 = 2 T 2 Q ( B T Q 1 + 1 ) = 2 T 2 [ B T + Q p g ( O ) 1 ] .
| b n σ b n 1 | 2.0.
| b n σ b n 1 | | ( 2 Q ) 1 / 2 a n | = | [ 2 Q p g ( O ) ] 1 / 2 a n | .
g ( x ) = T 1 { 1 + 2 k = 1 a k cos [ 2 π k T 1 ( x x o ) }
= T 1 [ 1 + 2 k = 1 a k cos ( 2 π k T 1 x o ) cos ( 2 π k T 1 x ) + 2 k = 1 a k sin ( 2 π k T 1 x o ) sin ( 2 π k T 1 x ) ] .
s n = 2 Q T 1 a n sin ( 2 π n T 1 x o ) ,
c n = 2 Q T 1 a n cos ( 2 π n T 1 x o ) .
b n = + ( s n 2 + c n 2 ) 1 / 2 = 2 Q T 1 | a n | .
x o n = T ( 2 π n ) 1 tan 1 ( s n c n 1 ) .
σ b n 2 = 2 T 2 Q [ B T Q 1 + 1 + a 2 n cos 2 ( 4 π n x o n T 1 ) ] .
σ x o n 2 = ( 2 Q T 2 ) 1 ( 2 π n a n ) 2 [ B T Q 1 + 1 + a 2 n cos 2 ( 4 π n T 1 x o ) ] = σ b n 2 ( 2 π n a n ) 2 ( 2 Q T 2 ) 2 .
σ x o 2 = ( 2 Q T 2 ) 1 ( 2 π n a n ) 2 .
X O = n = 1 N x o n σ x o n 2 ( n = 1 N σ x o n 2 ) 1 ,
σ x o 2 = ( n = 1 N σ x o n 2 ) 1 .
a n = d n exp ( γ τ n 2 ) .
c n = 2 T 1 x = 1 T f ( x ) cos ( 2 π n T 1 x ) ,
σ 2 c n = 4 T 2 x = 1 T f ( x ) cos 2 ( 2 π n T 1 x ) .
b n = I o a n = I 0 d n exp ( γ τ n 2 ) = 2 T 1 Q d n exp ( γ τ n 2 ) ,
U n = ln ( b n d n 1 ) = ln I o n 2 γ τ .
τ = γ 1 [ ( n N U n w n n N n 2 w n ) ( n N w n ) 1 n N U n n 2 w n ] × [ n N n 4 w n ( n N n 2 w n ) 2 ( n N w n ) 1 ] 1 .
I o = exp { n N U n w n ( n N w n ) 1 + n N n 2 w n ( n N w n ) 1 × [ n N n 4 w n ( n N n 2 w n ) 2 ( n N w n ) 1 ] 1 × [ n N U n w n n N n 2 w n ( n N w n ) 1 n N U n n 2 w n ] } ,
σ τ 2 = γ 2 [ n N n 4 w n ( n N n 2 w n ) 2 ( n N w n ) 1 ] 1 ,
σ I o 2 = I o 2 { ( n N w n ) 1 + [ n N n 2 w n ( n N w n ) 1 ] 2 × [ n N n 4 w n ( n N n 2 w n ) 2 ( n N w n ) 1 ] 1 } .
σ τ 2 = α τ 2.5 Q 1 ( B T Q 1 + 1 ) .
σ τ 2 = α ( P + 1 ) 2.5 τ 2.5 Q 1 ( B T Q 1 + 1 ) .
σ τ 2 = α ( P + 1 ) 1.5 τ 2.5 Q 1 ( B T Q 1 + 1 ) .
σ Q 2 = α ( P + 1 ) 1 / 2 τ 1 / 2 Q ( B T Q 1 + 1 ) ,
σ x o 2 = α ( P + 1 ) 1 / 2 τ 1.5 Q 1 ( B T Q 1 + 1 ) .
σ Q 2 = α ( P + 1 ) 1.5 τ 1 / 2 Q ( B T Q 1 + 1 ) ,
σ x o 2 = α ( P + 1 ) 1 / 2 τ 1.5 Q 1 ( B T Q 1 + 1 ) .
σ τ 2 = α ( B T Q 1 + 1 ) ( Q N 5 ) 1 ,
σ Q 2 = α Q ( B T Q 1 + 1 ) ( T 2 N ) 1 .

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