Abstract

The optimum parameters of operation for a Fabry-Perot spectrometer used in the equidistant equal-noise sampling method have been calculated. The results, expressed in terms of normalized halfwidths at half-height, are: etalon (a*) = 0.078 (0.095), linewidth (dg*) = 0.13 (0.19), and aperture (f*) = 0.10 (0.19) for temperature (wind) determinations. The etalon widths correspond to rather low reflectivities, namely, 0.62 and 0.56. The critical number of samples, required for unambiguous determinations of a measured profile, are found to be equal to 8 and 12. The usefulness of the equal-noise method in absorption measurements is discussed.

© 1985 Optical Society of America

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References

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  1. G. Hernandez, R. J. Sica, G. J. Romick, “Equal-Noise Spectroscopic Measurement,” Appl. Opt. 23, 915 (1984).
    [CrossRef] [PubMed]
  2. G. Hernandez, R. G. Roble, “Direct Measurements of Nighttime Thermospheric Winds and Temperatures 2. Geomagnetic Storms,” J. Geophys. Res. 81, 5173 (1976).
    [CrossRef]
  3. P. Jacquinot, C. Dufour, “Conditions optiques d’emploi des cellules photoélectriques dans les spectrographes et les interféromètres,” J. Res. CRNS 6, 91 (1948).
  4. L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961).
  5. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).
  6. G. Hernandez, “Analytical Description of a Fabry-Perot Spectrometer. 4: Signal Noise Limitations in Data Retrieval; Winds, Temperatures, and Emission Rate,” Appl. Opt. 17, 2967 (1978).
    [CrossRef] [PubMed]
  7. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
  8. G. Hernandez, “Analytical Description of a Fabry-Perot Spectrometer. 5: Optimization for Minimum Uncertainties in the Determination of Doppler Widths and Shifts,” Appl. Opt. 18, 3826 (1979).
    [PubMed]
  9. G. Hernandez, “Analytical Description of a Fabry-Perot Spectrometer. 5: Optimization for Minimum Uncertainties in the Determiniation of Doppler Widths and Shifts; corrigendum,” Appl. Opt. 21, 1538 (1982).
    [CrossRef] [PubMed]

1984 (1)

1982 (1)

1979 (1)

1978 (1)

1976 (1)

G. Hernandez, R. G. Roble, “Direct Measurements of Nighttime Thermospheric Winds and Temperatures 2. Geomagnetic Storms,” J. Geophys. Res. 81, 5173 (1976).
[CrossRef]

1948 (1)

P. Jacquinot, C. Dufour, “Conditions optiques d’emploi des cellules photoélectriques dans les spectrographes et les interféromètres,” J. Res. CRNS 6, 91 (1948).

Bevington, P. R.

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

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Dufour, C.

P. Jacquinot, C. Dufour, “Conditions optiques d’emploi des cellules photoélectriques dans les spectrographes et les interféromètres,” J. Res. CRNS 6, 91 (1948).

Hernandez, G.

Jacquinot, P.

P. Jacquinot, C. Dufour, “Conditions optiques d’emploi des cellules photoélectriques dans les spectrographes et les interféromètres,” J. Res. CRNS 6, 91 (1948).

Parratt, L. G.

L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961).

Roble, R. G.

G. Hernandez, R. G. Roble, “Direct Measurements of Nighttime Thermospheric Winds and Temperatures 2. Geomagnetic Storms,” J. Geophys. Res. 81, 5173 (1976).
[CrossRef]

Romick, G. J.

Sica, R. J.

Appl. Opt. (4)

J. Geophys. Res. (1)

G. Hernandez, R. G. Roble, “Direct Measurements of Nighttime Thermospheric Winds and Temperatures 2. Geomagnetic Storms,” J. Geophys. Res. 81, 5173 (1976).
[CrossRef]

J. Res. CRNS (1)

P. Jacquinot, C. Dufour, “Conditions optiques d’emploi des cellules photoélectriques dans les spectrographes et les interféromètres,” J. Res. CRNS 6, 91 (1948).

Other (3)

L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961).

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

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

Uncertainties of determination of the peak of a Fabry-Perot profile measured by the equal-noise method, expressed as contours of the standard deviation of the equivalent wind (109-m/sec units), as a function of the normalized source and etalon widths with the normalized aperture as a parameter. The last is indicated in the lower right corner of each panel. The increasing value contours are separated by a factor of 21/2 beginning with the marked value.

Fig. 2
Fig. 2

Uncertainties of determination of temperature from a Fabry-Perot profile measured by the equal-noise method. The uncertainties are expressed as the ratio of the standard deviation of determination to the temperature. Same conditions as Fig. 1.

Fig. 3
Fig. 3

Uncertainties of determination of the peak of a Fabry-Perot profile measured with the equal-noise method. Same conditions as Fig. 1, except the uncertainties are for unit time.

Fig. 4
Fig. 4

Uncertainties of determination of temperature from a Fabry-Perot profile measured by the equal-noise method. Same conditions as Fig. 1, except the uncertainties are for unit time.

Fig. 5
Fig. 5

Minimum uncertainties of determination for wind and temperature for an equal-noise measurement of a Fabry-Perot profile as a function of the normalized scanning aperture.

Fig. 6
Fig. 6

Optimum normalized widths for the etalon and source as a function of the normalized aperture for the equal-noise method.

Fig. 7
Fig. 7

Values of the nondimensional quantity f(ak) used to determine the time of measurement of an equal-noise experiment as a function of the normalized source and etalon widths with the normalized aperture as a parameter. The last is indicated in the lower right corner of each panel. The increasing value contours are separated by a factor of 21/2 beginning with the marked value.

Equations (31)

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P ( x ) = Q p ( x ) ,
Q = I A ε τ L 4 π f * n 0 1 [ 1 A ( 1 R ) 1 ] 2 ( 1 R ) ( 1 + R ) 1 ,
p ( x ) = 1 + J + 2 k = 1 k = N a k cos [ 2 π k ( x x 0 ) T 1 ] ,
J = B I 1 .
S = P ( x i ) t i = P ( x 0 ) t 0 = Q p ( x i ) t i ,
P ( x i ) = S t i 1 .
σ υ 2 = [ 8 π 2 c 2 n 0 2 T S k = 1 k = N e k ( a k k ) 2 ] 1 ,
σ τ 2 τ 2 = { 2 π 4 [ ln ( 2 ) ] 2 ( d g * ) 4 T S W τ } 1 ,
e k = [ ( 1 + J ) 2 2 ( 1 + J ) a 2 k + 2 l = 1 l = N a l 2 2 l = 1 l = N a 2 k + l a l l = 1 l = 2 k 1 a 2 k l a l ] 1 .
W τ = k = 1 k = N k 4 w k ( k = 1 k = N k 2 w k ) 2 ( k = 1 k = N w k ) 1 ,
w k = a k 2 [ ( 1 + J ) 2 + 2 ( 1 + J ) a 2 k + 2 l = 1 l = N a l 2 + l = 1 l = 2 k 1 a 2 k l a l + 2 l = 1 l = N a 2 k + l a l ] 1 .
σ υ 2 T S n 0 2 ( d g * ) 2 = [ 8 π 2 c 2 ( d g * ) 2 k = 1 k = N e k ( a k k ) 2 ] 1 ,
σ τ 2 τ 2 T S = [ 2 π 4 [ ln ( 2 ) ] 2 ( d g * ) 4 W τ ] 1 .
t i = p ( x 0 ) [ p ( x i ) ] 1 t 0 ,
i = 0 i = T t i = p ( x 0 ) t 0 i = 0 i = T [ p ( x i ) ] 1 .
S = Q p ( x 0 ) t 0 = Q i = 0 i = T t i { i = 0 i = T [ p ( x i ) ] 1 } 1 .
σ υ 2 I A τ L ε n 0 ( d g * ) 2 i = 0 i = T t i = T 1 i = 0 i = T [ p ( x i ) ] 1 × { 32 π 3 c 2 ( d g * ) 2 f * [ 1 A ( 1 R ) 1 ] 2 × ( 1 R ) ( 1 + R ) 1 k = 1 k = N e k ( a k k ) 2 } 1 ,
σ τ 2 τ 2 I A τ L ε n 0 1 i = 0 i = T t i = T 1 i = 0 i = T [ p ( x i ) ] 1 × { 8 π 5 [ ln ( 2 ) ] 2 ( d g * ) 4 f * [ 1 A ( 1 R ) 1 ] 2 × ( 1 R ) ( 1 + R ) 1 W τ } 1 .
T 1 i = 0 i = T [ p ( x i ) ] 1 = f ( a k ) ,
i = 0 i = T t i = T S Q 1 f ( a k ) ,
Y ( x i ) = Y o exp ( c l k x ) = Y 0 exp { c l k exp [ ( x i x 0 ) 2 / G 2 ] } ,
σ Y ( x i ) 2 = Y ( x i ) .
S = Y ( x i ) t i .
σ t i 2 = t i 2 S 1 .
Y ( x i ) = S t i 1 ,
σ Y ( x i ) 2 = [ Y ( x i ) ] 2 S 1 .
ln [ Y ( x i ) ] = ln ( Y 0 ) c l k exp [ ( x i x 0 ) 2 / G 2 ] ,
σ ln [ Y ( x i ) ] 2 = S 1 .
ln { Y 0 [ Y ( x i ) ] 1 } = c l k exp [ ( x x 0 ) 2 / G 2 ] ,
σ ln { Y 0 [ Y ( x i ) ] 1 } 2 = S 1 ,
σ ln { Y 0 [ Y ( x i ) ] 1 } 2 [ Y ( x i ) ] 1 ,

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