Abstract

A method for the optimization of a Fabry–Perot interferometer (FPI) designed for the measurement of Doppler-broadened emission lines is presented. Assuming that the measurement values (counts) are Poisson distributed, the likelihood function is derived. Maximization of the likelihood function yields optimal estimates of temperature T, wind velocity V, and line intensity I0 and is accomplished by an iterative procedure of the Newton type. An optimal FPI design is obtained by a minimization of the calculated estimation errors ΔT and ΔV that represent the measurement quality. The method is appropriate for short-time measurements of weak emission lines, especially in space applications.

© 1982 Optical Society of America

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References

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  1. J. E. Blamont and J. M. Luton, “Geomagnetic effect on the neutral temperature of the F region during the magnetic storm of September 1969,” J. Geophys. Res. 77, 3534–3556 (1972).
    [Crossref]
  2. G. Hernandez and R. G. Roble, “Direct measurements of nighttime thermospheric winds and temperatures 1. Seasonal variations during geomagnetic quiet periods,” J. Geophys. Res. 81, 2065–2074 (1976).
    [Crossref]
  3. P. B. Hays and J. W. Meriwether, “Nighttime thermospheric winds at high latitudes,” J. Geophys. Res. 84, 1905–1913 (1979).
    [Crossref]
  4. T. D. Cocks and F. Jacka, “Daytime thermospheric temperatures, wind velocities and emission intensities derived from ground based observations of the OI 630 nm airglow line profile,” J. Atmos. Terr. Phys. 41, 409–415(1979).
    [Crossref]
  5. G. Hernandez, “Analytical description of a Fabry–Perot spectrometer. 4: Signal noise limitations in data retrieval; winds, temperature and emission rate,” Appl. Opt. 17, 2967–2972 (1978).
    [Crossref] [PubMed]
  6. 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, 3826–3834 (1979).
    [PubMed]
  7. J. F. Walkup and J. W. Goodman, “Limitations of fringe-parameter estimation at low light levels,” J. Opt. Soc. Am. 63, 399–407 (1973).
    [Crossref]
  8. P. Jacquinot, “New developments in interference spectroscopy,” Rep. Progr. Phys. 23, 267–312 (1960).
    [Crossref]
  9. G. Hernandez, “Analytical description of a Fabry–Perot photoelectric spectrometer,” Appl. Opt. 5, 1745–1748 (1966).
    [Crossref] [PubMed]
  10. In a multiple-zone aperture variation of this design making use of N0additional rin gs of equal space angle Ω, the signal may be enhanced by a factor N0+ 1; see, e.g., S. Okano, J. S. Kim, and T. Ichikawa, “Design of a multiple-zone aperture and application to a Fabry–Perot interferometer,” Appl. Opt. 19, 1622–1629 (1980).
    [Crossref] [PubMed]
  11. P. B. Hays, R. G. Roble, G. R. Canigman, A. F. Nagy, and D. Rees, Report on Active and Planned Spacecraft and Experiments (National Aeronautics and Space Administration, Washington, D.C., August1979).
  12. H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968), pp. 52–86.
  13. D. M. Himmelblau, Applied Nonlinear Programming (McGraw-Hill, New York, 1972).
  14. R. Zielinski, Erzeugung von Zufallszahlen (VEB Fachbuchverlag, Leipzig, 1972).
  15. R. Ludwig, Methoden der Fehler- und Ausgleichsrechnung (Vieweg, Braunschweig, 1969).
    [Crossref]

1980 (1)

1979 (3)

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, 3826–3834 (1979).
[PubMed]

P. B. Hays and J. W. Meriwether, “Nighttime thermospheric winds at high latitudes,” J. Geophys. Res. 84, 1905–1913 (1979).
[Crossref]

T. D. Cocks and F. Jacka, “Daytime thermospheric temperatures, wind velocities and emission intensities derived from ground based observations of the OI 630 nm airglow line profile,” J. Atmos. Terr. Phys. 41, 409–415(1979).
[Crossref]

1978 (1)

1976 (1)

G. Hernandez and R. G. Roble, “Direct measurements of nighttime thermospheric winds and temperatures 1. Seasonal variations during geomagnetic quiet periods,” J. Geophys. Res. 81, 2065–2074 (1976).
[Crossref]

1973 (1)

1972 (1)

J. E. Blamont and J. M. Luton, “Geomagnetic effect on the neutral temperature of the F region during the magnetic storm of September 1969,” J. Geophys. Res. 77, 3534–3556 (1972).
[Crossref]

1966 (1)

1960 (1)

P. Jacquinot, “New developments in interference spectroscopy,” Rep. Progr. Phys. 23, 267–312 (1960).
[Crossref]

Blamont, J. E.

J. E. Blamont and J. M. Luton, “Geomagnetic effect on the neutral temperature of the F region during the magnetic storm of September 1969,” J. Geophys. Res. 77, 3534–3556 (1972).
[Crossref]

Canigman, G. R.

P. B. Hays, R. G. Roble, G. R. Canigman, A. F. Nagy, and D. Rees, Report on Active and Planned Spacecraft and Experiments (National Aeronautics and Space Administration, Washington, D.C., August1979).

Cocks, T. D.

T. D. Cocks and F. Jacka, “Daytime thermospheric temperatures, wind velocities and emission intensities derived from ground based observations of the OI 630 nm airglow line profile,” J. Atmos. Terr. Phys. 41, 409–415(1979).
[Crossref]

Goodman, J. W.

Hays, P. B.

P. B. Hays and J. W. Meriwether, “Nighttime thermospheric winds at high latitudes,” J. Geophys. Res. 84, 1905–1913 (1979).
[Crossref]

P. B. Hays, R. G. Roble, G. R. Canigman, A. F. Nagy, and D. Rees, Report on Active and Planned Spacecraft and Experiments (National Aeronautics and Space Administration, Washington, D.C., August1979).

Hernandez, G.

Himmelblau, D. M.

D. M. Himmelblau, Applied Nonlinear Programming (McGraw-Hill, New York, 1972).

Ichikawa, T.

Jacka, F.

T. D. Cocks and F. Jacka, “Daytime thermospheric temperatures, wind velocities and emission intensities derived from ground based observations of the OI 630 nm airglow line profile,” J. Atmos. Terr. Phys. 41, 409–415(1979).
[Crossref]

Jacquinot, P.

P. Jacquinot, “New developments in interference spectroscopy,” Rep. Progr. Phys. 23, 267–312 (1960).
[Crossref]

Kim, J. S.

Ludwig, R.

R. Ludwig, Methoden der Fehler- und Ausgleichsrechnung (Vieweg, Braunschweig, 1969).
[Crossref]

Luton, J. M.

J. E. Blamont and J. M. Luton, “Geomagnetic effect on the neutral temperature of the F region during the magnetic storm of September 1969,” J. Geophys. Res. 77, 3534–3556 (1972).
[Crossref]

Meriwether, J. W.

P. B. Hays and J. W. Meriwether, “Nighttime thermospheric winds at high latitudes,” J. Geophys. Res. 84, 1905–1913 (1979).
[Crossref]

Nagy, A. F.

P. B. Hays, R. G. Roble, G. R. Canigman, A. F. Nagy, and D. Rees, Report on Active and Planned Spacecraft and Experiments (National Aeronautics and Space Administration, Washington, D.C., August1979).

Okano, S.

Rees, D.

P. B. Hays, R. G. Roble, G. R. Canigman, A. F. Nagy, and D. Rees, Report on Active and Planned Spacecraft and Experiments (National Aeronautics and Space Administration, Washington, D.C., August1979).

Roble, R. G.

G. Hernandez and R. G. Roble, “Direct measurements of nighttime thermospheric winds and temperatures 1. Seasonal variations during geomagnetic quiet periods,” J. Geophys. Res. 81, 2065–2074 (1976).
[Crossref]

P. B. Hays, R. G. Roble, G. R. Canigman, A. F. Nagy, and D. Rees, Report on Active and Planned Spacecraft and Experiments (National Aeronautics and Space Administration, Washington, D.C., August1979).

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968), pp. 52–86.

Walkup, J. F.

Zielinski, R.

R. Zielinski, Erzeugung von Zufallszahlen (VEB Fachbuchverlag, Leipzig, 1972).

Appl. Opt. (4)

J. Atmos. Terr. Phys. (1)

T. D. Cocks and F. Jacka, “Daytime thermospheric temperatures, wind velocities and emission intensities derived from ground based observations of the OI 630 nm airglow line profile,” J. Atmos. Terr. Phys. 41, 409–415(1979).
[Crossref]

J. Geophys. Res. (3)

J. E. Blamont and J. M. Luton, “Geomagnetic effect on the neutral temperature of the F region during the magnetic storm of September 1969,” J. Geophys. Res. 77, 3534–3556 (1972).
[Crossref]

G. Hernandez and R. G. Roble, “Direct measurements of nighttime thermospheric winds and temperatures 1. Seasonal variations during geomagnetic quiet periods,” J. Geophys. Res. 81, 2065–2074 (1976).
[Crossref]

P. B. Hays and J. W. Meriwether, “Nighttime thermospheric winds at high latitudes,” J. Geophys. Res. 84, 1905–1913 (1979).
[Crossref]

J. Opt. Soc. Am. (1)

Rep. Progr. Phys. (1)

P. Jacquinot, “New developments in interference spectroscopy,” Rep. Progr. Phys. 23, 267–312 (1960).
[Crossref]

Other (5)

P. B. Hays, R. G. Roble, G. R. Canigman, A. F. Nagy, and D. Rees, Report on Active and Planned Spacecraft and Experiments (National Aeronautics and Space Administration, Washington, D.C., August1979).

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968), pp. 52–86.

D. M. Himmelblau, Applied Nonlinear Programming (McGraw-Hill, New York, 1972).

R. Zielinski, Erzeugung von Zufallszahlen (VEB Fachbuchverlag, Leipzig, 1972).

R. Ludwig, Methoden der Fehler- und Ausgleichsrechnung (Vieweg, Braunschweig, 1969).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Partition of the interference pattern: NP = 4, NR = 1, and N = 12.

Fig. 2
Fig. 2

Isolines of L function.

Fig. 3
Fig. 3

Estimation errors as functions of the FPI plate separation d: R = 0.7;—is ΔT; - - - - -is ΔV; and -·-·- is ΔI0/I0.

Fig. 4
Fig. 4

The variation of ΔT with the line intensity: T = 1000 K; —, Istray = 0; - - -, Istray = 1 R/cm−1; -·-, Istray = 10 R/cm−1; and …, Istray = 100 R/cm−1).

Fig. 5
Fig. 5

Results of Monte Carlo calculation of T and V: ○, true value.

Fig. 6
Fig. 6

Variations of ΔT and ΔV with the space angle Ω: d = 1.5 cm; —, ΔT; and - - -, ΔV.

Equations (39)

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I ( σ ) = I 0 χ 2 π exp [ - ( σ - σ L ) 2 / 2 χ 2 ] + I stray ,
J ( ϑ ) = μ opt F 0 H ( σ ) A ( σ , ϑ ) I ( σ ) d σ [ photons / ( sec - sr ) ] ,
A ( σ , ϑ ) = ( 1 - R - A abs ) 2 1 - R 2 × { 1 + 2 ν = 1 R ν cos [ ν ( 4 π σ d cos ϑ + ϕ ) ] } .
H ( σ ) = exp [ - ( σ - σ * ) 2 / 2 γ 2 ] ,
J ( ϑ ) = J line ( ϑ ) + J stray ( ϑ ) ,
J line ( ϑ ) = μ opt F ( 1 - R - A abs ) 2 1 - R 2 × I 0 { 1 + 2 ν = 1 R ν exp [ - 8 ( ν π d χ cos ϑ ) 2 ] × cos ( ν 4 π σ L d cos ϑ ) } ,
J stray ( ϑ ) = μ opt F ( 1 - R - A abs ) 2 1 - R 2 × I stray γ 2 π { 1 + 2 ν = 1 R ν exp [ - 8 ( ν π d γ × cos ϑ ) 2 ] cos ( ν 4 π σ * d cos ϑ ) } .
J stray ( ϑ ) = μ opt F ( 1 - R - A abs ) 2 1 - R 2 I stray γ 2 π .
S opt ( ϑ min , ϑ max ) = τ 2 π ϑ min ϑ max sin ϑ J ( ϑ ) d ϑ ( photons ) .
S opt line = I 0 F Ω τ μ opt ( 1 - R - A abs ) 2 1 - R 2 × { 1 + 2 ν = 1 R ν exp [ - 8 ( ν π d χ cos ϑ * ) 2 ] × sin ( ν σ L d Ω ) ν σ L d Ω cos ( ν 4 π σ L d cos ϑ * ) }
S opt stray = I stray F Ω τ μ opt ( 1 - R - A abs ) 2 1 - R 2 γ 2 π ,
S opt line = I 0 F Ω τ μ opt ( 1 - R - A abs ) 2 1 - R 2 × { 1 + 2 ν = 1 R ν exp [ - 8 ( ν π d χ ) 2 ] × sin ( ν σ 0 d Ω ) ν σ 0 d Ω cos ( ν 4 π σ L d cos ϑ * ) } .
S opt ( K ) = S opt line ( d K ) + S opt stray ,             K = 1 , 2 , N ,
Ω = π N p j 0 ,             j 0 = 2 σ 0 d .
S opt ( K ) = S opt line ( ϑ K * ) + S opt stray ,             K = 1 , , N ,
X K = μ q u S opt ( K ) + M D F D τ ( counts ) ,
X K = I 0 Z K + W ,             K = 1 , , N ,
P ( Y 1 Y N I 0 , T , V ) = K = 1 N X K Y K e - X K Y K ! .
L ( I 0 , T , V ) = K = 1 N { Y K ln [ I 0 Z K ( T , V ) + W ] - [ I 0 Z K ( T , V ) + W ] } .
u K + 1 = u K - H - 1 ( u K ) L ( u K ) ,
Δ u i = [ ( u ^ i - u i ) 2 ] 1 / 2 ,             ( i = 1 , 2 , 3 ) .
L ( u ) u ^ = 0
δ Y K × δ Y L = X K δ K , L
Δ I 0 = [ a 22 a 33 - a 23 2 det ( a i j ) ] 1 / 2 , Δ T = 1 I 0 [ a 11 a 33 - a 13 2 det ( a i j ) ] 1 / 2 , Δ V = 1 I 0 [ a 11 a 22 - a 12 2 det ( a i j ) ] 1 / 2 .
a 11 = K = 1 N Z K 2 X K , a 12 = K = 1 N Z K Z K T X K , a 13 = K = 1 N Z K Z K V X K , a 22 = K = 1 N ( Z K T ) 2 X K , a 23 = K = 1 N Z K T Z K V X K , a 33 = K = 1 N ( Z K V ) 2 X K .
( u ^ i - u i ) 2 ( J - 1 ) i i ,             ( i = 1 , 2 , 3 ) .
J i K = - 2 ln p ( Y Y N / u ) u i u k .
J = ( a 11 , I 0 a 12 , I 0 a 13 I 0 a 12 , I 0 2 a 22 , I 0 2 a 23 I 0 a 13 , I 0 2 a 23 , I 0 2 a 33 )
( J - 1 ) 11 = a 22 a 33 - a 23 2 det a , ( J - 1 ) 22 = 1 I 0 2 a 11 a 33 - a 13 2 det a , ( J - 1 ) 33 = 1 I 0 2 a 11 a 22 - a 12 2 det a .
Δ I 0 I 0 , Δ T , Δ V ~ ( I 0 F τ μ opt μ q u ) - 1 / 2 .
L I 0 | u ^ = K = 1 N [ Y K Z K I ( T ^ , V ^ ) Î 0 Z K ( T ^ , V ^ ) + W - Z K ( T ^ , V ^ ) ] = 0.
a 11 δ I 0 I 0 + a 12 δ T + a 13 δ V = 1 I 0 K = 1 N Z K ( T , V ) X K ( I 0 , T , V ) δ Y K = f 1 ,
a 12 δ I 0 I 0 + a 22 δ T + a 23 δ V = 1 I 0 K = 1 N Z K ( T , V ) δ T X K ( I 0 , T , V ) δ Y K = f 2 ,
a 13 δ I 0 I 0 + a 23 δ T + a 33 δ V = 1 I 0 K = 1 N Z K ( T , V ) δ T X K ( I 0 , T , V ) δ Y K = f 3 .
δ I 0 I 0 = ( a 22 a 33 - a 23 2 ) f 1 + ( a 13 a 23 - a 12 a 33 ) f 2 + ( a 12 a 23 - a 13 a 22 ) f 3 det a
f i f j = a i j I 0 2 .
f 1 2 = 1 I 0 2 K , e = 1 N Z K X K × Z e X e δ Y K δ Y e = 1 I 0 2 K , e = 1 N Z K X K × Z e X e X K δ K , e = 1 I 0 2 K = 1 N Z K 2 X K = a 11 I 0 2 .
( δ I 0 ) 2 = a 22 a 33 - a 23 2 det ( a i j )
Δ u i = ( δ u i ) 2 1 / 2 ,             ( i = 1 , 2 , 3 ) .