Abstract

The instrumental functions of the two-étalon system with a finite field of view have been examined in cases in which both étalons are inclined against the optical axis. The average transmittances of the rays in the finite field of view are obtained by numerical integration. It is shown that the line shape of the instrumental function becomes asymmetric, varying not only with the zenith angles of the symmetry axes of the two étalons but also with their azimuth angles. Experimental measurements of line shapes with the He–Ne laser show the asymmetric feature predicted by the calculation. The simulation calculations for the two-pass étalon system are also carried out and show a drastic discrepancy in transmittances between azimuth angle differences of 0° and 180°.

© 1996 Optical Society of America

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References

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  1. S. M. Lindsay, S. Burgess, I. W. Shepherd, “Correction of Brillouin linewidths measured by multipass Fabry–Perot spectroscopy,” Appl. Opt. 16, 1404–1407 (1977).
    [CrossRef] [PubMed]
  2. C. Roychoudhuri, M. Hercher, “Stable multipass Fabry–Perot interferometer: design and analysis,” Appl. Opt. 16, 2514–2520 (1977).
    [CrossRef] [PubMed]
  3. R. E. Loughhead, R. J. Bray, N. Brown, “Instrumental profiles of a triple Fabry–Perot interferometer for use in solar spectroscopy,” Appl. Opt. 17, 415–419 (1978).
    [CrossRef] [PubMed]
  4. L. N. Durvasula, R. W. Gammon, “Pressure-scanned three-pass Fabry–Perot interferometer,” Appl. Opt. 17, 3298–3303 (1978).
    [CrossRef] [PubMed]
  5. D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
    [CrossRef]
  6. G. Hernandez, Fabry-Perot Interferometer (Cambridge U.P., Cambridge, UK, 1986).
  7. G. Hernandez, “Afocal coupled étalons. DEM: a high-resolution double-étalon modulator spectrometer,” Appl. Opt. 26, 4857–4869 (1987).
    [CrossRef] [PubMed]
  8. W. R. Skinner, P. B. Hays, V. J. Abreu, “Optimization of a triple étalon interferometer,” Appl. Opt. 26, 2817–2827 (1987).
    [CrossRef] [PubMed]
  9. G. Hernandez, F. G. McCormac, “Afocal coupled étalons: experimental confirmation of a high-resolution double-étalon modulator,” Appl. Opt. 27, 3492–3495 (1988).
    [CrossRef] [PubMed]
  10. J. M. Alvarez, J. A. Valles, “Determination of a Fabry–Perot multipass interferometer instrumental function,” Appl. Opt. 28, 2191–2193 (1989).
    [CrossRef] [PubMed]
  11. T. Aoki, M. Fukabori, Te. Aoki, “Trace gas remote sounding from near IR sun glint observation with tunable étalons,” in Proceedings of the NATO Advanced Research Workshop: High Spectral Resolution Infrared Remote Sensing for Earth’s Weather and Climate Studies (Springer-Verlag, Berlin, 1994), pp. 309–322.
  12. E. D. Palik, H. Boukari, R. W. Gammon, “Line-shape studies for single- and triple-pass Fabry–Perot interferometer systems,” Appl. Opt. 34, 58–68 (1995).
    [CrossRef] [PubMed]
  13. H. Boukari, E. D. Palik, R. W. Gammon, “Closed-form expressions to fit data obtained with a multipass Fabry–Perot interferometer,” Appl. Opt. 34, 69–86 (1995).
    [CrossRef] [PubMed]
  14. T. L. Killeen, P. B. Hays, B. C. Kennedy, D. Rees, “Stable and rugged étalon for the Dynamics Explorer Fabry–Perot interferometer. 2: Performance,” Appl. Opt. 21, 3903–3912 (1982).
    [CrossRef] [PubMed]
  15. G. J. Sloggett, “Fringe broadening in Fabry–Perot interferometers,” Appl. Opt. 23, 2427–2432 (1984).
    [CrossRef] [PubMed]
  16. P. A. Wilksch, “Instrument function of the Fabry–Perot spectrometer,” Appl. Opt. 24, 1502–1511 (1985).
    [CrossRef] [PubMed]
  17. T. Martinez-Herrero, P. M. Mejias, E. Bernabeu, “Transmitted amplitude by a Fabry–Perot interferometer with random surface defects,” Appl. Opt. 24, 315–316 (1985).
    [CrossRef] [PubMed]
  18. D. P. Mahapatra, S. K. Mattoo, “Exact evaluation of the transmitted amplitude for a Fabry–Perot interferometer with surface defects,” Appl. Opt. 25, 1646–1649 (1986).
    [CrossRef] [PubMed]
  19. G. Hernandez, “Analytical description of a Fabry–Perot photoelectric spectrometer,” Appl. Opt. 5, 1745–1748 (1966).
    [CrossRef] [PubMed]
  20. G. Hernandez, “Analytical description of a Fabry–Perot photoelectric spectrometer. 3: Off-axis behavior and interference filters,” Appl. Opt. 13, 2654–2661 (1974).
    [CrossRef] [PubMed]
  21. T. L. Killeen, P. B. Hays, “Doppler line profile analysis for a multichannel Fabry–Perot interferometer,” Appl. Opt. 23, 612–620 (1984).
    [CrossRef] [PubMed]
  22. J. M. Vaugham, The Fabry–Perot interferometer; History, Theory, Practice and Applications (Hilger, Bristol, UK, 1989).
  23. T. Aoki, M. Fukabori, Te. Aoki, “Remote measurements of atmospheric constituents in the lower layers,” in Proceedings of the International Symposium on Global Cycles of Atmospheric Greenhouse Gases (Tohoku U., Sendai, Japan, 1994), pp. 165–170.

1995 (2)

1989 (1)

1988 (1)

1987 (2)

1986 (1)

1985 (2)

1984 (2)

1982 (2)

T. L. Killeen, P. B. Hays, B. C. Kennedy, D. Rees, “Stable and rugged étalon for the Dynamics Explorer Fabry–Perot interferometer. 2: Performance,” Appl. Opt. 21, 3903–3912 (1982).
[CrossRef] [PubMed]

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

1978 (2)

1977 (2)

1974 (1)

1966 (1)

Abreu, V. J.

Alvarez, J. M.

Aoki, T.

T. Aoki, M. Fukabori, Te. Aoki, “Trace gas remote sounding from near IR sun glint observation with tunable étalons,” in Proceedings of the NATO Advanced Research Workshop: High Spectral Resolution Infrared Remote Sensing for Earth’s Weather and Climate Studies (Springer-Verlag, Berlin, 1994), pp. 309–322.

T. Aoki, M. Fukabori, Te. Aoki, “Remote measurements of atmospheric constituents in the lower layers,” in Proceedings of the International Symposium on Global Cycles of Atmospheric Greenhouse Gases (Tohoku U., Sendai, Japan, 1994), pp. 165–170.

Aoki, Te.

T. Aoki, M. Fukabori, Te. Aoki, “Remote measurements of atmospheric constituents in the lower layers,” in Proceedings of the International Symposium on Global Cycles of Atmospheric Greenhouse Gases (Tohoku U., Sendai, Japan, 1994), pp. 165–170.

T. Aoki, M. Fukabori, Te. Aoki, “Trace gas remote sounding from near IR sun glint observation with tunable étalons,” in Proceedings of the NATO Advanced Research Workshop: High Spectral Resolution Infrared Remote Sensing for Earth’s Weather and Climate Studies (Springer-Verlag, Berlin, 1994), pp. 309–322.

Bernabeu, E.

Boukari, H.

Bray, R. J.

Brown, N.

Burgess, S.

Durvasula, L. N.

Fukabori, M.

T. Aoki, M. Fukabori, Te. Aoki, “Trace gas remote sounding from near IR sun glint observation with tunable étalons,” in Proceedings of the NATO Advanced Research Workshop: High Spectral Resolution Infrared Remote Sensing for Earth’s Weather and Climate Studies (Springer-Verlag, Berlin, 1994), pp. 309–322.

T. Aoki, M. Fukabori, Te. Aoki, “Remote measurements of atmospheric constituents in the lower layers,” in Proceedings of the International Symposium on Global Cycles of Atmospheric Greenhouse Gases (Tohoku U., Sendai, Japan, 1994), pp. 165–170.

Gammon, R. W.

Green-away, A. H.

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

Hays, P. B.

Hercher, M.

Hernandez, G.

Kennedy, B. C.

Killeen, T. L.

Lindsay, S. M.

Loughhead, R. E.

Mahapatra, D. P.

Martinez-Herrero, T.

Mattoo, S. K.

McCormac, F. G.

McWhirter, I.

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

Mejias, P. M.

Palik, E. D.

Rees, D.

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

T. L. Killeen, P. B. Hays, B. C. Kennedy, D. Rees, “Stable and rugged étalon for the Dynamics Explorer Fabry–Perot interferometer. 2: Performance,” Appl. Opt. 21, 3903–3912 (1982).
[CrossRef] [PubMed]

Rounce, P. A.

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

Roychoudhuri, C.

Scott, A. F. D.

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

Shepherd, I. W.

Skinner, W. R.

Sloggett, G. J.

Towison, W.

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

Valles, J. A.

Vaugham, J. M.

J. M. Vaugham, The Fabry–Perot interferometer; History, Theory, Practice and Applications (Hilger, Bristol, UK, 1989).

Wilksch, P. A.

Appl. Opt. (18)

S. M. Lindsay, S. Burgess, I. W. Shepherd, “Correction of Brillouin linewidths measured by multipass Fabry–Perot spectroscopy,” Appl. Opt. 16, 1404–1407 (1977).
[CrossRef] [PubMed]

C. Roychoudhuri, M. Hercher, “Stable multipass Fabry–Perot interferometer: design and analysis,” Appl. Opt. 16, 2514–2520 (1977).
[CrossRef] [PubMed]

R. E. Loughhead, R. J. Bray, N. Brown, “Instrumental profiles of a triple Fabry–Perot interferometer for use in solar spectroscopy,” Appl. Opt. 17, 415–419 (1978).
[CrossRef] [PubMed]

L. N. Durvasula, R. W. Gammon, “Pressure-scanned three-pass Fabry–Perot interferometer,” Appl. Opt. 17, 3298–3303 (1978).
[CrossRef] [PubMed]

G. Hernandez, “Afocal coupled étalons. DEM: a high-resolution double-étalon modulator spectrometer,” Appl. Opt. 26, 4857–4869 (1987).
[CrossRef] [PubMed]

W. R. Skinner, P. B. Hays, V. J. Abreu, “Optimization of a triple étalon interferometer,” Appl. Opt. 26, 2817–2827 (1987).
[CrossRef] [PubMed]

G. Hernandez, F. G. McCormac, “Afocal coupled étalons: experimental confirmation of a high-resolution double-étalon modulator,” Appl. Opt. 27, 3492–3495 (1988).
[CrossRef] [PubMed]

J. M. Alvarez, J. A. Valles, “Determination of a Fabry–Perot multipass interferometer instrumental function,” Appl. Opt. 28, 2191–2193 (1989).
[CrossRef] [PubMed]

E. D. Palik, H. Boukari, R. W. Gammon, “Line-shape studies for single- and triple-pass Fabry–Perot interferometer systems,” Appl. Opt. 34, 58–68 (1995).
[CrossRef] [PubMed]

H. Boukari, E. D. Palik, R. W. Gammon, “Closed-form expressions to fit data obtained with a multipass Fabry–Perot interferometer,” Appl. Opt. 34, 69–86 (1995).
[CrossRef] [PubMed]

T. L. Killeen, P. B. Hays, B. C. Kennedy, D. Rees, “Stable and rugged étalon for the Dynamics Explorer Fabry–Perot interferometer. 2: Performance,” Appl. Opt. 21, 3903–3912 (1982).
[CrossRef] [PubMed]

G. J. Sloggett, “Fringe broadening in Fabry–Perot interferometers,” Appl. Opt. 23, 2427–2432 (1984).
[CrossRef] [PubMed]

P. A. Wilksch, “Instrument function of the Fabry–Perot spectrometer,” Appl. Opt. 24, 1502–1511 (1985).
[CrossRef] [PubMed]

T. Martinez-Herrero, P. M. Mejias, E. Bernabeu, “Transmitted amplitude by a Fabry–Perot interferometer with random surface defects,” Appl. Opt. 24, 315–316 (1985).
[CrossRef] [PubMed]

D. P. Mahapatra, S. K. Mattoo, “Exact evaluation of the transmitted amplitude for a Fabry–Perot interferometer with surface defects,” Appl. Opt. 25, 1646–1649 (1986).
[CrossRef] [PubMed]

G. Hernandez, “Analytical description of a Fabry–Perot photoelectric spectrometer,” Appl. Opt. 5, 1745–1748 (1966).
[CrossRef] [PubMed]

G. Hernandez, “Analytical description of a Fabry–Perot photoelectric spectrometer. 3: Off-axis behavior and interference filters,” Appl. Opt. 13, 2654–2661 (1974).
[CrossRef] [PubMed]

T. L. Killeen, P. B. Hays, “Doppler line profile analysis for a multichannel Fabry–Perot interferometer,” Appl. Opt. 23, 612–620 (1984).
[CrossRef] [PubMed]

J. Phys. E (1)

D. Rees, P. A. Rounce, I. McWhirter, A. F. D. Scott, A. H. Green-away, W. Towison, “Observations of atmospheric absorption lines from a stabilized balloon platform and measurements of stratospheric winds,” J. Phys. E 15, 191–206 (1982).
[CrossRef]

Other (4)

G. Hernandez, Fabry-Perot Interferometer (Cambridge U.P., Cambridge, UK, 1986).

T. Aoki, M. Fukabori, Te. Aoki, “Trace gas remote sounding from near IR sun glint observation with tunable étalons,” in Proceedings of the NATO Advanced Research Workshop: High Spectral Resolution Infrared Remote Sensing for Earth’s Weather and Climate Studies (Springer-Verlag, Berlin, 1994), pp. 309–322.

J. M. Vaugham, The Fabry–Perot interferometer; History, Theory, Practice and Applications (Hilger, Bristol, UK, 1989).

T. Aoki, M. Fukabori, Te. Aoki, “Remote measurements of atmospheric constituents in the lower layers,” in Proceedings of the International Symposium on Global Cycles of Atmospheric Greenhouse Gases (Tohoku U., Sendai, Japan, 1994), pp. 165–170.

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Figures (10)

Fig. 1
Fig. 1

Definition of zenith and azimuth angles of the étalon (E) and tunable étalon (TE). The zenith is directed to the center of the FOV.

Fig. 2
Fig. 2

Example of the relationship between ζ and τ(ζ).

Fig. 3
Fig. 3

Procedure of dividing the FOV area for the numerical integration.

Fig. 4
Fig. 4

Instrumental function of the two-étalon system. The thick solid curves show the deviations of transmittances for φ1 = 0° from that for φ1 = 180° being shifted by 0.5 upward. The dashed curves are the instrumental functions of the single étalon system. The curves with five peaks are the instrumental functions of the two-étalon system divided by that of the single étalon system. The upper panel is for the θ2 = 0.5° case, and the lower is for the θ2 = 0.1° case.

Fig. 5
Fig. 5

Experimental setup for measuring the line shape of the two-étalon system: E, étalon; BF, blocking filter; TE, tunable étalon.

Fig. 6
Fig. 6

Measured transmittances of the two-étalon system at the wave number of the laser line, being represented as a function of the electric voltage applied to the piezoelements of the tunable étalon. The zenith angle of the étalon increases in the order of (A), (B), (C), (D), and (E). In panel (C) the band center of the étalon coincides with the laser line. The upper five panels are for the φ1 = 0° case, and the lower panels are for φ1 = 180°.

Fig. 7
Fig. 7

Calculated transmittances of the two-étalon system at the wave number of the laser line being represented as a function of the relative position of the line center of the tunable étalon. The upper five panels are for the φ1 = 0° case, and the lower panels are for φ1 = 180°. The numbers in the upper parts of the panels are line centers of the étalon in wave-number units.

Fig. 8
Fig. 8

Instrumental function of the two-pass étalon. The thick solid curves show the deviations in the transmittances calculated for φ = 0° from those for φ = 180° with θ1 = θ2 = 0.5°, being shifted by 0.5 upward. The thin solid curves are Y(ν, φ1 = 0°), and the dashed curves are Y(ν, φ1 = 180°). The upper panel is for the θ2 = 0.5° case, and the lower is for the θ2 = 0.1° case.

Fig. 9
Fig. 9

Instrumental function of the two-pass étalon in the case in which the image of the light source covers only half of the detector area. The thick solid curves show the deviations of the transmittances calculated for φ = 0° from that for φ = 180° with θ1 = θ2 = 0.5°, being shifted by 0.5 upward. The thin solid curves are Y(ν, φ1 = 0°), and the dashed curves are Y(ν, φ1 = 180°). The upper panel is for the θ2 = 0.5° case, and the lower is for the θ2 = 0.1° case.

Fig. 10
Fig. 10

Configuration of the image of the light source and the detector area assumed in the calculation of Fig. 9.

Equations (15)

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cos Θ 1 = cos θ 1 cos θ + sin θ 1 sin θ cos ( φ - φ 1 ) ,
cos Θ 2 = cos θ 2 cos θ + sin θ 2 sin θ cos ( φ ) ,
Y ( ν ) = 1 Ω 0 Ω 0 τ 1 ( ν , Θ 1 ) τ 2 ( ν , Θ 2 ) d Ω ,
τ = ( 1 - R - A 1 - R ) 2 1 - R 1 + R { 1 + 2 n = 1 R n exp × [ - ( n σ r β 1 2 ) 2 ] sinc ( n σ c β 1 ) cos ( n β 0 ν ) } ,
β 0 = 4 π μ d 0 cos Θ ,
β 1 = 4 π μ ν cos Θ ,
β 1 4 π μ ν 0 cos Θ 0 .
β 0 ν = 2 π m + x             ( - π 2 x π 2 ) ,
ζ = 1 - γ 2 γ 2 + x 2 ,
γ = γ 1 [ 1 + ( γ 2 γ 1 ) 1 / 2 + ( γ 3 γ 1 ) 1 / 2 ] 1 / 2 ,
γ 1 = 1 - R 2 π d 0 R ,
γ 2 = ν 0 ( σ r + σ c ) d 0 ,
γ 3 = 2 ν 0 sin ( θ ) sin ( Δ θ ) cos ( θ + Δ θ ) ,
N 0 = ln ( 0.0002 ) / ln ( R ) .
τ ¯ 1 ( ν ) = 1 Ω 0 Ω 0 τ 1 ( ν , Θ 1 ) d Ω .

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