Abstract

We need to examine the uncertainty added to the Doppler measurement process of atmospheric wind speeds of a practical incoherent detection lidar. For this application, the multibeam Fizeau wedge has the advantage over the Fabry-Perot interferometer of defining linear fringe patterns. Unfortunately, the convenience of using the transfer function for angular spectrum transmission has not been available because the non-parallel mirror geometry of Fizeau wedges. In this paper, we extent the spatial-frequency arguments used in Fabry-Perot etalons to the Fizeau geometry by using a generalized scattering matrix method based on the propagation of optical vortices. Our technique opens the door to consider complex, realistic configurations for any Fizeau-based instrument.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
  2. M. Endemann, P. Dubock, P. Ingmann, R. Wimmer, D. Morancais, D. Demuth, "The ADM-Aeolus Mission: The first wind-lidar in space," in Proceedings of 22nd International Laser Radar Conference, ILRC (ESA SP-561), pp. 953-956.
  3. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).
  4. G. Hernandez, Fabry-Perot Interferometers (Cambridge University Press, Cambridge, 1988).
  5. J. A. McKay, "Assessment of a Multibeam Fizeau Wedge Interferometer for Doppler Wind Lidar, " Appl. Opt. 41, 1760-1767 (2002).
    [CrossRef] [PubMed]
  6. A. Lázaro and A. Belmonte, "A unified approach to the analysis of incoherent Doppler lidars: Etalon-based systems," Opt. Express (to be published).
    [PubMed]
  7. J. Brossel, "Multipe-beam localized fringes. Part I. Intensity distribution and localization," Proc. Phys. Soc. London 59, 224-234 (1947).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. E. Stoykova, "Transmission of a Gaussian beam by a Fizeau interferential wedge," J. Opt. Soc. Am. A 22, 2756-2765 (2005).
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  12. J. R. Rogers, "Fringe shifts in multiple-beam Fizeau interferometry," J. Opt. Soc. Am. 72, 638-643 (1982)
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  13. P. H. Langenbeck, "Fizeau interferometer-fringe sharpening," Appl. Opt. 9, 2053-2058 (1970).
    [CrossRef] [PubMed]
  14. J. -M. Gagne, J. -P. Saint-Dizier, and M. Picard, "Methode d'echantillonnage des fonctions deterministes en spectroscopie: application a un spectrometre multicanal par comptage photonique," Appl. Opt. 13, 581-588 (1974)
    [CrossRef] [PubMed]
  15. B. J. Rye and R. M. Hardesty, "Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I: Spectral accumulation and the Cramer-Rao lower bound," IEEE Trans.Geosci. Remote Sens. 31,16-27 (1993).
    [CrossRef]

2005 (1)

2002 (1)

1994 (1)

1993 (2)

T. T. Kajava, H. M. Lauranto, and R. R. E. Salomaa, "Fizeau interferometer in spectral measurements," J. Opt. Soc. Am. B 10, 1980-1989 (1993).
[CrossRef]

B. J. Rye and R. M. Hardesty, "Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I: Spectral accumulation and the Cramer-Rao lower bound," IEEE Trans.Geosci. Remote Sens. 31,16-27 (1993).
[CrossRef]

1982 (1)

1981 (1)

1974 (1)

1970 (1)

1947 (1)

J. Brossel, "Multipe-beam localized fringes. Part I. Intensity distribution and localization," Proc. Phys. Soc. London 59, 224-234 (1947).
[CrossRef]

Belmonte, A.

A. Lázaro and A. Belmonte, "A unified approach to the analysis of incoherent Doppler lidars: Etalon-based systems," Opt. Express (to be published).
[PubMed]

Brossel, J.

J. Brossel, "Multipe-beam localized fringes. Part I. Intensity distribution and localization," Proc. Phys. Soc. London 59, 224-234 (1947).
[CrossRef]

Friberg, A. T.

Gagne, J. -M.

Hardesty, R. M.

B. J. Rye and R. M. Hardesty, "Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I: Spectral accumulation and the Cramer-Rao lower bound," IEEE Trans.Geosci. Remote Sens. 31,16-27 (1993).
[CrossRef]

Kajava, T. T.

Langenbeck, P. H.

Lauranto, H. M.

Lázaro, A.

A. Lázaro and A. Belmonte, "A unified approach to the analysis of incoherent Doppler lidars: Etalon-based systems," Opt. Express (to be published).
[PubMed]

McKay, J. A.

Meyer, Y. H.

Picard, M.

Rogers, J. R.

Rye, B. J.

B. J. Rye and R. M. Hardesty, "Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I: Spectral accumulation and the Cramer-Rao lower bound," IEEE Trans.Geosci. Remote Sens. 31,16-27 (1993).
[CrossRef]

Saint-Dizier, J. -P.

Salomaa, R. R. E.

Stoykova, E.

Appl. Opt. (3)

IEEE Trans.Geosci. Remote Sens. (1)

B. J. Rye and R. M. Hardesty, "Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I: Spectral accumulation and the Cramer-Rao lower bound," IEEE Trans.Geosci. Remote Sens. 31,16-27 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

A. Lázaro and A. Belmonte, "A unified approach to the analysis of incoherent Doppler lidars: Etalon-based systems," Opt. Express (to be published).
[PubMed]

Proc. Phys. Soc. London (1)

J. Brossel, "Multipe-beam localized fringes. Part I. Intensity distribution and localization," Proc. Phys. Soc. London 59, 224-234 (1947).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

M. Endemann, P. Dubock, P. Ingmann, R. Wimmer, D. Morancais, D. Demuth, "The ADM-Aeolus Mission: The first wind-lidar in space," in Proceedings of 22nd International Laser Radar Conference, ILRC (ESA SP-561), pp. 953-956.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).

G. Hernandez, Fabry-Perot Interferometers (Cambridge University Press, Cambridge, 1988).

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Figures (6)

Fig. 1.
Fig. 1.

Optical geometry of a Fizeau interferometer with wedge angle α, optical thickness L, and incidence angle θ. The interference pattern of the Fizeau detected behind the wedge at distance X shows unique phase and amplitude features (right). Multiple interferometer systems (left), combining both etalon and Fizeau filters, pose the considerable challenge of establishing analytical tools for evaluating their design and main spectrometric capabilities.

Fig. 2.
Fig. 2.

In our analysis, two high-resolution filters (left) based on interferometric devices are used two separate the Mie (aerosol) and the Rayleigh (molecular) channels. Although the Rayleigh channel is always based on a Fabry-Perot etalon, the Mie channel can be based on either a etalon or a Fizeau interferometer. Although we have considered both cases in this study, the signals transmitted through the interferometers and show in the figure (right) consider etalon-based systems. The bites in the spectrum received on the molecular channel are just evident on signals coming from the atmospheric boundary layer, where the aerosol load is more important. The atmospheric return signals (dashed lines) are also shown in the figures.

Fig. 3.
Fig. 3.

Normalized Doppler measurement errors for Fabry-Perot interferometric systems as a function of etalon separation mistuning. Middle-resolution (left) and high-resolution (right) etalon systems are considered along with several different etalon reflectivities (finesse).

Fig. 4.
Fig. 4.

Normalized Doppler measurement errors for Fabry-Perot interferometric systems as a function of etalon angular misalignments. Middle-resolution (left) and high-resolution (right) etalon systems are considered.

Fig. 5.
Fig. 5.

Doppler measurement uncertainty as a ratio to the Cramer-Rao limit of the measurement of the Gaussian centroid frequency with a perfect, lossless receiver. Modifying the angle of incidence 0 of the incoming light is possible to achieve most favorable conditions for any of the cases considered in the figure (left). Using the optimal incidence angles (see marker), we further optimize the uncertainty measurement by slightly changing the optical thickness of the Fizeau (right).

Fig. 6.
Fig. 6.

The analysis of normalized Doppler measurement uncertainty for a multiple interferometer system combining a Fizeau wedge (Mie channel, in blue) and a Fabry-Perot etalon (Rayleigh channel, in red) shows the coupling between both devices. Those incidence angle and optical thickness optimizing the performance of the Fizeau channel (see markers) increase the error measurement on the Fabry-Perot channel.

Equations (4)

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E i ( ρ , ϕ ) = m = M M n = 0 N a nm J m ( v n ρ ) e jmϕ ,
E k ( ρ , ϕ ) = { p = 1 N p Tℜ ( p 1 ) e j [ θ + 2 ( p 1 ) α ] } E i ( ρ , ϕ ) ,
( δf ) 2 = f i 2 S ( f i ) [ S ( f i ) ] 2
( δf ) 2 = Δ f 2 2 N 0

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