## Measurement uncertainty analysis in incoherent Doppler lidars by a new scattering approach

Optics Express, Vol. 14, Issue 17, pp. 7699-7708 (2006)

http://dx.doi.org/10.1364/OE.14.007699

Acrobat PDF (202 KB)

### 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

## 1. Introduction

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

## 2. Fizeau scattering matrix

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

8. Y. H. Meyer, “Fringe shape with an interferential wedge,” J. Opt. Soc. Am. **71**, 1255–1263 (1981). [CrossRef]

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

*and outgoing*

**a**_{i}*normalized waves at the ith port of the network. Although generally complex systems are multiports, a large number of optical elements (e.g., lens, filters, beam splitters) may be expressed as two-port elements described by the incident, transmitted, and reflected normalized waves. The scattering matrix S formulation represents the relationships among these normalized wave variables for any optical element as well as an entire system. By definition of scattering parameters,*

**b**_{i}*S*=

_{ki}*/*

**b**_{i}*equals the wave transformation between the kth port and the ith port of a multiport optical system and |*

**a**_{k}*S*|

_{ik}^{2}represents the power gain between the kth and the ith ports. For the simple two-port elements, the scattering parameters will define a

*2×2*complex scattering matrix.

*E*going into the optical system, a transform based on Bessel beams is a natural generalization of Fourier techniques:

_{i}*θ*to a Fizeau characterized by its wedge angle a. Because of multiple reflections at wedge surfaces, any incoming way is split into a set of waves, the propagation direction of which successively differ by twice the wedge angle

*α*. The transmitted light

*E*at kth port consists of a set of

_{k}*p*=l, 2,…,

*N*Bessel modes

_{P}*V*emerging with diverging angles

_{mn,v}*θ*and sharing the energy |

_{p}*a*|

_{nm}^{2}of each incident Bessel beam:

*N*that has to be taken into account depends on the mirror reflectivity ℜ [12

12. J. R. Rogers, “Fringe shifts in multiple-beam Fizeau interferometry,” J. Opt. Soc. Am. **72**, 638–643 (1982) [CrossRef]

13. P. H. Langenbeck, “Fizeau interferometer-fringe sharpening,” Appl. Opt. **9**, 2053–2058 (1970). [CrossRef] [PubMed]

## 3. Uncertainty in incoherent Doppler measurements

*δf*. Here, we just consider the intrinsic limitation on the determination of the center frequency

*f*of the signal that is due to the statistical nature of the sampling of the interferometer fringes [14

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]

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]

*(δf)*in the determination of the center frequency

^{2}*f*. and they are measured with respect to the center frequency of the spectral line. On the other hand, the Cramer-Rao lower bound for the maximum likelihood estimator of the Gaussian central frequency

_{i}*e*half-width Δ

*f*. The coefficient N

_{0}is the total signal (photons) transmitted through the interferometric device to be characterized. This assumes that just one spectral range of the interferometer is illuminated.

*d*=300 mm and symmetric mirror reflectivities of ℜ = 0.9. The molecular-driven light (1-GHz spectral width) reflected in the aerosol channel inpinges a second middle-resolution etalon whose mirrors (ℜ =0.74) are spaced a distance

*d*=16 mm. Realistically, and with the intention of blocking any solar background, at the input of this two-channel interferometric instrument a low resolution Fabry-Perot (2-mm spacing, 70% reflectivity) has been also added to the analyzed design. Working etalon apertures are made to match the instrument etendue. If etalon apertures are different, diffraction losses and spreading need to be consider in the uncertainty estimations. Our method takes these effects into account naturally.

*v*

_{mn}associated with the Bessel waves used in the modal analysis), this spatial free spectral range Δ is decreasing. As an increase of the etalon spacing d moves the etalon spectrum –used to estimate Doppler shift– on the opposite direction that a similar decrease of the spacing magnitude does, the asymmetry of the spectrum translate into an asymmetry of the system sensitivity to the mistuning. Our technique, based on a full electromagnetic estimation considering a realistic propagation of any spatial mode

*v*

_{mn}, is uniquely suited to deal accurately with this kind of critical aspects of etalon behavior scarcely pondered in previous analysis.

8. Y. H. Meyer, “Fringe shape with an interferential wedge,” J. Opt. Soc. Am. **71**, 1255–1263 (1981). [CrossRef]

*L*=15 mm (frequency spacing of adjacent modes of Δ

*v*=10 GHz and mode FWHM spectral width of Δ

_{PSR}*v*=333 MHz) to

*L*=45 mm (Δ

*v*=3.3 GHz and Δ

_{FSR}*v*=111MHz). In the analysis, with a working wavelength

*λ*of 355 nm, the Fizeau wedges act as optical filters with resolutions ranging from 0.15 to 0.05 pm. Such filters are suitable for spectral measurements of molecular-backscattered light, with typical linewidth Δ

*v*of a few GHz, or aerosol-backscattered light, where the atmospheric broadening of the laser source spectral linewidth can be neglected and Ave is in the order of 100 MHz.

_{e}*X*from the wedge and the angle of incidence is changed, a clear interference fringe is observed only within a narrow range of angles. This is illustrated in Fig. 5 (left), which shows that the most favorable conditions for constructive interference for our set of Fizeau wedges occur when the angle of incidence |

*θ*| is chosen between 1.0 and 1.2 mrad. For these angles, the fringe sharpness is at its maximum and the transfer of energy from the fringe peak to secondary fringes is minimized. This off-axis illumination compensates part of the optical path differences among the beams emerging from the wedge and whose interference tends to degrade the fringes. Once the ideal incidence angles are set in our analysis, we can further optimize the performance of our Fizeau filters by changing slightly the true effective thickness of the wedges (see Fig. 5, right). Small modifications (less than 10 nm in the examples considered in the figure) of the effective wedge thickness may improve significantly the probable performance of our interferometer by displacing the already sharp interference fringes from its ideal position.

*L*=60 mm; ℜ =0.92) defining an aerosol channel (Δ

*v*=2.5 GHz; Δ

_{FSR}*v*=66 MHz, Δ

*v*=30 MHz) is captured into a Fabry-Perot etalon (

_{e}*L*=6 mm; ℜ =0.75) with a much broader passband (Δ

*v*=25 GHz; Δ

_{FSR}*v*=2.5 GHz) to make possible Doppler estimations from molecular-backscattered light (Δ

*v*=1 GHz). Figure 6 shows normalized Doppler measurement uncertainty on both the aerosol and molecular channels as a function of the incidence angle

_{e}*θ*and optical thickness variations Δ

*L*on the Fizeau wedge. As happened in Fig. 5, by chosen pertinent Fizeau wedge parameters, we could optimize the performance of the aerosol channel. Once again, incidence angles slightly larger than 1 mrad (Fig. 3, left) and optical thickness a few nm smaller than the wedge nominal value (Fig. 6, right) translate into improved Doppler measurements. However, the coupling between Mie and Rayleigh channels translate into a suboptimal Fabry-Perot measurement uncertainty. By changing both the incidence angle and thickness of the wedge, we are also modifying the illumination pattern incoming into the molecular etalon in an uncertain way. Although most of the energy arriving into the Fabry-Perot comes from the first direct reflection on the Fizeau surface, a small but significant component of the incoming signal is due to multiple reflections on the wedge mirrors and whose propagation directions differ by twice the wedge angle

*α*. In most situations, the multibeam Fizeau effects appear on the Fabry-Perot interference fringes. Any realistic analysis of the performance of multiple interferometric incoherent lidar system needs to consider these genuine effects. Optimization of the Fabry-Perot based molecular channel will be strongly affected by the diverging beams coming from the Fizeau wedge.

## 4. Conclusions

*VBS*(Vortical Beam Spectra), a flexible software tool implementing our generalized scattering matrix theory.

## References and links

1. | M. Born and E. Wolf, |

2. | M. Endemann, P. Dubock, P. Ingmann, R. Wimmer, D. Morancais, and D. Demuth, “The ADM-Aeolus Mission: The first wind-lidar in space,” in |

3. | J. Goodman, |

4. | G. Hernandez, |

5. | J. A. McKay, “Assessment of a Multibeam Fizeau Wedge Interferometer for Doppler Wind Lidar,” Appl. Opt. |

6. | A. Läzaro 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 |

8. | Y. H. Meyer, “Fringe shape with an interferential wedge,” J. Opt. Soc. Am. |

9. | T. T. Kajava, H. M. Lauranto, and R. R. E. Salomaa, “Fizeau interferometer in spectral measurements,” J. Opt. Soc. Am. B |

10. | T. T. Kajava, H. M. Lauranto, and A. T. Friberg, “Interference pattern of a Fizeau interferometer,” J. Opt. Soc. Am. A |

11. | E. Stoykova, “Transmission of a Gaussian beam by a Fizeau interferential wedge,” J. Opt. Soc. Am. A |

12. | J. R. Rogers, “Fringe shifts in multiple-beam Fizeau interferometry,” J. Opt. Soc. Am. |

13. | P. H. Langenbeck, “Fizeau interferometer-fringe sharpening,” Appl. Opt. |

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. |

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. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(280.3640) Remote sensing and sensors : Lidar

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 19, 2006

Revised Manuscript: August 9, 2006

Manuscript Accepted: August 12, 2006

Published: August 21, 2006

**Citation**

Aniceto Belmonte and Antonio Lázaro, "Measurement uncertainty analysis in incoherent
Doppler lidars by a new scattering approach," Opt. Express **14**, 7699-7708 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7699

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### References

- 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).
- J. A. McKay, "Assessment of a Multibeam Fizeau Wedge Interferometer for Doppler Wind Lidar, " Appl. Opt. 41, 1760-1767 (2002). [CrossRef] [PubMed]
- A. Lázaro and A. Belmonte, "A unified approach to the analysis of incoherent Doppler lidars: Etalon-based systems," Opt. Express (to be published). [PubMed]
- J. Brossel, "Multipe-beam localized fringes. Part I. Intensity distribution and localization," Proc. Phys. Soc. London 59, 224-234 (1947). [CrossRef]
- Y. H. Meyer, "Fringe shape with an interferential wedge," J. Opt. Soc. Am. 71, 1255-1263 (1981). [CrossRef]
- 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]
- T. T. Kajava, H. M. Lauranto, and A. T. Friberg, "Interference pattern of a Fizeau interferometer," J. Opt. Soc. Am. A 11, 2045- 2054 (1994). [CrossRef]
- E. Stoykova, "Transmission of a Gaussian beam by a Fizeau interferential wedge," J. Opt. Soc. Am. A 22, 2756-2765 (2005). [CrossRef]
- J. R. Rogers, "Fringe shifts in multiple-beam Fizeau interferometry," J. Opt. Soc. Am. 72, 638-643 (1982) [CrossRef]
- P. H. Langenbeck, "Fizeau interferometer-fringe sharpening," Appl. Opt. 9, 2053-2058 (1970). [CrossRef] [PubMed]
- 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]
- 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]

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