## Polarization lidar: Corrections of instrumental effects

Optics Express, Vol. 7, Issue 12, pp. 427-435 (2000)

http://dx.doi.org/10.1364/OE.7.000427

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### Abstract

An algorithm for correcting instrumental effects in polarization lidar studies is discussed. Cross-talk between the perpendicular and parallel polarization channels and imperfect polarization of the transmitted laser beam are taken into account. On the basis of the Mueller formalism it is shown that - with certain assumptions - the combined effects of imperfect polarization of the transmitted laser pulse, non-ideal properties of transmitter and receiver optics and cross-talk between parallel and perpendicular polarization channels can be described by a single parameter, which is essentially the overall system depolarization.

© Optical Society of America

## 1 Introduction

1. S. R. Pal and A. I. Carswell, “Polarization properties of lidar backscattering from clouds,” Appl. Opt. **12**, 1530–1535 (1973) [CrossRef] [PubMed]

2. C. M. R. Platt, “Lidar observations of a mixed-phase altostratus cloud,” J. Appl. Meteorol. , **16**, 339–345 (1977) [CrossRef]

3. K. Sassen, “Depolarization of laser light backscattered by artificial clouds,” J. Appl. Meteorol. , **13**, 923–933 (1973) [CrossRef]

4. R. M. Schotland, R. M. Sassen, K. Stone, and R. J., “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. , **10**, 1011–1017 (1971) [CrossRef]

*I*,

*Q*,

*U*,

*V*), is defined as

*E*

_{‖,⊥}denote the two perpendicular components of the electric field with respect to the plane of polarization of the transmitted light, and 〈…〉 the temporal average. For simplicity normalized Stokes vectors and Mueller matrices are used except for the scattering matrix. For a scattering process the linear depolarization factor δ is defined [5]

*i*

_{‖,⊥}are the components of the scattered light with linear polarization parallel or perpendicular to the scattering plane. For backscattering the reference plane is not defined by the scattering geometry and is chosen to be parallel to the plane of polarization of the emitted light. It is now easy to express the depolarization factor by the Stokes parameters:

6. D.R. Bates, “Rayleigh Scattering by Air,” Planet. Space Sci. **32**, 785–790 (1984) [CrossRef]

7. A.T. Young, “Revised depolarization corrections for atmospheric extinction,” Appl. Opt. **19**, 3427–3428 (1980) [CrossRef] [PubMed]

*δ*=0.00365 at λ=532 nm and for Rayleigh scattering [8

8. A.T. Young, “Rayleigh Scattering,” Appl. Opt. **20**, 533–535 (1981) [CrossRef] [PubMed]

*δ*=0.0144 at λ=532 nm [6

6. D.R. Bates, “Rayleigh Scattering by Air,” Planet. Space Sci. **32**, 785–790 (1984) [CrossRef]

7. A.T. Young, “Revised depolarization corrections for atmospheric extinction,” Appl. Opt. **19**, 3427–3428 (1980) [CrossRef] [PubMed]

*β*and the backscatter signal intensity

*i*

*c*is a system constant,

*z*the scattering altitude, and

*T*the atmospheric transmission.

*S*is defined as the ratio of total (molecular plus aerosol) backscatter coefficient and molecular backscatter coefficient

*β*

_{‖},

*β*

_{⊥}and

*β*

_{T}≡

*β*

_{‖}+

*β*

_{⊥}, respectively.

*δ*

^{V}(≡

*δ*) and aerosol depolarization

*δ*

^{A}are given by the ratio of perpendicular and parallel backscatter signals

9. G. Beyerle, “Untersuchungen polarer stratosphärischer Aerosole vulkanischen Ursprungs und polarer stratosphärischer Wolken mit einem Mehrwellenlängen-Lidar auf Spitzbergen (79° N, 12° O),” Berichte zur Polarforschung 138/’94, Alfred-Wegener-Institut für Polar- und Meeresforschung, Bremerhaven (1994)

## 2 Method

*F*,

*F*

*F*can be calculated according to

*F*=

*F*

_{p}

*F*

_{s}

*F*

_{s}: Mueller matrix of the atmospheric scattering process

*F*

_{p}: Mueller matrix of the analyzer optics.

*α*, i.e.

*α*, 0,0) where (|

*α*|≪1). This includes to first order the effects of emitter optics, e.g. reflections on mirrors with polarization dependent reflectivity.

*F*

_{s}in backward direction is diagonal. It may be written as:

*B*

^{‖,⊥}describes the cross-talk factor.

*α*=

*B*

^{‖,⊥}=0) the intensities in the two polarization channels are given by the first component of the resulting Stokes vector

*i*

_{⊥}+

*i*

_{‖}=1. For a non-ideal instrument the measured intensities are

*α*with

*B*

^{⊥}resp.

*B*

^{‖}into two independent parameters,

δ ˜

^{C}and

*δ*

^{C}, defined as:

*i*

_{‖,⊥}is given by the combination of eqns. 12, 13 and 14:

*δ*

^{C},

δ ˜

^{C},

*α*,

*B*

^{‖,⊥}≪1). For nonchiral scatterers depolarization in the backward direction does not exceed 100% and thus

*β*

_{⊥}

*β*

_{‖}(see e.g. [10

10. M. Mishchenko and J. Hovenier, “Depolarization of light backscattered by randomly oriented non-spherical particles,” Optics Letters **20**, 1356–1358 (1995) [CrossRef] [PubMed]

δ ˜

^{C})

*β*

_{‖}≫

δ ˜

^{C}

*β*

_{⊥},

*δ*

^{C}.

### 2.1 Correction formula for cross-polarized backscatter ratio

*S*

_{⊥},

*S*

_{‖}is derived [11]. In terms of the measured backscatter coefficients

*δ*

^{C}

*δ*

^{R}and solving for

*S*

_{⊥}and

*S*

_{‖}we arrive at the final result

*δ*

^{C}we assume

*S*

_{⊥}=1, i.e. a cloud consisting of liquid (assumed to be spherical, i.e. non-depolarizing) particles is observed. From eqn. 18 it follows

*δ*

^{C}

*δ*

^{R}. The correction factor

*δ*

^{C}is obtained by linear regression of the measured, uncorrected backscatter ratios

*δ*

^{C}can be calculated from both, slope

*δ*

^{C}/(

*δ*

^{C}+

*δ*

^{R}) and intercept

*δ*

^{R}/(

*δ*

^{C}+

*δ*

^{R}). Alternatively, eqn. 21 can be solved for

*δ*

^{C},

*δ*

^{V}could be obtained from

*S*

_{‖,⊥}and eqn. 7 as well, in practice

*δ*

^{V}should be calculated from the definition

*k*is determined by imposing the condition

*δ*

^{R}at aerosol-free altitudes. Thus,

*k*=

*δ*

^{R}/(

*δ*

^{C}+(1-

*δ*

^{C})

*δ*

^{R}). Inserting this result

*δ*

^{C}is suficiently small, i.e. of the same order of magnitude as

*δ*

^{R}.

## 3 Example

*δ*

^{R}=1.44% as discussed above.

*δ*

^{C}using atmospheric observations as follows: data with a strictly linear correlation between minimum

*δ*

^{C}is then calculated from the instrumental contribution to the perpendicular channel. As an example, the (

*δ*

^{C}. Then, knowledge of

*δ*

^{C}is used to calculate

*S*

_{⊥}and

*δ*

^{A}for all data of a measurement period according to eqn. 20. Now all those data points which are consistent with

*S*

_{⊥}=1,

*δ*

^{A}=0 within the uncertainty limits are selected and the linear regression is applied to the restricted data set, yielding an improved value of

*δ*

^{C}; the procedure is iterated once more to check convergence. From this example we calculate a overall system depolarization of

*δ*

^{C}=2.17%. We conclude that manufacturers’ specifications for laser depolarization and imperfections of the polarizers should in general not be relied upon. Possible sources of uncertainty are (a) degradation of dielectric coatings leading to changes in optical properties, (b) unknown polarization-dependent properties of transmitter and detector optics (c) misalignment of the detector polarization plane with respect to the transmitter polarization plane. Note that other possible instrumental error sources (e.g. lidar signal contamination from other lidar beams) can be present in lidar polarization measurements that cannot be treated with the same correction factor such as done here, but can be avoided experimentally.

## 4 Summary and conclusions

## Acknowledgments

## References and links

1. | S. R. Pal and A. I. Carswell, “Polarization properties of lidar backscattering from clouds,” Appl. Opt. |

2. | C. M. R. Platt, “Lidar observations of a mixed-phase altostratus cloud,” J. Appl. Meteorol. , |

3. | K. Sassen, “Depolarization of laser light backscattered by artificial clouds,” J. Appl. Meteorol. , |

4. | R. M. Schotland, R. M. Sassen, K. Stone, and R. J., “Observations by lidar of linear depolarization ratios by hydrometeors,” J. Appl. Meteorol. , |

5. | H. C. van de Hulst, “Light Scattering by Small Particles,” Dover Publications, New York(1957) |

6. | D.R. Bates, “Rayleigh Scattering by Air,” Planet. Space Sci. |

7. | A.T. Young, “Revised depolarization corrections for atmospheric extinction,” Appl. Opt. |

8. | A.T. Young, “Rayleigh Scattering,” Appl. Opt. |

9. | G. Beyerle, “Untersuchungen polarer stratosphärischer Aerosole vulkanischen Ursprungs und polarer stratosphärischer Wolken mit einem Mehrwellenlängen-Lidar auf Spitzbergen (79° N, 12° O),” Berichte zur Polarforschung 138/’94, Alfred-Wegener-Institut für Polar- und Meeresforschung, Bremerhaven (1994) |

10. | M. Mishchenko and J. Hovenier, “Depolarization of light backscattered by randomly oriented non-spherical particles,” Optics Letters |

11. | G. Baumgarten, “Erste Messungen des Bonner Rayleigh/Mie/Raman-Lidar auf Esrange, Schweden, zur Untersuchung von dynamisch induzierten polaren Stratosphärenwolken im Januar 1997,” Diploma thesis, IB-97-26 University of Bonn, Germany (1997) |

12. | J. Biele, “Polare stratosphärische Wolken: Lidar-Beobachtungen, Charakterisierung von Entstehung und Entwicklung,” Berichte zur Polarforschung 03/’99, Alfred-Wegener-Institut für Polar- und Meeresforschung, Bremerhaven (1999) |

13. | J. Biele, A. Tsias, B. P. Luo, K. S. Carslaw, R. Neuber, G. Beyerle, and Th. Peter, “Non-equilibrium co-existence of Solid and Liquid Particles in Arctic Stratospheric Clouds,” J. Geophy. Res. submitted (2000) |

**OCIS Codes**

(010.3640) Atmospheric and oceanic optics : Lidar

(280.3640) Remote sensing and sensors : Lidar

(290.1350) Scattering : Backscattering

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 13, 2000

Published: December 4, 2000

**Citation**

Jens Biele, Georg Beyerle, and Gerd Baumgarten, "Polarization Lidar: Correction of instrumental effects," Opt. Express **7**, 427-435 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-12-427

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### References

- Pal, S. R. and Carswell, A. I., "Polarization properties of lidar backscattering from clouds," Appl. Opt. 12, 1530-1535 (1973) [CrossRef] [PubMed]
- Platt, C. M. R., "Lidar observations of a mixed-phase altostratus cloud," J. Appl. Meteorol. 16, 339-345 (1977). [CrossRef]
- Sassen, K., "Depolarization of laser light backscattered by arti_cial clouds," J. Appl. Meteorol. 13, 923-933 (1973). [CrossRef]
- R. M. Schotland, R. M., Sassen, K., and Stone, R. J., "Observations by lidar of linear depolariz- ation ratios by hydrometeors," J. Appl. Meteorol. 10, 1011-1017 (1971) [CrossRef]
- H. C. van de Hulst, "Light Scattering by Small Particles," (Dover Publications, New York, 1957).
- Bates, D.R., "Rayleigh Scattering by Air," Planet. Space Sci. 32, 785-790 (1984) [CrossRef]
- Young, A.T., "Revised depolarization corrections for atmospheric extinction," Appl. Opt. 19, 3427-3428 (1980) [CrossRef] [PubMed]
- Young, A.T., "Rayleigh Scattering," Appl. Opt. 20, 533-535 (1981). [CrossRef] [PubMed]
- Beyerle, G., "Untersuchungen polarer stratospharischer Aerosole vulkanischen Ursprungs und polarer stratosph�arischer Wolken mit einem Mehrwellenlangen-Lidar auf Spitzbergen (79 N, 12 O)," Berichte zur Polarforschung 138/'94, Alfred-Wegener-Institut fur Polar- und Meeresforschung, Bremerhaven (1994).
- Mishchenko, M., Hovenier, J., "Depolarization of light backscattered by randomly oriented nonspherical particles," Optics Letters 20, 1356-1358 (1995) [CrossRef] [PubMed]
- Baumgarten, G., "Erste Messungen des Bonner Rayleigh/Mie/Raman-Lidar auf Esrange, Schweden, zur Untersuchung von dynamisch induzierten polaren Stratosph�arenwolken im Januar 1997," Diploma thesis, IB-97-26 University of Bonn, Germany (1997).
- Biele, J., "Polare stratospharische Wolken: Lidar-Beobachtungen, Charakterisierung von Entstehung und Entwicklung," Berichte zur Polarforschung 03/'99, Alfred-Wegener-Institut fur Polarund Meeresforschung, Bremerhaven (1999).
- Biele, J., A. Tsias, B. P. Luo, K. S. Carslaw, R. Neuber, G. Beyerle, Th. Peter, "Non-equilibrium co-existence of Solid and Liquid Particles in Arctic Stratospheric Clouds," J. Geophy. Res. submitted (2000).

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