Abstract

An analytical solution to polarized lidar return, including multiple scattering from stratified media with large scatters, is developed and discussed.

© 1999 Optical Society of America

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References

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  1. E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B: Lasers Opt. 60, 345–353 (1995).
    [CrossRef]
  2. E. P. Zege, I. L. Katsev, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
    [CrossRef]
  3. B. V. Kaul, I. V. Samokhvalov, “Equation of the atmosphere laser sounding in the secondary scattering approximation with including polarization effects,” Izv. Vyssh. SSSR Fiz. N 1, 80–85 (1976).
  4. V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Investigation of polarization characteristics of backscatter signals in laser sounding of clouds,” in Laser Sounding of the Atmosphere (Nauka, Moscow, 1976), pp. 29–46.
  5. G. M. Krekov, M. M. Krekova, “Distinctive features of polarized laser sounding in the atmosphere–ocean system,” Opt. Atmosferi. 2, 73–79 (1989).
  6. P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
    [CrossRef]
  7. V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).
  8. C. Flesia, A. V. Starkov, “Polarization signals from mixed phase clouds for spaceborne lidar,” in Proceedings of the 9th International Workshop on Multiple Scattering Lidar Experiment (Hebrew University, Jerusalem, Israel, 1997), pp. 26–27.
  9. E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19–31 (1996).
    [CrossRef]
  10. I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
    [CrossRef]
  11. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, also Dover, New York, 1981).
  12. E. P. Zege, L. I. Chaikovskaya, “Approximation theory of linearly polarized radiation propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transf. (to be published).
  13. D. Deirmendjan, Electromagnetic Scattering on Spherical Polydispersion (Elsvier, New York, 1962).

1997 (2)

1996 (1)

E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19–31 (1996).
[CrossRef]

1995 (2)

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B: Lasers Opt. 60, 345–353 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
[CrossRef]

1989 (1)

G. M. Krekov, M. M. Krekova, “Distinctive features of polarized laser sounding in the atmosphere–ocean system,” Opt. Atmosferi. 2, 73–79 (1989).

1976 (1)

B. V. Kaul, I. V. Samokhvalov, “Equation of the atmosphere laser sounding in the secondary scattering approximation with including polarization effects,” Izv. Vyssh. SSSR Fiz. N 1, 80–85 (1976).

Arshinov, Yu.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Bruscaglioni, P.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
[CrossRef]

Chaikovskaya, L. I.

E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19–31 (1996).
[CrossRef]

E. P. Zege, L. I. Chaikovskaya, “Approximation theory of linearly polarized radiation propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

Deirmendjan, D.

D. Deirmendjan, Electromagnetic Scattering on Spherical Polydispersion (Elsvier, New York, 1962).

Flesia, C.

C. Flesia, A. V. Starkov, “Polarization signals from mixed phase clouds for spaceborne lidar,” in Proceedings of the 9th International Workshop on Multiple Scattering Lidar Experiment (Hebrew University, Jerusalem, Israel, 1997), pp. 26–27.

Gai, M.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
[CrossRef]

Gurioli, M.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
[CrossRef]

Herrmann, H.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Ismaelli, A.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
[CrossRef]

Katsev, I. L.

Kaul, B. V.

B. V. Kaul, I. V. Samokhvalov, “Equation of the atmosphere laser sounding in the secondary scattering approximation with including polarization effects,” Izv. Vyssh. SSSR Fiz. N 1, 80–85 (1976).

Kaul, V.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Krasting, H.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Krekov, G. M.

G. M. Krekov, M. M. Krekova, “Distinctive features of polarized laser sounding in the atmosphere–ocean system,” Opt. Atmosferi. 2, 73–79 (1989).

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Investigation of polarization characteristics of backscatter signals in laser sounding of clouds,” in Laser Sounding of the Atmosphere (Nauka, Moscow, 1976), pp. 29–46.

Krekova, M. M.

G. M. Krekov, M. M. Krekova, “Distinctive features of polarized laser sounding in the atmosphere–ocean system,” Opt. Atmosferi. 2, 73–79 (1989).

Matvienko, G. G.

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Investigation of polarization characteristics of backscatter signals in laser sounding of clouds,” in Laser Sounding of the Atmosphere (Nauka, Moscow, 1976), pp. 29–46.

Oppel, U.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Polonsky, I. N.

Popkov, A. I.

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Investigation of polarization characteristics of backscatter signals in laser sounding of clouds,” in Laser Sounding of the Atmosphere (Nauka, Moscow, 1976), pp. 29–46.

Prikhach, A. S.

Romashov, D.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Samokhvalov, I.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Samokhvalov, I. V.

B. V. Kaul, I. V. Samokhvalov, “Equation of the atmosphere laser sounding in the secondary scattering approximation with including polarization effects,” Izv. Vyssh. SSSR Fiz. N 1, 80–85 (1976).

Starkov, A. V.

C. Flesia, A. V. Starkov, “Polarization signals from mixed phase clouds for spaceborne lidar,” in Proceedings of the 9th International Workshop on Multiple Scattering Lidar Experiment (Hebrew University, Jerusalem, Israel, 1997), pp. 26–27.

Streicher, J.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, also Dover, New York, 1981).

Werner, Ch.

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

Zaccanti, G.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
[CrossRef]

Zege, E. P.

E. P. Zege, I. L. Katsev, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19–31 (1996).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B: Lasers Opt. 60, 345–353 (1995).
[CrossRef]

E. P. Zege, L. I. Chaikovskaya, “Approximation theory of linearly polarized radiation propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

Zuev, V. E.

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Investigation of polarization characteristics of backscatter signals in laser sounding of clouds,” in Laser Sounding of the Atmosphere (Nauka, Moscow, 1976), pp. 29–46.

Appl. Phys. B: Lasers Opt. (1)

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B: Lasers Opt. 60, 345–353 (1995).
[CrossRef]

Izv. Vyssh. SSSR Fiz. N (1)

B. V. Kaul, I. V. Samokhvalov, “Equation of the atmosphere laser sounding in the secondary scattering approximation with including polarization effects,” Izv. Vyssh. SSSR Fiz. N 1, 80–85 (1976).

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

J. Quant. Spectrosc. Radiat. Transf. (1)

E. P. Zege, L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19–31 (1996).
[CrossRef]

Opt. Atmosferi. (1)

G. M. Krekov, M. M. Krekova, “Distinctive features of polarized laser sounding in the atmosphere–ocean system,” Opt. Atmosferi. 2, 73–79 (1989).

Opt. Rev. (1)

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, M. Gai, M. Gurioli, “Polarization of lidar return from water clouds. Calculations and laboratory scaled measurements,” Opt. Rev. 2, 312–318 (1995).
[CrossRef]

Other (6)

V. Kaul, Yu. Arshinov, D. Romashov, I. Samokhvalov, Ch. Werner, J. Streicher, H. Herrmann, U. Oppel, H. Krasting, Crystal Clouds (Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk, Russia, 1997).

C. Flesia, A. V. Starkov, “Polarization signals from mixed phase clouds for spaceborne lidar,” in Proceedings of the 9th International Workshop on Multiple Scattering Lidar Experiment (Hebrew University, Jerusalem, Israel, 1997), pp. 26–27.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, also Dover, New York, 1981).

E. P. Zege, L. I. Chaikovskaya, “Approximation theory of linearly polarized radiation propagation through a scattering medium,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

D. Deirmendjan, Electromagnetic Scattering on Spherical Polydispersion (Elsvier, New York, 1962).

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Investigation of polarization characteristics of backscatter signals in laser sounding of clouds,” in Laser Sounding of the Atmosphere (Nauka, Moscow, 1976), pp. 29–46.

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Figures (2)

Fig. 1
Fig. 1

Geometry of (a) forward and (b) backward single scattering at a point R as projected onto the plane XOY perpendicular to the Z axis. Here |n|=sin ϑ1; (a) |n|=sin ϑ1, (b) |n|=sin(π-ϑ)1. Dashed lines denote projections of reference planes for the Stokes vectors S(n) and S(n), which are parallel to the XOZ plane. The angle of rotation of the reference plane for S(n) onto the scattering plane is α, and the angle of rotation of the scattering plane onto the reference plane for S(n) is (-α) and α [(a) and (b), respectively]. The rotation angle is positive when it is measured clockwise as viewed opposite the beam.

Fig. 2
Fig. 2

Degree of depolarization [Eq. (56)] of the lidar return from cloud C1 at λ=1.064 µm as a function of sounding depth computed through Eq. (51) (solid curves) in comparison with Monte Carlo data (symbols). The lidar–cloud distance is 1000 m, the cloud extinction coefficient is 0.02 m-1, and the receiver field of view is 0.005 (curve 1) and 0.0005 (curve 2).

Equations (70)

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Ssrc(R 0, n0)=PΦsrc(R0, n0)
Srec(R, n)=AΦrec(R, n),
W(t)=ATBˆ(t)P,
Fˆ(x)
=a1(x)b1(x)b1(x)a+(x)+a-(x)a+(x)-a-(x)-b2(x)b2(x)a4(x),
12-11a1(x)dx=1,
|Fik(x)|a1(x).
a±(x)=12[a2(x)±a3(x)],
bja1θ2,a-a1θ4ifθ1.
bja1(π-θ)2,j=1,2,a+a1(π-θ)4,
ifπ-θ1.
Dˆf(ϑ1, ϑ1, ψ-ψ, ψ)
=Lˆ(-α)Fˆ(x)Lˆ(α)=Fˆf0(x)+Lˆ(ψ)
×Lˆ(-i)0b1(x)00b1(x)a-(x)0000-a-(x)-b2(x)00b2(x)0Lˆ(i)×Lˆ(-ψ),
Fˆf0(x)=diag{a1(x), a+(x), a+(x), a4(x)}.
Dˆb(π-ϑ1, ϑ1, ψ-ψ, ψ)
=Lˆ(α)Fˆ(x)Lˆ(α)=Fˆb0(x)+Lˆ(-ψ)
×Lˆ(i)0b1(x)00b1(x)a+(x)0000a+(x)-b2(x)00b2(x)0Lˆ(i)
×Lˆ(-ψ),
Fˆb0(x)=diag{a1(x), a-(x),-a-(x), a4(x)},
x=cos ϑ cos ϑ+sin ϑ sin ϑ cos(ψ-ψ).
Lˆ(α)=10000cos 2α-sin 2α00sin 2αcos 2α00001
sin i=±sin ϑ1-x2sin(ψ-ψ),
12π02πd(ψ-ψ)Dˆb(π-ϑ1, ϑ1, ψ-ψ, ψ)
=Fˆb0(x)¯+Lˆ(-ψ)0B¯100B¯1A¯+0000A¯+-B¯200B¯20Lˆ(-ψ).
14π202πd(ψ-ψ)×02πdψDˆb(π-ϑ1, ϑ1, ψ-ψ, ψ)
=Fˆb0(x)¯=diag{a1(x)¯, a-(x)¯, -a-(x)¯, a4(x)¯},
S(z, r, n)=Jˆsrc(z, r, n)P,
Jˆsrc(z, r, n)=dr0dn 0Gˆ(z, r, n; z0=0, r0, n0)Φsrc(r0, n0),
Gˆ=G11G12G13G14G21G++G-G++G-G24G31-G++G-G+-G-G34G41G42G43G44.
dS(z, r, n)=σs(z)dzdnDˆb(z, n, n)S(z, r, n),
Wt=2zυ=υAT2drdnJˆrec(z, r, n)×dS(z, r, n)dz,
Jˆrec(z, r, n)=drdnΦrec(r, n)Gˆ(0, r, n; z, r, n)
Gˆ(0, r, n; z, r, n)=NˆGˆT(z, r,-n; 0, r,-n)Nˆ,
Jˆrec(z, r, n)=Nˆ[Jˆsrcrec(z, r, n)]TNˆ,
Jˆsrcrec(z, r, n)
=drdnΦsrcrec(r, n)Gˆ(z, r,-n; 0, r, n).
Bˆ(z)=υ2σsdrdndnNˆ[Jˆsrcrec(z, r, n)]T×NˆDˆb(-n, n)Jˆsrc(z, r, n).
Bˆ(z)=υ2σs exp-20z(z)dz×dr0dn0drdn×Φsrc(r0, n0)Φrec(r, n)Dˆb(n, n0),
Bˆ(z)=diag{B1(z), B-(z),-B-(z), B4(z)},
Bν(z)=υ2σs exp-20z(z)dz×dr0dn0drdn×Φsrc(r0, n0)Φrec(r, n)aν(x),ν=1,-, 4.
Jˆsrc=J1100J140J+J+00-J+J+0J4100J44+Lˆ(ψ)0J12J130J21J-J-J24J31J--J-J340J42J430Lˆ(-ψ),
Jˆsrc0(z, r, n)=diag{J11(z, r, n), J+(z, r, n),J+(z, r, n), J44(z, r, n)},
Jˆsrc0(z, r, n)=dr0dn0Gˆ0(z, r, n; 0, r0, n0)×Φsrc(r0, n0),
Gˆ0=diag{G1, G+, G+, G4},
Jˆsrc0,rec(z, r, n)=drdnGˆ0(z, r,-n; 0, r, n)×Φsrcrec(r, n),
R{Gν(z, r, n)}=σs4πdnaν(n-n)Gν(z, r, n)+δ(n-n0)δ(r-r0),
ν=1,+4,
R=nR+σe
Bˆ(z)=diag{B11(z), B-(z),-B-(z), B44(z)}.
Bˆ(z)=υ2σsdrdndnJˆsrc0,rec(z, r, n)×Fˆb0(-n, n)Jˆsrc0(z, r, n).
Wt=2zc=12(1, 1, 0, 0)Bˆ(z)1100.
Wt=2zc=12(1,-1, 0, 0)Bˆ(z)1100.
W+W=B11(z).
W-W=B-(z).
pLt=2zυ=W(t)-W(t)W(t)+W(t)=B-(z)B11(z).
Wrt=2zυ=12(1, 0, 0, 1)Bˆ(z)1001,
Wlt=2zυ=12(1, 0, 0,-1)Bˆ(z)1001,
Wr-Wl=B44(z).
pct=2zυ=Wr(t)-Wl(t)Wr(t)+Wl(t)=B44(z)B11(z).
Bii(z)=c2σsdr0dn 0Φsrc(r0, n0)×drdnΦsrcrec(r, n)drdndn×Gii0(z, r, n; 0, r, n)Fb,ii0(-n, n)×Gii0(z, r, n; 0, r0, n0), i=1, 2, 4,
Biit=2zυ=υσs2Jii,src0,eff(z, r=0, n)Fb,ii0(n)4πdn,
i=1, 2, 4,
Jii,src0,eff(z, r=0, n)
=dr0dn0Φsrceff(r0, n0)×Gii0,eff(z, r=0, n; 0; r0, n0).
Φsrceff(r0, n0)=drΦsrc(r, n)×Φrec(r+r, n+n)dn,
(nR+σeeff)Gii0,eff(z, r, n)
=σseff4πdnFf,ii0(n-n)Gii0,eff(z, r, n)+δ(n-n0)δ(r-r0), i=1, 2, 4.
σseff(z)=2σs(z),σeeff(z)=2σe(z)
δ(z)=W(z)W(z)=1-2B-(z)B11(z)+B-(z).

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