Abstract

The form factors for the atmospheric backscattering of laser light leaving the exit aperture of a lidar receiving telescope are calculated in the limiting cases of purely geometrical optics and pure TEM00 laser emission taking axial aperture displacement as well as misalignment of the transmitter axis into account. Some numerical results are plotted, displaying the effects of variations of the telescope's field of view, laser beam divergence, and misalignment.

© 1978 Optical Society of America

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References

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  1. W. Kroy, W. Rother, T. Halldórsson, “Reliable Lidar System for Automatic Measurement of Aerosol Distributions,” in Abstracts of the COST 1972 Technical Conference on Automatic Weather Stations, U. Reading, U.K. (September 1976).
  2. R. T. H. Collis, P. B. Russel, in Laser Monitoring of the Atmosphere (Springer, Berlin, 1976, Sec. 4.1, p. 76.
  3. J. Riegl, M. Bernhard, Appl. Opt. 13, 931 (1974).
    [CrossRef] [PubMed]
  4. G. Gaubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, Ed. (Pergamon, New York, 1963, p. 907.
  5. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  6. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]

1974 (1)

1966 (1)

1965 (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Bernhard, M.

Collis, R. T. H.

R. T. H. Collis, P. B. Russel, in Laser Monitoring of the Atmosphere (Springer, Berlin, 1976, Sec. 4.1, p. 76.

Gaubau, G.

G. Gaubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, Ed. (Pergamon, New York, 1963, p. 907.

Halldórsson, T.

W. Kroy, W. Rother, T. Halldórsson, “Reliable Lidar System for Automatic Measurement of Aerosol Distributions,” in Abstracts of the COST 1972 Technical Conference on Automatic Weather Stations, U. Reading, U.K. (September 1976).

Kogelnik, H.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Kroy, W.

W. Kroy, W. Rother, T. Halldórsson, “Reliable Lidar System for Automatic Measurement of Aerosol Distributions,” in Abstracts of the COST 1972 Technical Conference on Automatic Weather Stations, U. Reading, U.K. (September 1976).

Li, T.

Riegl, J.

Rother, W.

W. Kroy, W. Rother, T. Halldórsson, “Reliable Lidar System for Automatic Measurement of Aerosol Distributions,” in Abstracts of the COST 1972 Technical Conference on Automatic Weather Stations, U. Reading, U.K. (September 1976).

Russel, P. B.

R. T. H. Collis, P. B. Russel, in Laser Monitoring of the Atmosphere (Springer, Berlin, 1976, Sec. 4.1, p. 76.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Other (3)

W. Kroy, W. Rother, T. Halldórsson, “Reliable Lidar System for Automatic Measurement of Aerosol Distributions,” in Abstracts of the COST 1972 Technical Conference on Automatic Weather Stations, U. Reading, U.K. (September 1976).

R. T. H. Collis, P. B. Russel, in Laser Monitoring of the Atmosphere (Springer, Berlin, 1976, Sec. 4.1, p. 76.

G. Gaubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, Ed. (Pergamon, New York, 1963, p. 907.

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Figures (7)

Fig. 1
Fig. 1

The lidar arrangement.

Fig. 2
Fig. 2

Ray diagram for the calculation of ψ.

Fig. 3
Fig. 3

The effects of variation of the opening angle on the effective apertures, coaxial configuration.

Fig. 4
Fig. 4

The effects of variation of the opening angle on the effective apertures, biaxial configuration.

Fig. 5
Fig. 5

Effects of coplanar misalignment, biaxial configuration.

Fig. 6
Fig. 6

Effects of axial translation of the exit aperture from the focal plane.

Fig. 7
Fig. 7

The calculation of the overlap area.

Equations (61)

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P r ( z ) = P 0 c Δ τ 2 β ( z ) A ( z ) z 2 exp [ 2 0 z α ( ζ ) d ζ ]
A 0 ( z ) π R 2 A ( R + ϕ 1 z , w ( z ) ; δ ) π w 2 ( z ) ,
η c Δ τ 2 β = 1 I dP d Ω dA
dP = η · I · d Ω · dA
I ( r , z ) = P 0 π w 2 ( z ) exp [ r 2 / w 2 ( z ) ] = P 0 π w 2 ( z ) f [ r / w ( z ) ]
w 2 ( z ) = w 0 2 [ 1 + ( λ z π w 0 2 ) 2 ] = w 0 2 + ( ϕ 0 z ) 2 .
I ( r , z ) = P 0 π w 2 ( z ) θ [ w ( z ) r ] = P 0 π w 2 ( z ) f [ r / w ( z ) ] , w ( z ) = w 0 + ϕ 0 z ,
θ ( x ) 1 2 [ 1 + ( x ) ] , ( x ) = x / | x | .
d P r = η · Ω · I ( r , z ) · dA = η · π R 2 z 2 · P 0 π w 2 ( z ) f [ r / w ( z ) ] · rdrd ϑ ,
r = ( r 2 + δ 2 2 δ r cos ϑ ) 1 / 2
f ( x ) = exp ( x 2 ) , f ( x ) = θ ( 1 x ) .
A ( z ) z 2 η P 0 P r ( z ) = R 2 w 2 ( z ) 0 rdr 0 2 π d ϑ ψ ( r , z ) f [ r / w ( z ) ] .
r z = r f + z a : r = f + z a z r .
ρ f + z = ρ | z z a | : ρ = | f z z a ( z f zf ) | ρ ,
z = f 2 / ( z f ) .
r = f z r ; ρ = f z ρ ;
γ 1 + z a / f , ν | γ z a z f 2 | ,
r = γ fr z , ρ = ν f ρ z .
ψ ( r , z ) = [ A ( a , ν fR z ; γ fr z ) A ( a , ν fb z ; γ fr z ) ] / π ( ν fR z ) 2 = ( γ ν ) 2 [ A ( az γ f , ν γ R ; r ) A ( az γ f , ν γ b ; r ) ] / π R 2 .
A ( z | z a ; a , R , b ; w 0 , ϕ 0 , δ ) = ( γ / ν ) 2 A ( z | 0 ; a γ , ν R γ , ν b γ ; w 0 , ϕ 0 , δ ) .
A ( z | 0 ; λ a , λ R , λ b ; λ w 0 , λ ϕ 0 , λ δ ) = λ 4 A ( z | 0 ; a , R , b ; w 0 , ϕ 0 , δ ) ,
A ( z | z a ; a , R , b , w 0 , ϕ 0 , δ ) = 1 ( γ ν ) 2 A ( z | 0 ; a , ν R , ν b ; γ w 0 , γ ϕ 0 , γ δ ) .
A ( z ) = [ S ( R ) S ( b ) ] / w 2 ( z ) ,
S ( ρ ) 1 π 0 rdr A ( α , ρ ; r ) 0 2 π f [ r / w ( z ) ] d ϑ α az / f ϕ 1 z .
rdr 0 2 π d ϑ θ [ w ( z ) τ ] = d A [ r , w ( z ) ; δ ] ,
S ( ρ ) = 1 π 0 w ( z ) + δ A ( α , ρ ; r ) d A [ r , w ( z ) ; δ ] ,
S ( ρ ) = 0 if α + ρ δ w ( z ) ,
S ( ρ ) = π w 2 ( z ) min { α 2 , ρ 2 } if | α ρ | w ( z ) + δ ,
S ( ρ ) = w 2 ( z ) A [ α , ρ ; w ( z ) + δ ] + 1 π | α ρ | max | w + δ , α + ρ | A [ r , w ( z ) ; δ ] χ dr r ,
d A ( α , ρ ; r ) = { χ dr / r | α ρ | r α + ρ 0 otherwise ,
χ [ ( α + ρ ) 2 r 2 ] [ r 2 ( α ρ ) ] 2 = r 4 + 2 ( α 2 + ρ 2 ) ( α 2 ρ 2 ) 2 .
S ( ρ ) = 1 π 0 max | w , α + ρ | A ( α , ρ ; r ) d A ( r , w ; 0 ) = 0 r ¯ A ( α , ρ ; r ) d r 2 = r ̅ 2 A ( α , ρ ; r ̅ ) + | α ρ | r ̅ χ d r 2
S ( ρ ; δ = 0 ) = r ̅ 2 A ( α , ρ ; r ̅ ) + 1 4 ( r ̅ 2 α 2 ρ 2 ) χ + ( α ρ ) 2 Cos 1 ( α 2 + ρ 2 r ̅ 2 2 α ρ ) ,
r ̅ max { w ( z ) , α + ρ } .
1 π 0 2 π exp ( r 2 + δ 2 2 r δ cos ϑ w 2 ( z ) ) d ϑ = 2 exp [ r 2 + δ 2 w 2 ( z ) ] I 0 [ 2 r δ w 2 ( z ) ]
S ( ρ ) = 0 α + ρ d r 2 A ( α + ρ ; τ ) exp [ r 2 + δ 2 w 2 ( z ) ] I 0 [ 2 r δ w 2 ( z ) ] ,
S ( ρ ) = π w 2 ( z ) min { α 2 , ρ 2 } S [ ( α ρ ) 2 / w 2 , δ 2 / w 2 ] + | α ρ | α + ρ . . . ,
S ( x , y ) = 0 x d t exp ( t y ) I 0 [ 2 ( t y ) 1 / 2 ]
S ( x , y ) = k = 0 x k e x k ! l = 0 k y l e y l ! ,
( α + ρ ) | α ρ | 2 ρ 2 R .
δ ( z ) = [ ( δ + ϕ z ) 2 + ( ϕ z ) 2 ] 1 / 2 ,
δ ( z ) = [ ϕ + ϕ ] 1 / 2 z = z · Δ ϕ .
lim z A ( ϕ 1 z , ρ , ϕ z ) = π ρ 2 θ ( ϕ 1 ϕ )
lim z S ( ρ ) w 2 ( z ) = π ρ 2 ϕ = 0 ϕ 1 exp { [ ϕ 2 + ( Δ ϕ ) 2 ] / ϕ 0 2 } I 0 ( 2 ϕ Δ ϕ ϕ 0 2 ) d ( ϕ ϕ 0 ) 2
A ( ) = A tel exp [ ( Δ ϕ ϕ 0 ) 2 ] x = 0 ϕ 1 / ϕ 0 exp ( x 2 ) I 0 ( x Δ ϕ ϕ 0 ) d x 2 .
A ( , Δ ϕ = 0 ) = A tel { 1 exp [ ( ϕ 1 / ϕ 0 ) 2 ] } .
lim z S ( ρ ) w 2 ( z ) = ρ 2 ϕ 0 ϕ = 0 ϕ 1 d A ( ϕ , ϕ 0 ; Δ ϕ ) ,
A ( ) = A tel A ( ϕ 1 , ϕ 0 , Δ ϕ ) / π ϕ 0 2 .
A ( , Δ ϕ < | ϕ 1 ϕ 0 | ) = A tel min { ( ϕ 1 / ϕ 0 ) 2 , 1 } .
lim z z * ( γ ν ) 2 A ( α γ , ν ρ γ , r ) = lim ν 0 A ( α ν , ρ , γ r ν ) = π ρ 2 θ ( α γ r ) ,
lim z z * A ( z ) = A tel π w 2 ( z ) 0 α / γ r d r 0 2 π f [ r / w ( z ) ] d ϑ = A ( z * ) .
A ( z * ) = A tel π w 2 ( z * ) A [ α γ , w ( z * ) ; δ ] ,
A [ z * | δ < | α γ w ( z * ) | ] = A tel min { 1 , [ α γ w ( z * ) ] 2 } .
R = 0.175 m b = 0.04 m ϕ 1 = 7.5 × 10 4 rad δ = 0 m / 0.2 m w 0 = 0.01 m ϕ 0 = 5 × 10 4 rad .
A ( r 1 , r 2 ; r ) = 0 if r r 1 + r 2 A ( r 1 , r 2 ; r ) = π min { r 1 2 , r 2 2 } if r | r 1 r 2 | .
A 1 = r 1 2 Cos 1 ( x * r 1 ) x * y * ,
A ( r 1 , r 2 ; r ) = r 1 2 Cos 1 ( x * r 1 ) + r 2 2 Cos 1 ( r 1 x * r 2 ) r y * .
x * = ( r 2 + r 1 2 r 2 2 ) / 2 , y * = χ / 2 r ,
χ [ ( r 1 + r 2 ) 2 r 2 ] [ r 2 ( r 1 r 2 ) 2 ] = r 4 + 2 ( r 1 2 + r 2 2 ) r 2 ( r 1 2 r 2 2 ) 2
A ( r 1 , r 2 ; r ) = r 1 2 Cos 1 ( r 2 + r 1 2 r 2 2 2 r r 1 ) + r 2 2 Cos 1 ( r 2 + r 2 2 r 1 2 2 r r 2 ) 1 2 { [ ( r 1 + r 2 ) 2 r 2 ] [ r 2 ( r 1 r 2 ) 2 ] } 1 / 2 .
d d r A ( r 1 , r 2 , r ) = 2 r e [ y * ] = θ ( χ ) χ / r ,

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