Abstract

This paper analyzes the signal-to-noise ratio for a coaxial laser system that heterodynes the signal backscattered from the atmospheric aerosol. The laser radiation, which is assumed to have a wavefront with a gaussian amplitude distribution, is transmitted into the atmosphere through a telescope. Radiation is collected by the same telescope and directed onto a detector where it is mixed with a local oscillator beam originating from the same laser source. The signal-to-noise ratio at the output of the detector is calculated under shot noise limited conditions. The calculation is general and applies for both near and far fields and for focused and unfocused systems. Three specific cases are considered. These are a pulsed system, a cw system illuminating an infinite target, and a cw system illuminating a target of finite extent.

© 1971 Optical Society of America

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References

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  1. R. M. Huffaker, A. V. Jelalian, J. A. L. Thomson, Proc. IEEE 58, 322 (1970).
    [CrossRef]
  2. R. M. Huffaker, Appl. Opt. 9, 1026 (1970).
    [CrossRef] [PubMed]
  3. A. V. Jelalian, R. M. Huffaker, Specialist Conference on Molecular Radiation and Its Application to Diagnostic Techniques, NASA TM X-53711, p. 345, October5–6, 1967.
  4. D. E. Kerr, Propagation of Short Radio Waves (Boston Technical Publishers, Lexington, Mass., 1964).
  5. D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
    [CrossRef]
  6. D. L. Fried, J. Opt. Soc. Amer. 57, 169 (1967).
    [CrossRef]
  7. I. Goldstein, P. A. Miles, A. Chabot, Proe. IEEE 53, 1172 (1965).
    [CrossRef]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  9. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).
  10. M. Ross, Laser Receivers (Wiley, New York, 1966).
  11. K. M. van Vliet, Appl. Opt. 6, 1145 (1967).
    [CrossRef] [PubMed]

1970 (2)

R. M. Huffaker, A. V. Jelalian, J. A. L. Thomson, Proc. IEEE 58, 322 (1970).
[CrossRef]

R. M. Huffaker, Appl. Opt. 9, 1026 (1970).
[CrossRef] [PubMed]

1967 (3)

K. M. van Vliet, Appl. Opt. 6, 1145 (1967).
[CrossRef] [PubMed]

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

D. L. Fried, J. Opt. Soc. Amer. 57, 169 (1967).
[CrossRef]

1965 (1)

I. Goldstein, P. A. Miles, A. Chabot, Proe. IEEE 53, 1172 (1965).
[CrossRef]

Chabot, A.

I. Goldstein, P. A. Miles, A. Chabot, Proe. IEEE 53, 1172 (1965).
[CrossRef]

Fried, D. L.

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

D. L. Fried, J. Opt. Soc. Amer. 57, 169 (1967).
[CrossRef]

Goldstein, I.

I. Goldstein, P. A. Miles, A. Chabot, Proe. IEEE 53, 1172 (1965).
[CrossRef]

Huffaker, R. M.

R. M. Huffaker, A. V. Jelalian, J. A. L. Thomson, Proc. IEEE 58, 322 (1970).
[CrossRef]

R. M. Huffaker, Appl. Opt. 9, 1026 (1970).
[CrossRef] [PubMed]

A. V. Jelalian, R. M. Huffaker, Specialist Conference on Molecular Radiation and Its Application to Diagnostic Techniques, NASA TM X-53711, p. 345, October5–6, 1967.

Jelalian, A. V.

R. M. Huffaker, A. V. Jelalian, J. A. L. Thomson, Proc. IEEE 58, 322 (1970).
[CrossRef]

A. V. Jelalian, R. M. Huffaker, Specialist Conference on Molecular Radiation and Its Application to Diagnostic Techniques, NASA TM X-53711, p. 345, October5–6, 1967.

Kerr, D. E.

D. E. Kerr, Propagation of Short Radio Waves (Boston Technical Publishers, Lexington, Mass., 1964).

Miles, P. A.

I. Goldstein, P. A. Miles, A. Chabot, Proe. IEEE 53, 1172 (1965).
[CrossRef]

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

Ross, M.

M. Ross, Laser Receivers (Wiley, New York, 1966).

Seidman, J. B.

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

Thomson, J. A. L.

R. M. Huffaker, A. V. Jelalian, J. A. L. Thomson, Proc. IEEE 58, 322 (1970).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van Vliet, K. M.

Appl. Opt. (2)

J. Opt. Soc. Amer. (2)

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

D. L. Fried, J. Opt. Soc. Amer. 57, 169 (1967).
[CrossRef]

Proc. IEEE (1)

R. M. Huffaker, A. V. Jelalian, J. A. L. Thomson, Proc. IEEE 58, 322 (1970).
[CrossRef]

Proe. IEEE (1)

I. Goldstein, P. A. Miles, A. Chabot, Proe. IEEE 53, 1172 (1965).
[CrossRef]

Other (5)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

M. Ross, Laser Receivers (Wiley, New York, 1966).

A. V. Jelalian, R. M. Huffaker, Specialist Conference on Molecular Radiation and Its Application to Diagnostic Techniques, NASA TM X-53711, p. 345, October5–6, 1967.

D. E. Kerr, Propagation of Short Radio Waves (Boston Technical Publishers, Lexington, Mass., 1964).

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Figures (5)

Fig. 1
Fig. 1

Coordinate systems.

Fig. 2
Fig. 2

Heterodyne system configuration.

Fig. 3
Fig. 3

Alternate system configuration.

Fig. 4
Fig. 4

Variation of S/N with location of focus for pulsed system.

Fig. 5
Fig. 5

Variation of S/N with location of focus for cw system and an infinite scatterer.

Tables (1)

Tables Icon

Table I S/N Variation

Equations (32)

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ψ ( x , y , L ) = ( 2 n T ) 1 2 ( π L R λ ) 1 2 - - × exp [ i π λ L ( r - r ) 2 - i π r 2 λ f - ( r R ) 2 ] d x d y ,
- - ψ 2 d x d y = n T .
ψ ( x , y , L ) = ( 2 π n T ) 1 2 R λ L [ 1 - i π R 2 λ L ( 1 - L f ) ] exp [ - ( π R r λ L ) 2 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] × exp [ i ( k L - ω t + φ ) ] ,
φ = π r 2 λ L [ 1 - L f ( π R 2 λ L ) 2 ( 1 - L f ) 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] .
ψ R = R [ exp ( i k r ) / r ] .
R L = S 2 ψ inc L + S 3 ψ inc R ,
R R = S 4 ψ inc L + S 1 ψ inc R ,
S 3 = S 4 = 0 ,
S 1 ( π ) = S 2 ( π ) .
ψ R = S ( π ) ψ L exp { - i [ π λ f ( r ) 2 - π λ L ( r - r ) 2 - ( k L + Δ ω t ) ] } ,
i s = 2 η G e Re [ A R ψ R ψ * R E F dr ] ,
ψ R E F = ( η R E F ) 1 2 ( π R ) 1 2 exp { - [ i ω t + i φ R - ( r R ) 2 ] } ,
i s = 2 π ( 2 n T n R E F ) 1 2 R 2 S ( π ) η G e λ L 2 [ 1 + ( π R λ L ) ( 1 - L f ) 2 ] × exp { - 2 ( π R r λ L ) 2 [ 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] - 1 } × cos { 2 k L + Δ ω t - φ R + 2 φ + 2 tan - 1 [ π R 2 λ L ( 1 - L f ) ] } .
i s 2 = 4 π 2 n T n R E F R 4 S 2 ( π ) η 2 G 2 e 2 λ 2 L 2 [ 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] 2 × exp { - 4 ( π R r λ L ) 2 [ 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] - 1 } .
( i s 2 ) Total = π n T n R E F R 2 ρ S 2 ( π ) η 2 G 2 e 2 × 0 ( d L ) / L 2 [ 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] ,
β ( π ) = ρ S 2 ( π ) ,
i N 2 = 2 η G 2 e 2 n R E F B ,
S N = ( i s 2 ) Total ( i N ) 2 = η n T β ( π ) π R 2 2 B × 0 ( d L ) / L 2 [ 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] .
n T = P T / ( h ν ) ,
S N = η P T β ( π ) π R 2 2 B h ν × 0 ( d L ) / L 2 [ 1 + ( π R 2 λ L ) 2 ( 1 - L f ) 2 ] .
S N = η P T β ( π ) π R 2 Δ L 2 B h ν L A V 2 [ 1 + ( π R 2 λ L A V ) 2 ( 1 - L A V f ) 2 ] .
S N = η P T β ( π ) π R 2 Δ L 2 B h ν L A V 2 .
S N = η P T β ( π ) π R 2 Δ L 2 B h ν L A V 2 [ 1 + ( π R 2 λ L A V ) 2 ] .
S N = η P T β ( π ) λ 2 B h ν [ π 2 + tan - 1 ( π R 2 λ f ) ] .
S N = η P T β ( π ) λ π 4 B h ν .
S N = η P T β ( π ) λ 2 B h ν { tan - 1 [ λ L 2 π R 2 - π R 2 λ f ( 1 - L 2 f ) ] - tan - 1 [ λ L 1 π R 2 - π R 2 λ f ( 1 - L 1 f ) ] } .
S N = η P T β ( π ) λ 2 B h ν [ tan - 1 ( λ L 2 π R 2 ) - tan - 1 ( λ L 1 π R 2 ) ] .
( S / N ) F / ( S / N ) U n f = 1 + [ ( π R 2 ) / ( λ L A V ) ] 2 .
( S / N ) F / ( S / N ) U n f 1 + ( D F F / L A V ) 2 .
( S / N ) F / ( S / N ) U n f = 1 + 2 π tan - 1 [ ( π R 2 ) / ( λ f ) ] .
( S / N ) F / ( S / N ) U n f 1 + 2 π tan - 1 ( D F F / f ) .
S / N = { [ η P T β ( π ) λ ] / ( 2 B h ν ) } F ( R , λ , f ) ,

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