Abstract

Expressions for the frequency power spectral density and power spectral density SNR of a heterodyne detection system are derived assuming the signal radiation field results from a laser illuminated, moving rough object. The calculated spectral density is attenuated at large spectral spread frequencies (resulting from target rotation) by averaging of the high spatial frequency speckle pattern by the finite extent (aperture) of the local oscillator field. Optimum signal detection is obtained only at the Doppler shifted frequency. Spatial anisotropy of the radiation field mutual coherence function precludes exact calculations for arbitrary local oscillator field distributions.

© 1978 Optical Society of America

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References

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  1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).
  2. H. T. Yura, Appl. Opt. 13, 150 (1974).
    [CrossRef] [PubMed]
  3. J. C. Leader, J. Opt. Soc. Am. 67, 1091 (1977).
    [CrossRef]
  4. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  5. A. E. Siegman, Proc. IEEE 54, 1350 (1966).
    [CrossRef]

1977

1974

1967

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

1966

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).

Fried, D. L.

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Leader, J. C.

Siegman, A. E.

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[CrossRef]

Yura, H. T.

Appl. Opt.

J. Opt. Soc. Am.

Proc. IEEE

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[CrossRef]

Other

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).

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Figures (1)

Fig. 1
Fig. 1

Scattering geometry.

Equations (58)

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x ( t ) = η Ge h ν ̅ · 1 2 z × A d 2 r { [ E s ( r , t ) + E l ( r , t ) ] · [ E s ( r , t ) + E l ( r , t ) ] * } ,
P s = 1 2 A d 2 r Re ( E s × H s ) · n = 1 2 z A d 2 r | E s | 2 ,
P l = 1 2 A d 2 r Re ( E l × H l ) · n = 1 2 z A d 2 r | E l | 2 ,
x ( t ) = α { 2 [ P s ( t ) + P l ] + 1 z A d 2 r 2 Re [ E s ( r , t ) · E l ( r , t ) ] } .
x d c 2 α P l ,
α = ( η Ge 2 h ν ̅ )
x IF ( t ) = α z A d 2 r 2 Re [ E s ( r , t ) · E l ( r , t ) ] .
E s ( r , t ) = { [ E su ( r , t ) ] î + [ E s υ ( r , t ) ] ĵ } exp ( i ω ̅ t ) ,
E l ( r , t ) = [ E lu ( r ) ] î exp ( i ω t ) ,
r = u î + υ ĵ ,
Re [ E s ( r , t ) · E l ( r , t ) ] = | E su ( r , t ) | | E lu ( r ) | cos [ ( ω ̅ ω ) t + ϕ ( r , t ) ] ,
x IF ( t ) x IF ( t ) = 4 α 2 z 2 A d 2 r d 2 r | E su ( r , t ) | | E su ( r , t ) | | E lu ( r ) | | E lu ( r ) | × [ cos Δ ω D t cos Δ ω D t cos ϕ ( r , t ) cos ϕ ( r , t ) cos Δ ω D t sin Δ ω D t cos ϕ ( r , t ) sin ϕ ( r , t ) sin Δ ω D t cos Δ ω D t sin ϕ ( r , t ) cos ϕ ( r , t ) + sin Δ ω D t sin Δ ω D t sin ϕ ( r , t ) sin ϕ ( r , t ) ,
cos [ Δ ω D ( t + t ) ] = 0
sin [ Δ ω D ( t + t ) ] = 0 ,
[ . . . . ] = 1 2 [ cos Δ ω D τ cos ϕ ( r , t ) cos ϕ ( r , t ) + sin Δ ω D τ cos ϕ ( r , t ) sin ϕ ( r , t ) sin Δ ω D τ sin ϕ ( r , t ) cos ϕ ( r , t ) + cos Δ ω D τ sin ϕ ( r , t ) sin ϕ ( r , t ) ] = 1 2 cos [ Δ ω D τ + ϕ ( r , t ) ϕ ( r , t ) ] ,
τ = t t ,
uu ( r , r , t , t ) = E su ( r , t ) E su * ( r , t ) = | E su ( r , t ) | | E su ( r , t ) | cos [ ϕ ( r , t ) ϕ ( r , t ) ] ,
x IF ( t ) x IF ( t τ ) = 2 α 2 z 2 Re A d 2 r d 2 r uu ( r , τ ) × E lu ( r ) E lu * ( r ) exp ( i Δ ω D τ ) ,
uu ( r , r , t , t ) = uu ( r , τ ) .
A E ( r ) = A d 2 r + E lu ( r + + r / 2 ) E lu * ( r + r / 2 ) ,
r + = ( r + r ) / 2 = average coordinate ,
x IF ( t ) x IF ( t τ ) = Re [ exp ( i Δ ω D τ ) 2 α 2 z 2 d 2 r A E ( r ) uu ( r , τ ) ] .
xx δ ( δ ) = xx δ ( ω × R τ / R r / R ) ,
x IF ( t ) x IF ( t τ ) = Re [ exp ( i Δ ω D τ ) 2 α 2 R 2 z 2 d δ u d δ υ A E ( δ u , δ υ ) × uu δ ( ω υ τ δ u , ω u τ δ υ ) ] .
A E ( δ u , δ υ ) a E ( δ u ) a E ( δ υ ) ,
x IF ( t ) x IF ( t τ ) Re [ exp ( i Δ ω D τ ) 2 α 2 R 2 z 2 d δ u d δ υ × uu δ ( ω τ δ u , δ υ ) a E ( δ u ) a E ( δ υ ) ] .
L υ / R < Δ υ ,
uu δ ( δ u , δ υ ) = exp ( 1 ) uu δ ( δ u , 0 ) ,
d δ υ uu δ ( ω τ δ u , δ υ ) a E ( δ υ ) uu δ ( ω τ δ u , 0 ) d δ υ a E ( δ υ ) = xx δ ( ω τ δ u , 0 ) â E ( ζ υ = 0 ) ,
â E ( ζ υ ) = exp ( i ζ υ δ υ ) a E ( δ υ ) d δ υ .
uu δ ( δ u , 0 ) = d ζ u , d ζ υ exp ( + i ζ u δ u ) W uu ( ζ u , ζ υ ) .
S ( ω ) = d τ exp ( i ω τ ) x IF ( t ) x IF ( t τ ) = 2 α 2 R 2 ω z 2 δ ( ω Δ ω D ) â E ( ζ υ = 0 ) â E ( ζ u = ω / ω ) × d ζ υ W uu ( ω / ω , ζ υ ) ,
S ( ω ) = 2 α 2 R 2 ω z 2 1 4 π R 2 I δ ( ω Δ ω D ) â E ( ζ υ = 0 ) â E ( ζ u = ω / ω ) × λ d υ p d 2 σ u u ( ζ u = ω / ω , ζ υ = υ p / λ , ν ̅ ) | proj d u p d υ p ,
I = d ν Î ( ν ) d ν
S ( ω ) = 2 α 2 R 2 z 2 a E ( ζ υ = 0 ) â E ( ζ u = ω / ω ) W ( ω ) .
S ( ω ) = 2 α 2 z ( 2 P l ) A × λ I 4 π R 2 ω δ ( ω Δ ω D ) d u σ u ( ω ; ω ̅ ) du p × sinc 2 ( L u ω / ω R ) ,
S ( ω ) = 4 α 2 P l z A λ I 4 π R 2 ω δ ( ω Δ ω D ) d u σ u ( ω ; ω ̅ ) du p ψ ( ω ) ,
S shot ( ω ) d ω = G 2 [ 2 e ( x dc / G ) d ω ] = 2 eG × 2 α P l d ω ,
SNR ( ω ) = S sig ( ω ) / S shot ( ω ) ,
SNR ( ω ) = α eGz A λ I 4 π R 2 ω δ ( ω Δ ω D ) d u σ u ( ω ; ω ̅ ) du p ψ ( ω ) .
P i = dP i dS = | E i | 2 2 z = incident power / unit area = I / 2 z ,
P i = P t / π β 2 R 2 ,
SNR ( ω ) = η h ν ̅ A 4 π 2 λ P t β 2 R 4 ω δ ( ω Δ ω D ) d u σ u ( ω ; ω ̅ ) du p ψ ( ω ) .
ω max = D ω / λ ,
L u D / λ R < 1
R > L u D / λ
E l ( r ) = E l Rect ( u L u ) Rect ( υ L υ ) ,
Rect ( x ) = { 1 | x | < 1 0 | x | > 1 .
A E ( r ) = E l 2 du + d υ + × Rect [ ( u + + u / 2 ) / L ] Rect [ ( u + u / 2 ) / L ] × Rect [ ( υ + + υ / 2 ) / L ] Rect [ ( υ + + υ / 2 ) / L ] = E l 2 4 L u L υ ( 1 u / 2 L u ) ( 1 υ / 2 L υ ) = E l 2 4 L u L υ ( 1 δ u / 2 δ Lu ) ( 1 δ υ / 2 δ L υ ) = a E ( δ u ) a E ( δ υ ) ,
a E ( δ u ) = 2 E l L u ( 1 δ u / 2 δ Lu ) , a E ( δ υ ) = 2 E l L υ ( 1 δ υ / 2 δ L υ ) ,
δ u = u / R , δ υ = υ / R ,
δ Lu = L u / R , δ L υ = L υ / R .
â E ( ζ u ) = 2 E l L u × 2 δ Lu sinc 2 ( δ Lu ζ u ) ,
â E ( ζ υ ) = 2 E l L υ × 2 δ L υ sinc 2 ( δ L υ ζ υ ) ,
S ( ω ) = 2 α 2 z 2 R 2 W ( ω ) × 2 E l L υ × 2 δ L υ × 2 E l L u × 2 δ L υ sinc 2 ( δ Lu ω / ω ) .
A = 4 L u L υ ,
S ( ω ) = 2 α 2 z 2 ( 2 zP l ) AW ( ω ) sinc 2 ( L u ω / ω R ) ,
S ( ω ) = 2 α 2 z ( 2 P l ) A × λ I 4 π R 2 ω δ ( ω Δ ω D ) d u σ u ( ω ; ω ̅ ) du p × sinc 2 ( L u ω / ω R ) ,

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