Abstract

The optics of a coherent lidar system using incoherent backscatter is modeled in a way that allows restilts of the wave theory of aberrations to be applied directly. Parametric plots are obtained showing the aberration and range dependence of the effective antenna area using truncated Gaussian beams. The Rayleigh criterion is an adequate guide to the optical tolerances required for the far-field return, and in the near field returns are not necessarily degraded by aberrations. Examples are given for the range weighting of returns for various primary mirror configurations.

© 1982 Optical Society of America

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References

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  1. T. A. Nussmeier, S. H. Brewer, “Optical Analysis for Laser Heterodyne Communication System,” Hughes Research Laboratory Report on contract NAS 5-21898 (1974).
  2. A. E. Siegman, Appl. Opt. 5, 1588 (1966); Proc. IEEE 54, 1350 (1966).
    [CrossRef] [PubMed]
  3. B. J. Rye, Appl. Opt. 18, 1390 (1979).
    [CrossRef] [PubMed]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  6. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1979).
  7. B. J. Rye, J. Opt. Soc. Am. 71, 687 (1981).
    [CrossRef]
  8. J. T. Priestley, NOAA Wave Propagation Laboratory; unpublished.
  9. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950).
  10. The numerical computation of the 2-D autocorrelation function μ(b) was, of course, slow. It would be useful for anyone intending to repeat this exercise to note that because the integrand is Hermitian and the optical system symmetric about the tangential plane, it is necessary only to compute the contribution to A from a single quadrant in the b plane bounded by the tangential and sagittal planes and multiply the result by 4.
  11. A. Thomson, M. F. Dorian, “Heterodyne Detection of Monochromatic Light Scattered from a Cloud of Moving Particles,” General Dynamics Convair Division report GNC-ERR-Nov-1090 (1967).
  12. S. F. Clifford, T. R. Lawrence, G. R. Ochs, Ting-i Wang, “Study of a Pulsed Laser for Cross-wind Sensing,” NOAA Technical Memo ERL WPL-48 (1980).
  13. The complete mirror aberration function was computed numerically for the curves of Figs. (10) and (11), but in each case use of the primary aberration formulas was an adequate approximation.

1981 (1)

1979 (1)

1966 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1979).

Brewer, S. H.

T. A. Nussmeier, S. H. Brewer, “Optical Analysis for Laser Heterodyne Communication System,” Hughes Research Laboratory Report on contract NAS 5-21898 (1974).

Clifford, S. F.

S. F. Clifford, T. R. Lawrence, G. R. Ochs, Ting-i Wang, “Study of a Pulsed Laser for Cross-wind Sensing,” NOAA Technical Memo ERL WPL-48 (1980).

Dorian, M. F.

A. Thomson, M. F. Dorian, “Heterodyne Detection of Monochromatic Light Scattered from a Cloud of Moving Particles,” General Dynamics Convair Division report GNC-ERR-Nov-1090 (1967).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950).

Lawrence, T. R.

S. F. Clifford, T. R. Lawrence, G. R. Ochs, Ting-i Wang, “Study of a Pulsed Laser for Cross-wind Sensing,” NOAA Technical Memo ERL WPL-48 (1980).

Nussmeier, T. A.

T. A. Nussmeier, S. H. Brewer, “Optical Analysis for Laser Heterodyne Communication System,” Hughes Research Laboratory Report on contract NAS 5-21898 (1974).

Ochs, G. R.

S. F. Clifford, T. R. Lawrence, G. R. Ochs, Ting-i Wang, “Study of a Pulsed Laser for Cross-wind Sensing,” NOAA Technical Memo ERL WPL-48 (1980).

Priestley, J. T.

J. T. Priestley, NOAA Wave Propagation Laboratory; unpublished.

Rye, B. J.

Siegman, A. E.

Thomson, A.

A. Thomson, M. F. Dorian, “Heterodyne Detection of Monochromatic Light Scattered from a Cloud of Moving Particles,” General Dynamics Convair Division report GNC-ERR-Nov-1090 (1967).

Wang, Ting-i

S. F. Clifford, T. R. Lawrence, G. R. Ochs, Ting-i Wang, “Study of a Pulsed Laser for Cross-wind Sensing,” NOAA Technical Memo ERL WPL-48 (1980).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Other (10)

J. T. Priestley, NOAA Wave Propagation Laboratory; unpublished.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950).

The numerical computation of the 2-D autocorrelation function μ(b) was, of course, slow. It would be useful for anyone intending to repeat this exercise to note that because the integrand is Hermitian and the optical system symmetric about the tangential plane, it is necessary only to compute the contribution to A from a single quadrant in the b plane bounded by the tangential and sagittal planes and multiply the result by 4.

A. Thomson, M. F. Dorian, “Heterodyne Detection of Monochromatic Light Scattered from a Cloud of Moving Particles,” General Dynamics Convair Division report GNC-ERR-Nov-1090 (1967).

S. F. Clifford, T. R. Lawrence, G. R. Ochs, Ting-i Wang, “Study of a Pulsed Laser for Cross-wind Sensing,” NOAA Technical Memo ERL WPL-48 (1980).

The complete mirror aberration function was computed numerically for the curves of Figs. (10) and (11), but in each case use of the primary aberration formulas was an adequate approximation.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1979).

T. A. Nussmeier, S. H. Brewer, “Optical Analysis for Laser Heterodyne Communication System,” Hughes Research Laboratory Report on contract NAS 5-21898 (1974).

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Figures (11)

Fig. 1
Fig. 1

Reciprocal receiver geometry.

Fig. 2
Fig. 2

Bistatic lidar geometry.

Fig. 3
Fig. 3

η a for far-field return of aligned circular transmitter/receiver (same radius) with a Gaussian profile |u T (b)| as a function of γ T . BPLO profiles |u L (b)| are (1) uniform; (2) matched to transmitter; and (3) Gaussian with γ L = 119.

Fig. 4
Fig. 4

Signal reduction for OMGP far’-field return as a function of (1) α040 (spherical aberration); (2) α031 (coma); and (3) α022 (astigmatism).

Fig. 5
Fig. 5

(1) η a for OMGP as function of α120. (2) Notional η a (see text) for same profile in absence of aperture truncation.

Fig. 6
Fig. 6

η a for OMGP in the presence of spherical aberration and field curvature as a function of α120: (1) α040 = 0 (2) α040 = 0.5; (3) α040 = 1; and (4) α040 = 2.

Fig. 7
Fig. 7

As Fig. 6 but in the presence of astigmatism and field curvature: (1) α022 = 0, (2) α022 = 0.5; (3) α022 = 1; and (4) α022 = 2.

Fig. 8
Fig. 8

As Fig. 6 but in the presence of coma and field curvature: (1) α031 = 0; (2) α031 = 0.5; (3) α031 = 1; and (4) α031 = 2.

Fig. 9
Fig. 9

Mirror geometry.

Fig. 10
Fig. 10

Return T T A/r2 with OMGP at 10-μm wavelength and focused at infinity vs range for: (1) 25-cm f/8 off-center paraboloid properly aligned on-axis; and (2) 25-cm f/8 paraboloid aligned 35-mrad off-axis to avoid secondary obscuration.

Fig. 11
Fig. 11

Return T T A/r2 with OMGP at 10-μm wavelength and focused at infinity vs range for 30-cm f/1 off-center paraboloid: (1) properly aligned on axis (χ = 0); (2) χ = 2 mrad; (3) χ = 5 mrad; and (4) for 30-cm f/2 spherical mirror with χ = 0.

Tables (1)

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Table I Primary Aberrations

Equations (24)

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u L ( b ) = u L ( b ) exp [ j 2 π Φ L ( b ) / λ ] ,
I L ( s ) = T L P L F s [ μ L ( f ) ] ,
μ L ( f ) = ( λ r ) 2 T L P L u L * ( f ) u L ( f + f ) d 2 f
G R = I L ( 0 ) T L P L / ( 4 π r 2 ) = A R λ 2 / ( 4 π ) ,
A R = μ L ( b ) d 2 b
I T ( s ) = T T P 0 F s [ μ T ( f ) ] ,
μ T ( f ) = ( λ r ) 2 T T P 0 μ T * ( f ) u T ( f + f ) d 2 f .
u T ( b ) = u T ( b ) exp [ j 2 π Φ T ( b ) / λ ] .
I T ( s ) = T T P 0 F s [ μ T ( f ) exp ( j 2 π f · s 1 ) ] .
P R = 1 π T T A ρ s exp ( - 2 α r ) r 2 P 0 ,
P R = T T A ( r ) β s ( r ) exp ( - 2 α r ) r 2 P 0 ( r ) d r ,
P R = ½ T T A β s c exp ( - 2 α c ) r 2 E 0 ,
A = ( λ r ) 2 T L T T P L P 0 I T ( s ) I L ( s ) d 2 s .
A = ( λ r ) 2 μ T ( f ) μ L * ( f ) exp ( j 2 π f · s 1 ) d 2 f
A = μ T ( b ) μ L * ( b ) d 2 b .
A T = μ T ( b ) d 2 b
η a = T T A A T ,
u T ( b ) = ( 2 / π ) 1 / 2 P 0 γ T b T exp [ - b 2 / ( γ T b T ) 2 ] ,
u L ( b ) = ( 2 / π ) 1 / 2 P L γ L b R exp [ - b 2 / ( γ L b R ) 2 ] .
Φ T , L ( b , θ ) = λ n , m ( α l n m ) T , R ( b / b T , R ) n cos m θ ,
α 120 = ( α 120 ) L = b R 2 2 λ ( 1 u - 1 r ) .
ϕ T ( b ) = λ ( α 120 ) T ( b / b T ) 2 = ½ b 2 ( 1 / u - 1 / r ) .
A T = 2 π ( γ T b T ) 2 1 + [ 2 π γ T 2 ( α 120 ) T ] 2 = 2 π ( γ T b T ) 2 1 + [ ( π / λ ) ( γ T b T ) ( 1 / u - 1 / r ) ] 2 ,
1 A = 1 A T + 1 A R .

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