Abstract

The general target characterization for coherent laser radars is shown to be a two-frequency bistatic scattering-amplitude matrix. This matrix is used to develop target-signature expressions for pulsed imager and 3-D imager systems. The relationships between the scattering matrix and the more familiar bidirectional reflectance, diffuse reflectivity, and multiplicative target models are explored. Problems that may arise in test bed or reflectometer measurements of target reflectivity are discussed, and test bed calibration-plate measurement data are reported. The latter will indicate the viability of diffuse reflectivity for rough-surface targets.

© 1982 Optical Society of America

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  1. R. L. Del Boca, R. J. Mongeon, “Multifunction CO2 Heterodyning Laser Radar for Low Level Tactical Operations,” in 1979 National Aerospace and Electronics Conference (IEEE, New York, 1979), pp. 1079–1088.
  2. R. C. Harney, R. J. Hull, Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).
  3. J. M. Cruickshank, P. Pace, in Digest of Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), paper ThA2.
  4. O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).
  5. C. G. Bachman, Laser Radar Systems and Techniques (Artech, Dedham, 1979).
  6. J. H. Shapiro, in Digest of Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), paper ThA1.
  7. R. C. Harney, in Proc. Soc. Photo-Opt. Instrum. Eng. 300, 2 (1982).
  8. J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 20, 3292 (1981).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  10. Our convention for Fourier transforms is given in Sec. II.B.
  11. H. L. Van Trees, Detection Estimation and Modulation Theory, Part 3 (Wiley, New York, 1971), Chap. 11.
  12. For a stationary target, illumination at modulation frequency ωi yields a reflection only at modulation frequency ωi, hence the δ function in Eq. (9).
  13. The unit vector î may be complex valued, so as to allow for elliptical polarization. It obeys î · î* = 1.
  14. The Fourier transform uncertainty principle tells us that the spatial transform of an unresolved target is essentially constant over the spatial frequency region spanned by the transform of the spatial illumination pattern. Equation (10) is the mathematical embodiment of this result.
  15. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), p. 557.
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978), pp. 9–12.
    [CrossRef]
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978), pp. 27–35.
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 13.
  19. D. M. Papurt, J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 86 (1982).
  20. The local oscillator polarization has been chosen to agree with the λ/4 Brewster plate TR switch described in Sec. II.A.
  21. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 2, 3.
  22. R. J. Becherer, “System Design Study for Infrared Airborne Radar (IRAR),” Technical Note 1977–29, Lincoln Laboratory, MIT (Oct.1977).
  23. R. C. Harney, “Conceptual Design of a Multifunction Infrared Radar for the Tactical Aircraft Ground Attack Scenario,” Project Report TST-25, Lincoln Laboratory, MIT (Aug.1978).
  24. The AM–cw approach to range determination has received relatively little attention in the radar literature compared with the commonplace FM–cw approach: O. K. Nilssen, W. D. Boyer, IRE Trans. Aeronaut. Navig. Electron. ANE-9, 250 (1962) and M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 81–92.
    [CrossRef]
  25. For our formulation to be free from aliasing, Eq. (25) requires ωs ≪ ωIF. In practical AM–cw systems, there is often no local oscillator frequency shift (ωIF = 0). Instead the amplitude and phase shift of the frequency ωs component of the amplified detector photocurrent provide the target information. For such a system, the real part of Eq. (25) with s(t) = 2−1 cos(ωst) used in lieu of Eq. (23) characterizes the target return.
  26. W. L. Wolfe, “Radiation Theory,” in Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (ERIM, Ann Arbor, 1978).
  27. Data Compilation, Eleventh Supplement: Vol. 1, Bidirectional Reflectance: Definition, and Utilization; and Vol. 2, Bidirectional Reflectance: Graphic Data, AFAL-TR-72/226, (Target Signature Analysis Center, 1972).
  28. J. H. Shapiro, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Project Report TST-24, Lincoln Laboratory, MIT (Oct.1978).
  29. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975).
    [CrossRef]
  30. R. A. Brandewie, W. C. Davis, Appl. Opt. 11, 1526 (1972).
    [CrossRef] [PubMed]
  31. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 3 (Wiley, New York, 1971), pp. 308–313.
  32. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 411–420.

1982

O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).

R. C. Harney, in Proc. Soc. Photo-Opt. Instrum. Eng. 300, 2 (1982).

D. M. Papurt, J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 86 (1982).

1981

1980

R. C. Harney, R. J. Hull, Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

1972

1965

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

1962

The AM–cw approach to range determination has received relatively little attention in the radar literature compared with the commonplace FM–cw approach: O. K. Nilssen, W. D. Boyer, IRE Trans. Aeronaut. Navig. Electron. ANE-9, 250 (1962) and M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 81–92.
[CrossRef]

Bachman, C. G.

C. G. Bachman, Laser Radar Systems and Techniques (Artech, Dedham, 1979).

Becherer, R. J.

R. J. Becherer, “System Design Study for Infrared Airborne Radar (IRAR),” Technical Note 1977–29, Lincoln Laboratory, MIT (Oct.1977).

Bolander, G.

O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 13.

Boyer, W. D.

The AM–cw approach to range determination has received relatively little attention in the radar literature compared with the commonplace FM–cw approach: O. K. Nilssen, W. D. Boyer, IRE Trans. Aeronaut. Navig. Electron. ANE-9, 250 (1962) and M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 81–92.
[CrossRef]

Brandewie, R. A.

Capron, B. A.

Cruickshank, J. M.

J. M. Cruickshank, P. Pace, in Digest of Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), paper ThA2.

Davis, W. C.

Del Boca, R. L.

R. L. Del Boca, R. J. Mongeon, “Multifunction CO2 Heterodyning Laser Radar for Low Level Tactical Operations,” in 1979 National Aerospace and Electronics Conference (IEEE, New York, 1979), pp. 1079–1088.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975).
[CrossRef]

Gullberg, K.

O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).

Harney, R. C.

R. C. Harney, in Proc. Soc. Photo-Opt. Instrum. Eng. 300, 2 (1982).

D. M. Papurt, J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 86 (1982).

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 20, 3292 (1981).
[CrossRef] [PubMed]

R. C. Harney, R. J. Hull, Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

R. C. Harney, “Conceptual Design of a Multifunction Infrared Radar for the Tactical Aircraft Ground Attack Scenario,” Project Report TST-25, Lincoln Laboratory, MIT (Aug.1978).

Hull, R. J.

R. C. Harney, R. J. Hull, Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978), pp. 9–12.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978), pp. 27–35.

Mongeon, R. J.

R. L. Del Boca, R. J. Mongeon, “Multifunction CO2 Heterodyning Laser Radar for Low Level Tactical Operations,” in 1979 National Aerospace and Electronics Conference (IEEE, New York, 1979), pp. 1079–1088.

Nilssen, O. K.

The AM–cw approach to range determination has received relatively little attention in the radar literature compared with the commonplace FM–cw approach: O. K. Nilssen, W. D. Boyer, IRE Trans. Aeronaut. Navig. Electron. ANE-9, 250 (1962) and M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 81–92.
[CrossRef]

Pace, P.

J. M. Cruickshank, P. Pace, in Digest of Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), paper ThA2.

Papurt, D. M.

D. M. Papurt, J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 86 (1982).

Renhorn, I.

O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).

Shapiro, J. H.

D. M. Papurt, J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 86 (1982).

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 20, 3292 (1981).
[CrossRef] [PubMed]

J. H. Shapiro, in Digest of Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), paper ThA1.

J. H. Shapiro, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Project Report TST-24, Lincoln Laboratory, MIT (Oct.1978).

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), p. 557.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 411–420.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 2, 3.

Steinvall, O.

O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).

Van Trees, H. L.

H. L. Van Trees, Detection Estimation and Modulation Theory, Part 3 (Wiley, New York, 1971), Chap. 11.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 3 (Wiley, New York, 1971), pp. 308–313.

Widen, A.

O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 13.

Wolfe, W. L.

W. L. Wolfe, “Radiation Theory,” in Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (ERIM, Ann Arbor, 1978).

Appl. Opt.

IRE Trans. Aeronaut. Navig. Electron.

The AM–cw approach to range determination has received relatively little attention in the radar literature compared with the commonplace FM–cw approach: O. K. Nilssen, W. D. Boyer, IRE Trans. Aeronaut. Navig. Electron. ANE-9, 250 (1962) and M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 81–92.
[CrossRef]

Proc. IEEE

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

D. M. Papurt, J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 86 (1982).

R. C. Harney, in Proc. Soc. Photo-Opt. Instrum. Eng. 300, 2 (1982).

R. C. Harney, R. J. Hull, Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

O. Steinvall, G. Bolander, K. Gullberg, I. Renhorn, A. Widen, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 100 (1982).

Other

C. G. Bachman, Laser Radar Systems and Techniques (Artech, Dedham, 1979).

J. H. Shapiro, in Digest of Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), paper ThA1.

J. M. Cruickshank, P. Pace, in Digest of Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), paper ThA2.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 3 (Wiley, New York, 1971), pp. 308–313.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 411–420.

The local oscillator polarization has been chosen to agree with the λ/4 Brewster plate TR switch described in Sec. II.A.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), pp. 2, 3.

R. J. Becherer, “System Design Study for Infrared Airborne Radar (IRAR),” Technical Note 1977–29, Lincoln Laboratory, MIT (Oct.1977).

R. C. Harney, “Conceptual Design of a Multifunction Infrared Radar for the Tactical Aircraft Ground Attack Scenario,” Project Report TST-25, Lincoln Laboratory, MIT (Aug.1978).

For our formulation to be free from aliasing, Eq. (25) requires ωs ≪ ωIF. In practical AM–cw systems, there is often no local oscillator frequency shift (ωIF = 0). Instead the amplitude and phase shift of the frequency ωs component of the amplified detector photocurrent provide the target information. For such a system, the real part of Eq. (25) with s(t) = 2−1 cos(ωst) used in lieu of Eq. (23) characterizes the target return.

W. L. Wolfe, “Radiation Theory,” in Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (ERIM, Ann Arbor, 1978).

Data Compilation, Eleventh Supplement: Vol. 1, Bidirectional Reflectance: Definition, and Utilization; and Vol. 2, Bidirectional Reflectance: Graphic Data, AFAL-TR-72/226, (Target Signature Analysis Center, 1972).

J. H. Shapiro, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Project Report TST-24, Lincoln Laboratory, MIT (Oct.1978).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975).
[CrossRef]

R. L. Del Boca, R. J. Mongeon, “Multifunction CO2 Heterodyning Laser Radar for Low Level Tactical Operations,” in 1979 National Aerospace and Electronics Conference (IEEE, New York, 1979), pp. 1079–1088.

Our convention for Fourier transforms is given in Sec. II.B.

H. L. Van Trees, Detection Estimation and Modulation Theory, Part 3 (Wiley, New York, 1971), Chap. 11.

For a stationary target, illumination at modulation frequency ωi yields a reflection only at modulation frequency ωi, hence the δ function in Eq. (9).

The unit vector î may be complex valued, so as to allow for elliptical polarization. It obeys î · î* = 1.

The Fourier transform uncertainty principle tells us that the spatial transform of an unresolved target is essentially constant over the spatial frequency region spanned by the transform of the spatial illumination pattern. Equation (10) is the mathematical embodiment of this result.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980), p. 557.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978), pp. 9–12.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978), pp. 27–35.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 13.

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Figures (11)

Fig. 1
Fig. 1

Macroscopic scattering model for target–laser-beam interaction.

Fig. 2
Fig. 2

Target-return angular characteristics: (a) specular (glint) return from polished surface and diffuse (speckle) return from rough surface; (b) R ¯ ¯-matrix polar plots for glint and speckle target returns.

Fig. 3
Fig. 3

Target-return Doppler characteristics: (a) uniform Doppler shift from rigid-body motion and Doppler spread from a cluttered target; (b) R ¯ ¯-matrix frequency plots for Doppler-shifted and Doppler-spread target returns.

Fig. 4
Fig. 4

Biparaxial propagation geometry.

Fig. 5
Fig. 5

Direction conventions for illuminator and receiver angular beam patterns, ξ ˜ i ( s ¯ i ) and ξ ˜ r ( s ¯ r ), respectively.

Fig. 6
Fig. 6

Monostatic radar geometry.

Fig. 7
Fig. 7

Biparaxial geometry for defining the bidirectional reflectance ρ i i ( - s ¯ r , s ¯ i ; λ ).

Fig. 8
Fig. 8

Tilt invariance condition under which a multiplicative target model is valid.

Fig. 9
Fig. 9

Geometry for MIT Lincoln Laboratory calibration-plate measurements.

Fig. 10
Fig. 10

Sample of histogram data from calibration-plate measurements.

Fig. 11
Fig. 11

Laser radar test bed and incoherent reflectometer systems for measuring target reflectivity.

Tables (2)

Tables Icon

Table I Demonstrated Applications of Coherent IR Radar Waveforms

Tables Icon

Table II Calibration-Plate Data

Equations (38)

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CNR = ( P T / h ν B ) ( d 2 / 4 L 2 ) η ρ exp ( - 2 α L ) ,
E ¯ r ( Ω ¯ r , ω r ) = d ω i 2 π d Ω ¯ i ( ω o + ω i 2 π c ) R ¯ ¯ ( Ω ¯ r , ω r ; Ω ¯ i , ω i ; ω o ) E ¯ i ( Ω ¯ i , ω i ) .
R ¯ ¯ exp { - j [ ( ω o + ω i ) Ω ¯ i - ( ω o + ω r ) Ω ¯ r ] · r ¯ / c } ,
E ¯ i exp [ j ( ω o + ω i ) Ω ¯ i · r ¯ / c ]
d Ω ¯ i exp [ j ( ω o + ω i ) Ω ¯ i · ( r ¯ - r ¯ ) / c ] ,
E ¯ r ( s ¯ r , ω r ) = d ω i 2 π d s ¯ i λ R ¯ ¯ ( s ¯ r , ω r ; s ¯ i , ω i ; λ ) E ¯ i ( s ¯ i , ω i ) ,
E ¯ r ( s ¯ r , ω r ) = d t d ρ ¯ r λ E ¯ r ( ρ ¯ r , t ) exp [ j ( ω r t - k s ¯ r · ρ ¯ r ) ] ,
E ¯ i ( s ¯ i , ω i ) = d t d ρ ¯ i λ E ¯ i ( ρ ¯ i , t ) exp [ j ( ω i t - k s ¯ i · ρ ¯ i ) ] .
E ¯ i ( s ¯ i , ω i ) = P T 1 / 2 ξ ˜ i ( s ¯ i ) 2 π δ ( ω i ) i ^ ,
R ¯ ¯ ( s ¯ r , ω r ; s ¯ i , ω i ; λ ) = R ¯ ¯ ( s ¯ r , s ¯ i ; ω i ; λ ) 2 π δ ( ω r - ω i ) ,
d s ¯ i λ R ¯ ¯ ( s ¯ r , s ¯ i ; 0 ; λ ) i ^ ξ ˜ i ( s ¯ i ) = R ¯ ¯ ( s ¯ r , s ¯ i o ; λ ) i ^ d s ¯ i λ ξ ˜ i ( s ¯ i ) ,
E ˜ ¯ r ( s ¯ r ) = P T 1 / 2 R ¯ ¯ ( s ¯ r , s ¯ i o ; λ ) i ^ d s ¯ i λ ξ ˜ i ( s ¯ i ) .
P R = P T R i i ( - s ¯ r o , s ¯ i o ; λ ) 2 | d s ¯ r ξ ˜ r ( s ¯ r ) | 2 | d s ¯ i λ ξ ˜ i ( s ¯ i ) | 2 ,
P R = P T σ b i ( - s ¯ r o , s ¯ i o ) ( G T / 4 π L T 2 ) ( A R / 4 π L R 2 ) ,
σ b i ( - s ¯ r o , s ¯ i o ) = 4 π R i i ( - s ¯ r o , s ¯ i o ; λ ) 2
σ = 4 π i ^ R ¯ ¯ ( - s ¯ i o , s ¯ i o ; λ ) i ^ 2 = 4 π R i i ( - s ¯ i o , s ¯ i o ; λ ) 2
E ¯ T ( ρ ¯ , t ) = Re [ P T 1 / 2 s ( t ) F ( ρ ¯ ) exp ( - j ω o t ) i ^ ]
E ¯ l ( ρ ¯ , t ) = Re [ P l 1 / 2 F * ( ρ ¯ ) exp [ - j ( ω o - ω IF ) t ] i ^ * ] ,
y ( t ) = Re [ y ( t ) exp ( - j ω IF t ) ] ,
y ( t ) = P T 1 / 2 d s ¯ r d ω r 2 π d s ¯ i λ d ω i 2 π ξ ˜ ( s ¯ r ) R i i ( - s ¯ r , ω r ; s ¯ i , ω i ; λ ) × ξ ˜ ( s ¯ i ) S ( ω i ) exp [ j ( ω r + ω i ) L / c ] exp ( - j ω r t ) .
R i i ( - s ¯ r , ω r ; s ¯ i , ω i ; λ ) = R i i ( - s ¯ r , s ¯ i ; ω i ; λ ) 2 π × δ ( ω r - ω i - 2 ω o L ˙ / c )
d ω 2 π R i i ( - s ¯ r , s ¯ i ; ω ; λ ) S ( ω ) exp ( - j ω t ) = R i i ( - s ¯ r , s ¯ i ; λ ) s ( t )
y ( t ) = P T 1 / 2 d s ¯ r d s ¯ i λ ξ ˜ ( s ¯ r ) R i i ( - s ¯ r , s ¯ i ; λ ) ξ ˜ ( s ¯ i ) × s ( t - 2 L / c ) exp ( j 2 ω o L ˙ t / c + j ϕ ) .
s ( t ) = [ 1 + cos ( ω s t ) ] / 2 ,
R i i ( - s ¯ r , s ¯ i ; ω ; λ ) = R i i ( - s ¯ r , s ¯ i ; λ ) exp ( j 2 ω Δ L / c ) .
y ( t ) = P T 1 / 2 d s ¯ r d s ¯ i λ ξ ˜ ( s ¯ r ) R i i ( - s ¯ r , s ¯ i ; λ ) ξ ˜ ( s ¯ i ) × s [ t - 2 ( L + Δ L ) / c ] exp ( j 2 ω o L ˙ t / c + j ϕ ) ,
ρ i i ( - s ¯ r , s ¯ i ; λ ) = R i i ( - s ¯ r , s ¯ i ; λ ) 2 / A T cos θ ,
| d s ¯ r d s ¯ i λ ξ ˜ ( s ¯ r ) R i i ( - s ¯ r , s ¯ i ; λ ) ξ ˜ ( s ¯ i ) | 2 σ ( G T / 4 π L 2 ) ( A R / 4 π L 2 ) ,
R i i ( - s ¯ r , s ¯ i ; λ ) 2 = A T ρ / π .
E ¯ r ( ρ ¯ , t ) · i ^ * = T ( ρ ¯ ) E ¯ i ( ρ ¯ , t ) · i ^ *
E ¯ r ( s ¯ r , ω ) · i ^ * = d s ¯ i λ T ˜ ( - s ¯ r - s ¯ i ) E ¯ i ( s ¯ i , ω ) · i ^ * ,
E ¯ r ( s ¯ r , ω ) · i ^ * = d s ¯ i λ R i i ( - s ¯ r , s ¯ i ; λ ) E ¯ i ( s ¯ i , ω ) · i ^ * .
R i i ( - s ¯ r , s ¯ i ; λ ) R i i ( 0 ¯ , s ¯ i + s ¯ r ; λ ) = T ˜ ( - s ¯ r - s ¯ i )
d s ¯ r d s ¯ i λ ξ ˜ ( s ¯ r ) R i i ( - s ¯ r , s ¯ i ; λ ) ξ ˜ ( s i )
RR = d ρ ¯ Ψ TT ( ρ ¯ , - 2 ρ ¯ / L ) Ψ ξ 2 ξ 2 ( ρ ¯ , 2 ρ ¯ / L ) d ρ ¯ Ψ TT ( ρ ¯ , - ρ ¯ / l ) Ψ ζ ζ ( ρ ¯ , ρ ¯ / l ) l - 2 p ( - ρ ¯ / l ) .
Ψ g g ( ρ ¯ , s ¯ ) = d ρ ¯ λ g ( ρ ¯ + ρ ¯ / 2 ) g * ( ρ ¯ - ρ ¯ / 2 ) exp ( - j k s ¯ · ρ ¯ ) ;
RR Ψ ξ 2 ξ 2 ( 0 ¯ , 0 ¯ ) / Ψ ζ ζ ( 0 ¯ , 0 ¯ ) l - 2 p ( 0 ¯ ) .
R R = d ρ ¯ Ψ TT ( ρ ¯ , - 2 ρ ¯ / L ) Ψ ξ 2 ξ 2 ( ρ ¯ , 2 ρ ¯ / L ) d ρ ¯ Ψ TT ( ρ ¯ , - 2 ρ ¯ / L ) Ψ ξ 2 ξ 2 ( ρ ¯ , 2 ρ ¯ / L ) ,

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