Abstract

This paper reviews our studies on coherent and incoherent synthetic-aperture imaging ladars (SAILs). Using optical diffraction, a systematic theory of side-looking SAIL was mathematically formulated and the necessary conditions for assuring a correct phase history are established. Based on optical transformation and regulation of wavefront, a down-looking SAIL of two distinctive architectures was invented and the basic principle, systematic theory, design equations, and necessary conditions are presented. An incoherent spotlight-mode SAIL was proposed, and detailed mathematically. To validate the concepts, laboratory experiments were conducted. The spatially and temporally dependent laser speckles are analyzed by applying the partial coherence theorem, and proposals to reduce their effect are given. Optical antennas and their components are discussed. It is shown that for down-looking SAIL the width of the scanning strip may be greatly increased without loss of high resolution, and the influences from atmospheric turbulence and unmodeled line-of-sight motion can be automatically compensated.

© 2013 Optical Society of America

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  1. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
    [CrossRef]
  2. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
    [CrossRef]
  3. Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
    [CrossRef]
  4. L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
    [CrossRef]
  5. J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.
  6. B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.
  7. R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).
  8. L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).
  9. L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009).
    [CrossRef]
  10. L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).
  11. L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).
  12. L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010).
    [CrossRef]
  13. L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009).
    [CrossRef]
  14. L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011).
    [CrossRef]
  15. R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991).
    [CrossRef]
  16. Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
    [CrossRef]
  17. Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008).
    [CrossRef]
  18. Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004).
    [CrossRef]
  19. L. Liu, “Quasi-interferometry with coded correlation filtering,” Appl. Opt. 21, 2817–2826 (1982).
    [CrossRef]
  20. A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 54, 1350–1356 (1966).
    [CrossRef]
  21. T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996).
    [CrossRef]
  22. R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).
  23. A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001).
    [CrossRef]
  24. E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
    [CrossRef]
  25. Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
    [CrossRef]

2012 (3)

L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
[CrossRef]

2011 (3)

L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011).
[CrossRef]

2010 (1)

L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010).
[CrossRef]

2009 (4)

L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009).
[CrossRef]

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

2008 (2)

Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008).
[CrossRef]

L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

2005 (1)

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

2004 (1)

Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004).
[CrossRef]

2002 (1)

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
[CrossRef]

2001 (1)

A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001).
[CrossRef]

1996 (1)

T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996).
[CrossRef]

1991 (1)

R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991).
[CrossRef]

1988 (1)

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

1982 (1)

L. Liu, “Quasi-interferometry with coded correlation filtering,” Appl. Opt. 21, 2817–2826 (1982).
[CrossRef]

1966 (1)

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 54, 1350–1356 (1966).
[CrossRef]

Ashby, S.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Bashkansky, M.

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
[CrossRef]

R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).

Beck, S. M.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Benhamb, V. N.

A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001).
[CrossRef]

Buck, J.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Buck, J. R.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Buell, W. F.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Capes, R. N.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Craig, E. B.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Dai, E.

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Dickinson, R. P.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Dierking, M.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

Fuhrer, S.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

Funk, E.

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
[CrossRef]

R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).

Garreis, R.

R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991).
[CrossRef]

Gleason, A.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Hanesa, A. S.

A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001).
[CrossRef]

Hwang, D.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Jin, X.

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
[CrossRef]

Keicher, W. E.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Kondratko, P.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Kozlowski, D. A.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Krause, B.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Kulkarni, S. R.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Lasche, J. B.

A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001).
[CrossRef]

Liu, D.

Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008).
[CrossRef]

Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004).
[CrossRef]

Liu, L.

L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).

L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011).
[CrossRef]

L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009).
[CrossRef]

L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009).
[CrossRef]

Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008).
[CrossRef]

L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004).
[CrossRef]

L. Liu, “Quasi-interferometry with coded correlation filtering,” Appl. Opt. 21, 2817–2826 (1982).
[CrossRef]

Lu, W.

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Luan, Z.

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008).
[CrossRef]

Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004).
[CrossRef]

Lucke, R. L.

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
[CrossRef]

R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).

Malm, A.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Marechal, N. J.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Marino, R. M.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Parker, J. K.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Poon, T.-C.

T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996).
[CrossRef]

Reiley, M. F.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Reintjes, J.

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
[CrossRef]

R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).

Rickard, L. J.

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
[CrossRef]

R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).

Ricklin, J.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

Rowland, K. B.

A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001).
[CrossRef]

Ryan, C.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

Schumm, B.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

Senning, J. R.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Shinoda, K.

T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 54, 1350–1356 (1966).
[CrossRef]

Sun, J.

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
[CrossRef]

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

Suzuki, Y.

T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996).
[CrossRef]

Swezey, L. W.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

Teng, S.

Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004).
[CrossRef]

Tomlison, D.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

Wang, L.

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008).
[CrossRef]

Wright, T. J.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Wu, M.

T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996).
[CrossRef]

Wu, Y.

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Xu, N.

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

Yan, A.

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

Yan, Y.

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
[CrossRef]

Zeiss, C.

R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991).
[CrossRef]

Zhi, Y.

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

Zhou, Y.

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
[CrossRef]

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

Acta Opt. Sin. (9)

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009).
[CrossRef]

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011).
[CrossRef]

L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).

L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009).
[CrossRef]

L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011).
[CrossRef]

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009).
[CrossRef]

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012).
[CrossRef]

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012).
[CrossRef]

Appl. Opt. (3)

Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004).
[CrossRef]

L. Liu, “Quasi-interferometry with coded correlation filtering,” Appl. Opt. 21, 2817–2826 (1982).
[CrossRef]

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
[CrossRef]

Opt. Lett. (1)

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002).
[CrossRef]

Proc. IEEE (2)

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 54, 1350–1356 (1966).
[CrossRef]

T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996).
[CrossRef]

Proc. SPIE (7)

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001).
[CrossRef]

R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991).
[CrossRef]

L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010).
[CrossRef]

L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009).
[CrossRef]

Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008).
[CrossRef]

Other (3)

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).

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Figures (19)

Fig. 1.
Fig. 1.

SAIL with (a) strip-mode side-looking; (b) strip-mode down-looking; and (c) spotlight-mode side-looking.

Fig. 2.
Fig. 2.

Optical transmitting antennas based on (a) telescope; (b) imaging amplification; and (c) quasi-geometric projection.

Fig. 3.
Fig. 3.

Optical receiving antennas of (a) telescope for heterodyne detection; (b) focusing lens for heterodyne detection; and (c) telescope for self-heterodyne detection.

Fig. 4.
Fig. 4.

Component layout for a 2×4 90° free-space optical hybrid.

Fig. 5.
Fig. 5.

Schematic of a typical side-looking SAIL with key functional modules.

Fig. 6.
Fig. 6.

Geometry for strip-mode side-looking SAIL imaging.

Fig. 7.
Fig. 7.

General construction of down-looking SAIL with key functional modules.

Fig. 8.
Fig. 8.

Configuration (upper two) to produce a 2-D spherical wavefront, and configuration (lower two) to produce a 2-D paraboloidal wavefront.

Fig. 9.
Fig. 9.

Geometry for down-looking SAIL imaging.

Fig. 10.
Fig. 10.

Upper configuration to produce a 2-D divergent spherical wavefront, and lower configuration to produce a 2-D convergent spherical wavefront.

Fig. 11.
Fig. 11.

Schematic of typical spotlight-mode incoherent SAIL with key functional modules.

Fig. 12.
Fig. 12.

Coordinate systems for relative rotation between target and incoherent SAIL.

Fig. 13.
Fig. 13.

Normalized 1D integrated complex coherence functions at normalized coordinates, ratio of antenna width ACF to speckle width CCF: (a) 5:1; (b) 1:1, and (c) 1:5.

Fig. 14.
Fig. 14.

Design of a side-looking SAIL demonstrator and experimental arrangement for near-distance verification.

Fig. 15.
Fig. 15.

Reconstructed image from a side-looking SAIL experiment.

Fig. 16.
Fig. 16.

Experimental setup of down-looking SAIL.

Fig. 17.
Fig. 17.

Down-looking SAIL experiment: (a) amplitude distribution from in-phase channel; (b) amplitude distribution from quadrature channel and (c) reconstructed image.

Fig. 18.
Fig. 18.

Experimental setup of circular spotlight-mode incoherent SAIL.

Fig. 19.
Fig. 19.

Reconstructed images from a circular incoherent SAIL experiment with rotation coverage of (a) 360°, (b) 180°, (c) 90°, and (d) 45°.

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

eZ(x,y)=ejkZjλZexp(jπx2+y2λZ)et(α,β)exp(j2π(xαλZ+yβλZ)dαdβ.
eZ(x,y)=EejkZjλZexp(jπx2+y2λZ)DxDysinc(DxxλZ)sinc(DyyλZ).
eZ(x,y)=ejk(Z+2f1)Met(xM,yM)exp(jπx2+y2λZ).
eZ(x,y)=ejk(Z+f1)Met(xM,yM)exp(jπx2+y2λR).
er(x,y)=EtejkZlxlyjλZsinclx(xxp)λZsincly(yyp)λZexp(jπx2+y2λZ)exp(j2πxxp+yypλZ).
er(x,y)=EtejkZlxlyjλZKr,x(xxp)Kr,y(yyp)exp(jπx2+y2λZ)exp(j2πxxp+yypλZ),
I(x,y:t)=Eloc2+Er2+2ElocErcos[2π(xθx+yθy)φ(t)].
p(θx,θy:t)=(Eloc2+Er2)S+2ElocErSΘ(θx,θy)cosφ(t),
Θ(θx,θy)=Scos[2π(xθx+yθy)]dxdy/S.
Θ(θx,θy)=sinc(Dxθxλ)sinc(Dyθyλ).
I(t)=Er12+Er22+2Er1Er2cos[φr1(t)φr2(t)].
iIP(t)=KdE1E2Ad2cos[Δφ(t)+φrPBSφtPBSφpWP],
iQD(t)=KdE1E2Ad2cos[Δφ(t)φrPBS+φtPBSφvWP].
i(t)=iIP(t)+jiQD(t)=KdE1E2Ad2exp[jΔφ(t)].
x=αsinφ,y=β,z=αcosφ.
e0(x,y:tn,f)=E0rectxDxtrectyDyt,
φFM,n(tn,f)=2πf0tn,f+f˙2tn,f2+φn.
et(x,y:tn,f,nTs)=E0DxtDytjλZKt,x(x)Kt,y(y:nTs)exp[jπλZ(x2+(yvnTs)2)]×exp{j[φFM,n(tn,f+τp2)φFM,n(tn,f)]},
Kt,x(x)=sincDxtxλZ,Kt,y(y:nTs)=sincDyt(yvnTs)λZ,
er(x,y:xp,yp:tn,f,nTs)=E0DxtDytjλZρplxlyjλZKr,x(x:xp)Kr,y(y:yp:nTs)exp{jπλZ[xp2+(ypvnTs)2]}exp(jπλZ{(xxp)2+[y(ypvnTs)]2})exp{j[φn,FM(tn,f+τp)φn,FM(tn,f)]},
elo(x,y:)=EloMexp{j[φn,FM(tn,f+τlo)φn,FM(tn,f)]}.
IAC(x,y:xp,yp:tn,f,nTs)=2E0EloMDxtDytjλZρplxlyjλZη×Kr,x(0:xp)Kr,y(0:yp:nTs)Kt,x(xp)Kt,y(yp:nTs)×cos({πλZ/2[xp2+(ypvts)2]2πλZ[xxp+y(ypvts)]}+2πf˙Δτptn,f),
p(xp,yp:tn,f,nTs)=E0EloMDxtDytjλZρplxlyjλZDxrDyrη×Kr,x(0:xp)Kr,y(0:yp:nTs)Kt,x(xp)Kt,y(yp:nTs)Θx(xp)Θy(ypvnTs)cos{πλZ/2[xp2+(ypvnTs)2]+2πf˙Δτptn,f}.
Θ(xp,ypvnTs)=Θx(xp)Θy(ypvnTs)=sinc(DxrxpλZ)sinc[Dyr(ypvnTs)λZ].
i1D(xp,yp:tn,f,nTs)=KdE0EloKsKr,x(0:xp)Kr,y(0:yp:nTs)Kt,x(xp)Kt,y(yp:nTs)×Θx(xp)Θy(ypvnTs)rect[tn,f(τp+Tf/2)Tf]×exp{j[πλZ/2(ypvnTs)2+2πf˙Δτptn,f]},
Ks=DxtDytjλZρplxlyjλZDxrDyr2Mηexp(jπλZ/2xp2),
i2D(xp,yp:tn,f,nTs)=ni1D(xp,yp:tn,f,nTs).
I(ξ,mTs)=KdE0EloKsn(Kr,y(0:yp:nTs)Kt,y(yp:nTs)Θy(ypvnTs)×exp{j[πλZ/2(ypvnTs)2]}exp{jπλZ/2[(mn)vTs]}×Kr,x(0:xp)Kt,x(xp)Θx(xp)exp(j2πf˙Δτptn,f)rect[tn,f(τp+Tf/2)Tf]exp(j2πξtn,f)dtn,f).
I(ξ,mTs)=K[SR(ξ)*δ(ξf˙Δτp)][SA(m)*δ(mvTsyp)],
dx=c2Tff˙=c2B,
dy=λZ/2Dfp.
Irtel(d:D)=(hv2τpul)1/3[Pn(d×d)2]2/3(SnoiseSshot)2/32D2.
Ir(d:N)=Irtel(d:D)/N1/3.
Ir(1:N)=Irtel(d:N)N.
fx=f22R1in+R2in.
eHin(x,y:tn,f,nTs)=E0rect(x+vxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(x+vxintn,f)2λR1in+y2λR1in]},
eVin(x,y:tn,f,nTs)=E0rect(xvxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(xvxintn,f)2λR1iny2λR2in]},
eHT(x,y:tn,f,nTs)=E0ejkZMrect(xvxtn,fLx)rect(yvynTsLy)exp[jπΔf2λf22(vxtn,fM)2]×exp{jπλ[(xvxtn,f)2R1+(yvynTs)2R1]}exp{jπλZ[x2+(yvynTs)2]},
eVT(x,y:tn,f,nTs)=E0ejkZMrect(x+vxtn,fLx)rect(yvyTsLy)exp[jπΔf2λf22(vxtn,fM)2]×exp{jπλ[(x+vxtn,f)2R1(yvynTs)2R2]}exp{jπλZ[x2+(yvynTs)2]},
eHTR(x,y:xp,yp:tn,f,nTs)=E0Mρpej2kZjλZlxlyηKr,x(x:xp)Kr,y(y:yp:nTs)rect(xpvxtn,fLx)rect(ypvynTsLy)×exp{jπλ[(xpvxtn,f)2R1+(ypvynTs)2R1]}exp[jπΔf2λf22(vxtn,fM)2]exp(jφH)×exp{jπλZ[xp2+(ypvynTs)2]}exp(jπλZ{(xxp)2+[yΔY(ypvynTs)]2}),
eVTR(x,y:xp,yp:tn,f,nTs)=E0Mρpej2kZjλZlxlyηKr,x(x:xp)Kr,y(y:yp:nTs)rect(xp+vxtn,fLx)rect(ypvynTsLy)×exp{jπλ[(xp+vxtn,f)2R1(ypvynTs)2R2]}exp[jπΔf2λf22(vxtn,fM)2]exp(jφV)×exp{jπλZ[xp2+(ypvynTs)2]}exp(jπλZ{(xxp)2+[yΔY(ypvynTs)]2}).
ΔφTR(xp,yp:tn,f,nTs)=πλ[2xpvxtn,fR1/2+(ypvynTs)2R3]+φHφV,
IAC(x,y:xp,yp:tn,f,nTs)=2E02(ρplxlyηMλZ)Kr,x2(x:xp)Kr,y2(x:yp:nTs)Kt,x(xp:tn,f)Kt,y2(yp:nTs)cos[2πλR1/2xp(vxtn,f)+πλR3(ypvynTs)2]
Ks=DxrDyr(ρplxlyηMλZ)2,
Kt,x(xp:tn,f)=rect(xpvxtn,fLx)rect(xp+vxtn,fLx)Kt,y2(yp:nTs)=rect(ypvynTsLy).
i1D(xp,yp:tn,f,nTs)=I(xp,yp:tn,f,nTs)exp[j2πλR1/2xp(vxtn,f)+jπλR3(ypvynTs)2],
I(xp,yp:tn,f,nTs)=KdE02KsKr,x2(0:xp)Kr,y2(0:yp:nTs)Kt,x(xp:tn,f)Kt,y2(yp:nTs).
i2D(xp,yp:tn,f,nTs)=ni1D(xp,yp:tn,f,nTs).
I(ξ,mTs)=KdE02Ksn(Kr,y2(0:yp:nTs)Kt,y2(yp:nTs)×Kr,x2(0:xp)Kt,x(xp:tn,f)exp[j2πλR1/2xp(vxtn,f)]exp(j2πξtn,f)d(vxtn,f)×exp[jπλR3(ypvynTs)2]exp{jπλR3[vy(mn)Ts]}).
I(ξ,mTs)=K[Sot(ξ)*δ(ξ+xpλR1/2)][Stv(m)*δ(mvTsyp)],
dx=λR14kLx=λMR1in4kLxin.
dy=λR3Ly=MλR3inLyin.
fx=f22R1in+R2in.
θ˙V=f2R1in+R2inθ˙H.
eHin(x,y:tn,f,nTs)=E0rect(x+vxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(x+vxintn,f)2λR1in+y2λR1in]},
eVin(x,y:tn,f,nTs)=E0rect(x+vxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(x+vxintn,f)2λR2in+y2λR2in]}.
i1D(xp,yp:tn,f,nTs)=I(xp,yp:tn,f,nTs)exp{jπλR3[(xpvxtn,f)2+(ypvynTs)2]},
I(xp,yp:tn,f,nTs)=KdE02KsKr,x2(x:xp)Kr,y2(y:yp:nTs)Kt,x2(xp:tn,f)Kt,y2(yp:nTs).
i2D(xp,yp:tn,f,nTs)=ni1D(xp,yp:tn,f,nTs),
Kt,x(xp:tn,f)=rect(xpvxtn,fLx),Kt,y(yp:nTs)=rect(ypvynTsLy).
ΔφTR(xp,yp:tn,f,nTs)=πλR3[(xpvxtn,f)2+(ypvynTs)2]+φHφV.
I(ξ,mTs)=KdE02Ksn(Kr,y2(0:yp:nTs)Kt,y2(yp:nTs)×Kr,x2(0:xp)Kt,x2(xp:tn,f)exp[jπλR3(xpvxtn,f)2]exp[jπλR3(ξ+vxtn,f)2]d(vxtn,f)×exp[jπλR3(ypvynTs)2]exp{jπλR3[vy(mn)Ts]2}).
I(ξ,mTs)=K[Sot(ξ)*δ(ξ+xp)][Stv(m)*δ(mvTsyp)].
dx=λR12kLx=λMR1in2kLxin.
dy=λR3Ly=MλR3inLyin.
Ir(d:Kd)=2SnosPn(d×d)ND2.
x=x,y=ycosθ,
α=xsinφ,β=ycosθxctanφtanθ.
z=αcosφcosθβsinθ.
α=z/cosφcosθ+βtanθ/cosφ.
a=αcosγβsinγ,b=αsinγβcosγ,
tanγ=tanθcosφ.
a=αcosφcosθβsinθcosθtan2θ+cos2φ=z(α,β)cosθtan2θ+cos2φ.
GR(a:γ)=g(asinγ+bcosγ,acosγbsinγ)db.
gB(α,β)=iGR(αcosγiβsinγi)Δγ,
Δa(θ)=Δz(α,β)sin2θ+cos2θcos2φ.
dI=Δa(θ=0)=Δzcosφ.
Sx=λZlxcosφ;Sy=λZly.
μ(Δx,Δy)=[sinc(ΔxSx)*tri(ΔxDxt)][sinc(ΔySy)*tri(ΔyDyt)]sinc(αSx)tri(αDxt)dαsinc(βSy)tri(βDyt)dβ.

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