Abstract

This paper utilizes a combination of theory and simulations to examine synthetic aperture imaging across a wide range of turbulence conditions. Extensive wave optics simulations are used to validate existing theory and to investigate the use of a common measurement technique. It demonstrates the applicability of earlier synthetic aperture laser radar (ladar) (SAL) research across a wide range of turbulence conditions, and examines the metric approaches and limitations for the imaging conditions normally seen in practical SAL systems. To examine the full impact of turbulence on SAL, the derivations, simulations, and analyses include three different resolution metrics as well as a commonly used contrast metric: the integrated sidelobe ratio. This paper demonstrates the integrated effects of turbulence on SAL imaging. Finally, suggestions are given for measuring the true resolving power of operational SAL systems.

© 2017 Optical Society of America

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References

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  1. T. J. Karr, “Resolution of synthetic-aperture imaging through turbulence,” J. Opt. Soc. Am. A 20, 1067–1083 (2003).
    [Crossref]
  2. R. L. Lucke, “Synthetic aperture ladar simulations with phase screens and Fourier propagation,” in IEEE Aerospace Conference Proceedings (2004).
  3. M. A. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).
  4. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [Crossref]
  5. P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
    [Crossref]
  6. M. C. Roggeman and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
  7. J. W. Goodman, Statistical Optics (Wiley, 2000).
  8. G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Anntenas Propag. 15, 90–98 (1967).
    [Crossref]
  9. M. P. Dierking, “Multi-mode coherent ladar imaging via diverse periodic pseudo noise waveforms and code division multiple access apertures,” Ph.D. dissertation (University of Dayton, 2009).
  10. B. M. Welsh, “Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
    [Crossref]
  11. J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and P. Tomlinson, Synthetic Aperture Ladar for Tactical Imaging (SALTI) (Defense Technical Information Center, 2007).

2009 (1)

P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
[Crossref]

2003 (1)

1997 (1)

B. M. Welsh, “Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[Crossref]

1967 (1)

G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Anntenas Propag. 15, 90–98 (1967).
[Crossref]

1966 (1)

Bradford, B.

P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
[Crossref]

Dierking, M.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and P. Tomlinson, Synthetic Aperture Ladar for Tactical Imaging (SALTI) (Defense Technical Information Center, 2007).

Dierking, M. P.

M. P. Dierking, “Multi-mode coherent ladar imaging via diverse periodic pseudo noise waveforms and code division multiple access apertures,” Ph.D. dissertation (University of Dayton, 2009).

Fried, D. L.

Fuhrer, S.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and P. Tomlinson, Synthetic Aperture Ladar for Tactical Imaging (SALTI) (Defense Technical Information Center, 2007).

Gatt, P.

P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Heidbreder, G. R.

G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Anntenas Propag. 15, 90–98 (1967).
[Crossref]

Jacob, D.

P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
[Crossref]

Karr, T. J.

Krause, B.

P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
[Crossref]

Lucke, R. L.

R. L. Lucke, “Synthetic aperture ladar simulations with phase screens and Fourier propagation,” in IEEE Aerospace Conference Proceedings (2004).

Marron, J.

P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
[Crossref]

Richards, M. A.

M. A. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).

Ricklin, J.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and P. Tomlinson, Synthetic Aperture Ladar for Tactical Imaging (SALTI) (Defense Technical Information Center, 2007).

Roggeman, M. C.

M. C. Roggeman and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Schumm, B.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and P. Tomlinson, Synthetic Aperture Ladar for Tactical Imaging (SALTI) (Defense Technical Information Center, 2007).

Tomlinson, P.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and P. Tomlinson, Synthetic Aperture Ladar for Tactical Imaging (SALTI) (Defense Technical Information Center, 2007).

Welsh, B. M.

B. M. Welsh, “Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[Crossref]

M. C. Roggeman and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

IEEE Trans. Anntenas Propag. (1)

G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Anntenas Propag. 15, 90–98 (1967).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Proc. SPIE (2)

P. Gatt, D. Jacob, B. Bradford, J. Marron, and B. Krause, “Performance bounds of the phase gradient autofocus algorithm for synthetic aperture ladar,” Proc. SPIE 7323, 73230P (2009).
[Crossref]

B. M. Welsh, “Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[Crossref]

Other (6)

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and P. Tomlinson, Synthetic Aperture Ladar for Tactical Imaging (SALTI) (Defense Technical Information Center, 2007).

M. C. Roggeman and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

J. W. Goodman, Statistical Optics (Wiley, 2000).

R. L. Lucke, “Synthetic aperture ladar simulations with phase screens and Fourier propagation,” in IEEE Aerospace Conference Proceedings (2004).

M. A. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).

M. P. Dierking, “Multi-mode coherent ladar imaging via diverse periodic pseudo noise waveforms and code division multiple access apertures,” Ph.D. dissertation (University of Dayton, 2009).

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Figures (9)

Fig. 1.
Fig. 1.

SAL geometry.

Fig. 2.
Fig. 2.

Spherical wave phase structure function measured from the simulation showing excellent agreement with theory. The theoretical phase structure function is calculated using Eq. (14) and the coherence diameter measured from the phase screens.

Fig. 3.
Fig. 3.

Normalized resolution as a function of normalized SA length.

Fig. 4.
Fig. 4.

SAL coherence diameter as a function of Fried coherence diameter.

Fig. 5.
Fig. 5.

SAL WSF.

Fig. 6.
Fig. 6.

Average resolution and resolution calculated from an average MTF. This is resolution data for an r 0 value of 0.51 m.

Fig. 7.
Fig. 7.

Average MTF and OTF for a 3.3 m SA.

Fig. 8.
Fig. 8.

IPR of average MTF.

Fig. 9.
Fig. 9.

3 dB width and ISLR. Shows two different atmospheric conditions. The 3 dB width is normalized by the 3 dB width of a SA of length r ˜ 0 .

Tables (1)

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Table 1. Atmospheric Conditions Used in the Simulation

Equations (23)

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R ( t ) = [ ( x n v t ) 2 + z n 2 ] 1 / 2 ,
s r , 0 ( t ) = exp [ j k 2 R ( t ) ] = exp [ j k 0 2 ( z n + x n 2 2 z n x n v t z n + v 2 t 2 2 z n ) ] ,
s r ( t ) = s r , 0 ( t ) exp [ χ ( t ) + j ϕ ( t ) ] ,
s 0 ( t ) = exp [ j k 0 2 ( z n + x 0 2 2 z 0 x 0 v t z 0 + v 2 t 2 2 z 0 ) ]
s r c ( t ) = exp { j k 0 2 [ x 0 2 x n 2 2 z 0 ( x 0 x n ) v t z 0 ] + [ χ ( t ) + j ϕ ( t ) ] } .
K a = 4 π x λ R c ,
H ( K a ) = H 0 ( K a ) exp [ χ ( K a ) + j ϕ ( K a ) ] where    H 0 ( K a ) = { 1 for    L / 2 x ( t ) L / 2 0 otherwise ,
R 1 2 π d J a M ( J a ) ,
M ( J a ) = A d K a H ( K a ) H * ( K a J a ) ,
O ( J a ) = A d K a H ( K a ) H * ( K a J a )
R I 1 2 π d J a | O ( J a ) | .
R I = 1 2 π d J a | O ( J a ) | 1 2 π d J a O ( J a ) = R c .
O ( Δ x ) = A d x H ( x ) H * ( x Δ x ) ,
D ( Δ x ) = 6.88 ( | Δ x | r 0 ) 5 / 3 ,
Γ ( Δ x ) = Γ a p ( Δ x ) Γ p ( Δ x ) = Γ a p ( Δ x ) exp [ 1 2 D ( Δ x ) + 1 2 Δ a ¯ 2 Δ x 2 ] ,
D ( Δ x ) = [ ϕ ( x ) ϕ ( x + Δ x ) ] 2 + [ χ ( x ) χ ( x + Δ x ) ] 2 Δ a ¯ 2 Δ x 2 .
R c = 2 λ R c d Δ x Γ a p ( Δ x ) exp [ 1 2 D S A ( Δ x ) + 1 2 Δ a ( L ) ¯ 2 Δ x 2 ] .
Δ a ¯ 2 = min ( α β 1 , 1 ) L 2 D S A ( L ) .
D S A ( Δ x ) = 6.88 ( | Δ x | r ˜ 0 ) 5 3 ,
r ˜ 0 = r 0 2 6 / 5
ISLR = 10 log 10 ( a | h ( u ) | 2 d u + b | h ( u ) | 2 d u a b | h ( u ) | 2 d u ) ,
ISLR = 10 log 10 ( a | h ( u ) | 2 d u + b | h ( u ) | 2 d u a b | h ( u ) | 2 d u ) .
R 3 dB R 1 , 3 dB = w 0 w i ,

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