Jason W. Stafford University of Dayton Electro-Optics Program Bradley D. Duncan University of Dayton Electro-Optics Program Matthew P. Dierking EO Combat ID Branch, AFRL/ RYJM Wright-Patterson AFB
Holographic Aperture Ladar (HAL) uses an imaging sensor, not a point detector Accounts for synchronous motion of receiver aperture (RX) and transmitter (TX) Measures optical field indirectly via interferograms Assumes flood illumination and nominally planar target HAL transform relocates the transmitter to the origin of the receiver aperture plane for all shots Stripmap — transmitter is perpendicular to direction of flight and sweeps over the target
Complex Pupil Field go(xa,ya) Virtual Imaging Lens Multiple Cross Range Interferograms Collected Across the Synthetic Aperture (MO not shown)
Virtual Focal Plane Detector
ξ
xa xT
Ro TX
f(ξ,η)
RX Aperture
yT
ya
η
2π = g 0 ( xa + xT , ya + yT ) g sm ( xa , ya ) exp j [ xa xT + ya yT ] λ R0
For complete pupil plane coverage, shots must be collected at least every Dap/2
To avoid the problems associated with translating the RX and TX, the target can instead be translated while the TX and RX are held stationary The choice of a suitable target leads to other issues which we examine in more depth following.
High precision BB simulates a point target located at the focal point Nominally planar target assumption is not valid Excess phase is accumulated as distance from axis increases
Excess phase is D 2 φ =k
−
D
2
4
−ξ
2
( ( )) 2ξ
cos 2 sin
D
D + 2
−
D
2
4
−ξ
2
Paraxial approximation for a spherical mirror phase transformation is 2 k ξ φi =
( D / 2)
185μm offset (ξ0) for diffraction limited performance Creates an effective half-field of view for the reflected field obtained at the TX/RX plane For our setup the halffield of view will be 6.7º Max TX/RX separation of 11.6cm
As the target moves off-axis the focal point of the sphere rotates towards the TX. This leads to deterministic piston phase errors and longitudinal position errors. TX/RX separation of 11.6cm
∆φ piston = 2
2π
λ
z ∆ =
πD λ
(1 − cos θ 2 )
D ∆ξ = sin θ 2 4
where x −ξ C θ 2 = tan T R+D 4 −1
(
)
Standard Digital Holography setup Used an afocal telescope (3X magnification) Mixed pupil plane field with LO to obtain fringes at CCD Fourier transform, crop image, propagate back to pupil plane to obtain field segments λ = 633nm
Uncorrected and Corrected Off-Axis Phase Segments (xp = 1.127 cm) 3500 HAL Transformed Ideal Phase Raw Phase
3000
Phase [waves]
2500
2000
1500
1000
500
0 -0.1
-0.08
-0.06
-0.04
-0.02
0
x [m]
0.02
0.04
0.06
0.08
0.1
Uncorrected and Corrected On-Ax is Phase Segm ents (x p = 0 cm ) 3500 HAL Transformed Ideal Phase Raw Phase
3000
Phase [waves]
2500
2000
1500
1000
500
0 -0.1
-0.08
-0.06
-0.04
-0.02
0
x [m]
0.02
0.04
0.06
0.08
0.1
Uncorrected and Corrected Off-Ax is Phase Segm ents (x p = -1.127 cm ) 3500 HAL Transformed Ideal Phase Raw Phase
3000
Phase [waves]
2500
2000
1500
1000
500
0 -0.1
-0.08
-0.06
-0.04
-0.02
0
x [m]
0.02
0.04
0.06
0.08
0.1
The effective half-field of view due phase walk-off on a ball bearing target can be overcome with proper TX and RX separation. Deterministic piston phase and longitudinal error corrections have been derived. We have verified the effectiveness of the HAL transform in a laboratory demonstration. Future research will focus on relative piston phase errors between multiple shots.
This effort was supported in part by the U.S. Air Force through contract number FA8650-062-1081, and the University of Dayton Ladar and Optical Communications Institute (LOCI). The views expressed in this article are those of the authors and do not reflect on the official policy of the Air Force, Department of Defense or the U.S. Government.
B.D. Duncan., M.P. Dierking,: “Holographic Aperture Ladar”, Applied Optics, Vol. 48, No. 6, p. 1168 (2009)
J.W. Stafford., B.D. Duncan., M.P. Dierking,: “Monte Carlo simulation of the effects of pulse and platform jitter on Holographic Aperture Ladar systems”, Proceedings of the SPIE Conference on Defense, Security and Sensing, Vol. 7323 Laser Radar Technology and Applications XIV (2009)
A. Marechal,: “Study of the combined effects of diffraction and geometrical aberrations on the image of a luminous point”, Rev d’Optique, Vol. 26, pp. 257–77 (1947)
