Paul Merritt, Scott Peterson, Rastislav Telgarsky,. Shawn O«Keefe, Ralph Pringle, Dick Brunson. Boeing-SVS, Inc, 4411 The 25 Way, Suite 350, Albuquerque, ...
Performance of Tracking Algorithms under Airborne Turbulence Paul Merritt, Scott Peterson, Rastislav Telgarsky, Shawn O’Keefe, Ralph Pringle, Dick Brunson Boeing-SVS, Inc, 4411 The 25 Way, Suite 350, Albuquerque, NM 87109
ABSTRACT Tracking through a turbulent atmosphere poses several challenging problems. The authors have recently conducted a series of tracking tests at a MIT/Lincoln Laboratories facility where a complete tracking and adaptive optics system is available in a laboratory. The atmosphere is simulated using seven precision rotating phase screens. A great deal has been learned about tracking algorithms and their response under a scintillated atmosphere. Data will be shown to describe a key limitation to high bandwidth tracking. This effect, called the “Optical Frequency”, appears to be an upper bound on track bandwidth when using an image based tracking system. Keywords: Tracking, turbulence, scintillation, track algorithms, atmospheric tilt, optical frequency
1. Introduction Tracking through a long horizontal atmospheric path poses a series of challenges. The effects of the atmosphere include tilt due to large scale atmospheric lensing, scintillation due to intensity variations, and “higher order” aberrations due to small spatial scale atmospheric lensing. In support of an Air Force contract, Boeing has recently conducted a series of tracking tests at the Advanced Concepts Laboratory (ACL) of MIT/Lincoln Laboratory in Lexington, MA. This is an excellent facility that provides the experimenter with a complete simulated atmospheric propagation path, a complete tracking system, an adaptive optics system, and a target simulator. It also includes a high speed camera at the target end that permits accurate position measurements of the projected scoring beam. The data presented in this paper was taken at the ACL facility under a relatively simple configuration; a co-aligned point source target and scoring beam under open track loop operation. A schematic diagram for this configuration is shown as Figure 1.
Phase screens Scoring Beam
Imaging Camera
Scoring Camera Point Source Laser
Figure 1. Hardware Arrangement for Open Loop Testing
Laser Weapons Technology II, William E. Thompson, Paul H. Merritt, Editors, Proceedings of SPIE Vol. 4376 (2001) © 2001 SPIE · 0277-786X/01/$15.00
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As shown in Figure 1, the scoring beam laser is projected out of the aperture and traverses through up to seven phase screens. It is then measured as a focused spot on the scoring camera. At the same time, a point source laser is folded into the beam path very near to the scoring camera, it traverses the same path through the phase screens in the opposite direction and is measured by the imaging camera. It is usually assumed that the two beams are the reciprocal of each other and that the imaging camera would measure the same tilt as the scoring camera at the target end. This paper will show that this would not be an entirely correct assumption when the two beam are propagating through atmospheric turbulence. This paper will refer to a data reduction technique known as coherence analysis to examine the relationship between the two camera measurements. Figure 2 shows a block diagram showing how the camera outputs are related.
ntr(t) xtr(t)
H1(f) zturb(t)
Track sensor & processing
nsc(t) ysc (t)
H2(f) Scoring sensor & processing Figure 2. Outputs from two physical systems with noise added.
In most data analysis tasks we assume that we know the input to some mechanical or electrical system and that we can measure the outputs from the system. We also expect that noise has been added to the measured output signals. The coherence plot can be used to analyze the relationship between the two output signals. The coherence between the output records as a function of frequency is defined as equation (1).
γ x2−tr y − sc ( f ) = where: Gx-tr
G x −tr y −sc ( f )
2
G xx−tr ( f )G yy− sc ( f )
(1)
y-sc(f)
= the cross spectral density between the scoring beam camera and the tracker camera records Gxx-tr and Gyy-sc = the autospectral densities of the two outputs
The coherence plot provides information on the degree of independence between the two signals, and how that independence varies with frequency. If in Figure 2 above, the noise inputs, ntr and nsc, are small with respect to the signals, xtr and ysc, then the coherence will be close to 1 across the frequency range. On the other hand, if the noises are large with respect to the signals, the coherence will approach zero. The linear noise free systems processing the input, H1 and H2, do not influence the above result. For
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our measurements, we assume the input zturb(t) is the tilt on the tracker and scoring sensors due to the atmosphere. If there were no noise, the two output measurements would have a coherence of 1. However, effects of scintillation, camera noise, image processing noise, and the optical frequency will significantly decrease the similarity between the two measurements and hence lower the coherence.
2. Laboratory Results The measurements taken at Lincoln Laboratory compared the coherence of the tracked point in the tracking camera to the location of the transmitted laser beam measured on the scoring camera. We consider the scoring camera as the first output from the atmosphere, and we consider the tracking camera as the second output of the atmosphere. A sketch of this setup is shown as Figure 3.
Noise Scoring Output Y(t)
Scoring Sensor & Processor
Point Source
Atmosphere, simulated using Phase Screens (Z(t))
Noise
Laser Source Track Sensor & Processor
Track Output X(t)
Figure 3. Setup of the Scoring and Tracking Cameras at the Advanced Concepts Laboratory. As shown in Figure 3, the point source input to the tracker focal plane and the laser scoring beam are co-linear. There is no difference in the two atmospheric paths, or no anisoplanatic angle, involved in these measurements. Although the process is contaminated by sensor noise, we will show that the spatial frequency of the atmosphere is the main feature influencing the coherence between the two measured outputs. We propose that after a certain frequency there is an inherent inability to measure the tracking tilt and the measurements of the tracking sensor become dominated by the scintillation noise. We will now consider data acquired at the ACL from the setup described above. This data was taken in an open track loop configuration with the adaptive optics loop closed. The pupil aperture was 7.7 mm, the atmospheric velocity near the aperture is 11.86 mm/s and the Rytov Variance is a relatively low 0.088. A point source is set in the target plane and a centroid algorithm is used both in the track processor and scoring beam processor. The coherence function between the tracker and the scoring beam outputs is shown in Figure 4. Note that the coherence function in Figure 4 rapidly decreases at a point around 1.54 Hz. This observation is very important - this indicates that there is no coherence between the tracking sensor and the projected beam in the far-field above some frequency. This is an important observation in that it implies that increasing the tracking bandwidth beyond this limit will only amplify noise rather than accurately correct atmospheric tilt.
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1
v/D = 1.54 Hz
0.9
Coherence
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 -1
10 0
10 1
frequency (Hz)
10 2
Figure 4. Coherence plot taken with weak turbulence, Rytov Number = 0.088, Point Source, Adaptive Optics On. The coherence function for a second set of experimental data is shown in Figure 5. Here the velocity of the atmosphere near the aperture is 2.94 mm/s and the Rytov Variance is a stronger value of 0.38. Similar to the weak turbulence data, the stronger turbulence indicates a loss of coherence but at a lower frequency. Thus a scenario with stronger turbulence and a slower velocity shows a similar characteristic on the coherence function.
1 0.9
v/D = 0.38 Hz
0.8
Coherence
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 -1
10 0
10 1
10 2
frequency (Hz)
Figure 5. Coherence plot shown for medium to strong turbulence, Rytov Number = 0.38, Point Source, Adaptive Optics off There appears to be a common parameter in both of the above plots that defines where the corner frequency will occur. This parameter is the “optical frequency” and can be calculated as the velocity of the atmosphere near to the aperture divided by the diameter of the aperture. In Figure 4 above the optical frequency is 1.54 Hz and for Figure 5 it is 0.38 Hz. The calculation of v/D gives an estimate of the frequency where the coherence drops. In the next section we will discuss the physical explanation of the optical frequency.
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Historically, the tracking strategy has been to design a track loop that operates at a high enough bandwidth to reject the atmosphere to as small of a value as desired. This finding of the “optical frequency”, v/D, will impact that strategy.
3. Conceptual Description of Optical Frequency We now examine the physical phenomena responsible for this loss of coherence. Consider the mechanism that causes the tilt in the atmosphere; a non-uniformity of the index of refraction. The spatial structure of the atmospheric density changes with temperature and can have variations that measure from a few centimeters in diameter to several meters. The main feature of the atmosphere density change is that it acts like a lens. Consider first an atmospheric lens that is considerably longer than the diameter of the tracking telescope. This is shown as Figure 6.
10 meters 10 m/s 1m
Figure 6. An atmospheric tilt wave that is much longer than the telescope diameter. The effect of this atmospheric lensing will be to put a gradient on the wavefront entering the telescope. The telescope focuses the wavefront to a point on the focal plane, and a tilt gradient on the input has the effect of moving the location of this focused point around on the focal plane. By monitoring the centroid of this focused spot we determine the tilt of the incoming wave. If a density wave with a 10 meter repetition is passed over a 1 meter diameter telescope at 10 m/s the telescope will see a tilt across the aperture that varies at 1 Hz. The measurement of the tilt will be accurate and well correlated with the outgoing beam tilt. Figure 7 shows a situation where the density structure is the same size as the tracking aperture. In this case tilt is sensed over one complete cycle. The imaged point on the focal plane now will be distorted, but the location on the focal plane will not move. Therefore the average tilt value will be zero regardless of the velocity, a meaningful tilt value cannot be measured under these conditions. This would result in a decrease in the coherence between tracker measured tilt and the far field position because each would be zero and noise would decease the coherence.
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1 meters 1 m/s
1m
Figure 7. An atmospheric tilt wave that is the same spatial length as the telescope diameter. Finally, consider the case in which the density structure is significantly smaller than the tracking aperture diameter as shown in Figure 8. In this case, the image will be distorted due to the changing gradient on the input, and there will be some offset of the focused point because there is some addition tilt gradient in one direction. Since the imaged point will now include multiple cycles of the wavefront gradient, it is apparent that the tilt measured by the centroid will not correctly reflect the normal meaning of full aperture tilt. 0.75 meters 1 m/s
1m
Figure 8. An atmospheric tilt wave that is much shorter than the telescope diameter. It is easy to develop a simple mathematical relationship for the conditions shown in Figures 6 through 8. Let D represent the diameter of the tracker aperture and let L represent the length of one wave of the density structure of the atmosphere. The condition under which tilt can be uniquely measured is given by, D
