Coherent Laser Radar Efficiency and Power Variance with Gaussian Pointing Errors

In this paper we present analytic models of the CNR loss or efficiency due to Gaussian line-of-sight pointing errors with bias, as a function of the correlation coefficient between the transmit and back-propagated local oscillator beams of a coherent laser radar. We also present theoretical expressions for the normalized signal power variance (a.k.a., the scintillation index) including speckle noise. This theory is developed for Gaussian targets, which converges to the point and extended target solutions, under the appropriate small and large diameter target limits. Including correlation between the transmit and back propagated local oscillator (BPLO) beams allows one to predict performance as a function of target range for a monostatic ladar, since at zero range the two beam positions are fully correlated, whereas at infinite range they are fully uncorrelated. Numerical experiments were developed and the resulting measurements are shown to agree with the analytic theory. The validated simulation tool is then exercised against other targets (e.g., a disk), for which closed form solutions are elusive. Analysis of the best-fit Gaussian target to represent the efficiency of a uniform disk target is also explored. We seek in this work to develop analytic expressions for the impact of random pointing errors on coherent ladar performance. Pointing errors tend to reduce the signal power measured by a coherent ladar, as they cause the transmit beam to be misaligned with the receiver field-of-view or cause either of these to be misaligned with the target. A critical relationship in this effect would be the size of the pointing error, bias and jitter, compared to the size of the transmit beam and receiver field-of-view, the latter being defined by the back-propagated local oscillator (BPLO) beam. Our goal is to find an expression for the reduction in signal power that will be expressed in terms of these parameters. In addition we also develop theoretical expressions for the normalized signal power variance also known as the scintillation index. This is a critical parameter as it drives the SNR of a power measurement and detection statistics. The size of the target also plays a role in the impact of pointing errors on measured signal power. When operating against an essentially infinitely wide target, the transmitter and receiver are never misaligned with the target regardless of the size of the pointing error. However, the transmitter and receiver may be misaligned with respect to each other, and this will decrease measured signal power. In the case of a very small target, pointing errors have two effects. They create misalignment between the transmitter and receiver, as with the large target, and they create misalignment between these beams and the target. In the published literature there are results that address special cases of the scenarios we consider in this work. Reference (1) derives an expression for the reduction in mean signal power for the case of a point target and Gaussian pointing jitter. It assumes the pointing errors are identical between the transmitter and receiver, and its result agrees with the expression derived in this work under the same assumptions. In reference (2), an integral expression for the signal power is given that includes pointing errors, and it mirrors an integral expression used in this work. However, reference (2) also assumes identical pointing error between transmitter and receiver, while in this work we seek a solution for arbitrary correlation between the two. The published literature also contains results that address similar topics. Reference (3) examines the impact of pointing errors on a direct detection system (in fact, (1) extends the results of (3) to coherent ladar). However, (3) considers only direct detection and its focus is on probability of detecting rather than mean signal power that we seek here. There are published results that examine the effects of fixed pointing offsets on coherent ladar, such as (4) and (5). The loss in signal power due to misalignment of the receiver is given in these publications, but random pointing errors and correlation between transmitter and receiver pointing errors and not considered. Still, the methods used provide a starting point for our analysis.