1. Spotlight Holographic Aperture Ladar Transformation

The spotlight holographic aperture ladar (HAL) scenario involves a transmitter (TX) beam always being directed, through use of an appropriate beam-steering device, toward the center of a remote scene or target of interest. As in the stripmap case, we assume that the target is flood illuminated by a single coherent TX beam for all shots across the synthetic aperture, and that the scene remains unchanged over multiple looks. We also assume the target to be nominally planar and uniformly illuminated, and we ignore all atmospheric effects.

The geometry shown in Fig. 2 of the original article [1] applies to the spotlight HAL case. For convenience, this figure is reproduced in this erratum as Fig. 1. Shown in this figure is a single TX located at ($x_T$, $y_T$) in the receiver (RX) aperture plane coordinate system. The TX then illuminates the target $f(\xi ,\eta )$ at range $R_o$, and the returning complex field is collected by a single off-axis RX aperture, the location of which will not affect the HAL transformation. Notice also that the RX plane coordinates are given by variables ($x_a$, $y_a$), while the target plane coordinate variables are ($\xi$, $\eta$).

 

Fig. 1. HAL transformation geometry.

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Regrettably, an error has been found in Eq. (17) of the original article. As this is the starting point for the development of the spotlight HAL transformation, the mathematical analysis and conclusions following this equation are also in error. We seek to correct those errors here. To begin, we first reproduce Eq. (17) of the original article so that we might better examine the error it contains. In particular, the field $g_{\mathrm{sp}}(x_a,y_a)$, where the subscript sp indicates the spotlight case, in the plane of the RX aperture was incorrectly expressed as

Assuming for now that $f(\xi ,\eta )$ in Eq. (1) is separable, we can proceed with the details of the derivation in only one dimension [1]. Expanding and simplifying Eq. (1) in one dimension then yields

As in the stripmap HAL case, the transmitter would ideally be located at $x_T=0$ for all shots across the synthetic aperture, leading to an ideal received field $g_o(x_a,y_a)$ given in one dimension as

To then express $g_o$ in terms of the known or measured quantities $g_{\mathrm{sm}}(x_a)$ and $x_T$, we first add $x_T$ to $x_a$ in Eq. (5) and rearrange to yield

Interestingly, the stripmap and spotlight HAL transformations are identical, except for the first exponential piston phase factor (i.e., this factor is constant across the RX aperture plane) of Eq. (7), which varies only according to transmitter location. Since the piston phase term is constant with respect to the RX plane coordinates $(x_a,y_a)$, we should expect identical maximum sample spacings for contiguous field segments and identical image-sharpening capabilities (assuming synthetic apertures of equal length) for both spotlight and stripmap HAL. We will investigate this expectation through a one-dimensional, single point-target example.

To begin we set our target function to $f(\xi )=\delta (\xi -{\xi }_p)$, where $\delta$ indicates the Dirac delta function. After some simplification, inserting this target function into Eq. (2) and applying the sifting property yields

For our example, we will now assume, as in the original article, that the single point target is located at a longitudinal cross-range position of ${\xi }_p=0.25\mathrm{m}$. We will also assume a target range of $R_o=30\mathrm{km}$, a wavelength of $\lambda =1.5\mathrm{\mu m}$, and a real aperture diameter of $D_{\mathrm{ap}}=0.4\mathrm{m}$. We will furthermore assume that our HAL sensor is monostatic, with the TX centered on and exiting through the RX aperture, and that the synthetic aperture diameter is $D_{\mathrm{SAR}}=0.8\mathrm{m}$, extending from $x_a=-0.4\mathrm{m}$ to $x_a=0.4\mathrm{m}$. (Recall that the synthetic aperture is defined by the distance over which the TX travels.). Lastly, we will assume that raw field segments are collected at a uniform TX translation distance of $D_{\mathrm{ap}}/2$. With a synthetic aperture of 0.8 m, this means there will be a total of five equally spaced field-segment samples. The results are shown in Fig. 2.

 

Fig. 2. Example of the corrected spotlight HAL transformation applied to five sequentially collected phase-only field segments resulting from an off-axis point target. In this case, the RX aperture overlaps itself by half its diameter during each subsequent TX–RX cycle. Monostatic conditions apply.

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The black dots along the horizontal axis of Fig. 2 indicate the location of the TX during each field-segment collection cycle. Similarly, the black dot on the upper border of the figure indicates the longitudinal cross-range location of the point target (assuming, of course, that the target is also located at the proper range $R_o$). The ideal phase is then shown by the solid black curve. This was found by plotting Eq. (9) over an extended range of longitudinal cross-range values. Next, the raw phase segments were calculated by substituting appropriate values into Eq. (8), while each of the corrected phase segments, lying along the ideal phase curve, represent the results of applying the spotlight HAL transformation to the raw phase values shown with corresponding data-point markers. We see from Fig. 2 that the spotlight HAL transformation precisely corrects for the effects due to an off-axis TX. We also see that collecting data every $D_{\mathrm{ap}}/2$ results in contiguous field segments after application of the HAL transform. Notice that the raw field segment collected when $x_T=0$ requires no correction.

After careful consideration of Fig. 2, our example leads us to the following conclusions, assuming that we are able to design a spotlight HAL sensor capable of accurately back steering the beam as the sensor flies past the target:

Note that as expected the above three conclusions are identical to those for the stripmap HAL case. As with traditional stripmap SAR, the synthetic aperture in stripmap HAL is limited by the size of the illumination beam in the plane of the target. This in turn limits the maximum image-sharpening ratio. By contrast, the synthetic aperture in spotlight HAL, though potentially unlimited, is ultimately limited in practice by sensor’s ability to continuously steer the TX beam toward the center of the target. Nevertheless, once the synthetic aperture is established, the relationships for optimum sample spacing, effective aperture width, and image-sharpening ratio are identical for both stripmap and spotlight HAL. We should also note that as the stripmap and spotlight HAL transformations differ by only a simple piston phase factor, the data-processing and image-formation methods we have recently experimentally demonstrated for the stripmap HAL case also directly apply to the spotlight HAL case [2].