# Visualizing Small Objects Using Amplitude-Modulated Laser Light at Microwave Frequencies
## 2 Data Processing Methods  
Since laser light only works as a microwave signal carrier, signal processing is used to deal with the microwave signals. We consider a geometry in which a reference point set at the origin of the coordinate system, as shown in Figure 1.  
The transmitter/receiver, TX/TR, position is at \(\mathbf{r}_{0}\), and the point target is at \(\mathbf{r}_{t}\). The microwave-modulated laser light travels to the target over a distance of \(|\mathbf{r}_{0}-\mathbf{r}_{t}|\) and is scattered back to the transmitter/receiver. The scattered light signals are then detected and mixed with the reference microwaves, which have a waveform that is identical to the transmitted waveform. We chose to delay the reference signal by comparing the round-trip travel time to a reference, \(2|\mathbf{r}_{0}|/c\) The interference signal is given by  
\[\begin{array}{l}F(k)=A(\sigma)\exp[j2|k|\cdot(|\mathbf{r}_{0}-\mathbf{r}_{t}|-|\mathbf{r }_{0}|)]\\ =A(\sigma)\exp[j2\mathbf{k}\cdot\mathbf{r}_{t}],\ for\ |\mathbf{r}_{t}|\ll|\mathbf{r}_{0}| \end{array} \tag{1}\]  
where \(\mathbf{k}=(\omega/c)\cdot\mathbf{r}_{t}/|\mathbf{r}_{t}|\).  
The amplitude depends on the target-scattering cross-section, \(\sigma\).  
In the experiment presented here, the laser light was spotted on a continuous surface. The system then detects the phase \(\varphi=j2\mathbf{k}\cdot\mathbf{r}_{t}\). By scanning the focused spot, a 2D surface shape is found. The phase measurements have multiple-of-\(2\pi\) ambiguity. If the microwave frequency is chirped and \(F(\mathbf{k})\) is measured within a range of wavenumbers: from \(\mathbf{k}_{0}-\Delta k/2\) to \(\mathbf{k}_{0}+\Delta k/2\), then the Fourier transform of Equation (1) with respect to \(\mathbf{k}\) is  
\[S(\mathbf{r})=B(\sigma)\ since(\Delta k\cdot(\mathbf{r}-\mathbf{r}_{t}))\exp[j2\mathbf{k}_{0} \cdot\mathbf{r}_{t}]. \tag{2}\]  
The peak position of the sinc function gives an approximate value of \(\mathbf{r}\) with the resolution \(\pi/\Delta k\). Multiple-of-\(2\pi\) ambiguity is reduced. Two-dimensional surface measurements allow phase unwrapping processing to avoid \(2\pi\) ambiguity. In the LADAR experiment, we extracted the phase value at the maximum of the sinc function in Equation (2) to evaluate the range. It is also noted that the reference point employed in the LADAR experiment was set to the TX/RX position.  
Due to space limitations in the laboratory, the SAIL experiments were set in a way in which small targets were on a rotatable stage instead of moving transmitter/receiver. We chose the reference point at the center of stage, as shown in Figure 2.  
Figure 1: Coordinate system used for LADAR measurements.  
On the coordinate system fixed onto the stage, the direction of \(\mathbf{k}\) turns by \(-\theta\) when the stage turns by \(\theta\). The \(F(\mathbf{k})\) measurements are obtained by chirping the frequency, \(\omega\), and rotating the table; \(F(\mathbf{k})\) is obtained from the area of \(\mathbf{k}\), \((k_{0}-\frac{\Delta k}{2}<k<k_{0}+\frac{\Delta k}{2})\), \(\theta_{start}<\theta<\theta_{end})\). Interpolations of \(F(\mathbf{k})\) data on to the rectangular coordinate system using \(k_{x}=k\cos(\theta)\) and \(k_{y}=k\sin(\theta)\) allow for the digital Fourier transform, FFT, of \(F(\mathbf{k})\) to form 2D images. The \(F(\mathbf{k})\) data are in a fan-shaped area with \(k_{\rm x}\) and \(k_{y}\) spanning  
\[\begin{array}{c}K_{x}\ \sim\ k_{0}[\sin(\theta_{end})-\sin(\theta_{start})]\\ K_{y}\ \sim\ k_{0}[1-\cos(\theta_{start}\ or\ \theta_{end})]+\Delta k.\end{array} \tag{3}\]  
The image widths of a point target are approximately \(\pi/K_{x}\) in the \(x\) direction and \(\pi/K_{y}\) in the \(y\) direction and can be much smaller than the width given by the frequency chirp width \(\pi/\Delta k\).