# Measurement of Ultra-High Speed by Optical Multistage Cascade Frequency Reduction Technology
## 2 Principle  
### Light Path Construction  
In most cases, a DISAR illuminates a moving target with light with wavelength \(\lambda_{1}\) (velocity \(u\)), collects reflected Doppler signal light \(\lambda_{1}\), and mixes the signal light with reference light \(\lambda_{2}\). Therefore, the mixed beat frequency signal is as follows:  
\[v_{B}=\left|\frac{c}{\lambda_{1}^{\prime}}-\frac{c}{\lambda_{2}}\right|\approx \left|\frac{c}{\lambda_{1}}-\frac{c}{\lambda_{2}}+\frac{2u}{\lambda_{1}}\right| \tag{1}\]  
The size of the beat frequency signal is determined by the speed \(u\) of the target motion and the wavelength of the reference light. Normally, the beat optical signal is converted into an electrical signal by a rapid photodetector and recorded by a digital oscilloscope. In most devices, \(\lambda_{1}=\lambda_{2}\) is commonly set, which is DISAR or PDV technology. Because a DISAR generally uses a laser with a wavelength of 1550 nm, every 1 km/s speed requires an electronic bandwidth of about 1.3 GHz. It is unrealistic to track a moving target as high as 20 km/s using only a single-channel DISAR. A beat frequency signal is divided into multiple channels to reduce the bandwidth requirement of the detector and oscilloscope, so that the measured beat frequency signal can be \"relayed\" among multiple electronic channels. This technology can be called multi-stage cascade technology. Taking the two-stage cascade technology as an example, as shown in Figure 1, the main laser (wavelength: 1550.256 nm; power: 1 W) emits continuous single-frequency laser to the optical fiber probe through the second port of the optical fiber circulator, and the probe irradiates the measured surface and receives the reflected signal light, which is a heterodyne with reference laser 1 (wavelength: 1550.128 nm; power: 50 mW) and reference laser 2 (wavelength: 1550.000 nm; power: 50 mW). Figure 1: Schematic diagram of optical cascade frequency reduction optical path. Assuming that the wavelengths of the main laser, reference laser 1, and reference laser 2 are \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\), respectively, and the velocity of the target to be measured is \(u\), then the velocity \(u_{1}\) obtained by channel 1 after the first frequency reduction and the velocity \(u_{2}\) obtained by channel 2 after the second frequency reduction are, respectively, as follows:  
\[u_{1} =\frac{\lambda_{1}}{2}\bigg{|}f_{d}-\left(\frac{c}{\lambda_{2}}- \frac{c}{\lambda_{1}}\right)\bigg{|} \tag{2}\] \[u_{2} =\frac{\lambda_{1}}{2}\bigg{|}f_{d}-\left(\frac{c}{\lambda_{3}}- \frac{c}{\lambda_{1}}\right)\bigg{|} \tag{3}\]  
where \(f_{d}\) is Doppler shift. If the speed signal is a straight line motion with uniform acceleration, that is, a straight line signal with a fixed slope, as shown in Figure 2a, then Equations (2) and (3) correspond to the green and blue curves in Figure 2b, respectively. Due to the limitation of the bandwidth \(\mathrm{f}_{\mathrm{f}w}\) of the acquisition system, the actual channel 1 and channel 2 can only record the signals in the green and blue boxes, respectively. The final complete speed curve can be \"restored\" by \"flipping\" the two signals and \"splicing\" the end and end. This is the frequency reduction effect that can be achieved by a two-stage cascade. By selecting the appropriate reference laser wavelength, the recording system that originally needed quadruple bandwidth can be reduced to only one bandwidth. Each stage of frequency reduction can reduce the signal frequency by half, so when N-stage cascade frequency reduction technology is adopted, the original signal frequency can be reduced by 2N times, and there is no upper limit on the frequency reduction multiple of signal frequency in theory. However, it is worth noting that every time the frequency reduction is added, the original signal needs to be divided into one part for an optical heterodyne.