# A Traceable High-Accuracy Velocity Measurement by Electro-Optic Dual-Comb Interferometry
## 2 Measurement Principle  
To generate an optical comb, an electro-optic phase modulator is used to modulate the seed laser to produce different frequency sidebands. The modulated laser can be expressed as  
\[E=\sum_{m=-16}^{16}A_{m}\cos[2\pi(f_{0}+mf_{rup})+\varphi_{m}] \tag{1}\]  
where \(m\) is an integer (the range of \(m\) is [16; 16; 1], meaning that our experimental device produced 32 sidebands), \(A_{m}\) is the amplitude of the \(m^{th}\) sideband,\(f_{0}\) is the optical frequency of the seed laser,\(f_{rup}\) is the repetition frequency of the modulated laser, and \(\varphi_{m}\) is the initial phase. When this laser beam is incident on the moving target, the corresponding Doppler signal can be written as  
\[E_{dop}=\sum_{m=-16}^{16}A_{m}\cos[2\pi(f_{0}+mf_{rep})(1+\frac{2v}{c})+\varphi _{m}] \tag{2}\]  
where \(v\) is the velocity of the moving target and \(c\) is the velocity of light. The repetition frequency of the optical comb changes from \(f_{rup}\) to (\(1+2v\))/\(c\)/\(c\)/\(c\)/\(c\)/\(c\)/\(c\)/\(c\). Thus, the Doppler shift occurring at the repetition frequency reflects the information of velocity. When the target is close to the laser source, \(v\) is positive and the repetition frequency is increased. When the target is moving towards the laser source, \(v\) is negative and the repetition frequency is reduced. From Formula (1) and Formula (2), the velocity of the moving target can be expressed as \(v\)=\(\Delta f_{rup}\)/\(2f_{rup}\)/\(c\), where \(\Delta f_{rup}\) is the Doppler shift of the repetition frequency.  
As the velocity of the moving target is much smaller than that of light, in order to improve the resolution, it is better to increase the modulation repetition frequency. For the same moving velocity, the higher the repetition frequency, the greater the Doppler shift and the higher the resolution. However, detecting high repetition frequency requires very expensive detection equipment. Hence, we developed a method using a dual optical comb to detect the Doppler shift. One optical comb modulates the repetition frequency to _frrep.sig_ as a signal source and the other optical comb modulates the repetition frequency to _frrep.loc_ as a local oscillator.  
The signal source can be expressed as  
\[E_{sig\_i}=\sum_{m=-16}^{16}A_{m}\cos[2\pi(f_{0}+f_{AOM}+mf_{ rnp.sig})+\varphi_{m1}] \tag{3}\]  
The local oscillator can be expressed as  
\[E_{loc}=\sum_{m=-16}^{16}A_{m}\cos[2\pi(f_{0}+mf_{ rnp.loc})+\varphi_{m2}] \tag{4}\]  
Then, the beat signal manifests as a new comb in the frequency domain and can be detected by an oscilloscope. The detected beat signal can be expressed as  
\[I\propto\sum_{m=-16}^{16}\cos[2\pi(f_{AOM}+m(f_{ rnp.sig}-f_{ rnp.loc})+(\varphi_{m1}-\varphi_{m2})] \tag{5}\]  
After being reflected by the moving target, the signal source changes to  
\[E_{sig\_r}=\sum_{m=-16}^{16}A_{m}\cos[2\pi(f_{0}+f_{AOM}+mf_{ rnp.sig})(1+\frac{2v}{c})+\varphi_{m3}] \tag{6}\]  
Hence, the detected beat signal can be expressed as  
\[I^{\prime}\propto\sum_{m=-16}^{16}\cos[2\pi((1+\frac{2v}{c})f_{AOM}+\frac{2v}{ c}f_{0}+m(1+\frac{2v}{c})f_{rnp.sig}-mf_{ rnp.loc})+(\varphi_{m3}-\varphi_{m2})] \tag{7}\]  
When comparing Formula (5) and Formula (7), it can be seen that the repetition frequency changes from _frrep.sig_frrep.loc_ to _(1+2v)_frrep.sig_frrep.loc_. In this way, the velocity of the moving target can be calculated.