# Absolute Distance Measurement Based on Self-Mixing Interferometry Using Compressed Sensing
## 2 Theoretical Analysis  
### Self-Mixing Interferometry  
SMI can be interpreted using a three-facet Fabry-Perot cavity and simplified using a two-facet Fabry-Perot cavity [10]. Neglecting multiple reflections and under a weak feedback level, the modulated power \(P\) and the modulated frequency \(\
u_{F}\) can be expressed as  
\[\frac{4\pi L}{c}(\
u_{0}-\
u_{F})=C\sin(\frac{4\pi L}{c}\
u_{F}+\arctan\alpha), \tag{1}\]  
\[P=P_{0}[1+m\cos(\frac{4\pi L}{c}\
u_{F})], \tag{2}\]  
where \(L\) is the distance between the external target and the laser's front mirror, \(\
u_{0}\) is the optical frequency without feedback, \(m\) is the modulation coefficient, and \(c\) is the speed of light. \(C\) is the external feedback strength parameter related to external distance, length of the laser internal cavity, and reflectivity of the laser front mirror and the external target. \(C\) is an important parameter that affects the dynamics of the laser as well as its output power \(P\). A weak feedback level (\(C<1\)) gives a single solution in Equation (1) and causes \(\
u_{F}\) to approximate \(\
u_{0}\)[21]. When the laser is linearly modulated above its threshold by a direct current with a sawtooth wave current, the modulated power \(P\) in Equation (2) can be expressed as  
\[\begin{array}{rcl}P(t)&\approx&P_{0}+P_{0}m\cos\left\{\frac{4\pi L}{c}[\
u_ {0}+\gamma tr(t)]\right\}\\ &=&P_{0}+P_{0}m\cos\left\{\frac{4\pi L\
u(t)}{c}t+\frac{4\pi L}{c}\
u_{0} \right\}\\ &=&P_{0}+P_{0}m\cos(2\pi ft+\varphi)\end{array} \tag{3}\]  
where \(\gamma\) means the modulation relationship between lasing frequency and modulation current with a unit Hz/mA, which is always considered constant in distance measurement. The parameter \(r(t)\) represents the slope of the sawtooth wave current.  
Both the frequency and the phase of the SMI power signal contain external distance \(L\). However, the problem of phase ambiguity occurs when the phase is used to obtain the distance, so that the accurate distance cannot be distinguished. Therefore, the frequency of the SMI signal is usually extracted to obtain external distance via  
\[L=\frac{fc}{2\gamma r(t)}. \tag{4}\]Because of the excellent properties of the fast Fourier transform (FFT), it is commonly used to calculate frequency. Given that the accuracy of frequency measurement directly impacts the accuracy of distance measurement, some spectrum correction methods are used to overcome the picket fence effect. Interpolated FFT [20] is one of these methods, and is widely used due to its ease of implementation and high correction accuracy. The correction method for interpolated FFT using the Hanning window is defined as  
\[\hat{f}=\frac{f_{\text{s}}}{N}\bigg{(}k^{*}\pm\frac{2A_{k^{*}\pm 1}-A_{k^{*}}}{A_{k^{*} \pm 1}+A_{k^{*}}}\bigg{)}, \tag{5}\]  
where \(f_{\text{s}}\) is the sampling rate, \(N\) is the number of sampling points, \(k^{*}\) is the index of the maximum peak, and \(A_{k^{*}}\) and \(A_{k^{*}\pm 1}\) are the amplitudes of the maximum peak and lateral peaks respectively. The sign of the inner sum in Equation (5) depends on the amplitude of the lateral peaks. The sign \"+\" is used when \(A_{k^{*}+1}\) is greater than \(A_{k^{*}-1}\), and when vice versa, \"\(-\)\" is used.