# Frequency-Shifted Optical Feedback Measurement Technologies Using a Solid-State Microchip Laser
## 2 Experimental Setup and Theoretical Analysis  
### Laser Frequency-Shifted Optical Feedback  
The experimental setup of the laser frequency-shifted optical feedback is shown in Figure 1. A 3 mm \(\times\) 3 mm \(\times\) 0.75 mm Nd:YVO\({}_{4}\) crystal plate is employed to form a laser resonator with the coating on both surfaces. The left surface is coated to be antireflective at the pump wavelength of 808 nm and highly reflective (R \(>\) 99.8%) at the lasing wavelength of 1064 nm; the output surface is coated to be have 5% transmittance at a wavelength of 1064 nm. The pump light, produced using a fiber-coupled single-mode laser diode, is focused onto the center of the Nd:YVO\({}_{4}\) crystal. The threshold of Nd:YVO\({}_{4}\) is about 27 mW, and the range of the pump light is 0-200 mW. The output laser is split by the beam splitter (BS). The reflective part is detected by a photon detector (PD), and the transmitted part is collimated by the lens (L). Two acousto-optic modulators (AOMs) with working frequencies of \(\Omega_{1}\) and \(\Omega_{2}\) are utilized at a differential configuration to modulate the frequency of the output laser. By adjusting the AOMs and target (T), the laser frequency is shifted by \(\Omega=2\,\mathrm{\SIUnitSymbolO}_{1}-\,\Omega_{2}\,\mathrm{\SIUnitSymbolO}\) after a round-trip as the measuring light. The attenuator (ATT) is inserted in the optical path to modify the feedback level. In the weak feedback level, the laser power fluctuation under the effect of frequency-shifted feedback can be simulated using the modified Lang-Kobayashi equation [40; 41]:  
\[ \begin{array}{c}\frac{dN(t)}{dt}=\gamma(N_{0}-N(t))-BN(t){|E(t)|}^{2}\\ \frac{dE(t)}{dt}=\frac{1}{2}(BN(t)-\gamma_{C})E(t)+\gamma_{C}\kappa\cos(2\pi \Omega t-\omega\tau)E(t)\end{array}  \tag{1}\]  
where \(N(t)\) is the population inversion, \(E(t)\) is the amplitude of the laser electric field, \(N_{0}\) is the inversion particle number under a small signal, \(\gamma\) is the decay rate of the population inversion, \(B\) is the Einstein coefficient, \(\gamma_{C}\) is the laser cavity decay rate, \(\kappa\) is the effective laser feedback level, \(\omega\) is the optical running laser frequency, and \(\tau\) is the photon round-trip time between the laser and the target. The stationary solution of the Equation (1) can be obtained by setting the electric field \(E\) and population inversion \(N\) to be constant. Without feedback (\(\kappa=0\)), the steady laser solutions are:  
\[ \begin{array}{c}N_{\mathrm{s}}=\frac{\gamma_{C}}{B}\\ I_{\mathrm{s}}={|E_{\mathrm{s}}|}^{2}=\frac{1}{B}(\eta-1)\end{array}  \tag{2}\]  
where \(I_{\mathrm{s}}\) is the stationary intensity of the laser field, and \(\eta=\mathrm{BN}/\gamma_{C}\) is the normalized pumping rate. The power spectrum of the continuous pumped Nd: YVO\({}_{4}\) laser can be considered as slight fluctuations of the stable solutions. The electric field \(E(t)\) and the population inversion \(N(t)\) can be written as:  
\[ \begin{array}{c}N(t)=N_{S}+n(t)\\ E(t)=E_{S}+e(t)\end{array}  \tag{3}\]  
Substituting Equations (2) and (3) into Equation (1) and neglecting the second-order terms, we obtain:  
\[ \begin{array}{c}\frac{dn^{*}(t)}{dt}=-\gamma\eta n^{*}(t)-2\gamma(\eta-1)e( t)\\ \frac{de^{*}(t)}{dt}=\frac{1}{2}\gamma_{C}n^{*}(t)+\gamma_{C}\kappa\cos(2\pi \Omega t-\omega\tau)+\gamma_{C}\kappa\cos(2\pi\Omega t-\omega\tau)e^{*}(t)\end{array}  \tag{4}\]  
where \(n^{*}(t)=n(t)/N_{S}\) and \(e^{*}(t)=e(t)/E_{S}\) are the normalized variation of the population inversion and the electric field. The variation of laser intensity versus time can be derived according to Equation (3) as:  
\[\frac{\Delta I}{I}=\frac{2E(t)e(t)}{E(t)^{2}}\approx\frac{2e(t)}{E_{\mathrm{ s}}}=2e^{*}(t) \tag{5}\]  
Thus, the solution of Equation (4) denotes the power spectrum of a Nd: YVO\({}_{4}\) laser under frequency-shifted feedback. The results with different feedback levels using numerical solutions are shown in Figure 2. Three kinds of feedback levels are identified as weak, moderate, and strong feedback [42].