# Frequency-Shifted Optical Feedback Measurement Technologies Using a Solid-State Microchip Laser
## 3 Applications of the Laser Feedback Interferometer
### Refractive Index Measurement  
The refractive index \(n\) of a material is a physical quantity that describes how light propagates through the material. Thus, accurate measurement of material refractive index is significant for optical system design. Different methods have been developed. The minimum deviation method has the highest accuracy (\(10^{-6}\)), but the sample needs to be prism-shaped, which is complicated and costly. Figure 11: **(a)** Power spectra of the laser output observed when the laser light is emitted into the atmosphere. Power spectra of the modulated wave observed in drained (**b**) 7 \(\times\)\(10^{-3}\) wt% and (**c**) 7 \(\times\)\(10^{-5}\) wt% PLS-water mixture with PLS 250 nm in diameter. (Figure reproduced from Ref. [64]). Figure 12: Quasi-common-path laser feedback interferometer (QLFI) monitoring the liquid level. (Figure reproduced from Ref. [66]). An Abbe refractometer is the most commonly used instrument, but the measurement range is limited from 1.3 to 1.7, and the accuracy is 10\({}^{-4}\). In comparison, the interferometry methods have many advantages, such as easy sample processing, low cost, wide measuring range, high measurement accuracy, and they have been widely researched in the refractive index measurement field, including Michelson interferometry [67], Mach-Zehnder interferometry [68], Fabry-Perot interferometry [69] and plasmonic interferometry [70; 71], etc. The common principle of these interferometry methods is to measure the optical path change caused by the sample rotation or displacement. However, these interferometers cannot avoid the environmental disturbance, and the measurement accuracies are hard to be further improved. [PERSON] [72; 73] proposed a method to measure the refractive index and thickness simultaneously based on the laser feedback interferometry. Compared with other methods, the influence of environmental disturbance can be eliminated by the quasi-common path structure of the laser feedback interferometry, and the absolute uncertainty in the refractive index measurement reaches \(\approx\)10\({}^{-5}\). On the other hand, the refractive index measurement range is larger than the conventional interferometry due to the high sensitivity of the system. The system is shown in Figure 13.  
By rotating a transparent parallel sample inside the external feedback cavity, the optical path difference \(\Delta L\) can be derived as:  
\[ \begin{array}{ll}\Delta L&=(\Delta P_{m}-\Delta P_{r})\frac{\lambda}{2\pi} \\ &=d[\sqrt{n^{2}-{n_{0}}^{2}\sin^{2}\theta}-n_{0}\cos\theta-\sqrt{n^{2}-{n_{0}}^ {2}\sin^{2}\theta_{0}}+n_{0}\cos\theta_{0}]\end{array}  \tag{11}\]  
where \(\lambda\) is the laser wavelength; \(d\) is the thickness of the sample; \(n\) is the refractive index to be measured; \(n_{0}\) is the refractive index of the air; \(\theta_{0}\) and \(\theta\) are the angles between the laser beam and the normal of sample surface before and after rotation, respectively; and \(P_{m}\) and \(P_{r}\) are the phase variations of the measurement light and the reference light, respectively, which are demodulated by the phase meter. \(\theta\) and \(\Delta L\) are measured at multiple angles in the experiment, then the overdetermined equation can be solved, and the refractive index of the sample can be obtained together with the thickness of the sample. Due to the high sensitivity of the laser feedback interferometry, the method can be used to measure the low transmittance materials, including calcium fluoride (CaF\({}_{2}\)), fused silica, and zinc selenide (ZnSe). The refractive indexes cover a large range from 1.42847 to 2.48272. The results demonstrate that the system has absolute uncertainties in the refractive index measurement of \(\approx\)10\({}^{-5}\)  
Figure 13: Configuration of the refractive-index and thickness measurement using laser feedback interferometry.