# Frequency-Shifted Optical Feedback Measurement Technologies Using a Solid-State Microchip Laser
## 3 Applications of the Laser Feedback Interferometer
### Thermal Expansion Coefficient Measurement  
The thermal expansion coefficient is one of the most fundamental quantities of materials, which describes how the size of an object changes due to the temperature variation. The precise measurement of the thermal expansion coefficient is significant in basic scientific research and industrial application. [PERSON] [75; 76] proposed a non-contact method to measure the thermal expansion coefficient of materials utilizing a pair of Nd: YAG microchip laser feedback interferometers (MLFIs). The schematic diagram of the system is shown in Figure 15.  
Figure 14: Experimental setup of the liquid refractive index measurement. LD\({}_{1}\), LD\({}_{2}\), laser diodes; ML\({}_{1}\), ML\({}_{2}\), microchip lasers; BS, beam splitter; PIN\({}_{1}\), PIN\({}_{2}\), detectors; AOM\({}_{1}\), AOM\({}_{2}\), acousto-optics modulators; L\({}_{1}\), L\({}_{2}\), L\({}_{3}\), lenses; ATT, attenuator, M\({}_{1}\), M\({}_{2}\), M\({}_{\text{R}}\), mirrors. (Figure reproduced from Ref. [74]). The sample with a supporter is placed in the middle of the furnace chamber. Two symmetric laser feedback interferometers output two measurement beams, which are incident on each surface of the sample perpendicularly and coaxially, to compensate for the influence of the sample supporter distortion. Two M\({}_{\rm r}\)s are set close to the muffle furnace, generating the two reference signals, revealing the air flow and thermal lens effect disturbances outside the furnace chamber. By subtracting it, \(S_{1m}-S_{1r}\) and \(S_{2m}-S_{2r}\) are the compensated displacement of the two laser feedback interferometers, respectively. Thus, the length change of the sample from \(T_{0}\) to \(T_{1}\) can be expressed approximately as:  
\[\Delta S\approx S_{1m}+S_{2m}-(S_{1r}+S_{2r}) \tag{14}\]  
where \(S_{1m}\) and \(S_{2m}\) are the displacements of the two measurement beams, and \(S_{1r}\), and \(S_{2r}\) are the displacements of the two reference beams. Then, the thermal expansion coefficient can be obtained as:  
\[\alpha(T_{1};T_{0})=\frac{1}{L_{0}}\times\frac{\Delta S}{T_{1}-T_{0}} \tag{15}\]  
where \(L_{0}\) is the length of the sample at \(T_{0}\). The aluminum and the steel 45 samples are measured in the experiment from room temperature to 748 K, which proves that the measurement repeatability of thermal expansion coefficient is better than 0.6 \(\times\) 10\({}^{-6}\) (K\({}^{-1}\)) in the range 298 K-598 K and the high-sensitive non-contact measurement of the low reflectivity surface induced by the oxidization of the samples in the range of 598 K-748 K.  
The existing methods for the thermal expansion coefficient measurement mainly include a mechanical dilatometer [77], speckle pattern interferometry [78], Fabry-Perot interferometry [79], etc. The mechanical dilatometer is an old and frequently utilized method, which has a wide temperature range but cannot satisfy the requirement for high-precision measurement. The performance of speckle pattern interferometry mostly depends on the algorithm model and speckle size; the measurement range and accuracy usually come to a compromise. Optical interference has a great performance in resolution and precision, but the measurable materials and working temperature range are limited. The thermal expansion coefficient measurement performed at the high temperature in our system solves the issues in special applications, such as aerospace materials, where the temperature of the environment can be as high as thousands of degrees, and the parameter of materials needs to be calibrated. It also shows the excellent advantages of the setup, because many measurement methods cannot be utilized at such high temperatures. Figure 15: Schematic diagram of the measurement of thermal expansion coefficients based on the YAG MLFIs.