# The Dependence of Ultrasonic Velocity in Ultra-Low Expansion Glass on Temperature
## 1 Introduction  
Ultra-low expansion (ULE) glass is a SiO\({}_{2}\)-TiO\({}_{2}\) glass formed by flame hydrolysis and vapor deposition (nominal composition: 93 wt% SiO\({}_{2}\) and 7 wt% TiO\({}_{2}\)) that has found applications in large telescope mirror blanks because of its near-zero coefficient of thermal expansion (CTE) over the 5-35 \({}^{\circ}\)C temperature range [1, 2]. However, the uniformity of glass material has a significant impact on the CTE homogeneity of the final ULE glass products, which results in figure distortion and in the degradation of the optics. It is therefore necessary to know the CTE characteristics of ULE glass to better understand--and thus better control--the fabrication process for manufacturing the highest quality ULE boules. Commonly used methods for measuring the CTE of ULE glass include the pushing-rod dilatometer [3], interferometric [4, 5], and photoelastic analysis [6, 7, 8] methods. All involve destructive measurements, and the procedures are time-consuming and expensive, so they are not suitable for the detection of the CTE uniformity of large ULE glass. The ultrasonic velocity in ULE glass is proportional to its CTE, a fact that can be utilized to nondestructively assess the absolute and relative CTE of large ULE glass, and its feasibility has been demonstrated by researchers [9, 10]. In the process of using ultrasonic velocity to nondestructively detedmine CTE, there are two key points to focus on. One is how to guarantee high accuracy in the ultrasonic velocity measurements, and the second is thecorrespondence between CTE and ultrasonic velocity at certain given temperatures, mainly because both ultrasonic velocity and CTE are temperature-dependent characteristics. For the first point, the ultrasonic velocity can be measured by various effective methods, including the threshold method [11; 12], the zero-crossing method [13], the peak method [14; 15], and the correlation approach [16; 17]. The first three methods are based on using the local signal characteristics to measure physical quantities, and the measurement findings are strongly reliant on local signal characteristics and are easily influenced by human factors (such as subjectively selected thresholds) or noise. By contrast, the correlation method considers global signals, which contribute to its excellent noise robustness, and it is free of subjective effects. In this study, we use the correlation method to accurately measure the ultrasonic velocity in ULE glass. We now turn to the dependence of ultrasonic velocity on temperature. The use of ultrasonic velocity measurements to characterize the CTE of ULE glass was first discovered by researchers at Corning. In their research, some troublesome operations and correction models were used to achieve nondestructive CTE testing [10]. On the one hand, the internal physical mechanism is not very clear. On the other hand, the various cumbersome practical steps (including ensuring the constant temperature of the sample to reach thermal equilibrium, removal of the water bath, quick-drying, and ultrasonic velocity measurement [9]) can introduce uncertainty regarding the sample temperature, which, in turn, affects the accuracy of the CTE measurement results. In later research, the tested sample is not forcefully separated from the constant temperature water bath during ultrasonic CTE measurement [18], which reduces the CTE measurement error, but this also increases the time cost of the entire measurement. Herein, we have analytically modeled the influencing mechanisms, which will help to simplify the practical procedures. The paper is organized as follows. The principle and method of ultrasonic velocity measurement are mathematically stated in the following section. Section 3 summarizes the derivation of the dependence of ultrasonic velocity on temperature. Section 4 describes the composition of an ultrasonic velocity measurement system and the preparation of the tested samples in detail.