The Dependence of Ultrasonic Velocity in Ultra-Low Expansion Glass on Temperature

The coefficient of thermal expansion (CTE) is an important property of ultra-low expansion (ULE) glass, and the ultrasonic velocity method has shown excellent performance for the nondestructive measurement of CTE in large ULE glass. In this method, the accurate acquisition of the ultrasonic velocity in ULE glass is necessary. Herein, we present a correlation method to determine the ultrasonic TOF in ULE glass and to further obtain the ultrasonic longitudinal wave velocity (cL) indirectly. The performance of this method was verified by simulations. Considering the dependence of cL on temperature (T), we carried out the derivation of the analytical model between cL and T. Based on reasonable constant assumptions in the physical sense, a cL–T exponential model was produced, and some experimental results support this model. Additional experiments were carried out to validate the accuracy of the cL–T exponential model. The studies we conducted indicate that the cL–T exponential model can reliably predict the ultrasonic velocity in ULE glass at different temperatures, providing a means for the nondestructive CTE measurement of large ULE glass at a specified temperature.


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 • 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 understandand 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 pushingrod 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 the correspondence 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. In Section 5, the uncertainty and stability of the ultrasonic velocity measurement are discussed, the experimental establishment of the ultrasonic velocity-temperature exponential model is presented, and the comparison of the predicted ultrasonic velocity results are listed. Finally, the conclusions are presented in Section 6.

Principle of the Ultrasonic Velocity Measurement
Many instruments are now employed to measure ultrasonic velocity, most of which use the pulse reflection method [19][20][21]. This method usually involves time-domain analysis based on the ultrasonic time of flight (TOF). In this analysis, the ultrasonic velocity is obtained by the ratio of the material thickness to the ultrasonic TOF [13,22]. On this basis, we built a fully integrated high-precision system to measure the ultrasonic velocity in ULE glass using the immersion pulse reflection method. Figure 1 depicts the schematic diagram of the immersion pulse reflection method. The time of S F relative to the time base "0" point is called t 1 . The time interval between B 1 and S F is recorded as ∆t 1 , and the time interval between B 2 and B 1 is recorded as ∆t 2 , which have the following relationship: Using Equation (2), we can determine the L of the tested sample by first determining ∆t1 or ∆t2 of the ultrasonic waveform and the known thickness d.

The Correlation Calculation Principle
As the B1 and B2 signals of the ULE samples were readily available and were highly similar, the correlation method was used to determine the time interval between B1 and B2 as the ultrasonic TOF in the samples. In detail, the ultrasonic signal that was sampled and saved in the PC is designated as x( ). By separating the B1 and B2 signals and noting them as x1( ) and x2( ), respectively, the correlation coefficient ( ) can be represented as The ultrasonic signals that were collected by the data acquisition card and the PC, on the other hand, were two discrete signal arrays. As a result, for the discrete signals, the normalized R can be given as where is the computed length of the signal array, and is the location inside the signal array.
In the time domain of signal x( ), there is a time interval between the signals x1( ) and x2( ). The time interval between x1( ) and x2( ) is the round trip time of the ultrasonic wave propagated in the thickness direction, i.e., the TOF. Equation (4) is used to generate the correlation array, with the position having the largest correlation coefficient corresponding to the temporal position of . The corresponding TOF (Δt) is equal to / S if the sampling frequency is noted as S. Finally, the following equation can be used to calculate the ultrasonic velocity: The ultrasonic longitudinal wave velocity (c L ) of a sample can be calculated from Equation (1): Using Equation (2), we can determine the c L of the tested sample by first determining ∆t 1 or ∆t 2 of the ultrasonic waveform and the known thickness d.

The Correlation Calculation Principle
As the B 1 and B 2 signals of the ULE samples were readily available and were highly similar, the correlation method was used to determine the time interval between B 1 and B 2 as the ultrasonic TOF in the samples. In detail, the ultrasonic signal that was sampled and saved in the PC is designated as x(t). By separating the B 1 and B 2 signals and noting them as x 1 (t) and x 2 (t), respectively, the correlation coefficient (R) can be represented as The ultrasonic signals that were collected by the data acquisition card and the PC, on the other hand, were two discrete signal arrays. As a result, for the discrete signals, the normalized R can be given as where n is the computed length of the signal array, and i is the location inside the signal array.
In the time domain of signal x(t), there is a time interval between the signals x 1 (t) and x 2 (t). The time interval between x 1 (t) and x 2 (t) is the round trip time of the ultrasonic wave propagated in the thickness direction, i.e., the TOF. Equation (4) is used to generate the correlation array, with the position having the largest correlation coefficient corresponding to the temporal position of m. The corresponding TOF (∆t) is equal to m/f S if the sampling frequency is noted as f S . Finally, the following equation can be used to calculate the ultrasonic velocity:

Simulation of the Correlation Method
First, the reliability of the proposed algorithm needed to be validated by simulations in which the simulated ultrasonic signal is a declining sinusoidal function.
where α s is the declining coefficient of the sinusoidal function set as 9 × 10 6 Np/m; f c is the center frequency of the ultrasonic transducer set as 5 MHz; β is the amplitude of the signal; and the amplitudes of the initial and echo signals are set to 1 and 0.3 V, respectively. The sampling frequency of the signal was set as 2.5 GHz, and the TOF between the initial and echo signals was set as 17.36 µs. To come closer to the actual measurement signal, Gaussian noise with +15 dB SNR (Signal-to-Noise Ratio) was added to the ideal declining sinusoidal signal. The simulated signal waveform and the correlation distribution calculation results are shown in Figure 2a  First, the reliability of the proposed algorithm needed to be validated by simulations in which the simulated ultrasonic signal is a declining sinusoidal function.
where αs is the declining coefficient of the sinusoidal function set as 9 × 10 6 Np/m; c is the center frequency of the ultrasonic transducer set as 5 MHz; β is the amplitude of the signal; and the amplitudes of the initial and echo signals are set to 1 and 0.3 V, respectively. The sampling frequency of the signal was set as 2.5 GHz, and the TOF between the initial and echo signals was set as 17.36 μs. To come closer to the actual measurement signal, Gaussian noise with +15 dB SNR (Signal-to-Noise Ratio) was added to the ideal declining sinusoidal signal. The simulated signal waveform and the correlation distribution calculation results are shown in Figure 2a  Where the X value is the time corresponding to the largest correlation coefficient, and the Y value is the largest correlation coefficient.
In Figure 2, the calculated correlation coefficient has a clear peak on the time axis, with a TOF of 17.36 μs corresponding to the peak. The calculated results were in good agreement with the theoretical settings, so the algorithm is suitable for ultrasonic signal processing in ultrasonic velocity measurements.

Theoretical Model between Ultrasonic Velocity and Temperature
Ultrasound is defined as an elastic wave of high frequency that propagates in a medium. Therefore, when an elastic wave propagates in an isotropic medium without being affected by volume stress, the ultrasonic longitudinal wave velocity, denoted as L , can be found from the following equation: where is density, υ is Poisson's ratio, and is Young's modulus. Equation (7) indicates that the L in the material is mainly related to Young's modulus, density, and Poisson's ratio. For ULE glass, within the upper limit of 11.5 wt%, slight changes in the TiO2 content will not change the density or Poisson's ratio but will cause changes to Young's modulus [23]. Therefore, the difference in L is mainly determined by the difference in Young's modulus. Then, the effect of temperature on the ultrasonic velocity essentially reflects its effect on Young's modulus. The microscopic analysis of Young's modulus shows that the reaction of a solid to all forces comes from the potential energy of the interaction between atoms. The interaction potential U(r) of a pair of two atoms separated by r can be expressed as: Where the X value is the time corresponding to the largest correlation coefficient, and the Y value is the largest correlation coefficient.
In Figure 2, the calculated correlation coefficient has a clear peak on the time axis, with a TOF of 17.36 µs corresponding to the peak. The calculated results were in good agreement with the theoretical settings, so the algorithm is suitable for ultrasonic signal processing in ultrasonic velocity measurements.

Theoretical Model between Ultrasonic Velocity and Temperature
Ultrasound is defined as an elastic wave of high frequency that propagates in a medium. Therefore, when an elastic wave propagates in an isotropic medium without being affected by volume stress, the ultrasonic longitudinal wave velocity, denoted as c L , can be found from the following equation: where ρ is density, υ is Poisson's ratio, and E is Young's modulus. Equation (7) indicates that the c L in the material is mainly related to Young's modulus, density, and Poisson's ratio. For ULE glass, within the upper limit of 11.5 wt%, slight changes in the TiO 2 content will not change the density or Poisson's ratio but will cause changes to Young's modulus [23]. Therefore, the difference in c L is mainly determined by the difference in Young's modulus. Then, the effect of temperature on the ultrasonic velocity essentially reflects its effect on Young's modulus. The microscopic analysis of Young's modulus shows that the reaction of a solid to all forces comes from the potential energy of the interaction between atoms. The interaction potential U(r) of a pair of two atoms separated by r can be expressed as: where A, B, n, and m are all positive constants. The first term represents the energy of attraction, and the second term represents the energy of repulsion.
Assuming that the solid is stretched along the crystal axis when the tensile force changes df, the interatomic distance r changes to dr. At this time, the cross-sectional area r 2 of a unit cell is regarded as inconvenient. Then, Young's elastic modulus can be expressed as: where σ and ε represent the stress and strain, respectively.
Since the binding force f of the tensioned solid is only related to the first term of Equation (8), its magnitude is: Equation (10) is then substituted into Equation (9) by deriving the derivative for r to obtain Assume K = (n + 1) A and Q = n + 3. Equation (11) can be changed to: Equation (12) takes the derivative of T and divides both sides by Er Q at the same time, shifting the term to obtain: It is assumed that the distance between atoms still obeys the following rules when a solid is heated and expanded: where r 0 is the atomic distance when the absolute temperature T 0 = 0; α is the linear expansion coefficient of the solid, and its differential definition is: The η in Equation (15) is the temperature coefficient of the elastic modulus. Relevant studies have demonstrated that the elastic modulus of ULE glass increases with the increase in temperature, and the increment dE of E has a positive value [23]. When both sides of Equation (13) are multiplied by dT, Equation (14) is then substituted in and integrated, and we obtain: From Equation (16) and considering T 0 = 0, we obtain: From Equations (13) and (15), we see that: Therefore, Equation (17) is changed to: Equation (19) shows that the elastic modulus of ULE glass increases as the temperature increases, where η is the temperature coefficient of the elastic modulus. Therefore, it can be inferred that the c L in ULE glass also increases as T increases. A review of the data reveals that when considering the effects of thermal expansion, there is a specific equation relating the c L and T, which is as follows [24]: where α L is the temperature coefficient of the ultrasonic longitudinal wave velocity, a positive value of about 10 −4 orders of magnitude, and c T L , c T 0 L indicate the ultrasonic longitudinal wave velocity of the material at temperatures T and T 0 , respectively.
Although there are certain differences in α L at different temperature points, we can use the c L -T data in a small temperature range to fit and solve the average α L in this temperature range. The method of a differential equation is introduced here to obtain the mathematical relationship between c L and T in a small temperature range. Writing Equation (20) in a differential form, we obtain Solving this equation yields Equation (22) shows that the relationship between c L and T is theoretically exponential within a certain temperature range.

Materials and Sample Preparation
The selected ULE glass was Corning Code 7972 glass. Considering the boundary effect of ultrasonic wave propagation, the experimental samples were prepared in a cylindrical shape with a cross-sectional area that was larger than that of the ultrasonic transducer. To avoid the adverse effects of the scattering attenuation of ultrasound at the interface of the sample, the two end faces of the tested sample should be flat and parallel to each other. As shown in Figure 3, six cylindrical glass samples were cut with an equal thickness (50-0.1 mm) along the radial position of the ULE 7972 boule (No. 82714) using an abrasive water jet; then, these samples were finely ground and polished to achieve a flatness of 0.5λ and parallelism of 20 µm. The six samples were numbered 1#~6# from the inside to the outside of the boule according to the increasing CTE. The details of the ULE samples are summarized in Table 1.

Ultrasonic Measurement System
The schematic diagram of the c L . measurement experimental system is illustrated in Figure 4. A 75 MHz ultrasonic pulser/receiver (Olympus, Waltham, MA, USA, model 5073PR) was used, which was connected to a 400 MHz data acquisition card (Spectrum Instrumentation GmbH, Grosshansdorf, Germany, model M4i.2220-x8) that transmits the ultrasonic signals to a computer to be processed. All signals were captured with 2,500,000 points at a sampling rate of 2.5 GHz. After the acquisition, the data were properly processed to determine the involved ultrasonic velocities. The temperature of the tested sample was controlled by a thermostatic water tank (Hangzhou Qiwei, Hangzhou, China, model DHC-05-B) with a temperature control precision of ±0.05 • C.

Ultrasonic Measurement System
The schematic diagram of the L measurement experimental system is illustrated in Figure 4. A 75 MHz ultrasonic pulser/receiver (Olympus, Waltham, MA, USA, model 5073PR) was used, which was connected to a 400 MHz data acquisition card (Spectrum Instrumentation GmbH, Grosshansdorf, Germany, model M4i.2220-x8) that transmits the ultrasonic signals to a computer to be processed. All signals were captured with 2,500,000 points at a sampling rate of 2.5 GHz. After the acquisition, the data were properly processed to determine the involved ultrasonic velocities. The temperature of the tested sample was controlled by a thermostatic water tank (Hangzhou Qiwei, Hangzhou, China, model DHC-05-B) with a temperature control precision of ±0.05 °C.    Figure 5 illustrates the ultrasonic velocity measurement results for The self-generating and self-receiving transducer model was utilized to transmit and receive the ultrasonic waves. Considering the diameter and thickness of the tested sample, wideband focusing 19.05 mm diameter longitudinal wave immersion transducers (Olympus, Waltham MA, USA) with 3.5 and 5 MHz nominal center frequencies were employed in this investigation. Figure 5 illustrates the ultrasonic velocity measurement results for the samples 1#~6# at 20 • C using two different ultrasonic frequencies: 3.5 and 5 MHz. Each sample was subjected to five replicate measurements by each transducer. The ultrasonic frequency had little effect on the ultrasonic velocity of the sample, as the c L in the sample only depends on its inherent physical parameters, including its bulk modulus and density [25]. In addition, the repeatability of the c L measurement was essentially the same for both frequency transducers, both within 0.1 m/s. This suggests that any frequency could be used to characterize the CTE of ULE glass in engineering applications when using ultrasonic velocity measurement methods. The self-generating and self-receiving transducer model was utilized to transmit and receive the ultrasonic waves. Considering the diameter and thickness of the tested sample, wideband focusing 19.05 mm diameter longitudinal wave immersion transducers (Olympus, Waltham MA, USA) with 3.5 and 5 MHz nominal center frequencies were employed in this investigation. Figure 5 illustrates the ultrasonic velocity measurement results for the samples 1#~6# at 20 °C using two different ultrasonic frequencies: 3.5 and 5 MHz. Each sample was subjected to five replicate measurements by each transducer. The ultrasonic frequency had little effect on the ultrasonic velocity of the sample, as the L in the sample only depends on its inherent physical parameters, including its bulk modulus and density [25]. In addition, the repeatability of the L measurement was essentially the same for both frequency transducers, both within 0.1 m/s. This suggests that any frequency could be used to characterize the CTE of ULE glass in engineering applications when using ultrasonic velocity measurement methods.

Uncertainty Analysis
In this work, an ultrasonic transducer with a 5 MHz center frequency was employed to measure the ultrasonic velocities in all of the prepared ULE samples. Figure 6 gives an example of the received ultrasonic signals, which formed a series of ultrasonic longitudinal waves with a sampling rate of 2.5 GHz. High-frequency noise was found to be present in the actual ultrasonic signals. However, since the correlation of the noisy signal was very small, the effect of such noise was removed via the correlation calculation. The results of the TOF calculations obtained using this method are depicted in Figure 6. Again, there is a peak in the correlation coefficient distribution plot corresponding to a TOF of ∆t = 17.3383 µs.
The uncertainty in measuring ultrasonic velocity was also investigated, and the expression is shown in Equation (23).
where u c L , u d , and u ∆t are the uncertainties regarding the c L , d, and TOF in the ULE glass, respectively. As illustrated in the first item of Equation (23), the uncertainty related to the d was u c L (d) = (2/∆t)∆u d . A typical value of ∆t is 17.3562 µs. When measuring the d of the ULE sample, u d was evaluated by the measurement precision of a micrometer (0.001 mm), thus u c L (d) = 0.12 m/s. After substituting the measured values into the second item of Equation (23), the uncertainty introduced by the TOF (∆t) was obtained: u c L (∆t) = 0.332 × 10 9 u ∆t .
Since u ∆t was evaluated by the sampling period of the data acquisition card (0.4 ns), the typical u c L (∆t) was 0.13 m/s. Therefore, when measuring the ULE glass using the proposed experimental setup, the calculated total uncertainty was u c L = 0.2 m/s, ensuring that the ultrasonic velocity measurements at a single temperature had high reliability.

5.1.
Measurement at a Single Temperature

Uncertainty Analysis
In this work, an ultrasonic transducer with a 5 MHz center frequency was employed to measure the ultrasonic velocities in all of the prepared ULE samples. Figure 6 gives an example of the received ultrasonic signals, which formed a series of ultrasonic longitudinal waves with a sampling rate of 2.5 GHz. High-frequency noise was found to be present in the actual ultrasonic signals. However, since the correlation of the noisy signal was very small, the effect of such noise was removed via the correlation calculation. The results of the TOF calculations obtained using this method are depicted in Figure 6. Again, there is a peak in the correlation coefficient distribution plot corresponding to a TOF of Δt = 17.3383 μs. The uncertainty in measuring ultrasonic velocity was also investigated, and the expression is shown in Equation (23).
where , , and ∆ are the uncertainties regarding the L , d, and TOF in the ULE glass, respectively.
As illustrated in the first item of Equation (23) (∆ ) = 0.332 × 10 ∆ . Since ∆ was evaluated by the sampling period of the data acquisition card (0.4 ns), the typical (∆ ) was 0.13 m/s. Therefore, when measuring the ULE glass using the proposed experimental setup, the calculated total uncertainty was = 0.2 m/s, ensuring that the ultrasonic velocity measurements at a single temperature had high reliability.

Stability of L Measurement
To determine whether the proposed measurement system could provide reliable L measurements over long periods of time, the stability of the measured L value also needs to be considered when the tested samples reach thermal equilibrium. Long-term measurement of the ultrasonic velocity in samples 1#~6# with a high CTE was taken at the same temperature (20 °C), and the L data were recorded at 1 h intervals, for a total of seven measurements in one day. Figure 7 depicts the variation in L with the measurement time.

Stability of c L Measurement
To determine whether the proposed measurement system could provide reliable c L measurements over long periods of time, the stability of the measured c L value also needs to be considered when the tested samples reach thermal equilibrium. Long-term measurement of the ultrasonic velocity in samples 1#~6# with a high CTE was taken at the same temperature (20 • C), and the c L data were recorded at 1 h intervals, for a total of seven measurements in one day. Figure 7 depicts the variation in c L with the measurement time. The L measured in the same sample was almost constant at the different times while maintaining a constant temperature, and the RMSE (Root Mean Squared Error) of the L changes were all within 0.10 m/s, which indicate that the measurement system that was built in this paper has stable performance. This also provides a strong guarantee for the L measurement of a large batch of ULE glass samples.

Acquisition of -T Data
The tested sample and an ultrasonic transducer fixed above the sample were placed into a thermostatic water tank, and the temperature in the tank was steadily increased in the range of 10-30 °C. Here, the temperature range of 10-30 °C was chosen for two reasons: (1) this corresponds to the real temperature range that ULE glass is likely to experience when undergoing ultrasonic measurements, i.e., the range of room temperature throughout the year, and (2) it corresponds to the temperature range of the ultrasonic The c L measured in the same sample was almost constant at the different times while maintaining a constant temperature, and the RMSE (Root Mean Squared Error) of the c L changes were all within 0.10 m/s, which indicate that the measurement system that was built in this paper has stable performance. This also provides a strong guarantee for the c L measurement of a large batch of ULE glass samples.

Acquisition of c L -T Data
The tested sample and an ultrasonic transducer fixed above the sample were placed into a thermostatic water tank, and the temperature in the tank was steadily increased in the range of 10-30 • C. Here, the temperature range of 10-30 • C was chosen for two reasons: (1) this corresponds to the real temperature range that ULE glass is likely to experience when undergoing ultrasonic measurements, i.e., the range of room temperature throughout the year, and (2) it corresponds to the temperature range of the ultrasonic transducer in use. We were utilizing a standard immersion transducer, whose normal operating temperature range is 10-60 • C, beyond which the piezoelectric action of the transducer would be weakened, making high-amplitude data acquisition difficult. The ultrasonic echo signals were manually sampled and stored by the PC at a temperature interval of 1 • C, which was chosen to account for the maximum number of temperature points to be sampled and the required modeling time. The temperature was held constant within ±0.05 • C, as measured using a digital thermometer probe in the bath.
To ensure that the glass sample was in thermal equilibrium during the TOF measurement, the sample was immersed in a controlled water bath for at least 94 min, which was calculated according to the time t (hours) to reach equilibrium for a given glass thickness d (cm), which could be expressed as t = d 2 /16 in [26]. To reduce measurement errors, the echo signals were acquired three times at each temperature point. By using the described correlation algorithm method, the ultrasonic TOF was obtained, and the average values were used to calculate the c L of the tested ULE samples. Nevertheless, the experiment did not consider the change in the d and ρ of the tested samples.
The obtained ultrasonic velocities of samples 1#~6# are presented in Figure 8 in terms of temperature. The ultrasonic velocities are observed to increase as the temperature increases. Changes in c L are correlated with temperature changes, regardless of the absolute value of the CTE. Comparing Figure 8a-f, we can also observe that the ultrasonic velocity differs at the same temperature and that the c L increases with the CTE of ULE glass.

Analysis of the Change in the L with T
The data from Figure 8 were used to directly plot the changes in L as a function of T according to the exponential fitting in Equation (22). Table 2 displays the results of all six samples, with excellent exponential fits given for each case and with L mostly ranging from 0.000127 to 0.000129, which shows that for ULE materials with very small CTE differences, their L is almost constant. This means that the temperature effect pattern on ultrasonic velocity was essentially the same in ULE glasses with different CTE values. Therefore, the average value of these L coefficients could be taken as the temperature correction coefficient of ultrasonic velocity in ULE glass. Of course, because of the small difference in the CTE, the c  in Equation (22) showed relatively large fluctuations. In the physical sense, the c  indicates that L correpsonds to 0 °C. It was found that as the CTE of the ULE sample increased, c  increased correspondingly, which is consistent The data from Figure 8 were used to directly plot the changes in c L as a function of T according to the exponential fitting in Equation (22). Table 2 displays the results of all six samples, with excellent exponential fits given for each case and with α L mostly ranging from 0.000127 to 0.000129, which shows that for ULE materials with very small CTE differences, their α L is almost constant. This means that the temperature effect pattern on ultrasonic velocity was essentially the same in ULE glasses with different CTE values. Therefore, the average value of these α L coefficients could be taken as the temperature correction coefficient of ultrasonic velocity in ULE glass. Of course, because of the small difference in the CTE, the c in Equation (22) showed relatively large fluctuations. In the physical sense, the c indicates that c L correpsonds to 0 • C. It was found that as the CTE of the ULE sample increased, c increased correspondingly, which is consistent with the mechanism of the linear positive correlation between c L and the CTE. Table 2. Data from the six samples shown in Figure 8 were taken to exponentially fit the relationship between c L and T. To represent the more pronounced difference in the ultrasonic velocity with increasing temperature for the ULE samples with different CTE values, a new coefficient is defined as follows:

No
Here, we take the c L corresponding to 10 • C of the samples as the basis c L , and calculate the β L via Equation (24) with temperature increments of 5 • C, i.e., at 15 • C, 20 • C, 25 • C, and 30 • C. The β L for samples 1#, 2#, and 3# is shown in Figure 9. We noted that for the different temperature ranges, the speed of the ultrasonic velocity changes with the temperature was inconsistent. This is possibly because the elastic modulus also has a temperature coefficient β E , which varies in different temperature ranges. The PV value of β L for the three samples was 0.000098, 0.000056, and 0.000101. Considering that the difference was minor, we conclude that the multi-data point fitting method that was used to obtain α L in a given temperature range in this paper is a more feasible equivalent to an average function and has wider temperature applicability.
Here, we take the L corresponding to 10 °C of the samples as the basis L , and calculate the via Equation (24) with temperature increments of 5 °C, i.e., at 15 °C, 20 °C, 25 °C, and 30 °C. The for samples 1#, 2#, and 3# is shown in Figure 9. We noted that for the different temperature ranges, the speed of the ultrasonic velocity changes with the temperature was inconsistent. This is possibly because the elastic modulus also has a temperature coefficient , which varies in different temperature ranges. The PV value of L for the three samples was 0.000098, 0.000056, and 0.000101. Considering that the difference was minor, we conclude that the multi-data point fitting method that was used to obtain L in a given temperature range in this paper is a more feasible equivalent to an average function and has wider temperature applicability. Samples 4# and 5#, which had the same CTE, were also analyzed for differences in the L as the temperature increased, and the results are shown in Figure 10. We found that the two samples had similar patterns of variation in L . This implies that the changing trends in the ultrasonic velocity with temperature was, to some extent, consistent with the changing trend of the CTE with temperature; however, this assumes that the thickness of the ULE glass sample did not change over a wide temperature range. Samples 4# and 5#, which had the same CTE, were also analyzed for differences in the c L as the temperature increased, and the results are shown in Figure 10. We found that the two samples had similar patterns of variation in β L . This implies that the changing trends in the ultrasonic velocity with temperature was, to some extent, consistent with the changing trend of the CTE with temperature; however, this assumes that the thickness of the ULE glass sample did not change over a wide temperature range. Samples 4# and 5#, which had the same CTE, were also analyzed for differences in the L as the temperature increased, and the results are shown in Figure 10. We found that the two samples had similar patterns of variation in L . This implies that the changing trends in the ultrasonic velocity with temperature was, to some extent, consistent with the changing trend of the CTE with temperature; however, this assumes that the thickness of the ULE glass sample did not change over a wide temperature range. Figure 10. The relationship between and temperature for ULE samples with the same CTE. Figure 10. The relationship between β L and temperature for ULE samples with the same CTE.
The correlation between c L and T in ULE glass over a wide range of CTE values from −1 ppb/ • C to +3 ppb/ • C was investigated and discussed. We will next validate the accuracy of the c L -T exponential model for ULE samples with different CTE values through a series of experiments.

Accuracy Validation of c L -T Model
The accuracy of the c L -T model was decisive for its application in engineering. Samples 1#, 3#, and 5# were randomly selected, and their ultrasonic velocities were measured at ten random temperature points in the range of 10−30 • C. The measured values were compared with the model predictions in this paper, and the results are shown in Table 3.  Note: c m is the actual measured ultrasonic velocity, c p is the ultrasonic velocity predicted by the fitting model, and δ denotes the relative error. The standard deviations σ A and σ B are measures of these inaccuracies.
Considering that the measurement locations of the three samples in the model validation procedure may vary, it was reasonable to use a recalculation of c based on the standard ultrasonic velocity at 20 • C to determine the c L -T exponential model, which may differ slightly from the c obtained by modeling the above c L -T data. The c L values in the tested glasses at an average room temperature of 20 • C were substituted into the derived exponential model based on the average temperature coefficient from the above fitting analysis to determine a c L -T exponential model for each tested glass. As seen in Table 3, the standard deviation of the predicted values for three samples were all within 0.90 m/s, and the relative errors between measured and model-predicted values were mostly within 0.020%, which suggests that the models fitted in this work exhibit high precision.
Of course, it should be noted that increasing the sample thickness may further reduce the errors in c L measurement, but it also imposes more stringent requirements on the frequency of the transducer and time required for the sample to reach thermal equilibrium.

Conclusions
In this investigation, the aim was to theoretically analyze and validate the dependence of the ultrasonic velocity in ULE glass on temperature. Based on the simulation and experimental investigations, we can draw the following conclusions: 1.
The proposed pulse reflection immersion method provides reliable and stable measurements of the ultrasonic echo signal of ULE glass, and the calculation of the signal based on the correlation method can be used to conveniently and accurately extract the ultrasonic TOF of the tested sample and then obtain the ultrasonic velocity.

2.
The application of the proposed method for six ULE samples with different CTE values indicates that ultrasonic velocity increases as the experimental temperature is increased. Furthermore, the c L -T exponential model was theoretically analyzed and experimentally fitted. The predicted c L using the exponential model at ten random temperature points shows good agreement with the actual measured ultrasonic velocities at the same temperature.
These findings indicate the promising potential of the c L -T exponential model to determine the ultrasonic velocity at a specified temperature for ultrasonic nondestructive CTE measurement in large ULE glass, which will be critical for reliably evaluating the CTE homogeneity of large ULE glass at a specified temperature.
There are some comments that can be made about the general applicability of our method. In this paper, the derivation and verification of the c L -T relationship are only for isotropic ULE glass, but the method for deriving exponential model can also be applied to anisotropic materials, such as anisotropic crystals, and for obtaining the longitudinal wave velocity, transverse wave velocity, and then the Poisson's ratio. The only distinction is α L . As for the experimental verification method for the exponential model, the correlation method, it is also applicable for the measurement of the TOF, regardless of whether the material is isotropic.