**1. Introduction**

Dual-material lattices with tailorable coefficients of thermal expansion (CTE) have been widely used in many applications [1–3]. Various dual-material lattices have been proposed to obtain tailorable CTEs [4–8]. In order to guide their design and machining, the equivalent CTEs of the dual-material lattices must be accurately measured ahead. Usually, the CTE measuring process takes a long time due to the slow heating. During this long process, vibrations from the environment and the thermal deformation of the measuring device will cause micro-displacement between the measurement sensor and the test sample. The micro-displacement will generate unacceptable measurement error for high-precision measurement. Hence, developing a robust measurement method for testing the CTEs of dual-material lattices is meaningful to enhance measurement accuracy. However, the existing thermal expansion measurement CTE technologies are sensitive to the micro-displacement between the measurement sensor and the test sample.

Various techniques are available to measure the CTE of materials, including strain gauge technique [9,10], capacitance method [11,12], interferometric technique [13,14], and digital image correlation (DIC) method [1,15,16]. Among these, laser interferometric techniques and the DIC method are the most used to measure the CTEs of dual-material lattices. The laser interferometric technique uses interference fringe variation to measure thermal deformation with high accuracy. However, the laser interferometric fringe pattern is very sensitive to the vibration of the environment, and this method cannot eliminate the measurement error caused by the micro-displacement [13,14]. We measured the CTE of a dual-material lattice with negative thermal expansion using laser interferometry [17], but the measurement error was large, and we did not consider the measurement error caused by the micro-displacement. Digital image correlation provides a full-field thermal deformation by comparing the images captured before and after deformation [18]. It uses a high-resolution camera to capture the digital images of the test sample. This method can eliminate part of the measurement error caused by the micro-displacement via data processing. However, this method is sensitive to temperature variation, air turbulence, and out-of-plane displacement of the sample [19,20]. Therefore, developing a robust measurement method for testing the CTEs of dual-material lattices which can eliminate the measurement error caused by micro-displacement is still challenging and has never been reported (according to the authors' knowledge).

In this paper, we report on a robust laser interferometric measurement method for dual-material lattices to overcome the measurement error caused by micro-displacement. The presented method has high anti-interference capability by using a distance measuring system. The distance measuring system consists of a distance measuring set and two parallel plane lenses. The two parallel lenses can avoid the measurement error caused by the translational component of the micro-displacement. The right lens, working as a micro-rotation angle indicator, can measure the rotational angle of the micro-displacement. The measurement error caused by the rotational component of the micro-displacement is eliminated with the rotational angle. With this method, the measurement error caused by micro-displacement can be eliminated completely.

#### **2. Thermal Expansion Measurement System**

#### *2.1. Principle of the Measurement System*

The schematic diagram of the experimental system for CTE measurement is shown in Figure 1. It includes a temperature control system and distance measurement system. The temperature control system consists of a thermostat, an electric heating plate, a temperature controller, and a thermocouple thermometer. The thermostat is used to provide uniform ambient temperatures for the sample to reduce temperature inhomogeneity. The electric heating plate is heated by the electric resistance. It uses the temperature controller to realize the temperature control. This electric heating plate is used for rapid heating of the sample. The thermocouple thermometer has two thermocouple probes to monitor the temperature at di fferent locations of the sample. It can indicate whether the temperature of the test sample is uniform.

**Figure 1.** Schematic diagram of the measurement system. It consists of a temperature control system and distance measurement system.

The distance measuring system is composed of a distance measuring set and two plane optical lenses. The two lenses are mounted on both ends of the sample to reflect the laser beam. The right lens, near the laser, is used to measure the rotational angle between the sample and the laser beam.

#### *2.2. Establishment of the Experimental System*

The actual experimental setup for the CTE measurement is shown in Figure 2. In the actual experimental setup, an oven (Lichen-101BS, Shanghai, China) with temperature control was used as the thermostat. The temperature controller (HS-618F, Shanghai, China), with an accuracy of ±1 ◦C, was used to control the temperature of the heating plate. It allows non-contact temperature settings via an infrared remote controller. The thermocouple thermometer (UT320, Dongguan, China), with two thermocouple probes, was used to monitor the temperature at different locations of the sample.

**Figure 2.** Actual experimental setup. The oven, thermocouple thermometer heating plate, and temperature controller form the heating system. The Lenscan 600 (Nimes, France) was used to measure the distance.

The distance measuring set (LS600, Nimes, France), with an absolute accuracy of ±1 μm, was used to measure the thermal deformation of the sample. It can measure the length of the air gap between the two lenses and the thickness of the right lens based on low coherence interferometry [21]. The two plane lenses (N-BK7, Shanghai, China) were fixed on the sample by two mounting brackets. Each mounting bracket had three angle adjusting screws. Thus, the angles of the lenses could be adjusted to allow the reflected laser to coincide with the incident laser beam.

#### *2.3. Measurement Steps*

The following steps were the experimental measurement procedures as shown in Figure 3.

#### 2.3.1. Fixation of the Sample, Lenses, and Thermocouple Probes

The sample was fixed on the surface of the heating plate. The two thermocouple probes were fixed at different positions on the sample by high-temperature rubberized fabric. Then, the two lenses were fixed on the sample using the two mounting brackets.

**Figure 3.** Flowchart of the steps for coefficient of thermal expansion (CTE) measurement.

#### 2.3.2. Laser Path Adjustment

First, the height of the adjusting bracket was adjusted to ensure the center height of the laser probe was consistent with the center height of the two lenses. Second, the pitch angle and yaw angle of the laser probe were adjusted to ensure that the laser beam could pass through the center of the two lenses. Third, the angle adjusting screws of the two lenses were adjusted to reflect the light point to coincide with the incident point. Each lens had to be adjusted independently. Then, the length of the air gap between the two lenses was measured to determine whether the measurement quality was acceptable; if it was not, the laser path was readjusted.

#### 2.3.3. Heating and Heat Preservation

First, the temperature of the thermostat and the heating plate were set at the target temperatures and the sample was heated. When the thermostat and hot plate both reached their target temperatures, they were kept warm for approximately half an hour. Second, the thermocouple thermometer was used to monitor the temperature at di fferent positions of the sample. The readings of the two thermocouple probes were observed until the di fference between the two readings was less than 0.5 ◦C.

#### 2.3.4. Measurement and Data Collection

The sample was measured five times at each temperature point and the results recorded. Then, the temperature setting was changed, and Sections 2.3.3 and 2.3.4 were repeated until the measurements were completed.

#### **3. Measurement Error Analysis**

During the heating process, in order to guarantee the uniformity and accuracy of the sample temperature, the sample was heated slowly. Thus, the whole measurement process needed a long time. Over such a long period of time, the vibration of the environment and the thermal deformation of the measuring system caused uncertain micro-displacement between the sample and the measurement sensor. This micro-displacement can be decomposed into a translational component and a rotational component as shown in Figure 4. First, we assumed that these plane lenses were rigidly connected to the sample: the micro-displacement of the sample and the micro-displacement of the lenses were the same. Second, we assumed that the reflected laser coincided with the incident laser in the initial conditions: the initial angle between the sample and the laser was zero. Third, we assumed that the sample did not warp during the measurements. The following analyzes the measurement errors caused by these two components and how to eliminate these errors.

**Figure 4.** Schematic diagram of the micro-displacement decomposition. (**a**) Total displacement. (**b**) The translational component. (**c**) The rotational component.

During the measurement process, the temperature dependence of the refractive index must be considered. The distance measuring set measures the optical path based on a linear optical encoder [21]. Then, the actual air gap and thickness of the lens are calculated by the measurement software automatically. By setting the refractive index of each material in the initial measurement model, the measuring software can obtain the actual distance or thickness through calculation. However, in the process of thermal expansion measurement, the refractive indexes of the materials change with the temperature increase. This will lead to an inaccurate result. In order to improve the measurement accuracy, the influence of temperature on the refractive index should be considered.

#### *3.1. Influence of the Translational Component*

The translational component is one of the sources of measurement errors. In this presented CTE measurement system, it measures the CTE by measuring the optical path of the air gap between the two lenses. The optical path of the air gap was obtained by the di fference of the optical paths between the two lenses and the laser source as shown in Figure 5b. It can be expressed as:

$$D\_{AB} = D\_{LA} - D\_{LB} \tag{1}$$

In Equation (1), *DAB* is the optical path of the air gap, *DLA* is the optical path of the left lens and the laser source, and *DLB* is the optical path between of the right lens and the laser source. As shown in Figure 5b, the translational component can cause changes in the optical path between the lenses and the laser source.

**Figure 5.** Schematic diagram of the translation displacement decomposition. (**a**) Translation displacement decomposition along the axis. (**b**) Total translation displacement. (**c**) The *x* component. (**d**) The *y* and *z* component.

The translational component can be decomposed along the axis into three components: *dx*, *dy*, and *dz* as shown in Figure 5a. Through the laser path adjustment, the lens' surfaces are perpendicular to the laser beam. The two lenses are parallel to each other. Thus, *dy* and *dz* will not change distances between the lenses and the laser source as shown in Figure 5d. Therefore, *dy* and *dz* will not change

the optical path of the air gap. When the sample moves along the *x*-axis, as shown in Figure 5c, the optical path of *DLA* and *DLB* change to *DLA'* and *DLB'*. However, the lengths of *DLA* and *DLB* decrease by the same value, *dx*. Thus:

$$\frac{\partial D\_{AB}}{\partial \mathbf{x}} = 0 \tag{2}$$

In summary, the three translational components (i.e., *dx*, *dy*, and *dz*) will not change the optical path of the air gap. The translational component of the micro-displacement will not generate measurement errors for the CTE measurement. Thus, the measurement errors caused by the translational component can be avoid by the two parallel lenses.

#### *3.2. Measurement Error Analysis and Elimination of Rotational Component*

#### 3.2.1. Influence of Rotational Component

The rotational component is another source of measurement error. An excessive rotation angle will prevent the laser receiver from receiving reflected light. It will lead to measurement failure. The rotational component can be decomposed along the axis into three components in cartesian coordinates: around the *x*-axis, around the *y*-axis, and around the *z*-axis. The rotational component around the *x*-axis only makes the sample rotate around the laser beam. It does not cause extra optical path changes. Thus, it will not make measurement errors. Considering the rotational component around the *y*-axis and the rotational component around the *z*-axis, each component will cause the angle change between the sample and the laser beam. The situation will be more complex when the two components both occur. In order to simplify the calculation, the two components can be described in the polar coordinates. The total rotational component has just one direction and one angle to the laser beam. The direction of the total rotational component is circular symmetric. Thus, we can simplify the complicated situation into rotation around only one transverse axis. Take the rotation around the *y*-axis as an example for a small rotation angle; the incident and reflected rays no longer coincide as shown in Figure 6.

**Figure 6.** Schematic diagram of the laser transmission after the rotation. (**a**) The laser transmission at the right lens. (**b**) The laser transmission of the measurement system after rotation.

When the sample rotates for angle θ, the lenses are no longer perpendicular to the laser beam as shown in Figure 6b. According to the laws of reflection and refraction, the reflected rays will be deflected, and transmitted light will be refracted. Then, the optical path of the air gap changes. It can be expressed as:

$$D\_{\rm gup} = \frac{D\_{\rm AB} + D\_{\rm AC} + D\_{\rm CD}}{2} = n\_1 l / \cos \theta + n\_1 l \sin^2 \theta / \cos(2\theta) \tag{3}$$

where *Dgap* is the total optical path of the air gap between the two lenses, *DAB* is the optical path of the two points A and B, *DAC* is the optical path of the two points A and C, *DCD* is the optical path of the two points C and D, *n*1 is the refractive index of the air, *l* is the vertical distance of the two lenses, and θ is the rotation angle. If the rotation angle θ is zero, the optical path is minimized. Then:

$$D\_{\text{gup}} = n\_1 l \tag{4}$$

According to Equation (3), we can find the measurement result after the rotation is slightly larger. In Equation (3), considering the rotation angle θ is very small, the second term is:

$$n\_1 l \sin^2 \theta / \cos(2\theta) \approx n\_1 l \sin^2 \theta = o(\theta) \tag{5}$$

To simplify Equation (3), we ignore high-order small quantities. Thus:

$$D\_{\text{gap}} = n\_1 l / \cos \theta \tag{6}$$

$$\frac{\partial D\_{\text{gap}}}{\partial \theta} = \frac{\mathbf{n}\_1 l \sin \theta}{\cos^2 \theta} \tag{7}$$

According to Equation (7), when θ is zero, the derivative of *Dgap* is zero. Thus, the total optical path of the air gap reaches a minimum. In summary, no matter if the rotation angle is positive or negative, as long as rotation occurs, the optical path *Dgap* will become larger than the initial length after this rotation. This will cause the measurement result to be slightly larger and generate measurement error. To improve the accuracy of measurement, the measurement error caused by rotation must be reduced.

#### 3.2.2. Measurement Error Elimination

According to Equation (6), if we obtain the value of the rotation angle through measurement, the measurement error caused by the rotational component can be compensated. In the measurement system reported in this paper, considering the right lens is fixed on the sample rigidly, the rotational angle of the right lens is equal to the rotational angle of the sample.

$$t = \frac{D\_{\rm lens}}{n\_2} = \frac{D\_{\rm O\_1O\_2} + D\_{\rm O\_2O\_3} + D\_{\rm O\_3}E}{n\_2} = t\_0/\cos\theta + \frac{n\_1t\_0\sin^2\theta/\cos(2\theta)}{n\_2} \tag{8}$$

where *Dlens* is the optical path of the right lens, *t*0 is the original thickness of the right lens, *n*1 is the refractive index of the air, *n*2 is the refractive index of the right lens, and θ is the rotation angle. If the rotation angle θ is zero, the thickness of the right lens is *t* = *t*0.

In Equation (8), considering the rotational angle θ is very small, thus the second term is:

$$\frac{n\_1 t\_0 \sin^2 \theta / \cos(2\theta)}{n\_2} \approx \frac{n\_1 t\_0 \sin^2 \theta}{n\_2} = \text{o}(\theta) \tag{9}$$

To simplify Equation (8), we ignore high-order small quantities. Thus:

$$t = t\_0 / \cos \theta \tag{10}$$

Then, according to Equation (10), the rotational angle is:

$$\theta = \arccos(\frac{t}{t\_0}) \tag{11}$$

If the value of *t* is obtained, the rotational angle can be calculated according to Equation (11). In this measurement system, the thickness of the right lens and the air gap can be measured simultaneously using Lenscan 600. Thus, it can guarantee the rotational angle of the right lens, and the distance of the air gap is measured synchronously. When the sample rotates, the rotational angle of the sample can be

obtained by comparing the measured thickness of the right lens with the initial thickness. Thus, the measurement error of the rotational component can be eliminated by using Equation (6).

#### 3.2.3. Calculation of CTE

Generally, the variation in the sample's length during the heating process can be obtained by comparing the optical path of the air gap measured at different temperatures T1 and T2. To improve the accuracy of calculation, the change in the refractive index of the air and lens with the temperature must be considered. The CTE of the lens should be considered too.

$$
\Delta d = \frac{D\_{\text{gapr2}} - D\_{\text{gapr1}}}{n\_{1\_{T2}}} \cos \theta \tag{12}
$$

where Δ*d* is the deformation of the sample along the *x*-axis from temperature T1 to T2, *<sup>n</sup>*1*T2* is the refractive index of the air at temperature T2, *DgapT*1 and *DgapT*2 are the optical paths of the air gap between the two lenses at different temperatures, and the rotation angle θ is the parameter to be measured.

The rotational angle of the sample can be obtained by comparing the measured thickness of the right lens with the initial thickness if the temperature does not change. However, the temperature increases during CTE measurement. The measured thickness of the right lens will include thermal expansion deformation and optical path variation induced by refractive index of the right lens. These factors must be removed from the measurement result. Thus, the pure thickness of the right lens is:

$$t = \frac{D\_{\rm lens}}{n\_{2\,T2}} - \alpha\_{\rm lens} t\_0 (T\_2 - T\_1) \tag{13}$$

where *Dlens* is the optical path of the right lens, *<sup>n</sup>*2*T*2 is the refractive index of the right lens at temperature T2, and α*lens* is the CTE of the right lens. The rotational angle can be calculated according to Equation (11).

Then, combining Equations (11)–(13), the pure deformation of the sample can be obtained. Thus, the thermal expansion coefficient is:

$$a = \frac{\Delta d}{l(T\_2 - T\_1)}\tag{14}$$
