**5. Discussion**

Generally, the measurement accuracy of the thermal expansion coefficient depends on the accuracy of the temperature control and length measurement. The accuracy of the temperature control can be improved by advanced temperature control technology and high precision temperature sensor. In addition, the accuracy of the length measurement by laser interferometry has reached the nanometer scale. Thus, the measurement error will become the key factor to improving measurement precision. Vibration isolation is a good error control method in most cases. However, with temperature changes, it will bring many uncertain micro-displacements which cannot be suppressed by vibration isolation. This method can effectively remove the measurement error caused by the micro-displacements. It will reduce the isolation requirements for the environment and enhance the anti-interference ability. In addition, with the development of modern grating processing and high precision rail technology, interferometers with higher precision will be manufactured. The measurement accuracy of this method can be further improved.

This method has the potential to eliminate the measurement error caused by micro-displacement, and its effectiveness was verified by the experiment in Section 4. However, limited by the length measurement accuracy of the Lenscan 600, it cannot be used to eliminate the rotation error in actual measurements. In this presented measurement system, the rotational angle of the sample was obtained by detecting the thickness change of the right lens. Considering that the measurement accuracy of the Lenscan 600 is ±1 μm, the minimum detection angle of the lens with a thickness of 0.75 mm is 3◦. This angle is beyond the angular tolerance of Lenscan 600 and far more than the actual rotational angle. One possible approach to decrease the minimum detection angle is to increase the thickness of the right lens. If the thickness of the right lens is 10 mm, the minimum detection angle is 0.81◦. If the thickness of the lens is 656 mm, the minimum angle that can be measured is 0.1◦. This method requires a large thickness of lens in order to achieve a sufficient angular detection accuracy. Hence, limited to the length measurement accuracy in the experimental system, this method is difficult to apply in actual measurement.

With the development of high-precision measuring instruments, especially with the accuracy of interferometry reaching 1 nm, this method could achieve sufficient angular detection accuracy. Assuming the measurement accuracy of the distance measuring system is 1 nm, if the thickness of the lens is 1 mm, the minimum detection angle is 0.08◦. If the thickness of the lens is 20 mm, the minimum angle that can be measured is 1.5. Then, the angular detection accuracy would be sufficient. The measurement error caused by rotation can be eliminated by this method. This method can effectively improve the accuracy of thermal expansion measurement. This is meaningful for the

design and fabrication of zero thermal expansion lattice. Furthermore, it can be determined from theoretical analysis that rotational error can be eliminated as long as the rotational angle is obtained. There are several other approaches that can also achieve high-precision angle measurement such as high-precision electronic autocollimator and lens surface measurement. With these methods, we expect that the measurement error can be eliminated accurately. This method could greatly improve the anti-interference ability and accuracy by eliminating the measurement error.

The lattices with tailorable coe fficients of thermal expansion are realized using two or three kinds of materials to form complex truss structures. Some of the structures are symmetric and the thermal expansion of the structures are axial elongation or shrinkage, without a rotational component. Considering a more general situation where the truss structures are asymmetric, these would have di fferent sti ffnesses in each direction. Then, the thermal expansion of the designed structures would have an axial component and a rotational component. Even though this method can eliminate the measurement error, it still has some restrictions. It cannot be used to measure the complex thermal deformation when there is an axial component and a rotational component. This method tests the CTE by measuring the length variation of the lattice. It can only measure the change in distance between two points along the optical axis without rotation. If thermal deformation of the dual-material lattice contains both of the two components, the rotational component of the CTE will mix with the rotational error. This will lead to a wrong result. Thus, this method cannot be used to measure the CTE of the axial component and the rotational component simultaneously. One possible solution is to measure the CTE of each component alone using this method and then to obtain the total CTE though calculation.
