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The displacement of piezoelectric materials under an electric field is concerned as piezoelectric materials usually serve as actuators. However, the displacement is too small to be easily measured in a routine method. With very high resolution and no need for calibration on the length scale, optical interferometry provides the chance for the precise measurement of small displacements within units of nanometer [31–35]. Besides, optical interferometry can achieve the measurement without mechanical contact. Interferometry techniques for the detection of strains have been developed for nearly 50 years. However, at present, this method is still mainly used in research laboratories due to its high price and the need of vibration insulation system.

Piezoelectric coefficients can be measured using single beam, double beam, and heterodyne laser interferometers. For simplicity, the principle for the case based on a single beam laser interferometer is mainly discussed here.

Figure 1.6 depicts the schematic diagram of a Michelson interferometer. When a monochromatic light of wavelength λ interferes with a reference beam, the interference light intensity can be described as follows:

(1.17)

where I_p and I_r are the intensities of the probing and reference beams, respectively, Δd is the optical path‐length difference between the two beams, and k = 2π/λ is the wave number. Actually, the abovementioned relation can be converted into the formula with the parameters, the maximum and minimum interference light intensities I_max and I_min, which can be measured by a photo‐detector.

(1.18)

Figure 1.6 Schematic diagram of Michelson interferometer for the measurement of displacement.

In a photodetector, the corresponding photodiode output is determined by the light intensity related to the optical path‐length Δd, which is directly related with the sample displacement. An amplified output voltage signal corresponding to the displacement can be obtained, which is usually monitored by an oscilloscope. A definite relationship exists between the voltage output and the displacement. According to the relationship, interferometer sensitivity can be actually set to a specific value, say, 10 nm/V. The changes of the sample dimension are induced by the connection and the disconnection of voltage to the sample. The dimension changes by the identical applied voltage are measured multiple times. The averaged measured value and the connected voltage are used to calculate the piezoelectric charge constant d₃₃, which is governed by the following equation:

(1.19)

where U is the applied voltage and Δl is the change of length determined by multiplying the voltage output of the interferometer and its sensitivity.

Single Beam Michelson interferometers are widely used by different research groups to measure piezoelectric and electrostrictive strains. It should be noted that there are various factors, including sample shape and optical alignment, which can affect the measurement validity and accuracy. To obtain the actual value of the sample dimension change parallel to the laser beam, it should be ensured that only the front surface of a sample can move while the back surface of the sample is fixed, and the sample dimension along the lateral direction can expand and contract freely. Due to these requirements, the specimen shape and dimensions are limited. The bulk sample is usually in the form of a cylinder or a thin plate with a suitable ratio between the lateral dimension and the thickness. If this ratio is very large, the clamping effect and the sample bending effect will become significant. One should not neglect such an impact because the resultant errors can even surpass the longitudinal piezoelectric displacement. When measuring the d₃₃ of a thin film deposited on a very thick substrate, the clamping effect should be concerned as the substrate bonding is usually assumed to be infinitely rigid. The measured d₃₃ is actually the effective converse longitudinal piezoelectric coefficient d_33,eff, which can be determined as follows:

(1.20)

where S_ij is the mechanical compliance of the piezoelectric film and d₃₁ is the transverse piezoelectric coefficient.

Proper mounting of the sample is also very important to obtain the actual piezoelectric coefficients. It has been reported that the measured d₃₃ of the same disk specimen of PC5H (Morgan Electro Ceramics) showed different values ranging from 750 to 1250 pC/N when simply changing the way the sample was mounted [34]. The accuracy is also strongly affected by small vibrations or abnormal conditions during the measurement. Thus, the measurements using the laser interferometry should be carried out with great caution and patience, though this type of measurement method is simply based on acquiring the displacement value of the sample surface induced by the corresponding applied voltage.