**1. Introduction**

Microscopic interferometry is a combination of an electronic speckle pattern interferometry (ESPI) system and a microscopic optical path. It has the advantages of non-contact, full-field and high precision, and is widely used in dynamic and static measurement of micro devices [1–5]. ESPI has been developed for many years and is very popular in the field of non-contact measurement. By analyzing the interferograms, we can obtain the phase information of the measured object, and then the static morphology or dynamic deformation can be measured. Researchers have designed a variety of optical measurement configurations and phase processing algorithms for different measurement targets. In 1971, Butters, J.N. and Lecndcrtz, J.A. used a camera instead of a holographic dry plate to record the interference phenomenon of speckles. This series of speckle images can be electronically processed to compare the speckle images before and after deformation [6]. Jones, R. comprehensively analyzed the conditions for obtaining the best performance of the ESPI, such as camera characteristic, laser power, interferometer type and operation mode, which provide a general theoretical basis for the system design and optimization of various interferometers [7]. In reference [8], a detailed analysis of the vibration fringes obtained by phase stepping on a time-averaged electronic speckle pattern interferometer was presented. Moore, A.J. presented two phase-stepping algorithms that can calculate phase from ESPI fringes: the subtraction method and the addition method. Corresponding interference experiments were carried out, combining a piezoelectric transducer (PZT) and pulse laser, and the function of the algorithms was demonstrated through the collected interferograms [9]. One-shot phase-shifting optical and speckle interferometry with modulation of polarization was described in reference [10]. The system has the ability to record multiple phase-shifting optical or speckle patterns at the same time, so it can afford to measure rapid changes generated from rapid varying phenomena. However, this method reduced the spatial resolution while increasing the time resolution. Reference [11] reported a simple, compact ESPI incorporating a holographic optical system for the study of outof-plane vibration. The subtraction method was used to generate the fringe pattern. The

**Citation:** Gao, C.; Gao, Z.; Niu, Y.; Wang, X.; Zhao, J.; Deng, L. An Improved Large-Field Microscopic Speckle Interferometry System for Dynamic Displacement Measurementof MEMS. *Photonics* **2021**, *8*, 271. https://doi.org/10.3390/photonics 8070271

Received: 21 June 2021 Accepted: 7 July 2021 Published: 9 July 2021

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background speckle noise was eliminated by introducing a phase shift between consecutive images, and finally the amplitude and phase image were obtained through path difference modulation. D. Malacara summarized and described common phase-detection algorithms in interferometry, such as the least square method, quadrature phase method and discrete low-pass filtering, etc. [12]. These theories are very helpful for us to obtain the phase under a variety of different measurement conditions.

With the rapid development and wider application of MEMS, there are more and more studies on the measurement of micro devices. The measurement of micro devices can be achieved by introducing the microscopic optical path into the interferometry system. Reference [13] described an optoelectronic holography microscope (OEHM) for measuring static and dynamic modes of MEMS accelerometers with submicron accuracy. After obtaining the intensity patterns obtained by the phase-stepping algorithm, the phase value was obtained by the phase-shifting algorithm. Preliminary results indicate that the MEMS accelerometer considered in this study deforms by 1.48 μm. Kumar, U.P. presented a two-wavelength micro-interferometric setup for 3-D surface profile characterization of smooth and rough micro-specimens. The method removed ambiguity associated with the single-wavelength data and also extended the phase measurement range compared to the conventional single-wavelength interferometry. A seven-phase step algorithm was used for quantitative fringe analysis. The experimental results on rough silicon membrane and smooth sample were presented [14]. Reference [15] proposed a multiple-wavelength microscopic holographic configuration. This system used sequentially recorded phaseshifted frames at three different wavelengths to evaluate the relatively large deformation fields at the effective wavelengths. The phase distribution before and after loading the object was obtained by using the eight-phase step algorithm. The design of the system along with the experimental results on small-scale rough specimens under static load was presented.

In addition to the above several configurations, according to the optical path structure and the position of the microscope, the commonly used microscopic interferometry systems can be divided into three types: Michelson type [16–19], Mirau type [20–24] and Linnik type [25–33]. For example, Wiersma, J.T. used the Michelson microscopic system and synchronous phase sensor to realize high precision and repeatable measurement of common vibration [19]; Schmit, J. proposed an improved Mirau interferometer, which can generate orthogonally polarized output beams, and can obtain better fringe contrast by introducing achromatic phase shift [21]; Somekh, M.G. realized a plasmon microscope with sub-micron resolution by using the Linnik interferometer with speckle illumination [26]; Li, X.D. of Tsinghua University measured the thermal deformation of copper microbridges with different sizes (the maximum size is 2175 μm × 1009 μm) in real time using the Linnik microscopic interferometry system, and the accuracy reached submicron level [33].

Compared with ESPI, the microscopic interferometry system can achieve high-resolution measurement of micro devices. However, at the same time, the microscopic system is limited by the field of view of microscope, and the measurement range is very small. Considering that the size of MEMS is in the micron level to the millimeter level, especially after packaging, the size of many MEMS is in the millimeter level. For micro devices in this size range, the microscopic interferometry system is no longer sufficient to achieve full-field measurement. In this paper, a large-field microscopic speckle interferometry system based on an improved Mach–Zehnder structure is proposed. Our measurement system has two main advantages: 1. under the conditions of using the same microscope objective, compared with the traditional Linnik structure, the new system expands the field of view from a circular area with a diameter of 2.4 mm to a rectangular area of 6 mm × 8 mm; 2. it can reduce the reflected light beam in the optical path, thereby reducing the reflected coherent noise.

The phase extraction algorithm used in our work is the wavelet transform (WT) method, which is described in detail in our previous article [34]. Compared with the phase shifting method that appears many times in the above references, the WT does not need to

introduce multiple steps by PZT, which saves a lot of time. Additionally, WT can directly convert the phase change signal from the spatial domain to the frequency domain to realize real-time measurement of the displacement or deformation of the measured object. We used the new system and WT to perform real-time and large-field measurement on a large area of MEMS. Detailed principles and experiments will be introduced below.
