**2. Principle**

#### *2.1. Traditional Linnik Microscopic Interferometry System*

The Linnik microscopic interferometry system is a very common interferometry structure used to measure micro devices. We choose the Linnik microscopic interferometry system as a comparison for later analysis of the effectiveness of expanding the field of view of the new system. This system mainly includes the laser, Linnik microscopic interference structure and CCD. The Linnik microscopic interference structure is composed of a beam splitter and two identical microscope objectives. The laser beam is divided into two arms by the beam splitter. One beam enters the measurement arm and focuses on the measured object through the microscope objective, and the other beam enters the reference arm and focuses on the reference object. The speckle interferograms formed by the two beams is captured by the CCD. We can obtain the displacement or deformation information by processing and analyzing these interferograms. The diagram of the Linnik microscopic interferometry system is shown in Figure 1.

The intensity distribution of speckle interferograms can be expressed as:

$$I(\mathbf{x}, \mathbf{y}, \mathbf{t}\_0) = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \cos[q\_0 - q\_r] \tag{1}$$

where *I*1 and *I*2 are the light intensities of the object beam and the reference beam, respectively, and *ϕ*0 − *ϕr* represents the initial phase difference between them. In the measurement process, when the measured object undergoes continuous out-of-plane displacement, the optical path difference between the object arm and the reference arm changes correspondingly, and the light intensity distribution of the speckle interferogram becomes:

$$\mathbf{l}'(\mathbf{x}, \mathbf{y}, \mathbf{t}) = l\_1 + l\_2 + 2\sqrt{l\_1 l\_2} \cos[\varphi\_0 - \varphi\_r + \Delta \varphi] \tag{2}$$

where Δ*ϕ* indicates the phase change caused by the displacement of the measured object.

The speckle fringe pattern can be obtained by subtraction mode, and the output intensity can be expressed as:

$$\begin{array}{l} \mathbf{I}(\mathbf{x}, \mathbf{y}, \mathbf{t}) = \left| \mathbf{I}'(\mathbf{x}, \mathbf{y}, \mathbf{t}) - \mathbf{I}(\mathbf{x}, \mathbf{y}, \mathbf{t}\_0) \right| \\\mathbf{I} = 4\sqrt{I\_1 I\_2} \sin[(\varphi\_0 - \varphi\_r) + \Delta\varphi(\mathbf{t})/2] \cdot \sin[\Delta\varphi(\mathbf{t})/2] \end{array} \tag{3}$$

where sin[(*ϕ*0 − *ϕr*) + <sup>Δ</sup>*ϕ*(t)/2] is a high frequency term, which is far beyond the acquisition frequency of CCD; sin[<sup>Δ</sup>*ϕ*(t)/2] indicates the fringes related to the change in the measured object.

In our previous research, we introduced the WT to solve the wrapped phase. The continuous WT is defined as:

$$\mathcal{W}\_{I}(\mathbf{a}, \mathbf{b}) = \left\langle I(t), \Psi\_{a,b}(t) \right\rangle = |a|^{-\frac{1}{2}} \int\_{-\infty}^{+\infty} I(t) \Psi \* \left(\frac{t-b}{a}\right) dt \tag{4}$$

where *a* is the scale parameter, *b* is the shift parameter, *I*(*t*) is the signal to be analyzed, *ψ*(*t*) is the mother wavelet, and *ψ* ∗ *t*−*b a* is the conjugate function. The amplitude and the phase are given by:

$$A(\mathbf{a}, \mathbf{b}) = \sqrt{\{\text{Im}[\mathcal{W}\_I(\mathbf{a}, \mathbf{b})]\}^2 + \{\text{Re}[\mathcal{W}\_I(\mathbf{a}, \mathbf{b})]\}^2} \tag{5}$$

$$\varphi(a,b) = \arctan\{\text{Im}[\mathcal{W}\_l(\mathbf{a},\mathbf{b})]/\text{Re}[\mathcal{W}\_l(\mathbf{a},\mathbf{b})] \}\tag{6}$$

where Re[*WI*(a, b)] is the real part of the wavelet coefficient, and Im[*WI*(a, b)] is the imaginary part. Then, the final phase can be obtained by the phase unwrapping algorithm, and for:

$$
\Delta z(\mathbf{t}) = \left(\Delta \boldsymbol{\varphi} \times \boldsymbol{\lambda}\right) / 4\pi \,\tag{7}
$$

we can find that <sup>Δ</sup>*z*(t), that is, the real-time displacement information of the measured object.

#### *2.2. Large-Field Microscopic Speckle Interferometry System*

In this paper, we introduce a new type of large-field microscopic speckle interferometry system. The optical path diagram of the system is shown in Figure 2. In the process of measurement, the laser beam is expanded by the spatial filter and converted into parallel light by the first doublet lens L1. Then, it is split into the measurement beam and reference beam by beam splitter (BS) BS1. The measurement beam illuminates the measured object through BS2, and then reflects back to BS2, and converges into the field of view of microscope objective through doublet lens L2 and BS4. In the experiment, considering that the light is scattered on the rough surface of the measured object, the reflected light intensity gradually weakens as the distance increases, so the distance between BS2 and L2 needs to be as small as possible to ensure sufficient light intensity converges into L2. Therefore, the lens is placed in front of the BS3. Similarly, the reference beam illuminates the reference object through the BS3, and finally converges into the field of view of the microscope objective through doublet lens L3 and BS4. Two beams converge and interfere in front of the microscope objective, and the time-series interferograms amplified by the microscope objective are collected by CCD.

Our system effectively solves the two problems of small field of view and large reflected light coherent noise mentioned above.

**Figure 2.** The schematic diagram of large-field microscopic speckle interferometry system. L1, L2, and L3 represent lenses, BS1, BS2, BS3 and BS4 represent beam splitters.

The first step is to reduce the coherent noise of reflected light. The system is improved on the basis of the Mach–Zehnder structure and greatly reduces the influence of reflected light compared to the Linnik system (Michelson structure), for the Mach–Zehnder structure does not have the light returning to the light source, and the light beam only passes through the microscope objective once, which reduces the back-and-forth reflection phenomenon inside the microscope objective and further reduces the reflection coherent noise. Generally, the reduction in the multiple reflected light in the optical path can improve the image quality. However, it is difficult to carry out a comparative experiment to study how much the coherent noise of reflected light is reduced quantitatively.

Then, we need to expand the field of view while ensuring high imaging quality. The system introduces two doublet lens groups: L1 and L2 are one group, and L1 and L3 are the second group. Take the L1 and L2 group as an example. The two doublet lenses are the same. The light beam is converted into parallel light by the lens L1 to illuminate the surface of the measured object, and the reflected light is transmitted to the lens L2 in the manner of parallel light to be converged, then all the information enters the microscope objective. Similarly, the light beam modulated by another doublet lens group illuminates the surface of the reference object. Through the collimation and convergence process of the doublet lens group, more information will enter the microscope objective, and then the expansion of the field of view is realized.

In order to verify that the field of view is effectively expanded while ensuring the high imaging quality, we conducted three sets of comparative experiments. Three imaging modes are used to image the resolution board, respectively:


doublet lens group and the spatial resolution of the CCD together. It is a rectangular with an area of 6 mm × 8 mm, and its imaging area corresponds to the blue contour region in Figure 3. When we enlarge the selected area in the upper left corner, 60 stripes can also be clearly observed. Therefore, by introducing doublet lens groups, we not only expand the field of view, but also ensure the detailed information of the object.

**Figure 3.** (**a**) Resolution board and the pattern corresponding to number 60; (**b**) schematic diagram of imaging by CCD; (**c**) the captured image; (**d**) the enlarged image of the yellow contour region.

**Figure 4.** (**a**) Schematic diagram of imaging by CCD and objective; (**b**) the captured image corresponds to the yellow contour region in Figure 3c.

By imaging the resolution board, it is proved that the field of view is effectively enlarged by adding the doublet lens group while ensuring high imaging quality, without distortion or field curvature.

**F** 

**Figure 5.** (**a**) Schematic diagram of imaging by CCD, objective and the doublet lens group; (**b**) the captured image corresponds to the blue contour region in Figure 3c; (**c**) the enlarged image of the yellow contour region.
