# Micro Vibration Measurement with Microscopic Speckle Interferometry Based on Orthogonal Phase
## 3 Simulation  
In order to verify the rationality of the orthogonal characteristics and determine the range of the measurable vibration amplitude, simulations with a series of vibration amplitudes are conducted. In the simulation, when the measured object vibrates, the light intensity of these orthogonal points (the points of \(\varphi\) equal to \(\pi\)/2) can be expressed as:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos[\frac{\pi}{2}+\frac{4\pi A\cos(2\pi f\text {t})}{\lambda}], \tag{6}\]  
We set \(\lambda\) as 532 nm, which is equal to the laser wavelength used in the experiment; \(f\) has no effect on the distribution, and it is set to 1 Hz in the simulation. According to Equation (6), in order to satisfy the linear characteristics around \(\pi\)/2, the value of amplitude A should not be too large. We set \(A\) as 30 nm, 50 nm, 70 nm, and 90 nm, respectively, for simulation. The real-time position of measured object and the corresponding light intensity curves are shown in Figure 2.  
As shown in Figure 2, when the amplitude A is 30 nm and 50 nm, the light intensity curves (Figure 2e,f) are in good agreement with the vibration trajectory (Figure 2a,b). Additionally, when the amplitude A is 70 nm and 90 nm, the light intensity curves (Figure 2g,h) and the vibration trajectory (Figure 2c,d) become no longer consistent. Taking Figure 2h for example, when A is equal to 90 nm, the maximum value of the parameter in the cosine term exceeds the range of (0-\(\pi\)) and frequency doubling phenomenon occurs. At this time, we can get the maximum value I\({}_{\text{max}}\) and minimum value I\({}_{\text{min}}\). Figure 2g also shows frequency doubling phenomenon. As the amplitude of 70 nm is not particularly large, the frequency doubling phenomenon is not obvious. It can only be seen that the top and bottom of the light intensity curve become flat. On the basis of simulations and verifications, in order to meet the linear matching range and obtain accurate measurement results, the range of measurable amplitude is within 60 nm. Figure 2: (**a–d**) are the real-time position of the measured object when the amplitude A = 30 nm, 50 nm, 70 nm, 90 nm; (**e–h**) are the corresponding interference light intensity curves. Additionally, the following relationships can be obtained from Equation (6): when \(\frac{4\pi A\cos(2\pi f\ t)}{\lambda}\) takes the minimum value, the light intensity corresponds to the peak; when the vibration term takes the maximum value, the light intensity corresponds to the valley. The light intensity in these two cases can be expressed as:  
\[ \begin{array}{l}I_{peak}=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos[\frac{\pi}{2}+ \frac{4\pi(-A)}{\lambda}]\\ I_{valley}=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos[\frac{\pi}{2}+\frac{4\pi(A)}{ \lambda}]\end{array}  \tag{7}\]  
Compared with the Equation (4), which uses Bessel function, the expression is greatly simplified and no longer contains the double cosine function. As long as we get \(I_{1}\), \(I_{2}\) and the corresponding peak value of light intensity \(I_{peak}\) (or \(I_{valley}\)), we can get the amplitude \(A\). Then, we conduct the second simulation to analyze the influence of \(\varphi\) with a certain offset which is not equal to \(\pi/2\). In this case, the light intensity can be expressed as:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos[\frac{\pi}{2}+\sigma+\frac{4\pi A\cos(2 \pi f\ t)}{\lambda}], \tag{8}\]  
We take A to be 40 nm in the measurable range, \(f\) is still 1 Hz, and \(\sigma\) represents the magnitude of the offset. The real-time position of the measured object and the light intensity curves under different offset are shown in Figure 3:  
From Figure 3b-h, the absolute value of \(\sigma\) is reduced from \(3\pi/10\) to 0, and then increased from 0 to \(3\pi/10\). With the increase in the absolute value of \(\sigma\), the initial value in the cosine term gradually deviates from \(\pi/2\), then the light intensity curves and the vibration trajectory no longer match.