# Micro Vibration Measurement with Microscopic Speckle Interferometry Based on Orthogonal Phase
## 2 Principle  
In the optical path of interference measurement, the light intensity signal collected by the charge-coupled diode (CCD) can be expressed as:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos(\theta), \tag{1}\]  
where \(I_{1}\) and \(I_{2}\) are the light intensities of the object beam and the reference beam, respectively, and \(\theta\) represents the phase difference between them.  
When the measured object vibrates, its trajectory \(S\) can be described as:  
\[S=A\cos(2\pi f\ \text{t}), \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Frequency** & **Amplitude** \\ \hline Homodyne interferometry & kHz to GHz & pm \\ Heterodyne interferometry & kHz to GHz & nm \\ Sagnac interferometry & MHz to GHz & pm \\ Stroboscopic white light interferometry & MHz & nm \\ \hline \hline \end{tabular}
\end{table}
Table 1: The frequency and amplitude of the above methods.  
where \(A\) is the vibration amplitude and \(f\) is the vibration frequency. These two physical quantities are the physical quantities to be finally measured. At this time, the light intensity signal can be expressed as:  
\[\begin{array}{ll}I&=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos(\varphi+\frac{4\pi S}{ A})\\ &=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos[\varphi+\frac{4\pi A\cos(2\pi f\ \mathrm{t})}{ \lambda}]\end{array} \tag{3}\]  
where \(\varphi\) represents the initial phase difference.  
When processing light intensity signals with double cosine function in the vibration measurement, the Bessel function method is usually used [20; 21; 22]. In this method, the amplitude A is required to be far less than the wavelength, which limits the amplitude range of the measurement. By omitting the DC component of the above formula (for they do not contain vibration information) and expanding it according to the Bessel function, we can get:  
\[\begin{array}{ll}I&=2\sqrt{I_{1}I_{2}}\cos(\varphi+\frac{4\pi S}{\lambda}) \\ &=K\cos\varphi[J_{0}(\frac{4\pi A}{\lambda})-2J_{2}(\frac{4\pi A}{\lambda}) \cos 4\pi ft]\\ &-K\sin\varphi[2J_{1}(\frac{4\pi A}{\lambda})\sin 2\pi ft-2J_{3}(\frac{4\pi A}{ \lambda})\sin 6\pi ft]\end{array} \tag{4}\]  
where \(K\) represents the photoelectric constant before the cosine term, \(J_{0}\) is the zero-order Bessel function, and \(J_{1}\), \(J_{2}\), \(J_{3}\) correspond to the Bessel function of the corresponding order. It can be seen from Equation (4) that the spectrum of the photocurrent is composed of the fundamental frequency and its harmonics. We use \(P_{n}\) to represent the amplitude of the nth harmonic:  
\[P_{n}=2 KJ_{n}(\frac{4\pi A}{\lambda})\bigg{\{}\begin{array}{ll}\cos\varphi \ \text{when $n$ is even}\\ \sin\varphi\ \text{when $n$ is odd}\end{array}, \tag{5}\]  
In Equation (5), \(P_{n}\) is easy to be measured, but the constants \(K\) and \(\varphi\) are difficult to measure accurately. Therefore, the amplitude \(A\) is generally determined by the ratio of two different harmonic amplitudes or the special point (such as maximum value or zero point). This is the principle of the general Bessel function method.  
In this paper, the calculation method is more concise. According to the properties of the cosine function, the approximate linear distribution is satisfied near the position of \(\pi/2\), as shown in Figure 1. (Considering the symmetry of the cosine function, \(3\pi/2\) play the same role in (\(\pi\)-\(2\pi\)) as \(\pi/2\) in (0-\(\pi\)). For simplicity of expression, what we have been discussing in the article is \(\pi/2\).) When the initial phase difference \(\varphi\) equal to \(\pi/2\) and the light intensity, which is determined by the amplitude A, is not too large to exceed the linear region, the vibration trajectory and the light intensity change caused by the vibration are corresponding to each other. We use the orthogonal characteristics to obtain the vibration information from the light intensity signal. The detailed formula derivation will be shown below.  
Figure 1: The approximate linear distribution of the cosine function in the vicinity of \(\pi/2\).