# A Convenient Way to Determine the Optimum Angle of Incidence of Fizeau Interferometer
## 3 Determination of the Optimum Incident Angle  
Based on the influence of the thickness of parallel plates on the spatial interference intensity distribution analyzed in Section 2, it is found that there is an expression of a coordinate transformation based on the point of rotation \(\theta\) in the formula. In this case, the optimal incident angle when the beam passes through the parallel plate and interferes behind the parallel plate is equal to the optimal incident angle when the parallel plate does not exist. Unlike the symmetrical structure of a single fringe of the [PERSON] interferometer, the Fizeau interferometer's fringe has an asymmetric shape and substructure. The reason for fringe distortion is that the structure is unstable, making the interference beam move in the direction of increasing plate spacing so that the beam gradually dephases [20]. Figure 4: In order to illustrate the reproduction effect of the parallel plate on the spatial interference intensity, the spatial interference intensity is deliberately displayed in the form of contour lines. (**a**,**b**) are the spatial interference light intensity distribution in the absence of parallel plate modulation and the presence of parallel plate modulation, respectively. Figure 3: Under normal incidence conditions, the longitudinal period of lattice points in different media changes significantly. The period of longitudinal lattice point (**a**) in air is \(1/n\) that (**b**) in the parallel plate. However, there is no difference in the transverse period between the two cases. (\(\lambda\) = 1610 nm, \(\alpha\) = 0.1 mrad, \(n\) = 1.5). When \(\alpha\) is unchanged, with the increase of y, the sub-peak will gradually appear on one side of the main peak. During spectral inversion, the intensity of the side lobe will be superimposed on the intensity of the main peak of other wavelength interferograms, resulting in inversion distortion and reducing the accuracy of spectral measurement. This situation should be avoided in the field of spectral measurement. Observing the phase distribution formula, when the wedge air layer rotates around the wedge apex o, the fringe pattern remains unchanged, which is still formed by the interference between the same wavefront. Different portions of the interference light intensity are introduced into the focal plane by the change in the incident angle \(\theta\). Thus we need to find an incident angle \(\theta_{Opt}\), so that the focal plane is just coincident with the localized plane. Under the inclined illumination, the lattice period is not affected because it is only determined by the wedge \(\alpha\). In this paper, based on the research theory of [PERSON], a more accurate method of determining the optimum entry angle is proposed. Since the natural period of the lattice point in the y-direction is \(\lambda/2\alpha\), we select the peak value closest to the wedge apex in the y-direction as the reference strength. While \(\theta\) is changing, the intensity at different positions of the peak can always be introduced into the focal plane. When the intensity is maximum, the \(\theta\) is the optimum incident angle we are looking for. The specific approach is taking the straight line on the rear surface of the wedge-shaped cavity, which is also perpendicular to the ridge of the wedge-shaped cavity as the reference direction. A position with a phase difference of \(\pi/\alpha\) from the ridge of the wedge cavity is found in this direction. The sum of the amplitude vector here is used as the reference value. The amplitude derivative here must satisfy  
\[ \begin{array}{c}\frac{dE}{d\theta}\ =\ \sum\limits_{m\ =\ 0}^{M}R^{m}sin \left[\frac{\pi}{\alpha}\big{(}sin\big{(}\theta_{Opt}+2 ma\big{)}-sin \theta_{Opt}\big{)}\right]\\ \cdot\left[\big{(}cos\big{(}\theta_{Opt}+2 ma\big{)}-cos\theta_{Opt}\big{)} \right]\ =\ 0\end{array}  \tag{15}\]  
Due to the complexity of the Equation (13), the solution of the equation can be determined by the intersection of the function plot on the left side of the equation and the horizontal axis.