# A Convenient Way to Determine the Optimum Angle of Incidence of Fizeau Interferometer
## 3 Determination of the Optimum Incident Angle
Compared with [PERSON]'s method of indirectly determining the optimum incidence angle by graphing the relationship between amplitude vector and phase, the improved method proposed here can find the solution of the equation by determining the intersection point, which can more intuitively and accurately determine the optimum incidence angle. Figure 5 shows the optimum incident angle obtained by plotting when the reflectivity is 0.5, 0.7, and 0.9 under different wedge angles. The red dot marked in Figure 5 is the optimum incident angle obtained by plotting when the reflectivity is 0.5, 0.7, and 0.9 under different wedge angles. When determining the optimum incident angle, whether different graphing methods are accurate or not, the process is cumbersome. [PERSON] et al. tried to obtain the optimum incidence angle directly by formula method and gave an estimation formula of the optimum incidence angle. \[\theta_{Opt}\ =\ \frac{\sum_{m\ =\ 0}^{\infty}(2m\alpha)R^{m}}{\sum_{m\ =\ 0}^{\infty}R^{m}}\approx\frac{2\alpha R}{1-R} \tag{16}\]  
However, since the estimation formula derived by [PERSON] is quite different from the value of the optimum incidence angle obtained by the graphing method, the latter method to determine the optimum incidence angle is still the mainstream method at present. In order to solve the problem of insufficient accuracy of the formula method, the left side of the equation is expanded into a fourth-order term by using Taylor expansion, which can obtain a more accurate formula \(\theta_{Opt}\) estimator. \[\theta_{Opt}\ =\ \frac{\alpha\sum_{m\ =\ 0}^{\infty}m^{4}R^{m}}{\sum_{m\ =\ 0}^{\infty}m^{3}R^{m}} \tag{17}\]Formally, the estimation formula derived in this paper is consistent with that derived by [PERSON] et al. In essence, both formulas estimate the average direction of the intensity of the interference wavefront. By expanding  
\[\left(R^{m+4}\right)^{(4)} = \frac{(m+4)!}{m!}R^{m} \tag{18}\] \[\left(R^{m+3}\right)^{(3)} = \frac{(m+3)!}{m!}R^{m}\] (19) \[\left(R^{m+2}\right)^{(2)} = \frac{(m+2)!}{m!}R^{m}\] (20) \[\left(R^{m+1}\right)^{\prime} = (m+1)R^{m} \tag{21}\]  
and iterating into Equation (15), an expression after summation can be obtained  
\[\theta_{Opt} = \frac{-\alpha\left(R^{3}+11R^{2}+11R+1\right)}{(R^{3}+3R^{2}-3R-1)} \tag{22}\]  
Figure 5: The optimum incident angle is obtained by plotting when the reflectivity is 0.5, 0.7, and 0.9 under the condition of different wedge angles. When \(\theta\) changes, the intensities of the main peak and the sub-peak appearing in the focal plane also change. Nevertheless, regardless of the value of the incident angle, the total energy of the light on the focal plane array is constant. In other words, the lower energy of the main peak is due to the leakage of energy into the sub-peak, increasing the intensity of the sub-peak. Conversely, if the intensity of the main peak is on the increase, the power of the sub-peak will decrease. Figure 6 shows the trend of the intensity of the main peak with the incident angle \(\theta\). It can be found that when \(\theta\) is equal to \(\theta_{Opt}\), the sum of the amplitude vector of the main peak will be maximized, and the influence of the sub-peak will be minimized at this time. This plot verifies the correctness of the optimum incident angle obtained by our formula method. As shown in Figure 7, it is the curve of the optimum incident angle obtained by [PERSON]'s graphing method, [PERSON] formula method, and the formula method in this paper under different wedge angles and different reflectivity. It should be noted here that due to the different number of beams involved in an interference, the average direction of intensity obtained must not be consistent. Considering this factor, the number of effective beams in the three methods is unified as M = 100 in the figure. Through comparison, it is found that under different conditions, the optimum incidence angle obtained by [PERSON]'s method is in good agreement with the optimum incidence angle determined by the formula method proposed in this paper.