# A Convenient Way to Determine the Optimum Angle of Incidence of Fizeau Interferometer
## 1 Introduction  
Fizeau interferometer has the characteristics of high resolution and high light gathering ability, so spectroscopy workers have always valued it. In 2011, [PERSON] and [PERSON] proposed a static Fourier transform spectrometer using an optical wedge to generate optical path difference [1; 2; 3; 4; 5; 6; 7]. The structure integrates an infrared area array detector with a wedge prism and uses the principle of optical path difference generated by optical wedge for infrared spectrum detection. However, this structure still needs chemical etching after coating, and it requires high precision of the working surface. In order to realize the marketization and popularization of the spectrometer, based on the wedge structure, this paper proposes an improved Fizeau spectrometer structure with simple manufacture and lower cost. As shown in Figure 1, a parallel plate coated with high reflection film is pasted on the focal plane array. There is a wedge-shaped air chamber with angle \(\alpha\) composed by this parallel plate and another parallel plate also plated with high reflection film. Due to the sudden refractive index change at the air/plate interface, the light is reflected inside the wedge, and the multi-beams transmitted and reflected in the wedge will interfere. The focal plane array records the interference fringes, and then the incident spectrum is solved by corresponding algorithms. However, if the collimated light is directly incident on the Fizeau interference system, there is a sub-peak in the interference fringe received by the detector [8; 9; 10; 11; 12]. During spectral inversion, the intensity of the sub-peak will be superimposed on the intensity of the prominent peak of other wavelengths to reduce the accuracy of spectral inversion. Predecessors have done much research on eliminating the sub-peak. The general idea is to find an optimum incidence angle. When the collimated light is incident at this angle, the influence of the sub-peak will be reduced or even eliminated. In terms of determining the optimum incident angle, the graphing method of the sum of the amplitude vector proposed by [PERSON] and the estimation formula proposed by [PERSON] is mainly used to select the optimum incident angle [13; 14]. [PERSON] uses the relationship between the phase and the sum of the amplitude vector to make a plot. Then find the value of the incident angle when the sum of amplitude vector is the largest as the optimum incident angle. [PERSON] proposed a formula for the optimum incident angle in 1993 based on understanding the multi-beam intensity distribution involved in the interference in the Fizeau interferometer. The two methods have their own advantages and disadvantages. Firstly, the sum of amplitude vector graphing method has high accuracy in determining the optimum incidence angle, which is the most commonly used method to determine the optimum incidence angle at present. However, the process is cumbersome and complicated. The formula mentioned by [PERSON] can immediately obtain the value of the optimum incident angle by simply substituting the structural parameters. Nevertheless, the error of the optimum incident angle obtained by this formula is significant. Moreover, due to the special structure of the spectrometer mentioned in this paper, the incident beam is reflected many times in the wedge cavity and emitted through the parallel plate. The phase of each outgoing beam will be modulated by the thickness and refractive index of the parallel plate, resulting in the change of spatial interference intensity, which may lead to the deviation of the optimum incident angle. Furthermore, the theory of finding the optimum incident angle by predecessors such as [PERSON] and [PERSON] did not consider the influence of parallel plates. Generally, the models that can be used to describe the spatial distribution of light include the brightness model of random field and the model of plane wave superposition and interaction in space [15].