Based on the principle of the Michelson interferometer, the paper briefly describes the theoretical significance and calculates and deduces three expressions of the interference modulation depth. The influence of the surface shape error of plane mirror on modulation depth is analyzed, and the tolerance of error is also pointed out. Moreover, the dependence of modulation depth on the reflectance change of beam splitter interface is also analyzed, and the curve is given. It is concluded that this paper is of general significance for the Fourier transform spectrometer based on the principle of the Michelson two-beam interference.

The Fourier transform spectrometer is an instrument which is used to analyze the spectral distribution of the light source by means of the interference effect based on the Fourier transform technology. It has incomparably more advantages than conventional spectral instruments in acquiring higher spectral resolution. There are many kinds of Fourier transform spectrometers, including Michelson interferometer, Fabry-Perot interferometer, Sagnac interferometer, and double refraction interferometer [

The classical Michelson interferometer consists of a moving mirror and a fixed mirror. The two beams of lights, reflected by these two mirrors, make up the phase difference. When the moving mirror moves, the intensity of interference signals is modulated to form the interferogram. The modulation depth is also known as the interference modulation efficiency. It is one of the most important factors influencing the signal-to-noise ratio of the interferogram, and a change in it will significantly affect the sensitivity of the spectrometer [

The most classical Fourier transform spectrometer is the Michelson interferometer, which is composed of a fixed mirror, a moving mirror, a beam splitter, and a detector. Its basic structure is shown in Figure

Basic structure of Michelson interferometer.

In the Michelson interferometer, the fixed mirror and the moving mirror are perpendicular to each other, with an angle of 45° between the beam splitter and two mirrors. The collimated beam shines the interference system, and the beam splitter separates the incident light into two beams with roughly the same reflectance and transmittance. One reflected beam by the fixed mirror then goes through the beam splitter again, whereas the other transmitted beam is reflected by the moving mirror and divided by the beam splitter. Because these two beams of light are produced by the same beam, they have a constant phase difference. When these two beams of light travel through a convergent lens, the interference is formed on the detector.

Upon arrival at the detector, the two coherent beams have an optical path difference for the shift of the moving mirror. The optical path difference then shows periodic changes with round-trip translation of the moving mirror. According to the principle of interference spectroscopy, the interference signals can be obtained on the detector.

According to the interference theory, the intensity of interference signals generated by the two-beam interference is [

The definition of the modulation depth of interference signals, namely, the visibility of interference signals

From (

The fixed mirror and the moving mirror of Fourier transform spectrometer are composed of plane mirrors with a rectangular aperture. In the process of making plane mirrors, there are a number of factors modulating the surface error, such as polishing, coating, and mirror bending. It follows that these mirrors are not ideal surface shape, and there is always a certain surface error. Therefore, in a spectrometer composed of these mirrors, the modulation depth is often not ideal. In what follows, we will analyze the influence of curve error and polishing error of plane mirror on interference modulation depth. Suppose polishing error of mirror obeys normal distribution with zero mean, and its maximum fluctuation is

When there is a surface shape error of plane mirror, the reflection of two beams of coherent light will generate wavefront distortion, which results in a change in optical path difference of interferometer. The phase difference

When

It can be seen that the interference modulation depth is inversely proportional to the variance of polishing error of mirror, which means that a good polishing surface can improve the modulation depth. If the interference modulation depth is set at

In the same way, when the mirror surface is curved, the existence of the curve error causes the intensity distribution of the interference signals to be

Through the above analysis, we know that the surface error of mirror has an effect on the modulation depth. In order to achieve better interference effect, the surface error of mirror should meet error tolerance requirements.

In the Michelson interferometer shown in Figure

According to the Fresnel formula, the amplitude reflection coefficient and the amplitude transmission coefficient satisfy the following conditions:

Influence of reflectance change of beam splitter on modulation depth.

From Figure

The analysis of interference modulation depth indicates that, in order to realize the interference modulation depth

The surface shape of plane mirror.

The interferometer is placed on the testbed, and the interference is formed by using the He-Ne laser. The electrical signal of the laser interference can then be acquired. Figure

The electrical signal of the laser interference.

As can be seen from Figure

In the Fourier transform spectrometer, the Michelson interferometer is the core optical system. By focusing on the classic Michelson interferometer, we have conducted a theoretical analysis of the influence of surface shape of mirror and performance change of the beam splitter on the modulation depth. Theoretical deductions are obtained regarding tolerance requirements for the error of surface shape mirror and the reflection ratio of the beam splitter. The results obtained are applicable to the Fourier transform spectrometer within the framework of the Michelson dual beam interference.

The author declares that there is no conflict of interests regarding the publication of this paper.

The research was supported by the Program for Science and Technology funded by the Education Department of Jiangxi Province (GJJ14316).