# Multi-Beam Interference

The interferences above are actually approximate results based on the reflecting planes with low reflectance. For example, the reflectance of the fiber end is about 4%. It is suitable as concerns about only two beams according to the reason introduced in Section 1.2.1. If the reflectance of the reflecting plane increased, the result will be different. Similarly, if the intensity of the incident light is 100%, but the reflectance is 90%, the intensity of the first transmitted light is 1%, the second is 0.81%, the third is 0.66%, and so on. Although the intensities of these transmitted lights are decreased, the difference between the adjacent light is little enough. As a consequence, multi-beam interference has to be considered in this situation.

Consider the light shown successively reflected between the fiber ends of the FP cavity A and B in Figure 1.7. The incident ray *i* of unit amplitude stands for the propagating direction of the incident light and 0 represents the incident angle in the cavity as shown. A is the wavelength in vacuum, and the effective refractive index of the medium in the FP cavity is *n*. For convenience, the electric vector of the incident light is considered as unit amplitude, and linearly polarized either parallel or perpendicular to the incident plane; the OPD between two light beams is

**Figure 1.7 **Multi-beam interference.

The corresponding phase difference is

It is supposed that *E _{0i}* is the amplitude of incident light electric vector, and

*r, r, t,*and

*t'*are the reflection factors and the transmission factors when light goes in and out of the cavity. So, the amplitudes of the light reflecting from the FP cavity are

The complex optical vector of *P* is

According to the Fresnel formula, *r, r, t,* and *t'* satisfy the relations:
The reflectance *R* and the transmittance *T* also have relations:
Using the mathematical equation,

If the quantity of the reflected beams is big enough, Equation 1.24 can be simplified as

According to the equation above, the intensity of the reflected light after focusing is

Usually, the coefficient *F* can reflect the finesse of the interference fringe, which will be explained as detailed below.

Similarly, the intensity of the transmitted light after focusing is

Ignoring the absorption, *T =* 1 — *R,* the transmitted light can be simplified as

Equations 1.26 and 1.29, the intensity distribution formulas of the interferences by the reflected light beams or the transmitted light beams, are usually named *Airy formulas.* We also know *I _{r} + I_{t} = Ii,* which reflected the universal rule of energy conservation. If the reflected light is increased by interference, the transmitted light will be decreased and vice versa. It means that the distributions of the reflected light and the transmitted light are complementary.

According to Airy formulas, for reflected light, if ф = (2m + 1) П, *m =* 0, 1, 2, ... , the maximum intensity, *I _{rM}* satisfies

*I*(F/1 +

_{rM}=*F)I*and if ф = 2mn,

_{i }*m =*0, 1, 2, ... , the minimum intensity,

*I*satisfies

_{rm}*I*0. Similarly, the maximum and minimum of the transmitted light occur when ф = 2mn and ф = 2mn,

_{rm}=*m =*0, 1, 2, ... , separately, with

*I*I and

_{tM}=*I*(1/1 +

_{tm}=*F)I*Comparing with Section 1.2.1, that is, the two-beam interference, the conditions of the max or min interference intensities for reflected light or transmitted light are the same for both two-beam interference and multi-beam interference.

_{i}.With proper spacing of the reflectors, the transmittance of the FP cavity is high. Changing the spacing causes transmittance to drop. With high-reflectivity reflectors, the transmittance is very sensitive to changes in wavelength or reflector spacing. Overall, interferometer performance is frequently characterized by finesse *F* related by the reflectance R, shown in Equation 1.27. Transmission as a function of reflector spacing is shown in Figure 1.8 for various values of finesse, in which the x-coordinate presents the phase difference of the adjacent transmitted light and the y-coordinate presents the relative intensity. High-finesse interferometers are useful because of the precisely located spectral features; low-finesse interferometers provide linear operation with a wider range of the measurand without complex feedback schemes.

Taking the formulas of *I _{rM}, I_{rm}, I_{tM},* and

*I*back to Equation 1.7, the contrast ratios of reflection and transmission are

_{tm},

Obviously, *V _{t}* is less than 1; however, it does not mean that the transmitted spectrum is always worse than the reflected spectrum. As shown in Figure 1.8, the larger the

*R*is, the sharper the transmitted spectrum will be. According to the relationship of

*I, 1*and

_{1},*I, I*I-, the reflected spectrum will be wider if

_{r}+ I_{t}=*R*is larger.

We can find that the distribution of the light intensity is corresponding to the reflectance R. If *R* is really small, *F* is much less than 1. If carrying out Equations 1.25 and 1.27 and only reserving

**Figure 1.8 **Transmission spectrum of multi-beam interference.

the first item, the relative intensities of reflected light and transmitted light are

It can be proved that the equations above are just the intensity distributions of the two-beam interference.

The finesse of the spectrum is also relative to R. Higher finesse of the interference spectrum by transmitted light is just the most important factor of the multi-beam interference.

The finesse of the transmitted interference light can be reflected by the half-high width Дф introduced in Section 1.2. As shown in Figure 1.9, the phase of the half-high position is ф = 2mn ± (Дф/2). Taking it back to Equation 1.27, we can get an equation

**Figure 1.9 **Half-high width Дф.

If Дф is really small, sin (Дф/4) ~ Дф/4. The equation of the halfhigh width Дф can be derived, which is already shown in Equation 1.5.