通信向，F-P干涉仪__intro (for Optical_fiber Telecommunication)

论述 Fabry-Parot 干涉仪的原理以及其在光纤通信器件 OADM （光分插复用器）中的简单应用（This article is just for the application of the interferometer, in the Optical-fiber telecommunications.）

Filters! Please don't forget what are we talking about: The filter of F-P inferometer type!

Principle Of F-P Inferometer(Filter)

Which is also regarded as "Interferometric optical filter", and from multiple beam interference.

F-P etlon is made of two sheets (sometimes with K9 glass @《A novel tunable OADM based on F-P Filter》) and their surface (plated with high reverse film). For example, consist of two substrates coated with a high reflective film ( $15mm$ for the diameter) with 703 glue with $25\mu m$ thickness between them to control the distance.

Figure multiplied: @wikipedia&baidubaike,

As the graph tells, when the incident light has round trip phase different (in the Fabry-Parot etlon) as the integer multiplies of $2\pi$ , it will get the constructive interference, that is, form a strong light output through the cavity of the reflection film, as for the other can't get the phase different of $2n\pi$ , the light is difficult to be output through the light reflection membrane because of the destructive interference, this kind of interference is precisely the "Interferometric optical filter".

Now, the calculation below will tell you something about the interference characteristic exactly, that is, how can it be a filter? In quantitative calculation.

$\mathscr{A}$ .What about the extremely strong condition of the transmitted light（透射光强极大条件）?

You can see the enlaged drawing of the F-P Etlon, and called the specific point as (A,B,C,D,P-the focus). When the transmission rate is the maximum? The phase difference between two beams of light must satisfied $\delta=2n\pi$ , $n\in Z$ .

At the same time, we know that $\delta=\frac{2\pi\Delta }{\lambda}$ , where $\Delta$ is the path difference of adjacent transmitted light.

For F-P Inferometer calculation

We all know that $\Delta=n_2\ 2\overline{AB}-n_1\ \overline{AD}$ . How to describe $\Delta$ ? That is when the enlarged drawing be useful.

Make full use of the geometry proporty of the ray of light

$\overline{AB}=\frac{d}{cosr}$ , $\overline{AD}=\overline{AC}\ sini=2d\ tanr\ sini$ .

Hence we can have $\Delta=n_2\cdot\frac{2d}{cosr}-n_1\cdot2d\ tanr\ sini=\frac{2d}{cosr}(n_2-n_1sini\ sinr)$ ,

From Snell's law, $n_1sini=n_2sinr$ , Finally get $\color{green}{\Delta=2n_2d\ cosr}$ .

The phase difference can be represented by $\color{green}{\delta=\frac{4\pi n_2d\ cosr}{\lambda}}$ .

Here, might not give out the full derivation of the intensity of transmitted light but to show the equation of it directely $\color{green}{\frac{I_t}{I_i}=(1-\frac{A}{1-R})^2\frac{1}{1+Fsin^2\frac{\delta}{2}}}$ , where $A$ means the absorptivity of the membrane, and $R$ represent its reflectivity, $F=\frac{4R}{(1-R)^2}$ is called the Fineness coefficient. The figure below is for its distribution, I make the coefficient as $0.2、1、2、5、12、100$ from the top to the bottom.

The distribution diagram of relative light intensity of transmitted light (compiled from the mobile app Graph) for x-axis and for y-axis

Obviously, the high contrast ratio produced by the higher $F$ , it corresponding to higher $R$ , when it level off to $1$ , the interference fringes are extremely sharp, which is of high status of the multi-beam interference.

At the same time, so long as we can change the angle of incidence $i$ and change the distance of the substrates $h$ , we can achieve the purpose of filering.

$\mathscr{B}$ .The central wavelength（中心波长）?

Central wavelength is definited as "The wavelength with the maximum transmission ratio", to get it, let the phase difference equals $2n\pi$ , we can have

$\frac{4\pi n_2hcosr}{\lambda_c}=2n\pi$ ,

That is $\color{green}{\lambda_c=\frac{2n_2h}{n}}$ .

$\mathscr{C}$ .Perspective band half-wave width, FSR, and Q（透射光半波宽度"半带宽"，自由光谱程和精细度）

When the perspective rate is half to the maximum value, the wavelength is called the perspective band half-wave width. The smaller the width, the better the filter selection performance.

$\color{green}{\Delta\lambda_{3dB}=\frac{\lambda^2}{2\pi n_2d}\frac{1-R}{\sqrt R}}$ ;

FSR (Free spectrum range) showed how many wavelength channels the WDM signal can pass through the cavity, is a very important parameter in Optical-fiber telecommunication. To get the measure of it, get the wavelength interval between adjacent resonance peaks from the transform function.

$\color{green}{FSR=\frac{\lambda^2}{2n_2dcosr}}$ ;

As for Q of the F-P Inferometer, the ratio of the FSR and the bandwidth.

$\color{green}{S=\frac{\pi}{2}\sqrt{F}}$ .

Contribution to optical communications

Continue of the former article named from @通信向，光分插复用器__intro (What is Optical Add-Drop Mul... - 简书.

$\mathscr{D}$ .Apply F-P inferometer to OADM（F-P 腔在 OADM 中的应用）

The F-P Inferometer can be treated as the device "OADM downlink signal separation", ie. can realize the "drop".

F-P type of OADM @A Novel Tunable OADM Based on F-P Filter

$\mathscr{E}$ .Defect: The insertion loss（一个缺陷：插入损耗）

In telecommunications, insertion loss is the loss of signal power resulting from the insertion of a device in a transmission line or optical fiber and is usually expressed in decibels ( $dB$ ).

If the power transmitted to the load before insertion is $P_T$ and the power received by the load after insertion is $P_R$ , then the insertion loss in dB is given by, $IL(dB)=10log_{10}\frac{P_T}{P_R}$ ———— learning from @wikipedia

As for the OADM device, use $\diamond$ shape to represent the F-P inferometer, and the figure below shows you how the OADM of 4 channel works.

OADM for 4 channel, the direct channel will be reflected for 4 times

That means, if want more frenquency bands to be extraction from the WDM signal, you should make the input signal pass more $\diamond$ , that means, need to offer more insertion loss.

Which told us the applcation of F-P Inferometer will give out large loss, that is one of it defects.

$\mathscr{F}$ .Improvement strategy

x-skip-0 optical filter, which can be realized from the multi-beam interference, consist of multilayer cavity, to give out a more acute figure to the curve of the reflectivity, on the other hand, is to rise the rectangularity of the membane system passband to let x number of information channel go, next I will show a spectrum chart of 13 cavity film system, which can selected eight dense wavelenth division multiplexing channels, hence, the 8-skip-0 optical filter.

The 8-skip-0 filter's passband: more gentle @New Optical Thin Film Components Applied In Optical Communication

From this special characteristic, we can get to know if want to decrease the insertion loss, just to use the new type of F-P Inferometer to drop the information channel needed, and then, to make them seperate with common F-P etlon, the insertion loss will be $1/x$ times of former.

New idea of decrease the insertion loss

Finally, above are just a far-fetched argument of F-P Inferometer (For the Optical-fiber telecommuniction), some are too much detail and maybe some are not enough, wish more interesting idea come and communicate together.

通信向，F-P干涉仪__intro (for Optical_fiber Telecommunication)

Principle Of F-P Inferometer(Filter)

.What about the extremely strong condition of the transmitted light（透射光强极大条件）?

.The central wavelength（中心波长）?

.Perspective band half-wave width, FSR, and Q（透射光半波宽度"半带宽"，自由光谱程和精细度）

Contribution to optical communications

.Apply F-P inferometer to OADM（F-P 腔在 OADM 中的应用）

.Defect: The insertion loss（一个缺陷：插入损耗）

.Improvement strategy

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