Fiber Bragg grating (FBG) notes [1]: waveguide structure and Bragg wavelength derivation

introduction

This paper is about fiber Bragg grating （FBG） The notes 【1】, The waveguide structure of fiber Bragg grating is introduced , The expression of fiber Bragg wavelength is derived by using the conservation of momentum and energy

The most simple uniform fiber Bragg grating in ordinary fiber is that the refractive index changes periodically （FBG）, The waveguide structure is shown in the figure below .

The description of its sensing principle can be divided into the following three points ：

① The light propagating in the fiber core will be reflected and transmitted on each grating surface ;
② If Prague condition is not satisfied , The light reflected by the orderly arranged grating planes will gradually become different until they finally cancel each other ; At the same time, because the coefficients do not match , The reflection of light that does not match the Bragg wavelength in each grating plane is also very weak , Most of the light is transmitted in the optical fiber .
③ If Prague conditions are met , The light reflected from each grating plane is gradually accumulated , Finally, it will form a Reflection peak , The central wavelength is determined by the grating parameters .

We focus on the light that meets the Bragg condition , Temperature can be achieved by monitoring Bragg wavelength 、 Strain sensing . Using the conservation of momentum and energy, we can deduce the expression of Bragg wavelength ：

Conservation of energy ：$h{\nu }_r=h{\nu }_i$
This requires the frequency of incident light ${\nu }_i$ And reflected light ${\nu }_r$ frequency （ Light satisfying Bragg condition ） identical .

Conservation of momentum ：${\mathbf{P}}_i+\mathbf{P}={\mathbf{P}}_f$
This requires the incident wave vector ${\mathbf{P}}_i$ And grating wave vector $\mathbf{P}$ The sum is equal to the reflected wave vector ${\mathbf{P}}_f$. Because the momentum is $\frac{h}{\lambda }$, The wavelength of incident light and reflected light is the same （ Light satisfying Bragg condition ）, So the reflected wave vector ${\mathbf{P}}_f$ And the incident wave vector ${\mathbf{P}}_i$ Equal in size and opposite in direction , The amplitude of the wave vector of the grating is $\frac{2\pi }{\Lambda }$, So the momentum conservation condition can be rewritten as ：

$\frac{2\pi n_{eff}}{{\lambda }_B}-\frac{2\pi }{\Lambda }=-\frac{2\pi n_{eff}}{{\lambda }_B}$

That is to say ：
$2(\frac{2\pi n_{eff}}{{\lambda }_B})=\frac{2\pi }{\Lambda }$

Finally, the Bragg wavelength is obtained ${\lambda }_B$ The expression of ：
${\lambda }_B=2n_{eff}\Lambda$

In style , Fiber Bragg grating ${\lambda }_B$ Is the central wavelength of the incident light reflected from the fiber Bragg grating in free space ;$n_{eff}$ Is the refractive index of the fiber core for the central wavelength of free space .

reference ：

[1] Rao Yunjiang Wang Yiping . Principle and application of fiber Bragg grating [M]. Beijing : Science Press ,2006:136-137.