Fiber Bragg grating is a waveguide with periodically varying refractive index. From coupled-mode theory, it can be seen that when the waveguide is disturbed periodically, it will lead to the coupling of propagation modes. Fiber Bragg grating sensing principle diagram was shown in Figure 3.

Figure 3 Fiber Bragg grating sensing principle diagram

Maxwell’s equations can be expressed as the following Equations (5) and (6):

$$\begin{array}{r}\mathrm{\nabla}\times \overline{H}=\overline{J}+\frac{\mathrm{\partial}}{\mathrm{\partial}t}\left({\epsilon}_{0}\overline{E}+\overline{P}\right)\end{array}$$(5)$$\begin{array}{r}\mathrm{\nabla}\times \overline{E}=-\frac{\mathrm{\partial}}{\mathrm{\partial}t}(\mu \overline{H})\end{array}$$(6)For optical waveguides, there is no free charge, so $\overline{J}=$ 0. Equations (5) and (6), the wave equation of insulating medium can be, the expression can be written as the following Equation (7):

$$\begin{array}{r}{\mathrm{\nabla}}^{2}\overline{E}(\mathrm{\partial}\overline{r},t)=\mu {\epsilon}_{0}\frac{{\mathrm{\partial}}^{2}\overline{E}(\overline{r},t)}{\mathrm{\partial}{t}^{2}}+\mu \frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{t}^{2}}\overline{P}(\overline{r},t)\end{array}$$(7)Due to the non-uniformity of the medium fluctuation, it can be considered as perturbation, and the polarization intensity can be expressed as the following Equation (8):

$$\begin{array}{r}\overline{P}(\overline{r},t)=[\epsilon (\overline{r})-{\epsilon}_{0}]\overline{E}(\mathrm{\partial}\overline{r},t)+{\overline{P}}_{\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(\overline{r},t)\end{array}$$(8)Substituting (8) into (7), we can obtain the following equation (9):

$$\begin{array}{r}{\mathrm{\nabla}}^{2}{\overline{E}}_{y}-\mu \epsilon \frac{{\mathrm{\partial}}^{2}{\overline{E}}_{y}}{\mathrm{\partial}{t}^{2}}=\mu \frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{t}^{2}}[{\overline{P}}_{\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(\overline{r},t)]\end{array}$$(9)*E *_{x} and *E *_{z} are similar. For the TE mode, if you omit the guided mode and the radiation mode coupling, there is the following equation (10):

$$\begin{array}{r}{E}_{y}(r,t)=\frac{1}{2}\sum _{m}{A}_{m}(z){E}_{y}^{(m)}(x){e}^{i(\omega t-{\beta}_{m}z)}+c.c.\end{array}$$(10)where *m* is the number of the inherent mode, *c*.*c*. is the complex conjugate, and the inherent mode field satisfies the non-disrupting sex wave equation. The non-disrupting sex wave equation can be expressed as following:

$$\begin{array}{r}(\frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{x}^{2}}-{\beta}_{m}^{2}){E}_{y}^{(m)}(r)+{\omega}^{2}\mu \epsilon (r){E}_{y}^{(m)}(r)=0\end{array}$$(11)where *ε*(*r*) = *ε*_{0}*n*^{2}(*r*), *n*(*r*) is the medium refractive index. Substituting (10) into (11), there is the following Equation (12):

$$\begin{array}{rl}& {e}^{i\omega t}\sum _{m}\left[\frac{{A}_{m}}{2}\left(-{\beta}_{m}^{2}{E}_{y}^{(m)}\right.\right.\\ & \left.+\frac{{\mathrm{\partial}}^{2}{E}_{y}^{(m)}}{\mathrm{\partial}{x}^{2}}+{\omega}^{2}\mu \epsilon (r){E}_{y}^{(m)}\right){e}^{i{\beta}_{m}z}\\ & \left.+\frac{1}{2}\left(2i{\beta}_{m}\frac{d{A}_{m}}{dz}+\frac{{d}^{2}{A}_{m}}{d{z}^{2}}\right){E}_{y}^{(m)}{e}^{i{\beta}_{m}z}\right]+c.c.\\ & =\mu \frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{t}^{2}}[{P}_{\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(r,t)]\end{array}$$(12)The sum of the first three terms of the equation is equal to zero, and the slow amplitude changes are similar. Omitting the second derivative, because the second derivative satisfies the following Equation (13), we get:

$$\begin{array}{r}\left|\frac{{d}^{2}{A}_{m}}{d{z}^{2}}\right|\ll {\beta}_{m}\left|\frac{d{A}_{m}}{dz}\right|\end{array}$$(13)Substituting (13) into (12), there is the following Equation (14):

$$\begin{array}{rl}& \frac{d{A}_{s}^{-}}{dz}{e}^{i(\omega t+{\beta}_{m}z)}-\frac{d{A}_{s}^{+}}{dz}{e}^{i(\omega t+{\beta}_{m}z)}-c.c.\\ & =-\frac{i}{2\omega}\frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{t}^{2}}{\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}[{\overline{P}}_{\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(\overline{r},t)]{E}_{y}^{(x)}dx\end{array}$$(14)Where superscript − and + respectively represent propagation in the -z direction and the +z direction. The perturbation of the dielectric constant can be expressed as the periodic fluctuation of refractive index, which can be expressed as the following Equation (15):

$$\begin{array}{r}{\overline{P}}_{\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(\overline{r},t)=\mathrm{\Delta}{n}^{2}(\overline{r}){\epsilon}_{0}\overline{E}(\overline{r},t)\end{array}$$(15)Since $\mathrm{\Delta}{n}^{2}(\overline{r})$is a scalar, as can be seen from Equation (8), the periodic structure can only couple the TE mode to the TE mode or the TM mode to the TM mode. However, it cannot couple the TE mode.

For the TE mode propagation, substitute (10) into (15) and (14), there is the following Equation (16):

$$\begin{array}{rl}& \frac{d{A}_{s}^{-}}{dz}{e}^{i(\omega t+{\beta}_{m}z)}-\frac{d{A}_{s}^{+}}{dz}{e}^{i(\omega t+{\beta}_{m}z)}-c.c.\end{array}$$(16)$$\begin{array}{rl}& =-\frac{i{\epsilon}_{0}}{4\omega}\frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{t}^{2}}\sum _{m}[{A}_{m}{\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}\mathrm{\Delta}{n}^{2}(x,z)]{E}_{y}^{(m)}(x){E}_{y}^{(s)}(x)dx{e}^{i(\omega t+{\beta}_{m}z)}\\ & +c.c.]\end{array}$$Suppose the period of perturbation *Δn*^{2}(*x*, *z*) is *Δ*, and $\frac{l\pi}{\mathrm{\Lambda}}\approx {\beta}_{s},$where *l* is the integer. So the Equation (17) can be obtained, which can be expressed as the following:

$$\begin{array}{r}\mathrm{\Delta}{n}^{2}(x,z)\approx \mathrm{\Delta}{n}^{2}(x)\sum _{j=-\mathrm{\infty}}^{+\mathrm{\infty}}{a}_{j}{e}^{\left(\frac{i2j\pi}{\mathrm{\Lambda}}\right)z}\end{array}$$(17)Substituting (17) into (16), there is:

$$\begin{array}{r}\frac{d{A}_{s}^{-}}{dz}=-\frac{i{\epsilon}_{0}}{4\omega}{A}_{s}^{+}{\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}\mathrm{\Delta}{n}^{2}(x)]{\left[{E}_{y}^{(s)}(x)\right]}^{2}dx{e}^{i(\frac{2l\pi}{\mathrm{\Lambda}}-{\beta}_{s})}\end{array}$$(18)Then the coupling equation by backward wave and forward wave at *l*-th harmonic can be expressed as the following Equation (19):

$$\begin{array}{r}\frac{d{A}_{s}^{-}}{dz}={K}_{c}{A}_{s}^{+}{e}^{-i2(\mathrm{\Delta}\beta )z}\end{array}$$(19)Similarly, there is the following Equation (20)

$$\begin{array}{r}\frac{d{A}_{s}^{+}}{dz}={K}_{c}{A}_{s}^{-}{e}^{i2(\mathrm{\Delta}\beta )z}\end{array}$$(20)In the Equation (20), *K*_{c} and *Δβ* can be expressed as the following forms:

$$\begin{array}{r}{K}_{c}=-\frac{i\omega {\epsilon}_{0}}{4}{a}_{t}{\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}\mathrm{\Delta}{n}^{2}(x)]{\left[{E}_{y}^{(s)}(x)\right]}^{2}dx\end{array}$$(21)$$\begin{array}{r}\mathrm{\Delta}\beta \equiv {\beta}_{s}-\frac{l\pi}{\mathrm{\Lambda}}\equiv {\beta}_{s}-{\beta}_{0}\end{array}$$(22)Where *K*_{c} is called as the coupling coefficient. Equations (19) and (20) are the coupled mode equations for the forward and backward modes propagating in the periodic waveguide. Due to the agreement of two modes, so the power of these two modes is conserved.

For fiber Bragg grating, the waveguide structure is shown in Figure 4. Grating length is *L*, the grating period is *Δ*. The amplitude of light from the incident fiber, the amplitude of the backward wave at the boundary is 0. Assume that the fiber Bragg grating distribution is a strict sine function, the refractive index can be expressed as

Figure 4 Fiber Bragg grating waveguide structure diagram

$$\begin{array}{r}n(z)=\overline{n}+\mathrm{\Delta}n(z)=\overline{n}+\delta n\mathrm{sin}(\frac{2\pi z}{\mathrm{\Lambda}})\end{array}$$(23)The coupling coefficient of the grating is $K=\frac{\pi \delta n\eta}{{\lambda}_{B}}$where *η* is the coefficient related to the mode energy in the remaining core. Here *η* = −*V*^{2} is approximated as the value of the fiber, *V* characterizes the modulus of the fiber transmission. Solving for Equations (19) and (20), the Equations (24) and (25) can be obtained:

$$\begin{array}{rl}& {A}_{s}^{-}(z){e}^{i\beta z}\\ & ={A}_{s}^{+}(0)\frac{iK{e}^{i\beta z}}{-\mathrm{\Delta}\beta \mathrm{sinh}(SL)+iS\mathrm{cosh}(SL)}\mathrm{sinh}\left[S(z-l)\right]\end{array}$$(24)$$\begin{array}{rl}{A}_{s}^{-}(z){e}^{i\beta z}& ={A}_{s}^{+}(0)\frac{{e}^{-i\beta z}}{-\mathrm{\Delta}\beta \mathrm{sinh}(SL)+iS\mathrm{cosh}(SL)}\\ & \left\{\left(\mathrm{\Delta}\beta \mathrm{sinh}\left[S(z-l)\right]+iS\mathrm{cosh}\left[S(z-l)\right]\right)\right\}\end{array}$$(25)Where $S=\sqrt{{\left|K\right|}^{2}-\mathrm{\Delta}{\beta}^{2}}$so the fiber Bragg grating reflectivity is obtained from (24) and (25), which can be expressed as:

$$\begin{array}{r}R=\left\{\begin{array}{l}\frac{{K}^{2}{\mathrm{sinh}}^{2}(SL)}{\mathrm{\Delta}{\beta}^{2}{\mathrm{sinh}}^{2}(SL)+{S}^{2}{\mathrm{cosh}}^{2}(SL)}\phantom{\rule{1em}{0ex}}({K}^{2}>\mathrm{\Delta}{\beta}^{2})\\ \frac{{K}^{2}{\mathrm{sin}}^{2}(SL)}{\mathrm{\Delta}{\beta}^{2}-{K}^{2}{\mathrm{cos}}^{2}(SL)}\phantom{\rule{1em}{0ex}}({K}^{2}<\mathrm{\Delta}{\beta}^{2})\end{array}\right.\end{array}$$(26)When the incident light wavelength is equal to the center wavelength of the fiber Bragg grating [Bragg wavelength]. *Δβ* = 0, there is:

$$\begin{array}{r}{\lambda}_{B}=2{n}_{eff}\mathrm{\Lambda}\end{array}$$(27)$$\begin{array}{r}R({\lambda}_{B})={\mathrm{tanh}}^{2}(KL)\end{array}$$(28)Where *n*_{eff} is called as the core effective refractive index.

**Results and discussion**

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.