Stimulated Brillouin Scattering （SBS） is a process that results from a form of coherent light-sound coupling^［1-2］. SBS has attracted much attention and been investigated extensively since it was first observed experimentally in 1964^［3］. The SBS occurring in optical fibers plays a very important role in optical telecommunication and optical sensing system. A strong pump wave is scattered when transmitting along optical fibers， and the Stokes waves are produced which frequency is downshifted. Because of the electrostriction in the optical fibers， the acoustic waves are formed running along the optical fiber. Since the acoustic wave is a kind of mechanical waves， the periodic variation of the refractive index along the optical fiber is formed when the acoustic wave propagates in the optical fiber. This spatial and temporal periodic modulation of the refractive index in optical fibers is known as a Brillouin Dynamic Grating （BDG） running along the fiber. BDG in optical fibers has been extensively studied due to its great potential that exploits acousto-optic interactions to create optical fiber sensing^［4］， variable optical delay^［5］， photonic chip^［6］， all-optical signal processing^［7］， high-precision optical spectrum analyzer^［8］ and so on.

Up to now， a lot of research work on the BDG in optical fibers has been reported both theoretically and experimentally. In 2008， SONG K Y^［9-10］ experimentally observed BDG in the polarization maintaining fiber， then his team successfully produced BDG in a single mode fiber， which has a narrower reflection bandwidth， large reflectivity， and is easier to control. In 2012， TAKIGAWA S and HORIGUCHI T^［11］ built a moving fiber gratings model to derived an approximate expression and realized an experimental observation in a coiled single mode fiber， and then experimentally generated BDG in a few mode fiber^［12］. It was also reported that the BDG was produced by using chaotic laser in a polarization maintaining fiber^［13］. Through reading the previous reference papers， it is concluded that the reported research work on the BDG in optical fibers mainly focused on the experimental study. This is because the refractive index modulation depends on the acoustic modes in optical fibers， the acoustic modes are very different from the optical modes in optical fibers which are determined by the waveguide structure and the material. So， the distribution of the acoustic modes is necessary for studying the BDG. Because the acoustic wave exists in the optical waveguide， there are many acoustic modes existing in the optical fiber. Thus， it is very difficult to model and theoretically study the BDG.

In this work， the acoustic modes in the single mode optical fiber are investigated and the BDG model is theoretically derived based on the SBS and elastic acoustic theory， the reflection spectrum of the BDG in single mode fibers is simulated， and the results are in agreement with the reported experimental results.

The BDG in a single mode optical fiber can be treated as a moving refractive index grating along the optical fiber which is stimulated by a coherent acoustic wave field induced by SBS. Fig.1 shows the conceptual illustration of the BDG operation in an optical fiber. Counter propagating waves， pump1 and pump2， have the same polarization and meet in the optical fiber， the frequency offset between two pump waves is set to the Brillouin frequency shift of the fiber. The acoustic wave is driven by the beating between the pump and stokes waves； the power of pump1 is greater than that of pump2； and a probe wave with the orthogonal polarization propagates in the direction of pump1 and is reflected by the BDG for detection^［14］.

The optical fields in Fig.1 can be represented as

$\tilde{E}(z,t)={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}}{\tilde{E}}_i(z,t)$（1）

${\tilde{E}}_i(z,t)=A_i{\mathrm{e}}^{\mathrm{j}\left(k_{i}z-{\omega }_{i}t\right)}+\mathrm{c}.\mathrm{c}$（2）

where $\tilde{E}(z,t)$ means the optical field varying with time and space， $z$ represents the propagation direction of optical wave， $t$ is the time when the optical wave moves along the propagation direction； ${\tilde{E}}_{\mathrm{1}}(z,t)$， ${\tilde{E}}_{\mathrm{2}}(z,t)$， ${\tilde{E}}_{\mathrm{3}}(z,t)$， ${\tilde{E}}_{\mathrm{4}}(z,t)$ represent the optical fields of pump1， pump2， probe and reflection waves， respectively； $A_i$ is the amplitude of optical fields， $k_i$ represents the wave vectors， ${\omega }_i$ represents the frequency of pump1， pump2， probe， reflection waves， respectively； $\mathrm{c}.\mathrm{c}$ means complex conjugation. $\tilde{\rho }(z,t)$ is acoustic field， and can be expressed as^［15］

$\tilde{\rho }(z,t)=\mathrm{\Delta }\rho {\mathrm{e}}^{\mathrm{j}\left(qz-\Omega t\right)}+\mathrm{c}.\mathrm{c}$（3）

where $\mathrm{\Delta }\rho$， $q$， $\Omega$ are the amplitude， wave vector， frequency of acoustic wave， respectively.

The frequency of acoustic wave equals Brillouin frequency shift $\Omega$， which is given by

$\Omega =\frac{\mathrm{2}v}{c/n_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\omega }_{\mathrm{1}}$（4）

where $v$ is the velocity of acoustic wave， $c$ is the velocity of light in vacuum， ${\omega }_{\mathrm{1}}$ denotes the frequency of pump wave 1， and $n_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective refractive index at the pump wavelength.

Applying Newton's second law to the volume element of fiber material， mechanics equation can be expressed by

$\nabla P+\Gamma {\rho }_{\mathrm{0}}u+\rho \frac{\partial u}{\partial t}=\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\epsilon }}_{\mathrm{0}}{\mathrm{\gamma }}_{\mathrm{e}}\nabla ({\tilde{E}}^{\mathrm{2}})$（5）

where $P$ is the pressure per unit volume， $\Gamma$ means damping parameter， ${\rho }_{\mathrm{0}}$ is the material density of medium， $u$ is elastic velocity with propagation direction of the acoustic wave， $\rho$ means the density after the propagation of acoustic wave. ${\epsilon }_{\mathrm{0}}$ is permittivity of vacuum and ${\gamma }_{\mathrm{e}}$ is electrostriction constant. The source term on the right-hand side of Eq. （5） consists of the divergence of the electrostrictive force. The continuity equation of elasticity can be given by^［16］

$\nabla \cdot u+\frac{\mathrm{1}}{{\rho }_{\mathrm{0}}}\frac{\partial \rho }{\partial t}=\mathrm{0}$（6）

We substitute Eq. （6） into the Eq. （5）， one can obtain the result

${\nabla }^{\mathrm{2}}P-\Gamma \frac{\partial \rho }{\partial t}-\frac{{\partial }^{\mathrm{2}}\rho }{\partial t^{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{2}}{\epsilon }_{\mathrm{0}}{\gamma }_{\mathrm{e}}{\nabla }^{\mathrm{2}}({\tilde{E}}^{\mathrm{2}})$（7）

The adiabatic modulus $M$ is expressed as

$M={\rho }_{\mathrm{0}}\frac{\partial P}{\partial \rho }$（8）

Then Eq. （7） can be formed as

${\nabla }^{\mathrm{2}}\rho -\frac{{\rho }_{\mathrm{0}}}{M}\Gamma \frac{\partial \rho }{\partial t}-\frac{{\rho }_{\mathrm{0}}}{M}\frac{{\partial }^{\mathrm{2}}\rho }{\partial t^{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{2}}\frac{{\rho }_{\mathrm{0}}}{M}{\epsilon }_{\mathrm{0}}{\gamma }_e{\nabla }^{\mathrm{2}}({\tilde{E}}^{\mathrm{2}})$（9）

Based on the theory of elastic-acoustic， we can obtain

$\frac{{\rho }_{\mathrm{0}}}{M}=\frac{\mathrm{1}}{v^{\mathrm{2}}}$（10）

Then Eq. （7） can be written as

${\nabla }^{\mathrm{2}}\rho -\frac{\mathrm{1}}{v^{\mathrm{2}}}\Gamma \frac{\partial \rho }{\partial t}-\frac{\mathrm{1}}{v^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}\rho }{\partial t^{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{2}{v}^{\mathrm{2}}}{\epsilon }_{\mathrm{0}}{\gamma }_{\mathrm{e}}{\nabla }^{\mathrm{2}}({\tilde{E}}^{\mathrm{2}})$（11）

Suppose the two pump waves are all in x polarization， while the probe and the reflected waves are polarized in y direction， therefore there are only two terms left in the drive term of the acoustic field. The solution of equation under steady state condition is

$\mathrm{\Delta }\rho =\frac{\mathrm{1}}{v^{\mathrm{2}}}{\epsilon }_{\mathrm{0}}{\gamma }_{\mathrm{e}}{A}_{\mathrm{1}}{A}_{\mathrm{2}}$（12）

The equation for the velocity of acoustic wave is conveniently expressed in terms of the compressibility $C$

$v^{\mathrm{2}}=\frac{\mathrm{1}}{C{\rho }_{\mathrm{0}}}$（13）

Brillouin scattering of optical wave by acoustic wave can be treated theoretically by considering the time-varying change $\mathrm{\Delta }\tilde{\epsilon }$ in the dielectric constant induced by the density variation $\mathrm{\Delta }\tilde{\rho }$. It is usually assumed that $\mathrm{\Delta }\tilde{\epsilon }$ scales linearly with $\mathrm{\Delta }\tilde{\rho }$， so that

$\mathrm{\Delta }\tilde{\epsilon }=\frac{\partial \epsilon }{\partial \rho }\tilde{\rho }={\mathrm{\gamma }}_{\mathrm{e}}\frac{\tilde{\rho }}{{\rho }_{\mathrm{0}}}$（14）

Substituting Eq. （3）， Eq. （12） into Eq. （14）， one can obtain the result

$\mathrm{\Delta }\tilde{\epsilon }=(\mathrm{C}{\mathrm{\epsilon }}_{\mathrm{0}}{{\mathrm{\gamma }}_{\mathrm{e}}}^{\mathrm{2}}{A}_{\mathrm{1}}{A}_{\mathrm{2}}){\mathrm{e}}^{\mathrm{j}\left(qz-\Omega t\right)}$（15）

According to the third-order nonlinear effect， the refractive index can be defined as

$\tilde{n}=\sqrt[]{\mathrm{R}\mathrm{e}\left\{\mathrm{1}+{\chi }^{(\mathrm{1})}+{\chi }^{(\mathrm{3})}{\left|E(z,t)\right|}^{\mathrm{2}}\right\}}$（16）

The perturbation of refractive index associated with nonlinear effect can be expressed by

$\mathrm{\Delta }\tilde{n}=\frac{\mathrm{1}}{\mathrm{2}{n}_{\mathrm{e}\mathrm{f}\mathrm{f}}}\mathrm{R}\mathrm{e}({\chi }^{(\mathrm{3})})$（17）

The fluctuation in the susceptibility is then given by$\mathrm{\Delta }\tilde{\chi }=\mathrm{\Delta }\tilde{\epsilon }$， Eq. （17） can be rewrite as

$\mathrm{\Delta }\tilde{n}=\frac{\mathrm{1}}{\mathrm{2}{n}_{\mathrm{e}\mathrm{f}\mathrm{f}}}\mathrm{\Delta }\tilde{\epsilon }=\frac{{\mathrm{\epsilon }}_{\mathrm{0}}{{\mathrm{\gamma }}_{\mathrm{e}}}^{\mathrm{2}}C}{\mathrm{2}{n}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{A}_{\mathrm{1}}{A}_{\mathrm{2}}{\mathrm{e}}^{\mathrm{j}\left(qz-\Omega t\right)}$（18）

Ignoring the imaginary part and only using real quantities， the refractive index variation of BDG can be obtained

$\mathrm{\Delta }\tilde{n}=\frac{{\mathrm{\epsilon }}_{\mathrm{0}}{{\mathrm{\gamma }}_{\mathrm{e}}}^{\mathrm{2}}\mathrm{C}{A}_{\mathrm{1}}{A}_{\mathrm{2}}}{\mathrm{2}{n}_{\mathrm{e}\mathrm{f}\mathrm{f}}}\mathrm{c}\mathrm{o}\mathrm{s}(\Omega t\pm qz)$（19）

Then the effective refractive index of fiber can be expressed as

$n=n_{\mathrm{e}\mathrm{f}\mathrm{f}}+\mathrm{\Delta }\tilde{n}$（20）

It is simplified as

$n=n_{\mathrm{e}\mathrm{f}\mathrm{f}}+V\overline{\delta n_{\mathrm{e}\mathrm{f}\mathrm{f}}}\mathrm{c}\mathrm{o}\mathrm{s}(\Omega t\pm \frac{\mathrm{2}\mathrm{\pi }}{\Lambda }z)$（21）

Where $V$ is the fringe visibility of the refractive index change， $\overline{\delta n_{\mathrm{e}\mathrm{f}\mathrm{f}}}$ means the index change spatially averaged over a grating period， $\Lambda$ is grating period， which is actually wavelength of acoustic wave. It should be noted that the effective refractive index described by Eq. （21）， is identical to the fiber Bragg grating.

According to the coupled-mode theory of the fiber Bragg grating， the amplitude reflection coefficient can be simply given by

$r=\frac{-\kappa \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\sqrt[]{{\kappa }^{\mathrm{2}}-{\widehat{\sigma }}^{\mathrm{2}}}L)}{\widehat{\sigma }\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\sqrt[]{{\kappa }^{\mathrm{2}}-{\widehat{\sigma }}^{\mathrm{2}}}L)+i\sqrt[]{{\kappa }^{\mathrm{2}}-{\widehat{\sigma }}^{\mathrm{2}}}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\sqrt[]{{\kappa }^{\mathrm{2}}-{\widehat{\sigma }}^{\mathrm{2}}L})}$（22）

where $\kappa$， $\widehat{\sigma }$ are AC， DC self-coupling coefficients， $L$ means the effective length of the existing Brillouin grating^［17］. They are expressed as respectively

$L=\left(t_{\mathrm{P}\mathrm{1}}+t_{\mathrm{P}\mathrm{2}}\right)c/\mathrm{4}{n}_{\mathrm{e}\mathrm{f}\mathrm{f}}$（23）

$\kappa =\frac{\mathrm{\pi }}{\mathrm{2}\lambda n_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\epsilon }_{\mathrm{0}}{{\mathrm{\gamma }}_{\mathrm{e}}}^{\mathrm{2}}C{A}_{\mathrm{1}}{A}_{\mathrm{2}}$（24）

$\widehat{\sigma }=\mathrm{2}\mathrm{\pi }{n}_{\mathrm{e}\mathrm{f}\mathrm{f}}\mathrm{\Delta }f/c$（25）

where $t_{P\mathrm{1}}$， $t_{P\mathrm{2}}$ are the pulse widths of pump1 and pump2. $\mathrm{\Delta }f$ is the frequency shift of the probe wave from the center wavelength of reflection spectrum.

The reflectivity of fiber grating is expressed as

$R=|r{|}^{\mathrm{2}}=\frac{\mathrm{s}\mathrm{i}\mathrm{n}{\mathrm{h}}^{\mathrm{2}}(\sqrt[]{{\kappa }^{\mathrm{2}}-{\widehat{\sigma }}^{\mathrm{2}}}L)}{\mathrm{c}\mathrm{o}\mathrm{s}{\mathrm{h}}^{\mathrm{2}}(\sqrt[]{{\kappa }^{\mathrm{2}}-{\widehat{\sigma }}^{\mathrm{2}}}L)-\frac{{\widehat{\sigma }}^{\mathrm{2}}}{{\kappa }^{\mathrm{2}}}}$（26）

The maximum reflectivity can be expressed as

$R_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{t}\mathrm{a}\mathrm{n}{\mathrm{h}}^{\mathrm{2}}(\kappa L)$（27）

The BDG moves running along the optical fiber at the acoustic speed of v， then the reflective wave is influenced by the Doppler effect， thus the frequency of reflective wave downshifts which is given by

$\omega \mathrm{\text{'}}={\omega }_{\mathrm{p}}{\left[\raisebox{1ex}{$\left(\frac{c}{n_{\mathrm{e}\mathrm{f}\mathrm{f}}}+v\right)$}\!\left/ \!\raisebox{-1ex}{$\left(\frac{c}{n_{\mathrm{e}\mathrm{f}\mathrm{f}}}-v\right)$}\right.\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}$（28）

where $\omega \mathrm{\text{'}}$ is the frequency of reflection wave， ${\omega }_{\mathrm{p}}$ is the frequency of pump wave， when the grating is away from the observer， $v$ is negative， the frequency of stokes wave downshifts， on the contrary， $v$ is positive， the anti-stokes wave whose frequency upshifts is produced.

SBS occurs in a single mode fiber whose parameters are shown in Table 1. In our simulation， two short pulses are utilized as the pump waves and are initially with powers of 0.1 W and width of 2 ns. Fig. 2 shows the distribution of acoustic mode. It can be seen that all kinds of acoustic modes are mainly confined in the core^［18-19］. Thus， the optical and the acoustic waves are overlapped well in the core region of the single mode fiber to generate and enhance the BDG. Fig. 3 describes the reflection spectrum of the BDG in the single mode fiber. Because the BDG is a kind of weak fiber grating， the spectral width and the reflectivity are relatively small. The maximum reflectivity is 2.41×10^-11， with a spectral width of 3.5×10^-4 nm. The Brillouin frequency shift is 11.123 1 GHz； the Doppler frequency shift is 11.122 8 GHz； the difference between them is 0.319 MHz. The Main reason for the error is due to the calculation accuracy. Fig. 4 shows the reflection spectrum using Brillouin frequency shift and Doppler frequency shift as the frequency shifts of reflected wave. It is clear that the frequency of reflected wave is shifted downward due to the Doppler effect， and the values of the Doppler frequency shift is equal to the Brillouin frequency shift.

Fig. 5 depicts the temporal and spatial distribution of the refractive index perturbations. It clearly indicates that there are periodic variations of perturbation along with the temporal and spatial change. The spatial variation indicates that the distribution of the refractive index is similar to that of the Bragg grating， and the temporal variation shows that the BDG is a sort of moving fiber Bragg grating.

In the SBS， there is a Brillouin frequency shift between two pump waves， and the energy is transferred from the high frequency to the low frequency. According to the Eq. （19）， the multiplication of electric field amplitude of two pump waves modulates the refractive index. So， the change of pump1 power plays the same part as the pump2 power. The variation of BDG's reflectivity concerning the pump1 power is shown in Fig. 6 and Fig. 7. Fig. 6 depicts the reflectivity as a function of the pump power for a pulse width of 2 ns. It is obvious that the reflectivity of the BDG varies significantly with the pump power as more pump power is applied， but the change in the spectral width is negligible. The sidelobe of the reflection spectrum is caused by the non-uniform distribution of the refractive index in the optical fiber. The variation of refractive index at both ends of the BDG produces the sidelobe. The perturbation of the refractive index is relatively weak when the power of the pump wave is small. As the power increases， the perturbation becomes violent， which makes the sidelobe gradually increases. We can see from Fig. 7 that when the pump power increases to 30 W， the peak reflectivity also enhanced， but the relationship between the power and reflectivity is non-linear and grows rapidly. When the pump power reaches 30 W， the maximum reflectivity can be as high as 2.17×10^-6. It is because the stimulated Brillouin amplification becomes violent as the power of pump1 increases， the refractive index perturbations increase with the growth of pump power 1， the grating intensity is greatly extended， this enhances the reflection of BDG. These results are in agreement with the similar experiments^{［10-11，14］}.

The interaction time of the two pump waves extends as the pulse width increases， and the grating length also enhances. More energy of the incident wave is converted into the reflected wave as the grating length increases. In SBS， the increment in grating length induces actually a growth in the range of SBS effect. According to Eq. （23）， the pulse width of pump1 and pump2 affects the length of BDG. As shown in Fig. 8， the pulse width also affects the reflectivity and the bandwidth of the reflection spectrum. The grating length increases with the growth of the pulse width， and the increment of the grating length leads to an enhancement of reflectivity， but the spectral width decreases accordingly. The refractive index varies with the position of BDG. The growth of the grating length induced the non-uniform distribution of the refractive index. The sidelobe becomes more significant. Fig. 9（a） shows that the spectral width as a function of the pump pulse width， the spectral width decreases from 3.5×10^-4 nm to 1.2×10^-4 nm with growth of pulse width. Fig. 9（b） depicts that the peak reflectivity as a function of the pump pulse width， when the pulse width is 10 ns， the maximum reflectivity is 7.16×10^-9. The results agree well with the theory of the weak fiber grating^［20］. It also can be presumed that BDG is reinforced when continuous wave is utilized as pump wave， because the Brillouin grating can exist in the whole fiber.

Fig. 10 and Fig. 11 describe the reflectivity as a function of core diameter. It can be seen that the reflectivity decreases with the growth of core diameter， but the alteration of the spectral width can be ignored. BDG is also enhanced with the reduction of core diameter due to the tight confinement of both optical and acoustic fields in the fiber core. This also suggests that Brillouin interactions could be greatly magnified in the small core of the optical fiber.

It is known from the above analysis that the reflectivity of the BDG is enhanced by tailoring the pump power， the pulse width and the core diameter. The maximum reflectivity can reach up to 6.63×10^-5 with the pump power of 30 W， the pulse width of 10 ns and the core diameter of 8 μm. The reflectivity is still so small that it limits the application of BDG. For example， BDG is hard to introduce into the optical fiber amplifier because the amplifier needs fiber grating with high reflectivity to improve the pump conversion efficiency. But the BDG can be used as the optical fiber sensors in sensing. The BDG with low reflectivity can increase the capacity of the sensor， the wave at the same wavelength is not fully reflected and can be reused in a fiber.

In summary， the model of BDG is established， and the expression of refractive index perturbation is derived in this paper， the reflection spectrum of BDG in single mode fibers is investigated， the inference that Brillouin frequency shift is from the Doppler frequency shift is verified by the simulation result. The reflectivity is positive correlation with the pump power and pulse width， but it is negative correlation to the core diameter. The spectral width is not affected by the power of pump waves， but it is negative correlation to the pulse width and core diameter. The analysis results are in keeping with some experimental reports. BDG offers great promise for a range of applications， because it can be conveniently produced at any positions with variable length， which can be harnessed as optical communication， optical storage， and so on.