Influence of Refractive Index Distribution on Brillouin Gain Spectrum in GeO 2-Doped Optical Fibers

，*，，，，

1.Key Laboratory of Space Photoelectric Detection and Perception of Ministry of Industry and Information Technology，College of Astronautics，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，P.R，China；

2.College of Science，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，P.R.China；

3.Shanghai Aircraft Design and Research Institute，Shanghai200120，P.R.China

Abstract:GeO2 is commonly used as dopant to adjust the refractive index profile（RIP）and the acoustic velocity profile（AVP）in the fiber，thereby forming different Brillouin gain spectrum（BGS）characteristics such as Brillouin gain，acoustic mode number and peak intensity difference.When an optical fiber is used in optical fiber sensing or communication system，its BGS characteristics may play an important role in determining the performance of the system.In this paper，finite element analysis（FEA）method is used to study the influence of refractive index distribution and its corresponding AVP on the BGS in step-index，graded-index，and complex-index optical fibers.A new method has also been proposed to efficiently discriminate acoustic mode solution and obtain the new and full images of total Brillouin gain and acoustic modes number of the fiber as a function of the refractive index distribution，considering the influence of changing the refractive index difference and the geometric size simultaneously.For each type of optical fiber，the recommended parameter range is provided for optical fiber sensing and optical fiber communication.Moreover，the suitable optical fiber with close peak intensity in its multi-peak BGS is explored and achieved，which can be used in Brillouin beat spectrum detection systems to improve sensing accuracy.

Key words：Brillouin gain spectrum；optical fiber；acoustic mode；refractive index；finite element analysis（FEA）

0 Introduction

Stimulated Brillouin scattering（SBS）is a nonlinear phenomenon caused by the interaction between optical and acoustic waves［1-3］.Several acoustic modes can be excited in a single-mode fiber（SMF）which has a complex refractive index distribution.This determines the corresponding Brillouin gain spectrum（BGS）has multi-Brillouin peaks［4-5］.These properties effectively decrease the Brillouin gain in unit area within the core of the fiber，therefore，improving the SBS threshold which allows higher input optical power in the fiber to reach longer transmission distance and higher signal to noise ratio（SNR）.

Many studies have been carried out to research the characteristics of different acoustic modes or the interaction between acoustic modes and optical fundamental mode in different kinds of fibers［5-10］.Some of these characteristics have been applied beneficially in distributed fiber sensing systems［4-5，11-16］.For example，the large effective area fiber（LEAF，made by Corning Inc.）is a kind of non-zero dispersion-shifted fiber（NZDSF）with several acoustic modes，which can be utilized as the sensing fiber in Brillouin optical time domain analysis（BOTDA）system［4］and Brillouin optical time domain reflectometry（BOTDR）system［12］.In our previous study［12］，the homodyne BOTDR system is based on measuring the peak power of Brillouin beat spectrum（BBS）in LEAF.Actually，due to their diverse refractive index distribution，different kinds of fibers have distinct BGS characteristics such as Brillouin gain，acoustic mode number and peak intensity difference.That may affect the measurement accuracy and efficiency of the optical fiber sensing system，and the transmission performance of optical fiber communication systems.

With the advancement of fiber-optic communication and Brillouin sensing technology，it is necessary to know the comprehensive relationship between the refractive index distribution and the BGS.In the previous studies，the effect of dopant concentration on the BGS was studied.These dopants include commonly used GeO2and F［7，17-19］，as well as some special materials such as As2Se3，tellurite，and bismuth［3，20-22］.The BGSs of fibers have been explored by changing the geometry or refractive index difference［23-24］.However，there is no related research performed from the perspective of changing the fiber refractive index difference and geometric size simultaneously.To our knowledge，the effects of numerous possibilities for refractive index distribution on the shape of BGS have not been considered nor investigated.Until now，a full image of the effect of the refractive index profile on the BGS is still missing，although various commercial GeO2-doped fibers have been used in a large number of applications.

To this end，all the possibilities of the refractive index distribution of the GeO2-doped optical fibers are considered，including the influence of changing the refractive index difference and the geometric size simultaneously.We present the new and full images of Brillouin gain and acoustic modes number of the fiber as a function of the refractive index distribution.The influence of refractive index distribution on the BGS characteristics in common GeO2-doped fibers（including step-index，graded-index and complex-index optical fibers）are analyzed by using finite element analysis（FEA）method.

When using FEA software to solve the acoustic field distribution in the fiber，it is usually necessary to set a larger desired number of eigenvalues than the actual acoustic mode number.For example，when simulating an acoustic multimode fiber，it is usually need to set the desired number of eigenvalues to dozens to find two or three circularly symmetric solutions.When simulating a fiber with a specific structure，we can artificially observe these dozens of solutions and eliminate the pseudo-solutions to obtain circularly symmetric acoustic modes.However，if a study of a large quantity of fiber structure parameter combinations is required，artificially identifying the solutions is a very inefficient strategy.In order to solve this problem，an innovative method will be proposed that can be used to quickly determine the circular symmetry of the acoustic field.

For the three kinds of common fibers，the relationship between refractive index distribution and BGS characteristics（total Brillouin gain，acoustic mode number，and peak intensity difference）will be discussed.Furthermore，in order to realize more effective homodyne BOTDR system based on BBS detection［12］，the suitable fiber structural parameters to achieve multi-peak BGS with close peak intensity are also studied.The new results obtained in this paper may benefit in designing new optical fiber and choosing the optimal optical fiber to be used in sensing and communication systems.Besides，the analysis described in this paper may be of significant value in the search for high Brillouin gain fibers with multi-peak Brillouin spectrum，and provides a guide for designing suitable solid optical fibers applied in the Brillouin scattering based fiber sensing systems.

1 Principles

1.1 Brillouin gain spectrum

According to previous studies［10，25-26］，the occurrence of Brillouin scattering comes along with the propagation of acoustic modes during the SBS process.The SBS process is governed by the following set of two coupled equations［1，26］

whereISandIPare the Stokes and pump waves，respectively，andαis the linear loss coefficient of optical waves.When more than one Brillouin scattering active acoustic modes exist in fibers，each acoustic mode is responsible for a spectral feature in the BGS.The Brillouin gain spectrum corresponding to themth acoustic mode can be expressed as

whereνis the frequency of scattered light，νpis the incident light frequency，andνmandωmare the frequency shift and the full width at half maximum（FWHM）of themth peak in the BGS，respectively.The peak Brillouin gain of themth acoustic mode is

wherep12is the respective component of the electrostriction tensor，cthe velocity of light，λthe optical wavelength of the incident light in vacuum，neffthe effective refractive index of the fundamental optical mode，ρ0the mean value of the material density of the fiber，andfAmthe fraction by which the SBS gain is reduced if themth acoustic and optical modes do not fully overlap inside the fiber［27-28］.

whereAeffandAaomare named optical effective area and acousto-optic effective area，shown as

Indeed，dopant concentration in the fiber will affectp12，ρ0，andωm.However，the changes in these parameters are quite small and do not influence the shape of the BGS.For example，when the doping concentration of GeO2is from 2%to 10%，the correspondingωmis from 35 MHz to 45 MHz［18］.In this work 40 MHz is utilized as FWHM of each peak in BGS to calculate the peak Brillouin gain as the BGS linewidths obtained from the fiber structure parameters involved are around this value［29-30］.

As shown in Eq.（3），each acoustic mode in a fiber corresponds to a Brillouin gain peak with the Lorentz shape.If there is more than one acoustic mode in a fiber，the shape of the BGS maybe has multi-peaks.The total BGS of the fiber is the sum of the BGSs of individual acoustic modes，and can be expressed as

In Eq.（7），E（x，y）andum（x，y）are horizontal profiles of the fundamental optical and themth acoustic mode of the fiber，respectively.They can be calculated from the following 2-D scalar optical propagation Eq.（9）and mechanical Eq.（10）equations by using FEA method［8］.

where?2tis the transverse Laplacian operator in the（x，y）plane，nthe refractive index of the fiber，andβacothe propagation constant of the acoustic mode and given asβaco=4πneff/λ.ωacois the angular frequency of the acoustic wave，ωaco=2πνm，which can be calculated by FEA method.Vlis the longitudinal sound velocity which can be obtained from the following empirical relation［2］

where the 5 944 m/s is the longitudinal sound velocity in pure silica.When dopant is GeO2，the refractive index of the core is increased and the sound velocity in the longitudinal direction is lowered.The refractive indices of the fibers referred to herein are all adjusted by doping GeO2.So，the refractive index distribution we consider is actually determined by the material and geometry of the fiber［3，7］.

According to Eq.（3）to Eq.（11），when the refractive index distribution of a fiber is known，we can obtain the optical mode and acoustic modes distribution，as well as the number of acoustic modes and the BGS of the fiber.Therefore，the fiber parameters such as core radius and refractive index difference can be adjusted，and thus we can obtain a comprehensive view of the relationship between the number of acoustic modes and the Brillouin gain as a function of refractive index distribution.

The shape of BGS of some kind fibers has multi-peaks，and thus the commonly used peak Brillouin gain，gm，in Eq.（4）cannot effectively reflect the level of total Brillouin gain of fibers，as mentioned above.In this paper，a new parameter is defined，the total Brillouin gain，gB_total，to evaluate the total Brillouin gain of different GeO2-doped fibers.After obtaining the BGS of a fiber，G(ν)，the total Brillouin gain of the fiber can be obtained by calculating the integral of BGS on the frequency domain.It can be expressed as

It should be noted that both of the values and units ofgB_totalandgmare different.For example，when the optical wavelength of the incident light is 1 550 nm，the peak Brillouin gain of SMF-28 fiber，gm，obtained with Eq.（4）is 1.63×10-11m/W，while its corresponding total Brillouin gain，gB_total，is 1×10-3m/（W·s）which is obtained by Eq.（12）.

1.2 Acoustic mode solution discriminating

In the process of solving the acoustic mode solution by FEA，in order to effectively discriminate the actual acoustic mode solution from dozens of eigenvalues，a novel acoustic mode solution discriminating method is proposed.

In general，the cross section of the fiber is defined as thex-ycoordinate plane and the fiber axis is defined as the coordinate origin in FEA calculation.The obtained acoustic fieldu（x，y）is output in the form of a grid.Then we convertu（x，y）from a Cartesian coordinate system to a polar coordinate systemu（r，θ）byr=andθ=arctan(y/x).And we divide the polar coordinate plane from the origin intoNequal sectors.In a circularly symmetric acoustic mode，the sum ofu（r，θ）in each sector should be equal.In other words，the sum ofu（r，θ）in theith andjth regions（1≤i＜j≤N）should be nearly equal，which is

whereumax(r，θ)is the maximum of all acoustic mode solutions andδ1the small constant.We can take the value ofδ1as 10-5umax(r，θ)in the calculation process.

We can exclude some pseudo solutions ofu(r，θ)by the above method.In the remaining solutions，we estimate the degree of numerical dispersion over each radius and obtain the solution of circular symmetric distribution.We selectKradius（r1，r2，…，rl，…，r K，1≤l≤K）by a certain step size on the plane of the acoustic mode solution to calculate the standard deviation ofu（r l，θ）on each radius ofrl.In a circularly symmetric acoustic mode solution，these standard deviations should satisfy

whereMis the total number of acoustic field data points on the radiusr l，the average ofu(r l，θ)on the radiusr l，δ2the small constant and we take the value ofδ2as 10-4in the calculation process.If both Eqs.（13，14）are satisfied，it can be inferred that the acoustic field is circularly symmetric.

2 Influence of Refractive Index Distribution on BGS

According to refractive index distribution，solid optical fibers generally can be divided into three categories including step-index optical fiber（SIOF），graded-index optical fiber（GIOF），and complex-index optical fiber（CIOF）.The effects of dopant concentration of GeO2on the BGS in singlemode fibers have been studied［7，17-19］.However，these studies only discuss the cases where only one parameter changes and one parameter cannot describe the overall refractive index distribution.In this section，the effects of refractive index difference and geometry on the shape of the BGS are studied.We will consider the myriad possibilities of combining these two variables，and obtain the full image of the effect of the refractive index profile on the BGS.We will present the relationship between refractive index distribution and BGS characteristics（Brillouin gain，acoustic mode number，and peak intensity difference），and we also discuss whether there are suitable fiber structural parameters to achieve multi-peak BGS with close peak intensity in each fiber category.

2.1 Verification of simulation method

To illustrate the accuracy of our simulation method based on FEA method，we have simulated a graded-index fiber doped with GeO2in Ref.［10］.This paper gives the refractive index profile and the measurement results of the BGS of the fiber.Fig.1 shows the actual refractive index profile of the fiber.The measured BGS of the fiber and our simulation results are shown in Fig.2.

Fig.1 RIP of fiber in Ref.[10]

Fig.2 Simulated and measured BGSs of fiber

From Fig.2，we can see that the BGS of the fiber obtained by the FEA method is very consistent with the experimental measurement results.The frequency shift and relative power comparison of Peak 1—3 of BGSs in Fig.2 is given in Table 1.The relative error of the Brillouin frequency shift between simulation and experiment is within 0.5%，which indicate that our results are in good agreement with the experiment.The simulation errors relative to the measured relative power of the three peaks are 0%，7.1%and 8.8%，respectively.Therefore，the difference between the simulation and the experimental BGSs is very small.These errors may be caused by imperfections in the production process during fiber drawing.

Table 1 Comparison of simulated and measured BGSs

2.2 Step-index optical fiber（SIOF）

The standard single mode fiber（SMF-28）can be considered as a representative of SIOFs.The RIP and AVP of SMF-28 are shown in Fig.3.The parameters of SMF-28 at wavelength 1 550 nm can be obtained by referring to previous paper［31］.n1andn2in Fig.3 are the core and cladding refractive indices，respectively.ais the core radius.The AVP is determined by the RIP according to Eq.（11）.In the following，when talking about the change of the refractive index，the acoustic velocity of the fiber is regulated accordingly.

Fig.3 RIP and AVP of SMF-28 at wavelength 1 550 nm

In the case of general SIOFs，the refractive index distribution of the optical fiber depends on two factors，the core radius and the refractive index difference.In order to describe the changes in BGS with refractive index distribution，we firstly determine the suitable range of core radiusaand the refractive index differenceΔ，which can be expressed as

When considering the core radius of the fiber，taking into account the limitations of the manufacturing process and taking the SMF-28 as a reference，we set the core radiusafrom 1μm to 4.5μm.As for the range ofn1，sincen1must be greater thann2in a solid fiber，we set the minimum ofn1as 1.446 which is slightly larger thann2，as shown in Fig.3.To ensure that the fiber is a single mode fiber，when changing the core radius，the maximum value ofn1is limited based on the single mode cut-off conditionVc=2.405（Vcis optical fiber normalized frequency）at the corresponding radius.Furthermore，the refractive index differenceΔcan be determined by using the relationship between the refractive index difference and the fiber loss due to the Rayleigh scattering.The acceptable fiber loss is less than 0.5 dB/km.The fiber loss in a single-mode fiber doped with GeO2，αSR，can be expressed as［32］

According to Eq.（16），when the radius is less than 2μm，the core refractive indexn1can reach a maximum of 1.474 2.When the core radiusaincreases from 2μm，the maximum value ofn1decreases according to the single mode cut-off condition.Thus，the upper-right boundary of parameter space（a，n1）can be determined and it looks like an exponential decay curve.Therefore，the range of core radius and the core refractive index can be determined，as shown in Fig.4.By using the FEA method，the total Brillouin gaingB_totaland acoustic mode number can be obtained from Eq.（12）and Eq.（10），respectively.The profiles of total Brillouin gaingB_totaland acoustic mode number are shown in Fig.4（a）and Fig.4（b），respectively.We can obtain other parameters of BGS，such as Brillouin frequency shift and peak relative intensity.Since these results cannot be clearly presented in a 2-D graph，only the total Brillouin gain and the number of acoustic modes are displayed here.

Fig.4 Simulation results of SIOFs

From Fig.4（a），we can see that in the upper right region of（a，n1）combination，SIOFs have high Brillouin gain.Specially，the highest Brillouin gain area corresponds to the region where the core radius is from 1.4μm to 2μm and the core refractive index is from 1.467 to 1.474 2.In addition，as the core radius and refractive index increase，the number of acoustic modes guided in the fiber gradually increases.The distribution of the number of acoustic modes is basically consistent with the Brillouin gain distribution shown in Fig.4（a）.The numbers marked in each area in Fig.4（b）denote the numbers of acoustic modes guided in the fiber.It should be noted that，according to the single mode cut-off condition，the sector in the upper right corner is out of consideration.The lower left corner of Fig.4 shows the fiber with a small refractive index difference and a small diameter.From the simulation results，these fibers have only one acoustic mode，and the total Brillouin gain is also the smallest.In fact，the bending losses of these fibers are very large，so they have no practical value.In order to clearly show the relationship between the shape of BGS and the fiber parameters，we select one point in each region randomly as a representative example and present the BGS of these particular fibers except for the 4th spectrum，as shown in Fig.5.Since the total Brillouin gaingB_totalof the 4th spectrum is the largest in this paper，the first peak of the 4th spectrum is set as 0 d B，and all other BGSs in Fig.4 and below are plotted with respect to this value.

Fig.5 BGSs of SIOFs in four different areas

As can be seen from Fig.5，the 4th spectrum represents the BGS from the area④in Fig.4（b）.This BGS has the largest Brillouin peak gain.Since the frequency shift difference between the first acoustic mode and the second acoustic mode is small，the Brillouin peak contributed by the second acoustic mode（The corresponding frequency shift is 9.63 GHz）is enveloped by the first main peak of the BGS，and it only shows a minor protrusion.In this BGS，the third and fourth peaks are approximately 24 dB weaker than the first peak.For the 2nd and the 3rd spectra，we find that their total power gradually decreases compared to the 4th spectrum.As for the 1st spectrum，there is only one peak and the peak power is also the smallest.These results in Brillouin gain are consistent with that shown in Fig.4（a）.Specifically，the two peaks in the 2nd spectrum are closer in intensity，since its second peak is only 8.6 dB weaker than the first peak.The 3rd spectrum has three peaks，and its second and third peaks are 13.7 and 16.7 d B weaker than the first peak，respectively.

Generally，SBS is a harmful nonlinear optical phenomenon in the field of optical fiber communication，which attenuates the power of the signal，thereby reducing the communication distance and the system SNR［33-34］.For the application of step-index fiber in optical fiber communication，fibers with lower total Brillouin gaingB_totalperform better，as shown in the region from the yellow to light blue color in Fig.4（a）.This region corresponds to most of the area where the number of acoustic mode is 2，as shown in Fig.4（b）.Meanwhile，the fibers with high Brillouin gain，as shown in the upper right region in Fig.4（a），can be used as ideal sensing fibers.

2.3 Graded-index optical fiber（GIOF）

In GIOF，the refractive index of the core is variable.The refractive index decreases gradually from its maximum valuen1at the core center to its minimum valuen2at the core-cladding interface.The cladding usually has constant refractive index.Most GIOFs follow theg-type refractive index distribution which can be described as

whereais the core radius，andΔthe refractive index difference between the axis and the cladding.gdetermines the shape of the refractive index profile.Wheng→∞，the fiber is a step-index optical fiber.Wheng=1，2，the fiber types are triangular-index fiber and parabolic-index fiber，respectively.In this paper the parabolic index fibers are used as research object since the refractive index profile of most graded-index fibers is parabolic.RIP and AVP of parabolic index fiber are shown in Fig.6.

Fig.6 RIP and AVP of GIOF at wavelength 1 550 nm

As that shown in Fig.7，the ranges ofn1andaof GIOFs can be determined similarly as that discussed in SIOFs.The upper-right boundary of parameter space（a，n1）are calculated by considering the single mode cut-off conditionVc=3.518 at the corresponding radius and the acceptable maximum fiber loss of 0.5 dB/km［35］.The profiles of total Brillouin gaingB_totaland acoustic mode number are calculated from Eq.（12）and Eq.（10）and shown in Fig.7（a）and Fig.7（b），respectively.

Fig.7 Simulation results of GIOFs

From Fig.7（a），we can observe that the Brillouin gain distribution of graded-index fiber is similar to that of SIOFs.The GIOFs with core radius ranging from 2.25μm to 3μm and axis refractive index from 1.468 to 1.474 2 has a higher Brillouin gain.The numbers marked in each area in Fig.7（b）represent the numbers of acoustic modes guided in the fiber.Unlike the case of SIOFs，the single-mode cutoff condition isVc=3.518 in the parabolic-index fibers.The larger value ofVcresults in the larger range of the refractive index distribution in GIOFs.Combining Fig.7（a）and Fig.7（b），the fiber that guides more acoustic modes usually has a higher Brillouin gain.This is similar to the case of step-index optical fibers.Similar to the SIOFs in Section 2.2，the fibers corresponding to the lower left corner region have no practical value.

Similarly，in order to clearly show the relationship between the shape of BGS and the fiber parameters，we select one point in each region in a random way as a representative example and describe the BGS of these particular fibers.The five different BGSs are shown in Fig.8.

Fig.8 BGSs of GIOFs in six different areas

The 6th spectrum in Fig.8 is a typical BGS in the area⑥in Fig.7（b）.This BGS has six peaks.As the frequency shift increases，the intensity of the peaks decreases 2.9—7.3 d B one by one，and the intensity difference between two adjacent peaks decreases.The 5th spectrum has five peaks，and its shape is similar to that of the 6th spectrum.The 4th and the 3rd spectrums represent BGS in area④and area③in Fig.7（b），respectively.For the 2nd and the 1st spectra，we find that their total Brillouin gains are less than that of the 6th and 5th spectra.As for the 1st spectrum，there is only one peak and the peak power is the smallest.These results related to Brillouin gain are consistent with that in Fig.7（a）.

Similar to the case in Section 2.2，the optical fibers with low total Brillouin gaingB_totalare suitable to be utilized in the fiber communication system.As shown in Fig.7（b），this kind of fiber has two acoustic modes guided in the fiber core.Fibers with high Brillouin gain are more suitable to the Brillouin sensing system，and their parameter pairs（a，n1）are also in the upper right corner of Fig.7（a）.

2.4 Complex index optical fiber（CIOF）

In order to achieve the lower loss，the lower dispersion，the lower bending sensitivity or the larger optical mode field area，some CIOFs have appeared.Common CIOFs include dispersion compensating fiber，depressed cladding fiber，bend insensitive fiber，double-clad fiber，and large effective area fiber.In this section，Corning large effective area fiber（LEAF）and trench assisted fiber（TAF）are chosen as the typical fibers.

2.4.1 Large effective area fiber（LEAF）

The actual refractive index profile of LEAF is shown in Fig.9（a）［2］.For the convenience of calculation，RIP and AVP of LEAF are simplified as shown in Fig.9（b）.The simplified refractive index distribution of LEAF can be expressed as

Fig.9 RIP and AVP of LEAF

The difference between calculation results of the simplified and the actual situation is so small that can be neglected.

The triangle part is called inner core and has a radius ofa.The high refractive index difference outer-ring section ranges frombtoc.n1，n2，andn3are the refractive indices of the center，the outer-ring section，and the cladding，respectively.Whena=3.8μm，b=5μm，c=10μm，andn1，n2，andn3at wavelength 1 550 nm are 1.456 1，1.446 4，and 1.444（1.444 is the refractive index of fused silica at 1 550 nm［36］），respectively，we obtain the optical effective areaAeffand three acousto-optic effective areasand，as shown in Table 2.There are actually five peaks in our calculated BGS of LEAF，but the power of the 4th and 5th peaks is very low.Therefore，only the results of the first three peaks are given in Table 2 to compare with the results in Ref.［2］.We find that our results are consistent with that shown in Ref.［2］.

Table 2 Par ameter compar ison of actual and simulated fibers

In LEAF with a complex refractive index distribution，there are many parameters that can be adjusted，including three radiia，b，andcand two refractive indicesn1andn2.In order to make the discussion more reasonable and simpler，we expect to find some important parameters that have a greater impact on BGS in these parameters.We firstly explore the effect of the outer-ring section on BGS by changing its width and distance from the center.

The position of the outer-ring section is changed.Here the width 5μm remains unchanged.In other words，whenbchanges，c=（b+5）μm.Whenbchanges from 4μm to 12μm，the corresponding BGSs are shown in Fig.10.

Fig.10 BGSs of LEAFs when b changing from 4μm to 12μm

Then we change the width of the outer-ring section.The value ofcis changed while retaining the value ofbas 5μm.Whencchanges from 6μm to 15μm，the corresponding BGSs are similar to that shown in Fig.10，so the figures are not presented here.

Therefore，only changing the position and width of the outer-ring section has little effect on the entire BGS.The geometric parameters of the outer ring affect the BGS weakly due to a very small part of the optical mode energy in this area.

Based on the above discussion，we could find three parameters that have a significant effect on BGS in LEAFs，namely the refractive indicesn1andn2，and the radiusc.Whencchanges，aandbalso change according to the original ratio.According to the Fig.9（b），we setb=0.5canda=0.38c.In such a case，the refractive index distribution of the fiber is determined by the three variables ofn1，n2，andc.Here the total Brillouin gaingB_totaland the acoustic mode number actually are ternary functions.Therefore the result graph is a three-dimensional graph with the fourth dimension color-coded to indicate the magnitude of the value.The variation ranges of the three variables is shown in Table 3.

Table 3 Variation ranges of three variables

It can be seen from Table 3 that the range ofn1indicates that the inner core index difference changes from 0.07%to the value that causes the largest fiber loss of 0.5 d B/km.The range ofn2represents the outer-ring section grown out of nothing and its index difference increases from zero to 1.4%.The range ofcmeans that all radius sizes can be adjusted to between 50%and 150%of the original size.The total Brillouin gaingB_totalchanging withn1，n2，andcis shown in Fig.11.The acoustic mode number changing withn1，n2，andcis shown in Fig.12.The benefit of using a graph in the form of a cube is that it reflects the relationship of a function with three independent variables.These cube diagrams can fully demonstrate the effect of the refractive index distribution determined by the innumerable combination possibilities of these three variables on the total Brillouin gain and the number of acoustic modes.

Fig.11 Total Brillouin gain gB_total changing with n1,n2,and c

From surface of the cube shown in Fig.11（a），most areas have a high total Brillouin gaingB_total.This is because LEAFs have a larger core radius than that of SIOFs and GIOFs，resulting in a larger optical effective areaAeff.In the range ofcfrom 10μm to 15μm，n1from 1.472 to 1.475，andn2from 1.444 to 1.46，LEAFs have a higher total Brillouin gain.Fig.11（b）is the internal view of the three-dimensional figure，where the three orthogonal planes are corresponded toc=10μm，n1=1.46，andn2=1.454.We can see that the internal situation in total Brillouin gain is consistent with that on the surface，and the distribution of Brillouin gain is continuous without jumps or intervals.In the region ofn1=1.445 to 1.447 andn2=1.444 to 1.446，LEAFs have a very low Brillouin gain.The refractive index difference between the core and the cladding and the refractive index difference between the outer ring and the cladding are very small，and their ability to restrain the optical field is weak，resulting in the optical field leakage.Fibers with this defect cannot be actually used.In general，with respect to the refractive index difference，the total Brillouin gain is positively correlated withn1and negatively correlated withn2.

Fig.12 illustrates that in the case of this complex structure，most fibers have more than five acoustic modes.Less acoustic modes are possible if the refractive index difference in the inner-core is very small.It can be concluded that the LEAFs have multiple acoustic modes.This also makes LEAFs exhibit a higher total Brillouin gain than that of SIOFs and GIOFs.Due to the complexity of structure，the distribution of the acoustic mode number does not appear to be regular.Similarly，to clearly show the relationship between the shape of BGS and the fiber parameters，we select several examples of different combinations of parameters and the BGSs of these representative fibers.The results are shown in Fig.13 and Fig.14.In order to ensure the clarity of the figure，only six spectra are painted in Fig.13.The BGSs of fibers with more acoustic modes are shown in Fig.14.These BGSs here are normalized to 0 d B according to the maximum gain of the 4th spectrum in Fig.5.

Fig.12 Acoustic mode number changing with n1,n2,and c

Fig.13 BGSs of six LEAFs with different n1,n2,and c

Fig.14 BGSs with more acoustic modes

In Fig.13，the numbers marked on these spectra represent the acoustic mode number of these fibers.For example，the 6th spectrum corresponds to the fiber with six acoustic modes.The fifth peak is located in 10.64 GHz and it is almost submerged by the fourth peak.The shape of the 5th spectrum looks very similar to that of the 6th spectrum.The relative power of their first three peaks is pretty close.For the 4th，3rd，2nd and 1st spectra，their total power gradually decreases compared with the 5th and 6th spectra.As for the 1st spectrum，there is only one peak and the peak power is also the smallest.

In Fig.12，some LEAF-type fibers have more than six or even ten acoustic modes.However，the intensities of the latter peaks in these BGSs are very weak relative to the main peak.We select some examples to illustrate this point.The detailed results are shown in Fig.14.Although there are nine acoustic modes in the 9th spectrum，the 4th and 5th peaks are weaker by about 18 d B with respect to the first peak，and the 6th and subsequent peaks are weaker by about 30 dB.These weak peaks can be ignored in practical applications.

Since the LEAFs have a large core radius and usually support five or more acoustic modes guided in the fibers，these reasons lead to higher total Brillouin gain，even if their peak gain are lower than that of the SIOFs.In LEAFs withcfrom 10 to 15，n1from 1.472 to 1.475，andn2from 1.444 to 1.46，they have higher total Brillouin gain and are more suitable for optical fiber Brillouin sensing.The multi-peak BGS caused by the multi-acoustic mode number makes LEAFs a better candidate for multiparameter sensing.Optical fibers with the other parameter combinations in the cube shown in Fig.11（a）are better used in optical fiber communication.

2.4.2 Trench assisted fiber（TAF）

The requirement of bend-insensitive optical fibers has led to the emergence of the trench assisted fiber.Refractive index trench is introduced into the cladding section in the TAF.Different from the above-mentioned types of optical fibers，fluorine is usually added to the silica fiber to achieve the low refractive index of the trench.Its typical RIP and AVP are shown in Fig.15.

Fig.15 RIP and AVP of TAF

n1，n2andn3in Fig.15 represent the refractive index of the core，inner/outer cladding and trench，respectively.Anda，bandcin Fig.15 denote the core radius，inner cladding width and trench width，respectively.Therefore，the refractive index profile of the TAF can be expressed as

Three parameters，b，candn3，will change the refractive index distribution of trench，and thus affect BGSs of TAF.The same analysis method is adopted as that used in Section 2.4.1，changing the above three parameters separately to observe how TAF’s BGS will change.The related parameters and variation ranges of the three parameters are shown in Table 4.

Table 4 Parameters and variables of TAFs

First，we change the depth of the trench，which is its refractive indexn3.bandcare set to be 3.5μm and 6μm，respectively.Whenn3changes from 1.424 to 1.444，the corresponding BGSs are shown in Fig.16.

Fig.16 BGSs of TAFs when n3 changing from 1.424 to 1.444

Next the width of the inner cladding section is changed.Here we keepcandn3fixed atc=6μm andn3=1.434 respectively and change the value ofb.Whenbchanges from 1μm to 6μm，the corresponding BGSs are shown in Fig.17.

Fig.17 BGSs of TAFs when b changing from 1μm to 6μm

Finally，we setbandn3to be 3.5μm and 1.434 respectively and change thecfrom 1μm to 11μm.BGSs of TAFs are shown in Fig.18.

Fig.18 BGSs of TAFs when c changing from 1μm to 11μm

From the results shown in Fig.16 to Fig.18，changing the width of the inner claddingb，the width of the trenchcand the refractive index of the trenchn3does not make the BGS of the TAFs vary.The reason is that in the TAFs，the optical fundamental mode field and acoustic mode fields are both confined in the core and the inner cladding of TAFs，as shown in Fig.19.Therefore，no matter how the refractive index distribution of the trench is changed，it will not have any effect on the optical and acoustic mode fields，and TAF’s BGS will not change.

Fig.19 Calculation results of TAF model by FEA method

Therefore，under the condition that the core radius and refractive index remain unchanged，the TAFs has the only one BGS.Fig.20 shows the total BGS of TAFs and the contribution of each acoustic mode to the total BGS.

Fig.20 BGS of TAF

In general，changing the position，depth and width of the TAF’s trench has no effect on its BGS.TAF used in this section has four acoustic modes，but only a strong main peak and a small secondary peak are reflected on BGS.Therefore，TAF is not suitable for Brillouin fiber sensing，but more suitable for fiber communication，which is also the original intention of its design.

2.5 Comparison of three types of fibers

In the case of two simpler structures，SIOF and GIOF，we have found that the change in total Brillouin gain and acoustic mode number due to the change of core radius and refractive index difference.Briefly，there are up to four acoustic modes in SIOFs and up to six acoustic modes exist in GIOFs.Both of them tend to have high Brillouin gain where the core refractive index is high.The Brillouin gain distributions of SIOFs and GIOFs are roughly consistent with their respective acoustic mode number distributions.For CIOFs，LEAF and TAF are chosen as the typical optical fibers.In LEAFs simulation，we found that due to the existence of the outerring section，it usually guides more than five acoustic modes.In addition，LEAF-type optical fibers with a largergB_totalare in the range ofn1from 1.472 to 1.475，n2from 1.444 to 1.46，andcfrom 10μm to 15μm.In regard to the TAFs，the existence of the trench has no effect on their BGSs.In summary，for optical fiber communication，one should choose CIOFs as the working fiber，and the corresponding parameter combination is described in Section 2.4.The SIOFs with core radius from 1.5μm to 2μm and core refractive index from 1.465 to 1.474 2 are better for Brillouin sensing application.

3 An Example：Choosing a Fiber Used for Temperature Sensing Based on BBSPower Detection

In our previous work，distributed temperature measurement method based on Brillouin beat spectrum was proposed［12］.Compared with other Brillouin sensing systems that have the shortcomings such as time-consuming，expensive，and complicated，BBS has its advantage of fast，simple and convenient.This method detects the peak power of the Brillouin beat spectrum and uses the homodyne BOTDR system to achieve temperature measurement.From the previous content of this work，we know that multiple acoustic modes can exist in some fibers，corresponding to the different peaks of their BGSs.And the BBS corresponds to the beats between the main peak and the other peaks in BGSs.

The power of Brillouin scattering light in BOTDR is［37］

wherePP（0）is the incident light power，α=0.2 d B/km the fiber’s attenuation coefficient at the wavelength of 1.55μm，c'the velocity of light in vacuum，Wthe incident pulse width，αBthe Brillouin scattering-loss coefficient［38］andSthe backscatter capture fraction［39］.Now we substitute the Brillouin frequency shiftνm=2neffVl/λinto Eq.（4）and rewrite Eq.（20）as an expression with the Brillouin peak gaingm

whereA=c'2πωKTW/6Vl，ωis the FWHM of the corresponding peak in BGS，Kthe Boltzmann’s constant，andTthe absolute temperature.Longitudinal mode acoustic velocityVlcan be expressed as［40］

The peaks of BBS vary when the peak powers of different BGS peaks change caused by variation of temperature.The dependences of the BBS peak powerΔPBon temperature can be expressed as［12］

whereCPTis the temperature coefficient of the BBS peak power and theΔTthe change in temperature.

In the BBS of the fibers with several acoustic modes，the beat peak（i，j）is corresponding to the peakiand peakjin the BGS.If the peak power of the peakiand peakjare respectivelyPBiandPBj，the peak power of the beat peak（i，j）in the BBS is

If the power measurement uncertainty of the detection system isΔP，then the measurement uncertainty of BBS peak power in the form of relative percentage valueδPwill be

and the temperature measurement uncertainty is

In Eq.（25），the power measurement uncertaintyΔPoriginates from the detection or measurement error of the detectors and photoelectric instruments of the BBS system.When the BBS power detection works with a certain fiber as the sensor，then the smaller the value ofδP，the higher the accuracy of temperature measurement.It can be seen from the above equations that the measurement accuracy depends on the relative peak intensity of the multipeak BGS.Therefore，in order to enable the BBS system to have higher measurement accuracy，fibers with high Brillouin gain and（or）similar peak gain in their multimodal BGSs are needed.

Now，we name the fiber with the 1st BGS in Fig.5 as SIOF-1，the fiber with the 1st BGS in Fig.8 as GIOF-1，the fiber with the 1st BGS in Fig.13 as LEAF-1，and the fiber with BGS in Fig.20 as TAF，and so on.SIOF-3，SIOF-4，GIOF-5，GIOF-6，LEAF-5 to LEAF-9 and TAF are selected as the research objects to discuss the influence of total Brillouin gain and similar peak gain on the BBS measurement accuracy.

We continue to use the above setting for relative power，set the main peak power of the SIOF-4’s BGS to 0 d Bm，and the the temperature coefficient of all fibers to 29.5×10-4°C［12］.The power measurement uncertaintyΔPof the BBS detection system set to 0.1%of the maximum detection power.The simulation results are shown in Table 5.

Table 5 Temperature measurement accuracy of different fibers

From the Table 4，we can see that even though the SIOF-4 has the largest Brillouin main peak gaing1and the maximum detection powerPB1，the peak gain of its secondary peak is very low，making the corresponding detection power small，so the peak power of the beat peak（1，2）is small and its temperature measurement uncertainty is relatively poor.Meanwhile，LEAF-6 has a small peak intensity difference，so the beat power of peak 1 and peak 2 in its BGS is larger，and it shows the best temperature measurement accuracy.

Based on a comprehensive analysis of the simulation results of the above three types of fibers，the LEAF-type fiber represented by the 6th spectrum in Fig.13 has a large total Brillouin gain and the intensity of the three peaks in the BGS is the closest.Its second and third peaks are 2.4 and 8.1 dB weaker than the main peak，respectively.In such a fiber，the inner-core radius，inner-cladding radius，and outer-core radius are 3.04，4，8μm，respectively.Furthermore，the fiber axis refractive index，outercore refractive index，and cladding refractive index are 1.475，1.452，and 1.444，respectively.

The relative power of first three main peaks of the LEAF’s BGS in Ref.［12］are 0，-16.5 and-19 d B when utilized in BBS optical fiber sensing system.Assuming that the sum of the three peaks’power of the Corning’s LEAF is equal to that of the proposed new fiber（denoted asPA），then three peak powers of Corning’s LEAF are 0.97PA，0.02PAand 0.01PA，respectively.Similarly，the relative power of first three main peaks of the proposed fiber are 0，-2.4 and-8.1 d B and their peak powers can be expressed as 0.58PA，0.33PAand 0.09PA.According to Eq.（23），the ratio of Corning’s LEAF’s and new fiber’s power measurement uncertainty of the detection systemδPcan be expressed as

We once again assume that the Corning’s LEAF and the fiber we designed have the same power-temperature in sensing system，that is，compared with the original fiber，the SNR of the new fiber’s beat peak（1，2）is 3.1 times higher when used in BBS sensing system.It could be the most suitable fiber used in the optical fiber sensing system based on BBS.

4 Conclusions

The new and full images of Brillouin gain and acoustic modes number of the fiber as a function of the refractive index distribution are presented by using FEA method.When using the FEA method to solve the acoustic field in the fiber，a new method is also proposed that can be used to quickly obtain the circular symmetric acoustic mode solution，which greatly improves the simulation efficiency.The influence of refractive index distribution on the BGS in commonly used GeO2-doped optical fibers，including step-index，graded-index，and complex-index optical fibers，are discussed by considering the influence of changing the refractive index difference and the geometric size simultaneously.The refractive index distribution in fibers determines the total Brillouin gain，acoustic mode number，and peak intensity difference of the BGS.From the results of the SIOFs，we can conclude that fibers with more acoustic modes also exhibit higher Brillouin gain.And when the core radius is from 1.5μm to 2μm，and the refractive index difference is as large as possible，the total Brillouin gain is greater than that in other cases.For the GIOFs，the distribution of the total Brillouin gain is similar to that of SIOFs.And when the core radius is from 2μm to 3μm，and the refractive index difference is as large as possible，the total Brillouin gain is greater.Both of them tend to have high Brillouin gain where the core refractive index is high.And the Brillouin gain distributions of the SIOFs and the GIOFs are roughly consistent with their respective acoustic mode number distributions.As for LEAF-type complex index fibers，the number of acoustic modes is generally greater than or equal to five due to the presence of an outer-ring section.And the relative power difference between the adjacent peaks is small in LEAFs.For trench assisted fiber，the existence of trench has no effect on its BGS.We also obtain a special CIOF with a RIP in LEAF model as a sensing fiber used in sensing system based on BBS，and its fiber axis refractive index，outer-ring refractive index，cladding refractive index are 1.475，1.452，1.444，and its innercore radius，inner-cladding radius，and outer-core radius are 3.04，4，and 8μm，respectively.Compared with the original LEAF fiber，the SNR of the new fiber’s beat peak（1，2）is 3.1 times higher when used in BBS sensing system.