## Abstract

A numerical investigation is presented of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles. The equation determining the acoustic displacement in response to the electrostriction caused by the pump and Stokes waves reduces to the non-homogeneous Helmholtz equation for fibers with a uniform acoustic velocity profile. In this special case the acoustic displacement and subsequently the Brillouin gain are calculated using a Green's function. These results are then used to validate a finite-element solution of the same equation. This finite element method is then used to analyze a standard large mode area fiber as well as fibers incorporating four different acoustic velocity profiles with 5% variation in the acoustic velocity across the core. The profiles which suppress the peak Brillouin gain most effectively exhibit a maximum acoustic gradient near the midpoint between the center and boundary of the fiber core. These profiles produce 11 dB of suppression relative to standard large mode area fibers.

©2009 Optical Society of America

## 1. Introduction

The onset of stimulated Brillouin scattering (SBS) is the primary obstacle limiting power scaling of diffraction-limited single-frequency continuous wave fiber amplifiers [1–3]. Such sources have many important applications including atmospheric LIDAR [4], gravitational wave interferometry [5], and coherent beam combination [6]. Reported approaches for increasing the SBS threshold in these devices include increasing the mode field area (subject to the ability to maintain single-transverse mode operation [7], decreasing the fiber length [8], establishing longitudinal temperature [9,10] or strain [11] gradients within the fiber, and incorporating doping profile-dependent transverse acoustic velocity gradients within the core of the fiber [12–15].

Brillouin gain has been investigated experimentally and numerically in azimuthally-symmetric optical fibers with cores that are acoustically inhomogeneous due to the doping required to establish optical waveguides with the desired properties [16]. A design requiring no special control of the relative concentrations of multiple dopants was proposed for which 3dB suppression in the peak Brillouin gain was predicted. The use of multiple species of dopant allows the possibility of partially de-coupling the optical mode field intensity and acoustic velocity profiles to realize stronger suppression of the peak value of the Brillouin gain. Theoretical models of SBS suppression in such fibers, based on a generalization of the equations for Brillouin gain in media with homogeneous acoustic velocity [3,17] to inhomogeneous media characterized by multiple guided acoustic modes, have been presented [18,19,31,32]. The main effort of these investigations has focused on establishing a consistent relationship between the degree of overlap between the guided acoustic modes (phonons in a quantum mechanical context), and the fundamental optical mode field intensity profile, and the degree of SBS suppression observed in the fiber relative to standard fibers. This approach has resulted in varying degrees of agreement between theory and experiment. In some cases significant differences between observed and predicted thresholds have been observed [14,20] prompting investigations into alternative theoretical frameworks. One such framework is based on the application of a beam propagation method to the acoustic excitations in the fiber core [14,21].

In this paper we present a numerical investigation of the Brillouin gain spectrum (BGS) of large mode area fiber designs with non-uniform acoustic velocity profiles. We employ a finite element method to solve the non-homogeneous acoustic wave equation for the steady-state acoustic displacement amplitude with from which we derive Brillouin gain spectra for several fiber designs with different acoustic profiles.

## 2. Brillouin gain

We begin with the equation governing the propagation of the electromagnetic fields in the fiber:

*u*is the longitudinal acoustic displacement and $\gamma ={n}^{4}{\epsilon}_{0}{p}_{12}$is the electrostrictive constant in terms of the refractive index

_{z}*n,*the permittivity of free space ${\epsilon}_{0}$ and the Pockel coefficient ${p}_{12}$. The electric field is expressed in terms of pump and Stokes components with different frequencies and amplitudes:

*λ*, the vacuum speed of light

*c*, and the effective index of the fundamental optical mode ${n}_{e}$. The longitudinal acoustic displacement driven by electrostriction takes the form

*u*must satisfy the non-homogeneous acoustic wave Eq. (22).

*η*is the viscosity,

*ρ*is the mass density and are the transverse and longitudinal acoustic velocities given in terms of the Young’s modulus

*E =*73 GPa, the Poisson ratio ν = 0.17, and ρ = 2200 kg/m

^{3}where we have given the values for bulk fused silica. Thus the bulk acoustic velocities are

*V*5972 m/s and

_{l}=*V*= 3766 m/s. We have retained the usual symbol for the Young’s modulus; however, it is not to be confused with the symbol for the electric field.

_{t}In our treatment these velocities may vary throughout the fiber cross section due to doping with various compounds. The acoustic refractive index, characterizing the local acoustic velocities is given by

It is important to note that even though we are considering longitudinal acoustic displacements, the transverse acoustic velocity enters explicitly through the stress-strain constitutive relationships [23] evaluated for longitudinal displacements that vary throughout the fiber cross-section. This term is also clearly present in Eq. (33) of [22]. In Eq. (1) of [16] the different acoustic velocities are expressed through the Lamé constants λ and µ. This may be understood intuitively by the fact that while guided acoustic waves are travelling waves in the longitudinal direction, they are standing shear waves in the transverse direction. Thus the relationship between the temporal frequency of these waves and their transverse structure is governed by the shear wave velocity*V*. This wave equation is non-homogeneous in the sense that the term on the right hand side is a source term [24,25]. Substituting Eq. (3) and (4) into Eq. (6) yields

_{t}*G*with the source function Φ [25]

*κ*on

*ω*expressed through

_{s}*Ω*. In the case that the optical mode takes the form of a uniform plane wave, then the classic result [3,26] for the peak Brillouin gain coefficient is recovered.

## 3. Brillouin gain in a fiber with uniform acoustic profile

If the acoustic velocities are uniform throughout the fiber, then the Green's function for the Helmholtz operator can be derived [25]:

*K*is the modified Bessel function. We consider a step-index large mode area fiber whose fundamental mode field intensity profile is well approximated by a Gaussian

_{0}*d*is the 1/

*e*mode field diameter. The integral required to evaluate the acoustic displacement field is then

^{2}*θ*yields a 2-dimensional integral that must be evaluated numerically. This motivates the use of other methods to treat cases where the fiber has a non-uniform acoustic velocity profile.

## 4. Finite element analysis of Brillouin gain

Finite element analysis is a widely employed technique in the field of acoustics. The variational expression

*κ*and

*Φ*may be treated as constants within each element as shown in Fig. 1. The discrete field vector

*φ*corresponds to the values of $\phi \left(\colorbox[rgb]{}{$r$}\right)$at nodes on the boundary of each element. We employ quadratic Lagrange elements with three curvilinear sides and nodes at each vertex and at the midpoint of each side for a total of 6 nodes on each element boundary. The value of the integral over each element is calculated as a weighted sum of the values of the integrand at specified points on the interior of the element [27]. The values of the fields and the global coordinates at these interior points are interpolated from the field values at the nodes to yield:

*x*and

_{k}*y*are the global coordinates of the nodes and${\phi}_{k}$are the values of $\phi \left(\colorbox[rgb]{}{$r$}\right)$on the nodes. The shape functions are defined as

_{k}*L*and

_{1}*L*range from 0 to 1. The spatial derivatives of

_{2}*φ*at the integration points are then obtained by taking the derivatives of the shape functions.

**N**,

**N**

_{x}and

**N**

_{y}such that

*φ*within element

*i*at local coordinates

*L*and

_{1}*L*, likewise for its derivatives, and ${\phi}_{j}$ is the value of

_{2}*φ*at node

*j*when the nodes are numbered globally throughout the entire mesh. Thus the matrices

**N**,

**N**

_{x}and

**N**

_{y}have a number of rows equal to the number of elements in the mesh and a number of columns equal to the number of nodes in the mesh. These matrices are constructed by inserting the values given by Eq. (27) and (28) in the rows corresponding to each element in the six columns corresponding to the global indices of the six nodes on the boundary of that element. These matrices may then be used to write Eq. (25) in terms of the global vector

**φ**

*T*denotes the transpose of a matrix or vector,

*W*is the weighting factor of integration point

_{k}*k*,

*J*is a diagonal matrix containing the determinant of the Jacobian, Eq. (30), at the local coordinates of integration point

_{k}*k*,

**N**

_{k},

**N**

_{xk}and

**N**

_{yk}are the matrices defined in Eq. (31) evaluated at the local coordinates of integration point

*k*, and the diagonal matrix

**M**contains the values of κ

^{2}on each element in the mesh. Minimizing Eq. (32) with respect to

**φ**yields the set of linear equationswhich has the solutionThe Brillouin gain is then given by

*E*is the scalar electric field defined on the nodes of the mesh and the matrix

## 5. Validation of the finite element method

As a check on the validity of the finite element expression for the Brillouin gain given by Eq. (37), we may calculate the Brillouin gain spectrum for fibers with a conventional step optical index profile and a flat acoustic velocity profile. We investigate step-index fibers with core diameters of 20 µm, 25 µm, and 30 µm all with a numerical aperture of 0.06. The values of the electric field on the nodes for the fundamental mode were obtained using a finite element method described previously [28,29] for a wavelength of λ = 1.064 µm. The thermal Brillouin linewidth in fibers can vary substantially relative to the bulk value due to waveguide-induced broadening and variations in core composition [3,34]. The parameter entering into our model is the bulk value which is Γ/(2π) = 36 MHz at λ = 1.064 µm [34]. We evaluated Eq. (24) by first integrating symbolically over θ and then using 2-dimensional quadrature to evaluate the remaining integrals. Then we used Eq. (37) to calculate the BGS for the same three fibers. We calculated the optical mode field diameter used in Eq. (24) from the effective area of the fundamental mode obtained with the optical finite element calculation. The same compuational mesh, shown in Fig. 1 was used for the acoustic and electromagnetic calculations. Figure 2 shows the results for the fiber with the 20 µm diameter core. The results for the other two fibers were indistinguishable when plotted on the same axes indicating that the Brillouin gain in fibers such as these scales with the inverse of the non-linear effective area as expected. This also indicates that our implementation of the finite element method is capable of accurately describing the system. This particular example pertains to conventional large mode area fibers where the acoustic velocity difference between the core and cladding is approximately 1%. We note that the value we obtained for the peak Brillouin gain coefficient of 1.9×10^{−11} m/W is in excellent agreement with recently reported values for these types of fibers [14,30]. The imaginary part of the Brillouin gain coefficient can be related to the non-linear phase change of signals propagating in the fiber [22].

## 6. Fibers with an acoustic guiding layer and a linearly-ramped acoustic profile

Several detailed acoustic designs for SBS-suppressing fiber have been presented in the literature including fibers with an acoustic guiding layer (AGL) [15], fibers incorporating a negative acoustic lens [14], and fibers with an interface-free linearly-ramped acoustic velocity profile [13]. We calculated the Brillouin gain spectrum using the finite element method described above for an AGL fiber and a linearly-ramped fiber as well as a conventional fiber for reference. We checked the finite element calculations in these cases by employing different element sizes to ensure that the results remained consistent. For insufficient meshing density, oscillatory behavior in the BGS was typically observed in the upper frequencies as the optical field drove the acoustic displacements at a spatial frequency higher than the mesh could accurately approximate. Each example has identical optical index and fundamental mode field intensity profiles. A comparison between Figs. 2 and 3 indicates that the presence of a 1% acoustic index step causes the Brillouin gain spectrum to shift by approximately 160 MHz, but the spectrum width and peak value are unchanged. The frequency shift at which the Brillouin gain is at its maximum is related to the optical effective index, the pump wavelength, and the acoustic index through the relation

thus a 1% change in the acoustic velocity across the fiber core results in a 1% change in the Brillouin gain peak frequency.This line of reasoning has been extended to the case of non-uniform acoustic velocity profiles with the use of a localized approximation in which the transverse acoustic velocity is taken to be zero [21]. This provides a framework for interpreting the results shown in Fig. 4. This AGL fiber exhibits two peaks corresponding to the two different acoustic velocities present in the core region. The low-frequency peak corresponds to the guiding layer with its higher acoustic index while the high-frequency peak corresponds to the center of the core. By varying the thickness of the guiding layer, the relative heights of the two peaks can be changed. The SBS process is most effectively suppressed when the peak value of the Brillouin gain is minimized. This occurs in the AGL fiber when the height of the two peaks is roughly equal. The example design shown here has a peak Brillouin gain coefficient roughly 4 dB below that of the conventional fiber shown in Fig. 3.

This analysis implies that the peak value of the Brillouin gain spectrum may be suppressed more effectively by continuously varying the acoustic index profile throughout the fiber to broaden the spectrum. One example of this approach is the linearly-ramped acoustic profile shown in Fig. 5. This is similar to a fiber described previously [13]. The calculated BGS in this fiber is broadened relative to the conventional case, but still exhibits a clearly-defined peak. The Brillouin gain spectrum also exhibits oscillatory behavior at lower Brillouin frequency shifts. This is attributed to resonance of the electrostrictive driving term with acoustic modes guided within the regions of the core with an acoustic index > 1. This explains why the oscillations only occur at the lower end of the spectrum. Furthermore, acoustic profiles with no guiding regions exhibit no such oscillations. This design exhibits a peak Brillouin gain coefficient approximately 9 dB below that of the conventional fiber. Suppressions of 6 dB and 11 dB have been reported for similar fibers with acoustic index variations of 4% and 9% respectively [13,14]. Our BGS calculations for fiber designs with variations above ~7% have shown non-negligible dependence on meshing parameters indicating that further mesh refinement is required to obtain reliable results for such designs. These types of designs represent an improvement over the AGL fiber design, yet it is possible to suppress the peak Brillouin gain further by further refining the acoustic velocity profile.

## 7. Improved fiber designs

Although each of the acoustic designs considered so far have significantly different Brillouin gain spectra, they all exhibit the same area under the curve describing the real gain spectrum of approximately 1×10^{−12} GHz-m/W. This suggests that further suppression may be achieved by designing a fiber with a flat-top Brillouin gain spectrum. Combining a more rapid acoustic velocity variation mid-radius relative to the core with a less rapid variation at the center and boundary serves to flatten out the Brillouin gain spectrum thus achieving a smaller maximum Brillouin gain.

Figures 6 and 7 show the acoustic profile and Brillouin gain spectrum for two such fibers. These acoustic profiles were arrived at by iterative refinement. Typically the finite extent of the doped region within the fiber pre-form causes a return to an acoustic index of 1 at the core boundary outside of which the fiber cladding may consist of pure silica. This feature is incorporated in these designs. Since the absolute value of the slope of the acoustic index determines the Brillouin gain spectrum in the localized approximation, it is instructive to examine the Brillouin gain spectrum of the inverted profile. Figure 6 shows a profile with the minimum acoustic index at the center. The resulting BGS exhibits slight oscillatory behavior at lower frequencies due to resonance with acoustic modes guided in the region within the core near its boundary. The inverted profile shown in Fig. 7 exhibits enhanced low-frequency oscillatory behavior in the BGS due to increased overlap of the optical field with acoustic modes that are now guided in the center of the core and thus resonate more strongly when driven by the optical field. Each of these acoustic profiles produces a nearly flat-top BGS with a peak value that is approximately 11 dB below that of the conventional fiber. Figure 8 shows the acoustic displacement field within the fiber near the center of the BGS as one frame of the accompanying movie. We note that it is very similar to the acoustic displacement field reported in [14] for a similar acoustic profile.

## 8. Discussion and conclusion

An expression for the Brillouin gain spectrum in optical fibers with non-uniform acoustic profiles was derived previously [18] by expanding the acoustic displacement profile in a basis set comprised of the un-damped guided acoustic mode profiles. A finite element implementation of this method has been applied to single-mode, PANDA polarization-maintaining [32], and w-shaped triple layer fibers [31]. The approximation that all acoustic modes exhibit the same Brillouin frequency shift, used to derive this expression, bears further discussion. If the Brillouin frequency shifts of two separate acoustic modes differ by more than the width of the BGS corresponding to a single acoustic velocity, then each would contribute separately to the Brillouin gain at its respective Brillouin frequency shift, with amplification at the stronger frequency shift eventually dominating the SBS process. In this case, the uniform Brillouin frequency shift assumption [18] does not apply and the contributions to the Brillouin gain due to resonance with individual acoustic modes must be considered separately. The AGL fiber discussed above falls into this category. If a particular fiber supports multiple acoustic modes with Brillouin frequency shifts all grouped within the width of the BGS, then the uniform Brillouin frequency shift assumption applies, and the collective resonance with all of these modes contributes to the Brillouin gain spectrum, with an overall efficiency that depends on how tightly the frequency shifts of the various modes are grouped. The fiber with the linearly-ramped acoustic velocity profile falls into this category. Fibers also may exist somewhere in between these two limits. The improved fiber designs presented here fall into this category.

The main advantage of the approach presented in this work is that both the effects of the acousto-optical overlap and the distribution of modal Brillouin shifts are accounted for naturally through the direct solution of the non-homogeneous wave equation Eq. (6). Furthermore, no analysis of the spectrum and field profiles of propagating acoustic modes is required in order to calculate the Brillouin gain spectrum. Our finite element approach is fundamentally different from those presented previously [31,32] in that the acoustic displacement at a given Brillouin shift is obtained directly as the solution to a set of linear equations, Eq. (36), rather than as a sum of guided modes in varying degrees of resonance with the electrostrictive driving term. While both approaches employ finite elements to discretize the fiber cross-section, they do so to solve different equations which may be seen by comparing Eq. (7) of [31] where the right hand side is equal to zero to Eq. (9) of this work where it is not. A previous work [33] treated the spontaneous Brillouin scattering (SpBS) process as well as Brillouin gain. We have reserved further analysis of propagating acoustic modes, and their role the SpBS process, for future work. However, we have used the concept of resonance with guided acoustic modes to add insight to the interpretation of our results. To be clear, none of the results presented here required the calculation of any properties of acoustic modes or phonons.

Although all of the fiber designs treated here are axially symmetric and therefore the optical and acoustic partial differential equations can be reduced to one dimension for these cases, we have developed a two-dimensional computational method in anticipation of treating non axially-symmetric fibers such as photonic crystal fibers and coiled large mode area fibers. We anticipate that bending-induced mode distortion will affect the BGS spectrum in SBS-suppressing large mode area fibers due to modified acousto-optical overlap. Although we have generalized the computational method presented here to treat this case, this analysis is reserved for future works. One additional observation is that the minimum acoustic velocity variation within the core needed to achieve significant Brillouin gain suppression is that which corresponds to a separation of one bulk BGS width (36 MHz for the parameters employed here) or 0.2%. Also, longitudinal variations in the acoustic velocities caused by temperature or strain distributions can alter the effective BGS describing the amplification of the counter-propagating Stokes wave throughout the entire length of the fiber. In this case, knowledge of the longitudinal BGS variations may be employed to tailor the local BGS to further increase the SBS threshold for a particular fiber.

In conclusion, we have presented a new method for calculating the Brillouin gain spectrum in optical fibers with non-uniform acoustic velocity profiles that produces results in good agreement with experiment and used it to analyze four different SBS-suppressing large mode area fiber designs with a 5% variation in acoustic velocity across the core. The maximum suppression of the peak Brillouin gain was achieved by incorporating continuously-varying acoustic velocity profiles with the maximum slope occurring mid-radius and was found to be 11 dB relative to a standard large mode area fiber.

## Acknowledgments

The authors would like to thank the High Energy Laser Joint Technology Office for funding support, Steve Senator, United States Air Force Academy Modeling and Simulation Research Center, and Tom Cortese, Productivity Enhancement and Technology Transfer Team, for computational support. We would also like to thank Johann Nilsson, Josh Rothenberg, Almantas Galvanauskas, Craig Robin, Chris Vergien, David DiGiovanni, and Marc Mermelstein for helpful discussions.

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