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 OSA

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:

2E(x,y,z,t)n(x,y)2c22t2E(x,y,z,t)=μ02t2PNL(x,y,z,t)
where the scalar field approximation has been used and
PNL(x,y,z,t)=γ[uz(x,y,z,t)]E(x,y,z,t)
where uz is the longitudinal acoustic displacement and γ=n4ε0p12is the electrostrictive constant in terms of the refractive index n, the permittivity of free space ε0 and the Pockel coefficient p12. The electric field is expressed in terms of pump and Stokes components with different frequencies and amplitudes:
E(x,y,z,t)=12{ap(z)f(x,y)exp[i(kpzωpt)]+as(z)f(x,y)exp[i(ksz+ωst)]+c .c .}
where as,pare the pump and Stokes field amplitudes, f(x,y)is the transverse field profile, kp,s=2π/λp,sare the wave vectors, and ωp,s=ckp,s/neare the frequencies in terms of the vacuum wavelength of the pumpλ, the vacuum speed of light c, and the effective index of the fundamental optical mode ne. The longitudinal acoustic displacement driven by electrostriction takes the form
uz(x,y,z,t)=12{φ(x,y)exp[i(βzΩt)]+c .c.}
where Ω=ωpωsis the difference between the pump and Stokes frequencies and is the sum of the pump and Stokes wave vectors. Substituting Eq. (3) and (4) into Eq. (1), applying the slowly varying envelope approximation and the phase matching conditionβ=kp+ks, assuming single-transverse-mode fiber, and approximating β2ks(4πn)/λleads to the following equation for the evolution of the Stokes amplitudeas.
as(z,t)z=14ap(γε0)(2πλ)2f|φ|ff|f
where |indicates integration over the fiber cross-section. To determine the evolution of the Stokes intensity over the length of the fiber, the acoustic displacement u must satisfy the non-homogeneous acoustic wave Eq. (22).
[Vt2t2+(Vl2+ηρt)2z22t2]u(x,y,z,t)=12γρE(x,y,z,t)2
where η is the viscosity, ρ is the mass density and
Vl2=E(1ν)ρ(1+ν)(12ν)
Vt2=E2ρ(1+ν)
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/m3 where we have given the values for bulk fused silica. Thus the bulk acoustic velocities are Vl = 5972 m/s and Vt = 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.

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

nac=Vl,t,bulkVl,t(x,y)
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 Vt. 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
[Vt2t2+Ω2β2(Vl2+iηρΩ)]φ(x,y)=i2πnγλρapas(z)f(x,y)2
where again β(4πn)/λ. This may be simplified further by recognizing that the thermal Brillouin linewidth may be written Γ=(β2η)/ρ. Dividing all terms by a factor of Vt2yields
[t2+Ω2Vt2β2Vl2Vt2iΓΩVt2]φ(x,y)=i2πnγλρVt2apas(z)f(x,y)2.
This last step was taken to put the equation in the form of a generalized 2-dimensional non-homogeneous Helmholtz equation.
[t2κ2(x,y)]φ(x,y)=Φ(x,y)
where
κ2(x,y)1Vt2(β2Vl2Ω2iΓΩ)
Φ(x,y)i2πnγλρVt2apas(z)f(x,y)2
This equation may be expressed in linear operator notation
Lφ=Φ
and has a formal solution given by the convolution of the Green’s function G with the source function Φ [25]
φ(x,y)=G(x,y,x,y)Φ(x,y)dxdy(GΦ)
where the fiber cross-section is the domain of integration. Substituting Eq. (14) in Eq. (5) and expressing the Stokes and pump powers in terms of their amplitudes
Ps,p=12cnε0f|f|as,p|2
yields the equation for the evolution of the Stokes power in the absence of pump depletion
dPs(z)dz=gBAeffPsPp
where
gBAeff=(1ρc)(γε0)2(2πλ)3f|(Gg)|ff|f2,
g(x,y)f(x,y)2Vt(x,y)2
and
Aeff=f|f2f|f2|f
is the non-linear effective area of the fundamental optical mode. In general the Brillouin gain is a function of the Stokes frequency which enters through the dependence of κ on ωs expressed through 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.

gB,peak=2π2n7p122cλ2ρVlΓ.

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]:

G(r,r)=12πK0(κ|rr|)
where K0 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
f(r)2=8πd2exp[8r2d2]
where d is the 1/e2 mode field diameter. The integral required to evaluate the acoustic displacement field is then
φ(r)12π8πd2002πK0(κr2+r2+2rrcosθ)exp[8r2d2]rdrdθ
and the Brillouin gain is given by
gBAeff=(1ρVt2c)(γε0)2(2πλ)3(8πd2)2×0002πK0(κr2+r2+2rrcosθ)exp[8(r2+r2)d2]rrdrdrdθ
Even for this simple case, carrying out the integral over 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

S(φ)={12[tφ(r)]212κ(r)2φ(r)2Φ(r)φ(r)}d2r
leads to Eq. (6) and is the starting point for our analysis. We proceed by dividing the region of integration into elements small enough that 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 φ(r)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,y,φ}=k=16Ak(L1,L2){xk,yk,φk}
where the sum is over the nodes on the boundary of the element, xk and yk are the global coordinates of the nodes andφkare the values of φ(r)on the nodes. The shape functions are defined as
A1=L1(2L11)A2=L2(2L21)A3=(1L1L2)(12L12L2)A4=4L1L2A5=4L2(1L1L2)A6=4L1(1L1L2)
where the local coordinates L1 and L2 range from 0 to 1. The spatial derivatives of φ at the integration points are then obtained by taking the derivatives of the shape functions.
xφ=k=16xAk(L1,L2)φk
yφ=k=16yAk(L1,L2)φk
The Jacobian matrix relating the derivatives of the local and global coordinates is used to evaluate the derivatives of the shape functions in terms of the local coordinates of the integration points.
[xAkyAk]=J1[L1AkL2Ak]
with
J=[xL1xL2yL1yL2].
Equation (27) and (28) are then used to construct matrices N, N x and N y such that
φ˜i=Nij(L1,L2)φj
xφ˜i=Nx,ij(L1,L2)φj
yφ˜i=Ny,ij(L1,L2)φj
where φ˜iis the value of φ within element i at local coordinates L1 and L2, likewise for its derivatives, and φj is the value of 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
S(φ)=12φTKφRφ.
where
K=k=17Wk[NxkTJkNxkNykTJkNykNkTJkMNk]
and
R=k=17Wk[ΦkTNkTJkNk].
The sums are taken over the integration points within each element, T denotes the transpose of a matrix or vector, Wk is the weighting factor of integration point k, Jk is a diagonal matrix containing the determinant of the Jacobian, Eq. (30), at the local coordinates of integration point 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 κ2 on each element in the mesh. Minimizing Eq. (32) with respect to φ yields the set of linear equations
Kφ=R
which has the solution
φ=K1R.
The Brillouin gain is then given by
gBAeff=(1ρc)(γε0)2(2πλ)3ETV(K1R)×E(ETVE)2
where ×denotes elementwise multiplication, E is the scalar electric field defined on the nodes of the mesh and the matrix
V=k=17Wk[NkTJkNk]
facilitates the integration of the dot product of any two vectors defined on the nodes of the mesh over the fiber cross-section

 

Fig. 1 Central portion of the finite element mesh used for the electromagnetic and acoustic calculations.

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[φ(r)ψ(r)]d2r=φTVψ.
 

Fig. 2 Comparison between the Brillouin gain spectrum for a large mode area fiber with a uniform acoustic profile calculated using Eq. (24) and Eq. (37).

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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

νB=2neVl,bulknacλ
thus a 1% change in the acoustic velocity across the fiber core results in a 1% change in the Brillouin gain peak frequency.

 

Fig. 3 Acoustic index profile and Brillouin gain spectrum for a conventional large mode area fiber.

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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.

 

Fig. 4 Acoustic index profile and Brillouin gain spectrum for a fiber incorporating and acoustic guiding layer.

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Fig. 5 Acoustic index profile and Brillouin gain spectrum for a fiber with a linearly-ramped acoustic profile.

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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.

 

Fig. 6 Acoustic index profile and Brillouin gain spectrum for an improved fiber designed to produce a flat-top Brillouin gain spectrum. The region of high acoustic index is near the core boundary.

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Fig. 7 Acoustic index profile and Brillouin gain spectrum for an improved fiber designed to produce a flat-top Brillouin gain spectrum. The region of high acoustic index is near the core center.

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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.

 

Fig. 8 (Media 1) A single-frame excerpt from an animation showing how the Brillouin gain and acoustic displacement within the fiber core vary with Brillouin frequency shift for the improved fiber with the greatest acoustic velocity at the center of the core.

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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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11. J. E. Rothenberg, P. A. Thielen, M. Wickham, and C. P. Asman, “Suppression of Stimulated Brillouin Scattering in Single-Frequency Multi-Kilowatt Fiber Amplifiers,” Proc. SPIE 6873, 68,730O–1–7 (2008).

12. S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044. [CrossRef]  

13. M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290. [CrossRef]   [PubMed]  

14. M. D. Mermelstein, M. J. Andrejco, J. Fini, A. Yablon, C. Headley, and D. J. DiGiovanni, “11.2 dB SBS Gain Suppression in a Large Modea Area Yb-doped Optical Fiber,” Proc. SPIE 6873, 68,730N–1–68,730N–7 (2008).

15. P. D. Dragic, C. Liu, G. C. Papen, and A. Galvanauskas, “Optical Fiber with an Acoustic Guiding Layer for Stimulated Brillouin Scattering Suppression,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CThZ3. http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2005-CThZ3

16. A. H. McCurdy, “Modeling of Stimulated Brillouin Scattering in Optical Fibers with Arbitrary Radial Index Profile,” J. Lightwave Technol. 23(11), 3509–3516 (2005). [CrossRef]  

17. V. I. Kovalev and R. G. Harrison, “Threshold for stimulated Brillouin scattering in optical fiber,” Opt. Express 15(26), 17,625–17,630 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17625. [CrossRef]  

18. A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5338. [CrossRef]   [PubMed]  

19. P. D. Dragic, “Acoustical-optical fibers for control of stimulated Brillouin scattering,” in 2006 Digest of the LEOS Summer Topical Meetings, pp. 3–4. (2006).

20. M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952. [CrossRef]  

21. J. Nilsson, “SBS Supression at the kilowatt level,” presented at SPIE Photonics West, San Jose, California, USA, 24–29 Jan. (2009).

22. E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999). [CrossRef]  

23. R. D. Cook, Finite Element Modeling for Stress Analysis (Wiley, New York, 1995).

24. S. J. Farlow, Partial Differential Equations for Scientists and Engineers (Dover, New York, 1993).

25. M. D. Greenberg, Application of Green’s Functions in Science and Engineering (Prentice Hall, New Jersey,1971).

26. C. Tang, “Saturation and Spectral Characteristics of the Stokes Emission in the Stimulated Brillouin Process,” J. Appl. Phys. 37(8), 2945–2956 (1966), http://link.aip.org/link/?JAPIAU/37/2945/1. [CrossRef]  

27. P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956). [CrossRef]  

28. B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Inhomogeneous Anisotropic Refractive Index Tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008). [CrossRef]  

29. B. G. Ward, “Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture,” Opt. Express 16(12), 8532–8548 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-853. [CrossRef]   [PubMed]  

30. W. Torruellas, M. Alam, J. Edgecumbe, K. Tankala, J. Rothenberg, and M. Wickham, “Spectral SBS model for Yb:DCF with Discrete acoustic core designs,” presented at the SPIE Defense and Security Symposium, Orlando, Florida, USA, 13–17 Apr. 2009.

31. W. Zou, Z. He, and K. Hotate, “Acoustic modal analysis and control in w-shaped triple-layer optical fibers with highly-germanium-doped core and F-doped inner cladding,” Opt. Express 16(14), 10006–10017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10006. [CrossRef]   [PubMed]  

32. W. Zou, Z. He, and K. Hotate, “Two-dimensional finite element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. 18(23), 2487–2489 (2006). [CrossRef]  

33. Ward and Spring, “Brillouin gain in optical fibers with inhomogeneous acoustic velocity”, Proc. SPIE, 7195, 71951J–1–11 (2009).

34. V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(22), 2022–2024 (2002), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-22-2022. [CrossRef]  

References

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  1. Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-5-459 .
    [CrossRef] [PubMed]
  2. Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics, Third Edition (Academic, New York, 2001).
  4. C. G. Carlson, P. D. Dragic, B. W. Graf, R. K. Price, J. J. Coleman, and G. R. Swenson, “High power Ybdoped Fiber Laser-Based LIDAR for Space Weather,” Proc. SPIE 5891, 68,730K–1–68,730K–12 (2008).
  5. K. Takeno, T. Ozeki, S. Moriwaki, and N. Mio, “100 W, single-frequency operation of an injection-locked Nd:YAG laser,” Opt. Lett. 30(16), 2110–2112 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-16-2110 .
    [CrossRef] [PubMed]
  6. E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33(4), 354–356 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-4-354 .
    [CrossRef] [PubMed]
  7. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442 .
    [CrossRef]
  8. M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .
  9. M. D. Mermelstein, A. D. Yablon, and C. Headley, “Suppression of Stimulated Brillouin Scattering in an Er-Yb Fiber Amplifier Utilizing Temperature Segmentation,” in Optical Amplifiers and Their Applications Topical Meeting Technical Digest (CD) (Optical Society of America, 2005), paper TuD3.
  10. V. I. Kovalev and R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. 31(2), 161–163 (2006), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-2-161 .
    [CrossRef] [PubMed]
  11. J. E. Rothenberg, P. A. Thielen, M. Wickham, and C. P. Asman, “Suppression of Stimulated Brillouin Scattering in Single-Frequency Multi-Kilowatt Fiber Amplifiers,” Proc. SPIE 6873, 68,730O–1–7 (2008).
  12. S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
    [CrossRef]
  13. M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
    [CrossRef] [PubMed]
  14. M. D. Mermelstein, M. J. Andrejco, J. Fini, A. Yablon, C. Headley, and D. J. DiGiovanni, “11.2 dB SBS Gain Suppression in a Large Modea Area Yb-doped Optical Fiber,” Proc. SPIE 6873, 68,730N–1–68,730N–7 (2008).
  15. P. D. Dragic, C. Liu, G. C. Papen, and A. Galvanauskas, “Optical Fiber with an Acoustic Guiding Layer for Stimulated Brillouin Scattering Suppression,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CThZ3. http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2005-CThZ3
  16. A. H. McCurdy, “Modeling of Stimulated Brillouin Scattering in Optical Fibers with Arbitrary Radial Index Profile,” J. Lightwave Technol. 23(11), 3509–3516 (2005).
    [CrossRef]
  17. V. I. Kovalev and R. G. Harrison, “Threshold for stimulated Brillouin scattering in optical fiber,” Opt. Express 15(26), 17,625–17,630 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17625 .
    [CrossRef]
  18. A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5338 .
    [CrossRef] [PubMed]
  19. P. D. Dragic, “Acoustical-optical fibers for control of stimulated Brillouin scattering,” in 2006 Digest of the LEOS Summer Topical Meetings, pp. 3–4. (2006).
  20. M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 .
    [CrossRef]
  21. J. Nilsson, “SBS Supression at the kilowatt level,” presented at SPIE Photonics West, San Jose, California, USA, 24–29 Jan. (2009).
  22. E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999).
    [CrossRef]
  23. R. D. Cook, Finite Element Modeling for Stress Analysis (Wiley, New York, 1995).
  24. S. J. Farlow, Partial Differential Equations for Scientists and Engineers (Dover, New York, 1993).
  25. M. D. Greenberg, Application of Green’s Functions in Science and Engineering (Prentice Hall, New Jersey,1971).
  26. C. Tang, “Saturation and Spectral Characteristics of the Stokes Emission in the Stimulated Brillouin Process,” J. Appl. Phys. 37(8), 2945–2956 (1966), http://link.aip.org/link/?JAPIAU/37/2945/1 .
    [CrossRef]
  27. P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956).
    [CrossRef]
  28. B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Inhomogeneous Anisotropic Refractive Index Tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008).
    [CrossRef]
  29. B. G. Ward, “Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture,” Opt. Express 16(12), 8532–8548 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-853 .
    [CrossRef] [PubMed]
  30. W. Torruellas, M. Alam, J. Edgecumbe, K. Tankala, J. Rothenberg, and M. Wickham, “Spectral SBS model for Yb:DCF with Discrete acoustic core designs,” presented at the SPIE Defense and Security Symposium, Orlando, Florida, USA, 13–17 Apr. 2009.
  31. W. Zou, Z. He, and K. Hotate, “Acoustic modal analysis and control in w-shaped triple-layer optical fibers with highly-germanium-doped core and F-doped inner cladding,” Opt. Express 16(14), 10006–10017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10006 .
    [CrossRef] [PubMed]
  32. W. Zou, Z. He, and K. Hotate, “Two-dimensional finite element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. 18(23), 2487–2489 (2006).
    [CrossRef]
  33. Ward and Spring, “Brillouin gain in optical fibers with inhomogeneous acoustic velocity”, Proc. SPIE, 7195, 71951J–1–11 (2009).
  34. V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(22), 2022–2024 (2002), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-22-2022 .
    [CrossRef]

2008 (4)

2007 (5)

V. I. Kovalev and R. G. Harrison, “Threshold for stimulated Brillouin scattering in optical fiber,” Opt. Express 15(26), 17,625–17,630 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17625 .
[CrossRef]

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 .
[CrossRef]

Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

2006 (3)

V. I. Kovalev and R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. 31(2), 161–163 (2006), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-2-161 .
[CrossRef] [PubMed]

M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .

W. Zou, Z. He, and K. Hotate, “Two-dimensional finite element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. 18(23), 2487–2489 (2006).
[CrossRef]

2005 (4)

2002 (1)

2000 (1)

1999 (1)

E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999).
[CrossRef]

1966 (1)

C. Tang, “Saturation and Spectral Characteristics of the Stokes Emission in the Stimulated Brillouin Process,” J. Appl. Phys. 37(8), 2945–2956 (1966), http://link.aip.org/link/?JAPIAU/37/2945/1 .
[CrossRef]

1956 (1)

P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956).
[CrossRef]

Alegria, C.

Alvarez-Chavez, J. A.

Bickham, S. R.

Chen, X.

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

Cheung, E. C.

Chowdhury, D. Q.

Chryssou, C. E.

Codemard, C. A.

Crowley, A. M.

De Meritt, A. B. R. J. A.

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

Demeritt, J. A.

Dupriez, P.

Fini, J. M.

M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 .
[CrossRef]

Frede, M.

M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .

Ghalmi, S.

M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 .
[CrossRef]

Goldberg, L.

Goodno, G. D.

Gray, S.

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

Hammer, P. C.

P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956).
[CrossRef]

Harrison, R. G.

He, Z.

Hickey, L.

Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Hickey, L. M. B.

Hildebrandt, M.

M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .

Ho, J. G.

Horley, R.

Hotate, K.

Jeong, Y.

Kliner, D. A. V.

Kobyakov, A.

Koplow, J. P.

Kovalev, V. I.

Kracht, D.

M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .

Kumar, S.

Kwee, P.

M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .

Li, M.-J.

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

Liu, A.

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

Marlowe, O. J.

P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956).
[CrossRef]

McCurdy, A. H.

Mermelstein, M. D.

M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 .
[CrossRef]

Mio, N.

Mishra, R.

Moriwaki, S.

Nilsson, J.

Ozeki, T.

Payne, D.

Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Payne, D. N.

Peral, E.

E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999).
[CrossRef]

Ramachandran, S.

M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 .
[CrossRef]

Rice, R. R.

Rothenberg, J.

Ruffin, A. B.

Sahu, J.

Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Sahu, J. K.

Sauer, M.

Soh, D. B. S.

Stroud, A. H.

P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956).
[CrossRef]

Takeno, K.

Tang, C.

C. Tang, “Saturation and Spectral Characteristics of the Stokes Emission in the Stimulated Brillouin Process,” J. Appl. Phys. 37(8), 2945–2956 (1966), http://link.aip.org/link/?JAPIAU/37/2945/1 .
[CrossRef]

Thielen, P.

Turner, P.

Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Turner, P. W.

Walton, D. T.

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

Wang, J.

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

Wanzcyk, L.

Ward, B. G.

B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Inhomogeneous Anisotropic Refractive Index Tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008).
[CrossRef]

B. G. Ward, “Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture,” Opt. Express 16(12), 8532–8548 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-853 .
[CrossRef] [PubMed]

Weber, M.

Wickham, M.

Willke, B.

M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .

Yariv, A.

E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999).
[CrossRef]

Zenteno, L. A.

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

Zou, W.

IEEE J. Quantum Electron. (2)

E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999).
[CrossRef]

B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Inhomogeneous Anisotropic Refractive Index Tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. Zou, Z. He, and K. Hotate, “Two-dimensional finite element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. 18(23), 2487–2489 (2006).
[CrossRef]

J. Appl. Phys. (1)

C. Tang, “Saturation and Spectral Characteristics of the Stokes Emission in the Stimulated Brillouin Process,” J. Appl. Phys. 37(8), 2945–2956 (1966), http://link.aip.org/link/?JAPIAU/37/2945/1 .
[CrossRef]

J. Lightwave Technol. (1)

Math. Tables Other Aids Comput. (1)

P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956).
[CrossRef]

Opt. Express (8)

M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 .
[CrossRef]

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 .
[CrossRef]

M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .

M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 .
[CrossRef] [PubMed]

A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5338 .
[CrossRef] [PubMed]

B. G. Ward, “Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture,” Opt. Express 16(12), 8532–8548 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-853 .
[CrossRef] [PubMed]

W. Zou, Z. He, and K. Hotate, “Acoustic modal analysis and control in w-shaped triple-layer optical fibers with highly-germanium-doped core and F-doped inner cladding,” Opt. Express 16(14), 10006–10017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10006 .
[CrossRef] [PubMed]

V. I. Kovalev and R. G. Harrison, “Threshold for stimulated Brillouin scattering in optical fiber,” Opt. Express 15(26), 17,625–17,630 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17625 .
[CrossRef]

Opt. Lett. (6)

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V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(22), 2022–2024 (2002), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-22-2022 .
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Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-5-459 .
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W. Torruellas, M. Alam, J. Edgecumbe, K. Tankala, J. Rothenberg, and M. Wickham, “Spectral SBS model for Yb:DCF with Discrete acoustic core designs,” presented at the SPIE Defense and Security Symposium, Orlando, Florida, USA, 13–17 Apr. 2009.

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M. D. Mermelstein, A. D. Yablon, and C. Headley, “Suppression of Stimulated Brillouin Scattering in an Er-Yb Fiber Amplifier Utilizing Temperature Segmentation,” in Optical Amplifiers and Their Applications Topical Meeting Technical Digest (CD) (Optical Society of America, 2005), paper TuD3.

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P. D. Dragic, C. Liu, G. C. Papen, and A. Galvanauskas, “Optical Fiber with an Acoustic Guiding Layer for Stimulated Brillouin Scattering Suppression,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CThZ3. http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2005-CThZ3

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Supplementary Material (1)

» Media 1: MOV (197 KB)     

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Figures (8)

Fig. 1
Fig. 1

Central portion of the finite element mesh used for the electromagnetic and acoustic calculations.

Fig. 2
Fig. 2

Comparison between the Brillouin gain spectrum for a large mode area fiber with a uniform acoustic profile calculated using Eq. (24) and Eq. (37).

Fig. 3
Fig. 3

Acoustic index profile and Brillouin gain spectrum for a conventional large mode area fiber.

Fig. 4
Fig. 4

Acoustic index profile and Brillouin gain spectrum for a fiber incorporating and acoustic guiding layer.

Fig. 5
Fig. 5

Acoustic index profile and Brillouin gain spectrum for a fiber with a linearly-ramped acoustic profile.

Fig. 6
Fig. 6

Acoustic index profile and Brillouin gain spectrum for an improved fiber designed to produce a flat-top Brillouin gain spectrum. The region of high acoustic index is near the core boundary.

Fig. 7
Fig. 7

Acoustic index profile and Brillouin gain spectrum for an improved fiber designed to produce a flat-top Brillouin gain spectrum. The region of high acoustic index is near the core center.

Fig. 8
Fig. 8

(Media 1) A single-frame excerpt from an animation showing how the Brillouin gain and acoustic displacement within the fiber core vary with Brillouin frequency shift for the improved fiber with the greatest acoustic velocity at the center of the core.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

2E(x,y,z,t)n(x,y)2c22t2E(x,y,z,t)=μ02t2PNL(x,y,z,t)
PNL(x,y,z,t)=γ[uz(x,y,z,t)]E(x,y,z,t)
E(x,y,z,t)=12{ap(z)f(x,y)exp[i(kpzωpt)]+as(z)f(x,y)exp[i(ksz+ωst)]+c .c .}
uz(x,y,z,t)=12{φ(x,y)exp[i(βzΩt)]+c .c.}
as(z,t)z=14ap(γε0)(2πλ)2f|φ|ff|f
[Vt2t2+(Vl2+ηρt)2z22t2]u(x,y,z,t)=12γρE(x,y,z,t)2
Vl2=E(1ν)ρ(1+ν)(12ν)
Vt2=E2ρ(1+ν)
nac=Vl,t,bulkVl,t(x,y)
[Vt2t2+Ω2β2(Vl2+iηρΩ)]φ(x,y)=i2πnγλρapas(z)f(x,y)2
[t2+Ω2Vt2β2Vl2Vt2iΓΩVt2]φ(x,y)=i2πnγλρVt2apas(z)f(x,y)2.
[t2κ2(x,y)]φ(x,y)=Φ(x,y)
κ2(x,y)1Vt2(β2Vl2Ω2iΓΩ)
Φ(x,y)i2πnγλρVt2apas(z)f(x,y)2
Lφ=Φ
φ(x,y)=G(x,y,x,y)Φ(x,y)dxdy(GΦ)
Ps,p=12cnε0f|f|as,p|2
dPs(z)dz=gBAeffPsPp
gBAeff=(1ρc)(γε0)2(2πλ)3f|(Gg)|ff|f2,
g(x,y)f(x,y)2Vt(x,y)2
Aeff=f|f2f|f2|f
gB,peak=2π2n7p122cλ2ρVlΓ.
G(r,r)=12πK0(κ|rr|)
f(r)2=8πd2exp[8r2d2]
φ(r)12π8πd2002πK0(κr2+r2+2rrcosθ)exp[8r2d2]rdrdθ
gBAeff=(1ρVt2c)(γε0)2(2πλ)3(8πd2)2×0002πK0(κr2+r2+2rrcosθ)exp[8(r2+r2)d2]rrdrdrdθ
S(φ)={12[tφ(r)]212κ(r)2φ(r)2Φ(r)φ(r)}d2r
{x,y,φ}=k=16Ak(L1,L2){xk,yk,φk}
A1=L1(2L11)A2=L2(2L21)A3=(1L1L2)(12L12L2)A4=4L1L2A5=4L2(1L1L2)A6=4L1(1L1L2)
xφ=k=16xAk(L1,L2)φk
yφ=k=16yAk(L1,L2)φk
[xAkyAk]=J1[L1AkL2Ak]
J=[xL1xL2yL1yL2].
φ˜i=Nij(L1,L2)φj
xφ˜i=Nx,ij(L1,L2)φj
yφ˜i=Ny,ij(L1,L2)φj
S(φ)=12φTKφRφ.
K=k=17Wk[NxkTJkNxkNykTJkNykNkTJkMNk]
R=k=17Wk[ΦkTNkTJkNk].
Kφ=R
φ=K1R.
gBAeff=(1ρc)(γε0)2(2πλ)3ETV(K1R)×E(ETVE)2
V=k=17Wk[NkTJkNk]
[φ(r)ψ(r)]d2r=φTVψ.
νB=2neVl,bulknacλ

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