Brillouin gain in optical fibers with ... - SPIE Digital Library

Brillouin gain in optical fibers with inhomogeneous acoustic velocity Benjamin G. Ward and Justin B. Spring United States Air Force Academy Department of Physics, HQ USAFA/DFP, 2354 Fairchild Dr. Ste 2A167, USAF Academy CO 80840, USA ABSTRACT The power available from narrow-linewidth single-transverse-mode fiber amplifiers is primarily limited by the onset of stimulated Brillouin scattering. One approach for increasing the SBS threshold that has shown recent promise is to tailor the acoustic velocity within the fiber cross-section to suppress Brillouin gain.1–3 Relating the SBS threshold to an acousto-optic effective area has yielded a theory4 which contradicts experimental measurements that indicate the nonlinear optical effective area of the tested SBS suppressing and Higher Order Mode (HOM) fibers is of primary importance in the nonlinear process.5 In this work, we present a new formalism for determining the Brillouin gain in fibers with inhomogeneous acoustic velocity which may be implemented with a wide variety of computational methods. We find that the Brillouin gain amplitude and spectrum are independent of the acousto-optic effective area and that they reduce to the bulk result6, 7 for conventional step-index fibers. Implementing a finite-element method, we find that an SBS-suppressing design employing a negative focal length acoustic lens1 exhibits a broadened gain spectrum and reduced gain amplitude relative to step-index fibers. The SBS threshold of this fiber is increased by 8.4 dB relative to a standard large mode area fiber, each with an identical 6 meter length. Designs that further flatten the Brillouin gain spectrum have the potential to further increase the SBS threshold leading to higher single-frequency output power from devices incorporating these fibers. Keywords: Fiber Optics, Fiber Lasers, Stimulated Brillouin Scattering, Power Scaling

1. INTRODUCTION High-power single-frequency optical sources are useful for several applications including atmospheric LIDAR,8 gravitational wave interferometry,9 and coherent beam combination.10 Due primarily to their efficiency, ruggedness, compact size and beam quality, fiber sources are a leading candidate for these applications. Typically a low-power single-frequency diode-pumped solid-state, semiconductor, or fiber oscillator is amplified in a series of diode-pumped double-clad fiber amplifiers to form a multi-stage Master Oscillator Power Amplifier (MOPA) system to achieve the required output. The main drawback of this approach is that the long length of fiber required to absorb the pump in double-clad fibers and extremely high optical intensity within the core lead to relatively low thresholds for parasitic non-linear processes. For single-frequency sources, the process with the lowest threshold is the stimulated Brillouin scattering (SBS) process which leads to the amplification of a Stokes wave propagating in the opposite direction of the amplifier signal (commonly referred to as the pump wave in the context of SBS) which may lead to destabilization of the amplifier leading to optical damage due to spurious pulses. Although the maximum tolerable Stokes wave amplitude depends on the particular device in question, it has been proposed that the SBS threshold be defined as the amplifier input signal power level that results in a 1% reflection or an SBS reflectivity of -20 dB. The seed for the counter-propagating Stokes wave is due to pump radiation that is scattered from thermally-excited vibrations of the fused-silica matrix comprising the core of the fiber. These vibrations are also known as acoustic excitations or phonons and occur with a spectrum of frequencies. The peak frequencies occur when they are detuned from the pump frequency by an amount corresponding to an acoustic propagation velocity (speed of sound) within the fiber. This is otherwise known as a phase matching condition. Multiple acoustic velocities Further author information: (Send correspondence to B. G. W.) B. G. W. : E-mail: [email protected], Telephone: 1 719 333 3055 Fiber Lasers VI: Technology, Systems, and Applications, edited by Denis V. Gapontsev, Dahv A. Kliner, Jay W. Dawson, Kanishka Tankala, Proc. of SPIE Vol. 7195, 71951J © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.812882 Proc. of SPIE Vol. 7195 71951J-1

arise due to inhomogenaities within the fiber. The Brillouin resonances are broadened due to dissipative forces within the glass that damp out the vibrations. This type of scattering is commonly referred to as spontaneous Brillouin scattering or SpBS. The width of the SpBS spectrum in bulk glass is approximately 150 MHz.11 As it propagates counter to the signal, the Stokes wave experiences Brillouin gain which is the result of a non-linear polarization created through the electrostrictive effect by the interference of the pump and Stokes waves. This effect also requires the phase matching condition to be met. The Brillouin gain spectrum (BGS) determines how strongly a given frequency component of SpBS is amplified. The width of the BGS is typically approximately half the width of the SpBS spectrum. As the pump power is increased, the Brillouin gain spectrum dominates the SpBS spectrum resulting in a Stokes spectrum that is peaked at the BGS peak, although typically the SpBS spectrum and the BGS have their peaks at the same frequency. The simplest ways to increase the SBS threshold are to reduce the length and increase the mode field diameter of the fiber. However, as the length of the fiber decreases, so does the fraction of absorbed pump thus reducing optical conversion efficiency. As the mode field diameter is increased past a certain point, the beam quality suffers due to the amplification of higher order transverse modes. Modifying the BGS along the length of the fiber by varying the conditions imposed on the fiber such as the temperature12 and stress13 comprises another approach for increasing the SBS threshold. If the BGS peak is shifted in frequency along the length of the fiber such that its overlap with the Stokes spectrum is continually reduced, the SBS threshold is increased. This approach may be employed with discrete or continuous changes in the BGS peak frequency over the length of the fiber. The effectiveness of this strategy is limited by the maximum frequency shift that can be achieved over the length of the fiber. For thermal shifts, this is determined by the maximum temperature difference that can be maintained between different parts of the fiber. An alternative approach that has received increased attention lately is to attempt to decouple the Stokes optical wave and acoustic excitations spatially throughout the cross-section of the fiber core. This method is based on the assertion that the acoustic excitations can be expanded into a set of freely-propagating acoustic modes, analogous to transverse optical modes, and that only a small subset of these modes overlap with the optical mode.2–4, 14, 15 Minimizing the combined overlap of these acoustic modes with the optical mode through tailoring of the acoustic velocity profile throughout the fiber cross section degrades the efficiency of the Brillouin amplification leading to a higher SBS threshold. An alternative interpretation that has been put forward is that the acoustic velocity variations may be configured to produce the acoustic equivalent of an optical diverging lens that spreads out the amplitude of of the acoustic excitations as they propagate along the fiber core.1 In each of these cases the key design feature of these fibers is a tailored acoustic velocity profile. Several detailed acoustic designs for SBS-suppressing fiber have been presented in the literature including fibers with an acoustic guiding layer,3 fibers incorporating a negative acoustic lens,1 and fibers with an interfacefree linearly-ramped acoustic velocity profile.2 The experimentally observed SBS threshold of these fibers may be compared to conventional large mode area fibers which naturally exhibit some degree of inhomogeneity in their acoustic velocities,1, 2 as well as other fiber designs such as higher-order-mode fibers5 and microstructured fibers.16 Relative SBS suppression in some of these fibers appears to be reasonably attributable to a decrease in relative spatial overlap between acoustic and optical guided modes. For some fibers, however, this does not hold true. One example of this is a higher order mode (HOM) fiber where it proved impossible to explain the experimentally observed SBS reflectivity in terms of the acousto-optical overlap.5 The correspondence between acoustic velocity and the peak frequency of the BGS suggests that fibers incorporating substantial variations in the acoustic velocity would exhibit a broadened BGS compared to that of fibers with only small acoustic velocity variations. Taking this one step further, tailored acoustic velocity profiles may increase the SBS threshold by broadening the BGS achieving essentially the same effect as shifting the BGS peak along the fiber by thermomechanical methods. In this paper we present a new method of calculating the Stokes spectrum, BGS, SpBS spectrum and SBS reflectivity for fibers with arbitrary acoustic and optical index profiles. We then apply the method to a range of large mode area step index fiber designs with various acoustic velocity profiles designed to increase the SBS threshold.

Proc. of SPIE Vol. 7195 71951J-2

2. MODEL OF ACOUSTIC EXCITATIONS In the absence of external sources, longitudinal acoustic excitations may propagate along the axis of the fiber. Due to its amorphous structure, silicate glasses may be modeled as possessing linear isotropic elastic properties. One way of deriving the appropriate wave equation is to use a variational principal that relies on calculating the change in mechanical energy due to a longitudinal displacement field that varies throughout the fiber cross section. Carrying this out reveals that although the displacement is purely longitudinal, the shear velocity and thus the Poisson ratio of the glass plays a role. Assuming an acoustic mode that is localized in the region of the core (or guided acoustic mode), it may be viewed as a traveling longitudinal wave along the fiber axis and simultaneously a standing shear wave in the direction perpendicular to the fiber axis. The acoustic excitations also experience dissipation which can be characterized by the time constant for the exponential decay of their amplitude. Incorporating a source term created by the electrostrictive response to the interference of the pump and Stokes waves results leads to a partial differential equation (PDE) incorporating all of the properties of the fiber relevant to the SBS process.

Et

∂2 ∂2 + 2 2 ∂x ∂y

2 ∂ ∂ ∂2 1 − ρ + El + η u(r, t) = − γ∇E(r, t)2 2 2 ∂t ∂z ∂t 2

(1)

where γ is the electrostrictive constant, η is the viscosity, ρ is the mass density and the z axis is along the propagation axis of the fiber. The elastic constants may be expressed Et =

E(1 − ν) (1 + ν)(1 − 2ν)

(2)

E 2(1 + ν)

(3)

El =

in terms of the Young’s modulus E and the Poisson ratio ν. For single-frequency MOPAs, the spectrum of the pump wave is much narrower than that of the Stokes spectrum. Assuming exponential growth of each frequency component of the Stokes along the fiber and that the pump is not depleted results in a particularly simple form for the electrostrictive term in the equation that does not depend explicitly on the acoustic displacement field. E(r, t) = E(x, y) exp [−i(βz − Ωt)]

(4)

It is then evident that Equation is a non-homogeneous PDE the solution of which is broken into two parts according to the existence and uniqueness theorems pertaining to the solutions of such equations. These state that the solution is the sum of what is known as the particular solution and a superposition of solutions to the corresponding homogeneous equation (the same PDE with the source term omitted). This superposition allows the boundary and initial conditions of the PDE to be satisfied by the combined solution. The corresponding physical interpretation is that the solutions to the homogeneous equation are the freely propagating acoustic modes. The acoustic modes thus do not fully describe the acoustic excitations responsible for the SBS process. The particular solution is required as well. The most straightforward way to obtain particular solutions to nonhomogeneous PDEs is to integrate the source term against the Green’s function corresponding to the differential operator characterizing the PDE which depends on the acoustic velocity profile of the fiber. Carrying out this process leads to an expression for the Brillouin gain term in the equation describing the evolution of the Stokes wave, assuming the fiber supports a single transverse optical mode, from which the BGS may be extracted. 2 3 2π 2 γ E(x, y)|L−1 (ωs )E(x, y)2 |E(x, y) gB (ωs ) = (5) Aeff ρc ε0 λ E(x, y)|E(x, y)2 with L(ωs ) =

−ivt2

∂2 ∂2 + ∂x2 ∂y 2

+ iβ 2 vl2 − i(ωp − ωs )2 − Γ(ωp − ωs )


(6)

where c is the vacuum speed of light, ε0 is the permeability of free space, λ is the pump vacuum wavelength, ωp ωs are the optical frequencies of the pump and Stokes waves, Γ is the phonon decay rate,we have defined the spatially-inhomogeneous longitudinal and shear acoustic velocities vt,l =

Et,l ρ

we have used the notation f (x)|g(x)|f (x) ≡

1/2 ,

f (x)2 g(x) dx dy

for which the region of integration is the fiber cross section, −1 2 L (ωs )E(x, y) ≡ G(ωs , x, x , y, y )E(x , y )2 dx dy ,

(7)

(8)

(9)

and the Green’s function G(ωs , x, x , y, y ) is defined by L(ωs )G(ωs , x, x , y, y ) = δ 2 (x − x , y − y ).

(10)

The traditional definition of the nonlinear effective area Aeff does not play a role, however, its symbol is included in the definition of the Brillouin gain to facilitate comparison to the bulk result. The acoustic index is defined as nac (x, y) = vl,0 /vl (x, y) where vl,0 is the longitudinal acoustic velocity of bulk silica which works out to 5,944 m/s for a Young’s modulus of 73 GPa, Poisson ratio of 0.17, and mass density of 2200 kg/m3 . Equation 5 reduces to the bulk result6, 7 for acoustic plane waves in a homogeneous medium. Except for very simple, and therefore uninteresting acoustic profiles, the Green’s function is impossible to obtain analytically. However, if the PDE is discretized, then the linear operator L(ωs ) takes the form of a matrix and the particular solution may be obtained by multiplying the discretized source term E(x , y )2 by the inverse of the matrix. One advantage of this approach is that it holds for any discretization scheme including frequency and time domain finite difference methods as well as finite element methods. The SpBS spectrum may be obtained by considering the statistical mechanics of thermally excited acoustic modes. The stochastic nature of this process in which the homogeneous solutions (acoustic modes) come into play stands in contrast to the manner in which electrostriction leads to Brilloin gain as embodied in the particular solution to the non-homogeneous wave equation. Including N acoustic modes, the SpBS scattering coefficient is. κsp (ωs ) =

2 2 N 2π γ 1 kT Γ 2 v 2 Aao (Ω − ω + ω )2 + Γ2 16π ε λ ρn 0 m p s l,m m m=1

(11)

4 where vl,m , Aao m , and Ωm are the longitudinal velocity, acousto-optical effective area, and frequency of propagating acoustic mode m.

At this point,it is illustrative to discuss a driven, damped mechanical harmonic oscillator (mass with friction attached to a spring) as an analogy. Specifically, its frequency of oscillation. An the absence of a driving term, a slightly damped harmonic oscillator when moved from its equilibrium position and released will exhibit oscillation with gradually decaying amplitude at a frequency ω1 which depends on the mass, the spring constant, and the coefficient of friction the mass experiences. Each of these is a property of the oscillator itself. These relaxation oscillations are analogous to the propagating acoustic modes in the SBS case. If a force sinusoidal in time with a frequency ω2 is applied to the mechanical oscillator, after a time, the oscillator will oscillate with frequency ω2 which is completely independent of ω1 and a constant amplitude. Thus the oscillation frequency does not depend on the properties of the oscillator. However, the closer that ω2 is to ω1 , the larger the amplitude of the oscillation will be, that is to say the oscillator approaches resonance. Similarly, in the SBS case, the oscillation frequency of the acoustic excitations within the fiber leading to Brillouin gain does not depend on the frequency of freely propagating acoustic modes that are phase matched to the optical field, but rather the frequency of the optical field itself which is the difference between the pump frequency and the frequency component of the Stokes wave for which the value of the gain is sought. As in the case of the mechanical oscillator, if the Stokes


Figure 1. Geometry of the finite element mesh used for the optical calculations as well as the discretized Green’s function and the propagating acoustic modes. The diameter of the computational region is 55 µm and it incorporates 18,816 elements and 37,969 nodes.

frequency component is near that of a phase-matched propagating acoustic mode, resonance can occur leading to as SpBS spectrum that is similar to the BGS. In the SBS case, the transverse overlap integral between the optical mode and the acoustic excitation field also influences the amplification of the Stokes wave. Incorporating the SpBS and BGS results in the equation for the evolution of the power Ps (z, ωs ) carried by the spectral components of the counter propagating Stokes wave in the absence of pump depletion in a passive fiber. −

gB (ωs ) dPs (z, ωs ) = Pinp Ps (z, ωs ) + κsp (ωs ) dz Aeff

(12)

where Pinp is the launched pump power. This equation may be solved analytically and integrated over frequency to yield the SBS reflectivity ∞ Ps (0) exp[G(ωs )] − 1) = κsp (ωs )Lint (13) RSBS ≡ dωs Pinp G(ωs ) −∞ where G(ωs ) =

gB (ωs ) Lint Pinp Aeff

(14)

and Lint is the length of fiber. Analysis of active fibers requires the incorporation of additional terms, beyond the scope of this paper, that characterize the gain that the pump and Stokes wave experience.


1 0.8

acoustic index

optical intensity (arbitrary units)

1.1

0.6 0.4

1.05

1

0.2 0 −30

−20

−10

0 10 position (μm)

20

30

0.95 −30

−20

−10

(a)

0 10 position (μm)

20

30

(b)

Figure 2. (a) Optical mode field intensity profile for all fibers examined here (b) and acoustic index profile for the control fiber design examined here.

3. NUMERICAL RESULTS We implement a finite element scheme for the optical and acoustic calculations dividing a circular computational region into curvilinear triangular elements. For the fully-vectorial optical calculations, each field component is interpolated by a quadratic polynomial over the interior of the element to match values specified at 6 nodes corresponding to the three vertices of each element and the three midpoints of each curved boundary. For the acoustic calculations, the longitudinal displacement field is identically represented. Figure 1 depicts the structured symmetric geometry of the mesh. Thus E(x, y) is represented by a vector with 3N = 113, 907 elements and the local intensity E(x, y)2 a vector of N = 37, 969 elements. Likewise, the longitudinal acoustic displacement field uz (x, y) is represented by a vector of length N and the linear differential operator L is represented by a matrix of dimension N × N . The circular geometry of the mesh allows the step index core to be represented faithfully. Each element of the mesh assumes a constant values of refractive index and longitudinal and shear elastic velocities. One key assumption in our calculations is that the Poisson ratios for regions of different acoustic velocities are the same so that the ratio of shear and longitudinal velocities is constant throughout the fiber. Acoustic modes are first sought in the range (νm,min − 100 MHz) < νm < (νm,max + 200 MHz) where νm,min,max =

2vmin,maxneff λp

(15)

,vmin,max are the minimum and maximum longitudinal acoustic velocities in the fiber,neff is the effective index of the fundamental optical mode and Ω ≡ 2πν. These modes are then used to assemble κsp using Equation 11. The corresponding range of Stokes frequencies ωs = ωp − Ω is then divided into 80 frequency steps for computation of the BGS using Equation 5. Finally, the Stokes spectrum and reflectivity for a given length of fiber and input power are calculated using Equation 13. We examine fibers with a core diameter of 20 μm and a numerical aperture of 0.06, which are typical for large mode are fibers. The optical mode field intensity profile is shown in Figure 2 (a) for which the nonlinear effective area is 285μm2. For all of the results presented here, the fiber length Lint was taken to be 6 meters and the pump wavelength λ was taken to be 1083nm. Acoustic index differences of up to Δnac = 0.09 have been reported, so we restricted our designs so that nac falls within this range. Furthermore, we restrict our investigations to designs for which the acoustic profile extends throughout the entire core of the fiber but not beyond. In order to compare SBS-suppressing fiber designs to conventional large mode area fiber designs, we first calculate the SpBS spectrum, BGS, Stokes spectrum, and SBS threshold for a conventional fiber described by an acoustic profile shown in Figure 2 (b). We will refer to this fiber as the control fiber. The BGS peak occurs at a frequency of about 15.84 MHz and is well described by the Lorentzian functional form characteristic of the BGS in bulk silica. The SBS threshold power of 62.5 W corresponds to a Brillouin gain coefficient gB = 2.1 × 10−11m/W which is in reasonable agreement to previous results.12, 17


Fiber Length = 6 m

−9

x 10

gB/Aeff

1

0.04

0.8 0.6

0.02

0.4

−1

0.06

1.2

SBS Reflectivity (GHz )

κsp (m−1)

1.4

0.2 0 15.7

P = 62.5 W, R = −20 dB P = 59 W, R = −25 dB P = 56 W, R = −30 dB

0.6

κsp

−1

1.6

0.7

0.08

gB/Aeff(m−W)

1.8

0.5

0.4

0.3

0.2

0.1

15.8 15.9 Stokes Frequency (GHz)

0 16

0 15.7

15.8

15.9

16

16.1

16.2

16.3

Stokes Frequency Shift(GHz)

(a)

(b)

1.1

1.1

1.05

1.05

acoustic index

acoustic index

Figure 3. SpBS spectrum and BGS (a) and Stokes spectrum (b) for the control fiber examined here.

1

0.95 −30

−20

−10

0 10 position (μm)

20

30

1

0.95 −30

−20

(a)

−10

0 10 position (μm)

20

30

(b)

Figure 4. Acoustic index profile for the (a) V-shaped fiber and (b) acoustic guiding layer fiber designs examined here.

The first SBS-suppressing design considered exhibits an acoustic index which ramps from its minimum value at the center of the core to its maximum value at the boundary of the core as shown in Figure 2 (b). This design is similar to those reported previously.1, 2 For this design, both the SpBS spectrum and BGS exhibit a broad peak, however the peak frequencies differ by approximately 500 MHz as shown in Figure 5 (a). As the SBS threshold of -20dB reflectivity corresponding to a pump input power of 433 Watts is approached, the spectral peak at 15.7 GHz narrows but the overall spectrum is still 200 MHz wide as shown in Figure 5 (b). The acoustic guiding layer fiber examined here has the acoustic index boundary set arbitrarily at a radius of half of the core radius. Its profile is shown in Figure 4. This design is also similar to those reported previously.14 Its optical mode profile is the same as that of the V-shaped fiber. This fiber has a much lower SBS threshold than the V-shaped fiber, 129 Watts at -20 dB reflectivity. The SpBS spectrum and BGS each exhibit two widely separated peaks. The high frequency peak near 16.3 GHz matches the bulk Brillouin frequency of the glass at the center of the core. The low frequency peak corresponding to the acoustic index of the guiding layer in the outer portion of the core is seen at a frequency of 14.9 GHz very close to the bulk Brillouin frequency of the glass within the layer of 15.0 GHz. Interestingly the spontaneous scattering peak is stronger at the low frequency but the BGS peak is stronger at the high frequency and thus dominates the Stokes spectrum as power is increased. At threshold, the spectrum is less than 50 MHz wide. Apparently, the guiding layer boundary should be moved toward the outside of the fiber in order to promote increased acousto-optical interaction at lower frequencies to achieve better suppression performance. The results for the V-shaped fiber and the AGL fiber suggest that a fiber with flat BGS and SpBS spectrum may be created by combining the aspects of each. One such attempt has led to the results shown in Figure 7. Both the SpBS spectrum and BGS show a region of nearly constant amplitude over a range of 1.3 GHz, however there are still pronounced peaks at the ends of the spectra. Eventually the low frequency BGS


Fiber Length = 6 m

−10

3

x 10

0.1

0.012 gB/Aeff

2

0.008

1.5

0.006

1

0.004

0.5

0.002

0

15

15.5 16 Stokes Frequency (GHz)

16.5

−1

gB/Aeff(m−W)−1

κsp (m−1)

0.01


κsp

2.5

P = 433 W, R = −20 dB P = 410 W, R = −25 dB P = 387 W, R = −30 dB

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0

0 15.2

15.4

15.6

15.8

16

16.2


(a)

(b)

Figure 5. SpBS spectrum and BGS (a) and Stokes spectrum (b) for the V-shaped fiber design examined here.

Fiber Length = 6 m

−10

3.5

x 10

0.5

0.035 g /A

κsp (m−1)

0.025

2

0.02

1.5

0.015

1

0.01

0.5

−1

0.005 15


16.5


0.03

2.5

0


0.45

eff

gB/Aeff(m−W)−1

B

κsp

3

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0

0 16.2

16.25

16.3

16.35

16.4

16.45

16.5


(a)

(b)

Figure 6. SpBS spectrum and BGS (a) and Stokes spectrum (b) for the acoustic-guiding-layer fiber design examined here.

Fiber Length = 6 m

−10

3.5

x 10

0.5

0.012 gB/Aeff

0.006 1.5 0.004

1

0.002

0.5 0

15


16.5

0

gB/Aeff(m−W)−1

κsp (m−1)

0.008

2

−1

0.01

sp

2.5


0.45


κ

3

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 14.8

14.85

14.9

14.95

15

15.05

15.1

15.15

15.2


(a)

(b)

Figure 7. SpBS spectrum and BGS (a) and Stokes spectrum (b) for the hybrid design examined here.


15

10

10

5

5 y (μ m)

y (μ m)

15

0

0

−5

−5

−10

−10

−15

−15

−10

−5

0

5

10

15

−15

x (μ m)

−15 −10

−5

(a)

10

5

5 y (μ m)

10

y (μ m)

15

0

−5

−10

−10 0 x (μ m)

−15

10

−10

0 x (μ m)

15

15

10

10

5

5

0

−5

−10

−10

−10

−5

0

10

15

0

−5

−15

10

(d)

y (μ m)

y (μ m)

(c)

−15

15

0

−5

−10

10

(b)

15

−15

0 5 x (μ m)

5

10

15

−15

−15

−10

−5

0

x (μ m)

x (μ m)

(e)

(f)

5

Figure 8. Acoustic displacement profiles for (a) the control fiber at its BGS peak at 15.83 GHz, (b) the V-shaped fiber at its BGS peak at 15.7 GHz, (c) the AGL fiber at its BGS peak of 14.95 GHZ, (d) the AGL fiber at its BGS peak of 16.33 GHz, (e) the hybrid fiber at its BGS peak of 14.95 GHz, and (f) the hybrid fiber at its BGS peak of 16.33 GHz. Darker shading indicates increased displacement.

peak dominates the Stokes spectrum leading to a threshold power of 400W, slightly inferior to the V-shaped fiber Additional insight may be obtained by examining the 2-dimensional acoustic displacement profiles for each fiber at the peaks of the BGS. The control fiber exhibits an acoustic displacement that matches the optical mode field intensity to a high degree as is to be expected. The V-shaped fiber exhibits reduced response at the center of the core where the optical intensity is highest. The AGL fiber exhibits a high degree of uniform response across the center region of the core as well as within the guiding layer leading to efficient Brillouin amplification of the Stokes wave. The hybrid fiber exhibits a somewhat reduced response in the center and boundary regions


of the fiber leading to a higher SBS threshold relative to the AGL fiber. The spacing of the acoustic ripples also provides a visual indicator of the behavior of the acoustic index. For example, the V-shaped fiber exhibits ripples that become more closely spaced as the boundary of the core is approach reflecting the fact that the shear acoustic velocity is decreasing as it indeed is. The AGL fiber exhibits ripples that are constantly-spaced at a close interval within the AGL and then spaced at a larger constant interval outside of the fiber core reflecting the constant highly elevated acoustic index in the guiding layer and then a constant lower elevated acoustic index in the cladding.

4. CONCLUSION The results presented here support the conclusion that the SBS threshold may be increased through tailored acoustic profiles in optical fiber. V-shaped, acoustic guiding layer, and hybrid designs were calculated to exhibit SBS thresholds 8.4, 3.1, and 8.1 dB higher respectively than a standard design with an identical nonlinear effective optical area. The calculations predicted a threshold of 62.5 Watts for a 6 meter long standard large mode area fiber with a nonlinear effective area of 285 μm2 which is consistent with previously reported results. The theoretical method employed derives the spontaneous Brillouin scattering spectrum from the thermally excited population of propagating acoustic modes. The Brillouin gain spectrum, however, is found to be independent of these freely propagating modes and is determined instead by the electrostrictive response of the fiber to the propagating pump and Stokes waves as embodied in the solutions to a nonhomogeneous wave equation. To the authors’ knowledge, this result contradicts all previous reports on the BGS of fibers with inhomogeneous acoustic profiles. Further experimental and theoretical investigation is therefore warranted to resolve the issue.

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, for computational support, and Marc Mermelstein, OFS laboratories, for helpful discussions.

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