Abstract

Beam combining of phase-modulated kilowatt fiber amplifiers has generated considerable interest recently. We describe in the time domain how stimulated Brillouin scattering (SBS) is generated in an optical fiber under phase-modulated laser conditions, and we analyze different phase modulation techniques. The temporal and spatial evolutions of the acoustic phonon, laser, and Stokes fields are determined by solving the coupled three-wave interaction system. Numerical accuracy is verified through agreement with the analytical solution for the un-modulated case and through the standard photon conservation relation for counter-propagating optical fields. As a test for a modulated laser, a sinusoidal phase modulation is examined for a broad range of modulation amplitudes and frequencies. We show that, at high modulation frequencies, our simulations agree with the analytical results obtained from decomposing the optical power into its frequency components. At low modulation frequencies, there is a significant departure due to the appreciable cross talk among the laser and Stokes sidebands. We also examine SBS suppression for a white noise source and show significant departures for short fibers from analytically derived formulas. Finally, SBS suppression through the application of pseudo-random bit sequence modulation is examined for various patterns. It is shown that for a fiber length of 9 m the patterns at or near n=7 provide the best mitigation of SBS with suppression factors approaching 17 dB at a modulation frequency of 5 GHz.

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References

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  1. I. Dajani, C. Zeringue, and T. Shay, “Investigation of nonlinear effects in multitone-driven narrow linewidth high-power amplifiers,” IEEE J. Sel. Top. Quantum Electron.15(2), 406–414 (2009).
    [CrossRef]
  2. R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A42(9), 5514–5521 (1990).
    [CrossRef] [PubMed]
  3. 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. Express15(13), 8290–8299 (2007).
    [CrossRef] [PubMed]
  4. C. Robin and I. Dajani, “Acoustically segmented photonic crystal fiber for single-frequency high-power laser applications,” Opt. Lett.36(14), 2641–2643 (2011).
    [CrossRef] [PubMed]
  5. C. Zeringue, C. Vergien, and I. Dajani, “Pump-limited, 203 W single-frequency monolithic fiber amplifier based on laser gain competition,” Opt. Lett.36(5), 618–620 (2011).
    [CrossRef] [PubMed]
  6. F. W. Willems, W. Muys, and J. S. Leong, “Simultaneous Suppression of Stimulated Brillouin Scattering and interferometric noise in externally modulated lightwave AM-SCM systems,” IEEE Photon. Technol. Lett.6(12), 1476–1478 (1994).
    [CrossRef]
  7. G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett.35(10), 1542–1544 (2010).
    [CrossRef] [PubMed]
  8. C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett.36(14), 2686–2688 (2011).
    [CrossRef] [PubMed]
  9. M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A24(4), 1980–1993 (1981).
    [CrossRef]
  10. R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient stimulated Raman scattering,” Phys. Rev. A2(1), 60–72 (1970).
    [CrossRef]
  11. E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol.7(1), 171–174 (1989).
    [CrossRef]
  12. M. J. Damzen, V. I. Vlad, V. Babin, and A. Mocofanescu, Stimulated Brillouin Scattering Fundamentals and Applications (Institute of Physics Publishing, 2003).
  13. K. E. Gustafson, Partial Differential Equations and Hilbert Space Methods, 2nd ed. (Wiley, 1987).
  14. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).
  15. Y. Liu, Z. Lu, Y. Dong, and Q. Li, “Research on stimulated Brillouin scattering suppression based on multi-frequency phase modulation,” Chin. Opt. Lett.7(1), 29–31 (2009).
    [CrossRef]
  16. R. S. Williamson III, “Laser coherence control using homogeneous linewidth broadening.” U.S. Patent No. 7,280,568, Oct. 9 (2007).
  17. J. Edgecombe, 7 Airport Park Rd, East Granby, CT, 06026 (personal communication, 2011).
  18. D. Brown, M. Dennis, and W. Torruellas, “Improved phase modulation for SBS mitigation in kW-class fiber amplifiers,” Oral Presentation at Photonics West (2011).

2011 (3)

2010 (1)

2009 (2)

I. Dajani, C. Zeringue, and T. Shay, “Investigation of nonlinear effects in multitone-driven narrow linewidth high-power amplifiers,” IEEE J. Sel. Top. Quantum Electron.15(2), 406–414 (2009).
[CrossRef]

Y. Liu, Z. Lu, Y. Dong, and Q. Li, “Research on stimulated Brillouin scattering suppression based on multi-frequency phase modulation,” Chin. Opt. Lett.7(1), 29–31 (2009).
[CrossRef]

2007 (1)

1994 (1)

F. W. Willems, W. Muys, and J. S. Leong, “Simultaneous Suppression of Stimulated Brillouin Scattering and interferometric noise in externally modulated lightwave AM-SCM systems,” IEEE Photon. Technol. Lett.6(12), 1476–1478 (1994).
[CrossRef]

1990 (1)

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A42(9), 5514–5521 (1990).
[CrossRef] [PubMed]

1989 (1)

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol.7(1), 171–174 (1989).
[CrossRef]

1981 (1)

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A24(4), 1980–1993 (1981).
[CrossRef]

1970 (1)

R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient stimulated Raman scattering,” Phys. Rev. A2(1), 60–72 (1970).
[CrossRef]

Augst, S. J.

Bloembergen, N.

R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient stimulated Raman scattering,” Phys. Rev. A2(1), 60–72 (1970).
[CrossRef]

Boyd, R. W.

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A42(9), 5514–5521 (1990).
[CrossRef] [PubMed]

Carman, R. L.

R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient stimulated Raman scattering,” Phys. Rev. A2(1), 60–72 (1970).
[CrossRef]

Chen, X.

Crowley, A. M.

Dajani, I.

Demeritt, J. A.

Dong, Y.

Fan, T. Y.

Friesem, A. A.

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol.7(1), 171–174 (1989).
[CrossRef]

Goldizen, K. C.

Goodno, G. D.

Gray, S.

Leong, J. S.

F. W. Willems, W. Muys, and J. S. Leong, “Simultaneous Suppression of Stimulated Brillouin Scattering and interferometric noise in externally modulated lightwave AM-SCM systems,” IEEE Photon. Technol. Lett.6(12), 1476–1478 (1994).
[CrossRef]

Li, M. J.

Li, Q.

Lichtman, E.

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol.7(1), 171–174 (1989).
[CrossRef]

Liu, A.

Liu, Y.

Lu, Z.

McComb, T. S.

McNaught, S. J.

Mostowski, J.

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A24(4), 1980–1993 (1981).
[CrossRef]

Murphy, D. V.

Muys, W.

F. W. Willems, W. Muys, and J. S. Leong, “Simultaneous Suppression of Stimulated Brillouin Scattering and interferometric noise in externally modulated lightwave AM-SCM systems,” IEEE Photon. Technol. Lett.6(12), 1476–1478 (1994).
[CrossRef]

Narum, P.

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A42(9), 5514–5521 (1990).
[CrossRef] [PubMed]

Raymer, M. G.

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A24(4), 1980–1993 (1981).
[CrossRef]

Redmond, S. M.

Robin, C.

Rothenberg, J. E.

Ruffin, A. B.

Rzaewski, K.

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A42(9), 5514–5521 (1990).
[CrossRef] [PubMed]

Sanchez, A.

Shay, T.

I. Dajani, C. Zeringue, and T. Shay, “Investigation of nonlinear effects in multitone-driven narrow linewidth high-power amplifiers,” IEEE J. Sel. Top. Quantum Electron.15(2), 406–414 (2009).
[CrossRef]

Shimizu, F.

R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient stimulated Raman scattering,” Phys. Rev. A2(1), 60–72 (1970).
[CrossRef]

Thielen, P. A.

Vergien, C.

Waarts, R. G.

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol.7(1), 171–174 (1989).
[CrossRef]

Walton, D. T.

Wang, C. S.

R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient stimulated Raman scattering,” Phys. Rev. A2(1), 60–72 (1970).
[CrossRef]

Wang, J.

Weber, M. E.

Wickham, M. G.

Willems, F. W.

F. W. Willems, W. Muys, and J. S. Leong, “Simultaneous Suppression of Stimulated Brillouin Scattering and interferometric noise in externally modulated lightwave AM-SCM systems,” IEEE Photon. Technol. Lett.6(12), 1476–1478 (1994).
[CrossRef]

Yu, C. X.

Zenteno, L. A.

Zeringue, C.

C. Zeringue, C. Vergien, and I. Dajani, “Pump-limited, 203 W single-frequency monolithic fiber amplifier based on laser gain competition,” Opt. Lett.36(5), 618–620 (2011).
[CrossRef] [PubMed]

I. Dajani, C. Zeringue, and T. Shay, “Investigation of nonlinear effects in multitone-driven narrow linewidth high-power amplifiers,” IEEE J. Sel. Top. Quantum Electron.15(2), 406–414 (2009).
[CrossRef]

Chin. Opt. Lett. (1)

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

I. Dajani, C. Zeringue, and T. Shay, “Investigation of nonlinear effects in multitone-driven narrow linewidth high-power amplifiers,” IEEE J. Sel. Top. Quantum Electron.15(2), 406–414 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

F. W. Willems, W. Muys, and J. S. Leong, “Simultaneous Suppression of Stimulated Brillouin Scattering and interferometric noise in externally modulated lightwave AM-SCM systems,” IEEE Photon. Technol. Lett.6(12), 1476–1478 (1994).
[CrossRef]

J. Lightwave Technol. (1)

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol.7(1), 171–174 (1989).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (3)

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A24(4), 1980–1993 (1981).
[CrossRef]

R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient stimulated Raman scattering,” Phys. Rev. A2(1), 60–72 (1970).
[CrossRef]

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A42(9), 5514–5521 (1990).
[CrossRef] [PubMed]

Other (6)

M. J. Damzen, V. I. Vlad, V. Babin, and A. Mocofanescu, Stimulated Brillouin Scattering Fundamentals and Applications (Institute of Physics Publishing, 2003).

K. E. Gustafson, Partial Differential Equations and Hilbert Space Methods, 2nd ed. (Wiley, 1987).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).

R. S. Williamson III, “Laser coherence control using homogeneous linewidth broadening.” U.S. Patent No. 7,280,568, Oct. 9 (2007).

J. Edgecombe, 7 Airport Park Rd, East Granby, CT, 06026 (personal communication, 2011).

D. Brown, M. Dennis, and W. Torruellas, “Improved phase modulation for SBS mitigation in kW-class fiber amplifiers,” Oral Presentation at Photonics West (2011).

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Figures (12)

Fig. 1
Fig. 1

Normalized conservation equation along fiber length for different reflectivities using a) one-step Euler Method (top) and b) modified Euler method (bottom). <R> defines the time-averaged reflectivity over several transit times in the long time-limit Γ B t>>1 .

Fig. 2
Fig. 2

Comparison between the reflectivity for undepleted pump as obtained from Eq. (16) and numerical integration of the equations using a distributed fluctuating noise source. The reflectivity in the case of the numerical integration is averaged over several transit times in the long time limit.

Fig. 3
Fig. 3

Comparison between analytical SBS gain bandwidth for undepleted pump using Eq. (17) and numerical integration of the equations using a distributed fluctuating noise source. There is a departure as the single-pass SBS gain is increased. We also observe an asymptotic convergence to the spontaneous Brillouin bandwidth at low gain.

Fig. 4
Fig. 4

Numerical results (solid dots) of SBS threshold enhancement factor vs. normalized modulation frequency for various modulation amplitudes. Solid curves represent best fit for numerical results. As expected, the results indicate asymptotic convergence to the theoretical approximation of Eq. (21) in the large modulation frequency limit.

Fig. 5
Fig. 5

Results of the PSD for the laser and Stokes fields near SBS threshold for the case γ=1.435 with a) Δω=2 Γ B and b) Δω=20 Γ B . Both curves are normalized about their respective optical carrier frequencies. The results show that in the small modulation regime interactions among the various sidebands tend to feed the carrier frequency leading to a higher central Stokes mode. The PSD was generated using the total simulation time.

Fig. 6
Fig. 6

PSD of optical field driven with a white-noise source (WNS) passed through a 2 GHz band-pass filter and possessing a sinc 2 envelope. Figure 6(a) shows up to 5 nulls on each side of the carrier frequency. Figure 6(b) is the “zoomed-in” region of the primary envelope.

Fig. 7
Fig. 7

Resultant power spectral density (PSD) of optical field driven with a white-noise source (WNS) tailored to produce a Lorentzian lineshape with FWHM of 2 GHz.

Fig. 8
Fig. 8

Enhancement factor vs. normalized FWHM for the Lorentzian white noise at different lengths of fiber. The ten least steep lines correspond to fiber lengths in the range 2 m to 11 m in increments of 1 m. For long fiber, the enhancement factor approaches the theoretical limit provided by Eq. (24). For the parameters given in Table 1, a normalized linewidth of 25 corresponds to 1.4 GHz.

Fig. 9
Fig. 9

Enhancement factor vs. normalized linewidth for the sinc 2 white noise at different fiber lengths. For long fiber, the enhancement factor approaches the theoretical limit provided by Eq. (29).

Fig. 10
Fig. 10

a) Normalized power spectral density and b) “zoomed-in” region of optical field phase-modulated with PRBS pattern 2 3 1 with a modulation frequency of 2 GHz.

Fig. 11
Fig. 11

Enhancement factor vs. modulation frequency for PRBS pattern 2 3 1 for a 9 m fiber.

Fig. 12
Fig. 12

a) SBS threshold enhancement factor vs. modulation frequency for different PRBS patterns b) Sample phase as a function of time over several phonon lifetimes for n=7,17,31 at 2 GHz modulation frequency. The time is normalized to 2π τ p .

Tables (1)

Tables Icon

Table 1 Fiber simulation parameters. The density, index of refraction, Brillouin shift, and phonon lifetime parameters are based on approximate values for optical fiber at room temperature and a laser wavelength of 1064nm [14].

Equations (29)

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2 E ˜ n 2 c 2 2 E ˜ t 2 = 1 ε o c 2 2 P ˜ (nl) t 2
2 ρ ˜ t 2 Γ B q 2 2 ρ ˜ t v S 2 2 ρ ˜ = 1 2 ε o γ e 2 E ˜ 2 + f ˜
f ˜ =2iΩf(z,t) e i(qzΩt) +c.c.
f(z,t) f * (z,t) =Qδ(zz')δ(tt')
c n A L z + A L t = iω γ e 2 n 2 ρ o ρ A s
c n A S z + A S t = iω γ e 2 n 2 ρ o ρ A L
2 ρ t 2 +( Γ B 2iΩ ) ρ t +( Ω B 2 Ω 2 iΩ Γ B )ρ= ε o γ e q 2 A L A S 2iΩf
2 ρ t 2 +( Γ B 2i Ω B ) ρ t i Ω B Γ B ρ= ε o γ e q 2 A L A S 2i Ω B f
d A L dt =iσρ A s
d A S dt =iσ ρ A L
α Γ B d 2 ρ d t 2 +( αi ) dρ dt i Γ B 2 ρ=χ A L A S if
f j,k = nQ (Δt) 2 c S j,k
ρ j,k = nQ c Γ B S j,k
< | E L (z,t) | 2 >< | E S (z,t) | 2 >=C
A S (z,τ)= i γ e ω 2n ρ 0 c A L 0 τ dτ' 0 L dz' e Γ B (ττ')/2 f * (z',τ') I 0 ( [ G Γ B (zz')(ττ')/L ] 1/2 )
R= ( γ e ω 2 ρ o nc ) 2 QL Γ B e G/2 [ I o (G/2) I 1 (G/2) ]
P S (z=0,ω)[ exp( G (Γ / B 2) 2 ω 2 + ( Γ B /2 ) 2 )1 ]
φ(t)=γsin(Δωt)
E ˜ L (0,t)= A L 0 [ J 0 (γ)cos(ωt)+ J 1 (γ)cos(ω+Δω)t J 1 (γ)cos(ωΔω)t + J 2 (γ)cos(ω+2Δω)t J 2 (γ)cos(ω2Δω)t+..... ]
P L,n J n 2 (γ)
P th P th 0 = 1 J n,max 2 ( γ )
φ(t)φ( t ) =Cδ(t t )
φ(t) =0
P th P th 0 =1+ Δω Γ B
g eff ( ω S )= g 0 ( Γ B /2 )L I(ω)dω ( Γ B /2 ) 2 + (ω ω S Ω B ) 2
g eff,max = g 0 ( Γ B /2 )L A sin 2 [(ω ω L )π/Δω]dω [ ( Γ B /2) 2 + (ω ω L ) 2 ] [(ω ω L )π/Δω] 2
g eff,max = g 0 Γ B π I (0) L 2 (Δω) 2 a 2 ( sin c 2 w sin 2 w a 2 + w 2 ) dw
g eff,max = g 0 Γ B π 2 I (0) L 2 (Δω) 2 a 2 ( 1 1 2a + e 2a 2a )
P th P th 0 = π ( e π Γ B /Δω 1 ) Δω Γ B +π

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