Abstract

We present a propagation model for the dynamics of distributed feedback Brillouin lasers. The model is applied to the recently demonstrated DFB Brillouin laser based on a π-phase shifted grating in a highly nonlinear silica fiber. Steady state results agree with the experimental values for threshold and efficiency. We also simulate a DFB Brillouin laser in chalcogenide and find sub-milliwatt thresholds and the possibility of centimeter-long Brillouin-DFB’s.

© 2013 OSA

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  1. V. E. Perlin and H. G. Winful, “Distributed feedback fiber Raman laser,” IEEE J. Quantum Electron.37(1), 38–47 (2001).
    [CrossRef]
  2. V. E. Perlin and H. G. Winful, “Stimulated Raman scattering in nonlinear periodic structures,” Phys. Rev. A64(4), 043804 (2001).
    [CrossRef]
  3. Y. Hu and N. G. R. Broderick, “Improved design of a DFB Raman fibre laser,” Opt. Commun.282(16), 3356–3359 (2009).
    [CrossRef]
  4. P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Raman fiber distributed feedback lasers,” Opt. Lett.36(15), 2895–2897 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B8(2), 403–412 (1991).
    [CrossRef]
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    [CrossRef]
  17. N. J. Baker, M. A. F. Roelens, S. Madden, B. Luther-Davies, C. M. de Sterke, and B. J. Eggleton, “Pulse train generation by soliton fission in highly nonlinear chalcogenide (As2S3) waveguide Bragg grating,” Electron. Lett.45(15), 799–801 (2009).
    [CrossRef]
  18. H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
    [CrossRef] [PubMed]
  19. P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2(1), 011008 (2012).
    [CrossRef]

2013

T. Kremp, K. S. Abedin, and P. S. Westbrook, “Closed-form approximations to the threshold quantities of distributed-feedback lasers with varying phase shifts and positions,” IEEE J. Quantum Electron.49(3), 281–292 (2013).
[CrossRef]

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

2012

2011

2009

Y. Hu and N. G. R. Broderick, “Improved design of a DFB Raman fibre laser,” Opt. Commun.282(16), 3356–3359 (2009).
[CrossRef]

N. J. Baker, M. A. F. Roelens, S. Madden, B. Luther-Davies, C. M. de Sterke, and B. J. Eggleton, “Pulse train generation by soliton fission in highly nonlinear chalcogenide (As2S3) waveguide Bragg grating,” Electron. Lett.45(15), 799–801 (2009).
[CrossRef]

2006

2003

2001

V. E. Perlin and H. G. Winful, “Distributed feedback fiber Raman laser,” IEEE J. Quantum Electron.37(1), 38–47 (2001).
[CrossRef]

V. E. Perlin and H. G. Winful, “Stimulated Raman scattering in nonlinear periodic structures,” Phys. Rev. A64(4), 043804 (2001).
[CrossRef]

2000

1991

1990

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

Abedin, K. S.

Aggarwal, I.

Agrawal, G. P.

Alam, S.

Alam, S. U.

Baker, N. J.

N. J. Baker, M. A. F. Roelens, S. Madden, B. Luther-Davies, C. M. de Sterke, and B. J. Eggleton, “Pulse train generation by soliton fission in highly nonlinear chalcogenide (As2S3) waveguide Bragg grating,” Electron. Lett.45(15), 799–801 (2009).
[CrossRef]

M. Shokooh-Saremi, V. G. Ta’eed, N. J. Baker, I. C. M. Littler, D. J. Moss, B. J. Eggleton, Y. Ruan, and B. Luther-Davies, “High-performance Bragg gratings in chalcogenide rib waveguides written with a modified Sagnac interferometer,” J. Opt. Soc. Am. B23(7), 1323–1331 (2006).
[CrossRef]

Bashkansky, M.

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]

Broderick, N. G. R.

Y. Hu and N. G. R. Broderick, “Improved design of a DFB Raman fibre laser,” Opt. Commun.282(16), 3356–3359 (2009).
[CrossRef]

Camacho, R.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2(1), 011008 (2012).
[CrossRef]

Choi, D. Y.

Cox, J. A.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Davids, P.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2(1), 011008 (2012).
[CrossRef]

de Sterke, C. M.

N. J. Baker, M. A. F. Roelens, S. Madden, B. Luther-Davies, C. M. de Sterke, and B. J. Eggleton, “Pulse train generation by soliton fission in highly nonlinear chalcogenide (As2S3) waveguide Bragg grating,” Electron. Lett.45(15), 799–801 (2009).
[CrossRef]

C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B8(2), 403–412 (1991).
[CrossRef]

Dutton, Z.

Eggleton, B. J.

Florea, C.

Grujic, T.

Hile, S.

Hu, Y.

Y. Hu and N. G. R. Broderick, “Improved design of a DFB Raman fibre laser,” Opt. Commun.282(16), 3356–3359 (2009).
[CrossRef]

Ibsen, M.

Jackson, K. R.

Jarecki, R.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Kremp, T.

Lee, H.

Li, E. B.

Littler, I. C. M.

Liu, X.

Luther-Davies, B.

Madden, S.

N. J. Baker, M. A. F. Roelens, S. Madden, B. Luther-Davies, C. M. de Sterke, and B. J. Eggleton, “Pulse train generation by soliton fission in highly nonlinear chalcogenide (As2S3) waveguide Bragg grating,” Electron. Lett.45(15), 799–801 (2009).
[CrossRef]

Madden, S. J.

Mcfarlane, H.

Moss, D. J.

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]

Nicholson, J. W.

Ogusu, K.

Olsson, R. H.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Pant, R.

Perlin, V. E.

V. E. Perlin and H. G. Winful, “Distributed feedback fiber Raman laser,” IEEE J. Quantum Electron.37(1), 38–47 (2001).
[CrossRef]

V. E. Perlin and H. G. Winful, “Stimulated Raman scattering in nonlinear periodic structures,” Phys. Rev. A64(4), 043804 (2001).
[CrossRef]

Porque, J.

Poulton, C. G.

Pureza, P.

Qiu, W.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Rakich, P.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2(1), 011008 (2012).
[CrossRef]

Rakich, P. T.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Reinke, C.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2(1), 011008 (2012).
[CrossRef]

Robert, B. D.

Roelens, M. A. F.

N. J. Baker, M. A. F. Roelens, S. Madden, B. Luther-Davies, C. M. de Sterke, and B. J. Eggleton, “Pulse train generation by soliton fission in highly nonlinear chalcogenide (As2S3) waveguide Bragg grating,” Electron. Lett.45(15), 799–801 (2009).
[CrossRef]

Ruan, Y.

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]

Sanghera, J.

Shi, J.

Shin, H.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Shokooh-Saremi, M.

Starbuck, A.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Ta’eed, V. G.

Thévenaz, L.

Wang, Z.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2(1), 011008 (2012).
[CrossRef]

Westbrook, P. S.

Winful, H. G.

V. E. Perlin and H. G. Winful, “Distributed feedback fiber Raman laser,” IEEE J. Quantum Electron.37(1), 38–47 (2001).
[CrossRef]

V. E. Perlin and H. G. Winful, “Stimulated Raman scattering in nonlinear periodic structures,” Phys. Rev. A64(4), 043804 (2001).
[CrossRef]

Appl. Opt.

Electron. Lett.

N. J. Baker, M. A. F. Roelens, S. Madden, B. Luther-Davies, C. M. de Sterke, and B. J. Eggleton, “Pulse train generation by soliton fission in highly nonlinear chalcogenide (As2S3) waveguide Bragg grating,” Electron. Lett.45(15), 799–801 (2009).
[CrossRef]

IEEE J. Quantum Electron.

V. E. Perlin and H. G. Winful, “Distributed feedback fiber Raman laser,” IEEE J. Quantum Electron.37(1), 38–47 (2001).
[CrossRef]

T. Kremp, K. S. Abedin, and P. S. Westbrook, “Closed-form approximations to the threshold quantities of distributed-feedback lasers with varying phase shifts and positions,” IEEE J. Quantum Electron.49(3), 281–292 (2013).
[CrossRef]

J. Opt. Soc. Am. B

Nat Commun

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat Commun4, 1944 (2013), doi:.
[CrossRef] [PubMed]

Opt. Commun.

Y. Hu and N. G. R. Broderick, “Improved design of a DFB Raman fibre laser,” Opt. Commun.282(16), 3356–3359 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

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

V. E. Perlin and H. G. Winful, “Stimulated Raman scattering in nonlinear periodic structures,” Phys. Rev. A64(4), 043804 (2001).
[CrossRef]

Phys. Rev. X

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2(1), 011008 (2012).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of distributed feedback Brillouin laser with a π - phase shift in the grating.

Fig. 2
Fig. 2

Reflection spectrum of un-apodized Bragg grating with a π -phase shift in the middle (solid curve) and with an 8% offset from the middle of the grating (dashed curve). The pump laser is tuned by the Brillouin shift of 9.44 GHz above the Bragg resonance. The points labeled m = 1, 2, 3 are resonances outside the stop band.

Fig. 3
Fig. 3

Power in the transmitted pump, forward Stokes, and backward Stokes waves. (a) Result obtained from the full set of Eqs. (1) with pump reflections included. (b) Result from the reduced equations with pump reflections set to zero.

Fig. 4
Fig. 4

Spatial distribution of power in the grating. Note the peak at phase shift. The green curve is the forward power, the red dots represent the backward power, and the black curve is the total.

Fig. 5
Fig. 5

Pump, forward, and backward Stokes power as a function of time for a 1-cm-long uniform grating in a chalcogenide rib waveguide (a) in the presence of a Kerr nonlinearity and (b) with the nonlinear index set to zero. The inset in (a) shows the spatial distribution of the m = 1 band edge mode.

Fig. 6
Fig. 6

Transmitted pump, forward and backward Stokes output for a π - phase shifted DFB Brillouin laser in a chalcogenide single-mode fiber. The pump power is only 0.9 mW.

Equations (10)

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A p + z + n c A p + t = g B 2 A eff Q ˜ + A s +iκ A p +i δ p A p + α 2 A p + +iγ[ | A p + | 2 +2( | A p | 2 + | A s | 2 ) ] A p +
A p z + n c A p t = g B 2 A eff Q ˜ A s + +iκ A p + +i δ p A p α 2 A p +iγ[ | A p | 2 +2( | A p + | 2 + | A s | 2 ) ] A p
A s z + n c A s t = g B 2 A eff Q ˜ + A p + +iκ A s + +i δ s A s α 2 A s +iγ[ | A s | 2 +2( | A p | 2 + | A s + | 2 ) ] A s
A s + z + n c A s + t = g B 2 A eff Q ˜ A p +iκ A s +i δ s A s + α 2 A s + +iγ[ | A s + | 2 +2( | A p | 2 + | A s | 2 ) ] A s +
2 τ B Q ˜ + t + Q ˜ + = A p + A s * + f ˜ +
2 τ B Q ˜ t + Q ˜ = A p A s +* + f ˜
α th R 2κcosh(2κl)/cosh(κL)+α.
α th B 2[α+4κ e κL cosh(2κl)]
A p + (0,t)= P 0 A p (0,t)=0 A s + (0,t)=0 A s (0,t)=0 .
α th B 2[ π 2 /( κ 2 L 3 )+α ],

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