Abstract

Single-frequency laser sources have found a great number of applications, but are difficult to implement and suffer from poor robustness, poor quality (linewidth and stability), and are generally expensive to fabricate. One solution for a cheaper and simpler single-frequency source is a π-phase-shifted distributed feedback (DFB) fiber Bragg grating (FBG) based laser. Typically, such a laser usually uses a fiber with rare-earth dopants as an active medium for gain. However, its operating wavelength is limited to the emission bandwidth of the rare-earth dopant in the fiber. A proposed solution to overcome this limitation is to use Raman gain. Raman DFB fiber lasers have been successfully demonstrated, and a few simulations have been undertaken and reported. However, a thorough study of parameters and careful optimization has not been reported due to the long computation time and difficulty in the fabrication of long FBGs with known parameters. We demonstrate here, with the aid of a fast but exact method, a detailed optimization study on phase-shifted Raman DFB fiber lasers. These theoretical results are compared with the experimental operation of many fabricated FBGs thanks to a newly developed fabrication technique for the replication of FBGs. We show that fabricated lasers have poor performance compared to simulations of ideal lasers. We also show that the difference in performance is due to the high internal optical intensity induced nonlinear thermal gradient along the FBG.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
    [Crossref]
  2. J. T. Kringlebotn, J. L. Archambault, L. Reekie, and D. N. Payne, “Er3+:Yb3+-codoped fiber distributed-feedback laser,” Opt. Lett. 19, 2101–2103 (1994).
    [Crossref]
  3. V. E. Perlin and H. G. Winful, “Distributed feedback fiber Raman laser,” IEEE J. Quantum Electron. 37, 38–47 (2001).
    [Crossref]
  4. Y. Hu and N. G. R. Broderick, “Improved design of a DFB Raman fibre laser,” Opt. Commun. 282, 3356–3359 (2009).
    [Crossref]
  5. P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Demonstration of a Raman fiber distributed feedback laser,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA11.
  6. P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Raman fiber distributed feedback lasers,” Opt. Lett. 36, 2895–2897 (2011).
    [Crossref]
  7. J. Shi, S.-U. Alam, and M. Ibsen, “High power, low threshold, Raman DFB fibre lasers,” in Proceedings of the International Quantum Electronics Conference and Conference on Lasers and Electro-Optics Pacific Rim (Optical Society of America, 2011), paper C1174.
  8. J. Shi, S.-U. Alam, and M. Ibsen, “Sub-watt threshold, kilohertz-linewidth Raman distributed-feedback fiber laser,” Opt. Lett. 37, 1544–1546 (2012).
    [Crossref]
  9. S. Loranger, V. Karpov, G. W. Schinn, and R. Kashyap, “Single-frequency low threshold linearly polarized DFB Raman fiber lasers,” Opt. Lett. 42, 3864–3867 (2017).
    [Crossref]
  10. J. Shi, “Effects of phase and amplitude noise on? Phase-shifted DFB Raman fibre lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (Optical Society of America, 2011), paper JThA30.
  11. J. Shi, P. Horak, S.-U. Alam, and M. Ibsen, “Detailed study of four-wave mixing in Raman DFB fiber lasers,” Opt. Express 22, 22917–22924 (2014).
    [Crossref]
  12. J. Shi, S.-U. Alam, and M. Ibsen, “Highly efficient Raman distributed feedback fibre lasers,” Opt. Express 20, 5082–5091 (2012).
    [Crossref]
  13. 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, 281–292 (2013).
    [Crossref]
  14. P. S. Westbrook, K. S. Abedin, and T. Kremp, “Distributed feedback Raman and Brillouin fiber lasers,” in Raman Fiber Lasers, Y. Feng, ed. (Springer, 2017), pp. 235–271.
  15. T. Kremp, K. S. Abedin, and P. Westbrook, “Simulation of two-photon absorption in Raman DFB lasers,” in Advanced Photonics Congress, OSA Technical Digest (online) (Optical Society of America, 2012), p. BW3E.5.
  16. R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Academic, 2010).
  17. M. Gagné, S. Loranger, J. Lapointe, and R. Kashyap, “Fabrication of high quality, ultra-long fiber Bragg gratings: up to 2 million periods in phase,” Opt. Express 22, 387–398 (2014).
    [Crossref]
  18. S. Loranger, V. Lambin-Iezzi, and R. Kashyap, “Reproducible ultra-long FBGs in phase corrected non-uniform fibers,” Optica 4, 1143–1146 (2017).
    [Crossref]
  19. S. Loranger and R. Kashyap, “Are optical fibers really uniform? Measurement of refractive index on a centimeter scale,” Opt. Lett. 42, 1832–1835 (2017).
    [Crossref]
  20. K. S. Abedin, P. S. Westbrook, J. W. Nicholson, J. Porque, T. Kremp, and X. Liu, “Single-frequency Brillouin distributed feedback fiber laser,” Opt. Lett. 37, 605–607 (2012).
    [Crossref]
  21. S. Loranger, V. Lambin-Iezzi, M. Wahbeh, and R. Kashyap, “Stimulated Brillouin scattering in ultra-long distributed feedback Bragg gratings in standard optical fiber,” Opt. Lett. 41, 1797–1800 (2016).
    [Crossref]

2017 (3)

2016 (1)

2014 (2)

2013 (1)

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, 281–292 (2013).
[Crossref]

2012 (3)

2011 (1)

2009 (1)

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

2001 (1)

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

1995 (1)

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

1994 (1)

Abedin, K. S.

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, 281–292 (2013).
[Crossref]

K. S. Abedin, P. S. Westbrook, J. W. Nicholson, J. Porque, T. Kremp, and X. Liu, “Single-frequency Brillouin distributed feedback fiber laser,” Opt. Lett. 37, 605–607 (2012).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Raman fiber distributed feedback lasers,” Opt. Lett. 36, 2895–2897 (2011).
[Crossref]

P. S. Westbrook, K. S. Abedin, and T. Kremp, “Distributed feedback Raman and Brillouin fiber lasers,” in Raman Fiber Lasers, Y. Feng, ed. (Springer, 2017), pp. 235–271.

T. Kremp, K. S. Abedin, and P. Westbrook, “Simulation of two-photon absorption in Raman DFB lasers,” in Advanced Photonics Congress, OSA Technical Digest (online) (Optical Society of America, 2012), p. BW3E.5.

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Demonstration of a Raman fiber distributed feedback laser,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA11.

Alam, S.-U.

Archambault, J. L.

Asseh, H.

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

Broderick, N. G. R.

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

Gagné, M.

Horak, P.

Hu, Y.

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

Ibsen, M.

Karpov, V.

Kashyap, R.

Kremp, T.

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, 281–292 (2013).
[Crossref]

K. S. Abedin, P. S. Westbrook, J. W. Nicholson, J. Porque, T. Kremp, and X. Liu, “Single-frequency Brillouin distributed feedback fiber laser,” Opt. Lett. 37, 605–607 (2012).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Raman fiber distributed feedback lasers,” Opt. Lett. 36, 2895–2897 (2011).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Demonstration of a Raman fiber distributed feedback laser,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA11.

P. S. Westbrook, K. S. Abedin, and T. Kremp, “Distributed feedback Raman and Brillouin fiber lasers,” in Raman Fiber Lasers, Y. Feng, ed. (Springer, 2017), pp. 235–271.

T. Kremp, K. S. Abedin, and P. Westbrook, “Simulation of two-photon absorption in Raman DFB lasers,” in Advanced Photonics Congress, OSA Technical Digest (online) (Optical Society of America, 2012), p. BW3E.5.

Kringlebotn, J.

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

Kringlebotn, J. T.

Lambin-Iezzi, V.

Lapointe, J.

Liu, X.

Loranger, S.

Margulis, W.

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

Nicholson, J. W.

K. S. Abedin, P. S. Westbrook, J. W. Nicholson, J. Porque, T. Kremp, and X. Liu, “Single-frequency Brillouin distributed feedback fiber laser,” Opt. Lett. 37, 605–607 (2012).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Raman fiber distributed feedback lasers,” Opt. Lett. 36, 2895–2897 (2011).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Demonstration of a Raman fiber distributed feedback laser,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA11.

Payne, D. N.

Perlin, V. E.

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

Porque, J.

K. S. Abedin, P. S. Westbrook, J. W. Nicholson, J. Porque, T. Kremp, and X. Liu, “Single-frequency Brillouin distributed feedback fiber laser,” Opt. Lett. 37, 605–607 (2012).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Raman fiber distributed feedback lasers,” Opt. Lett. 36, 2895–2897 (2011).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Demonstration of a Raman fiber distributed feedback laser,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA11.

Reekie, L.

Sahlgren, B.

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

Sandgren, S.

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

Schinn, G. W.

Shi, J.

J. Shi, P. Horak, S.-U. Alam, and M. Ibsen, “Detailed study of four-wave mixing in Raman DFB fiber lasers,” Opt. Express 22, 22917–22924 (2014).
[Crossref]

J. Shi, S.-U. Alam, and M. Ibsen, “Highly efficient Raman distributed feedback fibre lasers,” Opt. Express 20, 5082–5091 (2012).
[Crossref]

J. Shi, S.-U. Alam, and M. Ibsen, “Sub-watt threshold, kilohertz-linewidth Raman distributed-feedback fiber laser,” Opt. Lett. 37, 1544–1546 (2012).
[Crossref]

J. Shi, “Effects of phase and amplitude noise on? Phase-shifted DFB Raman fibre lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (Optical Society of America, 2011), paper JThA30.

J. Shi, S.-U. Alam, and M. Ibsen, “High power, low threshold, Raman DFB fibre lasers,” in Proceedings of the International Quantum Electronics Conference and Conference on Lasers and Electro-Optics Pacific Rim (Optical Society of America, 2011), paper C1174.

Storoy, H.

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

Wahbeh, M.

Westbrook, P.

T. Kremp, K. S. Abedin, and P. Westbrook, “Simulation of two-photon absorption in Raman DFB lasers,” in Advanced Photonics Congress, OSA Technical Digest (online) (Optical Society of America, 2012), p. BW3E.5.

Westbrook, P. S.

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, 281–292 (2013).
[Crossref]

K. S. Abedin, P. S. Westbrook, J. W. Nicholson, J. Porque, T. Kremp, and X. Liu, “Single-frequency Brillouin distributed feedback fiber laser,” Opt. Lett. 37, 605–607 (2012).
[Crossref]

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Raman fiber distributed feedback lasers,” Opt. Lett. 36, 2895–2897 (2011).
[Crossref]

P. S. Westbrook, K. S. Abedin, and T. Kremp, “Distributed feedback Raman and Brillouin fiber lasers,” in Raman Fiber Lasers, Y. Feng, ed. (Springer, 2017), pp. 235–271.

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Demonstration of a Raman fiber distributed feedback laser,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA11.

Winful, H. G.

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

Electron. Lett. (1)

H. Asseh, H. Storoy, J. Kringlebotn, W. Margulis, B. Sahlgren, and S. Sandgren, “10  cm Yb3+ DFB fibre laser with permanent phase shifted grating,” Electron. Lett. 31, 969–970 (1995).
[Crossref]

IEEE J. Quantum Electron. (2)

V. E. Perlin and H. G. Winful, “Distributed feedback fiber Raman laser,” IEEE J. Quantum Electron. 37, 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, 281–292 (2013).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (3)

Opt. Lett. (7)

Optica (1)

Other (6)

P. S. Westbrook, K. S. Abedin, J. W. Nicholson, T. Kremp, and J. Porque, “Demonstration of a Raman fiber distributed feedback laser,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPA11.

J. Shi, S.-U. Alam, and M. Ibsen, “High power, low threshold, Raman DFB fibre lasers,” in Proceedings of the International Quantum Electronics Conference and Conference on Lasers and Electro-Optics Pacific Rim (Optical Society of America, 2011), paper C1174.

P. S. Westbrook, K. S. Abedin, and T. Kremp, “Distributed feedback Raman and Brillouin fiber lasers,” in Raman Fiber Lasers, Y. Feng, ed. (Springer, 2017), pp. 235–271.

T. Kremp, K. S. Abedin, and P. Westbrook, “Simulation of two-photon absorption in Raman DFB lasers,” in Advanced Photonics Congress, OSA Technical Digest (online) (Optical Society of America, 2012), p. BW3E.5.

R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Academic, 2010).

J. Shi, “Effects of phase and amplitude noise on? Phase-shifted DFB Raman fibre lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (Optical Society of America, 2011), paper JThA30.

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Figures (11)

Fig. 1.
Fig. 1. Schematic of a PS-DFB-FBG, including the system variables.
Fig. 2.
Fig. 2. Power distributions along the PS-DFB-FBG for comparison of field calculation using two methods: iterative approximated field fit (IAFF) used in this paper as a fast method, and a differential equation solution based on the shooting and Runge–Kutta (S and RK) algorithms as a standard method. The simulated grating is 300 mm long, with a κac of 40  m1, a loss of 0.03 dB/m, and a π-phase shift at 126 mm (24  mm off-center). The pump power is 3 W.
Fig. 3.
Fig. 3. Justification for the choice of PS-DFB-FBG length and strength. (a) The threshold power is shown versus FBG length for a constant κacL=12. (b) A limitation from nonlinearity is shown as the strength of FBG is increased for a constant pump power (8.8 W): the DFB mode displacement (δfDFB) is tuned towards the outside of the stop-band bandwidth (ΔfDFB). All FBGs simulated here have a π-phase shift at 5% of the length away from the center. The FBGs in (b) are 200 mm long.
Fig. 4.
Fig. 4. Effect of nonlinearity on the DFB mode. The DFB mode is detuned (red-shifted) toward the edge of the stop band as power inside the FBG is increased. Case of (a) a π-PS FBG and (b) a 0.75π-PS FBG. The lasing lines are represented by high gain lines (gain as seen by the input seed). The arrows with + and signs on top show a red- and blue-shift of the lines with increase and decrease of power, respectively. FBGs are 300 mm long with a centered phase shift. κac is 40  m1, and the loss is 0.03 dB/m.
Fig. 5.
Fig. 5. Effect of phase-shift value. (a) The activation threshold (when increasing the pump from 0 W initially), shut-down threshold (when decreasing the pump after achieving lasing state), and slope efficiency are shown with varying phase-shift values (b) An example of activation/shut-down hysteresis is shown with a phase shift of 0.6π. FBGs are 300 mm long with a phase shift at 8% from center. κac is 40  m1, and the loss is 0.03 dB/m.
Fig. 6.
Fig. 6. Optimization of phase shift (a) values and (b) positions at different pump levels. The optimization in (a) considers activation threshold. FBGs are 300 mm long. Phase shift is at 8% from center and κac is 40  m1 in (a). The loss is 0.03 dB/m.
Fig. 7.
Fig. 7. (a) Schematic of the fabrication system using a Talbot interferometer and moving fiber, (b) PS-DFB-FBG laser implementation with a 1480-nm pump. The FBG is placed on a cooled-down metal groove immersed in glycerin. (c) Typical example of written FBG of 250 mm in length. The dynamic range is limited at 40 dB for this high resolution (0.1 pm). The example is compared to a theoretical fit with κac of 65  m1.
Fig. 8.
Fig. 8. Output power characterization of a 250-mm-long π-PS fabricated FBG [Fig. 7(c)] considering different bases for thermal management. Base 1 is a thick metal cooled plate (at 10°C) in which the fiber is placed in a 1-mm-wide grove covered in glycerin. Base 2 is a thin metal plate in which the fiber is placed bare on the surface (no grove or glycerin). Base 3 is a thin plastic base in which the fiber is also placed bare on the surface. Results are compared to simulation with fitted parameters (α=0.03  dB/m, κac=65  m1). The pump was polarized in this case. (b) Simulation of efficiency versus effective nonlinearity ratio due to thermal gradient from nonuniform field for 300 mm, π-PS at 8% from center and 40  m1 FBGs.
Fig. 9.
Fig. 9. Temperature calculation based on partial differential equation (PDE) resolution of the heat equation for a core-heated optical fiber immersed in a temperature-controlled liquid. This example is for a 2.3-mW heat load in a 1-mm-long section of fiber.
Fig. 10.
Fig. 10. 0.75π phase-shifted FBG on base 1 where the output power shows hysteresis of activation/shutdown. This 250-mm-long FBG has its phase shift placed at 20  mm from center with a κac of 70  m1.
Fig. 11.
Fig. 11. Comparison of measurement and simulation results for series of fabricated FBGs using a large γeff (=25γ) for unpolarized pump (half Raman gain). (a) Pump power threshold and (b) slope efficiency of output power with phase-shift position. The theoretical results are fitted to the experimental data using a loss of 0.049 dB/m and a coupling constant of 70  m1. All FBGs are 250 mm, the current length limit of our temperature controlled base.

Equations (5)

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

dApdx=gp2(|Af|2+|Ab|2)Apα2Ap+iγp(|Ap|2+2|Af|2+2|Ab|2)Ap,
dAfdx=gs2|Ap|2Afα2Af+iκacAb+i(γp(2|Ap|2+|Af|2+2|Ab|2)+Δβ212dϕdx)Af,
dAbdx=gs2|Ap|2Abα2Ab+iκacAf+i(γp(2|Ap|2+2|Af|2+|Ab|2)+Δβ212dϕdx)Ab,
ϕ(x)={0x<xshiftϕshiftxxshift,
γγeff=γ+βTTI,

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