Abstract

A concise and effective model based on coupled mode theory to describe mode evolution in long tapered active fiber is presented in this manuscript. The mode coupling due to variation of core radius and slight perturbation have been analyzed and local gain with transverse spatial hole burning (TSHB) effect, loss and curvature have been taken into consideration in our model. On the base of this model, the mode evolution behaviors under different factors have been numerically investigated. Our model and results can provide instructive suggestions when designing long tapered fiber based laser and amplifiers.

© 2016 Optical Society of America

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References

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  1. M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
    [Crossref]
  2. V. Filippov, Y. K. Chamorovskii, K. M. Golant, A. Vorotynskii, and O. G. Okhotnikov, Optical amplifiers and lasers based on tapered fiber geometry for power and energy scaling with low signal distortion (2016), p. 97280V.
  3. V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “Single-mode 212 W tapered fiber laser pumped by a low-brightness source,” Opt. Lett. 33(13), 1416–1418 (2008).
    [Crossref] [PubMed]
  4. V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “600 W power scalable single transverse mode tapered double-clad fiber laser,” Opt. Express 17(3), 1203–1214 (2009).
    [Crossref] [PubMed]
  5. V. Filippov, J. Kerttula, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Highly efficient 750 W tapered double-clad ytterbium fiber laser,” Opt. Express 18(12), 12499–12512 (2010).
    [Crossref] [PubMed]
  6. J. Kerttula, V. Filippov, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Actively Q-switched 1.6-mJ tapered double-clad ytterbium-doped fiber laser,” Opt. Express 18(18), 18543–18549 (2010).
    [Crossref] [PubMed]
  7. J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012).
    [Crossref]
  8. A. I. Trikshev, A. S. Kurkov, V. B. Tsvetkov, S. A. Filatova, J. Kertulla, V. Filippov, Y. K. Chamorovskiy, and O. G. Okhotnikov, “160W single-frequency laser based on active tapered double-clad fiber amplifier,” in 2013 Conference on Lasers and Electro-Optics - International Quantum Electronics Conference(Optical Society of America, Munich, 2013), p. 33.
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  12. J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Principles and performance of tapered fiber lasers: from uniform to flared geometry,” Appl. Opt. 51(29), 7025–7038 (2012).
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  23. H. Yoda, P. Polynkin, and M. Mansuripur, “Beam Quality Factor of Higher Order Modes in a Step-Index Fiber,” J. Lightwave Technol. 24(3), 1350–1355 (2006).
    [Crossref]
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    [Crossref]
  25. O. G. Okhotnikov, Fiber lasers (John Wiley & Sons, 2012).
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    [Crossref] [PubMed]
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2016 (2)

H. Zhang, X. Du, P. Zhou, X. Wang, and X. Xu, “Tapered fiber based high power random laser,” Opt. Express 24(8), 9112–9118 (2016).
[Crossref] [PubMed]

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

2014 (1)

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

2012 (3)

2010 (2)

2009 (2)

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “600 W power scalable single transverse mode tapered double-clad fiber laser,” Opt. Express 17(3), 1203–1214 (2009).
[Crossref] [PubMed]

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009).
[Crossref]

2008 (2)

2007 (2)

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007).
[Crossref] [PubMed]

2006 (1)

2003 (1)

W. Yong and P. Hong, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” Lightwave Technology, Journalism 21, 2262–2270 (2003).

2000 (1)

1999 (1)

E. F. Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibres,” J. Opt. A, Pure Appl. Opt. 1(2), 197–200 (1999).
[Crossref]

1991 (1)

1982 (1)

1980 (1)

1972 (1)

Black, R. J.

Bures, J.

Chamorovskii, Y.

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012).
[Crossref]

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Principles and performance of tapered fiber lasers: from uniform to flared geometry,” Appl. Opt. 51(29), 7025–7038 (2012).
[Crossref] [PubMed]

J. Kerttula, V. Filippov, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Actively Q-switched 1.6-mJ tapered double-clad ytterbium-doped fiber laser,” Opt. Express 18(18), 18543–18549 (2010).
[Crossref] [PubMed]

V. Filippov, J. Kerttula, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Highly efficient 750 W tapered double-clad ytterbium fiber laser,” Opt. Express 18(12), 12499–12512 (2010).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “600 W power scalable single transverse mode tapered double-clad fiber laser,” Opt. Express 17(3), 1203–1214 (2009).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “Single-mode 212 W tapered fiber laser pumped by a low-brightness source,” Opt. Lett. 33(13), 1416–1418 (2008).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, K. Golant, M. Pessa, and O. G. Okhotnikov, “Double clad tapered fiber for high power applications,” Opt. Express 16(3), 1929–1944 (2008).
[Crossref] [PubMed]

Codemard, C. A.

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

Cole, J. H.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).

Diasty, E. F.

E. F. Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibres,” J. Opt. A, Pure Appl. Opt. 1(2), 197–200 (1999).
[Crossref]

Du, X.

Eickhoff, W.

Filippov, V.

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012).
[Crossref]

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Principles and performance of tapered fiber lasers: from uniform to flared geometry,” Appl. Opt. 51(29), 7025–7038 (2012).
[Crossref] [PubMed]

J. Kerttula, V. Filippov, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Actively Q-switched 1.6-mJ tapered double-clad ytterbium-doped fiber laser,” Opt. Express 18(18), 18543–18549 (2010).
[Crossref] [PubMed]

V. Filippov, J. Kerttula, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Highly efficient 750 W tapered double-clad ytterbium fiber laser,” Opt. Express 18(12), 12499–12512 (2010).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “600 W power scalable single transverse mode tapered double-clad fiber laser,” Opt. Express 17(3), 1203–1214 (2009).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “Single-mode 212 W tapered fiber laser pumped by a low-brightness source,” Opt. Lett. 33(13), 1416–1418 (2008).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, K. Golant, M. Pessa, and O. G. Okhotnikov, “Double clad tapered fiber for high power applications,” Opt. Express 16(3), 1929–1944 (2008).
[Crossref] [PubMed]

Golant, K.

Goldberg, L.

Gong, M.

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009).
[Crossref]

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007).
[Crossref] [PubMed]

Gonthier, F.

Hénault, A.

Hong, P.

W. Yong and P. Hong, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” Lightwave Technology, Journalism 21, 2262–2270 (2003).

Jiang, Z.

Kerttula, J.

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Principles and performance of tapered fiber lasers: from uniform to flared geometry,” Appl. Opt. 51(29), 7025–7038 (2012).
[Crossref] [PubMed]

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012).
[Crossref]

J. Kerttula, V. Filippov, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Actively Q-switched 1.6-mJ tapered double-clad ytterbium-doped fiber laser,” Opt. Express 18(18), 18543–18549 (2010).
[Crossref] [PubMed]

V. Filippov, J. Kerttula, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Highly efficient 750 W tapered double-clad ytterbium fiber laser,” Opt. Express 18(12), 12499–12512 (2010).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “600 W power scalable single transverse mode tapered double-clad fiber laser,” Opt. Express 17(3), 1203–1214 (2009).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “Single-mode 212 W tapered fiber laser pumped by a low-brightness source,” Opt. Lett. 33(13), 1416–1418 (2008).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, K. Golant, M. Pessa, and O. G. Okhotnikov, “Double clad tapered fiber for high power applications,” Opt. Express 16(3), 1929–1944 (2008).
[Crossref] [PubMed]

Kholodkov, A.

Kliner, D. A. V.

Koplow, J. P.

Lacroix, S.

Li, C.

Liao, S.

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009).
[Crossref]

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007).
[Crossref] [PubMed]

Lü, H.

Mansuripur, M.

Marcuse, D.

Okhotnikov, O. G.

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Principles and performance of tapered fiber lasers: from uniform to flared geometry,” Appl. Opt. 51(29), 7025–7038 (2012).
[Crossref] [PubMed]

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012).
[Crossref]

J. Kerttula, V. Filippov, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Actively Q-switched 1.6-mJ tapered double-clad ytterbium-doped fiber laser,” Opt. Express 18(18), 18543–18549 (2010).
[Crossref] [PubMed]

V. Filippov, J. Kerttula, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Highly efficient 750 W tapered double-clad ytterbium fiber laser,” Opt. Express 18(12), 12499–12512 (2010).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “600 W power scalable single transverse mode tapered double-clad fiber laser,” Opt. Express 17(3), 1203–1214 (2009).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “Single-mode 212 W tapered fiber laser pumped by a low-brightness source,” Opt. Lett. 33(13), 1416–1418 (2008).
[Crossref] [PubMed]

V. Filippov, Y. Chamorovskii, J. Kerttula, K. Golant, M. Pessa, and O. G. Okhotnikov, “Double clad tapered fiber for high power applications,” Opt. Express 16(3), 1929–1944 (2008).
[Crossref] [PubMed]

Pan, Z.

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

Pessa, M.

Polynkin, P.

Rashleigh, S. C.

Schermer, R. T.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).

Snyder, A. W.

Su, R.

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

Ulrich, R.

Ustimchik, V.

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012).
[Crossref]

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Principles and performance of tapered fiber lasers: from uniform to flared geometry,” Appl. Opt. 51(29), 7025–7038 (2012).
[Crossref] [PubMed]

Wang, X.

Xu, X.

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

H. Zhang, X. Du, P. Zhou, X. Wang, and X. Xu, “Tapered fiber based high power random laser,” Opt. Express 24(8), 9112–9118 (2016).
[Crossref] [PubMed]

Yan, P.

Yang, B.

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

Yoda, H.

Yong, W.

W. Yong and P. Hong, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” Lightwave Technology, Journalism 21, 2262–2270 (2003).

Yuan, Y.

Zervas, M. N.

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

Zhang, H.

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

H. Zhang, X. Du, P. Zhou, X. Wang, and X. Xu, “Tapered fiber based high power random laser,” Opt. Express 24(8), 9112–9118 (2016).
[Crossref] [PubMed]

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009).
[Crossref]

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007).
[Crossref] [PubMed]

Zhou, P.

Zhou, Z.

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).

IEEE Sel. Top. Quantum Electron. (1)

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. (1)

Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

E. F. Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibres,” J. Opt. A, Pure Appl. Opt. 1(2), 197–200 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Laser Phys. (2)

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009).
[Crossref]

J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012).
[Crossref]

Lightwave Technology, Journalism (1)

W. Yong and P. Hong, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” Lightwave Technology, Journalism 21, 2262–2270 (2003).

Opt. Express (7)

Opt. Lett. (3)

Other (5)

O. G. Okhotnikov, Fiber lasers (John Wiley & Sons, 2012).

A. I. Trikshev, A. S. Kurkov, V. B. Tsvetkov, S. A. Filatova, J. Kertulla, V. Filippov, Y. K. Chamorovskiy, and O. G. Okhotnikov, “160W single-frequency laser based on active tapered double-clad fiber amplifier,” in 2013 Conference on Lasers and Electro-Optics - International Quantum Electronics Conference(Optical Society of America, Munich, 2013), p. 33.
[Crossref]

V. Filippov, Y. K. Chamorovskii, K. M. Golant, A. Vorotynskii, and O. G. Okhotnikov, Optical amplifiers and lasers based on tapered fiber geometry for power and energy scaling with low signal distortion (2016), p. 97280V.

J. F. Nye, Physical properties of crystals: their representation by tensors and matrices (Oxford university press, 1985).

A. W. Snyder and J. Love, Optical waveguide theory (Springer Science & Business Media, 2012).

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Figures (12)

Fig. 1
Fig. 1 Illustration of long tapered fiber partitioning.
Fig. 2
Fig. 2 Radius profile for tapered fiber with different parabolic shape factors.
Fig. 3
Fig. 3 Illustration for different doping distribution; (a) flat doping with Γd = 0.5; (b) parabolic doping.
Fig. 4
Fig. 4 Simulation results of modal power (left column), M2 factor (middle column) and input/output field (right column) of tapered active fiber with different doping distributions and parameter b = b0 under near fundamental mode injection.
Fig. 5
Fig. 5 Effective gain area ratio of different modes with respect to Γd.
Fig. 6
Fig. 6 Simulation results of modal power (left column), M2 factor (middle column) and input/output field (right column) of tapered active fiber with flat doping Γd = 1 distributions; (a)concave profile with b = 0.1b0; (b)convex profile with b = 2b0.
Fig. 7
Fig. 7 Simulation results of modal power (left column), M2 factor (middle column) and input/output field (right column) of tapered active fiber with different doping distributions and parameter b = b0 under multimode injection.
Fig. 8
Fig. 8 Simulation results of modal power (left column), M2 factor (middle column) and input/output field (right column) of tapered active fiber with flat doping Γd = 1 distributions; (a)concave profile with b = 0.1b0; (b)convex profile with b = 2b0.
Fig. 9
Fig. 9 Loss difference between LP11 and LP01 mode with respect to core radius with core NA = 0.065
Fig. 10
Fig. 10 Simulation results of modal power (left column), M2 factor (middle column) and input/output field (right column) of tapered active fiber with curvature in first 1m length; (a)linear profile with b = b0; (b)concave profile with b = 0.1b0.
Fig. 11
Fig. 11 Simulation results of modal power (left column), M2 factor (middle column) and input/output field (right column) of tapered active fiber L-S configuration; (a)linear profile with b = b0, Γd = 1; (b)linear profile with b = b0, Γd = 0.5; (c)concave profile with b = 0.1b0, Γd = 1.
Fig. 12
Fig. 12 Simulation results of modal power (left column), M2 factor (middle column) and input/output field (right column) of tapered active fiber L-S configuration; (a)linear profile with b = b0, Γd = 0.5; (b)mode field on different position of fiber.

Equations (40)

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E( r,φ,z )= j b j ( z ) e ^ j ( r,φ,z )+radiation modes
{ E t = j ( b j + b j ) e ^ tj ( a ) H t = j ( b j b j ) h ^ tj ( b )
{ E t = 1 k 0 n 2 [ i μ 0 ε 0 z ^ × H t z + 1 k 0 t ×( t × E t ) ] ( a ) H t = 1 k 0 [ i ε 0 μ 0 z ^ × E t z 1 k 0 t ×( t × H t n 2 ) ] ( b )
{ e ^ tj = 1 k 0 n 2 [ μ 0 ε 0 β j z ^ × h ^ tj 1 k 0 t ×( t × e ^ tj ) ] ( a ) h ^ tj = 1 k 0 [ ε 0 μ 0 β j z ^ × e ^ tj + 1 k 0 t ×( t × h ^ tj n 2 ) ] ( b )
{ j [ ( b j b j ) z ^ × h ^ tj z i β j ( b j + b j ) z ^ × h ^ tj +( d b j dz d b j dz ) z ^ × h ^ tj ]=0 ( a ) j [ ( b j + b j ) z ^ × e ^ tj z i β j ( b j b j ) z ^ × e ^ tj +( d b j dz + d b j dz ) z ^ × e ^ tj ]=0 ( b )
d b j dz i β j z= k C jk b k
C jk = 1 4 A ( h ^ j × e ^ k z e ^ j × h ^ k z ) z ^ dA,jk
e ^ j ψ ^ j , h ^ j n co Z 0 z ^ × ψ ^ j
C jk = n co 2 Z 0 A ψ ^ j ψ ^ k z dA
( t 2 + k 0 2 n 2 β k 2 ) ψ ^ k =0
( t 2 + k 0 2 n 2 β k 2 ) ψ ^ k z +( k 0 2 n 2 z β k 2 z ) ψ ^ k =0
( ψ ^ j z t 2 ψ ^ k ψ ^ j t 2 ψ ^ k z )+( k 0 2 n 2 β k 2 )( ψ ^ j z ψ ^ k ψ ^ j ψ ^ k z ) =( k 0 2 n 2 z β k 2 z ) ψ ^ j ψ ^ k
( ψ ^ j z t 2 ψ ^ k ψ ^ k t 2 ψ ^ j z )+( ψ ^ k z t 2 ψ ^ j ψ ^ j t 2 ψ ^ k z ) +( β j 2 β k 2 )( ψ ^ j z ψ ^ k ψ ^ j ψ ^ k z ) =2 k 0 2 n 2 z ψ ^ j ψ ^ k ( β k 2 z + β j 2 z ) ψ ^ j ψ ^ k
( β j 2 β k 2 ) A ( ψ ^ j z ψ ^ k ψ ^ j ψ ^ k z )dA=2 k 0 2 A n 2 z ψ ^ j ψ ^ k dA
A ψ ^ j ψ ^ k z dA= 2 k 0 2 β j 2 β k 2 A n 2 z ψ ^ j ψ ^ k dA
C jk = k 0 2 Z 0 ( β j β k ) A n 2 z ψ ^ j ψ ^ k dA
n 2 ( r,z )= n co 2 N A 2 H[ rρ( z ) ]
n 2 z =N A 2 δ[ rρ( z ) ] dρ( z ) dz
F c =E× H ¯ * + E ¯ * ×H
E ¯ = e ^ j exp( i β j z ), H ¯ = h ^ j exp( i β j z )
z A F c z ^ dA= A F c dA
F c =i k 0 Z 0 ( n 2 n ¯ 2 )E E ¯ *
d b j dz i β j b j =i k 0 4 Z 0 A ( n 2 n ¯ 2 ) ψ ^ j EdA
d b j dz ( i β j + g j α j ) b j = C j I +i C j II
{ C j I = k 0 2 Z 0 kj 1 β j β k A n 2 z ψ ^ j ψ ^ k dA ( a ) C j II = k 0 4 Z 0 A ( n 2 n ¯ 2 ) ψ ^ j EdA ( b )
b j ( z+dz )=[ b j ( z )+ C j I +i C j II i β j + g j α j ] e ( i β j + g j α j )dz C j I +i C j II i β j + g j α j
η u ( r,φ,z )= I p ( z ) λ p σ a p + λ s σ a s k I s k ( r,φ,z ) hc τ + I p ( z ) λ p ( σ a p + σ e p )+ λ s ( σ a s + σ e s ) k I s k ( r,φ,z )
g( r,φ,z )= N dope ( r,φ,z )[ ( σ a s + σ e s ) η u ( r,φ,z ) σ a s ]
d P p ( z ) dz = I p ( z ) A co N dope ( r,φ,z )[ ( σ a p + σ e p ) η u σ a p ]dA α p P p ( z )
g j = 1 2 A co g( r,φ,z ) ψ ^ j dA
Δ n g =i g( r,φ,z ) 2 k 0
n eff = n co ( 1+ r R c cosφ )
{ σ x = E Y 2 R c 2 ( r 2 cos 2 φ a 2 ) σ y =0
{ Δ n i,ST = n 0 3 2 p ij ε j ( a ) ε j = ( 1+ν ) σ j / E Y ( b )
{ Δ n x,ST = 1 4 n co 3 p 11 1+ν R c 2 ( r 2 cos 2 φ a 2 ) Δ n y,ST = 1 4 n co 3 p 12 1+ν R c 2 ( r 2 cos 2 φ a 2 )
2 α j = π U j 2 exp( 2 W j 3 R c 3 a 3 β j 2 ) ξ l R c a W j 3 V 2 K l1 ( W j ) K l+1 ( W j )
{ V=NA k 0 a U j =a n co 2 k 0 2 β j 2 W j =a β j 2 ( n co 2 N A 2 ) k 0 2
M k 2 = 4 σ k 2 ( z 0 ) B k + A k 2
{ k ( z 0 )= A k | E( x,y, z 0 ) | 2 dA σ k 2 ( z 0 )= A [ k k ( z 0 ) ] 2 | E( x,y, z 0 ) | 2 dA A k = A [ k k ( z 0 ) ][ E( x,y, z 0 ) E * ( x,y, z 0 ) k c.c. ] dA B k = A | E( x,y, z 0 ) k | 2 dA + 1 4 { A [ E( x,y, z 0 ) E * ( x,y, z 0 ) k c.c. ] dA } 2
ρ( z )= b 0 b 2L z 2 + b 2 z+ 1 2 D 1

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