Abstract

We propose a model to calculate the thermally induced mode loss evolution in the coiled ytterbium doped large mode area (LMA) fiber. The mode loss evolution in the coiled conventional step index LMA 20/400 fiber is investigated. Meanwhile, a model of fiber amplifier considering thermally induced mode loss evolution is established. The higher order mode (HOM) suppression between a co-pumping scheme and counter-pumping scheme under the heat load are compared. The simulation shows that the HOM loss decreases quasi-exponentially with the heat load and the bending radius of the ytterbium doped fiber (YDF) should be optimized according to the heat load to achieve effectively single mode operation. Besides, the counter-pumping fiber amplifier shows much better HOM suppression than the co-pumping fiber amplifier. The results in this paper will provide guidance in the design of novel ytterbium doped LMA fiber and the optimization of the high power single mode fiber amplifier.

© 2017 Optical Society of America

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References

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  1. D. Richardson, J. Nilsson, and W. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” JOSA B 27(11), B63–B92 (2010).
    [Crossref]
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    [Crossref] [PubMed]
  3. C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
    [Crossref]
  4. A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010).
    [Crossref] [PubMed]
  5. M. N. Zervas and C. A. Codemard, “High power fiber lasers: a review,” IEEE J Sel Top Quant 20(5), 219–241 (2014).
    [Crossref]
  6. D. Jain, Y. Jung, P. Barua, S. Alam, and J. K. Sahu, “Demonstration of ultra-low NA rare-earth doped step index fiber for applications in high power fiber lasers,” Opt. Express 23(6), 7407–7415 (2015).
    [Crossref] [PubMed]
  7. F. Beier, C. Hupel, J. Nold, S. Kuhn, S. Hein, J. Ihring, B. Sattler, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Narrow linewidth, single mode 3 kW average power from a directly diode pumped ytterbium-doped low NA fiber amplifier,” Opt. Express 24(6), 6011–6020 (2016).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  13. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express 19(24), 23965–23980 (2011).
    [Crossref] [PubMed]
  14. F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012).
    [Crossref] [PubMed]
  15. E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
    [Crossref]
  16. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
    [Crossref]
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    [Crossref] [PubMed]
  18. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012).
    [Crossref] [PubMed]
  19. A. V. Smith and J. J. Smith, “Thermally induced mode instability in high power fiber amplifiers,” E. C. Honea, and S. T. Hendow, eds. (SPIE, California, USA, 2012), pp. 82370.
  20. H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Experimental Study of Mode Instabilities in High Power Fiber Amplifiers,” (Optical Society of America, 2012), p. AM4A.5.
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    [Crossref]
  22. L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
    [Crossref]
  23. A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19(12), 11318–11329 (2011).
    [Crossref] [PubMed]
  24. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012).
    [Crossref] [PubMed]

2016 (4)

F. Beier, C. Hupel, J. Nold, S. Kuhn, S. Hein, J. Ihring, B. Sattler, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Narrow linewidth, single mode 3 kW average power from a directly diode pumped ytterbium-doped low NA fiber amplifier,” Opt. Express 24(6), 6011–6020 (2016).
[Crossref] [PubMed]

X. Zheng, G. Ren, L. Huang, H. Li, B. Zhu, H. Zheng, and M. Cao, “Bending losses of trench-assisted few-mode optical fibers,” Appl. Opt. 55(10), 2639–2648 (2016).
[Crossref] [PubMed]

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

2015 (1)

2014 (2)

2013 (2)

2012 (4)

2011 (3)

2010 (2)

D. Richardson, J. Nilsson, and W. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” JOSA B 27(11), B63–B92 (2010).
[Crossref]

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010).
[Crossref] [PubMed]

2009 (1)

2007 (1)

2001 (1)

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Alam, S.

Alkeskjold, T. T.

Barankov, R. A.

Barua, P.

Beier, F.

Benoit, A.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Broeng, J.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Cao, M.

Chen, X.

Cheng, X.

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

Clarkson, W.

D. Richardson, J. Nilsson, and W. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” JOSA B 27(11), B63–B92 (2010).
[Crossref]

Codemard, C. A.

M. N. Zervas and C. A. Codemard, “High power fiber lasers: a review,” IEEE J Sel Top Quant 20(5), 219–241 (2014).
[Crossref]

Coscelli, E.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Cucinotta, A.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Dajani, I.

Darwich, D.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Dauliat, R.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Eberhardt, R.

Eidam, T.

Fini, J. M.

Gray, S.

Guo, S.

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

Haarlammert, N.

Hansen, K. R.

Hein, S.

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Huang, L.

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

X. Zheng, G. Ren, L. Huang, H. Li, B. Zhu, H. Zheng, and M. Cao, “Bending losses of trench-assisted few-mode optical fibers,” Appl. Opt. 55(10), 2639–2648 (2016).
[Crossref] [PubMed]

Hupel, C.

Ihring, J.

Jain, D.

Jamier, R.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Jansen, F.

Jauregui, C.

Jung, Y.

Kong, L.

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

Kuhn, S.

Lægsgaard, J.

Leng, J.

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

Li, H.

Li, M.-J.

Liem, A.

Limpert, J.

Liu, A.

Nilsson, J.

J. Nilsson and D. N. Payne, “High-Power Fiber Lasers,” Science 332(6032), 921–922 (2011).
[Crossref] [PubMed]

D. Richardson, J. Nilsson, and W. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” JOSA B 27(11), B63–B92 (2010).
[Crossref]

Nold, J.

Nunez-Velazquez, M.

Otto, H.-J.

Payne, D. N.

J. Nilsson and D. N. Payne, “High-Power Fiber Lasers,” Science 332(6032), 921–922 (2011).
[Crossref] [PubMed]

Poli, F.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Ramachandran, S.

Ren, G.

Richardson, D.

D. Richardson, J. Nilsson, and W. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” JOSA B 27(11), B63–B92 (2010).
[Crossref]

Robin, C.

Roy, P.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Sahu, J. K.

Salin, F.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Samson, B.

Sattler, B.

Schreiber, T.

Schuster, K.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Selleri, S.

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

Smith, A. V.

Smith, J. J.

Stutzki, F.

Tünnermann, A.

Walton, D. T.

Wang, J.

Ward, B.

Wei, K.

Zenteno, L. A.

Zervas, M. N.

M. N. Zervas and C. A. Codemard, “High power fiber lasers: a review,” IEEE J Sel Top Quant 20(5), 219–241 (2014).
[Crossref]

Zheng, H.

Zheng, X.

Zhou, P.

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

Zhu, B.

Appl. Opt. (2)

IEEE J Sel Top Quant (1)

M. N. Zervas and C. A. Codemard, “High power fiber lasers: a review,” IEEE J Sel Top Quant 20(5), 219–241 (2014).
[Crossref]

IEEE J. Quantum Electron. (1)

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

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

E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, and F. Salin, “Analysis of the modal content into large-mode-area photonic crystal fibers under heat load,” IEEE J. Sel. Top. Quantum Electron. 22(2), 323–330 (2016).
[Crossref]

J. Lightwave Technol. (1)

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

JOSA B (2)

D. Richardson, J. Nilsson, and W. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” JOSA B 27(11), B63–B92 (2010).
[Crossref]

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” JOSA B 33(6), 1030–1037 (2016).
[Crossref]

Nat. Photonics (1)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Opt. Express (9)

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express 19(24), 23965–23980 (2011).
[Crossref] [PubMed]

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012).
[Crossref] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19(12), 11318–11329 (2011).
[Crossref] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012).
[Crossref] [PubMed]

D. Jain, Y. Jung, M. Nunez-Velazquez, and J. K. Sahu, “Extending single mode performance of all-solid large-mode-area single trench fiber,” Opt. Express 22(25), 31078–31091 (2014).
[Crossref] [PubMed]

F. Beier, C. Hupel, J. Nold, S. Kuhn, S. Hein, J. Ihring, B. Sattler, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Narrow linewidth, single mode 3 kW average power from a directly diode pumped ytterbium-doped low NA fiber amplifier,” Opt. Express 24(6), 6011–6020 (2016).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012).
[Crossref] [PubMed]

D. Jain, Y. Jung, P. Barua, S. Alam, and J. K. Sahu, “Demonstration of ultra-low NA rare-earth doped step index fiber for applications in high power fiber lasers,” Opt. Express 23(6), 7407–7415 (2015).
[Crossref] [PubMed]

Opt. Lett. (1)

Science (1)

J. Nilsson and D. N. Payne, “High-Power Fiber Lasers,” Science 332(6032), 921–922 (2011).
[Crossref] [PubMed]

Other (3)

C.-H. Liu, G. Chang, N. Litchinitser, A. Galvanauskas, D. Guertin, N. Jabobson, and K. Tankala, “Effectively single-mode chirally-coupled core fiber,” in Advanced Solid-State Photonics(Optical Society of America, 2007), p. ME2.

A. V. Smith and J. J. Smith, “Thermally induced mode instability in high power fiber amplifiers,” E. C. Honea, and S. T. Hendow, eds. (SPIE, California, USA, 2012), pp. 82370.

H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Experimental Study of Mode Instabilities in High Power Fiber Amplifiers,” (Optical Society of America, 2012), p. AM4A.5.

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Figures (15)

Fig. 1
Fig. 1 Refractive index of the bending active fiber with the heat load.
Fig. 2
Fig. 2 Mode loss decrease with the heat load (a) FM (b) LP11o.
Fig. 3
Fig. 3 refractive index difference under zero heat load and 9.63 W/m heat load.
Fig. 4
Fig. 4 Relative Mode loss decrease with the heat load (a) FM (b) LP11o.
Fig. 5
Fig. 5 the mode loss evolution for LP11e mode and LP21e mode.
Fig. 6
Fig. 6 The FM loss and the LP11o loss evolution with heat load under different bending radius.
Fig. 7
Fig. 7 Pump power, FM power and HOM power along the active fiber for the co-pumping scheme (a) and the counter-pumping scheme (b).
Fig. 8
Fig. 8 Heat load and LP11O loss along the YDF for (a) the co-pumping scheme and (b) the counter-pumping scheme.
Fig. 9
Fig. 9 The PCE under different pump power for the case of different LP11o seed power in the (a) co-pumping scheme and (b) counter-pumping scheme.
Fig. 10
Fig. 10 Power distribution along the YDF for (a) 200 W pump and (b) 1000 W pump in co-pumping scheme.
Fig. 11
Fig. 11 Leakage power evolution with the pump power for (a) co-pumping scheme, (b) counter-pumping scheme.
Fig. 12
Fig. 12 The relatively HOM ratio under different pump power for the case of different LP11o seed power in the (a) co-pumping scheme and (b) counter-pumping scheme.
Fig. 13
Fig. 13 Common spiral coiled patterns for industrial fiber amplifier, in which the pump power and seed power are launched from (a) the outer ring (b) from the inner ring
Fig. 14
Fig. 14 (a) Heat load and loss distribution, (b) PCE evolution with the pump power when the pump and the signal are launching from the outer ring of the YDF.
Fig. 15
Fig. 15 (a) Heat load and loss distribution, (b) PCE evolution with the pump power when the pump and the signal are launching from the inner ring of the YDF, (c) power distribution along the YDF for 0.5 W input HOM.

Tables (2)

Tables Icon

Table 1 Parameters of the LMA 20/400 fiber

Tables Icon

Table 2 Parameters of the amplifier

Equations (10)

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

q( z )= Q( z ) π a 2 = 1 π a 2 ×( 1 λ p λ s )×( d P p (z) dz d P p + (z) dz )
T( r )={ T 0 + q a 2 2hc + q( a 2 r 2 ) 4 k si + q a 2 2 k si ln( b a )+ q a 2 2 k ac ln( c a ), 0ra T 0 + q a 2 2hc + q a 2 2 k si ln( b r )+ q a 2 2 k ac ln( c b ), arb T 0 + q a 2 2hc + q a 2 2 k ac ln( c r ), brc
Δ n T ( r )=β( T( r ) T 0 )
n T ( q,r )={ n 1 +β[ q( a 2 r 2 ) 4 k si + q a 2 2 k si ln( b a )+ q a 2 2 k ac ln( c a ) ],0ra n 2 +β[ q a 2 2 k si ln( b r )+ q a 2 2 k ac ln( c b ) ], arb
n eq 2 ( r,φ )= n 2 (r)×( 1+ 2r ρR cosφ )
n T ( q,r,φ )={ ( n 1 +β[ q( a 2 r 2 ) 4 k si + q a 2 2 k si ln( b a )+ q a 2 2 k ac ln( c a ) ] )× 1+ 2r ρR cosφ ,0ra ( n 2 +β[ q a 2 2 k si ln( b r )+ q a 2 2 k ac ln( c b ) ] )× 1+ 2r ρR cosφ , arb
RD=10 log 10 ( α(Q,r) α(0,r) )
n u ( x,y )= ( P p + + P p ) σ p α /h υ p A+ I s σ s α /h υ s ( P p + + P p )×( σ p α + σ p e )/h υ p A+ I s ( σ s α + σ s e )/h υ s +1/τ d P p ± dz =± P p ± A ( σ p e n u σ p α (1 n u ) ) N Yb dxdy Q=( d P p + dz + d P p dz )×( 1 λ p λ s ) d P s m dz = P s m [ ( σ s e n u σ s α (1 n u ) ) N Yb Φ m ( Q )dxdy ln10 10 α m (Q) ]
P leakage = m 0 L ln10 10 P s m α m dl
ξ=10 log 10 ( P HOM P Output )

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