Abstract

General analysis of the mode interaction in multimode fiber is presented in this paper. By taking local gain into consideration, the general coupled mode equations in multimode active fiber are deduced in the model and the effect of various factors can be analyzed based on the general coupled mode equations. Analytical expression of the beam quality factor is deduced for the optical field emerging from the multimode active fiber. The evolution of the mode power and M2 factor along the fiber are analyzed by numerical evaluations.

© 2012 OSA

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  1. Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
    [CrossRef] [PubMed]
  2. A. Liem, J. Limpert, H. Zellmer, and A. Tünnermann, “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. 28(17), 1537–1539 (2003).
    [CrossRef] [PubMed]
  3. Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
    [CrossRef]
  4. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
    [CrossRef] [PubMed]
  5. A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19(12), 11318–11329 (2011).
    [CrossRef] [PubMed]
  6. N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express 16(12), 8678–8684 (2008).
    [CrossRef] [PubMed]
  7. N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express 16(24), 20038–20046 (2008).
    [CrossRef] [PubMed]
  8. A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A 76(6), 063801 (2007).
    [CrossRef]
  9. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
    [CrossRef] [PubMed]
  10. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  11. N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, in Proceedings of the Fourth Conference on Finite difference methods: Theory and Application (University Press, Rousse, Bulgaria, 2007) 167–172.
  12. H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).
  13. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
    [CrossRef]
  14. 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]
  15. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. 27(10), 1248–1261 (2009).
    [CrossRef]
  16. D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. 52, 1423–1437 (1973).
  17. 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]
  18. X. Zhu, A. Schülzgen, H. Li, L. Li, L. Han, J. V. Moloney, and N. Peyghambarian, “Detailed investigation of self-imaging in large-core multimode optical fibers for application in fiber lasers and amplifiers,” Opt. Express 16(21), 16632–16645 (2008).
    [PubMed]

2011 (4)

2009 (1)

2008 (3)

2007 (3)

A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A 76(6), 063801 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[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]

2005 (1)

2003 (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. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

1973 (1)

D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. 52, 1423–1437 (1973).

Alegria, C.

Alkeskjold, T. T.

Alvarez-Chavez, J. A.

Andermahr, N.

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. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Chryssou, C. E.

Codemard, C. A.

Dupriez, P.

Eidam, T.

Fallnich, C.

Gong, M.

Han, L.

Hansen, K. R.

Hickey, L. M. B.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
[CrossRef] [PubMed]

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. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Horley, R.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
[CrossRef] [PubMed]

Jauregui, C.

Jeong, Y.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
[CrossRef] [PubMed]

Jiang, Z.

H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).

Kahn, J. M.

Lægsgaard, J.

Li, C.

Li, H.

Li, L.

Liao, S.

Liem, A.

Limpert, J.

Lü, H.

H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).

Mao, W.

Marcuse, D.

D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. 52, 1423–1437 (1973).

Moloney, J. V.

Napartovich, A. P.

A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A 76(6), 063801 (2007).
[CrossRef]

Nilsson, J.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
[CrossRef] [PubMed]

Panicker, R. A.

Payne, D. N.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
[CrossRef] [PubMed]

Peyghambarian, N.

Sahu, J. K.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
[CrossRef] [PubMed]

Schülzgen, A.

Shemirani, M. B.

Smith, A. V.

Smith, J. J.

Soh, D. B. S.

Tünnermann, A.

Turner, P. W.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005).
[CrossRef] [PubMed]

Vysotsky, D. V.

A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A 76(6), 063801 (2007).
[CrossRef]

Wang, X.

H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).

Wanzcyk, L.

Xiao, H.

H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).

Yan, P.

Yuan, Y.

Zellmer, H.

Zhang, H.

Zhou, P.

H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).

Zhu, X.

Bell Syst. Tech. J. (1)

D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. 52, 1423–1437 (1973).

IEEE J. Sel. Top. Quantum Electron. (2)

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

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[CrossRef]

J. Lightwave Technol. (1)

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

H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).

Opt. Express (8)

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]

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]

X. Zhu, A. Schülzgen, H. Li, L. Li, L. Han, J. V. Moloney, and N. Peyghambarian, “Detailed investigation of self-imaging in large-core multimode optical fibers for application in fiber lasers and amplifiers,” Opt. Express 16(21), 16632–16645 (2008).
[PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[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]

N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express 16(12), 8678–8684 (2008).
[CrossRef] [PubMed]

N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express 16(24), 20038–20046 (2008).
[CrossRef] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. A (1)

A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A 76(6), 063801 (2007).
[CrossRef]

Other (2)

W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, in Proceedings of the Fourth Conference on Finite difference methods: Theory and Application (University Press, Rousse, Bulgaria, 2007) 167–172.

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Figures (7)

Fig. 1
Fig. 1

Different doping distributions and corresponding results of mode power (left column) and M2 factor (right column); (a) different doping distributions; (b) Correspond to the left column in (a); (c) Correspond to the central column in (a); (d) Correspond to the right column in (a).

Fig. 2
Fig. 2

Accurate evolutions of mode power (left column) and M2 factor (right column) for different doping profiles; (a) flat doping, Г = 1; (b) flat doping, Г = 0.5; (c) parabolic doping.

Fig. 3
Fig. 3

Results of mode power (left column) and M2 factor (right column) for different pump power P and heat transfer coefficients hc where hc = 10 corresponds to no cooling while hc = 1000 water-cooling; (a) P = 20W, hc = 1000; (b) P = 20W, hc = 10; (c) P = 2000W, hc = 1000; (d) P = 2000W, hc = 10.

Fig. 4
Fig. 4

Results of mode power (left column) and M2 factor (right column) when there are LP011, LP111, LP110, LP211, LP210 and LP021 modes within the fiber.

Fig. 5
Fig. 5

Schematic illustration of the micro bends within the fiber.

Fig. 6
Fig. 6

Results of mode power (left column) and M2 factor (right column) for different bend amplitudes but with the same spatial frequency; (a) Ad = 0.05a, k’ = ∆β; (b) Ad = 0.1a, k’ = ∆β.

Fig. 7
Fig. 7

Results of mode power (left column) and M2 factor (right column) for different spatial frequencies but with the same bend amplitude; (a) Ad = 0.1a, k’ = 0.5∆β; (b) Ad = 0.1a, k’ = ∆β; (c) Ad = 0.1a, k’ = 2∆β.

Tables (1)

Tables Icon

Table 1 The Beat Lengths of any Two Modes

Equations (40)

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{ × E =i μ 0 ε 0 k H × H =i ε 0 μ 0 k n 2 E ,{ ( n 2 E )=0 H =0
n 2 =[ n x 2 0 0 0 n y 2 0 0 0 n z 2 ]
F = E × H ¯ E ¯ × H
{ × E ¯ =i μ 0 ε 0 k H ¯ × H ¯ =i ε 0 μ 0 k n ¯ 2 E ,{ ( n ¯ 2 E )=0 H =0
n ¯ 2 =[ n ¯ x 2 0 0 0 n ¯ y 2 0 0 0 n ¯ z 2 ]
F =( E × H ¯ )( E ¯ × H )=i ε 0 μ 0 k{ ( n ¯ 2 E ¯ ) E ( n 2 E ) E ¯ }
A F dA= z ( A F z ^ dA )
E t = j b j ( z ) e ^ tj + j 0 b j ( z,Q ) e ^ tj ( Q )dQ H t = j b j ( z ) h ^ tj + j 0 b j ( z,Q ) h ^ tj ( Q )dQ
E t = j b j ( z ) e ^ tj H t = j b j ( z ) h ^ tj
E z =i μ 0 ε 0 1 k n z 2 ( H y x H x y ) H z =i ε 0 μ 0 1 k ( E y x E x y )
e ^ zj =i μ 0 ε 0 1 k n ¯ z 2 ( h ^ yj x h ^ xj y )
E z =i μ 0 ε 0 1 k n z 2 j b j ( z )( h ^ yj x h ^ xj y )= n ¯ z 2 n z 2 j b j ( z ) e ^ zj
E ¯ = e ^ k e i β k z , H ¯ = h ^ k e i β k z .
e ^ k = e ^ t( k ) + e ^ z( k ) z ^ = e ^ tk e ^ zk z ^ h ^ k = h ^ t( k ) + h ^ z( k ) z ^ = h ^ tk + h ^ zk z ^
F z ^ ={ j b j ( z )( e ^ tj × h ^ tk + e ^ tk × h ^ tj ) } z ^ e i β k z
A F z ^ dA=4 b k ( z ) e i β k z
z A F z ^ dA= z { 4 b k ( z ) e i β k z }=4{ d b k ( z ) dz i β k b k ( z ) } e i β k z
A F dA=i ε 0 μ 0 k e i β k z A { ( n ¯ 2 e ^ k ) E ( n 2 E ) e ^ k } dA =i ε 0 μ 0 k e i β k z j b j ( z ){ A [ ( n ¯ x 2 n x 2 ) e ^ xj e ^ xk +( n ¯ y 2 n y 2 ) e ^ yj e ^ yk n ¯ z 2 n z 2 ( n ¯ z 2 n z 2 ) e ^ zj e ^ zk ] dA }
d b k ( z ) dz i β k b k ( z )= j=1 N C kj b j ( z ),k=1,2,...N
C kj = i 4 ε 0 μ 0 k A { ( n x 2 n ¯ x 2 ) e ^ xj e ^ xk +( n y 2 n ¯ y 2 ) e ^ yj e ^ yk n ¯ z 2 n z 2 ( n z 2 n ¯ z 2 ) e ^ zj e ^ zk } dA
g( r )= τ 1 Δ( I( r ), P p ) 1+ τ 1 λ s ( σ e ( λ s )+ σ a ( λ s ) ) hν I( r )
n eff =[ n ¯ 0x ig( r ,I( r )) 2 k 0 0 0 0 n ¯ 0y ig( r ,I( r )) 2 k 0 0 0 0 n ¯ 0z ig( r ,I( r )) 2 k 0 ]
n eff,T =[ n ¯ 0x +Δ n x,T 0 0 0 n ¯ 0y +Δ n y,T 0 0 0 n ¯ 0z +Δ n z,T ]
Δ n x,T =Δ n β (r,z)+Δ n ST,r (r,z)cosθΔ n ST,θ (r,z)sinθ Δ n y,T =Δ n β (r,z)+Δ n ST,r (r,z)sinθ+Δ n ST,θ (r,z)cosθ Δ n z,T =Δ n β (r,z)
n j = n ¯ j (xf(z),y,z),j=x,y,z
n ˜ j = n ¯ j (xf(z),y,z)+Δ n j,T (xf(z),y,z) ig(xf(z),y,z) 2 k 0 ,j=x,y,z
[ E x ' ( r ) E y ' ( r ) ]=[ cosϑ sinϑ sinϑ cosϑ ][ E x ( r ) E y ( r ) ]
C k ' = 1 2 A ( E t ' ( r ' )× h ^ tk ' ( r ' ) ) z ^ ' d x ' d y ' ,k=1,2,...,N
E x = j=1 N C xj Ψ j . E y = j=1 N C yj Ψ j
Ψ L P mnh ( r,θ )= A mn { J m ( u mn r )sin(mθ+h π 2 ),ra J m ( u mn a ) K m ( w mn a ) K m ( w mn r )sin(mθ+h π 2 ),r>a
Φ L P mnh =F{ Ψ L P mnh }= 2π (i) m A mn V 2 a( w mn 2 +4 π 2 ρ 2 )( u mn 2 4 π 2 ρ 2 ) { 2πρ J m ( u mn a) J m1 (2πaρ) u mn J m (2πaρ) J m1 ( u mn a) }sin(mφ+h π 2 )
M x 2 = 4π p x 2 I(x,y)dxdy ε 2 I ˜ (ε,η)dεdη = 2π n 0 Z 0 p [ (mnh)(m'n'h') Γ x 2 ,(mnh)(m'n'h') ( C x,(mnh) C x,(m'n'h') + C y,(mnh) C y,(m'n'h') ) ] 1/2 [ (mnh)(m'n'h') Γ ξ 2 ,(mnh)(m'n'h') ( C x,(mnh) C x,(m'n'h') + C y,(mnh) C y,(m'n'h') ) ] 1/2 M y 2 = 4π p y 2 I(x,y)dxdy η 2 I ˜ (ε,η)dεdη = 2π n 0 Z 0 p [ (mnh)(m'n'h') Γ y 2 ,(mnh)(m'n'h') ( C x,(mnh) C x,(m'n'h') + C y,(mnh) C y,(m'n'h') ) ] 1/2 [ (mnh)(m'n'h') Γ η 2 ,(mnh)(m'n'h') ( C x,(mnh) C x,(m'n'h') + C y,(mnh) C y,(m'n'h') ) ] 1/2
Γ x 2 ,(mnh)(m'n'h') = A mn A m'n' I (mn)(m'n') x 2 I mm' x 2 ,θ Γ ξ 2 ,(mnh)(m'n'h') = V 4 (j) m m ' A mn A m'n' I (mn)(m'n') ξ 2 I mm' ξ 2 ,φ Γ y 2 ,(mnh)(m'n'h') = A mn A m'n' I (mn)(m'n') y 2 I mm' y 2 ,θ Γ η 2 ,(mnh)(m'n'h') = V 4 (j) mm' A mn A m'n' I (mn)(m'n') η 2 I mm' η 2 ,φ
I (mn)(m'n') x 2 = I (mn)(m'n') y 2 = 0 a r 3 J m ( u mn r) J m' ( u m'n' r)dr+κ a r 3 K m ( w mn r) K m' ( w m'n' r)dr I (mn)(m'n') ξ 2 = I (mn)(m'n') η 2 = J m ( u mn a) J m' ( u m'n' a) 4 π 2 0 r 5 J m1 (r) J m'1 (r) ( w mn 2 a 2 + r 2 )( u mn 2 a 2 r 2 )( w m'n' 2 a 2 + r 2 )( u m'n' 2 a 2 r 2 ) dr u m'n' a J m ( u mn a) J m'1 ( u m'n' a) 4 π 2 0 r 4 J m1 (r) J m'1 (r) ( w mn 2 a 2 + r 2 )( u mn 2 a 2 r 2 )( w m'n' 2 a 2 + r 2 )( u m'n' 2 a 2 r 2 ) dr u mn a J m1 ( u mn a) J m' ( u m'n' a) 4 π 2 0 r 4 J m (r) J m'1 (r) ( w mn 2 a 2 + r 2 )( u mn 2 a 2 r 2 )( w m'n' 2 a 2 + r 2 )( u m'n' 2 a 2 r 2 ) dr+ u mn u m'n' a 2 J m1 ( u mn a) J m'1 ( u m'n' a) 4 π 2 0 r 3 J m (r) J m' (r) ( w mn 2 a 2 + r 2 )( u mn 2 a 2 r 2 )( w m'n' 2 a 2 + r 2 )( u m'n' 2 a 2 r 2 ) dr
I mm' x 2 ,θ = I mm' ξ 2 ,φ = 0 2π cos 2 θsin(mθ+h π 2 )sin(m'θ+h' π 2 )dθ ={ πsin(h π 2 )sin(h' π 2 ),(m,m')=(0,0) π 2 cos[ (hh') π 2 ] π 4 cos[ (h+h') π 2 ],(m,m')=(1,1) π 2 cos[ (hh') π 2 ],m=m'but0,1 π 2 sin(h π 2 )sin(h' π 2 ),(m,m')=(0,2)or(2,0) π 4 cos[ (hh') π 2 ],{ m=m'2,m'=3,4,... m=m'+2,m'=1,2,... 0,else
I mm' y 2 ,θ = I mm' η 2 ,φ = 0 2π sin 2 θsin(mθ+h π 2 )sin(m'θ+h' π 2 )dθ ={ πsin(h π 2 )sin(h' π 2 ),(m,m')=(0,0) π 2 cos[ (hh') π 2 ]+ π 4 cos[ (h+h') π 2 ],(m,m')=(1,1) π 2 cos[ (hh') π 2 ],m=m'but0,1 π 2 sin(h π 2 )sin(h' π 2 ),(m,m')=(0,2)or(2,0) π 4 cos[ (hh') π 2 ],{ m=m'2,m'=3,4,... m=m'+2,m'=1,2,... 0,else
n x 2 n ¯ x 2 = { n 0 (xf(z),y,z) i 2k g(xf(z),y,z)} 2 n 0 2 (x,y,z)
n x 2 n ¯ x 2 2 n 0 { n 0 (xf(z),y) n 0 (x,y) } i k g n 0 2 n 0 n 0 x f(z) i k g n 0
α= 1 4 ε 0 μ 0 k{ A n 0 2 x e ^ xj e ^ xk dA }
exp(iα A d sin(k'z))

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