Abstract

Acoustic waves that impinge transversely on a fiber Bragg grating (FBG) induce periodic microbends of the fiber, which modulate the phase index and lead to the changes of optical spectral characteristics of the FBG. We investigated the spectral characteristics of a FBG modulated by a transverse acoustic wave. The corresponding theoretical model is presented by modifying the multimode coupled equations. A fast algorithm based on the Newton–Raphson method is proposed to simulate numerically the spectral characteristics of such a FBG. Our numerical results are in excellent agreement with the known experimental results. For the first time, to our knowledge, the known experimental results have been reproduced by numerical simulations. Moreover, the optimization of the reflective spectra of such a FBG is also discussed. From the perspective of inherent physical mechanisms, the exceptional spectral characteristics of such a FBG are discussed as well.

© 2007 Optical Society of America

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References

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  1. C. C. Ye and R. P. Tatam, "Ultrasonic sensing using Yb3+/Er3+-codoped distributed feedback fibre grating lasers," Smart Mater. Struct. 14, 170-176 (2005).
    [CrossRef]
  2. P. A. Fomitchov and S. Krishnaswamy, "Fiber Bragg grating ultrasound sensor for process monitoring and NDE applications," Rev. Prog. Quant. Nondestr. Eval. 21, 937-944 (2002).
  3. D. W. Huang, W. F. Liu, C. W. Wu, and C. C. Yang, "Reflectivity-tunable fiber Bragg grating reflectors," IEEE Photon. Technol. Lett. 12, 176-178 (2000).
    [CrossRef]
  4. W. F. Liu, I. M. Liu, L. W. Chung, D. W. Huang, and C. C. Yang, "Acoustic-induced switching of reflection wavelength in fiber Bragg grating," Opt. Lett. 25, 1319-1321 (2000).
    [CrossRef]
  5. D. Yeom, H. S. Park, and B. Y. Kim, "Tunable narrow-bandwith optical filter based on acoustically modulated fiber Bragg grating," IEEE Photon. Technol. Lett. 16, 1313-1315 (2004).
    [CrossRef]
  6. W. F. Liu, P. S. J. Russell, and L. Dong, "Acousto-optic superlattice modulator using a fiber Bragg grating," Opt. Lett. 22, 1515-1517 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, "An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending," Opt. Fiber Technol. 13, 32-38 (2006).
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    [CrossRef]
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    [CrossRef]
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2007

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

2006

2005

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

C. C. Ye and R. P. Tatam, "Ultrasonic sensing using Yb3+/Er3+-codoped distributed feedback fibre grating lasers," Smart Mater. Struct. 14, 170-176 (2005).
[CrossRef]

2004

D. Yeom, H. S. Park, and B. Y. Kim, "Tunable narrow-bandwith optical filter based on acoustically modulated fiber Bragg grating," IEEE Photon. Technol. Lett. 16, 1313-1315 (2004).
[CrossRef]

2002

P. A. Fomitchov and S. Krishnaswamy, "Fiber Bragg grating ultrasound sensor for process monitoring and NDE applications," Rev. Prog. Quant. Nondestr. Eval. 21, 937-944 (2002).

N. H. Sun, C. C. Chou, M. J. Chang, C. N. Lin, C. C. Yang, Y. W. Kiang, and W. F. Liu, "Analysis of phase-matching conditions in flexural-wave modulated fiber Bragg grating," J. Lightwave Technol. 20, 311-315 (2002).
[CrossRef]

2000

D. W. Huang, W. F. Liu, C. W. Wu, and C. C. Yang, "Reflectivity-tunable fiber Bragg grating reflectors," IEEE Photon. Technol. Lett. 12, 176-178 (2000).
[CrossRef]

D. W. Huang, W. F. Liu, and C. C. Yang, "Q-switched all-fiber laser with an acoustically modulated fiber attenuator," IEEE Photon. Technol. Lett. 12, 1153-1155 (2000).
[CrossRef]

P. S. J. Russell and W. F. Liu, "Acousto-optic superlattice modulation in fiber Bragg gratings," J. Opt. Soc. Am. A 17, 1421-1429 (2000).
[CrossRef]

W. F. Liu, I. M. Liu, L. W. Chung, D. W. Huang, and C. C. Yang, "Acoustic-induced switching of reflection wavelength in fiber Bragg grating," Opt. Lett. 25, 1319-1321 (2000).
[CrossRef]

1998

1997

1996

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, "The acousto-optic effect in single-mode fiber tapers and couplers," J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

1992

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice Hall, 1992).

1991

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

1987

Abrishamian, F.

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, "An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending," Opt. Fiber Technol. 13, 32-38 (2006).
[CrossRef]

Andrés, M. V.

M. Delgado-Pinar, D. Zalvidea, A. Díez, P. Pérez-Millán, and M. V. Andrés, "Q-switching of an all-fiber laser by acousto-optic modulation of a fiber Bragg grating," Opt. Express 14, 1106-1112 (2006).
[CrossRef] [PubMed]

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Birks, T. A.

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, "The acousto-optic effect in single-mode fiber tapers and couplers," J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

Cai, Z. P.

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

Chang, M. J.

Chou, C. C.

Chung, L. W.

Cruz, J. L.

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Cui, Y. P.

Culverhouse, D. O.

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, "The acousto-optic effect in single-mode fiber tapers and couplers," J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

Delgado-Pinar, M.

M. Delgado-Pinar, D. Zalvidea, A. Díez, P. Pérez-Millán, and M. V. Andrés, "Q-switching of an all-fiber laser by acousto-optic modulation of a fiber Bragg grating," Opt. Express 14, 1106-1112 (2006).
[CrossRef] [PubMed]

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Díez, A.

M. Delgado-Pinar, D. Zalvidea, A. Díez, P. Pérez-Millán, and M. V. Andrés, "Q-switching of an all-fiber laser by acousto-optic modulation of a fiber Bragg grating," Opt. Express 14, 1106-1112 (2006).
[CrossRef] [PubMed]

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Dong, L.

Duchowicz, R.

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Erdogan, T.

Fink, K. D.

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice Hall, 1992).

Fomitchov, P. A.

P. A. Fomitchov and S. Krishnaswamy, "Fiber Bragg grating ultrasound sensor for process monitoring and NDE applications," Rev. Prog. Quant. Nondestr. Eval. 21, 937-944 (2002).

Huang, D. W.

W. F. Liu, I. M. Liu, L. W. Chung, D. W. Huang, and C. C. Yang, "Acoustic-induced switching of reflection wavelength in fiber Bragg grating," Opt. Lett. 25, 1319-1321 (2000).
[CrossRef]

D. W. Huang, W. F. Liu, C. W. Wu, and C. C. Yang, "Reflectivity-tunable fiber Bragg grating reflectors," IEEE Photon. Technol. Lett. 12, 176-178 (2000).
[CrossRef]

D. W. Huang, W. F. Liu, and C. C. Yang, "Q-switched all-fiber laser with an acoustically modulated fiber attenuator," IEEE Photon. Technol. Lett. 12, 1153-1155 (2000).
[CrossRef]

Imai, M.

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, "An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending," Opt. Fiber Technol. 13, 32-38 (2006).
[CrossRef]

Kiang, Y. W.

Kim, B. Y.

D. Yeom, H. S. Park, and B. Y. Kim, "Tunable narrow-bandwith optical filter based on acoustically modulated fiber Bragg grating," IEEE Photon. Technol. Lett. 16, 1313-1315 (2004).
[CrossRef]

Krishnaswamy, S.

P. A. Fomitchov and S. Krishnaswamy, "Fiber Bragg grating ultrasound sensor for process monitoring and NDE applications," Rev. Prog. Quant. Nondestr. Eval. 21, 937-944 (2002).

Li, Q.

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

Lin, C. N.

Liu, I. M.

Liu, W. F.

Lu, C. G.

Luo, Z. Q.

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

Mathews, J. H.

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice Hall, 1992).

Nakai, Y.

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, "An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending," Opt. Fiber Technol. 13, 32-38 (2006).
[CrossRef]

Park, H. S.

D. Yeom, H. S. Park, and B. Y. Kim, "Tunable narrow-bandwith optical filter based on acoustically modulated fiber Bragg grating," IEEE Photon. Technol. Lett. 16, 1313-1315 (2004).
[CrossRef]

Pérez-Millán, P.

Russell, P. S. J.

Russo, N. A.

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Sakuda, K.

Sato, S.

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, "An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending," Opt. Fiber Technol. 13, 32-38 (2006).
[CrossRef]

Si, M.

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

St. J. Russell, P.

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, "The acousto-optic effect in single-mode fiber tapers and couplers," J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

Sun, G.

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

Sun, N. H.

Tatam, R. P.

C. C. Ye and R. P. Tatam, "Ultrasonic sensing using Yb3+/Er3+-codoped distributed feedback fibre grating lasers," Smart Mater. Struct. 14, 170-176 (2005).
[CrossRef]

Wu, C. W.

D. W. Huang, W. F. Liu, C. W. Wu, and C. C. Yang, "Reflectivity-tunable fiber Bragg grating reflectors," IEEE Photon. Technol. Lett. 12, 176-178 (2000).
[CrossRef]

Yamada, M.

Yang, C. C.

N. H. Sun, C. C. Chou, M. J. Chang, C. N. Lin, C. C. Yang, Y. W. Kiang, and W. F. Liu, "Analysis of phase-matching conditions in flexural-wave modulated fiber Bragg grating," J. Lightwave Technol. 20, 311-315 (2002).
[CrossRef]

D. W. Huang, W. F. Liu, and C. C. Yang, "Q-switched all-fiber laser with an acoustically modulated fiber attenuator," IEEE Photon. Technol. Lett. 12, 1153-1155 (2000).
[CrossRef]

D. W. Huang, W. F. Liu, C. W. Wu, and C. C. Yang, "Reflectivity-tunable fiber Bragg grating reflectors," IEEE Photon. Technol. Lett. 12, 176-178 (2000).
[CrossRef]

W. F. Liu, I. M. Liu, L. W. Chung, D. W. Huang, and C. C. Yang, "Acoustic-induced switching of reflection wavelength in fiber Bragg grating," Opt. Lett. 25, 1319-1321 (2000).
[CrossRef]

Ye, C. C.

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

C. C. Ye and R. P. Tatam, "Ultrasonic sensing using Yb3+/Er3+-codoped distributed feedback fibre grating lasers," Smart Mater. Struct. 14, 170-176 (2005).
[CrossRef]

Yeom, D.

D. Yeom, H. S. Park, and B. Y. Kim, "Tunable narrow-bandwith optical filter based on acoustically modulated fiber Bragg grating," IEEE Photon. Technol. Lett. 16, 1313-1315 (2004).
[CrossRef]

Zalvidea, D.

M. Delgado-Pinar, D. Zalvidea, A. Díez, P. Pérez-Millán, and M. V. Andrés, "Q-switching of an all-fiber laser by acousto-optic modulation of a fiber Bragg grating," Opt. Express 14, 1106-1112 (2006).
[CrossRef] [PubMed]

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Appl. Opt.

IEEE Photon. Technol. Lett.

D. W. Huang, W. F. Liu, C. W. Wu, and C. C. Yang, "Reflectivity-tunable fiber Bragg grating reflectors," IEEE Photon. Technol. Lett. 12, 176-178 (2000).
[CrossRef]

D. Yeom, H. S. Park, and B. Y. Kim, "Tunable narrow-bandwith optical filter based on acoustically modulated fiber Bragg grating," IEEE Photon. Technol. Lett. 16, 1313-1315 (2004).
[CrossRef]

D. W. Huang, W. F. Liu, and C. C. Yang, "Q-switched all-fiber laser with an acoustically modulated fiber attenuator," IEEE Photon. Technol. Lett. 12, 1153-1155 (2000).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Opt. Commun.

Z. Q. Luo, C. C. Ye, G. Sun, Z. P. Cai, M. Si, and Q. Li, "Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers," Opt. Commun. 277, 118-124 (2007).
[CrossRef]

D. Zalvidea, N. A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Díez, J. L. Cruz, and M. V. Andrés, "High-repetition rate acoustic-induced Q-switched all-fiber laser," Opt. Commun. 244, 315-319 (2005).
[CrossRef]

Opt. Express

Opt. Fiber Technol.

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, "An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending," Opt. Fiber Technol. 13, 32-38 (2006).
[CrossRef]

Opt. Lett.

Rev. Prog. Quant. Nondestr. Eval.

P. A. Fomitchov and S. Krishnaswamy, "Fiber Bragg grating ultrasound sensor for process monitoring and NDE applications," Rev. Prog. Quant. Nondestr. Eval. 21, 937-944 (2002).

Smart Mater. Struct.

C. C. Ye and R. P. Tatam, "Ultrasonic sensing using Yb3+/Er3+-codoped distributed feedback fibre grating lasers," Smart Mater. Struct. 14, 170-176 (2005).
[CrossRef]

Other

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice Hall, 1992).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

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Figures (5)

Fig. 1
Fig. 1

Schematic of acoustic excitation of the transverse vibration of a FBG.

Fig. 2
Fig. 2

Acoustic-induced coupling parameters k 11 s L , k 31 s L , and k 51 s L as a function of the acoustic power P a in the FBG.

Fig. 3
Fig. 3

Reflective spectra of the uniform FBG under different values of the acoustic-induced coupling parameters: (a) k 11 s L = 0 , k 31 s L = 0 ; (b) k 11 s L = 0.60 , k 31 s L = 0.29 ; (c) k 11 s L = 1.10 , k 31 s L = 0.51 ; (d) k 11 s L = 2.10 , k 31 s L = 0.80 .

Fig. 4
Fig. 4

Calculated reflective spectra of the FBG when the different number of modes are considered in the coupled-mode equations: (a) two modes, (b) three modes, and (c) three modes.

Fig. 5
Fig. 5

Peak reflectivity as a function of k 11 s L : squares, at 1541.5 nm; triangles, at 1539.7   nm .

Tables (1)

Tables Icon

Table 1 Related Parameters for the Different Modes

Equations (222)

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

Er 3+
Yb 3+
Λ B
( Λ S )
( Λ B )
( Λ S )
    d A c o d z = j k c o B c o exp ( j 2 δ c o z ) + j μ ν k μ ν c l B μ ν c l × exp ( j 2 δ μ ν c l z ) + j μ ν k μ ν s A μ ν c l exp ( j 2 δ μ ν s z ) ,
d B c o d z = j k c o A c o exp ( j 2 δ c o z ) j μ ν k μ ν c l A μ ν c l exp ( j 2 δ μ ν c l z ) j μ ν k μ ν s B μ ν c l exp ( j 2 δ μ ν s z ) ,
μ , ν [ d A μ ν c l d z = j k μ ν c l B c o exp ( j 2 δ μ ν c l z ) + j k μ ν s A c o exp ( j 2 δ μ ν s z ) ] ,
μ , ν [ d B μ ν c l d z = j k μ ν c l A c o exp ( j 2 δ μ ν c l z ) j k μ ν s B c o exp ( j 2 δ μ ν s z ) ] .
A c o ( z = 0 ) = 1 , A μ ν c l ( z = 0 ) = 0 ,
  at   the   left   end   position   z = 0 ,
B c o ( z = L ) = 0 , B μ ν c l ( z = L ) = 0 ,
  at   the   right   end   position   z = L .
A c o
B c o
( LP 01 )
A μ ν c l
B μ ν c l
( LP μ ν )
δ c o = β c o π Λ B ,
δ μ ν c l = ( β c o + β μ ν c l 2 π Λ B ) / 2 ,
δ μ ν s = ( β c o β μ ν c l 2 π Λ s / n ) / 2 ,
β c o = 2 π n e f f c o / λ
β μ ν c l = 2 π n μ ν c l / λ
β c o
β μ ν c l
LP μ ν
n e f f c o
n μ ν c l
n ( n = 1 , 2 , 3   … )
k c o
k μ ν c l
k c o
k μ ν c l
k μ ν s
k μ ν s
k μ ν s = 8 2 j 01 j μ ν ( j μ ν 2 j 01 2 ) n e f f c o ( 1 + χ ) c e x t f λ P a 4 ρ ( π c e x t R 5 f 5 ) 1 / 2 ,
j 01
j μ ν
J 0
J μ
c ext
5760 ms 1
P a
2200   kg   m 3
χ = 0.22
Q 0 = ( B c o ( 0 ) B μ ν c l ( 0 ) ) T
Q 0 = ( 0.1 0.1 ) T
Q 0
f ( Q 0 ) = ( B c o ( L ) B μ ν c l ( L ) ) T | Q 0
Q 0
J ( i , j ) = f i / Q 0 j ( i , j = 1 , 2 , 3 )
Q 0
B c o ( 0 )
Q 0
Δ h
Δ h = 0.0001
f ( Q 0 )
Q 0 = ( B c o ( 0 ) + Δ h B μ ν c l ( 0 ) ) T
[ f ( Q 0 ) f ( Q 0 ) ] / Δ h
Q 0
Δ Q 0 = J 1 f ( Q 0 )
f ( Q 0 )
( 0 0 ) T
10 8
Q 0
15   μm
Λ B = 526.9   nm
δ = 0.35 × 10 4
f = 1.3   MHz
Λ s = 452   μm
Λ s = [ π R c ext / f ] 1 / 2
k c o
k μ ν c l
δ μ ν c l = 0
δ μ ν s = 0
n = 1
452   μm
452.85   μm
( LP 11 )
237.35   μm
( LP 31 )
158.39   μm
( LP 51 )
Λ s = 452   μm
Λ s / 2 = 226   μm
Λ s / 3 = 150.7   μm
LP 11
LP 31
LP 51
Λ s = 452   μm
k 11 s L
k 31 s L
k 51 s L
P a
0.01   mW
k 11 s L
k 31 s L
k 51 s L
1537.9   nm
LP 11
LP 31
LP 01
LP 01
LP 11
LP 31
P a
k 11 s L
k 31 s L
k 11 s L = k 31 s L = 0
1541.5   nm
k 11 s L
k 31 s L
1537.9   nm
1539.7   nm
k 11 s L
k 31 s L
1541.5   nm
20%
k 11 s L
k 31 s L
1539.7   nm
52%
( k μ ν s = 0 )
( LP 01 )
( LP 11
LP 31
LP 51  …
( LP 01 )
LP 01
LP 11
k 11 s L = 1.10
LP 01
LP 11
LP 31
LP 01
LP 11
LP 31
LP 51
k 11 s L = 1.10
k 31 s = 0.51
k 11 s L = 1.10
k 31 s = 0.51
k 51 s L = 0.32
1539.7   nm
1539.7   nm
LP 31
LP 51
LP 31
LP 51
k μ ν s L
k μ ν s L
LP 01
LP 11
1539.7   nm
1539.7   nm
k 11 s L
k 11 s L
1541.5   nm
1539.7   nm
k 11 s L
k 11 s L
k 11 s L
P a = 0.003   mW
1539.7   nm
k 11 s L
P a = 0.001   mW
k 11 s L
( Λ s = 452   μm )
1539.7   nm
( Λ s = 452   μm )
1541.5   nm
1541.5   nm
( LP 11 )
k 11 s L
1541.5   nm
1541.5   nm
k 11 s L
1539.7   nm
k 11 s L
k 11 s L
1539.7   nm
LP 11
( k 11 c l L )
( k 11 s L k 11 c l L )
1539.7   nm
1539.7   nm
k 11 s L
( k 11 s L k 11 c l L )
1539.7   nm
1539.7   nm
k 11 s L
k 11 s L
1539.7   nm
k 11 s L
Yb 3+
Er 3+
n e f f c o
n μ ν c l
k c o L
k μ ν c l L
LP 01
LP 11
LP 21
LP 31
LP 41
LP 51
k 11 s L
k 31 s L
k 51 s L
P a
k 11 s L = 0
k 31 s L = 0
k 11 s L = 0.60
k 31 s L = 0.29
k 11 s L = 1.10
k 31 s L = 0.51
k 11 s L = 2.10
k 31 s L = 0.80
k 11 s L
1539.7   nm

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