Abstract

We examine manipulating the electromagnetic (EM) wave with an external static magnetic field (ESMF) taking advantage of the versatility of the magnetic photonic crystal (PC). The effect of a nonuniform ESMF on the permeability of the constituent magnetic material in the PC is demonstrated to create a gradient of the effective optical index in the crystal, leading to the focusing of the EM wave, with a magnetically tunable focal length, focused waist radius, and the intensity at the focus.

© 2008 Optical Society of America

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2008 (5)

S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, and S. T. Chui, Phys. Rev. Lett. 101, 157407 (2008).
[CrossRef] [PubMed]

S. W. Chen, J. J. Du, S. Y. Liu, Z. F. Lin, and S. T. Chui, Phys. Rev. A 78, 043803 (2008).
[CrossRef]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

2007 (4)

C. Xu, D. Han, X. Wang, X. Liu, and J. Zi, Appl. Phys. Lett. 90, 061112 (2007).
[CrossRef]

Z. F. Lin and S. T. Chui, Opt. Lett. 32, 2288 (2007).
[CrossRef] [PubMed]

S. T. Chui and Z. F. Lin, J. Phys. 19, 406233 (2007).

F. J. Rachford, D. N. Armstead, V. G. Harris, and C. Vittoria, Phys. Rev. Lett. 99, 057202 (2007).
[CrossRef] [PubMed]

2006 (4)

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak B, Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

H. T. Chien and C. C. Chen, Opt. Express 14, 10759 (2006).
[CrossRef] [PubMed]

2005 (2)

E. Centeno and D. Cassagne, Opt. Lett. 30, 2278 (2005).
[CrossRef] [PubMed]

A. M. Merzlikin, A. P. Vinogradov, M. Inoue, and A. B. Granovsky, Phys. Rev. E 72, 046603 (2005).
[CrossRef]

2004 (1)

D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2004).

2003 (2)

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, J. Phys. D 36, R277 (2003).
[CrossRef]

2002 (2)

B. A. van Tiggelen and G. L. J. A. Rikken, in Optical Properties of Random Nanostructures, V.M.Shalaev, ed. (Springer-Verlag, 2002), p. 275.
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

1996 (2)

G. L. J. A. Rikken and B. A. van Tiggelen, Nature 381, 54 (1996).
[CrossRef]

C.M.Soukoulis, ed., Photonic Band Gap Materials, Vol. 315 of NATO Advanced Study Institute (Kluwer, 1996).

1995 (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

1987 (2)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

1978 (1)

C. P. Slichter, Principle of Magnetic Resonance (Springer-Verlag, 1978).

Aktsipetrov, O.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Albert, J.-P.

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Armstead, D. N.

F. J. Rachford, D. N. Armstead, V. G. Harris, and C. Vittoria, Phys. Rev. Lett. 99, 057202 (2007).
[CrossRef] [PubMed]

Baryshev, R. A.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Cassagne, D.

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

E. Centeno and D. Cassagne, Opt. Lett. 30, 2278 (2005).
[CrossRef] [PubMed]

Centeno, E.

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

E. Centeno and D. Cassagne, Opt. Lett. 30, 2278 (2005).
[CrossRef] [PubMed]

Chen, C. C.

Chen, S. W.

S. W. Chen, J. J. Du, S. Y. Liu, Z. F. Lin, and S. T. Chui, Phys. Rev. A 78, 043803 (2008).
[CrossRef]

Chen, W. K.

S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, and S. T. Chui, Phys. Rev. Lett. 101, 157407 (2008).
[CrossRef] [PubMed]

Chien, H. T.

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Chui, S. T.

S. W. Chen, J. J. Du, S. Y. Liu, Z. F. Lin, and S. T. Chui, Phys. Rev. A 78, 043803 (2008).
[CrossRef]

S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, and S. T. Chui, Phys. Rev. Lett. 101, 157407 (2008).
[CrossRef] [PubMed]

Z. F. Lin and S. T. Chui, Opt. Lett. 32, 2288 (2007).
[CrossRef] [PubMed]

S. T. Chui and Z. F. Lin, J. Phys. 19, 406233 (2007).

Dadoenkova, N. N.

I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, J. Phys. D 36, R277 (2003).
[CrossRef]

Decoopman, T.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak B, Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Du, J. J.

S. W. Chen, J. J. Du, S. Y. Liu, Z. F. Lin, and S. T. Chui, Phys. Rev. A 78, 043803 (2008).
[CrossRef]

S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, and S. T. Chui, Phys. Rev. Lett. 101, 157407 (2008).
[CrossRef] [PubMed]

Enoch, S.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak B, Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Fan, S.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Fedyanin, A.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Fujikawa, M. R.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Gralak B, B.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak B, Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Granovsky, A.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Granovsky, A. B.

A. M. Merzlikin, A. P. Vinogradov, M. Inoue, and A. B. Granovsky, Phys. Rev. E 72, 046603 (2005).
[CrossRef]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Han, D.

C. Xu, D. Han, X. Wang, X. Liu, and J. Zi, Appl. Phys. Lett. 90, 061112 (2007).
[CrossRef]

Harris, V. G.

F. J. Rachford, D. N. Armstead, V. G. Harris, and C. Vittoria, Phys. Rev. Lett. 99, 057202 (2007).
[CrossRef] [PubMed]

Inoue, M.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

A. M. Merzlikin, A. P. Vinogradov, M. Inoue, and A. B. Granovsky, Phys. Rev. E 72, 046603 (2005).
[CrossRef]

Joannopoulos, J. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Khanikaev, A.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Lim, P. B.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Lin, Z. F.

S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, and S. T. Chui, Phys. Rev. Lett. 101, 157407 (2008).
[CrossRef] [PubMed]

S. W. Chen, J. J. Du, S. Y. Liu, Z. F. Lin, and S. T. Chui, Phys. Rev. A 78, 043803 (2008).
[CrossRef]

Z. F. Lin and S. T. Chui, Opt. Lett. 32, 2288 (2007).
[CrossRef] [PubMed]

S. T. Chui and Z. F. Lin, J. Phys. 19, 406233 (2007).

Liu, S. Y.

S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, and S. T. Chui, Phys. Rev. Lett. 101, 157407 (2008).
[CrossRef] [PubMed]

S. W. Chen, J. J. Du, S. Y. Liu, Z. F. Lin, and S. T. Chui, Phys. Rev. A 78, 043803 (2008).
[CrossRef]

Liu, X.

C. Xu, D. Han, X. Wang, X. Liu, and J. Zi, Appl. Phys. Lett. 90, 061112 (2007).
[CrossRef]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Lyubchanskii, I. L.

I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, J. Phys. D 36, R277 (2003).
[CrossRef]

Lyubchanskii, M. I.

I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, J. Phys. D 36, R277 (2003).
[CrossRef]

Maystre, D.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak B, Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

Merzlikin, A. M.

A. M. Merzlikin, A. P. Vinogradov, M. Inoue, and A. B. Granovsky, Phys. Rev. E 72, 046603 (2005).
[CrossRef]

Murzina, T.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Pozar, D. M.

D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2004).

Rachford, F. J.

F. J. Rachford, D. N. Armstead, V. G. Harris, and C. Vittoria, Phys. Rev. Lett. 99, 057202 (2007).
[CrossRef] [PubMed]

Raghu, S.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Rasing, Th.

I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, J. Phys. D 36, R277 (2003).
[CrossRef]

Rikken, G. L. J. A.

B. A. van Tiggelen and G. L. J. A. Rikken, in Optical Properties of Random Nanostructures, V.M.Shalaev, ed. (Springer-Verlag, 2002), p. 275.
[CrossRef]

G. L. J. A. Rikken and B. A. van Tiggelen, Nature 381, 54 (1996).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

Shapovalov, E. A.

I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, J. Phys. D 36, R277 (2003).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

Slichter, C. P.

C. P. Slichter, Principle of Magnetic Resonance (Springer-Verlag, 1978).

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

Soljacic, M.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Tayeb, G.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak B, Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Uchida, H.

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

van Tiggelen, B. A.

B. A. van Tiggelen and G. L. J. A. Rikken, in Optical Properties of Random Nanostructures, V.M.Shalaev, ed. (Springer-Verlag, 2002), p. 275.
[CrossRef]

G. L. J. A. Rikken and B. A. van Tiggelen, Nature 381, 54 (1996).
[CrossRef]

Veronis, G.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Vinogradov, A. P.

A. M. Merzlikin, A. P. Vinogradov, M. Inoue, and A. B. Granovsky, Phys. Rev. E 72, 046603 (2005).
[CrossRef]

Vittoria, C.

F. J. Rachford, D. N. Armstead, V. G. Harris, and C. Vittoria, Phys. Rev. Lett. 99, 057202 (2007).
[CrossRef] [PubMed]

Wang, X.

C. Xu, D. Han, X. Wang, X. Liu, and J. Zi, Appl. Phys. Lett. 90, 061112 (2007).
[CrossRef]

Wang, Z.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

Xu, C.

C. Xu, D. Han, X. Wang, X. Liu, and J. Zi, Appl. Phys. Lett. 90, 061112 (2007).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

Yu, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Zi, J.

C. Xu, D. Han, X. Wang, X. Liu, and J. Zi, Appl. Phys. Lett. 90, 061112 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

C. Xu, D. Han, X. Wang, X. Liu, and J. Zi, Appl. Phys. Lett. 90, 061112 (2007).
[CrossRef]

J. Phys. (1)

S. T. Chui and Z. F. Lin, J. Phys. 19, 406233 (2007).

J. Phys. D (2)

I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, J. Phys. D 36, R277 (2003).
[CrossRef]

M. Inoue, M. R. Fujikawa, R. A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, J. Phys. D 39, R151 (2006).
[CrossRef]

Nature (1)

G. L. J. A. Rikken and B. A. van Tiggelen, Nature 381, 54 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

S. W. Chen, J. J. Du, S. Y. Liu, Z. F. Lin, and S. T. Chui, Phys. Rev. A 78, 043803 (2008).
[CrossRef]

Phys. Rev. B (2)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Phys. Rev. E (1)

A. M. Merzlikin, A. P. Vinogradov, M. Inoue, and A. B. Granovsky, Phys. Rev. E 72, 046603 (2005).
[CrossRef]

Phys. Rev. Lett. (9)

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak B, Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

F. J. Rachford, D. N. Armstead, V. G. Harris, and C. Vittoria, Phys. Rev. Lett. 99, 057202 (2007).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

(a), (b) Field intensity distribution for a PC illuminated by a Gaussian beam from the top, with incident angle θ = 0 ° and incident wavelength λ given by a = 0.505 λ . The waist radius of the incident beam is w 0 = 2 λ . ω m = 0.8 ( 2 π c a ) . ω 0 = 0.4 ( 2 π c a ) at the center of the PC. The ESMF gradient is g = ( a ) 0.0% and (b) 0.4%. (c) Field intensity at focal plane along the x direction, showing that the beam is focused, with w 0 shrunk by a factor of 2.

Fig. 2
Fig. 2

Photonic dispersion bands of the PC in two different uniform ESMFs with ω 0 = ( a ) 0.4 ( 2 π c a ) and (b) 0.42 ( 2 π c a ) . The red vertical lines indicate the incident frequency ω = 0.505 ( 2 π c a ) , corresponding to a = 0.505 λ in Fig. 1. The cyan (b) and magenta (a) curves are the tangents of the dispersion band at incident frequency. The insets in Fig. 2b serve to show that the slope of the tangent at ω = 0.505 ( 2 π c a ) is smaller for the case with smaller ω 0 ( H 0 ) .

Fig. 3
Fig. 3

(a) The field intensity at the focus and focal length versus number m r of rows. (b) The field intensity at the focus and focal length versus the ESMF gradient g. (c) The focused waist radius versus the ESMF gradient g. The number of columns m c = 13 and g = 0.4 % for (a). The numbers of columns and rows are m c = 13 and m r = 9 for (b) and (c). ω m = 0.8 ( 2 π c a ) and incident waist radius w 0 = 2 λ for all figures.

Fig. 4
Fig. 4

(a), (b) The same as Fig. 1 except that the PC illuminated by a beam with the incident angle θ = ( a ) 5° and (b) 10°, and the ESMF gradient g = 0.4 % for both (a) and (b). (c) Field intensity at focal plane versus distance ρ from the focus along the direction normal to the beam propagation for different values of g and θ, showing that the beam is focused with a nonuniform ESMF for oblique incidence as well.

Equations (3)

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μ ̂ = [ μ i μ 0 i μ μ 0 0 0 1 ] ,
μ = 1 + ω m ( ω 0 i α ω ) [ ( ω 0 i α ω ) 2 ω 2 ] ,
μ = ω m ω [ ( ω 0 i α ω ) 2 ω 2 ] ,

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