Abstract

A doubly clad fiber with a right-handed-material core and outer cladding but a left-handed-material inner cladding has been analyzed in weakly guiding approximation. The characteristic equations for four kinds of guided modes (HE, EH, TE, and TM) are derived and further simplified under far from cutoff condition. Dispersion curves for the lowest-order mode of each of the four modes have been calculated for different inner-cladding refractive index and thickness. Compared with corresponding curves for a common right-handed doubly clad fiber, these curves all behave with abnormal dispersion. Near cutoff, double degeneracy occurs for HE11 and EH11 modes.

© 2007 Optical Society of America

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References

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. D. R. Smith and N. Kroll, "Negative refraction index in lefthanded materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
    [CrossRef] [PubMed]
  3. L. G. Yang, P. F. Gu, B. Q. Huang, and J. P. Wang, "Transmission characters of optical waves at the interface between LHM and RHM," Acta Photonica Sin. 32, 1225-1227 (2003) (in Chinese).
  4. R. Marques and J. Baena, "Effect of losses and dispersion on the focusing properties of left-handed media," Microwave Opt. Technol. Lett. 41, 290-294 (2004).
    [CrossRef]
  5. H. Cory and C. Zach, "Wave propagation in metamaterial multi-layered structures," Microwave Opt. Technol. Lett. 40, 460-465 (2004).
    [CrossRef]
  6. H. Cory and T. Blum, "Surface-wave propagation along a metamaterial cylinder guide," Microwave Opt. Technol. Lett. 44, 31-35 (2005).
    [CrossRef]
  7. J. L. He, L. F. Shen, S. L. He, and Z. C. Ruan, "Abnormal characteristics of guided modes in a fiber formed by a medium with negative-refractive index," Acta Photonica Sin. 33, 1327-1329 (2004) (in Chinese).
  8. A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
    [CrossRef]
  9. S. Kawakami and S. Nishida, "Perturbation theory of a doubly clad optical fiber with a low-index inner cladding," IEEE J. Quantum Electron. 4, 130-138 (1975).
    [CrossRef]

2005 (2)

H. Cory and T. Blum, "Surface-wave propagation along a metamaterial cylinder guide," Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[CrossRef]

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

2004 (3)

J. L. He, L. F. Shen, S. L. He, and Z. C. Ruan, "Abnormal characteristics of guided modes in a fiber formed by a medium with negative-refractive index," Acta Photonica Sin. 33, 1327-1329 (2004) (in Chinese).

R. Marques and J. Baena, "Effect of losses and dispersion on the focusing properties of left-handed media," Microwave Opt. Technol. Lett. 41, 290-294 (2004).
[CrossRef]

H. Cory and C. Zach, "Wave propagation in metamaterial multi-layered structures," Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

2003 (1)

L. G. Yang, P. F. Gu, B. Q. Huang, and J. P. Wang, "Transmission characters of optical waves at the interface between LHM and RHM," Acta Photonica Sin. 32, 1225-1227 (2003) (in Chinese).

2000 (1)

D. R. Smith and N. Kroll, "Negative refraction index in lefthanded materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

1975 (1)

S. Kawakami and S. Nishida, "Perturbation theory of a doubly clad optical fiber with a low-index inner cladding," IEEE J. Quantum Electron. 4, 130-138 (1975).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Baena, J.

R. Marques and J. Baena, "Effect of losses and dispersion on the focusing properties of left-handed media," Microwave Opt. Technol. Lett. 41, 290-294 (2004).
[CrossRef]

Blum, T.

H. Cory and T. Blum, "Surface-wave propagation along a metamaterial cylinder guide," Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[CrossRef]

Boardman, A. D.

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

Cory, H.

H. Cory and T. Blum, "Surface-wave propagation along a metamaterial cylinder guide," Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[CrossRef]

H. Cory and C. Zach, "Wave propagation in metamaterial multi-layered structures," Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

Gu, P. F.

L. G. Yang, P. F. Gu, B. Q. Huang, and J. P. Wang, "Transmission characters of optical waves at the interface between LHM and RHM," Acta Photonica Sin. 32, 1225-1227 (2003) (in Chinese).

He, J. L.

J. L. He, L. F. Shen, S. L. He, and Z. C. Ruan, "Abnormal characteristics of guided modes in a fiber formed by a medium with negative-refractive index," Acta Photonica Sin. 33, 1327-1329 (2004) (in Chinese).

He, S. L.

J. L. He, L. F. Shen, S. L. He, and Z. C. Ruan, "Abnormal characteristics of guided modes in a fiber formed by a medium with negative-refractive index," Acta Photonica Sin. 33, 1327-1329 (2004) (in Chinese).

Huang, B. Q.

L. G. Yang, P. F. Gu, B. Q. Huang, and J. P. Wang, "Transmission characters of optical waves at the interface between LHM and RHM," Acta Photonica Sin. 32, 1225-1227 (2003) (in Chinese).

Kawakami, S.

S. Kawakami and S. Nishida, "Perturbation theory of a doubly clad optical fiber with a low-index inner cladding," IEEE J. Quantum Electron. 4, 130-138 (1975).
[CrossRef]

King, N.

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

Kroll, N.

D. R. Smith and N. Kroll, "Negative refraction index in lefthanded materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

Marques, R.

R. Marques and J. Baena, "Effect of losses and dispersion on the focusing properties of left-handed media," Microwave Opt. Technol. Lett. 41, 290-294 (2004).
[CrossRef]

Nishida, S.

S. Kawakami and S. Nishida, "Perturbation theory of a doubly clad optical fiber with a low-index inner cladding," IEEE J. Quantum Electron. 4, 130-138 (1975).
[CrossRef]

Ruan, Z. C.

J. L. He, L. F. Shen, S. L. He, and Z. C. Ruan, "Abnormal characteristics of guided modes in a fiber formed by a medium with negative-refractive index," Acta Photonica Sin. 33, 1327-1329 (2004) (in Chinese).

Shen, L. F.

J. L. He, L. F. Shen, S. L. He, and Z. C. Ruan, "Abnormal characteristics of guided modes in a fiber formed by a medium with negative-refractive index," Acta Photonica Sin. 33, 1327-1329 (2004) (in Chinese).

Smith, D. R.

D. R. Smith and N. Kroll, "Negative refraction index in lefthanded materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

Velasco, L.

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Wang, J. P.

L. G. Yang, P. F. Gu, B. Q. Huang, and J. P. Wang, "Transmission characters of optical waves at the interface between LHM and RHM," Acta Photonica Sin. 32, 1225-1227 (2003) (in Chinese).

Yang, L. G.

L. G. Yang, P. F. Gu, B. Q. Huang, and J. P. Wang, "Transmission characters of optical waves at the interface between LHM and RHM," Acta Photonica Sin. 32, 1225-1227 (2003) (in Chinese).

Zach, C.

H. Cory and C. Zach, "Wave propagation in metamaterial multi-layered structures," Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

Acta Photonica Sin. (2)

L. G. Yang, P. F. Gu, B. Q. Huang, and J. P. Wang, "Transmission characters of optical waves at the interface between LHM and RHM," Acta Photonica Sin. 32, 1225-1227 (2003) (in Chinese).

J. L. He, L. F. Shen, S. L. He, and Z. C. Ruan, "Abnormal characteristics of guided modes in a fiber formed by a medium with negative-refractive index," Acta Photonica Sin. 33, 1327-1329 (2004) (in Chinese).

Electromagnetics (1)

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Kawakami and S. Nishida, "Perturbation theory of a doubly clad optical fiber with a low-index inner cladding," IEEE J. Quantum Electron. 4, 130-138 (1975).
[CrossRef]

Microwave Opt. Technol. Lett. (3)

R. Marques and J. Baena, "Effect of losses and dispersion on the focusing properties of left-handed media," Microwave Opt. Technol. Lett. 41, 290-294 (2004).
[CrossRef]

H. Cory and C. Zach, "Wave propagation in metamaterial multi-layered structures," Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

H. Cory and T. Blum, "Surface-wave propagation along a metamaterial cylinder guide," Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

D. R. Smith and N. Kroll, "Negative refraction index in lefthanded materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Geometry of a doubly clad optical fiber.

Fig. 2
Fig. 2

u w curves of the HE 11 mode in DCF including LHM for the same b (0.99) and δ (0.3) but different a (0.96, 0.97, 0.98, respectively, along u).

Fig. 3
Fig. 3

u w curves of the HE 11 mode in the common DCF for the same b (0.99) and δ (0.3), but different a (0.98, 0.97, 0.96, respectively, along u).

Fig. 4
Fig. 4

u w curves of the HE 11 mode in the DCF including LHM for the same b (0.99) and a (0.98), but different δ (0.6, 0.4, 0.2, respectively, along u), shown by dashed curves. However, for the same a and b, a solid curve is plotted from Eq. (12).

Fig. 5
Fig. 5

u w curves of the HE 11 mode in the common DCF for the same b (0.99) and a (0.98), but different δ (0.2, 0.4, 0.6, respectively, along u), shown by dashed curves. However, for the same a and b, a solid curve is plotted from Eq. (16).

Fig. 6
Fig. 6

u w curves of the EH 11 mode in DCF including LHM for the same b (0.99) and δ (0.3) but different a (0.96, 0.97, 0.98, respectively, along u).

Fig. 7
Fig. 7

u w curves of the EH 11 mode in the DCF including LHM for the same b (0.99) and a (0.98), but different δ (1.2, 1.0, 0.8, 0.6, 0.4, respectively, along u), shown by dashed curves. For the same a and b, a solid curve is plotted from Eq. (13).

Fig. 8
Fig. 8

Common u w curves of TE 01 and TM 01 modes in DCF including LHM for the same b (0.99) and δ (0.3) but different a (0.96, 0.97, 0.98, respectively, along u).

Fig. 9
Fig. 9

u w curves of TE 01 and TM 01 modes in the DCF including, LHM for the same b (0.99) and a (0.98) but different δ (0.6, 0.5, 0.4, respectively, along u), shown by dashed curves. For the same a and b, a solid curve is plotted from Eq. (14).

Equations (18)

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E z 1 = A J m ( u r T ) cos m θ H z 1 = ± A n 1 ε 0 μ 0 J m ( u r T ) sin m θ ( r < T ) ,
E z 2 = [ B I m ( w ̂ r T ) + C K m ( w ̂ r T ) ] cos m θ H Z 2 = ± n 1 ε 0 μ 0 [ B I m ( w ̂ r T ) + C K m ( w ̂ r T ) ] sin m θ
( T < r < ( 1 + δ ) T ) ,
E z 3 = D K m ( w r T ) cos m θ H z 3 = ± D n 1 ε 0 μ 0 K m ( w r T ) sin m θ ( r > ( 1 + δ ) T ) ,
E θ 1 = j β A T 2 u 2 [ u T J m ( u r T ) + m r J m ( u r T ) ] sin m θ ( r < T ) ,
E θ 2 = j β T 2 w ̂ 2 { w ̂ T [ B I m ( w ̂ r T ) + C K m ( w ̂ r T ) ] m r [ B I m ( w ̂ r T ) + C K m ( w ̂ r T ) ] } sin m θ ( T < r < ( 1 + δ ) T ) ,
E θ 3 = j β D T 2 w 2 [ w T K m ( w r T ) + m r K m ( w r T ) ] sin m θ
( r > ( 1 + δ T ) ) .
[ K m ( w ̂ ) [ J m 1 ( u ) u J m ( u ) + K m + 1 ( w ̂ ) w ̂ K m ( w ̂ ) ] I m ( w ̂ ) [ J m 1 ( u ) u J m ( u ) I m + 1 ( w ̂ ) w ̂ I m ( w ̂ ) ] K m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) + K m + 1 ( s w ̂ ) s w ̂ K m ( s w ̂ ) ] I m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) K m + 1 ( s w ̂ ) s w ̂ K m ( s w ̂ ) ] ] [ B C ] = 0 .
K m ( w ̂ ) [ J m 1 ( u ) u J m ( u ) + K m + 1 ( w ̂ ) w ̂ K m ( w ̂ ) ] I m ( w ̂ ) [ J m 1 ( u ) u J m ( u ) I m + 1 ( w ̂ ) w ̂ I m ( w ̂ ) ] K m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) + K m + 1 ( s w ̂ ) s w ̂ K m ( s w ̂ ) ] I m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) I m + 1 ( s w ̂ ) s w ̂ I m ( s w ̂ ) ] = 0 .
K m ( w ̂ ) [ J m + 1 ( u ) u J m ( u ) K m 1 ( w ̂ ) w ̂ K m ( w ̂ ) ] I m ( w ̂ ) [ J m + 1 ( u ) u J m ( u ) + I m 1 ( w ̂ ) w ̂ I m ( w ̂ ) ] K m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) + K m + 1 ( s w ̂ ) s w ̂ K m ( s w ̂ ) ] I m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) I m + 1 ( s w ̂ ) s w ̂ I m ( s w ̂ ) ] = 0 .
K 0 ( w ̂ ) [ J 1 ( u ) u J 0 ( u ) K 1 ( w ̂ ) w ̂ K 0 ( w ̂ ) ] I 0 ( w ̂ ) [ J 1 ( u ) u J 0 ( u ) + I 1 ( w ̂ ) w ̂ I 0 ( w ̂ ) ] K 0 ( s w ̂ ) [ K 1 ( s w ) s w K 0 ( s w ) + K 1 ( s w ̂ ) s w ̂ K 0 ( s w ̂ ) ] I 0 ( s w ̂ ) [ K 1 ( s w ) s w K 0 ( s w ) I 1 ( s w ̂ ) s w ̂ I 0 ( s w ̂ ) ] = 0 ,
I n ( w ̂ ) K n ( s w ̂ ) I n ( s w ̂ ) K n ( w ̂ ) exp ( 2 δ w ̂ ) ;
J m 1 ( u ) u J m ( u ) + K m + 1 ( w ̂ ) w ̂ K m ( w ̂ ) = 0 ;
J m + 1 ( u ) u J m ( u ) K m 1 ( w ̂ ) w ̂ K m ( w ̂ ) = 0 ;
J 1 ( u ) u J 0 ( u ) K 1 ( w ̂ ) w ̂ K 0 ( w ̂ ) = 0 .
K m ( w ̂ ) [ J m 1 ( u ) u J m ( u ) + K m 1 ( w ̂ ) w ̂ K m ( w ̂ ) ] I m ( w ̂ ) [ J m 1 ( u ) u J m ( u ) I m 1 ( w ̂ ) w ̂ I m ( w ̂ ) ] K m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) + K m 1 ( s w ̂ ) s w ̂ K m ( s w ̂ ) ] I m ( s w ̂ ) [ K m 1 ( s w ) s w K m ( s w ) + I m + 1 ( s w ̂ ) s w ̂ I m ( s w ̂ ) ] = 0 ,
J m 1 ( u ) u J m ( u ) K m 1 ( w ̂ ) w ̂ K m ( w ̂ ) = 0 .

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