Abstract

The oscillating guided modes and surface modes in an optical fiber with left-handed material core and right-handed material cladding are discussed. Using the dispersion equations for TE, TM, EH, and HE guided modes, their dispersion curves are plotted under the consideration of material dispersion, which is based on an experimental model. Then the properties of the dispersion curves are discussed. For TE(TM) oscillating guided modes, their surface modes can connect with each other. However, the EH oscillating guided mode connects with that of the HE surface mode. Further, for the same frequency, the effective refractive index has two different values for oscillating guided modes and three different values for EH1surface mode. Double-value properties of guided modes in LHM slab waveguides are already known, but the three-value property of surface modes in LHM waveguides is found and reported here. We also investigate the normalized power fluxes of TE modes. New propagation characteristics for TE oscillating guided modes and surface modes are obtained.

© 2009 Optical Society of America

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References

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  1. D. R. Smith and N. Kroll, “Negative refraction index in left-handed materials,” Phys. Rev. Lett. 85, 2933-2936 (2000).
    [CrossRef] [PubMed]
  2. B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386-9388 (2003).
    [CrossRef]
  3. Z. Y. Xiao and Z. H. Wang, “Dispersion characteristics of asymmetric double-negative material slab waveguides,” J. Opt. Soc. Am. B 23, 1757-1760 (2006).
    [CrossRef]
  4. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  5. R. A. Silin, “Electromagnetic waves in artificial periodic structures,” Phys. Usp. 49, 542-545 (2006).
    [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. K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
    [CrossRef]
  8. K. Y. Kim, J. H. Lee, Y. K. Cho, and H. S. Tae, “Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole,” Opt. Express 13, 3653-3665 (2005).
    [CrossRef] [PubMed]
  9. A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibers,” J. Opt. A, Pure Appl. Opt. 72, S51-S56 (2005).
    [CrossRef]
  10. L. F. Shen and Z. H. Wang, “Guided modes characteristics in a fiber with left-handed material,” Microwave Opt. Technol. Lett. 49, 1039-1041 (2007).
    [CrossRef]
  11. V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic properties,” Sov. Phys. Usp. 10, 509-517 (1968).
    [CrossRef]

2007 (2)

K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
[CrossRef]

L. F. Shen and Z. H. Wang, “Guided modes characteristics in a fiber with left-handed material,” Microwave Opt. Technol. Lett. 49, 1039-1041 (2007).
[CrossRef]

2006 (2)

2005 (3)

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

K. Y. Kim, J. H. Lee, Y. K. Cho, and H. S. Tae, “Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole,” Opt. Express 13, 3653-3665 (2005).
[CrossRef] [PubMed]

A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibers,” J. Opt. A, Pure Appl. Opt. 72, S51-S56 (2005).
[CrossRef]

2003 (2)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386-9388 (2003).
[CrossRef]

2000 (1)

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

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic properties,” Sov. Phys. Usp. 10, 509-517 (1968).
[CrossRef]

Barkovsky, L. M.

A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibers,” J. Opt. A, Pure Appl. Opt. 72, S51-S56 (2005).
[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]

Cho, Y. K.

Cory, H.

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

Grzegorczyk, T. M.

B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386-9388 (2003).
[CrossRef]

Kim, K. Y.

K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
[CrossRef]

K. Y. Kim, J. H. Lee, Y. K. Cho, and H. S. Tae, “Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole,” Opt. Express 13, 3653-3665 (2005).
[CrossRef] [PubMed]

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

Kong, J. A.

B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386-9388 (2003).
[CrossRef]

Kroll, N.

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

Lee, J. H.

Novitsky, A. V.

A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibers,” J. Opt. A, Pure Appl. Opt. 72, S51-S56 (2005).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

Shen, L. F.

L. F. Shen and Z. H. Wang, “Guided modes characteristics in a fiber with left-handed material,” Microwave Opt. Technol. Lett. 49, 1039-1041 (2007).
[CrossRef]

Silin, R. A.

R. A. Silin, “Electromagnetic waves in artificial periodic structures,” Phys. Usp. 49, 542-545 (2006).
[CrossRef]

Smith, D. R.

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

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

Tae, H. S.

Veselago, V. G.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic properties,” Sov. Phys. Usp. 10, 509-517 (1968).
[CrossRef]

Wang, Z. H.

L. F. Shen and Z. H. Wang, “Guided modes characteristics in a fiber with left-handed material,” Microwave Opt. Technol. Lett. 49, 1039-1041 (2007).
[CrossRef]

Z. Y. Xiao and Z. H. Wang, “Dispersion characteristics of asymmetric double-negative material slab waveguides,” J. Opt. Soc. Am. B 23, 1757-1760 (2006).
[CrossRef]

Wu, B. L.

B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386-9388 (2003).
[CrossRef]

Xiao, Z. Y.

Zhang, Y.

B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386-9388 (2003).
[CrossRef]

J. Appl. Phys. (1)

B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386-9388 (2003).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
[CrossRef]

A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibers,” J. Opt. A, Pure Appl. Opt. 72, S51-S56 (2005).
[CrossRef]

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

Microwave Opt. Technol. Lett. (2)

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

L. F. Shen and Z. H. Wang, “Guided modes characteristics in a fiber with left-handed material,” Microwave Opt. Technol. Lett. 49, 1039-1041 (2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. E (1)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

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

Phys. Usp. (1)

R. A. Silin, “Electromagnetic waves in artificial periodic structures,” Phys. Usp. 49, 542-545 (2006).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic properties,” Sov. Phys. Usp. 10, 509-517 (1968).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry of an optical fiber.

Fig. 2
Fig. 2

Distribution curves of the dielectric permittivity, magnetic permeability, and refractive index with the frequency in the core.

Fig. 3
Fig. 3

Dispersion curves for all T E 0 n and T M 0 n oscillating guided modes as the frequency varries between 4.008 and 4.18 GHz . The solid curves show T E oscillating guided modes, the dashed curves, T M oscillating guided modes.

Fig. 4
Fig. 4

Dispersion curves for T E 06 and T M 06 oscillating guided modes and T E and T M surface modes at frequencies between 4.2 and 5.30 GHz . The solid curves show T E guided modes, the dashed curves, T M modes.

Fig. 5
Fig. 5

Dispersion curves for H E 1 n ( n 6 ) and H E 2 n oscillating guided modes at frequencies between 4.01 and 4.25 GHz . The solid curves show H E 1 n oscillating guided modes, dashed curves, H E 2 n oscillating guided modes.

Fig. 6
Fig. 6

Dispersion curves for E H 17 and E H 27 oscillating guided modes, E H 1 and E H 2 surface modes, and H E 1 and H E 2 surface modes at frequencies between 4.2 and 5.25 GHz . The solid curves stand for E H 1 n oscillating guided modes and E H 1 , H E 1 surface modes, dashed curves, for E H 2 n oscillating guided modes and E H 2 , H E 2 surface modes. The E H 1 dispersion curve near f = 4.9 GHz is magnified and shown in the inset.

Fig. 7
Fig. 7

Dispersion curves for E H 1 n ( n 6 ) and E H 2 n ( n 6 ) oscillating guided modes at frequencies between 4.01 and 4.12 GHz . The solid curves show E H 1 n oscillating guided modes, the dashed curves, E H 2 n oscillating guided modes.

Fig. 8
Fig. 8

Normalized power fluxes of T E oscillating guided modes and surface modes as a function of frequency. Amplified curves near f = 4 GHz are redrawn in the inset.

Equations (12)

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{ E Z = A J m ( U r a ) J m ( U ) sin m θ H Z = B J m ( U r a ) J m ( U ) cos m θ } , ( r a ) ,
{ E Z = A K m ( W r a ) K m ( W ) sin m θ H Z = B K m ( W r a ) K m ( W ) cos m θ } , ( r a ) ,
( ε 1 ε 2 J + K ) ( μ 1 μ 2 J + K ) = m 2 V 2 ( U 2 + ε 1 μ 1 ε 2 μ 2 W 2 ) ( U W ) 4 ,
ε 1 ε 2 J 1 ( U ) U J 0 ( U ) + K 1 ( W ) W K 0 ( W ) = 0 .
μ 1 μ 2 J 1 ( U ) U J 0 ( S ) + K 1 ( W ) W K 0 ( W ) = 0 .
2 ε 1 μ 1 ε 2 μ 2 J + ( ε 1 ε 2 + μ 1 μ ) K = ± ( ε 1 ε 2 μ 1 μ 2 ) 2 K 2 + 4 ε 1 μ 1 ε 2 μ 2 m 2 V 2 ( U 2 + ε 1 ε 2 μ 1 μ 2 W 2 ) ( U W ) 4 .
( ε 1 ε 2 I K ) ( μ 1 μ 2 I K ) = m 2 ( S 2 + W 2 ) ( S 2 + ε 1 μ 1 ε 2 μ 2 W 2 ) ( S W ) 4 ,
P i = β ω 1 μ i E y i 2 r d r d θ , ( i = 1 , 2 ) .
P 1 = π A 2 a 2 2 ε 1 μ 1 [ 1 + J 1 2 ( U ) J 0 2 ( U ) ] ,
P 2 = π A 2 a 2 2 ε 2 μ 2 [ K 1 2 ( W ) K 0 2 ( W ) 1 ] ,
P = P 1 + P 2 P 1 + P 2 .
ϵ 1 ( ω ) = 1 ω P 2 ω 2 , μ 1 ( ω ) = 1 F ω 2 ω 2 ω 0 2 ,

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