Abstract

Guided modes in a dielectric waveguide structure with a coaxial periodic multi-layer are investigated by using a matrix formula with Bessel functions. We show that guided modes exist in the structure, and that the field is confined in the core which consists of the optically thinner medium. The dispersion curves are discontinuous, so that the modes can exist only in particular wavelength bands corresponding to the stop bands of the periodic structure of the clad. It is possible that the waveguide structure can be applied to filters or optical fibers to reduce nonlinear effects.

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References

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  1. J. C. Knight, T. A. Birks, P. St. J. Russell and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
    [CrossRef] [PubMed]
  2. J. C. Knight, T. A. Birks, P. St. J. Russell and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding: errata," Opt. Lett. 22, 484-485 (1997).
    [CrossRef] [PubMed]
  3. T. A. Birks, J. C. Knight and P. St. J. Russell, "Endlessl single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
    [CrossRef] [PubMed]
  4. J. C. Knight, T. A. Birks, P. St. J. Russell and J. P. de Sandro, "Properties of photonic crystal fiber and the effective index model," J. Opt. Soc. Am. 15, 748-752 (1998).
    [CrossRef]
  5. J. C. Knight, T. A. Birks, R. F. regan, P. St. J. Russell and J. P. de Sandro, "Large mode area photonic crystal fibre," Electron. Lett. 34, 1347-1348 (1998).
    [CrossRef]
  6. J. Broeng, S. E.Barkou, A. Bjarklev, J. C.Knight, T. A. Birks and P. St. J.Russell, "Highly increased photonic band gaps in silica/air structures," Opt. ommun. 156, 240-244 (1998).
    [CrossRef]
  7. J. C. Knight, J. Broeng, T. A. Birks and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
    [CrossRef] [PubMed]
  8. S. E. Barkou, J. Broeng and A. Bjarklev, "Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect," Opt. Lett. 24, 46-48 (1999).
    [CrossRef]
  9. A. Ferrando, E. Silverstre, J. J.Miret, P. Andes and M. V. Andes "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999).
    [CrossRef]
  10. Z. Xue-Heng, "Theory of two-dimensional 'Fingerprint' resonators," Electron. Lett. 25, 1311-1312 (1989).
    [CrossRef]
  11. M. Toda, "Single-mode behavior of a circular grating for potential disk-shaped DFB lasers," IEEE J. Quantum Electron. 26, 473-481 (1990).
    [CrossRef]
  12. T. Erdogan and D. G. Hall, "Circularl symmetric distribution feedback semiconductor laser: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
    [CrossRef]
  13. X. H. Zheng and S. Lacroix, "Mode coupling in circular-cylindrical system and its application to fingerprint resonators," J. Lightwave Technol. 8, 1509-1516 (1990).
    [CrossRef]
  14. C. Wu, T. Makino, J. Glinski, R. Maciejko and S. I. Najafi, "Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides," J.Lightwave Technol. 8, 1264-1277 (1991).
    [CrossRef]
  15. C. Wu, T. Makino, R. Maciejko, S. I. Najafi and M. Svilans, "Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides," J. Lightwave Technol. 10, 1575-1589 (1992).
    [CrossRef]
  16. S. Noda, T. Ishikawa, M. Imada and A. Sasaki, "Surface-emitting device with embedded circular grating coupler for possible application on optoelectronic integrated devices," IEEE J. Photon. Technol. Lett 7, 1397-1399 (1995).
    [CrossRef]
  17. A. Yariv and A. Gover, "Equivalence of the coupled-mode and Floquet-Bloch formalisms in periodic optical waveguides," Appl. Phys. Lett. 26, 537-539 (1975).
    [CrossRef]
  18. A. Y. Cho, A. Yariv and P. Yeh, "Observation of confined propagation in Bragg waveguides," Appl. Phys. Lett. 30, 471-472 (1977).
    [CrossRef]
  19. P. Yeh, A. Yariv and A. Y. Cho, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
    [CrossRef]
  20. W. Ng, P. Yeh, P. C. Chen and A. Yariv, "Optical surface waves in periodic la ered medium grown by liquid phase epitax," Appl. Phys. Lett. 32, 370-371 (1978).
    [CrossRef]

Other

J. C. Knight, T. A. Birks, P. St. J. Russell and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding: errata," Opt. Lett. 22, 484-485 (1997).
[CrossRef] [PubMed]

T. A. Birks, J. C. Knight and P. St. J. Russell, "Endlessl single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell and J. P. de Sandro, "Properties of photonic crystal fiber and the effective index model," J. Opt. Soc. Am. 15, 748-752 (1998).
[CrossRef]

J. C. Knight, T. A. Birks, R. F. regan, P. St. J. Russell and J. P. de Sandro, "Large mode area photonic crystal fibre," Electron. Lett. 34, 1347-1348 (1998).
[CrossRef]

J. Broeng, S. E.Barkou, A. Bjarklev, J. C.Knight, T. A. Birks and P. St. J.Russell, "Highly increased photonic band gaps in silica/air structures," Opt. ommun. 156, 240-244 (1998).
[CrossRef]

J. C. Knight, J. Broeng, T. A. Birks and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

S. E. Barkou, J. Broeng and A. Bjarklev, "Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect," Opt. Lett. 24, 46-48 (1999).
[CrossRef]

A. Ferrando, E. Silverstre, J. J.Miret, P. Andes and M. V. Andes "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999).
[CrossRef]

Z. Xue-Heng, "Theory of two-dimensional 'Fingerprint' resonators," Electron. Lett. 25, 1311-1312 (1989).
[CrossRef]

M. Toda, "Single-mode behavior of a circular grating for potential disk-shaped DFB lasers," IEEE J. Quantum Electron. 26, 473-481 (1990).
[CrossRef]

T. Erdogan and D. G. Hall, "Circularl symmetric distribution feedback semiconductor laser: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
[CrossRef]

X. H. Zheng and S. Lacroix, "Mode coupling in circular-cylindrical system and its application to fingerprint resonators," J. Lightwave Technol. 8, 1509-1516 (1990).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko and S. I. Najafi, "Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides," J.Lightwave Technol. 8, 1264-1277 (1991).
[CrossRef]

C. Wu, T. Makino, R. Maciejko, S. I. Najafi and M. Svilans, "Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides," J. Lightwave Technol. 10, 1575-1589 (1992).
[CrossRef]

S. Noda, T. Ishikawa, M. Imada and A. Sasaki, "Surface-emitting device with embedded circular grating coupler for possible application on optoelectronic integrated devices," IEEE J. Photon. Technol. Lett 7, 1397-1399 (1995).
[CrossRef]

A. Yariv and A. Gover, "Equivalence of the coupled-mode and Floquet-Bloch formalisms in periodic optical waveguides," Appl. Phys. Lett. 26, 537-539 (1975).
[CrossRef]

A. Y. Cho, A. Yariv and P. Yeh, "Observation of confined propagation in Bragg waveguides," Appl. Phys. Lett. 30, 471-472 (1977).
[CrossRef]

P. Yeh, A. Yariv and A. Y. Cho, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

W. Ng, P. Yeh, P. C. Chen and A. Yariv, "Optical surface waves in periodic la ered medium grown by liquid phase epitax," Appl. Phys. Lett. 32, 370-371 (1978).
[CrossRef]

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Figures (14)

Fig. 1.
Fig. 1.

A waveguide structure with one-dimensional periodic structures.

Fig. 2.
Fig. 2.

Definition of propagation direction of a wave in a periodic structure.

Fig. 3.
Fig. 3.

Coaxial periodic optical waveguide.

Fig. 4.
Fig. 4.

Cross-section.

Fig. 5.
Fig. 5.

Dispersion curves for k 0 c=1.0

Fig. 6.
Fig. 6.

Dispersion curves for k 0 c=5.0

Fig. 7.
Fig. 7.

Stopband of a coaxial periodic optical waveguide.

Fig. 8.
Fig. 8.

Stopband of a one-dimensonal photonic band gap structure.

Fig. 9.
Fig. 9.

z-direction components of magnetic field of TE-polarized mode and electric field of TM-polarized mode, where k=1.2k 0 and β=0.7859[TE], 0.5270[TM].

Fig. 10.
Fig. 10.

z-direction components of magnetic field of TE-polarized mode and electric field of TM-polarized mode, where k=3.0k 0 and β=0.9672[TE], 0.9672[TM].

Fig. 11.
Fig. 11.

Intensity of EM wave field in CPOW, excited by TE01 mode. |T|=0.986, |R|=0.007.

Fig. 12.
Fig. 12.

Intensity of EM wave field in CPOW, excited by TE21 mode. |T|=0.479, |R|=0.049.

Fig. 13.
Fig. 13.

Intensity of EM wave field in CPOW, excited by TM01 mode. |T|=0.440, |R|=0.106.

Fig. 14.
Fig. 14.

Intensity of EM wave field in CPOW, excited by TM21 mode. |T|=0.049, |R|=0.493.

Equations (16)

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cos k TE ( a + b ) = n 1 2 cos 2 θ 1 + n 2 2 cos 2 θ 2 2 n 1 n 2 cos θ 1 cos θ 2 sin ( n 1 cos θ 1 k a ) sin ( n 2 cos θ 2 k b )
+ cos ( n 1 cos θ 1 k a ) cos ( n 2 cos θ 2 k b )
cos k TM ( a + b ) = n 2 2 cos 2 θ 1 + n 1 2 cos 2 θ 2 2 n 1 n 2 cos θ 1 cos θ 2 sin ( n 1 cos θ 1 k a ) sin ( n 2 cos θ 2 k b )
+ cos ( n 1 cos θ 1 k a ) cos ( n 2 cos θ 2 k b ) ,
cos k TE ( a + b ) > 1 ,
cos k TM ( a + b ) > 1 ,
r i = { n ( b + a ) + c i = 2 n n ( b + a ) + b + c i = 2 n + 1 ( n = 0 , 1 , 2 , 3 , ) .
i = { I i = 2 n II i = 2 n + 1 ( n = 0 , 1 , 2 , 3 , ) .
E z ( r ) = { A i J m ( q i r ) + B i Y m ( q i r ) } sin ( m ϕ + θ m )
H z ( r ) = { C i J m ( q i r ) + D i Y m ( q i r ) } cos ( m ϕ + θ m ) ,
[ E z ( r ) H z ( r ) E ϕ ( r ) H ϕ ( r ) ] = U · T ( r , i ) · u i
U [ sin ( m ϕ + θ m ) 0 0 0 0 cos ( m ϕ + θ m ) 0 0 0 0 cos ( m ϕ + θ m ) 0 0 0 0 sin ( m ϕ + θ m ) ]
T ( r , i ) [ J m ( q i r ) Y m ( q i r ) 0 0 0 0 J m ( q i r ) Y m ( q i r ) j β m J m ( q i r ) q i 2 r j β m Y m ( q i r ) q i 2 r + j ω μ 0 J ' m ( q i r ) q i + j ω μ 0 Y ' m ( q i r ) q i j ω i J ' m ( q i r ) q i j ω i Y ' m ( q i r ) q i + j β m J m ( q i r ) q i 2 r + j β m Y m ( q i r ) q i 2 r ]
u i = [ A i , B i , C i , D i ] t .
T ( r i , i ) · u i = T ( r i , i + 1 ) · u i + 1 .
u 0 = i = 0 n R i · u n + 1 , R i T 1 ( r i , i ) · T ( r i , i + 1 ) ,

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