Abstract

It is demonstrated that Bragg reflection waveguides, either planar or cylindrical, can be designed to support a symmetric mode with a specified core field distribution, by adjusting the first layer width. Analytic expressions are given for this matching layer, which matches between the electromagnetic field in the core, and a Bragg mirror optimally designed for the mode. This adjustment may change significantly the characteristics of the waveguide. At the particular wavelength for which the waveguide is designed, the electromagnetic field is identical to that of a partially dielectric loaded metallic or perfect magnetic waveguide, rather than a pure metallic waveguide. Either a planar or coaxial Bragg waveguide is shown to support a mode that has a TEM field distribution in the hollow region.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. P. Yeh and A. Yariv, ???Bragg reflection waveguides,??? Opt Commun. 19, 427???430 (1976).
    [CrossRef]
  2. P. Yeh, A. Yariv, and E. Marom, ???Theory of Bragg fiber,??? J. Opt. Soc. Am. 68, 1196???1201 (1978).
    [CrossRef]
  3. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, ???Guiding optical light in air using an all-dielectric structure,??? J. Lightwave Technol. 17, 2039???2041 (1999).
    [CrossRef]
  4. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, ???Wavelength-scalable hollow optical fibres with large Photonic bandgaps for CO2 laser transmission,??? Nature 420, 650???653 (2002).
    [CrossRef] [PubMed]
  5. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Solja??i??, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, ???Low-loss asymptotically single-mode propagation in large-core Omniguide fibers,??? Opt. Express 9, 748???779 (2001). <a href="http://www.opticsexpressorg/abstract.cfm?URI=OPEX-9-13-748.">http://www.opticsexpressorg/abstract.cfm?URI=OPEX-9-13-748</a>
    [CrossRef] [PubMed]
  6. M. Ibanescu, S. G. Johnson, M. Solja??i??, J. D. Joannopoulos, Y. Fink, O.Weisberg, T. D. Engeness, S. A. Jacobs, and M. Skorobogatiy, ???Analysis of mode structure in hollow dielectric waveguide fibers,??? Phys. Rev. E 67, 046,608 (2003).
    [CrossRef]
  7. Y. Xu, R. K. Lee, and A. Yariv, ???Asymptotic analysis of Bragg fibers,??? Opt. Lett. 25, 1756???1758 (2000).
    [CrossRef]
  8. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, ???Asymptotic matrix theory of Bragg fibers,??? J. Lightwave Tech. 20, 428???440 (2002).
    [CrossRef]
  9. Y. Xu and A. Yariv, ???Asymptotic analysis of Silicon based Bragg fibers,??? Opt. Express 11, 1039???1049 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039</a>
    [CrossRef] [PubMed]
  10. G. Ouyang, Y. Xu, and A. Yariv, ???Theoretical study on dispersion compensation in air-core Bragg fibers,??? Opt. Express 10, 899???908 (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-899">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-899</a>
    [CrossRef] [PubMed]
  11. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, ???An all-dielectric coaxial waveguide,??? Science 289, 415???419 (2000).
    [CrossRef] [PubMed]
  12. Y. Xu, R. K. Lee, and A. Yariv, ???Asymptotic analysis of dielectric coaxial fibers,??? Opt. Lett. 27, 1019???1021 (2002).
    [CrossRef]
  13. G. Ouyang, Y. Xu, and A. Yariv, ???Comparative study of air-core and coaxial Bragg fibers: single-mode transmission and dispersion characteristics,??? Opt. Express 9, 733???747 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733</a>
    [CrossRef] [PubMed]
  14. T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, ???Dispersion tailoring and compensation by modal interactions in Omniguide fibers,??? Opt. Express 11, 1175???1196 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1175">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1175</a>
    [CrossRef] [PubMed]
  15. A. Mizrahi and L. Schächter, ???Optical Bragg accelerators,??? Phys. Rev. E (to be published) (2004).
    [CrossRef]
  16. F. Benabid, J. C. Knight, and P. S. J. Russell, ???Particle levitation and guidance in hollow-core photonic crystal fiber,??? Opt. Express 10, 1195???1203 (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1195">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1195</a>
    [CrossRef] [PubMed]
  17. G. Lenz, E. Baruch, and J. Salzman, ???Polarization discrimination properties of Bragg-reflection waveguides,??? Opt. Lett. 15, 1288???1290 (1990).
    [CrossRef] [PubMed]
  18. C. M. de Sterke, I. M. Bassett, and A. G. Street, ???Differential losses in Bragg fibers,??? J. Appl. Phys. 76, 680???688 (1994).
    [CrossRef]
  19. P. Yeh, ???Electromagnetic propagation in birefringent layered media,??? J. Opt. Soc. Am. 65, 742???756 (1979).
    [CrossRef]

J. Appl. Phys. (1)

C. M. de Sterke, I. M. Bassett, and A. G. Street, ???Differential losses in Bragg fibers,??? J. Appl. Phys. 76, 680???688 (1994).
[CrossRef]

J. Lightwave Tech. (1)

Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, ???Asymptotic matrix theory of Bragg fibers,??? J. Lightwave Tech. 20, 428???440 (2002).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (2)

Nature (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, ???Wavelength-scalable hollow optical fibres with large Photonic bandgaps for CO2 laser transmission,??? Nature 420, 650???653 (2002).
[CrossRef] [PubMed]

Opt Commun. (1)

P. Yeh and A. Yariv, ???Bragg reflection waveguides,??? Opt Commun. 19, 427???430 (1976).
[CrossRef]

Opt. Express (6)

G. Ouyang, Y. Xu, and A. Yariv, ???Comparative study of air-core and coaxial Bragg fibers: single-mode transmission and dispersion characteristics,??? Opt. Express 9, 733???747 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733</a>
[CrossRef] [PubMed]

T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, ???Dispersion tailoring and compensation by modal interactions in Omniguide fibers,??? Opt. Express 11, 1175???1196 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1175">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1175</a>
[CrossRef] [PubMed]

F. Benabid, J. C. Knight, and P. S. J. Russell, ???Particle levitation and guidance in hollow-core photonic crystal fiber,??? Opt. Express 10, 1195???1203 (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1195">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1195</a>
[CrossRef] [PubMed]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Solja??i??, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, ???Low-loss asymptotically single-mode propagation in large-core Omniguide fibers,??? Opt. Express 9, 748???779 (2001). <a href="http://www.opticsexpressorg/abstract.cfm?URI=OPEX-9-13-748.">http://www.opticsexpressorg/abstract.cfm?URI=OPEX-9-13-748</a>
[CrossRef] [PubMed]

Y. Xu and A. Yariv, ???Asymptotic analysis of Silicon based Bragg fibers,??? Opt. Express 11, 1039???1049 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039</a>
[CrossRef] [PubMed]

G. Ouyang, Y. Xu, and A. Yariv, ???Theoretical study on dispersion compensation in air-core Bragg fibers,??? Opt. Express 10, 899???908 (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-899">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-899</a>
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. E (2)

A. Mizrahi and L. Schächter, ???Optical Bragg accelerators,??? Phys. Rev. E (to be published) (2004).
[CrossRef]

M. Ibanescu, S. G. Johnson, M. Solja??i??, J. D. Joannopoulos, Y. Fink, O.Weisberg, T. D. Engeness, S. A. Jacobs, and M. Skorobogatiy, ???Analysis of mode structure in hollow dielectric waveguide fibers,??? Phys. Rev. E 67, 046,608 (2003).
[CrossRef]

Science (1)

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, ???An all-dielectric coaxial waveguide,??? Science 289, 415???419 (2000).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Planar and cylindrical Bragg reflection waveguides.

Fig. 2.
Fig. 2.

First layer width for v ph=c, normalized by Δ q λ 0 / ( 4 ε 1 1 ) . The layer adjacent to the core has a refractive index of n 1=1.6 and the other material has n 2=4.6.

Fig. 3.
Fig. 3.

Planar TM profiles. (a) kxD int=0 low refractive index first (b) kxD int=0 high refractive index first (c) kxD int=π/3 (d) kxD int=3π/4 (e) kxD int=π/2 (metallic-like walls) (f) kxD int=π (magnetic-like walls).

Fig. 4.
Fig. 4.

Symmetric TM mode dispersion diagram for planar waveguide with D int=0.3λ0 (left) and cylindrical waveguide with R int=0.3λ0 (right). In both cases the red (dashed) curves are obtained with no design procedure, and the blue (solid) curves correspond to a v ph=c design procedure.

Fig. 5.
Fig. 5.

Symmetric TM mode dispersion diagram for planar waveguide with D int=1.0λ0 (left) and cylindrical waveguide with R int=1.0λ0 (right). In both cases the red (dashed) curves are obtained with no design procedure, and the blue (solid) curves correspond to a v ph=c design procedure.

Fig. 6.
Fig. 6.

Group velocity dispersion for the v ph=c cylindrical waveguide.

Fig. 7.
Fig. 7.

Planar odd TM power profiles: kxD int=π/4 (solid blue) and kxD int=/4 (dashed red).

Fig. 8.
Fig. 8.

Planar TEM-TM profiles: higher refractive index first (top) and lower refractive index first (bottom).

Fig. 9.
Fig. 9.

Coaxial TEM-TM profiles: the higher refractive index is first at the hollow region outer boundary (top) and the lower refractive index first (bottom).

Tables (2)

Tables Icon

Table 1. Hollow core symmetric modes. The transverse wavenumbers are k x , r = ω 2 / c 2 k z 2 .

Tables Icon

Table 2. Comparison between the waveguide parameters that correspond to the field distributions of Fig. 3 and the analogous TE cases.

Equations (21)

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

A 1 / E 0 = ( B 1 / E 0 ) * = 1 2 e j k 1 D int cos ( k x D int ) j k 1 2 ε 1 k x e j k 1 D int sin ( k x D int ) ;
Δ v = π 2 ω 0 2 c 2 ε v k z 2 .
{ E z ( x = D int + Δ 1 ) = 0 Z 1 > Z 2 E z x ( x = D int + Δ 1 ) = 0 Z 1 < Z 2 ,
Δ 1 ( TM ) = { 1 k 1 arctan [ ε 1 k x k 1 cot ( k x D int ) ] Z 1 > Z 2 1 k 1 arctan [ k 1 ε 1 k x tan ( k x D int ) ] Z 1 < Z 2 .
Δ 1 ( TM ) = { 1 k 1 arctan [ ( Z 1 η 0 ω 0 c D int ) 1 ] Z 1 > Z 2 1 k 1 arctan ( Z 1 η 0 ω 0 c D int ) Z 1 < Z 2 .
Δ 1 ( TE ) = { 1 k 1 arctan [ k x k 1 cot ( k x D int ) ] Y 1 > Y 2 1 k 1 arctan [ k 1 k x tan ( k x D int ) ] Y 1 < Y 2 .
{ E z = 0 Z v > Z v + 1 E z r = 0 Z v < Z v + 1 .
U ( TM ) ε 1 k 1 Y 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r Y 0 ( k 1 R int ) J 1 ( k r R int ) ε 1 k 1 J 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r J 0 ( k 1 R int ) J 1 ( k r R int ) ,
Δ 1 ( TM ) = { 1 k 1 arcBess tan 0 ( U ( TM ) ) R int Z 1 > Z 2 1 k 1 arcBess tan 1 ( U ( TM ) ) R int Z 1 < Z 2 .
U ( TM ) = ε 1 k 1 Y 1 ( k 1 R int ) R int 2 Y 0 ( k 1 R int ) ε 1 k 1 J 1 ( k 1 R int ) R int 2 J 0 ( k 1 R int ) .
U ( TE ) 1 k 1 Y 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r Y 0 ( k 1 R int ) J 1 ( k r R int ) 1 k 1 J 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r J 0 ( k 1 R int ) J 1 ( k r R int )
α = 2 ω 0 tan δ W E , clad ( z ) P ( z ) .
E z = E 0 sin ( k x x ) e j k z z ,
E x = j k z k x E 0 cos ( k x x ) e j k z ,
H y = j ω 0 ε 0 k x E 0 cos ( k x x ) e j k z z .
Δ 1 ( TM ) = { 1 k 1 arctan [ ε 1 k x k 1 tan ( k x D int ) ] Z 1 > Z 2 1 k 1 arctan [ k 1 ε 1 k x cot ( k x D int ) ] Z 1 < Z 2 .
E z = E 0 J 0 ( k r r ) e j k z z ,
E r = j η 0 ε r ε r 1 E 0 J 1 ( k r r ) e j k z z ,
H ϕ = j ε r 1 E 0 J 1 ( k r r ) e j k z z .
E r = C η 0 1 r e j k z z ,
H ϕ = C r e j k z z .

Metrics