Abstract

We introduce an extended transfer matrix method (TMM) for solving guided modes in leaky optical fibers with layered cladding. The method can deal with fibers with circular but nonconcentric material interfaces. Validity of the method is verified by two full-vector numerical methods. The TMM is then used to investigate the guidance property of perturbed Bragg fibers. Our analysis reveals that the core modes will interact with each other when a perturbed Bragg fiber has only C 1 symmetry. Special attention is paid to the first transverse-electric (TE01) mode, which is found to experience severe degradation around spectral regions where its dispersion curve supposedly crosses a transverse-magnetic (TM) or hybrid mode.

© 2006 Optical Society of America

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  1. P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
    [CrossRef]
  2. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. M. Solja ci`c, 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-748.
    [CrossRef] [PubMed]
  3. K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, "Hollow multilayer photonic bandgap fibers for NIR applications," Opt. Express 12, 1510-1517 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1510.
    [CrossRef] [PubMed]
  4. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
    [CrossRef]
  5. T. Okoshi, Optical Fibers, 1st ed. (Academic Press, 1982).
  6. P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides — I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
    [CrossRef]
  7. S. Guo, F. Wu, S. Albin, H. Tai, and R. S. Rogowski, "Loss and dispersion analysis of microstructured fibers by finite-difference method," Opt. Express 12, 3341-3352 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-15-3341.
    [CrossRef] [PubMed]
  8. T. Katagiri, Y. Matsuura, and M. Miyagi, "Photonic bandgap fiber with a silica core and multilayer dielectric cladding," Opt. Lett. 29, 557-559 (2004).
    [CrossRef] [PubMed]
  9. G. Vienne, Y. Xu, C. Jakobsen, H.-J. Deyerl, J. B. Jensen, T. Sorensen, T. P. Hansen, Y. Huang, M. Terrel, R. K. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, "Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports," Opt. Express 12, 3500-3508 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-15-3500.
    [CrossRef] [PubMed]
  10. M. Yan, J. Canning, G. Vienne, and P. Shum, "Investigation of practical air-silica Bragg fiber," in Australian Conference on Optical Fibre Technology (ACOFT) (Sydney, Australia, 2005).
  11. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, and J. D. Joannopoulos, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066,611 (2002).
    [CrossRef]
  12. M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, "Dielectric profile variations in high-index-contrastwaveguides, coupled mode theory, and perturbation expansions," Phys. Rev. E 67, 046,613 (2003).
    [CrossRef]

2004

2003

M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, "Dielectric profile variations in high-index-contrastwaveguides, coupled mode theory, and perturbation expansions," Phys. Rev. E 67, 046,613 (2003).
[CrossRef]

2002

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, and J. D. Joannopoulos, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066,611 (2002).
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
[CrossRef]

2001

1978

1975

P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides — I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

Albin, S.

Bayindir, M.

Benoit, G.

Bjarklev, A.

Botten, L. C.

Broeng, J.

de Sterke, C. M.

Deyerl, H.-J.

Engeness, T. D.

Fink, Y.

K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, "Hollow multilayer photonic bandgap fibers for NIR applications," Opt. Express 12, 1510-1517 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1510.
[CrossRef] [PubMed]

M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, "Dielectric profile variations in high-index-contrastwaveguides, coupled mode theory, and perturbation expansions," Phys. Rev. E 67, 046,613 (2003).
[CrossRef]

Guo, S.

Hansen, T. P.

Hart, S. D.

Huang, Y.

Ibanescu, M.

Jacobs, S. A.

M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, "Dielectric profile variations in high-index-contrastwaveguides, coupled mode theory, and perturbation expansions," Phys. Rev. E 67, 046,613 (2003).
[CrossRef]

Jakobsen, C.

Jensen, J. B.

Joannopoulos, J. D.

Johnson, S. G.

M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, "Dielectric profile variations in high-index-contrastwaveguides, coupled mode theory, and perturbation expansions," Phys. Rev. E 67, 046,613 (2003).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, and J. D. Joannopoulos, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066,611 (2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. M. Solja ci`c, 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-748.
[CrossRef] [PubMed]

Katagiri, T.

Kuhlmey, B. T.

Kuriki, K.

Kuriki, Y.

Lee, R. K.

Marom, E.

Matsuura, Y.

Maystre, D.

McIsaac, P.

P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides — I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

McPhedran, R. C.

Miyagi, M.

Mortensen, N. A.

Renversez, G.

Rogowski, R. S.

Shapira, O.

Simonsen, H.

Skorobogatiy, M.

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, and J. D. Joannopoulos, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066,611 (2002).
[CrossRef]

Sorensen, T.

Tai, H.

Terrel, M.

Vienne, G.

Viens, J. F.

Weisberg, O.

White, T. P.

Wu, F.

Xu, Y.

Yariv, A.

Yeh, P.

IEEE Trans. Microwave Theory Tech.

P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides — I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. E

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, and J. D. Joannopoulos, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066,611 (2002).
[CrossRef]

M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, "Dielectric profile variations in high-index-contrastwaveguides, coupled mode theory, and perturbation expansions," Phys. Rev. E 67, 046,613 (2003).
[CrossRef]

Other

M. Yan, J. Canning, G. Vienne, and P. Shum, "Investigation of practical air-silica Bragg fiber," in Australian Conference on Optical Fibre Technology (ACOFT) (Sydney, Australia, 2005).

T. Okoshi, Optical Fibers, 1st ed. (Academic Press, 1982).

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Figures (16)

Fig. 1.
Fig. 1.

(a) A fiber with multilayered cladding. The dielectric interfaces are circular, but not concentric. (b) Two general nonconcentric interfaces of the fiber in (a).

Fig. 2.
Fig. 2.

Convergence of the n eff value with respect to m. Inset, the fiber under test. The core has index n 1 = 1.45. The first cladding layer has index n 2 = 1.42. The outer cladding shares the same index as the core. The inner two interfaces have radii at 5μm and 10μm respectively, and their centers are displaced by ΔO = 3μm in y (vertical) direction.

Fig. 3.
Fig. 3.

Comparison of n eff values of the HE11y-like mode computed using TMM, FDM and FEM. (a) Real part; (b) Imaginary part.

Fig. 4.
Fig. 4.

Dispersion (a) and loss (b) curves for ten modes found in the unperturbed Bragg fiber whose structure is described in the text. In (a), the light-gray region is the TE cladding bandgap; the dark-gray region is the TM cladding bandgap. Notice the TM bandgap is “imersed” in the TE bandgap. Omni-reflection wavelength range (1.560~1.841μm) for the cladding stack is also indicated.

Fig. 5.
Fig. 5.

Deformed Bragg fibers (in part). (a) ΔO = 0.02μm; (b) ΔO = 0.04μm. Axis unit: μm. The indices for near-black, gray, and light-gray regions are 1.0, 1.5, and 3.0, respectively.

Fig. 6.
Fig. 6.

Dispersion (a) and loss (b) curves for the TE01 mode as ΔO changes. In (a), the TM and TE cladding bandgaps for the unperturbed fiber cladding are shaded in light- and dark-gray, respectively, m = 8 is used for all calculations, (c), (d), and (e) are loss spectra resulted from fine scans for the TE01 mode in the perturbed Bragg fiber with ΔO = 0.04μm.

Fig. 7.
Fig. 7.

The TE01 modes at 1.516064μm (a) and 1.571936μm (b) wavelengths. Et field is in quiver plot, and E z field is in contour plot (red: positive; blue: negative).

Fig. 8.
Fig. 8.

Dispersions and losses for the TE01 and two MP12-like modes in the perturbed Bragg fiber around λ= 1.578μm.

Fig. 9.
Fig. 9.

The TE01 modes at wavelengths of 1.5775μm (a), 1.57769μm (b), 1.577726μm (c), 1.577738μm (d), 1.57775μm (e), 1.57777μm (f), 1.5778μm (g), and 1.5779μm (h), as indicated by the black markers in Fig. 8(b). The E z field is also shown in (b).

Fig. 10.
Fig. 10.

One of the MP12 modes at wavelengths of 1.5775μm (a), 1.5777μm (b), 1.578μm (c), and 1.5788μm (d), as indicated by the blue markers in Fig. 8(b).

Fig. 11.
Fig. 11.

Radiation loss of the TE01 mode at 1.55μm and 1.75μm wavelengths as ΔO increases.

Fig. 12.
Fig. 12.

Dispersion (a) and loss (b) curves for the TE01 mode as ΔO changes. In (a), TM cladding bandgap is shaded in light-gray; TE cladding bandgap is shaded in dark-gray. m = 3 is used for calculation when ΔO = 0.05μm, while m = 5 is used for calculation when ΔO = 0.10 μm.

Fig. 13.
Fig. 13.

Distorted air-silica Bragg fibers. (a) ΔO = 0.05μm; (b) ΔO = 0.10μm. White regions are silica.

Fig. 14.
Fig. 14.

Schematic diagram of dispersion crossings among the second-order modes.

Fig. 15.
Fig. 15.

Loss curve resulted from a fine mode trace. The mode transforms from TE01 to MP21, and then TM01. Inset shows the detailed mode transitions between two MP21 modes.

Fig. 16.
Fig. 16.

Modes as specified by the markers in Fig. 15. The wavelengths are at 1.46μm (a), 1.524μm (b), 1.53μm (c), 1.54μm (d), 1.565μm (e), 1.6μm (f), 1.63μm (g), and 1.675μm (h).

Equations (11)

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E z = m [ A m Ei , i J m ( k t , i r i ) + B m Ei , i H m ( k t , i r i ) ] exp ( im θ i ) ,
K z = m [ A m Ki , i J m ( k t , i r i ) + B m Ki , i H m ( k t , i r i ) ] exp ( im θ i ) .
E z , 2 = m ̂ [ A m ̂ E 2 , 1 J m ̂ ( k t , 2 r 1 ) + B m ̂ E 2 , 1 H m ̂ ( k t , 2 r 1 ) ] exp ( i m ̂ θ 1 ) ,
A m ̂ E 2 , 1 J m ̂ ( k t , 2 r 1 ) exp ( i m ̂ θ 1 ) = m J 12 ( m ̂ , m ) A m E 2,2 J m ( k t , 2 r 2 ) exp ( i m θ 2 ) ,
B m ̂ E 2 , 1 H m ̂ ( k t , 2 r 1 ) exp ( i m ̂ θ 1 ) = m J 12 ( m ̂ , m ) B m E 2,2 H m ( k t , 2 r 2 ) exp ( i m θ 2 ) ,
J 12 ( m ̂ , m ) = J m ̂ m ( k t , 2 r 12 exp ) [ i ( m ̂ m ) arg ( r 12 ) ] .
[ A n E 2,1 A n E 2,1 ] = J 12 [ A n E 2,2 A n E 2,2 ] , [ B n E 2,1 B n E 2,1 ] = J 12 [ B n E 2,2 B n E 2,2 ] .
[ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] [ J 12 0 0 0 0 J 12 0 0 0 0 J 12 0 0 0 0 J 12 ] [ A E 2 , 2 B E 2,2 A K 2,2 B K 2,2 ] = [ A E 1 , 1 B E 1,1 A K 1,1 B K 1,1 ] ,
M 11 = [ m 11 n 0 0 0 0 0 0 m 11 n ] , A E 2,2 = [ A n E 2,2 A n E 2,2 ] .
M 0 [ B EN , N B KN , N ] = 0 ,
k 0 d i n i 2 1 = π 2 +qπ, (q=0,1,...),

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