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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 1 — Jan. 12, 2004
  • pp: 198–207
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Comparative analysis of Bragg fibers

Shangping Guo, Sacharia Albin, and Robert S. Rogowski  »View Author Affiliations


Optics Express, Vol. 12, Issue 1, pp. 198-207 (2004)
http://dx.doi.org/10.1364/OPEX.12.000198


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Abstract

In this paper, we compare three analysis methods for Bragg fibers, viz. the transfer matrix method, the asymptotic method and the Galerkin method. We also show that with minor modifications, the transfer matrix method is able to calculate exactly the leakage loss of Bragg fibers due to a finite number of H/L layers. This approach is more straightforward than the commonly used Chew’s method. It is shown that the asymptotic approximation condition should be satisfied in order to get accurate results. The TE and TM modes, and the band gap structures are analyzed using Galerkin method.

© 2004 Optical Society of America

1. Introduction

The confined modes in a circular Bragg fiber with air core and periodic coaxial claddings were first analyzed by Yeh et al. [1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]

]. This kind of fiber is of particular interest in guiding light through a hollow waveguide in UV and far-infrared regions [2

2. N. Croitoru, J. Dror, and I. Gannot, “Characterization of hollow fibers for the transmission of infrared radiation,” Appl. Opt. 29, 1805 (1990). [CrossRef] [PubMed]

]. Recently it has attracted much attention in many applications because of its extraordinary properties and significant experimental and theoretical studies. Unlike the conventional index-guiding fiber, light in Bragg fiber is confined in a low index core due to a photonic band gap (PBG) by Bragg reflection. The fundamental mode is a non-degenerate TE01 mode without azimuthal dependence. The TE01 mode has the lowest loss since the Fresnel reflection for TE components is larger than TM components, and the mode confinement is always the best in the case of TE01. In addition, it effectively eliminates the polarization mode dispersion [3

3. T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

, 4

4. C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993). [CrossRef]

]. By employing a hollow core, the absorption loss due to fiber materials is reduced significantly. However, the leakage loss introduced by imperfect PBG confinement or by a finite number of layers is not negligible. The leakage loss can be controlled by proper design of the structure, and it is largely different for different modes, which can effectively differentiate various modes [3

3. T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

, 4

4. C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993). [CrossRef]

]. Hence asymptotic single mode propagation can be achieved in a Bragg fiber even with a very large air core [5

5. S. G. Johnsonet al., “low-loss asymptotically single-mode propagation in large core omniguide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

].

The invention of perfect dielectric mirror [6

6. Y. Finket al., “A dielectric omnidirectional reflector,” Science 282, 1679 (1998). [CrossRef] [PubMed]

9

9. M. Ibanescuet al., “An all-dielectric coaxial waveguide,” Science 289, 415 (2000). [CrossRef] [PubMed]

] enables great improvements in the properties of Bragg fibers, especially in reducing the leakage loss by several orders of magnitude. By applying high index/low index (H/L) materials with a high index contrast (both refractive indices are much higher than the refractive index of air), one avoids the Brewster angle where no reflection occurs for TM components, and a complete band gap for all polarizations is formed. In the band gap, light with any polarization at any incident angle will be reflected perfectly. Such Bragg fibers, also called Omniguide fibers, have been fabricated successfully [6

6. Y. Finket al., “A dielectric omnidirectional reflector,” Science 282, 1679 (1998). [CrossRef] [PubMed]

] and used for CO2 laser guiding at 10.6µm [10

10. B. Temelkuranet al., “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002). [CrossRef] [PubMed]

], with a loss <1.0dB/m, which is much lower than the intrinsic material losses. Theoretical study shows that in hollow Omniguide fiber, the leakage loss can be reduced further than the lowest loss (0.2dB/km) in conventional glass fibers [5

5. S. G. Johnsonet al., “low-loss asymptotically single-mode propagation in large core omniguide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

], leading to potential applications in long distance communications.

In this paper, we briefly review the calculation methods for Bragg fibers including transfer matrix method, asymptotic method and Galerkin method. Even though Chew’s method [17

17. W. C. Chew, Waves and fields in inhomogeneous media (New York: Van Nostrand Reinhold, 1990).

] has been prevalent in treating the leakage loss in Bragg fibers [4

4. C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993). [CrossRef]

, 5

5. S. G. Johnsonet al., “low-loss asymptotically single-mode propagation in large core omniguide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

, 18

18. C. Themistoset al., “Loss/gain characterization of optical waveguides,” J. lightwave. Technol. 13, 1760(1995). [CrossRef]

22

22. N. A. Issaet al., “Identifying hollow waveguide guidance in air-cored microstructured optical fibres,” Opt. Express 11, 996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996. [CrossRef] [PubMed]

], we will show that the transfer matrix method can be modified for this purpose in a more straightforward way. Our work on Galerkin method is extended to include the calculations of the modes and the band gap structure.

2. Transfer matrix method

In each uniform cylindrical layer, the four field components (Ez, Hϕ, Hz, Eϕ) can be expressed as a linear combination of any two types of Bessel functions [1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]

]. In matrix form, they can be expressed as:

[EzHϕHzEϕ]=[Jm(kir)Ym(kir)00iωεkiJm(kir)iωεkiYm(kir)mβki2rJm(kir)mβki2rYm(kir)00Jm(kir)Ym(kir)mβki2rJm(kir)mβki2rYm(kir)iωμkiJm(kir)iωμkiYm(kir)][ABCD]
(1)

2.1 Bragg fibers with infinite number of H/L layers

If we assume the Bragg fiber has an infinite number of layers, the propagation constant β and the coefficients A, B, C and D are all real numbers, leading to perfectly confined modes. In each layer, the field is a standing wave and the power flux of the incoming and outgoing waves are equal [1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]

].

The four field components are continuous across the interfaces. Using this boundary condition, a transfer matrix [T] for a β-ω pair (or λ-neff) is obtained to relate the coefficients in the innermost layer (i=1) and the outermost layer (i=N) [1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]

, 11

11. T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10. [CrossRef] [PubMed]

]:

[A1B1C1D1]=[T11T12T13T14T21T22T23T24T31T32T33T34T41T42T43T44][ANBNCNDN]
(2)

Giving arbitrary values to the coefficients for the 1st layer or the last layers, the coefficients for the last layer or first layer can be easily evaluated. It is also known that B1=D1=0 since Y m(r) goes to infinite at the origin.

Since the effective indices (neff) of Bragg modes are in the range of [0,1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]

], for a given wavelength, all possible modes with neff in this range are evaluated. If the mode is inside the band gap, the mode increases rapidly due to the increasing numerical error.

As an example, we use the same parameters in [11

11. T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10. [CrossRef] [PubMed]

]; the fiber has an air core with a radius of 5.0µm, refractive indices of the alternate layers are nH=2.0 and nL=1.0, along with widths dH=1.0µm and dL=1.0µm, respectively. Fifteen periods (N=30) are used in the calculation. To confirm if a state is inside the gap or not, we use the value of A12+B12AN2+BN2 to ascertain whether the field increases fast. A rough value to compare with is (nH/nL)N/2 for TE (~105) and a smaller number (~102) for TM, based on their different Fresnel reflection. The band gap and the Bragg modes for TE and TM polarizations are shown in Fig. 1. The curves with ‘x’ in Fig.1 are allowed modes in the cylindrical fiber that satisfy the condition of B1=0 and D1=0. The white area is the band gap. A TE01 and TM01 modes are supported in the fundamental gap, and one TM and three TE modes are supported in the second gap.

Fig. 1. Bragg modes and band gap structures calculated by transfer matrix method. Left: TE, Right: TM.

The calculation of mode field of the Bragg modes in the gap is sensitive to the propagation constants. In order to get a decaying mode field, the propagation constant should have a high accuracy, which can be obtained by finding the root to satisfy B1=0 and D1=0. When k0=1.2, the effective indices are 0.7859081 for TE01 and 0.5270 for TM01. The calculated fields for k0=1.2 are shown in Fig. 2, which are normalized to its maximum. Both the effective indices and the mode fields agree well with results in [11

11. T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10. [CrossRef] [PubMed]

]1.

Fig. 2. TE01 and TM01 at k=1.2 in the Bragg fiber, calculated by transfer matrix method. Left: TE, Right: TM.

2.2 Bragg fibers with finite number of H/L layers

Bragg fibers, in practice, have a finite number of H/L layers, and the outermost layer is a large uniform cladding. In this case, the guiding modes are leaky since the confinement is not perfect, and leakage loss induced by the imperfect structure is a very important parameter for practical applications.

A complex propagation constant is introduced to take into account of the leakage loss. In most published works, Chew’s method [17

17. W. C. Chew, Waves and fields in inhomogeneous media (New York: Van Nostrand Reinhold, 1990).

] was used [4

4. C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993). [CrossRef]

, 5

5. S. G. Johnsonet al., “low-loss asymptotically single-mode propagation in large core omniguide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

, 19

19. A. Argyros, “Guided modes and loss in Bragg fibers,” Opt. Express 10, 1411 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef] [PubMed]

, 20

20. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

], where only Ez and Hz were used to track the field in each layer. A 2×2 reflection matrix and a 2×2 transmission matrix were used to relate adjacent layers. This approach is not so straightforward. Here we show that with minor modification of the transfer matrix method, the complex propagation constants can be calculated accurately.

All layers except the outermost cladding are treated the same way as in Eq. (1). The coefficients and propagation constants are not real numbers any more, indicating a nonzero net power flux across the interfaces. In the outermost cladding, there will be no incoming waves since there is no reflection from outside, and we rewrite the four field components equivalently in Hankel functions:

[EzHϕHzEϕ]=[HmI(kir)HmII(kir)00iωεkiHmI(kir)iωεkiHmII(kir)mβki2rHmI(kir)mβki2rHmII(kir)00HmI(kir)HmII(kir)mβki2rHmI(kir)mβki2rHmI(kir)iωμkiHmi(kir)iωμkiHmII(kir)][ANBNCNDN]
(3)

where HI =J+iY and HII =J-iY. HI and HII represent the outgoing and incoming wave, respectively. A similar transfer matrix [T’] is obtained to relate the coefficients in the first and last layers:

[A1B1C1D1]=[T11T12T13T14T21T22T23T24T31T32T33T34T41T42T43T44][ANBNCNDN]
(4)
[T]=[T]×[1100ii00001100ii]
(5)

where [T] is the transfer matrix obtained in the same way as in the case of infinite fiber.

Using B 1=D 1=0 and B N=D N=0, we obtain:

[T21T23T41T43][ANCN]=0
(6)

The nontrivial solution condition is:

det[T21T23T41T43]=0.
(7)

The complex propagation constant is the root of above equation. Since the imaginary part is much less than the real part, we expect that the real part of the propagation constant to be close to that of the infinite fiber. Using it as the initial value in a multi-variable root finder, for example, the ‘fsolve’ function in MATLAB’s optimization toolbox, the complex propagation constant can be calculated easily.

Using the example of Fiber A in [20

20. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

], which has an air core (n=1.0) of radius 1.3278µm, followed by 16 pairs of alternating H/L layers (nH=1.49, dH=0.2133µm and nL=1.17, dL=0.346 µm), and an infinite medium of the higher index (n=1.49) as the outer cladding, we have obtained identical results as by Chew’s method. At λ=1.0µm, the values of neff are 0.8910672175+1.422605×10-8i and 0.7920859031+1.819323×10-3i for TE01 and TE02 modes, respectively2. The loss in dB/m can be calculated from the imaginary part of the effective index using the relation [22

22. N. A. Issaet al., “Identifying hollow waveguide guidance in air-cored microstructured optical fibres,” Opt. Express 11, 996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996. [CrossRef] [PubMed]

]:

Loss=40πλln10Im(neff)
(8)

And the loss is 7.76387×10-1dB/m for TE01 and 9.92897×103dB/m for TE02.

3. Asymptotic method

According to Erdogan et al. in [3

3. T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

], TE/TM and other low order azimuthal hybrid modes are very close to modes in 1D planar stacks. Asymptotic method approximates the Bessel functions for the cladding layers with their asymptotic expressions [3

3. T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

, 14

14. Y. Xuet al., “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428 (2002). [CrossRef]

, 15

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

, 23

23. Y. Xu, R.K. Lee, and A. Yariv, “Asymptotic analysis of dielectric coaxial fibers,” Opt. Lett. 27, 1019 (2002). [CrossRef]

]; therefore, the periodic cylindrical claddings are approximated as planar Bragg stacks. In this approach, only the core and the first several layers are evaluated by analytical form, which is able to reduce computation time significantly. The advantage of this method is that it provides an analytical eigen-equation for the propagation constant in the case of air core Bragg fibers. The calculated results at k0=1.2 (λ=5.2360µm) for the same Bragg fiber as in section 2.1 are: neff=0.79935 (TE01), 0.57850 (TM01).

Fig. 3. Mode field of TE01, TM01 at k=1.2 by asymptotic method. Left: TE, Right: TM.

The calculated mode fields are shown in Fig. 3. Eϕ or Hϕ from Hz or Ez has some discontinuities at the first several interfaces that might lead to incorrect results. In this example, nL of the low-index material is close to the index of air core, and kir in the first low-index layer is not large enough (~5 for TE mode) and asymptotic approximation of Bessel functions introduces an error of ~4%. When the asymptotic approximation condition is satisfied well, for example the Omniguide fiber with nL much higher than the index of air, this method is generally able to provide reliable mode properties.

In Ref. [24

24. Y. Xuet al., “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039. [CrossRef] [PubMed]

], Xu et al. extended the asymptotic method to include the leakage loss. For the same Fiber A as in [20

20. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

], the asymptotic method gives: neff=0.9091923789-6.86126×10-9i and 0.7869166381-3.56600×10-3i for TE01 and TE02 respectively. Though the real parts are close to the accurate values obtained by transfer matrix method, the imaginary parts have much large errors with even a negative sign. The same reason as mentioned above introduced this error.

4. Galerkin method

Assuming Eϕ (r,ϕ)=f(r), for TE modes, and Hϕ (r,ϕ)=g(r), for TM modes, the wave equations in cylindrical coordinate are [32

32. A. W. Snyder and J. D. Love, Optical Waveguide theory (New York: Chapman Hall, 1983).

]:

d2fdr2+1rdfdr+(k02n2β21r2)f=0
(9)
d2gdr2+1rdgdr+(k02n2β21r2)gdlnn2dr(dgdr+1rg)=0
(10)

Since the Bragg modes decay fast in the first several layers, we terminate the periodic cladding after a finite number of layers with an infinite uniform cladding. The refractive index ncl of the imaginary cladding is close to 0, so that all the modes with effective indices larger than ncl will be confined by total internal reflection. Thus, we expect that it can obtain the gap structure in addition to the Bragg modes. Figure 4 shows the same Bragg fiber as in section 2.1 terminated by an imaginary cladding after 10 periods. The core radius is a and the core includes all microstructures. The index profile of the fiber is n(r). The refractive indices of the core and cladding are nco and ncl. nco could be an arbitrary value different from ncl.

Fig. 4. The index profile of a Bragg fiber in Galerkin method.

We define several normalized parameters:

x=σr2/a2,h(r)=n2(r)ncl2nco2ncl2,
(11)
V2=k02a2(nco2ncl2),b=(β/k0)2ncl2nco2ncl2
(12)

where σ is an arbitrary positive number, and h(r) is the normalized profile. V number and normalized propagation constant b are defined as in the case of conventional optical fiber. V2 could be negative if nco is chosen to be less than ncl. h(r) is zero in the cladding region. The choice of σ may affect the computation and convergence. A larger σ leads to a longer computation time but a better convergence, therefore there is a tradeoff. σ is chosen to be a in our cases and a satisfactory speed and convergence are achieved.

Expanding the mode fields using orthonormalized m th order (m=1 for TE/TM modes) associated Laguerre-Gauss functions [33

33. G. Arfken, Mathematical methods for physicists (Orlando, FL: Academic Press, 1985).

],

f(x)=i=0N1aiφi(x),g(x)=i=0N1biφi(x)
(13)
φi(x)=i!(i+m)!ex/2xm/2Li(m)(x)
(14)
Li(m)(x)=k=0i(i+m)!(ik)!(k+m)!k!(x)k
(15)

where φ(x) is the Laguerre-Gauss function, Li(m) (x) is the associated Laguerre polynomial, and N is the number of Laguerre-Gauss functions used.

The Galerkin method [16

16. S. Guoet al., “Analysis of circular fibers with arbitrary index profiles by Galerkin method,” To appear in Opt. Lett. 29, 32 (Jan. 2004).

] transforms Eq (9) or (10) into a system of eigenvalue equations:

[M][A]=b[A]
(16)

where [A] is the coefficient eigenvector and [M] is a square matrix with a dimension of NxN. For a given wavelength, the normalized propagation constants of all allowed modes are calculated. The profile dependent matrices are calculated only once for different wavelengths.

One important advantage of this method is its versatility in treating circular fibers with arbitrary index profiles, and it is not limited in treating step index profile. The calculated band gap structure and Bragg modes are shown in Fig. 5 using 300 functions. Both the gap and Bragg modes have a good agreement with the results from transfer matrix method (See Fig 1). When the mode is close to the cutoff, which is shown in the figure as the region of low effective index, the mode is close to the radiation mode of the introduced waveguide and Galerkin method gives complex propagation constants.

Fig. 5. Band gap and Bragg modes obtained by Galerkin method. Left: TE, Right: TM.

The mode fields for TE and TM modes at k0=1.2 are also calculated as shown in Fig. 6. The TE mode matches very well with the result obtained using transfer matrix method. Hϕ of TM mode also has a good match in the core and the first cladding, but it decays faster than it should. This is because Hϕ is not smooth and the high frequency components are not fully included when using a finite number of functions.

Fig. 6. Mode fields of Bragg mode by Galerkin method. Left: TE, Right: TM.

We illustrate the calculated effective indices of the TE/TM modes for the fiber in section 2.1 by these three methods in Table 1. The errors are evaluated with respect to the results by transfer matrix method.

Table 1. Comparison of calculated effective indices by three methods at k 0 =1.2

table-icon
View This Table

5. Conclusions

In conclusion, we have reviewed and compared the calculation methods for Bragg fibers. Transfer matrix method is most accurate, but it has no explicit form to obtain the propagation constants. In order to obtain the field distribution of Bragg modes, highly accurate propagation constants may be needed. Asymptotic method provides an explicit form for the propagation constants, and is generally stable if the asymptotic condition is satisfied. The transfer matrix method can be modified to calculate the leakage loss due to a finite number of H/L layers, which is more straightforward than the commonly used Chew’s method. Galerkin method is generally stable and gives good results when the mode is away from the cutoff region. The main advantage of this method is that it is able to analyze circular fibers with arbitrary index profiles without modifications.

Acknowledgments

The research at Old Dominion University is supported by NASA Langley Research Center through NASA-University Photonics Education and Research Consortium (NUPERC). We acknowledge the fruitful discussions with Feng Wu and Khalid Ikram of Photonics Laboratory. We also thank Yan Ming of Nanyang Technology University for his help on the complex root finder.

Footnotes

11 Note: In both Fig. 9 and 10 of [11

11. T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10. [CrossRef] [PubMed]

], there are several minor mistakes: the solid lines are actually for r1/2Hz, and dashed lines are for r1/2Ez, and the β values given in [11

11. T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10. [CrossRef] [PubMed]

] are actually the effective indices neff.
22 Two extra digits are given here to differentiate from the results in Ref. [20

20. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

]. The loss in dB/m for TE02 in Table 1a of Ref. [20

20. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

] has a typo.

References and links

1.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]

2.

N. Croitoru, J. Dror, and I. Gannot, “Characterization of hollow fibers for the transmission of infrared radiation,” Appl. Opt. 29, 1805 (1990). [CrossRef] [PubMed]

3.

T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

4.

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993). [CrossRef]

5.

S. G. Johnsonet al., “low-loss asymptotically single-mode propagation in large core omniguide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

6.

Y. Finket al., “A dielectric omnidirectional reflector,” Science 282, 1679 (1998). [CrossRef] [PubMed]

7.

J. N. Winnet al., “Omnidirectional reflection from a one-dimensional photonic crystals,” Opt. Lett. 23, 1573 (1998). [CrossRef]

8.

Y. Finket al., “Guiding optical light in air using an all dielectric structure,” J. Lightwave Technol. 17, 2039 (1999). [CrossRef]

9.

M. Ibanescuet al., “An all-dielectric coaxial waveguide,” Science 289, 415 (2000). [CrossRef] [PubMed]

10.

B. Temelkuranet al., “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002). [CrossRef] [PubMed]

11.

T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10. [CrossRef] [PubMed]

12.

Y. Xu, R.K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756 (2000). [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 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733. [CrossRef] [PubMed]

14.

Y. Xuet al., “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428 (2002). [CrossRef]

15.

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

16.

S. Guoet al., “Analysis of circular fibers with arbitrary index profiles by Galerkin method,” To appear in Opt. Lett. 29, 32 (Jan. 2004).

17.

W. C. Chew, Waves and fields in inhomogeneous media (New York: Van Nostrand Reinhold, 1990).

18.

C. Themistoset al., “Loss/gain characterization of optical waveguides,” J. lightwave. Technol. 13, 1760(1995). [CrossRef]

19.

A. Argyros, “Guided modes and loss in Bragg fibers,” Opt. Express 10, 1411 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef] [PubMed]

20.

I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

21.

N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341. [CrossRef] [PubMed]

22.

N. A. Issaet al., “Identifying hollow waveguide guidance in air-cored microstructured optical fibres,” Opt. Express 11, 996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996. [CrossRef] [PubMed]

23.

Y. Xu, R.K. Lee, and A. Yariv, “Asymptotic analysis of dielectric coaxial fibers,” Opt. Lett. 27, 1019 (2002). [CrossRef]

24.

Y. Xuet al., “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039. [CrossRef] [PubMed]

25.

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 71 (1981). [CrossRef]

26.

P. K. Mishraet al., “Matrix method for determining propagation characteristics of optical waveguides,” Optica Acta 31, 1041 (1984). [CrossRef]

27.

H. Etzkom and T. Heun, “Highly accurate numerical method for determination of propagation dispersion-flattened fibers,” Opt. Quantum Electron. 18, 1 (1986). [CrossRef]

28.

A. Sharma and S. Banarjee, “Chromatic dispersion in single mode fibers with arbitrary index profiles: a simple method for exact numerical evaluation,” J. Lightwave Technol. 7, 1919 (1989). [CrossRef]

29.

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 27, 459 (1992). [CrossRef]

30.

A. Weisshaaret al., “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795 (1995). [CrossRef]

31.

W. Zhiet al., “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express 11, 3542–3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542. [CrossRef] [PubMed]

32.

A. W. Snyder and J. D. Love, Optical Waveguide theory (New York: Chapman Hall, 1983).

33.

G. Arfken, Mathematical methods for physicists (Orlando, FL: Academic Press, 1985).

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2310) Fiber optics and optical communications : Fiber optics
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Research Papers

History
Original Manuscript: November 25, 2003
Revised Manuscript: December 31, 2003
Published: January 12, 2004

Citation
Shangping Guo, Sacharia Albin, and Robert Rogowski, "Comparative analysis of Bragg fibers," Opt. Express 12, 198-207 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-198


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References

  1. P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]
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  6. Y. Fink et al., "A dielectric omnidirectional reflector," Science 282, 1679 (1998). [CrossRef] [PubMed]
  7. J. N. Winn et al., "Omnidirectional reflection from a one-dimensional photonic crystal," Opt. Lett. 23, 1573 (1998). [CrossRef]
  8. Y. Fink et al., "Guiding optical light in air using an all dielectric structure," J. Lightwave Technol. 17, 2039 (1999). [CrossRef]
  9. M. Ibanescu et al., "An all-dielectric coaxial waveguide," Science 289, 415 (2000). [CrossRef] [PubMed]
  10. B. Temelkuran et al., "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002). [CrossRef] [PubMed]
  11. T. Kawanishi, and M. Izutsu, "Coaxial periodic optical waveguide," Opt. Express 7, 10 (2000), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10</a> [CrossRef] [PubMed]
  12. Y. Xu, R.K. Lee and A. Yariv, "Asymptotic analysis of Bragg fibers," Opt. Lett. 25, 1756 (2000). [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 (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. Y. Xu et al., "Asymptotic matrix theory of Bragg fibers," J. Lightwave Technol. 20, 428 (2002). [CrossRef]
  15. G. Ouyang, Y. Xu, and A. Yariv, "Theoretical study on dispersion compensation in air-core Bragg fibers," Opt. Express 10, 899 (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]
  16. S. Guo et al., "Analysis of circular fibers with arbitrary index profiles by Galerkin method," Opt. Lett. 29, 32 (2004).
  17. W. C. Chew, Waves and fields in inhomogeneous media (New York: Van Nostrand Reinhold, 1990).
  18. C. Themistos et al., "Loss/gain characterization of optical waveguides," J. Lightwave. Technol. 13, 1760 (1995). [CrossRef]
  19. A. Argyros, "Guided modes and loss in Bragg fibers," Opt. Express 10, 1411 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411</a> [CrossRef] [PubMed]
  20. I. M. Bassett and A. Argyros, "Elimination of polarization degeneracy in round waveguides," Opt. Express 10, 1342 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342</a> [CrossRef] [PubMed]
  21. N. A. Mortensen, "Effective area of photonic crystal fibers," Opt. Express 10, 341 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a> [CrossRef] [PubMed]
  22. N. A. Issa et al., "Identifying hollow waveguide guidance in air-cored microstructured optical fibres," Opt. Express 11, 996 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996</a> [CrossRef] [PubMed]
  23. Y. Xu, R.K. Lee and A. Yariv, "Asymptotic analysis of dielectric coaxial fibers," Opt. Lett. 27, 1019 (2002). [CrossRef]
  24. Y. Xu et al., "Asymptotic analysis of silicon based Bragg fibers," Opt. Express 11, 1039 (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]
  25. J. P. Meunier, J. Pigeon., and J. N. Massot, "A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers," Opt. Quantum Electron. 13, 71 (1981). [CrossRef]
  26. P. K. Mishra et al., "Matrix method for determining propagation characteristics of optical waveguides," Optica Acta 31, 1041 (1984). [CrossRef]
  27. H. Etzkom and T. Heun, "Highly accurate numerical method for determination of propagation dispersion-flattened fibers," Opt. Quantum Electron. 18, 1 (1986). [CrossRef]
  28. A. Sharma and S. Banarjee, "Chromatic dispersion in single mode fibers with arbitrary index profiles: a simple method for exact numerical evaluation," J. Lightwave Technol. 7, 1919 (1989). [CrossRef]
  29. D. Marcuse, �??Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,�?? IEEE J. Quantum Electron. 27, 459 (1992). [CrossRef]
  30. A. Weisshaar et al., "Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions," J. Lightwave Technol. 13, 1795 (1995). [CrossRef]
  31. W. Zhi et al., "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express 11, 3542-3549 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542</a> [CrossRef] [PubMed]
  32. A. W. Snyder and J. D. Love, Optical Waveguide Theory (New York: Chapman Hall, 1983).
  33. G. Arfken, Mathematical methods for physicists (Orlando, FL: Academic Press, 1985).

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