## Guided modes and loss in Bragg fibres

Optics Express, Vol. 10, Issue 24, pp. 1411-1417 (2002)

http://dx.doi.org/10.1364/OE.10.001411

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### Abstract

In this paper we investigate Bragg fibres and compare calculations on the exact fibre structures with calculations based on band diagrams and a simplified model involving multilayers. We show how the number of layers and the core size affect the wavelengths guided, the loss and the effective singlemodedness. An approximate relation between the real and imaginary parts of the effective mode indices is derived. The general design considered has a TE mode as the least lossy mode providing effectively single polarisation non-degenerate mode guidance.

© 2002 Optical Society of America

## 1. Introduction

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

3. G. Ouyang, Y. Xu, and A. Yariv, “Theoretical study on dispersion compensation in air-core Bragg fibres,” Opt. Express **10**, 889–908 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-899. [CrossRef]

*n*

_{co},

*n*

_{1},

*n*

_{2},

*r*

_{co},

*d*

_{1},

*d*

_{2}, and

*N*(see Fig. 1), these being the refractive indices of the core and the alternating high-low index layers, the core radius, layer thicknesses and the number of layers respectively. The effects of changing one or more of these parameters have been considered previously (e.g. [5

5. S.G. Johnson et al., “Low-loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express **9**, 748–779, (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

*n*

_{1},

*n*

_{2},

*d*

_{1},

*d*

_{2}} with

*N*= ∞. Comparison of the results obtained using this approach to those obtained from calculations on the actual fibre structure highlight the importance of the size of the core and the number of layers in determining the behaviour of these fibres, information that cannot be inferred from the band diagram alone. The results allow for better understanding of the role of these parameters in Bragg fibres, which is important when designing such fibres for particular applications.

## 2. Results from calculations on the fibre structure

*n*

_{co}= 1.0,

*n*

_{1}= 1.49,

*n*

_{2}= 1.17,

*d*

_{1}= 0.2133 μm,

*d*

_{2}= 0.346 μm (Fig. 1). Justification for the choice of these parameters is given in [8

8. I. Bassett and A. Argyros, “Elimination of polarisation degeneracy in round waveguides,” Opt. Express **10**, 1342–1346, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

*n*

_{eff}, were calculated for the least lossy modes of fibres with such layers surrounding the air core, but with different values of

*r*

_{co}and

*N*. Results of these calculations appear in Fig. 2.

*N*= 32 and

*r*

_{co}is varied from 1.3278 to 2.3278 μm. For each value of

*r*

_{co}only the TE

_{01}and TE

_{02}modes are shown (analogous to the TE

_{0x}modes of step-index fibres) as these have the lowest confinement loss. Modes of other polarisations or higher order have a higher loss and are not discussed in this work. In Fig. 2(c) and (d), the effects of reducing

*N*to 26 are shown. There are clear patterns in the response of the fibres’ behaviour to these changes, which are explained by simple models below.

## 3. Model for the cavity condition

5. S.G. Johnson et al., “Low-loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express **9**, 748–779, (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

9. F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. **36**, 514, (2000). [CrossRef]

*n*

_{co}=1.0 and the modes of interest are TE modes. The electric field

**E**is given in cylindrical polar coordinates (

*r*,

*ϕ*,

*z*) by [11]

_{1}is a Bessel function of the first kind of order 1,

*k*is the free-space wavenumber,

*ω*is the angular frequency and

*κ*=

*k*(

^{1/2}=

*k*(1-

^{1/2}is the radial wavenumber. For

**E**to be zero at

*r*=

*r*

_{co}the product

*κr*

_{co}must satisfy

*κr*

_{co}=

*j*

_{1,i}, for

*j*

_{1,i}a zero of J

_{1}. Each value of

*j*

_{1,i}corresponds to a mode of different order: the

*i*

^{th}zero corresponds to the TE

_{0i}mode. This and the expression for

*κ*give the

*n*

^{eff}of the

*i*

^{th}mode (

*n*

_{eff}is real due to the assumption made):

*n*

_{eff}increases with

*r*

_{co}as in Fig. 2(a) and is independent of

*N*as in Fig. 2(c) (in that ΔRe{

*n*

_{eff}} is small near the low loss wavelengths). The sharp minima in Fig. 2(c) occur at wavelengths slightly larger than the least loss wavelengths and observations indicated that at these minima the phase change for external reflection off a planar stack equivalent to the alternating layers of the fibre was approximately zero, though this was not investigated further. As

*n*

_{eff}increases for a particular wavelength,

*κ*decreases, meaning that the light strikes the interfaces with a larger angle of incidence. However, a fixed value of

*κ*is defined by the Bragg condition. From the expression for

*κ*we can see that as

*n*

_{eff}increases, a larger value of

*k*must be used to give the

*κ*required for Bragg reflection, hence for larger

*r*

_{co}maximum reflection occurs at shorter wavelengths. The reflections become stronger (and hence the loss is reduced) at these shorter wavelengths due to a larger angle of incidence leading to a stronger Fresnel reflection off each individual interface in the fibre. This agrees with Fig. 2(b), showing lower loss minima at lower wavelengths.

*N*increases as expected. The amount it decreases depends on whether the particular mode is in the band-gap. Inside the band-gap increasing

*N*increases the number of coherent reflections, which has a larger effect on the reflected power than increasing the number of incoherent reflections, as is the case outside the band-gap. This explains the peaks in Fig. 2(d) at the least loss wavelength for each design.

_{02}mode for

*r*

_{co}= 1.3278 μm displays different characteristics to the rest of the modes. It has a high loss, no distinct loss minimum [Fig. 2(b)] and shows a large change in Re{

*n*

_{eff}} and constant change in Im{

*n*

_{eff}} when

*N*is changed [Figs. 2(c) and (d)]. The properties predicted for band-gap modes are not observed in this mode’s behaviour, which leads to the conclusion that this mode is not guided by a Bragg effect but by partial reflection like in an antiguide. Equation (2) does not apply to this mode as is clear in Fig. 3.

*κr*

_{co}=

*j*

_{1,i}also defines the maximum number of low-loss modes the fibre can have i.e. modes that lie in the band-gap. Since

*κ*is a component of the wavevector, the largest value it can take is

*k*. Therefore if

*i*- 1 times at most. If there is a band-gap at all wavelengths, the fibre will have

*i*- 1 modes. Since in practice there will not be a band-gap at all wavelengths, a fibre will have a maximum of

*i*- 1 modes. Setting

*i*= 2 gives

*r*

_{co}and

*λ*required for a maximum of one mode, making the fibre effectively single-moded in the sense that it will have only one low loss mode. Examining the case of

*r*

_{co}= 1.3278 μm, we have

*j*

_{2}≈ 7.0156, giving that

*λ*> 1.19 μm is required for singlemodedness. In the discussion above it is concluded that the second (TE

_{02}) mode for this core radius is not in the band-gap even for wavelengths smaller than 1.19 μm. This occurs because there is no band-gap at the limit given by Eq. (4) - in general the absence of a band-gap can effectively lower the limit given by Eq. (4). This can be shown directly on a band diagram.

## 4. Band diagram calculations

_{01}and TE

_{02}for

*r*

_{co}= 1.5278, 2.0278 and 2.5278 μm) are marked on this graph as curves. The intercept of each of these curves on the frequency axis is what can be determined using Eq. (4) (where applicable): above a certain wavelength, i.e. below a certain frequency, a mode does not exist. Above that frequency the mode needs to be in the band-gap to be useful. The least loss point for each mode of each design is also marked and these trace out a curve [black curve in Fig. 4(a)]. If the lowest loss wavelength of a design needs to have a particular value, this “lowest loss” curve gives the

*n*

_{eff}that the mode must have and the core radius can be determined from Eq. (2). Calculations indicated that the position of this curve was largely insensitive to

*N*when the loss is low.

*r*

_{co}= 1.3278 μm, the TE

_{01}mode falls in the band-gap and the TE

_{02}does not. This greatly affects the dependence of singlemodedness on

*N*, as will be discussed further in section 6.

## 5. Finite multilayer stacks

*N*[Fig. 2(b) and (d)] a ray picture is used and the alternating high-low layers of the fibre are assumed to have the same reflection properties as their planar equivalent. The ray picture analogy is valid under the assumption of strong reflection, and for a sufficiently large

*N*, as the layers in the fibre asymptotically approach a planar regime with increasing

*N*.

*θ*

_{i}is taken to be the complement of the angle between the wavevector and the fibre axis, so

*θ*

_{i}= sin

^{-1}(Re{

*n*

_{eff}}/

*n*

_{co}) = sin

^{-1}(Re{

*n*

_{eff}}). The number of reflections per unit length is 1/2

*r*

_{co}tan

*θ*

_{i}, and

*r*

^{2}of the power remains after each reflection;

*r*is the reflection coefficient for the planar stack, calculated using standard transfer matrix techniques [12]. Upon comparison with Eq. (1) and the cavity condition

*κr*

_{co}=

*j*

_{1,i}this gives

_{01}for

*r*

_{co}= 1.3278 μm and for TE

_{02}

*r*

_{co}= 2.3278 μm). The only term that depends on the number of layers is the reflection coefficient

*r*, and hence the behaviour of the loss as

*N*changes can be inferred from the behaviour of multilayer stacks [12]. In Fig. 5, Eq. (5) is compared to the results of direct calculations (from Fig. 2 and additional results).

## 6. Effects on singlemodedness

_{01}mode has a lower loss than the remaining modes [see [8

8. I. Bassett and A. Argyros, “Elimination of polarisation degeneracy in round waveguides,” Opt. Express **10**, 1342–1346, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

8. I. Bassett and A. Argyros, “Elimination of polarisation degeneracy in round waveguides,” Opt. Express **10**, 1342–1346, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef] [PubMed]

_{sm}. Since the TE

_{01}mode is also leaky, at a longer length L

_{max}it will also be attenuated significantly and the fibre will no longer usefully guide. For example, the designs considered here have L

_{sm}~ 1 μm and L

_{max}~ 10 m for

*r*

_{co}= 1.3278 μm and L

_{sm}~ 1 m and L

_{max}~ 1 km for

*r*

_{co}= 2.3278 μm, when

*N*= 32. In some cases, both the TE

_{01}and TE

_{02}modes are in the band-gap at the wavelength for which the TE

_{01}mode has the lowest loss, hence increasing

*N*can decrease the loss of both these modes by large amounts [Fig. 2(d)], which can result in both L

_{sm}and L

_{max}increasing. In other cases the TE

_{02}mode is outside the band-gap at the least loss wavelength of the TE

_{01}mode, in which case increasing

*N*affects the least lossy mode much more than the second least lossy mode. This allows one to increase L

_{max}almost independently of L

_{sm}As an illustration, when

*N*is increased from 26 to 32 the ratio L

_{max}/L

_{sm}increases by twice as much for

*r*

_{co}= 1.3278 μm with only one mode is in the band-gap, compared to

*r*

_{co}= 2.3278 μm where both modes are in the band-gap.

## 7. Conclusion

## Acknowledgements

## References and Links

1. | P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. |

2. | K. Sakoda, |

3. | G. Ouyang, Y. Xu, and A. Yariv, “Theoretical study on dispersion compensation in air-core Bragg fibres,” Opt. Express |

4. | S.D. Hart et al., “External Reflection from Omnidirectional Dielectric Mirror Fibers,” Science |

5. | S.G. Johnson et al., “Low-loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express |

6. | A. Argyros et al, “Ring structures in microstructured optical fibres,” Opt. Express |

7. | G. Ouyang, Y. Xu, and A. Yariv, “Comparative study of air-core and coaxial Bragg fibres: single-mode transmission and dispersion characteristics,” Opt. Express |

8. | I. Bassett and A. Argyros, “Elimination of polarisation degeneracy in round waveguides,” Opt. Express |

9. | F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. |

10. | A. Argyros and I. Bassett, “Counting Modes in Optical Fibres with Leaky Modes,” in |

11. | W.C. Chew, |

12. | M. Born and E. Wolf, |

**OCIS Codes**

(060.2430) Fiber optics and optical communications : Fibers, single-mode

(230.1480) Optical devices : Bragg reflectors

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 5, 2002

Revised Manuscript: November 20, 2002

Published: December 2, 2002

**Citation**

Alexander Argyros, "Guided modes and loss in Bragg fibres," Opt. Express **10**, 1411-1417 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-24-1411

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### References

- P.Yeh, A. Yariv, and E. Marom, �??Theory of Bragg fiber,�?? J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals, Chapter 2 (Springer, Berlin, 2001).
- G. Ouyang, Y. Xu, and A. Yariv, �??Theoretical study on dispersion compensation in air-core Bragg fibres,�?? Opt. Express 10, 889-908 (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]
- S.D. Hart et al., �??External Reflection from Omnidirectional Dielectric Mirror Fibers,�?? Science 296, 510, (2002). [CrossRef] [PubMed]
- S.G. Johnson et al., �??Low-loss asymptotically single-mode propagation in large-core Omniguide fibers,�?? Opt. Express 9, 748-779, (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a>. [CrossRef] [PubMed]
- A.Argyros et al, �??Ring structures in microstructured optical fibres,�?? Opt. Express 9, 813-820, (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-813">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-813</a>. [CrossRef] [PubMed]
- G. Ouyang, Y. Xu, and A. Yariv, �??Comparative study of air-core and coaxial Bragg fibres: 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]
- I. Bassett, and A. Argyros, �??Elimination of polarisation degeneracy in round waveguides,�?? Opt. Express 10, 1342-1346, (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]
- F. Brechet, P. Roy, J. Marcou and D. Pagnoux, �??Singlemode propagation into depressed-core-index photonicbandgap fibre designed for zero-dispersion propagation at short wavelengths,�?? Electron. Lett. 36, 514, (2000). [CrossRef]
- A. Argyros, and I. Bassett, �??Counting Modes in Optical Fibres with Leaky Modes,�?? in Symposium on Optical Fiber Measurements SOFM 2002, (National Institute of Standards and Technology, Colorado, 2002)
- W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York, 1990).
- M. Born and E. Wolf, Principles of Optics, Chapter 1.6 (Pergamon Press, Oxford, 1980).

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