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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 24 — Dec. 2, 2002
  • pp: 1411–1417
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Guided modes and loss in Bragg fibres

Alexander Argyros  »View Author Affiliations


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

In general, a Bragg fibre can be described by seven parameters: 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]

]) but only over a limited range and not in the context of singlemodedness.

As these fibres have a one-dimensional band gap, information can be obtained through band diagrams, which rely on the parameters {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

The design we have investigated is described by the parameters 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]

]. Using Chew’s method [11

11. W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York, 1990).

] the complex mode effective indices, 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.

Fig. 1. Schematic diagram showing the refractive index profile of the Bragg fibre design investigated, with the various parameters and values thereof indicated. The parameters r co; and the number of layers N are treated as variables in this work.

In Fig. 2(a) and (b) N = 32 and r co is varied from 1.3278 to 2.3278 μm. For each value of r co only the TE01 and TE02 modes are shown (analogous to the TE0x 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.

Fig. 2. (a) Re{neff} vs wavelength for various values of r co as indicated, with n co = 1.0, n 1 = 1.49, n 2 = 1.17, d 1 = 0.2133 μm, d 2 = 0.346 μm and N= 32. (b) Loss = 40πIm{n eff}/(ln(10)λ) vs wavelength for the same. (c) Absolute value of the change in Re{n eff}when N is decreased from 32 to 26. (d) Ratio of the loss at N = 26 to N = 32. Some of the data values used to plot the curves are shown in (c) as circles.

3. Model for the cavity condition

A model for the cavity condition that must be satisfied by all modes is presented (i.e. they must resonate with the core to form a standing wave). At the lowest loss wavelength for each fibre design, the alternating high-low layers reflect very strongly and are approximated by a metallic layer at the edge of the core, which requires the electric fields at the edge of the core to be zero, as observed before [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]

, 9

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]

]. Here 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

11. W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York, 1990).

]

ErϕzJ1(κr)ei(neffkzωt)φ̂,
(1)

where J1 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(nco2 -neff2)1/2 =k(1-neff2)1/2 is the radial wavenumber. For E to be zero at r = r co the product κr co must satisfy κr co =j1,i , for j1,i a zero of J1. Each value of j1,i corresponds to a mode of different order: the i th zero corresponds to the TE0i 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):

neff,TE0i=1j1,i2k2rco2.
(2)

Away from these least loss wavelengths the approximation becomes less applicable as the alternating layers become less reflective. The extreme is for a mode to be supported by Fresnel reflection off (predominantly) the first interface, like in an antiguide (waveguide with a depressed core and high index cladding). The mode will have a high loss and the approximation will be inapplicable. Equation (2) was verified by direct comparison to the results of calculations, as shown in Fig. 3. Excellent agreement is seen at and around the least loss wavelengths, with the agreement decreasing away from these, as expected.

Fig. 3. Plot of Re{n eff} against wavelength [same parameters as Fig. 2(a)] as calculated using Eq. (2) (solid lines) and from calculations on the fibre structure (points). The lowest loss points are indicated in red. The case where the fibre behaves like an antiguide (r co = 1.3278 μm, TE02 mode) and Eq. (2) is inapplicable is indicated in blue.

Figure 2(d) illustrates that the loss decreases as 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.

It is noted that the TE02 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.

Continuing with the model, the expression κr co = j1,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

krco=2πrcoλ<j1,i,
(3)

the cavity condition will be satisfied 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

2πrcoj1,2<λ,
(4)

i.e. the condition on 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 (TE02) 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

Following the method outlined in [2

2. K. Sakoda, Optical Properties of Photonic Crystals, Chapter 2 (Springer, Berlin, 2001).

] a band diagram for the TE polarisation was constructed for the Bragg fibre from Sec. 2 and is shown in Fig. 4(a). The modes from Fig. 2(a) and additional ones (TE01 and TE02 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.

Fig. 4. (a) Band diagram (TE polarisation) for the alternating layers used in the fibre designs, defined by {n 1 = 1.49, n 2 = 1.17, d 1 = 0.2133 μm, d 2 = 0.346 μm}. The axes are in dimensionless units of frequency ω and propagation constant ω = n eff k, a = d 1 + d 2 is the periodicity. White represents band-gap regions and the position of the modes (from Fig. 2 plus additional ones as in the text) is indicated by coloured curves. The “lowest loss” curve is shown in black. The TE02 mode for r co = 1.3278 μm is the left-most mode and is entirely outside the band-gap. (b) A contour plot of loss (blue represents low loss) superimposed on the band diagram to show that the loss of a mode can be inferred from its position relative to the band-gap and the light line. (c) Band diagram for TM polarisation.

As the core radius is increased, the curves representing the modes move to the right towards the light line and, as observed above, the loss of the least lossy modes decreases. In Fig. 4(b) a contour plot of loss is superimposed on the band diagram. The ability to show loss in this way, when more than one mode of more than one value of r co is considered, and the observation from Fig. 2 that two modes (r co = 1.3278 μm, TE01 and r co = 2.3278 μm, TE02) that have similar values of Re{n eff} also have similar values of Im{n eff} indicates that Re{n eff} and Im{n eff} are related, and that this relation is independent of mode order.

5. Finite multilayer stacks

To examine the calculated losses and their response to varying 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.

In the reflections off the planar stack the angle of incidence 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 1/2r cotanθ 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

12. M. Born and E. Wolf, Principles of Optics, Chapter 1.6 (Pergamon Press, Oxford, 1980).

]. Upon comparison with Eq. (1) and the cavity condition κr co =j1,i this gives

Im{neff}=ln(r)2krcotanθi=ln(r)2j1,itanθisecθi.
(5)
Fig. 5. Comparison between the results for Im{n eff} from direct calculations and Eq. (5) for various modes (TE01 and TE02 modes for r co = 1.5278, 1.8278, 2.0278, 2.3278 and 2.5278 μm and TE01 for r co = 1.3278 μm with N = 26 and 32, remaining parameters as in Fig. 1). The comparison is made around the lowest loss wavelength for each mode. As expected from the assumptions made, the agreement increases with decreasing Im{n eff}.

The first expression in Eq. (5) shows that the loss is independent of the mode order as observed in Fig. 2(b) (TE01 for r co = 1.3278 μm and for TE02 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

12. M. Born and E. Wolf, Principles of Optics, Chapter 1.6 (Pergamon Press, Oxford, 1980).

]. In Fig. 5, Eq. (5) is compared to the results of direct calculations (from Fig. 2 and additional results).

6. Effects on singlemodedness

For the design considered, the TE01 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]

] and Fig. 2. (b)], lower by several orders of magnitude. This means that, as formalised 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]

, 10

10. 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)

], after a certain fibre length, all but one mode will be severely attenuated and the fibre will become effectively single-moded. This length is denoted by Lsm. Since the TE01 mode is also leaky, at a longer length Lmax it will also be attenuated significantly and the fibre will no longer usefully guide. For example, the designs considered here have Lsm ~ 1 μm and Lmax ~ 10 m for r co = 1.3278 μm and Lsm ~ 1 m and Lmax ~ 1 km for r co = 2.3278 μm, when N = 32. In some cases, both the TE01 and TE02 modes are in the band-gap at the wavelength for which the TE01 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 Lsm and Lmax increasing. In other cases the TE02 mode is outside the band-gap at the least loss wavelength of the TE01 mode, in which case increasing N affects the least lossy mode much more than the second least lossy mode. This allows one to increase Lmax almost independently of Lsm As an illustration, when N is increased from 26 to 32 the ratio Lmax/Lsm 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

The response of the guided modes of a Bragg fibre to changes in the core radius and the number of layers was examined and explained in terms of simple models for the cavity condition and the guiding mechanism. The results obtained allow for the role of the core radius and number of layers to be better incorporated into the design and optimisation process, when designing such fibres for particular applications. Further optimisation of these single polarisation non-degenerate mode fibre designs, especially relating to minimizing confinement loss, will be investigated in the future. The simple models used to explain the design’s behaviour should be equally well applicable to band-gap fibres of other geometries.

Acknowledgements

Ian M. Bassett, Ross C. McPhedran, Martijn A. van Eijkelenborg and Maryanne C.J. Large are acknowledged for useful discussions. Steven Manos is acknowledged for assisting with the graphics.

References and Links

1.

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

2.

K. Sakoda, Optical Properties of Photonic Crystals, Chapter 2 (Springer, Berlin, 2001).

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]

4.

S.D. Hart et al., “External Reflection from Omnidirectional Dielectric Mirror Fibers,” Science 296, 510, (2002). [CrossRef] [PubMed]

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]

6.

A. Argyros et al, “Ring structures in microstructured optical fibres,” Opt. Express 9, 813–820, (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-813. [CrossRef] [PubMed]

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 9, 733–747, (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733. [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]

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]

10.

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)

11.

W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York, 1990).

12.

M. Born and E. Wolf, Principles of Optics, Chapter 1.6 (Pergamon Press, Oxford, 1980).

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

  1. P.Yeh, A. Yariv, and E. Marom, �??Theory of Bragg fiber,�?? J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
  2. K. Sakoda, Optical Properties of Photonic Crystals, Chapter 2 (Springer, Berlin, 2001).
  3. 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]
  4. S.D. Hart et al., �??External Reflection from Omnidirectional Dielectric Mirror Fibers,�?? Science 296, 510, (2002). [CrossRef] [PubMed]
  5. 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]
  6. 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]
  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 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]
  8. 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]
  9. 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]
  10. 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)
  11. W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York, 1990).
  12. M. Born and E. Wolf, Principles of Optics, Chapter 1.6 (Pergamon Press, Oxford, 1980).

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