## Approximate band structure calculation for photonic bandgap fibres

Optics Express, Vol. 14, Issue 20, pp. 9483-9490 (2006)

http://dx.doi.org/10.1364/OE.14.009483

Acrobat PDF (1386 KB)

### Abstract

An approximate method for finding the band structure of simple photonic bandgap fibres is presented. Our simple model is an isolated high-index rod in a circular unit cell with two alternative boundary conditions. Band plots calculated this way are found to correspond closely to calculations using an accurate numerical method.

© 2006 Optical Society of America

## 1. Introduction

11. J. Lægsgaard, “Gap formation and guided modes in photonic band gap fibres with high-index rods,” J. Opt. A: Pure Appl. Opt. **6**, 798–804 (2004). [CrossRef]

3. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St.J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express **13**, 309–314 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-309. [CrossRef] [PubMed]

5. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express **14**, 5688–5698 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5688. [CrossRef] [PubMed]

4. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (<20 dB/km) around 1550 nm,” Opt. Express **13**, 8452–8459 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-21-8452. [CrossRef] [PubMed]

7. A. Wang, A. K. George, and J. C. Knight, “Three-level neodymium fiber laser incorporating photonic bandgap fiber,” Opt. Lett. **31**, 1388–1390 (2006). [CrossRef] [PubMed]

8. C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, “A 158 fs 5.3 nJ fiber-laser system at 1 µm using photonic bandgap fibers for dispersion control and pulse compression,” Opt. Express **14**, 6063–6068 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6063. [CrossRef] [PubMed]

*ck*and propagation constant β where photonic states exist (photonic bands) or do not exist (photonic bandgaps) can be mapped on a band plot. Guided modes in a core introduced as a defect can be represented as curves on the same plot. Such plots contain detailed information about the fibre.

9. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. **27**, 1977–1979 (2002). [CrossRef]

11. J. Lægsgaard, “Gap formation and guided modes in photonic band gap fibres with high-index rods,” J. Opt. A: Pure Appl. Opt. **6**, 798–804 (2004). [CrossRef]

## 2. Method of calculation

*a*=

*d*/2 and uniform refractive index

*n*

_{hi}in a background of lower index

*n*

_{lo}, Fig. 2(a). The difference

*n*

_{hi}-

*n*

_{lo}is small in all-solid-silica bandgap fibres [3

3. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St.J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express **13**, 309–314 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-309. [CrossRef] [PubMed]

*f*=(

*π*/2√3)(

*d*/Λ)

^{2}=(

*a*/

*b*)

^{2}of the unit cell, the radius of the circular unit cell is

*b*=(√3/2π)

^{1/2}Λ. The approximation is justified not only by the rough similarity between a circle and a hexagon, but also by the key finding of the ARROW model that the bandgap frequencies are primarily determined by the properties of the individual rods rather than their arrangement in a lattice [9

9. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. **27**, 1977–1979 (2002). [CrossRef]

11. J. Lægsgaard, “Gap formation and guided modes in photonic band gap fibres with high-index rods,” J. Opt. A: Pure Appl. Opt. **6**, 798–804 (2004). [CrossRef]

15. T. A. Birks, J. C. Knight, and P. St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

*d*/Λ is not too big.

*l*and

*m*defining the azimuthal and radial variations respectively. Although these labels cannot properly be used either in a square or hexagonal lattice, they are well defined for highly-confined states above cutoff because these are simply the weakly-coupled LP

_{lm}modes of the rods. The method therefore enables us to track these states as they move towards and below cut-off when frequency is reduced.

*l*and

*m*are presented in the Appendix. The region between the two β(ω) curves satisfying these equations is filled with states derived from the LP

_{lm}rod mode. By mapping such regions on a band plot for all relevant values of

*l*and

*m*, we can therefore identify the bandgaps (the spaces left over) as well as the states making up the bands.

## 3. Comparison with exact numerical calculation

*d*/Λ=0.41,

*n*

_{lo}=1.458 and

*n*

_{hi}=1.48716) is compared to an “exact” numerical calculation using a vector plane-wave method [16

16.
sections I and II of
G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B **71**, 195108 (2005). [CrossRef]

*kn*

_{lo})Λ representing β and

*k*Λ representing ω. As in previous papers [5

5. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express **14**, 5688–5698 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5688. [CrossRef] [PubMed]

6. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express **14**, 6291–6296 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6291. [CrossRef] [PubMed]

17. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St.J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express **12**, 69–74 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-1-69. [CrossRef] [PubMed]

17. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St.J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express **12**, 69–74 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-1-69. [CrossRef] [PubMed]

5. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express **14**, 5688–5698 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5688. [CrossRef] [PubMed]

*l*and

*m*values for the bands calculated by the exact method.

17. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St.J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express **12**, 69–74 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-1-69. [CrossRef] [PubMed]

*w*

^{2}from [17

**12**, 69–74 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-1-69. [CrossRef] [PubMed]

_{04}band are plotted in Fig. 4 for

*k*Λ=115 and 170 (where both band edges follow the steep high-index trend) and

*k*Λ=145 (in between, where they follow the flatter low-index trend). Whereas in the former cases the light is concentrated in the high-index rod, in the latter case it is concentrated in the low-index background, showing that the LP

_{04}rod states have indeed become strongly hybridised with resonant states concentrated in the background regions between rods. This behaviour is typical of the bands we examined.

*b*while keeping the rod size

*a*fixed is shown in Fig. 5. As

*b*(and hence the rod separation) increases there is: a scaling in frequency so that

*ka*is invariant (as expected from the ARROW picture [9

9. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. **27**, 1977–1979 (2002). [CrossRef]

**12**, 69–74 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-1-69. [CrossRef] [PubMed]

*kn*

_{lo}(because the LP

_{lm}modes of adjacent rods are becoming more weakly coupled).

*f*and the V-value

*V*(representing frequency) are defined in Section 2 and the Appendix respectively. As expected [5

**14**, 5688–5698 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5688. [CrossRef] [PubMed]

**6**, 798–804 (2004). [CrossRef]

*f*the width decreases rapidly with increasing

*l*, and more gradually with increasing frequency for a given

*l*. Table 1 gives the widths of the first five

*m*=2 bands for the fibre of Fig. 3 from both Eq. (1) and a plane-wave calculation (scalar, to avoid confusion due to polarisation splitting), showing a good match and also reflecting our previous findings on

*l*and bend loss [5

**14**, 5688–5698 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5688. [CrossRef] [PubMed]

## 4. Extensions and limitations

**14**, 5688–5698 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5688. [CrossRef] [PubMed]

*r*and simple numerical techniques for solving ordinary differential equations can be applied.

*n*(

*r*) of several uniform regions bounded by steps, such as a high-index ring [6

6. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express **14**, 6291–6296 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6291. [CrossRef] [PubMed]

18. T. A. Birks, Y. W. Li, and C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. **83**, 203–209 (1991). [CrossRef]

*r*=

*a*the same index).

*d*/Λ=0.4, for a profile volume matching a ring of thickness 0.1Λ and index

*n*

_{hi}=1.48716 in a background of index

*n*

_{lo}=1.458. It illustrates the limit of the behaviour described in [6

6. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express **14**, 6291–6296 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6291. [CrossRef] [PubMed]

*d*/Λ), with all

*m*>1 rod modes (and hence the bands derived from them) pushed below cutoff. Hence the low-loss frequency ranges of core-guided modes are delimited only by

*m*=1 modes of the rings. For comparison Fig. 6(b) shows the LP

_{lm}mode cutoff V-values for an isolated thick ring versus the ratio of the inner and outer radii

*c*and

*a*, where

*V*

^{2}=

*k*

^{2}(

*a*

^{2}-

*c*

^{2})(

*c*/

*a*→1) the cutoffs of the

*m*=1 modes approach

*V*=2√

*l*while those of the

*m*>1 modes are banished to infinity.

*d*/Λ: the circular unit cell becomes less appropriate as the rod approaches the unit cell boundary and the corners of the hexagonal unit cell become a more significant fraction of the background.

## 5. Conclusions

*d*/Λ is not too large. We expect this method to be of value where speed of calculation is more important than accuracy, and as an aid to intuition.

## Appendix

_{lm}mode of the rod and assuming that β<

*kn*

_{hi}, is

*l*θ) and sin(

*l*θ).

*J*

_{l},

*Y*

_{l},

*I*

_{l}and

*K*

_{l}are Bessel functions,

*A*to

*H*are unknown constants and

*U*,

*W*and

*V*are the familiar parameters of waveguide theory [12] applied to the rod

*Q*being useful where β<

*kn*

_{lo}.

*r*=

*b*and the continuity of Ψ and Ψ′ at

*r*=

*a*as boundary conditions gives analytical characteristic equations satisfied by β for given ω,

*l*and

*m*, which can be written as

*J*

_{l}etc, α=

*b*/

*a*=1/√

*f*represents the relative size of the rods, and the coefficients

*A*to

*H*in Eqs. (2), (6) and (7) are

*W*

^{2}. The particular forms of the equations are chosen so that the

*g*(

*V*,

*W*

^{2}) are continuous and finite as

*W*

^{2}varies through

*W*

^{2}=0, which simplifies the application of root-finding algorithms. The functions are well-behaved where

*U*

^{2}>0, which is sufficient for our purposes since there are no propagating states with

*U*

^{2}≤0(β≥

*kn*

_{hi}).

*W*

^{2}=0 and α large.

## Acknowledgements

## References and links

1. | R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in |

2. | F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St.J. Russell, “All-solid photonic band gap fiber,” Opt. Lett. |

3. | A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St.J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express |

4. | G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (<20 dB/km) around 1550 nm,” Opt. Express |

5. | T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express |

6. | J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express |

7. | A. Wang, A. K. George, and J. C. Knight, “Three-level neodymium fiber laser incorporating photonic bandgap fiber,” Opt. Lett. |

8. | C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, “A 158 fs 5.3 nJ fiber-laser system at 1 µm using photonic bandgap fibers for dispersion control and pulse compression,” Opt. Express |

9. | T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. |

10. | N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express |

11. | J. Lægsgaard, “Gap formation and guided modes in photonic band gap fibres with high-index rods,” J. Opt. A: Pure Appl. Opt. |

12. | A. W. Snyder and J. D. Love, |

13. | S. L. Altmann, |

14. | P. W. Atkins, |

15. | T. A. Birks, J. C. Knight, and P. St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

16. |
sections I and II of
G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B |

17. | T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St.J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express |

18. | T. A. Birks, Y. W. Li, and C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: July 11, 2006

Revised Manuscript: September 12, 2006

Manuscript Accepted: September 14, 2006

Published: October 2, 2006

**Citation**

Timothy A. Birks, Greg J. Pearce, and David M. Bird, "Approximate band structure calculation for photonic bandgap fibres," Opt. Express **14**, 9483-9490 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9483

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### References

- R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, D. J. Trevor, "Tunable photonic band gap fiber," in Proceedings of the Optical Fiber Communications Conference (2002), 466-468.
- F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, P. St.J. Russell, "All-solid photonic band gap fiber," Opt. Lett. 29, 2369-2371 (2004). [CrossRef] [PubMed]
- A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, P. St.J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005). [CrossRef] [PubMed]
- G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005). [CrossRef] [PubMed]
- T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, D. M. Bird, "Bend loss in all-solid bandgap fibres," Opt. Express 14, 5688-5698 (2006). [CrossRef] [PubMed]
- J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, D. M. Bird, "An improved photonic bandgap fiber based on an array of rings," Opt. Express 14, 6291-6296 (2006). [CrossRef] [PubMed]
- A. Wang, A. K. George, J. C. Knight, "Three-level neodymium fiber laser incorporating photonic bandgap fiber," Opt. Lett. 31, 1388-1390 (2006). [CrossRef] [PubMed]
- C. K. Nielsen, K. G. Jespersen, S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 µm using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 6063-6068 (2006). [CrossRef] [PubMed]
- T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, B. J. Eggleton, "Resonance and scattering in microstructured optical fibers," Opt. Lett. 27, 1977-1979 (2002). [CrossRef]
- N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, C. M. de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003). [CrossRef] [PubMed]
- J. Lægsgaard, "Gap formation and guided modes in photonic band gap fibres with high-index rods," J. Opt. A: Pure Appl. Opt. 6, 798-804 (2004). [CrossRef]
- A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
- S. L. Altmann, Band Theory of Solids: An Introduction from the Point of View of Symmetry (Clarendon Press, 1994).
- P. W. Atkins, Molecular Quantum Mechanics (Oxford University Press, 1983).
- T. A. Birks, J. C. Knight, P. St.J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
- sections I and II of G. J. Pearce, T. D. Hedley, D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell's equations in photonic crystals," Phys. Rev. B 71, 195108 (2005). [CrossRef]
- T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, P. St.J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004). [CrossRef] [PubMed]
- T. A. Birks, Y. W. Li, C. D. Hussey, "Waveguides with delta function layers," Opt. Commun. 83, 203-209 (1991). [CrossRef]

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