## Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure |

Optics Express, Vol. 19, Issue 7, pp. 6945-6956 (2011)

http://dx.doi.org/10.1364/OE.19.006945

Acrobat PDF (3885 KB)

### Abstract

Propagation of light in a square-lattice hollow-core photonic crystal fibre is analysed as a model of guidance in a class of photonic crystal fibres that exhibit broad-band guidance without photonic bandgaps. A scalar governing equation is used and analytic solutions based on transfer matrices are developed for the full set of modes. It is found that an exponentially localised fundamental mode exists for a wide range of frequencies. These analytic solutions of an idealised structure will form the basis for analysis of guidance in a realistic structure in a following paper.

© 2011 OSA

## 1. Introduction

1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science **285**, 1537–1539 (1999). [CrossRef] [PubMed]

2. P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express **13**, 236–244 (2005). [CrossRef] [PubMed]

3. D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science **301**, 1702–1704 (2003). [CrossRef] [PubMed]

4. A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express **16**, 5035–5047 (2008). [CrossRef] [PubMed]

5. F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. **31**, 3574–3576 (2006). [CrossRef] [PubMed]

7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science **318**, 1118–1121 (2007). [CrossRef] [PubMed]

*π*/3 to each other, as shown in Fig. 1(a). Figure 1(b) shows a square-lattice hollow-core PCF, whose cladding structure has two sets of orthogonal glass strips. These PCFs have a large pitch, where Λ is typically 12

*μ*m for Kagome PCFs [5

5. F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. **31**, 3574–3576 (2006). [CrossRef] [PubMed]

*μ*m for square-lattice PCFs [6

6. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express **16**, 20626–20636 (2008). [CrossRef] [PubMed]

5. F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. **31**, 3574–3576 (2006). [CrossRef] [PubMed]

6. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express **16**, 20626–20636 (2008). [CrossRef] [PubMed]

7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science **318**, 1118–1121 (2007). [CrossRef] [PubMed]

**31**, 3574–3576 (2006). [CrossRef] [PubMed]

7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science **318**, 1118–1121 (2007). [CrossRef] [PubMed]

**31**, 3574–3576 (2006). [CrossRef] [PubMed]

**318**, 1118–1121 (2007). [CrossRef] [PubMed]

8. A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express **15**, 7713–7719 (2007). [CrossRef] [PubMed]

9. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express **15**, 12680–12685 (2007). [CrossRef] [PubMed]

10. S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express **18**, 5142–5150 (2010). [CrossRef] [PubMed]

*k*

_{0}= 2

*π*/

*λ*and

*λ*is the vacuum wavevector,

*β*is the propagation constant,

*n*

^{2}is the dielectric function which describes the transverse structure of the PCF, and ∇

*is the transverse gradient operator. The vector character of Eq. (1) is contained in the right-hand side. Although numerical tools, such as the finite element method [12*

_{t}12. S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express **17**, 13050–13058 (2009). [CrossRef] [PubMed]

**h**

*is either*

_{t}*x*or

*y*polarised in the transverse plane. Although this approximation has a limited quantitative accuracy, it provides a useful insight into a wide range of PCFs [15

15. T. Birks, D. Bird, T. Hedley, J. Pottage, and P. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express **12**, 69–74 (2004). [CrossRef] [PubMed]

*x*and

*y*axes. The two model structures differ only at the intersections of the glass strips. Figure 1(c) shows a structure which, in principle, can be fabricated. However, it still does not allow for a simple solution. We therefore use the structure shown in Fig. 1(d) as our ideal model structure. It differs from Fig. 1(c) in having a higher refractive index at the intersections of the glass strips. The key point is that this design makes the total dielectric function separable into identical sets of parallel glass slabs along the two axes and allows us to obtain analytical solutions. A similar approach has been used in previous work on waveguides [16

16. S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. **20**, 1644–1650 (2002). [CrossRef]

17. A. Kumar, A. N. Kaul, and A. K. Ghatak, “Prediction of coupling length in a rectangular-core directional coupler: an accurate analysis,” Opt. Lett. **10**, 86–88 (1985). [CrossRef] [PubMed]

16. S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. **20**, 1644–1650 (2002). [CrossRef]

## 2. Solution for scalar model structure

### 2.1. Separation of variables

*x*and

*y*directions in the transverse plane. This feature makes the two-dimensional dielectric function separable as where

*n*

^{2}gives the difference of the dielectric constant between the glass strips and air holes in each direction. Thus, ▵

*n*

^{2}takes the value

*h*(

*x, y*) in the scalar governing equation can be separated as

*h*(

*x,y*) =

*X*(

*x*)

*Y*(

*y*). Substituting this into Eq. (2) leads to the dimensionless equations and where a constant pitch Λ is introduced and

*ξ*acts as a separation constant. The position variables,

*x*and

*y*, are now dimensionless through division by Λ. At the high-index intersections, the magnitude of the normalised transverse wavevector

*K*takes the value:

_{g}*g*represents the values of

*p*and

_{x}*p*in the glass regions. A trigonometric function can thus be introduced to replace

_{y}*ξ*and we write separable transverse wavevectors as where

*θ*represents the angle relative to the

*x*axis of the transverse wavevector in the high-index intersections. For simplicity a variable

*C*= cos(2

*θ*) is used to replace

*θ*. The wavevector components can then be written as The equivalent expressions in the air regions become and

*β*and

*C*. By looking at Eq. (7) or Eqs. (8) and (9), the solutions can be sorted into two types: ‘symmetric modes’ with

*C*= 0 and ‘non-symmetric modes’ with

*C*≠ 0. These have the same or different transverse wavevector components in the

*x*and

*y*directions, respectively. The symmetric modes are important because, as we will show later, all the modes of the model structure can be derived from solutions with

*C*= 0.

### 2.2. Matrix expressions for the fields

*N*and (

^{th}*N*+ 1)

*layers (shown by the red line in Fig. 2 (a)), both the fields and their derivatives are continuous, leading to the matrix equation*

^{th}### 2.3. Modes in a supercell geometry

20. G. Pearce, J. Pottage, D. Bird, P. Roberts, J. Knight, and P. Russell, “Hollow-core pcf for guidance in the mid to far infra-red,” Opt. Express **13**, 6937–6946 (2005). [CrossRef] [PubMed]

*N × N*primitive cells in the transverse plane. In our analysis, we will use a supercell geometry because this leads to a finite number of calculated modes and, most importantly, the modes can be normalised, whether or not they are localised. This will be important in the perturbation analysis in the following paper. We focus only on the modes at the Γ point of the supercell Brillouin zone, which implies that the field coefficients in the centres of neighbouring supercells are the same for each mode. It is expected that the modes at the Γ point can effectively represent all the solutions within a supercell. With an increase of the supercell size, the area of the first Brillouin zone correspondingly decreases. The accuracy of Γ point sampling can be examined through tests of the convergence of solutions with respect to the size of the supercell.

*x*and

*y*directions, i.e.

*C*= 0. By choosing the centre of the central defect of a supercell as the origin, the arrangement of the glass layers is symmetric along both axes. Owing to this structural symmetry, the permitted one-dimensional fields have either even or odd symmetry with respect to the centre of the central defect; two examples are shown in Figs. 2(b) and 2(c). To find these two types of modes, a particular value of normalised frequency

*k*

_{0}Λ is first chosen. For the symmetric solutions we begin at the origin with the field coefficients

*a*= (1,0)

*. A trial value of*

^{T}*β*Λ is chosen and transfer matrices are used to determine the field at the origin of the next supercell. The

*β*Λ value is then scanned to determine values for which the field returns to (1,0)

*. Similarly, odd modes are found by starting with*

^{T}*a*= (0,1)

*. We note that the search for allowed*

^{T}*β*Λ values is the only numerical part of the calculation. Once the

*β*Λ value is determined, the field coefficients are given by analytical expressions. It is also straightforward to normalise the modes, again by using analytical expressions [13].

## 3. Results in the scalar approximation

### 3.1. Guidance properties of rectangular hollow-core PCFs

*×*8 supercell is shown in Figs. 2(b) and 2(c).

21. J. Pottage, D. Bird, T. Hedley, J. Knight, T. Birks, P. Russell, and P. Roberts, “Robust photonic band gaps for hollow core guidance in pcf made from high index glass,” Opt. Express **11**, 2854–2861 (2003). [CrossRef] [PubMed]

23. 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). [CrossRef] [PubMed]

*k*

_{0}Λ and the range of normalised propagation constants close to the air-line, i.e. (

*β*–

*k*

_{0})Λ. The yellow and white regions represent photonic bands and bandgaps, respectively. It is observed that there is a sequence of bandgaps that cross the air-line.

*×*28 supercell. It is not surprising that a guided mode should exist within the bandgaps of the perfect cladding structure; an example of the fundamental mode at

*k*

_{0}Λ = 16.5 is given in Fig. 3(b). More surprisingly, we find that an exponentially localised mode exists for all normalised frequencies within the given range, whether or not there is a bandgap. An example is given in Fig. 3(c), and Fig. 3(d) shows a cladding state with the same frequency and propagation constant (to within 6 d.p.) as the guided mode in Fig. 3(c). As discussed in the introduction, this coexistence of localised and delocalised modes has been previously noted for our model structure in Ref. [16

16. S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. **20**, 1644–1650 (2002). [CrossRef]

*k*

_{0}Λ = 56) where there is no guided mode; this occurs when a new one-dimensional mode is just trapped in the transverse direction across the glass strips [24

24. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. **27**, 1592–1594 (2002). [CrossRef]

**31**, 3574–3576 (2006). [CrossRef] [PubMed]

**318**, 1118–1121 (2007). [CrossRef] [PubMed]

6. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express **16**, 20626–20636 (2008). [CrossRef] [PubMed]

### 3.2. Guided modes in the supercell geometry

*k*

_{0}Λ = 40, which lies within a band of cladding states. For convenience, in discussing the modes, we begin with a relatively small 8

*×*8 supercell.

*x*and

*y*directions (i.e. those with

*C*= 0). It is convenient to consider these modes as being of three distinct types in terms of their propagation constants. The first is the set of air-guided modes just below the air-line. This set contains eight modes (which is associated with the size of the supercell chosen). These air-guided modes are well separated from other modes with lower values of

*β*Λ. The transverse fields of this group of modes nearest to the air-line are shown in Fig. 4; among them is the fundamental guided mode, as shown by Fig. 4(a). The wavefunctions are both even in modes (a) to (e) and both odd in modes (f) to (h) in Fig. 4.

*β*Λ)

^{2}= 3611.54, with extremely small differences in the (

*β*Λ)

^{2}values of less than 10

^{−7}. Plots of these modes are shown in Fig. 5, where it can be seen that the fields are located in the high-index intersections of the glass strips. Owing to this property, these modes are referred to as ‘high-index modes’.

*C*= 0 with

*β*Λ values much lower than the air-line. Three examples of these modes are shown in Fig. 6. As mentioned above, solutions with negative (

*β*Λ)

^{2}values are important in determining the full set of non-symmetric modes with positive (

*β*Λ)

^{2}. This also explains why we are interested in the three modes in Fig. 6: they can generate non-symmetric modes with

*β*Λ values close to the fundamental mode. Figure 6 shows that, unlike the previous two types, the fields of these ‘delocalised modes’ are no longer confined in a specific region but instead spread throughout the whole transverse plane. Moreover, they exhibit rapid transverse oscillation, as indicated by the alternation of the red and blue colours in these plots.

*C*values. In practice, we are interested only in modes with propagation constants close to the fundamental guided mode. Therefore, there are only two relevant choices of combination of symmetric modes. One is the internal combination of the air-guided modes shown in Fig. 4; the other involves a high-index mode and a delocalised mode, as shown in Figs. 5 and 6, respectively.

*π*/2 in the transverse plane because of the opposite

*C*values. The field intensity is still mainly confined in the air holes. Figures 7(c) and 7(d) display examples of the modes formed by a combination of high-index modes and delocalised modes. These modes are glass-guided, and their fields are concentrated along the glass strips. Because of the large number of delocalised modes, these glass-guided modes evenly cover a broad region of

*β*Λ values and are therefore likely to become very close to the fundamental mode. As observed in Kagome and square-lattice hollow-core PCFs [6

**16**, 20626–20636 (2008). [CrossRef] [PubMed]

**318**, 1118–1121 (2007). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science |

2. | P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express |

3. | D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science |

4. | A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express |

5. | F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. |

6. | F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express |

7. | F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science |

8. | A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express |

9. | G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express |

10. | S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express |

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

12. | S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express |

13. | L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009). |

14. | L. Chen and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres II: Perturbation theory for a realistic fibre structure,” Submitted to Opt. Express (2011). |

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

16. | S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. |

17. | A. Kumar, A. N. Kaul, and A. K. Ghatak, “Prediction of coupling length in a rectangular-core directional coupler: an accurate analysis,” Opt. Lett. |

18. | P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic bloch waves and photonic band gaps,” in |

19. | 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 |

20. | G. Pearce, J. Pottage, D. Bird, P. Roberts, J. Knight, and P. Russell, “Hollow-core pcf for guidance in the mid to far infra-red,” Opt. Express |

21. | J. Pottage, D. Bird, T. Hedley, J. Knight, T. Birks, P. Russell, and P. Roberts, “Robust photonic band gaps for hollow core guidance in pcf made from high index glass,” Opt. Express |

22. | T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express |

23. | 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 |

24. | N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. |

**OCIS Codes**

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

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 27, 2011

Manuscript Accepted: March 9, 2011

Published: March 25, 2011

**Citation**

Lei Chen, Greg J. Pearce, Timothy A. Birks, and David M. Bird, "Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure," Opt. Express **19**, 6945-6956 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6945

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### References

- R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]
- P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005). [CrossRef] [PubMed]
- D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003). [CrossRef] [PubMed]
- A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express 16, 5035–5047 (2008). [CrossRef] [PubMed]
- F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006). [CrossRef] [PubMed]
- F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008). [CrossRef] [PubMed]
- F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007). [CrossRef] [PubMed]
- A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007). [CrossRef] [PubMed]
- G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007). [CrossRef] [PubMed]
- S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18, 5142–5150 (2010). [CrossRef] [PubMed]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
- S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17, 13050–13058 (2009). [CrossRef] [PubMed]
- L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009).
- L. Chen and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres II: Perturbation theory for a realistic fibre structure,” Submitted to Opt. Express (2011).
- T. Birks, D. Bird, T. Hedley, J. Pottage, and P. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express 12, 69–74 (2004). [CrossRef] [PubMed]
- S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. 20, 1644–1650 (2002). [CrossRef]
- A. Kumar, A. N. Kaul, and A. K. Ghatak, “Prediction of coupling length in a rectangular-core directional coupler: an accurate analysis,” Opt. Lett. 10, 86–88 (1985). [CrossRef] [PubMed]
- P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Application , E. Burstein and C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.
- 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).
- G. Pearce, J. Pottage, D. Bird, P. Roberts, J. Knight, and P. Russell, “Hollow-core pcf for guidance in the mid to far infra-red,” Opt. Express 13, 6937–6946 (2005). [CrossRef] [PubMed]
- J. Pottage, D. Bird, T. Hedley, J. Knight, T. Birks, P. Russell, and P. Roberts, “Robust photonic band gaps for hollow core guidance in pcf made from high index glass,” Opt. Express 11, 2854–2861 (2003). [CrossRef] [PubMed]
- T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14, 9483–9490 (2006). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

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