## Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes

Optics Express, Vol. 15, Issue 21, pp. 13783-13795 (2007)

http://dx.doi.org/10.1364/OE.15.013783

Acrobat PDF (3749 KB)

### Abstract

In this article, we deal with new properties of a Solid Core Photonic Bandgap (SC-PBGF) fiber with intersticial air holes (IAHs) in its transverse structure. It has been shown recently, that IAH enlarges its bandgaps (BG), compared to what is observed in a regular SC-PBGF. We shall describe the mechanisms that account for this BG opening, which has not been explained in detail yet. It is then interesting to discuss the role of air holes in the modification of the Bloch modes, at the boundaries of the BG. In particular, we will use a simple method to compute the *exact* BG diagrams in a faster way, than what is done usually, drawing some parallels between structured fibers and physics of photonic crystals. The very peculiar influence of IAHs on the upper/lower boundaries of the bandgaps will be explained thanks to the difference between mode profiles excited on both boundaries, and linked to the symmetry / asymmetry of the modes. We will observe a modification of the highest index band (*n*_{FSM}) due to IAHs, that will enable us to propose a fiber design to guide by Total Internal Reflection (TIR) effect, as well as by a more common BG confinement. The transmission zone is deeply enlarged, compared to regular photonic bandgap fibers, and consists in the juxtaposition of (almost non overlapping) BG guiding zones and TIR zone.

© 2007 Optical Society of America

## 1. Introduction

1. F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, “Identification of Bloch-modes in hollow-core photonic crystal fiber cladding,” Opt. Express **15**, 325 (2007). [CrossRef] [PubMed]

3. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. B. Cordeiro, F. Luan, and Russell, “Photonic bandgap with an index step of one percent,” Opt. Express **13**, 309 (2005). [CrossRef] [PubMed]

8. G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express **14**, 5682 (2006). [CrossRef] [PubMed]

7. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express **14**, 9483 (2006). [CrossRef] [PubMed]

*isolated*inclusion, getting rid of coupling effects between rods. This eventually lead the community to think a cladding design in terms of the index profile of a unique inclusion, embedded in a background. In particular, many research focused on coated inclusions [9

9. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express **14**, 10851 (2006). [CrossRef] [PubMed]

10. 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 (2006). [CrossRef] [PubMed]

11. G. Ren, P. Shum, L. Zhang, M. Yan, X. Yu, W. Tong, and J. Luo, “Design of all-solid Bandgap fiber with improved confinement and bend losses,” IEEE Photon. Technol. Lett. , **18**, 24 (2006). [CrossRef]

12. A. Bétourné, V. Pureur, G. Bouwmans, Y. Quiquempois, L. Bigot, M. Perrin, and M. Douay, “Solid photonic bandgap fiber assisted by and extra air-clad structure for low-loss operation around 1.5 *μ*m,” Opt. Express **15**, 316 (2007). [CrossRef] [PubMed]

13. 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 (2005). [CrossRef] [PubMed]

15. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. , **32**, N 12 (2007). [CrossRef]

16. J. Laegsgaard and A. Bjarklev, “Doped photonic bandgap fibers for short-wavelength nonlinear devices,” Opt. Lett. **28**, 783 (2003). [CrossRef] [PubMed]

18. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fibres”, J. Opt. A : Pure Appl. Opt. **1**, 477 (1999). [CrossRef]

14. A. Cerqueira S. Jr., F. Luan, C. M. B. Cordeiro, A. K. George, and J. C. Knight, “Hybrid Photonic crystal fiber,” Opt. Express **14**, 926 (2006). [CrossRef] [PubMed]

*d*/Λ ratio – where

*d*is the high index rod diameter and Λ the pitch of the hexagonal structure – than what we reported from our previous studies [15

15. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. , **32**, N 12 (2007). [CrossRef]

*cladding*structure with IAHs – structure A – and the very same structure, but where IAHs have been removed – structure B. There, we shall remain general, without considering the core of a fiber structure, but being interested only in the properties of the infinite cladding. This will lead us to show the possibility of existence of a TIR mode for fibers that would use this cladding. At this point, we will describe fibers, that consist in a silica core surrounded by a finite size cladding. The paper will end by a discussion on the conditions of existence of such mode(s), and conclude on the unusual broad transmission band of this fiber.

## 2. Influence of IAHs on the Bloch modes at BG boundaries

15. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. , **32**, N 12 (2007). [CrossRef]

*in section 2*have been performed using MPB [17

17. MPB software, URL: http://ab-initio.mit.edu/mpb/

*k*⃗ = (

*k*,

_{x}*k*,

_{y}*β*), and gets at the output the eigen-frequencies

*ω*, from which one can compute the effective index of the mode,

*n*=

_{eff}*cβ/ω*. Note, that the MPB method used here is analogous to the one presented in [18

18. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fibres”, J. Opt. A : Pure Appl. Opt. **1**, 477 (1999). [CrossRef]

### 2.1. Numerical observation of the influence of IAHs

*n*= 1.45), whereas they broaden at their tips. Their width consequently becomes constant on a much broader wavelength range than what can be observed without IAH, which is interesting, to enlarge transmission zones, to reduce confinement losses and bending losses, as recently shown experimentally [15

_{c}**32**, N 12 (2007). [CrossRef]

*n*, both borders of any BG remain the same whether there is IAH or not – as if the fields were unaffected by the presence of IAHs. On the contrary, for lower values of the effective index, in particular, below the cut-off index, both borders do not behave the same way. The upper border remains unaffected by IAHs, whereas the lower border is shifted, towards lower values of the effective index.

_{eff}19. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic Band Gap Guidance in Optical Fibers”, Science **282**, 1476 (1998). [CrossRef] [PubMed]

### 2.2. Computation ofBG boundaries

20. J. C. Knight, F. Luan, G. J. Pearce, A. Wang, T. A. Birks, and D. M. Birds, “Solid Photonic Badgap Fibres and Applications”, Jpn. J. Appl. Phys. **45**, 6059 (2006). [CrossRef]

21. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, T. A. Birks, and D. M. Birds, “Bend loss in all-solid bandgap fibres”, Opt. Express **14**, 5688 (2006). [CrossRef] [PubMed]

3. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. B. Cordeiro, F. Luan, and Russell, “Photonic bandgap with an index step of one percent,” Opt. Express **13**, 309 (2005). [CrossRef] [PubMed]

8. G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express **14**, 5682 (2006). [CrossRef] [PubMed]

_{lm}notation of the isolated rod modes. However, when

*λ*/Λ increases, to approach the cut-off of isolated inclusions, light spreads more and more in the low index background, leading to an increase of coupling between rods. This explains the degeneracy lift appearing for each group of bands, as its effective index decreases. In fact, a very similar behavior has been observed in the simpler case of two weakly coupled single mode waveguides, for which modes of the coupled structure, described in terms of odd and even supermodes, have effective index which separate farther apart as coupling increases [22]. Approximate models have been developed to describe the resulting band diagram, where the coupling between rods, even weak, alters dramatically the isolated rod mode [7

7. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express **14**, 9483 (2006). [CrossRef] [PubMed]

*boundaries*, for any type of inclusion or periodic lattice.

*k*⃗

_{⊥}= (

*k*,

_{x}*k*) always corresponds to some particular high symmetry points of the transverse Brillouin zone – see the insert, Fig. 2-, (Γ,

_{y}*K*or

*M*). Such results should not be surprising : thanks to symmetry considerations, well known in solid state physics [23], and transposed in the community of photonic crystals [2, 18

18. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fibres”, J. Opt. A : Pure Appl. Opt. **1**, 477 (1999). [CrossRef]

*only*compute the bands that corresponds to 3 particular transverse wavevectors.

*β*. One deduces that, in the most general case, there are three type of boundaries, denoted Γ,

*K*and

*M*for an hexagonal lattice. Such method has the advantage (i) to be exact – if one is interested by BG boundaries only –, (ii) to require much less computational time than commonly used methods, which computes the whole density of state, and (iii) to shed some light on the connection between photonic materials and microstructured fibers.

_{11}mode, delimits

*both*the upper boundary of the second BG (Γ-type boundary) and the lower boundary of the first BG (

*M*-type boundary). This is not always the case, as, for higher order BG, cf. Fig. 2, several nearby modes of isolated rod can mix in, for example LP

_{02}and LP

_{21}. Besides, note that one of the three particular type of boundary – Γ,

*K*,

*M*– can correspond either to upper or lower boundaries. The interpretation of such behavior will be detailed in the subsection (2.3). This information has been gathered on Tab. 1, for the structure we study, where the rod index profile is parabolic.

### 2.3. Bloch theorem interpretation of the constructive or destructive interaction between rods

*β*Λ = 3. For the lower border, cf. Fig. 3(b), we have superimposed on the intensity profile a set of contour plots. They represent iso-intensity lines, in logarithmic scale. One notice that intensity drops down to zero at zones between the high index rods. The zero-intensity is precisely at the position of the IAH, if they were there. Such zones are absent for the upper band, cf. Fig. 3(a). Let us interpret these observations.

*E*(

_{x}*x,y*) =

*E*(

_{o}*x,y*)exp(

*ik*⃗

_{⊥}∙

*R*⃗), where

*R*⃗ is the position vector of the center of elementary cells, on the real lattice, and

*k*⃗

_{⊥}the vector of the reciprocal lattice. One understands then the interplay between the mode profile of one elementary cell – described by

*E*– and the influence of the lattice, given by the phase term. Especially, depending on the type of the boundary, respectively Γ,

_{o}*K*or

*M*, that is excited, we shall get different phase shifts between two neighboring rods : respectively 0, 2

*π*/3,

*π*. Moreover, if we consider that the rod modes are sufficiently weakly coupled to one another, one can write that the field profile over an elementary cell,

*E*(

_{o}*x,y*), is not too different from the field profile for an

*isolated*individual rod LP mode.

*K*⃗ direction –, that the phase of

*E*evolves periodically, by steps of 2

_{x}*π*/3. Note that the norm of the arrow has been chosen so that it corresponds to a 2

*π*phase factor, i.e., one transverse wavelength. The phase shift between one rod and its six closer neighbors is therefore ±2

*π*/3. Zones of zero real part and imaginary part – white rings around the rods, on Figs. 3(c), 3(d)– lead to zero intensity for the

*x*component, |

*E*|, at their intersection. This corresponds to the observation made on Fig. 3(b). Therefore, the intensity profiles shown on Figs. 3(b), 3(a) can be seen as an interference pattern of LP

_{x}_{01}profiles centered on each rod with a null (resp. 2

*π*/3) phase difference between neighboring rods for the upper (resp. lower) border corresponding to a Γ(resp.

*K*)-type boundary.

7. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express **14**, 9483 (2006). [CrossRef] [PubMed]

_{11}, cf. Fig. 4. Note that we represent here only the real part, as the imaginary part is almost identical (the whole map is multiplied by –1), and the intensity map has the same shape too. Because of the odd nature of this mode, a line of zero-field now goes through the center of each rods, separating them in two poles (corresponding to positive and negative value of the real part of the field). Due to the symmetry imposed by the lattice , these lines align along one direction of the hexagonal structure. For the Γ boundary, cf. Fig. 4(a), all rods should be in phase. The juxtaposition without phase shift imposes one positive pole to be close from one negative pole of the nearby cell. For continuity reasons, the field has then to go through zero between two neighboring rows of rods, perpendicularly to Γ

*M*⃗ direction. On Fig. 4(b), on the contrary, the

*π*phase shifts corresponding to the

*M*boundary mode, inverts the poles periodically, so that two poles of the same sign are face to face. The field between the rods, in the Γ

*M*⃗ direction, then remains non-zero.

_{01}mode, the upper boundary is characterized by non-zero intensity values between the rods, whereas see zero intensity lines running between the inclusions for the lower boundary mode.

_{lm}mode has an upper limit given by a mode where constructing interferences occur between neighboring rods, whereas its lower border is limited by a mode presenting destructive interferences between adjacent high index inclusions. The nature (Γ,

*K*or

*M*) of the upper/lower limiting bands then depends on the symmetry of the LP

_{lm}mode, i.e. on the

*l*number. Indeed, for symmetric modes (even value of

*l*), the upper border has to be of Γ-type (inclusions all in phase), to obtain constructive interferences, whereas the lower band is a

*M*or

*K*type band (inclusions out of phase). For modes with an odd value of

*l*, this will be the opposite. The above remarks have been gathered in Tab. 2.

### 2.4. Field profile in structure A and B

*λ*/Λ, light leaks out of the high index rods, and one has to distinguish two type of behaviors, according to what has been described in previous section.

- for lower bandgaps borders, the situation is rather different as constructive interferences are expected to occur between the rods. So, by inserting IAHs at these positions, one will modify drastically the mode profile. Indeed, as it can be clearly seen from Figs. 5(a), 5(c) light is expelled out of these regions by the presence of IAH. This is due to the fact that the effective indexes of the modes of interest are notably higher than the air index. Adding IAHs will also then lead to a significant decrease of the effective indexes of lower bandgap borders as observed on Fig. 1(b).

## 3. Existence of a total refraction mode

24. J. P. Bérenger,“A perfectly matched layer for the absorption of electomagnetic waves”, J. Comp. Phys. **114**, 185 (1994). [CrossRef]

*r*/Λ > 4.5.10

_{IAH}^{-2}–, the effective index of points of the FSM line,

*n*

_{FSM}, goes below the value

*n*= 1.45. One can thus expect to build a fiber made of an undoped silica core – index

_{c}*n*–, surrounded by structure A – with both high index parabolic rods, and IAH in the cladding –, which could display TIR guiding above some wavelength, and BG guiding below.

_{c}16. J. Laegsgaard and A. Bjarklev, “Doped photonic bandgap fibers for short-wavelength nonlinear devices,” Opt. Lett. **28**, 783 (2003). [CrossRef] [PubMed]

*as well as the first ring ofIAHs*. Besides, note that we consider an 8 rings structure, in the figures of section (3), and to compute the confinement losses. The movie, available online, displays a 3 rings structure so as to render the mode shape more visible.

*λ*/Λ. Both TIR and FSM lines diverges when

_{vac}*λ*/Λ decreases, up to a maximum index difference, where (

_{vac}*n*

_{FSM}–

*n*

_{TIR}) is as high as 2.5 × 10

^{-3}, around

*λ*/Λ ≈ 0.7. Note that this is of the order of magnitude of a typical index contrast for a conventional telecom single mode fiber. For further decrease of

_{vac}*λ*/Λ, both lines get closer, and finally merge around

_{vac}*λ*/Λ ≈ 0.3, into an ensemble of six isolated rod LP

_{vac}_{01}modes. Guiding by Total Internal Reflection is possible, in the pure silica core, when TIR effective index lies between

*n*

_{FSM}and the core index

*n*=1 .45. Moreover, one can conjecture that still a larger core defect would broaden the range of existence of the TIR mode, possibly letting appear higher order TIR modes.

_{c}*λ*/Λ = 1, the intensity profile, cf. Fig. 7(a) has a Gaussian shape, centered on the fibre core, with a tail that extends into the cladding. In this regime, the light can propagate in the silica core (

*n*= 1.4420 <

_{eff}*n*) and is evanescent in the cladding (

_{c}*n*>

_{eff}*n*) as it shall be, for any conventional step index fibre guiding by TIR. The index contrast between effective core index and cladding-FSM-index is quite weak, so that losses are important:

_{FSM}*n*

_{1}= 1.4420 + 6.52 ∙ 10

^{-6}

*i*, which corresponds, at 1.5μm to 2.37 ∙ 10

^{5}dB/km. For slightly smaller wavelength, for example

*λ*/Λ = 0.752, cf. Fig. 7(b), the effective index of TIR mode increases up to

*n*

_{2}= 1.4457 + 1.25 ∙ 10

^{-7}

*i*. The intensity profile expands a bit less in the cladding, but is at the same time a bit more perturbed from a Gaussian distribution, and energy begin to accumulate in the high index rods. This is visible, as some modulation of the Gaussian tail appears at the rod position – see the cut-view. At some point, Fig. 7(c), obtained for

*λ*/Λ = 0.535, the effective index becomes

*n*

_{3}= 1.4499+2.24∙10

^{-10}

*i*, relatively close from the “theoretical” limit of existence of confinement by total refraction, for the material (silica) we consider in the core

*n*=

_{eff}*n*. There, the mode profile is flat at the center of the core, which is a sign that the curvature of the radial energy field density changes sign, going through zero, when one crosses the point

_{c}*n*=

_{eff}*n*. The corresponding losses are 8.15 dB/km. Above the theoretical limit of existence of TIR, light cannot propagate in the core anymore, as

_{c}*n*>

_{eff}*n*. The mode remains confined in the six high index rods around the core, and decays exponentially towards the center of the core. The Fig. 7(d), obtained for

_{c}*λ*/Λ = 0.308 illustrates such behavior, where there is almost no energy in the fibre core. The effective index is then

*n*

_{1}= 1.4573 + 5.50 ∙ 10

^{-13}

*i*, corresponding to very low confinement losses (about 2.∙10

^{-2}dB/km). What low they could be, such losses are reachable, however, at the expense of a mode shape very far from a Gaussian. One can note that the mode profile, cf. Fig. 7(c), 7(d), seems analogous to surface modes, found commonly in hollow core fibres. Besides, in our fibre design, because of the symmetry breaking induced by the removal of the first ring of IAH, such modes can be expected. As the symmetry breaking is due to the replacement of small IAH by a higher index material we can expect that their index will be relatively close to the mode of the infinite cladding without defect [26

26. M. J. F. Digonnet, H. K. Kim, J. Shin, S. Fan, and G. S. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers,” Opt. Express **12**, 1864 (2004). [CrossRef] [PubMed]

*on*the BG boundary, and is therefore of negligible impact.

*n*<

_{eff}*n*) could be, if required, significantly reduced either by increasing the number of rings, the diameter of IAHs or adding an extra air-clad [12

_{c}12. A. Bétourné, V. Pureur, G. Bouwmans, Y. Quiquempois, L. Bigot, M. Perrin, and M. Douay, “Solid photonic bandgap fiber assisted by and extra air-clad structure for low-loss operation around 1.5 *μ*m,” Opt. Express **15**, 316 (2007). [CrossRef] [PubMed]

## 4. Conclusion

*guess*– instead of

*computing*a band diagram – what can be the influence of air holes, at, possibly different positions than what we have chosen here.

## Acknowledgments

## References and links

1. | F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, “Identification of Bloch-modes in hollow-core photonic crystal fiber cladding,” Opt. Express |

2. | J. D. Joannopoulos, R. D. Meade, and J.N. Winn, “Photonic Crystals: molding the flow of light,” Princeton: Princeton University Press. |

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

4. | N. M. Litchinister, 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 |

5. | A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express |

6. | T. P. White, R. C. McPhedran, C. Martijn de Sterke, N. M. Litchinister, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibres,” Opt. Lett. |

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

8. | G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express |

9. | B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express |

10. | 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 |

11. | G. Ren, P. Shum, L. Zhang, M. Yan, X. Yu, W. Tong, and J. Luo, “Design of all-solid Bandgap fiber with improved confinement and bend losses,” IEEE Photon. Technol. Lett. , |

12. | A. Bétourné, V. Pureur, G. Bouwmans, Y. Quiquempois, L. Bigot, M. Perrin, and M. Douay, “Solid photonic bandgap fiber assisted by and extra air-clad structure for low-loss operation around 1.5 |

13. | 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 |

14. | A. Cerqueira S. Jr., F. Luan, C. M. B. Cordeiro, A. K. George, and J. C. Knight, “Hybrid Photonic crystal fiber,” Opt. Express |

15. | A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. , |

16. | J. Laegsgaard and A. Bjarklev, “Doped photonic bandgap fibers for short-wavelength nonlinear devices,” Opt. Lett. |

17. | MPB software, URL: http://ab-initio.mit.edu/mpb/ |

18. | J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fibres”, J. Opt. A : Pure Appl. Opt. |

19. | J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic Band Gap Guidance in Optical Fibers”, Science |

20. | J. C. Knight, F. Luan, G. J. Pearce, A. Wang, T. A. Birks, and D. M. Birds, “Solid Photonic Badgap Fibres and Applications”, Jpn. J. Appl. Phys. |

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

22. | A. Yariv, “Quantum Electronics”, 3rd edition John Wiley&Sons1988 (Chapitre 22.8 627–640). |

23. | C. Kittel, |

24. | J. P. Bérenger,“A perfectly matched layer for the absorption of electomagnetic waves”, J. Comp. Phys. |

25. | A. Bjarklev, J. Broeng, and A. S. Bjarklev, “Photonic Crystal Fibers”, Kluwer Academic Publishers, see section 6.4.2.2. |

26. | M. J. F. Digonnet, H. K. Kim, J. Shin, S. Fan, and G. S. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers,” Opt. Express |

**OCIS Codes**

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

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: April 16, 2007

Revised Manuscript: June 7, 2007

Manuscript Accepted: June 8, 2007

Published: October 5, 2007

**Citation**

Mathias Perrin, Yves Quiquempois, Géraud Bouwmans, and Marc Douay, "Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes," Opt. Express **15**, 13783-13795 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13783

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### References

- F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, S. A. Maier, "Identification of Bloch-modes in hollow-core photonic crystal fiber cladding," Opt. Express 15, 325 (2007). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: molding the flow of light, (Princeton: Princeton University Press).
- A. Argyros, T. A. Birks, S. G. Leon-Saval, C. B. Cordeiro, F. Luan, and Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309 (2005). [CrossRef] [PubMed]
- N. M. Litchinister, 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, 1243 (2003). [CrossRef]
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