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
The cladding of a microstructured fiber is a periodic array of inclusions, whether they are of high or low index. Lattice periodicity imposes that the modes of field excited in the cladding should be Bloch modes, defined as the exact solution of the eigenvalue problem that corresponds to an infinite periodic cladding (see especially  for a description of Bloch modes in Hollow Fibres). Consequently, the field profile in the whole structure should be described using the Bloch theorem , where both the individual cell and the lattice periodicity are taken into account. In the particular case where isolated inclusions have their own modes, i.e. if they can confine fields by their high dielectric constant, a naive picture, but powerful model – the ARROW description – -, is very useful. It teaches us that the isolated inclusion characteristics do determine by themselves, albeit roughly, BG position and width, generated by a cladding made of many rods. This model, and some extensions to it  are known to be valid in quite a large range of parameters, provided that energy does not leak too much out of the high index inclusions. Such result is conceptually extremely important, as it simplifies design problems and shrinks their complexity down to the study of the modes of an 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 , whether they are annular , ‘W-shaped’ , or cylindrical, with a parabolic profile [12, 13], and gave numerous results. However, beyond the ARROW model, and especially at large wavelength, when fields leak out of the high index inclusions, Bloch modes have to be considered.
In this paper, we study a Photonic Band Gap Fibre (PBGF) structure that contains Intersticial Air Holes (IAHs) [15, 16, 18]. As we will see, this is a typical case where the properties we shall focus on cannot be modeled well by the ARROW model. However, we have been able to build up a very simple method that allows us to compute exact BG boundaries easily, with much less numerical load than what is often done in the literature. We will describe the influence of IAHs on the gap broadening and deepening, and notice that it can lead to new ideas to tailor fibres properties. Previous work, that studied anisotropic fiber structures, mixing air holes and high index inclusions, , has shown that light confinement could occur through different mechanisms (TIR or Arrow guiding) depending on the direction, at a given wavelength. We shall present here a fiber structure whose cladding is made of an elementary cell, identical in all directions of the transverse plane. Then, we shall see that the type of confinement, identical for all directions, depends on the injected wavelength. Note that, for the sake of consistency, we have chosen a fiber design with the same 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 .
First, we shall detail the mechanism that is responsible for IAH influence, by comparing a 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
We now detail the mechanism of influence of the IAHs on the BGs of the structure. The presence of IAHs between high index inclusions changes the index profile of the elementary cell, cf. Fig. 1(a). Then, if one considers an infinite cladding, one would find an IAH at the center of each equilateral triangle, defined by three high index rods. Note that the shape of an IAH can be described in first approximation by a circle – instead of a triangle – . All the numerical computations in section 2 have been performed using MPB . In particular, in order to compute BG diagram, one inputs the wavevector in the cladding, k⃗ = (kx, ky, β), and gets at the output the eigen-frequencies ω, from which one can compute the effective index of the mode, neff = cβ/ω. Note, that the MPB method used here is analogous to the one presented in .
2.1. Numerical observation of the influence of IAHs
The Fig. 1(b) shows the transformation of the first four BGs, induced by the presence of IAHs, in a SC-PBGF, made of high index – Ge-doped – inclusions of parabolic profile.
Comparing, for a given BG, structure B (grey line) and structure A (black lines), we notice clearly a deepening (the tip of each BG is at lower index values), and a change of shape, due to the presence of IAHs.
Besides, the BGs are not modified by IAHs near the cut-off index (nc = 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 . Moreover, a peculiar behavior is to be noticed, for each of the four first BGs. Indeed, at high enough values of neff, 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.
In the attempt to explain these observations, let us make some preliminary remarks.
A first and naive approach, would be to consider that the average background index is modified. In our case, it shall be lowered by the presence of IAHs. However, from this model, the shape of the BGs border near the cut-off should change, which is not the case.
In a second approach, we can notice that the index profile of our cladding can be seen as the superposition of a honeycomb structure,  and a common PBGF, without IAH. Is it possible to describe the resulting BG diagram as a superposition too ? If one examines the band diagram of the corresponding honeycomb structure, one would see that it only displays a single BG, for this particular index contrast, so that a superposition of BG of both structures cannot account for the effect observed Fig. 1(b).
Note that we have observed that the superposition of two index profiles can sometimes results in a superposition of the permitted bands, relative to both structures. This happens when both superposed inclusions have a higher index than the background, and confine fields. Explanation of the impact of IAHs has to be sought elsewhere.
2.2. Computation ofBG boundaries
In order to do so, we shall focus on the mode profile in the whole structure. More precisely, the field profile around the position of IAH will prove to be determinant, to understand the way the band structure is modified.
First, let us remind some general features concerning BG boundaries, for structure B.
Considering values of the effective index sufficiently above cut-off, one observes that permitted bands gather in groups. This phenomenon is well known and understood, [20, 21] as in the large effective index regime, light is mainly confined in high index regions so that Bloch modes can be described quite accurately in terms of isolated rods modes. This property has been intensively used to predict roughly the low loss transmission bands, thanks to the ARROW model -. We will thus label each band group, in the following, using the conventional LPlm 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 . Approximate models have been developed to describe the resulting band diagram, where the coupling between rods, even weak, alters dramatically the isolated rod mode . However, it is useful to present a fast but exact method, that goes beyond ARROW model, and helps to determine the BG boundaries, for any type of inclusion or periodic lattice.
Among all the permitted modes, that comes out of one isolated rod mode, through a lift of degeneracy, one can find the BG upper and lower boundaries. We have noticed that, on these boundaries, the transverse wavevector k⃗⊥ = (kx, ky) always corresponds to some particular high symmetry points of the transverse Brillouin zone – see the insert, Fig. 2-, (Γ, K or M). Such results should not be surprising : thanks to symmetry considerations, well known in solid state physics , and transposed in the community of photonic crystals [2, 18], a local extrema of an eigen-frequency diagram has to correspond to a high symmetry points. Therefore, in order to plot a BG diagram quickly, one should only compute the bands that corresponds to 3 particular transverse wavevectors.
The Fig. 2 shows a magnification of Fig. 1(b), where permitted bands for high symmetry points (line with symbols) clearly correspond to BG boundaries, for any value of 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.
One can wonder what is the link between BG boundaries and the border of permitted modes, associated to a specific LP mode. In simple situations, a given beam of permitted modes that stems, say from the LP11 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 LP02 and LP21. 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.
Nevertheless, our previous observations on the different types of boundary remain valid.
For the sake of clarity, we shall consider in the following both borders of a given beam of permitted mode, stemming from some isolated rod mode, when a lift of degeneracy occurs, and study low order BG. In particular, we shall see that the difference between upper and lower borders are the consequence of a complex interplay between the single rod properties and lattice characteristics.
2.3. Bloch theorem interpretation of the constructive or destructive interaction between rods
Intensity and field profiles of the upper and lower borders of the first beam of permitted bands are plotted on Fig. 2.3. One can observe that light intensity is at its maximum in each rod center and spreads significantly out of the high index regions as expected in this low effective index regime – for βΛ = 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.
The general description of fields in a periodic medium is given by the Bloch theorem, which reads, Ex(x,y) = Eo(x,y)exp(ik⃗⊥ ∙ R⃗), where R⃗ is the position vector of the center of elementary cells, on the real lattice, and k⃗⊥ the vector of the reciprocal lattice. One understands then the interplay between the mode profile of one elementary cell – described by Eo – and the influence of the lattice, given by the phase term. Especially, depending on the type of the boundary, respectively Γ, K or M, that is excited, we shall get different phase shifts between two neighboring rods : respectively 0, 2π/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, Eo(x,y), is not too different from the field profile for an isolated individual rod LP mode.
Looking at the field profile of the lower boundary, cf. Figs. 3(c), 3(d), one can see, along the arrow – which materializes the ΓK⃗ direction –, that the phase of Ex evolves periodically, by steps of 2π/3. Note that the norm of the arrow has been chosen so that it corresponds to a 2π phase factor, i.e., one transverse wavelength. The phase shift between one rod and its six closer neighbors is therefore ±2π/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, |Ex|, 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 LP01 profiles centered on each rod with a null (resp. 2π/3) phase difference between neighboring rods for the upper (resp. lower) border corresponding to a Γ(resp. K)-type boundary.
Therefore, the upper and lower border profiles can be described as, respectively, constructive and destructive interferences between rods. Note that, as for the even and odd supermodes of two weakly coupled waveguides , the mode presenting zero intensities between the high index inclusions has a lower effective index than the other mode. This decomposition in odd and even modes is reminiscent of ref. , where two classes of modes have been distinguished, depending on the zero value of the field or its derivative, at the boundary of an elementary cell.
Let us now consider the next beam of permitted modes, associated to LP11, 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 ΓM⃗ 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 ΓM⃗ direction, then remains non-zero.
All these observations point that, similarly to the case of the LP01 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.
More generally, a set of borders associated to a specific LPlm 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 LPlm 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
Let us now come back to structure A. When effective index is high enough, the light is well confined in the high index rods for both mode borders, and one can understand that IAHs do not play any role.
For higher values of 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).
We have thus shown that the influence of the IAH is to enlarge the gaps at their tips, leaving them unchanged at their heel. The explanation resides in the presence / absence of zero field at the position of IAH.
A complete explanation has been given, that take into account both the isolated rod mode profile, and the lattice influence. A link has been drawn with both the theory of couplers , and the Bloch theorem [23, 2].
3. Existence of a total refraction mode
Having shed some light on the IAH effect concerning infinite cladding, we shall now discuss their influence on a finite size system : a real fiber. For the sake of simplicity, we shall consider that its core is made of pure silica. One can use a finite element method to compute the fiber eigenmodes. Precisely, we have used the commercial software Comsol, to solve Maxwell equations on the fiber structure, and output the complex eigenvalues, as well as the modes profiles. In particular, this enables us, through the use of Perfectly Matched Layers , to compute the imaginary parts of the effective index, and then, the confinement losses of the fibre.
If one starts from the infinite cladding behavior described section 2, and if one considers that the first cladding mode (also called Fundamental Space filling Mode) can be seen as the lower boundary of the zeroth order BG, one may guess that it is possible to lower the effective index of this line. In this section, we will see that such a modification of the FSM line of the cladding will permit us to predict the existence of a Total Internal Refraction mode for some well chosen fibre profiles. We will detail all the necessary conditions.
The effect of IAH on the FSM line can be inferred from Fig. 6(b) – look at the curves without symbols –, where the evolution of the effective index, for different size of IAH is shown. One notice that, for large enough sizes – in our case, rIAH/Λ > 4.5.10-2 –, the effective index of points of the FSM line, n FSM, goes below the value nc = 1.45. One can thus expect to build a fiber made of an undoped silica core – index nc –, 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.
Nevertheless, our first attempt has shown that some condition on the core size has to be fulfilled, in order to see some TIR mode. In previous literature , a “one rod defect” core has been used, and no TIR mode observation have been reported. This is not surprising, since, in this case, the core effective index  is lower than the FSM effective index. Hence, no TIR mode can be seen, because removing only the central high index rod – keeping the first ring of IAH unplugged – leads to an index lowering defect. To support a TIR mode, the core effective index need to be increased, to reach values above the FSM line. Such configuration can be obtained by increasing the amount of pure silica inside the core. Indeed, in order to find a convenient design, cf. Fig. 6(a), we have removed the central high index rod, 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.
One can see on Fig. 6(b), that the TIR mode comes out of the FSM line, at high λvac/Λ. Both TIR and FSM lines diverges when λvac/Λ decreases, up to a maximum index difference, where (n FSM – n TIR) is as high as 2.5 × 10-3, around λvac/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 λvac/Λ, both lines get closer, and finally merge around λvac/Λ ≈ 0.3, into an ensemble of six isolated rod LP01 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 nc=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.
It is interesting to study more in detail how energy is spatially distributed in this TIR mode, when it evolves from large wavelength (close to the FSM line), down to the isolated rod LP mode. This information is contained in Fig. 7(a)–7(d), for the paper version of this article, and in a movie, available online.
At high wavelength, 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 (neff = 1.4420 < nc) and is evanescent in the cladding (neff > nFSM) 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: n 1 = 1.4420 + 6.52 ∙ 10-6 i, which corresponds, at 1.5μm to 2.37 ∙ 105dB/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 neff = nc. 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 neff = nc. The corresponding losses are 8.15 dB/km. Above the theoretical limit of existence of TIR, light cannot propagate in the core anymore, as neff > nc. 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 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.∙10-2dB/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 . Actually, the computed effective index of the first mode below TIR lies on the BG boundary, and is therefore of negligible impact.
Finally, let us remark that the confinement losses of this TIR mode increase with the normalized wavelength , as for any conventional microstructured fibre guiding by TIR. However these relatively high losses of the TIR guiding mode (i.e. when neff < nc) could be, if required, significantly reduced either by increasing the number of rings, the diameter of IAHs or adding an extra air-clad . Therefore, the structure proposed in this section allows to confine the light with relatively low losses, preserving a Gaussian-like intensity profile, in different bands of transmission, one being at largest wavelength, corresponding to TIR guiding, whereas the others, at shorter wavelengths correspond to Photonic Band Gap guiding effect.
In this paper, we have focused on the effect of IAHs in PBGF. First, we have shown that IAHs do not perturb BG around cut-off, but only at the tip of the Bandgaps, so that the generally useful ARROW model is of no help here. More precisely, we have pointed out that only the lower boundaries of the BG are drastically modified below the cut-off line. To clarify this effect, we have computed the exact BG boundaries exploiting the symmetry of the crystal, without having to deal with the whole density of state, that sometimes contain too much information. We have shown that, at the lower (upper) limit of the bandgaps, destructive (constructive) interferences between modes of the high index rods occurs. These phenomena have been described in detail and shown to be a direct consequence of the symmetry of isolated rod modes and the nature of the transverse vector excited in the cladding, in agreement with Bloch theorem. The presence or absence of the field at the position of IAH (which is directly link to the nature of the BG border) has proven to be the physical explanation for the strong or weak influence of IAH on the BG diagram of the structure. Such reasoning could be used as a clue, so as to 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.
Besides, we have noticed that the FSM line of the cladding can be affected similarly, which can lead to a fibre design, that displays two distinct guiding mechanisms. Both correspond to nearby range of frequencies, which gives the fiber the property to guide on a very broad range of frequency, still preserving its bandgap properties.
This work was supported by the “Conseil Régional Nord-Pas de Calais”, the “Fonds Européen de Développement Economique des Régions”, and the “Agence National de la Recherche” (ANR-05-BLAN-0080).
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