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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 21 — Oct. 17, 2007
  • pp: 13783–13795
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Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes

Mathias Perrin, Yves Quiquempois, Géraud Bouwmans, and Marc Douay  »View Author Affiliations


Optics Express, Vol. 15, Issue 21, pp. 13783-13795 (2007)
http://dx.doi.org/10.1364/OE.15.013783


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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 (nFSM) 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

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 [1

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]

] for a description of Bloch modes in Hollow Fibres). Consequently, the field profile in the whole structure should be described using the Bloch theorem [2

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

], 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 – [3

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

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]

], 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 [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]

] 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 [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]

], whether they are annular [10

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]

], ‘W-shaped’ [11

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]

], or cylindrical, with a parabolic profile [12

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 μm,” Opt. Express 15, 316 (2007). [CrossRef] [PubMed]

, 13

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]

], 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.

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 – [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]

]. All the numerical computations in section 2 have been performed using MPB [17

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

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 [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]

]. 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.

Fig. 1. Panel (a) shows the cladding we study. In an elementary cell (inside the white bordered parallelogram), two IAHs (in blue) surround a parabolic radial index profile, that model a high index rod (in dark red), with d/Λ = 0.725. The maximum index gradient, between the rod center and the background is Δn = 3.2∙10-2. IAHs have a diameter dair/Λ = 0.152. Panel (b) shows the BG diagram for the cladding with IAHs (full thick black line and dashed thick black line, which represents the FSM line), and without IAH (grey line with symbols). The thin horizontal black line represents the silica index nc = 1.45.

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, [19

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]

] 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

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

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]

] 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 [3

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

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]

]. 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 [22

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

]. 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]

]. 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 k = (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 [23

23. C. Kittel, Introduction to solid state physics, Wiley.

], and transposed in the community of photonic crystals [2

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

, 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]

], 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.

Fig. 2. Band diagram for structure B cladding. Only the lines corresponding to Γ, M or K have been represented. The BG have been shaded, and isolated rod LP modes are labeled. An insert represents the Brillouin zone of an hexagonal lattice. Particular values of β have been pointed out, in units of Λ-1.

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.

Table 1. Summary of the characteristics of the first four BG boundaries.

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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.

Fig. 3. Intensity and field profile for both borders of the first beam of permitted modes, associated to the LP01 mode, computed near the tip of first BG, at βΛ = 3, cf. Fig. 2, for structure B. Color bar at the right of the panels refer to the normalized field or intensity value. A contour plot on Fig. 3(b) shows iso-intensity lines, in logarithmic scale. The corresponding color bar is on the left of panel (b). On every panel, the dotted circles represent the high index inclusions. The arrow on Fig. 3(c) indicate ΓK direction.

The general description of fields in a periodic medium is given by the Bloch theorem, which reads, Ex(x,y) = Eo(x,y)exp(ikR⃗), where R⃗ 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 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 [22

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

], 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. [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]

], 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.

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.

Fig. 4. Real part of one transverse component of the electric field, for both borders of the second beam of permitted mode, associated to the LP11 mode, computed at βΛ = 7, cf. Fig. 2, for structure B. The arrow on Fig. 4(b) indicates the ΓM direction.

Table 2. Synthetic presentation of the properties of permitted mode borders. The nature of the border, as well as the influence of the presence (or absence) of IAH have been mentioned, due to constructive (or destructive) interference between the rods.

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  1. for upper bandgaps boundaries, there are no significative field at IAH locations, whether they exist or not, as there are destructive interferences at these points: thus adding IAHs does not perturb notably the mode profiles (cf. Fig. 5(b), 5(d).
  2. 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.

Fig. 5. Real part of one transverse component of electric field, Ex, for both borders of the second beam of permitted mode, associated to the LP11 mode, computed at βΛ = 7, cf. Fig. 2, for structure A and B. The dotted circles represent the geometry of the structure.

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 [24

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

], 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.

Fig. 6. Panel (a) shows a zone around the core of the fibre we study. The color code refers to the different index of the constituent – Ge-doped silica, silica, air. The boundaries are those of the periodic cladding without defect. On panel(b), one can see (lines without symbols) the dispersion diagram of the lowest frequency band index – n FSM – of the cladding structure. Different sizes of IAH have been used. From top to bottom, the dashed lines correspond to IAH radius of rIAH = 2.5.10-2 and rIAH = 4.5.10-2, in Λ units. The full black line corresponds to structure A, rIAH = 7.5.10-2, as used in section (2). The line with symbols corresponds to the TIR mode observed in the fiber we study.

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 FSMn 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.

Fig. 7. Modes intensity profiles at different values of λ/Λ, for an 8 rings structure. In each panel, two insets represents a transverse and a unidimensional cut view. [Media 1]

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 [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]

]. 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 [12

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 μm,” Opt. Express 15, 316 (2007). [CrossRef] [PubMed]

]. 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.

4. Conclusion

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.

Acknowledgments

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).

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 15, 325 (2007). [CrossRef] [PubMed]

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 13, 309 (2005). [CrossRef] [PubMed]

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 11, 1243 (2003). [CrossRef]

5.

A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320 (1999).

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. 27, 1977 (2002). [CrossRef]

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]

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]

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 μm,” 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]

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]

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]

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. 1, 477 (1999). [CrossRef]

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]

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]

22.

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

23.

C. Kittel, Introduction to solid state physics, Wiley.

24.

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

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 12, 1864 (2004). [CrossRef] [PubMed]

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

  1. 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]
  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 13, 309 (2005). [CrossRef] [PubMed]
  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 11, 1243 (2003). [CrossRef]
  5. A. K. Abeeluck, N. M. Litchinitser, C. Headley, B. J. Eggleton, "Analysis of spectral characteristics of photonic bandgap waveguides," Opt. Express 10, 1320 (1999).
  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. 27, 1977 (2002). [CrossRef]
  7. T. A. Birks, G. J. Pearce, D.M. Bird, "Approximate band structure calculation for photonic bandgap fibres," Opt. Express 14, 9483 (2006). [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]
  9. B. T. Kuhlmey, K. Pathmanandavel, 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, 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, J. Luo, " Design of all-solid Bandgap fiber with improved confinement and bend losses," IEEE Photon. Technol. Lett.  18, 24 (2006). [CrossRef]
  12. A. Betourne, V. Pureur, G. Bouwmans, Y. Quiquempois, L. Bigot, M. Perrin, M. Douay, "Solid photonic bandgap fiber assisted by and extra air-clad structure for low-loss operation around 1.5 ?m," Opt. Express 15, 316 (2007). [CrossRef] [PubMed]
  13. 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 (2005). [CrossRef] [PubMed]
  14. A. Cerqueira S. Jr., F. Luan, C. M. B. Cordeiro, A. K. George, J. C. Knight, "Hybrid Photonic crystal fiber," Opt. Express 14, 926 (2006). [CrossRef] [PubMed]
  15. A. Betourne, G. Bouwmans, Y. Quiquempois, M. Perrin, 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]
  17. MPB software, URL: http://ab-initio.mit.edu/mpb/>
  18. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, A. Bjarklev, "Waveguidance by the photonic bandgap effect in optical fibres," J. Opt. A : Pure Appl. Opt. 1, 477 (1999). [CrossRef]
  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]
  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]
  22. A. Yariv, Quantum Electronics, 3rd ed., (John Wiley & Sons 1988) Chap. 22.8 627-640.
  23. C. Kittel, Introduction to Solid State Physics, (Wiley, 2004).
  24. J. P. B’erenger,"A perfectly matched layer for the absorption of electomagnetic waves," J. Comp. Phys. 114, 185 (1994). [CrossRef]
  25. A. Bjarklev, J. Broeng, 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, 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]

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