## Abstract

We introduce an extended transfer matrix method (TMM) for solving guided modes in leaky optical fibers with layered cladding. The method can deal with fibers with circular but nonconcentric material interfaces. Validity of the method is verified by two full-vector numerical methods. The TMM is then used to investigate the guidance property of perturbed Bragg fibers. Our analysis reveals that the core modes will interact with each other when a perturbed Bragg fiber has only *C*
_{1} symmetry. Special attention is paid to the first transverse-electric (TE_{01}) mode, which is found to experience severe degradation around spectral regions where its dispersion curve supposedly crosses a transverse-magnetic (TM) or hybrid mode.

©2006 Optical Society of America

## 1. Introduction

Transfer matrix method (TMM) for analyzing modal properties of optical fibers with layered cladding structure was first introduced by Yeh *et al*. [1]. Johnson *et al*. have modified the method to enable direct leaky mode analysis by expressing field in the outermost layer in terms of Hankel functions [2]. Such method is extremely efficient in treating multi-clad optical fibers, which would otherwise be very complex, if not computationally forbidding, to all numerical methods. This is especially true to the Bragg fibers whose feature sizes are sometimes in tens of nanometers [3]. On another hand, the discrepancy between theoretical TMM results and experimental results has been noticed to be quite large [3]. It suggests that Bragg fibers might be extremely sensitive to cladding perturbations. In this paper, we introduce an extended TMM that is able to treat Bragg fibers in a deviated form, where the cladding interfaces are circular but not concentric. Each cladding layer in such a perturbed Bragg fiber no longer has a uniform thickness. With the proposed method, we will show that the TE_{01} mode in an ideal Bragg fiber experiences significant degradation as its cladding layers are deformed. Most evidently, in some spectral regions within the cladding’s photonic bandgap, the TE_{01} mode abruptly becomes less confined. Such deterioration in lightwave confinement is difficult to be compensated by merely adding the number of cladding layer pairs, as they are due to the interactions between the TE_{01} mode and other TM or hybrid core modes whose radiation losses are several orders higher.

## 2. Extended transfer matrix method

#### 2.1. Formulation

Figure 1(a) shows schematic cross section of the fiber under study. We number the layers from the innermost one (core) to the outermost one (cladding, which has infinite radius) as layer 1 to *N*. Each layer has a refractive index *n*_{i}
, with *i* = 1, ⋯ ,*N*. The interfaces are numbered, also from inside to outside, as Γ_{1} to Γ_{N-1}, each of which is a circle with radius *r*_{i}
(*i* = 1, ⋯ ,*N* - 1). It is known that in each layer, both *E*_{z}
and *K*_{z}
(*K*_{z}
= *E**H*_{z}
is normalized magnetic field, and $L=\sqrt{{\mu}_{0}/{\epsilon}_{0}}$
) fields satisfy the Helmholtz equation. The general solution of the wave equation in *i*^{th}
layer can be expressed as a Fourier-Bessel series, in the layer’s own cylindrical coordinate, as [4]

In Eqs. (1) and (2), *J*_{m}
is a Bessel function of the first kind of order *m*, and *H*_{m}
is a Hankel function of the first kind of order *m*. *m* is an integer from - ∞ to ∞. *k*_{t,i}
= $\sqrt{{k}_{0}^{2}{n}_{i}^{2}-{\beta}_{2}}$, and *β* is the propagation constant to be derived. *A*s and *B*s are coefficients, also to be derived. **r**
_{i} is a position vector in layer *i*’s coordinate, and it can be decomposed into (|**r**
_{i}|, *θ*_{i}
). When all material interfaces are concentric, all layers share the same cylindrical coordinate. The only chance to ensure continuity of both *E*_{z}
and *K*_{z}
fields across interfaces is to let coefficients of the Fourier-Bessel terms of the same order in all layers be equal. Hence for such a waveguide, we can fix an azimuthal quantum number (denoted here as *m*) prior to the radial mode field calculation [5]. Notice such classification of modes in a fiber with *C*
_{∞} symmetry according to the azimuthal quantum number is also a direct result from group theory [6]. However, for the fiber shown in Fig. 1(a), the continuous rotational symmetry is broken. Hence local Fourier-Bessel terms of different orders are not term-by-term equatable for two adjacent layers. That is, the coefficients of all terms in Eqs. (1) and (2) need to be solved collectively.

We choose the core’s coordinate as the global coordinate system. Figure 1(b) depicts two general nonconcentric interfaces in the fiber. It is not difficult to realize that, if we express fields in both layers (of indices *n*
_{1} and *n*
_{2}) in the same cylindrical coordinate as characterized by their interface Γ_{1}, term-by-term equation can be done. What we should do is hence to re-express the fields in layer 2 in layer 1’s coordinate system. This can be achieved using the Graf’s addition theorem [4].

For the *E*_{z}
field in layer 2, we assume it can be written in layer 1’s coordinate system as

where *m̂* is an integer from -∞ to ∞. Each *J* (or *H*) term in Eq. (3) can be written as a series of *J* (or *H*) terms in layer 2’s coordinate system as

where **J**
_{12} is the projection matrix, whose elements are calculated as [4]

**r**
_{12} is a vector pointing from the center of interface Γ_{1} to that of Γ_{2} [Fig. 1(b)].

In order to make the problem solvable, we truncate both *m* and *m̂* so that they all have a finite integer value from -*n* to *n*. Equations (4) and (5), as *m̂* changes from -*n* to *n*, can be put into matrix forms

K_{z} field in each layer can be transformed from one coordinate system to another in exactly the same way. Fields in two adjacent layers across interface Γ_{1} can then be related as

where some representative submatrices/subvectors are defined as

Derivation of matrix elements like *m*
_{11} in Eq. (9) can be done similarly as in [1], though they have chosen a slightly different field expansion basis. Equation (8) can be used iteratively to relate the field components in the innermost layer to those in the outermost layer for an *N*-layered Bragg fiber. Finally, with the boundary conditions of **B**
^{E1,1} = **B**
^{K1,1} = 0 and **A**
^{EN,N} = **A**
^{KN,N} = 0, we can deduce a matrix equation as

where matrix **M**
_{0} has a size of 2(2*n*+1)×2(2*n* + 1), and its elements depend on *β*. Equation (10) has nontrivial solution only when the determinant of **M**
_{0} goes to zero, hence *β* solutions (valid modes) can be found by searching for roots of the function ℱ(*β*) = det|**M**
_{0}|. The corresponding mode field components **A**
^{Ei,i}, **B**
^{Ei,i}, **A**
^{Ki,i}, **B**
^{Ki,i} can be solved in turn.

#### 2.2. Convergence test and comparison

We have carried out a convergence test for a W-fiber with a displaced core (Fig. 2, inset). The physical parameters of the fiber are described in the caption of Fig. 2. As the outer cladding has the same refractive index as the core, modes supported by the fiber are leaky due to evanescent coupling. We focus on the *y*-polarized HE_{11}-like fundamental mode. Figure 2 shows that both real and imaginary parts of the mode effective index (*n*
_{eff} = *β*/*k*
_{0}) converge quickly as *m* increases. Result calculated with *m* = 6 has a relative error of 6.6370e-9 in real part and 0.0017 in imaginary part as compared to the converged value. Wavelength is at 1.55*μm* for the test.

To validate the TMM, a full-vector finite-difference method (FDM) [7] is used to derive the *y*-polarized HE_{11}-like mode in the same W-fiber, with the offset between two interfaces Δ*O* varying from 0 to 3*μ*m. The results are compared with those calculated using the TMM in Fig. 3. The real part has a consistent relative error of 1.9e-5 between the two methods [Fig. 3(a)]; the imaginary part has a relative error of 0.024 to 0.095 between the two methods [Fig. 3(b)]. The nearly constant discrepancy in real part suggests that we might have scaled the fiber structure when the index profile is input into the FDM program as an image. *m* = 11 has been used for TMM calculations. For FDM calculations, domain is at 27×27*μ*m, meshed with 300×300 grid points. Perfectly-matched layer is of a width equivalent to 12 grid points. Wavelength is at 1.55*μ*m. Results from a commercial full-vector mode solver (FemSIM v.1.0 from RSoft Design Group, Inc.) based on the finite-element method (FEM) are also presented in Fig. 3. The FEM results agree in general with those calculated using TMM and FDM, despite there are some convergence jerks observed in the real part.

## 3. Simulation results

Although a Bragg fiber can be made of any number of materials, research efforts in past few years are particularly put upon those made of two or three materials. Three-material Bragg fibers are preferable in realizing lightwave propagation in air [3], as the core can be left empty while two solid materials are stacked in pairs to form the cladding. In a two-material Bragg fiber, the core material is the same as one of the cladding materials [8]. Though ideally speaking, such fiber can’t be fabricated with an air core, Vienne *et al*. have nevertheless demonstrated a hollow-core air-silica fiber in close resemblance to a Bragg fiber [9]. However, both types of fibers have been observed to experience much higher leakage losses than theoretical expectations [3, 8, 9], even when material absorption is taken into account [3]. In this section, we give separate studies on both types of Bragg fibers. Their waveguiding abilities are examined when their cladding layers are no longer uniform along the azimuthal direction. Such deformation is created by displacing the cladding material interfaces while the interfaces are kept circular. Attention will be given to the most promising TE_{01} mode.

#### 3.1. Three-material Bragg fiber

In this subsection, we study the modal property of a hollow-core Bragg fiber whose cladding is formed by ten bi-layers of two dielectric materials. The refractive indices are chosen as 3.0 and 1.5. Bragg reflection condition,

is used to estimate the layer thickness *d*_{i}
. We set *q* = 0 to facilitate guidance with the primary cladding bandgap. It can be calculated using Eq. (11) that widths of the high- and low-index cladding layers should be around 140nm and 350nm, respectively, in order for the minimum leakage loss to occur at 1.55*μ*m wavelength. Core radius is set as 10*μ*m.

First we use the traditional TMM [2] to solve for ten modes guided by the unperturbed Bragg fiber. In this paper, we categorize the modes in such a waveguide into three major groups: transverse-electric (TE) modes, transverse-magnetic (TM) modes, and mixed-polarization (MP) or hybrid modes. All modes are denoted with two subscripts *mn*, where *m* represents the az-imuthal quantum number, and *n* represents its order in terms of energy (from low to high) in the particular mode class as characterized by *m*. TE and TM modes automatically have *m* = 0. The dispersion curves (*n*
_{eff} versus *λ*) and loss curves of the ten modes are shown in Fig. 4(a) and (b), respectively. It is found that the leakage loss of the TE_{01} mode (~0.066dB/km at 1.56*μ*m wavelength) is about five orders smaller as compared to the TM and MP modes. The next-lossier mode is the TE_{02} mode, which still experiences 2.9 times higher leakage loss than the TE_{01} mode. Therefore, theoretically speaking it is possible to achieve single-mode, single-polarization transmission in such a Bragg fiber.

However, it is also observed in Fig. 4(a) that the TE_{01} mode crosses (or accidentally degenerates with) a few MP modes in their dispersion curves, namely the MP_{41} mode at 1.495*μ*m, the MP_{31} mode at 1.544*μ*m, and the MP_{12} at 1.578*μ*m. They do not interact with each other at the crossings as long as the waveguide has continuous rotational symmetry. This is inarguable, since the group theory allows them to be solved separately in different classes as if they belong to different physical problems [6]. In the following we will introduce a perturbed Bragg fiber which has only *C*
_{1} symmetry. Based on group theory, modes supported by such a waveguide cannot be separated into more than one class. Will accidental degeneracies be allowed for modes in the same McIsaac class? So far we can’t find a definite analytical answer to this question. Here we only provide our findings by explicitly solving the Maxwell’s equations. And the results are intriguing.

Perturbation is introduced by offsetting interface rings (except the first interface) by a constant amount Δ*O*. The direction of the offsets, from the second interface onwards, is along +*x*, +*y*, -*x*, -*y*, +*x* …, so on and so forth. Such a perturbed Bragg fiber has only *C*
_{1} rotational symmetry. Firstly, we let Δ*O* = 20nm, which is about 5.7% as compared to the cladding period. One half of the perturbed Bragg fiber can be observed in Fig. 5(a). Dispersion and loss curves of the TE_{01} mode^{1} are calculated using the extended TMM method with *m* = 8. Convergence is assured in all calculations in this paper. As shown in Fig. 6, whereas the effective index for the TE_{01} mode has almost no apparent change, its radiation loss increases by 53% (examined at *λ* = 1.75*μ*m). Besides, there are certain spectral regions where the radiation loss increases sharply. Three most obvious spikes are noticed around 1.35*μ*m, 1.52*μ*m, 1.58*μ*m, and 1.90*μ*m wavelengths. We then increase Δ*O* to 40nm. One half of the perturbed fiber is plotted in Fig. 5(b). The displacement of cladding interfaces has severely spoiled the cladding periodicity. In Fig. 6, still, there isn’t any apparent shift in the dispersion curve. But the overall leakage loss is observed to increase by 660% (examined at *λ* = 1.15*μ*m), and those abrupt high-loss regions appear more prominent. From Fig. 6(b), it is worth mentioning that, when Δ*O* increases, the bandgap region remains in its position, though it shrinks in width.

We are not surprised by the overall rising of the leakage loss at the presence of a non-zero Δ*O*. Qualitatively, it is owing to the degradation of the cladding bandgap. However, sudden increases in loss value by several orders at various wavelength positions may require some different explanation. Two possible factors may contribute to such loss spikes: (1) extra cladding modes appear, which break the previously continuous cladding bandgap into pieces; (2) interactions happen between the well-confined TE_{01} mode and other heavily leaky TM or MP modes. We concentrate on the loss spikes within the TM (full) bandgap region. In fact, a scan with a 0.2nm step size in wavelength around 1.58*μ*m reveals there are in fact two loss spikes, as shown in Fig. 6(c). Three loss spikes at 1.516*μ*m, 1.572*μ*m and 1.578*μ*m are reasonably close to the dispersion crossings between the TE mode and three MP modes (the MP_{41}, MP_{31} and MP_{12} modes) of the unperturbed fiber in Fig. 4(a). However, to confirm it is the second factor which causes the loss spikes, we need to carry out some finer mode traces. We focus on the perturbed fiber with Δ*O* = 0.04*μ*m. The wavelength step size of the scans is now as small as 0.002nm in order to well resolute the tip regions of the loss spikes. Fig. 6(d) and (e) show that the loss curves around 1.516*μ*m and 1.572*μ*m wavelengths are smooth. This dismisses the anticrossing behavior of the TE_{01} mode with another mode at these two wavelengths. In Fig. 7(a) and (b), we show the electric field components of the TE_{01} mode at two loss summits in Fig. 6(d) and (e), respectively. The E_{t} fields are observed not strictly parallel to the core-cladding interface [not quite obvious in Fig. 7(a) under the current presentation]. A non-zero *E*_{z}
component also appears. In fact, a close study of this minor longitudinal field component can tell us what is behind the appearance of loss spikes. In Fig. 7(a), we notice the *E*_{z}
field has an 8*π* azimuthal phase variation, which corresponds to an azimuthal quantum number of 4. This confirms that it is the MP_{41} mode which causes the high loss near 1.516*μ*m wavelength. Similarly we notice the *E*_{z}
field in Fig. 7(b) has an azimuthal quantum number of 3, which confirms that it is the MP_{31} mode which causes the high loss near 1.572*μ*m wavelength.

Now we focus on the loss spike at 1.578*μ*m wavelength. At this point the TE_{01} mode experiences the highest loss within the cladding’s full bandgap. Therefore we expect its interaction with some other mode (supposedly the MP_{12} mode) is also the strongest. The dispersion as well as the loss curves of the TE01 mode around 1.578*μ*m are plotted in Fig. 8(a) and (b), respectively. Behaviors of the two MP_{12} modes (they are previously a degenerate pair corresponding to the MP_{12} mode in Fig. 4) are also traced for a clear knowledge of the interaction. It is quite obvious that the dispersion crossings between the TE_{01} mode and the two nearly degenerate MP_{12} modes are not avoided. However, such hard-crossings do not leave the TE_{01} mode undisturbed. The dispersion curve of the TE_{01} mode undergos a slight fluctuation as it crosses one of the MP_{12} modes (solid-blue curve), and their loss spectra allure each other at the point of dispersion crossing. It is interesting to notice that the other MP_{12} mode (dashed-blue curve) does not cause any obvious fluctuation on the TE_{01} mode, either in the dispersion or loss spectrum. In Fig. 9, we have plotted eight TE_{01} modes at wavelength positions as indicated by the black markers in Fig. 8(b). The TE_{01} mode does an “amusing” transformation, and turns itself from a left-circularly-polarized mode (mode *a*) into a right-circularly-polarized mode (mode *h*). TE_{01} mode aside, the two nearly-degenerate MP_{12} modes are observed to switch their roles around the same wavelength position. Notice the two blue dispersion lines in Fig. 8(a) are not parallel. They become closer around 1.5777*μ*m wavelength. Such intermodal behavior is known as modal anticrossing. In Fig. 10, we have shown four MP_{12} modes as indicated by the blue markers in Fig. 8(b).

Figure 11 shows the variation of radiation loss of the TEoi mode at 1.55*μ*m and 1.75*μ*m wavelengths as Δ*O* increases. These two wavelengths are not at dispersion crossings, therefore the curves in Fig. 11 reflect the degree of cladding bandgap degradation as perturbation increases.

#### 3.2. Two-material Bragg fiber

Though air-silica Bragg fiber is only possible in practice in an imitated form [9], we nevertheless in this subsection pay attention to such a “single-material” waveguide, owing to the fact that it might be possible to achieve low-loss air guiding. The fiber considered, for operation around 1.55*μ*m wavelength and in an unperturbed form, has an air core with 10*μ*m radius, and three silica walls with 0.37*μ*m thickness, which are separated by air gaps with 4.13*μ*m thickness. Equation (11) can still be used to predict the silica cladding layer width. However, it fails for the air cladding layers. In fact, in planar-approximation, the air layers should have half of the core width in order to meet the anti-resonance condition. In a circular Bragg fiber, the air layer width for the least-leaky TE guidance is found to be much smaller by trial-and-error. The number of cladding period is three, with the outermost infinite cladding layer be silica. Silica and air indices are assumed to be 1.45 and 1 respectively. Using the traditional TMM, the leakage loss for the TE_{01} mode in the unperturbed Bragg fiber is found to be as small as 0.8dB/km at 1.14*μ*m wavelength [Fig. 12(b)].

As a first perturbation analysis, we likewise offset the interfaces (except the innermost interface), by a Δ*O* value of 50nm. The resulted fiber cross section is shown in Fig. 13(a). The offset is only about 1.1% of the cladding period, and 13.5% of the silica wall thickness. In Fig. 12(b), it is observed that the transmission wavelength range for the TE_{01} mode becomes narrower, obviously due to shrinking of the cladding bandgap. However, it is interesting to know that the shrinkage is more severe on the short-wavelength side. The lowest loss is found to be 1.25dB/km at 1.2*μ*m wavelength. Similar to that in the three-material Bragg fiber, we have noticed an apparent high-loss spectral region near to 1.53*μ*m wavelength. There is another weak spike at 1.58*μ*m. We then increase Δ*O* to 100nm in a second calculation. The resulted fiber cross section is shown in Fig. 13(b). The cladding bandgap is noticed to experience further shrinking (Fig. 12). The nominal minimum loss is found to be 5.9dB/km at 1.34*μ*m wavelength. Two loss spikes at 1.53*μ*m and 1.58*μ*m become more prominent.

It has been noticed that in such a two-material unperturbed Bragg fiber, there exist crossings among dispersion curves of the TE_{01}, MP_{21} (a degenerate pair) and TM_{01} modes within the cladding bandgap [10]. These modes are called the second-order modes in conventional step-index fibers. The dispersion curves of these four modes are almost overlapping with each other. We have presented the crossings schematically in Fig. 14 for the current air-silica Bragg fiber. The TE_{01} mode encounters the MP_{21} (a degenerate pair) at 1.522*μ*m, and then the TM_{01} at 1.579*μ*m. A close look at the loss spectrum of TE_{01} mode of the perturbed fiber (Δ*O*= 0.10*μ*m) reveals that the high-loss regions are very close to these two dispersion crossing points. Evocatively, we suspect that, when the fiber lack of *C*
_{∞} symmetry, the TE_{01} mode might interact with the MP_{21} and TM_{01} modes at wavelengths where their dispersion curves are supposed to cross, and therefore sudden high loss spikes are resulted.

To confirm this, we made a finer trace of the TE_{01} mode at a 0.2nm step size in wavelength. We have focused on the perturbed fiber with Δ*O* = 0.10*μ*m. The trace starts from a TE_{01} mode at 1.45*μ*m wavelength, and ends with a TM_{01} mode at 1.7*μ*m wavelength. Figure 15 shows the loss spectrum of the trace. An examination of the modal fields tells the mode undergoes two major transitions, i.e., from a TE_{01} mode to a MP_{21} mode around 1.524*μ*m wavelength, and from a MP_{21} mode to a TM_{01} mode around 1.640*μ*m wavelength. The transition positions agree quite well with the crossing points in Fig.14. Such mode transitions imply that anticrossings happen among the second-order modes. Besides the two major transitions, we have also noticed two MP_{21} modes anticross between themselves for three times (Fig. 15 inset). In Fig. 16, we give eight modes as indicated in by the markers in Fig. 15. At major anticrossing points, the mode appears to be a linear combination of two anticrossed modes, which is quite evident in modes shown by Fig. 16(b) and (g).

## 4. Discussion and conclusion

In conclusion, we have extended the transfer matrix method to deal with multilayered circular fibers whose dielectric interfaces are circular, but not concentric. We believe the extended method is computationally much less expensive as compared to full numerical methods. With this method, we have studied guidance property of the TE_{01} mode in Bragg fibers which lack of continuous rotational symmetry. It is noticed that, when the waveguide has only *C*
_{1} symmetry, the least-leaky TE_{01} mode interacts with other modes which have TM field components, at wavelength regions where their dispersion curves are supposed to cross each other. Such interactions, appearing either as a hard-crossing or an anticrossing in dispersion curves, will result in very high losses at those particular spectral regions.

We have noticed that in the three-material Bragg fiber three loss spikes of the TE_{01} mode due to crossings by MP_{41}, MP_{31} and MP_{12} modes are increasing in height. It might imply that, for a perturbed fiber the interaction at a dispersion crossing is strong for two modes with very close azimuthal quantum number, and such interaction gets weak for the two modes with distinct azimuthal quantum numbers. However, we can’t predict at this moment if the interaction strength varies with the radial field distribution according to a similar rule. Further study need to be carried out to confirm all these.

Two terms, “anticrossing” and “hard-crossing”, are used to address the interactions between two modes. However, special attention should be paid to the “hard-crossing”. Modes studied in this paper are all leaky. A crossing in the real part of their *n*
_{eff} (or *β*) values does not necessarily mean they are degenerate to each other. A true degeneracy happens only when both real and imaginary parts of the *n*
_{eff} values are the same for two modes at a particular wavelength.

Our study has concentrated on, as theoretically limited by the proposed transfer matrix method, Bragg fibers with circular but nonconcentric interfaces. The effects of exact deformations in fabricated Bragg fibers are not studied, which would be at the cost of much more intensive numerical calculations. In fact, as there are so many types of deformations which can possibly happen during fabrication, analysis of any particular structure usually does not provide more heuristic information than what we have presented. It is also noticed that some perturbation methods exist for analyzing electromagnetic field propagation in fibers with shifted material interfaces [2, 11, 12]. A perturbation method normally first solves for the solution of an unperturbed problem, which is then corrected by some additional manipulation of the obtained ideal solution and the structural irregularities. Such methods only help when the problem lacks of analytical solution, and at the same time is too complex for a numerical calculation. Here we directly solve for the propagating modes in aperiodic Bragg fibers with a semi-analytical method, at almost any desired accuracy. We believe the result reported in this paper cannot be achieved, at least not at the same efficiency, by using the perturbation methods.

## Acknowledgement

This work is partially supported by the Agency for Science, Technology and Research (A^{٭} Star), Singapore.

## Footnotes

^{} | ^{1} In fact, the mode is now a hybrid one. However, for simplicity we borrow names of the modes in an unperturbed Bragg fiber to denote those in a perturbed one owing to their proximity in modes appearances. |

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