1. Introduction

A number of papers published in recent years [1–6] present the results of both experimental and theoretical investigations of macro-bending losses in the index-guided photonic crystal fibers (PCFs). The simplest method of estimating bending losses in PCFs uses the so called equivalent step index (ESI) approximation of the holey fiber [1–5]. In this approach, the complicated structure of the index-guided PCF is replaced by an equivalent step index fiber and then the methods developed for conventional fibers are used to calculate the macro-bending losses in PCFs. The precision of this approach is limited due to difficulties in choosing the parameters necessary to assign the appropriate ESI profile [1–4]. To avoid this problem, a theoretical model based on Hermite-Gauss approximation of the mode profile has been derived in [6, 7]. It retains full information on the complicated structure of PCFs, and therefore allows to study the effect of fiber angular orientation on the bending losses [7].

In the theoretical investigations of bending losses in PCFs reported so far, it is assumed that the holey region surrounding the core extends to infinity. Such an assumption causes that the theoretical bending losses change monotonically vs. 1/λ and 1/R, where λ and R stand respectively for the wavelength and the bend radius. The assumption of unlimited holey cladding in PCFs is far from reality. In fact, three regions can be distinguished in the cross section of the typical index-guided PCF: the core, the holey part of the cladding surrounding the core, and the outer part of the cladding made of solid silica. The more adequate structure of the ESI fiber should therefore contain two-layer cladding with lower effective index in the inner region. Very recently, such a model of equivalent double-clad SI fiber has been successfully applied to estimate confinement losses in PCFs using simple analytical formulas [8].

There have been many papers published so far [9–13] devoted to the analysis and measurements of the bending losses in conventional double-clad fibers. The bending losses in such fibers experience oscillations superimposed on monotonic loss increase versus λ and 1/R. It is well documented that oscillatory character of the bending losses in the double-clad conventional fibers is caused by light coupling between the fundamental mode and the gallery of the cladding modes that arise due to light reflection from the interface between outer and inner cladding.

In this article we present the results of numerical simulations indicating for the first time that similar phenomenon exists in the PCFs due to reflection of the radiative component of the fundamental mode from the solid part of the cladding. To demonstrate the effect of the oscillations, we calculated the bending losses within a short wavelength bend loss edge in the large core PCF. Our results show that both the depth and the period of loss oscillations depend significantly upon angular orientation of the bent fiber.

2. Calculation method

To illustrate the impact of the coupling effect on the bending loss characteristic, we have performed the calculations for a large core PCF (LMA-20) produced by Crystal Fibre S/A. According to the information provided by the manufacturer, we assumed the following geometrical parameters of the analyzed fiber: pitch distance Λ= 13.2μm, fill factor d/Λ = 0.485, and number of holes’ layers equal to 7. The bending losses for identical fiber were determined earlier using equivalent step index fiber method [5]. In order to demonstrate the influence of the fiber angular orientation on loss characteristics, the calculations were carried out for the fiber bent respectively in xz- and yz-plane, see Fig. 1.

In our analysis, we used a fully vectorial modal approach employing the finite element method (FEM) based on the edge/nodal hybrid elements and anisotropic perfectly matched layers (PML) boundary conditions. The curved fiber was replaced by a straight one with equivalent refractive index distribution [14]:

where p = x or y, depending on the bending direction, and R stands for the effective bend radius, which differs from the actual (experimental) bend radius R_exp by a material dependent elastooptic correction factor that represents the change in refractive index of silica glass induced by axial stress [15, 16].

As the equivalent refractive index in the bent fibers is no longer constant across the absorbing layer, the conventional anisotropic PML approach [17] has to be modified to be applicable to this problem. Recently, it has been demonstrated in [18, 19] that the full-vectorial wave equation in the anisotropic PML region with space dependent refractive index may be presented as:

where k ₀ is a free space wave number, and E̿ is a new electric field vector given by:

and r̄ represents a coordinate converted position vector:

where s_x, s_y, and s_z are complex stretching variables and ^T denotes a transpose. The matrices [S], [μ]_PML, and [ε]_PML are defined as follows:

where n(r̄) and μ(r̄) indicate respectively the coordinate converted equivalent refractive index and the coordinate converted magnetic permittivity. One should notice that, if n and μ don’t change in the direction perpendicular to the PML, the relations (6) and (7) are the same as in a conventional anisotropic PML.

Applying the approach summarized above, we surrounded rectangular computational domain with the PML composed of three regions with different complex stretching variables _sx, s_y, and s_z, see Fig. 1. For the fiber bent in xz-plane and the parabolic profile of attenuation, the complex stretching variables in the PML can be expressed in the following way:

and

respectively in the regions I and II that are indicated in Fig. 1(a), where α is the attenuation parameter. In the corners of the PML (region III) s_x and s_y are both complex and depend upon respective spatial coordinate similarly as in the above equations.

For the fiber bent in xz-plane, the gradient of equivalent refractive index is parallel to x-axis, see Fig. 1. In such a case, according to eqs. (1) and (9), the following relation holds in the region II of the PML:

while in the regions I and III, we substitute into eq. (7) the coordinate converted distribution of equivalent refractive index:

Corresponding relations for bending in yz-plane can be easily derived after replacing x with y coordinate in eqs. (8–11).

Dividing one half of the fiber cross section into a mesh of 100K triangular elements and using the finite element method with appropriate symmetry conditions, we determined the complex propagation constant β for the fundamental modes of respective polarization, and finally the bending losses expressed in dB/m:

The attenuation coefficient α and the PML thickness d_PML used in our numerical analysis were adjusted by gradually increasing each parameter until the value of the calculated loss became stable. The optimum values of the PML parameters found in this way were α = 5 and d_PML = 6 μm. Furthermore, each PML region was divided into 6 layers (each 1 μm thick) parallel to the respective edge of the computational domain. It was assumed that the space dependent complex part of the stretching variable appearing in eqs.(8, 9) is constant within each layer.

As it is shown in Fig. 1, the PML is in contact with the solid part of the cladding. Such a location of the rectangular PML causes that in the present analysis we disregard the effect of the coupling between the fundamental mode and the cladding modes arising in the solid part of the cylindrical cladding. This simplification is justified by the fact that, due to greater distance, the coupling to modes located in the solid part of the cladding is much weaker than to modes propagating in the holey region.

 

Fig. 1. Computational domain limited with rectangular PML for the fiber bent in the xz-(a) and yz-plane (b), z-axis overlaps with the fiber symmetry axis. The geometrical parameters of the analyzed fiber are: pitch distance Λ = 13.2 μm and fill factor d/Λ= 0.485.

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3. Results of calculations

The results of the loss calculations for bends in xz- and yz-plane vs. both the bending radius and the wavelength are displayed in Fig. 2 and Fig. 3, respectively. The calculations carried out for orthogonal polarizations of the fundamental mode showed that, due to hexagonal symmetry of the analyzed fiber, the bending losses are practically polarization independent. Therefore, in all figures, we display only the results obtained for the y-polarized fundamental mode.

In our vectorial analysis, we take into account limited dimensions of the holey region. Therefore, the obtained results are significantly different from those reported in earlier publications [1–6] and indicate the existence of loss oscillations instead of the monotonic loss increase vs. 1/R and 1/λ predicted by simplified theoretical models. As it is shown in Fig. 2, the oscillations are especially well pronounced for the bends in xz-plane.

 

Fig. 2. Bending losses for the y-polarized fundamental mode calculated versus bending radius using the fully vectorial method. The calculations were performed for the xz (a) and yz bending plane (b) at λ = 0.83 μm. Green line indicates losses determined using simplified analytical formula from ref. [5]. Solid lines 1, 4 and 9 correspond to the cladding modes with m = 1 symmetry arising for xz-bent.

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Fig. 3. Bending losses for the y-polarized fundamental mode calculated versus wavelength using the fully vectorial method. The calculations were performed for the xz (a) and yz bending plane (b) at R = 80 mm. Green line indicates losses determined using simplified analytical formula from ref. [5].

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The physical origin of the oscillations of bending losses in PCFs is related to the coupling between the fundamental mode and the gallery of the cladding modes (symmetrical with respect to the bending plane) arising due to light reflection from the boundary of the holey part of the cladding. For both directions of the bending, we identified and enumerated the first 10 loss peaks (Fig. 2) and the corresponding cladding modes responsible for the coupling losses. The distribution of the dominant component of the electric field in some of those cladding modes is shown in Fig. 4. The field distributions were calculated for λ = 0.83 μm and the bend radii corresponding to peak losses. As it is shown in Fig. 4(a), for the bends in xz-plane, the radiative component of the fundamental mode is mostly reflected from the flat part of the boundary of the holey cladding. In such a case, the cladding modes of the lower order have twofold plane symmetry and, for the sake of comprehensiveness, we will refer to them as CL_km, where k, m represent the number of field extremes respectively in x- and y-direction.

The highest coupling efficiency occurs when the fundamental and the cladding modes propagate with the same velocity. In Fig. 5, we show the dependence of the effective indices of the fundamental mode and 10 cladding modes of the lowest order calculated vs. the bending radius at λ = 0.83 μm. One can easily notice that indeed the highest peaks in the loss characteristics shown in Fig. 2 occur for such bending radii at which the effective indices of the cladding modes match that of the fundamental mode.

 

Fig. 4. Distribution of the dominant component of the electric vector in selected y-polarized cladding modes. The calculations were performed for the xz (a) and yz-bending plane (b) for radii assuring peak losses, λ = 0.83 μm. Modes’ numeration is the same as in Fig. 2.

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For the bends in xz-plane, the coupling between the guided mode and the cladding modes of the class CL_k1 is the most efficient. By increasing k, the field distribution in the cladding modes approaches the fiber core, thus increasing the overlap integral with the fundamental mode, see Fig. 6(a). As shown in Fig. 5, the phase matching condition can also be met for the cladding modes with m>1, however, there is no significant coupling observed at these bending radii, as for example for the modes CL₁₃ (2) and CL₁₅ (3) indicated in Fig. 2(a). Small coupling to these modes is caused by the field oscillations in y-direction that results in diminishing the overlap integral with the fundamental mode. Therefore, the principal peaks in the loss characteristic for bend in the xz-plane, indicated in Fig. 2(a) with solid lines, are caused by light coupling between the fundamental mode and the cladding modes with an m = 1 symmetry (CL_k1). The coupling to other cladding modes results only in secondary oscillations of the loss curve.

 

Fig. 5. Effective indices of the fundamental mode (n_f) and the cladding modes calculated vs. bending radius for bent in the xz-plane (a) and yz-plane (b), λ = 0.83 μm. Modes’ numeration is the same as in Fig. 2.

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Fig. 6. Intensity distribution in the fundamental mode (lower half of the fiber cross-section) and in the cladding modes number 9 and 6 (upper half) calculated respectively for bent in the xz-plane (a) and yz-plane (b). The calculations were carried out for λ = 0.83 μm and radii assuring peak losses. A logarithmic scale with a step of 1.5 dB between successive contour lines and independent normalization of both intensity distributions to 0 dB were used to better show the overlap regions.

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When the fiber is bent in the yz-plane, the radiative component of the fundamental mode is reflected from the corner of the holey region. As shown in Fig. 4(b) and Fig. 6(b), in such a case, the cladding modes don’t hold twofold plane symmetry, have lower size, and arise in greater distance from the core compared to the xz-bends. It causes that the overlap integral and in consequence the coupling coefficient between the fundamental mode and the cladding modes are much lower than for the xz-bends. As a result, for the bends in the yz-plane, we observe only relatively weak oscillations in the loss characteristics.

For comparison, we display in Fig. 2 and Fig. 3 the bending losses calculated using the simplified analytical formula from ref. [5] that was derived using the ESI fiber approach. In general, the bending losses predicted by the fully vectorial method are higher than those predicted by the ESI fiber model. This discrepancy is most probably related to the fact that the ESI model neglects the loss mechanism related to the coupling to the cladding modes, while our method accounts for this effect.

4. Conclusions

Using the fully vectorial FEM combined with the PML boundary conditions, we have demonstrated that the coupling between the fundamental mode and the gallery of the cladding modes causes oscillations in the dependence of the bending losses upon 1/R and 1/λ within the short wavelength bending loss edge in the large core PCFs. The amplitude and periodicity of these oscillations depend on the angular orientation of the fiber bent and is particularly well pronounced when the radiative component of the fundamental mode is reflected from the flat boundary of the holey cladding.

The oscillatory behavior of the bending losses in conventional double-clad fibers has been well known for many years from several theoretical and experimental publications [9–13]. To our knowledge, this is the first report indicating the existence of similar effects in PCFs. The results of our theoretical analysis require experimental confirmation. As the amplitude of the loss oscillations and their periodicity depend on the direction of the bending, one should carefully control in future experiments the angular orientation of the fiber in order to avoid averaging of the loss peaks over the fiber length. It is worth mentioning that at least in some recently published works presenting the results of the bending loss measurements in PCFs, for example in ref. [4], Fig. 2(d), there are clearly visible, unexplained loss oscillations that are similar to the predictions of our analysis.