## Abstract

The cause of fundamental core-mode leakage in a tapered photonic crystal fiber is examined in terms of modal coupling from the core-mode to the ring modes. Experimental data are compared to predictions. The main features of the transmission are convincingly explained by the proposed model.

© 2007 Optical Society of America

## 1. Introduction

The potential of microstructured Optical Fibers (MOFs) was first investigated more than three decades ago [1]. Then, following the successful fabrication of the photonic crystal fiber (PCF) [2], a wide range of MOFs with different configurations, sizes and shapes of holes have been designed. Light guidance is provided either by a total internal reflection (TIR) or by a photonic bandgap effect. MOFs guiding by the TIR effect consist of a solid core, usually silica, the holes of the surrounding layer resulting in an average lower index cladding. These fibers have proven useful as a platform for building a range of novel devices [3]. However, the fuse and tapering technology, which is extensively used to manufacture components from standard two-layer fibers, appeared to be inadequate in the case of MOFs. It was indeed observed that, while tapering, the power leaked from the fundamental core mode (FCM) even for gentle slopes such as those used to manufacture fibre couplers [4, 7]. In these articles, the leakage was qualitatively explained by the coupling of the FCM with high index ring modes. However, the exact mechanism behind the coupling and also the identity of the high index ring modes involved was not specified. Up to now, tapering MOFs has been mainly used to precisely tailor the fiber properties, such as the mode field diameter [5, 6].

The power leakage was later explained more in details in terms of the FCM cutoff using a model which has been extensively developped in a recent article [8]. According to this model, the leakage would be due to the coupling of the FCM with the continuum of radiation modes of an infinite extent outer silica layer. Although this model correctly predicts the critical wavelength *λ*
_{c} beyond which a significant amount of power is lost by the FCM, it fails to explain the oscillations which are observed in the tapered fiber transmission, in particular at wavelengths longer than *λ*
_{c}. This is due to the fact that this model does not take into account the limited extent of the fiber cladding, which is known to play an essential role in the behavior of tapered standard fiber. In this paper, we provide a more detailed description of the FCM power leakage of a tapered PCF, taking into account that the tapered region is stripped of the protective jacket and immersed in the air.

In the following section, we first introduce, for the sake of simplicity, an equivalent depressed inner-cladding (DIC) fiber to approximate the PCF index profile and explain the leakage phenomenon in terms of power splitting between modes through coupling phenomena. Section 3 presents a calculation of the modal effective indices and adiabacity criteria of the microstructured fiber under test, using a finite difference method. In Section 4, we show an example transmission of a tapered PCF calculated by using the coupled equations formalism. Section 5 presents experimental results regarding the tapering of a real PCF with a geometry and an index profile close to the ones used in the simulations. Finally, in the last section, we discuss the differences between the simulated and experimental results.

## 2. The DIC approximation

A Depressed Inner Cladding (DIC) fiber model is used to represent our PCF. Being inherently a circular fiber, this ideal fiber is not expected to predict all the PCF properties, but it gives useful insight before more accurate but also more lengthy calculations are done [9]. Figure 1(a) shows the actual PCF used in our experiments and Fig. 1(b) the equivalent DIC index profile used as an approximation. The core radius has been chosen to be Λ/√3=2.3*μ*m according to Ref. 9; the inner and outer cladding radii are 29.1*μ*m and 62.5*μ*m respectively, as shown in Fig. 1(b). The core, being of silica is assumed to have a refractive index value *n*
_{co}=1.444 at *λ*=1550nm. The equivalent inner cladding index value, *n*
_{ic}, is not easy to determine in our case as it depends on the mode field confinement, due to the non uniformity of the holes of the actual fiber (see Fig. 1(a)). The holes radii *d* vary from 0.9*μ*m (for the first hole layer close to the core) to 1.1*μ*m (close to the outer cladding). Assuming all the holes identical to the smallest ones, one would obtain *n*
_{ic}=1.372, whereas assuming they are identical to the largest ones would lead to a value *n*
_{ic}=1.338. On the other hand, the exact value used for calculation is not of crucial importance as we only expect from the DIC fiber model a qualitative prediction of the actual fiber behavior. An intermediate *n*
_{ic}=1.359 was chosen.

Strictly speaking, this type of fiber is not a single-mode structure. It is sometimes referred to as a coaxial coupler [11]. In our case the external medium is the air and the structure as a whole guides supermodes [12]. The supermodes of circular symmetry are labeled LP_{0m} according to the number of radial nodes.

Figure 2(a) is a plot of the effective indices of these LP_{0m} supermodes as functions of the Inverse Taper Ratio, defined to be ITR=*ρ*/*ρ*
_{0}, where *ρ* and *ρ*
_{0} stand respectively for the fiber current and initial radii. Also plotted in this graph is the effective index of the LP_{01} fundamental core-mode (FCM) of a two-layer fiber having its core radius and index identical to those of the DIC fiber and an infinite single cladding of index *n*
_{ic}. As seen from this graph, when one excites this core-mode of an untapered fiber, i.e., for ITR=1, one actually mainly excites the LP_{09} supermode, which shape and propagation constant coincide with the two-layer fiber LP_{01} FCM for ITR values ranging from 1 to ≈ 0.53. This graph may alternatively be viewed as the crossing of the effective indices of the ring structure with that of the FCM of the two-layer fiber. The presence of the core perturbs the ring modes and lifts the degeneracy, which results in turn in anticrossing instead of crossing of the modal effective indices.

When the fiber is tapered, the FCM remains LP_{09} until the first coupling region LP_{08}-LP_{09} is attained for ITR ≈ 0.53. The two modal effective indices are almost identical, making the supermodes almost degenerate for this particular value of the ITR-parameter. In other words, the coupling is so strong that the core-mode evolves as if there was no outer cladding ring and its effective index follows that of the two-layer fiber. In terms of supermodes, the so-called FCM is then labeled LP_{08}.

Tapering further the fiber makes it experience similar situations whenever anticrossing of the modal indices are encountered. However, as seen in Fig. 2(a), the gap between adjacent modal effectives indices in the coupling regions becomes larger and larger as the ITR decreases. This may be viewed as the consequence of the FCM spreading toward the outer cladding region. To give a more quantitative description of these coupling effects, we calculated the normalized coupling coefficients *C*¯_{jℓ} between the LP_{0j} and LP_{0ℓ} modes

where *β*
_{j} and *β*
_{ℓ} are the propagation constants of the modes with *ϕ*̂_{j} and *ϕ*̂_{ℓ} their respective normalized fields. Only LP_{0ℓ} modes having the same circular symmetry as the core-mode may be involved in the coupling process and only adjacent modes can efficiently couple (ℓ=*j*+1). The coupling coefficient *C*
_{jℓ} is equal to the normalized coupling coefficient multiplied by the normalized slope :

The corresponding weak power transfer adiabacity criteria is respected whenever [13]

The delineation curves for the relevant pairs of modes, i.e., the value of
$\frac{{\beta}_{j}-{\beta}_{\mathit{\ell}}}{{\overline{C}}_{\mathit{j\ell}}}$
as functions of ITR are plotted in Fig. 2(b). As expected from the anticrossing regions of the effective index curves of Fig. 2(a), the resonant coupling phenomena are less and less effective as ITR decreases. In Fig. 2(b) is also represented the normalized slope of a typical tapered PCF fabricated in our lab. The situation for the three-layer DIC fiber is similar to that of a two-layer fiber as long as the FCM does not expand into the outer cladding: although changing its label (from 09 to 08, then to 07…), the FCM evolves ignoring the ring structure. For ITR values smaller than 0.3, it clearly can no longer be the case and transfer of power to the ring is unavoidable. In terms of supermodes, the transfer of power from LP_{0j} to LP_{0j-1} is no longer complete: part of the power is still transfered to LP_{0j-1} but part of it may still be guided by LP_{0j}. This phenomenom has been known for a while and, from a practical viewpoint, prevents the fabrication of fused-tapered couplers from the DIC fibers, as the adiabacity criterion is difficult to obey in this case [13].

In short, and quite paradoxically, a tapered DIC fiber preserves the FCM shape as long as complete transfer of power is possible from the LP_{0j} to the LP_{0j-1} mode, i.e., as long as the FCM is confined in the core and can “ignore” the presence of the ring. But for smaller ITR values, due to the largely multimode outer cladding with a raised index able to trap the power, the power may leak to the ring modes.

## 3. Modal analysis of the PCF

The PCF used in our experiments (see Fig. 1(a)) has a 6-fold symmetry. The fiber profile used for the calculations is shown in Fig. 3(a). It represents the actual Fig. 1(a) fiber with its 7 layers of holes with a diameter varying linearly from 1.8*μ*m to 2.2*μ*m, from center to periphery. Holes are separated by a constant pitch Λ= 4*μ*m. The external fiber contour was assumed to be circular with a diameter of 121*μ*m, the largest dimension of the fiber. The core-mode, having also the 6-fold symmetry, can only couple with modes having a 6*n*-fold symmetry (6, 12, 18, …). The effective indices *n*
_{eff} of the modes of interest and the delineation adiabacity criteria were calculated as functions of ITR, using again a finite difference method, in the scalar approximation. Calculations were restricted to modes having not only the relevant symmetry but also the value of their effective index *n*
_{eff} close to that of the FCM of the two-layer fiber. In addition, only ITR values ranging from 0.09 to 0.22 were considered. For ITR values higher than 0.22, the FCM power leakage is very small because the narrowness and deepness of the adiabacity criteria peaks prevent any significant transfer of power to the cladding modes. Also, at ITR=0.09, all the features reponsible for the FCM major power drops have already been attained and from our perspective, further investigations at lower ITR values do not justify any additional calculation time. Figures 3(b) and 3(d) show the corresponding curves. Supermodes are identified by capital letters and Fig. 3(c) shows their field amplitude distribution represented at their lowest ITR value shown in Fig. 3(b). In Fig. 3(b), mode K at ITR=0.22 is the FCM. For lower ITR values, the FCM label changes but its effective index can be traced down to ITR=0.09 by following the dotted line. Figure 3(d) shows the adiabaticity criterion delineation curves corresponding to adjacent pairs of modes. As in Fig. 2(b) the normalized slope of a typical tapered fiber is represented for reference. All calculations were made at *λ*=1550nm.

The situation, although more complex, is clearly similar to that of the DIC fiber. The complexity comes from the 6-fold symmetry of the fiber and thus of the FCM, which allows it to couple with a number of modes which are not permitted in the case of circular symmetry. As in the circular case, during tapering, the FCM is expected to transit through the anticrossing regions of Fig. 3(b) without significant leakage provided the corresponding dips in Fig. 3(d) are sufficiently deep, as is the case for J-K, I-J, H-I, G-H, and F-G pairs. The situation for the E-F pair is radically different as the peak is larger and more shallow : leakage is therefore expected around an ITR value of 0.155. This partial power transfer could also have been predicted from the significant gap between the E and F curves in Fig. 3(b). Moreover, if some power is transferred to the E-mode and the fiber further tapered, the same phenomenon is expected to occur at ITR ≈ 0.136, resulting in a power splitting between the D and E modes, and thus in additional leakage.

As the ITR decreases, the power initially concentrated in the core is transferred to the ring. Once several modes are excited, mode beating is expected to produce oscillatory responses as functions of elongation and wavelength because tapered fibers may be viewed as modal interferometers [14, 15].

## 4. Calculation of the PCF taper transmission

By using the modal parameters of Figs. 3(b) and 3(d), we calculated the transmission of a taper made with a PCF having the index profile of Fig. 3(a), using the coupled equations formalism [10]. However, as one coupled equation must be added for each additional mode considered, a problem involving many modes can become very demanding in terms of calculation time. Therefore, we carefully selected the most relevant modes and considered only them in our calculation.

In Fig. 3(d), we can see that the peaks associated to the I-J and H-I pairs of modes are somewhat deeper and narrower than the others. Consequently, it is reasonnable to assume that the power leakage at these peaks is smaller than at the other wider and shallower ones. Taking this in consideration, we decided to neglect the power leakage at the I-J and H-I anticrossing points and merge all these three modes H, I and J into one fictitious mode “L” able to transfer power with the K- and with the G-modes in the same manner as the J- and H-modes, respectively.

In a further simplification, we only considered coupling of adjacent modes along the FCM effective index curve of Fig. 3(b). For example, the equation for mode F is

where *A*
_{E}, *A*
_{F} and *A*
_{G} represent the modal amplitudes of the E, F and G modes, respectively. We therefore neglect possible coupling between ring modes, e.g. from the F-mode to higher order modes which, in our case, could occur for ITR values smaller than 0.16. This effect drains the power to ring modes with very little chance of recovery in the FCM.

Calculations of the modal effective indices, coupling coefficients and adiabaticity criteria have been made for a limited range of ITR. As mentioned before, the adiabacity criteria peaks calculated at *λ*=1550nm, are extremely deep and narrow for ITR values higher than 0.22 and consequently prevent any significant power leakage from the FCM. This value has however to be adjusted according to the wavelength of operation. For the upper value of the wavelength range of interest (1250–1650nm), one has to increase the ITR value to 0.24.

The solution of the coupled mode equation system, restricted to modes K, “L”, G, F, E and D, is presented as a function of wavelength in Fig. 4(a) for a tapered fiber profile shown in Fig. 4(b). Figure 4(a) shows the power distribution between the modes at the exit of the tapered PCF. At the entrance and exit of the tapered fiber, the FCM being the K-mode, the tapered fiber transmission is that of the K-mode. At a wavelength slightly longer than *λ* ≈ 1300nm, the FCM transmission suddently drops to about -15 dB from the initial ≈ 0 dB value. It is clearly seen to come from the occurence of splitting the power with the E-mode resulting in large amplitude sinusoidal oscillations in the FCM-transmission at longer wavelengths. Although with a smaller extent, another relatively large power transfer also occurs with the D-mode, visible from *λ* ≈ 1525nm and the region beyond. Other modes considered for this calculation, namely the L-, G-, and F- modes, play an almost negligible role. Only the G-mode influence appears with small amplitude oscillations at wavelengths smaller than *λ* ≈ 1300nm. This remark retrospectively justifies the approximation which consisted of neglecting the I-J and H-I coupling effects.

## 5. Experimental results

The experiment consisted in tapering the fiber shown in Fig. 1(a). This fiber was designed so that its FCM is matched to that of a standard telecommunication fiber at wavelength *λ*=1550nm. This allowed us to splice a strand of this PCF in between two strands of standard single-mode fiber without significant loss. It was connected at the entrance to a laser/wideband source and at the exit to a photodetector/OSA. This strategy, apart from saving PCF helps prevent undesirable collapse of the holes during the tapering process.

Figure 5(a) shows the transmission of a taper as a function of elongation, using a laser of wavelength *λ*=1550nm as a source. The essential feature to observe is that the power transmission starts to drop at an elongation of ≈ 2.4mm before abruptly falling for an elongation of ≈ 2.55mm. (The small amplitude oscillations observed for very small elongation may be due to unperfect splicing, which would allow ring modes to beat with the core-mode.) In Fig. 5(b) is shown the same tapered fiber transmission but as a function of wavelength after completion of the stretching process. The corresponding longitudinal profile was measured and is shown in Fig. 5(c). As in Fig. 5(a), there is a dramatical fall of the transmitted power occuring for *λ* > 1300nm. As for tapered standard fibers, the wavelength response reproduces the elongation response. Several fibers were tapered using the same recipe. When we monitored the transmission as a function of wavelength during the stretching process, we observed first a nearly constant transmission with small oscillations before the abrupt fall takes place, the pattern evolving towards short wavelengths as the structure is more and more tapered.

The cleaved endface at the waist of the taper of Fig. 5(c) is shown in Fig. 5(d). The measurements were made using a SEM and showed us that the PCF holes at the waist did not experience a significant collapse.

## 6. Discussion

We observed that the transmissions of tapered PCFs as functions of elongation and wavelength exhibit the same pattern: small amplitude oscillations followed by a dramatical fall of the transmission with large amplitude oscillations. From the experimental point of view, these observations were reproducible provided the tapering recipe was the same. From the theoretical point of view, simulations were able to reproduce the main features of the experimental responses. In particular, the model, as it couples the FCM to ring modes rather than a continuum of radiation modes, is able to predict not only the abrupt drop but also the oscillatory behavior of the FCM transmission. Furthermore, the main modes involved in the coupling leakage process have been clearly identified as the E-, D- and G-modes having the transverse field distribution shown in Fig. 3(c).

However, from a quantitative viewpoint, there are some discrepancies between the experimental results and the simulation predictions. First, as the calculation neglected power coupling from the E-, D- and G-modes to higher order modes, the power loss is underestimated by our simulations and the amplitude of the oscillations following the power drop are therefore overestimated as they mainly involve two modes (K and E) of roughly equal amplitudes. Second, if one compares the longitudinal profile used for simulation to the measured one, they are quite different in terms of both slopes and waists. The profile used for simulation was chosen to roughly reproduce the onset of the power drop and the oscillation amplitudes and periods. Indeed, the experimental profile was also used to simulate its elongation and wavelength responses but the results were not satisfactorily reproducing the order of magnitude of the experimental features. This is of course due to an imperfect calculation of the coupling coefficients, which comes from both the imperfect description of the transverse profile and of the modal fields theirselves. We had to make trade-off between the accurate description of the modes and the computing time. To correctly decribe tiny details such as the holes of the transverse profile of Fig. 1(a), one has to reduce the finite difference grid periodicity, which in turn would prohibitively increase the computing time. Beside, even though the actual profile presents a 6-fold symmetry, the grid we used for the modal calculation inherently presents the x- and y-symmetries which tend to distort the calculated field shapes. We also observed that for a slight variation of the index profile, the calculated modal effective indices and the coupling coefficients of mode pairs vary significantly. Finally, we made several assumptions for the calculations, such as the transverse profile invariance up to a scaling ITR factor along the taper, which may not be true. Although the hole collapse the tapered fiber waist was verified to be limited in our case (compare Fig. 5(d) to Fig. 1(a)), Ref. 8 reports that *d*/Λ along the taper can vary by as much as 38 percents. All these above mentioned imperfections in the model can possibly explain the observed discrepancies between the experiment and the simulation.

In summary, due to the imperfect description of the transverse and longitudinal profile of the tapered fiber (including Λ and *d* variations), our calculations fail to predict accurately the details of the experimental transmission. Our finite difference mode solver is also imperfect and its results should be compared to others (such as those based on finite elements or multipole methods) to evaluate its accuracy. Besides, more modes should ideally be taken into account in the resolution of the coupled mode equations, and the predicted transmission compared to a brute force method such as a BPM. However, all these verifications and improvements would require a larger investment in terms of i) geometrical characterisation tools, ii) both hardware and software computational tools, and iii) time. Despite our model imperfections, the overall behavior of the transmission is well predicted. As in the experimental responses, the power initially remains constant and later experiences slight oscillations which are followed by an abrupt fall. Ultimately, the transmitted power returns in an oscillatory manner to higher values. All these important features are satisfactorily explained by the partial transfer of power from the FCM to ring modes. As it gives physical insight of what is going on along the tapered structure regarding the transfer of power between modes, our approach is a step beyond the cutoff wavelength prediction derived in Ref. 8. In particular, our calculation points out to the possibility to recover power in the FCM such as for the tapering of standard fibers. As long as only a loss criterion is needed, the Ref. 8 approach may be sufficient but the design of devices made of tapered microstructured fibers would require to use the coupled mode approach. For practical purposes, more than an insight would then be needed and the improvements suggested above should be made.

## 7. Conclusion

In this paper, we presented an analysis of the power leakage phenomenon observed when one tapers a Photonic Crystal Fiber (PCF). We first proposed a qualitative interpretation using a equivalent Depressed Inner Cladding fiber, which highlights the role of the infinite air layer surrounding the tapered fiber. We then presented modal calculations for a more realistic index profile of the PCF using a finite difference scheme. Tapered fiber transmissions were calculated using the coupled equations formalism with a limited number of modes. Predictions from this model were then compared with experimental results. Despite the discrepancies between the two, attributed to the imperfect description of the fiber index profile, the main features of the transmission – namely the initial slight oscillations, the abrupt power drop and its partial recovery into the fundamental core-mode – are convincingly explained by our model.

## Acknowledgments

We would like to thank INO (Institut National d’Optique) for providing the fiber used for the experiments as well as the Canadian Institute for Photonic Innovations (CIPI) and the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support.

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