Rare-earth doped photonic crystal fibers rely ideally on an index matching of the doped core to the surrounding glass to work properly. Obtaining a perfect index matching is technologically very challenging, and fiber manufacturers opt for targeting an index depression instead, which still ensures the influence of the photonic structure on the light propagation. In this paper the analysis of the influence of this core index depression on the higher-order mode discrimination and on the beam quality of the fundamental mode of different designs of core-pumped active large pitch photonic crystal fibers is discussed. The most promising design is evaluated in terms of mode area scaling with a view to mode field diameters above 100µm. Detailed requirements on the accuracy of the core index matching are deduced.
© 2010 OSA
Single-mode Large-Mode-Area (LMA) fibers have been instrumental in the extremely rapid increase of the performance of fiber lasers seen in recent years. In order to be able to scale the pulse energy, peak power and average output power, LMA fibers are required to overcome the limitations imposed by nonlinear effects. Additionally, only single-transverse mode operation grants the desired high beam pointing stability and focusability.
Single-mode operation in LMA fibers can be achieved by various techniques such as modified excitation , differential bend loss for higher-order modes (HOMs) [2,3], resonant out-coupling of HOMs , mode filtering with tapers , modified spatial dopant profiles  and gain-guiding while index-antiguiding . Further approaches are the single-mode propagation in HOMs  and core pumping with the fundamental mode (FM) to achieve a Gaussian inversion profile that delivers preferential gain for the FM .
In any case, the approach with the highest scaling potential is the use of fiber designs that provide single-mode operation, which is the topic of this paper. Several advanced fiber designs have been proposed recently. Among them the most versatile are the so-called photonic crystal fibers (PCF) . In contrast to step-index fibers, PCFs are not single-mode in the strict sense. However, due to the high degree of freedom offered by the PCFs, they can be designed to offer differential propagation losses to the different transverse modes. Accordingly, novel active fiber designs have been proposed that provide high losses for the HOMs, i.e. high modal discrimination and, therefore, effective single mode operation [11,12]. These designs can be classified in two different categories, namely the Rod-type fiber  and the leakage channel fiber [12,13], that preserve the classical hexagonal PCF structure (Fig. 1 ). Both designs can in turn be labelled as Large Pitch Photonic Crystal Fibers, short Large Pitch Fibers (LPF) , as their pitch Λ is larger than 10 times the wavelength λ.
For the mode discrimination in active PCFs to be effective, the active core has to be (ideally) index-matched to the surrounding cladding material. This represents a technical challenge that is quite difficult to meet in practice. As a consequence, the core index of active PCFs is typically slightly different to that of the cladding. If the core index were higher than that of the surrounding glass, the light would be trapped in a step index structure (being almost not influenced by the microstructured cladding) which, in turn, would result in a strong decrease of the mode field diameter and in strongly reduced mode discrimination capabilities. Thus, in order to avoid these negative effects, the manufacturers of rare-earth-doped PCFs typically target core-index depressions, i.e. core indices below that of the surrounding cladding material. However, obtaining index depressions smaller than 10-4 in a controllable and repeatable fashion still represents a technological challenge.
In this paper we study the influence of this core index depression on the mode discrimination capabilities and beam quality (M²) of different LPF designs (Fig.1a). It will be shown that for propagation losses of 1 dB/m for the Fundamental Mode (FM), the two ring structure offers the highest mode discrimination. Therefore, this structure is further investigated in terms of mode area scaling.
The presented study evaluates different fiber designs in terms of mode discrimination, i.e. differential propagation losses. Obviously, all modes are lossless if the leaky modal sieve is surrounded by a step index or by an air cladding for pump light guidance. Thus, the evaluation of mode discrimination is only strictly valid for core-pumped active fibers. However, statements concerning the effective area and the beam quality factor M² are valid for double clad fibers, as well.
The paper is divided as follows: in section 2, the simulation technique and strategy is outlined. In section 3 it is explained how to counteract the influence of index depressions on the modal properties of LPFs. Then, section 4 compares the sensitivity of the different fiber designs to index depressions. Finally, before concluding, section 5 evaluates the mode field area scaling prospects of the most promising design.
2. Simulation technique and background
All the fiber designs considered in this paper have a hexagonal PCF structure, with the core formed by one missing air-hole. This core is surrounded by one, two or three rings of air holes [Fig. 1(b)]. In these structures, the pitch Λ, i.e. the hole-to-hole distance, and the normalized hole-diameter d/Λ define the guiding properties. To simulate a real active core, a central index-depressed region (with index depression Δn) is added to our model (as shown in Fig. 1(a). This region is hexagonally shaped with its vertices pointing to the spaces between the air holes and represents the unit cell of a photonic crystal fiber produced by the stack-and-draw technique.
As a design parameter, we set the propagation loss of the fundamental mode to 1 dB/m at 1030 nm wavelength (implicitly assuming an Ytterbium-doped fiber), which is a reasonable value for short high power fiber amplifiers . To adjust the propagation loss of the FM, exclusively the inner guiding structure (pitch Λ and normalized hole size d/Λ) was modified. The propagation losses of 1 dB/m for the FM endorse effective single-mode operation in straight fibers, which is required for extremely large mode areas (since, because of the collapse of the mode area spacing, differential bend losses cannot be exploited anymore). These design guidelines mark a clear departure from the work on leakage channel fibers presented for example in [12,13]. The mode discrimination can be then defined as the difference between the FM loss and the LP11-like HOM loss (the first HOM). Thus, the mode discrimination is used to quantify to which extent a certain fiber of a defined length is single-mode.
The simulations were done using a mode solver based on a full-vectorial finite-differences approach similar to that described in . For each mode, effective index, propagation loss (obtained by using a perfectly matched layer) and field amplitude and phase are determined. The size of the simulation area was chosen carefully such that the studied modal losses were not influenced by this parameter. By using a sufficiently thick perfectly matched layer at the simulation boundary, the cladding was made effectively infinite.
Evaluating the simulation results, we paid particular attention to the first higher order mode (LP11-like) and its damping in comparison to the fundamental mode. As a figure of merit, the 1st higher order mode (HOM) discrimination factor was introduced as the difference between the losses (in dB/m) of the 1st HOM and the FM. However, it is worth noting that future investigations will have to clarify the role of other higher order modes such as the LP31-like modes. These modes are particularly dominant in the one ring design, due to its much larger hole size compared to the two or three ring designs. These modes, nevertheless, may have very low propagation losses but they are hard to excite in a fiber amplifier. Therefore, their impact on the system is expected to be very small.
Additionally, the analysis presented in  provides a means to calculate the M² of the fundamental mode from its simulated electric field by employing the second moment method. At this point it is important to highlight that all the M² values given in this paper correspond to the fundamental mode only. Hence, any deviation from M² = 1 is due to the fact that the fundamental mode of the studied fiber is not Gaussian . This is a consequence of the hexagonal fiber design combined with the weak guidance required for effective single mode operation. In other words, a fundamental mode with losses of about 1dB/m ensures that all significant HOMs undergo substantially higher losses. Thus, while the losses of the FM can be easily compensated for by gain in short fiber amplifiers, the higher order modes will be effectively suppressed. Unfortunately this also implies that the FM has a tendency to leak out of the core becoming slightly hexagonal, which results in a degradation of the M2. In spite of this, the pointing and mode stability will remain excellent as the slightly higher M² is not due to higher order mode content.
The main focus of this paper is thus the evaluation of the additional influence of an index depression on the HOM propagation loss and the M² of the fundamental mode. For different depressions and number of rings, the designs can only be properly compared when their fundamental modes possess the same properties, i.e. propagation loss (1dB/m) and effective area (2000-10000µm²). To keep these values constant, the pitch and the d/Λ have been adapted accordingly.
3. Counteracting index depressions
For a fixed LPF structure (fixed Λ and d/Λ), an increasing central index depression influences severely FM properties such as propagation loss, effective index and calculated M². This is illustrated in Fig. 2(a) for perfect index-matching (i.e. no index depression of the active region) as well as for increasing index depressions in Fig. 2(b) and Fig. 2(c). It can be seen that index depressions progressively “push” the FM out of the core, leading to a larger mode field area (increased by 40%, Fig. 2(b), and 220%, Fig. 2(c), for index depressions of 5∙10-5 and 10∙10-5, respectively), but also to significantly higher propagation losses and M² (increased from 1.2 to 1.7 and 3, respectively, Fig. 2(a)-(c).
To counteract the negative influence of the index depression, the relative hole size d/Λ can be increased. This measure “pulls” the fundamental mode back into the core. However, when the fundamental mode is pushed out of the core by the index depression and retained in the core by increased hole sizes, it becomes slightly deformed acquiring an hexagonal shape, which in turn increases its M2. This can be seen in Fig. 3(a)-(c) , where the hole sizes have been increased (compared to Fig. 2(a)-(c) to maintain a constant FM propagation loss of 1dB/m.
In these examples only the FM propagation loss was kept constant. For further comparability, the mode area needs to be fixed as well. Thus, for each fiber design (i.e. number of rings and index depression Δn) there is only one combination of pitch Λ and normalized hole size d/Λ that leads to a fixed combination of FM propagation loss of 1dB/m and mode field area of 2000µm², which is a typical value where single mode operation gets difficult to achieve with conventional fiber designs.
The different values of the d/Λ and Λ required to obtain the constant fundamental mode properties are displayed in Fig. 4(a) and Fig. 4(b). For each design, the pitch Λ requires only slight adjustments over the whole depression range and, therefore, can be treated as approximately constant [Fig. 4(a)]. This way, for one ring of surrounding air holes, the pitch was ~45µm, for two rings ~34µm and for three rings ~28µm.
Adding more rings increases the guidance, which leads to smaller hole sizes required to maintain the mode properties constant. However, smaller holes lead to a higher leakage and, consequently, to a mode that is more prone to distortion by an index depression. Therefore, it is understandable that, in order to counteract the effects of the index depression by increasing the hole size, the number of rings plays an important role. In this way, the more rings, the higher the relative hole size increase required to keep the properties of the FM constant. Thus, while in a one ring design the d/Λ needs just to be increased from 0.54 (no depression) to 0.71 (highest simulated depression, i.e. 20·10-5), in a three ring design it increases from 0.18 to 0.46.
Now that the fundamental mode properties have been kept constant, it is time to evaluate the comparison criteria between the different designs. In this work we have primarily focused on the M² of the fundamental mode and the propagation losses of the HOMs, as they are the two parameters of outmost practical relevance.
4. Simulations on mode discrimination and beam quality
In this section both the 1st HOM discrimination and the beam quality of the FM are examined for an index depression between 0 and 20·10-5. Fig. 5 shows that the two-ring design exhibits the highest mode discrimination. While in the one-ring design the discrimination reaches a hardly acceptable value of 20 dB/m without depression, it drops below 10 dB/m for higher depressions. This value can be easily compensated in high gain fibers (specially if transversal hole burning takes place ). Contrarily, multiple ring designs offer a significantly improved 1st HOM discrimination with values that stay above 60 dB/m even for substantial depressions.
To understand this 1st HOM discriminating behaviour, the d/Λ needs to be considered. As explained before, for a growing number of rings, the designs require lower pitches and lower d/Λ-values. In general, a low d/Λ-value (smaller holes) causes the modes to leak out of the core more strongly. However, different modes leak out at different rates , being the leakage of HOMs (including the 1st HOM) considerably higher than that of the FM. Consequently, the discrimination increases significantly when stepping from one to two rings. Due to the smaller d/Λ, the 1st HOM is not guided anymore in the core but between the 1st and 2nd ring instead (see upper insets in Fig. 5), and from this second ring it radiates out to the cladding. Adding a third ring, on the contrary, improves the guidance for this mode, yielding lower losses and, therefore, a worsened discrimination. However, note that this behaviour is particular for our choice of parameters (and especially of the loss of 1 dB/m of the FM) and cannot be generalized. A different choice of the FM loss might yield a different optimum design. For example, a FM propagation loss of 0.01 dB/m (provided by a larger d/Λ) would mean a stronger guidance not only for the FM, but also for the HOMs. The HOMs of different designs react differently to this stronger guidance. The more rings are used, the smaller the required hole sizes for a certain FM loss. Smaller hole sizes, in turn, allow a stronger leakage of the HOMs, leading to the three ring design as the optimum in this example. However, this working point is out of the scope for this paper, as the overall mode discrimination would be significantly decreased.
The calculated M² of the fundamental mode is displayed in Fig. 6 for the fiber designs with the structural parameters of Fig. 4. For higher number of rings, at a fixed depression, the fundamental mode penetrates the photonic structure more deeply and gets deformed more strongly. This, in turn, leads to higher M² values. For all designs, the M² degrades with higher index depressions because the FM departs more and more from the ideal Gaussian profile and obtains a hexagonal shape (see insets of Fig. 6). The one ring design offers the best M² of <1.4 even in presence of a strong index depression. On the opposite end, the three ring design shows a high M² (~1.5) even without index depression and a further significant degradation of this parameter with index depressions. This is again due to the small hole sizes at the point of operation of 1dB/m propagation loss.
5. Mode Area Scaling
In this section the impact of index depression on the mode area scaling of LPFs is explored. Therefore, the fixed FM mode area of previous sections will be dropped here and we will scale this parameter up to 10000µm². However, the FM loss will still be fixed to 1 dB/m. Firstly, we compare the designs for perfect index matching and an index depression of 1·10-4, respectively. This is done in Fig. 7 for fiber designs with one to three rings of surrounding air holes. In every case, the first higher order mode losses drop significantly in presence of an index depression, which is due to the larger d/Λ required to obtain the constant FM properties. Furthermore, as a result of the collapsing mode spacing, the first HOM losses decrease even further for larger mode field diameters. In spite of this, whenever an index depression is considered, the two ring design preserves the highest mode discrimination and is, therefore, the best suited for short high power fiber amplifiers.
At this point, the influence of the index depression on the mode discrimination and M² of the FM will be examined in more detail for the most promising design, i.e. the two ring design. Thus, according to the simulations, the mode discrimination (Fig. 8 ) of the two-ring LPFs in the case of perfect index matching (no index depression) is always larger than 40 dB/m for mode field areas of up to 10000 µm², i.e. such a fiber would operate effectively single-mode for lengths longer than 1m.
Thus, for example, for a ~1m long high power fiber amplifier with a gain of ~20dB, a 30dB/m HOM discrimination should suffice to ensure effective single-mode operation. According to Fig. 8(a), for an effective mode area of ~8000µm², this requires an index depression of about 5∙10-5, which is within the limits of what is achievable nowadays (albeit at the edge of feasibility). The corresponding mode field diameter amounts to >100µm. In comparison, a step-index fiber with 100µm mode field diameter fabricated with the same index difference of 5∙10-5 would have a V-parameter of 3.8 and, thus, be multimode without any discrimination due to its step-index nature.
Fig. 8(b) shows the beam quality as a function of the FM area and the index depression. It can be seen that two-ring LPFs with perfect index matching maintain a nearly diffraction limited beam quality (M²<1.4) up to 10000 µm² mode field area. However, the beam quality degrades quite fast once index depressions are considered. Even though, as said before, the mode discrimination also decreases with higher index depressions, the rapidly degrading beam quality is the main parameter of concern in the context of high power fiber amplifiers. As an example, assuming an index depression of 5·10-5, the FM M²-value exceeds 1.7 for a mode field area of 4000 µm². In the worst case considered herein, i.e. a two-ring LPF with index depression of 10·10-5, a M²-value of 1.7 corresponds to a mode field area of just 2500 µm².
As a conclusion, it can be seen that the one-ring LPF is not appropriate for real-world active core-pumped short LMA high power fibers because its mode discrimination capabilities become unacceptably small even for small index depressions. On the other hand, the two-ring design is the one that offers the highest mode discrimination combined with only a moderate loss of beam quality of the FM. Here, an increased M² is attributed to an increasingly hexagonally shaped FM. As a result, it can be stated that the two-ring design appears to be the best qualified for a new class of single-mode Very Large Mode Area fibers. However, in order to scale the mode field diameter to 100µm and beyond, a core-cladding index-matching better than 5·10-5 is required to maintain both the effective single mode operation and an acceptable beam quality. This value is an important reference for fiber manufactures.
The research leading to these results has received funding from the German Federal Ministry of Education and Research (BMBF), the Helmholtz-Institute Jena (HIJ) and the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement n°  "PECS". Additionally, F. Jansen acknowledges financial support by the Abbe School of Photonics Jena.
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