The impact of avoided crossings (also known as anti-crossings) in single and double-clad large mode area Photonic Crystal Fibers (PCFs) suitable for high-power laser systems is evaluated numerically. It is pointed out that an inappropriate choice of pump core diameter, bending radius and/or index depression may lead to avoided crossings that manifest themselves in unwanted deformations of the output beam.
©2011 Optical Society of America
Ytterbium-doped fibers have been a major player in the extremely rapid increase of pulse energy  and average output power  of high power femtosecond fiber laser systems. This has been possible due to the high gain and efficiency provided by Yb-doped media, but also due to the development of large mode area effective single-mode photonic crystal fibers .
To outperform current systems, Very Large Mode Area (VLMA) fibers (i.e. fibers with >50 μm mode field diameter) need to be further up-scaled to enable even higher pulse energies at high repetition frequencies. However, preserving single-transverse-mode operation in VLMA fibers is very challenging. One approach that provides very large mode areas and single-mode operation is the leakage channel fiber concept [4,5]. This fiber design has already demonstrated Mode Field Diameters (MFD) of up to 62 µm at moderate output power levels of 30 W. This approach works by presenting lower leakage losses to the Fundamental Mode (FM) than to the Higher Order Modes (HOMs), a concept introduced as “modal sieve” by Russell et.al. in .
Nevertheless, in our opinion, the most promising fiber design to date that satisfies the requirements for high average output power and exhibits very large mode areas, is the so-called Large-Pitch photonic crystal Fiber (LPF) [7,8]. LPFs are characterized by a hole-to-hole-distance (pitch) larger than ten times the wavelength. These fibers work by delocalizing the HOMs from the doped core. This results in a reduced excitation of HOMs by an incident Gaussian beam and, simultaneously, in a reduced overlap of HOMs with the doped region and/or increased confinement loss for HOMs. The delocalization of the HOMs in an exemplary LPF can be seen in Fig. 1 . The top row of Fig. 1 displays the modes of single-clad LPF together with their respective confinement losses. Details on the simulation can be found in section 2. Confinement loss only serves as a quality measure for single-mode operation in single-clad fibers as all modes are lossless in double-clad fibers. This latter case is displayed in the bottom row, where the modes corresponding to the same PCF structure as before but surrounded by an air-clad can be seen. Here, the overlap with the doped core was calculated and this parameter is used to determine the most relevant HOMs.
Recently, double-clad LPFs have demonstrated high average output powers of ~300 W with very large mode areas (a prerequisite for high peak powers) of 62 µm in robust single-mode operation . Furthermore, a record peak power has been recently extracted from an Ytterbium-doped LPF with 105 µm MFD . However, in this experiment the output average power was limited by the presence of an avoided crossing that led to deformations of the output beam. This experiment emphasized the importance of a study on avoided crossings in LPFs, as their open core structure (Fig. 1) enables the interaction of core and cladding modes. Thus, a deep knowledge of avoided crossings is a fundamental requisite for fiber designers in order to avoid them (in terms of modal deformations of the FM) or even to exploit them (in terms of modal deformations for unwanted HOMs) as proposed in  for the so-called leakage channel fibers.
After commenting on some basic concepts of LPFs and avoided crossings in section 2, the simulation procedure will be summarized. Section 3, 4 and 5 describe avoided crossings caused by various parameters such as the air-clad diameter, bending radius and core-cladding index-mismatch. In section 6 the results are summarized and conclusions are drawn.
2. Large-pitch fibers and avoided crossings
To simplify the understanding of avoided crossings in LPFs we have selected a set of fiber parameters that ensures their occurrence in all the situations that will be discussed in the following. The simulations for this paper were carried out using a full-vectorial finite-differences frequency domain mode solver delivering the full field information of the modes. The confinement loss was calculated by making use of a perfectly matched layer. In all simulations, the wavelength was set to 1030 nm. The schematic structure of this LPF is displayed in Fig. 2 . This design is characterized by a core created by a single missing hole in a silica matrix. The core is surrounded by two rings of air holes in a hexagonal lattice. The choice of two surrounding rings of air holes as the optimum design was discussed in detail in . Suffice to say here that two rings provide the highest HOM delocalization while preserving an acceptable beam quality of the FM. The concrete design parameters are a pitch Λ = 30 µm and a normalized hole diameter d/Λ = 0.30. This parameter set leads to a mode field diameter (MFD) of ~50 µm when no index depression is assumed in the fiber core. Save for section 3, throughout the paper a fixed air-clad diameter of 170µm will be used.
Since the waveguide structure of LPFs is very open, a clear differentiation between core and cladding modes is not possible. This is illustrated in Fig. 3 , where a qualitative scheme of the modal effective indices in a step index fiber and a PCF with a widely open structure (e.g. a LPF) is shown. It can be seen that while core and cladding modes are clearly separated in a step index fiber, this is not the case for widely open PCFs. In particular, in PCFs the FM may not even be the mode with the highest effective index. However, a coarse classification into core or cladding modes can still be carried out by calculating the overlap of the modes with the core (doped) region, i.e. by evaluating the degree of localization of each mode. This way, the FM will be defined as the mode combining the highest overlap with the core region with - most importantly - the highest beam quality factor M2 (which is a clear indicator of this mode being the most Gaussian-like). Thus, throughout the paper we will use the terms core and cladding modes (when attending to their degree of localization in the core), as well as FM and HOMs, where the last one encompasses all core and cladding modes except the FM.
Crossings and avoided crossings are a well understood phenomenon in photonic bandgap fibers [9–11], where they lead to a reduced transmission within the bandgap. At a crossing, two modes have identical effective indices but their modal overlap remains zero, i.e. they are still orthogonal to each other. Therefore, there is no interaction between the crossing modes, and the crossing can be simply described as a fortuitous coincidence of the index of refraction of two modes with different dispersion characteristics without further physical consequences. On the contrary, when avoided crossings (or anti-crossings) occur, two orthogonal modes evolve to show nearly identical transverse profiles (in spite of their effective refractive index not being equal, hence avoided-crossing). Progressively, during the avoided-crossing, the modes exchange their roles and their effective indices diverge.
Crossings and avoided crossings have been shown to exist in various types of PCFs such as index-guiding PCFs  or PCFs with an “open structure”, namely in LPFs. However, the open structure that guarantees the preferential amplification of the FM (as HOMs possess a reduced overlap with the doped region) in LPFs also imposes some important constraints on the fiber design as it enhances the interaction between core and cladding modes. In this study it is shown that avoided crossings need to be considered as an integral part of the design process of PCFs with open structures, i.e. the whole waveguide has to be considered and not just the region containing air holes.
In the following, several parameters are investigated that can cause avoided crossings. Thus, a short review of avoided crossings caused by changing the air-clad diameter is presented. Additionally, we will comment on bending-induced avoided crossings. Finally another parameter that creates avoided crossings is systematically analyzed: the index depression of the doped region in active fibers. As doping the fiber core with rare earths increases its refractive index, co-doping with other materials (such as fluorine) is necessary to re-establish an index-matching between the core material and silica (which is required for the proper operation of PCFs). Achieving a perfect index matching is technologically not feasible, reason for which this index matching process typically results in an index depression of the doped region compared to the silica matrix. The influence of index depressions on LPFs with a special view on mode area scaling was discussed in detail in . In this paper we show that these index depressions can potentially lead to avoided crossings.
When plotting all the modes over the design parameter sweep, it is necessary to distinguish between crossings and avoided crossings. In order to do this, the overlap integral of all modes with a reference mode was calculated. As a fixed reference mode, an undistorted mode, i.e. a mode far away from any avoided crossings (specifically for the smallest possible air-clad or, respectively without bending or index-depression), was chosen. This reference mode (one mode for each section) is compared to all modes of each fiber structure. Note that in a “standard” waveguide all modes are orthogonal and, therefore, the overlap between two different modes is always zero. This is not the case for the calculation of the overlap of these orthogonal modes with a reference mode of a slightly different waveguide. As soon as an avoided-crossing occurs and deforms participating modes, the overlap integral with the reference mode will differ from zero. This technique ensures the reliable identification of avoided crossings at first glance.
3. Air-clad-induced avoided crossings
Surrounding the core structure of a PCF with a pump core leads to a completely new structure, in which core and cladding modes can interact. With growing air-clad diameters, the effective index of a core guided mode basically remains constant while that of the cladding modes grows. Therefore, at some point, the effective index of the HOMs will reach the effective index of the FM. When this happens two different effects can be observed. Some HOMs only cross the effective index of the FM (situation denoted as crossing as mentioned above), while other HOMs deform and an avoided-crossing can be observed. In the latter case, the shapes of both modes converge and, finally, the former HOM takes over the role of the FM while the former FM is delocalized from the core and becomes a cladding mode (Figs. 4 and 5 ). In  it was shown that FM avoided crossings can be circumvented and HOM avoided crossings can be exploited by the proper choice of air-clad diameter.
To illustrate the principle of avoided crossings, Fig. 4 shows, for the LPF with the design parameters detailed above, a region of air-clad diameters containing one main (broad) avoided-crossing of the fundamental core mode with a cladding mode, and two further successive narrow avoided crossings, each indicated by vertical green lines. In the upper half of Fig. 4, the effective index of all modes is plotted versus the air-clad diameter. To distinguish between crossings and avoided crossings, as mentioned before, we propose the calculation of the overlap between each mode and a fixed reference FM. In this way, when sweeping the air-clad diameter, the reference mode is the FM of the waveguide with the smallest air-clad diameter. The position of the reference mode (mode 1 with lowest air-clad diameter) to which all other modes are compared via overlap calculation is highlighted in Fig. 4. The result of this overlap integral is shown in the lower half of Fig. 4. Three avoided crossings can be clearly identified in this region. Figure 5 displays the changing modal profiles of the modes involved in the first (main) avoided-crossing with growing air-clad diameter. The avoided-crossing at 188.5µm air-clad diameter is apparent through the deformations of the crossing modes in Fig. 5. Additionally, it can be seen in the upper half of Fig. 4 that the effective indexes of the two modes involved in the avoided-crossing repel each other. Furthermore, it can be observed in Figs. 4 and 5 that in a certain region of air-clad diameters no Gaussian-like FM exists. For larger air-clad diameters, the former HOM (here, mode 4) evolves into a Gaussian-like FM.
Figure 6 expands the simulation window of Fig. 4. It can be seen that the FM effective index remains constant while the effective indices of the cladding modes increase strongly with growing air-clad diameters. This is a typical behavior and prerequisite for the existence of crossings and avoided crossings, as can be seen in hollow core photonic bandgap fibers . It is clearly visible that the effective index plot is not sufficient to identify avoided crossings as soon as more modes come into play, while the overlap plot is unambiguous. Here, the air-clad diameter was increased from 140 µm to 440 µm, revealing several broad avoided crossings on the one hand and large regions without any broad avoided crossings on the other hand. It is important to note that avoided crossings do not generally occur more often with larger air-clad diameters, although this is what would be expected from the effective index plot since it shows an increasing density of cladding modes.
In Fig. 7 we study the avoided-crossing around 190 µm air-clad diameter more closely for a constant pitch Λ = 30 µm but with three different values of the normalized hole diameter d/Λ. It can be observed that the avoided-crossing of the FM becomes broader the smaller the d/Λ. For smaller hole sizes the FM is significantly larger and extends further into the cladding. Therefore, the open structure enables a stronger interaction between core and cladding modes and, thus, this leads to avoided crossings extending over a larger range of air-clad diameters. Furthermore, larger hole sizes cause a small shift in the air-clad diameter at which the avoided-crossing occurs. For fiber designers this is important to know when choosing the air-clad diameter sufficiently far away from any avoided crossings to mitigate deformations of the FM.
4. Bend-induced avoided crossings
In this section it is shown that even the right choice of air-clad diameter is not enough to ensure that an avoided-crossing for the FM is circumvented. Therefore, employing the same inner guiding structure as above (pitch Λ = 30 µm and d/Λ = 0.30) and a fixed air-clad diameter of 170 µm, we simulated the impact of bending on avoided crossings. Furthermore, we apply the same analysis to the same structure without air-clad to demonstrate two different kinds of avoided crossings: ones involving cladding and core modes and others involving only core modes. Starting with a straight fiber, we successively decrease the bend radius down to 30 cm in both cases (the smallest bending radius with limited FM deformation).
The results of our study are shown in Fig. 8 . The effective index and the overlap with the FM of the straight fiber are plotted against the bend radius. To determine the position of bend-induced avoided crossings, every mode of every bend-radius is compared to the FM of the straight fiber (reference FM) by calculating their overlap integral. This enables the identification of the FM even under strong deformation. Deformations of the FM by bending manifest themselves by a decreasing maximal overlap for smaller bending radius (for bending radii far away from avoided crossings). At every bend-radius of Fig. 8, the mode with the highest overlap with the Gaussian-like FM of the straight fiber is shown in black while all other modes are grayed out in both graphs. Five avoided crossings are visible in the effective index plot and in the overlap plot. To illustrate the modal deformations taking place across one of these avoided crossings, Fig. 9 depicts the two modes involved in the avoided-crossing around 2.2m bend-radius.
Note that the presence of an air-clad enables the existence of avoided crossings between core and cladding modes, while the same inner structure without air-clad will only exhibit avoided crossings between core modes (and, thus, significantly fewer avoided crossings). This is depicted in Fig. 10 for the same interval as in Fig. 8. While now two broad avoided crossings appear at a bending radius of 0.3 m and 0.7 m instead of 0.3 m, 0.65 m and 2.2 m, all narrower (core-cladding) avoided crossings have vanished.
Here, two different situations have to be explained separately. Either two core modes or a core and a cladding mode may interact. A core-cladding avoided crossings occurs due to the fact that cladding modes are much more sensitive to bending and, thus, have a steeper slope for decreasing bending radius than core modes. In core-core avoided crossings, however, a core-guided HOM is deformed by bending. This eventually leads to the point where the HOM effective index approaches the effective index of the FM and both modes interact. In Fig. 9 a LP11-like HOM (bottom, already deformed by bending) was shown to interact with the FM (top).
5. Index depression-induced avoided crossings
Another waveguide parameter that might vary for different active fiber pulls is the central index depression (Fig. 2) of the rare-earth-doped region compared to the silica matrix . Typically, this mismatch is <10−4. We will study the influence of an index depression of this order of magnitude for the exemplary guiding structure introduced above (Λ = 30 µm, d/Λ = 0.30 and an air-clad diameter of 170 µm) and only in the case of an active straight fiber.
Without considering avoided crossings, with a higher index depression one would expect a steadily decreasing effective index of the FM and higher order core modes, whereas that of the cladding modes is basically not influenced by the index depression. But in fact, in this open waveguide structure (compare Fig. 3) core and cladding modes react very differently to the index depression since a distinct separation between them is not always possible. On the one hand, pure cladding modes will remain uninfluenced by the index depression and, therefore, possess a constant effective index for increasing central index depression. On the other hand, all modes with intensity in the core will be influenced by the index depression at one point or another, which is accompanied by a decreasing effective index for increasing index depression. Moreover, as different core modes are influenced differently by the index depression, their effective indices are also able to cross. Note that the presence of the air-clad is not a prerequisite for these core-core mode crossings and avoided crossings.
The situation in air-clad LPFs is illustrated in Fig. 11 , where the effective indices of the three modes involved in an avoided-crossing (FM and HOMs) are plotted versus the index depression as black, red and blue lines, respectively. All other modes, i.e. those not involved in the main two avoided crossings, are shown in gray. At a depression of 6·10−5 the first avoided-crossing between the FM (mode 1) and one HOM (mode 2) can be seen. At a depression around 17·10−5 a second avoided-crossing follows between the “new” FM (mode 2) and mode 3. For further clarification, Fig. 11 also shows the overlap between the fundamental mode without index depression (taken as the reference mode in this case) and all simulated modes. Although not perceptible in the effective index plot, several more narrow avoided crossings are revealed by the overlap plot around an index depression of 6.5·10−5, 12.5·10−5 and 19.5·10−5.
Figure 11 also illustrates that for larger index depressions the overlap with the initial FM never reaches values close to one because the index depression deforms the modes. While mode 1 and mode 2 reach a nearly Gaussian-like profile in the center of the waveguide (i.e. in the doped region) for larger index depressions, they never lose their characteristic higher order mode features, i.e. one or two surrounding intensity rings (Fig. 12 ).
For further illustration, Fig. 12 shows the modal pictures of the modes involved in the avoided crossings with increasing index depressions. It is clearly visible that the FM (mode 1) gets pushed out of the core while, at first, mode 2 and then mode 3 become more and more FM-like.
Note that the index depression-induced avoided crossings shown in Figs. 11 and 12 and the quantitative values of the index depression at which they occur are to be understood as an example. Furthermore, it can be stated that avoided crossings occur at smaller values of index depression the larger the mode field diameter of the FM. A sufficiently “good” index-matching (lets say 5·10−5) for a fiber with the parameters discussed above might lead to significant modal deformations through avoided crossings in an up-scaled version of the same structure. Thus, while the first avoided-crossing is centered at an index depression of 6.1·10−5 for a pitch Λ = 30 µm, this value drops to 2.5·10−5 for a pitch of 45 µm and 1.6·10−5 for Λ = 60 µm.
In the previous sections we have shown that avoided crossings may be caused by various changes in the fiber structure such as the air-clad diameter, the bend-radius and the central index depression. The existence of avoided crossings is a characteristic property of all PCFs, being it especially pronounced in fibers with an open core structure such as LPFs, where core and cladding modes can interact easily (Fig. 3). This is particularly important for LPFs where a large delocalization of HOM is exploited to ensure single-mode operation at high average output powers and very large mode areas [1,8]. In conclusion, all changes of the waveguide structure affecting one group of modes, e.g. core or cladding modes, stronger than other groups of modes will potentially lead to avoided crossings between this group and all other modes.
Thus, VLMA PCF designs have to consider avoided crossings between core and cladding modes not only by carefully choosing the pump cladding diameter, but also by taking into account bending and a possible index depression. Most likely, a combination of the mentioned effects needs to be evaluated for a specific design target. In some cases it may even be possible to take advantage of avoided crossings for HOM discrimination .
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 no.  “PECS”. Additionally, F.J. acknowledges financial support by the Abbe School of Photonics Jena.
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