Thermally induced waveguide changes become significant for very large mode area fibers. This results in a reduction of the mode-field diameter, but simultaneously in an improvement of the beam quality. In this work the first systematic experimental characterization of the reduction of the mode-field diameter in various fibers during high-power operation is carried out. It is shown that the reduction of the mode-field diameter shows a characteristic behavior that scales with the core size but that is independent of the particular fiber design. Furthermore, the strength of the actual index change is experimentally estimated, and its use to overcome avoided crossings is discussed and experimentally demonstrated.
©2012 Optical Society of America
The increasing appeal and broad presence of fiber lasers in science and industry is the direct consequence of their exceptional properties such as excellent beam quality, high efficiency and low maintenance costs, just to mention a few. In recent decades the output powers of continuous-wave and pulsed fiber lasers have increased from a few Watts all the way up to the kilowatt regime. Due to the confinement of the energy within small areas over long lengths, nonlinear effects tend to play a limiting role in these devices. Therefore, the mode area of fibers had (and still has) to be increased especially in the case of pulsed operation. Several sophisticated approaches have been successfully realized to enforce single-mode operation while increasing the mode-field area of active fibers [1–4].
One of the most beneficial advantages of fiber lasers and amplifiers is their ability to cope with thermal effects. This attribute is mainly due to their large surface compared to their active volume. However, it is only natural that with the need for ever growing mode-field areas and core diameters, at some point or another, thermal effects should become measurable even in fibers. Recently, one manifestation of these effects has been discussed in the literature where thermally induced long-period gratings probably play a major role in the threshold-like appearance of mode instabilities in active high power fibers [5–7].
The thermal effects discussed in this paper are connected to the parabolic transversal thermal profile in the core (and the logarithmic decay in the cladding), which, through the thermo-optic effect, leads to a mode-field area shrinking or even to a complete change of the guided mode set. Note that these thermal effects occur in all single- and multimode fibers during high-power operation regardless of their inner structure (step index or photonic crystal). For small mode-field diameters, power dependent waveguide changes during active operation can be mostly neglected based on simple approximations . For moderate output powers their impact is below a few percent of the mode-field diameter, which is hard to notice in practice taking into account the typical measurement accuracies. However, for larger mode-field diameters the impact of index changes in the range of 10−5 becomes significant [9–12]. For active high power operation these index changes can be induced by temperature , inversion [13, 14] or the Kerr effect . The absolute index change due to the inversion level is typically limited to 0.5∙10−5 (for doping concentrations of ~3⋅1025 ions/m3). The Kerr effect typically can be neglected in low or moderate pulse energy systems. On the contrary, the thermo-optical effect can raise the refractive index by 1∙10−5/K. During high power operation a temperature gradient (measured from the core center to the core edge) of several Kelvin can be expected, thus, the resulting index change can be about one order of magnitude stronger than inversion or Kerr-effect induced index changes.
Thus, it is important to consider the implications of thermally induced index changes. On the one hand, in the following the reduction of the effective mode area will be discussed. The knowledge about this mode shrinking is of utmost importance when low nonlinearity is the reason for mode area scaling. On the other hand, it will be shown that thermally induced index changes have to be considered when designing Very Large Mode Area (VLMA) fiber.
For the first time, to the best of our knowledge, thermal waveguide changes in VLMA fibers are experimentally studied during high power operation. To get a feeling about the relevance of these changes, several Ytterbium-doped fibers with the same structure but different sizes are compared. In order to separate different effects, the discussion is split in two main parts: after describing the experimental setup in section 2, firstly, the effect of mode shrinking is discussed in section 3. Then, section 4 describes a beneficial effect of thermal waveguide changes thanks to which index-depression-induced avoided crossings  are overcome leading to high average output power with VLMA and excellent beam quality. Additionally, this enables a rough experimental estimation of the strength of the thermally induced index change. Section 5 summarizes the results.
2. Experimental setup
The test setup comprised a state-of-the-art high power chirped-pulse amplification system. Consisting of an Yb:KYW oscillator, a stretcher and two fiber amplifiers, the system delivered stretched pulses with a pulse duration of 630 ps and a pulse repetition rate of 39 MHz centered at a wavelength of 1040 nm. For our series of experiments this system was used with ~10 W of average output power to seed the main amplifier. This final stage consisted of the Fiber Under Test (FUT) in a counter-pumped amplifier configuration with a pump wavelength of 976 nm.
As test fibers for this study on waveguide changes, Ytterbium-doped Large-Pitch photonic-crystal Fibers (LPFs) were chosen as their unique properties have recently enabled a significant increase of single-mode output power in fiber laser systems with mode-field diameters (MFDs) exceeding 50 µm [4, 17]. The key factor for the robust single-mode operation of these fibers is their open waveguide structure (Fig. 1 ) that provides a strong delocalization of higher order modes . These delocalized modes experience a strongly reduced gain. Furthermore, this delocalization reduces the excitation of the higher order modes of these fibers.
For our study several LPFs with hole-to-hole distances (pitches) between 30 µm and 75 µm have been tested, leading to mode-field diameters between 53 µm and approximately 130 µm in passive operation (from now on referred to as 'cold' state). All LPFs are rod-type fibers, meaning that they cannot be bent to ensure very large mode-field diameters over their whole length and, therefore, low nonlinearity. The length of all the LPFs was around 1.2 m. Additionally, two commercially available (NKT Photonics A/S) standard photonic crystal fibers (PCFs) were studied for comparison: the bendable DC-200-40-PZ-Yb-03 (3 m fiber length) and the rod-type DC-285-100-PM-Yb-ROD (1.2 m fiber length), designated in the following as 200/40 and 285/100, respectively. Table 1 summarizes all FUT.
The extracted output power was calculated by subtracting the average seed power from the measured average output power. To monitor the evolution of the output mode and the mode-field diameter with increasing output power, the fiber end facets were imaged onto a CMOS camera using just a fraction of the beam power. The camera image was calibrated using the known geometric dimensions of the fiber as reference (i.e. hole-to-hole distance and air-clad diameter). Then, the effective mode-field area was calculated by integrating the intensity of a measured near field profile as follows :
To reduce the influence of stray light, intensities smaller than 5% of the maximal value were neglected. This, of course, slightly underestimates the mode-field area (by ~5%) but ensures a realistic lower limit without the influence of stray light. Finally, the average mode-field diameter was approximated from the effective area:
In the presence of avoided crossings (see section 4), the core area had to be additionally cropped by a circular aperture to determine the appropriate mode-field area. Furthermore, high quality optics are required to rule out any defocusing effect on the lenses and beam-splitters, which might influence the near-field image. The reliability of our imaging setup during high power operation has been verified by comparing the near-field fiber structure at different output powers without measuring any deviation.
3. Thermal index gradient in high power VLMA fibers
This section focuses on the formation of a thermally induced parabolic index gradient in the fiber core and its impact on the mode properties. This index gradient is often referred to as thermal lens. However, it is important to note that the calculation of an equivalent focal length for this 'thermal lens' makes no physical sense inside a fiber . Furthermore, the term 'thermal lens' is usually associated with detrimental effects concerning beam quality and system stability for bulk solid-state lasers, which is not the case in the context of optical fibers. Therefore, we have intentionally chosen not to use this term and, instead, refer to name it after its physical origin, the parabolic index profile. In  the parabolic profile was numerically studied and considered as a 'limiting factor' for very large mode areas at high-power operation. Nevertheless, we think that this point of view is too simplified and that, as it is shown in the following, this thermal profile might even result to be advantageous to a certain degree. Furthermore, since this thermal index profile can substantially modify the waveguide characteristics of the fiber, it has to be considered as an integral part of the fiber design.
The most obvious consequence of a thermally induced parabolic index profile is a decrease of the mode-field diameter, denoted as mode shrinking in the following. Additionally, in PCFs, depending on the design employed, the hexagonal features of the fundamental mode become weaker as the parabolic index profile becomes successively more dominant. Figure 2 illustrates this by showing the measured evolution of the fundamental mode of a LPF30 with increasing extracted average output power. This figure clearly demonstrates the progressive fading of the hexagonal features as the thermal index profile takes over as the main guidance mechanism. Furthermore, a shrinking of the fundamental mode diameter by ~20% for the output beam mode profile has been measured for this fiber at 350 W extracted output power when compared to the cold setup.
The measured mode-field diameter for different fibers depending on the extracted output power is depicted in Fig. 3 and summarized in Table 2 . It has to be noted that the measurements are limited by the onset of mode instabilities . All given MFD values refer to the MFD at the fiber end facet where the thermal load is largest in a counter-pumped amplifier. Along the fiber length, the mode-field diameter remains larger. The effect of mode shrinking can be seen in all FUT, but it becomes more pronounced the larger the initial MFD (MFDcold). For LPF30, LPF35, LPF45 and the 200/40 the measurements can be approximately fitted by a linear regression. This allows for the calculation of an average (linear) mode shrinking per 100 W extracted output power.
However, the largest FUT (LPF60 and LPF75) cannot be linearly fitted mainly due to the strong mode deformations induced by avoided crossings (see section 4). They show an exponential-like decrease of the MFD and, thus, a non-constant slope. Please note that the large statistical spread of the MFD for LPF45, LPF60 and LPF75 in low power operation is basically due to the presence of avoided crossings for the fundamental mode. On the one hand, avoided crossings cause an interaction of at least two FM-like modes leading to an output profile that strongly depends on the seed excitation and pump conditions. On the other hand, a strong deformation of the fundamental mode takes place within a large pump power region in which, even though the pump power is increased, the signal output power remains more or less constant. These effects are discussed in detail in section 4.
In Fig. 4 , the slope of the mode shrinking is displayed versus the (fitted) cold MFD. The physical origin of the mode shrinking is the thermal load deposited in the material due to power conversion. Therefore, it is proportional to the absorbed pump power or, for saturated amplifiers, it is proportional to the extracted output power. Hence, the given values for mode shrinking are normalized to the extracted output power (per 100 W) per fiber length (in m). This ensures that the results are comparable between different fibers and fiber lengths. Correspondingly, for a 1.2 m long LPF30 with 6.5% mode shrinking per 100 W/m of extracted power an absolute mode shrinking of 19% has to be expected for 350 W of extracted output power. For MFDs below 40 µm the mode shrinking can be neglected, but it increases to values of more than 20% per 100 W/m for cold MFDs of up to 80 µm. For cold MFDs approaching 100 µm, the mode shrinking increases further, but the specification of a constant slope makes no physical sense, at least not in the presence of avoided crossings (see next section). Seeing Fig. 4, it can be deduced that the mode shrinking increases exponentially with the mode-field diameter. The LPFs and the commercial index-guiding PCFs studied for comparison both fit into this general picture, indicating the general validity of this analysis for all types of fibers including step-index fibers.
The effect of mode shrinking has been discussed theoretically in the past [8–10]. Based on  the index profile can be approximated by a parabolic shape inside the doped area (core) and by a logarithmic decay in the undoped region where no heat is generated. For the experiment with the LPF30 the thermal load was estimated based on numerical simulations of the laser system. At 350 W of extracted output power, a thermal load of ~20-40 W/m can be expected in the last ~5 cm of a counter-pumped amplifier. The resulting transversal temperature gradient inside the fiber core is ~1.5-4 K. This, in combination with a thermo-optic coefficient of 1.25⋅10−5 K−1 , yields a parabolic index gradient of ~2-5⋅10−5 for an output power of 350 W. It is difficult to measure index changes in the order of some 10−5 directly, but, a first experimental estimation based on avoided crossings is provided in the next section.
It has to be noted that the mode shrinking caused by the thermal load in the fiber core becomes stronger in the region closer to the fiber output facet of an amplifier in counter-pumped configuration. For short fibers, e.g. the FUT, this region extends over the last tens of cm. Even though the thermal load along the fiber depends on different parameters, e.g. on the pump-configuration (co- vs. counter-pumping) and the pump absorption an appropriate fiber design may compensate for the index gradient inside the core and, therefore, for mode shrinking. For example, the index profile may be pre-compensated as proposed for bend fibers just recently . Another promising idea to reduce mode-shrinking is based on the doping geometry which, in turn, affects the thermal-load distribution along the fiber. For instance, reduced mode overlap fibers have been proposed to reduce photodarkening , but it can be shown that the thermally induced index gradient is reduced as well. Finally, in the context of ultra-short pulse amplification the impact on the accumulated nonlinearity (B-integral) can change significantly depending on the mode-shrinking .
In conclusion, the impact of thermal effects in active high power fibers exponentially scales with the mode-field diameter. The comparison of two different fiber designs (index-guiding PCFs and LPFs) shows a similar sensitivity to thermal effects. The core size of VLMA fibers has reached a dimension at which thermal effects have to be considered in the fiber design process. Some fiber designs, like index-guiding PCFs or LPFs, can cope with these thermal effects without seeing any significant degradation of their performance or beam quality. However, thermal effects may become critical for VLMA fiber designs relying on resonant effects, such as photonic bandgap fibers  and fibers with resonant filtering mechanisms, namely distributed mode filter fibers  or chirally-coupled core fibers . In these fiber designs a waveguide change during high power operation can severely affect the resonance condition.
4. Avoided crossings in active fibers
In this section the influence of thermal effects on avoided crossings (also known as anti-crossings) is discussed. This effect enables a first experimental estimation of the actual strength of thermally induced index changes. Just recently, avoided crossings have been observed in  causing a severe beam quality degradation approaching a certain output power level. However, the following investigation shows that the thermally induced index profile is sufficiently strong to overcome this beam quality degradation.
For crossings and avoided crossings to appear, the effective refractive indices of at least two modes have to approach each other while changing a waveguide parameter or the wavelength. At a mode crossing, the effective indices of two modes simply cross, leading to a point where they have identical effective indices but completely different intensity profiles. During an avoided crossing, on the contrary, the participating modes repel each other in the effective index while their intensity profiles converge. After the avoided crossing, their roles/intensity profiles are interchanged.
Avoided crossings have been studied in detail for example in the context of hollow-core photonic bandgap fibers , where core and surface modes can interact. Furthermore, they were shown to exist in leakage channel fibers  and index-guiding PCF . The open waveguide structure of LPFs that ensures robust single-mode operation also facilitates the interaction between core and cladding modes. The parameters that cause avoided crossings in LPFs have been investigated in . A careful choice of air-clad diameter, bending radius and index mismatching between the actively doped core and the silica matrix guarantees that the Fundamental Mode (FM) of the waveguide is not influenced by avoided crossings.
Avoided crossings caused by an index mismatch between the rare-earth doped core and the silica matrix are of particular interest for this work. During the production of a PCF it has to be avoided that the doped core material has a higher index than the silica matrix in which it is embedded, since this would result in step-index guidance. Thus, the refractive index of the doped material is typically lower than that of the surrounding silica. The resulting index depression (Δn = nsilica - ndoped) has to be smaller than 5·10−5 for VLMA fibers . In spite of its small value, this index depression facilitates the interaction of core and cladding modes.
The impact of avoided crossings caused by index depressions depends on the mode area of the LPF. Fibers with larger mode-field areas are more sensitive to index depressions, which is due to the fact that the effective indices of the guided modes converge with larger mode-field areas. Therefore, the same absolute value of the index depression can cause FM deformations by avoided crossings in up-scaled fibers while smaller fibers are not affected. This relationship is highlighted in Fig. 5 . There, the position of FM avoided crossings depending on the hole-to-hole distance (pitch) Λ is depicted. A LPF design with two rings, a relative hole size d/Λ of 0.22 and an air-clad diameter of 5.7Λ has been simulated (corresponding to the LPFs under test), further details can be found in . A characteristic feature of index-depression-induced avoided crossings is that the FM (the mode that best resembles a Gaussian mode) exhibits an increasing number of peripheric ring-like features after each avoided crossing. For index depressions below the first avoided crossing (black curve), the fundamental mode maintains its Gaussian-like shape. In the vicinity of the black curve the FM is deformed by the first avoided crossing. A higher order mode with a single ring-like feature interacts with the fundamental mode of the waveguide and they exchange their roles while passing the first avoided crossing. Above the black curve the most Gaussian-like mode exhibits one faint ring-like feature while the former FM is delocalized in the air-clad. Close to and above the second avoided crossing the fundamental mode acquires a second ring-like feature, etc. From this point of view, fundamental mode operation with high beam quality can only be guaranteed below the black curve of the first avoided crossing in Fig. 5, resulting in a seemingly strict limit for mode-field area scaling in these fibers. However, thermal effects can compensate these index depressions and, consequently, avoided crossings that appear in an unpumped LPF with a certain index depression can be overcome during active operation. An increasing strength of the thermal profile in active operation can be interpreted as a downward movement in Fig. 5, as we will show in the following.
In a real fiber the index depression is defined during the fiber manufacturing process. Even for index depressions <5⋅10−5, avoided crossings become hard to eliminate for very large LPFs. Assuming a proportional scaling of the geometric parameters, one or more avoided crossings affect the output mode above a certain size. All LPFs in this publication have been drawn from the same preform to guarantee that they are as similar as possible. Their relative size is indicated in Fig. 5. In cold operation, for LPF30 and LPF35 no avoided crossings can be observed, while, on the other hand, LPF45 shows one avoided crossing. At the same time both the LPF60 and the LPF75 are affected by two avoided crossings. These observations lead to the conclusion, that the index depression of these LPFs is around (2.3 ± 0.3)⋅10−5, as indicated by the shaded gray area in Fig. 5.
Figure 6 illustrates two typical examples of the output beam profile of a LPF affected by avoided crossings. For the LPF45 (Fig. 6a) the avoided crossing region is overcome at about 15 W, while in the LPF60 it is far more pronounced (Fig. 6b). It is worth mentioning that the actual modal shape in the region of avoided crossings (low power operation) is strongly dependant on the excitation since the output beam is always a coherent superposition of the modes taking part in the corresponding avoided crossing.
The thermal impact is not homogeneous along the fiber and it depends on the particular amplifier configuration. In the counter-pumped configuration the point with the highest thermal load is at the output end of the fiber. Thus, any waveguide change by thermal load occurs first (i.e. at lower powers) at the output end and it then propagates with increasing pump power in the direction of the seed side. Finally, even the coupling conditions may be modified by thermal waveguide changes. All these effects combined make a detailed simulation difficult. Therefore, in the following only a qualitative understanding is provided.
The initial output beam profile of the LPF45 (first picture in Fig. 6a) is a superposition of two FM-like modes being affected by an avoided crossing (compare with Fig. 5, black curve). However, with increasing pump and extracted output power (going to the right in Fig. 6a) the thermally induced index profile becomes successively stronger and the ring-like feature fades. Finally, above some 10 Watts of extracted output power the avoided crossing has been overcome and a fundamental mode with good beam quality is emitted.
The situation is more complicated for the LPF60 (Fig. 6b). Here, in passive operation a fundamental mode can be excited. This fundamental mode is slightly distorted by some faint peripheric ring-like features, which can be explained as a tail of the second fundamental mode avoided crossing (red curve in Fig. 5). The beam quality is increased by pumping this fiber, but after some Watts of average output power at first a strong ring-like feature emerges. This ring-like feature can be explained by the first avoided crossing (black curve in Fig. 5), which is reached due to the thermally induced index profile.
The strong deformations of the output beam profile are also reflected in the slope of this amplifier (Fig. 7 ). In the presence of a strong avoided crossing, the output power does not increase linearly with the pump power as the overlap with the doped region is dramatically reduced at some power levels (fourth image in Fig. 6b). Furthermore, it has to be noted that, theoretically, modes involved in an avoided crossing do not exchange energy as they remain orthogonal to each other . Thus, the power seeded into a FM before an avoided crossing might not be transferred completely into the FM after the avoided crossing. Both effects combine to cast a nonlinear slope.
Further pumping increases the thermally induced gradient (upward movement in Fig. 5) and shifts the avoided crossing region along the fiber towards the seed input end. Finally, these strong deformations are overcome and the extracted output power increases fast until a linear slope is reached once again. For extracted average output powers above ~30 W robust single mode operation was observed. However, as shown in section 3, the effect of mode shrinking decreases the mode-field diameter to ~70 µm at an average output power of 110 W. Simultaneously, the beam quality factor (measured with second moments method) improves from M2 < 1.4 at ~40 W to M2 < 1.2 at average output powers above 70 W.
While the general behavior of the LPF75 is similar to that of the LPF60, the deformations due to avoided crossings are even stronger. The avoided crossings are only overcome beyond ~50 W of output power. Above this regime an average output power of 100 W with robust single-mode operation and a (minimal) MFD of ~100 µm can be observed.
Lastly, these basic considerations give a qualitative understanding of the behavior of the output beam profile. By increasing the output power (i.e. the thermal load), a thermally induced index-profile is formed, which effectively reduces the index depression in the fiber core. From the comparison between numerical simulations (Fig. 5) and experimental observations (Fig. 6) it can be inferred that the thermally induced index profile overcompensates the index depression. Therefore, the thermally induced index profile has to be stronger than 2⋅10−5 for extracted output powers of only 50 W.
In this publication a first systematic experimental study of thermally induced waveguide changes that occur in all fibers was presented. LPFs were employed as model fibers due to the fact that they enable robust single-mode operation at high average output powers and very large mode-field diameters. Furthermore, they can easily be up-scaled as they do not rely on resonant effects.
It was shown that thermally induced waveguide changes have to be considered for VLMA fibers with mode-field diameters beyond 50 µm. A parabolic thermally induced index profile manifests itself in a decreased mode-field diameter in high power operation. This effect is negligible for fibers with mode-field diameters of some ten microns, but it becomes significant for VLMA fibers (MFD > 50 µm). For example the fundamental mode of a fiber with a 'cold' mode-field diameter of 130 µm reduces to approximately 100 µm at ~100 W extracted average output power. The mode shrinking is strongest where the thermal load is largest, i.e. the output end for a counter-pumped amplifier. The values given in this paper only refer to the shrinking at the fiber output. Several fibers have been studied to show that these thermal effects are a common property of VLMA fibers and that their impact strongly scales with the mode-field diameter. Simultaneously, in PCFs with slight hexagonal features of the fundamental mode, these thermal index gradients result in a beam cleanup and improved beam quality.
It was pointed out that the thermally induced waveguide changes can be expected for all fiber designs. Therefore, this effect may become increasingly challenging especially for VLMA fiber designs that rely on resonant effects, i.e. photonic bandgap fibers, distributed mode filter fibers or chirally-coupled core fibers.
The guiding properties of fibers enable fiber designs that address these thermal effects. This will pave the way for active fibers with mode-field diameters far beyond the 100 µm limit during high-power operation. For example, the thermal index profile may be pre-compensated, similarly to a pre-compensation for bending-induced index changes . Furthermore, the heat load may be tailored by changing the doping geometry. This can be accomplished, for example, with reduced mode overlap fibers, which have been proposed for reduced photo darkening effects .
The research leading to these results has received funding from the German Federal Ministry of Education and Research (BMBF) and the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no.  “PECS” and the Thuringian Ministry for Economy, Labour and Technology (TMWAT, Project No. 2011 FGR 0103) with a European Social Fund (ESF) grant. Additionally, F. Jansen acknowledges financial support by the Abbe School of Photonics Jena.
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