Recent theoretical researches reveal that the self-focusing critical power in the fiber waveguide is identical to that in the bulk medium. However, the delivery of peak power much higher than the self-focusing critical power has been demonstrated experimentally in ultra-large-mode-area fiber (ULMAF). And no experimental observation of self-focusing effect has been reported in recent pulsed fiber laser system whose peak power has reached or even exceeded the critical power. In this paper, we try to address this issue by studying the self-focusing length theoretically in the ULMAF which is highly multimode. Nonlinear beam propagation method employing PÁDE(2,2) approximation is applied in the numerical simulation. The results show that the self-focusing length of the fundamental mode is typically a few millimeters which is almost identical to that in the bulk medium. However, the self-focusing length of the summation of numerous modes can be as long as a few meters.
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In the last decades, high power fiber laser technologies developed rapidly driven by the demands in industry, scientific research and defense . The output average power has reached 10kW  with diffraction limited beam quality, while the output peak power can be as high as a few MWs . To suppress the nonlinearities in the fiber laser which are the limiting factors for further power scaling, large-mode-area fiber (LMAF) is applied to reduce the intensity. For typical step-index LMAF, single-mode operation can be achieved by applying mode filtering methods such as bending . However, the core scaling of the ultra-large-mode-area fiber (ULMAF) is not acceptable because of the large bending loss . The mode-field-diameter (MFD) can be further scaled by applying photonic crystal fiber (PCF) which can work in single-mode operation for core diameter up to 80μm . The ultimate power limit in ULMAF is believed to be the self-focusing (SF) effect  whose critical power is independent of the mode field size .
As is pointed out by A. V. Smith et al. , for MFD smaller than 50μm, the bulk damage critical intensity is reached under the SF critical power. On the contrary, for MFD larger than that, SF will be the ultimate limitation. Recent theoretical researches   reveal that the SF critical power in the fiber waveguide is almost the same with that in the bulk. For higher order mode (HOM) which has multiple lobes, the lobes will rotate and merge into a single one after a distance of propagation without enhancing the SF critical power . In experimental researches, A. Galvanauskas et al.  achieve output peak power of 6MW@0.11ns in UMLAF in a 80μm-core fiber and 4.5MW@0.5ns in a 200μm-core fiber . Fabio Di Teodoro et al. achieve 4.5MW@0.45ns in a 140μm-core fiber . Actually, earlier experiments have demonstrated power delivery of 10MW  and 20MW  in ULMAF system. Why there is no obvious SF effect observed in these laser systems employing ULMAF? A simple but convincing answer is that the SF length is longer than the length of the fiber. We try to explain this problem by studying the self-focusing length (SFL) in the fiber waveguide. Nonlinear beam propagation method (NBPM) is applied for the numerical simulation. The results show that, the SFL of the fundamental mode in the fiber waveguide is similar to that in the bulk. However, for the HOMs and the summation of numerous modes, SF length can be increased substantially.
2. Nonlinear beam propagation method
For the weak waveguide we are interested in, the propagation of the field can be described by the scalar wave equation given as 16]
To take the Kerr effect into account, the local modification of the refractive index according to the intensity should be employed 
For a more accurate simulation of the beam converging process, PÁDE(2,2) is employed . The bi-conjugate-gradient (BICG) algorithm is used for algebraic solution. An iteration procedure is introduced for local nonlinearity modification Eq. (4) to modify the nonlinear refractive index and get the calculation result . If the relative error is smaller than the tolerance, the iteration terminates. Otherwise, repeat the previous procedure until the error convergences. is defined as . For PÁDE(2,2) approximation where a single step is split into two half steps, the iteration is applied in both half steps.
3. Self-focusing length of the fundamental mode
For weak waveguide where the cladding diameter is much larger than the core diameter, the field distribution of the linear polarized (LP) mode is given as follows 
Two types of fibers are used in our simulation, whose core diameters are 80μm and 200μm, respectively. The parameters of the fibers and numerical simulation are listed in Table 1 . The normalized frequency is defined as , where is the core radius and is the numerical aperture. The values of V for the 80μm-core and 200μm-core fiber are 14.2 and 35.4, which can support around 100 and 630 guided modes, respectively.
Theoretical researches [8,9] reveal that the SF critical power of the fundamental mode in fiber waveguide is the same with the that in the bulk. The critical power for linearly polarized light in the bulk is given as 19] which corresponds to a critical power of 5.1MW. The value of is used in the computation. For circularly polarized light, the value of n2 reduces to 2/3 of that for the linearly polarized light . For beam power higher than Pcr, catastrophic SF will happen after a propagation length which is denoted as self-focusing length (Lsf). In the bulk medium, the SFL is given as 20]. However, due to the aperture diameter as large as 400mm, the SFL is a few kilometers long which is much longer than the thickness of the nonlinear media used in the facility. Small scale self-focusing  instead of the whole beam self-focusing dominates in this case and a limit of B integral, which is the accumulated phase shift due to the Kerr nonlinearity, is introduced to guarantee that this nonlinearity doesn’t degrade the beam quality too much. Similar to the case in the bulk medium, we think that the SFL is an important parameter for the nonlinear propagation of the beam with peak power higher than the critical power. In this section, we will study the nonlinear propagation of the LP01 mode which is required in many applications in the ULMAF and compare the SFL in the waveguide with that in the bulk.
The intensity distributions of the LP01 mode in the 80μm-core and 200μm-core fiber are shown in Fig. 1(a) and 1(b), respectively. The mode field is basically confined in the core region, while the leaky field is almost negligible. The Gaussian fit of the field profile is also shown in Fig. 1, whose radii (1/e2 intensity) of the 80μm-core and 200μm-core fiber are 35μm and 82μm, respectively. The Gaussian fit profile coincides with the LP01 profile well within the full-width-half-magnitude region. But the Gaussian fit profile extends to the cladding. The effective radii of the LP01 mode in 80μm-core and 200μm-core fiber are 29.5μm and 71.0μm which are obviously smaller than that of the Gaussian fit beam.
In Fig. 2 , the evolution of the peak intensity along the propagation direction is given under the power of 2Pcr for Gaussian fit beam in the bulk, LP01 mode in the bulk and LP01 mode in the waveguide. The peak intensity goes through a few oscillations and the position of the first peak characterizes the SFL . In the 80μm-core fiber, the SFLs of the Gaussian fit beam in the bulk, LP01 mode in the bulk and LP01 mode in the waveguide are 3.37mm, 2.28mm and 2.36mm, respectively. In 200μm-core fiber, the SFLs of the Gaussian fit beam in the bulk, LP01 mode in the bulk and LP01 mode in the waveguide are 18.4mm, 13.1mm and 13.6mm, respectively. The results of the Gaussian fit beam in the bulk coincide well with the prediction of Eq. (7) by substituting the fitting radius into the formula. The SFL for the LP01 mode in the bulk is slightly smaller than that in the waveguide. The ratio of the SFL of the Gaussian fit beam to that of the LP01 mode (both in the bulk and in the waveguide) is close to the ratio of the square of the beam radius which is around 1.41 for the 80μm-core fiber and 1.33 for 200μm-core fiber.
As is shown in Fig. 3 , SFL decreases with the increasing power. The SFL in the bulk is close to that in the waveguide for different beam power. By substituting the effective radius of LP01 mode into Eq. (7), the numerical simulation coincides with the theoretical fit perfectly. According to these results, a few conclusions can be drawn as follows: 1) the self-focusing behavior of the LP01 mode in the bulk is similar to that in the waveguide which also implies a same critical power in both media; 2) Eq. (7) can be applied to predict the SFL in the waveguide provided that the effective radius is used; 3) for beam power larger than 2Pcr, the SFL is only a few millimeters in the ULMAF.
4. Self-focusing length of the higher order mode
It’s difficult to realize single-mode operation in ULMAF when hundreds of guided modes can be supported. Therefore, it’s important to study the self-focusing behavior of the HOM. According to the symmetry of the LP modes, we can divide them into two categories, i.e. LP0m mode and LPlm mode (). For LP0m mode, the peak intensity locates at the center. There are m-1 rings outside of the central lobe. Because the self-focusing effect is due to the nonlinear refractive index proportional to the intensity profile, the central lobe is critical for the self-focusing evolution. As is shown in Fig. 4 , the ratio of the peak intensity increases almost linearly with the order m. On the contrary, the radius of the central lobe decreases with m.
As is shown in Fig. 5(a) and 5(b), the SFLs of the LP02 mode under 2Pcr in the 80μm-core and 200μm-core fiber are 1.65mm and 14.0mm, respectively, which are still comparable to that of the LP01 mode. The central lobe dominates the self-focusing process. As is shown in Fig. 5(c) and 5(d), the SFLs of the LP04 mode under 2Pcr in the 80μm-core and 200μm-core fiber are 9.50mm and 40.0mm, respectively, which are much larger than that of the LP01 mode. For LP0m modes, the decrease of the central lobe radius and increase of the peak intensity tend to reduce the SFL. However, the ring-type nonlinear refractive index due to the intensity ring tends to enlarge the mode field which will also increase the SFL. For small m, these two effects are comparable, which makes the SFL comparable to that of the LP01 mode. For larger m, the latter effect dominates which leads to a larger SFL.
For LPlm modes (), there are separate lobes. A. V. Smith et al.  pointed out that the two lobes of the LP11 mode in a 25μm-core fiber will rotate and merge into a single lobe in a few millimeters. However, according to Ref , the fiber with this core diameter cannot deliver power higher than Pcr. In the ULMAF, the separation between the two lobes is larger and their interaction becomes weak. As is shown in Fig. 6 , in both of the 80μm-core and the 200μm-core fiber, the two lobes of the LP11 mode evolves almost independently. The corresponding SFLs under the power of 2Pcr in the 80μm-core and the 200μm-core fiber are 3.80mm and 19.0mm, respectively, which are 1.6 and 1.4 times of that of the fundamental mode under the same power.
5. Self-focusing length of the summation of numerous modes
The practical laser system employing ULMAF usually works in multimode operation. The output beam quality degrades because of the excitation of HOMs. The beam quality can be calculated with the methods reported in Ref . The summation of N LP modes can be written as
The nonlinear propagation of the summation of multimode in the 80μm-core fiber is shown in Fig. 7 under power of 2Pcr and 6MW. Due to the interference of the numerous modes, obvious speckles pattern is shown in the input field. For power of 2Pcr, catastrophic self-focusing happens after a propagation distance of 432mm. On the contrary, for power of 6MW, no catastrophic self-focusing happens within 1m propagation although the speckles seem to merge into larger ones after 600mm propagation. Because of the difference in propagation constants for various LP modes, the speckle pattern changes rapidly during the propagation which means the nonlinear refractive index cross section is not stable. The variation of the refractive index interrupts the self-focusing process which leads to a much larger SFL.
The nonlinear propagation of the summation of multimode in the 200μm-core fiber is shown in Fig. 8 under power of 2Pcr. No catastrophic self-focusing happens after a propagation of 2m. The speckles of the input field merge into hotter ones (red speckles) in the output field. The beam power reduces by around 0.7MW due to the leakage of radiation mode during the nonlinear propagation. The peak intensity oscillates randomly but the maximum does not exceeds twice of the mean value which is slightly larger than . The beam quality factors in both x direction and y direction are larger than 10 although they reduce slightly by 3. As is revealed by the simulation, the 200μm-core fiber can deliver optical power larger than Pcr safely for a fiber length which can be applied in practical fiber system provided that the numerous modes are excited in the fiber.
5. Conclusions and discussions
Nonlinear beam propagation method employing PÁDE(2,2) approximation is applied to study the SFL in the ULMAF. The self-focusing process of the LP01 mode in the bulk and waveguide are similar and the SFL can be characterized by the effective beam radius. In large aperture bulk laser driver systems, flat-top beam shape is applied to increase the spatial duty cycle and relieve the whole beam self-focusing. However, the flat-top beam cannot maintain its shape during the propagation in the multimode fiber if it’s a summation of the numerous modes. The SFL of the HOM can be lengthened substantially under the same power compared to that of the fundamental mode. The varying speckle pattern due to the multimode interference interrupts the self-focusing process which makes the delivery of optical power much higher than Pcr in the long ULMAF possible. Linear polarization and continuous wave or long pulse are assumed in the computation, which means that the critical power can be further enhanced by injecting circularly polarized light or ultrashort pulse. For applications where near single-mode operation is required, Pcr is the ultimate power limit and only a few millimeters propagation is allowed for beam power higher than Pcr. On the contrary, for applications where the beam quality requirement is low, much higher power can be delivered by exciting numerous modes in the fiber. If the fiber is divided into segments shorter than the SFL and the self-focusing process is interrupted in the joint of neighboring segments, e.g. by expanding the core, the effective SFL may be increased further.
This research is supported by Foundation of Science and Technology of Plasma Physics Laboratory under grant 9140C6803010905 and National Natural Science Foundation of China under grant 60878058.
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