Yb-doped double-cladding large mode area rod-type photonic crystal fibers are a key component for power scaling in fiber laser systems. Recently, designs with 19-cell core defect, that is with 19 missing air-holes in the center of the photonic crystal cladding, have been proposed, with reported core diameter up to 100 μm. In this paper an analysis of the cut-off wavelength of the first high-order mode in such low-NA fibers is reported, accounting for different approaches for the definition of the cladding effective index. Results have shown that taking into account the finite fiber cross-section and considering the first cladding mode of the actual fiber is mandatory to obtain a correct estimate of the cut-off wavelength.
©2011 Optical Society of America
In recent years a great interest has been focused on fiber lasers, leading to a dramatic increase of the performances of fiber laser systems in terms of output power, beam quality, efficiency and available emission wavelengths. Remarkably, the maximum reported output power in continuous-wave operation has been increased by a factor of 1.7 per year in the last decade , up to the current state-of-the-art of 10 kW average power, diffraction-limited output beam laser . Such a constant growth of fiber laser performances has been supported by an equally fast improvement of the optical fiber technology, that constantly supplied innovative solutions for power scaling, better nonlinear effect suppression, higher pump absorption, and access to new wavelengths through different rare-earth dopants .
Yb-doped double cladding large mode area fibers have been the key component to obtain such high powers, while avoiding nonlinear and thermal effects. The beam quality is another very important issue for many applications, and therefore the suppression of high-order modes is often a mandatory requirement for these fibers. The unique properties of Photonic Crystal Fibers (PCFs) can be successfully exploited to realize ultra low-NA designs which combine large effective area and single-mode guiding. However, the minimum required bending diameter increases with the core size, and this limits the practical usable core diameter for flexible fibers. By instead using inflexible rod-type PCFs core diameters up to 100 μm have been obtained , by removing 19 air-holes from the center of the triangular lattice in the photonic crystal cladding. In this kind of doped fibers the air-cladding provides guiding of the pump light from high-power low-brightness multimode diodes .
So far, a systematic study of the single-mode regime of fibers with a 19-cell core has not been reported. Cut-off analysis have been presented in literature for 1-cell triangular PCFs [5–8], that is for fibers with only 1 missing air-hole in the cross-section, and for 7-cell PCFs [9, 10], where the detrimental effect of the core enlargement on the single-mode regime is clearly shown. In most of these studies the classical parameters used for cut-off analysis for step-index fibers, such as the V-number, have been adapted to the PCF case. In particular, the effective index of the Fundamental Space-filling Mode (FSM) of the photonic crystal cladding, assuming its extension infinite, is considered as the refractive index of the cladding in conventional fibers. Therefore, it is usually supposed that the cut-off condition of the PCF guided modes is reached when their effective index equals the FSM one.
In this paper the cut-off properties of 19-cell Yb-doped double-cladding PCFs have been investigated for the first time, to the authors’ knowledge. In particular, a full-vector modal solver, based on the Finite Element Method (FEM) , has been used to calculate the Fundamental Mode (FM) and the first High Order Mode (HOM) for PCFs with different number of air-hole rings in the inner cladding, and with different core refractive index. The cut-off condition has been estimated by taking into account the dispersion curves of the guided modes and of the cladding one. The effective index of the PCF cladding has been computed considering different approaches. Besides FSM, the effective index of the first cladding mode of the fibers with finite dimension, due to the air-cladding presence, has been calculated, first removing, and then adding the core defect. Moreover, the overlap integral on the fiber doped core  has been considered as the figure of merit to evaluate the confinement of the modes at cut-off, and results have been compared to the ones obtained for a standard single-mode fiber. The analysis of the values obtained with different approaches has demonstrated that both the methods based on the FSM and the finite cladding, neglecting the core presence, are not reliable for studying the single-mode regime of these fibers. In fact, the HOM confinement is still too strong at the cutoff condition evaluated. On the contrary, by considering the first cladding mode of the actual fiber, fairly more accurate cut-off wavelengths have been obtained for each PCFs, as well as overlap integral values closer to the one calculated for the standard step-index fiber.
2. PCF characteristics
The 19-cell Yb-doped double-cladding PCFs considered in the present analysis are all characterized by air-holes with diameter d = 0.1 Λ, being Λ = 16.6 μm the hole-to-hole spacing. The inner-cladding of the fibers analyzed is formed by 5, 4 and 3 air-hole rings, as shown by the PCF cross-section reported in Fig. 1(a), (b) and (c), respectively. Notice that the Yb-doped core has a hexagonal shape, with edge equal to 5/2Λ, that is 41.5 μm, due to the stack-and-draw fabrication process. It is important to underline that the air-cladding position has been scaled according to the air-hole ring number, while keeping fixed to 7 μm its width. In particular, the air-cladding inner radius is 141.5 μm, 122 μm and 99.5 μm for the 5, 4 and 3 air-hole ring inner-cladding, respectively. Three values of the core refractive index have been considered, that is 1.44985, 1.44990 and 1.44995, which are lower than the silica one, equal to 1.45. Core down-doping, which is necessary to reduce the refractive index increase due to the presence of Yb ions, provides also an improvement of the HOM suppression, as it has been already demonstrated for large-mode-area double-cladding rod-type PCFs .
3. Numerical modelling
In order to analyze the cut-off regime of the 19-cell Yb-doped double-cladding PCFs, a FEM-based full-vector modal solver has been applied . The guiding properties of the FM and of the first HOM have been studied. As shown in Fig. 1(b) and (c), a quarter of the fiber cross-section has been considered for the numerical analysis. An electric wall along the vertical boundary and a magnetic one along the horizontal edge of the computational domain have been used to calculate the FM. Otherwise the same boundary condition, that is an electric wall, has been applied to both the horizontal and vertical axis to obtain the first HOM. The effective index n eff and the overlap integral Γhex of the guided-mode field on the hexagonal Yb-doped core have been calculated as a function of the wavelength. In particular, the overlap integral has been evaluated as Γhex = ∫∫hex i(x, y)dxdy, being i(x, y) the guided-mode normalized intensity distribution and hex the core hexagonal active section . Notice that the overlap integral is a significant parameter to describe the properties of large mode area Yb-doped PCFs, since it not only provides a figure of the interaction between the Yb ions and the signals, and, in turn, of the amount of the achievable amplification , but it also describes the tightness of the confinement of the modes into the fiber core, helping to discriminate if they are guided or not. As prior mentioned, the bending loss of very large core fibers is very high , and thus the present analysis is limited to rod-type inflexible PCF designs . As a consequence, the bending effects have been neglected in this study.
The effective index of the FSM has been obtained using a freely available software package, taking into account an infinite cladding . On the contrary, the dispersion curve of the first mode of the finite cladding, with or without the core defect, has been calculated by applying the FEM-based full-vector modal solver.
4. Crossing based cut-off analysis
The dispersion curves of the FM and the HOM of the 19-cell Yb-doped double-cladding PCFs with 5, 4 and 3 air-hole rings and a fixed core refractive index n core = 1.44985 are reported in Fig. 2(a), (b) and (c), respectively. The effective index n eff of the FSM, calculated for an infinite cladding, and of the first mode of the finite cladding, evaluated in the absence of the core, are also shown. Notice that, as expected, the difference between the n eff values of the FSM and of the finite cladding mode becomes higher when the air-hole ring number decreases, especially at longer wavelengths in the range considered, that is 1000–1500 nm. In fact, the guiding properties of the cladding mode are strongly influenced by the inner cladding dimension and, when it is formed by few air-hole rings, by the silica region before the air-cladding. As demonstrated by the magnetic field modulus distributions of the first cladding mode at 1250 nm reported in Fig. 3(a), (b) and (c) for the PCF with 5, 4 and 3 air-hole rings, respectively, there is a significant field component in the silica area between inner and air-cladding for the PCF with only 3 air-hole rings.
The cut-off wavelengths of the FM and of the HOM, called λ c , FM and λ c , HOM respectively, have been calculated, firstly, as the intersection between the dispersion curves of each guided mode and the FSM one. Notice that the HOM cut-off wavelength has a significant red-shift of about 50 nm as the inner-cladding dimension decreases, being about 1276 nm and 1325 nm for the PCF with 5 and 3 air-hole rings, respectively. After that, λ c , FM and λ c , HOM have been evaluated again by considering the crossing with the dispersion curve of the finite cladding mode. Notice that the cut-off wavelengths obtained with this approach are similar to the ones calculated with the FSM method only for the PCF with the largest inner-cladding. On the contrary, the difference becomes significant as the air-hole ring number decreases. In particular, an opposite behaviour of the HOM cut-off wavelength has been demonstrated as the PCF inner-cladding becomes smaller, since λ c , HOM is blue-shifted of about 30 nm, being about 1266 nm and 1232 nm for the 5 and 3 air-hole fibers, respectively. The same comments apply to the FM cut-off wavelength. It is important to underline the presence of anti-crossings in the dispersion curves of the FM for the PCFs with 4 and 3 air-hole rings, as shown in Fig. 2(b) and (c). This is due to the coupling, which is easier for thinner inner-cladding, of the guided mode with the one confined in the silica region before the air-cladding, as demonstrated by the FM magnetic field modulus distributions around the anti-crossing at 1235 nm for the fiber with 4 air-hole rings, reported in Fig. 4(a)–(e) for the wavelengths between 1225 nm and 1245 nm.
The anti-crossings in the FM dispersion curves correspond to deep minima in the overlap integral Γhex spectra, which are shown in Fig. 5(a), (b) and (c) for both the guided modes of the 19-cell PCFs with different inner-cladding dimension. The FM and HOM cut-off wavelengths previously calculated with the FSM and the finite cladding mode approach are shown with black and pink vertical lines, respectively, in order to clearly demonstrate the width of the PCF single-mode region Δλ SM, which is the wavelength range between λ c , FM and λ c , HOM. Notice that Δλ SM is only slightly affected by the number of air-hole rings in the inner-cladding of the 19-cell fiber. However, the single-mode wavelength range obtained with the FSM-based approach is red-shifted with respect to the one calculated using the finite cladding mode, especially for the 3 air-hole ring case.
The cut-off wavelengths of the FM and the first HOM, as well as the single-mode wavelength range calculated for the 19-cell PCFs with 5, 4 and 3 air-hole rings with the method based on the mode of the infinite cladding, and the one of the finite cladding without the core defect, are summarized in Tab. 1.
4.1. Overlap integral analysis
The values of the FM and HOM overlap integral at the cut-off wavelengths estimated with both the methods are shown with symbols in Fig. 5. For what concerns the results obtained with the approach based on the FSM effective index, notice that the HOM Γhex at λ c , HOM is quite high and it increases as the inner-cladding becomes smaller, being, respectively, 0.41 and 0.61 for the double-cladding PCF with 5 and 3 air-hole rings. The same behaviour has been obtained with the method based on the finite cladding mode. However, the values of the HOM overlap integral at cut-off are slightly lower in this case, but still higher than 0.39. Similar comments apply to the FM Γhex at the cut-off wavelength λ c , FM.
It is important to underline that the quite high values of the HOM Γhex calculated at λ c , HOM demonstrate that the confinement into the fiber core of the first high-order mode is not completely compromised at cut-off condition. Moreover, previous analysis results have demonstrated that HOMs with such values of the overlap integral on the Yb-doped core are not effectively suppressed in the gain competition with the FM, thus worsening the performances of the high-power PCF-based amplifiers [13, 16]. A comparison with the overlap integral of the first HOM in a conventional Step-Index Fiber (SIF), calculated at the cut-off wavelength, which can be evaluated according to a well-established definition, is useful in order to estimate the accuracy of the two approaches used so far for the 19-cell PCF cut-off analysis. The HOM overlap integral on the SIF core at λ c , HOM will be considered as the reference one hereafter. A SIF very similar to the Corning®SMF-28™ optical fiber , that is with a core diameter of 8.2 μm and a numerical aperture of about 0.12, has been taken into account. Fig. 6(a) shows the FM and HOM dispersion curves and the cladding refractive index. As expected from the V-number calculation, the first high-order mode cut-off wavelength λ c , HOM, where the HOM n eff equals the cladding index, is around 1250 nm. The HOM overlap integral, reported in Fig. 6(b) together with the FM one, is about 0.19 at cut-off condition. Notice that this value is significantly lower than the ones previously calculated for the 19-cell fibers.
In summary, this comparison has demonstrated that the HOM cut-off wavelength values evaluated for the 19-cell double-cladding PCFs by applying the approaches based, respectively, on the effective index of the FSM and of the finite cladding mode, calculated without the core defect, are not accurate enough. As a consequence, a different method suitable for the cut-off analysis of these large mode area fibers with finite cladding dimension should be considered.
5. Avoided-crossing based cut-off analysis
A new approach for the cut-off analysis of 19-cell double-cladding PCFs is proposed in the present work, by making a different choice for the cladding effective index definition. In particular, the FEM-based full-vector modal solver has been applied to calculate the n eff of the first cladding mode in the actual fiber structure, by taking into account the finite inner-cladding dimension and the presence of the Yb-doped core. As demonstrated by the dispersion curves reported in Fig. 7(a) for the 19-cell fiber with 5 air-hole rings, the cladding mode effective index is lower than the FM and HOM n eff in all the considered wavelength range, that is between 1000 nm and 1500 nm, so the crossing, taken into account to estimate the cut-off wavelength with the approaches previously described, no longer occurs. However, it is important to underline that an avoided crossing exists between the dispersion curves of both FM and HOM, and the cladding mode, where the difference between the n eff values is minimum. In order to investigate what happens to the guided modes in this condition, the magnetic field distributions of the FM and of the HOM at their respective avoided-crossing wavelength, that is about 1078 nm and 1149 nm, are shown in Fig. 8(a) and (c) for the 19-cell Yb-doped double-cladding PCF with 5 air-hole rings and core refractive index of 1.44985. At the same time, Fig. 8(b) and (d) report the magnetic field distributions of the finite cladding mode at the avoided-crossing wavelength of FM and HOM, respectively. Notice that both the guided modes are no longer confined into the fiber core and are significantly spread in the inner-cladding, where the cladding mode is mainly distributed. Starting from these considerations, the avoided-crossing wavelengths are considered as the cut-off ones. It is important to underline that the HOM cut-off wavelength evaluated according to this approach is blue-shifted of about 120 nm with respect to the one calculated with the methods based on the FSM and the finite cladding mode, neglecting the core presence, for the same PCF. The new values of λ c , FM and λ c , HOM are reported with vertical black lines in Fig. 7(b), where the overlap integral spectra of the FM, HOM and cladding mode of the PCF with 5 air-hole ring are shown. In particular, a single-mode wavelength range Δλ SM of about 71 nm has been calculated with this new approach. Moreover, it is important to underline that the HOM Γhex at λ c , HOM, evaluated with the avoided-crossing method, is about 0.16, which is very similar to the value obtained for the SIF previously described. Therefore, as confirmed by the Γhex behaviour, the cut-off wavelength values calculated by taking into account the avoided-crossing condition between guided mode and cladding one are more accurate than the ones obtained with the methods described in the previous Section. Moreover, it is interesting to notice that the overlap integral of the cladding mode, shown in Fig. 7(b), is low, such as around 0.09, and almost constant for the wavelengths longer than 1300 nm, that is in the range where both the guided modes are well confined into the PCF doped core. As the wavelength decreases, Γhex of the finite cladding mode becomes higher, reaching a maximum of about 0.4 just after the avoided-crossing with the FM, and then it decreases again.
The dispersion curve of the finite cladding mode has been calculated also for the 4 and 3 air-hole ring fibers, and it is reported, together with the FM and HOM effective index values, in Fig. 9(a) and (b), respectively. Notice that λ c , HOM is blue-shifted of about 30 nm for the 19-cell PCF with the inner cladding formed by 4 air-hole rings. A more significant shift of about 130 nm towards the shorter wavelengths has been obtained for the fiber with the smaller cladding, being λ c , HOM ≃ 1016 nm. Fig. 10(a) and (b) show with vertical black lines the values of the FM and HOM cut-off wavelengths evaluated with the method based on the finite cladding mode in the actual fiber structure with 4 and 3 air-hole rings, respectively. Notice that Δλ SM is almost halved when the air-hole ring number decreases from 5 to 3, being, respectively, 71 nm and 40 nm. This means that in such a small cladding fiber the limit condition for light guiding is almost reached. On the contrary, a small reduction, that is about 6 nm, of the single-mode wavelength range has been obtained by removing only the outermost ring of the 19-cell double-cladding PCF. It is important to underline that the overlap integral of the HOM at cut-off condition is only slightly influenced by the the finite cladding dimension, being about 0.22 for both the 4 and 3 air-hole ring PCFs. Notice that this value is similar to the one calculated with the same approach for the 19-cell fiber with 5 air-hole rings, and to the one obtained for the SIF, thus providing a further validation of the method proposed in the present analysis.
A summary of the cut-off analysis results obtained with the avoided-crossing method is reported in Tab. 2.
5.1. Core refractive index tolerance analysis
The new approach based on the finite cladding mode in the fiber with core defect has been applied to analyze the tolerance of the cut-off properties to core refractive index changes. In particular, different values of the core refractive index n core have been considered for the 19-cell Yb-doped double-cladding PCFs with 5, 4, and 3 air-hole rings in the inner cladding. The influence of n core and of cladding dimension on four important parameters to describe the properties of Yb-doped fibers with large mode area, such as the HOM cut-off wavelength λ c , HOM, the single-mode wavelength range Δλ SM, and the overlap integral of the HOM and the FM at λ c , HOM, is demonstrated by simulation results reported in Fig. 11. Notice that, as the core refractive index increases, the HOM cut-off wavelength, shown in Fig. 11(a), is blue-shifted regardless of the cladding dimension. However, the λ c , HOM values obtained for the PCFs with 3 air-hole rings are significantly lower than the others, especially for n core = 1.44985. It is interesting to underline that, for a fixed inner cladding dimension, the HOM Γhex at cut-off condition is almost unaffected by the core index. On the contrary, by choosing a certain n core, slightly lower values of the HOM overlap integral have been calculated for the 5-ring fibers, as shown in Fig. 11(b). However, the most important result is that all the values obtained with the cladding mode avoided-crossing approach are similar to the one evaluated for the SIF. Moreover, as expected, a significant decrease of Δλ SM, shown in Fig. 11(c), is obtained when n core becomes higher, regardless of the air-hole ring number. The extremely reduced single-mode wavelength range of 19-cell PCFs with 3 air-hole rings is a further confirmation of the light guiding worsening as the inner-cladding dimension decreases. This is demonstrated also by the FM overlap integral at the HOM cut-off wavelength, shown in Fig. 11(d). In fact, a FM Γhex of only 0.4 has been obtained for the fiber with 3 air-hole rings, being about 0.55 for the ones with a larger inner cladding. Finally, notice that also the latter value of the FM overlap integral at the HOM cut-off wavelength is quite low. This potentially represents a critical aspect for the performances of the high-power amplifiers based on 19-cell Yb-doped double-cladding PCFs, which is worth further studying.
Finally, for comparison purposes, Fig. 11 reports the results obtained for 5-ring PCFs which differ from the ones considered so far only for the smaller air-hole diameter d = 0.05Λ. Notice that the values of FM and HOM overlap integral at λ c , HOM are slightly affected by the air-hole size change, as shown in Fig 11(b) and (d). With respect to the results obtained for the PCFs with d/Λ = 0.1, the air-hole diameter decrease causes a significant red-shift of λ c , HOM, which is about 290 nm when n core = 1.44985, and a broadening of the single-mode wavelength range, which almost doubles, regardless of the core refractive index. It is important to underline that almost the same HOM cut-off wavelength has been calculated for the 5-ring PCF with d/Λ = 0.1 and n core = 1.44985, and the one with d/Λ = 0.05 and n core = 1.44990, but the latter design provides a wider single-mode wavelength range.
The cut-off properties of 19-cell Yb-doped double-cladding PCFs have been investigated for the first time to the authors’ knowledge. A FEM-based full-vector modal solver has been used to calculate the cut-off wavelength of the first HOM and its overlap integral on the doped core of fibers with different inner-cladding dimension and different core refractive index. The comparison with the cut-off properties of the HOM of a conventional step-index fiber has demonstrated the lack of accuracy of the method based on the FSM of the infinite cladding, and of the one based on the first mode of the finite cladding, calculated without considering the core defect, when applied to these large mode area PCFs with air-cladding. Therefore, a new approach, which takes into account the avoided-crossing between the dispersion curves of the guided mode and of the finite cladding one, evaluated for the actual fiber structure, must be applied. Simulation results obtained for 19-cell PCFs with 5, 4 and 3 air-hole rings and different core refractive index values have demonstrated the accuracy of the proposed method.
The Authors acknowledge the support of the EU funded FP7 ALPINE Project, n. 229231.
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