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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7263–7273
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Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier

Mette Marie Jørgensen, Sidsel Rübner Petersen, Marko Laurila, Jesper Lægsgaard, and Thomas Tanggaard Alkeskjold  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7263-7273 (2012)
http://dx.doi.org/10.1364/OE.20.007263


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Abstract

High-power fiber amplifiers for pulsed applications require large mode area (LMA) fibers having high pump absorption and near diffraction limited output. Photonic crystal fibers allow realization of short LMA fiber amplifiers having high pump absorption through a pump cladding that is decoupled from the outer fiber diameter. However, achieving ultra low NA for single mode (SM) guidance is challenging, thus different design strategies must be applied. The distributed modal filtering (DMF) design enables SM guidance in ultra low NA fibers with very large cores, where large preform tolerances can be compensated during the fiber draw. Design optimization of the SM bandwidth of the DMF rod fiber is presented. Analysis of band gap properties results in a fourfold increase of the SM bandwidth compared to previous results, achieved by utilizing the first band of cladding modes, which can cover a large fraction of the Yb emission band including wavelengths of 1030 nm and 1064 nm. Design parameters tolerating refractive index fabrication uncertainties of ± 10−4 are targeted to yield stable SM bandwidths.

© 2012 OSA

1. Introduction

Rare-earth-doped fiber optical amplifiers allow high brightness and excellent mode quality due to fiber beam confinement [1

1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]

4

4. J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12(7), 1313–1319 (2004). [CrossRef] [PubMed]

]. In addition, good thermal handling is achieved due to the large ratio of surface area to active volume. Nonlinearities set the main limitations for increasing peak power and pulse energy, motivating the possibility to increase the effective mode area. However, the fiber core area cannot be increased unlimited, since single mode (SM) performance is crucial for maintaining diffraction limited focus ability and high beam pointing stability. Degradation of beam quality is a critical factor for future power scaling of fiber optical lasers and amplifiers. An all solid fiber with chirally-coupled cores (CCC) has been demonstrated to increase the core diameter up to 33.5 μm and have effective SM operation by coupling all higher order modes (HOMs) of the central core into high-loss helix side modes [5

5. C.-H. Liu, G. Chang, N. Litchinitser, D. Guertin, N. Jacobsen, K. Tankala, and A. Galvanauskas, “Chirally coupled core fibers at 1550-nm and 1064-nm for effectively single-mode core size scaling,” OSA Technical Digest Series (CD), paper CTuBB3 (2007).

, 6

6. S. Lefrancois, T. S. Sosnowski, C.-H. Liu, A. Galvanauskas, and F. W. Wise, “Energy scaling of mode-locked fiber lasers with chirally-coupled core fiber,” Opt. Express 19(4), 3464–3470 (2011). [CrossRef] [PubMed]

]. The CCC fiber is dependent on fabrication parameters of the helix side core and further core scaling has yet to be seen.

2. Distributed modal filtering rod fiber

Recently, it has been shown that a temperature induced index change cause the SM bandwidth to slightly blueshift approximately 5 nm with a 157 W signal power [15

15. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012). [CrossRef]

]. This magnitude of the blueshifting is relatively small compared to the SM bandwidth, and this paper therefore focuses on the SM properties of the passive DMF rod fiber, and does not include any thermal effects. Including thermal effects is scope of further work.

3. Computational model

The quarter of the DMF rod fiber cross-section is modeled numerically with the finite element method, where each subdomain is specified by the refractive index [16

16. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001). [CrossRef]

]. The quadrilateral at the center is used as the core for the computations, although it has a slightly smaller area, than the internal region surrounded by the inner cladding structure, see Fig. 1
Fig. 1 The model of a quarter of the cross-section of the distributed mode filtered (DMF) rod fiber for numerical simulations. The core of the DMF rod fiber is defined by the quadrilateral at the center. The smallest circles represent air holes in the cladding structure with a refractive index of nair, where some are surrounded by up-doped silica placed in a kagome-type lattice with refractive index ndope. The outer ring represents the air cladding, and the pitch, Λ, is indicated as the center to center distance between two adjacent air holes.
. However for the active DMF rod fiber the area corresponds to the doped region of the core. The outer silica ring of the DMF rod fiber is not considered, since essentially the field is zero in this region. The air cladding of the DMF rod fiber is represented by a ring instead of a large number of air holes, and has a numerical aperture of 0.6 relative to silica, see Fig. 1. The field will decrease towards zero in the air cladding, meaning that the actual structure of the air cladding is unimportant. In the fabricated DMF rod fiber the air cladding confine all modes to the inner cladding structure yielding low loss. Therefore it becomes important to consider core overlap for double clad PCFs instead of differential loss, since a simulated PML as the outer fiber boundary does not represent the physical fiber structure.

A filling fraction, f, is introduced to ensure mass conservation of the DMF elements, yielding the ability to change the pitch and air hole diameter without increasing or decreasing the mass of the up-doped Ge-rings. The filling fraction, f, is given by the radius of the up-doped Ge-ring, rdope, the air hole diameter, d, and the pitch, Λ:

f=(2rdopeΛ)2(dΛ)2.
(1)

If the pitch or air hole diameter changes in Eq. (1), rdope must change as well to keep f constant. This allows the simulations to match variations in the fabricated DMF rod fiber, without adding or reducing the volume of the up-doped Ge-rings.

4. Results for original design

The cross-section of the DMF rod fiber is modeled with a pitch of 14.5 μm yielding a hexagonal core with a maximum width of 72.5 μm and a maximum width within the inner cladding structure of approximately 84 μm. The air hole and resonator diameter relative to the pitch is 0.1 and 0.6 respectively, and the refractive index difference between the silica background and the up-doping of the Ge-rings is set to 2.5 × 10−3.

In [12

12. T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-Mode ytterbium-doped Large-Mode-Area photonic bandgap rod fiber amplifier,” Opt. Express 19(8), 7398–7409 (2011). [CrossRef] [PubMed]

] Alkeskjold et al. measured the transmission spectrum of a passive DMF rod fiber with a pitch of 14.7 μm, an index contrast between silica and the up-doped Ge-rings of 2.5 × 10−3, and an air hole diameter of approximately 0.07 – 0.10 relative to pitch. The DMF rod fiber is identified to be SM from 1050 nm to 1070 nm, yielding a relative SM bandwidth of 1.9%, corresponding well with the simulations above. This indicates that the applied criterion for quantifying the SM bandwidth is reasonable. The relative air hole diameters of the fabricated DMF rod fiber varies in the fiber cross-section and some are smaller than the simulated air hole diameter. Together with the slightly larger pitch this contributes to the red-shift in the SM bandwidth of approximately 10%. Uncertainties in the refractive index of the core and the up-doped regions are expected for the fabricated DMF rod fiber. These influence the width of the band of cladding modes that creates the SM bandwidth, the mode spacing between the FM and first HOM and the spectral location of the band gap created by the resonators. Therefore differences on the calculated and measured relative SM bandwidth are expected.

5. Design optimization

The up-doped Ge-rings in the inner cladding structure functions as resonators, which can couple with core modes due to index matching. An effective V-parameter for the resonators is introduced with the characteristic length as the width of the up-doped Ge-rings, wdope, instead of the radius. This is to omit the air hole at the center, since it does not confine the light.

Vres=2πwdopeλNA.
(2)

For the first band of cladding modes the center wavelength is 1532 nm, which yields Vres = 1.28. Keeping Vres constant and decreasing the center wavelength to 1030 nm yields a refractive index contrast for the resonators of 1.1 × 10−3, when setting the surrounding refractive index to the index of silica. There are six air holes around each resonator which reduces the effective refractive index of the surrounding silica. In reality the NA in Eq. (2) should be calculated with this effective refractive index, which must be lower at 1520 nm than at 1030 nm, due to the increment in wavelength. Therefore the index contrast of the up-doped Ge-rings is set to 1.2 × 10−3 to compensate the increase in the effective refractive index surrounding the resonators upon scaling from 1532 nm to 1030 nm center wavelength.

The power spectrum showing the fraction of power within the core as a function of wavelength for the first band of cladding modes is seen in Fig. 4
Fig. 4 The fraction of power within the core for the fundamental mode (FM) and first higher order mode (LP11) as a function of wavelength. Within the SM bandwidth (light blue) the power level of two additional LP11-like mode increase (LP(2)11 and LP(3)11). One normalized transverse component of the electrical field distribution of the modes at A, B, C and D are seen to the right.
. The DMF rod fiber is simulated SM from 990 nm to 1061 nm, which yields a relative SM bandwidth of 6.9%. The center wavelength for the optimized design is at 1026 nm, i.e. close to the maximum gain of Yb. Again two additional LP11-like modes, LP11(2) and LP11(3), increases in power level within the SM bandwidth, as for the original design with a refractive index contrast of 2.5 × 10−3 utilizing the first band of cladding modes, seen in Fig. 4 as mode C and D. However, their fraction of power within the core is approximately equal to the ones of the original design, and thereby must be dependent on the cladding band number i.e. the band gap number. In Fig. 4 the field distribution within the DMF elements resembles LP01-like modes from a step index fiber if omitting the air hole, verifying that the first band of cladding modes is utilized to create the SM bandwidth.

The relative SM bandwidth is reduced from 7.9% to 6.9% as the index contrast between the up-doped inclusions and silica is reduced. The confinement of the field within the resonators depends on the index contrast between the up-doped inclusions and silica. Lowering the index contrast causes the field to extend further into the surrounding cladding, and thereby affects the coupling between two adjacent resonators. The smaller relative SM bandwidth for the optimized design compared to the original design is attributed to this change in resonator coupling. Compared to the original DMF rod fiber design the relative SM bandwidth is increased from 1.8% to 6.9% in the simulations, i.e. with almost a factor of four.

6. Fabrication uncertainties

The fabrication uncertainty of the core refractive index of the active DMF rod fiber is approximately ±10−4 in production. However, the SM bandwidth must have approximately the same center wavelength in a fabricated DMF rod fiber despite uncertainties of the core refractive index. Therefore the air hole diameter is varied during the fiber draw to compensate for core refractive index changes. A lower core refractive index than expected corresponds to a redshift of the SM bandwidth, whereas a higher core refractive index corresponds to a blueshift. The former is compensated by increasing the air hole diameter, i.e. blueshifting the SM bandwidth, and the latter is compensated by decreasing the air hole diameter. A minimum air hole diameter of ~1.5 μm corresponding to ~0.1Λ is required due to reproducibility of the fibers. Thus the optimized fiber design in Sec. 5 represents the ideal case.

This section presents parameters to be targeted during the fiber draw taking limiting fabrication uncertainties into consideration. Both a positive and negative core refractive index contrast must be compensated; therefore the relative air hole diameter is increased to 0.15 for the case of a core refractive index equal to the refractive index of silica. This shifts the SM bandwidth to shorter wavelengths, which is compensated by increasing the relative resonator diameter from 0.60 to 0.69. Three index contrasts between the core refractive index and the refractive index of silica are considered: Δn = 10−4, Δn = 0 and Δn = −10−4. The relative air hole diameter is altered accordingly in three steps: d/Λ = 0.1, d/Λ = 0.15 and d/Λ = 0.2, while all other parameters are kept constant. The power spectra are seen in Figs. 5(a)
Fig. 5 Power spectra for estimating fabrication parameters to be targeted during the fiber pull. Three values of the refractive index contrast, Δn, is considered, where the relative air hole diameter is altered accordingly: (a) Δn = 10−4 and d/Λ = 0.1, (b) Δn = 0 and d/Λ = 0.15, and (c) Δn = −10−4 and d/Λ = 0.2.
, 5(b), and 5(c) with SM bandwidths of 5.1%, 5.4% and 5.8% respectively, seen as the light blue area, having center wavelengths of 1024 nm, 1016 nm and 1028 nm respectively. This is summarized in Table 1

Table 1. Design Parameters and Obtained Single Mode Bandwidths for Three Values of the Core Refractive Index and the Air Hole Diameter

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. The design of the DMF rod fiber has the ability to compensate fabrication uncertainties in the core refractive index while maintaining the SM properties. The relative resonator diameter of the optimized design in Sec. 5 must be increased to 0.69 and relative air hole diameters in the range 0.1 to 0.2 should be targeted during the fiber draw to guarantee an active DMF rod fiber with a SM bandwidth having a center wavelength close to the maximum Yb gain.

Also, noticed in Fig. 5, the power levels of the additional LP11-like modes decrease as the core refractive index decreases. Therefore it could be an advantage to decrease the core refractive index in the fabricated DMF rod fiber to increase the SM bandwidth and to lower the power level of other LP11-like modes that might disturb the SM properties. The notch of the FM around 883 nm in Fig. 5(c) corresponds to an avoided crossing.

7. Pitch scaling

Changing the pitch of the DMF rod fiber corresponds to scaling Maxwell's equations, causing the SM bandwidth to scale accordingly. This allows the spectral location of the SM bandwidth to be moved to shorter or longer wavelengths, which is utilized during the fiber draw. However, nothing is gained or lost in the relative size of the SM bandwidth, since the band gap width and spectral location change accordingly. In this section the effect of the pitch on the relative size of the SM bandwidth is investigated, when keeping the physical size of air holes and up-doped Ge-rings constant. The air hole diameter is 1.45 μm and the Ge-ring radius is determined from Eq. (1). Thereby investigating if the core area can be increased by increasing the pitch and also maintain the SM properties of the DMF rod fiber. Increasing the pitch increases the distance between the air holes and up-doped Ge-rings. The optimized design in Sec. 5 with a pitch of 14.5 μm is considered, changing only the pitch in three steps: 11.5 μm, 17.5 μm, and 20.5 μm, corresponding to core diameters of 67 μm, 103 μm and 120 μm, respectively.

Figure 6
Fig. 6 The fraction of power within the core as a function of wavelength for the optimized design with three different values of the pitch: (a) 11.5 μm, (b) 17.5 μm, and (c) 20.5 μm, having relative single mode bandwidths of 6.9%, 6.9%, and 7.0% respectively.
shows the fraction of power in the core as a function of wavelength for the three values of pitch. The relative SM bandwidth is approximately constant at 6.9%, 6.9%, and 7.0% for a pitch of 11.5 μm, 17.5 μm, and 20.5 μm respectively. Even though the pitch is significant changed yielding a maximum core diameter of 120 μm, the SM bandwidth stays approximately constant. The physical size of the resonators and air holes are constant, this means that the spectral location of the band gap should remain unchanged. As the pitch increases the effective NA between core and cladding is reduced and the design becomes very sensitive to refractive index uncertainties. There is a slight shift of the center wavelength of the SM bandwidth to longer wavelengths as the pitch increases in Fig. 6. Increasing the pitch moves the air holes away from the resonators, which results in a lower field overlap with the air holes of the cladding modes confined to the resonators. This increases the effective refractive index of the cladding modes moving the avoided crossings of the cladding modes with core modes to longer wavelengths. Also, the fraction of core power decreases slightly as the pitch increases, since the core modes become less confined to the core. The absolute distance between the core and the first surrounding air holes increases as the pitch increases, allowing the core modes to penetrate deeper into the cladding yielding a lower core overlap.

8. Conclusion

References and links

1.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]

2.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007). [CrossRef]

3.

W. Wadsworth, R. Percival, G. Bouwmans, J. Knight, and P. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express 11(1), 48–53 (2003). [CrossRef] [PubMed]

4.

J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12(7), 1313–1319 (2004). [CrossRef] [PubMed]

5.

C.-H. Liu, G. Chang, N. Litchinitser, D. Guertin, N. Jacobsen, K. Tankala, and A. Galvanauskas, “Chirally coupled core fibers at 1550-nm and 1064-nm for effectively single-mode core size scaling,” OSA Technical Digest Series (CD), paper CTuBB3 (2007).

6.

S. Lefrancois, T. S. Sosnowski, C.-H. Liu, A. Galvanauskas, and F. W. Wise, “Energy scaling of mode-locked fiber lasers with chirally-coupled core fiber,” Opt. Express 19(4), 3464–3470 (2011). [CrossRef] [PubMed]

7.

F. Jansen, F. Stutzki, H.-J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, and A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express 18(26), 26834–26842 (2010). [CrossRef] [PubMed]

8.

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012). [CrossRef]

9.

F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Avoided crossings in photonic crystal fibers,” Opt. Express 19(14), 13578–13589 (2011). [CrossRef] [PubMed]

10.

J. Fini, “Design of solid and microstructure fibers for suppression of higher-order modes,” Opt. Express 13(9), 3477–3490 (2005). [CrossRef] [PubMed]

11.

T. Murao, K. Saitoh, and M. Koshiba, “Multiple resonant coupling mechanism for suppression of higher-order modes in all-solid photonic bandgap fibers with heterostructured cladding,” Opt. Express 19(3), 1713–1727 (2011). [CrossRef] [PubMed]

12.

T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-Mode ytterbium-doped Large-Mode-Area photonic bandgap rod fiber amplifier,” Opt. Express 19(8), 7398–7409 (2011). [CrossRef] [PubMed]

13.

M. Laurila, J. Saby, T. T. Alkeskjold, L. Scolari, B. Cocquelin, F. Salin, J. Broeng, and J. Lægsgaard, “Q-switching and efficient harmonic generation from a single-mode LMA photonic bandgap rod fiber laser,” Opt. Express 19(11), 10824–10833 (2011). [CrossRef] [PubMed]

14.

J. Lægsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A 6(8), 798–804 (2004). [CrossRef]

15.

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012). [CrossRef]

16.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: January 18, 2012
Revised Manuscript: March 2, 2012
Manuscript Accepted: March 9, 2012
Published: March 14, 2012

Citation
Mette Marie Jørgensen, Sidsel Rübner Petersen, Marko Laurila, Jesper Lægsgaard, and Thomas Tanggaard Alkeskjold, "Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier," Opt. Express 20, 7263-7273 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7263


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References

  1. D. J. Richardson, J. Nilsson, W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]
  2. J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007). [CrossRef]
  3. W. Wadsworth, R. Percival, G. Bouwmans, J. Knight, P. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express 11(1), 48–53 (2003). [CrossRef] [PubMed]
  4. J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12(7), 1313–1319 (2004). [CrossRef] [PubMed]
  5. C.-H. Liu, G. Chang, N. Litchinitser, D. Guertin, N. Jacobsen, K. Tankala, and A. Galvanauskas, “Chirally coupled core fibers at 1550-nm and 1064-nm for effectively single-mode core size scaling,” OSA Technical Digest Series (CD), paper CTuBB3 (2007).
  6. S. Lefrancois, T. S. Sosnowski, C.-H. Liu, A. Galvanauskas, F. W. Wise, “Energy scaling of mode-locked fiber lasers with chirally-coupled core fiber,” Opt. Express 19(4), 3464–3470 (2011). [CrossRef] [PubMed]
  7. F. Jansen, F. Stutzki, H.-J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express 18(26), 26834–26842 (2010). [CrossRef] [PubMed]
  8. F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012). [CrossRef]
  9. F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, A. Tünnermann, “Avoided crossings in photonic crystal fibers,” Opt. Express 19(14), 13578–13589 (2011). [CrossRef] [PubMed]
  10. J. Fini, “Design of solid and microstructure fibers for suppression of higher-order modes,” Opt. Express 13(9), 3477–3490 (2005). [CrossRef] [PubMed]
  11. T. Murao, K. Saitoh, M. Koshiba, “Multiple resonant coupling mechanism for suppression of higher-order modes in all-solid photonic bandgap fibers with heterostructured cladding,” Opt. Express 19(3), 1713–1727 (2011). [CrossRef] [PubMed]
  12. T. T. Alkeskjold, M. Laurila, L. Scolari, J. Broeng, “Single-Mode ytterbium-doped Large-Mode-Area photonic bandgap rod fiber amplifier,” Opt. Express 19(8), 7398–7409 (2011). [CrossRef] [PubMed]
  13. M. Laurila, J. Saby, T. T. Alkeskjold, L. Scolari, B. Cocquelin, F. Salin, J. Broeng, J. Lægsgaard, “Q-switching and efficient harmonic generation from a single-mode LMA photonic bandgap rod fiber laser,” Opt. Express 19(11), 10824–10833 (2011). [CrossRef] [PubMed]
  14. J. Lægsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A 6(8), 798–804 (2004). [CrossRef]
  15. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012). [CrossRef]
  16. S. Selleri, L. Vincetti, A. Cucinotta, M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001). [CrossRef]

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