Non-hexagonal Large-Pitch Fibers for enhanced mode discrimination

doi:10.1364/OE.19.012081

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Non-hexagonal Large-Pitch Fibers for enhanced mode discrimination

Fabian Stutzki, Florian Jansen, Cesar Jauregui, Jens Limpert, and Andreas Tünnermann »View Author Affiliations

Optics Express, Vol. 19, Issue 13, pp. 12081-12086 (2011)
http://dx.doi.org/10.1364/OE.19.012081

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– Abstract
1. Introduction
2. Design considerations
3. Non-hexagonal designs as alternative to hexagonal structures
4. Conclusion
– References and links

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Abstract

Photonic-Crystal Fibers (PCF) are among the most promising concepts to achieve large mode field areas suitable for the reduction of nonlinearities in fibers. Differential mode propagation loss is the cornerstone of effective single-mode behavior in passive and core-pumped active PCFs. In this work, we explore non-hexagonal PCF designs with increased mode discrimination in comparison to the classical hexagonal PCF designs. It is shown that a pentagonal design can increase the mode discrimination and, simultaneously, also improve the beam quality of Large-Pitch Fibers with mode field diameters well beyond 100 µm.

1. Introduction

Fiber lasers and amplifiers have demonstrated outstanding performance in different laser applications ranging from high average power cw-operation to ultra-short pulse laser systems. Nevertheless, the progress of power scaling of fiber lasers is currently challenged by the onset of two effects: parasitic nonlinear effects for high pulse peak powers and mode instabilities at high average powers. To circumvent the onset of nonlinear effects, short fibers with large Mode Field Areas (MFA) have to be considered. Fibers with mode field diameters larger than 50 µm, so called Very Large Mode Area fibers (VLMA), have already led to ultrashort pulse laser systems delivering pulse energies above 2 mJ [1

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255. [CrossRef] [PubMed]

J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14(7), 2715–2720 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2715. [CrossRef] [PubMed]

]. However, in general, larger cores allow the propagation of higher order modes, which can finally result in mode instabilities at high average powers. These mode instabilities have a threshold-like behavior and they lead to the distortion of the output beam of a fiber laser system at high average powers [3

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36(5), 689–691 (2011). [CrossRef] [PubMed]

]. The exact threshold values depend on the fiber design and on the experimental conditions. Thus, while kilowatt output powers have already been demonstrated by fibers with mode field diameters below 30 µm, VLMA fibers have only achieved 300 W of average power [3

] to date (albeit with a much higher pulse energy).

Single-mode operation in LMA fibers can be achieved by various techniques such as modified matching [4

M. E. Fermann, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23(1), 52–54 (1998), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-23-1-52. [CrossRef]

], differential bend loss for higher-order modes (HOMs) [5

D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21(23), 4208–4213 (1982), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-21-23-4208. [CrossRef] [PubMed]

J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442. [CrossRef]

], resonant out-coupling of HOMs [7

C. 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,” Conference on Lasers and Electro-Optics (CLEO), paper CTuBB3 (2007).

], mode filtering with tapers [8

D. Donlagic, “In-Line higher order mode filters based on long highly uniform fiber tapers,” J. Lightwave Technol. 24(9), 3532–3539 (2006), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-24-9-3532. [CrossRef]

], confined doping [9

T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B 19(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539. [CrossRef]

] and gain-guiding index-antiguiding [10

A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24(8), 1677–1682 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=josab-24-8-1677. [CrossRef]

]. In any case, all these techniques have their own drawbacks and, to date, the largest mode field diameter combined with the highest average power has been demonstrated using Photonic Crystal Fibers (PCFs) [1

–3

Even though PCFs have played a fundamental role in enabling this scaling in mode area and power/pulse energy in fiber laser systems, they are not single-mode in the strict analytical sense [11

H. P. Uranus, “Theoretical study on the multimodeness of a commercial endlessly single-mode PCF,” Opt. Commun. 283(23), 4649–4654 (2010). [CrossRef]

]. This might potentially lead to performance degradation in high power operation as mentioned above. However, it is possible to achieve effective single-transverse-mode operation by using fiber designs that offer higher confinement losses for HOMs compared to the Fundamental Mode (FM), i.e. by exploiting mode discrimination. This concept was introduced by P. Russell in [12

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003). [CrossRef] [PubMed]

] as “modal sieve” in the context of endlessly single-mode fibers. VLMA fibers using this effect are also known as leakage channel fibers [13

L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-24-8-1689. [CrossRef]

]. However, this labeling is misleading for double clad structures and, therefore, the more general definition as Large-Pitch photonic crystal Fiber (LPF) [14

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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-26834. [CrossRef]

] is preferred. This term makes reference to the fact that the hole-to-hole distance (pitch Λ) is at least 10 times larger than the wavelength to be guided.

Mode area scaling with effective single mode operation is the key to further power scaling. Unfortunately the mode discrimination capabilities of conventional PCF designs decrease with larger core sizes. Therefore, there is an urgent need for new design proposals to increase the mode discrimination at large mode areas. The intention of this paper is to point out that fiber geometries other than the hexagonal lattice might be advantageous rather than give a comprehensive study of all possible directions. In that sense, non-hexagonal designs are proposed as a means to enhance mode discrimination, allowing for both shorter fiber lengths and larger mode field areas.

2. Design considerations

A hexagonal hole arrangement for a PCF is characterized by the hole-to-hole distance (pitch) Λ and the relative hole diameter d/Λ, as depicted in Fig. 1 . The number of missing air holes forming the signal core and the number of rings around it lead to a well-defined structure.

Fig. 1 Schematic design of a hexagonal two ring Large-Pitch Fiber with one hole missing forming the signal core.

All the fiber designs considered in the following have been evaluated using a full-vectorial finite-difference mode solver based on [15

Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10(17), 853–864 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853. [PubMed]

]. A perfectly matched layer is implemented for an effectively infinite simulation area, thus enabling the calculation of the modal confinement losses. Please note that this loss calculation is only applicable for single clad fibers since in double-clad structures the modal confinement loss is in good approximation zero for all modes.

As all PCFs are multimode, the fundamental mode has to be defined based on its properties, such as the highest effective refractive index, the highest beam quality (the most Gaussian-like mode) and, most importantly, the lowest propagation loss. For the proposed structures in this paper all definitions lead to the same fundamental mode. Furthermore, the relevance of different HOMs is dependent on application-specific parameters. Thus, the definition of the most harming HOM may lead to a non-unique decision. Typically the HOM with lowest propagation loss is considered to be the most harming HOM, but its bend resistance, its likelihood to be excited and its core overlap should also be considered. In some designs there are fine-structured higher order modes exhibiting the lowest propagation losses. However, these modes will be excluded from the calculation of mode discrimination, because they are very difficult to excite in a practical setup and, consequently, they are usually not observable as the dominant mode in experiments. Nevertheless, for the sake of completeness, these modes will be shown for all proposed designs.

The level of mode discrimination required to obtain effectively single mode behavior has to be defined based on the application. For example, in an active fiber the HOM gain has to be compensated by its propagation loss. This leads to a mode discrimination of at least 30 dB for a fiber amplifier with about 1 m length, a typical dimension of a rod-type fiber amplifier [2

Different designs can only be properly compared for similar properties of the fundamental mode. Therefore, the evaluation of different designs is performed for a fundamental mode field area of 4000 µm². For a Gaussian-like mode profile this corresponds to a mode field diameter of 71 µm. The scaling capabilities are discussed for mode field areas as large as 12000 µm², corresponding to 124 µm MFD. Furthermore, a fundamental mode propagation loss of 1 dB/m is acceptable for a short high power fiber laser, as a gain of >20 dB (easily achievable in this type of fibers) can compensate for this extra loss [2

]. Both fundamental mode parameters (mode field diameter and propagation loss) can be adjusted simultaneously by the proper choice of Λ and d/Λ.

Another key parameter of a fiber laser is the beam quality of its output beam. Thus, in this work the M² value of the FM has been calculated by the second moment method from the electric field obtained as a solution of the mode solver [16

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009). [CrossRef]

]. Note that the beam quality M² is calculated from the field components of the FM only. Thus, it measures to which extent the FM is Gaussian-like. Therefore, the M² value given is no measurement of the HOM content. For a high power fiber laser a FM with M² < 1.5 is an acceptable trade-off, as long as the fiber operates in single transversal mode. A multimode fiber will additionally degrade the resulting output beam quality and, even worse, cause beam pointing instabilities.

A detailed analysis of different hexagonal designs with one, two and three rings can be found in [14

]. For a FM propagation loss of 1 dB/m the two rings design was pinpointed as the design with the highest mode discrimination and a good beam quality for the given parameters. This results as a tradeoff between high confinement for all modes (one ring design due to large hole sizes) and stronger guiding properties for the HOMs in the three ring design. This fundamental conclusion has been verified for the designs proposed in this work. Accordingly, we focus only on two rings designs and will further improve the mode discrimination possibilities.

3. Non-hexagonal designs as alternative to hexagonal structures

In the following section the mode discrimination capabilities of non-hexagonal designs will be discussed. Therefore, a general building instruction is developed and different designs are compared.

The hexagonal structure of most PCFs results from the close-packing of circular rods. For non-hexagonal structures a more general building instruction is needed. Nevertheless, this general instruction can emerge from the hexagonal structure by having another view on it. Interleaving two hexagons, one having the double radius of the other, produces the basis. The air holes have to be positioned on each edge of the two hexagons. Now one additional air hole has to be placed at half the distance between two outer edges of the hexagon. This leads to a two ring hexagonal structure. This building instruction can be used for equilateral polygons with more than four edges in general. As an example the pentagonal design is constructed in Fig. 2 .

Fig. 2 Building instruction for a pentagonal two-ring LPF.

Figure 3 shows four different designs, named by their inner geometry as (a) square, (b) pentagonal, (c) hexagonal and (d) heptagonal. On the left hand side the structure is illustrated in the same scale as the mode pictures, the white dots represent the air holes in the glass matrix (light blue). The structural parameters are chosen in a way that the FM MFA is approximately 4000 µm² and the FM propagation loss is 1 dB/m. The modes are sorted from left to right in ascending propagation loss, all mode fields are scaled to their maximum power. As mentioned before, for the calculation of the mode discrimination the first higher order mode with large feature size is used (disregarded HOMs have grayed text in Fig. 3).

Fig. 3 Simulated mode pictures of (a) square, (b) pentagonal, (c) hexagonal and (d) heptagonal designs for a FM area of 4000 µm² and FM loss of 1 dB/m. The higher order modes are sorted from left to right in ascending propagation loss. The propagation loss is depicted for each higher order mode (disregarded HOMs have grayed and smaller text).

The comparison of all four designs is difficult as different effects counteract each other. The hexagonal and square designs favor the LP₁₁-like HOMs due to the presence of two diagonally opposed air holes. Additionally, the square design offers weak mode discrimination, resulting from the high d/Λ values (~0.4) given by the requirement for a fundamental mode loss of 1 dB/m. On the other hand, the heptagonal design approaches a circular guiding structure with small d/Λ values (<0.2), accordingly the mode discrimination induced by mode deformation is reduced. In summary this leaves the pentagonal structure as the design with the highest mode discrimination.

The mode discrimination depending on the mode field area for all four designs is illustrated in Fig. 4 . Generally, the mode discrimination decreases with larger MFAs. Nevertheless, the pentagonal design allows for a mode discrimination exceeding 45 dB/m even at mode field areas of up to 12000 µm².

Fig. 4 Mode discrimination between FM and first relevant HOM depending on the mode field area for a FM loss of 1 dB/m.

Besides the mode discrimination, the beam quality of the fundamental mode has to be investigated (Fig. 5 ). The M² value of all designs increases with the mode field area. It can be seen that the beam quality of the fundamental mode is increased with lower number of air holes, since in this case the hole diameter has to be increased to maintain the FM loss constant at 1 dB/m. The square design offers the best M² value with M² < 1.35 for MFAs of up to 10000 µm². For the same fundamental mode area the pentagonal design has a M² = 1.38, the hexagonal and heptagonal designs exceed M² = 1.42. In conclusion the highest beam quality is offered by the square design. But the pentagonal design allows for an increased beam quality combined with enhanced mode discrimination, both compared to the commonly used hexagonal design.

Fig. 5 Beam quality factor of the fundamental mode depending on the mode field area for a FM loss of 1 dB/m.

4. Conclusion

A pentagonal design can enhance the mode discrimination of LPFs. It lacks structures favoring the guidance of the first HOM. In conclusion the mode discrimination exceeds 45 dB/m even for mode field diameters of 125 µm. Besides the higher mode discrimination the pentagonal design offers an increased beam quality compared to a hexagonal design. An even higher beam quality (M² ≈1.35) can be achieved with a square design at the expense of mode discrimination capability.

Acknowledgments

The research leading to these results has received funding from the German Federal Ministry of Education and Research (BMBF), the Helmholtz-Institute Jena (HIJ) and the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013) / ERC Grant agreement n° [240460] “PECS”. Additionally, F.J. acknowledges financial support by the Abbe School of Photonics Jena.

References and links

1.	T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255. [CrossRef] [PubMed]
2.	J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14(7), 2715–2720 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2715. [CrossRef] [PubMed]
3.	F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36(5), 689–691 (2011). [CrossRef] [PubMed]
4.	M. E. Fermann, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23(1), 52–54 (1998), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-23-1-52. [CrossRef]
5.	D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21(23), 4208–4213 (1982), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-21-23-4208. [CrossRef] [PubMed]
6.	J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442. [CrossRef]
7.	C. 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,” Conference on Lasers and Electro-Optics (CLEO), paper CTuBB3 (2007).
8.	D. Donlagic, “In-Line higher order mode filters based on long highly uniform fiber tapers,” J. Lightwave Technol. 24(9), 3532–3539 (2006), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-24-9-3532. [CrossRef]
9.	T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B 19(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539. [CrossRef]
10.	A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24(8), 1677–1682 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=josab-24-8-1677. [CrossRef]
11.	H. P. Uranus, “Theoretical study on the multimodeness of a commercial endlessly single-mode PCF,” Opt. Commun. 283(23), 4649–4654 (2010). [CrossRef]
12.	P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003). [CrossRef] [PubMed]
13.	L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-24-8-1689. [CrossRef]
14.	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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-26834. [CrossRef]
15.	Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10(17), 853–864 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853. [PubMed]
16.	S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.5295) Fiber optics and optical communications : Photonic crystal fibers
(060.3510) Fiber optics and optical communications : Lasers, fiber

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 28, 2011
Revised Manuscript: May 20, 2011
Manuscript Accepted: May 29, 2011
Published: June 7, 2011

Citation
Fabian Stutzki, Florian Jansen, Cesar Jauregui, Jens Limpert, and Andreas Tünnermann, "Non-hexagonal Large-Pitch Fibers for enhanced mode discrimination," Opt. Express 19, 12081-12086 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12081

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References

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255 . [CrossRef] [PubMed]
J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14(7), 2715–2720 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2715 . [CrossRef] [PubMed]
F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36(5), 689–691 (2011). [CrossRef] [PubMed]
M. E. Fermann, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23(1), 52–54 (1998), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-23-1-52 . [CrossRef]
D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21(23), 4208–4213 (1982), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-21-23-4208 . [CrossRef] [PubMed]
J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442 . [CrossRef]
C. 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,” Conference on Lasers and Electro-Optics (CLEO), paper CTuBB3 (2007).
D. Donlagic, “In-Line higher order mode filters based on long highly uniform fiber tapers,” J. Lightwave Technol. 24(9), 3532–3539 (2006), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-24-9-3532 . [CrossRef]
T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B 19(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539 . [CrossRef]
A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24(8), 1677–1682 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=josab-24-8-1677 . [CrossRef]
H. P. Uranus, “Theoretical study on the multimodeness of a commercial endlessly single-mode PCF,” Opt. Commun. 283(23), 4649–4654 (2010). [CrossRef]
P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003). [CrossRef] [PubMed]
L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-24-8-1689 . [CrossRef]
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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-26834 . [CrossRef]
Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10(17), 853–864 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853 . [PubMed]
S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009). [CrossRef]

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