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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 14 — Jul. 4, 2011
  • pp: 13578–13589
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Avoided crossings in photonic crystal fibers

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


Optics Express, Vol. 19, Issue 14, pp. 13578-13589 (2011)
http://dx.doi.org/10.1364/OE.19.013578


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Abstract

The impact of avoided crossings (also known as anti-crossings) in single and double-clad large mode area Photonic Crystal Fibers (PCFs) suitable for high-power laser systems is evaluated numerically. It is pointed out that an inappropriate choice of pump core diameter, bending radius and/or index depression may lead to avoided crossings that manifest themselves in unwanted deformations of the output beam.

© 2011 OSA

1. Introduction

Ytterbium-doped fibers have been a major player in the extremely rapid increase of pulse energy [1

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]

] and average output power [2

2. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94. [CrossRef] [PubMed]

] of high power femtosecond fiber laser systems. This has been possible due to the high gain and efficiency provided by Yb-doped media, but also due to the development of large mode area effective single-mode photonic crystal fibers [3

3. A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-49-25-F71. [CrossRef] [PubMed]

].

To outperform current systems, Very Large Mode Area (VLMA) fibers (i.e. fibers with >50 μm mode field diameter) need to be further up-scaled to enable even higher pulse energies at high repetition frequencies. However, preserving single-transverse-mode operation in VLMA fibers is very challenging. One approach that provides very large mode areas and single-mode operation is the leakage channel fiber concept [4

4. L. Dong, H. A. McKay, L. Fu, M. Ohta, A. Marcinkevicius, S. Suzuki, and M. E. Fermann, “Ytterbium-doped all glass leakage channel fibers with highly fluorine-doped silica pump cladding,” Opt. Express 17(11), 8962–8969 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-11-8962. [CrossRef] [PubMed]

,5

5. L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27(11), 1565–1570 (2009), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-27-11-1565. [CrossRef]

]. This fiber design has already demonstrated Mode Field Diameters (MFD) of up to 62 µm at moderate output power levels of 30 W. This approach works by presenting lower leakage losses to the Fundamental Mode (FM) than to the Higher Order Modes (HOMs), a concept introduced as “modal sieve” by Russell et.al. in [6

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

].

Nevertheless, in our opinion, the most promising fiber design to date that satisfies the requirements for high average output power and exhibits very large mode areas, is the so-called Large-Pitch photonic crystal Fiber (LPF) [7

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

,8

8. 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), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-689. [CrossRef] [PubMed]

]. LPFs are characterized by a hole-to-hole-distance (pitch) larger than ten times the wavelength. These fibers work by delocalizing the HOMs from the doped core. This results in a reduced excitation of HOMs by an incident Gaussian beam and, simultaneously, in a reduced overlap of HOMs with the doped region and/or increased confinement loss for HOMs. The delocalization of the HOMs in an exemplary LPF can be seen in Fig. 1
Fig. 1 Exemplary large pitch fiber and calculated transverse mode profiles: without (upper half) and with (lower half) air-clad. The modes are sorted by ascending loss and, respectively, descending overlap with the doped region.
. The top row of Fig. 1 displays the modes of single-clad LPF together with their respective confinement losses. Details on the simulation can be found in section 2. Confinement loss only serves as a quality measure for single-mode operation in single-clad fibers as all modes are lossless in double-clad fibers. This latter case is displayed in the bottom row, where the modes corresponding to the same PCF structure as before but surrounded by an air-clad can be seen. Here, the overlap with the doped core was calculated and this parameter is used to determine the most relevant HOMs.

After commenting on some basic concepts of LPFs and avoided crossings in section 2, the simulation procedure will be summarized. Section 3, 4 and 5 describe avoided crossings caused by various parameters such as the air-clad diameter, bending radius and core-cladding index-mismatch. In section 6 the results are summarized and conclusions are drawn.

2. Large-pitch fibers and avoided crossings

To simplify the understanding of avoided crossings in LPFs we have selected a set of fiber parameters that ensures their occurrence in all the situations that will be discussed in the following. The simulations for this paper were carried out using a full-vectorial finite-differences frequency domain mode solver delivering the full field information of the modes. The confinement loss was calculated by making use of a perfectly matched layer. In all simulations, the wavelength was set to 1030 nm. The schematic structure of this LPF is displayed in Fig. 2
Fig. 2 Schematic structure of a rare-earth doped (green) double-clad LPF.
. This design is characterized by a core created by a single missing hole in a silica matrix. The core is surrounded by two rings of air holes in a hexagonal lattice. The choice of two surrounding rings of air holes as the optimum design was discussed in detail in [7

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

]. Suffice to say here that two rings provide the highest HOM delocalization while preserving an acceptable beam quality of the FM. The concrete design parameters are a pitch Λ = 30 µm and a normalized hole diameter d/Λ = 0.30. This parameter set leads to a mode field diameter (MFD) of ~50 µm when no index depression is assumed in the fiber core. Save for section 3, throughout the paper a fixed air-clad diameter of 170µm will be used.

Crossings and avoided crossings have been shown to exist in various types of PCFs such as index-guiding PCFs [12

12. F. Poli, E. Coscelli, T. T. Alkeskjold, D. Passaro, A. Cucinotta, L. Leick, J. Broeng, and S. Selleri, “Cut-off analysis of 19-cell Yb-doped double-cladding rod-type photonic crystal fibers,” Opt. Express 19(10), 9896–9907 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9896. [CrossRef] [PubMed]

] or PCFs with an “open structure”, namely in LPFs. However, the open structure that guarantees the preferential amplification of the FM (as HOMs possess a reduced overlap with the doped region) in LPFs also imposes some important constraints on the fiber design as it enhances the interaction between core and cladding modes. In this study it is shown that avoided crossings need to be considered as an integral part of the design process of PCFs with open structures, i.e. the whole waveguide has to be considered and not just the region containing air holes.

In the following, several parameters are investigated that can cause avoided crossings. Thus, a short review of avoided crossings caused by changing the air-clad diameter is presented. Additionally, we will comment on bending-induced avoided crossings. Finally another parameter that creates avoided crossings is systematically analyzed: the index depression of the doped region in active fibers. As doping the fiber core with rare earths increases its refractive index, co-doping with other materials (such as fluorine) is necessary to re-establish an index-matching between the core material and silica (which is required for the proper operation of PCFs). Achieving a perfect index matching is technologically not feasible, reason for which this index matching process typically results in an index depression of the doped region compared to the silica matrix. The influence of index depressions on LPFs with a special view on mode area scaling was discussed in detail in [7

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

]. In this paper we show that these index depressions can potentially lead to avoided crossings.

When plotting all the modes over the design parameter sweep, it is necessary to distinguish between crossings and avoided crossings. In order to do this, the overlap integral of all modes with a reference mode was calculated. As a fixed reference mode, an undistorted mode, i.e. a mode far away from any avoided crossings (specifically for the smallest possible air-clad or, respectively without bending or index-depression), was chosen. This reference mode (one mode for each section) is compared to all modes of each fiber structure. Note that in a “standard” waveguide all modes are orthogonal and, therefore, the overlap between two different modes is always zero. This is not the case for the calculation of the overlap of these orthogonal modes with a reference mode of a slightly different waveguide. As soon as an avoided-crossing occurs and deforms participating modes, the overlap integral with the reference mode will differ from zero. This technique ensures the reliable identification of avoided crossings at first glance.

3. Air-clad-induced avoided crossings

Surrounding the core structure of a PCF with a pump core leads to a completely new structure, in which core and cladding modes can interact. With growing air-clad diameters, the effective index of a core guided mode basically remains constant while that of the cladding modes grows. Therefore, at some point, the effective index of the HOMs will reach the effective index of the FM. When this happens two different effects can be observed. Some HOMs only cross the effective index of the FM (situation denoted as crossing as mentioned above), while other HOMs deform and an avoided-crossing can be observed. In the latter case, the shapes of both modes converge and, finally, the former HOM takes over the role of the FM while the former FM is delocalized from the core and becomes a cladding mode (Figs. 4
Fig. 4 Detailed plot of one broad and two narrow avoided crossings. The modes involved are shown in black, red, green and blue, respectively. Crossing modes are grayed. The position of the fixed reference mode to which all other modes are compared is highlighted.
and 5
Fig. 5 Transverse mode profiles of the modes involved in the broad avoided-crossing of Fig. 4 around 188.5µm. The narrow avoided crossings were left out for clarity. Across the avoided crossing the former FM evolves into a HOM, and a HOM takes over the role as the FM.
). In [5

5. L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27(11), 1565–1570 (2009), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-27-11-1565. [CrossRef]

] it was shown that FM avoided crossings can be circumvented and HOM avoided crossings can be exploited by the proper choice of air-clad diameter.

Figure 6
Fig. 6 Effective index of all modes in the displayed effective index interval versus the air-clad diameter (upper half) and the corresponding overlap of each mode with a well confined FM of the waveguide structure (lower half). The core-guided fundamental mode of each waveguide (i.e. of the fibers with the different air-clad diameters) is represented as a black dot in both graphs. The green vertical lines indicate the position of the seven resolved avoided crossings.
expands the simulation window of Fig. 4. It can be seen that the FM effective index remains constant while the effective indices of the cladding modes increase strongly with growing air-clad diameters. This is a typical behavior and prerequisite for the existence of crossings and avoided crossings, as can be seen in hollow core photonic bandgap fibers [10

10. J. West, C. Smith, N. Borrelli, D. Allan, and K. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1485. [CrossRef] [PubMed]

]. It is clearly visible that the effective index plot is not sufficient to identify avoided crossings as soon as more modes come into play, while the overlap plot is unambiguous. Here, the air-clad diameter was increased from 140 µm to 440 µm, revealing several broad avoided crossings on the one hand and large regions without any broad avoided crossings on the other hand. It is important to note that avoided crossings do not generally occur more often with larger air-clad diameters, although this is what would be expected from the effective index plot since it shows an increasing density of cladding modes.

In Fig. 7
Fig. 7 One broad avoided-crossing for a constant pitch Λ = 30µm but with three different d/Λ-values.
we study the avoided-crossing around 190 µm air-clad diameter more closely for a constant pitch Λ = 30 µm but with three different values of the normalized hole diameter d/Λ. It can be observed that the avoided-crossing of the FM becomes broader the smaller the d/Λ. For smaller hole sizes the FM is significantly larger and extends further into the cladding. Therefore, the open structure enables a stronger interaction between core and cladding modes and, thus, this leads to avoided crossings extending over a larger range of air-clad diameters. Furthermore, larger hole sizes cause a small shift in the air-clad diameter at which the avoided-crossing occurs. For fiber designers this is important to know when choosing the air-clad diameter sufficiently far away from any avoided crossings to mitigate deformations of the FM.

4. Bend-induced avoided crossings

In this section it is shown that even the right choice of air-clad diameter is not enough to ensure that an avoided-crossing for the FM is circumvented. Therefore, employing the same inner guiding structure as above (pitch Λ = 30 µm and d/Λ = 0.30) and a fixed air-clad diameter of 170 µm, we simulated the impact of bending on avoided crossings. Furthermore, we apply the same analysis to the same structure without air-clad to demonstrate two different kinds of avoided crossings: ones involving cladding and core modes and others involving only core modes. Starting with a straight fiber, we successively decrease the bend radius down to 30 cm in both cases (the smallest bending radius with limited FM deformation).

Note that the presence of an air-clad enables the existence of avoided crossings between core and cladding modes, while the same inner structure without air-clad will only exhibit avoided crossings between core modes (and, thus, significantly fewer avoided crossings). This is depicted in Fig. 10
Fig. 10 Avoided crossings in a LPF without air-clad. Both the effective index (upper graph) and the overlap (lower graph) with the FM of the straight fiber are plotted against the bend radius. For each bend radius the mode with the highest overlap with the FM of the straight fiber is shown in black, all other modes are grayed out. Vertical green lines indicate the position of avoided crossings.
for the same interval as in Fig. 8. While now two broad avoided crossings appear at a bending radius of 0.3 m and 0.7 m instead of 0.3 m, 0.65 m and 2.2 m, all narrower (core-cladding) avoided crossings have vanished.

Here, two different situations have to be explained separately. Either two core modes or a core and a cladding mode may interact. A core-cladding avoided crossings occurs due to the fact that cladding modes are much more sensitive to bending and, thus, have a steeper slope for decreasing bending radius than core modes. In core-core avoided crossings, however, a core-guided HOM is deformed by bending. This eventually leads to the point where the HOM effective index approaches the effective index of the FM and both modes interact. In Fig. 9 a LP11-like HOM (bottom, already deformed by bending) was shown to interact with the FM (top).

5. Index depression-induced avoided crossings

Another waveguide parameter that might vary for different active fiber pulls is the central index depression (Fig. 2) of the rare-earth-doped region compared to the silica matrix [7

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

]. Typically, this mismatch is <10−4. We will study the influence of an index depression of this order of magnitude for the exemplary guiding structure introduced above (Λ = 30 µm, d/Λ = 0.30 and an air-clad diameter of 170 µm) and only in the case of an active straight fiber.

Without considering avoided crossings, with a higher index depression one would expect a steadily decreasing effective index of the FM and higher order core modes, whereas that of the cladding modes is basically not influenced by the index depression. But in fact, in this open waveguide structure (compare Fig. 3) core and cladding modes react very differently to the index depression since a distinct separation between them is not always possible. On the one hand, pure cladding modes will remain uninfluenced by the index depression and, therefore, possess a constant effective index for increasing central index depression. On the other hand, all modes with intensity in the core will be influenced by the index depression at one point or another, which is accompanied by a decreasing effective index for increasing index depression. Moreover, as different core modes are influenced differently by the index depression, their effective indices are also able to cross. Note that the presence of the air-clad is not a prerequisite for these core-core mode crossings and avoided crossings.

The situation in air-clad LPFs is illustrated in Fig. 11
Fig. 11 Effective index (upper graph) and overlap (lower graph) of the mode 1 (black), the HOMs that lead to the first two avoided crossings (mode 2 in red and mode 3 in blue) and all other modes in this effective index region (gray). The position of the two broadest avoided crossings between a HOM and the FM are indicated with green vertical lines.
, where the effective indices of the three modes involved in an avoided-crossing (FM and HOMs) are plotted versus the index depression as black, red and blue lines, respectively. All other modes, i.e. those not involved in the main two avoided crossings, are shown in gray. At a depression of 6·10−5 the first avoided-crossing between the FM (mode 1) and one HOM (mode 2) can be seen. At a depression around 17·10−5 a second avoided-crossing follows between the “new” FM (mode 2) and mode 3. For further clarification, Fig. 11 also shows the overlap between the fundamental mode without index depression (taken as the reference mode in this case) and all simulated modes. Although not perceptible in the effective index plot, several more narrow avoided crossings are revealed by the overlap plot around an index depression of 6.5·10−5, 12.5·10−5 and 19.5·10−5.

Figure 11 also illustrates that for larger index depressions the overlap with the initial FM never reaches values close to one because the index depression deforms the modes. While mode 1 and mode 2 reach a nearly Gaussian-like profile in the center of the waveguide (i.e. in the doped region) for larger index depressions, they never lose their characteristic higher order mode features, i.e. one or two surrounding intensity rings (Fig. 12
Fig. 12 Modal pictures of the modes involved in three avoided-crossing. It can be seen that these modes exchange their role as the most Gaussian-like mode of the waveguide with changing index depression.
).

6. Conclusion

Thus, VLMA PCF designs have to consider avoided crossings between core and cladding modes not only by carefully choosing the pump cladding diameter, but also by taking into account bending and a possible index depression. Most likely, a combination of the mentioned effects needs to be evaluated for a specific design target. In some cases it may even be possible to take advantage of avoided crossings for HOM discrimination [5

5. L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27(11), 1565–1570 (2009), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-27-11-1565. [CrossRef]

].

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 Programme (FP7/2007-2013) / ERC Grant agreement no. [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.

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94. [CrossRef] [PubMed]

3.

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-49-25-F71. [CrossRef] [PubMed]

4.

L. Dong, H. A. McKay, L. Fu, M. Ohta, A. Marcinkevicius, S. Suzuki, and M. E. Fermann, “Ytterbium-doped all glass leakage channel fibers with highly fluorine-doped silica pump cladding,” Opt. Express 17(11), 8962–8969 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-11-8962. [CrossRef] [PubMed]

5.

L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27(11), 1565–1570 (2009), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-27-11-1565. [CrossRef]

6.

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003). [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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-26834. [CrossRef] [PubMed]

8.

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), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-689. [CrossRef] [PubMed]

9.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003). [CrossRef] [PubMed]

10.

J. West, C. Smith, N. Borrelli, D. Allan, and K. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1485. [CrossRef] [PubMed]

11.

D. Noordegraaf, L. Scolari, J. Laegsgaard, T. Tanggaard Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S.-T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. 33(9), 986–988 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-9-986. [CrossRef] [PubMed]

12.

F. Poli, E. Coscelli, T. T. Alkeskjold, D. Passaro, A. Cucinotta, L. Leick, J. Broeng, and S. Selleri, “Cut-off analysis of 19-cell Yb-doped double-cladding rod-type photonic crystal fibers,” Opt. Express 19(10), 9896–9907 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9896. [CrossRef] [PubMed]

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: April 26, 2011
Revised Manuscript: June 14, 2011
Manuscript Accepted: June 17, 2011
Published: June 29, 2011

Citation
Florian Jansen, Fabian Stutzki, Cesar Jauregui, Jens Limpert, and Andreas Tünnermann, "Avoided crossings in photonic crystal fibers," Opt. Express 19, 13578-13589 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13578


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References

  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. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 . [CrossRef] [PubMed]
  3. A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-49-25-F71 . [CrossRef] [PubMed]
  4. L. Dong, H. A. McKay, L. Fu, M. Ohta, A. Marcinkevicius, S. Suzuki, and M. E. Fermann, “Ytterbium-doped all glass leakage channel fibers with highly fluorine-doped silica pump cladding,” Opt. Express 17(11), 8962–8969 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-11-8962 . [CrossRef] [PubMed]
  5. L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27(11), 1565–1570 (2009), http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-27-11-1565 . [CrossRef]
  6. P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003). [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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-26834 . [CrossRef] [PubMed]
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