OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 12 — Jun. 6, 2011
  • pp: 11852–11866
« Show journal navigation

Solid-core photonic bandgap fibers for cladding-pumped Raman amplification

Benjamin Ward  »View Author Affiliations


Optics Express, Vol. 19, Issue 12, pp. 11852-11866 (2011)
http://dx.doi.org/10.1364/OE.19.011852


View Full Text Article

Acrobat PDF (1446 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Cladding-pumped solid-core photonic bandgap Raman fiber amplifiers are analyzed theoretically as possible candidates for power scaling. An example fiber design with a mode field diameter of 46 µm and a cladding diameter of 250 µm is calculated to exhibit 0.12 dB/km of confinement loss at the first Stokes wavelength and >10 dB/m at the second Stokes wavelength for a pump wavelength of 1.567 µm while maintaining low loss and large mode area in realistic coiling configurations. A Raman amplifier based on this fiber with 85 kW of output power, 78% optical conversion efficiency, a maximum heat load of 130 W/m, and a length of 235 m is simulated.

© 2011 OSA

1. Introduction

Recently an upper limit of ~40 kW has been proposed for the average power achievable using double-clad fiber lasers under a set of reasonable assumptions [1

1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13240. [CrossRef] [PubMed]

]. The power was found to be limited by thermal lensing and stimulated Raman scattering (SRS) regardless of the achievable mode field area. Solutions mitigating these limits would in theory enable further power scaling.

W-type, dual-hole assisted, and solid-core photonic bandgap fibers (SCPBGF) have been investigated as possible means to increase the SRS threshold in fibers [2

2. J. S. Kim, C. Codemard, Y. Jeong, J. Nilsson, and J. K. Sahu, “High power continuous-wave Yb-doped fiber laser with true single-mode output using W-type structure,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2006). http://dx.doi.org/10.1109/CLEO.2006.4628264.

4

4. S. Blin, L. Provino, N. Traynor, A. Mugnier, D. Pureur, and T. Chartier, “Design of all-solid photonic-bandgap fibers for Raman-free propagation,” Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe - EQEC 2009. European Conference on, pp.1, 14–19 June 2009.

]. Of these approaches, SCPBGF have enabled the largest mode field area of 413 µm2 [4

4. S. Blin, L. Provino, N. Traynor, A. Mugnier, D. Pureur, and T. Chartier, “Design of all-solid photonic-bandgap fibers for Raman-free propagation,” Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe - EQEC 2009. European Conference on, pp.1, 14–19 June 2009.

] compared to 155 µm2 for a proposed w-type fiber [2

2. J. S. Kim, C. Codemard, Y. Jeong, J. Nilsson, and J. K. Sahu, “High power continuous-wave Yb-doped fiber laser with true single-mode output using W-type structure,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2006). http://dx.doi.org/10.1109/CLEO.2006.4628264.

] and 78 µm2 for the dual hole-assisted fiber [3

3. L. Zenteno, J. Wang, D. Walton, B. Ruffin, M. Li, S. Gray, A. Crowley, and X. Chen, “Suppression of Raman gain in single-transverse-mode dual-hole-assisted fiber,” Opt. Express 13(22), 8921–8926 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-22-8921. [CrossRef] [PubMed]

]. Optimization of w-type fiber designs for SRS suppression depends on balancing mode field area against spectral filtering cutoff sharpness [2

2. J. S. Kim, C. Codemard, Y. Jeong, J. Nilsson, and J. K. Sahu, “High power continuous-wave Yb-doped fiber laser with true single-mode output using W-type structure,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2006). http://dx.doi.org/10.1109/CLEO.2006.4628264.

]. In contrast, core size does not affect the spectral filtering cutoff properties of PBGF which are rather determined by the resonance properties of high index inclusions comprising the cladding structure [5

5. A. Argyros, T. Birks, S. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13(7), 2503–2511 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2503. [CrossRef] [PubMed]

]. Therefore SCPBGF represent an opportunity for mode field area scaling in SRS-suppressing fibers.

Mitigation of thermal lensing has received less attention due to its relative unimportance so far. One proposed approach has been to depress the core refractive index in a photonic crystal fiber relative to the cladding so that the thermal lens causes the core index to approach that of the cladding at high thermal loads [6

6. B. G. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE 6453, 645307, 645307-9 (2007), http://dx.doi.org/10.1117/12.717326. [CrossRef]

]. In order to reach the thermal lens and SRS limits, a mode field diameter of 90 µm was determined necessary [1

1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13240. [CrossRef] [PubMed]

]. Single transverse mode operation is difficult to achieve for mode field diameters this large. One of the main reasons is that it is difficult to achieve a highly uniform refractive index profile in a doped core [1

1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13240. [CrossRef] [PubMed]

]. Minimum index variations on the order of 10−4 achievable with state-of-the-art manufacturing methods [7

7. T. Kokki, J. Koponen, M. Laurila, and C. Ye, “Fiber amplifier utilizing an Yb-doped large-mode-area fiber with confined doping and tailored refractive index profile,” Proc. SPIE 7580, 758016, 758016-9 (2010), http://dx.doi.org/10.1117/12.842404. [CrossRef]

] cannot ensure the stability of the fundamental mode in such large cores.

Raman fiber amplifiers (RFA), in contrast, do not require doped cores. While core-pumped Raman fiber amplifiers are useful for generating wavelengths inaccessible with rare earth fibers, they offer no increase in brightness beyond that of the pump source unless pumped from both ends in which case a modest brightness gain of less than 2 is possible. Cladding-pumped Raman fiber amplifiers (CPRFA) do however enable an increase in brightness [8

8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

10

10. J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786. [CrossRef]

]. CPRFA rely on a dual-waveguide structure wherein the Raman pump propagates in the outer waveguide, or pump cladding, while the Stokes is amplified within the inner waveguide, or core. The brightness gain advantage of CPRFA over core-pumped RFA arises from the ability to couple more pump power to the larger pump cladding which also typically has a high numerical aperture.

The growth of the second Stokes wave, which is seeded by noise and core-pumped, limits the brightness enhancement as well as the total amount of pump power that can be converted into the first Stokes wave [8

8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

10

10. J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786. [CrossRef]

]. One method of extending the brightness enhancement limit is to employ a fiber design that exhibits high loss at the second Stokes wavelength and low loss at the first Stokes wavelength [9

9. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14705. [CrossRef] [PubMed]

,10

10. J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786. [CrossRef]

]. However, this approach is subject to the same tradeoffs between mode field area and induced Stokes loss as rare earth doped SRS suppressing fibers. One additional benefit of CPRFA is that they require very long fiber lengths therefore they create a modest heat load even for very high average powers thus mitigating thermal lensing [10

10. J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786. [CrossRef]

].

This work argues that solid-core photonic bandgap fibers with low loss at the first Stokes wavelength and very high loss at the second Stokes wavelength can enable power scaling of cladding-pumped Raman fiber lasers and amplifiers as proposed previously [8

8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

]. General considerations of this approach are discussed and then a candidate SCPBGF design is analyzed theoretically and evaluated with respect to its anticipated capabilities and usefulness in a CPRFA.

2. General considerations

Desired characteristics of fiber sources include high power, high efficiency and good beam quality. The optical conversion efficiency of a CPRFA is determined by pump and Stokes material and waveguide losses and completeness of pump conversion [8

8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

,9

9. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14705. [CrossRef] [PubMed]

]. For sufficiently strong second Stokes suppression, long fibers enable a large fraction of the pump to be converted to the first Stokes wavelength [8

8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

10

10. J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786. [CrossRef]

]. The present study is focused on fibers with a pump wavelength of 1.567 µm with first and second Stokes wavelengths of 1.683 µm and 1.818 µm respectively based on the Raman wavenumber shift of 440 cm−1 for fused silica [11

11. N. Shibata, M. Horigudhi, and T. Edahiro, “Raman spectra of binary high-silica glasses and fibers containing GeO2, P2O5 and B2O3,” J. Non-Cryst. Solids 45(1), 115–126 (1981), http://dx.doi.org/10.1016/0022-3093(81)90096-X. [CrossRef]

] and the signal wavelength of a reported high-power Er-Yb fiber source [12

12. Y. Jeong, S. Yoo, C. A. Codemard, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, P. W. Turner, L. Hickey, A. Harker, M. Lovelady, and A. Piper, “Erbium:ytterbium codoped large-core fiber laser with 297-W continuous-wave output power,” IEEE J. Sel. Top. Quantum Electron. 13(3), 573–579 (2007), http://dx.doi.org/10.1109/JSTQE.2007.897178. [CrossRef]

]. Fused silica fibers exhibit the lowest material loss in the telecommunications band near 1.550 µm [13

13. T. Miya, Y. Terunuma, T. Hosaka, and T. Miyashita, “Ultimate low-loss single-mode fibre at 1.55 µm,” Electron. Lett. 15(4), 106–108 (1979), http://dx.doi.org/10.1049/el:19790077. [CrossRef]

]. These wavelengths also present a reduced risk of eye injury [14

14. S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005), http://dx.doi.org/10.1109/JSTQE.2005.850249. [CrossRef]

].

The maximum pump power that can be launched into the fiber cladding is a function of the pump source brightness and the diameter and numerical aperture of the pump cladding. Diode-pumped double-clad rare earth fiber lasers are attractive candidates for pump sources. The power that can be coupled from multiple single transverse mode (STM) fiber sources into a multi-mode pump cladding may be derived by considering the divergence angle of the freely propagating output of a STM fiber. Brightness conversion then limits the pumping power to
Pcl(πdclNAcl2λ)2Pp,
(1)
where d cl and NA cl are the diameter and numerical aperture of the pump cladding, λ is the pump wavelength, and P p is the power available from each STM fiber pump source. It is important to realize that this expression does not assume coherent combination of the STM fiber sources. If the Raman amplification process converts this power to a STM Stokes beam with conversion efficiency η c then the brightness enhancement is limited toBηc(πdclNAcl/2λ)2as shown previously [9

9. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14705. [CrossRef] [PubMed]

].

In SCPBGF used in CPRFA not all of the power launched into the cladding is available for Raman pumping. Some pump light is confined to modes guided in the high index cladding structure that have poor overlap with the core region. The fraction of trapped light increases as the total area and refractive index of all high index cladding features increases. Assuming that the cladding is uniformly-pumped at its maximum numerical aperture and that the photonic band gap (PBG) structure comprised of a triangular array of cylindrical inclusions occupies the majority of the pump cladding results in the following expression for a lower limit for the pump injection efficiency:
ηp1NiAiAclNAi2NAcl2=1π3(dΛ)2nΔnNAcl2,
(2)
where N i is the number of inclusions, A i is the area occupied by one inclusion, NA i is the numerical aperture of the inclusions, A cl is the cladding area, d is the inclusion diameter, Λ is the lattice pitch, n is the background refractive index, Δn is the index step of the inclusions, and NA cl is the numerical aperture of the pump cladding. This expression predicts the pump efficiency to be at least 78% for the cladding-pumped Yb-doped PBGF described in [15

15. A. Shirakawa, H. Maruyama, K. Ueda, C. B. Olausson, J. K. Lyngsø, and J. Broeng, “High-power Yb-doped photonic bandgap fiber amplifier at 1150-1200 nm,” Opt. Express 17(2), 447–454 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-2-447. [CrossRef] [PubMed]

] while 88% was observed. The better efficiency is attributable to the pump cladding region beyond the PBG structure. In this case, the high pump cladding numerical aperture significantly enhances the efficiency.

To achieve high efficiency, the signal core of the fiber must exhibit low signal loss at the first Stokes wavelength. Waveguide confinement loss and material loss contribute to the total loss. Although theoretical confinement losses below 10−5 dB/km have been reported for SCPBGF [16

16. T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615. [CrossRef] [PubMed]

], the best experimental result to date has been 2 dB/km at 1.310 µm [17

17. W. Tong, H. Wei, J. Li, H. Wang, Q. Han, J. Luo, G. Ren, X. Yu, and P. Shum, “Investigation of all-solid photonic bandgap fiber with low losses in low-order bandgaps,” Opt. Quant. Electron. 39, 1071 (2008), http://dx.doi.org/10.1007/s11082-007-9145-x.

] of which 1.2 dB/km was attributed to material losses due to water contamination. It is believed that improvements in the manufacturing process can decrease the loss further [17

17. W. Tong, H. Wei, J. Li, H. Wang, Q. Han, J. Luo, G. Ren, X. Yu, and P. Shum, “Investigation of all-solid photonic bandgap fiber with low losses in low-order bandgaps,” Opt. Quant. Electron. 39, 1071 (2008), http://dx.doi.org/10.1007/s11082-007-9145-x.

]. Low-loss fibers have also incorporated graded index (GRIN) high index regions [17

17. W. Tong, H. Wei, J. Li, H. Wang, Q. Han, J. Luo, G. Ren, X. Yu, and P. Shum, “Investigation of all-solid photonic bandgap fiber with low losses in low-order bandgaps,” Opt. Quant. Electron. 39, 1071 (2008), http://dx.doi.org/10.1007/s11082-007-9145-x.

,18

18. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8452. [CrossRef] [PubMed]

]. Improvements in the manufacturing process of index guiding photonic crystal fibers have decreased experimentally observed loss to 0.37 dB/km near 1.550 µm [19

19. K. Tajima, J. Zhou, K. Nakajima, and K. Sato, “Ultralow loss and long length photonic crystal fiber,” J. Lightwave Technol. 22(1), 7–10 (2004), http://dx.doi.org/10.1109/JLT.2003.822143. [CrossRef]

]. One of the keys to low loss in these fibers is the minimization of surface roughness at the boundaries of the air holes. This suggests that GRIN inclusions in SCPBGF may help decrease loss by minimizing roughness at the inclusion boundaries. An additional benefit of employing parabolic GRIN inclusions is that they enable highly reduced bend-induced mode field distortion within each inclusion [20

20. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-69. [CrossRef] [PubMed]

] thus rendering the cladding photonic band structure more stable overall against bending.

One prerequisite for low confinement loss in SCPBGF is adequate cladding thickness [16

16. T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615. [CrossRef] [PubMed]

]. Increasing the thickness of the cladding decreases the pump intensity for a given pump brightness however. Furthermore, it necessitates a larger pump cladding thus placing a more stringent high loss requirement on the second Stokes [10

10. J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786. [CrossRef]

]. The depth and order of the chosen bandgap also influences the confinement loss. Signals guided in higher-order bandgaps exhibit lower loss [16

16. T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615. [CrossRef] [PubMed]

]. For high order bandgaps to exist in the cladding, the inclusions forming the PBG structure must support a large number of guided modes which requires that they have relatively high numerical aperture and large diameter. Another method of decreasing loss is to introduce air holes into the PBG structure [21

21. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32(12), 1719–1721 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-12-1719. [CrossRef] [PubMed]

]. Considering these factors, it is reasonable to expect that a loss of 1 dB/km for a SCPBGF operating further into the infra-red material loss regime at 1.683 µm is possible. This was also the loss value assumed in a recent theoretical analysis of CPRFLs employing a w-type fiber [9

9. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14705. [CrossRef] [PubMed]

].

The possibility that the PBG cladding structure would exhibit SRS thus depleting the pump and decreasing efficiency also requires attention. The array of cylindrical high index inclusions may be thought of as a multi-core fiber that can exhibit amplification of noise through the Raman process. This can compete with the amplification of the desired signal. This situation is exacerbated by the fact that germanosilicate glass typically used to form the high-index inclusions generally exhibits a significantly higher Raman gain coefficient than pure fused silica [22

22. J. Bromage, K. Rottwitt, and M. E. Lines, “A method to predict the Raman gain spectra of germanosilicate fibers with arbitrary index profiles,” IEEE Photon. Technol. Lett. 14(1), 24–26 (2002), http://dx.doi.org/10.1109/68.974149. [CrossRef]

].

Fibers with increasingly large core diameters support increasing numbers of guided modes that are detrimental to output beam quality. Furthermore, they become affected by bend-induced mode distortion [20

20. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-69. [CrossRef] [PubMed]

]. Also, unintended variations in the core refractive index of doped fibers limit the degree to which the profile may be manipulated to reduce the number of guided modes [1

1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13240. [CrossRef] [PubMed]

]. It is important to understand how removing this last limitation can affect the capabilities of SCPBGF-based CPRFL. One method of improving the beam quality of few-moded fibers is to attempt to induce preferential loss in the higher order modes. Alternatively an input beam may be matched to the fundamental mode of the core and steps taken to reduce coupling into other guided modes.

One of the primary causes of mode coupling in large-core fibers is bend-induced effective refractive index profile change. This mechanism has been studied extensively in the context of microbend losses in single-mode fibers [25

25. K. Petermann and R. Kühne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4(1), 2–7 (1986), http://dx.doi.org/10.1109/JLT.1986.1074620. [CrossRef]

]. If the variation of the curvature radius is gradual enough, then the adiabatic approximation applies and mode coupling will be absent. The conditions under which this is the case may be found by solving the scalar paraxial wave equation
iψz=12β[t2+(n(x,y)2k02β2)]ψ=12β[t2+((n0(x,y)+Δn(x,y))2k02β2)]ψ,
(3)
in the limit that the refractive index profile varies gradually along the fiber. In Eq. (3), ψ is a function of position and represents the slowly-varying envelope of the electric field within the fiber, β is a chosen propagation constant, k0=ω/cis the propagation constant in a vacuum, and n(x,y) is the refractive index profile. This equation is analogous to the time dependent Schrödinger equation with time being replaced by propagation distance and the value of ħ being set to 1. The detailed procedure for deriving the adiabatic limit is described elsewhere [26

26. B. H. Bransden, and C. J. Joachain, Introduction to Quantum Mechanics (Wiley, 1989).

]. Restricting the system to two transverse modes, the adiabatic condition thus arrived at is
|βi(βiβj)2ϕi|x|ϕjddz(1R)|1,
(4)
where βi and βj are the propagation constants of the two modes, ϕiandϕjare their normalized field profiles in the scalar approximation, the angular brackets represent integration over the fiber cross-section, R is the effective radius of curvature, z is the axis of propagation, and the effective index profile
n(x,y)=n0(x,y)(1+xR)
(5)
has been assumed [16

16. T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615. [CrossRef] [PubMed]

,27

27. D. Marcuse, Theory of Dielectric Optical Waveguides. (Academic, 1991).

,36

36. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for Optical Fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://dx.doi.org/10.1109/JQE.2007.903364. [CrossRef]

]. It is evident from Eq. (3) and (5) that the role of the time dependent part of the quantum mechanical Hamiltonian operator is represented in the present case by
δH(x,y,z)(n02(x,y)k02β)(xR(z))βxR(z).
(6)
Equation (4) may be simplified considerably if the two modes are assumed to be LP01 and LP11 in which case the term with the integral may be approximated roughly by the diameter a of the fiber core. Furthermore, assuming that the two values of β are separated by a small valueΔβ=(2π/λ)Δneff, where n eff is the effective index, the adiabatic condition becomes
|ddz(1R)|2πΔneff2neffaλ.
(7)
If the fiber can be maintained in a configuration satisfying Eq. (7), then it is reasonable to expect that it may be free from bending induced mode coupling.

1Acmax(x,y)[ϕi(x,y)ϕj(x,y)].
(9)

3. Fiber design

Given that a relatively sharp increase in loss with wavelength is required to achieve simultaneously low first Stokes loss and high second Stokes loss the third bandgap was chosen to guide the first Stokes. While higher order bandgaps are more susceptible to bending-induced loss variability this is less of an issue for laser fibers that can be arranged deliberately to control the bending radius along the length in contrast to telecommunications fibers that must tolerate a range of configurations. Based on prior reported results [16

16. T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615. [CrossRef] [PubMed]

18

18. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8452. [CrossRef] [PubMed]

] it is evident that a Δn value of ~0.03 and an inclusion diameter d of ~10 µm will place a pure silica core defect mode in the 3rd bandgap at a 1.683 µm wavelength.

Furthermore, the candidate design incorporates GRIN inclusions arranged in a triangular lattice. The maximum index increase over background is 0.0342 and the diameter of the doped regions is 10.4 µm. The lattice pitch is 20 µm resulting in a diameter to pitch ratio of 0.52. The central core is formed by the omission of 7 inclusions. While this causes multiple transverse modes to be guided, the alternative of shrinking the core would reduce the maximum output power due to the bulk optical damage threshold while increasing the pitch would lead to an unacceptable amount of confinement loss [30

30. S. L. Semjonov, O. N. Egorova, A. F. Kosolapov, A. E. Levchenko, V. V. Velmiskin, A. D. Pryamikov, M. Y. Salganskiy, V. F. Khopin, M. V. Yashkov, A. N. Guryanov, and E. M. Dianov, “LMA fibers based on two-dimensional solid-core photonic bandgap fiber design,” Proc. SPIE 7580, 758018, 758018-9 (2010), http://dx.doi.org/10.1117/12.841265. [CrossRef]

]. Therefore this design is not inherently single-mode and effectively single-mode operation may or may not be possible. The cladding incorporates three complete rings of defects and one incomplete ring to better fit within a uniform diameter pump cladding. Figure 1
Fig. 1 Solid-core photonic bandgap fiber incorporating parabolic graded-index inclusions. Darker regions represent higher refractive index. The inset shows the index profile of each inclusion.
depicts the fiber cross section.

To analyze the photonic band structure of the PBG cladding, fully-vectorial eigenmodes of Maxwell's equations with periodic boundary conditions were computed by preconditioned conjugate-gradient minimization of the block Rayleigh quotient in a planewave basis, using a freely available software package [35

35. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-173. [CrossRef] [PubMed]

]. Figure 3
Fig. 3 Photonic band structure of the candidate fiber design sampled at a discrete set of wave vectors describing the Bloch states of the periodic lattice. The blue line represents the index of the silica background at 1.683 µm and the red line represents the index of the fundamental guided core mode at wavelengths spanning 1.6 µm - 1.86 µm. The insets depict the intensity profiles of the cladding modes bounding the guided mode.
shows the relationship between the effective index of the guided fundamental mode and the cladding modes for a range of wavelengths. The band structure thus obtained confirms that the loss increases as the fundamental guided core mode approaches the band edge. The high losses at wavelengths beyond 1.820 µm occur as the mode is guided in a minor gap within the band between the second and third primary band gaps. The discontinuity in the effective index occurs at the red edge of the band.

At wavelengths beyond 1.840 µm the core and cladding modes are approximately degenerate rendering core mode loss an ill-defined quantity. The upper and lower numerical apertures relative to the cladding mode indices [30

30. S. L. Semjonov, O. N. Egorova, A. F. Kosolapov, A. E. Levchenko, V. V. Velmiskin, A. D. Pryamikov, M. Y. Salganskiy, V. F. Khopin, M. V. Yashkov, A. N. Guryanov, and E. M. Dianov, “LMA fibers based on two-dimensional solid-core photonic bandgap fiber design,” Proc. SPIE 7580, 758018, 758018-9 (2010), http://dx.doi.org/10.1117/12.841265. [CrossRef]

] at 1.683 µm are 0.066 and 0.054 respectively yielding V parameters of 5.7 and 4.6 confirming that the fiber is multi-mode. Evaluating Eq. (7) yields an adiabatic mode coupling limit of 2250 m−2 on the radius of curvature change rate for this fiber.

It is interesting to compare the photonic band structure of the GRIN-type SCPBGF to that of a SCPBGF with step-index (SI) inclusions. Figure 4
Fig. 4 Calculated band structure for triangular lattices of parabolic GRIN inclusions (a) and step-index inclusions (b). The properties of the GRIN inclusions match those of the SCPBGF described above. The properties of the step-index inclusions were chosen to place the red edge of the third band-gap at approximately the same place as the GRIN-based fiber.
shows the band structure in the neighborhood of the third bandgap for fibers of each type. The gaps for the SI-type fiber are not as broad and the bands are broader relative to the GRIN-type fiber. This demonstrates that inclusions with approximately the same diameter and index step can exhibit significantly different band structure leading to different guiding properties. A comprehensive comparison between SI-type and GRIN-type SCPBGF, though perhaps warranted, is outside the scope of this paper.

The fiber must be coiled to some reasonable diameter for use in a practical device. The two primary concerns for coiling large-mode-area fibers are excessive fundamental mode bend loss that causes low efficiency and mode field distortion that reduces the effective area thus lowering the power handling capacity. The bend-induced confinement loss increase and mode field area decrease were investigated using the finite element scheme discussed above.

One previously discussed method of treating the effects of coiling is to employ the equivalent straight waveguide (ESW) approximation [16

16. T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615. [CrossRef] [PubMed]

,36

36. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for Optical Fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://dx.doi.org/10.1109/JQE.2007.903364. [CrossRef]

]. In the calculations presented here, the refractive index tensor is modified due to coiling-induced stress, and the absorbing boundary conditions. Therefore it is found more convenient to keep the original index profile, modify the trial solution to describe propagation in a curved fiber
E(x,y,z,t)=f(x,y)exp[i(β[1+xR]zωt)]
(12)
and employ the finite element method to solve for the field profile at z = 0. The validity of this procedure stems from the fact that no modification of the optical wave equation occurs due to the substitution specified by Eq. (12), the observation that this functional form maintains the orientation of the optical phase relative to the curved fiber, and the local nature of the optical wave equation. This method for calculating the bending loss also does not require further artificial manipulation of the effective index profile beyond the caustic radius [36

36. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for Optical Fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://dx.doi.org/10.1109/JQE.2007.903364. [CrossRef]

]. Furthermore, it has also been verified against published experimental and theoretical results for bending losses in step index fibers [36

36. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for Optical Fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://dx.doi.org/10.1109/JQE.2007.903364. [CrossRef]

].

Figure 5
Fig. 5 Confinement loss and effective area as a function of inverse bending radius at 1.683 µm wavelength.
shows the bend-induced confinement loss and mode field area as a function of inverse bending radius. It is evident that larger coils are beneficial for both power and efficiency. The maximum tolerable coil diameter will then be restricted by packaging considerations. If a 2 meter diameter is acceptable, then the mode field area will be reduced by less than 1% while the confinement loss will increase to only 0.2 dB/km. The loss near the second Stokes wavelength also increases with bending.

The inner-cladding should be as small as possible to provide maximum pumping intensity for a given available pump power. The size is limited by the requirement for this cladding to accommodate the PBG structure. The diameter of the PBG structure is approximately 200 µm as shown in Fig. 1 therefore a pump clad diameter of 250 µm is sufficient to contain the entire structure leaving some room to ease the fabrication process. A fluorosilicate glass outer cladding is assumed resulting in a pumping numerical aperture of 0.30. In summary, the proposed fiber has a core diameter of 46 µm with a 0.054 numerical aperture and a pump cladding of diameter 250 µm with a 0.30 numerical aperture.

4. Amplifier simulation

A SCPBGF CPRFA incorporating the fiber described above in a co-pumped configuration is considered. The coupled set of differential equations describing this system is [8

8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

]

dPpdz=αpPpg0Aclλs1λpPpPs1g1Aclλs1cλpPpPs1c,
(13a)
dPs1dz=αs1Ps1+g0AclPs1Ppg0Acoλs2λs1Ps1Ps2,
(13b)
dPs1cdz=αs1cPs1c+g1AclPs1cPp,
(13c)
dPs2dz=αs2Ps2+g0AcoPs2Ps1.
(13d)

In these equations, P denotes power, z is propagation distance, α is loss, g 0 is the Raman gain coefficient in pure silica, g 1 is the Raman gain coefficient in the germanosilicate inclusions, A is area, and λ is wavelength. The subscript p denotes the Raman pump, s1 the first Stokes signal in the core, s1c the first Stokes signal in the inclusions, s2 the second Stokes signal in the core. The subscripts cl and co refer to the cladding and the core respectively. The pump and first Stokes signal are assumed to be launched with a given power in the cladding and core respectively while the second Stokes in the core and the first Stokes in the inclusions build up from noise. The effective seed power in this case is taken to be the product of the energy of one Stokes photon and the Raman gain bandwidth [39

39. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

]. The bandwidth of the seed source is assumed to be broad enough that stimulated Brillouin scattering does not affect the operation of the amplifier. Finally counter-pumped Stokes waves are neglected as discussed below.

One prominent feature of the fiber discussed here is that to accommodate the guiding structure, the pump cladding diameter must be approximately 250 µm. This large value reduces the pump intensity. This leads to long fibers or high required pump powers. Additionally, the length of large diameter fiber that can be manufactured within tolerance from a single pre-form is limited. In one case, a 387 meter long draw was achieved [18

18. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8452. [CrossRef] [PubMed]

]. A practical fiber length limit of 500 meters is assumed here. The effective Raman gain coefficient in the core is taken to be 3.2 × 10−14 m/W based on scaling the value at a 1 µm wavelength [39

39. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

] for the pump wavelength of 1.567 µm and the un-polarized pump (factor of 0.5 [39

39. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

]). The Raman gain coefficient in the cladding inclusions is taken to be 6.3 × 10−14 m/W based on a prior reported value [8

8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

]. Furthermore, the modes within the inclusions are well confined so that their amplification is governed only by this second Raman gain coefficient. The total confinement and material loss within the core at 1.683 µm is 1 dB/km while at 1.818 µm it is 10 dB/m based on the finite element calculations. Pump loss is 2 dB/km and loss for the fundamental mode guiding in the cladding structure is 0.2 dB/km. The first Stokes in the inclusions is shifted by 425 cm−1 from the pump resulting in a 1.679 µm wavelength. A seed power of 10 W provided by a core-pumped fiber Raman oscillator is presumed. Additionally, fundamental mode propagation within the core is assumed.

Equation (13a)-(13d) were solved using a Runge-Kutta numerical method for a range of launched pump powers. The results are summarized in Fig. 6
Fig. 6 Simulation results for a SCPBGF CPRFA including the evolution of pump and signal over the length of the fiber (a), peak first Stokes power (b), optimum fiber length (c), maximum heat load (d), location along the fiber of the maximum heat load (e), and optical conversion efficiency (f). All results take into account the trapping of 8% of the pump power within the cladding structure.
. According to these calculations, the launched pump power must be several tens of kilowatts in order to obtain good efficiency. A scheme for combining 637 individual pump sources using fused fiber bundles has been reported [37

37. M. H. Muendel, R. Farrow, K. Liao, D. Woll, J. Luu, C. Zhang, J. J. Morehead, J. Segall, J. Gregg, K. Tai, B. Kharlamov, H. Yu, and L. Myers, “Fused fiber pump and signal combiners for a 4-kW ytterbium fiber laser,” Proc. SPIE 7914, 791431, 791431-7 (2011), http://dx.doi.org/10.1117/12.877572. [CrossRef]

]. To reach 50 kW of total pump power, such a scheme would require 79 W from each source. While this would be very challenging to accomplish with single-emitter diode pump sources it is relatively easy to achieve with STM fiber lasers. If 100 W were available from a STM pump source, then brightness conservation would limit the maximum available pump power to 565 kW according to Eq. (1); however, this would require the combination of 5,650 individual sources. Combining such a large number of fibers may prove to be impractical, in which case the engineering of fused fiber combiners would determine the true limit.

As the pumping power is increased the maximum heat load increases as well. The maximum heating occurs along the fiber where the pump is being depleted most rapidly. In the absence of sufficient seed power, spontaneous Raman scattering is amplified in the cladding structure first by virtue of its higher Raman gain coefficient thus depleting the pump and preventing amplification in the core. Furthermore, the high loss at the second Stokes wavelength enabled the by the SCPBGF completely suppresses amplification at this wavelength. At the highest launched pump power, the first Stokes power in the inclusions that has built up from noise is calculated to be 265 W representing a small fraction of the total output power. The counter-propagating first Stokes power in the inclusions is even weaker due to the much lower pump power at the output end of the fibers therefore it is omitted in the solutions of Eq. (13a)-(13d).

Typically a shooting algorithm would be used to treat the counter-propagating second Stokes [9

9. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14705. [CrossRef] [PubMed]

] seeded by spontaneous Raman scattering. In the present case however, the suppression of the second Stokes by the fiber is so strong that the counter-propagating Stokes field is so weak at the pumped end of the fiber that its representation in the computer program would be below the minimum double precision real value. Rather, the worst case scenario of the second Stokes being pumped by the full output value of the first Stokes was considered to see if the second Stokes can be amplified. Using Eq. (13d) this condition may be expressed
αs2>g0Ps1(L)Aco.
(14)
Evaluating this at the maximum calculated output power of 85 kW yielded 7 dB/m of required second Stokes attenuation.

For such high pump powers, heating of the fiber can become an issue even for such long fibers. An upper limit on the heat generated in the fiber is given by the total optical power lost per unit length from the four signals described by Eq. (13a)-(13d). The maximum value and the position along the fiber at which this maximum occurs are shown in Fig. 6(d) and Fig. 6(e). The heating obtained by this method overestimates the true deposited heat because it assumes all loss is converted to heat while some signal loss, such as confinement loss and Rayleigh scattering, transfers power optically.

It is also important to consider possible temperature-induced radial refractive index profile perturbations within the inclusions due to additional absorption caused by the varying germanium doping level. The doping adds approximately 0.1 dB/km of additional loss over pure silica. The For 110 kW of pump power evenly distributed throughout the pump cladding, this amounts to 0.004 W/m of additional heat load in each inclusion. Comparison to a prior thermal study of fiber lasers indicates that the index profile perturbation due to this heat load is negligible [38

38. D. C. Brown, and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quant. Electron. 37, 207–217 (2001), http://dx.doi.org/10.1109/3.903070.

]. Quantum defect heating caused by Raman process concentrates the heat load in the core. In this case, the temperature profile assumes a maximum value in the center of the core and decays rapidly within the core heating region and then more gradually throughout the cladding [38

38. D. C. Brown, and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quant. Electron. 37, 207–217 (2001), http://dx.doi.org/10.1109/3.903070.

]. The predicted maximum heating of 130 W/m is far below a previously reported thermal lensing threshold [1

1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13240. [CrossRef] [PubMed]

].

At such high intensities the third order nonlinearity of the fused silica may cause spectral broadening through self phase modulation, cross phase modulation, and four wave mixing. If a single frequency seed source is considered, the anomalous dispersion of the guided core mode gives rise to modulation instability [39

39. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

] however as previously mentioned the seed source must be broad enough to mitigate SBS. The calculated dispersion of this fiber is shown in Fig. 7
Fig. 7 Calculated material and total dispersion for the fundamental core mode of the fiber.
. For wavelengths near the center of the band gap, the waveguide contribution to the dispersion is negligible as expected for a weakly guiding fiber. At the first Stokes wavelength the total dispersion is slightly higher than the material dispersion. Approaching the band gap, the total dispersion rises severely as the waveguide contribution begins to dominate.

Techniques developed to treat continuous wave fiber supercontinuum sources apply to the present situation. The main differences are that the spectrum of the first Stokes is too narrow to cause intra-pulse Raman amplification of spontaneously generated solitons and the ~50 nm width of the Raman gain spectrum limits the maximum amplified spectral width. CW supercontinuum generation is highly dependent on the characteristics of the pump, which in the present case corresponds to the first Stokes seed [41

41. J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16(19), 14435–14447 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14435. [CrossRef] [PubMed]

]. This suggests that the temporal and spectral characteristics of the seed source may be optimized to mitigate broadening of the signal. Optical power that is shifted outside of the Raman gain bandwidth may still be guided within the core as long as it remains within the band gap. Optical power that is shifted outside of the bandgap is lost into cladding modes thus reducing the power in the core. Ultimately such non-linear effects will lead to some additional efficiency loss. A computational analysis of non-linear spectral broadening in SCPBGF, though beyond the scope of this paper, may be required in the future.

5. Conclusion

Solid core photonic bandgap fibers incorporating graded index inclusions enable robust suppression of the second Stokes wavelength in the core of cladding-pumped Raman fiber amplifiers enabling efficient optical conversion of the pump to the first Stokes wavelength. Additional requirements for realizing such amplifiers include high power single mode pump sources, an efficient method of launching the pump and seed, careful attention to the packaging of the fiber, and very low optical loss in the fiber. An amplifier simulation incorporating this type of fiber exhibited an output power of 85 kW with 78% optical conversion efficiency under realistic assumptions.

Acknowledgments

This work was supported in part by a grant of computer time from the DOD High Performance Computing Modernization Program at the Maui High Performance Computing Center. The author gratefully acknowledges Dr. Iyad Dajani for helpful discussions, Dr. Kyle Gresham for incorporating this work into the research electives portion of the Air Command and Staff College curriculum and the High Energy Laser Joint Technology Office, the Air Force Research Laboratory Directed Energy Directorate, and the Laser and Optics Research Center at the United States Air Force Academy for support.

References and links

1.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13240. [CrossRef] [PubMed]

2.

J. S. Kim, C. Codemard, Y. Jeong, J. Nilsson, and J. K. Sahu, “High power continuous-wave Yb-doped fiber laser with true single-mode output using W-type structure,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2006). http://dx.doi.org/10.1109/CLEO.2006.4628264.

3.

L. Zenteno, J. Wang, D. Walton, B. Ruffin, M. Li, S. Gray, A. Crowley, and X. Chen, “Suppression of Raman gain in single-transverse-mode dual-hole-assisted fiber,” Opt. Express 13(22), 8921–8926 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-22-8921. [CrossRef] [PubMed]

4.

S. Blin, L. Provino, N. Traynor, A. Mugnier, D. Pureur, and T. Chartier, “Design of all-solid photonic-bandgap fibers for Raman-free propagation,” Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe - EQEC 2009. European Conference on, pp.1, 14–19 June 2009.

5.

A. Argyros, T. Birks, S. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13(7), 2503–2511 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2503. [CrossRef] [PubMed]

6.

B. G. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE 6453, 645307, 645307-9 (2007), http://dx.doi.org/10.1117/12.717326. [CrossRef]

7.

T. Kokki, J. Koponen, M. Laurila, and C. Ye, “Fiber amplifier utilizing an Yb-doped large-mode-area fiber with confined doping and tailored refractive index profile,” Proc. SPIE 7580, 758016, 758016-9 (2010), http://dx.doi.org/10.1117/12.842404. [CrossRef]

8.

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229. [CrossRef]

9.

J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14705. [CrossRef] [PubMed]

10.

J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786. [CrossRef]

11.

N. Shibata, M. Horigudhi, and T. Edahiro, “Raman spectra of binary high-silica glasses and fibers containing GeO2, P2O5 and B2O3,” J. Non-Cryst. Solids 45(1), 115–126 (1981), http://dx.doi.org/10.1016/0022-3093(81)90096-X. [CrossRef]

12.

Y. Jeong, S. Yoo, C. A. Codemard, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, P. W. Turner, L. Hickey, A. Harker, M. Lovelady, and A. Piper, “Erbium:ytterbium codoped large-core fiber laser with 297-W continuous-wave output power,” IEEE J. Sel. Top. Quantum Electron. 13(3), 573–579 (2007), http://dx.doi.org/10.1109/JSTQE.2007.897178. [CrossRef]

13.

T. Miya, Y. Terunuma, T. Hosaka, and T. Miyashita, “Ultimate low-loss single-mode fibre at 1.55 µm,” Electron. Lett. 15(4), 106–108 (1979), http://dx.doi.org/10.1049/el:19790077. [CrossRef]

14.

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005), http://dx.doi.org/10.1109/JSTQE.2005.850249. [CrossRef]

15.

A. Shirakawa, H. Maruyama, K. Ueda, C. B. Olausson, J. K. Lyngsø, and J. Broeng, “High-power Yb-doped photonic bandgap fiber amplifier at 1150-1200 nm,” Opt. Express 17(2), 447–454 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-2-447. [CrossRef] [PubMed]

16.

T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615. [CrossRef] [PubMed]

17.

W. Tong, H. Wei, J. Li, H. Wang, Q. Han, J. Luo, G. Ren, X. Yu, and P. Shum, “Investigation of all-solid photonic bandgap fiber with low losses in low-order bandgaps,” Opt. Quant. Electron. 39, 1071 (2008), http://dx.doi.org/10.1007/s11082-007-9145-x.

18.

G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8452. [CrossRef] [PubMed]

19.

K. Tajima, J. Zhou, K. Nakajima, and K. Sato, “Ultralow loss and long length photonic crystal fiber,” J. Lightwave Technol. 22(1), 7–10 (2004), http://dx.doi.org/10.1109/JLT.2003.822143. [CrossRef]

20.

J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-69. [CrossRef] [PubMed]

21.

A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32(12), 1719–1721 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-12-1719. [CrossRef] [PubMed]

22.

J. Bromage, K. Rottwitt, and M. E. Lines, “A method to predict the Raman gain spectra of germanosilicate fibers with arbitrary index profiles,” IEEE Photon. Technol. Lett. 14(1), 24–26 (2002), http://dx.doi.org/10.1109/68.974149. [CrossRef]

23.

C. Brooks and F. Di Teodoro, “1-mJ energy, 1-MW peak-power, 10-W average-power, spectrally narrow, diffraction-limited pulses from a photonic-crystal fiber amplifier,” Opt. Express 13(22), 8999–9002 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-22-8999. [CrossRef] [PubMed]

24.

A. V. Smith and B. T. Do, “Bulk and surface laser damage of silica by picosecond and nanosecond pulses at 1064 nm,” Appl. Opt. 47(26), 4812–4832 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=ao-47-26-4812. [CrossRef] [PubMed]

25.

K. Petermann and R. Kühne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4(1), 2–7 (1986), http://dx.doi.org/10.1109/JLT.1986.1074620. [CrossRef]

26.

B. H. Bransden, and C. J. Joachain, Introduction to Quantum Mechanics (Wiley, 1989).

27.

D. Marcuse, Theory of Dielectric Optical Waveguides. (Academic, 1991).

28.

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-4-3258. [CrossRef] [PubMed]

29.

B. Ward and M. Mermelstein, “Modeling of inter-modal Brillouin gain in higher-order-mode fibers,” Opt. Express 18(3), 1952–1958 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-1952. [CrossRef] [PubMed]

30.

S. L. Semjonov, O. N. Egorova, A. F. Kosolapov, A. E. Levchenko, V. V. Velmiskin, A. D. Pryamikov, M. Y. Salganskiy, V. F. Khopin, M. V. Yashkov, A. N. Guryanov, and E. M. Dianov, “LMA fibers based on two-dimensional solid-core photonic bandgap fiber design,” Proc. SPIE 7580, 758018, 758018-9 (2010), http://dx.doi.org/10.1117/12.841265. [CrossRef]

31.

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18(5), 737–743 (2000), http://dx.doi.org/10.1109/50.842091. [CrossRef]

32.

F. L. Teixeira, and W. C. Chew, “General closed-form pml constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guided Wave Lett. 8, 223–225 (1998), http://dx.doi.org/10.1109/75.678571.

33.

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of Eigenvalue problems,” ACM Trans. Math. Softw. 31(3), 351–362 (2005), http://dx.doi.org/10.1109/75.678571. [CrossRef]

34.

E. Coscelli, F. Poli, T. T. Alkeskjold, D. Passaro, A. Cucinotta, L. Leick, J. Broeng, and S. Selleri, “Single-mode analysis of Yb-doped double-cladding distributed spectral filtering photonic crystal fibers,” Opt. Express 18(26), 27197–27204 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-26-27197. [CrossRef]

35.

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-173. [CrossRef] [PubMed]

36.

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for Optical Fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://dx.doi.org/10.1109/JQE.2007.903364. [CrossRef]

37.

M. H. Muendel, R. Farrow, K. Liao, D. Woll, J. Luu, C. Zhang, J. J. Morehead, J. Segall, J. Gregg, K. Tai, B. Kharlamov, H. Yu, and L. Myers, “Fused fiber pump and signal combiners for a 4-kW ytterbium fiber laser,” Proc. SPIE 7914, 791431, 791431-7 (2011), http://dx.doi.org/10.1117/12.877572. [CrossRef]

38.

D. C. Brown, and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quant. Electron. 37, 207–217 (2001), http://dx.doi.org/10.1109/3.903070.

39.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

40.

L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002), http://dx.doi.org/10.1117/12.461727. [CrossRef]

41.

J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16(19), 14435–14447 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14435. [CrossRef] [PubMed]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(140.3510) Lasers and laser optics : Lasers, fiber
(140.3550) Lasers and laser optics : Lasers, Raman
(140.4480) Lasers and laser optics : Optical amplifiers
(290.5910) Scattering : Scattering, stimulated Raman
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 15, 2011
Revised Manuscript: May 4, 2011
Manuscript Accepted: May 6, 2011
Published: June 3, 2011

Citation
Benjamin Ward, "Solid-core photonic bandgap fibers for cladding-pumped Raman amplification," Opt. Express 19, 11852-11866 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11852


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13240 . [CrossRef] [PubMed]
  2. J. S. Kim, C. Codemard, Y. Jeong, J. Nilsson, and J. K. Sahu, “High power continuous-wave Yb-doped fiber laser with true single-mode output using W-type structure,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2006). http://dx.doi.org/10.1109/CLEO.2006.4628264 .
  3. L. Zenteno, J. Wang, D. Walton, B. Ruffin, M. Li, S. Gray, A. Crowley, and X. Chen, “Suppression of Raman gain in single-transverse-mode dual-hole-assisted fiber,” Opt. Express 13(22), 8921–8926 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-22-8921 . [CrossRef] [PubMed]
  4. S. Blin, L. Provino, N. Traynor, A. Mugnier, D. Pureur, and T. Chartier, “Design of all-solid photonic-bandgap fibers for Raman-free propagation,” Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe - EQEC 2009. European Conference on, pp.1, 14–19 June 2009.
  5. A. Argyros, T. Birks, S. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13(7), 2503–2511 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2503 . [CrossRef] [PubMed]
  6. B. G. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE 6453, 645307, 645307-9 (2007), http://dx.doi.org/10.1117/12.717326 . [CrossRef]
  7. T. Kokki, J. Koponen, M. Laurila, and C. Ye, “Fiber amplifier utilizing an Yb-doped large-mode-area fiber with confined doping and tailored refractive index profile,” Proc. SPIE 7580, 758016, 758016-9 (2010), http://dx.doi.org/10.1117/12.842404 . [CrossRef]
  8. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009), http://dx.doi.org/10.1109/JSTQE.2008.2010229 . [CrossRef]
  9. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14705 . [CrossRef] [PubMed]
  10. J. Ji, C. A. Codemard, and J. Nilsson, “Analysis of spectral bendloss filtering in a cladding-pumped W-type fiber Raman amplifier,” J. Lightwave Technol. 28(15), 2179–2186 (2010), http://dx.doi.org/10.1109/JLT.2010.2052786 . [CrossRef]
  11. N. Shibata, M. Horigudhi, and T. Edahiro, “Raman spectra of binary high-silica glasses and fibers containing GeO2, P2O5 and B2O3,” J. Non-Cryst. Solids 45(1), 115–126 (1981), http://dx.doi.org/10.1016/0022-3093(81)90096-X . [CrossRef]
  12. Y. Jeong, S. Yoo, C. A. Codemard, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, P. W. Turner, L. Hickey, A. Harker, M. Lovelady, and A. Piper, “Erbium:ytterbium codoped large-core fiber laser with 297-W continuous-wave output power,” IEEE J. Sel. Top. Quantum Electron. 13(3), 573–579 (2007), http://dx.doi.org/10.1109/JSTQE.2007.897178 . [CrossRef]
  13. T. Miya, Y. Terunuma, T. Hosaka, and T. Miyashita, “Ultimate low-loss single-mode fibre at 1.55 µm,” Electron. Lett. 15(4), 106–108 (1979), http://dx.doi.org/10.1049/el:19790077 . [CrossRef]
  14. S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005), http://dx.doi.org/10.1109/JSTQE.2005.850249 . [CrossRef]
  15. A. Shirakawa, H. Maruyama, K. Ueda, C. B. Olausson, J. K. Lyngsø, and J. Broeng, “High-power Yb-doped photonic bandgap fiber amplifier at 1150-1200 nm,” Opt. Express 17(2), 447–454 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-2-447 . [CrossRef] [PubMed]
  16. T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7615 . [CrossRef] [PubMed]
  17. W. Tong, H. Wei, J. Li, H. Wang, Q. Han, J. Luo, G. Ren, X. Yu, and P. Shum, “Investigation of all-solid photonic bandgap fiber with low losses in low-order bandgaps,” Opt. Quant. Electron. 39, 1071 (2008), http://dx.doi.org/10.1007/s11082-007-9145-x .
  18. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8452 . [CrossRef] [PubMed]
  19. K. Tajima, J. Zhou, K. Nakajima, and K. Sato, “Ultralow loss and long length photonic crystal fiber,” J. Lightwave Technol. 22(1), 7–10 (2004), http://dx.doi.org/10.1109/JLT.2003.822143 . [CrossRef]
  20. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-69 . [CrossRef] [PubMed]
  21. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32(12), 1719–1721 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-12-1719 . [CrossRef] [PubMed]
  22. J. Bromage, K. Rottwitt, and M. E. Lines, “A method to predict the Raman gain spectra of germanosilicate fibers with arbitrary index profiles,” IEEE Photon. Technol. Lett. 14(1), 24–26 (2002), http://dx.doi.org/10.1109/68.974149 . [CrossRef]
  23. C. Brooks and F. Di Teodoro, “1-mJ energy, 1-MW peak-power, 10-W average-power, spectrally narrow, diffraction-limited pulses from a photonic-crystal fiber amplifier,” Opt. Express 13(22), 8999–9002 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-22-8999 . [CrossRef] [PubMed]
  24. A. V. Smith and B. T. Do, “Bulk and surface laser damage of silica by picosecond and nanosecond pulses at 1064 nm,” Appl. Opt. 47(26), 4812–4832 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=ao-47-26-4812 . [CrossRef] [PubMed]
  25. K. Petermann and R. Kühne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4(1), 2–7 (1986), http://dx.doi.org/10.1109/JLT.1986.1074620 . [CrossRef]
  26. B. H. Bransden, and C. J. Joachain, Introduction to Quantum Mechanics (Wiley, 1989).
  27. D. Marcuse, Theory of Dielectric Optical Waveguides. (Academic, 1991).
  28. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-4-3258 . [CrossRef] [PubMed]
  29. B. Ward and M. Mermelstein, “Modeling of inter-modal Brillouin gain in higher-order-mode fibers,” Opt. Express 18(3), 1952–1958 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-1952 . [CrossRef] [PubMed]
  30. S. L. Semjonov, O. N. Egorova, A. F. Kosolapov, A. E. Levchenko, V. V. Velmiskin, A. D. Pryamikov, M. Y. Salganskiy, V. F. Khopin, M. V. Yashkov, A. N. Guryanov, and E. M. Dianov, “LMA fibers based on two-dimensional solid-core photonic bandgap fiber design,” Proc. SPIE 7580, 758018, 758018-9 (2010), http://dx.doi.org/10.1117/12.841265 . [CrossRef]
  31. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18(5), 737–743 (2000), http://dx.doi.org/10.1109/50.842091 . [CrossRef]
  32. F. L. Teixeira, and W. C. Chew, “General closed-form pml constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guided Wave Lett. 8, 223–225 (1998), http://dx.doi.org/10.1109/75.678571 .
  33. V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of Eigenvalue problems,” ACM Trans. Math. Softw. 31(3), 351–362 (2005), http://dx.doi.org/10.1109/75.678571 . [CrossRef]
  34. E. Coscelli, F. Poli, T. T. Alkeskjold, D. Passaro, A. Cucinotta, L. Leick, J. Broeng, and S. Selleri, “Single-mode analysis of Yb-doped double-cladding distributed spectral filtering photonic crystal fibers,” Opt. Express 18(26), 27197–27204 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-26-27197 . [CrossRef]
  35. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-173 . [CrossRef] [PubMed]
  36. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for Optical Fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://dx.doi.org/10.1109/JQE.2007.903364 . [CrossRef]
  37. M. H. Muendel, R. Farrow, K. Liao, D. Woll, J. Luu, C. Zhang, J. J. Morehead, J. Segall, J. Gregg, K. Tai, B. Kharlamov, H. Yu, and L. Myers, “Fused fiber pump and signal combiners for a 4-kW ytterbium fiber laser,” Proc. SPIE 7914, 791431, 791431-7 (2011), http://dx.doi.org/10.1117/12.877572 . [CrossRef]
  38. D. C. Brown, and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quant. Electron. 37, 207–217 (2001), http://dx.doi.org/10.1109/3.903070 .
  39. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).
  40. L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002), http://dx.doi.org/10.1117/12.461727 . [CrossRef]
  41. J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16(19), 14435–14447 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14435 . [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited