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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 15 — Jul. 18, 2011
  • pp: 13940–13948
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On the tunability of a narrow-linewidth Yb-fiber laser from three- to four-level lasing behaviour

Peter Zeil and Fredrik Laurell  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 13940-13948 (2011)
http://dx.doi.org/10.1364/OE.19.013940


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Abstract

We report on a tunable multi-watt ytterbium-doped fiber laser bridging the gap from three-level lasing around 980 nm to true four-level lasing at 1100 nm. Wavelength-locking and -tuning was achieved by using an external volume-Bragg grating(VBG) as the cavity end mirror. The results are compared with detailed numerical calculations based on a spectrally resolved rate equation analysis, taking competing emission at other wavelengths into account.

© 2011 OSA

1. Introduction

Broadly tunable laser sources are readily used for research purposes in spectroscopy and photochemistry. Yb-doped silica fiber lasers comprise a wide pumping range as well as a broad fluorescence spectrum, making them uniquely suitable for continously tunable lasers. The inherently high spatial and spectral beam-quality together with the extensive power scaling abilities (> 1 kW) [1

1. Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088–6092 (2004). [CrossRef] [PubMed]

] make Yb-doped silica fiber lasers excellent pump sources for nonlinear frequency converters [2

2. P. Gross, M. Klein, T. Walde, K. Boller, M. Auerbach, P. Wessels, and C. Fallnich, “Fiber-laser-pumped continuous-wave singly resonant optical parametric oscillator,” Opt. Lett. 27, 418–420 (2002). [CrossRef]

], thereby facilitating their use for example in mid-infrared spectroscopy around 3 μm, typically interesting for gas sensing.

In general, transitions between the two Stark splitted manifolds in the Yb-ion allow for lasing at wavelengths around 980 nm and between 1020-1200 nm [3

3. H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2 m region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995). [CrossRef]

], corresponding to three-level lasing and quasi-three level to four-level lasing, respectively.

Although wide narrow linewidth wavelength tuning between 1032-1124 nm [4

4. M. Auerbach, P. Adel, D. Wandt, C. Fallnich, S. Unger, S. Jetschke, and H. Mueller, “10 W widely tunable narrow linewidth double-clad fiber ring laser,” Opt. Express 10, 139–144 (2002). [PubMed]

] for continous wave Yb-doped fiber lasers has been demonstrated, extensive tuning over the entire emission spectrum from 980 nm to wavelengths above 1 μm has not been shown. This contribution reports on the viability of such an unprecendented tuning range for a laser with excellent spatial and spectral beam quality.

2. Tuning range limitations for Yb-doped silica fiber lasers

Ytterbium fiber lasers operating below 1 μm require a high population inversion (> 50%), which entails three challenges for achieving efficient and stable lasing: photodarkening, competing emission of quasi-three level transition above 1 μm and life-time quenching.

2.1. Photodarkening

Photodarkening in Yb-doped silica fibers is attributed to color centers which absorb light in the UV und visible spectral region but also extend an absorption tail to the near-infrared. With photodarkening being highly susceptible to the density of excited Yb-ions [5

5. J. Koponen, M. Sderlund, H. Hoffman, D. Kliner, J. Koplow, and M. Hotoleanu, “Photodarkening rate in Yb-doped silica fibers,” Appl. Opt. 47, 1247–1256 (2008). [CrossRef] [PubMed]

], it results in significant transmission losses for three-level Yb fiber lasers. However, this degragation in output power can be counteracted by Ce-codoping [6

6. M. Engholm, P. Jelger, F. Laurell, and L. Norin, “Improved photodarkening resistivity in ytterbium-doped fiber lasers by cerium codoping,” Opt. Lett. 34, 1285–1287 (2009). [CrossRef] [PubMed]

], which was adressed by adding 0.06 at. % to the preform of the in-house drawn fiber used in our experiments. Yb- and Al-contents were 0.22 at. % and 2.18 at. %, respectively.

2.2. Competing emission

Competing emission of the quasi-three level transition in Yb-doped silica around 1030 nm, where the lower level is only partially populated, starts at considerably lower excited ion levels and therefore needs to be suppressed. As wavelength selective elements will not provide more than approximately 50 dB suppression of undesired gain, the applicable fiber length will be limited, because increased fiber length favors lasing at longer wavelengths. To estimate the maximum fiber length, Nilsson et al. suggested to consider the interdependencies of gains at different wavelengths [7

7. J. Nilsson, J. Minelly, R. Paschotta, A. Tropper, and D. Hanna, “Ring-doped cladding-pumped single-mode three-level fiber laser,” Opt. Lett. 23, 355–357 (1998). [CrossRef]

]. They showed that gain at any third wavelength can be quantified in terms of the gain at two other wavelengths, in this case undesired gain at 1030 nm relates to the negative gain at the pump wavelength (915 nm) and the desired oscillation wavelength. With the help of wavelength-dependent cross-section data ( σia, σie absorption and emission cross-section at wavelength λi) from [8

8. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]

] this yields
G1030=σ1030eσ980eG980Γ1030Γ915σ1030eσ980eσ980aσ915a.G915=0.22G980Γ1030Γ9150.7G915,
(1)
Gλ and Γλ being the gain and the overlap factor between dopant distribution and normalized optical mode in the fiber at the respective wavelengths. The contribution from the gain at 980 nm is minor (<1.6 dB), as a common cavity with 96 % outcoupling loss per round trip requires 7 dB gain single pass at threshold.

In a typical double clad pumping configuration the ratio of the overlap factors in (1) can be approximated by the ratio of cladding area and core area Acla/Acor (typically between 50 and 100), which means that for each dB absorbed pump the second term in (1) amounts in significant parasitic gain at 1030 nm. It is therefore imperative to use fibers with low cladding/core ratio to ensure efficient lasing [7

7. J. Nilsson, J. Minelly, R. Paschotta, A. Tropper, and D. Hanna, “Ring-doped cladding-pumped single-mode three-level fiber laser,” Opt. Lett. 23, 355–357 (1998). [CrossRef]

]. The maximum fiber length can then be estimated by limiting gain at 1030 nm to the available wavelength selective suppression. Assuming spatially constant pump absorption, the fiber used in the experiments (cladding area: 70000 μm2, core area, i.e. doped area: 800 μm2) with a cladding/core ratio of <90 and 2 dB/m pump absorption i.e. negative gain at 915 nm is theoretically limited to a length of 40 cm, given a parasitic gain suppression of 50 dB.

2.3. Life-time quenching

Paschotta et al. have discovered that Yb-doped fibers may exhibit an unbleachable loss especially when a large fraction of the ions is excited [9

9. R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5–6), 375–378 (1997). [CrossRef]

]. They attributed this to a small number of ions featuring a shorter excited state life-time which are available for reabsorption of the signal after their fast deexcitation. Quantitatively, Nilsson et al. described the wavelength dependent loss αλq with the help of cross section data [8

8. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]

]
αλqξGpΓλΓpσλa(σλa+σλe)(σpaσλeσpeσλa)1
(2)
with ξ being the fraction of ions that have a shorter excited state life-time. Figure 1 displays the spectral dependence of αλq for different fractions of quenched ions for a fiber with the same geometry as the one used in the experiments. Evidently, even low quenching results in considerable loss for three-level lasing transitions i.e. only a fraction of 0.5 % already entails a 3 dB loss at 975 nm for each dB absorbed pump. As the losses decrease substantially for slightly longer wavelength, the lower limit of the experimental tuning range was increased to 980 nm.

Fig. 1 Spectrally resolved quenching losses per dB pump absorption for several quenched ion fractions in a fiber with cladding/core ratio of 90.

3. Modelling of the laser process - rate equation analysis

The estimation of the optimum fiber length presented in section 2 presumes a spatially constant excited ion population over the entire fiber, thus disregarding important factors such as pump saturation, signal/pump cross saturation or cavity configuration. To obtain a more accurate understanding of the laser process, a spectrally resolved rate equation analysis taking spatially non-constant Yb-ion populations into account was conducted. The analysis is based on the model presented in [10

10. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Numerical and experimental analysis of erbium-doped fiber linear cavity lasers,” Opt. Commun. 156(4–6), 264–270 (1998). [CrossRef]

], which had to be adapted for cladding-pumped Yb-doped fiber lasers and said spatially non-constant excited ion populations.

By dividing the amplified spontaneaous emission (ASE) spectrum into N slots with bandwidth Δλ, rate equations [Eq. (3)] governing the evolution of pump and signal powers along the fiber can be expressed.
dPi±dz=±Γi(σien2σian1)Pi±±2σien2Aihc2λi3ΔλαiPi±n2=ntot(iNΓiλihcAσia(Pi++Pi))(iNΓiλihcA(σia+σie)(Pi++Pi)+τ1)1
(3)
Pi is the power at wavelength λi, n 1(n 2) and n the coordinate dependent lower(upper) state population densities and total Yb-ion concentration with ntot = n 1 + n 2. Ai denotes the effective transversal mode area, which can be approximated with core area and cladding area for signal and pump radiation, respectively. A is the doped area of the fiber and τ the excited-state lifetime. The term αi accounts for losses and the signs ± refer to either forward or backward propagation of the beams. The described system of non-linear coupled rate with its corresponding set of boundary conditions [Eq. (4)] for a linear cavity (Rm,i and ηm,i mirror reflectivity and coupling loss, λp and Pp pump wavelength and launched pump power)
Pi+(z=0)=R1,iη1,iPi(z=0)Pi(z=L)=R2,iη2,iPi+(z=L)Pλp+(z=0)=Pp+R1,iη1,iPi(z=0)
(4)
can be numerically solved with the Runge-Kutta method making use of a shooting method which effectively reduces the solution of a boundary value problem to the solution of an initial value problem [11

11. Michael T. Heath, Scientific Computing, 2nd ed. (McGraw-Hill, 2002).

]. In this configuration the developed numerical tool allows to simulate spectral particularities of the laser in detail, as all variables (transmission losses, coupling losses, mirror reflectivities) can be spectrally resolved, it does however not regard other transversal modes than the fundamental mode and is not sensitive to longitudinal mode selection of the linear cavity, which proved to be a valid simplification as experimental and theoretical results are in good agreement.

4. Competing emission in laser cavity

As a first step the introduced model was used to examine the validity of the estimation made in section 2, especially whether the made assumptions of spatially constant excited ion population levels and constant pump absorption are sustainable. Therefore, the lasing process of a fiber laser using the formerly suggested optimal fiber length of 40 cm of the in-house drawn fiber was modelled. The assumed single-pass pumped linear cavity was comprised of one narrowband (0.4 nm) cavity delimiter that provides 100% reflection at 980 nm with 50 dB suppression of spurious reflections and the second outcoupling delimiter with 4% broadband reflection, which is comparable to a combination of a VBG and the Fresnel reflection of a cleaved fiber end. To study the influence of the cavity configuration, simulations were performed with and without (delimiter reflectivities = 0) the cavity at pump power 5.7 W (915 nm), which corresponds to the calculated pump power at laser threshold. Results are displayed in Fig. 2, where Fig. 2(a) depicts the change in pump power as well as the change in excited ion population along the fiber and Fig. 2(b) displays the signal outputs at both fiber ends. First of all, it has to be noted that pump saturation significantly reduces pump absorption with respect to small signal absorption (0.8 dB for 0.4 m with 2 dB/m) and would therefore give rise to a possibly increased fiber length, according to the mentioned remarks on gain suppression (section 2). Secondly, the introduction of the cavity and thereby feedback from the broadband Fresnel reflection apparently increases the ratio between ASE levels at 980 nm and 1030 nm. In contrast to the estimation based solely on gain considerations, here the demand for stronger suppression of ASE around 1030 nm and for an ultimately shorter maximal available fiber length becomes necessary. However, as both effects counteract each other the simulated maximum applicable fiber length is with 50 cm (at 50 dB suppression of ASE) slightly longer. In addition, it was determined that, increasing the suppression of ASE (also 50 dB) even on the outcoupling side (compare Fig. 3), a maximum fiber length of 80 cm might be feasible.

Fig. 2 Dependence of inversion level and pump absorption (a) and fiber output signals (b) when pumping the fiber inside or outside a cavity.
Fig. 3 Setup for angle-tuning with VBG.

5. Experimental setup - cavity configuration

Tuning of an Ytterbium fiber laser from 980 nm to 1100 nm was carried out by the setup depicted in Fig. 3. Pump radiation at 915 nm was launched into the fiber with approxametely 75 % efficiency. As parasatic reflections, especially of ASE around 1030 nm, would prevent lasing at shorter and longer wavelengths, both fiber end facets were angle-polished. Wavelength selection and suppression of parasitic gain was provided by an intra-cavity VBG. For the short wavelength tuning range (< 1000 nm) ASE was further suppressed by the use of an dichroic ASE filter on the outcoupling side (40 dB suppression >1000 nm).

5.1. Wavelength selection

5.2. Choice of outcoupling mirror

As mentioned before in section 4 the available fiber length was theoretically limited to 80 cm, however the eventual experimental length could not exceed 65 cm without losing sufficient suppression of gain at 1030 nm and still overcoming additional cavity losses (propagation loss in fiber, coupling loss, etc.).

Also, the limited fiber length considerably limits the gain for wavelengths between the two main emission peaks in Ytterbium, requesting that special care is given to the amount of outcoupling. With the help of the rate equation analysis different arrangements have been simulated to determine the optimal outcoupling mirror reflectivity. Simulations were performed for a linear cavity comprising a high-reflective VBG tuned to the desired outcoupling wavelength and a broadband outcoupling mirror with wavelength independent reflectivity which was swept from 0 to 50 %. Coupling efficiencies of 90 % each at both cavity delimeters, as well as estimated 0.2 dB/m transmission losses in the fiber were also taken into account and the results are displayed in Fig. 4. Disregarding more elaborate mirror reflectivity designs that optimize output power for all wavelengths, an outcoupling mirror with spectrally constant reflectivity of approximately 10 % maximizes output power in the critical wavelength region between 980 nm and 1000 nm, predicting output powers over 2 W while pumping with 25 W at 915 nm. For the experiments a mirror exhibiting a constant reflectivity of (12 ± 1) % for wavelengths between 900 and 1200 nm was chosen in the actual experimental setup (Fig. 3) to make sure the gain would exceed any additional losses in the cavity.

Fig. 4 Study on optimal cavity configuration at 25 W launched pump for 65 cm Ce-codoped Yb-fiber with 16 μm core and 150 μm cladding diameter.

6. Results and discussion

The experimental results for the afore mentioned setup are disclosed in Fig. 5. Output powers within the investigated tuning interval (980-1100 nm) range from 1.3 W at 980 nm to 4.5 W at 1030 nm when pumping with 25 W power at 915 nm. Spectral properties (Fig. 6) of the output signals were recorded with an optical spectrum analyzer (OSA). Resolution limited (OSA resolution 0.06 nm) bandwidth measurements determined the FWHM to be 0.06 nm for all signals. The suppression of parasitic ASE compared to the signal exceeded 40 dB across the tuning range. However, the ratio of integrated spectral power fractions in the undesired spectral regions to the spectral power contained in the emission peak has a maximum of −22 dB at 980 nm und decreased towards longer wavelength to less than −35 dB, thus giving further experimental indication that longer fiber lengths are not favorable, when including the 980 nm emission peak in the tuning range.

Fig. 5 Comparison experimental and numerical data for 25 W launched pump power.
Fig. 6 Output spectra of tunable laser, wavelength-locked with four VBGs (997 nm - blue, 1030 nm - green, 1066 nm - red, 1100 nm - cyan).

Moreover, the wavelength dependency of the maximal output power is compared with simulated data applying the same assumptions as in section 5.2. The solid agreement between experimental data and simulated results for lasing wavelengths close to design wavelength of the respective VBGs drops moderately when the VBGs are tuned to shorter wavelengths and therefore used at larger angles. This seems to be a consequence of an introduced cavity loss at the VBGs that occurs due to the slowly increasing mismatch between their angular bandwidth and the angular divergence of the laser beam. This effect is especially pronounced for the VBG with design wavelength 1030 nm, as there was just one VBG with a bandwidth of 0.20 nm available. Additionally, the simulated data imparts information on the lower and upper bound of the tuning range of the investigated fiber gain medium, which are modelled to be 972 and 1120 nm. Although the prediction of the upper bound could not be verified, as no VBG with wavelengths longer than 1100 nm was available, the expected continued decrease in output power when tuning to longer wavelengths could be reproduced experimentally. At the lower bound the experimental results differ considerably from modelled data, which suggests that a fraction of Yb-ions in the used fiber displays life-time quenching. In order to verify, output powers for a gain medium with partially quenched ions have been simulated by introducing a spectrally resolved loss according to Eq. (2). The comparison of experimental results with those simulations hint to a quenched ion fraction of approxametely 2 %. So not only poses the lack of gain for wavelength shorter than 972 nm a limitation for continued tuning to shorter wavelengths but also the strong unbleachable loss induced by life-time quenching.

7. Conclusion and outlook

We have demonstrated a narrow-bandwidth Ytterbium fiber laser covering an unprecedented tuning range of 120 nm from 980 nm to 1100 nm, applying a very simple cavity configuration. By pumping this fiber at 915 nm we obtained watt-level continous wave output powers across the whole investigated spectrum. We achieved this by using angle-tuned VBGs as narrow-band filters that delivered wavelength-selection and eliminated spurious ASE. In addition, we devised a numerical tool to simulate the lasing process and put it to use in designing the laser cavity and predicting output signals. The results of the simulations are in good agreement with the experiments and deepened the understanding of the spectral dynamics and the tuning limitations of the fiber gain medium. Futhermore it allowed to estimate the degrading impact of life-time quenching on the laser performance.

Even though higher output powers might be achieved by employing fibers with a smaller cladding/core ratio, to date those fibers will still exhibit significant photodarkening which in this case was circumvented by utilizing a Ce-codoped fiber. Further improvement in output power should be feasible by using Ce-codoped photonic crystal fibers that either feature a large single-mode core which effectively reduces unwanted ASE by allowing for low core/cladding ratios or even photonic crystal fibers that possess a photonic bandgap suppressing ASE additionally.

Acknowledgments

The authors thank the Linneus Centre ADOPT, the Swedish Research Council (VR), and the Acreo Optic Fiber Center (AFOC) for financial support as well as Magnus Engholm and Fiber Optic Valley for supplying the doped fiber. Additionally, we would like to thank P. Jelger and B. Jacobsson for fruitful discussions concerning the experimental work.

References and links

1.

Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088–6092 (2004). [CrossRef] [PubMed]

2.

P. Gross, M. Klein, T. Walde, K. Boller, M. Auerbach, P. Wessels, and C. Fallnich, “Fiber-laser-pumped continuous-wave singly resonant optical parametric oscillator,” Opt. Lett. 27, 418–420 (2002). [CrossRef]

3.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2 m region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995). [CrossRef]

4.

M. Auerbach, P. Adel, D. Wandt, C. Fallnich, S. Unger, S. Jetschke, and H. Mueller, “10 W widely tunable narrow linewidth double-clad fiber ring laser,” Opt. Express 10, 139–144 (2002). [PubMed]

5.

J. Koponen, M. Sderlund, H. Hoffman, D. Kliner, J. Koplow, and M. Hotoleanu, “Photodarkening rate in Yb-doped silica fibers,” Appl. Opt. 47, 1247–1256 (2008). [CrossRef] [PubMed]

6.

M. Engholm, P. Jelger, F. Laurell, and L. Norin, “Improved photodarkening resistivity in ytterbium-doped fiber lasers by cerium codoping,” Opt. Lett. 34, 1285–1287 (2009). [CrossRef] [PubMed]

7.

J. Nilsson, J. Minelly, R. Paschotta, A. Tropper, and D. Hanna, “Ring-doped cladding-pumped single-mode three-level fiber laser,” Opt. Lett. 23, 355–357 (1998). [CrossRef]

8.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]

9.

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5–6), 375–378 (1997). [CrossRef]

10.

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Numerical and experimental analysis of erbium-doped fiber linear cavity lasers,” Opt. Commun. 156(4–6), 264–270 (1998). [CrossRef]

11.

Michael T. Heath, Scientific Computing, 2nd ed. (McGraw-Hill, 2002).

12.

B. Jacobsson, M. Tiihonen, V. Pasiskevicius, and F. Laurell, “Narrowband bulk Bragg grating optical parametric oscillator,” Opt. Lett. 30, 2281–2283 (2005). [CrossRef] [PubMed]

13.

B. Jacobsson, J. Hellstrom, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007). [CrossRef] [PubMed]

14.

P. Jelger and F. Laurell, “Efficient narrow-linewidth volume-Bragg grating-locked Nd:fiber laser,” Opt. Express 15, 11336–11340 (2007). [CrossRef] [PubMed]

15.

J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44(1), 81–89 (2008). [CrossRef]

OCIS Codes
(140.3615) Lasers and laser optics : Lasers, ytterbium
(060.3510) Fiber optics and optical communications : Lasers, fiber

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: May 23, 2011
Revised Manuscript: June 17, 2011
Manuscript Accepted: June 17, 2011
Published: July 6, 2011

Citation
Peter Zeil and Fredrik Laurell, "On the tunability of a narrow-linewidth Yb-fiber laser from three- to four-level lasing behaviour," Opt. Express 19, 13940-13948 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-13940


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References

  1. Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088–6092 (2004). [CrossRef] [PubMed]
  2. P. Gross, M. Klein, T. Walde, K. Boller, M. Auerbach, P. Wessels, and C. Fallnich, “Fiber-laser-pumped continuous-wave singly resonant optical parametric oscillator,” Opt. Lett. 27, 418–420 (2002). [CrossRef]
  3. H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2 m region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995). [CrossRef]
  4. M. Auerbach, P. Adel, D. Wandt, C. Fallnich, S. Unger, S. Jetschke, and H. Mueller, “10 W widely tunable narrow linewidth double-clad fiber ring laser,” Opt. Express 10, 139–144 (2002). [PubMed]
  5. J. Koponen, M. Sderlund, H. Hoffman, D. Kliner, J. Koplow, and M. Hotoleanu, “Photodarkening rate in Yb-doped silica fibers,” Appl. Opt. 47, 1247–1256 (2008). [CrossRef] [PubMed]
  6. M. Engholm, P. Jelger, F. Laurell, and L. Norin, “Improved photodarkening resistivity in ytterbium-doped fiber lasers by cerium codoping,” Opt. Lett. 34, 1285–1287 (2009). [CrossRef] [PubMed]
  7. J. Nilsson, J. Minelly, R. Paschotta, A. Tropper, and D. Hanna, “Ring-doped cladding-pumped single-mode three-level fiber laser,” Opt. Lett. 23, 355–357 (1998). [CrossRef]
  8. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]
  9. R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5–6), 375–378 (1997). [CrossRef]
  10. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Numerical and experimental analysis of erbium-doped fiber linear cavity lasers,” Opt. Commun. 156(4–6), 264–270 (1998). [CrossRef]
  11. Michael T. Heath, Scientific Computing , 2nd ed. (McGraw-Hill, 2002).
  12. B. Jacobsson, M. Tiihonen, V. Pasiskevicius, and F. Laurell, “Narrowband bulk Bragg grating optical parametric oscillator,” Opt. Lett. 30, 2281–2283 (2005). [CrossRef] [PubMed]
  13. B. Jacobsson, J. Hellstrom, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007). [CrossRef] [PubMed]
  14. P. Jelger and F. Laurell, “Efficient narrow-linewidth volume-Bragg grating-locked Nd:fiber laser,” Opt. Express 15, 11336–11340 (2007). [CrossRef] [PubMed]
  15. J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44(1), 81–89 (2008). [CrossRef]

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