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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12658–12663
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Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers

Andrei A. Fotiadi, Oleg L. Antipov, and Patrice Mégret  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12658-12663 (2008)
http://dx.doi.org/10.1364/OE.16.012658


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Abstract

We quantify the refractive index changes (RIC) in single-mode ytterbium-doped optical fibers induced by optical pulses at 980 nm and, for the first time, report details of the effect dynamics. The RIC dynamics is shown to follow that of the population of the excited/unexcited ion states with a factor proportional to their polarizability difference (PD). The absolute PD value is evaluated in the spectral range of 1460–1620 nm for different fiber samples and is found to be independent on the fiber geometry and on the ion concentration. The PD dispersion profile indicates to a predominant far-resonance UV rather than near-resonant IR transitions contribution to the RIC.

© 2008 Optical Society of America

1. Introduction

One of the main mechanisms of the refractive index change (RIC) in pumped rare-earth-doped optical fibers is associated with changes of population of the ion states with different polarizabilities [1

1. M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly Enhanced Nonlinearity in Doped Fibers for Low-Power All-Optical Switching: A Review,” Opt. Fiber Technol. 3, 44–64 (1997). [CrossRef]

,2

2. E. Desurvire, Erbium-doped fiber amplifiers: Principles and Applications (Willey, New York, 1994).

]. This electronic effect, which is important both for laser crystals and glasses, is intensively investigated during several years [3–10

3. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M.J. F. Digonnet, “Experimental and Theoretical Analysis of the Resonant Nonlinearity in Ytterbium-Doped Fiber,” J. Lightwave Technol. 16, 798–806 (1998). [CrossRef]

]. The origin of the polarizability difference (PD) in rear-earth ions used for doping the laser fibers is also widely discussed. Some authors believe that the main contribution to the PD comes from located far from the resonance strong UV transitions [1

1. M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly Enhanced Nonlinearity in Doped Fibers for Low-Power All-Optical Switching: A Review,” Opt. Fiber Technol. 3, 44–64 (1997). [CrossRef]

], similar to those observed in laser crystals [4

4. O.L. Antipov, O.N. Eremeykin, A.P. Savikin, V.A. Vorob’ev, D.V. Bredikhin, and M.S. Kuznetsov, “Electronic Changes of Refractive Index in Intensively Pumped Nd:YAG Laser Crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003). [CrossRef]

,8

8. O.L. Antipov, D.V. Bredikhin, O.N. Eremeykin, A.P. Savikin, E.V. Ivakin, and A.V. Sukhadolau, “Electronic mechanism of refractive index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006). [CrossRef] [PubMed]

]. An alternative model suggests the predominant contribution from the near-resonance IR transitions [2

2. E. Desurvire, Erbium-doped fiber amplifiers: Principles and Applications (Willey, New York, 1994).

,5–7

5. E. Bochove, “Nonlinear refractive index of rare-earth-doped fiber laser,” Opt. Lett. 29, 2414–2416 (2004). [CrossRef] [PubMed]

]. Correct understanding of these electronic phenomena in Yb-doped fibers [3

3. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M.J. F. Digonnet, “Experimental and Theoretical Analysis of the Resonant Nonlinearity in Ytterbium-Doped Fiber,” J. Lightwave Technol. 16, 798–806 (1998). [CrossRef]

,5

5. E. Bochove, “Nonlinear refractive index of rare-earth-doped fiber laser,” Opt. Lett. 29, 2414–2416 (2004). [CrossRef] [PubMed]

,7

7. H. Garsia, A.M. Johnson, F.A. Oguama, and S. Trivedi, “Pump-induced nonlinear refractive index change in erbium and ytterbium-doped fibers: theory and experiment,” Opt. Lett. 30, 1261–1263 (2005). [CrossRef]

,11

11. S.I. Stepanov, A.A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064nm,” Opt. Express 15, 8832–8837 (2007). [CrossRef] [PubMed]

] is very important because the pump-induced index changes influence dynamics of the fiber lasers and amplifiers, fiber Bragg gratings, etc. significantly. Besides, the enhanced nonlinear phase shift can be used for optical switching [1

1. M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly Enhanced Nonlinearity in Doped Fibers for Low-Power All-Optical Switching: A Review,” Opt. Fiber Technol. 3, 44–64 (1997). [CrossRef]

] and coherent beam combining [5

5. E. Bochove, “Nonlinear refractive index of rare-earth-doped fiber laser,” Opt. Lett. 29, 2414–2416 (2004). [CrossRef] [PubMed]

,12–14

12. H. Bruesselbach, D. C. Jones, M. S. Mangir, M Minden, and J.L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30, 13–15 (2005). [CrossRef]

].

In this paper we report original results on observation of RIC in standard single-mode Yb-doped optical fibers under diode pumping at 980 nm in configuration of Mach-Zehnder interferometer operating in IR far from the absorption and emission Yb3+ ion resonances. The main objective is to explore the RIC dynamics and to determine the values of the PD in standard Yb-doped fibers in spectral range 1460–1620 nm.

2. Experimental setup

The Yb-doped fibers under investigation ware included in one arm of the all-fiber spliced Mach-Zehnder interferometer (Fig. 1(a)). The CW-radiation of diode laser “Tunics” with the coherence length ~10 m was used as a probe wave and was detected at the interferometer output by fast photodiode. The probe wavelength λT could be continuously tuned from 1460 to 1620 nm. The fibers under test were pumped by a standard pumping laser diode operating at λP≈980 nm with the power up to ~145 mW. The RIC was evaluated from response to the rectangular pump pulse of 10 µs - 10 ms duration (Fig. 1(b)).

Fig. 1. Experimental setup for testing of Yb-doped fibers (a); (b): the pump pulse profile (red), recorded oscilloscope trace (black), and reconstructed phase trace (blue).

3. Experimental results and their interpretation

The phase shifts (up to ~4π) shown in Fig. 2 indicate strong RICs induced in Fiber 1 by different pump pulses. Being dependent on the total absorbed energy, the effect exhibits smooth saturation at some steady-state level that also depends on the pulse amplitude (Fig. 2(a)) and on the pulse duration (Fig. 2(b)) in a similar manner. Decaying parts of the phase traces describe the refractive index relaxation after the end of the excitation pumping pulse. They are perfectly fitted by the exponential function φ(t)~exp(-t/τsp) with the relaxation constant corresponding to the excited state life-time of Yb-ion τsp≈850 µs (nearly the same for all Fibers 1–4). All normalized curves shown in the inserts in Fig.2 seem to be similar, despite they correspond to different pumping conditions. A multi-exponential relaxation behavior that might be attributed to the temperature induced RIC has not been observed in our experiments. The above-presented relaxation behavior indicates to the electronic mechanism of RIC associated with population redistribution between the Yb-ion levels (2F5/2 and 2F7/2) possessing different polarizabilities.

Fig. 2. Phase shifts at 1550 nm induced in Fiber 1 of 2m length by pulses of different (0.02–2.5 ms) pulse duration (a) and of different (2–145 mW) pulse amplitude (b). Inserts show the same phase shifts normalized to their maximal levels (only decaying parts are presented).

In simplified two-level approximation, the electronic RIC δn caused by the excited state population (2F7/2) change δN 2 can be expressed as [3

3. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M.J. F. Digonnet, “Experimental and Theoretical Analysis of the Resonant Nonlinearity in Ytterbium-Doped Fiber,” J. Lightwave Technol. 16, 798–806 (1998). [CrossRef]

,4

4. O.L. Antipov, O.N. Eremeykin, A.P. Savikin, V.A. Vorob’ev, D.V. Bredikhin, and M.S. Kuznetsov, “Electronic Changes of Refractive Index in Intensively Pumped Nd:YAG Laser Crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003). [CrossRef]

]:

δn=2πFL2n0ΔpδN2
(1)

where n 0 is the unperturbed refractive index; FL=(n 0 2+2)/3 is the Lorentz factor; Δp=p 2-p 1, p 1 and p 2 are the polarizabilities of Yb3+ ions in the ground (2F7/2) and excited state (2F5/2), respectively.

The corresponding phase shift detected in the probe wave at λT in the fiber length L can be evaluated from Eq. (1) by integration with ρT(r) as a weight function:

δφ=4π2λT0L0δn(z,r)ρT(r)rdrdzη¯ρT(0)λT[4π2FL2n0Δp]δN2
(2)

where δN2=2π0L0N2(z,r)rdrdz is the pump-induced change in the number of the excited Yb3+ ions in the whole fiber volume, ρ T(r) is the normalized power radial distribution of the probe light. Parameter η¯ ρT(0) approximates the efficiency of the probe mode interaction with the population changes δN 2(r) induced in the fiber core. Here we intentionally separated the factor ρT(0)/λT that includes the major part of the dependence on λT and the correction factor η¯ accounting for a distributed character of the population changes δN 2(r) within the doped area core. If all changes are considered to occur near the fiber axis η̄→1 and for uniform Yb-ion distribution on the fiber core η≈0.7. For a Gaussian ion distribution the factor η¯ ρT(0)~ā -2 has been quantified from a step-index fiber approximation with the core radius ā=w[1.3+0.864(λS/λc)3/2+0.0298(λS/λc)6]-1 [15

15. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

,16

16. L. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, New. York, 1983).

] evaluated from Fiber 1–4 specifications.

Equation (2) predicts that the phase shift is proportional to the pump-induced change in the whole number of the excited ions. Therefore, the dynamics of both of them is governed by the same rate equation:

dδN2dt=PinPouthvPPASEhvsδN2τsp
(3)

where h is the Plank constant, νP, νS are the average frequencies of the pump and the amplified spontaneous emission (ASE) (or lasing), Pin, Pout are the input and output (residual) powers at νP, and PASE is the emitted power at νS.

In case of total pump absorption and low spontaneous emission, Eqs. (2,3) reduce to the following simple expression that is validated for highly doped Fibers 1–3 below:

δφ(t)=Kτsp[1exp(tτsp)]P0
(4)

Here K=2πF 2 LΔpη¯ ρT(0)λP/hcλT and P 0 is the pump pulse amplitude.

Fig. 3. Phase shifts at 1550 nm induced by 4ms pulses in 2 m long Fiber 2 as functions of time (a), normalized time τ=1-exp(-t/τsp) (b), pulse amplitude (c). (d) shows the slope δφ/δt|t→0.

In highly doped fibers the pump power is completely absorbed by Yb-ions. The experimental traces shown in Fig. 3 follow the key features predicted by Eq. (4) very well. The exponential character of the phase growth to its steady-state (Fig. 3(b)), and the linear dependence on the pump pulse amplitude (Fig. 3(c)) are observed. If the pass gain and ASE level remain low, the slopes δφ/δt|t→0 can be measured for different pulse amplitudes (Fig. 3(d)), and their linear fit allows one to determine the factor K=0.056π rad ms-1mW-1 that proves to be the only fitting parameter in Eqs. (2–4).

To validate the model at higher excitation level, the output powers detected at both fiber ends during the pulse excitation were recorded together with the phase traces. The spectra of the optical emission at both fiber ends highlights ASE as the reason of saturation of the monotonic growth of the phase shift at high pump energies (Fig. 4(a,b)) (ripples are attributed to a specific coupling with the analyzer). Notice that due to absorption saturation and reabsorption processes, the ASE pulse observed in the forward direction is delayed with respect to the backward pulse providing a break in the leading edge of the total power (Fig. 4(c)). Using Eqs. (2,3) we can present the ASE power by the following equation:

PASE(t)PinK1[dδφ(t)dt+δφ(t)τsp]
(5)

which is in a good quantitative agreement with the recorded traces (Fig. 4(c,d)). Note that the reconstructed pulse edge also exhibits a break as the experimentally observed ASE pulse.

Fig. 4. ASE from Fiber 2 in backward (B) and forward (F) directions during the excitation pulse: optical spectra (a,b), and detected power (c). Curve (d) presents the reconstructed power.

Factor K is the only parameter relating the induced phase shift and the pump power in the fiber samples and at the probe wavelengths used in our experiments. Phase dynamics observed in two different samples of Fiber 3 demonstrates the factor K to be independent on the fiber length (Fig. 5(a)). Reduction in the fiber length just causes reduction of the saturation energy, but does not affect the slope in phase changes. On other hand, the phase shift dependences observed in Fibers 1–4, provided that both the fiber length and other conditions are the same, reveal different slopes and other distinct features (Fig. 5(b)). In particular, Fiber 4 with lowest Yb-ion concentration exhibits lower saturation power than others fibers, but slope in this fiber is nearly the same as that in Fiber 3 and is larger than the slope in Fiber 2 and Fiber 1. Approximation by Eq. (5) gives K 4K 3≈1.21K 2≈1.57K 1≈0.067π rad ms-1mW-1 that highlights also the ratios between the parameters K for Fibers 4-1, respectively. These results are in good agreement with Eqs. (2–4) that predict that factor K is to be independent on Yb-ion concentration and to be inversely proportional to the square of the fiber core radius ā. They allow us estimation of the polarizability difference at 1550 nm which proves to be the same for all tested fibers. Evaluation of η¯ ρT(0) for a step-index fiber and Gaussian Yb-ion distribution with diameter equal to ā(η¯≈0.85) gives Δp 1550≈7.5·10-26 cm3. We expect ~20% error in this value due to uncertainty of the doped area size. Notice that above mentioned PD value is comparable with those reported for Yb-doped laser crystals earlier [4

4. O.L. Antipov, O.N. Eremeykin, A.P. Savikin, V.A. Vorob’ev, D.V. Bredikhin, and M.S. Kuznetsov, “Electronic Changes of Refractive Index in Intensively Pumped Nd:YAG Laser Crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003). [CrossRef]

].

Fig.5. Phase shift induced by 145-mW pulses: in different lengths of Fiber 3 (a), in 2 m long Fibers 1–4 (b), and in Fiber 2 at different probe wavelengths (c). Figure (d) shows the relative polarizability difference (points) in comparison with the resonance and non-resonance PD contributions at ~1µm (blue), and ~0.4 µm (black), and the dependence ~ρT(0)/λT (red).

To characterize the PD dispersion in spectral range of 1450–1620 nm, the Fiber 2 was also additionally tested at different probe wavelengths λT. As it was reported for the two-core fibers earlier [3

3. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M.J. F. Digonnet, “Experimental and Theoretical Analysis of the Resonant Nonlinearity in Ytterbium-Doped Fiber,” J. Lightwave Technol. 16, 798–806 (1998). [CrossRef]

], significantly different slopes δφ/δτ (Fig. 5(c)) are observed at different λT. In our case this difference is explained by the dependence ~ρT(0)/λT calculated for Fiber 2 (Fig. 5(d)) mainly. The experimentally observed dispersion Δp(λT) is to be compared with the Lorentz-line dispersion profiles associated with near-IR and UV transitions of Yb3+ ions at ~1µm and ~0.4 µm, respectively. The reconstructed PD profile matches the UV-line wing and has the same value in all investigated spectral range.

4. Conclusion

Summarizing, we have reported observation of a strong RIC at 1460–1620 nm in standard single-mode Yb-doped fibers pumped in the absorption line of Yb3+ ions. The RIC exhibits a typical excited population dynamics and is explained in two-level approximation. These results clearly indicate to the electronic mechanism of PD between the excited and unexcited Yb3+ ion states and allow us to estimate its value. The PD dispersion curve indicates the dominant nonresonant contribution of the UV transitions.

Acknowledgements

This work was supported by “Interuniversity Attraction Pole program VI/10” of the Belgian Science Policy program and by “Nonlinear optics of unique laser system” program of Russian Academy of Science and Russian Foundation of Basic Research (grant №07-02-92184).

References

1.

M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly Enhanced Nonlinearity in Doped Fibers for Low-Power All-Optical Switching: A Review,” Opt. Fiber Technol. 3, 44–64 (1997). [CrossRef]

2.

E. Desurvire, Erbium-doped fiber amplifiers: Principles and Applications (Willey, New York, 1994).

3.

J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M.J. F. Digonnet, “Experimental and Theoretical Analysis of the Resonant Nonlinearity in Ytterbium-Doped Fiber,” J. Lightwave Technol. 16, 798–806 (1998). [CrossRef]

4.

O.L. Antipov, O.N. Eremeykin, A.P. Savikin, V.A. Vorob’ev, D.V. Bredikhin, and M.S. Kuznetsov, “Electronic Changes of Refractive Index in Intensively Pumped Nd:YAG Laser Crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003). [CrossRef]

5.

E. Bochove, “Nonlinear refractive index of rare-earth-doped fiber laser,” Opt. Lett. 29, 2414–2416 (2004). [CrossRef] [PubMed]

6.

Yu.O. Barmenkov, A.V. Kir’yanov, and M.V. Andres, “Resonant and thermal changes of refractive index in a heavily doped erbium fiber pumped at wavelength 980 nm,” Appl. Phys. Lett. 85, 2466–2468 (2004). [CrossRef]

7.

H. Garsia, A.M. Johnson, F.A. Oguama, and S. Trivedi, “Pump-induced nonlinear refractive index change in erbium and ytterbium-doped fibers: theory and experiment,” Opt. Lett. 30, 1261–1263 (2005). [CrossRef]

8.

O.L. Antipov, D.V. Bredikhin, O.N. Eremeykin, A.P. Savikin, E.V. Ivakin, and A.V. Sukhadolau, “Electronic mechanism of refractive index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006). [CrossRef] [PubMed]

9.

D.N. Messias, T. Catunda, J.D. Myers, and M.J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32, 665–667 (2007). [CrossRef] [PubMed]

10.

J. Margerie, R. Moncorgé, and P. Nagtegaele, “Spectroscopic investigation of the refractive index variations in the Nd:YAG laser crystal,” Phys. Rev. B 74, 235108–10 (2006). [CrossRef]

11.

S.I. Stepanov, A.A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064nm,” Opt. Express 15, 8832–8837 (2007). [CrossRef] [PubMed]

12.

H. Bruesselbach, D. C. Jones, M. S. Mangir, M Minden, and J.L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30, 13–15 (2005). [CrossRef]

13.

T.Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005). [CrossRef]

14.

H. Bruesselbach, Sh. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B22, 347–353 (2005).

15.

W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

16.

L. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, New. York, 1983).

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.3510) Fiber optics and optical communications : Lasers, fiber

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: June 23, 2008
Revised Manuscript: August 3, 2008
Manuscript Accepted: August 3, 2008
Published: August 6, 2008

Citation
Andrei A. Fotiadi, Oleg L. Antipov, and Patrice Mégret, "Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers," Opt. Express 16, 12658-12663 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12658


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References

  1. M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, "Resonantly Enhanced Nonlinearity in Doped Fibers for Low-Power All-Optical Switching: A Review," Opt. Fiber Technol. 3, 44-64 (1997). [CrossRef]
  2. E. Desurvire, Erbium-doped fiber amplifiers: Principles and Applications (Willey, New York, 1994).
  3. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M.J. F. Digonnet, "Experimental and Theoretical Analysis of the Resonant Nonlinearity in Ytterbium-Doped Fiber," J. Lightwave Technol. 16, 798-806 (1998). [CrossRef]
  4. O.L. Antipov, O.N. Eremeykin, A.P. Savikin, V.A. Vorob�??ev, D.V. Bredikhin, M.S. Kuznetsov, "Electronic Changes of Refractive Index in Intensively Pumped Nd:YAG Laser Crystals," IEEE J. Quantum Electron. 39, 910-918 (2003). [CrossRef]
  5. E. Bochove, "Nonlinear refractive index of rare-earth-doped fiber laser," Opt. Lett. 29, 2414-2416 (2004). [CrossRef] [PubMed]
  6. Yu.O. Barmenkov, A.V. Kir�??yanov, M.V. Andres, "Resonant and thermal changes of refractive index in a heavily doped erbium fiber pumped at wavelength 980 nm," Appl. Phys. Lett. 85, 2466-2468 (2004). [CrossRef]
  7. H. Garsia, A.M. Johnson, F.A. Oguama, S. Trivedi, "Pump-induced nonlinear refractive index change in erbium and ytterbium-doped fibers: theory and experiment," Opt. Lett. 30, 1261-1263 (2005). [CrossRef]
  8. O.L.  Antipov, D.V.  Bredikhin, O.N.  Eremeykin, A.P.  Savikin, E.V.  Ivakin, A.V.  Sukhadolau, "Electronic mechanism of refractive index changes in intensively pumped Yb:YAG laser crystals," Opt. Lett. 31, 763-765 (2006). [CrossRef] [PubMed]
  9. D.N. Messias, T. Catunda, J.D. Myers, M.J. Myers, "Nonlinear electronic line shape determination in Yb3+-doped phosphate glass," Opt. Lett. 32, 665-667 (2007). [CrossRef] [PubMed]
  10. J. Margerie, R. Moncorgé, P. Nagtegaele, "Spectroscopic investigation of the refractive index variations in the Nd:YAG laser crystal," Phys. Rev. B 74, 235108-10 (2006) [CrossRef]
  11. S.I. Stepanov, A.A. Fotiadi, P. Mégret, "Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064nm," Opt. Express 15, 8832-8837 (2007). [CrossRef] [PubMed]
  12. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J.L. Rogers, "Self-organized coherence in fiber laser arrays," Opt. Lett. 30, 13-15 (2005). [CrossRef]
  13. T.Y. Fan, "Laser Beam Combining for High-Power, High-Radiance Sources," IEEE J. Sel. Top. Quantum Electron. 11, 567-577 (2005). [CrossRef]
  14. H. Bruesselbach, Sh. Wang, M. Minden, D.C. Jones, and M. Mangir, "Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere," J. Opt. Soc. Am. B 22, 347-353 (2005).
  15. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).
  16. L. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, New. York, 1983).

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