OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 12 — Jun. 6, 2011
  • pp: 11318–11329
« Show journal navigation

Mode competition in high power fiber amplifiers

Arlee V. Smith and Jesse J. Smith  »View Author Affiliations


Optics Express, Vol. 19, Issue 12, pp. 11318-11329 (2011)
http://dx.doi.org/10.1364/OE.19.011318


View Full Text Article

Acrobat PDF (1097 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Using a beam propagation model of Yb3+ doped, CW fiber amplifiers we show that gain saturation by a strong fundamental mode significantly suppresses the growth of higher order modes with parallel polarization, but enhances the growth of higher order modes with perpendicular polarization. We quantify this effect in straight and bent fibers, with full core or restricted area doping.

© 2011 OSA

1. Introduction

Field interference can occur only if the light in the two modes is sufficiently spectrally/temporally coherent. Coherence between two modes is strong if they have the same spectrum and if the group velocity induced temporal walk off between them is much less than the coherence time of the light. Walk off times are of order 1 ps/m in large mode area fibers typical of high power amplifiers [14

14. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Gaalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16, 7233–7243 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-10-7233. [CrossRef] [PubMed]

], so for spectral widths of a few GHz, coherence remains high over the full length of a 10 m long amplifier.

Numerous modeling papers have addressed the growth of higher order modes in large mode area, Yb-doped fiber amplifiers in the incoherent case [4

4. Z. Jiang and J. R. Marciante, “Impact of transverse spatial-hole burning on beam quality in large-mode-area Yb-doped fibers,” J. Opt. Soc. Am. B25, 247–254 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=josab-25-2-247. [CrossRef]

10

10. J. Sousa and O. Okhotnikov, “Multimode Er-doped fiber for single-transverse-mode amplification,” Appl. Phys. Lett. 74, 1528–1530 (1999). [CrossRef]

]. They usually assume that the launched signal light is predominantly in the fundamental mode, but with a specified small fraction accidentally injected or scattered into some of the higher order modes. The most commonly employed models compute the local upper state populations based on the irradiances of the pump and certain populated signal modes, neglecting interference. The upper state population distribution is then used to compute the power gain for each signal mode, along with the pump loss. The gains of the higher order modes are often found to be greater than that of the fundamental mode. Such models have been applied to straight fibers with step index cores and uniform doping across the core [4

4. Z. Jiang and J. R. Marciante, “Impact of transverse spatial-hole burning on beam quality in large-mode-area Yb-doped fibers,” J. Opt. Soc. Am. B25, 247–254 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=josab-25-2-247. [CrossRef]

], where it is found that the output power fraction in the higher order modes is increased by gain saturation. This is sometimes cited as an important cause of beam degradation in high power amplifiers. Similar conclusions are drawn for bent fiber, with even higher relative gains in the higher order modes due to the bend-induced displacement and compression of the fundamental mode [5

5. S. Liao, M. Gong, and H. Zhang, “Influence of mode distortion on the transverse mode competition in large-mode-area amplifiers,” Opt. Commun.282, 406–412 (2009), http://linkinghub.elsevier.com/retrieve/pii/S0030401808010067. [CrossRef]

]. To counteract this effect, doping of the active ion can be confined to the center of the core so it overlaps best with the fundamental mode [6

6. J. Marciante, “Gain filtering for single-spatial-mode operation of large-mode-area fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron.15, 30–36 (2009), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4773298. [CrossRef]

10

10. J. Sousa and O. Okhotnikov, “Multimode Er-doped fiber for single-transverse-mode amplification,” Appl. Phys. Lett. 74, 1528–1530 (1999). [CrossRef]

, 15

15. T. Eidam, S. Hadrich, F. Jansen, F. Stutzki, J. Rothhardt, H. Carstens, C. Jauregui, J. Limpert, and A. Tunnermann, “Preferential gain photonic-crystal fiber for mode stabilization at high average powers,” Opt. Express19, 8656–8661 (2011). http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-9-8656. [CrossRef] [PubMed]

]. However, it is noted that confined doping is less effective in bent than in straight fibers [5

5. S. Liao, M. Gong, and H. Zhang, “Influence of mode distortion on the transverse mode competition in large-mode-area amplifiers,” Opt. Commun.282, 406–412 (2009), http://linkinghub.elsevier.com/retrieve/pii/S0030401808010067. [CrossRef]

].

A few studies have included intermodal coherence [11

11. N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express16, 8678–8684 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-8678. [CrossRef] [PubMed]

13

13. D. Vysotsky and A. Napartovich, “Mode competition in steady-state optical waveguide amplifiers,” J. Exp. Theor. Phys.108, 547–556 (2009). http://dx.doi.org/10.1134/S1063776109040013. [CrossRef]

]. They showed that, rather than experiencing enhanced gain, a weak higher order mode in the presence of a strong fundamental mode experiences depressed gain relative to the fundamental mode. Andermahr and Fallnich showed in experimental measurements [11

11. N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express16, 8678–8684 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-8678. [CrossRef] [PubMed]

] and in numerical modeling using a beam propagation method [12

12. N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers.” Opt. Express16, 20038–20046 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-24-20038. [CrossRef] [PubMed]

] that parallel polarized higher order mode light was gain depressed while orthogonally polarized light was gain enhanced. In their model they assumed the step index fiber was straight, with full core doping. Their model did not include pump depletion. Similarly, using analytical methods to study planar waveguides, Vysotsky and Napartovich [13

13. D. Vysotsky and A. Napartovich, “Mode competition in steady-state optical waveguide amplifiers,” J. Exp. Theor. Phys.108, 547–556 (2009). http://dx.doi.org/10.1134/S1063776109040013. [CrossRef]

] found that the inclusion of intermodal interference suppressed the gain of higher order modes in the presence of a strong fundamental.

In this paper we present modeling results for a test high power fiber amplifier, illustrating the important difference in behavior of the two polarizations in higher order modes, for both straight and bent fibers, with and without confined doping. In one set of model runs we assume there is no mixing of the two polarizations, as should be true of high quality PM fiber. In another set of runs for non-PM fibers we assume the fundamental and higher order mode polarizations evolve at different rates. We simulate this by periodically swapping the fields of the two polarizations in the higher order modes, with an arbitrarily chosen period of 40 mm.

2. The test fiber amplifier

For our study we chose the amplifier parameters listed in Table 1. This design does not represent any actual design, but it is representative of high power fiber amplifiers, while also allowing efficient modeling. We usually inject 30 W in the fundamental mode (LP01) and 0.3 W total in select higher order modes. When the doping diameter is reduced from 30 μm to study the effects of confined doping, we increase the doping density by the ratio of doped areas to keep the pump absorption constant without increasing the fiber length. Longer fibers would lead to longer modeling run times. We model both a straight fiber and one bent to a 50 mm radius, studying the influence of doping confinement in both.

Table 1. Parameters of Test Amplifier

table-icon
View This Table

3. Our models

3.1. Modes

We use a matrix eigen value method to compute the modes [16

16. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991).

]. The basis functions comprise sine expansions with twenty terms in x and y directions. The three modes we will use are illustrated in Fig. 1. For short we call the LP01 mode m1, the LP11 h mode polarized parallel to m1 is called m2||, its orthogonally polarized counterpart is m2, and mode LP02 is m6.

Fig. 1 Mode irradiance profiles in straight fiber (top) and fiber bent to 50 mm radius (bottom). We call them m1 (left), m2 (center), and m6 (right).

3.2. Irradiance based model

This model is appropriate for a single higher order mode polarized orthogonal to the fundamental mode. The irradiance profile is used in the steady state expression for the upper state population fraction,
nu(x,y)=Ppσpa/hνpA+Isσsa/hνsPp(σpa+σpe)/hνpA+Is(σsa+σse)/hνs+1/τ,
(1)
to compute the upper state profile across the fiber at each z location. Here Pp is the pump power, σpa is the pump absorption cross section, σep is the pump emission cross section, σsa is the signal absorption cross section, σse is the signal emission cross section, A is the pump cladding area, Is is the signal irradiance formed by adding the irradiances of the signal modes, and τ is the upper state radiative lifetime. The irradiance profile Is is computed by adding the irradiance profiles of the two modes. We use the usual approximation that the pump light is uniformly distributed across the pump cladding including the core. When Is = 0 the upper state fraction is approximately 0.5 in amplifiers strongly pumped with 976 nm light. When IsPp(σpa+σpe)/(Aσse) the upper state fraction is reduced to approximately 0.25. This reduction in nu results in gain saturation, and this value of Is indicates the signal level associated with saturation. Note that it is proportional to the local value of the pump power Pp.

3.3. Field based models

3.4. Modeling notes

To allay doubts whether the first modeling method correctly models orthogonally polarized modes, we validated it by modeling the amplifier using the BPM model with the two orthogonally polarized modes propagated separately so there is no interference between them. The upper state population was computed from the pump irradiance and sum of the irradiances of the two oppositely polarized signal fields. We found negligible difference between this and the irradiance based model. We use the same two-field BPM model to simulate a non-PM fiber by periodically swapping the two orthogonally polarized higher order mode fields. First one field and then the other interferes with the fundamental mode. We do not claim this is necessarily a realistic treatment. It treats the limiting case of a large difference in the birefringence of the fundamental and higher order mode, which is useful given our lack of detailed knowledge of actual polarization evolution.

We verified convergence of the models by doubling the grid resolution in each of the dimensions, x, y, (from 1.2×1.2 μm) and z (from 2 μm). We also verified that mixtures of modes propagate as expected if the doping concentration is set to zero, or if the pump power is set to zero.

We also showed that doubling the doping density while keeping the amplifier length constant was nearly equivalent to keeping the doping constant and doubling the length. This is used below to facilitate modeling of restricted area doping.

4. Modeled amplifier performance

For later comparisons we present in Fig. 2 the baseline performance of co- and counter-pumped, straight, fully doped, amplifiers with the parameters of Table 1. The power in all higher order modes is zero.

Fig. 2 Baseline model results for co-pumped amplifier (top) and counter-pumped amplifier (bottom) using the parameters in Table 1. Higher order modal input powers are set to zero.

4.1. Full core doping

Our first example clearly illustrates the difference between the behavior of higher order modes of parallel and orthogonal polarizations. Mode competition suppresses the growth of the || polarized modes relative to the strong fundamental mode, but enhances the growth of the ⊥ polarized modes. Fig. 3 shows plots of the power in modes m2|| and m2 relative to the m1 power, for a co-pumped amplifier (top) and for a counter-pumped amplifier (bottom). The power in m2 is initially 1% of the power in m1. The irradiance based model is used to compute m2, while the field based BPM model is used for m2|| and for non-PM fiber. Because our model for non-PM fiber switches the polarization many times at regular intervals, the light in either polarization state at any position has propagated nearly equal distances in each polarization state. This implies the non-PM curve should approximate the geometric mean of the parallel and orthogonal curves, and we verified this.

Fig. 3 Power in m2|| (solid) or m2 (dashed) or the sum of m2|| and m2 (chain) divided by power in m1 for a co-pumped amplifier (top) and a counter-pumped amplifier (bottom). The fiber is straight and the full core is doped. For the non-PM fiber the m2 power is launched half in each polarization, and the fields for the two polarizations of m2 are swapped every 20 mm. Other properties are listed in Table 1 and Fig. 1, and the pump and m1 powers are nearly identical to those in Fig. 2.

In the co-pumped case most of the separation in power between m2|| and m2 occurs before z = 2 m. For greater distances the pump is weak and the signal strong (see upper plot in Fig. 2) so the upper state population is strongly suppressed across the full core, and m2 with either polarization experiences approximately the same gain as m1. In the counter-pumped case the separation between m2 and m2|| continues to grow over the full length of the fiber because the signal and pump powers are comparable the whole way, causing a relatively constant, moderate gain saturation.

Fig. 4 Ratio of power in m2|| to power in m1 for Ps1=300W and Ps2=3W. The beat length is 3.63 mm for these two modes. Depending on the phase between the two modes, m2|| growth relative to m1 is either positive or negative, but the net change in this ratio over a full cycle is negative.

4.2. Confined doping

Fig. 5 summarizes our results for m2 (top) and m6 (bottom) in unbent fiber. The gain Gm is defined as the output power of mode m divided by its input power. The doping diameter is varied from 10 μm up to the full core diameter of 30 μm. As expected, the gain of m2 can be reduced below that of the fundamental by reducing the doping diameter below 24 μm. Similarly, the gain of m6 can be reduced below the fundamental by reducing the doping diameter below 28 μm. One important point is that, according to these plots, confined doping influences the || polarized modes in nearly the same way as the ⊥ modes. The curves for a non-PM fiber in all four graphs can be estimated by taking the geometric mean of the parallel and orthogonal curves.

Fig. 5 A comparison of the relative gains vs. doping diameter for co-pumping. Other parameters are listed in Table 1. The gain Gm is the output power divided by the input power in the m th mode. The upper plot is the gain of modes m2 and m2|| relative to m1; the lower is the gain of modes m6 and m6|| relative to m1. For non-PM fiber the gain of mode 2 is reasonably approximated by the geometric mean of the two curves shown.

Fig. 6 summarized our results for a similar calculation using fiber bent to a 50 mm radius. Bending increases the gains of m2 and m6 of both polarizations relative to m1. The ratio of powers for the two polarizations of m2 and m6 is approximately the same as in unbent fiber, and the effect of confined doping is also similar.

Fig. 6 A comparison of the relative gain vs. doping diameter for fiber bent to a radius of 50 mm in co-pumped amplifier (see Fig. 1 for bent fiber mode profiles). Other parameters are listed in Table 1. The gain Gm is the output power divided by the input power in the m th mode. The upper plot is the gain of modes m2 and m2|| relative to m1; the lower is the gain of modes m6 and m6|| relative to m1. For non-PM fiber the gain of mode 2 is reasonably approximated by the geometric mean of the two curves shown.

5. Discussion

In the results presented here we have used 1% of the total power in the higher order mode. However, our modeling shows that the degree of suppression is quite constant for higher power ratios, up to 25% power in the higher order mode. Further, if several higher order modes are populated at the 1% level we find each interacts with the fundamental mode nearly independently, and each is suppressed as though the other modes were absent.

We note that the fundamental mode is not unique in creating a differential gain between the two polarizations of a much weaker mode. It appears that any strong mode can produce this effect on any weak mode.

6. Conclusions

In coherent beam combining a pure fundamental (LP01) mode is desired from each fiber amplifier. The second mode (LP11) probably poses the biggest problem because it adds variable uncorrected tilts to the beam which degrades efficiency of the combination process. This mode is also easily populated at launch by slight tilts and displacements of the amplifier fiber relative to the input beam. Only the light in LP 11 polarized parallel to the fundamental at the output end is of real concern because the orthogonally polarized light, while it wastes a certain amount of pump power, does not interfere with beam combination.

As we have demonstrated, in a PM fiber the parallel polarized light of the non-dominant mode is substantially suppressed by gain saturation. Differential gain between the fundamental and higher order modes due to mode competition does not worsen the beam quality as is sometimes claimed, but rather helps purify the output beam. In non-PM fiber the polarizations of the fundamental and higher order mode do not evolve together, so at the fiber output the power in a higher order mode is on average nearly equal in the two polarizations, and its power is intermediate between the suppressed and enhanced levels. Here again, however, mode competition does not degrade mode purity. Only in tightly bent fiber does the mode purity degrade due to mode competition.

If greater suppression of higher order modes is required than is available from the mode competition effect, confined doping can be used. We have shown that this suppression adds with that provided by competition. However, confined doping may be incompatible with tight bending.

The overall conclusion must be that the effects of mode competition on beam quality from high power fiber amplifiers are modest, and except for tightly bent fiber are unlikely to strongly impact the beam quality. However, it is noteworthy that to the extent it does have an influence, it is usually positive, contradicting the notion of degradation due to mode competition. In any case, we have demonstrated that the effects can be modeled realistically in reasonable run times, so further exploration through modeling is straightforward.

Finally we note the well known method of bending the fiber to smaller radii than we used can be an effective way to suppress higher order modes. The differential bend loss between the fundamental and higher order modes may be sufficient to suppress higher order modes without excess loss of the fundamental [17

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

]. Done properly, this can have a much stronger influence on beam quality than does mode competition.

References and links

1.

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19, 10180–10192 (2011). http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-11-10180. [CrossRef] [PubMed]

2.

J. B. Spring, T. H. Russell, T. M. Shay, R. W. Berdine, A. D. Sanchez, B. G. Ward, and W. B Roh, “Comparison of stimulated Brillouin scattering thresholds and spectra in nonpolarization-maintaining and polarization-maintaining passive fibers,” Proc. SPIE 5709, 147–156 (2005). [CrossRef]

3.

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1161 (1979). [CrossRef]

4.

Z. Jiang and J. R. Marciante, “Impact of transverse spatial-hole burning on beam quality in large-mode-area Yb-doped fibers,” J. Opt. Soc. Am. B25, 247–254 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=josab-25-2-247. [CrossRef]

5.

S. Liao, M. Gong, and H. Zhang, “Influence of mode distortion on the transverse mode competition in large-mode-area amplifiers,” Opt. Commun.282, 406–412 (2009), http://linkinghub.elsevier.com/retrieve/pii/S0030401808010067. [CrossRef]

6.

J. Marciante, “Gain filtering for single-spatial-mode operation of large-mode-area fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron.15, 30–36 (2009), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4773298. [CrossRef]

7.

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys.19, 437–444 (2009), http://www.springerlink.com/index/10.1134/S1054660X09030141. [CrossRef]

8.

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express15, 3236–3246 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3236. [CrossRef] [PubMed]

9.

T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multi-mode waveguides,” J. Opt. Soc. Am. B19, 1539–1543 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=josab-19-7-1539. [CrossRef]

10.

J. Sousa and O. Okhotnikov, “Multimode Er-doped fiber for single-transverse-mode amplification,” Appl. Phys. Lett. 74, 1528–1530 (1999). [CrossRef]

11.

N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express16, 8678–8684 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-8678. [CrossRef] [PubMed]

12.

N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers.” Opt. Express16, 20038–20046 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-24-20038. [CrossRef] [PubMed]

13.

D. Vysotsky and A. Napartovich, “Mode competition in steady-state optical waveguide amplifiers,” J. Exp. Theor. Phys.108, 547–556 (2009). http://dx.doi.org/10.1134/S1063776109040013. [CrossRef]

14.

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Gaalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16, 7233–7243 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-10-7233. [CrossRef] [PubMed]

15.

T. Eidam, S. Hadrich, F. Jansen, F. Stutzki, J. Rothhardt, H. Carstens, C. Jauregui, J. Limpert, and A. Tunnermann, “Preferential gain photonic-crystal fiber for mode stabilization at high average powers,” Opt. Express19, 8656–8661 (2011). http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-9-8656. [CrossRef] [PubMed]

16.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991).

17.

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

OCIS Codes
(140.3280) Lasers and laser optics : Laser amplifiers
(140.3510) Lasers and laser optics : Lasers, fiber
(140.3615) Lasers and laser optics : Lasers, ytterbium

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 11, 2011
Revised Manuscript: May 17, 2011
Manuscript Accepted: May 18, 2011
Published: May 25, 2011

Citation
Arlee V. Smith and Jesse J. Smith, "Mode competition in high power fiber amplifiers," Opt. Express 19, 11318-11329 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11318


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011). http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-11-10180 . [CrossRef] [PubMed]
  2. J. B. Spring, T. H. Russell, T. M. Shay, R. W. Berdine, A. D. Sanchez, B. G. Ward, and W. B Roh, “Comparison of stimulated Brillouin scattering thresholds and spectra in nonpolarization-maintaining and polarization-maintaining passive fibers,” Proc. SPIE 5709, 147–156 (2005). [CrossRef]
  3. R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1161 (1979). [CrossRef]
  4. Z. Jiang and J. R. Marciante, “Impact of transverse spatial-hole burning on beam quality in large-mode-area Yb-doped fibers,” J. Opt. Soc. Am. B 25, 247–254 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=josab-25-2-247 . [CrossRef]
  5. S. Liao, M. Gong, and H. Zhang, “Influence of mode distortion on the transverse mode competition in large-mode-area amplifiers,” Opt. Commun. 282, 406–412 (2009), http://linkinghub.elsevier.com/retrieve/pii/S0030401808010067 . [CrossRef]
  6. J. Marciante, “Gain filtering for single-spatial-mode operation of large-mode-area fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15, 30–36 (2009), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4773298 . [CrossRef]
  7. S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19, 437–444 (2009), http://www.springerlink.com/index/10.1134/S1054660X09030141 . [CrossRef]
  8. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15, 3236–3246 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3236 . [CrossRef] [PubMed]
  9. T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multi-mode waveguides,” J. Opt. Soc. Am. B 19, 1539–1543 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=josab-19-7-1539 . [CrossRef]
  10. J. Sousa and O. Okhotnikov, “Multimode Er-doped fiber for single-transverse-mode amplification,” Appl. Phys. Lett. 74, 1528–1530 (1999). [CrossRef]
  11. N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express 16, 8678–8684 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-8678 . [CrossRef] [PubMed]
  12. N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers.” Opt. Express 16, 20038–20046 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-24-20038 . [CrossRef] [PubMed]
  13. D. Vysotsky and A. Napartovich, “Mode competition in steady-state optical waveguide amplifiers,” J. Exp. Theor. Phys. 108, 547–556 (2009). http://dx.doi.org/10.1134/S1063776109040013 . [CrossRef]
  14. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Gaalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-10-7233 . [CrossRef] [PubMed]
  15. T. Eidam, S. Hadrich, F. Jansen, F. Stutzki, J. Rothhardt, H. Carstens, C. Jauregui, J. Limpert, and A. Tunnermann, “Preferential gain photonic-crystal fiber for mode stabilization at high average powers,” Opt. Express 19, 8656–8661 (2011). http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-9-8656 . [CrossRef] [PubMed]
  16. D. Marcuse, Theory of Dielectric Optical Waveguides , 2nd ed. (Academic Press, 1991).
  17. J. Koplow, D. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-7-442 . [CrossRef]

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