Abstract

Transverse mode instability (TMI) is compared to thermal lensing (TL) power threshold and used to derive power scaling limits in high-power fiber amplifiers. The TMI power threshold is shown to be ~65% of the TL one and dominates power scaling. In addition to commonly used limiting effects, we introduce a bend-induced mechanical reliability criterion, which limits the maximum allowable cladding diameter to ~600μm. This also results in the introduction of a critical pump brightness, the minimum required pump brightness at which the maximum signal power is achieved. The maximum achievable power depends primarily on the choice of pumping wavelength, amplifier gain and heat coefficient. Maximum signal powers of ~28kW to ~38kW, for diode pumping (λp = 976nm), and ~35kW to ~52kW, for tandem pumping (λp = 1018nm), are predicted for single-mode fiber amplifiers operating at signal wavelength λs = 1070nm, when the amplifier gain is increased from 10dB to 20dB. For an amplifier gain of 10dB, the maximum achievable signal power varies from 85kW to 25kW for tandem pumping and 35kW to 20kW for diode pumping, when the heat coefficient varies from 1% to 15% and 5.5% to 20%, respectively. The corresponding critical pump brightness varies from ~0.50 W/(μm2 sr) to ~0.14 W/(μm2 sr) for tandem pumping and ~0.25 W/(μm2 sr) to ~0.13 W/(μm2 sr) for diode pumping.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. Introduction

Cladding-pumped Yb3+-doped high-power fiber lasers (HPFLs), in master-oscillator power amplifier (MOPA) configurations, have shown record single-moded diffraction-limited output powers in the range of 10kW [1,2]. Output power limits in cladding-pumped high-power fiber amplifiers (HPFAs) have been shown to be set by the pump brightness, optical nonlinearities (such as stimulated Raman (SRS) or Brillouin (SBS) scattering), optical damage, glass rapture or melting and thermal lensing (TL) [‎3,4].

The TL power threshold has been set arbitrarily to a level that results in pumped “hot” fiber mode-field diameter (MFD) ωTL ≈0.7 - 0.8 ω0, where ω0 is the corresponding un-pumped (“cold”) fiber MFD [3]. Therefore, its impact on power scalability is to some extend arbitrary and artificial. It has been shown that in diode-pumped HPFAs the output power limit is set to ~37kW, by a combination of pump brightness, SRS and TL limitations [3,4]. In the case of tandem pumping, the predicted power limit is increased to 70kW [5] and 68-97kW [6], limited by optical damage, SRS and TL effects.

More recently, transverse mode instability (TMI) [7] has been identified as a severely limiting power-dependent effect and its impact on power scaling has been considered [8–10]. In these studies, different TMI power threshold formulae were considered resulting in substantially different power scaling limits. In [8] the TMI threshold was considered to be reached at a fixed average heat load Q0 = 34W/m, independent of core diameter [11], and predicted a diode-pumped power limit of 70kW, defined though by pump brightness, SRS and TL. The power limit, however, was reduced to 10kW, when pump brightness, TMI and TL were considered. In [9] the TMI threshold scaled inversely with the core area and the diode-pumped power limit was set by pump brightness, SRS and TMI at ~28kW.

It should be noted that the aforementioned increased power limits were achieved with fiber-length/core-diameter combinations of ~20m/60μm [5], ~80m/130-220μm [6] and ~120m/260μm [8]. To date, however, the largest demonstrated active core is 135μm [12], and passive core diameter is 205μm [13], both achieved with microstructured large-pitch fibers and have to be kept straight over limited lengths of 1-2m. In addition, all previous power scaling investigations consider a fixed core pump absorption (Yb3+ concentration) and pump cladding absorption, which result also in large cladding diameters (>1mm). However, in practical HPFL systems to accommodate such ultra-long lengths the fibers should be bent in compact packages and the ramifications of bending such extra-large core and cladding diameter fibers have not been fully considered so far.

Following these results, it becomes apparent that efforts should be made to establish first a relation between the thermally driven TL and TMI thresholds, as well as, consider additional constraints imposed by the required long fiber lengths and large core diameters. Given that the majority of practical industrial or directed energy HPFL systems should preferably have minimum footprint, the additional constraints of mechanical reliability and mode field deformation of appropriately bent fibers should also be considered.

In this work, in Section 2 we first derive a simplified expression of the MFD shrinkage due to heat load and TL in HPFAs and compare it with published experimental results [14]. We also recast a recently derived simple TMI power threshold formula [‎15-17], in terms of laser efficiency (ηlaser) and heat generation coefficient (ηheat), and compare it to the widely used TL power threshold [3]. In Section 3 we show representative power scaling results limited by the pump brightness, the onset of SRS/TMI and mechanical reliability, for diode- (DP) [18] and tandem-pumped (TP) [2,19,20] Yb3+-doped HPFAs. In Section 4, we introduce the concept of critical pump brightness and derive the maximum achievable signal power as a function of pump brightness for different amplifier gains, laser efficiencies and dopant concentrations. In Section 5, we discuss the results and conclude. In Appendix A, we consider the bending-induced mechanical reliability and establish the maximum allowable cladding diameter. In Appendix B, we consider the bending-induced mode field deformation, and in Appendix C we summarize the different power limits due nonlinear, thermal and other effects. Finally, in Appendix D, we consider the heat-induced fundamental mode MFD shrinkage in optical fibers and compare it to the free-space case.

2. TL and TMI effects in optical fibers

In this section, we first consider the relation between thermal lensing, which is traditionally considered as one of the main power scaling limiting effects in HPFAs, and TMI.

2.1. TL-induced MFD shrinkage in optical fibers

We first derive a simplified formula for the MFD shrinkage due to thermal lensing and compare it with experimental results from the literature. The MFD diameter ω0 in a step-index (SI) fiber is approximated by [21]:

ω0d0=0.65+1.619V3/2+2.879V6
where d0 = 2r0 is the core diameter, V=π(d0/λ0)n12n22π(d0/λ0)2n2Δn, Δn=n1n2, with n1 (n2) being the core (cladding) refractive index, and λ0 the signal wavelength (see Fig. 1(a)).

 figure: Fig. 1

Fig. 1 Schematic of (a) “cold” step-index fiber, and (b) “hot” fiber quasi-parabolic refractive index profile; (c) “hot”/”cold” SI fiber MFD ratio.

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Steady-state thermal effects in HPFAs are described by the simplified steady-state heat equationκT2ΔT(z)=Q0(z), where T2 is the transverse Laplacian, ΔTis the temperature change, κ is the silica thermal conductivity. Q0 is the heat power density (volumetric heat source), which in the case of an optically pumped amplifier is given by [22]:

Q0qDdIs/dz+αsIs=(qDgs+αs)Is
where qD=(λs/λp)1is the quantum defect parameter, gs is the saturated signal gain coefficient, αs is the signal background loss, λs (λp) is the signal (pump) wavelength, and Is the signal intensity. Thermal effects can also be quantified by the average heat load Q¯heatapproximated by [3]:
Q¯heat=PheatL=ηheatPpumpabsorbedL=(ηheatηlaser)PsignalextractedL=(ηheatηlaser)Ps(L)LG1G
where Pheatis the power dissipated into heat, L is the amplifier length, Ppumpabsorbed is the absorbed pump power, ηheat is the heat coefficient, ηlaser is the laser efficiency and Psignalextracted is the extracted signal power (Psignalextracted=Ps(L)Ps(0)) and G=Ps(L)/Ps(0) is the amplifier gain. For HPFAs for which G1, Q¯heat(ηheat/ηlaser)(Ps/L).

Assuming a uniform heat load distribution across the fiber core, the thermally-induced core refractive index acquires a quasi-parabolic profile (see Fig. 1(b)) with maximum RI variation approximated by [23]:

ΔnthQ¯heat(dn/dT)4πκ
where (dn/dT) is the thermo-optic coefficient. In the case of thermally-induced quasi-parabolic RI profile, the fundamental mode MFD diameter ωth is approximated by [21]:
ωthd0=2Vth1/2+0.23Vth3/2+18.01Vth6
where Vth=π(d0/λ0)2n2Δn(1+Δnth/Δn)=V1+Δnth/Δn. The “hot” to “cold” MFD ratio (ωth/ω0) versus the heat load is plotted in Fig. 1(c), for different core diameters and V = 2.2. It is shown that for all core diameters the MFD “shrinks” as the heat load increases. The “hot” fiber MFD shrinkage is more pronounced in larger core diameter fibers. This implies that large mode area (LMA) fibers are expected to be more prone to thermal effects.

2.2. TL and TMI power limits

In the literature [3], TL limit is considered as the power above which a Gaussian beam propagating in free space negates diffraction and starts self-guiding under the influence of a thermally-induced quasi-parabolic RI profile. Using an ABCD matrix analysis, the derived TL power limit is then given by Eq. (24), namely:

PoutTL=2πκ[ηheat/(ηlaserL)](dn/dT)(λ0d0)2
and corresponds to a reduction of the beam initial MFD by a factor of 0.7-0.8. κ is the silica thermal conductivity and dn/dT the thermo-optic coefficient.

A TMI power threshold has been derived by carrying out a stability analysis of the fundamental (LP01) mode amplification process, in the presence of heat load induced by the amplifier quantum defect. The TMI limit or threshold, in this case, corresponds to the power above which a small spatial perturbation (e.g. in the form of LP11 mode) increases exponentially [15–17]. The derived TMI power limit, in this case, is given by Eq. (23). In HPFAs, heat generation is dominated by the quantum-defect contribution and, therefore, we can set αsqDgs(αsqDgs). Expressing Q¯heat=Q0πr02and using Eqs. (2) and (3), we approximateαsηheat/(ηlaserL). This approximation is more appropriate for the case of backward pumping. From Eq. (24) then the TMI power limit takes the simplified form:

PoutTMI=κUε2(Uε2Us2)2πneff[ηheat/(ηlaserL)](dn/dT)(λ0d0)2,
which now can be compared directly with the TL power limit in Eq. (6). neff is the fundamental mode effective index and Us(Uε) is the normalized transverse wavenumber for the fundamental (higher-order) fiber mode. Despite their fundamentally different physical character and derivation method, both the TL and TMI limits show the same dependence [(λ0/d0)2] on signal wavelength (λ0), core diameter (d0), laser efficiency (ηlaser), heat generation coefficient (ηheat) and fiber length L. In addition to core diameter and the parameters shown in Eq. (7), it has been shown experimentally (see Fig. 1 in [16]) that the TMI threshold depends strongly also on the signal linewidth. The simplified TMI threshold formula in Eq. (7) applies to broad linewidth operation.

However, it should be emphasized that TMI results in detrimental beam break-up, while TL results simply in 20-30% MFD shrinkage. The latter is not damaging, although it can decrease the threshold of other nonlinear effects, such as SRS. In some cases, though, like in large-pitch fibers, TL can be beneficial to some extend as it can initially stabilize the beam shape and contribute to more robust guiding [24,25].

From Eqs. (6) and (7), it is deduced that the TMI/TL power limit ratio is given by

PoutTMIPoutTL=Uε2(Uε2Us2)4π2neff
and depends only on the fiber core V-number. For V = 3, PoutTMI/PoutTL0.6. Figure 2 compares TMI and TL power limits as a function of fiber core diameter, for V = 3, ηheat = 0.3 and ηlaser = 0.7. The other parameters are given in Appendix C. As expected, PoutTMIis ~40% lower than PoutTL. This is an important finding and it is expected to redefine the limits in HPFA power scaling (see Section 3).

 figure: Fig. 2

Fig. 2 TMI and TL power limits in fibers with V = 3, ηheat = 0.3 and ηlaser = 0.7. The other parameters are given in Appendix C. The inset zooms into the dashed area and superimposes experimental data (open circles) [14].

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2.3. MFD shrinkage & TL / TMI power limits comparison - experimental validation

As already mentioned, in previous studies the TL power limit (see Eq. (6)) corresponds to ~20-30% MFD reduction of a Gaussian beam propagating in free space [3]. Using the simplified analysis and results of Sections 2.1 and 2.2 in Appendix D we have shown that at the TL power limit (i.e.Ps=PoutTL) and typical “cold” fiber V = 3-4, the “hot”-to-“cold” MFD ratio of the fundamental fiber mode is approximated by ωthTL/ω02/(V2)0.70.8, which is similar to the free-space case.

To verify the theoretical predictions regarding TMI and TL power limits, given by Eqs, (6)–(8), and the dependence of “hot” fiber fundamental mode MFD shrinkage on extracted power, or equivalently, on average heat load given by Eqs. (3)–(5), we have compared our findings with published experimental results. In Ref [14], MFD variations with extracted signal power and TMI power thresholds have been measured in a range of large-pitch microstructured fibers (LPFs) with different core diameters. The LPFs are drawn from the same preform and, therefore, have the same Yb3+ ion concentration, core/cladding ratio and length of L = 1.2m.

Figure 3 replots the results for three LPFs with fiber core diameters of 136μm (LPF75-blue open circles), 80μm (LPF45-green open circles) and 63μm (LPF35-red open circles)) [14]. It is shown that the output MFD diameter decreases with extracted power, due to increasing level of thermal lensing. For each fiber, the measurements are limited by the onset of TMI, beyond which the output beam is distorted dynamically by the presence of higher-order modes (HOMs) and the MFD cannot be defined. The corresponding fundamental mode MFD diameter (ωth) variation with extracted power (Eqs.. (3)–(5)) for each fiber (solid lines) is also plotted, showing a very good agreement. The used parameters were V = 3, ηheat = 0.3 and ηlaser = 0.7. The other parameters are given in Table 1 in Appendix C. The dashed lines denote the 30% reduction from the “cold” MFD value (i.e.ωthTL/ω00.7), which corresponds to the TL limit. The red arrows denote the corresponding TL powers (PoutTL), while the blue arrows denote the expected TMI limits (PoutTMI) with PoutTMI0.65PoutTL. It is observed that for LPF75 and LPF45 the predicted TMI powers agree very well with the experimentally obtained ones. Also, the used ratio PoutTMI/PoutTL0.65 is in very good agreement with the theoretically predicted value of 0.6. In the case of LP35, the experimental TMI limit is slightly lower than the predicted value.

 figure: Fig. 3

Fig. 3 MFD shrinkage with extracted power. Open circles: experimental data [14]; blue: LPF75, green: LPF45, red:LPF35. Solid lines: Theory (Eq. (5)) with V = 3, ηheat = 0.3 and ηlaser = 0.7 and d0 = 136μm (blue); 80μm (green); 63μm (blue). Dashed lines correspond to 70% “hot”-to”-cold” MFD shrinkage (ωth/ω0 = 0.7). Red and blue arrows denote the TL and TMI powers, respectively, for which PTMI/PTL = 0.65.

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Tables Icon

Table 1. List of parameters, symbols used in the text and values used in the calculations

In all cases, it is observed that TMI sets in well before the conventional TL limit is reached, following closely a (1/d0)2 dependence. The inset of Fig. 2 zooms into the dashed area and superimposes experimental data (stars) [14], showing again a very good agreement for LPF75 and LPF45. For LPF35 (d0 = 63μm), however, the experimental TMI power limit is again lower than the theoretical prediction. In addition to thermal load, TMI threshold is known to be lowered by a number of other factors such as additional backreflections [26], amount of the input higher-order mode [27] and photodarkening [28]. These effects are not included in our simplified analysis and could contribute to the observed discrepancy. At this point, it should be mentioned that using Eq. (3), the alternative TMI power threshold, which is considered to occur at a fixed average heat load of Q¯heat = 34W/m [11] and was used in [8] to evaluate the impact of TMI on power scaling, takes the simplified form PTMI2 = 34(ηlaser/ηheat)L. In this case, the predicted TMI power threshold would have been ~105W, independent of the core diameter and not matching the observed experimental trend.

Finally, independently in [29] M.M. Johansen et.al. using a semi-analytical model have calculated numerically both the MFD shrinkage and TMI power threshold as a function of heat load. Using again ωthTL/ω00.70.8as the thermal lensing threshold criterion, from Figs. 4 and 6 in [29] it is deduced that the TMI/TL average-heat-load ratio is Q¯heatTMI/Q¯heatTL75/120=0.63. Given the linearity between signal power and average heat load, this corresponds to PoutTMI/PoutTL0.63in excellent agreement with our results in this section.

 figure: Fig. 4

Fig. 4 Contour plots of (a) power lower limits (in kW) due to pump brightness (blue area), SRS (green area) and TMI (orange area); and (b) required cladding diameter (in μm) as a function of amplifier length and core diameter. The parameters are similar to Ref [3]. (see Table 1 in Appendix C and Bp = 0.02W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 10dB).

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Experimental data in Fig. 3 were used primarily to show the relation between TL and TMI thresholds and compare them with the simplified formulae (6) and (7). The TMI threshold is inversely proportional to the saturated amplifier gain gs through αsqDgsηheat/(ηlaserL) and, therefore, it is a multi-dimensional function of the pump/signal wavelength, input signal power, dopant concentration, and cladding diameter. In Fig. 5 of Ref [16], it is shown that the simplified TMI threshold formula agrees with experimental data up to multi-kW level for a large range of core diameters and amplifier gains.

 figure: Fig. 5

Fig. 5 Contour plots of power limits (in kW) due to pump brightness (blue area), SRS (green area) and TMI (orange area). The mechanical-reliability dominated parameter space (grey area) is also shown. (a) DP with Bp = 0.02W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 10dB; (b) DP with Bp = 0.18W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 10dB; (c) DP with Bp = 0.2W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 23dB; (d) TP with Bp = 0.3W/μm2/sr; ηlaser = 0.85; ηheat = 0.06, G = 10dB. Core pump absorption: (a)-(c) αcore = 250dB/m, (d) 100dB/m.

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3. Power scaling limits

We now investigate the power scaling limits in HPFAs, taking into account the newly developed TMI threshold formula and the constraints into maximum allowable cladding diameter due to fiber mechanical reliability, in addition to the nonlinear and other limits summarized in Appendix C. Throughout the analysis the fiber V-number is fixed to V = 3.

Figure 4(a) shows contour plots of the power scaling limits (in kW) due to the nonlinear and other effects with parameters similar to [3], i.e. Bp = 0.02W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, all related to 976nm diode pumping and lasing at ~1070nm. It is shown that the power scaling in this case is limited by pump power (blue area), SRS (green area), and TMI (orange area) in place of the previously dominating thermal lensing. It is noted that due to the lower TMI threshold, the maximum achievable power is reduced from ~37kW in [3] to ~28kW.

From Fig. 4(a), three border (dashed) lines are identifiable. The border line between pump- and SRS-limited powers is obtained by settingPoutpump=PoutSRS(see Eqs. (22) and (26)), and it is given by:

LpumpSRS=4ΓNAln(G)ApπgRηlaserBpαcore
which is independent of the core diameter. The border line between pump- and TMI-limited powers is obtained by setting Poutpump=PoutTMI(see Eqs. (22) and (6)), and it is given by:
dpumpTMI=2κUε2(Uε2Us2)λ02Apπ3neffηheat(dn/dT)BpαcoreNA24
which is independent of the amplifier length. For L > Lpump-SRS and d0 > dpump-TMI, the border line between TMI- and SRS-limited powers corresponds to PoutTMI=PoutSRS (see Eqs. (6) and (26)), and it is given by:
LTMISRS=2πd02λ02neffηheat(dn/dT)Γ2ln(G)ηlaserκUε2(Uε2Us2)gR
Along this border the maximum achievable power is obtained, given by:

PTMISRSmax=λ02ηlaserκUε2(Uε2Us2)Γ2ln(G)neffηheat(dn/dT)gR

The required minimum amplifier length and core diameter to obtain the maximum achievable output power are given by Eqs. (9) and (10), respectively. In the case of Fig. 4(a), the corresponding values are 41m and 78μm, respectively. Figure 4(b), on the other hand, plots the required cladding diameter (in μm), as given by Eq. (23), required for the power levels plotted in Fig. 4(a) to be achieved. It is shown that the maximum achievable power of ~28kW requires a minimum cladding diameter of 1.6mm. With such cladding dimensions these structures can hardly qualify as “standard” bendable optical fibers and the implications on power scaling will be considered next.

3.1 Fiber mechanical reliability considerations

In all the previous power scaling investigations, fixed core (αcore) and total (Ap) pump absorption were considered and as a result large core diameters were accompanied by large cladding diameters (>1mm) and long fiber lengths (>>10m) [3–6,8]. Given that most practical industrial or directed energy HPFL systems should preferably have minimum footprint, the additional constraints of mechanical reliability of bent large-cladding-diameter fibers should be considered [30]. Figure 9 in Appendix A shows the calculated failures-in-time (FIT) as a function of fiber bending diameter (Dbend), for different cladding diameters. The fiber is assumed to have undergone minimum proof testing and the failure time is 20years [31,32]. Assuming a maximum bending diameter Dbend = 1m, the maximum allowable cladding diameter is dmech = 600μm. This additional limit will be used in the subsequent calculations. From Eq. (23), the mechanical reliability border line can be defined as:

Lmech=Apdmech2αcored02

When within the pump-limited (blue) region, the maximum achievable signal power, along the mechanical reliability border line, is given by:

Ppumpmax=14ηlaserBpπ2NA2dmech2

Figure 5(a) replots the power limits (in kW) shown in Fig. 4(a). However, when we consider the additional mechanical reliability constraint, the maximum power reduces from 28kW to 3kW! As shown in Fig. 5(b), increasing the pump brightness to Bp = 0.18W/(μm2 sr) reduces considerably the fiber length and core diameter requirements and results in a maximum power of ~28kW even when the mechanical reliability constraint is considered. It should be pointed out that at this pump brightness the dominant power limits (i.e. pump/SRS, pump/TMI, SRS/TMI and mechanical reliability) all acquire the same maximum achievable power level. This leads to the critical pump brightness introduced in Section 4.

In Fig. 5(c) it is shown that increasing the amplifier gain to 23dB increases the SRS/TMI-limited power to ~42kW. However, when we impose the mechanical reliability limit the maximum power reduces to ~15kW. It should be mentioned that in this case, a 20μm core fiber, ~30m long amplifier provides a maximum of ~5kW SRS limited, in close agreement with recent experimental results with similar parameters [33].

Finally, in the case of tandem pumping, with parameters Bp = 0.3W/(μm2 sr); ηlaser = 0.85; ηheat = 0.06 λp = 1018nm, the maximum achievable power is increased to ~36kW limited by SRS/TMI. In this case, imposition of the mechanical reliability limit has no effect on the maximum achievable power, which is achieved with a minimum fiber length of ~17m and a minimum core diameter of ~57μm. However, as discussed in Appendix B, for a more practical core diameter of ~35-40μm the maximum achievable power reduces to ~15-20kW. It should be mentioned also that a 30μm core fiber, 10dB gain, ~15m long amplifier is expected to provide a maximum of ~10kW pump/SRS limited signal power, in close agreement with the record experimental result in [2].

4. Pump brightness requirements

Pump brightness, along with the mechanical reliability imposed maximum cladding diameter, are among the most important parameters in defining the power scaling limits in HPFAs. While the TMI/SRS-limited power, given by Eq. (12), is independent of pump brightness (Bp), the pump-limited power, given by Eq. (14), is proportional to Bp. We can define a critical pump brightness Bpcrit, which is the minimum brightness for which Ppumpmax=PTMISRSmax. This corresponds to the case shown in Fig. 5(b), where the pump/SRS, pump/TMI, SRS/TMI and mechanical reliability power limits coincide. From Eqs. (12) and (14), the critical pump brightness is then given by:

Bpcrit=4λ0UεΓπ2NA2dmech22κ(Uε2Us2)ln(G)neffηlaserηheat(dn/dT)gR
The critical pump brightness depends on the maximum allowable cladding diameter dmech. It also depends on the amplifier gain, laser efficiency and heat coefficient, in addition to other fiber parameters and thermo-optic constants.

The critical pump brightness defines the dependence of maximum achievable signal power and the required minimum core diameter, fiber length and cladding diameter on pump brightness.

A. Maximum Signal Power (Psmax): It is easily deduced that the maximum achievable signal power is given by:

Psmax={Ppumpmax;Bp<BpcritPTMISRSmax;BpBpcrit
where Ppumpmaxand PTMISRSmaxare given by Eqs.. (14) and (12), respectively.

B. Minimum Core Diameter (d0min): When Bp<Bpcrit, the mechanical reliability border line overlaps with the pump-limited region (see Figs. 5(a) and (c)) and there is a range of fiber core-diameter/length combinations available. The minimum core diameter dmechSRS, however, is obtained by setting Lmech=LpumpSRS, which when using Eqs. (9) and (13) takes the form:

dmechSRS=dmechπηlasergRNA2ApBp16Γ2ln(G)αcore4

When BpBpcrit, on the other hand, the mechanical reliability border line does not overlap with the pump-limited region and instead it intersects the TMI/SRS border line LTMISRS(see Eq. (11) and Fig. 5(d)). However, since the maximum signal power is constant along this border, the minimum core diameter is given by the corresponding dpumpTMI, given by Eq. (10). Overall, the minimum core diameter is given by:

d0min={dmechSRS;Bp<BpcritdpumpTMI;BpBpcrit
C. Minimum Fiber Length (Lmin): Following the same reasoning, for any pump brightness the corresponding minimum fiber length is obtained by settingLmin=LpumpSRS and it is given by Eq. (9).

D. Minimum Cladding Diameter (dcladmin): The minimum cladding diameter is given by Eq. (23) by substituting d0minfrom Eq. (18) and Lmin=LpumpSRS from Eq. (9). It is then straightforward to show that dcladminis given by:

dcladmin={dmech;Bp<Bpcritdclad;BpBpcrit
where

dclad=32κΓ2Uε2(Uε2Us2)λ02ln(G)neffgRηlaserηheat(dn/dT)π4NA4Bp24

Figure 6(a) plots the maximum achievable signal power (right axis) and minimum fiber length (right axis) as a function of pump brightness for diode pumping (DP – red lines) and tandem pumping (TP-blue lines). For DP the parameters were ηheat = 0.10, ηlaser = 0.85, αcore = 250dB/m, while for TP they were ηheat = 0.06, ηlaser = 0.90, and αcore = 100dB/m. In both cases, the amplifier gain was G = 10dB. The rest of the parameters are shown in Table 1. For Bp<Bpcrit, the maximum signal power is pump limited and increases linearly with the pump brightness. For BpBpcrit, on the other hand, it is TMI/SRS limited and it is independent of the pump brightness. Tandem pumping shows larger critical pump brightness due to smaller ηheat, (see Eq. (15), and larger slope due to larger ηlaser (see Eq. (14). As a consequence, TP results in larger maximum signal power (~37kW) compared to DP (~28kW). The minimum required fiber length, on the other hand, decreases monotonically with Bp (Bp0.5), for both TP and DP. TP results in longer fiber lengths because of the smaller absorption cross-section at 1018nm as compared to 976nm pumping.

 figure: Fig. 6

Fig. 6 (a), Maximum signal power (right axis), minimum fiber length (left axis), and (b) Minimum core diameter (left axis), minimum cladding diameter (right axis). Diode pumping (DP): ηheat = 0.10, ηlaser = 0.85, αcore = 250dB/m; Tandem pumping (TP): ηheat = 0.06, ηlaser = 0.90, αcore = 100dB/m. Amplifier gain G = 10dB (vertical dashed lines show the corresponding critical pump brightness).

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Figure 6(b) plots the required minimum cladding diameter (left axis) and minimum core diameter (right axis) for TP (red lines) and DP (blue lines), as a function of pump brightness. The required minimum core/cladding diameters are shown to vary non-monotonically with Bp. ForBp<Bpcrit, the minimum core diameter increases with Bp (Bp0.25). ForBpBpcrit, on the other hand, the minimum core diameter decreases with Bp (Bp0.25). The minimum required core diameter peaks at a higher value for TP (~62μm), as compared to DP (~46μm). In Fig. 6(b), it also shown that forBp<Bpcrit the minimum cladding diameter remains constant at dmech, for both DP and TP. ForBpBpcrit, on the other hand, the minimum cladding diameter decreases with Bp (Bp0.5), for both DP and TP. From a physical point of view, since forBp<Bpcrit the cladding diameter remains fixed and the fiber length decreases with Bp, the core diameter increases in order to maintain a total cladding pump absorption of Ap = 20dB. For the same reason whenBpBpcrit both fiber length and cladding diameter decrease and result in an appropriate core diameter decrease. This can also be verified analytically by considering Ap=αcore(r0/rclad)2Land substituting Eqs. (9), (18) and (19).

From Figs. 6(a) and (b) it is deduced that although increasing the pump brightness above the critical value does not result in further signal power increase, nevertheless it reduces considerably the requirements in fiber core/cladding diameters and length leading to more robust and manufacturable HPFLs.

Figure 7(a) plots the maximum signal power (right axis) and minimum fiber length (left axis), while Fig. 7(b) plots the required minimum cladding diameter (left axis) and minimum core diameter, respectively, for diode pumping (ηheat = 0.10, ηlaser = 0.80, αcore = 250dB/m). Figures 7(c) and (d) plots the corresponding parameters for tandem pumping (ηheat = 0.06, ηlaser = 0.85, αcore = 100dB/m). In both cases, the amplifier gain (G) is set to 10dB (blue lines), 15dB (green lines) and 20dB (red lines). In the DP case, increasing the amplifier gain from 10dB to 20dB increases the maximum signal power from ~27kW to ~38kW. The critical pump brightness increases from 0.18 W/(μm2 sr) to 0.26 W/(μm2 sr), while the required core diameter reduces from ~46μm to ~42μm. In the TP case, the same amplifier gain variation increases the maximum signal power from ~37kW to ~52kW. The critical pump brightness increases from 0.22 W/(μm2 sr) to 0.32 W/(μm2 sr), while the required core diameter reduces from ~62μm to ~55μm. These results show that increasing the amplifier gain increases the maximum achievable signal power, as a direct result of the resulting increase in critical pump brightness (Bpcrit[ln(G)]+0.5,(η)heat0.5 - see Eq. (15)).

 figure: Fig. 7

Fig. 7 (a) maximum signal power, (b) minimum fiber length/core diameter for diode pumping (ηheat = 0.10, ηlaser = 0.80, αcore = 250dB/m); (c) maximum signal power, (d) minimum fiber length/core diameter for tandem pumping (ηheat = 0.06, ηlaser = 0.85, αcore = 100dB/m). Amplifier gain G = 10dB (blue lines), 15dB (green lines) and 20dB (red lines).

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From Eq. (12), it is obvious that the maximum achievable signal power depends explicitly on the heat coefficient ηheat. The heat coefficient depends on the choice of the pump and signal wavelengths [34], as well as, additional effects such as the signal background loss due to glass impurities or photodarkening [28,35]. The heat coefficient, therefore, can vary from a minimum value determined by the quantum defect (ηheatmin=qDmin) to a maximum value ηheatmax=1ηlaser (assuming negligible signal scattering, HOM excitation and other pump losses).

Figure 8 plots (a) the maximum achievable signal power, (b) critical pump brightness, (c) corresponding core diameter and (d) fiber length as a function of heat coefficient, for diode pumping (red lines) and tandem pumping (blue lines). In the DP case (λp = 976nm), ηheatmin=qDminvaries from ~5.5% to ~12% when the signal varies from λs = 1030nm to 1090nm (indicated by red vertical arrows). Assuming ηlaser. = 0.80 results in ηheatmax = 0.20. In the TP case (λp = 1018nm), ηheatminvaries from ~1.2% to ~7% when the signal varies from λs = 1030nm to 1090nm (indicated by blue vertical arrows). In this case we assume ηlaser. = 0.85 and, therefore, ηheatmax = 0.15. In both cases, the amplifier gain is G = 10dB and the cladding diameter is 600μm. Under these conditions, the maximum achievable signal power varies from 85kW to 25kW for TP, and 35kW to 20kW for DP. The corresponding critical pump brightness varies from ~0.50 W/(μm2 sr) to ~0.14 W/(μm2 sr) for TP, and ~0.26 W/(μm2 sr) to ~0.13 W/(μm2 sr) for DP.

 figure: Fig. 8

Fig. 8 (a) the maximum achievable signal power, (b) the critical pump brightness, (c) corresponding core diameter and (d) the fiber length as a function of heat coefficient, for diode pumping (ηlaser = 0.80 - red lines) and tandem pumping (ηlaser = 0.85 - blue lines). G = 10dB.

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Variations in laser efficiency (ηlaser) result in smaller variations in maximum laser power and the required core diameters. For DP and G = 10dB, varying the laser efficiency from 75% to 85% increases the maximum signal power from ~26kW to ~28kW. Similarly, for TP varying the laser efficiency from 80% to 90% increases the maximum signal power from ~35kW to ~37.5kW. The impact on the critical pump brightness and minimum required length, core/cladding diameter is also minimal. Finally, varying the pump core absorption (αcore), or equivalently the Yb3+ concentration, does not change the maximum signal power (see Eq. (12)) and affects only the required core radius (see Eq. (17)) and fiber length (see Eq. (9)).

5. Summary - discussion

In this work, in addition to the commonly used thermal and nonlinear power limiting effects in HPFA power scaling, we have considered the impact of TMI, as well as, further limitations due to bend-induced mechanical reliability and modal deformation.

We have first introduced a simple formula, which provides the MFD shrinkage due to the average heat load in Yb3+-doped fiber amplifiers, and have used it to compare traditional TL power limits to TMI ones. The TMI power limit is shown to be ~35-40% smaller than the TL limit, in very good agreement with experimental results. This demonstrates that TMI occurs before the traditional TL limit is reached and, therefore, dominates power scaling. Using a FIT analysis and considering 1 FIT over 20 years, we have introduced a maximum allowable cladding outer diameter of dmech≈600μm as a new power limiting factor related to the mechanical reliability of bent fibers. This corresponds to a bending diameter of 1m.

In deriving the TMI threshold we have considered primarily the effects of heat load due to quantum defect. Other parameters, such as gain saturation, higher-order mode excitation at the input and excess signal loss (e.g. due to photodarkening), which are known to affect the TMI threshold have not been taken into account in this study. Any additional factor that reduces the TMI threshold will impact the calculated maximum achievable powers.

We have also introduced the concept of critical pump brightness, which defines the minimum pump brightness needed to reach the maximum achievable signal power. The achievable signal power increases linearly with pump brightness when below the critical value, and remains unchanged above it.

We have used these new findings to investigate the power scaling limits in HPFAs under diode and tandem pumping. In the case of diode pumping, increasing the amplifier gain from 10dB to 20dB increases the maximum signal power from ~27kW to ~38kW. The critical pump brightness in this case increases from 0.18 W/(μm2 sr) to 0.26 W/(μm2 sr), and the required core diameter reduces from ~46μm to ~42μm. In the case of tandem pumping, the same amplifier gain variation increases the maximum signal power from ~37kW to ~52kW. The critical pump brightness increases from 0.22 W/(μm2 sr) to 0.32 W/(μm2 sr), while the required core diameter reduces from ~62μm to ~55μm. While increasing the amplifier gain results in substantial increase of the maximum achievable signal power, it should be taken into account that high gain amplifiers are more susceptible to back reflections and other external perturbations. As a result, a compromise between highest possible signal power and laser sensitivity to external perturbations should be sought in a robust practical HFFL design.

Although increasing the pump brightness above the critical value does not result in further signal power increase, it reduces considerably the requirements in fiber core/cladding diameters and fiber length leading to more compact and manufacturable HPFLs. Reduced core diameters, e.g. Dcore < 40-45μm, are more manufacturable with current fiber fabrication techniques and can provide fibers with lower V number and modality. This is beneficial for splicing to SM fibers and for increasing the overall laser efficiency. In addition, as shown in the Fig. 10 inset, reducing the core diameter to Dcore < 40-45μm results in minimal mode deformation and effective area reduction even with bend diameters in the range of 10-50cm. Reduced cladding diameters, on the other hand, relax the mechanical reliability requirements and allow also for smaller bend diameters (see Fig. 9). This is also known to facilitate bend-induced distributing filtering of HOMs [36], proven to increase the TMI threshold [37] and efficiency of HPFAs with SM output [38].

 figure: Fig. 9

Fig. 9 Failures-in-Time (FIT) versus bending diameter Dbend., for different cladding diameters.

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For an amplifier gain G = 10dB, the maximum achievable signal powers (up to 35kW for DP, and up to 80kW for TP) will require multi-kW pump powers delivered in fibers with maximum cladding diameter of ~600μm and brightness in the range of 0.13-0.26 W/(μm2 sr) for DP, and 0.14-0.50 W/(μm2 sr) for TP. Multi-kW pump powers are achieved by combining a number of lower power modules through different incoherent combination techniques [1], such as spatial/polarization geometric combination [39,40], and wavelength beam combination (WBC) [41] or combination of the two [42]. High power pump modules can be potentially further combined by cascaded Nx1 fused tapered fiber bundles [43].

Due to brightness conservation requirements, the pump brightness at the output of an incoherent spatial combiner is always smaller than the brightness of the input pump modules. For DP, current semiconductor diode laser technology can provide geometrically combined multi-element modules with 340W delivered in 105um/0.15NA fibers with brightness of ~0.7 W/(μm2 sr) [44]. It is questionable though whether spatial incoherent combination of such diode pump modules can provide multi-kW pump power with the required brightness to achieve the predicted maximum signal powers. Alternatively, recent WBC diode technology has demonstrated modules with outputs of 4,680W in 100μm/0.08 NA fiber and record brightness of ~36 W/(μm2 sr) [41]. This is a promising diode technology for providing the required multi-kW pumps with the brightness to achieve the aforementioned maximum signal powers.

TP, on the other hand, relies on incoherently combined SM 1018nm fiber lasers to pump a HPFA operating in the 1060nm-1080nm region [2]. It has already been demonstrated that kW-level industrial SM fiber lasers incoherently combined through a cascade of Nx1 fused fiber tapered bundles, can provide 100kW power in 300μm/0.105 NA output fiber with a brightness of ~50 W/(μm2 sr) [45]. Recently, a 1018nm laser has been demonstrated providing 1,150W output in a 30μm/0.05 NA fiber with brightness of ~207 W/(μm2 sr) [46]. These results indicate that TP is a mature technology that can provide the required pumping power and brightness to reach the maximum signal powers derived in this work.

Finally, the observed dependencies of minimum core, cladding and fiber length on pump brightness (Bp) (see Fig. 6 and 7) are due to the assumption of constant cladding pump absorption (Ap) and core pump absorption (αcore) (or dopant concentration) and a scaling given by Eq. (23). If we assume a different core/cladding scaling, the dopant concentration has to be adjusted appropriately, so that the required Ap and αcore are obtained.

Appendix A Bending-induced mechanical reliability

In all previous power scaling investigations, a fixed core pump absorption (Yb3+ concentration) was considered and as a result large core diameters were accompanied by large cladding diameters (>1mm) and long fiber lengths (>>10m) [2–6]. Given that most practical industrial or directed energy HPFL systems should preferably have minimum footprint, the additional constraints of mechanical reliability of bent fibers should be considered. Failure in time (FIT) is calculated by the failure probability, given by [31,32]

FIT=αγNpL(ents)βepnptp
where
γ=(Bp/E2)/(B/E2)β,e=0.83dclad/Dbendα=m/(np2),β=(np2)/(n2)
with m, Np, L, E, dclad and Dbend being a Weibull distribution constant, failure number per unit amplifier length during proof testing, amplifier length, the fiber elastic modulus, cladding diameter and amplifier bending diameter, respectively. B, n, Bp and np, on the other hand, are parameters determined from the crack growth under certain environmental conditions, while e and ep are applied strain applied and ts and tp are times the strain was applied during proof-testing. Typical parameters, used in the calculations, are m = 2.42, Np = 0.1, L = 10m, n = 20, np = 24, ep = 0.0015, tp = 1s, ts = 20years ( = 20x31.536x10+6s), and γ = 1000.

Figure 9 shows the calculated failures-in-time (FIT) as a function of fiber bending diameter (Dbend), for different cladding diameters. The fiber is assumed to have undergone minimum proof testing and the failure time is 20years [31,32]. Assuming a maximum bending diameter of 1m, the maximum allowable cladding diameter is ~600μm. This limit will be used in the subsequent calculations.

Appendix B Bend-induced mode field deformation

Bending induced mode field deformation was calculated as a function of the core diameter for different bending diameters. Fiber bending and bend-induced refractive index changes both tend to distort the mode field distributions, and ultimately push the field away from the center of curvature [47].

Figure 10(a) plots the ratio of the fundamental-mode effective area of bent fibers (Aeff) over the one of straight fiber (Aeff) as a function of the core diameter, for different bend diameters. It is shown that increasing the core diameter makes the fiber more susceptible to bending. For a bending diameter of Dbend=100cm, a fiber with core diameter Dcore=50μm suffers an effective area reduction of ~5%, while with Dcore=100μm suffers a reduction of ~50%. Fig. 10(b) shows representative mode field deformation for core diameter Dcore=40μm and bending diameter Dbend=40cm, 50cm, 70cm and 100cm. They correspond to effective mode area reduction, (defined as Reff=1(Aeff/Aeff), of about 9%, 6%, 3% and 1%, respectively.

 figure: Fig. 10

Fig. 10 (a) Mode field deformation for core diameter Dcore = 40μm and bending diameter Dbend = 40cm, 50cm, 70cm and 100cm; (b) Mode effective area ratio versus core diameter for different bending diameters (V = 6). The inset zooms into the top-left highlighted area.

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Bend-induced modal deformation and effective mode area reduction in multimode fibers is more pronounced for the fundamental mode and has negative impact on the modal competition and overall amplifier efficiency [48]. By setting the effective mode area reduction limit to Reff=10%, from the inset of Fig. 9(a) we conclude that, for a bend diameter Dbend=100cm, the maximum tolerable core diameter is ~55μm. If we consider smaller core diameters in the range of 25μm-45μm, on the other hand, the allowable bending diameter reduces to Dbend=10cm-50cm. This potentially results in more compact designs with smaller footprint.

Appendix C Nonlinear and other power limit formulae [J. W. Dawson, Opt. Express 16(17), 13240–13266 (2008)]

C.1. Pump power limit

Poutpump=ηlaserBp(πrclad2)(πNA2)=ηlaserBpπ2NA2r02αcoreApL,

where Ap=αcladL (in dB) is the small-signal total pump absorption and αclad=αcore(r0/rclad)2small-signal cladding pump absorption coefficient (in dB/m) and αcore the small-signal pump core absorption coefficient (in dB/m). r0 and rclad are the core and cladding radii. ηlaser is the pump-to-signal conversion efficiency. L is the fiber length.

For given αcore, αclad (in dB/m) and L, the required cladding diameter dclad is given by:

dclad=d0αcore/αclad=d0αcoreL/Ap
where dclad = 2rclad and d0 = 2r0.

C.2. TMI Power limit [M. N. Zervas, Proc. SPIE 10083, 100830M (2017)]

PoutTMI=κUε2(Uε2Us2)4πneffαs(dn/dT)(λ0d0)2

where αs=αs+0.5qDgsis a heat generation coefficient. αsincludes the signal background loss and other contributions, such as phorodarkening, while qDgsdescribes the heat generation coefficient due to quantum defect (qD=λs/λp1) and gain saturation (gs). κ is the silica thermal conductivity and dn/dT the thermo-optic coefficient. d0 = 2r0 is the core diameter. neff is the fundamental mode effective index and Us(Uε) is the normalized transverse wavenumber for the fundamental (higher-order) fiber mode, approximated by Us(ε)(V)=Us(ε)()exp(1/V), where the asymptotic values for LP01 and LP11 mores are Us(∞) = 2.405 and Uε(∞) = 3.832, respectively, and V is the core V-number [49].

C.3. Thermal lens power limit

PoutTL=2πκ[ηheat/(ηlaserL)](dn/dT)(λ0d0)2

where ηheat is the fraction of power turned into heat inside the core, due to the quantum defect and other contributions (such as background signal loss and photodarkening).

C.4. Stimulated Raman scattering (SRS) power limit

PoutSRS16AeffGgRLeff=16πr02Γ2ln(G)gRL,

where Aeff=Γ2πr02=πrMF2, Leff=1gs(egsL1), gs=ln(G)/Land G is the amplifier single-pass saturated average gain. rMF=r0Γ is the mode field radius.

C.5. Optical damage power limit

Poutdamage=IdamageΓ2πr02

where Idamage is the damage intensity.

C.6. Thermal rapture power limit

Poutrupture=ηlaserηheat4πRm1r022rclad2L

where Rm is the rupture modulus of silica glass and rclad is the cladding diameter.

C.7. Core melting power limit

Poutmelting=ηlaserηheat4πκ(TmTc)1+2κrcladh+2ln(rcladr0)L

where Tm is the melt temperature of fused silica, Tc is the coolant temperature and h is the convective coefficient.

Typical parameter values are given in Table 1

Appendix D Heat-induced fundamental mode MFD shrinkage in optical fibers

As already mentioned, in previous studies [3], the TL power limit (see Eq. (6)) corresponds to ~20-30% MFD reduction of a Gaussian beam propagating in free space. We can now use the simplified analysis and results of Sections 2.1 and 2.2 to derive the corresponding MFD reduction in optical fibers at the TL power limit.

From Eqs.. (3), (4) and (6), setting Ps=PoutTL, the peak of the thermally-induced quasi-parabolic RI profile in the “hot” fiber is given by:

ΔnthTL=12(λ0d0)2
From Eq. (5), the corresponding “hot” fiber normalized MFD is approximated by:
ωthTLd02VthTL
where VthTL=V1+ΔnthTL/Δn. From the “cold” fiber V-number definition, on the other hand, we obtain Δn=(V2/2n2π2)(λ0/d0)2, which for typical V = 3-4 is approximated by Δn(1/2)(λ0/d0)2=ΔnthTL and results in VthTLV2. For V = 3-4, on the other hand, Eq. (1) results in (ω0/d0)1 for the “cold” fiber normalized MFD.

Under these approximations, the “hot”-to-“cold” fiber fundamental mode MFD ratio is given by:

ωthTLω02V
which for V = 3-4 results in ωthTL/ω00.70.8. It is shown that at the TL power limit, given by Eq. (6), the optical fiber fundamental mode MFD shrinkage is similar to the one considered in Gaussian mode free-space propagation [3].

Funding

EPSRC (EP/P027644/1; EP/N00762X/1); Royal Academy of Engineering (Research Chairs and Senior Research Fellowships Scheme).

Acknowledgements

Special thanks go to Dr. Natasha Vukovic for the bending-induced mode field deformation calculations, and Dr. C. A. Codemard for useful discussions. The data can be found in https://doi.org/10.5258/SOTON/D0923.

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36. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000). [CrossRef]   [PubMed]  

37. A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013). [CrossRef]   [PubMed]  

38. L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” J. Opt. Soc. Am. B 33(6), 1030–1037 (2016). [CrossRef]  

39. T. Y. Fan, “Efficient coupling of multiple diode laser arrays to an optical fiber by geometric multiplexing,” Appl. Opt. 30(6), 630–632 (1991). [CrossRef]   [PubMed]  

40. K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010). [CrossRef]  

41. R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

42. M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019). [CrossRef]  

43. C. Headley III, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005). [CrossRef]  

44. M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

45. E.A. Shcherbakov, V.V. Fomin, A.A. Abramov, A.A. Ferin, D.V. Mochalov, and V. P. Gapontsev, “Industrial grade 100 kW power CW fiber laser,” in Advanced Solid-State Lasers Congress (Optical Society of America, 2013), paper ATh4A.2.

46. P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018). [CrossRef]  

47. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]  

48. J. M. Fini, “Design of large-mode-area amplifier fibers resistant to bend-induced distortion,” J. Opt. Soc. Am. B 24(8), 1669–1676 (2007). [CrossRef]  

49. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech. 17(12), 1130–1138 (1969). [CrossRef]  

References

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  20. C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
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  23. D. C. Brown and H. J. Hoffman, “Thermal, Stress, and Thermo-Optic Effects in High Average Power Double-Clad Silica Fiber Lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
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  24. F. Stutzki, F. Jansen, H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Designing advanced very-large mode-area fibers for power scaling of fiber-laser systems,” Optica 1(4), 233–242 (2014).
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  26. O. Antipov, M. Kuznetsov, D. Alekseev, and V. Tyrtyshnyy, “Influence of a backward reflection on low-threshold mode instability in Yb3+-doped few-mode fiber amplifiers,” Opt. Express 24(13), 14871–14879 (2016).
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  27. Z. Li, Z. Huang, X. Xiang, X. Liang, H. Lin, S. Xu, Z. Yang, J. Wang, and F. Jing, “Experimental demonstration of transverse mode instability enhancement by a counter-pumped scheme in a 2 kW all-fiberized laser,” Photon. Res. 5(2), 77–81 (2017).
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  29. M. M. Johansen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Estimating modal instability threshold for photonic crystal rod fiber amplifiers,” Opt. Express 21(13), 15409–15417 (2013).
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  30. M. Walorny, J. Abramczyk, N. Jacobson, and K. Tankala, “Mechanical reliability of double clad fibers in typical fiber laser deployment conditions,” Proc. SPIE 9728, 97283A (2016).
    [Crossref]
  31. Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, and Y. Ishida, “Failure prediction for long length optical fiber based on proof testing,” J. Appl. Phys. 53(7), 4847–4853 (1982).
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  32. M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
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  34. T. Yao, J. Ji, and J. Nilsson, “Ultra-low quantum defect heating in Ytterbium-doped aluminosilicate fibers,” J. Lightwave Technol. 32(3), 429–434 (2014).
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  36. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000).
    [Crossref] [PubMed]
  37. A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
    [Crossref] [PubMed]
  38. L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” J. Opt. Soc. Am. B 33(6), 1030–1037 (2016).
    [Crossref]
  39. T. Y. Fan, “Efficient coupling of multiple diode laser arrays to an optical fiber by geometric multiplexing,” Appl. Opt. 30(6), 630–632 (1991).
    [Crossref] [PubMed]
  40. K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010).
    [Crossref]
  41. R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).
  42. M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
    [Crossref]
  43. C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
    [Crossref]
  44. M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).
  45. E.A. Shcherbakov, V.V. Fomin, A.A. Abramov, A.A. Ferin, D.V. Mochalov, and V. P. Gapontsev, “Industrial grade 100 kW power CW fiber laser,” in Advanced Solid-State Lasers Congress (Optical Society of America, 2013), paper ATh4A.2.
  46. P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
    [Crossref]
  47. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976).
    [Crossref]
  48. J. M. Fini, “Design of large-mode-area amplifier fibers resistant to bend-induced distortion,” J. Opt. Soc. Am. B 24(8), 1669–1676 (2007).
    [Crossref]
  49. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech. 17(12), 1130–1138 (1969).
    [Crossref]

2019 (4)

A. Steinkopff, C. Jauregui, F. Stutzki, J. Nold, C. Hupel, N. Haarlammert, J. Bierlich, A. Tünnermann, and J. Limpert, “Transverse single-mode operation in a passive large pitch fiber with more than 200 μm mode-field diameter,” Opt. Lett. 44(3), 650–653 (2019).
[Crossref] [PubMed]

M. N. Zervas, “Transverse-modal-instability gain in high power fiber amplifiers: effect of the perturbation relative phase,” APL Photonics 4(2), 022802 (2019).
[Crossref]

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

2018 (2)

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

M. N. Zervas, “Power scaling limits in high power fiber amplifiers due to transverse mode instability, thermal lensing, and fiber mechanical reliability,” Proc. SPIE 10512, 10501205 (2018).
[Crossref]

2017 (3)

2016 (6)

M. Walorny, J. Abramczyk, N. Jacobson, and K. Tankala, “Mechanical reliability of double clad fibers in typical fiber laser deployment conditions,” Proc. SPIE 9728, 97283A (2016).
[Crossref]

O. Antipov, M. Kuznetsov, D. Alekseev, and V. Tyrtyshnyy, “Influence of a backward reflection on low-threshold mode instability in Yb3+-doped few-mode fiber amplifiers,” Opt. Express 24(13), 14871–14879 (2016).
[Crossref] [PubMed]

H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Average power limit of Ytterbium-doped fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016).

C. Jauregui, H.-J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016).
[Crossref] [PubMed]

L. Huang, L. Kong, J. Leng, P. Zhou, S. Guo, and X. Cheng, “Impact of high-order-mode loss on high-power fiber amplifiers,” J. Opt. Soc. Am. B 33(6), 1030–1037 (2016).
[Crossref]

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

2015 (1)

2014 (3)

2013 (4)

2012 (5)

2011 (2)

2010 (3)

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
[Crossref]

K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010).
[Crossref]

2008 (1)

2007 (1)

2005 (1)

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

2004 (1)

2001 (1)

D. C. Brown and H. J. Hoffman, “Thermal, Stress, and Thermo-Optic Effects in High Average Power Double-Clad Silica Fiber Lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

2000 (1)

1991 (1)

1982 (1)

Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, and Y. Ishida, “Failure prediction for long length optical fiber based on proof testing,” J. Appl. Phys. 53(7), 4847–4853 (1982).
[Crossref]

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

1976 (1)

1969 (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech. 17(12), 1130–1138 (1969).
[Crossref]

Abramczyk, J.

M. Walorny, J. Abramczyk, N. Jacobson, and K. Tankala, “Mechanical reliability of double clad fibers in typical fiber laser deployment conditions,” Proc. SPIE 9728, 97283A (2016).
[Crossref]

Alekseev, D.

Alkeskjold, T. T.

Andrejco, M. J.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Antipov, O.

Bai, C.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Bao, L.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

Bao, X.-F.

Barty, C. P. J.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Beach, R. J.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Beier, F.

Biekert, N.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Bierlich, J.

Brar, K.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Breitkopf, S.

Broeng, J.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, Stress, and Thermo-Optic Effects in High Average Power Double-Clad Silica Fiber Lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Bullington, A. L.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

Burgess, J.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Chann, B.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Chen, Z.

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Cheng, X.

Codemard, C. A.

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0904123 (2014).
[Crossref]

C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
[Crossref]

Cook, R.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Cruz, M.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Dawson, D.

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

Dawson, J. W.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

DeFranza, M.

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

DeVito, M.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

DiGiovanni, D. J.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Dubinskii, M.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

Dugmore, D.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Eberhardt, R.

Eidam, T.

Fan, T. Y.

Fini, J. M.

Fishteyn, M.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Fortier, K.

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Goldberg, L.

Gong, M.

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

Grimshaw, M.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Guan, X.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Guo, S.

Haarlammert, N.

Hansen, K. R.

Hanzawa, N.

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

Headley, C.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Heebner, J. E.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Hein, S.

Hemenway, M.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, Stress, and Thermo-Optic Effects in High Average Power Double-Clad Silica Fiber Lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Huang, L.

Huang, R. K.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Huang, Y.

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

Huang, Z.

Hupel, C.

Ichii, K.

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

Ishida, Y.

Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, and Y. Ishida, “Failure prediction for long length optical fiber based on proof testing,” J. Appl. Phys. 53(7), 4847–4853 (1982).
[Crossref]

Jacobson, N.

M. Walorny, J. Abramczyk, N. Jacobson, and K. Tankala, “Mechanical reliability of double clad fibers in typical fiber laser deployment conditions,” Proc. SPIE 9728, 97283A (2016).
[Crossref]

Jansen, F.

Jauregui, C.

A. Steinkopff, C. Jauregui, F. Stutzki, J. Nold, C. Hupel, N. Haarlammert, J. Bierlich, A. Tünnermann, and J. Limpert, “Transverse single-mode operation in a passive large pitch fiber with more than 200 μm mode-field diameter,” Opt. Lett. 44(3), 650–653 (2019).
[Crossref] [PubMed]

F. Beier, C. Hupel, S. Kuhn, S. Hein, J. Nold, F. Proske, B. Sattler, A. Liem, C. Jauregui, J. Limpert, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Single mode 4.3 kW output power from a diode-pumped Yb-doped fiber amplifier,” Opt. Express 25(13), 14892–14899 (2017).
[Crossref] [PubMed]

C. Jauregui, H.-J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016).
[Crossref] [PubMed]

H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Average power limit of Ytterbium-doped fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016).

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Designing advanced very-large mode-area fibers for power scaling of fiber-laser systems,” Optica 1(4), 233–242 (2014).
[Crossref]

F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012).
[Crossref] [PubMed]

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

Jeong, Y.

Ji, J.

Jing, F.

Johansen, M. M.

Jørgensen, M. M.

Kanskar, M.

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Karlsen, S.

K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010).
[Crossref]

Katsuyama, Y.

Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, and Y. Ishida, “Failure prediction for long length optical fiber based on proof testing,” J. Appl. Phys. 53(7), 4847–4853 (1982).
[Crossref]

Ke, W.-W.

Kliner, D. A. V.

Kobayashi, H.

Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, and Y. Ishida, “Failure prediction for long length optical fiber based on proof testing,” J. Appl. Phys. 53(7), 4847–4853 (1982).
[Crossref]

Kong, L.

Koplow, J. P.

Kuhn, S.

Kuznetsov, M.

Lægsgaard, J.

Laurila, M.

Leisher, P.

K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010).
[Crossref]

Leng, J.

Li, D.

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

Li, S.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Li, Z.

Liang, X.

Liem, A.

Limpert, J.

A. Steinkopff, C. Jauregui, F. Stutzki, J. Nold, C. Hupel, N. Haarlammert, J. Bierlich, A. Tünnermann, and J. Limpert, “Transverse single-mode operation in a passive large pitch fiber with more than 200 μm mode-field diameter,” Opt. Lett. 44(3), 650–653 (2019).
[Crossref] [PubMed]

F. Beier, C. Hupel, S. Kuhn, S. Hein, J. Nold, F. Proske, B. Sattler, A. Liem, C. Jauregui, J. Limpert, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Single mode 4.3 kW output power from a diode-pumped Yb-doped fiber amplifier,” Opt. Express 25(13), 14892–14899 (2017).
[Crossref] [PubMed]

C. Jauregui, H.-J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016).
[Crossref] [PubMed]

H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Average power limit of Ytterbium-doped fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016).

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Designing advanced very-large mode-area fibers for power scaling of fiber-laser systems,” Optica 1(4), 233–242 (2014).
[Crossref]

F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012).
[Crossref] [PubMed]

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

Lin, H.

Liu, Z.

Lochman, B.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Ma, L.

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

Ma, Y.

Mann, J.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976).
[Crossref]

Martin, E.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Martinsen, R.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010).
[Crossref]

Matsuo, S.

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

Mermelstein, M. D.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Messerly, M. J.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Mitsunaga, Y.

Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, and Y. Ishida, “Failure prediction for long length optical fiber based on proof testing,” J. Appl. Phys. 53(7), 4847–4853 (1982).
[Crossref]

Modsching, N.

Nilsson, J.

Nold, J.

Ono, H.

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

Otto, H.-J.

Pax, P. H.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Payne, D.

Price, K.

K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010).
[Crossref]

Proske, F.

Prunty, T.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Sahu, J.

Sahu, J. K.

C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
[Crossref]

Sattler, B.

Schmidt, O.

Schreiber, T.

Shattuck, J.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Shu, X.-J.

Shverdin, M. Y.

Siders, C. W.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Smith, A. V.

A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref] [PubMed]

Smith, J. J.

A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref] [PubMed]

Snyder, A. W.

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech. 17(12), 1130–1138 (1969).
[Crossref]

Sridharan, A. K.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Stappaerts, E. A.

Steinkopff, A.

Stutzki, F.

Tankala, K.

M. Walorny, J. Abramczyk, N. Jacobson, and K. Tankala, “Mechanical reliability of double clad fibers in typical fiber laser deployment conditions,” Proc. SPIE 9728, 97283A (2016).
[Crossref]

Tayebati, P.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Tsujikawa, K.

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

Tünnermann, A.

A. Steinkopff, C. Jauregui, F. Stutzki, J. Nold, C. Hupel, N. Haarlammert, J. Bierlich, A. Tünnermann, and J. Limpert, “Transverse single-mode operation in a passive large pitch fiber with more than 200 μm mode-field diameter,” Opt. Lett. 44(3), 650–653 (2019).
[Crossref] [PubMed]

F. Beier, C. Hupel, S. Kuhn, S. Hein, J. Nold, F. Proske, B. Sattler, A. Liem, C. Jauregui, J. Limpert, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Single mode 4.3 kW output power from a diode-pumped Yb-doped fiber amplifier,” Opt. Express 25(13), 14892–14899 (2017).
[Crossref] [PubMed]

H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Average power limit of Ytterbium-doped fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016).

C. Jauregui, H.-J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016).
[Crossref] [PubMed]

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Designing advanced very-large mode-area fibers for power scaling of fiber-laser systems,” Optica 1(4), 233–242 (2014).
[Crossref]

F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012).
[Crossref] [PubMed]

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

Tyrtyshnyy, V.

Urbanek, W.

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Walorny, M.

M. Walorny, J. Abramczyk, N. Jacobson, and K. Tankala, “Mechanical reliability of double clad fibers in typical fiber laser deployment conditions,” Proc. SPIE 9728, 97283A (2016).
[Crossref]

Wang, J.

Wang, X.

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

Wang, X.-J.

Wang, Z.

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

Welch, K.

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

Wilkins, B.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Wirth, C.

Xiang, X.

Xiao, Q.

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

Xu, S.

Xu, X.

Yablon, A. D.

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

Yamada, M.

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

Yan, P.

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

Yang, Z.

Yao, T.

Zervas, M. N.

M. N. Zervas, “Transverse-modal-instability gain in high power fiber amplifiers: effect of the perturbation relative phase,” APL Photonics 4(2), 022802 (2019).
[Crossref]

M. N. Zervas, “Power scaling limits in high power fiber amplifiers due to transverse mode instability, thermal lensing, and fiber mechanical reliability,” Proc. SPIE 10512, 10501205 (2018).
[Crossref]

M. N. Zervas, “Transverse mode instability analysis in fiber amplifiers,” Proc. SPIE 10083, 100830M (2017).
[Crossref]

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0904123 (2014).
[Crossref]

M. N. Zervas, “Power scalability in high power fiber amplifiers,” in Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2017) paper CJ-6.1.

Zhang, J.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Zhang, S.

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Zhou, P.

Zhou, W.

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

Zhu, J.

APL Photonics (1)

M. N. Zervas, “Transverse-modal-instability gain in high power fiber amplifiers: effect of the perturbation relative phase,” APL Photonics 4(2), 022802 (2019).
[Crossref]

Appl. Opt. (1)

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D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
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IEEE J. Quantum Electron. (2)

D. C. Brown and H. J. Hoffman, “Thermal, Stress, and Thermo-Optic Effects in High Average Power Double-Clad Silica Fiber Lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0904123 (2014).
[Crossref]

P. Yan, X. Wang, Z. Wang, Y. Huang, D. Li, Q. Xiao, and M. Gong, “A 1150-W 1018-nm fiber laser bidirectional pumped by wavelength-stabilized laser diodes,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0902506 (2018).
[Crossref]

IEEE Photonics Technol. Lett. (1)

M. Yamada, K. Tsujikawa, L. Ma, K. Ichii, S. Matsuo, N. Hanzawa, and H. Ono, “Optical fiber amplifier employing a bundle of reduced cladding erbium-doped fibers,” IEEE Photonics Technol. Lett. 24(21), 1910–1913 (2012).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech. 17(12), 1130–1138 (1969).
[Crossref]

J. Appl. Phys. (1)

Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, and Y. Ishida, “Failure prediction for long length optical fiber based on proof testing,” J. Appl. Phys. 53(7), 4847–4853 (1982).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Express (13)

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015).
[Crossref] [PubMed]

F. Beier, C. Hupel, S. Kuhn, S. Hein, J. Nold, F. Proske, B. Sattler, A. Liem, C. Jauregui, J. Limpert, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Single mode 4.3 kW output power from a diode-pumped Yb-doped fiber amplifier,” Opt. Express 25(13), 14892–14899 (2017).
[Crossref] [PubMed]

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012).
[Crossref] [PubMed]

M. M. Johansen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Estimating modal instability threshold for photonic crystal rod fiber amplifiers,” Opt. Express 21(13), 15409–15417 (2013).
[Crossref] [PubMed]

O. Antipov, M. Kuznetsov, D. Alekseev, and V. Tyrtyshnyy, “Influence of a backward reflection on low-threshold mode instability in Yb3+-doped few-mode fiber amplifiers,” Opt. Express 24(13), 14871–14879 (2016).
[Crossref] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

J. Zhu, P. Zhou, Y. Ma, X. Xu, and Z. Liu, “Power scaling analysis of tandem-pumped Yb-doped fiber lasers and amplifiers,” Opt. Express 19(19), 18645–18654 (2011).
[Crossref] [PubMed]

W.-W. Ke, X.-J. Wang, X.-F. Bao, and X.-J. Shu, “Thermally induced mode distortion and its limit to power scaling of fiber lasers,” Opt. Express 21(12), 14272–14281 (2013).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

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(25), 6088–6092 (2004).
[Crossref] [PubMed]

C. Jauregui, H.-J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016).
[Crossref] [PubMed]

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012).
[Crossref] [PubMed]

Opt. Lett. (4)

Optica (1)

Photon. Res. (1)

Proc. SPIE (11)

A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref] [PubMed]

M. Walorny, J. Abramczyk, N. Jacobson, and K. Tankala, “Mechanical reliability of double clad fibers in typical fiber laser deployment conditions,” Proc. SPIE 9728, 97283A (2016).
[Crossref]

M. N. Zervas, “Power scaling limits in high power fiber amplifiers due to transverse mode instability, thermal lensing, and fiber mechanical reliability,” Proc. SPIE 10512, 10501205 (2018).
[Crossref]

M. N. Zervas, “Transverse mode instability analysis in fiber amplifiers,” Proc. SPIE 10083, 100830M (2017).
[Crossref]

H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Average power limit of Ytterbium-doped fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016).

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

K. Price, S. Karlsen, P. Leisher, and R. Martinsen, “High-brightness fiber-coupled pump laser development,” Proc. SPIE 7583, 758308 (2010).
[Crossref]

R. K. Huang, B. Chann, J. Burgess, B. Lochman, W. Zhou, M. Cruz, R. Cook, D. Dugmore, J. Shattuck, and P. Tayebati, “TeraDiode’s high brightness semiconductor lasers,” Proc. SPIE 9730, 97300C (2016).

M. Hemenway, Z. Chen, M. Kanskar, W. Urbanek, D. Dawson, L. Bao, M. DeFranza, M. DeVito, K. Fortier, R. Martinsen, and K. Welch, “976nm high brightness fiber-coupled laser modules for ytterbium fiber laser pumping,” Proc. SPIE 10900, 109000D (2019).
[Crossref]

C. Headley, M. Fishteyn, A. D. Yablon, M. J. Andrejco, K. Brar, J. Mann, M. D. Mermelstein, and D. J. DiGiovanni, “Tapered fiber bundles for combining laser pumps,” Proc. SPIE 5709, 263–272 (2005).
[Crossref]

M. Kanskar, C. Bai, L. Bao, N. Biekert, Z. Chen, M. DeFranza, M. DeVito, K. Fortier, M. Grimshaw, X. Guan, M. Hemenway, S. Li, E. Martin, R. Martinsen, T. Prunty, W. Urbanek, B. Wilkins, J. Zhang, and S. Zhang, “High Brightness Diodes and 600 W & 60% Efficient Low SWaP Fiber-coupled Package Enabled by Reduced-mode (REM) Diodes,” Proc. SPIE 10900, 109000H (2019).

Other (5)

E.A. Shcherbakov, V.V. Fomin, A.A. Abramov, A.A. Ferin, D.V. Mochalov, and V. P. Gapontsev, “Industrial grade 100 kW power CW fiber laser,” in Advanced Solid-State Lasers Congress (Optical Society of America, 2013), paper ATh4A.2.

M. O’Connor and B. Shiner, “High power fiber lasers for industry and defense,” in High-Power Laser Handbook, H. Injeyan, G.D. Goodno (Eds), (McGraw Hill, 2011), Ch. 18.

M. N. Zervas, “Power scalability in high power fiber amplifiers,” in Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2017) paper CJ-6.1.

J. D. Minelly, R. I. Laming, J. E. Townsend, W. L. Barnes, E. R. Taylor, K. P. Jedrzejewski, and D. N. Payne, “High gain fibre power amplifier tandem-pumped by a 3 W multistripe diode,” in Conference on Optical Fiber Communication (Optical Society of America, 1992), pp. 32–33.

M. N. Zervas, “TMI threshold in high power fiber amplifiers,” in OSA Advanced Photonics Congress (Optical Society for America, 2016), paper SoW2H.2.
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Schematic of (a) “cold” step-index fiber, and (b) “hot” fiber quasi-parabolic refractive index profile; (c) “hot”/”cold” SI fiber MFD ratio.
Fig. 2
Fig. 2 TMI and TL power limits in fibers with V = 3, ηheat = 0.3 and ηlaser = 0.7. The other parameters are given in Appendix C. The inset zooms into the dashed area and superimposes experimental data (open circles) [14].
Fig. 3
Fig. 3 MFD shrinkage with extracted power. Open circles: experimental data [14]; blue: LPF75, green: LPF45, red:LPF35. Solid lines: Theory (Eq. (5)) with V = 3, ηheat = 0.3 and ηlaser = 0.7 and d0 = 136μm (blue); 80μm (green); 63μm (blue). Dashed lines correspond to 70% “hot”-to”-cold” MFD shrinkage (ωth/ω0 = 0.7). Red and blue arrows denote the TL and TMI powers, respectively, for which PTMI/PTL = 0.65.
Fig. 4
Fig. 4 Contour plots of (a) power lower limits (in kW) due to pump brightness (blue area), SRS (green area) and TMI (orange area); and (b) required cladding diameter (in μm) as a function of amplifier length and core diameter. The parameters are similar to Ref [3]. (see Table 1 in Appendix C and Bp = 0.02W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 10dB).
Fig. 5
Fig. 5 Contour plots of power limits (in kW) due to pump brightness (blue area), SRS (green area) and TMI (orange area). The mechanical-reliability dominated parameter space (grey area) is also shown. (a) DP with Bp = 0.02W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 10dB; (b) DP with Bp = 0.18W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 10dB; (c) DP with Bp = 0.2W/μm2/sr; ηlaser = 0.85; ηheat = 0.10, G = 23dB; (d) TP with Bp = 0.3W/μm2/sr; ηlaser = 0.85; ηheat = 0.06, G = 10dB. Core pump absorption: (a)-(c) αcore = 250dB/m, (d) 100dB/m.
Fig. 6
Fig. 6 (a), Maximum signal power (right axis), minimum fiber length (left axis), and (b) Minimum core diameter (left axis), minimum cladding diameter (right axis). Diode pumping (DP): ηheat = 0.10, ηlaser = 0.85, αcore = 250dB/m; Tandem pumping (TP): ηheat = 0.06, ηlaser = 0.90, αcore = 100dB/m. Amplifier gain G = 10dB (vertical dashed lines show the corresponding critical pump brightness).
Fig. 7
Fig. 7 (a) maximum signal power, (b) minimum fiber length/core diameter for diode pumping (ηheat = 0.10, ηlaser = 0.80, αcore = 250dB/m); (c) maximum signal power, (d) minimum fiber length/core diameter for tandem pumping (ηheat = 0.06, ηlaser = 0.85, αcore = 100dB/m). Amplifier gain G = 10dB (blue lines), 15dB (green lines) and 20dB (red lines).
Fig. 8
Fig. 8 (a) the maximum achievable signal power, (b) the critical pump brightness, (c) corresponding core diameter and (d) the fiber length as a function of heat coefficient, for diode pumping (ηlaser = 0.80 - red lines) and tandem pumping (ηlaser = 0.85 - blue lines). G = 10dB.
Fig. 9
Fig. 9 Failures-in-Time (FIT) versus bending diameter Dbend., for different cladding diameters.
Fig. 10
Fig. 10 (a) Mode field deformation for core diameter Dcore = 40μm and bending diameter Dbend = 40cm, 50cm, 70cm and 100cm; (b) Mode effective area ratio versus core diameter for different bending diameters (V = 6). The inset zooms into the top-left highlighted area.

Tables (1)

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Table 1 List of parameters, symbols used in the text and values used in the calculations

Equations (33)

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ω 0 d 0 =0.65+ 1.619 V 3/2 + 2.879 V 6
Q 0 q D d I s / dz + α s I s =( q D g s + α s ) I s
Q ¯ heat = P heat L = η heat P pump absorbed L =( η heat η laser ) P signal extracted L =( η heat η laser ) P s (L) L G1 G
Δ n th Q ¯ heat ( dn/ dT ) 4πκ
ω th d 0 = 2 V th 1/2 + 0.23 V th 3/2 + 18.01 V th 6
P out TL = 2πκ [ η heat / ( η laser L ) ]( dn/ dT ) ( λ 0 d 0 ) 2
P out TMI = κ U ε 2 ( U ε 2 U s 2 ) 2π n eff [ η heat / ( η laser L ) ]( dn/ dT ) ( λ 0 d 0 ) 2 ,
P out TMI P out TL = U ε 2 ( U ε 2 U s 2 ) 4 π 2 n eff
L pumpSRS = 4Γ NA ln( G ) A p π g R η laser B p α core
d pumpTMI = 2κ U ε 2 ( U ε 2 U s 2 ) λ 0 2 A p π 3 n eff η heat ( dn/ dT ) B p α core N A 2 4
L TMISRS = 2π d 0 2 λ 0 2 n eff η heat ( dn/ dT ) Γ 2 ln( G ) η laser κ U ε 2 ( U ε 2 U s 2 ) g R
P TMISRS max = λ 0 2 η laser κ U ε 2 ( U ε 2 U s 2 ) Γ 2 ln(G) n eff η heat ( dn/ dT ) g R
L mech = A p d mech 2 α core d 0 2
P pump max = 1 4 η laser B p π 2 N A 2 d mech 2
B p crit = 4 λ 0 U ε Γ π 2 N A 2 d mech 2 2κ( U ε 2 U s 2 )ln(G) n eff η laser η heat ( dn/ dT ) g R
P s max ={ P pump max ; B p < B p crit P TMISRS max ; B p B p crit
d mechSRS = d mech π η laser g R N A 2 A p B p 16 Γ 2 ln(G) α core 4
d 0 min ={ d mechSRS ; B p < B p crit d pumpTMI ; B p B p crit
d clad min ={ d mech ; B p < B p crit d clad ; B p B p crit
d clad = 32κ Γ 2 U ε 2 ( U ε 2 U s 2 ) λ 0 2 ln(G) n eff g R η laser η heat ( dn/ dT ) π 4 N A 4 B p 2 4
FIT=αγ N p L ( e n t s ) β e p n p t p
γ= ( B p / E 2 )/ ( B/ E 2 ) β ,e= 0.83 d clad / D bend α=m/ ( n p 2 ) ,β= ( n p 2 )/ ( n2 )
P out pump = η laser B p ( π r clad 2 )( πN A 2 )= η laser B p π 2 N A 2 r 0 2 α core A p L,
d clad = d 0 α core / α clad = d 0 α core L/ A p
P out TMI = κ U ε 2 ( U ε 2 U s 2 ) 4π n eff α s ( dn/ dT ) ( λ 0 d 0 ) 2
P out TL = 2πκ [ η heat / ( η laser L ) ]( dn/ dT ) ( λ 0 d 0 ) 2
P out SRS 16 A eff G g R L eff = 16π r 0 2 Γ 2 ln(G) g R L ,
P out damage = I damage Γ 2 π r 0 2
P out rupture = η laser η heat 4π R m 1 r 0 2 2 r clad 2 L
P out melting = η laser η heat 4πκ( T m T c ) 1+ 2κ r clad h +2ln( r clad r 0 ) L
Δ n th TL = 1 2 ( λ 0 d 0 ) 2
ω th TL d 0 2 V th TL
ω th TL ω 0 2 V

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