Abstract

Spatio-temporal instability of the fundamental mode in Yb3+-doped few-mode PM fiber amplifiers with a core diameter of 8.5 μm was registered at 2-30 Watts pump power. Both experimental and theoretical analysis revealed the nonlinear power transformation of the LP01 fundamental mode into high-order modes. Numerical simulation revealed self-consistent growth of the higher-order mode and traveling electronic index grating accompanying the population grating induced by the mode interference field (due to different polarizability of the excited and unexcited Yb3+ ions). Experimental results and numerical calculations showed the increase of the instability threshold along with an increase of the signal frequency bandwidth.

© 2014 Optical Society of America

1. Introduction

An average-power scaling of Yb-doped fiber amplifiers is limited by a parasitic phenomenon, known as a “mode instability” (MI) - a dynamic energy transfer from the fundamental transverse mode to higher order modes. It was shown that this effect results in the decrease of an output beam quality and randomizes temporal dynamics [14]. The mode instability was reported to occur in large mode area fibers with core diameter of about 30-50 μm and in photonic crystal fibers with similar mode field diameters at pump powers ranging from several hundred watts to kilowatt, both in narrow bandwidth continuous wave and in nanosecond broadband pulsed amplifiers and laser systems [14].

The physical mechanism of the MI was explained by the beat of different transverse modes, which results in the periodic intensity modulation along the fiber and subsequently in the formation of the long-period refractive index gratings (RIGs). These gratings, in turn, enable energy transfer between the transverse modes due to the mode coupling [48]. A hot debate around the origin of the induced RIGs responsible for MI was finalized to the moment by these opinions: 1). the thermo-optical effect (or stimulated thermal Rayleigh scattering) is liable for the RIGs formation and the MI [610]; 2). the electronic RIG accompanying the population grating due to the different polarizability of the excited and unexcited Yb3+ions (or “Kramers-Kronig enhanced” effect [11,12]) gives a negligibly small impact on the MI behavior [57,13].

An effect similar to the MI has been reported recently but occurring at lower pump levels of only several Watts in the Yb3+-doped fiber amplifier with the relatively-small core diameter of 8-10 μm and the numerical aperture (NA) of 0.21 [14]. This active fiber supported few guided modes, and the significant part of fundamental mode power was nonlinearly transferred to higher order modes once a pump power threshold was achieved. This paper is devoted to the experimental and theoretical analysis of the MI effect in the amplifiers based on few mode fibers (FMFs). The numerical simulation of the MI effect that considers both the population (electronic) and thermal RIGs is introduced. Our assessment was based on the measured difference of the polarizability of Yb-ions in the doped fibers [12,15]. The theoretical analysis was conducted with an assumption that the MI effect in the fiber is similar to the well-studied “small-scale” instability effect in the bulk nonlinear media [1618].

2. Experimental setup, and measurement results

The polarization maintaining (PM) Yb-doped fiber with 7 μm MFD (produced by “NTO IRE-Polus”) was used as an amplifier of a CW or pulsed (with the pulse duration ~1.5 ns and repartition rate 100...1000 kHz) linearly polarized signal at 1064 nm (Fig. 1). The passive losses in the Yb-doped fiber (measured at 1.15 µm) were less than 20 dB/km. CW laser diodes (λ~975 nm, average power up to 30 W) were used as a pump source for the amplifier (co- or contra-directed to the signal). The parameters of the active fiber, pump and signal are summarized in Table 1.

 

Fig. 1 Experimental setup.

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Tables Icon

Table 1. Experimental Parameters of Active Fiber, Pump and Input Signal

The experimental setup consisted of the Yb-doped fiber spliced at both ends with two passive single mode fibers (that played a role of the fundamental mode filters). The fundamental mode of the active fiber was registered by the photodiode PD1, and the higher-order modes, which had high losses at the output splice, were registered by the photodiode PD2. The threshold-like MI effect was observed at the output power level of about few Watts (at CW input power of about 50 mW, and signal bandwidth of Δλ~0.1…1.0 nm): the modulation of the registered output power occurred above threshold (Fig. 2); a drop of the linear dependence of power of the output signal vs pump was registered (Fig. 3).

 

Fig. 2 Oscillograms of the fundamental mode (bright blue, from PD 1), and the higher-order mode (dark blue, from PD 2) below (a), slightly above (b), and far above (c) MI threshold.

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Fig. 3 Output power dependence on pump power (a), and dependence of the mode instability threshold (for output power of the signal) on input power (b). Input signal bandwidth was less than 0.05 nm, active fiber length was 6 m.

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Typical frequencies of the observed modulation were in the range of few kHz. In separate experiments with different amplifiers various modulation kinetics were observed up to chaotic power modulation. It should be noticed that detailed study of modulation kinetics above the mode instability threshold is complicated besides there is a high risk of amplifier damage.

Nevertheless the threshold value can be measured safely. The measured MI threshold increased with the input-signal power (Fig. 3b).

The MI effect was registered for the fiber amplifier with length varied from 1 to 10 m. The threshold power for each fiber length was measured to depend on the power and bandwidth of the input signal, and had the same order of magnitude as in the 6-m fiber amplifier for the same power and bandwidth of the signal.

It was observed earlier that the input signal bandwidth broadening results in increase of the mode instability threshold. For detailed experiments we manufactured a CW laser at 1064 nm with tunable spectral bandwidth and stable output power up to 50 mW. Measured MI threshold dependence (for the output signal power) on the input signal bandwidth is presented in Fig. 4.

 

Fig. 4 Mode instability threshold dependence on the input signal bandwidth (the signal input power is 50 mW).

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The same MI effect was observed during the amplification of the pulsed optical signal. The average power of MI threshold was similar to the threshold in CW regime. Typical waveform of the amplified 1.5-ns pulses (at the repetition rate of 500 kHz) above the MI threshold is presented in Fig. 5.

 

Fig. 5 Oscillogram of the fundamental mode above MI threshold.

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In order to obtain the picture of the mode composition in an active fiber, the experiment without splicing of the amplifier with the output passive fiber was performed. The image of the active fiber end was guided directly to the CMOS matrix. Below the MI threshold a round Gaussian beam with low-brightness inclusions of the higher-order mode was observed [Fig. 6(a)]; above the threshold the image considerably changed: additional spot appeared [Fig. 6(b)]. It is worth noting that 90 degree rotation of the input signal polarization didn’t change the observed image. The registered above-threshold images clearly indicated the fundamental mode degradation and the generation of the higher-order modes, however they can’t be simply identified by the LP11 or another mode (the last fact can be explained by partial leakage of the mode to the fiber clad, and distortion of the image transfer from the fiber output to the CMOS matrix). The nonlinear generation of the higher-order leaky modes resulted also in a fiber burning at the above-threshold pump power in the experiments with splicing of amplifier output with a single-mode passive fiber.

 

Fig. 6 Beam at the output of the active fiber below (a) and above (b) the MI threshold, and the red-light fiber images (c) at the same space scale.

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Note, that the registered instability threshold was in hundreds times lower than in the previous experimental observations of the similar effect in the LMA fibers [14]. From our point of view, the main difference in experimental conditions determining the MI-threshold decrease is the smaller core diameter (of 8.5 μm) of our PM Yb-doped fiber and the relatively narrowband signal.

3. Theoretical model of MI

In order to analyze the MI, the Yb-doped PM phosphor-silicate double-clad fiber was modeled as the infinite composite cylinder consisting of a Yb3+-doped core, a silica-glass pumping cladding and a polymer cladding in air with the parameters similar to the experimental ones (Tables 1 and 2). The fiber was pumped directly into the cladding, or it was side-pumped by an additional transporting fiber in the GT-wave configuration [19].

Tables Icon

Table 2. Fiber Parameters Used for Calculation

Proposed model assumed the existence of the fundamental mode LP01 and a small seed of the second mode LP11 at the input of the fiber amplifier. The complex amplitudes of the linear-polarized modes were described by the expressions:

E01=A0(z,t)ei(2πνstk0z)ψ0(r),
E11=(A1s(z,t)eiΩt+iϕ+A1as(z,t)eiΩtiϕ)ei(2πνstk1z)ψ1(r),
where νs is the signal frequency, t is the time, k0 and k1 are the propagation constants of the modes, z is the coordinate along the fiber, r is the transverse coordinate of the fiber, φ is the polar angle, Ω is the frequency shift of the Stokes and anti-Stokes components of the LP11 mode with complex amplitudes A1sandA1as, respectively (note, that the Stokes and anti-Stokes wave interaction can give strong correction to the instability conditions, as it is well known for the bulk media [1618]); ψ0 and ψ1 are the radial distributions of the modes [24]: ψ0,1(r)=С0,1J0,1(u0,1r/r0)/J0,1(u0,1) if rr0, and ψ0,1(r)=С0,1K0,1(w0,1r/r0)/K0,1(w0,1), if r>r0, C0,1 are the normalizing constants, Ji and Ki are the Bessel and Macdonald functions of order i, the constants u0,1 and w0,1 are defined by the Eqs.:u0,12=r02(n02(2π/λs)2k0,12),w0,12=r02(k0,12(n0Δn)2(2π/λs)2), and linked by the characteristic Eqs.:u0,1J1,2(u0,1)/J0,1(u0,1)=w0,1K1,2(w0,1)/K0,1(w0,1). The mode spatial structures were assumed to be unchangeable due to nonlinear self-action and mode interaction.

The system of Eqs. for the complex amplitudes of the quasi-monochromatic modes in weakly guiding approximation (after averaging across the fiber for the each mode) was as follows:

A0z+1υ0A0t=i2πn0λsk0(A0ψ02δH+A1seiΩt+iqzψ1ψ0δHeiϕ+A1aseiΩt+iqzψ1ψ0δHeiϕ),
A1sz+1υ1A1st=i2πn0λsk1(A1sψ12δH+A0eiΩtiqzψ0ψ1δHeiϕ),
A1asz+1υ1A1ast=i2πn0λsk1(A1asψ12δH+A0eiΩtiqzψ0ψ1δHeiϕ),
where υ0,1 is the mode speeds, q=k0k1(|q/k0|<<1), ...=1πr02002π...rdrdϕ, ψ0,12=1 (from the normalization),
δH=2πλs(nT)δT+i2((σems+σabs)(1+iβ)δNexσabsNd),
where δNex is the population of the exited state 2F5/2, β=8π2λsn0FL2Δpσems+σabs, FL=n02+23 is the local-field Lorentz factor.

The expression (6) assumed the refractive index change due to the temperature change δT (the first summand) and population change (the summand with the β parameter). The population-enhanced index change is caused by the different polarizability of the excited and unexcited Yb3+ ions in the fiber core and resulted in the real part of the nonlinear dielectric susceptibility [12,15,21].

The excited-state population change was described by the following Eq.:

Next+Nexτ+Nex(σabp+σemp)PphνpScl=σabpNdPphνpScl(σems+σabs)Ishνs(NexσabsNdσems+σabs),
where Scl = πr12, Pp and νp are the pump power and frequency, respectively, Is is total signal intensity (including both the mode intensities and their interference field).

The temperature distribution inside the core was described by the following Eq.:

TtΚ1ρ1C1p2T=hνTρ1C1pNexτ+νpνsνs(σems+σabs)Isρ1C1p(NexσabsNdσems+σabs),
where 2 is the Laplasian, T is the energy of nonradioactive transitions between sublevels of the ground state 2F7/2 after spontaneous emission.

Each of both temperature distribution δT and exited state population δNex were separated into three components:

(δTδNex)=(T0(z,r,t)δNex0(z,r,t))+(δTs(z,r,t)δNexs(z,r,t))eiΩt+iϕ+iqz+(δTas(z,r,t)δNexas(z,r,t))eiΩtiϕ+iqz.

After substitution of expressions (10) in (7) a number of the following correlators aroused in right part of the Eqs. (3)-(5):

(T00N00T11N11T01sN01sT01asN01as)(T0(z,r,t)ψ02δNex0(z,r,t)ψ02T0(z,r,t)ψ12δNex0(z,r,t)ψ12δTs(z,r,t)ψ0ψ1eiϕδNexs(z,r,t)ψ0ψ1eiϕδTas(z,r,t)ψ1ψ0eiϕδNexas(z,r,t)ψ1ψ0eiϕ).

The correlators (10) were found by averaging Eqs. (7) and (8) with substitution of expressions (9). The higher-order correlators arousing in the right part of the Eqs. for the components (10) during the averaging procedure were reduced to the lower-order correlators by the following manner:

δNex0ψ0,14N00,11ψ0,14(r)ψ0,12(r),δNex0ψ02ψ12N00ψ02(r)ψ12(r)ψ02(r),δNexs,asψ0,13ψ1,0N01s,asψ0,14(r)ψ1,02(r)ψ02(r)ψ12(r).

The correlators with the spatial Laplasian in the left side of the thermal conductivity equation (arising due to averaging) were also reduced to lower-order correlators (in the thermal grating approximation δTs,as~ψ1ψ0e±iϕiqz~J1(u1r/r0)J0(u0r/r0)e±iϕiqz). These reductions for the thermal gratings were as follows: ψ0(r)ψ1(r)eiϕ±iqz2(δTs,as(z,r,t))(b2+q2)T01s.as(z,t), where constant b1 was estimated by the following expression:

b2=u02+u12r02+2u0u10r0J1(u1r/r0)J0(u0r/r0)(J1(u0r/r0)J0(u1r/r0)rJ1(u1r/r0)J1(u0r/r0)r0/u1)drr020r0J12(u1r/r0)J02(u0r/r0)rdr13r02.

In this way the system of equations for the population and thermal correlators was obtained. For example, the equations for the “grating correlators” were the following:

T01s,astiΩT01s,asΚ1ρ1C1p(b2+q2)T01s=(σems+σabs)(νpνs)ρ1C1pνs[N01s,asψ04ψ12|A0|2+(|A1s|2+|A1as|2)ψ14ψ02ψ02ψ12++(N00ψ02σabsNdσems+σabs)ψ02ψ12(A0*A1,s,as+A1as,s*A0e2iqz)]сn08π+hνTρ1C1pτN01s,as,
N01s,astiΩN01s,as+N01s,asτ+N01s,as(σemp+σabp)Ppπr12hνp=σems+σabshνscn08π[N01s,asψ04ψ12|A0|2+(|A1s|2+|A1as|2)ψ14ψ02ψ12ψ02++(N00ψ02σabsNdσems+σabs)ψ12ψ02(A0*A1s,as+A1as,s*A0e2iqz)],
where c is the light speed in vacuum.

The full equation system was completed by equations for the pumping power in the active fiber Pp and in the auxiliary fiber Pax (in the case of GT-wave fiber):

Ppz=(Nd(σemp+σabpab)+σempδNex)r02r12Pp+γ(PaxPp),
Paxz=γ(PaxPp),
where γ is the transformation coefficient of the pump from the auxiliary to the active fiber.

In the simulations the signal had been switched on before the pump was switched on (the switch duration was 1…50 μs). The ratio of power of the LP01 and LP11 modes at the input of the fiber amplifier varied from 40 to 104. The initial perturbations of the temperature and population were assumed to be zero. The system of the partial derivative equations (with t and z fluents) was solved using predictor-corrected method.

To summarize the model, both the thermal and electronic gratings accompanying the population gratings induced by the mode-interference field, possible interaction of the Stokes-anti-Stokes components of the nonlinear growing modes, and walk-off of the LP01 and LP11 modes were taken into consideration.

4. Modeling results for narrow signal bandwidth

For the narrow-bandwidth signal (when the mode walk-off time on the fiber length L was less than the signal coherence time: (1/υ01/υ1)L<<Δνs1 (17), where Δνs is the signal bandwidth) the partial derivatives with respect to time in the left side of the Eqs. (3)-(5) were neglected.

The gain of the anti-Stokes shifted LP11 mode obtained from numerical simulation was found to be stronger than the gains of the fundamental mode LP01 and Stokes shifted LP11 mode in these experimental conditions (fiber numerical aperture, length and diameters, Yb3+ doping concentration and so on). The typical distribution of the mode powers inside the fiber amplifier, and the mode waveforms on the fiber output is shown in the Fig. 7.

 

Fig. 7 Powers of the fundamental mode LP01 (red), anti-Stokes (blue) and Stokes (green) shifted LP11 mode and the pump inside the active fiber (violet) and the auxiliary fiber (black) on the fiber length at the time t = 2 ms after switch on of the pump with power 5 mW (the signal with power mW was switched on in 20 μs before the pump) (a), and on the time in the fiber output (b). The input ratio of the LP01 and LP11 mode power was 40, the frequency detuning Ω = 4 kHz.

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The relative gain (determined as ratio of the amplification of the anti-Stokes LP11 mode (P11as(z=l,tst)/P11as(z=0,tst)) to the LP01 mode amplification (P01(z=l,tst)/P01(z=0,tst)) at a time of gain stabilizing tst = 2 ms) was found to have maximum at frequency shift Ω ≈4-5 kHz for pump power of 1-1.5 W (Fig. 8). The optimal frequency shift Ω grew with pump and fundamental mode power, and was almost independent on the fiber NA (varied from 0.16 to 0.23) and ratio of the input power of the LP11 and LP01 modes (varied from 40 to 103 at the fixed input-signal power). Note that the optimal frequency shift and its power dependence are in good qualitative agreement with the experimentally measured oscillation frequency of the output beam power that can be explained by modulation of the interference of the frequency-shifted modes. According to the numerical analysis, the real frequency shift of the growing mode can be also determined by a random mechanical vibration of the fiber-amplifier input.

 

Fig. 8 The relative gain of the anti-Stokes shifted LP11 mode (with respect to the fundamental mode) on the frequency detuning Ω for different pump and signal powers (Pp and PS) in the fiber input, numerical aperture (NA) and the time.

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For the transient condition, when the pump pulse duration is less than τ, the energy transfer from the LP01 to LP11 mode occurs even without any additional frequency shift (see orange dashed curve on the Fig. 8).

The additional gain of the anti-Stokes LP11 mode can be explained by the nonlinear energy transfer from the fundamental mode due to scattering on the dynamic “electronic” RIGs accompanying the population gratings (δNexs,as) induced by the interference field of the modes LP01 and LP11. The RIG caused by the polarizability difference was found to grow (along the fiber amplifier and in time at the output) much stronger than that caused by the thermal grating (Fig. 9). In our experiments with the relatively small fiber-core diameter the domination of the electronic RIC over the thermal grating don’t radically contradict the opposite conclusion for the LMA fiber amplifiers [57]: the steady-state thermal grating amplitude increases with grow of the core diameter, but the electronic grating amplitude decreases due to saturation at high pumping and signal powers. The similar nonlinear effect of the optical wave interaction by the dynamic “electronic” RIGs accompanying the population gratings is known to exist in the bulk crystal amplifiers [2527], and resonantly absorbing fibers [28]. The relatively high difference of the longitudinal wavenumber q in the experimental fiber prevented the Stokes - ant-Stokes interaction, and only the anti-Stokes wave was found to grow due to nonlinear effect.

 

Fig. 9 Amplitudes of the RIGs caused by the temperature change (red) and polarizability difference (brown, violet, green) on the fiber length at the time 2 ms after signal switch on (left) and the time in the fiber output (right). The input power of the LP01 mode is 5 mW (the LP01 and LP11 mode power ratio on the input is 40), the input pump power (in the auxiliary fiber) is Pax(0) = 2.5 W and the frequency shift is Ω = 6 kHz (green and red), Pax(0) = 1.5 W and Ω = 4.25 kHz (brown), Pax(0) = 0.75 W and Ω = 4.25 kHz (violet).

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Numerical calculations reviled that the output power of the fundamental mode increased with the pump power up to a “threshold”, and decreased after this point due to energy transfer to the anti-Stokes component of the LP11 mode (Fig. 10). The “threshold” output power grew with increase of the input signal power and the ratio of the input power of the LP11 and LP01 modes.

 

Fig. 10 Output power of the LP01 mode (solid lines) and the optimal-shifted LP11 mode (dashed lines) on the pump power for the input LP01-mode power 5 mW (blue and green) or 60 mW (red), and the input power ratio of the LP01 and LP11 modes 40 (for blue and red) or 200 (for green).

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5. Modeling results of the broadband signal instability

For the broadband input signal (when the mode walk-off time is more than the coherence time and the condition (17) is not fulfilled) the partial derivative in time in Eqs. (3)-(5) has to be taken into consideration. The broadband LP01 and LP11 transverse modes were assumed to consist of a number of the longitudinal modes:

A0,1s,as(z,t)=m=MMB0,1ms,as(z,t)eimΔt+iφm,
where B0,1ms,as are the complex amplitudes, Δ is a intermode frequency interval, ϕm is a phase of the m-th longitudinal mode. The intermode interval Δ was assumed to be much more than all characteristic frequencies of the nonlinear interaction (Δ >> τeff-1, τther−1, where τeff is the effective lifetime of the excited level, τther is the relaxation time of the temperature perturbations).

The Eqs. (3) and (5) for the complex amplitudes (18) were rewritten in following forms (the Stokes components were neglected due to qL>>1):

B0mz+imΔυ0B0m=σems+σabs2(1+iβ)(N00B0m+N01B1mas)σabs2NdB0mi2πλs(nT)(T00B0m+T01B1mas),
B1masz+imΔυ1B1mas=σems+σabs2(1+iβ)(N00B1mas+N01*B0m)σabs2NdB1masi2πλs(nT)(T00B1mas+T01*B0m),
where the population and temperature components Nij and Tij depended on the average intensity <IΣ>=m=MM(|Bom|2+|B1mas|2), and the interference field in the form m=MMBBom1mas*.

Numerical calculation of the system including Eqs. (19) and (20) for the complex amplitudes (2M + 1 equations for each mode LP01 and LP11), and Eqs. for the pump, population, temperature and their gratings showed the nonlinear power transformation from the LP01 mode to the anti-Stokes-shifted LP11 mode. The additional nonlinear gain of the LP11 mode was found to depend on the signal bandwidth Δνs = 2MΔ, and the signal and pump powers. The power of the output signal in the LP01 mode increased up to a “threshold” [Fig. 11(a)], and the threshold was found to depend on the signal bandwidth and the mode-power ratio at the fiber input [Fig. 11(b)]. These numerical results are in good qualitative accord with the experimental dependences (see Figs. 3,4), however, the MI threshold power in the theory is less than in experiments. The quantitative disagreement can be explained by the usage of the two-mode model in the theory, as the experimentally-used fiber supports a number of modes.

 

Fig. 11 Output power of the LP01 mode (solid curves) and the optimal-shifted LP11 mode (dushed curves) on the pump power for the different longitudinal mode numbers M (a); the threshold LP mode power on the signal bandwidth (b) (the input LP01-mode power is 5 mW; the ratio of the input mode-power is 350 (a), and is varied from 40 to 103 (b).

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6. Conclusion

The experiments showed the spatio-temporal instability of the fundamental mode in few-mode Yb3+-doped fiber amplifier with the core diameter of 8.5 μm at few-Watts pump power level. Experimentally observed instability threshold grew with both signal input power and the frequency bandwidth. The numerical simulation indicated the nonlinear interaction of the fundamental LP01 and higher-order LP11 modes by mutual scattering on the population gratings induced in the fiber by the mode interference field. Dynamic RIG caused by different polarizability of the excited and unexcited Yb3+ ions gave main contribution to the energy transfer from the fundamental mode to LP11 mode in the experimental conditions. The anti-Stokes kHz frequency shift provided the nonlinear increment of the LP11 mode for both narrowband and broadband signal. The calculation revealed increase of the instability threshold with increasing of the signal frequency bandwidth, of the input signal power, and of the ratio of the input mode power.

Acknowledgments

This work was supported in part by the program of Russian Academy of Sciences “Nonlinear-optical materials and methods for development of novel laser systems” and grant of the Ministry of Education and Science of the Russian Federation for Nizhniy Novgorod State University (agreement Nº02.B.49.21.0003).

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17. R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated Four-Photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17(22), 1158–1161 (1966). [CrossRef]  

18. S. N. Vlasov and V. I. Talanov, Wave Self-Focusing (IAP RAS, 1997).

19. C. Codemard, K. Yla-Jarkko, J. Singleton, P. W. Turner, I. Godfrey, S.-U. Alam, J. Nolssson, J. Sahu, and A. B. Grudinin, in Proceeding of European Conference on Optical Communication (ECOC'2002, Copenhagen, Denmark, 2002), PD1.6.

20. M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

21. A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193 – 230.

22. M. Bass, E. Van Stryland, D. Williams, and W. Wolfe, Handbook for Optics, 2nd ed. (MGH, 1995).

23. V. Privalko, Handbook for Physical Chemistry of Polymers (Naukova Dumka, 1984).

24. H.-G. Unger, Planar Optical Waveguides and Fibres (Oxford University, 1977).

25. O. L. Antipov, S. I. Belyaev, and A. S. Kuzhelev, “Stimulated resonant scattering of optical waves in laser crystals with population inversion,” JETP Lett. 63(1), 13–18 (1996). [CrossRef]  

26. O. L. Antipov, S. I. Belyaev, A. S. Kuzhelev, and D. V. Chausov, “Resonant two-wave mixing of optical beams by refractive index and gain gratings in inverted Nd:YAG,” J. Opt. Soc. Am. B 15(8), 2276–2281 (1998). [CrossRef]  

27. M. Chi, J.-P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” JOSA B 26(8), 1578–1584 (2009). [CrossRef]  

28. S. Stepanov, A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064nm,” Opt. Express 15(14), 8832–8837 (2007). [CrossRef]   [PubMed]  

References

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  • |

  1. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
    [Crossref] [PubMed]
  2. 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]
  3. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36(23), 4572–4574 (2011).
    [Crossref] [PubMed]
  4. N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20(12), 13274–13283 (2012).
    [Crossref] [PubMed]
  5. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
    [Crossref] [PubMed]
  6. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20(1), 440–451 (2012).
    [Crossref] [PubMed]
  7. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012).
    [Crossref] [PubMed]
  8. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012).
    [Crossref] [PubMed]
  9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012).
    [Crossref] [PubMed]
  10. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013).
    [Crossref] [PubMed]
  11. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M. J. F. Digonnet, “Experimental and theoretical analysis of the resonant nonlinearity in ytterbium-doped fiber,” J. Lightwave Technol. 16(5), 798–806 (1998).
    [Crossref]
  12. A. A. Fotiadi, O. L. Antipov, and P. Mégret, “Resonantly induced refractive index changes in Yb-doped fibers:the origin, properties and application for all-fiber coherent beam combining,” in Frontiers in Guided Wave Opticsand Optoelectronics, B. Pal, ed. (Intec, 2010), pp. 209–234.
  13. A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21(13), 15168–15182 (2013).
    [Crossref] [PubMed]
  14. V. Tyrtyshnyy, O. Vershnin, and S. Larin, “Influence of the radiation spectral parameters on the nonlinear interaction of modes in active fiber,” in Technical digests of International Symposium “High-Power Fiber Lasers and Their Applications,” (S-Petersburg, Russia, 2010), paper TuSy, p. 04.M.
  15. M. S. Kuznetsov, O. L. Antipov, A. A. Fotiadi, and P. Mégret, “Electronic and thermal refractive index changes in ytterbium-doped fiber amplifiers,” Opt. Express 21(19), 22374–22388 (2013).
    [Crossref] [PubMed]
  16. V. I. Bespalov and V. I. Talanov, “About filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(12), 307–310 (1966).
  17. R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated Four-Photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17(22), 1158–1161 (1966).
    [Crossref]
  18. S. N. Vlasov and V. I. Talanov, Wave Self-Focusing (IAP RAS, 1997).
  19. C. Codemard, K. Yla-Jarkko, J. Singleton, P. W. Turner, I. Godfrey, S.-U. Alam, J. Nolssson, J. Sahu, and A. B. Grudinin, in Proceeding of European Conference on Optical Communication (ECOC'2002, Copenhagen, Denmark, 2002), PD1.6.
  20. M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).
  21. A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193 – 230.
  22. M. Bass, E. Van Stryland, D. Williams, and W. Wolfe, Handbook for Optics, 2nd ed. (MGH, 1995).
  23. V. Privalko, Handbook for Physical Chemistry of Polymers (Naukova Dumka, 1984).
  24. H.-G. Unger, Planar Optical Waveguides and Fibres (Oxford University, 1977).
  25. O. L. Antipov, S. I. Belyaev, and A. S. Kuzhelev, “Stimulated resonant scattering of optical waves in laser crystals with population inversion,” JETP Lett. 63(1), 13–18 (1996).
    [Crossref]
  26. O. L. Antipov, S. I. Belyaev, A. S. Kuzhelev, and D. V. Chausov, “Resonant two-wave mixing of optical beams by refractive index and gain gratings in inverted Nd:YAG,” J. Opt. Soc. Am. B 15(8), 2276–2281 (1998).
    [Crossref]
  27. M. Chi, J.-P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” JOSA B 26(8), 1578–1584 (2009).
    [Crossref]
  28. S. Stepanov, A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064nm,” Opt. Express 15(14), 8832–8837 (2007).
    [Crossref] [PubMed]

2013 (3)

2012 (5)

2011 (4)

2009 (1)

M. Chi, J.-P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” JOSA B 26(8), 1578–1584 (2009).
[Crossref]

2007 (1)

1998 (2)

1996 (1)

O. L. Antipov, S. I. Belyaev, and A. S. Kuzhelev, “Stimulated resonant scattering of optical waves in laser crystals with population inversion,” JETP Lett. 63(1), 13–18 (1996).
[Crossref]

1966 (2)

V. I. Bespalov and V. I. Talanov, “About filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(12), 307–310 (1966).

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated Four-Photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17(22), 1158–1161 (1966).
[Crossref]

Alkeskjold, T. T.

Antipov, O. L.

Arkwright, J. W.

Atkins, G. R.

Belyaev, S. I.

O. L. Antipov, S. I. Belyaev, A. S. Kuzhelev, and D. V. Chausov, “Resonant two-wave mixing of optical beams by refractive index and gain gratings in inverted Nd:YAG,” J. Opt. Soc. Am. B 15(8), 2276–2281 (1998).
[Crossref]

O. L. Antipov, S. I. Belyaev, and A. S. Kuzhelev, “Stimulated resonant scattering of optical waves in laser crystals with population inversion,” JETP Lett. 63(1), 13–18 (1996).
[Crossref]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “About filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(12), 307–310 (1966).

Broeng, J.

Chausov, D. V.

Chi, M.

M. Chi, J.-P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” JOSA B 26(8), 1578–1584 (2009).
[Crossref]

Chiao, R. Y.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated Four-Photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17(22), 1158–1161 (1966).
[Crossref]

Dajani, I.

de Vries, O.

Digonnet, M. J. F.

Dong, L.

Eberhardt, R.

Eidam, T.

Elango, P.

Fotiadi, A.

Fotiadi, A. A.

Gaida, C.

Garmire, E.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated Four-Photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17(22), 1158–1161 (1966).
[Crossref]

Haarlammert, N.

Hansen, K. R.

Huignard, J.-P.

M. Chi, J.-P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” JOSA B 26(8), 1578–1584 (2009).
[Crossref]

Jansen, F.

Jauregui, C.

Kelley, P. L.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated Four-Photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17(22), 1158–1161 (1966).
[Crossref]

Kliner, A.

Kuzhelev, A. S.

O. L. Antipov, S. I. Belyaev, A. S. Kuzhelev, and D. V. Chausov, “Resonant two-wave mixing of optical beams by refractive index and gain gratings in inverted Nd:YAG,” J. Opt. Soc. Am. B 15(8), 2276–2281 (1998).
[Crossref]

O. L. Antipov, S. I. Belyaev, and A. S. Kuzhelev, “Stimulated resonant scattering of optical waves in laser crystals with population inversion,” JETP Lett. 63(1), 13–18 (1996).
[Crossref]

Kuznetsov, M. S.

Lægsgaard, J.

Liem, A.

Limpert, J.

Mégret, P.

Otto, H.-J.

Peschel, T.

Petersen, P. M.

M. Chi, J.-P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” JOSA B 26(8), 1578–1584 (2009).
[Crossref]

Robin, C.

Schmidt, O.

Schreiber, T.

Smith, A. V.

Smith, J. J.

Stepanov, S.

Stutzki, F.

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “About filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(12), 307–310 (1966).

Tünnermann, A.

Ward, B.

Whitbread, T.

Wirth, C.

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

JETP Lett. (2)

O. L. Antipov, S. I. Belyaev, and A. S. Kuzhelev, “Stimulated resonant scattering of optical waves in laser crystals with population inversion,” JETP Lett. 63(1), 13–18 (1996).
[Crossref]

V. I. Bespalov and V. I. Talanov, “About filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(12), 307–310 (1966).

JOSA B (1)

M. Chi, J.-P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” JOSA B 26(8), 1578–1584 (2009).
[Crossref]

Opt. Express (11)

S. Stepanov, A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064nm,” Opt. Express 15(14), 8832–8837 (2007).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21(13), 15168–15182 (2013).
[Crossref] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013).
[Crossref] [PubMed]

M. S. Kuznetsov, O. L. Antipov, A. A. Fotiadi, and P. Mégret, “Electronic and thermal refractive index changes in ytterbium-doped fiber amplifiers,” Opt. Express 21(19), 22374–22388 (2013).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
[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]

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20(12), 13274–13283 (2012).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20(1), 440–451 (2012).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012).
[Crossref] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated Four-Photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17(22), 1158–1161 (1966).
[Crossref]

Other (9)

S. N. Vlasov and V. I. Talanov, Wave Self-Focusing (IAP RAS, 1997).

C. Codemard, K. Yla-Jarkko, J. Singleton, P. W. Turner, I. Godfrey, S.-U. Alam, J. Nolssson, J. Sahu, and A. B. Grudinin, in Proceeding of European Conference on Optical Communication (ECOC'2002, Copenhagen, Denmark, 2002), PD1.6.

M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193 – 230.

M. Bass, E. Van Stryland, D. Williams, and W. Wolfe, Handbook for Optics, 2nd ed. (MGH, 1995).

V. Privalko, Handbook for Physical Chemistry of Polymers (Naukova Dumka, 1984).

H.-G. Unger, Planar Optical Waveguides and Fibres (Oxford University, 1977).

A. A. Fotiadi, O. L. Antipov, and P. Mégret, “Resonantly induced refractive index changes in Yb-doped fibers:the origin, properties and application for all-fiber coherent beam combining,” in Frontiers in Guided Wave Opticsand Optoelectronics, B. Pal, ed. (Intec, 2010), pp. 209–234.

V. Tyrtyshnyy, O. Vershnin, and S. Larin, “Influence of the radiation spectral parameters on the nonlinear interaction of modes in active fiber,” in Technical digests of International Symposium “High-Power Fiber Lasers and Their Applications,” (S-Petersburg, Russia, 2010), paper TuSy, p. 04.M.

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Figures (11)

Fig. 1
Fig. 1 Experimental setup.
Fig. 2
Fig. 2 Oscillograms of the fundamental mode (bright blue, from PD 1), and the higher-order mode (dark blue, from PD 2) below (a), slightly above (b), and far above (c) MI threshold.
Fig. 3
Fig. 3 Output power dependence on pump power (a), and dependence of the mode instability threshold (for output power of the signal) on input power (b). Input signal bandwidth was less than 0.05 nm, active fiber length was 6 m.
Fig. 4
Fig. 4 Mode instability threshold dependence on the input signal bandwidth (the signal input power is 50 mW).
Fig. 5
Fig. 5 Oscillogram of the fundamental mode above MI threshold.
Fig. 6
Fig. 6 Beam at the output of the active fiber below (a) and above (b) the MI threshold, and the red-light fiber images (c) at the same space scale.
Fig. 7
Fig. 7 Powers of the fundamental mode LP01 (red), anti-Stokes (blue) and Stokes (green) shifted LP11 mode and the pump inside the active fiber (violet) and the auxiliary fiber (black) on the fiber length at the time t = 2 ms after switch on of the pump with power 5 mW (the signal with power mW was switched on in 20 μs before the pump) (a), and on the time in the fiber output (b). The input ratio of the LP01 and LP11 mode power was 40, the frequency detuning Ω = 4 kHz.
Fig. 8
Fig. 8 The relative gain of the anti-Stokes shifted LP11 mode (with respect to the fundamental mode) on the frequency detuning Ω for different pump and signal powers (Pp and PS) in the fiber input, numerical aperture (NA) and the time.
Fig. 9
Fig. 9 Amplitudes of the RIGs caused by the temperature change (red) and polarizability difference (brown, violet, green) on the fiber length at the time 2 ms after signal switch on (left) and the time in the fiber output (right). The input power of the LP01 mode is 5 mW (the LP01 and LP11 mode power ratio on the input is 40), the input pump power (in the auxiliary fiber) is Pax(0) = 2.5 W and the frequency shift is Ω = 6 kHz (green and red), Pax(0) = 1.5 W and Ω = 4.25 kHz (brown), Pax(0) = 0.75 W and Ω = 4.25 kHz (violet).
Fig. 10
Fig. 10 Output power of the LP01 mode (solid lines) and the optimal-shifted LP11 mode (dashed lines) on the pump power for the input LP01-mode power 5 mW (blue and green) or 60 mW (red), and the input power ratio of the LP01 and LP11 modes 40 (for blue and red) or 200 (for green).
Fig. 11
Fig. 11 Output power of the LP01 mode (solid curves) and the optimal-shifted LP11 mode (dushed curves) on the pump power for the different longitudinal mode numbers M (a); the threshold LP mode power on the signal bandwidth (b) (the input LP01-mode power is 5 mW; the ratio of the input mode-power is 350 (a), and is varied from 40 to 103 (b).

Tables (2)

Tables Icon

Table 1 Experimental Parameters of Active Fiber, Pump and Input Signal

Tables Icon

Table 2 Fiber Parameters Used for Calculation

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E 01 = A 0 (z,t) e i(2π ν s t k 0 z) ψ 0 (r),
E 11 =( A 1 s (z,t) e iΩt+iϕ + A 1 as (z,t) e iΩtiϕ ) e i(2π ν s t k 1 z) ψ 1 (r),
A 0 z + 1 υ 0 A 0 t =i 2π n 0 λ s k 0 ( A 0 ψ 0 2 δH+ A 1 s e iΩt+iqz ψ 1 ψ 0 δH e iϕ + A 1 as e iΩt+iqz ψ 1 ψ 0 δH e iϕ ),
A 1 s z + 1 υ 1 A 1 s t =i 2π n 0 λ s k 1 ( A 1 s ψ 1 2 δH+ A 0 e iΩtiqz ψ 0 ψ 1 δH e iϕ ),
A 1 as z + 1 υ 1 A 1 as t =i 2π n 0 λ s k 1 ( A 1 as ψ 1 2 δH+ A 0 e iΩtiqz ψ 0 ψ 1 δH e iϕ ),
δH= 2π λ s ( n T )δT+ i 2 (( σ em s + σ ab s )(1+iβ)δ N ex σ ab s N d ),
N ex t + N ex τ + N ex ( σ ab p + σ em p ) P p h ν p S cl = σ ab p N d P p h ν p S cl ( σ em s + σ ab s ) I s h ν s ( N ex σ ab s N d σ em s + σ ab s ),
T t Κ 1 ρ 1 C 1p 2 T= h ν T ρ 1 C 1p N ex τ + ν p ν s ν s ( σ em s + σ ab s ) I s ρ 1 C 1p ( N ex σ ab s N d σ em s + σ ab s ),
( δT δ N ex )=( T 0 (z,r,t) δ N ex 0 (z,r,t) )+( δ T s (z,r,t) δ N ex s (z,r,t) ) e iΩt+iϕ+iqz +( δ T as (z,r,t) δ N ex as (z,r,t) ) e iΩtiϕ+iqz .
( T 00 N 00 T 11 N 11 T 01 s N 01 s T 01 as N 01 as )( T 0 (z,r,t) ψ 0 2 δ N ex 0 (z,r,t) ψ 0 2 T 0 (z,r,t) ψ 1 2 δ N ex 0 (z,r,t) ψ 1 2 δ T s (z,r,t) ψ 0 ψ 1 e iϕ δ N ex s (z,r,t) ψ 0 ψ 1 e iϕ δ T as (z,r,t) ψ 1 ψ 0 e iϕ δ N ex as (z,r,t) ψ 1 ψ 0 e iϕ ).
δ N ex 0 ψ 0,1 4 N 00,11 ψ 0,1 4 (r) ψ 0,1 2 (r) , δ N ex 0 ψ 0 2 ψ 1 2 N 00 ψ 0 2 (r) ψ 1 2 (r) ψ 0 2 (r) , δ N ex s,as ψ 0,1 3 ψ 1,0 N 01 s,as ψ 0,1 4 (r) ψ 1,0 2 (r) ψ 0 2 (r) ψ 1 2 (r) .
b 2 = u 0 2 + u 1 2 r 0 2 + 2 u 0 u 1 0 r 0 J 1 ( u 1 r/ r 0 ) J 0 ( u 0 r/ r 0 ) ( J 1 ( u 0 r/ r 0 ) J 0 ( u 1 r/ r 0 ) r J 1 ( u 1 r/ r 0 ) J 1 ( u 0 r/ r 0 ) r 0 / u 1 )dr r 0 2 0 r 0 J 1 2 ( u 1 r/ r 0 ) J 0 2 ( u 0 r/ r 0 ) rdr 13 r 0 2 .
T 01 s,as t iΩ T 01 s,as Κ 1 ρ 1 C 1p ( b 2 + q 2 ) T 01 s = ( σ em s + σ ab s )( ν p ν s ) ρ 1 C 1p ν s [ N 01 s,as ψ 0 4 ψ 1 2 | A 0 | 2 +( | A 1 s | 2 + | A 1 as | 2 ) ψ 1 4 ψ 0 2 ψ 0 2 ψ 1 2 + +( N 00 ψ 0 2 σ ab s N d σ em s + σ ab s ) ψ 0 2 ψ 1 2 ( A 0 * A 1 ,s,as + A 1 as,s* A 0 e 2iqz ) ] с n 0 8π + h ν T ρ 1 C 1p τ N 01 s,as ,
N 01 s,as t iΩ N 01 s,as + N 01 s,as τ + N 01 s,as ( σ em p + σ ab p ) P p π r 1 2 h ν p = σ em s + σ ab s h ν s c n 0 8π [ N 01 s,as ψ 0 4 ψ 1 2 | A 0 | 2 +( | A 1 s | 2 + | A 1 as | 2 ) ψ 1 4 ψ 0 2 ψ 1 2 ψ 0 2 + +( N 00 ψ 0 2 σ ab s N d σ em s + σ ab s ) ψ 1 2 ψ 0 2 ( A 0 * A 1 s,as + A 1 as,s* A 0 e 2iqz ) ],
P p z =( N d ( σ em p + σ ab p ab )+ σ em p δ N ex ) r 0 2 r 1 2 P p +γ( P ax P p ),
P ax z =γ( P ax P p ),
A 0,1 s,as (z,t)= m=M M B 0,1m s,as (z,t) e imΔt+i φ m ,
B 0m z + imΔ υ 0 B 0m = σ em s + σ ab s 2 (1+iβ)( N 00 B 0m + N 01 B 1m as ) σ ab s 2 N d B 0m i 2π λ s ( n T )( T 00 B 0m + T 01 B 1m as ),
B 1m as z + imΔ υ 1 B 1m as = σ em s + σ ab s 2 (1+iβ)( N 00 B 1m as + N 01 * B 0m ) σ ab s 2 N d B 1m as i 2π λ s ( n T )( T 00 B 1m as + T 01 * B 0m ),

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