Influence of a backward reflection on low-threshold mode instability in Yb<sup>3+</sup>-doped few-mode fiber amplifiers

Oleg Antipov; Maxim Kuznetsov; Dmitriy Alekseev; Valentin Tyrtyshnyy

doi:10.1364/OE.24.014871

1. Introduction

Mode instability (MI) in Yb-doped fiber amplifiers attracted great interest in the last years due to its limiting effect for the power scaling. The MI effect was referred to a nonlinear transformation of power from the intense fundamental mode LP₀₁ to the initially-weak higher-order modes [1,2]. It was shown that this effect results in the decrease of an output beam quality, randomization of temporal dynamics and damage of fiber splicer. The MI was observed in large mode area fibers, in photonic crystal fibers at pump powers ranging from several hundred watts to kilowatt, and in few-mode fibers with a relatively small core (10 μm) and high Yb dopant concentration at the pump power of just few watts [3]. 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 [1–6]. Two mechanisms of formation of RIGs caused by population inversion (due to polarizability difference, Δp, of the excited and unexcited Yb³⁺ ions) and temperature distributions induced by the modes interference field were discussed as the main reason for the MI. The thermal gratings were found to be the main mechanism of the MI in the large mode area fibers [4–6]. According to the vice version the RIGs engendered by the polarizability difference are the dominant mechanism for the low-threshold MI effect in the few-mode fibers with the 10-μm core diameter [3].

This paper is devoted to the detailed experimental investigation and theoretical modeling of the MI effect in Yb-doped few-mode fibers with core diameter of 10 um in the presence of a backward propagating wave (due to a backward reflection on the fiber end).

2. Experimental setup and measurement results

The polarization maintaining Yb-doped fiber (PANDA) was used as an amplifier of a linearly polarized CW signal at 1064 nm (Fig. 1). As the input signal, we used a radiation of an amplified single-frequency laser with the output power 1-500 mW and a bandwidth of Δλ < 0.04 nm or a fiber laser with a bandwidth varied from 0.14 to 0.5 nm. The seed radiation was injected into the 6 meters long fiber amplifier after passing through the optical isolator. The passive loss level of the active fiber was measured to be less than 20 dB/km at 1.15 µm. The experimental parameters are summarized in Table 1.

Fig. 1 Optical scheme of the experimental setup.

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

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An image of the active fiber end and the beam profile (from the “Mode Visualization Plane”) were registered using a CMOS-camera (Fig. 2(a-d)). The amplified beam became unstable when the pump power exceeded a critical level: a transformation of the fundamental mode into the LP₁₁ mode was clearly observed (compare Fig. 2(b) and 2(c,d)). The mode instability power threshold (MIPT) was determined as the maximum output power registered by power meter PM1. The MIPT was measured to be less in a perpendicular cleaved fiber amplifier (<1 W) in comparison with an angle-cleaved active fiber (~14 W). Changing the amplifier pump direction (from counter- to co-propagating to the signal) resulted in a small difference (~10%) of the MIPT at the instrumental uncertainty level.

Fig. 2 Image of the active fiber end (a), beam shape at the end of the active fiber without MI (b), and with MI (c,d). The LP₀₁-mode output power vs pump power at input signal power of 43 mW (e). Photodiode signal with MI (f).

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In the main experimental series, input and output of the few-mode active fiber were spliced to the single-mode fibers. This introduces a selectivity of the fundamental mode in the few-mode active fiber and allows the measurement of the output power without high order modes using the power meter PM1. A photodiode near the spliced area at the amplifier output was used to detect the intensity of the high order modes. The backward reflection (BR) from the output passive fiber end (with variable angled cleave) provided the power controllable backward-propagating wave, which was measured using a fiber coupler by power meter PM2. The slope efficiency of the amplifier was excellent (> 80%) below MIPT, and dropped abruptly at a power level above MIPT (Fig. 2(e)). MIPT was also registered by sharp increase of the photodiode signal and appearance of the signal oscillations with modulation depth of 20-40% (Fig. 2(f)).

MIPT dependence on the input signal power for several bandwidths is close to linear on a logarithmic scale (Fig. 3(a)). Its extrapolation shows the possibility of a significant increase of MIPT at strong input signal with wide bandwidth. However, in the experiments we had chosen the input signal parameters (power 43 mW, bandwidth 0.14 and 0.25 nm) which provide a relatively small MIPT (~10 W) to eliminate induced stress of the isolator. The strong MIPT decrease was found even at low level of BR (Fig. 3(b)). MIPT dependence on the backward reflection coefficient was also linear on the logarithmic scale.

Fig. 3 MIPT vs input signal power for different signal bandwidths (at −60 dB of BR) (a), and vs BR coefficient (b).

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3. Numerical modeling of MI in the presence of a backward-reflected wave

In order to analyze the MI, the Yb-doped polarization maintaining phosphosilicate double-clad fiber was modeled as the composite cylinder (with a small diameter comparing to the length of a fiber) consisting of an Yb³⁺-doped core, a silica-glass pump cladding and a polymer cladding with the parameters similar to the experimental ones (Tables 1 and 2). The fiber was side-pumped by an additional transporting fiber in the GT-wave configuration.

Table 2. Fiber Parameters Used for Calculation

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Proposed model assumed the existence of the fundamental mode LP₀₁ and an initially small seed of the second mode LP₁₁ in the fiber amplifier. The counter-propagating waves (in the forward direction with symbol “+”, and in the backward direction with symbol “-“) were taken into consideration. The complex amplitudes of the linear-polarized modes were described by the expressions:

E_{01}^{\pm} = A_{0}^{\pm} (z, t) ψ_{0} (r) e^{i (ω t \mp k_{0} z)},

E_{11}^{\pm} = (A_{1}^{\pm, s} (z, t) e^{- i (Ω t - φ)} + A_{1}^{\pm, a s} (z, t) e^{i (Ω t - φ)}) ψ_{1} (r) e^{i (ω t \mp k_{1} z)},

where ω and

k_{0, 1}

are the frequency and propagation constants of the modes, t is time, 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

A_{1}^{\pm, s}

and

A_{1}^{\pm, a s}

, respectively;

ψ_{0, 1}

are the radial distributions of the modes:

ψ_{0, 1} (r) = С_{0, 1} J_{0, 1} (u_{0, 1} r / r_{0}) / J_{0, 1} (u_{0, 1})

if r≤r₀, and

ψ_{0, 1} (r) = С_{0, 1} K_{0, 1} (w_{0, 1} r / r_{0}) / K_{0, 1} (w_{0, 1})

, if r>r₀, C_0,1 are the normalizing constants, J_i and K_i are the Bessel and Macdonald functions of order i, the constants

u_{0, 1}^{\pm}

and

w_{0, 1}^{\pm}

are defined by the Eqs.:

u_{0, 1}^{2} = r_{0}^{2} (n_{0}^{2} ({2 π / λ)}^{2} - k_{0, 1}^{2}), w_{0, 1}^{2} = r_{0}^{2} (k_{0, 1}^{2} - {(n_{0} - Δ n)}^{2} ({2 π / λ)}^{2})

(these parameters depend on the signal wavelength, λ), and linked by the characteristic Eqs.:

u_{0, 1} J_{1, 2} (u_{0, 1}) / J_{0, 1} (u_{0, 1}) = w_{0, 1} K_{1, 2} (w_{0, 1}) / K_{0, 1} (w_{0, 1})

[11,12]. 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:

\begin{array}{l} (\pm \frac{\partial}{\partial z} + \frac{1}{v_{0}} \frac{\partial}{\partial t}) A_{0}^{\pm} = A_{0}^{\pm} < ψ_{0}^{2} δ H_{0} > + A_{0}^{\mp} < ψ_{0}^{2} δ H_{0} e^{\pm i 2 k_{0} z} > + A_{1}^{\pm, s} < ψ_{0} ψ_{1} δ H_{0} e^{i φ \pm i q z} > +, \\ + A_{1}^{\mp, s} < ψ_{0} ψ_{1} δ H_{0} e^{i φ \pm i s z - i Ω t} > + A_{1}^{\pm, a s} < ψ_{0} ψ_{1} δ H_{0} e^{- i φ \pm i q z + i Ω t} > + A_{1}^{\mp, a s} < ψ_{0} ψ_{1} δ H_{0} e^{- i φ \pm i s z + i Ω t} > \end{array}

\begin{array}{l} (\pm \frac{\partial}{\partial z} + \frac{1}{v_{1}} \frac{\partial}{\partial t}) A_{1}^{\pm, s} = A_{1}^{\pm, s} < ψ_{1}^{2} δ H_{1} > + A_{1}^{\mp, s} < ψ_{1}^{2} δ H_{1} e^{\pm i 2 k_{1} z} > + A_{1}^{\pm, a s} < ψ_{1}^{2} δ H_{1} e^{\mp 2 i φ + 2 i Ω t} > +, \\ + A_{1}^{\mp, a s} < ψ_{1}^{2} δ H_{1} e^{\mp 2 i φ \mp i 2 k_{1} z + i 2 Ω t} > + A_{0}^{\pm} < ψ_{1} ψ_{0} δ H_{1} e^{- i φ \mp i q z + i Ω t} > + A_{0}^{\mp} < ψ_{1} ψ_{0} δ H_{1} e^{- i φ \pm i s z + i Ω t} > \end{array}

\begin{array}{l} (\pm \frac{\partial}{\partial z} + \frac{1}{v_{1}} \frac{\partial}{\partial t}) A_{1}^{\pm, a s} = A_{1}^{\pm, a s} < ψ_{1}^{2} δ H_{1} > + A_{1}^{\mp, a s} < ψ_{1}^{2} δ H_{1} e^{\pm i 2 k_{1} z} > + A_{1}^{\pm, s} < ψ_{1}^{2} δ H_{1} e^{\pm i 2 φ - 2 i Ω t} > +, \\ + A_{1}^{\mp, s} < ψ_{1}^{2} δ H_{1} e^{\mp i 2 k_{1} z \pm i 2 φ - i 2 Ω t} > + A_{0}^{\pm} < ψ_{1} ψ_{0} δ H_{1} e^{i φ \mp i q z - i Ω t} > + A_{0}^{\mp} < ψ_{1} ψ_{0} δ H_{1} e^{i φ \pm i s z - i Ω t} > \end{array}

where υ_0,1 is the mode speeds,

q = k_{0} - k_{1} (| q / k_{0} | < < 1)

,

s = k_{0} + k_{1}

, $〈 ... 〉 = \frac{1}{π r_{0}^{2}} \int_{0}^{\infty} \int_{0}^{2 π} ... r d r d ϕ$ ,

〈 ψ_{0, 1}^{2} 〉 = 1

(from the normalization),

δ H_{0, 1} = - i k_{0, 1} δ T \frac{\partial n}{\partial T} + \frac{1}{2} ((σ_{e m}^{s} + σ_{a b}^{s}) δ N_{e x} (1 + i β) - σ_{a b}^{s} N_{d}),

where δN_ex is the population change of the exited state ²F_5/2,

β = \frac{8 π^{2}}{λ_{s} n_{0}} F_{L}^{2} \frac{Δ p}{σ_{e m}^{s} + σ_{a b}^{s}}

,

F_{L} = \frac{n_{0}^{2} + 2}{3}

is the Lorentz local-field factor.

The expression (6) assumed the refractive index change due to the temperature change δT (the first summand) and the population change (the summand with the β parameter) due to Δp in the fiber core [8,9,13]. We neglected any other nonlinear effects (stimulated Brillouin and Raman scatterings, Kerr nonlinearity and others). To bolster this argument, the increment of the stimulated backward Brillouin scattering was estimated to be < 1.

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

\frac{\partial N_{e x}}{\partial t} + \frac{N_{e x}}{τ} + \frac{N_{e x} (σ_{a b}^{p} + σ_{e m}^{p}) P_{p}}{h ν_{p} S_{c l}} = \frac{σ_{a b}^{p} N_{d} P_{p}}{h ν_{p} S_{c l}} - \frac{(σ_{e m}^{s} + σ_{a b}^{s}) I_{s}}{h ν_{s}} (N_{e x} - \frac{σ_{a b}^{s} N_{d}}{σ_{e m}^{s} + σ_{a b}^{s}}),

where S_cl = πr₁², P_p and ν_p are the pump power and frequency, respectively; I_s is total signal intensity (including both intensities of all modes propagating in the both directions and their interference field).

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

\frac{\partial T}{\partial t} - \frac{Κ_{1}}{ρ_{1} C_{1 p}} \nabla^{2} T = \frac{h ν_{T}}{ρ_{1} C_{1 p}} \frac{N_{e x}}{τ} + \frac{ν_{p} - ν_{s}}{ν_{s}} \frac{(σ_{e m}^{s} + σ_{a b}^{s}) I_{s}}{ρ_{1} C_{1 p}} (N_{e x} - \frac{σ_{a b}^{s} N_{d}}{σ_{e m}^{s} + σ_{a b}^{s}}),

where

\nabla^{2}

is the Laplasian, hν_T is the energy of nonradiative transitions between sublevels of the ground state ²F_7/2 after spontaneous emission.

The full equation system was completed by equations for the pumping power in the active fiber P_p and in the auxiliary fiber P_ax (in the case of GT-wave fiber [14]):

\frac{\partial P_{p}}{\partial z} = (- N_{d} σ_{a b}^{p} + (σ_{e m}^{p} + σ_{a b}^{p}) 〈 δ N_{e x} 〉) \frac{r_{0}^{2}}{r_{1}^{2}} P_{p} + γ (P_{a x} - P_{p}), \frac{\partial P_{a x}}{\partial z} = - γ (P_{a x} - P_{p}),

where γ is the transformation coefficient of the pump from the auxiliary to the active fiber.

The boundary conditions assumed the existence of an initial signal in the fundamental mode LP₀₁ and an initially small seed of the LP₁₁ mode on the input of the fiber amplifier (z = 0), and the reflection of the modes on the output boundary (z = L): $A_{0, 1}^{-} (z = L) = \sqrt{R_{0, 1}} A_{0, 1}^{+} (z = L)$ , where R_0,1 are the reflection coefficients. In the simulations (similar to the experiments) the signal had been switched on before the pump was switched on (the signal and pump switch-on times t_s and t_p, and the delay time t_d are presented in the Table 1). The ratio of power of the LP₀₁ and LP₁₁ modes at the input of the fiber amplifier was varied from 40 to 10⁴. The initial perturbations of the temperature and population were assumed to be zero. To simplify the Eq. system the procedure of transformation of the higher-order correlators to the lower-order correlators (described in our previous paper [3]) was used. The final system of the partial derivative Eqs. (with t and z fluents) was solved using predictor-corrected method for the coordinate and time steps.

Considering the theory of the grating formation, the LP₀₁ and LP₁₁ mode interference field induces the population gratings and RIGs: the long-period RIGs are induced by interference of the co-propagating modes, and the short-period gratings are induced by the counter-propagating mode interference. Both the long-period and short-period RIGs can provide the nonlinear four-wave mixing and energy transfer, similar to the bulk media [15,16]. In the experimental conditions the signal bandwidth, Δν, was wide enough, and the signal coherence length was less than the fiber amplifier length v_0,1/Δν << 2L. Hence, the interaction of the counter-propagating waves due to the short-period RIGs was negligibly small (we neglected the short-period RIGs for the simulations).

The first calculation series were made for the medium-narrow bandwidth (v_0,1/2L << Δν < L⁻¹(1/v₀ - 1/v₁)⁻¹), when the LP₀₁ and LP₁₁ mode walk-off time on the fiber length L is less than the signal coherence time. In that case the partial derivatives with respect to time in the left side of the Eqs. (3)-(5) were neglected.

The numerical modeling showed that the LP₁₁ mode can have a bigger gain than the gain of the LP₀₁ mode (Figs. 4(a) and (b)), and this effect is stronger in comparison with the fiber amplifier without BR. The additional gain of the LP₁₁ mode is explained by the nonlinear energy transfer from the fundamental mode due to scattering on the common long-period dynamic RIGs accompanying the population gratings induced by the interference field components: $(A_{0}^{+} A_{1}^{+ a s *} + A_{0}^{- *} A_{1}^{- s}) ψ_{0} ψ_{1} e x p (- i Ω t - i q z + i φ)$ ). The positive feedback between growth of the RIG amplitude and the LP₁₁ mode amplitude is determined by an appropriate phase shift (depended on the frequency detuning, Ω). The forward propagating anti-Stokes and backward propagating Stokes components of the LP₁₁ mode have the stronger gain (indicating their cooperative contribution to RIG), than another wave pair. The RIG caused by Δp was found to grow (along the fiber amplifier and in time at the output) much stronger than the thermal RIG (Fig. 4(c) and (d)).

Fig. 4 Power of the LP₀₁ mode (green), anti-Stokes at the optimal frequency shift, Ω (blue), and Stokes (red) shifted LP₁₁ mode, the pump inside the active fiber (violet) and the auxiliary fiber (black) for forward (solid lines) and backward (dashed lines) propagating waves on the fiber length z at the time t = 10 ms (a), and on the time in the fiber output (b) (R₀ = R₁ = R = 9·10⁻², D = 10 µm, NA = 0.21). The amplitude of the Δp-enhanced (pink) and thermal (orange) RIGs on the fiber length at the time t = 10 ms (c), and on the time in the fiber output (d) (the solid lines correspond to the reflection R = 9·10⁻², the dashed lines correspond to R = 0). The pump power is 1.1 W, the input LP₀₁ mode power P₀₁⁺(z = 0) is 50 mW, the LP₀₁ and LP₁₁ mode power ratio (P₀₁⁺(0)/P₁₁⁺(0)) on the fiber input is 10³.

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In our experiments with the fiber-core diameter of ~10 μm the domination of the electronic RIG over the thermal grating can be explained by rather fast erasing of the thermal RIG due to thermal conductivity across the fiber core. The increase of the core diameter (for the fibers with 30-μm core diameter, for example) will result in an increase of the steady-state thermal grating amplitude, but the electronic grating amplitude decreases due to a gain saturation. The similar nonlinear effect of the four-wave interaction by the competing electronic and thermal RIGs accompanying the dynamic population gratings is known to exist in the bulk crystal amplifiers and “self-starting” lasers [17,18].

The LP₀₁–mode gain was found to strongly depend on the BR coefficient: an increase of the reflection coefficient resulted in an increase of the higher mode increment, and the MIPT was determined as the maximum of the LP₀₁-mode output power before roll over (Fig. 5(a)). The calculated MIPT grew with increase of the LP₀₁ mode input power and with decrease of the LP₁₁ mode seed (Fig. 5(b)). The MIPT grew also with decrease of the core diameter and NA. The relative gain of the LP₁₁ mode had a maximum vs the frequency shift Ω (~few kHz) at a low pump power. At the higher pump power this dependence had a number of maxima, in that case the output power starts to oscillate (Fig. 6(a)). The chaotic pulsations of all interacting waves were calculated at the bigger pump power (Fig. 6(b)).

Fig. 5 The output power of the LP₀₁ mode (solid lines) and the LP₁₁ mode (dashed lines) vs pump power for the different reflection coefficient R at the time t (a) (P₀₁⁺(0) = 5 mW; P₀₁⁺(0)/P₁₁⁺(0) = 40); MIPT vs logarithmic reflection coefficient for the different input power of the LP₀₁ mode and the mode power ratio (t = 10 ms) (b). NA = 0.21; D = 10 µm. The LP₁₁ mode has the optimal frequency shift.

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Fig. 6 Waveform of the output powers of the fundamental LP₀₁ mode (green), the anti-Stokes (blue) and Stokes (red) shifted LP₁₁ mode in the fiber output with the pump power P_ax(0) = 2.8W (a) and 4.3W (b); P₀₁⁺(0) = 5 mW; P₀₁⁺(0)/P₁₁⁺(0) = 10³; R = 10⁻⁴; NA = 0.21; D = 10 µm; at the optimal frequency shift, Ω = 10.5 kHz.

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For the broadband input signal when the mode walk-off time in the amplifier is more than the coherence time (Δν > L⁻¹(1/v₀ - 1/v₁)⁻¹) the partial derivative in time in Eqs. (3)-(5) has to be taken into consideration. The counter-propagating broadband LP₀₁ and LP₁₁ transverse modes were assumed to consist of a number of the longitudinal modes. Numerical calculation of Eqs. for the complex amplitudes of each longitudinal mode, 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 Δν, and the signal and pump powers. The output signal power in the LP₀₁ mode increased up to a “threshold” (Fig. 7), and the MIPT was found to grow with increase of Δν and the mode-power ratio at the fiber input.

Fig. 7 Output power of the LP₀₁ mode (solid lines) and the LP₁₁ modes (the dashed and doted lines are the anti-Stokes and Stokes components, respectively, with optimal frequency shift, Ω) vs pump power (a), and MIPT vs the signal bandwidth, Δν (b). R = 10⁻²; t = 10 ms; P₁₁⁺(0)/P₀₁⁺(0) = 10⁻³; P₀₁⁺(0) = 5 mW; D = 10 µm.

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4. Discussion

The numerical results for dependence of the MIPT on the reflection coefficient, the input signal power and the signal bandwidth are in good qualitative accord with the experimental results (compare Fig. 2(e) and Figs. 5(a) and 7(a), Fig. 3(b) and Fig. 5(b)). However, the equal reflection coefficient R₀ = R₁ corresponded only to the perpendicular cleave of the fiber (with the anti-reflecting coating, for example). Moreover, the BR coefficient of the LP₁₁ mode was negligibly small due to its illumination out of the core on the splice with the single mode fiber on the amplifier output.

The numerical simulation for different BR coefficients (R₀ ≠ R₁) showed that the reflection coefficient of the narrow-band LP₁₁ mode (R₁) doesn’t strongly affect the MIPT at the fixed R₀ coefficient. The numerical simulation of MI was also made for two counter-propagating mutually-incoherent LP₀₁ modes without any reflection (R₁ = R₀ = 0), but boundary conditions at both amplifier ends assumed the existence of the LP₀₁ mode and the LP₁₁ mode seeds. The MIPT of two mutually-incoherent counter-propagating waves was found to be the same as the MIPT in the amplifier with BR, when the BR power of the LP₀₁ mode was equal to the power of the backward-propagating mode at the amplifier boundary z = L (compare the green and violet curves on Fig. 8). The independence of the MIPT of the narrow-band signal on the LP₁₁ mode reflection coefficient R₁ is explained by the mutual RIG formation of the forward-propagating anti-Stokes and backward propagating Stokes waves (interfering with the fundamental modes) assuming that the reflection of the Stokes and anti-Stokes waves occurs independently.

Fig. 8 Output power of the LP₀₁ mode (solid lines) and the LP₁₁ modes (the dashed and doted lines are the anti-Stokes and Stokes components, respectively, with optimal frequency shift, Ω) vs pump power for the backward reflection R₁ = R₂ = 10⁻² (the violet curve), or the incoherent backward-propagating LP₀₁ mode with power of 0.6 mW (at z = L) without any reflection but with the LP₁₁ seed P₁₁^-(0)/P₀₁^-(0) = 10⁻³ (the green curve). t = 10 ms; P₁₁⁺(0)/P₀₁⁺(0) = 10⁻³; P₀₁⁺(0) = 5 mW; D = 10 μm, NA = 0.18 (in both cases).

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5. Summary

The experiments showed the strong influence of the backward reflection on the spatio-temporal instability of the fundamental mode in the few-mode Yb³⁺-doped fiber amplifier with the core diameter of 10 μm: the MIPT decreases with increase of the reflection coefficient, and decrease of the signal input power and frequency bandwidth. The numerical simulation indicates the nonlinear interaction of the fundamental LP₀₁ and higher-order LP₁₁ modes (with anti-Stokes and Stokes frequency shift) by mutual scattering (four-wave mixing) on the common long-period RIGs induced in the fiber by the interference of the counter-propagating mode pairs. The well-known results of four-wave mixing and instabilities of the counter-propagating waves in bulk media indicate that the MIPT decreases in presence of the backward propagating waves for variety of the fiber core diameters due to optical nonlinearities of different origins (electronic, thermal or strictional nonlinearities) [15,16,19,20].

Acknowledgments

This work was supported in part by the program of Russian Academy of Sciences “Novel nonlinear-optical materials, structures and methods for development of laser systems with unique characteristics”.

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Parameter	Value	Reference
Emission cross section of the amplified signal, σ^s_em, pm²	0.13	[7]
Absorption cross section of the amplified signal, σ^s_ab, pm²	0.0016	[7]
Pump absorption cross section, σ^p_ab, pm²	1.38	[7]
Emission cross section at pump wavelength, σ^p_em, pm²	1.46	[7]
Life time of the upper ²F_5/2 level, τ, ms	1.276; 1.5	[7]
Polarizability difference of Yb³⁺ ions at wavelength 1064 nm, Δp, cm³	1.3 × 10⁻²⁶	[8]
Ratio of the real and imaginary part of susceptibility at 1064 nm, β	9.8	[9]
Energy allocated to heat from the therm ²F_7/2 of Yb³⁺ ions, hν_T, cm⁻¹	740	[7]
Glass refractive index, n₀	1.47	[10]
Derivative of refractive index with respect to temperature, ∂n/∂T, K⁻¹	1.2 × 10⁻⁵	[8]
Glass density, ρ₁, g/cm³	2.2	[10]
Specific heat capacity of glass, C_p₁, cal/(g K),	0.188	[10]
Thermal conductivity of glass, K₁, W/(cm·K) Heat transfer coefficient from plastic to air, H, cal/(cm²sK)	0.014 1.2 × 10⁻⁴	[10] [8]
External plastic-clad radius, R_pl, μm	125
Core diameter, D = 2r₀, μm Core NA Signal switch-on time, t_s, μs Delay time of the pump switch on with respect to signal, t_d, μs Pump switch-on time, t_p, μs	8.5 - 10 0.17 - 0.21 10 200 5

Parameter	Value	Reference
Emission cross section of the amplified signal, σ^s_em, pm²	0.13	[7]
Absorption cross section of the amplified signal, σ^s_ab, pm²	0.0016	[7]
Pump absorption cross section, σ^p_ab, pm²	1.38	[7]
Emission cross section at pump wavelength, σ^p_em, pm²	1.46	[7]
Life time of the upper ²F_5/2 level, τ, ms	1.276; 1.5	[7]
Polarizability difference of Yb³⁺ ions at wavelength 1064 nm, Δp, cm³	1.3 × 10⁻²⁶	[8]
Ratio of the real and imaginary part of susceptibility at 1064 nm, β	9.8	[9]
Energy allocated to heat from the therm ²F_7/2 of Yb³⁺ ions, hν_T, cm⁻¹	740	[7]
Glass refractive index, n₀	1.47	[10]
Derivative of refractive index with respect to temperature, ∂n/∂T, K⁻¹	1.2 × 10⁻⁵	[8]
Glass density, ρ₁, g/cm³	2.2	[10]
Specific heat capacity of glass, C_p₁, cal/(g K),	0.188	[10]
Thermal conductivity of glass, K₁, W/(cm·K) Heat transfer coefficient from plastic to air, H, cal/(cm²sK)	0.014 1.2 × 10⁻⁴	[10] [8]
External plastic-clad radius, R_pl, μm	125
Core diameter, D = 2r₀, μm Core NA Signal switch-on time, t_s, μs Delay time of the pump switch on with respect to signal, t_d, μs Pump switch-on time, t_p, μs	8.5 - 10 0.17 - 0.21 10 200 5

Influence of a backward reflection on low-threshold mode instability in Yb³⁺-doped few-mode fiber amplifiers

Abstract

1. Introduction

2. Experimental setup and measurement results

3. Numerical modeling of MI in the presence of a backward-reflected wave

4. Discussion

5. Summary

Acknowledgments

References and links

Cited By

Figures (8)

Tables (2)

Equations (9)

Optics Express