Low-threshold mode instability in Yb<sup>3+</sup>-doped few-mode fiber amplifiers

Maxim Kuznetsov; Oleg Vershinin; Valentin Tyrtyshnyy; Oleg Antipov

doi:10.1364/OE.22.029714

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 [1–4]. 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 [1–4].

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 [4–8]. 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 [6–10]; 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 [5–7,13].

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 [16–18].

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

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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 [1–4]. 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].

Table 2. Fiber Parameters Used for Calculation

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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:

E_{01} = A_{0} (z, t) e^{i (2 π ν_{s} t - k_{0} z)} ψ_{0} (r),

E_{11} = (A_{1}^{s} (z, t) e^{- i Ω t + i ϕ} + A_{1}^{a s} (z, t) e^{i Ω t - i ϕ}) e^{i (2 π ν_{s} t - k_{1} z)} ψ_{1} (r),

where ν_s 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

A_{1}^{s}

and

A_{1}^{a s}

, 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 [16–18]); ψ₀ and ψ₁ are the radial distributions of the modes [24]:

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 and w_0,1 are defined by the Eqs.:

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

, 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})

. 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:

\frac{\partial A_{0}}{\partial z} + \frac{1}{υ_{0}} \frac{\partial A_{0}}{\partial t} = - i \frac{2 π n_{0}}{λ_{s} k_{0}} (A_{0} 〈 ψ_{0}^{2} δ H 〉 + A_{1}^{s} e^{- i Ω t + i q z} 〈 ψ_{1} ψ_{0} δ H e^{i ϕ} 〉 + A_{1}^{a s} e^{i Ω t + i q z} 〈 ψ_{1} ψ_{0} δ H e^{- i ϕ} 〉),

\frac{\partial A_{1}^{s}}{\partial z} + \frac{1}{υ_{1}} \frac{\partial A_{1}^{s}}{\partial t} = - i \frac{2 π n_{0}}{λ_{s} k_{1}} (A_{1}^{s} 〈 ψ_{1}^{2} δ H 〉 + A_{0} e^{i Ω t - i q z} 〈 ψ_{0} ψ_{1} δ H e^{- i ϕ} 〉),

\frac{\partial A_{1}^{a s}}{\partial z} + \frac{1}{υ_{1}} \frac{\partial A_{1}^{a s}}{\partial t} = - i \frac{2 π n_{0}}{λ_{s} k_{1}} (A_{1}^{a s} 〈 ψ_{1}^{2} δ H 〉 + A_{0} e^{- i Ω t - i q z} 〈 ψ_{0} ψ_{1} δ H e^{i ϕ} 〉),

where υ_0,1 is the mode speeds,

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

where δN_ex is the population 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 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.:

\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 the mode intensities 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 nonradioactive transitions between sublevels of the ground state ²F_7/2 after spontaneous emission.

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

(\begin{matrix} δ T \\ δ N_{e x} \end{matrix}) = (\begin{matrix} T_{0} (z, r, t) \\ δ N_{e x}^{0} (z, r, t) \end{matrix}) + (\begin{matrix} δ T^{s} (z, r, t) \\ δ N_{e x}^{s} (z, r, t) \end{matrix}) e^{- i Ω t + i ϕ + i q z} + (\begin{matrix} δ T^{a s} (z, r, t) \\ δ N_{e x}^{a s} (z, r, t) \end{matrix}) e^{i Ω t - i ϕ + i q z} .

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

(\begin{matrix} \begin{matrix} T_{00} & N_{00} \end{matrix} \\ \begin{matrix} T_{11} & N_{11} \end{matrix} \\ \begin{matrix} T_{01}^{s} & N_{01}^{s} \end{matrix} \\ \begin{matrix} T_{01}^{a s} & N_{01}^{a s} \end{matrix} \end{matrix}) \equiv (\begin{matrix} \begin{matrix} 〈 T_{0} (z, r, t) ψ_{0}^{2} 〉 & 〈 δ N_{e x}^{0} (z, r, t) ψ_{0}^{2} 〉 \end{matrix} \\ \begin{matrix} 〈 T_{0} (z, r, t) ψ_{1}^{2} 〉 & 〈 δ N_{e x}^{0} (z, r, t) ψ_{1}^{2} 〉 \end{matrix} \\ \begin{matrix} 〈 δ T^{s} (z, r, t) ψ_{0} ψ_{1} e^{- i ϕ} 〉 & 〈 δ N_{e x}^{s} (z, r, t) ψ_{0} ψ_{1} e^{- i ϕ} 〉 \end{matrix} \\ \begin{matrix} 〈 δ T^{a s} (z, r, t) ψ_{1} ψ_{0} e^{i ϕ} 〉 & 〈 δ N_{e x}^{a s} (z, r, t) ψ_{1} ψ_{0} e^{i ϕ} 〉 \end{matrix} \end{matrix}) .

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:

\begin{array}{l} 〈 δ N_{e x}^{0} ψ_{0, 1}^{4} 〉 \approx N_{00, 11} \frac{〈 ψ_{0, 1}^{4} (r) 〉}{〈 ψ_{0, 1}^{2} (r) 〉}, 〈 δ N_{e x}^{0} ψ_{0}^{2} ψ_{1}^{2} 〉 \approx N_{00} \frac{〈 ψ_{0}^{2} (r) ψ_{1}^{2} (r) 〉}{〈 ψ_{0}^{2} (r) 〉}, \\ 〈 δ N_{e x}^{s, a s} ψ_{0, 1}^{3} ψ_{1, 0} 〉 \approx N_{01}^{s, a s} \frac{〈 ψ_{0, 1}^{4} (r) ψ_{1, 0}^{2} (r) 〉}{〈 ψ_{0}^{2} (r) ψ_{1}^{2} (r) 〉} . \end{array}

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 $δ T^{s,as} ~ ψ_{1} ψ_{0} e^{\pm i ϕ \mp i q z} ~ J_{1} (u_{1} r / r_{0}) J_{0} (u_{0} r / r_{0}) e^{\pm i ϕ \mp i q z}$ ). These reductions for the thermal gratings were as follows: $〈 ψ_{0} (r) ψ_{1} (r) e^{\mp i ϕ \pm i q z} \nabla^{2} (δ T^{s, a s} (z, r, t)) 〉 \approx - (b^{2} + q^{2}) T_{01}^{s . a s} (z, t)$ _, where constant b₁ was estimated by the following expression:

b^{2} = \frac{u_{0}^{2} + u_{1}^{2}}{r_{0}^{2}} + \frac{2 u_{0} u_{1} \int_{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}) d r}{r_{0}^{2} \int_{0}^{r_{0}} J_{1}^{2} (u_{1} r / r_{0}) J_{0}^{2} (u_{0} r / r_{0}) r d r} \approx \frac{13}{r_{0}^{2}} .

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:

\begin{array}{l} \frac{\partial T_{01}^{s, a s}}{\partial t} \mp i Ω T_{01}^{s, a s} - \frac{Κ_{1}}{ρ_{1} C_{1 p}} (b^{2} + q^{2}) T_{01}^{s} = \frac{(σ_{e m}^{s} + σ_{a b}^{s}) (ν_{p} - ν_{s})}{ρ_{1} C_{1 p} ν_{s}} [N_{01}^{s, a s} \frac{〈 ψ_{0}^{4} ψ_{1}^{2} 〉 {| A_{0} |}^{2} + ({| A_{1}^{s} |}^{2} + {| A_{1}^{a s} |}^{2}) 〈 ψ_{1}^{4} ψ_{0}^{2} 〉}{〈 ψ_{0}^{2} ψ_{1}^{2} 〉} + \\ + (\frac{N_{00}}{〈 ψ_{0}^{2} 〉} - \frac{σ_{a b}^{s} N_{d}}{σ_{e m}^{s} + σ_{a b}^{s}}) 〈 ψ_{0}^{2} ψ_{1}^{2} 〉 (A_{0}^{*} A_{1}^{, s, a s} + A_{1}^{a s, s *} A_{0} e^{\mp 2 i q z})] \frac{с n_{0}}{8 π} + \frac{h ν_{T}}{ρ_{1} C_{1 p} τ} N_{01}^{s, a s}, \end{array}

\begin{array}{l} \frac{\partial N_{01}^{s, a s}}{\partial t} \mp i Ω N_{01}^{s, a s} + \frac{N_{01}^{s, a s}}{τ} + \frac{N_{01}^{s, a s} (σ_{e m}^{p} + σ_{a b}^{p}) P_{p}}{π r_{1}^{2} h ν_{p}} = \frac{σ_{e m}^{s} + σ_{a b}^{s}}{h ν_{s}} \frac{c n_{0}}{8 π} [N_{01}^{s, a s} \frac{〈 ψ_{0}^{4} ψ_{1}^{2} 〉 {| A_{0} |}^{2} + ({| A_{1}^{s} |}^{2} + {| A_{1}^{a s} |}^{2}) 〈 ψ_{1}^{4} ψ_{0}^{2} 〉}{〈 ψ_{1}^{2} ψ_{0}^{2} 〉} + \\ + (\frac{N_{00}}{〈 ψ_{0}^{2} 〉} - \frac{σ_{a b}^{s} N_{d}}{σ_{e m}^{s} + σ_{a b}^{s}}) 〈 ψ_{1}^{2} ψ_{0}^{2} 〉 (A_{0}^{*} A_{1}^{s, a s} + A_{1}^{a s, s *} A_{0} e^{\mp 2 i q z})], \end{array}

where c is the light speed in vacuum.

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):

\frac{\partial P_{p}}{\partial z} = (- N_{d} (σ_{e m}^{p} + σ_{a b}^{p}_{a b}) + σ_{e m}^{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.

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: $(1 / υ_{0} - 1 / υ_{1}) L < < Δ ν_{s}^{- 1}$ (17), where Δν_s 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 ( $P_{11}^{a s} (z = l, t_{s t}) / P_{11}^{a s} (z = 0, t_{s t})$ ) to the LP₀₁ mode amplification ( $P_{01} (z = l, t_{s t}) / P_{01} (z = 0, t_{s t})$ ) at a time of gain stabilizing t_st = 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 (P_p and P_S) 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 ( $δ N_{e x}^{s, a s}$ ) 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 [5–7]: 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 [25–27], 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 P_ax(0) = 2.5 W and the frequency shift is Ω = 6 kHz (green and red), P_ax(0) = 1.5 W and Ω = 4.25 kHz (brown), P_ax(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:

A_{0, 1}^{s, a s} (z, t) = \sum_{m = - M}^{M} B_{0, 1 m}^{s, a s} (z, t) e^{i m Δ t + i φ_{m}},

where

B_{0, 1 m}^{s, a s}

are the complex amplitudes, Δ is a intermode frequency interval, ϕ_m 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 >> τ_eff^-¹, τ_ther⁻¹, where τ_eff is the effective lifetime of the excited level, τ_ther 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):

\frac{\partial B_{0 m}}{\partial z} + \frac{i m Δ}{υ_{0}} B_{0 m} = \frac{σ_{e m}^{s} + σ_{a b}^{s}}{2} (1 + i β) (N_{00} B_{0 m} + N_{01}^{} B_{1 m}^{a s}) - \frac{σ_{a b}^{s}}{2} N_{d} B_{0 m} - i \frac{2 π}{λ_{s}} (\frac{\partial n}{\partial T}) (T_{00} B_{0 m} + T_{01} B_{1 m}^{a s}),

\frac{\partial B_{1 m}^{a s}}{\partial z} + \frac{i m Δ}{υ_{1}} B_{1 m}^{a s} = \frac{σ_{e m}^{s} + σ_{a b}^{s}}{2} (1 + i β) (N_{00} B_{1 m}^{a s} + N_{01}^{*} B_{0 m}) - \frac{σ_{a b}^{s}}{2} N_{d} B_{1 m}^{a s} - i \frac{2 π}{λ_{s}} (\frac{\partial n}{\partial T}) (T_{00} B_{1 m}^{a s} + T_{01}^{*} B_{0 m}),

where the population and temperature components N_ij and T_ij depended on the average intensity

< I_{Σ} > = \sum_{m = - M}^{M} ({| B_{o m} |}^{2} + {| B_{1 m}^{a s} |}^{2})

, and the interference field in the form

\sum_{m = - M}^{M} B {}_{o m}B_{1 m}^{a s *}

.

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 Δν_s = 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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Parameter	Value	Reference
Emission cross section of the amplified signal, $σ_{e m}^{s}$ , pm²	0.13	[20]
Absorption cross section of the amplified signal, $σ_{a b}^{s}$ , pm²	0.0016	[20]
Pump absorption cross section, $σ_{a b}^{p}$ , pm²	1.38	[20]
Emission cross section at pump wavelength, $σ_{e m}^{p}$ , pm²	1.46	[20]
Life time of the upper ²F_5/2 level, τ, ms	1.276	[20]
Polarizability difference of Yb³⁺ ions at wavelength 1064 nm, Δp, cm³	1.3 × 10⁻²⁶	[15]
Ratio of the real and imaginary part of susceptibility at 1064 nm, β	9.8	[21]
Energy allocated to heat from the therm ²F_7/2 of Yb³⁺ ions, hν_T, cm⁻¹	740	[20]
Glass refractive index, n₀	1.47	[22]
Derivative of refractive index with respect to temperature, ∂n/∂T, K⁻¹	1.2 × 10⁻⁵	[15]
Glass density, ρ₁, g/cm³	2.2	[22]
Specific heat capacity of glass, C_p₁, cal/(g K),	0.188	[22]
Thermal conductivity of glass, K₁, W/(cm·K)	0.014	[22]
Plastic density, ρ₂, g/cm³	1.19	[22]
Specific heat capacity of plastic, C_p₂, cal/(g K),	0.3	[23]
Thermal conductivity of plastic, K₂, W/(cm·K)	2 × 10⁻³	[23]
External plastic-clad radius, R, μm	125
Heat transfer coefficient from plastic to air, H, cal/(cm²sK)	1.2 × 10⁻⁴

Parameter	Value	Reference
Emission cross section of the amplified signal, $σ_{e m}^{s}$ , pm²	0.13	[20]
Absorption cross section of the amplified signal, $σ_{a b}^{s}$ , pm²	0.0016	[20]
Pump absorption cross section, $σ_{a b}^{p}$ , pm²	1.38	[20]
Emission cross section at pump wavelength, $σ_{e m}^{p}$ , pm²	1.46	[20]
Life time of the upper ²F_5/2 level, τ, ms	1.276	[20]
Polarizability difference of Yb³⁺ ions at wavelength 1064 nm, Δp, cm³	1.3 × 10⁻²⁶	[15]
Ratio of the real and imaginary part of susceptibility at 1064 nm, β	9.8	[21]
Energy allocated to heat from the therm ²F_7/2 of Yb³⁺ ions, hν_T, cm⁻¹	740	[20]
Glass refractive index, n₀	1.47	[22]
Derivative of refractive index with respect to temperature, ∂n/∂T, K⁻¹	1.2 × 10⁻⁵	[15]
Glass density, ρ₁, g/cm³	2.2	[22]
Specific heat capacity of glass, C_p₁, cal/(g K),	0.188	[22]
Thermal conductivity of glass, K₁, W/(cm·K)	0.014	[22]
Plastic density, ρ₂, g/cm³	1.19	[22]
Specific heat capacity of plastic, C_p₂, cal/(g K),	0.3	[23]
Thermal conductivity of plastic, K₂, W/(cm·K)	2 × 10⁻³	[23]
External plastic-clad radius, R, μm	125
Heat transfer coefficient from plastic to air, H, cal/(cm²sK)	1.2 × 10⁻⁴

Low-threshold mode instability in Yb³⁺-doped few-mode fiber amplifiers

Abstract

1. Introduction

2. Experimental setup, and measurement results

3. Theoretical model of MI

4. Modeling results for narrow signal bandwidth

5. Modeling results of the broadband signal instability

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Tables (2)

Equations (19)

Optics Express