Theoretical analysis of mode instability in high-power fiber amplifiers

Kristian Rymann Hansen; Thomas Tanggaard Alkeskjold; Jes Broeng; Jesper Lægsgaard

doi:10.1364/OE.21.001944

1. Introduction

Recently, a phenomenon known as Mode Instability, or Transverse Mode Instability (TMI), has emerged as one of the greatest limitations on the power scalability of large mode area (LMA) ytterbium-doped fiber amplifiers [1, 2]. The phenomenon manifests itself as a temporal fluctuation, typically on a ms timescale, of the output beam profile as the output power reaches a certain threshold. Detailed experimental investigations of TMI have shown that the power and relative phase of the light in the fundamental and higher-order modes of the amplifier fluctuates on a timescale which depends strongly on the core diameter of the fiber, and that only the first higher-order mode (HOM) is involved, except for fibers with very large core diameters [3, 4]. While the onset of TMI does not in itself prevent the amplification of the signal beyond the instability threshold, the resulting dramatic decrease in beam quality renders the amplifier useless for applications that require a stable output beam.

An initial attempt to understand the physical mechanism responsible for TMI proposed that a self-induced long-period grating (LPG) could cause a transfer of power from the fundamental mode (FM) to a HOM of the fiber [5]. This LPG is induced by mode beating between the light in the FM and a small amount of light unintentionally coupled into the HOM, since the resulting spatial intensity oscillation creates a matching index grating due to the ytterbium doping in the fiber core. The ytterbium ions can cause the required refractive index perturbation either directly, since their contribution to the refractive index of the doped core depends on the population inversion, which again depends on the local intensity, or indirectly though the thermo-optic effect. In the latter case, the intensity oscillation of the signal leads to spatially varying stimulated emission, which due to the quantum defect creates a spatially oscillating temperature profile in the fiber. Since the refractive index of fused silica depends on temperature, the mode beating between the FM and the HOM again leads to a LPG, which has the correct period to couple the two modes [6]. However, it was shown in [7] that a phase lag between the intensity oscillation and the LPG must exist in order to have an efficient coupling between the modes. Using a beam propagation model, it was shown that such a phase lag can appear if the light in the HOM is slightly redshifted relative to the light in the FM and the slow response time of the thermal nonlinearity is taken into account. By seeding the model with a small amount of light in the HOM redshifted by a few kHz, it was shown that the light in the HOM would experience a large nonlinear gain. Later numerical simulations by Ward et al.[8] showed that thermally induced TMI could be initiated by a transient signal and proposed that longitudinal heat flow in the fiber is responsible for the onset of TMI. However, additional simulations by the authors, in which the longitudinal heat flow was neglected, failed to support this hypothesis.

In a recent paper, we developed a semi-analytical model of the thermal nonlinearity in fiber amplifiers and showed that quantum noise could act as a seed for TMI [9]. However, that model did not allow us to study the temporal and spectral characteristics of TMI. In this paper, we extend the method to a full coupled-mode model of thermally induced TMI. While our model makes a number of simplifying assumptions, we nonetheless believe that it explains the most important features of TMI. Our paper is organized as follows: In section 2 we present our theoretical model and discuss the approximations made. In section 3 we consider operation at the threshold and derive an approximate analytic solution of the coupled-mode equations, which is valid when the HOM is only weakly excited. Using this approximate solution, we consider seeding of TMI by quantum noise and signal intensity noise. Through a numerical solution, we investigate the temporal and spectral features of TMI near the threshold as well as the dependence of the threshold power on the spectral width of the signal. In section 4 we consider operation beyond the stability threshold, and study the temporal and spectral dynamics in this case, as well as investigate how the average HOM content of the output signal varies as power is increased. Finally, in section 5, we consider an amplitude modulated input signal and show how the thermally induced nonlinear interaction between the modes leads to the generation of additional sidebands in the spectral characteristics of the TMI phenomenon.

2. Theory

2.1. Coupled-mode equations

The quasi-monochromatic electric field of the signal propagating in the fiber is written in terms of a slowly-varying envelope

E (r, t) = \frac{1}{2} u (E (r, t) e^{i ω_{0} t} + c . c .),

where u is the polarization unit vector, E is the temporally slowly-varying field envelope and ω₀ is the carrier angular frequency of the signal. Using an analogous expression for the induced polarization, the frequency domain wave equation in the scalar approximation can be written as

\nabla^{2} E (r, ω - ω_{0}) + \frac{ω^{2}}{c^{2}} ε (r) E (r, ω - ω_{0}) = - μ_{0} ω^{2} P_{N L} (r, ω - ω_{0}),

where E(r, ω) is the Fourier transform of E(r, t), ε is the complex relative permittivity of the fiber and P_NL(r, ω) is the slowly-varying nonlinear induced polarization due to the heating of the fiber. The relative permittivity is written in terms of its real and imaginary parts as

ε (r) = ε_{f} (r_{⊥}) - i \frac{g (r) \sqrt{ε_{f} (r_{⊥})}}{k_{0}},

where ε_f is the real relative permittivity of the fiber, g is the bulk gain coefficient due to the rare-earth doping of the fiber core, k₀ = ω₀/c is the vacuum wave number and the subscript ⊥ denotes the transverse coordinates x, y. We have disregarded material dispersion of the fiber, since we shall restrict ourselves to considering signals with a linewidth of less than a few tens of kHz in this paper.

The nonlinear induced polarization is related to the change in temperature of the fiber ΔT and the electric field by

P_{N L} (r, t) = ε_{0} η Δ T (r, t) E (r, t),

where η is a thermo-optic coefficient, which relates the change in relative permittivity of the fiber to the change in temperature through

Δ ε (r, t) = η Δ t (r, t) .

Taking the Fourier transform of Eq. (4) and inserting the result into Eq. (2) yields

\nabla^{2} E (r, Ω) + k^{2} ε (r) E (r, Ω) = - \frac{η k^{2}}{2 π} \int_{- \infty}^{\infty} Δ T (r, Ω') E (r, Ω - Ω^{'}) d Ω^{'},

where Ω = ω − ω₀ and k = ω/c. The change in temperature ΔT obeys the heat equation

ρ C \frac{\partial Δ T}{\partial t} - κ \nabla_{⊥}^{2} Δ T (r, t) = Q (r, t),

where ρ is the density, C is the specific heat capacity, and κ is the thermal conductivity of the fiber material, all of which are assumed to be constant throughout the fiber cross section and independent of temperature. We have assumed that the longitudinal heat diffusion is negligible compared to the transverse heat diffusion, and hence we have omitted the z derivative part of the Laplacian in the heat equation. The heat source is due to the quantum defect associated with the gain medium, and the heat power density Q is related to the signal intensity I by

Q (r, t) = (\frac{λ_{s}}{λ_{p}} - 1) g (r) I (r, t),

where λ_s and λ_p are the signal and pump wavelengths, respectively. Fourier transforming Eq. (7) with respect to time yields

\nabla_{⊥}^{2} Δ T (r, ω) - q (ω) Δ T (r, ω) = - \frac{Q (r, ω)}{κ},

where q = iρCω/κ. Eq. (9) can be solved by an appropriate Green’s function G[10], and ΔT in the frequency domain is given by

Δ T (r, ω) = \frac{1}{κ} \iint G (r_{⊥}, r_{⊥}^{'}, ω) Q (r^{'}, ω) d^{2} r_{⊥}^{'},

where the Green’s function satisfies the differential equation

\nabla_{⊥}^{2} G (r_{⊥}, r_{⊥}^{'}, ω) - q (ω) G (r_{⊥}, r_{⊥}^{'}, ω) = - δ (r_{⊥} - r_{⊥}^{'}) .

The signal intensity is given by the slowly varying electric field as

I (r, t) = \frac{1}{2} \sqrt{ε_{f} (r_{⊥})} ε_{0} c E (r, t) E {(r, t)}^{*},

which upon insertion into Eq. (8) and taking the Fourier transform yields

Δ T (r, ω) = \frac{n_{c} ε_{0} c}{4 π κ} (\frac{λ_{s}}{λ_{p}} - 1) g (z) \iint_{S_{d}} G (r_{⊥}, r_{⊥}^{'}, ω) \int_{- \infty}^{\infty} E (r_{⊥}^{'}, ω + ω^{'}) E {(r^{'}, ω^{'})}^{*} d ω^{'} d {r^{'}}_{⊥} .

In the derivation of Eq. (13) we have assumed that the gain coefficient g is independent of time, and that it is uniform within the rare-earth doped region of the fiber core, which we denote S_d. Both these assumptions are approximations, since g is given by the population inversion of the gain medium, which depends on the signal intensity. If the signal field is composed of multiple transverse modes and frequency components, the intensity will oscillate in both space and time, and this will result in spatio-temporal oscillations in g if the signal intensity is high compared to the saturation intensity of the gain medium as shown in [5, 6]. It is clear that the minima of g will coincide with the maxima of I, and by considering Eq. (8) we see that neglecting the spatio-temporal oscillations of g leads to an overestimate of the oscillations in ΔT. Including the effects of gain saturation in our model leads to a much more complicated formalism, which is beyond the scope of this paper. Nevertheless, we shall see that our simplified model explains the major qualitative features of TMI, and also provides quantitative predictions that agree reasonably well with experiments.

To derive coupled-mode equations, we expand the electric field in a set of orthogonal transverse modes

E (r, Ω) = \sum_{m} A_{m} (z, Ω) ψ_{m} (r_{⊥}) e^{- i β_{0, m} z},

where the A_m are the slowly-varying mode amplitudes, β_0,m are the propagation constants of the modes at ω = ω₀ and the normalized mode functions satisfy the eigenvalue equation

\nabla_{⊥}^{2} ψ_{m} (r_{⊥}) + k^{2} ε_{f} (r_{⊥}) ψ_{m} (r_{⊥}) = β_{m} {(ω)}^{2} ψ_{m} (r_{⊥}),

where β_m(ω) is the propagation constant for mode m. Note that the thermal perturbation of the refractive index is neglected in the calculation of the mode functions. This approximation breaks down in rod-type fiber amplifiers with very large cores, in which the mode-field diameter decreases with increasing average power [11]. By inserting the mode expansion in Eq. (14) into Eq. (6) and Eq. (13) and invoking the paraxial approximation, we can obtain a set of coupled-mode equations for the mode amplitudes

\begin{array}{l} 2 i β_{0, n} \frac{\partial A_{n}}{\partial z} & = β_{1, n} Ω A_{n} (z, Ω) + i k_{0} \sum_{m} α_{n m} A_{m} (z, Ω) e^{i Δ β_{n m} z} + B \sum_{k l m} e^{i (Δ β_{n m} - Δ β_{k l}) z} \\ \times \int_{- \infty}^{\infty} A_{m} (z, Ω - Ω^{'}) G_{n m k l} (Ω^{'}) \int_{- \infty}^{\infty} A_{k} (z, Ω^{'} + Ω^{″}) A_{l} (z, Ω^{″}) * d Ω^{″} d Ω^{'} . \end{array}

In this expression, we have introduced the quantities Δβ_nm = β_0,n − β_0,m and the inverse group velocity β_1,n = 1/v_g,n. Terms involving the group velocity dispersion and higher-order dispersion are neglected, since we consider signals of narrow linewidth in this paper. Furthermore, we have introduced

α_{n m} = n_{c} g (z) \iint_{S_{d}} ψ_{n} {(r_{⊥})}^{*} ψ_{m} (r_{⊥}) d^{2} r_{⊥}

and

B (z) = \frac{η k_{0}^{2} n_{c} ε_{0} c}{8 π^{2} κ} (\frac{λ_{s}}{λ_{p}} - 1) g (z) .

Finally we have introduced the coupling coefficients defined by

G_{n m k l} (Ω) = \iint ψ_{n} {(r_{⊥})}^{*} ψ_{m} (r_{⊥}) \iint_{S_{d}} G (r_{⊥}, r_{⊥}^{'}, Ω) ψ_{k} (r_{⊥}^{'}) ψ_{l} {(r_{⊥}^{'})}^{*} d r_{⊥}^{'} d r_{⊥},

where the outer integral is over the entire fiber cross section. These coupling coefficients can be evaluated numerically using standard quadrature methods for any given set of modes.

Although our model can include an arbitrary number of transverse modes, the detailed experimental analysis of TMI has shown that only the FM and the first higher-order mode of LMA fibers are involved, except for fibers with very large mode areas [4]. In this paper we therefore include only the fundamental LP₀₁ mode and one of the two degenerate LP₁₁ modes of a simple step-index fiber (SIF). We also assume that the fiber is water cooled and the appropriate boundary condition for the heat equation at the fiber surface is therefore

κ \frac{\partial Δ T}{\partial r} + h_{q} Δ T = 0,

where h_q is the convection coefficient for the cooling fluid. The Green’s function G is in this case given by the expansion

G (r_{⊥}, r_{⊥}^{'}, Ω) = \frac{1}{2 π} \sum_{m = - \infty}^{\infty} g_{m} (r, r^{'}, Ω) e^{i m (ϕ - ϕ^{'})} .

Here r, ϕ are the usual cylinder coordinates and the radial Green’s functions g_m are given by

g_{n} (r, r^{'}, Ω) = {\begin{array}{l} I_{n} (\sqrt{q} r) [C_{n} I_{n} (\sqrt{q} r^{'}) + K_{n} (\sqrt{q} r^{'})] & for 0 \leq r \leq r^{'} \\ I_{n} (\sqrt{q} r^{'}) [C_{n} I_{n} (\sqrt{q} r) + K_{n} (\sqrt{q} r)] & for r^{'} \leq r \leq R \end{array},

where I_n and K_n are the modified Bessel functions of the first and second kind, respectively, q = iρCΩ/κ and the coefficients C_n are given by

C_{n} = \frac{K_{n + 1} (\sqrt{q} R) + K_{n - 1} (\sqrt{q} R) - a K_{n} (\sqrt{q} R)}{I_{n + 1} (\sqrt{q} R) + I_{n - 1} (\sqrt{q} R) + a I_{n} (\sqrt{q} R)},

with R being the outer radius of the fiber and

a = 2 h_{q} / \sqrt{q} κ

.

Introducing scaled mode amplitudes $p_{i} = \sqrt{n_{c} ε_{0} c / 2} A_{i}$ and keeping only phase-matched terms, we obtain the coupled-mode equations

\begin{array}{l} \frac{\partial p_{1}}{\partial z} = & (\frac{n_{c} Γ_{1} g}{2 n_{eff, 1}} - \frac{i Ω}{v_{g, 1}}) p_{1} (z, Ω) - i K_{1 g} \times ( \\ \int_{- \infty}^{\infty} p_{1} (z, Ω - Ω^{'}) G_{1111} (Ω^{'}) C_{11} (z, Ω^{'}) d Ω^{'} + \\ \int_{- \infty}^{\infty} p_{1} (z, Ω - Ω^{'}) G_{1122} (Ω^{'}) C_{22} (z, Ω^{'}) d Ω^{'} + \\ \int_{- \infty}^{\infty} p_{2} (z, Ω - Ω^{'}) G_{1212} (Ω^{'}) C_{12} (z, Ω^{'}) d Ω^{'}), \end{array}

\begin{array}{l} \frac{\partial p_{2}}{\partial z} = & (\frac{n_{c} Γ_{2 g}}{2 n_{eff, 2}} - \frac{i Ω}{v_{g, 2}}) p_{2} (z, Ω) - i K_{2 g} \times ( \\ \int_{- \infty}^{\infty} p_{2} (z, Ω - Ω^{'}) G_{2222} (Ω^{'}) C_{22} (z, Ω^{'}) d Ω^{'} + \\ \int_{- \infty}^{\infty} p_{2} (z, Ω - Ω^{'}) G_{2211} (Ω^{'}) C_{11} (z, Ω^{'}) d Ω^{'} + \\ \int_{- \infty}^{\infty} p_{1} (z, Ω - Ω^{'}) G_{2121} (Ω^{'}) C_{21} (z, Ω^{'}) d Ω^{'}), \end{array}

where we have introduced the effective index n_eff,n = β_0,n/k₀ and the overlap integrals

Γ_{n} = \iint_{S_{d}} ψ_{n} {(r_{⊥})}^{*} ψ_{n} (r_{⊥}) d^{2} r_{⊥} .

The constants K_n are given by

K_{n} = \frac{η (λ_{s} - λ_{p})}{4 π κ n_{eff, n} λ_{s} λ_{p}},

and C_ij are given by the correlations

C_{i j} (z, Ω) = \int_{- \infty}^{\infty} p_{i} (z, Ω + Ω^{'}) p_{j} {(z, Ω^{'})}^{*} d Ω^{'} .

The first nonlinear term on the rhs. of Eq. (24) and Eq. (25) gives rise to intra-modal effects such as self-phase modulation (SPM) and four-wave mixing (FWM). Both of these effects are governed by the real part of G_nnnn. The imaginary part of G_nnnn is responsible for a nonlinear gain on the Stokes side of the spectrum. Due to the long response time of the thermal nonlinearity, caused by the slow heat diffusion in the fiber, the spectrum of G_nnnn is extremely narrow, typically on the order of 100 Hz or less, depending on fiber design.

The second nonlinear term gives rise to a cross-phase modulation (XPM) effect between the light in the two modes, and the last term is responsible, through the imaginary part of G_nmnm, for transfer of power between the modes, as we shall see later. Interestingly, it turns out that G_nmnm has a significantly wider spectrum compared to the other Green’s function overlap integrals, typically on the order of a kHz, again depending on fiber design.

The physical mechanism responsible for the power transfer between the modes is the presence of a thermally induced LPG when light is propagating in both modes. The mode beating pattern creates a spatially and temporally varying temperature grating due to quantum defect heating, which results in the aforementioned LPG through the thermo-optic effect. Because of the long thermal diffusion time, the thermally induced LPG will be out of phase with the mode beating pattern, unless the latter is stationary. It has been argued in [7] that a phase lag between the mode beating pattern and the thermally induced LPG is required for power transfer between the modes to occur, and our coupled-mode equations should therefore have steady-state solutions. This is indeed the case, as we shall show next.

Finally, we note that by keeping only phase-matched terms in the coupled-mode equations, we have neglected the effect of a FWM interaction between the modes, which could also potentially transfer power between them. However, this effect is highly suppressed since the group-velocity dispersion is far too low to provide phase-matching for the narrow-band signals we are considering.

2.2. Steady-state solution

It is easy to see that the coupled-mode equations admit a steady-state solution given by

p_{n} (z, Ω) = 2 π \sqrt{P_{0, n}} exp (\frac{n_{c} Γ_{n}}{2 n_{eff, n}} \int_{0}^{z} g (z^{'}) d z^{'}) e^{i Φ_{n} (z)} δ (Ω),

where P_0,_n is the initial power in mode n. Φ_n is the nonlinear phase

Φ_{1} (z) = Φ_{1} (0) - [γ_{11} (P_{1} (z) - P_{0, 1}) + γ_{12} (P_{2} (z) - P_{0, 2})],

Φ_{2} (z) = Φ_{2} (0) - [γ_{22} (P_{2} (z) - P_{0, 2}) + γ_{21} (P_{1} (z) - P_{0, 1})],

with P_n being the power in mode n

P_{n} (z) = P_{0, n} exp (\frac{n_{c} Γ_{n}}{n_{eff, n}} \int_{0}^{z} g (z^{'}) d z^{'})

and the nonlinear parameters γ_nm are given by

γ_{11} = \frac{4 π^{2} K_{1} n_{eff, 1}}{n_{c} Γ_{1}} G_{1111} (0), γ_{22} = \frac{4 π^{2} K_{2} n_{eff, 2}}{n_{c} Γ_{2}} G_{2222} (0),

γ_{12} = \frac{4 π^{2} K_{1} n_{eff, 2}}{n_{c} Γ_{2}} [G_{1122} (0) + G_{1212} (0)], γ_{21} = \frac{4 π^{2} K_{2} n_{eff, 1}}{n_{c} Γ_{1}} [G_{2211} (0) + G_{2121} (0)] .

From Eq. (11) it is seen that G(r_⊥, r′_⊥, −ω) = G(r_⊥, r′_⊥, ω)^*, from which it immediately follows that the nonlinear parameters γ_nm are all real. It is therefore clear that there is no transfer of power between the modes in our steady-state solution, and the only effect of the thermal nonlinearity is to cause SPM and XPM. One might therefore naively expect that our simple model is unable to explain TMI, and that more elaborate models with fewer approximations or additional physical mechanisms are required. However, as we shall show in the following, the steady-state solution is not stable, and the presence of amplitude noise will lead to a nonlinear transfer of power between the modes.

The steady-state solution given in Eq. (29) is of course highly idealized in the sense that the bandwidth of the signal is infinitesimally small. Actual CW laser sources have a finite bandwidth, and it is therefore of interest to examine whether a steady-state solution with a finite bandwidth exists. We consider a solution which in the time domain has the form

p_{n} (z, t) = \sqrt{P_{0, n}} exp (\frac{n_{c} Γ_{n}}{2 n_{eff, n}} \int_{0}^{z} g (z^{'}) d z^{'}) e^{i [Φ_{n} (z) + θ (t)]},

where θ(t) is a stochastic phase which gives rise to a finite bandwidth of the CW field. It is easy to show that Eq. (35) is indeed a solution to the coupled-mode equations, where the deterministic phase Φ_n is given by Eqs. (30) and (31). Phase noise alone is thus not sufficient to rule out a steady-state solution, but when amplitude noise is included, it is no longer possible to find a stable steady-state solution, and we therefore expect that amplitude noise, either due to quantum fluctuations or due to intensity noise of the seed laser, is responsible for TMI.

3. Operation at threshold

In this section, we derive approximate analytical solutions of the coupled-mode equations, which are valid in cases where the average output power does not exceed the threshold for TMI. These solutions show that both quantum noise and intensity noise of the input signal can act as a seed for TMI, and thus lead to transfer of power between the FM and a HOM of the fiber. By solving the coupled-mode equations numerically, we verify the validity of the approximate analytical solutions and also study the temporal dynamics of TMI.

3.1. Quantum noise seeding

We first consider the case where a perfectly monochromatic signal is launched in the FM of the fiber amplifier, with no signal launched in the HOM. We can then show by solving Eqs. (24) and (25) to first order in p₂ that the presence of quantum noise in the HOM leads to a nonlinear transfer of power from the FM to the HOM, and that this transfer exhibits a threshold-like dependence on output power. In the following we have assumed that n_eff,1 ≈ n_eff,2 ≈ n_c and v_g,1 ≈ v_g,2, in which case the coupled-mode equations to first order in p₂ become

\frac{\partial p_{1}}{\partial z} = \frac{Γ_{1}}{2} g (z) p_{1} (z, Ω) - i K g (z) \int_{- \infty}^{\infty} p_{1} (z, Ω - Ω^{'}) G_{1111} (Ω^{'}) C_{11} (z, Ω^{'}) d Ω^{'},

\begin{array}{l} \frac{\partial p_{2}}{\partial z} & = \frac{Γ_{2}}{2} g (z) p_{2} (z, Ω) - i K g (z) (\int_{- \infty}^{\infty} p_{2} (z, Ω - Ω^{'}) G_{2211} (Ω^{'}) C_{11} (z, Ω^{'}) d Ω^{'} \\ + \int_{- \infty}^{\infty} p_{1} (z, Ω - Ω^{'}) G_{2121} (Ω^{'}) C_{21} (z, Ω^{'}) d Ω^{'}), \end{array}

where K = K₁ ≈ K₂ and the group velocity term has been transformed away by shifting to a retarded time frame. We take the solution of Eq. (36) to be the CW solution given by Eq. (29). Inserting this solution into Eq. (37) we obtain

\frac{\partial {| p_{2} |}^{2}}{\partial z} = [Γ_{2} + χ (Ω) P_{1} (z)] g (z) {| p_{2} (z, Ω) |}^{2},

where χ(Ω) = 4π²KIm[G₂₁₂₁(Ω)] and P₁(z) is given by Eq. (32). We can solve this differential equation to obtain the energy spectral density in the HOM at the output

{| p_{2} (L, Ω) |}^{2} = {| p_{2} (0, Ω) |}^{2} exp (Γ_{2} g_{a v} L) exp [\frac{χ (Ω)}{Γ_{1}} (P_{1} (L) - P_{0, 1})],

where L is the length of the fiber and the average gain coefficient is given by

g_{a v} = \frac{1}{L} \int_{0}^{L} g (z) d z .

It is clear from the solution given by Eq. (39) that any frequency components present in the HOM for which χ > 0 will experience a nonlinear gain in addition to the gain provided by the rare-earth doping. While we have assumed that no signal is launched into the HOM, quantum fluctuations of the field will always be present, and certain frequency components of this quantum noise can thus experience nonlinear gain. Writing the solution in terms of the power spectral density (PSD) S_n of mode n, we have

S_{2} (L, Ω) = S_{2} (0, Ω) exp (Γ_{2} g_{a v} L) exp [\frac{χ (Ω)}{Γ_{1}} (P_{1} (L) - P_{0, 1})] .

To model the influence of quantum noise, we use the approach in [12] and define an equivalent input PSD of the quantum noise as S₂(0, Ω) = h̄(ω₀ + Ω). The total output power in the HOM is thus given by

P_{2} (L) = exp (Γ_{2} g_{a v} L) \int_{- \infty}^{\infty} \bar{h} (ω_{0} + Ω) exp [\frac{χ (Ω)}{Γ_{1}} (P_{1} (L) - P_{0, 1})] d Ω .

As a specific example, we consider an Yb-doped SIF with a core radius R_c = 20 μm, a core refractive index n_c = 1.45 and with a V parameter of 3. This fiber thus supports the fundamental LP₀₁ mode as well as the degenerate LP₁₁ modes. Although actual double-clad fiber amplifiers have both an inner and outer cladding, we assume for simplicity that the radius of the inner cladding is sufficiently large that the index step associated with the inner/outer cladding boundary has a negligible impact on the modes guided in the core. The outer radius of the fiber R = 500 μm and the convection coefficient of the cooling fluid h_q = 1000 W/(m²K). Since we are considering an Yb-doped fiber amplifier, we take the pump wavelength to be 975 nm and the launched signal wavelength to be 1032 nm. We shall refer to this fiber amplifier as Fiber A, and the parameters are summarized in Table 1.

Table 1. Parameters of Fiber A.

View Table

To calculate the nonlinear coupling coefficient χ we insert ψ₁(r, ϕ) = R₁(r) and ψ₂(r, ϕ) = R₂(r) cosϕ into Eq. (19), where R₁ and R₂ are the radial mode functions for the LP₀₁ and LP₁₁ modes, respectively. Using the expansion for the Green’s function given in Eq. (21) we obtain an expression for G₂₁₂₁

G_{2121} (Ω) = π \int_{0}^{R} R_{1} (r) R_{2} (r) r \int_{0}^{R_{c}} R_{1} (r^{'}) R_{2} (r^{'}) g_{1} (r, r^{'}, Ω) r^{'} d r^{'} d r .

This double integral is evaluated by standard numerical quadrature methods for a range of frequencies, and the nonlinear coupling coefficient χ is calculated using η = 3.5 × 10⁻⁵ K⁻¹, κ = 1.4 W/(Km) and ρC = 1.67 × 10⁶ J/(Km³). The result, shown in Fig. 1, shows that χ is positive for negative values of Ω and has a sharp peak at Ω/2π ≈ 1 kHz. The overlap integrals given by Eq. (26) are Γ₁ ≈ 0.90 for LP₀₁ and Γ₂ ≈ 0.66 for LP₁₁.

Fig. 1 Nonlinear coupling coefficient χ for LP₀₁ − LP₁₁ coupling as a function of Ω for Fiber A.

R_c	20 μm
R	500 μm
n_c	1.45
V	3
λ_s	1032 nm
λ_p	975 nm
h_q	1000 W/(m²K)
η	3.5 × 10⁻⁵ K⁻¹
κ	1.4 W/(Km)
ρC	1.67 × 10⁶ J/(Km³)

Abstract

1. Introduction

2. Theory

2.1. Coupled-mode equations

2.2. Steady-state solution

3. Operation at threshold

3.1. Quantum noise seeding

3.2. Intensity noise seeding

3.3. LP01 - LP02 coupling

3.4. Numerical results

4. Operation beyond threshold

5. Amplitude modulated input signal

5.1. Perturbative calculation

5.2. Numerical results

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (20)

Tables (1)

Equations (58)

Optics Express

3.3. LP₀₁ - LP₀₂ coupling