General analysis of the mode interaction in multimode active fiber

Haibin Lü; Pu Zhou; Xiaolin Wang; Zongfu Jiang

doi:10.1364/OE.20.006456

1. Introduction

Fiber lasers have widely application potential due to their advantages in good beam quality, high optical to optical efficiency and superior thermal-optical properties. Nevertheless, with the increased output power, thermo-optical effects, surface damage and nonlinear effects will arise inevitably that constraints further power scaling of the fiber lasers. The master oscillator power amplifier (MOPA) system offers an effective solution to the problem, which boost the output power of the master oscillator by employing cascaded structure configuration. For the sake of obtaining good beam quality of the output laser beam, single-mode active fibers are always employed in the master oscillator [1–3]. Meanwhile, multimode active fibers with large mode area (LMA) have been utilized to decrease the high power density within the core in the amplifier sections [2, 3]. Therefore, the examination of transverse mode behavior is very important to get the knowledge of the gain mechanism in the multimode active fiber and how the beam quality and polarization characteristics evolve in the MOPA system. In general, the mode interaction in the multimode active fiber is resulted from various factors including local gain, temperature and bend. In the previous papers, local gain is mainly taken into account when analyzing the transverse mode interaction [4–9].To the best of our knowledge, there is hardly a comprehensive model suitable to analyze the mode interaction induced by the factors mentioned above. Therefore, a general analysis of mode interaction in the multimode active fiber is required.

In this paper, we will present a general model to investigate the mode interaction in the multimode active fiber under different factors. General coupled mode equations are deduced in the model and various factors are analyzed. Moreover, formulation of the beam quality factor is deduced in the paper. On the base of this model, the evolution of the mode power and M² factor along the fiber are investigated by numerical simulations.

2. Models and theoretical analysis

This model presented here starts with the unconjugated reciprocity theorem and describes the propagation of the light along the optical waveguide with nonuniformity that varies with distance z along the waveguide, i.e. the waveguide is no longer translationally invariant.

The perturbation method is available in the nonuniform waveguide provided that the perturbation-induced refractive index changes are slight perturbations of the uniform waveguide, and this model is based on the statement mentioned above. Hence, the description here can employ the complete set of bound eigenmodes of the unperturbed fiber and leads to a set of linear, coupled mode equations for the nonuniform waveguide.

2.1 The general coupled mode equation

Assume that the waveguide is anisotropic; hence the Maxwell equations for a source-free and anisotropic medium are:

{\begin{cases} \nabla \times \overset{⇀}{E} = i \sqrt{\frac{μ_{0}}{ε_{0}}} k \overset{⇀}{H} \\ \nabla \times \overset{⇀}{H} = - i \sqrt{\frac{ε_{0}}{μ_{0}}} k {\overset{\leftrightarrow}{n}}^{2} \overset{⇀}{E} \end{cases}, {\begin{cases} \nabla \cdot ({\overset{\leftrightarrow}{n}}^{2} \cdot \overset{⇀}{E}) = 0 \\ \nabla \cdot \overset{⇀}{H} = 0 \end{cases}

where

{\overset{\leftrightarrow}{n}}^{2} = [\begin{matrix} n_{x}^{2} & 0 & 0 \\ 0 & n_{y}^{2} & 0 \\ 0 & 0 & n_{z}^{2} \end{matrix}]

and n_x≠n_y≠n_z. Meanwhile, assume that all vector quantities in the model contain the implicit time dependence

e^{- j ω t}

, where

ω

is the angular frequency. Define a vector function

\overset{⇀}{F}

by:

\overset{⇀}{F} = \overset{⇀}{E} \times \bar{\overset{⇀}{H}} - \bar{\overset{⇀}{E}} \times \overset{⇀}{H}

where

\bar{\overset{⇀}{E}}

and

\bar{\overset{⇀}{H}}

satisfy the Maxwell equations in the same way:

{\begin{cases} \nabla \times \bar{\overset{⇀}{E}} = i \sqrt{\frac{μ_{0}}{ε_{0}}} k \bar{\overset{⇀}{H}} \\ \nabla \times \bar{\overset{⇀}{H}} = - i \sqrt{\frac{ε_{0}}{μ_{0}}} k {\bar{\overset{\leftrightarrow}{n}}}^{2} \overset{⇀}{E} \end{cases}, {\begin{cases} \nabla \cdot ({\bar{\overset{\leftrightarrow}{n}}}^{2} \cdot \overset{⇀}{E}) = 0 \\ \nabla \cdot \overset{⇀}{H} = 0 \end{cases}

where

{\bar{\overset{\leftrightarrow}{n}}}^{2} = [\begin{matrix} {\bar{n}}_{x}^{2} & 0 & 0 \\ 0 & {\bar{n}}_{y}^{2} & 0 \\ 0 & 0 & {\bar{n}}_{z}^{2} \end{matrix}]

Hence

\nabla \cdot \overset{⇀}{F} = \nabla \cdot (\overset{⇀}{E} \times \bar{\overset{⇀}{H}}) - \nabla \cdot (\bar{\overset{⇀}{E}} \times \overset{⇀}{H}) = i \sqrt{\frac{ε_{0}}{μ_{0}}} k {({\bar{\overset{\leftrightarrow}{n}}}^{2} \cdot \bar{\overset{⇀}{E}}) \cdot \overset{⇀}{E} - ({\overset{\leftrightarrow}{n}}^{2} \cdot \overset{⇀}{E}) \cdot \bar{\overset{⇀}{E}}}

According to the two-dimensional form of the divergence theorem:

\int_{A_{\infty}} \nabla \cdot \overset{⇀}{F} d A = \frac{\partial}{\partial z} (\int_{A_{\infty}} \overset{⇀}{F} \cdot \hat{z} d A)

Assume that $\bar{\overset{\leftrightarrow}{n}} (x, y, z)$ is the refractive index distribution of the unperturbed waveguide while $\overset{\leftrightarrow}{n} (x, y, z)$ represents the refractive index distribution of the slightly perturbed waveguide; hence $\overset{⇀}{E}$ and $\overset{⇀}{H}$ represent the electromagnetic fields in the perturbed waveguide. The perturbation theory suggests that the transverse components of $\overset{⇀}{E}$ and $\overset{⇀}{H}$ can be expressed as the linear combination of bound eigenmodes and radiation modes in the unperturbed waveguide [10]:

\begin{array}{l} {\overset{⇀}{E}}_{t} = \sum_{j} b_{j} (z) {\overset{⇀}{\hat{e}}}_{t j} + \sum_{j} \int_{0}^{\infty} b_{j} (z, Q) {\overset{⇀}{\hat{e}}}_{t j} (Q) d Q \\ {\overset{⇀}{H}}_{t} = \sum_{j} b_{j} (z) {\overset{⇀}{\hat{h}}}_{t j} + \sum_{j} \int_{0}^{\infty} b_{j} (z, Q) {\overset{⇀}{\hat{h}}}_{t j} (Q) d Q \end{array}

where

({\overset{⇀}{\hat{e}}}_{j}, {\overset{⇀}{\hat{h}}}_{j})

represent the electromagnetic fields of the bound eigenmodes in the unperturbed waveguide. However, transformation of bound egienmodes into radiation ones is a negligible effect for typical parameters of fiber amplifiers, as was confirmed by Beam Propagation Method (BPM) calculations [11]. On the other hand, the radiation modes experience gain much less than the bound egienmodes in the multimode active fiber. For these reasons we neglect the radiation modes in Eq. (8). Then:

\begin{array}{l} {\overset{⇀}{E}}_{t} = \sum_{j} b_{j} (z) {\overset{⇀}{\hat{e}}}_{t j} \\ {\overset{⇀}{H}}_{t} = \sum_{j} b_{j} (z) {\overset{⇀}{\hat{h}}}_{t j} \end{array}

Here the backward-propagating modes are not taken into account due to emphasis on the fiber amplifier.

From the Maxwell equations, the longitudinal component E_z in the perturbed waveguide is given by:

\begin{array}{l} E_{z} = i \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{1}{k n_{z}^{2}} (\frac{\partial H_{y}}{\partial x} - \frac{\partial H_{x}}{\partial y}) \\ H_{z} = - i \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{1}{k} (\frac{\partial E_{y}}{\partial x} - \frac{\partial E_{x}}{\partial y}) \end{array}

In the same way, due that $({\overset{⇀}{\hat{e}}}_{j}, {\overset{⇀}{\hat{h}}}_{j})$ are the bound eigenmodes in the unperturbed waveguide, so:

{\hat{e}}_{z j} = i \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{1}{k {\bar{n}}_{z}^{2}} (\frac{\partial {\hat{h}}_{y j}}{\partial x} - \frac{\partial {\hat{h}}_{x j}}{\partial y})

with

{\overset{⇀}{H}}_{t} = \sum_{j} b_{j} (z) {\overset{⇀}{\hat{h}}}_{t j}

, then:

E_{z} = i \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{1}{k n_{z}^{2}} \sum_{j} b_{j} (z) (\frac{\partial {\hat{h}}_{y j}}{\partial x} - \frac{\partial {\hat{h}}_{x j}}{\partial y}) = \frac{{\bar{n}}_{z}^{2}}{n_{z}^{2}} \sum_{j} b_{j} (z) {\hat{e}}_{z j}

In the same way, $H_{z} = \sum_{j} b_{j} (z) {\hat{h}}_{z j}$ .

We choose $(\bar{\overset{⇀}{E}}, \bar{\overset{⇀}{H}})$ as the kth backward-propagating bound eigenmodes in the unperturbed waveguide, so:

\bar{\overset{⇀}{E}} = {\overset{⇀}{\hat{e}}}_{- k} e^{- i β_{k} z}, \bar{\overset{⇀}{H}} = {\overset{⇀}{\hat{h}}}_{- k} e^{- i β_{k} z} .

Considering the property of waveguide, hence:

\begin{array}{l} {\overset{⇀}{\hat{e}}}_{- k} = {\overset{⇀}{\hat{e}}}_{t (- k)} + {\hat{e}}_{z (- k)} \hat{z} = {\overset{⇀}{\hat{e}}}_{t k} - {\hat{e}}_{z k} \hat{z} \\ {\overset{⇀}{\hat{h}}}_{- k} = {\overset{⇀}{\hat{h}}}_{t (- k)} + {\hat{h}}_{z (- k)} \hat{z} = - {\overset{⇀}{\hat{h}}}_{t k} + {\hat{h}}_{z k} \hat{z} \end{array}

with

{\overset{⇀}{E}}_{t} = \sum_{j} b_{j} (z) {\overset{⇀}{\hat{e}}}_{t j}, {\overset{⇀}{H}}_{t} = \sum_{j} b_{j} (z) {\overset{⇀}{\hat{h}}}_{t j}

, and then:

\overset{⇀}{F} \cdot \hat{z} = - {\sum_{j} b_{j} (z) ({\overset{⇀}{\hat{e}}}_{t j} \times {\overset{⇀}{\hat{h}}}_{t k} + {\overset{⇀}{\hat{e}}}_{t k} \times {\overset{⇀}{\hat{h}}}_{t j})} \cdot \hat{z} e^{- i β_{k} z}

Applying the orthogonality conditions of the bound eigenmodes in the unperturbed waveguide, so:

\int_{A_{\infty}} \overset{⇀}{F} \cdot \hat{z} d A = - 4 b_{k} (z) e^{- i β_{k} z}

Hence

\frac{\partial}{\partial z} \int_{A_{\infty}} \overset{⇀}{F} \cdot \hat{z} d A = \frac{\partial}{\partial z} {- 4 b_{k} (z) e^{- i β_{k} z}} = - 4 {\frac{d b_{k} (z)}{d z} - i β_{k} b_{k} (z)} e^{- i β_{k} z}

From Eq. (2), (3), (6), (9), (12), (13) and (14), we can obtain that:

\begin{array}{l} \int_{A_{\infty}} \nabla \cdot \overset{⇀}{F} d A = i \sqrt{\frac{ε_{0}}{μ_{0}}} k e^{- i β_{k} z} \int_{A_{\infty}} {({\bar{\overset{\leftrightarrow}{n}}}^{2} \cdot {\overset{⇀}{\hat{e}}}_{- k}) \cdot \overset{⇀}{E} - ({\overset{\leftrightarrow}{n}}^{2} \cdot \overset{⇀}{E}) \cdot {\overset{⇀}{\hat{e}}}_{- k}} d A \\ = i \sqrt{\frac{ε_{0}}{μ_{0}}} k e^{- i β_{k} z} \sum_{j} b_{j} (z) {\int_{A_{\infty}} [({\bar{n}}_{x}^{2} - n_{x}^{2}) {\hat{e}}_{x j} {\hat{e}}_{x k} + ({\bar{n}}_{y}^{2} - n_{y}^{2}) {\hat{e}}_{y j} {\hat{e}}_{y k} - \frac{{\bar{n}}_{z}^{2}}{n_{z}^{2}} ({\bar{n}}_{z}^{2} - n_{z}^{2}) {\hat{e}}_{z j} {\hat{e}}_{z k}] d A} \end{array}

Substituting Eq. (17) and (18) into Eq. (7) leads to the coupled mode equation for the kth forward-propagating bound eigenmode:

\frac{d b_{k} (z)}{d z} - i β_{k} b_{k} (z) = \sum_{j = 1}^{N} C_{k j} b_{j} (z), k = 1, 2, ... N

where the coupling coefficients are given by:

C_{k j} = \frac{i}{4} \sqrt{\frac{ε_{0}}{μ_{0}}} k \iint_{A_{\infty}} {(n_{x}^{2} - {\bar{n}}_{x}^{2}) {\hat{e}}_{x j} {\hat{e}}_{x k} + (n_{y}^{2} - {\bar{n}}_{y}^{2}) {\hat{e}}_{y j} {\hat{e}}_{y k} - \frac{{\bar{n}}_{z}^{2}}{n_{z}^{2}} (n_{z}^{2} - {\bar{n}}_{z}^{2}) {\hat{e}}_{z j} {\hat{e}}_{z k}} d A

Hence, mode interaction within the perturbed optical waveguides has been converted to a series of only z-dependent ordinary differential equations which can be solved using the forth-order Runge-Kutta method. Compared with the BPM, this model reduces the amount of calculation significantly, and describes the mode interaction intuitionisticly.

2.2 Local gain

In the active fiber, due to gain saturation, the gain distribution is nonuniform within the core of fiber. For the Tm-doped fiber laser and amplifier emitting at near 2μm, the formulation of gain in the fiber is given by [12]:

g (\overset{⇀}{r}) = \frac{τ_{1} Δ (I (\overset{⇀}{r}), P_{p})}{1 + \frac{τ_{1} λ_{s} (σ_{e} (λ_{s}) + σ_{a} (λ_{s}))}{h ν} I (\overset{⇀}{r})}

where

I (\overset{⇀}{r})

is the intensity distribution within the core in the active fiber;

τ_{1}

is the lifetime of the upper level of laser; P_p is the pump power; σ_a(λ_s) and σ_e(λ_s) is the stimulated absorption and emission cross-sections of the laser transition at the wavelength λ_s;

ν

is the speed of light in the active fiber. Due to the Cross Relaxation (CR), ∆(I(

\overset{⇀}{r}

),P_p) is a function of

I (\overset{⇀}{r})

and pump power P_p and different with that of other doped active fiber. It is worth noting that ∆(I(

\overset{⇀}{r}

),P_p) is too complicated to be listed here. As

I (\overset{⇀}{r})

increases, the gain tends to the saturation.

Nonuniformity in gain distribution leads to the interaction among different bound eigenmodes, and this impact on the mode coupling can be treated as a change of refractive index within the core. Assume that the local gain distribution is isotropic so that the impact on the refractive index can be equivalent with the same imaginary part for every orientation, and hence the effective refractive index can be given by:

{\overset{\leftrightarrow}{n}}_{e f f} = [\begin{matrix} {\bar{n}}_{0 x} - \frac{i g (\overset{⇀}{r}, I (\overset{⇀}{r}))}{2 k_{0}} & 0 & 0 \\ 0 & {\bar{n}}_{0 y} - \frac{i g (\overset{⇀}{r}, I (\overset{⇀}{r}))}{2 k_{0}} & 0 \\ 0 & 0 & {\bar{n}}_{0 z} - \frac{i g (\overset{⇀}{r}, I (\overset{⇀}{r}))}{2 k_{0}} \end{matrix}]

where k₀ is the vacuum wave number, and

{\bar{n}}_{0 x} = {\bar{n}}_{0 y} = {\bar{n}}_{0 z} = n_{0}

for the step-index fiber.

2.3 Temperature

Virtually it is available to employ the mode coupling theory to describe the thermally induced mode interaction in the waveguide provided that the thermally induced changes of refractive index can be treated as slight perturbations for the uniform waveguide. Hence we can utilize the coupled mode equations deduced in Section 2.1 to depict the thermally induced modal coupling.

Change of temperature will lead to the birefringence, and the refractive index changes can be described as [13, 14]:

{\overset{\leftrightarrow}{n}}_{e f f, T} = [\begin{matrix} {\bar{n}}_{0 x} + Δ n_{x, T} & 0 & 0 \\ 0 & {\bar{n}}_{0 y} + Δ n_{y, T} & 0 \\ 0 & 0 & {\bar{n}}_{0 z} + Δ n_{z, T} \end{matrix}]

where

\begin{array}{l} Δ n_{x, T} = Δ n_{β} (r, z) + Δ n_{S T, r} (r, z) \cos θ - Δ n_{S T, θ} (r, z) \sin θ \\ Δ n_{y, T} = Δ n_{β} (r, z) + Δ n_{S T, r} (r, z) \sin θ + Δ n_{S T, θ} (r, z) \cos θ \\ Δ n_{z, T} = Δ n_{β} (r, z) \end{array}

Where ∆n_β(r,z) represents the index change caused by temperature only; ∆n_ST,r,θ(r,z) are the index changes caused by the thermal stress. Note that all the three refractive index changes are dependent on the critical heat transfer coefficient h_c, and the concrete formulations can be found in references [13, 14].

2.4 Bend

Bends along the fiber lead to both the birefringence and spatial mode coupling. In general, the approach available to deal with the mode interaction caused by curvature in the fiber is the multi-section model [15]. In the multi-section model, the fiber in each section is assumed to be in the same plane, and the fibers in different sections have different refractive index principal axes, leading to the polarization-dependent mode coupling.

In each section, deviation of the fiber from its perfect geometry due to the curvature can be described by the change of the refractive index distribution. Assume that bends only occur in x-z plane, and then the refractive index changes of the anisotropic media in each section can be given by [16]:

n_{j} = {\bar{n}}_{j} (x - f (z), y, z), j = x, y, z

where f(z) = R(1-cos(z/R)) for the macro bends; R is the macro curvature radius of the fiber in this section. Provided that

R ≫ a

where a is radius of the core, it is available to neglect the stress-induced birefringence due to the curvature. Hence, the total changes caused by local gain, temperature and curvature can be collected and expressed as:

{\tilde{n}}_{j} = {\bar{n}}_{j} (x - f (z), y, z) + Δ n_{j, T} (x - f (z), y, z) - \frac{i g (x - f (z), y, z)}{2 k_{0}}, j = x, y, z

And the total impacts on the mode coupling within the fiber can be treated as a slight perturbation of the uniform fiber under the condition mentioned at the beginning of Section 2.

Meanwhile, bend-induced rotations of the refractive-index principal axes between the adjacent sections lead to the polarization-dependent mode coupling. Assume that the principal axes between the adjacent sections are rotated by an angle $ϑ$ , and hence the transverse components of the electric field in the frontal section relative to the principal axes of the next section are expressed by:

[\begin{matrix} E_{x}^{'} (\overset{⇀}{r}) \\ E_{y}^{'} (\overset{⇀}{r}) \end{matrix}] = [\begin{matrix} \cos ϑ & \sin ϑ \\ - \sin ϑ & \cos ϑ \end{matrix}] [\begin{matrix} E_{x} (\overset{⇀}{r}) \\ E_{y} (\overset{⇀}{r}) \end{matrix}]

where

E_{x} (\overset{⇀}{r}) = \sum_{i = 1}^{N} C_{i} {\hat{e}}_{x i} (\overset{⇀}{r}), E_{y} (\overset{⇀}{r}) = \sum_{i = 1}^{N} C_{i} {\hat{e}}_{y i} (\overset{⇀}{r})

relative to the principal axes of the frontal section.

At the junction between the adjacent sections, axis rotation leads to energy transform between the bound and leaky modes, and the remaining energy is redistributed on the bound modes within the next section. So the mode coefficients in the next section are given by:

C_{k}^{'} = \frac{1}{2} \iint_{A_{\infty}} ({\overset{⇀}{E}}_{t}^{'} ({\overset{⇀}{r}}^{'}) \times {\overset{⇀}{\hat{h}}}_{t k}^{'} ({\overset{⇀}{r}}^{'})) \cdot {\hat{z}}^{'} d x^{'} d y^{'}, k = 1, 2, ..., N

where

{\overset{⇀}{E}}_{t}^{'} ({\overset{⇀}{r}}^{'}) = {[\begin{matrix} E_{x}^{'} ({\overset{⇀}{r}}^{'}) & E_{y}^{'} ({\overset{⇀}{r}}^{'}) & 0 \end{matrix}]}^{T}

.

2.5 Beam quality factor

The beam quality factor of the laser beam emerging from the multimode active fiber can be calculated provided that mode distribution at the output end of the fiber is known. As follows, the beam factor for the step-index multimode active fiber is calculated. The results thus obtained can easily be extended to the general case.

Considering the weak-guidance approximation, there are a series of linearly polarized (LP) eigenmodes which constitute a group of complete orthogonal basis for the steady fields in the step-index fiber. Due to the degeneracy in x-y polarizations, the transverse components of the electrical field can be decomposed into the linear combination of the corresponding polarized LP eigenmodes, respectively:

\begin{array}{l} E_{x} = \sum_{j = 1}^{N} C_{x j} Ψ_{j} . \\ E_{y} = \sum_{j = 1}^{N} C_{y j} Ψ_{j} \end{array}

Where {Ψ_j,j = 1,2,…,N} represent the total LP eigenmodes in the fiber, and when j = (m,n,h)

Ψ_{L P_{m n h}} (r, θ) = A_{m n} {\begin{cases} J_{m} (u_{m n} r) \sin (m θ + h \frac{π}{2}), r \leq a \\ \frac{J_{m} (u_{m n} a)}{K_{m} (w_{m n} a)} K_{m} (w_{m n} r) \sin (m θ + h \frac{π}{2}), r > a \end{cases}

where

u_{m n} = \sqrt{n_{1}^{2} k_{0}^{2} - β_{m n}^{2}}, w_{m n} = \sqrt{β_{m n}^{2} - n_{2}^{2} k_{0}^{2}}

; n₁ and n₂ are the refractive indices of core and inner cladding, respectively; a is radius of the core. Hence, the corresponding Fourier transform of

Ψ_{L P_{m n h}} (ρ, φ)

is given by:

\begin{array}{l} Φ_{L P_{m n h}} = F {Ψ_{L P_{m n h}}} = \frac{2 π {(- i)}^{m} A_{m n} V^{2}}{a (w_{m n}^{2} + 4 π^{2} ρ^{2}) (u_{m n}^{2} - 4 π^{2} ρ^{2})} \\ {2 π ρ J_{m} (u_{m n} a) J_{m - 1} (2 π a ρ) - u_{m n} J_{m} (2 π a ρ) J_{m - 1} (u_{m n} a)} \sin (m φ + h \frac{π}{2}) \end{array}

Hence the M² factor can be calculated as follows:

\begin{array}{l} M_{x}^{2} = \frac{4 π}{p} \sqrt{\iint x^{2} I (x, y) d x d y \iint ε^{2} \tilde{I} (ε, η) d ε d η} = \\ \frac{2 π n_{0}}{Z_{0} p} {[\sum_{(m n h) (m' n' h')} Γ_{x^{2}, (m n h) (m' n' h')} (C_{x, (m n h)} C_{x, (m' n' h')}^{*} + C_{y, (m n h)} C_{y, (m' n' h')}^{*})]}^{1 / 2} \cdot \\ {[\sum_{(m n h) (m' n' h')} Γ_{ξ^{2}, (m n h) (m' n' h')} (C_{x, (m n h)} C_{x, (m' n' h')}^{*} + C_{y, (m n h)} C_{y, (m' n' h')}^{*})]}^{1 / 2} \\ M_{y}^{2} = \frac{4 π}{p} \sqrt{\iint y^{2} I (x, y) d x d y \iint η^{2} \tilde{I} (ε, η) d ε d η} = \\ \frac{2 π n_{0}}{Z_{0} p} {[\sum_{(m n h) (m' n' h')} Γ_{y^{2}, (m n h) (m' n' h')} (C_{x, (m n h)} C_{x, (m' n' h')}^{*} + C_{y, (m n h)} C_{y, (m' n' h')}^{*})]}^{1 / 2} \cdot \\ {[\sum_{(m n h) (m' n' h')} Γ_{η^{2}, (m n h) (m' n' h')} (C_{x, (m n h)} C_{x, (m' n' h')}^{*} + C_{y, (m n h)} C_{y, (m' n' h')}^{*})]}^{1 / 2} \end{array}

where

\begin{array}{l} Γ_{x^{2}, (m n h) (m' n' h')} = A_{m n} A_{m' n'}^{*} I_{(m n) (m' n')}^{x^{2}} I_{m m'}^{x^{2}, θ} \\ Γ_{ξ^{2}, (m n h) (m' n' h')} = V^{4} (^{- j) m - m^{'}} A_{m n} A_{m' n'}^{*} I_{(m n) (m' n')}^{ξ^{2}} I_{m m'}^{ξ^{2}, φ} \\ Γ_{y^{2}, (m n h) (m' n' h')} = A_{m n} A_{m' n'}^{*} I_{(m n) (m' n')}^{y^{2}} I_{m m'}^{y^{2}, θ} \\ Γ_{η^{2}, (m n h) (m' n' h')} = V^{4} (^{- j) m - m'} A_{m n} A_{m' n'}^{*} I_{(m n) (m' n')}^{η^{2}} I_{m m'}^{η^{2}, φ} \end{array}

where

\begin{array}{l} I_{(m n) (m' n')}^{x^{2}} = I_{(m n) (m' n')}^{y^{2}} = \int_{0}^{a} r^{3} J_{m} (u_{m n} r) J_{m'} (u_{m' n'} r) d r + κ \int_{a}^{\infty} r^{3} K_{m} (w_{m n} r) K_{m'} (w_{m' n'} r) d r \\ I_{(m n) (m' n')}^{ξ^{2}} = I_{(m n) (m' n')}^{η^{2}} = \frac{J_{m} (u_{m n} a) J_{m'} (u_{m' n'} a)}{4 π^{2}} \int_{0}^{\infty} \frac{r^{5} J_{m - 1} (r) J_{m' - 1} (r)}{(w_{m n}^{2} a^{2} + r^{2}) (u_{m n}^{2} a^{2} - r^{2}) (w_{m' n'}^{2} a^{2} + r^{2}) (u_{m' n'}^{2} a^{2} - r^{2})} d r - \\ \frac{u_{m' n'} a J_{m} (u_{m n} a) J_{m' - 1} (u_{m' n'} a)}{4 π^{2}} \int_{0}^{\infty} \frac{r^{4} J_{m - 1} (r) J_{m' - 1} (r)}{(w_{m n}^{2} a^{2} + r^{2}) (u_{m n}^{2} a^{2} - r^{2}) (w_{m' n'}^{2} a^{2} + r^{2}) (u_{m' n'}^{2} a^{2} - r^{2})} d r - \\ \frac{u_{m n} a J_{m - 1} (u_{m n} a) J_{m'} (u_{m' n'} a)}{4 π^{2}} \int_{0}^{\infty} \frac{r^{4} J_{m} (r) J_{m' - 1} (r)}{(w_{m n}^{2} a^{2} + r^{2}) (u_{m n}^{2} a^{2} - r^{2}) (w_{m' n'}^{2} a^{2} + r^{2}) (u_{m' n'}^{2} a^{2} - r^{2})} d r + \\ \frac{u_{m n} u_{m' n'} a^{2} J_{m - 1} (u_{m n} a) J_{m' - 1} (u_{m' n'} a)}{4 π^{2}} \int_{0}^{\infty} \frac{r^{3} J_{m} (r) J_{m'} (r)}{(w_{m n}^{2} a^{2} + r^{2}) (u_{m n}^{2} a^{2} - r^{2}) (w_{m' n'}^{2} a^{2} + r^{2}) (u_{m' n'}^{2} a^{2} - r^{2})} d r \end{array}

where

κ = \frac{J_{m} (u_{m n} a) J_{m'} (u_{m' n}' a)}{K_{m} (w_{m n} a) K_{m'} (w_{m' n'} a)}

,

p = \iint I (x, y) d x d y

;

\tilde{I} (ε, η)

is the intensity distribution of the plane of spectrum, and Z₀ is the impedance of the fiber.

And

\begin{array}{l} I_{m m'}^{x^{2}, θ} = I_{m m'}^{ξ^{2}, φ} = \int_{0}^{2 π} \cos^{2} θ \sin (m θ + h \frac{π}{2}) \sin (m' θ + h' \frac{π}{2}) d θ \\ = {\begin{cases} π \sin (h \frac{π}{2}) \sin (h' \frac{π}{2}), (m, m') = (0, 0) \\ \frac{π}{2} \cos [(h - h') \frac{π}{2}] - \frac{π}{4} \cos [(h + h') \frac{π}{2}], (m, m') = (1, 1) \\ \frac{π}{2} \cos [(h - h') \frac{π}{2}], m = m' b u t \neq 0, 1 \\ \frac{π}{2} \sin (h \frac{π}{2}) \sin (h' \frac{π}{2}), (m, m') = (0, 2) o r (2, 0) \\ \frac{π}{4} \cos [(h - h') \frac{π}{2}], {\begin{cases} m = m' - 2, m' = 3, 4, ... \\ m = m' + 2, m' = 1, 2, ... \end{cases} \\ 0, e l s e \end{cases} \end{array}

\begin{array}{l} I_{m m'}^{y^{2}, θ} = I_{m m'}^{η^{2}, φ} = \int_{0}^{2 π} \sin^{2} θ \sin (m θ + h \frac{π}{2}) \sin (m' θ + h' \frac{π}{2}) d θ \\ = {\begin{cases} π \sin (h \frac{π}{2}) \sin (h' \frac{π}{2}), (m, m') = (0, 0) \\ \frac{π}{2} \cos [(h - h') \frac{π}{2}] + \frac{π}{4} \cos [(h + h') \frac{π}{2}], (m, m') = (1, 1) \\ \frac{π}{2} \cos [(h - h') \frac{π}{2}], m = m' b u t \neq 0, 1 \\ - \frac{π}{2} \sin (h \frac{π}{2}) \sin (h' \frac{π}{2}), (m, m') = (0, 2) o r (2, 0) \\ - \frac{π}{4} \cos [(h - h') \frac{π}{2}], {\begin{cases} m = m' - 2, m' = 3, 4, ... \\ m = m' + 2, m' = 1, 2, ... \end{cases} \\ 0, e l s e \end{cases} \end{array}

3. Numerical simulation

In this section, we simulated the mode interaction in Tm-doped multimode fiber amplifiers under various factors numerically on the base of the model presented above. The laser wavelength is 1950nm. The fiber is assumed to be the double-clad fiber, and the radius of core and inner cladding are about 10μm and 200μm, respectively. The numerical aperture (NA) of the core is about 0.1. So the normalized frequency is about 3 and hence there are LP₀₁ and LP₁₁ modes in the multimode fiber while each mode has two orthogonal orientations. Considering the degeneracy of different azimuthal functions, hence there are LP_011x, LP_011y, LP_111x, LP_111y, LP_110x and LP_110y modes in the fiber. The beat length L_β^01-11 between the LP₀₁ and LP₁₁ mode is about 0.0011m. Assume that the initial signal power injected into the amplifier is 0.01W, and different input modes possess the same power ratio with respect to the total signal power.

3.1 Local gain

The mode interaction in the multimode Tm-doped active fiber with different doping distribution is investigated here. The injected pump power is assumed to be 20W. Different doping profiles are listed in Fig. 1(a) and the corresponding dependences of mode power and M² factor on the length of fiber are depicted in Fig. 1(b)-1(d).

Fig. 1 Different doping distributions and corresponding results of mode power (left column) and M² factor (right column); (a) different doping distributions; (b) Correspond to the left column in (a); (c) Correspond to the central column in (a); (d) Correspond to the right column in (a).

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As is shown in Fig. 1(a), there are three kinds of doping distributions [17]. The first two belong to flat doping where Г is the doping confinement factor defined as the ratio of the maximum gain radius to the core radius, and the third one is parabolic doping which satisfies N(r) = N(1-r²), r<1, where r is the normalized radius and N is assumed to be 2.43х10²⁶ which is the same with those of the first two.

Figure 1(b)-1(d) presents the corresponding three results of mode power and M² factor. When the doping distribution is flat doping with Г = 0.5, the LP₁₁ mode is suppressed significantly by the fundamental mode because the fundamental mode experiences larger gain than higher-order modes. As a result, the beam quality is improved with the increase of power of the fundamental mode. The same results are also presented when the doping distribution is parabolic doping. Inspired by this, various kinds of beam can be obtained provided that the corresponding doping distribution is designed reasonably.

Actually the phase difference between the LP₀₁ and LP₁₁ modes achieves self-imaging [18] within a range of the beat length. Therefore the M² factor should oscillate with a period of the beat length along the fiber. However, the oscillation of M² factor cannot be presented subtly in the Fig. 1(b)-1(d) due that the sampling step is much larger than the beat length. When the sampling step is far less than the beat length, the accurate evolution of mode power and M² factor for flat doping with Г = 1 are depicted in Fig. 2 , as well as other doping distributions.

Fig. 2 Accurate evolutions of mode power (left column) and M² factor (right column) for different doping profiles; (a) flat doping, Г = 1; (b) flat doping, Г = 0.5; (c) parabolic doping.

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As is shown in Fig. 2, energy transformation among different modes within a range of the beat length is quite weak due to the small pump power and hence the oscillation amplitude of M² factor is about 0.05 and not prominent.

3.2 Temperature

Generally speaking, the thermal effect becomes prominent as the pump power _P increases or different cooling methods are employed leading to different heat transfer coefficients h_c. Impacts of different pump powers and heat transfer coefficients on mode interaction are discussed in this section. The fiber parameters are the same with those in section 3.1 while the doping profile is the flat doping with Г = 1, and the simulated results of mode power and M² factor are shown in Fig. 3 .

Fig. 3 Results of mode power (left column) and M² factor (right column) for different pump power _P and heat transfer coefficients h_c where h_c = 10 corresponds to no cooling while h_c = 1000 water-cooling; (a) P = 20W, h_c = 1000; (b) P = 20W, h_c = 10; (c) P = 2000W, h_c = 1000; (d) P = 2000W, h_c = 10.

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As is shown in Fig. 3(a)-3(d), when water-cooling is employed, the thermal effect is suppressed effectively and the thermal-induced change of refractive index is weak so that mode interaction among different modes is not strong. However, when no cooling is employed, as the pump power increases, the perturbation of refractive index caused by the thermal effect becomes larger, and leads to stronger energy transformation among different modes, as well as the M² factor. The figures show that the period of mode interaction is about 1mm, which is equal to the beat length L_β^01-11. Because there are only two different propagation constants among the total LP modes, the phase distribution of the field achieves self-imaging after experiencing the beat length. Hence, the thermally induced refractive index change shows periodicity with a period of the beat length L_β^01-11, which can be interpreted as a series of micro wedges of higher refractive index [4]. The wedge transfers a small fraction of the light from the fundamental mode to the LP₁₁ mode. If there are more LP modes with different propagation constants within the fiber, the periodicity of the thermally induced index change will change approximately as the least common multiple of all the different beat lengths within the fiber. In order to verify the viewpoint mentioned above, we assume that there are LP₀₁, LP₁₁, LP₂₁and LP₀₂ modes within the fiber without consideration of the mode orientation. The beat lengths of any two modes are listed in Table 1 . These modes occupy the same power fraction and the other parameters remain unchanged. The corresponding results are depicted in Fig. 4 .

Table 1. The Beat Lengths of any Two Modes

View Table

Fig. 4 Results of mode power (left column) and M² factor (right column) when there are LP₀₁₁, LP₁₁₁, LP₁₁₀, LP₂₁₁, LP₂₁₀ and LP₀₂₁ modes within the fiber.

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As is shown in Table 1, the least common multiple of all the different beat lengths within the fiber is about the beat length L_β^21-02. The curve of mode power (left) in Fig. 4 depicts that the period of mode interaction caused by the thermally induced index change is about 0.0057m, which is equal to the beat length L_β^21-02, the least common multiple of all the different beat lengths within the fiber. That is, the phase distribution of the field within the fiber achieves self-imaging after propagating through a length of L_β^21-02.

3.3 Micro bends

Assume that the orientation of the micro bends within the fiber, as is shown in Fig. 5 , is parallel to LP₁₁₁ mode and satisfies f(z) = A_dsin(k’z) along the fiber where A_d is the bend amplitude and $k'$ is the spatial frequency of the micro bends. It is assumed to neglect the stress-induced birefringence due to the curvature. Therefore, without loss of generality, it is reasonable to consider the bound modes with one polarization orientation only. Impacts of different bend amplitudes and spatial frequencies of the micro bends on mode interaction are discussed in this section. The fiber parameters are the same with those in section 3.1 while the doping profile is the flat doping with Г = 1. Figure 6 presents the simulated results for different bend amplitudes with the same frequency $k^{'} = Δ β$ .

Fig. 5 Schematic illustration of the micro bends within the fiber.

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Fig. 6 Results of mode power (left column) and M² factor (right column) for different bend amplitudes but with the same spatial frequency; (a) A_d = 0.05a, k’ = ∆β; (b) A_d = 0.1a, k’ = ∆β.

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As is shown in Fig. 6(a), when the bend amplitude A_d is twentieth of the radius of core, the oscillation period of mode power and M² factor is about 0.02m, which is two-tenfold compared with the beat length. Likewise, the similar result is presented in Fig. 6(b), where A_d is a tenth of the radius of core. Moreover, by numerical simulations, the same result also applies with other bend amplitudes. It suggests that the spatial period of mode interaction within the fiber, in which energy transfer among different modes completes a cycle, cuts as the bend amplitude of micro bends increases. This phenomenon can be explained by an in-depth study on Eq. (20). Considering the micro bends within the fiber, so the $n_{x}^{2} - {\bar{n}}_{x}^{2}$ in Eq. (20) can be given by:

n_{x}^{2} - {\bar{n}}_{x}^{2} = {n_{0} (x - f (z), y, z) - \frac{i}{2 k} g (x - f (z), y, z)}^{2} - n_{0}^{2} (x, y, z)

Simplifying the equation above, then

n_{x}^{2} - {\bar{n}}_{x}^{2} \approx 2 n_{0} {n_{0} (x - f (z), y) - n_{0} (x, y)} - \frac{i}{k} g n_{0} \approx - 2 n_{0} \frac{\partial n_{0}}{\partial x} f (z) - \frac{i}{k} g n_{0}

After integrating, the z-dependent item of the first section in C_kj, as well as other sections, presents as iαf(z), where

α = - \frac{1}{4} \sqrt{\frac{ε_{0}}{μ_{0}}} k {\iint_{A_{\infty}} \frac{\partial n_{0}^{2}}{\partial x} {\hat{e}}_{x j} {\hat{e}}_{x k} d A}

Hence there should be an oscillating item exp(iαf(z)) within the mode coefficients b_j(z) and b_k(z). When f(z) = A_dsin(k’z), this item is expressed as:

\exp (i α A_{d} \sin (k' z))

which oscillates approximately with a frequency proportional to A_d. Hence the spatial period of mode interaction within the fiber cuts as the bend amplitude of micro bends increases. Because that the orientation of the micro bends is parallel to the LP₁₁₁ mode, the level of energy transfer between the LP₀₁₁ and LP₁₁₁ mode is much larger than that between the LP₀₁₁ and LP₁₁₀ mode, as well as the M² factor. Figure 7 shows the simulated results of mode power and M² factor for different spatial frequencies while maintaining the same bend amplitude.

Fig. 7 Results of mode power (left column) and M² factor (right column) for different spatial frequencies but with the same bend amplitude; (a) A_d = 0.1a, k’ = 0.5∆β; (b) A_d = 0.1a, k’ = ∆β; (c) A_d = 0.1a, k’ = 2∆β.

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It suggests that only when the spatial frequency of micro bends is equal to the beat frequency between the LP₀₁ and LP₁₁ modes, the mode interaction with each other becomes prominent and hence the M² factor oscillates in magnitude. Therefore, this conclusion can be generalized into the fiber supporting more bound modes. That is, mode interaction between a pair of modes is more prominent than that of others provided that the beat frequency between them is closest to the spatial frequency of micro bends.

4. Conclusion

In this paper, we have built a model to investigate the mode interaction in the multimode active fiber under different factors. The general coupled mode equations have been deduced in the model and various factors have been analyzed based on the general coupled mode equations. Moreover, the analytical expression of the beam quality factor has been calculated for the optical field in the multimode active fiber. On the base of this model, the evolution of the mode power and the M² factor along the fiber have been analyzed by numerical simulations and the simulation results are consistent with the physical insights well. Hence, the model in the paper can deal with the problem of mode interaction under various factors and provide instructive suggestions when designing the fiber lasers and amplifiers.

References and links

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Beat Length	L_β^01-11	L_β^01-21	L_β^01-02	L_β^11-21	L_β^11-02	L_β^21-02
Value	0.0018	0.0008	0.0007	0.0014	0.0012	0.0057
L_β^21-02/-	~3	~7	~8	~4	~5	1

General analysis of the mode interaction in multimode active fiber

Abstract

1. Introduction

2. Models and theoretical analysis

2.1 The general coupled mode equation

2.2 Local gain

2.3 Temperature

2.4 Bend

2.5 Beam quality factor

3. Numerical simulation

3.1 Local gain

3.2 Temperature

3.3 Micro bends

4. Conclusion

References and links

Cited By

Figures (7)

Tables (1)

Equations (40)

Optics Express