Theoretical study of mode evolution in active long tapered multimode fiber

Chen Shi; Xiaolin Wang; Pu Zhou; Xiaojun Xu; Qisheng Lu

doi:10.1364/OE.24.019473

1. Introduction

For the advantage of high slope efficiency, good beam quality, superior thermal management property and compactness, fiber lasers are widely used in the field of laser marking, material processing, medical, communication and many other industrial applications [1]. Compared with traditional fiber laser oscillator, main oscillator power amplifier (MOPA) system, which can boost the progress of fiber laser power scaling, offers an effective way to acquire high power fiber laser source with excellent beam quality by employing cascaded structure. In amplification stage of MOPA system, large mode area (LMA) fiber has been utilized to decrease the power density inside the fiber core in order to eliminate the possible nonlinear effects which can degrade, even destroy the whole system. In comparison with uniformed LMA active fiber, tapered double clad fiber (T-DCF) shows numerous unique advantages when being employed as gain medium of optical amplifier, such as larger mode area, higher pump absorption, suppression to nonlinear effects, maintaining good beam quality and so on [2]. There are a number of applications which using T-DCF as gain medium of fiber oscillator, fiber amplifier in both CW and pulsed regime have been reported [3–10] and theoretical analysis mainly based on ray theory is usually employed [11, 12]. Other than uniformed LMA active fiber, there is hardly a concise and effective model to describe mode evolution behavior in long tapered active fiber. Therefore, the theoretical analysis of mode evolution in long tapered active fiber is required.

In this manuscript, we will present a model based on coupled mode theory to describe mode evolution in long tapered multimode fiber with respect to different factors. The coupled mode equation which contains different factors will be given in section 2. In section 3, the mode evolution of long tapered active fiber will be numerically investigated and discussed based on the model.

2. Models and theoretical analysis

In this section, we will present the model that we used in our simulation to describe the propagation of field along the optical fiber with non-uniformity that varies with distance z along the waveguide. It is worth noting that weak-guidance approximation is employed when acquiring the final result, which is valid in weakly guiding fiber.

In long tapered fiber, there are two major factors that will lead to mode coupling, the variation of fiber core radius and the slight-perturbation-induced refractive index change. In order to employ the local complete bound eigenmodes of unperturbed and uniform fiber, we have to partition the whole fiber into pieces of sections as shown in Fig. 1. Because of that the radius of long tapered fiber is varying very slowly, we can consider that local eigenmodes inside each section will remain unchanged, and the total field in previous section will excite mode field in next section at junction of adjacent sections. Based on the statements above, a set of coupled mode equations for long tapered fibers were deduced.

Fig. 1 Illustration of long tapered fiber partitioning.

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2.1 Mode coupling induced by radius variation

Because of the orthogonality of eigenmodes, the total field inside the fiber can be expressed as the superposition of all eigenmodes in fiber core, i.e [13].

E (r, φ, z) = \sum_{j} b_{j} (z) {\hat{e}}_{j} (r, φ, z) + radiation modes

where

{\hat{e}}_{j}

is the j-th normalized bound eigenmodes in position z and b_j is complex amplitude which describes both amplitude and phase of corresponding mode. However, because field transformation of bound eigenmodes into radiation ones is a negligible effect for typical fiber parameters, we will ignore radiation modes in the rest of our discussion.

Considering about the properties of waveguide [14], the transverse field of waveguide can be expressed as

{\begin{matrix} E_{t} = \sum_{j} (b_{j} + b_{- j}) {\hat{e}}_{t j} & (a) \\ H_{t} = \sum_{j} (b_{j} - b_{- j}) {\hat{h}}_{t j} & (b) \end{matrix}

According to Maxwell’s equations, the total transverse field will satisfy the following relations

{\begin{matrix} E_{t} = \frac{1}{k_{0} n^{2}} [i \sqrt{\frac{μ_{0}}{ε_{0}}} \hat{z} \times \frac{\partial H_{t}}{\partial z} + \frac{1}{k_{0}} \nabla_{t} \times (\nabla_{t} \times E_{t})] & (a) \\ H_{t} = - \frac{1}{k_{0}} [i \sqrt{\frac{ε_{0}}{μ_{0}}} \hat{z} \times \frac{\partial E_{t}}{\partial z} - \frac{1}{k_{0}} \nabla_{t} \times (\frac{\nabla_{t} \times H_{t}}{n^{2}})] & (b) \end{matrix}

while transverse component of each eigenmode satisfies

{\begin{matrix} {\hat{e}}_{t j} = - \frac{1}{k_{0} n^{2}} [\sqrt{\frac{μ_{0}}{ε_{0}}} β_{j} \hat{z} \times {\hat{h}}_{t j} - \frac{1}{k_{0}} \nabla_{t} \times (\nabla_{t} \times {\hat{e}}_{t j})] & (a) \\ {\hat{h}}_{t j} = \frac{1}{k_{0}} [\sqrt{\frac{ε_{0}}{μ_{0}}} β_{j} \hat{z} \times {\hat{e}}_{t j} + \frac{1}{k_{0}} \nabla_{t} \times (\frac{\nabla_{t} \times {\hat{h}}_{t j}}{n^{2}})] & (b) \end{matrix}

where β_j is the propagating constant of j-th mode and k₀ is the vacuum wave number of the field. Using (2) and put (4) into (3), we will get

{\begin{matrix} \sum_{j} [(b_{j} - b_{- j}) \hat{z} \times \frac{\partial {\hat{h}}_{t j}}{\partial z} - i β_{j} (b_{j} + b_{- j}) \hat{z} \times {\hat{h}}_{t j} + (\frac{d b_{j}}{d z} - \frac{d b_{- j}}{d z}) \hat{z} \times {\hat{h}}_{t j}] = 0 & (a) \\ \sum_{j} [(b_{j} + b_{- j}) \hat{z} \times \frac{\partial {\hat{e}}_{t j}}{\partial z} - i β_{j} (b_{j} - b_{- j}) \hat{z} \times {\hat{e}}_{t j} + (\frac{d b_{j}}{d z} + \frac{d b_{- j}}{d z}) \hat{z} \times {\hat{e}}_{t j}] = 0 & (b) \end{matrix}

Dot multiply (5)(a) with

{\hat{e}}_{t k}

, (5)(b) with

{\hat{h}}_{t k}

and integrate over the infinite cross section. By using the orthogonality and assuming the waveguide is non-absorbing, we can obtain the coupled mode equation for local eigenmodes

\frac{d b_{j}}{d z} - i β_{j} z = \sum_{k} C_{j k} b_{k}

where the coupling coefficients are given by

C_{j k} = \frac{1}{4} \int_{A_{\infty}} ({\hat{h}}_{j} \times \frac{\partial {\hat{e}}_{k}}{\partial z} - {\hat{e}}_{j} \times \frac{\partial {\hat{h}}_{k}}{\partial z}) \cdot \hat{z} d A, j \neq k

Here we throw the subscript t because only transverse components of the field contribute to the surface integration we have done in previous step. Considering about the weak-guidance approximation, there are a series of linearly polarized (LP) modes that constitute a set of complete orthogonal basis for field of weakly guiding fiber. We denote the normalized electric field of j-th LP eigenmode as

{\hat{ψ}}_{j}

. By applying following approximation [15]

{\hat{e}}_{j} ≅ {\hat{ψ}}_{j}, {\hat{h}}_{j} ≅ \frac{n_{c o}}{Z_{0}} \hat{z} \times {\hat{ψ}}_{j}

where

Z_{0} = \sqrt{μ_{0} / ε_{0}} \approx 376.73 Ω

is the vacuum wave impedence and n_co is the refractive index of fiber core. Then the coupling coefficients in (7) can be expressed in weak-guidance approximation

C_{j k} = - \frac{n_{c o}}{2 Z_{0}} \int_{A_{\infty}} {\hat{ψ}}_{j} \frac{\partial {\hat{ψ}}_{k}}{\partial z} d A

For each single LP eigenmode, it will fulfill the scalar Helmholtz equation

(\nabla_{t}^{2} + k_{0}^{2} n^{2} - β_{k}^{2}) {\hat{ψ}}_{k} = 0

By taking differential with respect to z for (10)

(\nabla_{t}^{2} + k_{0}^{2} n^{2} - β_{k}^{2}) \frac{\partial {\hat{ψ}}_{k}}{\partial z} + (k_{0}^{2} \frac{\partial n^{2}}{\partial z} - \frac{\partial β_{k}^{2}}{\partial z}) {\hat{ψ}}_{k} = 0

Here n is the refractive index distribution of waveguide. Through (10) ×

\frac{\partial {\hat{ψ}}_{j}}{\partial z}

−(11) ×

{\hat{ψ}}_{j}

, we can obtain

\begin{matrix} (\frac{\partial {\hat{ψ}}_{j}}{\partial z} \nabla_{t}^{2} {\hat{ψ}}_{k} - {\hat{ψ}}_{j} \nabla_{t}^{2} \frac{\partial {\hat{ψ}}_{k}}{\partial z}) + (k_{0}^{2} n^{2} - β_{k}^{2}) (\frac{\partial {\hat{ψ}}_{j}}{\partial z} {\hat{ψ}}_{k} - {\hat{ψ}}_{j} \frac{\partial {\hat{ψ}}_{k}}{\partial z}) \\ = (k_{0}^{2} \frac{\partial n^{2}}{\partial z} - \frac{\partial β_{k}^{2}}{\partial z}) {\hat{ψ}}_{j} {\hat{ψ}}_{k} \end{matrix}

We swap the roles of j and k in (12) to form a new equation and add it with (12)

\begin{array}{l} (\frac{\partial {\hat{ψ}}_{j}}{\partial z} \nabla_{t}^{2} {\hat{ψ}}_{k} - {\hat{ψ}}_{k} \nabla_{t}^{2} \frac{\partial {\hat{ψ}}_{j}}{\partial z}) + (\frac{\partial {\hat{ψ}}_{k}}{\partial z} \nabla_{t}^{2} {\hat{ψ}}_{j} - {\hat{ψ}}_{j} \nabla_{t}^{2} \frac{\partial {\hat{ψ}}_{k}}{\partial z}) \\ + (β_{j}^{2} - β_{k}^{2}) (\frac{\partial {\hat{ψ}}_{j}}{\partial z} {\hat{ψ}}_{k} - {\hat{ψ}}_{j} \frac{\partial {\hat{ψ}}_{k}}{\partial z}) \\ = 2 k_{0}^{2} \frac{\partial n^{2}}{\partial z} {\hat{ψ}}_{j} {\hat{ψ}}_{k} - (\frac{\partial β_{k}^{2}}{\partial z} + \frac{\partial β_{j}^{2}}{\partial z}) {\hat{ψ}}_{j} {\hat{ψ}}_{k} \end{array}

Integrate (13) over infinite cross section, by using orthogonality and Green’s theorem which indicate that all items contains ∇_t² will vanish, we obtain the following equation

(β_{j}^{2} - β_{k}^{2}) \int_{A_{\infty}} (\frac{\partial {\hat{ψ}}_{j}}{\partial z} {\hat{ψ}}_{k} - {\hat{ψ}}_{j} \frac{\partial {\hat{ψ}}_{k}}{\partial z}) d A = 2 k_{0}^{2} \int_{A_{\infty}} \frac{\partial n^{2}}{\partial z} {\hat{ψ}}_{j} {\hat{ψ}}_{k} d A

By replacing j with –j in (14) to form a new equation and add it with (14), the following relationship is revealed

\int_{A_{\infty}} {\hat{ψ}}_{j} \frac{\partial {\hat{ψ}}_{k}}{\partial z} d A = - \frac{2 k_{0}^{2}}{β_{j}^{2} - β_{k}^{2}} \int_{A_{\infty}} \frac{\partial n^{2}}{\partial z} {\hat{ψ}}_{j} {\hat{ψ}}_{k} d A

Let’s recall (9) and using condition β_j + β_k≈2k₀n_co in weakly guiding fiber, we can get the final form of coupling coefficient under weak-guidance approximation

C_{j k} = \frac{k_{0}}{2 Z_{0} (β_{j} - β_{k})} \int_{A_{\infty}} \frac{\partial n^{2}}{\partial z} {\hat{ψ}}_{j} {\hat{ψ}}_{k} d A

For step-index long tapered fiber, the refractive index distribution can be expressed in the following form

n^{2} (r, z) = n_{c o}^{2} - N A^{2} H [r - ρ (z)]

where NA is the numerical aperture of the fiber core, H(x) is the Heaviside step function and ρ(z) is the radius profile of fiber core. Then z-derivative of n² can be expressed as

\frac{\partial n^{2}}{\partial z} = N A^{2} δ [r - ρ (z)] \frac{d ρ (z)}{d z}

2.2 Mode coupling induced by slight perturbation

Except for core radius variation, there are many slight perturbations that will lead to mode interaction in fiber section. In this part, we will derive the coupling equation induced by the slight-perturbation-induced refractive index change. Here we define a vector function F_c by

F_{c} = E \times {\bar{H}}^{*} + {\bar{E}}^{*} \times H

where the barred fields describe the field in unperturbed waveguide and unbarred fields describe the field in slightly perturbed waveguide. Both of barred and unbarred field satisfy the Maxwell’s equation. Here we set the barred field equal to j-th bound eigenmode in unperturbed waveguide

\bar{E} = {\hat{e}}_{j} \exp (i β_{j} z), \bar{H} = {\hat{h}}_{j} \exp (i β_{j} z)

and unbarred field as the total field in perturbed waveguide. We assume that the waveguide is non-absorbing and source free, by using the conjugate form of reciprocity theorem [16]

\frac{\partial}{\partial z} \int_{A_{\infty}} F_{c} \cdot \hat{z} d A = \int_{A_{\infty}} \nabla \cdot F_{c} d A

\nabla \cdot F_{c} = i \frac{k_{0}}{Z_{0}} (n^{2} - {\bar{n}}^{2}) E \cdot {\bar{E}}^{*}

where

n

and

\bar{n}

represents the refractive index distributions of perturbed and unperturbed waveguide, respectively. Substitute (19), (20) and (22) into (21), through applying orthogonal relationship and rearrange items and using weak-guidance approximation, we can obtain the coupled mode equation for slight perturbation

\frac{d b_{j}}{d z} - i β_{j} b_{j} = i \frac{k_{0}}{4 Z_{0}} \int_{A_{\infty}} (n^{2} - {\bar{n}}^{2}) {\hat{ψ}}_{j} \cdot E d A

When recalling (6), we can find that the coupled mode equation of radius variation and slight perturbation has the same form. Considering about gain and loss of every eigenmode, we denote g_j and α_j as gain and loss of j-th eigenmode, respectively. The total coupled mode equation for long tapered multimode active fiber can be written as

\frac{d b_{j}}{d z} - (i β_{j} + g_{j} - α_{j}) b_{j} = C_{j}^{I} + i C_{j}^{I I}

where

{\begin{matrix} C_{j}^{I} = \frac{k_{0}}{2 Z_{0}} \sum_{k \neq j} \frac{1}{β_{j} - β_{k}} \int_{A_{\infty}} \frac{\partial n^{2}}{\partial z} {\hat{ψ}}_{j} {\hat{ψ}}_{k} d A & (a) \\ C_{j}^{I I} = \frac{k_{0}}{4 Z_{0}} \int_{A_{\infty}} (n^{2} - {\bar{n}}^{2}) {\hat{ψ}}_{j} \cdot E d A & (b) \end{matrix}

In each section, differential Eq. (24) can be calculated as

b_{j} (z + d z) = [b_{j} (z) + \frac{C_{j}^{I} + i C_{j}^{I I}}{i β_{j} + g_{j} - α_{j}}] e^{(i β_{j} + g_{j} - α_{j}) d z} - \frac{C_{j}^{I} + i C_{j}^{I I}}{i β_{j} + g_{j} - α_{j}}

Hence, coupled mode equation in active long tapered multimode fiber has been converted to a recursion formula of z-coordinate which is easy to solve. In following part of section 2, we will discuss about some slight perturbations exist in active long tapered fiber that lead to spatial mode coupling.

2.3 Local gain and gain-induced refractive index change

Here we assume that active long tapered fiber is Yb-doped in our discussion and it can be easily extended to other type of active fibers. Here we use incoherent field model to take TSHB effect into consideration. We assume that rare-earth ions are only doped in fiber core and pump laser will distribute homogeneously in fiber core cross section. With steady-state two-level rate equation [17], the upper level population fraction of fiber cross section can be expressed as

η_{u} (r, φ, z) = \frac{I_{p} (z) λ_{p} σ_{a}^{p} + λ_{s} σ_{a}^{s} \sum_{k} I_{s}^{k} (r, φ, z)}{\frac{h c}{τ} + I_{p} (z) λ_{p} (σ_{a}^{p} + σ_{e}^{p}) + λ_{s} (σ_{a}^{s} + σ_{e}^{s}) \sum_{k} I_{s}^{k} (r, φ, z)}

where h is Planck’s constant; c is velocity of light in vacuum; τ is upper-level lifetime of Yb-ions; I_p is intensity of pump laser; I_s^k is intensity distribution of k-th eigenmode; λ_p and λ_s are pump and signal wavelength, respectively; σ_a and σ_e are absorption and emission cross section of Yb-ions and superscript p and s means pump and signal, respectively. Hence, the gain distribution in core cross section can be calculated as

g (r, φ, z) = N_{d o p e} (r, φ, z) [(σ_{a}^{s} + σ_{e}^{s}) η_{u} (r, φ, z) - σ_{a}^{s}]

The variation of pump laser can be written as

\frac{d P_{p} (z)}{d z} = I_{p} (z) \int_{A_{c o}} N_{d o p e} (r, φ, z) [(σ_{a}^{p} + σ_{e}^{p}) η_{u} - σ_{a}^{p}] d A - α_{p} P_{p} (z)

where α_p is background loss of pump laser. The gain for each eigenmode can be calculated as

g_{j} = \frac{1}{2} \int_{A_{c o}} g (r, φ, z) {\hat{ψ}}_{j} d A

From (28), the gain-induced refractive index change can be expressed as

Δ n_{g} = - i \frac{g (r, φ, z)}{2 k_{0}}

2.4 Bend-induced refractive index change and curvature loss

In order to make the whole fiber system more compact, bend is always inevitably applied in most practical applications. Bends along the fiber will lead to both the spatial mode coupling and bend-induced birefringence. For simplicity, we assume that bends will only happens in x-z plane and we firstly consider about refractive index change induced by curvature itself. Assuming the curvature radius in fiber section is R_c and corresponding core radius is a, the curved waveguide can be equivalent to a straight waveguide with following refractive index distribution [16, 18]

n_{e f f} = n_{c o} (1 + \frac{r}{R_{c}} \cos φ)

Through Eq. (32), we can treat bend fiber as a straight fiber and the refractive index distribution describes spatial mode coupling and field deformation due to curvature simultaneously. As for the birefringence induced by bend, because of that bends only happen in x-z plane, the curvature stress of fiber core can be described by following relations [19, 20]

{\begin{matrix} σ_{x} = \frac{E_{Y}}{2 R_{c}^{2}} (r^{2} \cos^{2} φ - a^{2}) \\ σ_{y} = 0 \end{matrix}

where E_Y is the Young’s modulus of fiber core. The elasto-optic index changes Δn_ST can be evaluated from the stress by following relations [21]

{\begin{matrix} Δ n_{i, S T} = - \frac{n_{0}^{3}}{2} \sum p_{i j} ε_{j} & (a) \\ ε_{j} = (1 + ν) σ_{j} / E_{Y} & (b) \end{matrix}

where p_ij denotes strain-optics coefficients, ε_j denotes the strain components and ν = 0.16 is the Poisson ratio of material. Hence, the bend stress induced refractive index change can be expressed by

{\begin{matrix} Δ n_{x, S T} = - \frac{1}{4} n_{c o}^{3} p_{11} \frac{1 + ν}{R_{c}^{2}} (r^{2} \cos^{2} φ - a^{2}) \\ Δ n_{y, S T} = - \frac{1}{4} n_{c o}^{3} p_{12} \frac{1 + ν}{R_{c}^{2}} (r^{2} \cos^{2} φ - a^{2}) \end{matrix}

Due to this birefringence effect, refractive index has to be expressed by a tensor and polarization related mode coupling is employed.

Except for refractive index change, curvature of waveguide also leads to power loss for corresponding bound eigenmodes. The curvature loss for eigenmodes can be described by [22]

2 α_{j} = \frac{\sqrt{π} U_{j}^{2} \exp (- \frac{2 W_{j}^{3} R_{c}}{3 a^{3} β_{j}^{2}})}{ξ_{l} \sqrt{R_{c} a W_{j}^{3}} V^{2} K_{l - 1} (W_{j}) K_{l + 1} (W_{j})}

where l is the azimuthal number of corresponding eigenmode, ξ_l takes 2 when l = 0 and takes 1 when l equals to positive integers and K represents modified Bessel function of the second kind. Parameter V, W_j and U_j are defined as follows

{\begin{matrix} V = N A \cdot k_{0} \cdot a \\ U_{j} = a \sqrt{n_{c o}^{2} k_{0}^{2} - β_{j}^{2}} \\ W_{j} = a \sqrt{β_{j}^{2} - (n_{c o}^{2} - N A^{2}) k_{0}^{2}} \end{matrix}

2.5 Beam quality factor

The beam quality factor of the laser beam output from the long tapered multimode active fiber can be evaluated by the field distribution at the output facet. Here we are using M²-parameter factor which has become a universal standard of the laser beam quality. From second-order intensity moment calculation method for M²-parameter [23, 24], the beam quality factor can be calculated using following equations (notation k can represent both x or y coordinate and z₀ refers to the position of fiber output facet).

M_{k}^{2} = \sqrt{4 σ_{k}^{2} (z_{0}) B_{k} + A_{k}^{2}}

{\begin{matrix} 〈 k 〉 (z_{0}) = \int_{A_{\infty}} k {| E (x, y, z_{0}) |}^{2} d A \\ σ_{k}^{2} (z_{0}) = \int_{A_{\infty}} {[k - 〈 k 〉 (z_{0})]}^{2} {| E (x, y, z_{0}) |}^{2} d A \\ A_{k} = \int_{A_{\infty}} [k - 〈 k 〉 (z_{0})] [E (x, y, z_{0}) \frac{\partial E^{*} (x, y, z_{0})}{\partial k} - c . c .] d A \\ B_{k} = \int_{A_{\infty}} {| \frac{\partial E (x, y, z_{0})}{\partial k} |}^{2} d A + \frac{1}{4} {\int_{A_{\infty}} [E (x, y, z_{0}) \frac{\partial E^{*} (x, y, z_{0})}{\partial k} - c . c .] d A}^{2} \end{matrix}

where

〈 k 〉 (z_{0})

and

σ_{k}^{2} (z_{0})

are the center and variance of beam with respect to corresponding coordinates, respectively. A_k and B_k are parameters in calculation.

3. Numerical simulation and discussion

In this section, we will simulate mode evolution in Yb-doped long tapered active fiber based fiber amplifier under different parameters numerically on the base of the model we demonstrated above. Instead of investigating active fiber with arbitrary radius profile, we can use the profile definition in Ref [25]. Assuming the core diameter of small and large end and length of tapered fiber are D₁, D₂ and L, respectively. Then radius profile of tapered fiber can be given by

ρ (z) = \frac{b_{0} - b}{2 L} z^{2} + \frac{b}{2} z + \frac{1}{2} D_{1}

where

b_{0} = (D_{2} - D_{1}) / L

is the average tapering angle and b is the parabolic shape factor. Figure 2 shows the radius profile with different b value.

Fig. 2 Radius profile for tapered fiber with different parabolic shape factors.

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According to the relative value of b and b₀, radius profile can be divided into three categories, b>b₀, b = b₀ and b<b₀ correspond to convex, linear and concave radius profile, respectively. All long tapered fibers with more complex profile can be approximated by these three type of profiles. Based on the statement above, our numerical simulation will concern on these three types of tapered fibers.

In our simulation, the signal laser wavelength is set to 1064nm while the pumping wavelength is 976nm. The total pump power is 20W forward pump. The tapered fiber is assumed to be doubly clad, and the core-inner-clad radius ratio and the core-outer-clad ratio are 0.05 and 0.04, respectively. The numerical aperture of fiber core is 0.065. We assume these ratios and numerical aperture will stay unchanged along with fiber length z. The total fiber length is 4.5m and the small and large end core diameters are 20μm and 45μm, respectively.

3.1 Local gain

Theoretically investigation of mode evolution in Yb-doped long tapered active fiber with different transverse doping distribution will be presented here. The flat doping and parabolic doping distribution [26] in fiber core are illustrated in Fig. 3. For flat doping, we can define an overlapping factor Γ_d of doping area in fiber core as the ratio of doping area to the total core area. The parabolic doping concentration fulfills the relation of N(r) = N₀(1−r²/a²), where N₀ is the maximum doping concentration which is the same as in flat doping distribution.

Fig. 3 Illustration for different doping distribution; (a) flat doping with Γ_d = 0.5; (b) parabolic doping.

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In this part of our simulation, the tapered fiber is supposed to be coiled with 30cm curvature radius which is commonly used in experiments. Both the signal and pump power are injected into small end of the fiber.

Near fundamental mode injection

According to the parameters we have shown above, we know that even the small end of the long tapered fiber supports LP₀₁, LP₁₁ and LP₂₁ modes at the same time, and the supported core bound modes will increase along with the increment of core radius. Therefore, we define near fundamental mode injection as that the injecting power in fundamental mode is tenfold higher of which is injected to higher-order-modes (HOMs), and multimode injection as injecting power evenly to every supported bound mode. Here we assume that signal power is 0.01W near fundamental mode injection.

Figs 4(a)~4(c) shows the corresponding results for different doping distributions of linear profile long tapered fiber. Compared with flat doping with Γ_d = 1, both parabolic and less overlapped flat doping can effectively suppress the power in HOMs which result in beam quality improvement that can be observed in M² factor curves and output field figure. This is mainly because the gain of which the modes can experience in core is related to effective overlapping doping area of modes. Here we calculate the effective overlapping doping area as a function of Γ_d for first 6 LP modes (LP₀₁, LP₁₁, LP₂₁, LP₀₂, LP₃₁, LP₁₂) in large end of long tapered fiber and show the results in Fig. 5.

Fig. 4 Simulation results of modal power (left column), M² factor (middle column) and input/output field (right column) of tapered active fiber with different doping distributions and parameter b = b₀ under near fundamental mode injection.

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Fig. 5 Effective gain area ratio of different modes with respect to Γ_d.

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As shown in Fig. 5, when Γ_d is close to 1 which is fully flat doping, many HOMs have larger overlapping doping area than LP₀₁ mode, and this will become a severe problem which lead to degradation of beam quality as we will show in the following multimode injection part. We can simply draw a conclusion from Fig. 5 that the best choice of Γ_d is between 0.2 and 0.4 if leave out of efficiency.

The middle column of Fig. 4 shows the variation of beam quality factor along with fiber length. The beam quality was getting better due to LP₀₁ mode has more power and extracts more energy from pump at the beginning part of active fiber. Then M² factor will degrade slightly along with the increment of fiber core radius. It can be obviously observed there are some “loose points” where the curve is sparser than other positions on curves of M² factors. These “loose points” are induced by the slow varying core radius which causes slightly change of propagating constants along with fiber length. The slightly change of propagating constants generates re-imaging with beat length in a much larger scale.

Figure 6 plots the simulation results of concave(b = 0.1b₀) and convex(b = 2b₀) profile fiber with flat doping and Γ_d = 1. Benefit from higher average pump intensity in core, LP₀₁ mode can extract more energy from concave profile fiber which result in weaker mode coupling compared with linear and convex fiber which reflects on M² factor curve.

Fig. 6 Simulation results of modal power (left column), M² factor (middle column) and input/output field (right column) of tapered active fiber with flat doping Γ_d = 1 distributions; (a)concave profile with b = 0.1b₀; (b)convex profile with b = 2b_0.

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Multimode injection

In this part, our simulation is based on multimode injection which means the injected power will distribute evenly to every supported bound mode at small end of tapered fiber. The fiber parameters are still the same as we used in previous part. Assume that total injecting signal power is 0.01W.

Different from the results in Fig. 4, the different doping distribution shows totally different mode evolution behaviors as shown in Figs. 7(a)~7(c). Because of that the quantity of HOMs is four times as many as LP₀₁ mode, the initial power of HOMs is higher than fundamental mode under multimode injection. As we discussed in Fig. 5, the effective mode gain area of HOMs is larger when Γ_d = 1 which result in that HOMs become the prominent component in output field as shown in Fig. 7(a). In Fig. 7(b) we selected Γ_d = 0.25 which effectively suppress the HOMs and obtained near fundamental mode output. On the other hand, because of the initial power of fundamental mode is too small to effectively extract energy from active fiber, longer fiber length is needed in this case to compensate smaller effective doping area in order to increase efficiency. The results of parabolic doping profile fell in between Γ_d = 1 and Γ_d = 0.25 results. It was comprehensible because we can estimate the equivalent overlapping factor for parabolic doping through integration in fiber core as Γ_d^parabolic≈0.5.

Fig. 7 Simulation results of modal power (left column), M² factor (middle column) and input/output field (right column) of tapered active fiber with different doping distributions and parameter b = b₀ under multimode injection.

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Figure 8 plots the simulation results of concave(b = 0.1b₀) and convex(b = 2b₀) profile fiber with flat doping and Γ_d = 1 under multimode injection. Although neither concave nor convex profile tapered fiber can effectively suppress the HOMs, LP₀₁ mode still can extract more energy from concave fiber due to higher average pumping intensity and slightly improve beam quality.

Fig. 8 Simulation results of modal power (left column), M² factor (middle column) and input/output field (right column) of tapered active fiber with flat doping Γ_d = 1 distributions; (a)concave profile with b = 0.1b₀; (b)convex profile with b = 2b_0.

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To sum up of section 3.1, active tapered fiber can efficiently maintain near fundamental field and good output beam quality with near fundamental signal injection. However, special doping distribution has to be applied in order to suppress HOMs with multimode injection to obtain desired beam quality of output field. The concave tapered fiber has more stable beam quality character due to higher average pump intensity compared with other profiles.

3.2 Macro curvature

In this section, we will discuss about curvature induced modal loss in long tapered active fiber. It is well known that curvature induced loss will be different between modes under certain core radius and curvature, and mode selecting technology using coiling multimode fiber has already been applied [27]. Because of that LP₁₁ mode will become well guided along with the increase of core radius, it usually has to employ smaller curvature in LMA fiber to achieve single mode operation.

We define Δα as the loss difference between LP₁₁ and LP₀₁ mode. Figure 9 shows the relationship between Δα and fiber core radius. The figure indicate that the curvature will be more effective at small end of long tapered fiber. This inspires us that we can apply effective curvature loss only at small end of long tapered fiber which is much easier than applying to the whole piece.

Fig. 9 Loss difference between LP₁₁ and LP₀₁ mode with respect to core radius with core NA = 0.065

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Figure 10 shows the simulation results for long tapered active fiber coiled with first 1m length at small end. R_c = 7cm is selected according to Fig. 9. Only linear and concave profile tapered fiber is investigated because it is obvious convex fiber will be ineffective in this case due to larger core radius at small end. Compared with the results in Fig. 7(a) and Fig. 8(a), the curvature significantly improves the output beam quality. Both linear and concave profile fiber show effective suppression effect to HOMs under multimode injection. The curvatures are only applied to small end of the fiber which is much easier and safe in real experimental setup, and the results shows the potential of effective curvature mode filtering using long tapered active fiber.

Fig. 10 Simulation results of modal power (left column), M² factor (middle column) and input/output field (right column) of tapered active fiber with curvature in first 1m length; (a)linear profile with b = b₀; (b)concave profile with b = 0.1b_0.

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3.3 Large-to-small performance

Before this section, all our simulations are based on the assumption that signal power is transmitted from small end to the large end of long tapered fiber. Due to longitudinal asymmetry, we will theoretically investigate the performance of large-to-small (L-S) usage of long tapered active fiber in this section.

Different from small-to-large (S-L) situation, L-S configuration has a decreased fiber core profile in which well-guided HOMs will become leaky and couple more modal power into inner clad and gradually vanish. Intuitively, L-S usage seems to be inefficient according to the statement above. However, we still numerically investigate the case under near fundamental mode and multimode injections to reveal the truth of L-S performance of tapered fiber. The simulation parameters will keep the same as section 3.1. We firstly investigate near fundamental mode injection.

Figure 11 shows the simulation results of long tapered active fiber in L-S configuration with near fundamental mode injection. Due to small end output, good beam quality has achieved in all three cases but with different efficiency. Figure 11(b) has the best performance in both M² factor and amplification efficiency. Different from S-L configuration, concave (b<b₀) profile fiber has the worst performance which even worse than linear in Fig. 11(a). This result can be explained by mode coupling induced by radius variation. By recalling (16) and (18), we can know that coupling coefficient is related to z derivative of radius profile and quantities of propagating constant. In L-S usage, concave profile will generate larger coupling coefficients from fundamental modes to HOMs which degrade the performance of whole amplification progress. The simulation result of convex profile is not listed here because it shows no advantage in performance. According to the analysis we presented above, only linear profile and flat doping Γ_d<1 is needed to be taken into consideration in multimode injection case.

Fig. 11 Simulation results of modal power (left column), M² factor (middle column) and input/output field (right column) of tapered active fiber L-S configuration; (a)linear profile with b = b₀, Γ_d = 1; (b)linear profile with b = b₀, Γ_d = 0.5; (c)concave profile with b = 0.1b₀, Γ_d = 1.

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Figure 12 shows the result of linear profile fiber with L-S configuration and Γ_d = 0.5 doping distribution. The output beam quality is relatively good as expected, but the efficiency of the amplifier has been reduced to <40%. In fact, the highest efficiency of L-S configuration was just ~50% which showed in Fig. 11(b). This reveals the mechanism to obtain good beam quality by HOMs leakage of L-S configuration of long tapered active fiber. Figure 12(b) shows the gradual procedure of higher order mode leakage which reflect on the variation of field distribution.

Fig. 12 Simulation results of modal power (left column), M² factor (middle column) and input/output field (right column) of tapered active fiber L-S configuration; (a)linear profile with b = b₀, Γ_d = 0.5; (b)mode field on different position of fiber.

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To sum up of section 3.3, the L-S configuration of long tapered active fiber is inapplicable in building up fiber amplifier due to low efficiency. Our simulation reveals that the HOMs should be stripped as early as possible to obtain desired output which can only be achieved by confined doping management. But the leakage of HOMs of L-S shows the potential in applications of transverse mode filtering in fiber laser oscillator which has already been reported experimentally [6].

4. Conclusion

In this manuscript, we have presented a concise and effective model to investigate mode evolution in long tapered active fiber. The mode coupling due to variation of core radius and slight perturbation have been deduced and local gain with TSHB effect, loss and curvature have been taken into consideration in our model. On the base of this model, the mode evolution behaviors under different factors have been numerically investigated. From the simulation results we have get, we can come to following conclusions: 1) modal excitation can greatly affect the performance to long tapered active fiber. S-L configuration can effectively maintain the good beam quality if near fundamental mode field is injected; 2) concave (b<b₀) profile fiber has better performance in S-L configuration with forward pump due to higher average pump intensity; 3) in multimode injection case, confined doping can be an effective solution for HOMs suppression; 4) L-S configuration is inapplicable in building up fiber amplifier due to leakage induced low efficiency character. Moreover, our model can be easily extended to contain more factors according to the investigation purpose and can provide instructive suggestions when designing the system based on long tapered active fiber.

Acknowledgment

The authors would like to thank Dr. Haibin Lyu for enlightening discussion, Dr. Liangjin Huang for the help in M² factor calculation and Prof. John E. Sipe for valuable suggestions. This work is supported by National Natural Science Foundation of China (Grant No. 61505260).

References and links

1. M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE Sel. Top. Quantum Electron. 20(5), 219–241 (2014). [CrossRef]

2. V. Filippov, Y. K. Chamorovskii, K. M. Golant, A. Vorotynskii, and O. G. Okhotnikov, Optical amplifiers and lasers based on tapered fiber geometry for power and energy scaling with low signal distortion (2016), p. 97280V.

3. V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “Single-mode 212 W tapered fiber laser pumped by a low-brightness source,” Opt. Lett. 33(13), 1416–1418 (2008). [CrossRef] [PubMed]

4. V. Filippov, Y. Chamorovskii, J. Kerttula, A. Kholodkov, and O. G. Okhotnikov, “600 W power scalable single transverse mode tapered double-clad fiber laser,” Opt. Express 17(3), 1203–1214 (2009). [CrossRef] [PubMed]

5. V. Filippov, J. Kerttula, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Highly efficient 750 W tapered double-clad ytterbium fiber laser,” Opt. Express 18(12), 12499–12512 (2010). [CrossRef] [PubMed]

6. J. Kerttula, V. Filippov, Y. Chamorovskii, K. Golant, and O. G. Okhotnikov, “Actively Q-switched 1.6-mJ tapered double-clad ytterbium-doped fiber laser,” Opt. Express 18(18), 18543–18549 (2010). [CrossRef] [PubMed]

7. J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Tapered fiber amplifier with high gain and output power,” Laser Phys. 22(11), 1734–1738 (2012). [CrossRef]

8. A. I. Trikshev, A. S. Kurkov, V. B. Tsvetkov, S. A. Filatova, J. Kertulla, V. Filippov, Y. K. Chamorovskiy, and O. G. Okhotnikov, “160W single-frequency laser based on active tapered double-clad fiber amplifier,” in 2013 Conference on Lasers and Electro-Optics - International Quantum Electronics Conference(Optical Society of America, Munich, 2013), p. 33. [CrossRef]

9. H. Zhang, X. Du, P. Zhou, X. Wang, and X. Xu, “Tapered fiber based high power random laser,” Opt. Express 24(8), 9112–9118 (2016). [CrossRef] [PubMed]

10. Z. Zhou, H. Zhang, X. Wang, Z. Pan, R. Su, B. Yang, P. Zhou, and X. Xu, “All-fiber-integrated single frequency tapered fiber amplifier with near diffraction limited output,” J. Opt. 18(6), 065504 (2016). [CrossRef]

11. V. Filippov, Y. Chamorovskii, J. Kerttula, K. Golant, M. Pessa, and O. G. Okhotnikov, “Double clad tapered fiber for high power applications,” Opt. Express 16(3), 1929–1944 (2008). [CrossRef] [PubMed]

12. J. Kerttula, V. Filippov, Y. Chamorovskii, V. Ustimchik, K. Golant, and O. G. Okhotnikov, “Principles and performance of tapered fiber lasers: from uniform to flared geometry,” Appl. Opt. 51(29), 7025–7038 (2012). [CrossRef] [PubMed]

13. F. Gonthier, A. Hénault, S. Lacroix, R. J. Black, and J. Bures, “Mode coupling in nonuniform fibers: comparison between coupled-mode theory and finite-difference beam-propagation method simulations,” J. Opt. Soc. Am. B 8(2), 416–421 (1991). [CrossRef]

14. H. Lü, P. Zhou, X. Wang, and Z. Jiang, “General analysis of the mode interaction in multimode active fiber,” Opt. Express 20(6), 6456–6471 (2012). [CrossRef] [PubMed]

15. A. W. Snyder and J. Love, Optical waveguide theory (Springer Science & Business Media, 2012).

16. A. W. Snyder, “Coupled-Mode Theory for Optical Fibers,” J. Opt. Soc. Am. 62(11), 1267–1277 (1972). [CrossRef]

17. W. Yong and P. Hong, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” Lightwave Technology, Journalism 21, 2262–2270 (2003).

18. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21(23), 4208–4213 (1982). [CrossRef] [PubMed]

19. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5(6), 273–275 (1980). [CrossRef] [PubMed]

20. E. F. Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibres,” J. Opt. A, Pure Appl. Opt. 1(2), 197–200 (1999). [CrossRef]

21. J. F. Nye, Physical properties of crystals: their representation by tensors and matrices (Oxford university press, 1985).

22. R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).

23. H. Yoda, P. Polynkin, and M. Mansuripur, “Beam Quality Factor of Higher Order Modes in a Step-Index Fiber,” J. Lightwave Technol. 24(3), 1350–1355 (2006). [CrossRef]

24. S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Phys. 19(3), 437–444 (2009). [CrossRef]

25. O. G. Okhotnikov, Fiber lasers (John Wiley & Sons, 2012).

26. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

27. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000).

Theoretical study of mode evolution in active long tapered multimode fiber

Abstract

1. Introduction

2. Models and theoretical analysis

2.1 Mode coupling induced by radius variation

2.2 Mode coupling induced by slight perturbation

2.3 Local gain and gain-induced refractive index change

2.4 Bend-induced refractive index change and curvature loss

2.5 Beam quality factor

3. Numerical simulation and discussion

3.1 Local gain

Near fundamental mode injection

Multimode injection

3.2 Macro curvature

3.3 Large-to-small performance

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (12)

Equations (40)

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