Mode instability in ytterbium-doped non-circular fibers

N. Xia; S. Yoo

doi:10.1364/OE.25.013230

1. Introduction

High average power fiber laser with good beam quality is found as an attractive source in many applications for industrial, defense and scientific research [1]. Output powers of fiber lasers surpass bulk solid-state lasers thanks to its larger surface-to-active-volume ratio which efficiently extracts heat and makes robust single-mode operation [1,2]. J. W. Dawson et al. analyzed the power scalability of diffraction-limited fiber lasers and amplifiers [3], which shows that the highest achievable output power is determined by a combined effect of thermal lens and nonlinear scatterings such as stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS). H. J. Otto et al. included transverse mode instability (TMI) in a similar analysis, and found that the TMI can occur before the combined limitation, hence setting another limitation for power scalability in commercial fibers with a V-parameter less than 14 [4]. The TMI was firstly reported in a large mode area (LMA) fiber amplifier in 2011 [5], and it is recognized as the major limitation for high average power scalability till now. Upon an onset of the TMI, the fundamental mode (FM) power transfers to a higher-order mode (HOM), and consequently the TMI degrades output beam quality as well as limits the FM power scalability. Soon after the observation of TMI phenomenon in high power fiber operation, C. Jauregui et al. suggested that the TMI is caused by the changed refractive index in core, which is induced by variations in the population of upper level lasing ions [6]. Shortly afterwards, A. Smith et al. considered the temporal evolution of the upper level ions population and temperature-induced index grating [7]. Now it is widely agreed that the TMI is caused by following three conditions [8]: a) FM and HOM interfere each other to produce an irradiance pattern that has a period equal to the beating length between modes; b) The irradiance pattern in fiber core induces different population inversion levels along the fiber, producing different heat generation through a quantum defect. The variation of generated heat leads to the formation of thermal grating via the thermo-optic effect. This temperature-induced refractive index grating accounts for the modal power transfer. c) A phase shift necessary for the TMI between irradiance pattern and thermal grating is induced by the thermal diffusion time across the core, frequency offset and different propagation constant between the associated modes.

A rectangular (or ribbon) core fiber design has been proposed for the effective FM area scaling above 1000 μm² [9–14]. The asymmetric core offers bending insensitivity [11], robust higher-order mode area scaling [13], and better heat dissipation [9] than a circular core fiber, suggesting better TMI performance. However, there has been no investigation on TMI in this type of fiber, to the best of our knowledge. In this paper, we use a numerical model to present theoretical investigation of TMI in a rectangular core fiber. Compared to the semi-analytical model, the numerical model offers more flexible platform to adopt the asymmetric fiber geometry [15]. The influence of core shapes on the TMI is investigated by varying the core aspect ratio (AR.) and we present important parameters governing the TMI including the frequency offset between FM and HOM, TMI threshold pump power and temperature distribution in the non-circular fiber geometries. The benefit of better heat dissipation in the non-circular fibers is examined against the TMI threshold pump power, and we also employ a power coupling coefficient to attain more insight of the TMI behaviors in non-circular fibers. Hence, this paper contributes to the investigation of TMI in non-circular fibers, which hasn’t been examined in prior studies, with an emphasis on the influence of temperature, gain saturation effect, thermal lensing effect and power coupling between modes which is associated with frequency offset.

The paper is structured as follows: Section 2 describes the highly numeric model we employed. In section 3, we calculate the frequency offset that makes the most efficient power transfer from FM to HOM in rectangular core fibers as well as circular core fiber. In section 4, we compare the TMI threshold pump power between circular core and rectangular core fibers, and theoretically analyze the TMI behaviors in rectangular core fibers. The investigation on different cladding shapes is presented in section 5 and the summary of paper is provided in section 6.

2. TMI numerical model

The TMI in a fiber can be simulated via numerical models [15–18] or semi-analytical models [19–21]. A. Smith et al. proposed the highly numeric model in [15], and such model can handle a variety of fiber core shapes, refractive index profiles and doping profiles. The model intrinsically includes gain saturation effect and a thermal lensing effect that influence the TMI behavior. Furthermore, it allows including additional physical effects, such as photodarkening effect and bending effect. However, the highly numeric model needs longer computing time compared with other analytical models. K. R. Hansen et al. presented the semi-analytical model in [19,20] and the advantages of such model include shorter running time and ability to portray important physics of TMI with analytical expressions such as a power coupling coefficient. However, the semi-analytical model employs approximations and is usually formulated to solve a cylindrical symmetric fiber, or photonic crystal fiber [16]. To accommodate the asymmetric core designs of our interest and various contributing effects of the TMI, we use the highly numerical model, which is similar to the model in [15], with the fast Fourier transform (FFT) based beam propagation method (BPM). We briefly describe our model in the following, and here we only consider the LP₀₁ and LP₁₁ to define the TMI because the power transfer is strongest from LP₀₁ to LP₁₁ [8]. An input signal field is presented by:

E_{s} (x, y, t) = \sqrt{P_{01}} E_{01} (x, y) + \sqrt{P_{11}} E_{11} (x, y) e^{- i Δ ω t}

where P₀₁ is the LP₀₁ mode input power and P₁₁ is the LP₁₁ mode input power, E₀₁ and E₁₁ are the normalized LP₀₁ and LP₁₁ mode amplitudes, respectively, Δω is the frequency offset between modes. The beam propagation equation is described as follows:

\frac{\partial E_{s} (x, y, z, t)}{\partial z} = - \frac{i}{2 k_{c}} \nabla_{⊥}^{2} E_{s} (x, y, z, t) - \frac{i [k^{2} (x, y, z, t) - k_{c}^{2}]}{2 k_{c}} E_{s} (x, y, z, t) + g (x, y, z, t) E_{s} (x, y, z, t)

where

\nabla_{⊥}^{2}

is the Laplacian operator in the x and y dimensions, g(x,y,z,t) is the laser gain, k(x,y,z,t) and k_c are carrier wave numbers in the core and cladding, respectively. The wave number in the core and laser gain can be expressed as:

k (x, y, z, t) = \frac{ω_{c} n_{c o r e} (x, y, z, t)}{c}

g (x, y, z, t) = \frac{1}{2} [- σ_{s}^{a} + (σ_{s}^{a} + σ_{s}^{e}) n_{u} (x, y, z, t)] N_{Y b} (x, y)

where ω_c is the carrier frequency, n_clad is the refractive index of cladding, σ_s^a and σ_s^e are the signal absorption and emission cross-sections, N_Yb(x,y) is the doping profile. The core refractive index during amplification, n_core(x,y,z,t), and upper state population, n_u(x,y,z,t), are described through:

n_{c o r e} (x, y, z, t) = n_{c o r e} + \frac{d n}{d T} Δ T (x, y, z, t)

n_{u} (x, y, z, t) = \frac{\frac{P_{p} (z, t) σ_{p}^{a}}{h ν_{p} A_{p}} + \frac{I_{s} (x, y, z, t) σ_{s}^{a}}{h ν_{s}}}{\frac{P_{p} (z, t) (σ_{p}^{a} + σ_{p}^{e})}{h ν_{p} A_{p}} + \frac{I_{s} (x, y, z, t) (σ_{s}^{a} + σ_{s}^{e})}{h ν_{s}} + \frac{1}{τ}}

where n_core is a pristine refractive index of core, dn/dT is the thermo-optic coefficient, ΔT(x,y,z,t) is the increased temperature distribution in fiber, σ_p^a and σ_p^e are the pump absorption and emission cross-sections, P_p(z,t) is the pump power, I_s(x,y,z,t) is the signal intensity, A_p is the pumping area, ν_s and ν_p are the signal and pump frequencies, h is the Planck constant and τ is the excited dopant ion lifetime.

We utilize the split-step FFT method to solve the BPM shown in Eq. (2), which has been proposed in [15,22–24]. In our simulation, each step size Δz consists of three sub-steps, and in the first and third sub-step, the field is advanced by Δz/2 in a homogeneous medium with a refractive index equal to the cladding index. In these two sub-steps, the BPM keeps only the diffractive term on right-hand-side as shown below and it is solved in the k space.

\frac{\partial E_{s} (x, y, z, t)}{\partial z} = - \frac{i}{2 k_{c}} \nabla_{⊥}^{2} E_{s} (x, y, z, t)

the field E_s(x,y,z,t) is transformed to the k space by using the FFT presented as:

E_{s} (x, y, z, t) = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{s} (k_{x}, k_{y}, z, t) e^{i k_{x} x} e^{i k_{y} y} d k_{x} d k_{y}

By substituting Eq. (8) into Eq. (7), we obtain:

\frac{\partial E_{s} (k_{x}, k_{y}, z, t)}{\partial z} = i \frac{k_{x}^{2}}{2 k_{c}} E_{s} (k_{x}, k_{y}, z, t) + i \frac{k_{y}^{2}}{2 k_{c}} E_{s} (k_{x}, k_{y}, z, t)

Subsequently, the k space field E_s(k_x,k_y,z,t) is advanced by Δz/2 consisting of a phase shift at each (k_x,k_y) plane as:

ϕ (k_{x}, k_{y}) = - \frac{Δ z}{2} \frac{(k_{x}^{2} + k_{y}^{2})}{2 k_{c}}

Lastly, the IFFT is used to convert the k space field E_s(k_x,k_y,_z,_t) back to E_s(x,y,z,t).

In the second sub-step, the field is advanced by Δz, keeping only the phases induced by the thermally generated index plus guiding index, together with a signal gain on the right-hand-side of Eq. (2). Then the BPM for this step becomes:

\frac{\partial E_{s} (x, y, z, t)}{\partial z} = - \frac{i [k^{2} (x, y, z, t) - k_{c}^{2}]}{2 k_{c}} E_{s} (x, y, z, t) + g (x, y, z, t) E_{s} (x, y, z, t)

with the phase of the field advanced over Δz as:

ϕ (x, y, t) = Δ z \frac{k^{2} (x, y, t) - k_{c}^{2}}{2 k_{c}}

To find the changed refractive index n_core(x,y,z,t) in Eq. (5), the increased temperature distribution ΔT(x,y,z,t) is calculated by using the Green’s function method presented in [15,25,26]. Additionally, we use a Dirichlet boundary condition for our thermal boundary, implying that temperature at the fiber surface is set at the room temperature, which is the same consideration as in [7,19]. When the temperature is fixed at the thermal boundary as indicated with dashed lines in Fig. 1, the Green’s function can be presented as:

G (x, y, ω | x', y') = \sum_{n = 0}^{\infty} F_{n} (y, y') P_{n} (x, x', ω)

F_{n} (y, y') = \frac{1}{W K} \sin (\frac{n π}{W} y) \sin (\frac{n π}{W} y')

\begin{matrix} P_{n} (x, x', ω) = {\exp [- σ_{n} (2 H - | x - x' |)] - \exp [- σ_{n} (2 H - x - x')] \\ + \exp [- σ_{n} | x - x' |] - \exp [- σ_{n} (x + x')]} / σ_{n} (1 - \exp [- 2 σ_{n} H]) \end{matrix}

σ_{n}^{2} = {(\frac{n π}{W})}^{2} + i ω \frac{ρ C}{K}

where 0 ≤ x ≤ H, 0 ≤ y ≤ W, and ω is the frequency of heat. The temperature can be calculated over the entire (x,y) grid with the contribution of periodically heated point (x’,y’) via the Green’s function. The heat equation to solve ΔT(x,y,z,t) is presented as follows:

ρ C \frac{\partial Δ T (x, y, z, t)}{\partial t} = Q (x, y, z, t) + K (\frac{\partial^{2} Δ T (x, y, z, t)}{\partial x^{2}} + \frac{\partial^{2} Δ T (x, y, z, t)}{\partial y^{2}})

where K is the thermal conductivity, ρ is the density, C is the heat capacity. The thermal load Q(x,y,z,t) in fiber core is expressed as:

Fig. 1 Intensity of FM (LP₀₁) and HOM (LP₁₁) in (a) 50 μm circular core (Fiber 1), (b) square core 44.3 μm × 44.3 μm (AR. 1:1) (Fiber 2), (c) rectangular core 88.8 μm × 22.2 μm (AR. 4:1) (Fiber 3) and (d) rectangular core 140 μm × 14 μm (AR. 10:1) (Fiber 4). The dashed lines indicate the thermal boundary.

Parameter	Fiber 1	Fiber 2	Fiber 3	Fiber 4
Core radius	25 μm	-	-	-
Core width	-	44.3 μm	88.8 μm	140 μm
Core height	-	44.3 μm	22.2 μm	14 μm
AR.	-	1:1	4:1	10:1
Cladding radius	200 μm	200 μm	200 μm	200 μm
Core area	1963 μm²	1963 μm²	1963 μm²	1960 μm²
Cladding index	1.45	1.45	1.45	1.45
Core NA	0.06	0.06	0.06	0.06
LP₀₁ effective mode area	1167 μm²	1083 μm²	1232 μm²	1368 μm²
LP₁₁ effective mode area	1062 μm²	1015 μm²	1196 μm²	1345 μm²
Signal effective mode area	1167 μm²	1083 μm²	1232 μm²	1368 μm²
LP₀₁ effective refractive index	1.45117	1.45116	1.45110	1.45099
LP₁₁ effective refractive index	1.45105	1.45104	1.45107	1.45097
Ion concentration	3.5 × 10²⁵ m⁻³	3.5 × 10²⁵ m⁻³	3.5 × 10²⁵ m⁻³	3.5 × 10²⁵ m⁻³
Doping area	Uniform core doping	Uniform core doping	Uniform core doping	Uniform core doping
Fiber length	2 m	2 m	2 m	2 m
LP₀₁ overlap with doping area	0.9826	0.9811	0.9627	0.9229
LP₁₁ overlap with doping area	0.9525	0.9508	0.9619	0.9223
LP₀₁ overlap with LP₁₁	0.8576	0.8598	0.8645	0.8692

	Fiber 1	Fiber 2	Fiber 3	Fiber 4
Output LP₀₁ power	148.62 W	147.71 W	141.59 W	107.6 W
Output LP₁₁ power	1.02 W	0.46 W	7.29 W	39.45 W
Output total signal power	149.71 W	148.51 W	149.34 W	148.6 W
Pump efficiency	> 95%	> 95%	> 95%	> 95%

Abstract

1. Introduction

2. TMI numerical model

3. Calculation of frequency offset for TMI

4. TMI in rectangular core fibers

4.1 Gain saturation effect

4.2 Power coupling strength

4.3 Thermal lensing effect

4.4 TMI threshold power versus core area and effective mode area

5. TMI in D-shaped cladding fibers

5.1 TMI in circular core D-shaped cladding fiber

5.2 TMI in rectangular core D-shaped cladding fiber

6. Conclusion

Funding

Acknowledgment

References and links

Cited By

Figures (20)

Tables (2)

Equations (24)

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