Gain guided and index alternate-guided fibers designed for large-mode-area and single-mode laser with higher output power and slope efficiency

Xiao Shen; Wei Wei

doi:10.1364/OE.24.001089

1. Introduction

High power fiber lasers have many important applications in industry, defense and commercial fields. However, the major limitations to high power lasers in conventional single mode fibers are undesired optical nonlinear effects [1–4]. The most direct method to solve these problems is to reduce the optical power density [5] by increasing the mode area. In 2003, Siegman proposed a new type of gain-guided and index-antiguided fiber (GG-IAGF) with large mode area [6, 7], and the maximal mode field diameter reported is up to 400 μm [1]. However, the fiber was deliberately devised to leak light power to maintain single mode, both the output laser power and slope efficiency are relatively low [3, 8]. In 2011, McComb et al. proposed several improved models of the GG-IAGF (named as hybrid gain guiding), which would be better applied in the related fields [9].

In this work, an improved model of gain-guided and index alternate-guided fiber (GGIA-GF) is proposed, which features two-part structure. The first part is gain-guided and index-guided fiber; the second is gain-guided and index anti-guided fiber. The single mode oscillation conditions and output characteristics of the laser based on GGIA-GF are analyzed based on the improved rate equations. The results show that the output laser power and slope efficiency are both greatly improved compared with GG-IAGF lasers.

2. Structure and single mode characteristics of the GGIA-GF

The structure of the proposed GGIA-GF is shown in figure (Fig. 1). Compared with GG-IAGF, the GGIA-GF is composed of part A and part B. The fiber length of part A and part B is L₁ and L₂, respectively, and thus the whole fiber length is L = L₁ + L₂. The fiber core of the two parts is made of the same material with all the same parameters, such as the core refractive index n_C,core radius a and optical power gain factor g. The fiber cladding of the two parts is intentionally designed with different refractive index of n_A and n_B, respectively. And the refractive index of each part meets the following relationship: . Herein, Part A can be called gain-guided and index-guided fiber (GG-IGF), part B can be called gain-guided and index anti-guided fiber (GG-IAGF) [2], and the whole fiber is named gain-guided and index alternate-guided fiber (GGIA-GF).

Fig. 1 The structure schematic diagram of GGIA-GF fiber.

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For the part B, that is GG-IAGF, the threshold gain coefficients for the establishment of the LP01 and LP11 modes in the fiber can be expressed as [4]:

g_{01}^{t h} = {λ^{2} / [{(2 π)}^{2} \cdot a^{3}]} \cdot \sqrt{133.8 / (- 2 n_{C}^{3} Δ n)}

g_{11}^{t h} = {λ^{2} / [{(2 π)}^{2} \cdot a^{3}]} \cdot \sqrt{862.2 / (- 2 n_{C}^{3} Δ n)}

Where, $λ$ is light wavelength, $Δ n = n_{B} - n_{C}$ . For the whole fiber, the average threshold gain coefficients of LP01 mode and LP11 mode can be expressed as $g_{01}^{t h}' = (L_{2} / L) \cdot g_{01}^{t h}$ and $g_{11}^{t h}' = (L_{2} / L) \cdot g_{11}^{t h}$ , respectively.

The threshold gain for laser oscillation can be given by

g_{t h}^{o s c} = - \ln (R_{1} R_{2}) / (2 L) + α_{s}

Where, $α_{s}$ is the signal light loss coefficient in the cavity, R₁ and R₂ are the cavity mirrors reflectivity at signal light wavelength, and L is the length of the active laser medium.

To assure that only the LP01 mode oscillates, the requirement is [4]:

g_{01}^{t h'} < g_{t h}^{o s c} < g_{11}^{t h'} - g_{01}^{t h'}

That is:

(L_{2} / L) \cdot g_{01}^{t h} < g_{t h}^{o s c} < (L_{2} / L) \cdot (g_{11}^{t h} - g_{01}^{t h})

Supposing that the reflectivity of total reflection mirror and output mirror is R₁ and R₂, respectively, when R₁ = 1, the following inequality is obtained:

\exp {- 2 \cdot [L_{2} \cdot (g_{11}^{t h} - g_{01}^{t h}) - L \cdot α_{s}]} < R_{2} < \exp {- 2 \cdot [L_{2} \cdot g_{01}^{t h} - L \cdot α_{s}]}

That is:

{\begin{cases} \exp {- 2 \cdot [L_{2} \cdot (g_{11}^{t h} - g_{01}^{t h}) - L \cdot α_{s}]} < \exp {- 2 \cdot [L_{2} \cdot g_{01}^{t h} - L \cdot α_{s}]} \\ \exp {- 2 \cdot [L_{2} \cdot (g_{11}^{t h} - g_{01}^{t h}) - L \cdot α_{s}]} < 1 \end{cases}

From the inequalities (7), the following inequality can be calculated as:

L_{2} / L > α_{s} / (g_{11}^{t h} - g_{01}^{t h})

That is to say, there is a minimum ratio between L2 and L, expressed as Min (L₂ /L), which is determined by $α_{s}$ , $g_{11}^{t h}$ and $g_{01}^{t h}$ . Otherwise, there may be multimode oscillation in the cavity.

Figure 2 shows the relationship between Min (L₂ /L) and fiber core radius a with different refractive index difference $Δ n$ under the conditions of single mode. The value of Min (L₂ /L) decreases with the decrease of a, because $g_{11}^{t h}$ increases with the decrease of a, shorter fiber of part B is needed to leak LP11 and higher-order modes. For example: $Δ n = - 0.005$ , a = 70μm,then Min (L₂ /L) = 19.95%, if L = 20cm, the needed fiber length of part B is 3.99cm. Also, the value of Min (L₂ /L) decreases with the decrease of $Δ n$ , since $g_{11}^{t h}$ increases with the decrease of $Δ n$ , then shorter fiber of part B is demanded to leak LP11 and higher-order modes.

Fig. 2 Relationship between Min (L₂/L) and fiber core radius with different $Δ n$ .

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After Min (L₂/L) and L are determined, assuming that L = 20cm, a = 70μm and $Δ n = - 0.005$ , from inequality (6), the relationship between the range of R₂ and the value of L₂/L can be obtained, as shown in Fig. 3 with grey shadow. When the value of (L₂/L) changes, the corresponding range of R₂ also varies to maintain single mode oscillation.

Fig. 3 The relationship between R₂ and L₂:L.

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When the value of (L₂/L) is determined, supposing that L = 20cm, L₂/L = 25%, a = 70μm and $Δ n = - 0.005$ , under the conditions of single mode oscillation, the relationship between the allowed range of R₂ and L is obtained. Figure 4 shows that the allowed range of R₂ increases with the increase of the fiber length L.

Fig. 4 The relationship between R₂ and L.

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If GGIA-GF works as a laser amplifier, supposing that the injected signal is multimode light, and the power of LP01 and LP11 modes is P₀₁ and P₁₁, the output power of LP01 and LP11 modes is $P_{01}^{}'$ and $P_{11}'$ .Then there are:

P_{01}' = P_{01} \cdot \exp [(g - α_{s}) \cdot L_{1}] \cdot \exp [(g - α_{s} - g_{01}^{t h}) \cdot L_{2}]

P_{11}' = P_{11} \cdot \exp [(g - α_{s}) \cdot L_{1}] \cdot \exp [(g - α_{s} - g_{11}^{t h}) \cdot L_{2}]

In order to amplify LP01 mode and suppress LP11 and other higher-order modes, the requirements are $P_{01}' > P_{01}$ and $P_{11}' < P_{11}$ , Then the calculation results are:

{\begin{cases} (g - α_{s}) / g_{11}^{t h} < L_{2} / (L_{1} + L_{2}) < (g - α_{s}) / g_{01}^{t h} \\ 0 < (g - α_{s}) / g_{11}^{t h} < 1 \end{cases}

Thus the value of $L_{1} / L_{2}$ is decided by $g$ , $α_{s}$ , $g_{01}^{t h}$ and $g_{11}^{t h}$ .

3. The output laser characteristics of the GGIA-GF

Figure 5 shows the schematic diagram of one-end-pumped fiber laser based on GGIA-GF. $P_{p}^{}$ is pump light injected into the fiber. $P_{p}^{+} (z)$ and $P_{p}^{-} (z)$ are forward and backward pump light power, respectively. $P_{s}^{+} (z)$ and $P_{s}^{-} (z)$ are forward and backward signal light power, respectively.

Fig. 5 The schematic diagram of one-end-pumped GGIA-GF fiber laser.

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For the Nd³⁺-doped one-end-pumped GGIA-GF fiber laser, rate equations (Eqs.) can be expressed as (12), (13), (14):

\frac{N_{2} (z)}{N} = \frac{\frac{[P_{P}^{+} (z) + P_{P}^{-} (z)] σ_{a p} Γ_{P}}{h ν_{P} A} + \frac{[P_{s}^{+} (z) + P_{s}^{-} (z)] σ_{a s} Γ_{S}}{h ν_{s} A}}{\frac{[P_{P}^{+} (z) + P_{P}^{-} (z)] (σ_{a p} + σ_{e p}) Γ_{P}}{h ν_{P} A} + \frac{1}{τ} + \frac{[P_{s}^{+} (z) + P_{s}^{-} (z)] (σ_{a s} + σ_{e s}) Γ_{S}}{h ν_{s} A}}

\pm \frac{d P_{P}^{\pm} (z)}{d z} = - Γ_{P} [σ_{a p} N - (σ_{a p} + σ_{e p}) N_{2} (z)] P_{P}^{\pm} (z) - (α_{p} + l_{p}) P_{P}^{\pm} (z)

\pm \frac{d P_{S}^{\pm} (z)}{d z} = Γ_{S} [(σ_{e s} + σ_{a s}) N_{2} (z) - σ_{a s} N] P_{S}^{\pm} (z) - (α_{s} + l_{s}) P_{S}^{\pm} (z)

Where, N₂ (z) is the upper level particle number density, N is the concentration of doped Nd³⁺. Γ_p and Γ_s are pump and signal field overlaps. $ν_{p}$ and $ν_{s}$ are pump and signal frequencies, respectively. A is the cross-sectional area of the core. $τ$ is the spontaneous lifetime and the parameter h is Planck’s constant. c is the speed of light in vacuum. $σ_{a p}$ and $σ_{e p}$ are the absorption and emission cross sections of pump light, respectively. $σ_{a s}$ and $σ_{e s}$ are the absorption and emission cross sections of signal light respectively. $α_{p}$ and $α_{s}$ are the host glass absorption coefficients of the pump and signal light respectively. $l_{p}$ and $l_{s}$ are the additional leakage losses of the pump and signal light respectively, and they are [10]:

{\begin{cases} l_{p} = (L {}_{2}/ L) \cdot [λ_{p}^{2} / {(2 π)}^{2} \cdot a^{3}] \cdot \sqrt{133.8 / (- 2 n_{0}^{3} Δ n}) \\ l_{s} = (L {}_{2}/ L) \cdot [λ_{s}^{2} / {(2 π)}^{2} \cdot a^{3}] \cdot \sqrt{133.8 / (- 2 n_{0}^{3} Δ n}) \end{cases}

According to the boundary conditions:

{\begin{cases} P_{s}^{+} (0) = R_{1} P_{s}^{-} (0) \\ P_{s}^{-} (L) = R_{2} P_{s}^{+} (L) \end{cases}

The output laser power $P_{o u t}$ and slope efficiency $η$ under one-end-pumped condition are expressed by Eqs. (17) and (18), respectively, which are derived from Eqs. (12)-(14). The parameters used in the numerical calculations are listed in Table 1.

Table 1. The parameters used in numerical calculations

View Table

\begin{array}{l} P_{o u t} = \frac{(1 - R_{2}) \sqrt{R_{1}}}{(1 - R_{1}) \sqrt{R_{2}} + (1 - R_{2}) \sqrt{R_{1}}} \cdot \frac{h ν_{s} A}{τ Γ_{s} (σ_{a s} + σ_{e s})} \cdot \\ {P_{p}^{} \frac{ν_{s}}{ν_{p}} (1 - \frac{α_{p}}{α}) [1 - \exp (- α L)] \cdot \frac{τ Γ_{s} (σ_{a s} + σ_{e s})}{h ν_{s} A} - [(N Γ_{s} σ_{a s} + α_{s}) L + \ln \frac{1}{\sqrt{R_{1} R_{2}}}]} \end{array}

η = \frac{(1 - R_{2}) \sqrt{R_{1}}}{(1 - R_{1}) \sqrt{R_{2}} + (1 - R_{2}) \sqrt{R_{1}}} \cdot {\frac{ν_{s}}{ν_{p}} (1 - \frac{α_{p}}{α}) [1 - \exp (- α L)]}

Where, $α = Γ_{p} σ_{a p} N - Γ_{p} (σ_{a p} + σ_{e p}) K L + α_{p}, K = [- \ln (R_{1} R_{2}) / (2 L) + (σ_{a s} Γ_{s} N + α_{s})] / [Γ_{s} (σ_{a s} + σ_{e s})]$ .

The relationship between output laser power and pump power is shown in Fig. 6, when L₁: L = 1, 0 and 0.75, respectively, the whole fiber is changed into GG-IGF, GG-IAGF and GGIA-GF, respectively. Figure 6 shows that the laser threshold of GG-IGF, GG-IAGF and GGIA-GF is 2.96W, 3.84W and 6.48W, respectively. By calculation, 3.84W:6.48W is 59.26%, which shows the threshold of GGIA-GF is 59.26% of GG-IAGF’s threshold. Furthermore, (6.48W-3.84W):(3.84W-2.96W) is 3:1, which just equals to L₁:L₂. That is to say, compared with the GG-IAGF, the reduction proportion of the GGIA-GF laser threshold is decided by L₁:L₂.

Fig. 6 The relationship between Pout and Pp.

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The slope efficiency of GG-IGF, GG-IAGF and GGIA-GF lasers is shown in Fig. 7. The saturated slope efficiency of GG-IGF, GG-IAGF and GGIA-GF is 76.33%, 76.14% and 75.57%, and the difference among them is very small, but (76.14%-75.57%):(76.33%-76.14%) is also 3:1, which equals to L₁:L₂. That is to say, the difference among them is decided by L₁:L₂ as well.

Fig. 7 The relationship between $η$ and L.

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When fiber radius a becomes smaller, as shown in Fig. 2, the corresponding allowed value of L₁:L₂ becomes bigger, then the laser threshold and slope efficiency of GGIA-GF become closer to that of GG-IGF.

4. Conclusion

In this work, an improved model of GGIA-GF is proposed and numerically demonstrated. Single mode conditions and output laser characteristics of the GGIA-GF are obtained. Under the conditions of single mode oscillation, the value of Min (L₂/L) decreases with the decrease of a and $Δ n$ .When $Δ n = - 0.005$ , a = 70μm, the Min (L₂/L) is 19.95%. When the value of L₂/L changes, the allowed range of R₂ varies in order to maintain single mode oscillation. After the value of (L₂/L) is determined, the allowed range of R₂ increases with the increase of the fiber length L. The output laser power and slope efficiency are both derived from the improved rate equations. Compared with the GG-IAGF, the reduction proportion of the GGIA-GF laser threshold is decided by L₁/L₂. The difference of the laser threshold and slope efficiency among GG-IGF, GG-IAGF and GGIA-GF are both decided by L₁/L₂. The smaller the a and $Δ n$ , the bigger the allowed value of L₁/L₂, then the laser threshold and slope efficiency of GGIA-GF become closer to that of GG-IGF. In conclusion, GGIA-GF will be better applied in the field of large mode area and single mode fiber lasers through the rational design.

Acknowledgments

This work was supported by: the National Natural Science Foundation of China (No. 61077070), the Open Research Fund of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences (No. SKLST201405), the Jiangsu Province Universities Graduate Student Research and Innovation Program of China (No. CXZZ13_0467), and Nanjing University of Posts and Telecommunications Research Fund (No. NY214125).

References and links

1. Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32(17), 2505–2507 (2007). [CrossRef] [PubMed]

2. A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006). [CrossRef]

3. V. Sudesh, T. McComb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008). [CrossRef]

4. W. Hageman, Y. Chen, X. Wang, L. Gao, G. U. Kim, M. Richardson, and M. Bass, “Scalable side-pumped, gain-guided index-antiguided fiber laser,” J. Opt. Soc. Am. B 27(12), 2451–2459 (2010). [CrossRef]

5. H. S. Kim, T. McComb, V. Sudesh, and M. C. Richardson, “Numerical Investigation of Beam Propagation inside an Index Antiguided Fiber Laser” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, VOLS: 1–9(Optical Society of America, 2008), paper JTuA80.

6. A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20(8), 1617–1628 (2003). [CrossRef] [PubMed]

7. A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24(8), 1677–1682 (2007). [CrossRef]

8. R.Sims, V.Sudesh, T.McComb, Y.Chen, M.Bass, M.Richardson. “Diode-pumped very large core, gain guided, index antiguided single mode fiber laser” in Advanced Solid-State Photonics, (Optical Society of America, 2009), paper WB3.

9. T. McComb, M. Richardson, and V. Sudesh, “Hybrid gain guiding in laser resonators” US Patent, Patent No., US 7,881,347 B2 (2011).

10. X. Shen, H. Zou, H. T. Tang, and W. Wei, “Threshold characteristics analysis of a uniformly side-pumped Yb3+-doped gain-guided and index-antiguided fiber laser,” Opt. Laser Technol. 68, 1–5 (2015). [CrossRef]

Parameters	values	Parameters	values
Γ_p	0.5	τ	5.92 × 10⁻⁴s
Γ_s	0.5	σ_ap	3.69 × 10⁻²⁰cm²
$λ$ _s	1053nm	σ_ep	6.67 × 10⁻²²cm²
$λ$ _p	808nm	σ_as	3.23 × 10⁻²²cm²
L	0-20cm	σ_es	1.87 × 10⁻²⁰cm²
a	50-100μm	α_p	0.015cm⁻¹
n₀	1.530	α_s	0.015cm⁻¹
$Δ n$	−0.003–-0.007	R₁	1
N	1.554 × 10²⁰ (cm⁻³)	R₂	0.6-0.95
$h$	6.6260775 × 10⁻³⁴J·s	c	3 × 10⁸m/s

Gain guided and index alternate-guided fibers designed for large-mode-area and single-mode laser with higher output power and slope efficiency

Abstract

1. Introduction

2. Structure and single mode characteristics of the GGIA-GF

3. The output laser characteristics of the GGIA-GF

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (18)

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