Self-organization of arrays of two mutually-injected fiber lasers: theoretical investigation

Jianqiu Cao; Qisheng Lu; Jing Hou; Xiaojun Xu

doi:10.1364/OE.17.007694

1. Introduction

Coherent beam combining of fiber laser arrays is a promising method for breaking the power limit of single fiber lasers and realizing laser beams with both high power and good beam quality, and therefore, has attached much attention [1–3]. This method requires that the elementary beams, exported from fiber laser array, should be coherent temporally and spatially. However, because of the sensitivity of fiber lasers and the shortness of laser wavelength, it is really a puzzling problem to make elementary beams of laser array coherent spatially (i.e., make these elementary beams phase-locked). Many attempts were carried out to solve this problem, e.g., active phase controlling [4–6], transverse-mode selection [7–11], self-organized fiber laser arrays [12–20] and so on.

Among these attempts, one scheme was to make elementary fiber lasers injected mutually with each other [19, 20]. This scheme was also used to build solid-state laser arrays [21]. This scheme can make elementary lasers phase-locked (also called “mutual injection-locking”) without any active control [19–21]. This phenomenon is considered as a self-organization process, and thus, this array is one kind of self-organized fiber laser arrays. Furthermore, this kind of arrays can be easily scaled. Thus, this scheme is attractive. Currently, mutual injection-locking of two-fiber laser array was studied. The phase locked state was experimentally observed and the output spectral property of the array was discussed. Some preliminary analyses were also carried out.

Phase-locking is a key to understand the array of mutually-injected lasers. To analyze the phase locking of this array, a mutually injection-locking theory was presented in Ref. 21. With this theory, it was predicted that there should be one phase-locked state with π/2 phase difference in two-laser array of this sort. Later, the model of coupled class B laser array was used to analyze phase locking of mutually-injected fiber laser arrays [19, 20]. With this model, it was predicted that only one phase-locking state (i.e., out-of-phase mode) could be output steadily from two-laser array of this sort. However, the experimental observation (will be shown in Section 3) indicates that the output of mutually-injected fiber laser array is not one pure and stable phase-locking state, which does not agree with the predictions of these theoretical models. It means that these models can not make an effective analysis on the phase-locking mechanism (or self-organization mechanism) of mutually-injected fiber laser array. The mechanism of self-organization of this array is still not very clear.

In this paper, with the help of the model presented in Ref. 22, the array of two mutually-injected fiber lasers is theoretically investigated. By analyzing the steady state of this array, it is found that phase locking of this array is induced by the longitudinal-mode competition in compound cavity of this array. The phase-locked states of this array are studied. Furthermore, the performance of this array is also discussed.

2. Description of the array of mutually-injected fiber lasers

The interest of this paper is focused on the array with the configuration shown in Fig. 1. Gain is provided by two ion-doped fibers, each core pumped through a wavelength division multiplexer (WDM). Fiber Bragg gratings (FBG) are used as the high reflectors. The cavity of each fiber laser is defined by FBG and the output face of the output port. C-1 and C-2 are two 1×2 couplers which make two fiber lasers (I & II) mutually injected.

Fig. 1. Scheme of the array of two mutually-injected fiber lasers.

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The configuration of this array can be divided into three sections: gain section, coupling-output section and feedback section. Here, with the purpose of studying the self-organization mechanism of this array, we merely care about the steady state of this array which can be described by a set of differential Eqs. with boundary conditions [22]. These Eqs. are shown as follows:

c \frac{d A_{m}^{(+)} (x)}{dx} = g_{m} (x) A_{m}^{(+)} (x)

- c \frac{d A_{m}^{(-)} (x)}{dx} = g_{m} (x) A_{m}^{(-)} (x)

c \frac{d φ_{m}^{(+)} (x)}{dx} = - (ω_{cm} - ω) - \frac{(ϖ - ω) g_{m} (x)}{γ_{⊥}}

- c \frac{d φ_{m}^{(-)} (x)}{dx} = - (ω_{cm} - ω) - \frac{(ϖ - ω) g_{m} (x)}{γ_{⊥}}

- γ_{/ /} [g_{m} (x) - g_{0 m}] - g_{m} (x) \cdot {{[A_{m}^{(+)} (x)]}^{2} + {[A_{m}^{(-)} (x)]}^{2}}

where A_m ^(±) (x) and φ_m ^(±)(x) are the amplitude (arbitrary unit) and phase (unit: rad) of the complex slow-varying envelope of laser fields. The superscripts “+” and “ − ” correspond, respectively, to right-going and left-going fields propagating in the array. The subscript m=1, 2 represent two fiber lasers. g_m (x) is the gain coefficient (unit: s^-1), and g _0m is the pump coefficient (unit: s^-1). ω is the laser frequency (unit: rad/s), ϖ is the atomic frequency (unit:rad/s). ω_cm − ck_m is the resonance frequency (unit: rad/s) of the mth cavity, C is the velocity of light in vacuum (unit: m/s), k_m is the wave vector (unit: rad/m). γ _⊥ and γ_∥ are transverse and longitudinal relaxation rates (unit: s^-1), respectively.

The functions of two couplers (C-1 and C-2) can be described as

F_{cm} = (\begin{matrix} \sqrt{1 - ε_{m}} & i \sqrt{ε_{m}} \\ i \sqrt{ε_{m}} & \sqrt{1 - ε_{m}} \end{matrix}), (m = 1,2)

where ε_m is the percentage of the input power coupled crossly into the other port [23]. Then, two boundary conditions corresponding to coupling-output and feedback sections are

(\begin{matrix} A_{1}^{(-)} (L) \\ A_{2}^{(-)} (L) \end{matrix}) = F_{co} (\begin{matrix} A_{1}^{(+)} (L) \\ A_{2}^{(+)} (L) \end{matrix})

(\begin{matrix} A_{1}^{(+)} (0) \\ A_{2}^{(+)} (0) \end{matrix}) = F_{fb} (\begin{matrix} A_{1}^{(-)} (0) \\ A_{2}^{(-)} (0) \end{matrix})

where

F_{co} = (\begin{matrix} r_{1} (1 - ε_{1}) \exp [i \frac{2 ω}{c} (l_{1}^{(c)} + l_{1}^{(r)}) + iδ ϕ_{R 1}] & - \sqrt{ε_{1} ε_{2}} \exp [i \frac{ω}{c} (l_{cc} + l_{1}^{(c)} + l_{2}^{(c)})] \\ - \sqrt{ε_{1} ε_{2}} \exp [i \frac{ω}{c} (l_{cc} + l_{1}^{(c)} + l_{2}^{(c)})] & r_{2} (1 - ε_{2}) \exp [i \frac{2 ω}{c} (l_{2}^{(c)} + l_{2}^{(r)}) + iδ φ_{R 2}] \end{matrix})

F_{fb} = (\begin{matrix} {r'}_{1} \exp (i \frac{2 ω}{c} {l'}_{1}^{(r)} + iδ {ϕ'}_{R 1}) & 0 \\ 0 & {r'}_{2} \exp (i \frac{2 ω}{c} {l'}_{2}^{(r)} + iδ {ϕ'}_{R 2}) \end{matrix})

Here, r_m (or r'_m) is the reflectivity coefficient of the output face (or FBG) of the mth laser cavity; δϕ_Rm (δϕ'_Rm) are the phase differences caused by the reflecting of the output face (FBG).l_m ^(c) is the optical path length (OPL) between x = L and the coupler of mth cavity; l_m ^(r) is the OPL between the coupler and the output face; l'_m ^(r) is the OPL between x = 0 and FBG; l_cc is the OPL between two couplers (i.e., C-1 and C-2). Note that the fields A _m ^(±) (0) and A _m ^(±) (L) should be expressed as

A_{m}^{(\pm)} (0) = A_{m}^{(\pm)} (0) \exp [i φ_{m}^{(\pm)} (0)], A_{m}^{(\pm)} (L) = A_{m}^{(\pm)} (L) \exp [i φ_{m}^{(\pm)} (L)] \exp (\pm i k_{m} L)

respectively. By solving Eqs. (1)–(5) with the boundary conditions (7) and (8), the steady state of this array will be given. In the following section, the results will be given and the steady phase state of this array will be discussed.

3. Results and discussions

Firstly, let’s consider the equivalent reflectivity coefficient f_m and the phase difference ϑ_m caused by the coupler-output section. Here, f_m and ϑ_m are defined as

f_{m} = \frac{A_{m}^{(-)} (L)}{A_{m}^{(+)} (L)}, ϑ_{m} = φ_{m}^{(-)} (L) - φ_{m}^{(+)} (L), (m = 1,2)

where ϕ_m ^(±) (L) = φ_m ^(±) (L) ± km_L.

From Eqs. (1)–(4) and (7), f_m and ϑ_m can be obtained as

f_{1} = {{[r_{1} (1 - ε_{1})]}^{2} + {[\sqrt{ε_{1} ε_{2}} \frac{A_{2}^{(+)} (L)}{A_{1}^{(+)} (L)}]}^{2} + 2 r_{1} (1 - ε_{1}) \sqrt{ε_{1} ε_{2}} \frac{A_{2}^{(+)} (L)}{A_{1}^{(+)} (L)} \cos θ_{1}}^{\frac{1}{2}}

f_{2} = {{[r_{2} (1 - ε_{2})]}^{2} + {[\sqrt{ε_{1} ε_{2}} \frac{A_{1}^{(+)} (L)}{A_{2}^{(+)} (L)}]}^{2} + 2 r_{2} (1 - ε_{2}) \sqrt{ε_{1} ε_{2}} \frac{A_{1}^{(+)} (L)}{A_{2}^{(+)} (L)} \cos θ_{2}}^{\frac{1}{2}}

ϑ_{m} = \frac{2 ω}{c} (l_{m}^{(c)} + l_{m}^{(r)}) + δ ϕ_{R m} + Θ_{m}, (m = 1,2)

where

θ_{1} = \frac{ω}{c} (l_{cc} + l_{2}^{(c)} - l_{1}^{(c)} - 2 l_{1}^{(r)}) + ϕ_{2}^{(+)} (L) - ϕ_{1}^{(+)} (L) + π - δ ϕ_{R 1}

θ_{2} = \frac{ω}{c} (l_{cc} + l_{1}^{(c)} - l_{2}^{(c)} - 2 l_{2}^{(r)}) + ϕ_{1}^{(+)} (L) - ϕ_{2}^{(+)} (L) + π - δ ϕ_{R 2}

Θ_{1} = \arctan {\frac{\sqrt{ε_{1} ε_{2}} \frac{A_{2}^{(+)} (L, t)}{A_{1}^{(+)} (L, t)} \sin (θ_{1})}{r_{1} (1 - ε_{1}) + \sqrt{ε_{1} ε_{2}} \frac{A_{2}^{(+)} (L, t)}{A_{1}^{(+)} (L, t)} \cos (θ_{1})}}

Θ_{2} = \arctan {\frac{\sqrt{ε_{1} ε_{2}} \frac{A_{1}^{(+)} (L)}{A_{2}^{(+)} (L)} \sin (θ_{2})}{r_{2} (1 - ε_{2}) + \sqrt{ε_{1} ε_{2}} \frac{A_{1}^{(+)} (L)}{A_{2}^{(+)} (L)} \cos (θ_{2})}}

From Eqs. (13) and (14) and Eqs. (16) and (17), it can be found that the equivalent reflectivity f_m varies with the laser frequency ω, because θ_m is the function of ω. It means that the longitudinal modes own different losses in the coupler-output section. According to the mode-competition theory, the longitudinal mode(s) corresponding to the minimum loss (i.e., the maximum equivalent reflectivity) will be dominant in the mode-competition and oscillate in the array. Therefore, the longitudinal mode(s) should satisfy the maximum conditions of f ₁ and f ₂ , i.e.,

θ_{1} + θ_{2} = 2 nπ, θ_{1} = 2 n_{1} π, (n, n_{1} = 0, \pm 1, \pm 2, \dots)

simultaneously. Combining Eqs. (20) with Eqs. (16) and (17), we can obtain that

\frac{ω}{c} (2 l_{cc} - 2 l_{1}^{(r)} - 2 l_{2}^{(r)}) - δ ϕ_{R 1} - δ ϕ_{R 2} = 2 n π

\frac{ω}{c} (l_{cc} + l_{2}^{(c)} - l_{1}^{(c)} - 2 l_{1}^{(r)}) + ϕ_{2}^{(+)} (L) - ϕ_{1}^{(+)} (L) + π - δ ϕ_{R 1} = 2 n_{1} π

From Eqs. (21) and (22), it can be got that

Δ θ_{21} = θ_{2} - θ_{1} = q_{1} π - \frac{1}{2} (δ ϕ_{R 2} - δ ϕ_{R 1}), (q_{1} = 0, \pm 1, \pm 2, \dots)

where $θ_{1} = \frac{ω}{c} (l_{1}^{(c)} + l_{1}^{(r)}) + ϕ_{1}^{(+)} (L), θ_{2} = \frac{ω}{c} (l_{2}^{(r)} + l_{2}^{(c)}) + ϕ_{2}^{(+)} (L)$ From Eq. (23), it is very interesting to find that Δθ ₂₁ is just the phase difference of two elementary lasers at the output faces. Meanwhile, the phase difference is a constant which does not vary with the OPL of fibers in the array. It means that two elementary lasers are phase-locked at the output faces.

From above discussion, it is found that the self-organization of this array is virtually a process of longitudinal-mode competition. As a result, the dominant mode(s), which own(s) constant phase difference(s) at the output faces, will ultimately oscillate in the array and lead(s) to the output of phase-locked state(s).

From Eq. (23), it can be seen that there are two phase-locked states corresponding to

q_{1} = {\begin{matrix} 0, \pm 2, \pm 4, \pm 6, \pm 8, \dots (State I) \\ \pm 1, \pm 3, \pm 5, \pm 7, \dots (State II) \end{matrix}

respectively. If δϕ _R2 is equal to δϕ _R1, State I is the in-phase state and State II is the out-of-phase state. From above analysis, it is difficult to tell which state is more dominant. Two states seem to be of the same importance, and which one can eventually output is determined by the state of laser cavity. Then, because of the irregular variation of fibers in the array (caused by the unavoidable disturbance in the circumstance), the two states should transfer from one to the other in experiment. Actually, in our preliminary experiment with the same system shown in Fig. 1 of Ref. 20, we really find that there are two, and only two, phase-locked states (see Fig. 2) which transfer irregularly from one to the other. Frankly speaking, we can not tell which state is in-phase (or out-of-phase) state because of the unavoidable errors of arrangements of output faces. In spite of that, we can see clearly that the phase difference between two elementary lasers in one state is π different from that in the other state. This observation is identical with the theoretical prediction (see Eq. (23)).

Fig. 2. Fringe patterns of interference of two elementary lasers. Both of fringes are recorded with the same experimental arrangement. Each fringe corresponds to a phase-locked state. As is marked by the white dashed line, the position of maximum intensity of fringe (a) is identical with that of the minimum intensity of fringe (b), and vice versa.

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As is discussed above, the array can be self-organized because of the dependence of f_m on ω which is determined by the cosine term of θ_m in the expression of fm (see Eqs. (13) and (14) and Eqs. (16) and (17)). Therefore, the weight of the cosine term will determine the longitudinal-mode discrimination of the array, and ultimately, influence the output state of the array. If the weight of the cosine term is too small, the longitudinal-mode discrimination will be too weak to realize effective self-organization. As a result, the coherence of elementary beams will decrease and the visibility of the interference fringe will be lowered.

Define that

R t_{1} = \frac{2 r_{1} (1 - ε_{1}) \sqrt{ε_{1} ε_{2}} \frac{A_{2}^{(+)} (L)}{A_{1}^{(+)} (L)}}{{[r_{1} (1 - ε_{1})]}^{2} + {[\sqrt{ε_{1} ε_{2}} \frac{A_{2}^{(+)} (L)}{A_{1}^{(+)} (L)}]}^{2}}, R t_{2} = \frac{2 r_{2} (1 - ε_{2}) \sqrt{ε_{1} ε_{2}} \frac{A_{1}^{(+)} (L)}{A_{2}^{(+)} (L)}}{{[r_{2} (1 - ε_{2})]}^{2} + {[\sqrt{ε_{1} ε_{2}} \frac{A_{1}^{(+)} (L)}{A_{2}^{(+)} (L)}]}^{2}}

Then, when Rt ₁ and Rt ₂ get the maximum values (i.e., 1) simultaneously, the array has the best performance of longitudinal-mode discrimination. To make Rt_m maximum, following conditions should be satisfied, i.e.,

r_{1} (1 - ε_{1}) = \sqrt{ε_{1} ε_{2}} \frac{A_{2}^{(+)} (L)}{A_{1}^{(+)} (L)}, r_{2} (1 - ε_{2}) = \sqrt{ε_{1} ε_{2}} \frac{A_{1}^{(+)} (L)}{A_{2}^{(+)} (L)}

From Eqs. (12)–(14), it can be found that, when Eqs. (26) are met, A_m ⁽⁻⁾ (L) are induced by interference of two fields with the same intensity. Therefore, A_m ⁽⁻⁾(L) (or f_m) are most sensitive to the laser frequency (or longitudinal modes) at this time. Then, the array performs the longitudinal-mode discrimination best.

If the configuration of the array is symmetrical, i.e., ε = ε ₁ = ε ₂ , r = r ₁ = r ₂ , and A ₁ ⁽⁺⁾ (L) = A ₂ ⁽⁺⁾ (L), we can get the relationship between ε and r from Eqs. (26). That is

ε = \frac{r}{1 + r}

Here, it should be noticed that r ≠ 0,1, otherwise, the laser system is not an array of two mutually-injected lasers any more.

At last, it should be noticed that Eqs. (20) are not sufficient to determine the stationary longitudinal modes. In addition, three other condition is also needed, i.e.,

Δ Φ_{1} = \frac{ω}{c} {\bar{L}}_{1} + δ ϕ_{R 1} + Θ_{1} + \frac{(ϖ - ω)}{γ_{⊥}} \ln [f_{1} r_{1}'] + δ {ϕ'}_{R 1} = 2 n_{3} π

Δ Φ_{2} = \frac{ω}{c} {\bar{L}}_{2} + δ ϕ_{R 2} + Θ_{2} + \frac{(ϖ - ω)}{γ_{⊥}} \ln [f_{2} r_{2}'] + δ {ϕ'}_{R 2} = 2 n_{4} π

Δ Φ_{2} - Δ Φ_{1} = 2 n_{5} π, (n_{3}, n_{4}, n_{5} = 0, \pm 1, \pm 2, \pm 3, \dots)

where L̄₁ =2(L+l ₁ ^(c) + l ₁^(r) + l ₁'^(r)) and L̄₂ =2(L+l ₂ ^(c) + l ₂^(r) + l ₂'^(r)). Here, ΔΦ_m is the phase difference per round-trip of the mth elementary laser. Of course, there are two independent Eqs. in Eqs. (28)–(30). Then, there should be some separated stationary modes satisfying these conditions within a given spectrum, as long as the bandwidth of the spectrum is broad enough. Furthermore, Eqs. (21) and (30) also indicate that increasing optical path differences (OPDs) of (l_cc − l ₁ ^(r)) and (L̄₂−L̄₁) , will reduce the frequnecy spacing between adjacent stationary modes (corresponding to adjacent phase-locked states). Thus, there are two way to ensure the existence of phase-locking states in experiment, i.e., using broadband couplers and reflectors, and designing the compound cavity of this array reasonably to reducing the frequency spacing between adjacent phase-locked states.

4. Conclusion

In conclusion, the array of two mutually-injected fiber lasers is theoretically investigated. The self-organization mechanism of this array is demonstrated as the longitudinal-mode competition in the compound laser cavity. Although the mechanism is investigated by studying a two-laser array, it is believed that this mechanism has relevance to other arrays of this sort. It is also found that two phase-locked states can be formed at the output faces. This theoretical result agrees well with the experimental observation. Furthermore, our analysis shows that there is an optimal configuration of the array, which will be of help for the design of this array. Aiming at ensuring the existence of phase-locked states, some advices are also given.

From our discussions, it should be noticed that two phase-locked states will emerge alternately because of the irregular variation of laser cavity in experiment. Thus, it is impossible for this array to keep only one of these states exported steadily. To realize single-state output, some additional control should be added to this array. Certainly, the array of two fiber lasers is the simplest case. Detailed study on more complicated arrays with more-than-two lasers will be carried out in the future.

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Self-organization of arrays of two mutually-injected fiber lasers: theoretical investigation

Abstract

1. Introduction

2. Description of the array of mutually-injected fiber lasers

3. Results and discussions

4. Conclusion

References and links

Cited By

Figures (2)

Equations (31)

Optics Express