Dynamical model for self-organized fiber laser arrays

Jianqiu Cao; Qisheng Lu; Jing Hou; Xiaojun Xu

doi:10.1364/OE.17.005402

1. Introduction

Coherent beam combining of fiber laser arrays is a promising scheme to realize high-power, high-radiance lasers [1–3]. This scheme aims at combining the elementary beams exported from fiber laser arrays coherently and has two advantages: 1) this scheme can make full use of the outstanding properties of fiber lasers (e.g., heat dispersion, compactness, good beam quality, etc.); 2) this scheme can break down the power limitations of single fiber lasers. Therefore, this scheme has attached much attention.

Coherent beam combining requires that the elementary beams of fiber laser arrays should be coherent temporally and spatially. To satisfy this requirement, many attempts were carried out [1–3]. Among these attempts, one method is coupling elementary lasers of array with each other [4–14]. It is found experimentally that the spectrum and phases of elementary lasers can be synchronized by optical-field coupling between elementary lasers. Because this kind of arrays can output coherent beams without any active control, they are also called self-organized laser arrays [8–10]. Although many studies have been carried out on this kind of arrays [15–19], current theoretical understanding of the arrays is still in its infancy.

To make a further investigation of the self-organized fiber laser arrays, a dynamics model is needed. Fortunately, there is a useful analogue, i.e., the model of arrays of coupled solid-state lasers [20–24]. In this model, the optical field was assumed to change very little between consecutive round-trips (the standard slowly varying wave approximation). This assumption is reasonable for solid-state lasers which own small-gain, small-loss and short cavities, but not suitable for fiber lasers, because the gain and loss of fiber lasers are higher than that of solid-state lasers and the cavities of fiber lasers are also much longer. Therefore, this model is not appropriate for fiber laser arrays.

There was also a model, developed by Rogers et al [17, 18], for high-gain fiber laser array. This model, presented as a set of iterative maps, does not rely on the standard slowly varying wave approximation. Nevertheless, the rate equation is used in this model to describe the interaction between laser field and gain medium. As is known to all, the rate equation is useful to predict the intensity of optical field, but not appropriate for analyzing phenomena relative to the phase of laser field. Actually, in coherent beam combining, we care mostly about the relationship of phases of elementary lasers.

In this paper, an improved dynamical model is presented. This dynamical model does not only abandon the standard slowly varying wave approximation, but also use the Maxwell-Bloch equations to describe the interaction between field and gain medium in this dynamical model. Furthermore, instead of giving the iterative maps, the improved model just consists of partial differential equations, which can describe the whole system more precisely.

In the Section 2, the dynamical model will be introduced. In the Section 3, this model will be applied to 2-laser interferometric array. Because the phase of field is cared more in fiber laser arrays, we focus our discussions on the phase dynamics. The results agree well with the reported experimental results.

2. Introduction of the dynamical model

Our interest is focus on the arrays with the configuration shown in Fig. 1. This configuration can be divided into three sections: feedback section, gain section, and coupling-output section. The feedback section consists of reflectors with high reflectivity, e.g., holophotes or fiber Bragg gratings (FBGs). The gain section is the gain medium, e.g., ion-doped fibers. The coupling-output section is composed of couplers and output faces. The couplers are used to make elementary lasers interact with each other. The output face is often performed by the polished fiber faces or reflectors with low reflectivity.

Fig. 1. Scheme of fiber laser arrays

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2.1 Evolution of fields in gain section

Firstly, we will introduce the evolution of fields in the gain section. In fiber lasers, the interaction between the field (i.e., the electric field) and gain medium can be described by the Maxwell-Bloch equations [25]. Then, in the mth elementary laser of an N-laser array, the set of equations is

{\dot{E}}_{m} (x, t) - i \frac{c^{2}}{2 ω} \frac{\partial^{2} E_{m} (x, t)}{\partial x^{2}} - i \frac{ω}{2} E_{m} (x, t) = \frac{iω}{2 ε_{0}} P_{m} (x, t)

{\dot{P}}_{m} (x, t) = [- i (ϖ - ω) - γ_{⊥}] P_{m} (x, t) + \frac{1}{iħ} E_{m} (x, t) {∣ θ_{12} ∣}^{2} D_{m} (x, t)

{\dot{D}}_{m} (x, t) = - γ_{∥} (D_{m} - D_{0 m}) - \frac{2}{iħ} [E_{m} (x, t) P_{m}^{*} (x, t) - E_{m}^{*} (x, t) P_{m} (x, t)]

where E(x,t) is the complex field envelope, P(x,t) is the atomic polarization envelope and D(x,t) is the population difference, D ₀ is the population difference in the absence of the optical field (i.e., only decided by pump ), and the dot represents the partial derivative with respect to time. ω is the laser frequency, ϖ is the atomic frequency. c is the velocity of light in vacuum, ε ₀ is the dielectric constant in vacuum, θ ₁₂ is the atomic dipole moment matrix element. γ _⊥ and γ _∥ are transverse and longitudinal relaxation rates, respectively.

E _m ^*(x, t) and P _m ^*(x, t) are conjugate values of E _m(x, t) and P _m(x, t), respectively.

In spite of the multi-longitudinal-mode nature of fiber lasers, the model is also built on the basis of the single-longitudinal-mode assumption, which has been used in former studies on laser arrays [17–24]. Generally, the field oscillating in Fabry-Perot cavity is assumed as standing waves in the low-gain and low-loss resonators. Nevertheless, this assumption is not suitable for fiber lasers with high gain and high loss. Thus, in this model, the electric field is decomposed into two counter-propagating traveling-waves, i.e.,

E_{m} (x, t) = E_{m}^{(+)} (x, t) \exp (i k_{m} x) + E_{m}^{(-)} (x, t) \exp (- i k_{m} x)

where E _m(x, t) and E _m ^(-)(x, t) are complex amplitudes of two counter-propagating waves, k_m is the wave vector. Eq. (2) will return to the form of standing wave with standard slowly varying wave approximation. Correspondingly, P _m(x, t) can also be written as

P_{m} (x, t) = P_{m}^{(+)} (x, t) \exp (i k_{m} x) + P_{m}^{(-)} (x, t) \exp (- i k_{m} x)

Substituting (2) and (3) into Eq. (1), one can obtain that

c \frac{\partial E_{m}^{(+)} (x, t)}{\partial x} + {\dot{E}}_{m}^{(+)} (x, t) = - i (ω_{cm} - ω) E_{m}^{(+)} (x, t) + \frac{iω}{2 ε_{0}} P_{m}^{(+)} (x, t)

- c \frac{\partial E_{m}^{(-)} (x, t)}{\partial x} + {\dot{E}}_{m}^{(-)} (x, t) = - i (ω_{cm} - ω) E_{m}^{(-)} (x, t) + \frac{iω}{2 ε_{0}} P_{m}^{(-)} (x, t)

{\dot{P}}_{m}^{(+)} (x, t) = [- i (ϖ - ω) - γ_{⊥}] P_{m}^{(+)} (x, t) + \frac{1}{iħ} E_{m}^{(+)} (x, t) {∣ θ_{12} ∣}^{2} D_{m} (x, t)

{\dot{P}}_{m}^{(-)} (x, t) = [- i (ϖ - ω) - γ_{⊥}] P_{m}^{(-)} (x, t) + \frac{1}{iħ} E_{m}^{(-)} (x, t) {∣ θ_{12} ∣}^{2} D_{m} (x, t)

{\dot{D}}_{m} (x, t) = - γ_{∥} (D_{m} - D_{0 m}) - \frac{2}{iħ} [E_{m} (x, y) P_{m}^{*} (x, t) - E_{m}^{*} (x, t) P_{m} (x, t)]

where ω_cm - ck_m is the resonance frequency of the mth cavity. Here, the terms ∂ ² E _m ⁽⁺⁾(x,t)/∂x ² and ∂ ² E _m ^(-)(x,t)/∂x ² are neglected (assuming that ∂ ² E _m ^(±)(x,t)/∂x ² ≪ k⌊∂ E _m ^(±)(x,t)/∂x⌋).

In fiber lasers, generally, γ _⊥ ≫ γ _∥. Therefore, the polarization variables can be adiabatically eliminated. Demand that P˙ _m ⁽⁺⁾(x,t) = 0 and P˙ _m(x, t) = 0. Then, we can eliminate the equations of polarization (i.e., Eqs. 4c & 4d), and Eqs. (4) can be reduced as

c \frac{\partial E_{m}^{(+)} (x, t)}{\partial x} + E_{m}^{(+)} (x, t) = - i (ω_{cm} - ω) E_{m}^{(+)} (x, t) + g_{m} (x, t) E_{m}^{(+)} (x, t) - \frac{i (ϖ - ω) g (x, t)}{γ_{⊥}} E_{m}^{(+)} (x, t)

- c \frac{\partial E_{m}^{(-)} (x, t)}{\partial x} + {\dot{E}}_{m}^{(-)} (x, t) = - i (ω_{cm} - ω) E_{m}^{(-)} (x, t) + g_{m} (x, t) E_{m}^{(-)} (x, t) - \frac{i (ϖ - ω) g (x, t)}{γ_{⊥}} E_{m}^{(-)} (x, t)

{\dot{g}}_{m} (x, t) = - γ_{∥} [g_{m} (x, t) - g_{0 m} (x, t)] - \frac{4 c σ ε_{0}}{ωħ} g_{m} (x, t) \times

where $σ = \frac{ω γ_{⊥} {∥ θ_{12} ∥}^{2}}{ε_{0} cħ [{(ϖ - ω)}^{2} + {γ_{⊥}}^{2}]}$ is the stimulated emission cross section, and g_m(x,t) = σcD_m(x,t)/2 is the gain coefficient. Because g _0m(x,t )= σcD _0m(x,t)/2 is only determined by pump of the mth elementary laser, g _0m(x,t) is the pump coefficient.

Assuming that

E_{m}^{(+)} (x, t) = E_{m}^{(+)} (x, t) \exp [i ϕ_{m}^{(+)} (x, t)]

E_{m}^{(-)} (x, t) = E_{m}^{(-)} (x, t) \exp [i ϕ_{m}^{(-)} (x, t)]

where E _m ⁽⁺⁾(x,t) and ϕ_m ^(-)(x,t) are the magnitude and phase of E _m ^(-)(x,t); E _m ⁽⁺⁾(x,t) and ϕ_m ^(-)(x,t) are the magnitude and phase of E _m(x,t). Substituting Eqs. (6, 7) into Eq. (5) and assuming that

A_{m}^{(+)} (x, t) = \sqrt{\frac{4 cσ ε_{0}}{ωħ}} E_{m}^{(+)} (x, t), A_{m}^{(-)} (x, t) = \sqrt{\frac{4 cσ ε_{0}}{ωħ}} E_{m}^{(-)} (x, t)

we can get that

c \frac{\partial A_{m}^{(+)} (x, t)}{\partial x} + \frac{\partial A_{m}^{(+)} (x, t)}{\partial t} = g_{m} (x, t) A_{m}^{(+)} (x, t)

- c \frac{\partial A_{m}^{(-)} (x, t)}{\partial x} + \frac{\partial A_{m}^{(-)} (x, t)}{\partial t} = g_{m} (x, t) A_{m}^{(-)} (x, t)

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

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

{\dot{g}}_{m} (x, t) = - γ_{∥} [g_{m} (x, t) - g_{0 m} (x, t)] - g_{m} (x, t) \times

Comparing Eqs. (9) with Eqs. (8–10) in Ref. 17, we find that the iterative maps in Ref. 17 can be derived from Eqs. (9) with two main approximations:

Ignoring the time derivative terms $\frac{\partial A_{m}^{(+)} (x, t)}{\partial t} and \frac{\partial A_{m}^{(-)} (x, t)}{\partial t}$ in Eqs. (9.a & b). It means that the effect of the temporal variation of A_m ⁽⁺⁾(x,t) (A_m ^(-)(x,t)) on the evolution of the gain coefficient g_m(x,t) is neglected. This approximation is reasonable to predict the steady-state of laser arrays, but too crude to analyze the dynamics of laser arrays;
Ignoring the temporal variation of ϕ_m ⁽⁺⁾(x,t) and ϕ_m(x,t), i.e., assuming that the Eqs. (10.c&d) approximate zero. This approximation is reasonable when ω_cm is close to ϖ and the gain g_m(x,t) is not high. Otherwise, this approximation should be re-examined.

Therefore, compared with Eqs.(8–10) in Ref. 17, Eqs.(9) are more suitable to analyze the interaction between the laser field and gain medium, especially when the gain is high.

2.2 Evolution of fields in the coupling-output section and feedback section

Firstly, we will introduce the evolution of fields in the coupling-output section. As is discussed above, the field of the mth laser exported from the gain section (x = L) can be expressed as (see Eq. (2))

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

Then, when the field propagates to the coupler, the field should be

A_{m}^{(+)} (L_{c}, t) = A_{m}^{(+)} (L, t) \exp (i \frac{ω}{c} l_{m}^{(c)})

where x = L_c is the position of the coupler and l_m ^(c) is the optical path length (OPL) between L and L_c. Here, the function of the coupler is represented by the operator

F_{c} = {C_{m, n}}_{m, n = 1}^{m, n = N}

Then, after passing through the coupler, the output field of the mth ports is

{A'}_{m}^{(+)} (L_{c}, t) = \sum_{n = 1}^{N} C_{m, n} A_{n}^{(+)} (L_{c}, t)

Subsequently, the field will reach to the output face. After reflected by the output face, the field will propagate to the coupler, again. At this time, the field will be

A_{m}^{(-)} (L_{c}, t) = r_{m} e^{iδ φ_{Rm}} \exp (i \frac{2 ω}{c} l_{m}^{(r)}) {A'}_{m}^{(+)} (L_{c}, t)

where r_m is the magnitude of the reflectivity of the output face and δφ_Rm is the phase difference caused by reflecting of the output face, and l_m ^(r) is the OPL between the coupler and the output face. After secondly passing through the coupler, the field will be

{A'}_{m}^{(-)} (L_{c}, t) = \sum_{n = 1}^{N} C_{m, n} A_{n}^{(-)} (L_{c}, t)

When the field arrives at the boundary of the gain section (x = L), the field will become

A_{m}^{(-)} (L, t) = {A'}_{m}^{(-)} (L_{c}, t) \exp (i \frac{ω}{c} l_{m}^{(c)})

Here, it should be noticed that

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

From above derivation, it can be found that the evolution of fields in the coupling-output section can be expressed as

(\begin{matrix} A_{1}^{(-)} (L, t) \\ A_{2}^{(-)} (L, t) \\ ⋮ \\ A_{m}^{(-)} (L, t) \\ ⋮ \\ A_{N}^{(-)} (L, t) \end{matrix}) = F (\begin{matrix} A_{1}^{(+)} (L, t) \\ A_{2}^{(+)} (L, t) \\ ⋮ \\ A_{m}^{(+)} (L, t) \\ ⋮ \\ A_{N}^{(+)} (L, t) \end{matrix})

where F is a N × N matrix and can be written as

F = F_{P} F_{c} F_{R} F_{c} F_{p}

{(F_{p})}_{mm} = \exp (i \frac{ω}{c} l_{m}^{(c)}), {(F_{p})}_{jm} = 0, (j, m = 1,2, \dots N; j \neq m)

{(F_{R})}_{mm} = r_{m} e^{iδ φ_{Rm}} \exp (i \frac{2 ω}{c} l_{m}^{(r)}), {(F_{R})}_{jm} = 0, (j, m = 1,2, \dots, N; j \neq m)

With the similar derivation, we can get the evolution of fields in the feedback section. That is

(\begin{matrix} A_{1}^{(+)} (0, t) \\ A_{2}^{(+)} (0, t) \\ ⋮ \\ A_{m}^{(+)} (0, t) \\ ⋮ \\ A_{N}^{(+)} (0, t) \end{matrix}) = F (\begin{matrix} A_{1}^{(-)} (0, t) \\ A_{2}^{(-)} (0, t) \\ ⋮ \\ A_{m}^{(-)} (0, t) \\ ⋮ \\ A_{N}^{(-)} (0, t) \end{matrix})

where

{(F')}_{mm} = {r'}_{m} e^{iδ {φ'}_{Rm}} \exp (i \frac{2 ω}{c} {l'}_{m}^{(r)}), {(F')}_{jm} = 0, (j, m = 1,2, \dots, N; j \neq m)

r′_m is the magnitude of the reflectivity of the reflector (i.e., holophotes or FBGs) and δφ′_Rm is the phase difference caused by reflecting, and l′_m ^(r) is the OPL between the coupler and the reflector. Here, it should also be noted that

A_{m}^{(+)} (0, t) = A_{m}^{(+)} (0, t) \exp [i ϕ_{m}^{(+)} (0, t)], A_{m}^{(-)} (0, t) = A_{m}^{(-)} (0, t) \exp [i ϕ_{m}^{(-)} (0, t)]

By solving Eqs. (9) with the boundary conditions (18) and (22) (see Appendix A), the dynamics of the fiber laser array will be revealed. In the next section, this model will be applied to analyze the phase dynamics of 2-laser interferometric array.

3. Fiber laser arrays of interferometer configuration

Now, we apply this model to a 2-laser interferometric array. The configuration of this array is shown in Fig. 2. The fiber face of the leakage port is cleaved to get a very low reflectivity (assumed to be zero in the following discussion). The polished fiber face is used as the output face of the output port, and its reflectivity is assumed to be r . A 2×2 coupler is used for coupling between elementary lasers.

Fig. 2. Scheme of 2-laser interferometric array

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Because the leakage port is involved in this section, the operator F_R should be written as

F_{R} = \exp (i \frac{2 ω}{c} l_{1}^{(r)}) (\begin{matrix} {re}^{iδ φ_{R}} & 0 \\ 0 & 0 \end{matrix})

where r is the magnitude of the reflectivity of the polished fiber face and δφ_R is the phase shift caused by the reflect of the fiber face. The function of the coupler can be expressed as [27]

F_{c} = (\begin{matrix} \sqrt{1 - ε} & i \sqrt{ε} \\ i \sqrt{ε} & \sqrt{1 - ε} \end{matrix})

where ε is the percentage of the input power of coupled crossly into the other port. Here, the loss of the coupler is negligible because it is much smaller than the loss of the output face. Substitute the Eqs. (25–26) into Eq. (19), and we can obtain that

F = r \exp [i (δ ϕ_{R} + \frac{2 ω}{c} l_{1}^{(r)})] (\begin{matrix} (1 - ε) \exp (i \frac{2 ω}{c} l_{1}^{(c)}) & \sqrt{ε (1 - ε)} \exp [i \frac{ω}{c} (l_{1}^{(c)} + l_{2}^{(c)}) + i \frac{π}{2}] \\ \sqrt{ε (1 - ε)} \exp [i \frac{ω}{c} (l_{1}^{(c)} + l_{2}^{(c)}) + i \frac{π}{2}] & ε \exp [i (\frac{2 ω}{c} l_{2}^{(c)} - π)] \end{matrix})

Assume that

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

Substituting Eq. (27) into Eq. (18), we can get the expressions of f ₁, f ₂,ϑ ₁, ϑ ₂ (see Appendix B).

In fiber laser arrays, the phase difference between two elementary lasers is often cared more about. Next, the evolution of the phase difference will be discussed. From Eqs. (9), we can obtain the evolution of the phase difference between two lasers in the steady state, i.e.,

c \frac{\partial Δ ϕ_{21 st}^{(+)} (x)}{\partial x} = - (ω_{c 2} - ω_{c 1}) - \frac{(ϖ - ω)}{γ_{⊥}} [g_{2 st} (x) - g_{1 st} (x)]

- c \frac{\partial Δ ϕ_{21 st}^{(+)} (x, t)}{\partial x} = - (ω_{c 2} - ω_{c 1}) - \frac{(ϖ - ω)}{γ_{⊥}} [g_{2 st} (x) - g_{1 st} (x)]

where ∆ϕ _21st ⁽⁺⁾(x) = ϕ _2st ⁽⁺⁾(x) - ϕ _1st ⁽⁺⁾(x) and ∆ϕ _21st⁽⁻⁾(x) = ϕ _2st ^(-)(x) - ϕ _1st⁽⁻⁾(x). Then, it can be obtained that

Δ ϕ_{21 st}^{(+)} (L) - Δ ϕ_{21 st}^{(+)} (0) = - \frac{(ω_{c 2} - ω_{c 1})}{c} L - \frac{(ϖ - ω)}{γ_{⊥}} {\ln [\frac{A_{2 st}^{(+)} (L)}{A_{2 st}^{(+)} (0)}] - \ln [\frac{A_{1 st}^{(+)} (L)}{A_{1 st}^{(+)} (0)}]}

Δ ϕ_{21 st}^{(-)} (0) - Δ ϕ_{21 st}^{(-)} (L) = - \frac{(ω_{c 2} - ω_{c 1})}{c} L + \frac{(ϖ - ω)}{γ_{⊥}} {\ln [\frac{A_{2 st}^{(-)} (L)}{A_{2 st}^{(-)} (0)}] - \ln [\frac{A_{1 st}^{(-)} (L)}{A_{1 st}^{(-)} (0)}]}

Note that two elementary lasers come from the same port of the coupler (see Fig. 2). Then, when two elementary lasers are exported from the coupler, the phase difference should be

Δ ϕ_{21 st}^{(-)} (L) = \frac{ω}{c} (l_{2}^{(c)} - l_{1}^{(c)}) + \frac{π}{2} - \frac{ω_{c 2} l_{2}^{(c)}}{c} - \frac{ω_{c 1} l_{1}^{(c)}}{c}

Here, the term [(ω _c2 l ₂ ^(c) - ω _c1 l ₁ ^(c))/c] is the phase difference caused by the difference between ω _c1 and ω _c2. Consider that

Δ ϕ_{21 st}^{(+)} (0) = Δ ϕ_{21 st}^{(-)} (0) = \frac{2 ω}{c} ({l'}_{2}^{(r)} - {l'}_{1}^{(r)}) - \frac{{2 ω}_{c 2} {l'}_{2}^{(r)}}{c} + \frac{{2 ω}_{c 1} {l'}_{1}^{(r)}}{c} + (δ {φ'}_{R 2} - δ {φ'}_{R 1})

which can be obtained from Eq. (22). Thus, we can get that

Δ ϕ_{21 st}^{(+)} (L) = \frac{ω}{c} [(l_{2}^{(c)} + 2 {l'}_{2}^{(r)}) - (l_{1}^{(c)} + 2 {l'}_{1}^{(r)})] -

where L̄′_m = 2L_m + l_m ^(c) + 2l′_m ^(r), (m = 1,2).

After two elementary lasers pass through the coupler, the field in the output port should be

A_{st}^{(+)} (L_{c}) = ⌊ \sqrt{1 - ε} A_{1 st}^{(+)} (L) + \sqrt{ε} A_{2 st}^{(+)} (L) e^{Δ φ_{21}^{(+)}} ⌋

Here

Δ φ_{21}^{(+)} = \frac{2 ω}{c} [(l_{2}^{(c)} + {l'}_{2}^{(r)}) - (l_{1}^{(c)} + {l'}_{1}^{(r)})] + \frac{(ϖ - ω)}{γ_{⊥}} \ln (\frac{{r'}_{2}}{{r'}_{1}} \sqrt{\frac{ε}{1 - ε}}) + π + (δ {φ'}_{R 2} - δ {φ'}_{R 1})

Equation (B.10) is used in the derivation of Eq.(37). The effect of the detuning (ϖ - ω)is revealed by the second term of Eq. (37). From Eq. (37), it is very interesting to find that ε can only effect ∆φ ₂₁ ⁽⁺⁾ by the detuning term. Note that ε determines the coupling strength of two elementary lasers. When the detuning term can be ignored (i.e., ω is very close to ϖ), the phase difference ∆φ ₂₁ ⁽⁺⁾ will not vary with ε (i.e., coupling strength). In other word, the coupling strength will not affect the steady phase state (i.e., phase-locked state) of this array. This conclusion is very different from that of solid-state laser arrays with evanescent coupling where the coupling strength plays an important role in the phase-locked state [20–22].

Now, consider the situation of reported experiments. 3dB coupler is used (i.e., ε = 1/2) and both of the elementary lasers are pumped with the same power. Two identical fiber Bragg gratings (with ~ 1 reflectivity) are used as the reflectors of the feedback section. The fiber face (with a low reflectivity) is used as the output face. Then, the values of parameters are

ε = \frac{1}{2}, {r'}_{1} = {r'}_{2} = 1, δ φ_{R} = δ {φ'}_{R 1} = δ {φ'}_{R 2} = π

Substituting these values into Eq. (37), we can obtain that

Δ φ_{21}^{(+)} = \frac{2 ω}{c} [(l_{2}^{(c)} + {l'}_{2}^{(r)}) - (l_{1}^{(c)} + {l'}_{1}^{(r)})] + π

From Eq. (39), it can be seen that ∆φ ₂₁ ⁽⁺⁾ varies with ω. Then, the intensity of field in the output port will also change with ω (see Eq. (36)). Therefore, if there is a number of longitudinal modes (corresponding to different ω s) competing in an array (considering the multi-longitudinal-mode nature of fiber lasers), only the mode(s) with the largest intensity of A _st ⁽⁺⁾ (L_c) can be dominant in the mode-competition and oscillate in the array. When |A _st ⁽⁺⁾(L_c)| gets the maximum value, ω should satisfy that

Δ φ_{21}^{(+)} = 2 qπ, (q = 0, \pm 1, \pm 2, \dots)

Then, the frequency difference between neighbor oscillating modes is

Δ ω = \frac{πc}{(l_{2}^{(c)} + {l'}_{2}^{(r)}) - (l_{1}^{(c)} + {l'}_{1}^{(r)})}

Note that the term [(l ₂ ^(c) + l′₂ ^(r)) - (l ₁ ^(c) + l′₁ ^(r))] is the difference between OPLs of two elementary laser cavities. This result agrees with the reported experimental results [4, 5].

4. Conclusion

In this paper, a dynamics model of self-organized fiber laser arrays is presented. In this model, the field oscillating in the Fabry-Perot laser cavity is decomposed into two counter-propagating waves. Then, the standard slowly varying wave approximation, used in the model of solid-state laser arrays, is abandoned in this model. Furthermore, in this model, the Maxwell-Bloch equations are used to describe the interaction between laser field and gain medium. It is revealed that the set of iterative maps, presented in Ref. 17, is an approximated case of this model. Thus, this model can describe the array in a more general way than the iterative maps in Ref. 17.

By analyzing the phase dynamics of interferometric fiber laser arrays, the model is verified. A general expression of the phase difference in the 2-laser interferometric array is also given. It is found that the coupling strength will not influence the phase-locked state of 2-laser interferometric array if the detuning between laser frequency and atomic frequency is ignored. This result is different from that of array of spatially coupled solid-state lasers. Here, it should be noticed that this model can also be used to study the dynamics of other kinds of self-organized fiber laser arrays by changing the equations of boundary conditions. We believe that this model will be of great help in understanding the self-organized fiber laser arrays.

Appendix A: Discussion about the boundary conditions (18) and (22)

The boundary conditions (18) and (22) are only approximate ones. More precisely, the time-delay caused in coupling-output and feedback sections should be taken into account. Take Eq. (18) for example, the precise expression should be

(\begin{matrix} A_{1}^{(-)} (L, t + τ_{1}) \\ A_{2}^{(-)} (L, t + τ_{2}) \\ ⋮ \\ A_{m}^{(-)} (L, t + τ_{m}) \\ ⋮ \\ A_{N}^{(-)} (L, t + τ_{N}) \end{matrix}) = F (\begin{matrix} A_{1}^{(+)} (L, t) \\ A_{2}^{(+)} (L, t) \\ ⋮ \\ A_{m}^{(+)} (L, t) \\ ⋮ \\ A_{N}^{(+)} (L, t) \end{matrix})

where τ_m = (2l _m ^(c) + 2l _m ^(r))/c is the time-delay caused by the coupling-output section.

However, although (A.1) is more precise than Eq. (18), (A.1) will make Eqs. (9) difficult to be solved because of the time delay. Furthermore, when (2l _m ^(r) + 2l _m ^(r)) is so short that the slow-varying amplitude of field (i.e., E _m ⁽⁺⁾(x,t) or E _m ^(-)(x,t)) changes very little within the time τ_m, the time delay will only have a negligible effect on the whole system. Meanwhile, the tine delay has no effect on the steady state, because A _m ^(±)(x,t + τ_m) should be equal to A _m ^(±)(x,t) for all m when the array operates in the steady state. Therefore, the time delay is ignored in the Eq. (18). The same approximate is also used in Eq. (22). If the variation of the slow-varying amplitude of field is too large within τ_m to be neglected, this approximate should be re-examined.

Appendix B: Derivation of f₁, f₂, ϑ₁ and ϑ₂

Substituting Eq. (27) into Eq. (18), we can get that

A_{1}^{(-)} (L, t) = B_{1} ⌊ \sqrt{1 - ε} A_{1}^{(+)} (L, t) + i \sqrt{ε} A_{2}^{(+)} (L, t) \exp (iθ) ⌋

A_{2}^{(-)} (L, t) = {iB}_{2} ⌊ \sqrt{1 - ε} A_{1}^{(+)} (L, t) + i \sqrt{ε} A_{2}^{(+)} (L, t) \exp (iθ) ⌋

where

B_{1} = r \sqrt{1 - ε} \exp {i [δ ϕ_{R} + \frac{2 ω}{c} (l_{1}^{(r)} + l_{1}^{(c)}) + \frac{ω_{c 1}}{c} L + ϕ_{1}^{(+)} (L, t)]}

B_{2} = r \sqrt{ε} \exp {i [δ ϕ_{R} + \frac{ω}{c} (l_{1}^{(c)} + l_{2}^{(c)} + 2 l_{1}^{(r)}) + \frac{ω_{c 1}}{c} L + ϕ_{1}^{(+)} (L, t)]}

θ = \frac{ω}{c} (l_{2}^{(c)} - l_{1}^{(c)}) + ϕ_{2}^{(+)} (L, t) - ϕ_{1}^{(+)} (L, t) + \frac{(ω_{c 2} - ω_{c 1})}{c} L

Then,

f_{1} = r (1 - ε) {1 - 2 \sqrt{\frac{ε}{1 - ε}} [\frac{A_{2}^{(+)} (L, t)}{A_{1}^{(+)} (L, t)}] \sin θ + (\frac{ε}{1 - ε}) {[\frac{A_{2}^{(+)} (L, t)}{A_{1}^{(+)} (L, t)}]}^{2}}^{\frac{1}{2}}

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

f_{2} = r \sqrt{ε (1 - ε)} {1 - 2 \sqrt{\frac{ε}{1 - ε}} [\frac{A_{2}^{(+)} (L, t)}{A_{1}^{(+)} (L, t)}] \sin θ + (\frac{ε}{1 - ε}) {[\frac{A_{2}^{(+)} (L, t)}{A_{1}^{(+)} (L, t)}]}^{2}}^{\frac{1}{2}}

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

where $[\frac{\sqrt{ε} A_{2}^{(+)} (L, t) \cos θ}{\sqrt{1 - ε} A_{1}^{(+)} - \sqrt{ε} A_{2}^{(+)} (L, t) \sin θ}]$ .

From Eq. (B.6) and (B.8), it can be found that

\frac{f_{2}}{f_{1}} = \sqrt{\frac{ε}{1 - ε}}

which means that f ₂/f ₁ is only determined by ε.

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Dynamical model for self-organized fiber laser arrays

Abstract

1. Introduction

2. Introduction of the dynamical model

2.1 Evolution of fields in gain section

2.2 Evolution of fields in the coupling-output section and feedback section

3. Fiber laser arrays of interferometer configuration

4. Conclusion

Appendix A: Discussion about the boundary conditions (18) and (22)

Appendix B: Derivation of f₁, f₂, ϑ₁ and ϑ₂

References and links

Cited By

Figures (2)

Equations (68)

Optics Express

Abstract

1. Introduction

2. Introduction of the dynamical model

2.1 Evolution of fields in gain section

2.2 Evolution of fields in the coupling-output section and feedback section

3. Fiber laser arrays of interferometer configuration

4. Conclusion

Appendix A: Discussion about the boundary conditions (18) and (22)

Appendix B: Derivation of f1, f2, ϑ1 and ϑ2

References and links

Cited By

Figures (2)

Equations (68)

Optics Express

Appendix B: Derivation of f₁, f₂, ϑ₁ and ϑ₂