Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays

Tsai-wei Wu; Wei-zung Chang; Almantas Galvanauskas; Herbert G. Winful

doi:10.1364/OE.18.025873

1. Introduction

The possibility of multi-kW power scaling through passive coherent phasing of fiber laser arrays has spurred intense research into the physics and technology of beam combining. Several groups have successfully demonstrated highly efficient coherent combining of up to eight discretely coupled fiber lasers by 50:50 directional couplers [1–5]. Theoretical work has addressed various steady state aspects of beam combining, such as the scaling of combining efficiency with array size [6–10]. However there have been few detailed dynamical studies that probe the process in which the independent fiber amplifiers organize themselves to produce a mutually coherent output. We recently presented a theoretical model of fiber laser coherent beam combining that showed clearly that the mechanism of beam combining is the selection of the composite cavity mode that satisfies the condition of minimum loss [11]. In that work a ring cavity was assumed in order to avoid the computational complexities of counter-propagating waves while still gaining insight into the nature of the beam combining process. Although there have been some demonstrations of passive coherent phasing in ring cavity geometries, all the attempts to scale up beyond two coupled lasers have involved standing wave cavities. It would be desirable to have a simulation tool that permits detailed studies of beam combining with Fabry-Perot cavities.

In this paper we present a full bidirectional model of passive coherent phasing based on mutually-coupled nonlinear Schrödinger equations for counterpropagating waves. These equations are coupled to dynamic rate equations that include the effects of cross saturation by the forward and backward waves. The model allows a detailed look at how the composite cavity modes evolve from noise and how the phase-difference between coupled amplifiers locks to a fixed value. It is evident from the model that the coherent phasing results from the selection of the composite cavity mode with the lowest loss. We show how the spacing between array modes is determined and examine the controversial role of the nonlinear refractive index. Our results yield insight into the mechanisms that limit beam combining efficiency.

2. Model

A two-channel fiber laser array is depicted in Fig. 1 with two independent single mode fibers coupled discretely by a directional coupler. The continuous-wave pump beams, launched into each fiber by a wavelength division multiplexer (WDM) at z = 0, excite active ions and give rise to gain at longer wavelengths. The fiber Bragg gratings (FBG) provide ~100% feedback at the right ends of the fibers, while differential power reflectivities, R₁ and R₂ in Fig. 1, are applied to the output ports of the 50:50 coupler at the left-hand sides. Assuming single polarization, the coherent waves propagating in + z and −z directions in each fiber laser are governed by the nonlinear Schrödinger equation together with the rate equation [12, 13]:

\frac{\partial E_{j}^{f}}{\partial z} = \frac{1}{2} (g_{j} - α) E_{j}^{f} - β_{1} \frac{\partial E_{j}^{f}}{\partial t} + \frac{1}{2} (b - i β_{2}) \frac{\partial^{2} E_{j}^{f}}{\partial t^{2}} + i γ ({| E_{j}^{f} |}^{2} + 2 {| E_{j}^{b} |}^{2}) E_{j}^{f}

\frac{\partial E_{j}^{b}}{\partial z} = - \frac{1}{2} (g_{j} - α) E_{j}^{b} + β_{1} \frac{\partial E_{j}^{b}}{\partial t} - \frac{1}{2} (b - i β_{2}) \frac{\partial^{2} E_{j}^{b}}{\partial t^{2}} - i γ (2 {| E_{j}^{f} |}^{2} + {| E_{j}^{b} |}^{2}) E_{j}^{b}

τ \frac{d g_{j} (z, t)}{d t} = g_{0 j} - g_{j} (z, t) - \frac{{| E_{j}^{f} |}^{2} + {| E_{j}^{b} |}^{2}}{P_{s a t}} g_{j} (z, t)

E_{j}^{f}

and

E_{j}^{b}

refer to the slowly varying envelopes of the forward and backward electromagnetic waves in the first (j = 1) and second (j = 2) fiber respectively. Various effects including linear gain g_j, fiber losses α, the inverse of the group velocity β₁, the frequency-dependent losses b, the group velocity dispersion β₂, and the nonresonant Kerr nonlinearity γ are all incorporated in Eqs. (1) and (2). In Eq. (3) for gain dynamics, g_0j specifies the unsaturated gain while the second and third terms respectively describe the process of excited population relaxation with upper-state lifetime τ and laser gain saturation at high intensity fields. We normalize the electric field amplitudes so

| E_{j} |^{2}

corresponds to the power distribution. Based on the coupled mode theory [14], the 50:50 directional coupler connecting the inputs

E_{1}^{b}, E_{2}^{b}

to the outputs

A_{1}, A_{2}

is represented by a linear matrix.

Fig. 1 A two-channel fiber laser array.

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(\begin{matrix} A_{1} \\ A_{2} \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & - j \\ - j & 1 \end{matrix}) (\begin{matrix} E_{1}^{b} \\ E_{2}^{b} \end{matrix})

It is clear that the forward and backward waves are coupled through cross-phase modulation (the last terms in Eqs. (1) and (2)) as well as by the co-saturation of the gain fields described in Eq. (3) [13]. In our previous work on unidirectional fiber laser arrays, straightforward integration by the split step Fourier method (SSFM) was applied since there was no coupling between forward and backward waves to contend with [15]. Here, however the nonlinearly-coupled differential Eqs. (1-3) require some self-consistent solutions of $E_{j}^{f}$ and $E_{j}^{b}$ existing for all $z \in [0, L]$ as well as for all t within the computation window. We handle this complication by using an iterative SSFM [16] such that $E_{j}^{f}, E_{j}^{b}$ are in turn integrated along + z and –z directions respectively while the information of the updated field is stored and used to compute the other one at a later time. The iteration continues until certain convergences are reached.

In the dynamical aspects of the array, rather than turning on g_j abruptly as assumed in many cases [15, 17, 18], we allow the gain to build up gradually and retain its dependence on both time and position. Due to the discrete nature of numerical computation, the fiber length is partitioned into segments and we will deal with an integrated gain variable ${\tilde{g}}_{j} = \int_{z_{0}}^{z_{0} + ℓ} g_{j} d z$ over each segment instead of $g_{j} (z, t)$ in Eq. (3). The product term involving the gain variable and the intensity field on the right hand side of Eq. (3) is, however, difficult to integrate without losing accuracy; we thus follow Ref [9]. in replacing that term in the manner described below.

Based on Beer’s law $d | E_{j}^{f} |^{2} / d z = + g_{j} | E_{j}^{f} |^{2}$ and $d | E_{j}^{b} |^{2} / d z = - g_{j} | E_{j}^{b} |^{2},$ the rate equation can be written as

τ \frac{d g_{j} (z, t)}{d t} = g_{0 j} - g_{j} (z, t) - \frac{1}{P_{s a t}} (\frac{d {| E_{j}^{f} |}^{2}}{d z} - \frac{d {| E_{j}^{b} |}^{2}}{d z})

By integrating

g_{j}

along

z \in [z_{0}, z_{0} + ℓ]

and defining

{\tilde{g}}_{j} = \int_{z_{0}}^{z_{0} + ℓ} g_{j} d z,

we find that Eq. (5) becomes

τ \frac{d {\tilde{g}}_{j} (t)}{d t} = {\tilde{g}}_{0 j} - {\tilde{g}}_{j} (t) - \frac{1}{P_{s a t}} ({| E_{j}^{f} (z_{0} + ℓ) |}^{2} - {| E_{j}^{f} (z_{0}) |}^{2} - {| E_{j}^{b} (z_{0} + ℓ) |}^{2} + {| E_{j}^{b} (z_{0}) |}^{2})

Further approximation can be applied with the relation

| E_{j}^{f} (z_{0} + ℓ) |^{2} = e^{{\tilde{g}}_{j} (z_{0})} | E_{j}^{f} (z_{0}) |^{2}

and

| E_{j}^{b} (z_{0}) |^{2} = e^{{\tilde{g}}_{j} (z_{0} + ℓ)} | E_{j}^{b} (z_{0} + ℓ) |^{2}

for sufficiently small length ℓ. Its substitution into Eq. (6) produces the final expression for the rate equation.

τ \frac{d {\tilde{g}}_{j} (t)}{d t} = {\tilde{g}}_{0 j} - {\tilde{g}}_{j} (t) - \frac{1}{P_{s a t}} [{| E_{j}^{f} (z_{0}) |}^{2} (e^{{\tilde{g}}_{j} (z_{0})} - 1) + {| E_{j}^{b} (z_{0} + ℓ) |}^{2} (e^{{\tilde{g}}_{j} (z_{0} + ℓ)} - 1)]

Equations (1-2) and (7) are then solved together by iterative SSFM and the Euler method, where the latter is adopted for connecting the integrated gain variable at current and next time incidents, i.e.

{\tilde{g}}_{j}

and

{\tilde{g}}_{j},

by

{\tilde{g}}_{j}' = {\tilde{g}}_{j} + \frac{d {\tilde{g}}_{j}}{d t} Δ t

We set the increment step Δt to be four times the roundtrip duration in the simulation since numerical accuracy is ensured when Δt (~1 μs) is much smaller than the population relaxation time constant τ (10 ms). To validate our derivation of the dynamical model, an example is given for a two-channel fiber laser array of fiber lengths 24.3 and 24.0 m. Here by setting γ = 0 W⁻¹m⁻¹ the array is assumed to be linear so the nonlinear phases will not modify the distinct modes of the laser cavity [11]. Each of the active fibers is partitioned into 70 segments (rtstps = 70.) Other parameter values used are

λ_{0} = 1.545 μ m,

α = 0.058 m^{- 1},

β_{2} = - 0.003 {ps}^{2} m^{-1},

n_{1} = 1.5,

b = 0.13 {ps}^{2} m^{-1}, g_{0} = 2.67 m^{-1}

and

P_{s a t} = 0.6 mW .

3. Simulation results

The simulation results are shown in Fig. 2 to Fig. 4 . In addition to the standard SSFM outputs (temporal and spectral domain profiles), the spatial distributions and dynamical evolutions of the array can also be retrieved from our model. The spatial distributions refer to the self-consistent solutions of the coupled equations and they are plotted with red circles in Fig. 2 for (a) both forward and backward waves and (b) the gain field within one of the fiber lasers (L₁ = 24.3 m) at steady state. It is clear that the backward signal dominates in this efficient backward pumping configuration [19] and the resulting gain exhibits stronger saturation near the front end of the fiber. To verify the numerical solutions of the dynamic equations, the time derivatives and the nonlinear index terms of Eqs. (1-3) are set to zero since they do not affect the field distributions at steady states. We then solve the simplified ODEs together with the boundary conditions using the built-in BVP (boundary value problems) solver of Matlab. The solid black line represents such solution in Fig. 2(a), 2(b) and its agreement with the red circles supports our simulation results. As for array dynamics, the time evolutions of (c) the output power, coming out of the partially-reflected port, and (d) the gain variable (averaged over z) in both fiber lasers are clearly observed in Fig. 2. The array exhibits transient oscillations in the beginning of the excitation and settles eventually after a few milliseconds. At steady state, the averaged gain variables ${\bar{g}}_{1, 2}$ amount to 0.1244 and 0.1251 m⁻¹ and these equal the roundtrip losses of $α - \ln (R) / 2 / L_{1, 2}$ as expected from fundamental laser theory.We now turn to the beam combining properties of the array. The temporal (left) and spectral (right) domain outputs are shown in Fig. 3 . As expected, almost all the power comes out of the upper, partially reflected (R₁ = 0.04) port, while a negligible amount leaks through the lower, angle-cleaved (R₂ = 0) one. Taking $P_{o u t}$ as the output of the straight-cleaved end and $P_{i}$ as the power from the $i^{t h}$ single laser if uncoupled, we define the power combining efficiency for an N - channel array as

Fig. 2 The spatial distributions of one of the fiber laser (L1 = 24.3 m) are plotted as an example for (a) both propagating waves and (b) the gain field along the z axis. The three curves consisting of red circles present the self-consistent steady-state solutions obtained from our model, while that of solid black lines are calculated from Matlab with its built-in BVP solver. As for array dynamics, the time evolution of the output power and the averaged gain variable (over z) of each fiber are displayed in (c) and (d). The output power refers to the combined power coming out of the partially-reflected, R1, port as seen in Fig. 1.

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Fig. 4 Evolution diagram of the output power spectrum for (a) the array modes, (b) the zoom-in longitudinal modes and (c) the relative phase difference $Δ ϕ$ (π) between two incident (backward) waves at z = 0. All of them start from random and noisy spontaneous emissions. The free spectral range in (b) is 4.1MHz.

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Fig. 3 An Er-doped fiber laser array in Fig. 1 with L₁ 24.3 and L₂ 24.0 m. The output powers from (a) upper port with partial reflectivity and (b) lower, angle-cleaved, port are plotted for time (left) and frequency (right) domains respectively. The separation between spikes in the frequency domain is 0.333 GHz.

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N - channel combined power efficiency = P_{o u t} / \sum_{1}^{N} P_{i}

The calculated combining efficiency is close to 100% and is consistent with experimental observations and with our previous calculations for unidirectional laser cavities. To understand the role of counterpropagating waves in array combining, we examine the modulated power spectrum in which a series of equi-distant spikes appears as a result of a Vernier effect relating to the overlap of the frequency combs corresponding to two laser cavities of different length. The modulation period, found to be 0.333 GHz from the spectral plots of Fig. 3, is in agreement with the theoretical prediction of $Δ v = c / (2 n_{1} Δ L)$ [20]. Compared with the $Δ v = 0.667$ GHz obtained for unidirectional fiber laser arrays of the same parameters [11], the factor of two difference can be readily understood by the fact that the optical path lengths double in the bidirectional configurations and the period halves accordingly. The simulation results suggest that, besides giving rise to additional phases through propagation, the backward waves merely serve to co-saturate the population inversion and they do not influence the coherent combining mechanism, in which the key component is the multilongitudinal modes of fiber lasers. The pseudo-random shape of the temporal profiles in Fig. 3 is the result of the complex beating between the many modes.

To fully characterize the array dynamics, the model is used to study the formation process of the coincident modes and also the associated phase-locked states. Figure 4 illustrates the evolutionary spectrum for (a) the array modes, (b) the longitudinal modes and (c) the relative phase difference $Δ ϕ$ between two incident (backward) waves at z = 0 before the 50:50 coupler, where $Δ ϕ$ is defined as ( $ϕ_{1} - ϕ_{2}$ ) modulo 2π. The initial noise in the frequency domain is modeled as uniformly distributed complex numbers with both signs of the real and imaginary parts of the field assigned stochastically to be positive or negative. The serial snapshots show that the array output grows out of noisy spontaneous emission and is continually filtered due to the interferometric nature of the composite cavity (Fig. 4(a)). After a few milliseconds the initial random spectrum is transformed into a set of discrete mode clusters spaced by $Δ v = c / (2 n_{1} Δ L) = 0.333 GHz
.$ (We show in the Appendix that for larger arrays the spacing between the mode clusters is given by $Δ v = c_{0} / (2 n_{1} Δ L_{\gcd})$ , where gcd stands for Greatest Common Divisor.) As time increases, the width of these clusters shrinks considerably owing to gain competition. Zooming in the spectrum further (Fig. 4(b)) shows that each mode cluster consists of the Fabry-Perot resonances of a single cavity with a free spectral range of 4.1 MHz, i.e. $c / (2 n_{1} L) .$ The result is particularly appealing since the fundamental characteristic of the laser cavity manifests itself naturally out of the model as anticipated. The phase difference between the two lasers also evolves from an initial random distribution to a linear function of frequency centered at the peak of the mode cluster. This indicates that those modes are phase locked. Note these oblique lines center around 1.5π on the vertical axes of Fig. 4(c) because this particular phase difference yields constructive interferences in the upper output port of the directional coupler and destructive ones in the other. Finally, cross-referencing the time orders in Fig. 2(c) and Fig. 4 reveals that both mode formation and the phase-locking behavior are established very early before the transient oscillation begins. The transient relaxation oscillations are thus a collective phenomenon of the coupled lasers.

4. Nonlinearity

The role of an intensity-dependent nonlinear phase shift in beam combining or in the phase locking process remains controversial. Some groups have claimed finding support for the enforcement of self locking through a non-resonant nonlinear index n₂ [21–23], while at the same time opposing evidence is demonstrated experimentally with high power, > 50W, fiber laser arrays [23, 24]. This confusing state of affairs needs to be clarified and so we utilize the bidirectional model to study the role of nonlinear phases in coherent combining.

At low operating powers, our previous simulations have shown that the small electronic nonlinear coefficient n₂ has no apparent effect on the combining efficiency in a two-channel fiber laser array [11]. Here we increase the nonlinear coefficient γ by more than two orders of magnitude and thereby force the effects of nonlinearity to be manifested at much lower power levels. Assuming γ to be 0.9 W⁻¹m⁻¹, the previous simulation is repeated without changing other parameters. The two array outputs are plotted in Fig. 5(a) and 5(b) respectively for both temporal (left) and spectral (right) domains. Compared to Fig. 3, several apparent differences can be observed. Firstly, the FWHMs of each spectral packet broaden considerably. Second, in contrast to the centered power spectrum of the linear arrays, the nonlinearity causes the frequency components to spread and so the outermost packets are most intense. The resultant spectrum is not constrained to the parabolic loss profile and looks similar to that of pulse propagation in the presence of self-phase modulation. Third, the combining efficiency reduces to 85.5% with a significant amount of power leaking from the lossy port. The final comparison is made to the phase spectrum of the circled spectral packet in Fig. 5(a) and its linear counterpart of Fig. 3(a). The relative phase difference plots show that the range of $Δ ϕ$ expands from ± 0.1π in Fig. 5(d) to ± 0.5π in (c) as the nonlinear coefficient is turned on.

Fig. 5 A two-channel fiber laser array is simulated with γ = 0.9 W⁻¹m⁻¹. The array outputs are plotted in (a) and (b) respectively for both temporal (left) and spectral (right) domains. The relative phase difference of the circled spectral packets (in (a)) is plotted in (c) for nonlinear and in (d) for linear fiber laser arrays.

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To completely account for the decrease of the combining efficiency owing to nonlinearity, we calculate the output powers using Eq. (4) assuming equal amplitudes of the incident waves before the coupler. Their power ratio is expressed as a function of their phase difference $Δ ϕ$ by [14]

\frac{P_{1}}{P_{2}} = \frac{1 - \sin (Δ ϕ)}{1 + \sin (Δ ϕ)}

A logarithmic plot of Eq. (9) is shown in Fig. 6 to illustrate how rapidly the powers transfer between one port to the other when

Δ ϕ

changes. It is clear that the singularities occur at 1.5π and 0.5π representing the two extremes of power combining. Observe when

Δ ϕ

is confined within 1.4π and 1.6π (in the case of the linear arrays), the power ratio is high and most of the power resides in P₁. On the other hand, when

Δ ϕ

deviates far from 1.5π and approaches π or 2π, P₁ decreases and more power emerges out of the lower, angle-cleaved port as is evident in Fig. 5. The drop in combining efficiency is thus seen to be a result of the increasing bandwidth of the power spectrum, in particular, the broadening of each spectral packet under modulation.

Fig. 6 The logarithmic plot of the output power ratio in terms of relative phase $Δ ϕ .$

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5. Array lasing frequencies - the minimum loss

As mentioned in Ref [11], the nonzero loss dispersion coefficient b may give rise to reduced combining efficiency as well as shifted lasing frequencies in a two-channel fiber laser array. It is essential to understand how the resonant frequencies are determined and why the frequency shift occurs in the presence of additional loss sources. A reasonable expectation is that the array chooses to lase at the frequency that experiences the least overall losses. In order to verify this point, we present a simple loss analysis based on the unidirectional two-channel fiber laser array. The ring cavity configuration is adopted here as seen in Fig. 7 since it is simpler and we have shown earlier that the coherent beam combining is not affected by the backward propagating waves of standing-wave cavities.

Fig. 7 A unidirectional two-channel fiber laser array.

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To derive the frequency-dependent array loss, the circulating power within each fiber laser before the coupler is assumed to be $P .$ The coupler output powers are calculated, according to Eq. (9), as $P (1 + \sin (Δ ϕ))$ and $P (1 - \sin (Δ ϕ))$ depending on the accumulated phase difference of $Δ ϕ = ω / c n_{1} (L_{1} - L_{2})$ between the two incident fields. Only one of the output powers is reflected and fed back into the other end of the two fibers. Take $P (1 + \sin (Δ ϕ))$ for example; the steady-state laser oscillation requires that the power be restored to P at the reference plane just before the coupler, so we can write

P (1 + \sin (Δ ϕ)) \times R \times \frac{1}{2} \exp [(g - α - b ω^{2}) L] = P

where R is the power reflectivity, g is the saturated gain, α is the linear loss and b is the loss dispersion coefficient. We take the logarithm of Eq. (10) and the expression for the loss is given as the right hand side of Eq. (11).

g = α + b ω^{2} - \frac{\log (R (1 + \sin (Δ ϕ)) / 2)}{L}

The frequency dependent loss profile can thus be readily plotted by plugging in

Δ ϕ = ω / c n_{1} (L_{1} - L_{2}) .

Consider an example of a two-channel fiber laser array of lengths 24.0005 m and 24.0 m, with the simulated power spectra shown in Fig. 8 for (a) b = 0 ps²m⁻¹ and (b) b = 0.13 ps²m⁻¹ respectively. (The very small length difference is chosen to ensure the spike separation and the frequency shifts are large enough for clear visualization.) It is clear the combining efficiency drops considerably and the lasing frequency shifts from 126.5 GHz to 45.79 GHz in the presence of nonzero loss dispersion. Utilizing Eq. (11), we plot the frequency-dependent loss on a logarithmic scale with blue solid lines and further overlap them with the output lasing frequencies (red solid lines) in Fig. 9 for better visualization. The loss curves exhibit minimum values near −300 GHz and 100 GHz in Fig. 9(a) and around 50 GHz in Fig. 9(b). In both cases, the good agreement between array resonant modes and the location of the minimum losses validates the hypothesis that the coupled array finds the mode with minimum overall losses.

Fig. 8 Power spectra of a two channel fiber laser array with fiber lengths 24.0005m and 24.0m for (a) b = 0 ps²m⁻¹ and (b) b = 0.13 ps²m⁻¹.

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Fig. 9 The frequency dependent losses (m⁻¹), plotted in the log scale with blue lines, are overlapped with the lasing spectrum of the output fields (red spikes) for (a) zero and (b) nonzero b coefficients respectively.

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6. Conclusion

To conclude, we have extended our dynamic model of passive beam combining in fiber lasers to include transient gain dynamics and the interaction of counterpropagating waves. The model allows us to study the process in which composite cavity modes are selected and the establishment of a fixed phase relationship between the coupled amplifiers. We find that the phase locked state is established relatively soon within several hundred roundtrips and that the amplifiers exhibit collective transient relaxation oscillations upon turn-on. The unsettled issue of nonlinearity is also studied. Our simulation suggests the nonresonant n₂ induces spectral broadening and reduces the combining efficiency at high power levels. We explore the working principle of the array and demonstrate that it is based on the selection of composite cavity modes with the minimum overall losses.

Appendix

Array mode spacing – the greatest common divisor

It is well known that when two lasers of length $L_{1}$ and $L_{2}$ are combined in a composite cavity, the individual Fabry-Perot frequency combs become modulated with an envelope whose peaks are separated by $Δ v = c / (2 n_{1} Δ L)$ , where $Δ L = L_{2} - L_{1}$ . For N coupled lasers the separation between the maxima of the modulation envelope can be found by examining the condition for constructive interference at all the 50:50 couplers. Since lasing of the composite cavity should occur near these maxima (corresponding to frequencies of minimum loss) this analysis helps to make sense of the complicated spectra observed in multi-element arrays.

Fig. A1 A four-channel fiber laser array. The figure is taken and modified from Ref [25].

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Consider a four-channel fiber laser array in Fig. A1, where the coherent combining is governed by the 50:50 directional couplers and the linear coupling matrix of Eq. (4). For any frequency $f,$ constructive interferences can occur at the lower output port of the couplers M₁ and the upper output ports of M₂ and M₃ respectively when

\begin{array}{l} \frac{2 π n_{1} f}{c_{0}} 2 (L_{2} - L_{1}) = \frac{3 π}{2} + m_{1} 2 π \\ \frac{2 π n_{1} f}{c_{0}} 2 (L_{3} - L_{4}) = \frac{3 π}{2} + m_{2} 2 π \\ \frac{2 π n_{1} f}{c_{0}} 2 (L_{3} - L_{2}) = \frac{3 π}{2} + m_{3} 2 π \end{array}

Here n₁ is refractive index of the fiber and $m_{1}, m_{2}, m_{3}$ are integers. Note the third equation of Eq. (A1) describes the power addition criterion in M₃ and it depends only on the fiber lengths L₂ and L₃. This can be understood by calculating the phase of the output fields from the coupler M₁ by

\frac{1}{\sqrt{2}} [\begin{matrix} 1 & - j \\ - j & 1 \end{matrix}] [\begin{matrix} e^{j 2 k L_{1}} \\ e^{j 2 k L_{2}} \end{matrix}] = \frac{1}{\sqrt{2}} [\begin{matrix} e^{j 2 k L_{1}} - j e^{j 2 k L_{2}} \\ - j e^{j 2 k L_{1}} + e^{j 2 k L_{2}} \end{matrix}]

Assuming the input waves interfere destructively at the upper output port of M₁ such that $e^{j 2 k L_{1}} - j e^{j 2 k L_{2}} = 0,$ the emerging field of the lower port $- j e^{j 2 k L_{1}} + e^{j 2 k L_{2}}$ can then be written as $2 e^{j 2 k L_{2}}$ if we replace $e^{j 2 k L_{1}}$ by $j e^{j 2 k L_{2}} .$ Similar calculation can be applied to M₂ with the result of $2 e^{j 2 k L_{3}}$ and so the phases of the two input fields into M₃ are merely characterized by fiber lengths L₂ and L₃ in Eq. (A2).

Given random combinations of lengths L₁ to L₄, the exact solution f generally does not exist for all three equations in Eq. (A1) even with the degrees of freedom provided by $m_{1}, m_{2} and m_{3} .$ In most cases only an optimal frequency $\bar{f}$ can be obtained. Let us assume $\bar{f}$ satisfies the following conditions:

\begin{array}{l} \frac{2 π n_{1} \bar{f}}{c_{0}} 2 (L_{2} - L_{1}) = \frac{3 π}{2} + m_{1} 2 π + Δ ϕ_{1} \\ \frac{2 π n_{1} \bar{f}}{c_{0}} 2 (L_{3} - L_{4}) = \frac{3 π}{2} + m_{2} 2 π + Δ ϕ_{2} \\ \frac{2 π n_{1} \bar{f}}{c_{0}} 2 (L_{3} - L_{2}) = \frac{3 π}{2} + m_{3} 2 π + Δ ϕ_{3} \end{array}

where $Δ ϕ_{k},$ k = 1⋯3, indicates the deviations of $\bar{f}$ away from the exact solution of each equation in Eq. (A1) and is responsible for the imperfect power combining due to the residual phase mismatch. The optimal solution $\bar{f}$ is recognized as the frequency of minimum coupling loss in the four-channel array simulation as shown in Ref [11]. We can then calculate the period of these modes $Δ v$ by substituting $\bar{f} + Δ v$ into $\bar{f}$ in Eq. (A3);

\begin{array}{l} \frac{2 π n_{1} (\bar{f} + Δ v)}{c_{0}} 2 (L_{2} - L_{1}) = \frac{3 π}{2} + (m_{1} + p_{1}) 2 π + Δ ϕ_{1} \\ \frac{2 π n_{1} (\bar{f} + Δ v)}{c_{0}} 2 (L_{3} - L_{4}) = \frac{3 π}{2} + (m_{2} + p_{2}) 2 π + Δ ϕ_{2} \\ \frac{2 π n_{1} (\bar{f} + Δ v)}{c_{0}} 2 (L_{2} - L_{3}) = \frac{3 π}{2} + (m_{3} + p_{3}) 2 π + Δ ϕ_{3} \end{array}

where $p_{k}, k = 1 \dots 3$ are new integers. Upon eliminating $\bar{f}$ from the above two sets of equations we find

\begin{array}{l} \frac{2 π n_{1} Δ v}{c_{0}} 2 (L_{2} - L_{1}) = p_{1} 2 π \\ \frac{2 π n_{1} Δ v}{c_{0}} 2 (L_{3} - L_{4}) = p_{2} 2 π \\ \frac{2 π n_{1} Δ v}{c_{0}} 2 (L_{2} - L_{3}) = p_{3} 2 π \end{array}

The maxima of the modulation envelope are thus spaced by a frequency given by

Δ ν = \frac{c_{0}}{2 n_{1} Δ L_{c}} = LCM[\frac{c_{0}}{2 n_{1} (L_{1} - L_{2})}, \frac{c_{0}}{2 n_{1} (L_{2} - L_{3})}, \frac{c_{0}}{2 n_{1} (L_{3} - L_{4})}]

where LCM represents the least common multiple of the arguments and $Δ L_{c}$ represents some equivalent path length difference. For this solution $p_{1}, p_{2}, p_{3}$ are integers with no common factor. From this result it can also be shown that $Δ ν$ is determined by the greatest common divisor of the length differences. Thus the frequencies of minimum loss are spaced by

Δ v = \frac{c_{0}}{2 n_{1} Δ L_{\gcd}}

where $Δ L_{\gcd} = GCD[L_{1} - L_{2}, L_{2} - L_{3}, L_{3} - L_{4}] = Δ L_{c} .$ This result can be generalized to any number of lasers in the tree architecture described here.

It is important to note that the composite cavity modes have to satisfy the condition that the field at any point reproduces itself after a round trip, within a phase shift of an integral multiple of $2 π .$ These modes form a very dense comb structure which is modulated by an envelope representing the transfer function of the multiple-coupler interferometer. Maximum combining efficiency occurs where these modes coincide with maxima of the transfer function.

To support the analysis, a numerical example is given for a unidirectional four-channel fiber laser array of lengths 24.0, 24.3, 23.733 and 24.633 m. The greatest common divisor of their length differences is 3 mm and the mode periodicity is calculated to be 66.7 GHz according to $c_{0} / (n_{1} Δ L_{\gcd})$ where the factor of two differences from Eq. (A7) is due to the single pass nature of the laser cavity in this case. The simulation result is shown in Fig. A2 with $Δ v$ seen to be 66.7 GHz and is consistent with the theoretical prediction.

Fig. A2 Coherent combining of a four-channel fiber laser array with lengths 24.0, 24.3, 23.733 and 24.633 m. The period of the power spectrum pattern indicated by the red arrow is measured to be 66.7 GHz.

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Acknowledgments

Partial funding for this research is provided by the Office of Naval Research under grant No. N00014-07-1-1155.

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Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays

Abstract

1. Introduction

2. Model

3. Simulation results

4. Nonlinearity

5. Array lasing frequencies - the minimum loss

6. Conclusion

Appendix

Array mode spacing – the greatest common divisor

Acknowledgments

References and links

Cited By

Figures (11)

Equations (19)

Optics Express