Phase-locking of arrays of weakly coupled semiconductor lasers

Niketh Nair; Erik Bochove; Yehuda Braiman

doi:10.1364/OE.26.020040

1. Introduction

Semiconductor laser diodes are employed for a wide variety of applications. Such lasers can emit light in wide range of wavelengths, exhibit very high (in the range of 60-70%) electro-optical efficiency, are compact, and are low cost. However, a single diode’s emission power is in the range of Watts or lower. Consequently, beam combining of many diodes is required in order to provide high emission radiance from an array [1,2]. The master oscillator power amplifier (MOPA) designs have been shown to allow for almost perfect semiconductor diode phase locking with arrays as large as 900 lasers [3–5]. For external cavity designs, while excellent beam quality from single mode and broad-area diode arrays has been demonstrated [6–11], the scalability to very large arrays and stacked-arrays still remains an open matter. It is important to elucidate whether phase locking of large semiconductor diode arrays is possible and, if possible, what type of external cavity designs allow for phase synchrony. We would like to note that phase synchrony should not be very sensitive to disorder diode and cavity parameters given the natural heterogeneity of commercial laser diodes used in the experiments [6,11,12].

Spatial mode selection in laser arrays has been widely studied as a plausible mechanism of passive phasing of laser arrays. Modal analysis has also been applied to single resonators [13] as well as compound-resonators [8,10,14–18]. In order to study the stability of spatial modes, we applied a modified version of Master Stability Function (MSF) theory [19–23] to an array of weakly coupled semiconductor lasers described by the Lang-Kobayashi equations [24,25]. MSF theory is essentially a type of modal analysis for coupled nonlinear oscillators.

Using the Lang-Kobayashi equations [24] it is possible to describe a semiconductor laser array as a network of coupled nonlinear oscillators [25]. Lang-Kobayashi equations have been extensively tested both theoretically and experimentally [12,25–28]. It is known that small arrays of semiconductor lasers can be phase synchronized when nearest neighbor coupled [25,26], and even chaotically synchronized for small numbers of lasers [25, and references therein]. For large arrays, as the number of lasers increase the in-phase solutions destabilize in favor of anti-phase, traveling-wave, and chaotic solutions. The coupling strength at which destabilization occurs seems to decrease with array size [26]. This destabilization occurs because the number of fixed-frequency solutions increases as coupling strength increases and the coupled lasers begin to chaotically hop between these solutions [25,27]. In the case of perfect global coupling (also referred to as all-to-all or mean-field coupling) the common perception is that most large systems including semiconductor lasers [28] will synchronize with appropriate parameters and sufficiently low disorder [29]. However, perfect global coupling for diode arrays cannot be experimentally implemented.

In this paper we propose and study an external cavity design where coupling strength between any two lasers decays as the distance between them increases. This type of nonlocal coupling can be implemented, for example, in a V-shape Talbot [6,7], self-Fourier and intra-Talbot second harmonic generation [8,9] cavities. The geometry of the external cavity of the coupled laser array determines the form of the coupling matrix [17], however for Lang-Kobayashi type systems, the coupling matrix can only be specified phenomenologically, as there is no rigorous way to derive the matrix from first principles in the weak-coupling limit under which the Lang-Kobayashi equations are specified. However, if the lasers are identical, then the modes of the external cavity (described by the eigenvectors of the coupling matrix) should be identical to the modes of the gain-free (’cold’) compound resonator system (the lasers coupled with the external cavity) [13,17]. Employing a V-shape cavity, an almost perfect diffraction limited beam (meaning almost perfect phase synchrony) has been experimentally realized [7].

2. Analysis

We start by presenting the dynamical equations for an array of non-locally coupled semiconductor diode lasers. An array of $M$ semiconductor lasers can be modeled using an equation of the general form [24,30, 31]:

{\dot{X}}_{i} (t) = F (X_{i} (t)) + η (X_{i} (t), t) + \frac{κ^{f}}{M} \sum_{j = 1}^{M} K_{i j} C (X_{j} (t - τ), X_{i} (t))

where

κ^{f}

is the feedback strength. Here

F (X_{i} (t))

represents the noiseless laser dynamics,

η (X_{i}, t)

represents phase and carrier noise, and the term proportional to

κ^{f}

represents feedback from the array. Diode laser equations can be viewed as a representation of general system of oscillators coupled through a network with delayed feedback. The state of a laser

X_{i} = {(r_{i}, ϕ_{i}, N_{i})}^{T}

can be described by a set of the following equations [24,30–34]:

\begin{array}{l} {\dot{r}}_{i} (t) = \frac{1}{2} (g \frac{N_{i} (t) - N_{0}}{1 + s r_{i}^{2} (t)} - γ) r_{i} (t) + \sqrt{R_{s p}} ℜ (η_{E}^{†} e^{i ϕ_{i} (t)}) \\ + \frac{κ^{f}}{M} \sum_{j = 1}^{M} K_{i j} r_{j} (t - τ) \cos (ϕ_{j} (t - τ) - ϕ_{i} (t)) \end{array}

\begin{array}{l} {\dot{ϕ}}_{i} (t) = \frac{α}{2} (g \frac{N_{i} (t) - N_{0}}{1 + s r_{i}^{2} (t)} - γ) + ω_{i} - \frac{{\sqrt{R}}_{s p}}{r_{i} (t)} ℑ (η_{E}^{†} e^{i ϕ_{i} (t)}) \\ + \frac{κ^{f}}{M} \sum_{j = 1}^{M} K_{i j} \frac{r_{j} (t - τ)}{r_{i} (t)} \sin (ϕ_{j} (t - τ) - ϕ_{i} (t)) \end{array}

{\dot{N}}_{i} (t) = J_{0} - γ_{n} N_{i} (t) - g \frac{N_{i} (t) - N_{0}}{1 + s r_{i}^{2} (t)} r_{i}^{2} (t) + \sqrt{γ_{n} N_{i} (t)} η_{N} (t)

where

r_{i}

is the field magnitude,

ϕ_{i}

is the phase, and

N_{i}

is the number of carriers in the gain medium. The electric field in the

i

th laser is

E_{i} = r_{i} (t) e^{i ϕ_{i} (t)}

. The parameters we choose are typical for Lang-Kobayashi type models of single-mode semiconductor lasers [35–37]. Here

N_{0} = 1.5 * 10^{8}

is the number of carriers at transparency,

g = 1.5 * 10^{- 8} {ps}^{- 1}

is the differential gain coefficient,

s = 2 * 10^{- 7}

is the gain saturation coefficient,

γ = 0.5 {ps}^{- 1}

is the loss,

γ_{n} = .5 {ns}^{- 1}

is the loss term for the carriers, and

α = 5

is the linewidth enhancement factor [30]. The delay time for feedback is

τ = 3 ns

. The large delay-time and line-width enhancement factors are selected so that feedback-induced instabilities occur at lower feedback-strength values [38]. These feedback-induced instabilities seem to be part of the reason semiconductor laser arrays are difficult to synchronize [39], so studying this system in a parameter range where they occur is useful. We use the saturable gain model in this system because we are modeling lasers pumped well above the threshold current where gain saturation becomes apparent in the dynamics [31].

J_{0} = a γ_{n} (N_{0} - \frac{γ}{g})

is the pump current, where

a = 4

is a scalar multiplier denoting the ratio between

J_{0}

and the (lasing) threshold current. We choose the large value of pump current to simulate behavior far above threshold for higher power applications. This also implies that the low-frequency-fluctuation region of parameter space is avoided [40].

ω_{i}

is the frequency detuning for the

i

th laser and is distributed according to a Gaussian distribution with mean zero and standard deviation

2 π σ

. The phase noise and carrier noise are modeled as in [32,33]. The phase noise terms depend on the spontaneous emission rate

R_{s p} = 10 {ns}^{- 1}

and the carrier noise is proportional to the carrier lifetime

γ_{n}

and the number of carriers.

η_{E}

is complex uncorrelated Gaussian white noise and

η_{N}

is real uncorrelated Gaussian white noise. The sum terms represent feedback from the other lasers in the array to the

i

th laser. The decayed non-local coupling scheme can be described using a matrix

K

whose

i j

element is:

K_{i j} = d_{x}^{| i - j |}

where

d_{x} \in [0, 1]

. We use the scaling of

1 / M

in Eq. (1) to ensure that

κ^{f} \sum_{j} K_{i j} < γ

for the laser cavities. The reason for such scaling is that the fields in the gain-free system (neglecting delay)

{\dot{E}}_{i} = - γ E_{i} + κ^{f} \sum_{j = 1}^{M} K_{i j} E_{j}

should decay to

E_{i} = 0

for all

i

to guarantee that energy conservation is not violated. This is also the basis of the weak coupling approximation;

κ^{f}

is small enough that the feedback from the external cavity to an individual laser can be treated as a linear addition. Note that

K

is a symmetric positive definite matrix. So its eigenvalue decomposition is

K = \sum_{i = 1}^{M} λ_{i} {\vec{V}}_{i} {\vec{V}}_{i}^{T}

, where

λ_{1} > λ_{2} > ... > λ_{M}

are the eigenvalues and

{\vec{V}}_{1}, ..., {\vec{V}}_{M}

are the corresponding eigenvectors.

We can linearize this system (Eq. (1)) and go into the modal basis using a procedure similar to finding the Master Stability Function for a system of coupled oscillators (i.e. linearizing about the synchronous solution and then left-multiplying by a matrix whose row-vectors are the eigenvectors of the coupling matrix) [19]. Our linearization procedure is not identical to the original procedure in [19] and is described in detail in [41]. We present here a condensed version of this derivation. We begin by linearizing to first order about the perfectly synchronous solution ${\vec{X}}^{*}$ where $X_{i}^{*} = X_{j}^{*}$ . We let $\vec{ξ} = \vec{X} - {\vec{X}}^{*}$ . Then Eq. (1) becomes:

\begin{array}{l} \dot{\vec{ξ}} (t) = [I_{M} \otimes \frac{\partial F (X_{i} (t))}{\partial X_{i} (t)} |_{X_{i} (t) = X^{*} (t)} \\ + \frac{κ^{f}}{M} Γ \otimes \frac{\partial C (X_{i} (t), X_{j} (t - τ))}{\partial X_{i} (t)} |_{X_{i} (t) = X^{*} (t)}] \vec{ξ} (t) \\ +[\frac{κ^{f}}{M} K \otimes \frac{\partial C (X_{i} (t), X_{j} (t - τ))}{\partial X_{j} (t - τ)} |_{X_{i} (t) = X^{*} (t)}] \vec{ξ} (t - τ) \end{array}

where

Γ

is a diagonal matrix whose

i i

th entry is the

i

th row-sum of

K

. This system has a

3 * M

dimensional space. This space is spanned by a set of vectors

V_{i}^{n} = {\vec{V}}_{i} \otimes e_{n}

(where

e_{n}

is the

n

th unit vector in a 3-dimensional space). We can then decompose

\vec{ξ}

as

\vec{ξ} = \sum_{i = 1}^{M} \sum_{n = 1}^{3} δ_{i}^{n} V_{i}^{n} .

Here,

δ_{i} = (δ_{i}^{1}, δ_{i}^{2}, δ_{i}^{3})

represents the linearized amplitude of the array in the direction of the first mode. If we then left-multiply Eq. (6) by

U^{T} \otimes I_{3}

where

U

is a matrix whose

i

th column is

{\vec{V}}_{i}

. This yields the equation for the modal amplitude:

\begin{array}{l} {\dot{δ}}_{i} (t) = [\frac{\partial}{\partial X_{i} (t)} F + \frac{κ^{f}}{M} λ_{1} \frac{\partial}{\partial X_{i} (t)} C] δ_{i} (t) \\ +[\frac{κ^{f}}{M} λ_{i} \frac{\partial}{\partial X_{j} (t - τ)} C] δ_{i} (t - τ) \end{array}

Recall that $λ_{i}$ is the i th eigenvalue for the eigenvector ${\vec{V}}_{i}$ of the matrix $K$ . We approximate row-sum by $λ_{1}$ . This approximation is valid because the off-diagonal terms of the matrix $U^{T} Γ U$ are small. For additional details about the approximation we refer to the reference [41].

For decayed non-local coupling matrix, the first eigenvector (corresponding to the largest eigenvalue) ${\vec{V}}_{1}$ is the principal Gaussian type mode. Letting $κ^{'} = \frac{κ^{f}}{M} λ_{1}$ , we can write the modal amplitude equation for the first mode as:

\begin{array}{l} {\dot{δ}}_{1} (t) = [\frac{\partial}{\partial X_{i} (t)} F + κ^{'} \frac{\partial}{\partial X_{i} (t)} C] δ_{i} (t) \\ +[κ^{'} \frac{\partial}{\partial X_{j} (t - τ)} C] δ_{i} (t - τ) \end{array}

Equation (8) is identical to the linearized equation for a single laser with only self-feedback, where $ξ (t)$ is the first-order deviation of the laser dynamics from the solution about which it is linearized (this is verified by linearizing Eq. (1) for a system with a single laser, $M = 1$ , and a coupling matrix $K = 1$ ). Consequently, the linearized equation for the first order deviation of a single diode laser $ξ (t)$ can be written as:

\begin{array}{l} \dot{ξ} (t) = [\frac{\partial}{\partial X_{i} (t)} F + κ^{f} \frac{\partial}{\partial X_{i} (t)} C] ξ (t) \\ +[κ^{f} \frac{\partial}{\partial X_{j} (t - τ)} C] ξ (t - τ) \end{array}

The implication of this scaling of stability relation is that the solution set and corresponding analysis for the single laser are valid for the first mode of the array. The solutions for the single laser can be found by solving Eq. (1) with $K = 1$ and $M = 1$ for a set of fixed-frequency, fixed-intensity solutions with frequency $Ω$ such that $ϕ (t) = Ω t$ :

\begin{matrix} Ω = - κ^{f} α \cos (Ω τ) - κ^{f} \sin (Ω τ) \\ r = \sqrt{g \frac{J_{0} - γ_{n} N_{0}}{(s γ_{n} + g) (γ - 2 κ^{f} \cos (Ω τ))} - \frac{γ_{n}}{s γ_{n} + g}} \\ N = \frac{J_{0} + \frac{g N_{0} r^{2}}{1 + s r^{2}}}{γ_{n} + \frac{g r^{2}}{1 + s r^{2}}} \end{matrix}

Solutions to this system have been studied in detail for similar Lang-Kobayashi type systems (with various nonlinearities and gain terms but similar dynamics nonetheless) [25,40,42–44]. As $κ^{f}$ is increased, the number of solutions (known as external cavity modes) to the above system increases. Each solution has a unique frequency $Ω$ and the solutions have overlapping regions of stability, making the system highly multistable. When $κ^{f}$ is low, the single laser solution stays on the continuous-wave solution with a fixed frequency $Ω$ . However, once $κ^{f}$ increases, a series of period-doubling bifurcations leads to quasiperiodic and then chaotic dynamics around the fixed-point solution. Further increase of $κ^{f}$ leads to a chaotic state where the system hops between the chaotic attractors formed around the solutions. This is known as the ‘coherence collapse’ region [40,43]. Since this is a delay system, the stability of these solutions cannot easily be found (the characteristic equation is transcendental). However, since the equations used to determine stability of the scaled $M$ -laser system are the same as the equations to determine stability of a single laser (Eq. (8) and Eq. (9)), consequently these solutions should be the same. We show bifurcation diagrams that illustrate this behavior in Fig. 1 for a single laser and a 10-laser array. It is clear from this diagram that the dynamics of the 10-laser array very well resembles the dynamics of the single laser. The yellow line in the figure corresponds to the central solution $Ω$ computed from Eq. (10) for the varying value of $κ^{f}$ (or $κ^{f} = κ^{'}$ for Fig. 1(b)). The bifurcation structure in this system is similar to that in [43].

Fig. 1 Poincare sections of $(ϕ (t) - ϕ (t - τ)) / τ$ are plotted for a single laser and a 10-laser array as a function of feedback strength. Each trajectory was generated from a continuation simulation started with random initial conditions and $κ^{f} = 0.01 {ns}^{- 1}$ . Every 1000ns, $κ^{f}$ was increased by $0.01 {ns}^{- 1}$ . The yellow line in each plot is the central solution frequency $Ω$ from Eq. (11) for a single laser with $κ^{f} = κ^{'} .$ It is clear from these diagrams that the dynamics are almost identical except for the scaling.

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3. Results

We find that the effective coupling constant $κ^{'}$ can be utilized to predict whether the first mode decays (so that there is perfect synchronization rather than first-mode-selection) or is neutrally stable (so that the first mode is selected). Numerical experiments show that this prediction holds even for very large arrays of lasers.

In Fig. 2 we summarize how phase synchronization S scales with array size when coupled through nonlocal decayed coupling. We use $S = < | \sum_{i = 1}^{M} E_{i} |^{2} / M \sum_{i = 1}^{M} | E_{i} |^{2} >$ as a measure of phase synchrony of the array. As coupling constant $κ^{f}$ and size of the array $M$ vary, we observe different levels of phase synchrony including CW, quasi-periodic, and chaotic synchrony. We have also observed spatiotemporal chaos leading to poor phase synchrony. Following the figure, levels of synchronization are sectioned by the effective coupling $κ^{'}$ value. When the value of the effective coupling constant $κ^{'}$ is below 6ns⁻¹, the array exhibits close-to perfect synchronization. At around $κ^{'} =$ 6ns⁻¹ the dynamics become quasiperiodic, as shown in Fig. 1. When $κ^{'}$ is between 6ns⁻¹ and 16ns⁻¹ there are areas of both chaotic and quasiperiodic synchronization. The bifurcation diagram shown in Fig. 1 corroborates this behavior. Because the equation even for the single laser is highly multi-stable in this area, it is rather tricky to find exact regions of quasiperiodic and chaotic behaviors. Once $κ^{'}$ exceeds 16ns⁻¹ phase synchrony destabilizes. We further corroborate this behavior in Fig. 3 where we present the values of $\cos ϕ_{i}$ plotted for 10-laser arrays of identical lasers with varying feedback strength in the presence of carrier and phase noise.

Fig. 2 Average phase synchrony of large arrays of lasers. Each point corresponds to a simulation of an array with randomized initial conditions (we started averaging process after 800ns simulation for convergence to occur). Lines denoting regions of effective coupling are related to the degree of synchronization in the system. For very large arrays achieving phase synchronization requires either weak or very strong (though possibly unrealistic) coupling $κ^{f}$ between the diodes in the array.

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Fig. 3 $\cos ϕ_{i}$ are plotted for 10-laser arrays of identical lasers with varying feedback strength in the presence of carrier and phase noise. When $κ^{f} = 5 {ns}^{- 1}$ , the effective coupling $κ^{'} = 3.65 {ns}^{- 1}$ and the behavior is CW as predicted in Fig. 1. When $κ^{f} = 20 {ns}^{- 1}$ , the effective coupling $κ^{'} = 14.6 {ns}^{- 1}$ , and the behavior is chaotic with a high degree of synchrony. When $κ^{f} = 30 {ns}^{- 1}$ , the effective coupling $κ^{'} = 21.9 {ns}^{- 1}$ , the behavior is chaotic, and the lasers begin to de-synchronize.

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Figure 3 presents clear evidence that as the coupling strength increases, the behavior of a synchronized array becomes chaotic. For a single laser, the onset of chaos is related to the linear increase of the number of fixed-point solutions of the equations in the laser [25]. Since the equation for the stability of the leading transverse mode of the array is the same as that of the single laser with $κ^{f} = κ^{'}$ , it is likely that the onset of chaos for the array of lasers can be attributed to the same reason as the onset of chaos for a single laser. The fact that the dynamics of the laser array can be predicted using the effective coupling scaling corroborates this relationship and shows that the synchrony in the array follows from the transverse mode selection.

To show that modal discrimination takes place in the array of lasers, we consider the modal inner products ${\vec{V}}_{i}^{T} \vec{E}$ . This product represents the magnitude of the array field parallel to the $i$ th transverse mode. When ${\vec{V}}_{1}^{T} \vec{E} (t) > {\vec{V}}_{j}^{T} \vec{E} (t)$ for all $j$ (for a given time interval), it is reasonable to state that for that given time interval the array dynamics have converged to the first spatial mode.

In Figs. 4(a)-4(c), we show modal inner products ${\vec{V}}_{i}^{T} \vec{E} (t)$ for a 30-laser array with various coupling strengths in the presence of noise and disorder. This inner product represents the level of mode discrimination in the array. In Figs. 4(b) and 4(c) we observe that even though modal amplitudes are oscillating, the discrimination is still high (the first mode still is fully separated in amplitude from the rest). We observe that in spite of significant noise, disorder and even chaotic behavior, mode selection still takes place and phase synchrony is high. We also plot the far-field behavior of an array of $M = 100$ lasers with various levels of disorder in Fig. 3(d). We observe that for $σ = 0.1 / τ$ , there is not much of a difference in the central peak intensity compared to the case of $σ = 0$ . However, as the disorder increases, the peak intensity degrades.

Fig. 4 (a-c) The inner products of the first four eigenvectors ${\vec{V}}_{i}^{T} \vec{E} (t)$ ( $i = 1, 2, 3, 4$ ) are plotted for an array of 30 lasers with $d_{x} = .8$ at various values of coupling strength. The simulations have realistic amounts of noise and a frequency disorder of $σ τ = 0.3$ . The synchronization level $S$ is also given, showing that there is indeed synchronization in the quasiperiodic (b) and chaotic (c) regimes. (d) The central far-field lobes of an array of 100 lasers subject to noise and disorder are plotted. For $σ = 0$ , the synchrony level is $S = 1.0$ . For $σ = 0.1 / τ$ , $S = .97$ . For $σ = 0.5 / τ$ , $S = .63$ . For $σ = 1.0 / τ$ , $S = .39$ .

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4. Conclusions

In summary, in this manuscript we have studied and presented conditions for almost perfect phase synchronization of large arrays of semiconductor diode lasers. This phase synchronization can be achieved by employing nonlocal decayed diode coupling structure and choosing array parameters that allow diode laser array to settle at the first spatial mode. We believe examining almost-perfect phase synchronization in the form of mode selection rather than studying perfect in-phase synchronization (as common for global coupling designs) can be of fundamental advantage in order to predict, experimentally achieve and understand the causes of large array phase locking and especially chaotic phase locking.

Our approach may also shed light on how to design a scalable external cavity to phase lock large arrays of semiconductor diode lasers. Indeed we have numerically shown that almost perfect phase synchrony can be achieved for one-dimensional arrays consisting of O (100) coupled diode lasers (see Figs. 2 and 3(d)). Projecting this result to two-dimensional arrays implies O (10,000) diodes (or perhaps more) could be, in principle, coherently phase synchronized provided external cavity (described by the coupling matrix) is properly designed. We would like to note though that engineering challenges in designing such diode array phase locking experiments may be significant. However, it has been demonstrated that using diffractive coupling [45] or optical fiber networks [12] one can design reconfigurable semiconductor laser networks that can show a wide variety of array behaviors so it should be possible to test specific network structures experimentally.

Funding

This research was supported in part by the Office of Naval Research and the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory. Oak Ridge National Laboratory is managed by UT- Battelle, LLC for the U.S. Department of Energy under Contract DE-AC05-00OR22725.

Acknowledgment

The authors would like to thank Alejandro Aceves of Southern Methodist University, Department of Mathematics and Brendan Neschke of Raytheon for valuable discussions that were important for the outcome of this work. Opinions, interpretations, and conclusions, and recommendations are those of the authors and are not necessarily endorsed by the U.S. government.

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Phase-locking of arrays of weakly coupled semiconductor lasers

Abstract

1. Introduction

2. Analysis

3. Results

4. Conclusions

Funding

Acknowledgment

References and links

Cited By

Figures (4)

Equations (10)

Optics Express