# Stability Boundaries in Laterally-Coupled Pairs of Semiconductor Lasers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{s}, ν

_{a}, of the symmetric and antisymmetric normal modes by η = (ν

_{s}− ν

_{a})/2. Expressions for η in the case of purely real index guidance (i.e., ignoring any effects of gain or loss) have been derived for one-dimensional step-index (slab) waveguides [27] and for circular optical fibres that are weakly guiding (i.e., the difference between core and cladding refractive indices is much less than either index) [28]. A simplified expression for the latter that is valid for multimode fibre couplers has been given by Ogawa [29]. A more general expression for the coupling coefficient between circular cross-section VCSELs with real guidance is also available [30]. Comparison of the dynamics of a coupled pair of lasers modelled by slab waveguides indicated generally good agreement between coupled-mode and normal-mode treatments for edge-to-edge spacings greater than the waveguide width [17]. The only experimental test (to the best of our knowledge) of coupled-mode limits for laser pairs was performed by comparing predicted and measured far-field visibility of optically-pumped VCSELs [31]; the results indicated that coupled-mode theory was inaccurate for a spacing between the two pump spots of less than 13 μm when the modal radius of a solitary VCSEL was estimated as 3.5 ± 0.5 μm. In general, a clear definition of the ranges of parameters where coupled-mode theory is sufficiently accurate has not been given as yet.

## 2. Model

_{core}(n

_{clad}) and the edge-to-edge separation is 2d. The slab half-width and cylinder radius are each a. With this notation, the conventional definitions of the normalised frequency v and the normalised decay constant of the fields in the cladding w for single solitary guides are:

_{eff}is the effective index of the single solitary guide.

_{k}is the frequency of mode k, E

_{k}, Φ

_{k}are the time-dependent and spatial-dependent field components, respectively, and t is time. For the one-dimensional slab waveguide of Figure 1a, the confinement is in the x direction, so that the dependence on y can be neglected.

_{k}is then:

_{p}) where τ

_{p}is the photon lifetime), c is the speed of light, n

_{g}is the group index of the cavity, α is the linewidth enhancement factor, $\overline{{g}_{1}},\overline{{g}_{2}}$ are the mean gains per unit length in guides 1 and 2, and Γ

_{1kk’}, Γ

_{2kk’}are overlap factors in guides 1 and 2, defined as:

_{j}in guide j is:

_{j}is the pumping rate in the jth guide and γ is the carrier recombination rate. The conventional linear relationship between gain, $\overline{{g}_{j}}$, and carrier concentration is assumed:

_{diff}is the differential gain and N

_{0}is the transparency concentration.

## 3. Results

_{core}≈ n

_{clad}= 3.4. Three values of the difference (n

_{core}− n

_{clad}) are chosen as 0.000971, 0.002 and 0.0055. From Equation (1), these correspond to normalised frequency v = 1.571, 2.255 and 3.740, respectively. These values have been chosen to explore different regions of operation of the coupled guides in terms of the transverse modes supported by each solitary guide: the cut-offs for the first and second higher-order modes of the slab are v = π/2 and π, and those for the circular cylinder are 2.405 and 3.832, respectively. Hence, these values include operating regions, where 1, 2 or 3 transverse modes of the slab and 1 or 2 modes of the circular cylinder are present. The values of the other parameters appearing in the rate equations are κ = 327 ns

^{−1}, α = 2, γ = 1 ns

^{−1}, a

_{diff}= 1 × 10

^{−15}cm

^{2}and N

_{0}= 1 × 10

^{18}cm

^{−3}.

_{s}− ν

_{a})/2. The dashed lines are fitted to these results using the approximation due to Ogawa [29] of the form:

_{S}for the lowest-order mode (LP01) of the corresponding solitary single guide, given by [32]:

_{0}, K

_{1}are modified Bessel functions. This ratio occurs when rate Equations (4) and (7) are cast into normalised form for computational purposes. In these figures, the subscript denoting the guide has been dropped and the values for subscripts $k\ne k\u2019$ are given as |Γ

_{sa}|, since this quantity has a different sign in each guide. The symbols (circles, triangles, diamonds) in these figures refer to numerically calculated points, whilst the broken lines are empirical fits used later in the numerical solutions of the rate equations. It is noteworthy that there is significant structure in the variation of these overlap factors for low values of d/a in the operating regions considered here, and that the region of d/a where this structure occurs reduces with increasing v. Plots of the overlap factors for a specific slab waveguide have been presented in [17], including the detuning between resonant frequencies of the two lasers, and show somewhat less variation at corresponding d/a values for the case of zero detuning. In the limit of large d/a, all the ratios of overlap to confinement factors shown in Figure 3, Figure 4 and Figure 5 tend to a value of 0.5. In this limit, it can be shown that rate Equations (4) and (7) reduce to the corresponding equations for the coupled-mode approximation [21]; details of this reduction as well as other aspects of the normal-mode treatment will be presented elsewhere.

_{th}vs. d/a, where P

_{th}is the threshold value of P for a solitary laser.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of two coupled slab waveguides. (

**b**) Schematic of two coupled circular cylindrical waveguides.

**Figure 2.**Variation of coupling rate with d/a for pairs of circular cylindrical guides with three values of v and for a slab guide with v = 1.571. Symbols (circles, squares, triangles, diamonds) are calculated from normal-mode theory; dashed and dotted lines are fitted from analytic results given in the text.

**Figure 6.**Stability boundaries in the plane of P/P

_{th}versus d/a for coupled circular cylindrical guides with v = 1.571. Curves labelled ‘inf’ are obtained using the values of overlap factors in the limit of large d/a.

**Figure 7.**Stability boundaries in the plane of P/P

_{th}versus d/a for coupled circular cylindrical guides with v = 2.255.

**Figure 8.**Stability boundaries in the plane of P/P

_{th}versus d/a for coupled circular cylindrical guides with v = 3.740.

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**MDPI and ACS Style**

Vaughan, M.; Susanto, H.; Li, N.; Henning, I.; Adams, M.
Stability Boundaries in Laterally-Coupled Pairs of Semiconductor Lasers. *Photonics* **2019**, *6*, 74.
https://doi.org/10.3390/photonics6020074

**AMA Style**

Vaughan M, Susanto H, Li N, Henning I, Adams M.
Stability Boundaries in Laterally-Coupled Pairs of Semiconductor Lasers. *Photonics*. 2019; 6(2):74.
https://doi.org/10.3390/photonics6020074

**Chicago/Turabian Style**

Vaughan, Martin, Hadi Susanto, Nianqiang Li, Ian Henning, and Mike Adams.
2019. "Stability Boundaries in Laterally-Coupled Pairs of Semiconductor Lasers" *Photonics* 6, no. 2: 74.
https://doi.org/10.3390/photonics6020074