The split-density model: a unified description of polarization and array dynamics for vertical-cavity surface-emitting lasers

Abstract

We present a generalized rate equation model for the array and polarization dynamics of general one or two dimensional arrays of Vertical Cavity Surface Emitting Lasers (VCSELs). It is demonstrated that our model includes both the previous theory for edge emitting laser arrays and the theory of polarization dynamics in quantum well VCSELs in a single unified description. The model is based on the physical assumption of separated carrier densitiy pools individually coupled to different light field modes. These modes, defined by the separate gain profile of each pool, interact through the coherent dynamics of the light field derived from Maxwell's equations and represented by the coefficients for index and loss guiding. The special case of two densities and two light field modes is solved and the implications of the results for large VCSEL arrays are discussed. Our analytic results show that typical solutions of the split density model range from phase locking to chaos, depending on the magnitude of the frequency coefficients. For weak coupling, the stable supermode is always the mode of highest frequency. This indicates that anti-phase locking is the only stable phase locking possible in arrays.

Spatial hole-burning effects in the amplified-spontaneous-emission spectrum of the nonlasing supermode in semiconductor laser arrays

“…For the case of s = 0 equation ( 14) reduces to the expressions derived in previous studies [4,5]. Since an instability is most likely to occur at low intensities I 0 , however, a closer examination of the stability boundary is of interest in the present context.…”

“…This type of anti-phase locking is commonly observed in vertical cavity surface emitting laser arrays [7][8][9][10]. In particular, the presence of carrier diffusion leading to a stochastical exchange of carriers between the lasers was found to enhance the stability of the anti-phase locking considerably [4]. In the case of stable anti-phase locked operation, the special features of the model are still present in the dynamics of the quantum fluctuations causing amplified spontaneous emission in the non-lasing modes of the array.…”

“…Note that the consideration of carrier diffusion effects represents an extension of the original model of Winful and Wang [4,5]. The dissipative optical coupling term s represents additional optical losses in the region between the two lasers, where the two field modes interfere.…”

“…The carrier density dynamics of semiconductor lasers couple not only to the lasing modes, but also modify the amplified spontaneous emission into the non-lasing modes. It is therefore possible to gain interesting insights into holeburning effects by studying the light field fluctuations in the non-lasing modes [1][2][3][4]. The most simple system which can exhibit holeburning effects is a laser array in which each of the individual lasers can be considered to operate in a single mode which is coupled to the local carrier density of that individual laser.…”

“…The most simple system which can exhibit holeburning effects is a laser array in which each of the individual lasers can be considered to operate in a single mode which is coupled to the local carrier density of that individual laser. A rate equation description for this type of device was developed by Winful and Wang [5,6] and has been discussed in a more general context by the authors [4]. In the following, this model will be referred to as the split density model since the characteristic feature of the model is the separation of the carrier density into distinct density pools associated with the individual lasers in the array.…”