Theory and Modeling of Lasing Modes in Vertical Cavity Surface Emitting Lasers

The problem of obtaining the lasing modes and corresponding threshold conditions for vertical cavity surface emitting lasers (VCSELs) is formulated as a frequency-dependent eigenvalue problem in required gain amplitudes and corresponding fields. Both index and gain guiding are treated on an equal footing. The complex gain eigenvalues define necessary but not sufficient conditions for lasing. The actual lasing frequencies and modes that the VCSEL can support are then determined by matching the gain necessary for the optical system in both magnitude and phase to the gain available from the laser’s electronic system. Examples are provided.


INTRODUCTION
In the simulation of edge emitting lasers, the optical field can usually be modeled simply by a predetermined set of Fox-Li quasimodes of the passive cavity [1].The threshold condition can then be expressed as mode gain times mode lifetime equals unity.No laser previously has presented the challenge to optical simulation now presented by vertical cavity surface emitting lasers (VCSELs).
The challenge to VCSEL simulation is not just a much more complicated optical cavity geometry for analysis-although this problem alone is significant-but that quasi-mode analysis, itself, can no longer be relied upon.The mirrors are distributed and lateral confinement may be pro- duced by a combination of index and gain guiding.
(See, for example, Ref. [2].) Thus the VCSEL optical cavity boundaries, along with the passive cavity modes and conventional parameters such as the photon lifetime, are often poorly defined.

THEORY
At the threshold for lasing, the optical field will be self-supporting in the presence of the gain supplied by the laser.Consider an open VCSEL optical cavity with tensor electric susceptibility (', co) Xcav(', co) + Xg(r, co) where Xg(r, co) is the necessa- rily complex susceptibility representing the laser gain inside the active region provided under bias, and cav(Y, co) is the potentially complex suscept- ibility representing the rest of the VCSEL.Within the semiclassical approximation, lasing requires [3] g(r, f J where (Y,)is the electric component of the lasing field, Gcav(Y',) is the tensor Green's function for radiation from a current source within the open VCSEL cavity defined by Xcav, and jeoX(r' ) (' ) acts as an equivalent cur- rent source.
If the spatial distribution of the gain is known to good approximation, then Eq. ( 1) reduces to an eigenvalue problem in the necessary gain ampli- tudes for lasing and the corresponding lasing (0) fields.That is, if Xg(r,) g()Xg (r) where Xg () is known, then Eq. ( where () is an integral operator defined by (3) Note that this eigenvalue problem need.gnlybe evaluated over regions for which Xg (') is significant, that is, over the active region of the laser.The solutions to this eigenvalue problem will be complex gain eigenvalues and the corresponding fields as a continuous function of frequency, in contrast to the more familiar case of finding the discrete real frequency eigenvalues and modes of a closed passive cavity.
It is also possible to repartition the VCSEL susceptibility to take part of Xcav out of the Green's function and put it in Eq. ( 2) as a second source term.This may assist in the solution of problems for which the Green's function is not easily obtained.Extending the formulation in this way changes the ordinary eigenvalue problem of Eq. ( 2) into a generalized eigenvalue problem.
The solutions to Eq. ( 2) establish necessary but not sufficient conditions for the VCSEL to lase.The frequencies at which the VCSEL can actually lase are those for which the complex gain susceptibility necessary for the optical system can be matched to the complex gain susceptibility available from the laser electronic system in both magnitude and phase.

EXAMPLE
To illustrate the use of this formulation, lasing in the VCSEL cavity diagrammed in Figure was considered [2].This cavity allows for relatively easy analysis while allowing much of the essential physics of VCSELs to be modeled.Xcav(',co)was taken as that of the planarly layered structure including the nominally lossy quantum well layer, but with no lateral confinment.Note, for simpli- city, all susceptibilities were treated as scalars.The spatial distribution of the gain susceptibility was approximated as uniform inside the active region and zero elsewhere.Thus for later convenience we set X(g ) (Y) -j within the 9 pm 10 gm3 nm quantum well active region, and X(g)(f ") -0 else- where.
The Green's functions for such a planarly layered structure are well known [4], and only those with both source and field coordinates within the thin quantum well active layer were required.The Green's functions were obtained in Fourier- space where the only portion of the calculation that requires a computer is the calculation of the distributed-Bragg-reflector (DBR) reflectivities vs. frequency, incident angle and polarization, which may be performed using several equivalent meth- ods [5,6].Of course, calculation of the Green's function for more complex cavity structures could require computationally intensive numerical meth- ods, while taking advantage of the cylindrical symmetry in many VCSELs could significantly reduce the computational load.However, while of obvious practical importance, how the Green's function is obtained or what coordinate system is used is of no conceptual importance in this formulation.Note that the potentially difficult problem of finding the Green's function would only need to be solved once for all bias conditions, as the Green's functions are independent of bias as long as effects such as thermal expansion of the cavity are ignored.
Once the Green's function was calculated, the gain eigenvalue problem of Eq. ( 4) was discretized and solved.Using a moment method, the field, approximated as constant over the width of the well, was expanded in rectangular testing functions with unknown complex coefficients in the (x-y) plane of the well, i.e.

'(ff)rect x-nxdx
)/\ ff(x, y) rect ( y nydy (4) where rect[(r/-rio)/d] is a rectangular pulse of amplitude one and full width d centered at r/o, and the discrete variable ff (nx, ny) labels the grid sites.
Figure (2) shows the first three gain eigenvalues n(co) (two are essentially degenerate) plotted at Real(K) FIGURE 2 First three complex gain eigenvalues at discrete intervals in frequency; two are essentially degenerate.Lasing is possible only where the eigenvalue curves crosses the real axis, with the values of the gain eigenvalues at these crossings defining the threshold condition for lasing.
constant intervals in frequency.Approximating the gain susceptibility available from the electronic system of the laser under bias as purely imaginary, lasing is only possible where one of the gain eigenvalue curves crosses the real axis, such that Xg(F, o) ()X(g ) (F) is also purely imaginary.
The gain eigenvalue at the crossing frequency defines the threshold gain susceptibility.Approximating the gain susceptibility as purely imaginary is done here for illustrative purposes only.In reality the actual lasing modes of the system would be determined by the intersection of the gain eigenvalues with a curve describing the Kramers-Kronig relation between the real and imaginary parts of the gain susceptibility.The large area active region considered in this example produces a very tight spacing of gain eigenvalues in frequency, but the gain amplitudes required for lasing are significantly different because of the different overlaps of the self-consistently calcu- lated lasing modes with the active region.Figure (3) shows the field patterns of the first and third lasing modes inside the gain region.
x (microns) FIGURE 3 Lasing fields at (a) A _ _ _ 1.0003 Ao and (b) A _ 1.0001 Ao as a function of position within the gain region.

CONCLUSION
The problem of obtaining the lasing modes and corresponding threshold conditions for VCSELs has been formulated as a frequency-dependent eigenvalue problem in required gain amplitudes and corresponding fields.Index and gain guiding are treated on an equal footing.The complex gain eigenvalues define necessary but not sufficient conditions for lasing.The actual lasing frequencies and modes that the VCSEL can support are then determined by matching the gain necessary for the optical system in both magnitude and phase to that available from the laser electronic system.With this formulation the lasing modes and corresponding threshold conditions are well defined even when the optical cavity boundaries and conventional cavity parameters such as the photon lifetime are not.
This formulation also has the practical advan- tage that the problem of obtaining the self- supporting lasing fields of a VCSEL cavity is separated into two distinct and already thoroughly studied problems: that of finding the optical Green's functions for a fixed radiation source in an open cavity and that of solving a complex eigenvalue problem.Thus, this formulation pro- vides a framework for the application of a large preexisting knowledge base to the relatively new challenge of modeling VCSELs.For example, in the specific implementation used for this work, the cavity Green's function was obtained by textbook methods [4-6] and the generalized complex eigenvalue problem was solved using a commer- cially available numerical routine.