## Abstract

The semiconductor laser simulator MINILASE is being extended to simulate vertical cavity surface emitting lasers (VCSELs). The electronic system analysis for VCSELs is identical to that for edge emitting lasers. A brief discussion of the capabilities of MINILASE in this domain will be presented. In order to simulate VCSELs, the optical mode solver in MINILASE must be extended to handle the reduced index guiding and significant gain guiding typical of many VCSEL structures. A new approach to solving the optical problem which employs active cavity modes rather than the standard passive cavity modes is developed. This new approach results in an integral eigenvalue equation in required gain amplitudes and corresponding modal fields. Sample results from an early implementation of a gain eigenvalue solver are shown to clarify the possibilities of this approach.

© Optical Society of America

## 1. Introduction

Previous attempts to simulate vertical cavity surface emitting lasers (VCSELs) have often concentrated on the optical problem (see, for example, [1,2]), as the small cavities are expected to exhibit interesting optical properties. There also have been several sophisticated optical models coupled to rudimentary electronic models (for example [3]), which attempt to capture the essential physics of the operation of the laser with a minimum of overhead. Finally, there have been a few attempts to comprehensively model VCSELs (for example [4]) using precise optical and electronic models. However, previous electronic models have lacked the full dynamic capabilities of the edge emitting laser simulator MINILASE II, which are necessary for obtaining accurate modulation response curves [5]. Also, previous optical models have involved approximations which are not valid for all VCSELs [4]. We describe in this paper an ongoing effort to comprehensively model VCSELs through an extension of MINILASE. The extension will include a new optical mode solver, partially based on a similar solver previously used by Lin and Deppe [2]. This full vector three dimensional mode solver finds active rather than passive cavity modes, treats both index and gain guiding equally well, and doesn’t require that the cavity geometry be separable. A simple non-separable index guided structure is analyzed to demonstrate the capabilities of the mode solver.

## 2. Electronic simulation

In MINILASE, the solution of the two dimensional electronic equations that determine electron and hole supply to the active lasing region is accomplished along the lines of silicon device simulation [6] with the addition of specific provisions that describe transport over abrupt heterojunctions [7] and capture in the active quantum well region [5]. The quantum wells are coupled to the classical transport equations by Bethe’s thermionic emission theory and by the inclusion of ballistic regions [5]. Within the quantum wells we solve a Boltzmann transport equation in energy space and a rate equation for phonon heating [8]. The resulting electrical simulator has been described in detail in [9] and represents the core of the MINILASE II simulation package that has been extensively tested through simulations of edge emitting lasers and subsequent comparisons with experimental results [9]. It provides the gain function including all known nonlinear effects that arise from transport phenomena: spectral hole burning, carrier and phonon heating, and finite capture rates, for example. The electrical simulator is expected to be unchanged for VCSELs and provides us with the carrier densities and gain that are required by the optical solver that is described as the main topic of this paper.

## 3. Optical simulation

#### 3.1 Theory

MINILASE solves the photon rate equation for the intensity of a given optical mode. The photon rate equation for a single mode is

where *S* is the mode intensity, *G* is the gain
(due to stimulated emission) for the mode under consideration,
*τ*_{phot}
is the photon lifetime of
the mode, and *R*_{spon}
is the amount of spontaneous
emission which couples into the mode. In order to find *G*,
*τ*_{phot}
, and
*R*_{spon}
for the mode under consideration, the
optical modes must be obtained from the Maxwell equations without
field-independent sources (spontaneous emission). In the conventional
semiconductor laser theory [10], a set of simplifying assumptions are made in order to
find the optical modes. The cavity geometry is usually assumed to be separable
to some degree, and the gain is often treated as a small perturbation to the
permittivity *∊*(*r*⃗,
*ω*). In practice, the optical cavity is often
closed in order to find the modes, and the effect of the gain and cavity loss on
the spatial mode pattern is neglected altogether [9]. The mode patterns calculated in this way are known as
the passive or cold cavity modes. The assumption that the gain distribution has
little effect on the lasing mode pattern is well justified for strongly index
confined devices [10]. However, many VCSEL devices may be strongly gain
guided. Therefore it is desirable to include the gain in
*∊*(*r*⃗,
*ω*) and find instead the *active*
cavity modes of the VCSEL structure.

In order to solve Maxwell’s equations for the active cavity modes, we
first separate
∊(*r*⃗,*ω*) into
two parts as

where ∊_{cav}(*r*⃗, *ω*) is the
potentially complex relative permittivity representing the passive cavity, and
*χ*_{gain}
(*r*⃗, *ω*) is the
necessarily complex susceptibility representing the gain region of the laser. A
susceptibility rather than a permittivity is used to represent the gain in order
to clearly separate it from *∊*_{cav}
, for
reasons that will become apparent shortly. We can write the vector Helmholtz
equation for the electric field as

$$\phantom{\rule{16em}{0ex}}=\mathit{i\omega}{\mu}_{o}{\overrightarrow{J}}_{\mathit{stim}}(\overrightarrow{r},\omega )$$

where *J*⃗_{stim},(*r*⃗,*ω*) =
-*iω*
*∊*_{o}
*χ*_{gain}
(*r*⃗,*ω*)*E*⃗(*r*⃗,*ω*)
is the field-dependent current representing stimulated emission in the gain
region. The solution of the inhomogeneous wave equation Eq. (3) can be represented as

$$\phantom{\rule{2.8em}{0ex}}=\int {d}^{3}\overrightarrow{r}\prime {\overleftrightarrow{G}}_{\mathit{cav}}(\overrightarrow{r},\overrightarrow{r}\prime \omega )\xb7\left[i\omega {\u220a}_{o}{\chi}_{\mathit{gain}}(\overrightarrow{r}\prime ,\omega )\overrightarrow{E}(\overrightarrow{r\prime},\omega )\right],$$

where *G*⃡_{cav}
(*r*⃗,*r*⃗′,*ω*)
is the tensor Green’s function for radiation from a current source
within the open VCSEL cavity defined by
*∊*_{cav}
.

Eq. (4) can be viewed as a condition that the gain
susceptibility *χ*_{gain}
must satisfy in
order that the corresponding mode *E*⃗ lase (be
self-supporting) in the absence of spontaneous emission. With spontaneous
emission present, the gain (and the mode) must approach this condition
asymptotically. In order to arrive at a more useful form of Eq. (4), we note that spatial distribution of the gain
susceptibility *χ*_{gain}
can be calculated
from the electronic system using MINILASE. We therefore write
*χ*^{gain}
(*r*⃗,*ω*)
=
*K*(*ω*)${\chi}_{\mathit{\text{gain}}}^{\left(0\right)}$(*r*⃗,*ω*),
where ${\chi}_{\mathit{\text{gain}}}^{\left(0\right)}$ is the
normalized gain susceptibility pattern, and *K* is a complex
constant that allows *χ*_{gain}
to meet the
lasing condition given as Eq. (4). Now Eq. (4) becomes

where *G*⃡ (*ω*) is an
integral operator defined by

Eq. (5) is now an eigenvalue problem; the eigenfunctions
*E*⃡ are the mode field patterns and the
eigenvalues *K* are the gain amplitudes which define the lasing
condition for the corresponding modes. This eigenmode equation is unusual as the
eigenvalues are complex gain eigenvalues rather than complex frequency
eigenvalues.

The complex gain eigenvalues *K* are a continuous function of
frequency. In order to obtain the frequencies at which the VCSEL can actually
lase, the complex gain susceptibility necessary for the optical system must be
matched to the complex gain susceptibility available from the laser electronic
system in both magnitude and phase. This match will only occur at discrete
frequencies. This determines the lasing frequencies of the laser. In addition,
the required photon rate equation parameters may be calculated for each mode
obtained in this fashion through use of the lasing mode pattern
*E*⃗ and the gain susceptibility required to lase
*χ*_{gain}
. For example, we may obtain
the photon lifetime by using the fact that gain equals loss at threshold; hence
1/*τ*_{phot}
will equal the total gain
of the active cavity mode at threshold, which may be calculated as the overlap
integral of the mode’s normalized field intensity pattern times the
gain susceptibility required to lase.

It is possible to repartition the VCSEL susceptiblility to take part of
*∊*_{cav}
out of the
Green’s function and put it into the eigenvalue problem instead.
Suppose we have a localized passive cavity feature which we don’t
want to include in the passive cavity Green’s function *G*⃡_{cav} We may then split the electric permittivity further as

where the new term *χ*_{cav}
represents a
localized deviation from the cavity structure defined by
*∊*_{cav}
. This susceptibility then
turns up in Eq. (3) as a second source term, and Eq. (5) now takes the form of a generalized eigenvalue equation:

There are two primary reasons to use this generalized formulation. First, if some
localized regions of the laser cavity change with increasing bias (e.g.,
localized temperature related index changes) it would be useful to put these
changes into the eigenvalue problem rather than the Green’s function,
to avoid the need to recalculate Green’s functions. Second, there are
a wide range of VCSELs which can be effectively modeled as very simple cavity
structures containing localized deviations from the simple structure. The simple
cavity structure will have an easily calculable Green’s function. The
localized deviations from the simple structure can be partitioned into
*χ*_{cav}
Thus we greatly simplify the
task of finding the Green’s function with only a modest corresponding
increase in the difficulty of the eigenvalue problem. Making use of this allows
greater computational speed and may in fact produce greater accuracy as well, by
avoiding the complicated and slow numerical techniques needed to find the tensor
Green’s function for a generalized cavity structure.

#### 3.2 Implementation

We present here some sample results from our current implementation of the gain eigenvalue method. Although these simulations did not include the electronic solver, useful information and insight can be gained by examining the optical modes and corresponding gain eigenvalues.

We used the gain eigenvalue method to find the optical modes of a simple but
non-separable problem, the index guided VCSEL diagrammed in Fig. 1. The underlying cavity structure
*∊*_{cav}
is planar, and therefore an
analytic Green’s function is available [11]. The wavelength-tall optical cavity is bounded above
and below by distributed Bragg reflectors (DBRs). There is an oxide layer
surrounding the active region in the plane of the active region (similar to the
model of Lin and Deppe [2]). The oxide layer is slightly lossy in order to enforce
the radiation condition as well as to physically model optical losses in the
VCSEL structure. For our initial simulations, we set the normalized gain
susceptibility pattern
${\chi}_{\mathit{\text{gain}}}^{\left(0\right)}$ to be uniform in a
rectangular active region, and zero elsewhere. Thus for later convenience we set
${\chi}_{\mathit{\text{gain}}}^{\left(0\right)}$(*r*⃗)
≡ -*i* within the 9*μ*m
×10*μ*m×3nm quantum well
active region, and
${\chi}_{\mathit{\text{gain}}}^{\left(0\right)}$(*r*⃗)
≡ 0 elsewhere. The entire layer which contains the oxide and the
active region is given the background index of the oxide, which is effectively
raised to the index of *GaAs* inside the active region by setting
*χ*_{cav}
inside the active region to
be equal to the difference between the two permittivities.

Once the Green’s function for sources within the active region was calculated, the gain eigenvalue problem of Eq. (8) was discretized and solved. Using a moment method, the field was represented as a sum of known expansion functions with unknown coefficients, and the inner product of both sides of Eq. (8) was taken with a set of known testing functions. The resulting dense matrix eigenvalue problem was solved using a commercially available eigenvalue solver (IMSL).

Figure 2 shows the real and imaginary parts of the first
six gain eigenvalues *K*(*ω*) for this
structure (the first two are essentially degenerate) plotted at constant
intervals in frequency. Approximating the gain susceptibility available from the
electronic system of the laser under bias as purely imaginary (neglecting the
Kramers-Kronig relation for now), lasing is only possible at frequencies at
which one of the gain eigenvalue curves crosses the real axis, such that
*χ*_{gain}
(*r*⃗,*ω*)
=
*K*(*ω*)${\chi}_{\mathit{\text{gain}}}^{\left(0\right)}$(*r*⃗)
is also purely imaginary. The gain eigenvalue at the crossing frequency defines
the gain susceptibility required to lase. The large area active region
considered in this example produces a very tight spacing of lasing modes in
frequency, but the gain amplitudes required for lasing are significantly
different because of the different overlaps of the self-consistently calculated
lasing modes with the active region. Figure 3 shows the field patterns of the first and third
lasing modes inside the active region. These modes are full-vector modes of the
three dimensional cavity.

## Acknowledgement

The authors gratefully acknowledge the support of the Office of Naval Research.

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