## Abstract

We present calculations of the modification of the spontaneous emission rate from a point source dipole in a Fabry-Perot microcavity containing an optically thin dielectric aperture. The dielectric aperture is described as a passive current source which is driven by the spontaneous point source. The spontaneous emission rate is shown to depend on the details of the aperture design, and shows a strong enhancement on resonance due to 3-dimensional optical confinement by the dielectric aperture.

© Optical Society of America

## 1. Introduction

Low loss, planar Fabry-Perot microcavities can be realized using epitaxial crystal growth to form highly reflecting distributed Bragg reflectors. These semiconductor microcavities have been studied extensively to determine how they might be used to control spontaneous emission. [1–6] Past studies have shown that while the spontaneous radiation pattern from a dipole confined in the planar microcavity can be modified, the spontaneous emission rate cannot be increased too significantly. For idealized cavity systems based on single, highly reflecting interfaces, the largest increase that can be achieved in the planar system is found for a half-wave cavity spacer and is about a factor of two. [7] The limitation in lifetime control for such a system stems from coupling to waveguide modes that decrease the amount of optical feedback due to the cavity. [6–9]

Recently, Fabry-Perot microcavities with current and optical confining dielectric apertures [10] formed by “wet” oxidation [11] have been demonstrated to have interesting effects on spontaneous emission. [12] The similar kind of structure has attracted a great deal of attention because of the many improvements it has made in the performance of vertical-cavity surface-emitting lasers. [10] For spontaneous emission control, the dielectrically apertured Fabry-Perot microcavity is interesting because of the 3-dimensional confinement exerted on the optical mode. Recently we showed that this system can be analyzed in its ideal form by treating the apertured region as a passive current source driven by a gain region, and we have presented a detailed study on the lasing eigenmode in such a system including the boundary condition of the active gain region. [13,14] The self-consistent analysis shows that a resonance shift associated with the aperture leads to cutoff of lateral loss that would occur due to waveguide modes, and provides the 3-dimensional confinement in the otherwise planar microcavity. [15] Because of the rigorous inclusion of the active source’s electromagnetic boundary conditions, a similar approach can be used to evaluate spontaneous emission from a spectrally narrow-band point source in such a system. We present such an analysis for a non-zero loss cavity below, and show that a sizable change over the “open-space” spontaneous emission rate is possible. Both spontaneous enhancement and inhibition effects are calculated.

## 2. Calculation

We consider an idealized Fabry-Perot microcavity with an optically thin dielectric
aperture as shown is Fig. 1. The coordinate system chosen is also plotted in the
figure. The cavity consists of two metallic reflectors with equal field reflectivity
ρ that are separated by the cavity length L. There is a thin dielectric
aperture of thickness Δz_{R} in the center of the cavity. The
dielectric constant inside and outside the cavity is taken as that of free space,
which simplifies the notation. However, we note that the same treatment can be
applied to the semiconductor dielectric microcavity, although the complexity of the
math is increased.

The spontaneous point source emitter can be written as

where the time dependence J_{sp}(t) sets the frequency extent. In general, a
range of frequencies should be used to study the spontaneous emission. However, for
a spectrally narrow-band source, a single frequency accurately predicts the lifetime
modification due to the cavity, and more clearly treats the transverse mode coupling
that exists due to the nonunity mirror reflectivity and the dielectric aperture.
Multifrequency emission can also be treated by expansion into a summation over the
single frequency emissions. To the first order approximation, the spontaneous point
source is considered as unchanged by the cavity, even though it interacts with its
emitted field and the incoming vacuum field. If the point source is in free space,
we know that the radiated field can be described by spherical waves, and that the
spherical waves can be expanded into plane wave modes. [16] When the spontaneous point source is placed in a
microcavity, the radiated field will be modified by both the cavity mirrors [9] and the dielectric aperture, and the modified field is most
conveniently treated in terms of the plane wave modes as well. The spontaneous point
source acts as an excitation source to the dielectric aperture (Fig. 1), which is described by a real susceptibility function
χ_{R}(x,y,ω) whose z dependence is removed in the
limit that the aperture is optically thin. The induced polarization current due to
the aperture is related to the total electric field at the aperture’s
position and the real susceptibility χ_{R} (x, y, ω)
by

The total electric field includes the field directly radiated from the spontaneous point source, the radiated fields from the aperture and cavity mirrors as driven by the spontaneous source. Under the planar cavity boundary conditions, it is very convenient to use the plane wave modes which forms a complete and orthogonal basis. Any radiation field in the passive cavity can be expanded into the spatial plane wave mode set. In general, when a single frequency source, either passive or active, is placed inside the planar microcavity, it is included in the Fourier transformed Maxwell’s equations as

where ε(**r**,ω) is the dielectric constant that
satisfies the planar cavity boundary conditions. For the case of Fig. 1, **J**(**r**,ω) has two
contributions which are the spontaneous point source
**J**
_{sp}(**r**,ω) and the induced
polarization current in the aperture **J**
_{d}(**r**,
ω). The radiation field from **J**(**r**, ω)
can be calculated directly from Maxwell’s equations with the precaution
that reflections from the cavity mirrors must be considered. Taking the spontaneous
source as having vector amplitude lying in the x-y plane, for a symmetrical system
as shown in Fig 1 the z component of the radiation field in the center of
the cavity is zero while the x and y components take the Fourier transformed form [13,14]

$$\left\{\mathbf{k}\times \left[\mathbf{k}\times \left(\underset{-\infty}{\overset{\infty}{\int}}{\mathrm{dk}}_{x}^{\prime}\underset{-\infty}{\overset{\infty}{\int}}{\mathrm{dk}}_{y}^{\prime}\left(\frac{-\mathrm{i\omega}{\epsilon}_{o}}{4{\pi}^{2}}\right)\Delta {z}_{R}{\chi}_{R}({k}_{x}^{\prime},{k}_{y}^{\prime},\omega )\mathbf{E}\left({k}_{x}-{k}_{x}^{\prime},{k}_{y}-{k}_{y}^{\prime},0,\omega \right)\right)+{\mathbf{J}}_{\mathrm{sp}}{\Delta}^{3}{\mathbf{r}}_{n}\right]\right\}$$

In solving Eq. (4) a self-consistent solution can be obtained through numerical
iteration by starting with the calculated field due to the spontaneous point source
without the presence of dielectric aperture to get the first order approximation of
**J**
_{d} (**r**, ω) from Eq. (2). The solution is then reinserted back into the right-hand
side of (4) to calculate the next higher order approximation. This process is
continued through enough iterations to obtain the convergent spontaneous field in
the presence of the dielectric aperture.

## 3. Results

We apply the analysis to calculate the spontaneous lifetime for a half wave cavity
with the cavity length tuned for resonance at 1 μm wavelength. Once the
single frequency, self-consistent field for a point source is found, the spontaneous
emission rate is found either by summing the emission rate into each plane wave
mode, [1,2] or using the strength of the field at the
emitter’s position to find the radiated power from
∫d^{3}r**E**
^{*}(**r**, t)
· **J**
_{sp}(**r**, t) with the time dependence
approximated as e^{-iωt} . The two cavity mirrors have the same
field reflectivity of 0.995. To obtain rapid convergence in Eq. (4), we assume that the thin dielectric disk has a Gaussian
distribution for
χ_{R}(x,y,ω)Δz_{R}. The amount of
dielectric confinement is then characterized by the index step
χ_{R}(0,0,ω)Δz_{R} at the
center of the cavity and the radius of the disk is taken as
w_{χR} = 2 μm. The influence of the aperture on the
spontaneous emission rate is studied by choosing different values of
χ_{R}(0,0,ω)Δz_{R} (real
values) and finding the self-consistent solutions from Eq. (4) for a range of *ω*
Figure 2 shows the calculated results for three values of
susceptibility given as (a)
χ_{R}(0,0,ω)Δz_{R} = 0, (b)
χ_{R}(0,0,ω)Δz_{R}=18Å,
and (c) χ_{R}(0,0,ω)Δz_{R}
=36Å. The case of (a)
χ_{R}(0,0,ω)Δz_{R} = 0 (planar
half-wave cavity), in particular, has been studied previously with the results
fairly well understood. [7,8] For the purely planar cavity, waveguide modes in general
play a major role in establishing the spontaneous lifetime from a point source
emitter and are rather insensitive to the cavity length. [6] The emission into the angular range around the cavity
normal, on the other hand, is quite sensitive to the cavity length. For the
half-wave cavity the coupling to the waveguide modes both for ideal and low loss
dielectric cavities is reduced, and the spontaneous emission rate is much more
sensitive to small variations in the cavity length for a fixed emitter frequency, [8] or to the emitter frequency for a fixed cavity length.

The result as shown in curve (a) is a spontaneous lifetime that, for the somewhat idealized cavity and emitter of Fig. 1, varies by nearly an order of magnitude as the emitter frequency is changed in a 2% range from below to above resonance (Fig. 2). This change in the spontaneous emission rate is caused by inhibition for the smaller frequencies as compared to resonance, and enhancement for the larger frequencies into the angular range of emission near the cavity normal.

Curve (b) of Fig. 2 shows the spontaneous emission rate dependence on
frequency when a thin dielectric aperture is introduced into the cavity, and the
horizontal emitter is placed at the aperture center, with
χ_{R}(0,0,ω)Δz_{R}=18Å.
The cut-off of the waveguide modes due to the aperture leads to 3-dimensional
confinement, and we see a peak form in the spontaneous emission rate close to
resonance. As expected for a 3-dimensionally confined mode, a reduction in the
spontaneous emission rate is obtained for frequencies either too far above or below
resonance due to detuning. For a 3-dimensionally confined mode, a loss rate
dependence is expected for the resonant frequency and this is also observed. The
3-dimensionally confined mode suffers loss due to both mirror transmission and
waveguide propagation. [9] Increasing the aperture susceptibility decreases the loss
rate due to waveguide propagation, and both sharpens the peak in the spontaneous
emission rate and increases the peak value at resonance. This is seen in Fig. 2(c) as compared to (b) for the increased aperture
susceptibility of
χ_{R}(0,0,ω)Δz_{R}=36Å.

We note that two additional regimes can also be considered. Dipole dephasing generally occurs at a high rate in semiconductors, and leads to the spontaneous emission occurring into a range of frequencies. The dipole dephasing can occur due to collisions such as phonon scattering or carrier-carrier scattering, or due to the spontaneous emission itself that leads to amplitude decay of the spontaneous current source. Therefore the actual emission rate from a spontaneous source will have a spectral bandwidth set by a Lorentzian lineshape, and the total spontaneous emission rate will be due to a summation over of the rates for each frequency in the Lorentzian. In the limit that dipole dephasing tends to zero, the nonlinear response due to Rabi oscillations must be included.

The normalized self-consistent field profiles calculated from Eq. (4) for the resonant frequencies and three radii of
susceptibilities are shown in Fig. 3. The actual calculated amplitudes can be related to
the spontaneous source amplitude that generates the field. Of interest in the curves
of Fig. 3 is how the field profiles depend on the radius of the
dielectric aperture. The solid curve shows the case for
w_{χR}->∞, and like
χ_{R}(0,0,ω)Δz_{R} =0, describes
a fully planar half-wave cavity. One might expect the field profile to increase in
spatial extent as the aperture radius is increased. However, with respect to the
complete and orthogonal set of modes of the cavity, the field generated by the point
source is in general a multi-spatial mode field. As the aperture size is increased,
numerous transverse modes are excited in phase by the point source, and lead to the
field being strongly peaked at the position of the source. Reducing the aperture
size restricts the number of transverse modes that can be excited. The longer dashed
curve of Fig. 3 shows the calculated field profile for an aperture
radius of w_{χR}=^{2} μm, which excites
predominantly a single transverse mode set by the aperture size, and is broadened
over the planar cavity case. Once the aperture size is reduced to what is required
to enter the single mode regime, we then see that the field profile does indeed
reduce with reducing aperture size over a range of sizes. In Fig. 3 the field profile has a smaller spatial extent for the
1 μm radius for w_{χR}(short dashed curve) as compared
to the 2 μm radius.

## 4. Summary

Although the planar Fabry-Perot microcavity suffers a significant drawback in optical confinement due to the continuous range of waveguide modes, we show above that the dielectrically-apertured Fabry-Perot microcavity can in large part correct this problem. Recent results on ultra-low threshold oxide-confined vertical-cavity surface-emitting lasers show that such systems are readily fabricated from III-V semiconductors. Although dipole dephasing must be controlled to fully realize the benefits of the 3-dimensional mode confinement, new epitaxial growth techniques to realize quantum dot emitters as well as excitons can yield narrow spectral linewidth semiconductor sources, and might be used to demonstrate novel cavity effects in apertured microcavities. We further note that while the analysis above treats rather idealized cavities, the plane wave mode basis set is also suitable for analyzing systems based on distributed Bragg reflectors. [14]

## Acknowledgments

This work is supported by the Air Force Office of Sponsored Research under Grant No. F49620-96-1-0336, and the Joint Services Electronics Program under contract No. F49620-95-C-0045.

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