# Quantum Electrodynamic Modeling of  Silicon-Based Active Devices

Authors: Shouyuan Shi; Brandon Redding; Tim Creazzo; Elton Marchena; Dennis W. Prather
Journal: Advances in Optical Technologies
 (2008)

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## Abstract

We propose a time-domain analysis of an active medium based on a coupled quantum mechanical and electromagnetic model to accurately simulate the dynamics of silicon-based photonic devices. To fully account for the nonlinearity of an active medium, the rate equations of a four-level atomic system are introduced into the electromagnetic polarization vector. With these auxiliary differential equations, we solve the time evolution of the electromagnetic waves and atomic population densities using the FDTD method. The developed simulation approach has been used to model light amplification and amplified spontaneous emission in silicon nanocrystals, as well as the lasing dynamics in a novel photonic crystal-based silicon microcavity.


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## Body

## 1. Introduction

Silicon, owing to its excellent electronic material properties,
availability, and efficient processing, has played major roles in
microelectronics during past decades and promises to be the key material in the
foreseeable future. However, the rapid progress of microprocessors will soon be
limited by the transmission bandwidth capability of electronic connections. To
eliminate the bottleneck of electronic circuits and establish interconnection
links between circuit boards, between chips on a board, or even with a single
chip, research into silicon microphotonics, which merges the power of silicon
microeletronics with the advantages of photonics, has attracted increasing attention
in recent years [1, 2].While a wide variety of passive silicon photonic devices have
been developed, recent activities have focused on achieving active
functionalities, particularly light amplification and generation, in silicon
photonic devices [3–5]. However, due to the
fundamental limitation related to the indirect nature of the bulk Si band gap,
this semiconductor has very low-light emission efficiency. As a result, the
development of silicon light emitters, amplifiers, and lasers has become one of
the central goals in the advancement of silicon photonics and optoelectronics. Another
driving force stimulating research is the need for low-cost photonic devices for
applications related to future computing and communication systems. In the
past, a number of different approaches have been taken to overcome silicon’s
fundamental limitations including silicon nanocrystals, Er-doped Si–Ge, Si–Ge
quantum dots, and Er-doped silicon nanocrystals [6–10]. More recently, stimulated
Raman scattering has been used to demonstrate light amplification and lasing in
silicon, both in pulse and continuous-wave operation [11, 12].As a result, there is a need for optimal design solutions and
physical insight in the field of quantum optics, encompassing applications such
as laser systems, amplifiers, laser cooling of atoms, quantum computing,
plasmons, polaritons, and enhanced spontaneous emission in microcavities. To address
this, various approaches based on quantum electrodynamics, in which the atoms
in active systems are treated quantum mechanically but the electromagnetic wave
is treated classically, have been developed. In this way, light interaction
with an active medium can be studied using a classical harmonic oscillator model
and the rate equations of electron population density. Although, simple
analytic expressions can be derived for well-defined problems, precise
solutions for a realistic system require a numerical solution, particularly in
nanophotonic devices.In addition, over the past several years, numerous methods have
been proposed to account for material dispersion, absorption, and gain using
the finite-difference and time-domain (FDTD) method [13–17]. One of the most commonly
used and the memory-efficient
algorithms is the auxiliary differential equation (ADE) FDTD scheme which
accounts for the rate equations that determine the electron energy states'
population densities during the propagation of pump and input signals. The approach
has been widely used to characterize the absorption in a two-level atomic
system [13], gain in a four-level atomic
system [14–16, 18], and more complicated lasing
dynamics in a four-level two-electron
atomic system [19].In this paper, we present a time-domain analysis of
amplification, amplified spontaneous emission (ASE), and lasing dynamics in
silicon-based nanophotonic devices based on a coupled quantum mechanical and
electromagnetic model. In order to simulate quantum electrodynamics, we
incorporate the rate equations of a four-level atomic system to characterize
the gain and absorption of an active material. Both amplification and ASE with the
assistance of a microcavity are investigated in one-dimensional (1D) and
two-dimensional (2D) silicon nanocrystals (Si-ncs). To study lasing dynamics, a
novel silicon microcavity based on the unique dispersion properties of photonic
crystals is introduced.


## 2. Coupled Rate Equations and Electromagnetic Dynamics Model

To achieve light amplification and lasing behavior in silicon photonic
devices, the development of gain media is essential for the successful design
of next generation integrated photonics [4, 5]. As many material systems,
such as nanocrystals and Er-doped glasses, have been extensively explored and
found to not only experimentally demonstrate photoluminescence for the
application of a silicon-based emitter, but also exhibit net gain for the
development of a silicon based laser [3, 6–8, 20, 21]. Due to strong coupling
between the quantum state and nanoscale electromagnetic waves, a mathematical model
is necessary to intimately link quantum mechanical effects to the electromagnetic
interaction. To this end, we take both
field evolution and material dynamics, such as the dispersive and nonlinear
properties, into modeling consideration.The classical electron oscillator (CEO) model has been widely
used to bridge the link between the quantum world and electromagnetic waves. Such
quantum-based dynamics are dictated by rate equations which simulate the time evolution
of atomic-energy-level
populations in the gain medium. The
electromagnetic dynamics are dictated by the electromagnetic wave equations [22]. The CEO model allows us to
couple these domains and study devices in which the material properties affect the
electromagnetic field and vice versa. Depending on the material system, various
distinguished energy levels are used to represent its quantum processes,
stimulated, and spontaneous emission. As an example, Er-doped glass has been applied as an
application of optical amplifier in the field of fiber optics, that is,
Er-doped fiber amplifier (EDFA) [18, 23]. A simple four-level model of
rate equations is employed to successfully characterize the light
amplification. In the presented work, we will apply a similar model for the
design of a silicon photonic microcavity.Recently, the direct observation of photoluminescence (PL) reveals
the potential for optical gain in Si-ncs. Although a full theoretical model for
the stimulated emission is still not enough, conventional rate equations can be employed to
describe the active and nonlinear properties of Si-ncs [9, 10, 24]. Under the first order of
approximation, the Auger process is neglected to phenomenologically study the
gain medium. A more general model
interested in analyzing gain saturation and other higher-order effects could
incorporate additional transitions into the rate equation model to represent
Auger-type processes, as discussed in [21]. As a consequence, we can further simplify the rate equations to a
four-level atomic system. Coupled with Maxwell’s equations, we are able to
model the time evolution of the atomic-energy-level population density as well as the optical
signal propagation, amplification, and absorption in candidate devices.As shown in Figure1, time-domain population dynamics can be
expressed with a four-level rate equation formulism: (1)dN3(t)dt=−N3(t)τ32+WpN0,dN2(t)dt=N3(t)τ32−N2(t)τ21+1ℏωsE(t)⋅dP(t)dt,dN1(t)dt=N2(t)τ21−N1(t)τ10−1ℏωsE(t)⋅dP(t)dt,dN0(t)dt=N1(t)τ10−WpN0, where Ni(i=0,1,2,3) are the transition population densities for
different atomic levels, which are related to the total concentration Ntot=∑i=03Ni.τij  are the lifetimes associated with the
transitions from energy Ei to Ej.ωs is the central frequency of radiation of the
materials related to the atomic transition energy levels through ωs=(E2−E1)/ℏ and (1/ℏωs)E(t)⋅dP(t)/dt is the induced radiation rate or excitation
rate depending on its sign. The pump rate Wp is defined as Wp=σpIp/ℏωp,
where σp is the absorption cross section, Ip/ℏωp is the photon flux, and Ip is the intensity of the pumping signal.Figure 1
An effective four-atomic-level system to qualitatively model the recombination
dynamics under the gain condition.Based on the CEO model, the net macroscopic polarizationP(t) induced in the presence of an applied electric
field E(t) for an isotropic medium can be described by
the following equations [22]: (2)d2P(t)dt2+ΔωsdP(t)dt+ωs2P(t)=κΔN12(t)E(t), where ΔN12(t) is the instantaneous population density
difference between energy levels 1 and 2 of an atomic transition, given by ΔN12(t)=N1(t)−N2(t).
The coupling coefficient, κ,
determines the strength of the interaction between the quantum and electromagnetic
world. It can be easily derived from the above equation that the amplification
line shape is Lorentzian and homogeneously broadened, and can be considered as
a quantum mechanically correct equation for the induced polarization density in
a real atomic system. To excite the system, an optical pumping source that is
homogeneously distributed across the device is introduced into the system. Amplification
takes place when the external pumping mechanism produces a population inversion ΔN12(t)<0.In addition, in the frequency domain, the polarization due to
the gain or absorption materials can be derived from (2) as (3)[(ω2−ωs2)+jωΔωs]P=κΔN12E. The material susceptibility can then be introduced as P=ε0χE.
Assuming the host material has a relative dielectric constant of εr,
the complex dielectric constant can be expressed as (4)εs=ε0(εr+χs)=ε0(εr+κΔN12/ε0[(ω2−ωs2)+jωΔωs]). If a plane wave is propagating in such a material, then the
imaginary part of the wave vector can be considered as the amplification or
absorption coefficient, given by (5)αs(ω)=πλεrIm(χs). On the other hand, if we know the radiative cross-section, σs,
of the gain medium at peak gain frequency ωs,
the amplification coefficient can be written as (6)αs=−σsΔN122. As a result, we can relate the coupling coefficient with the
radiative cross-section through the amplification coefficient as (7)κ=cε0εrσsΔωs. In this manner, we can obtain a coupling coefficient based on
experimental observations.


## 3. Auxiliary Differential Equations FDTD Analysis

With coupled rate and Maxwell’s equations, we introduce FDTD to
numerically simulate amplification and lasing dynamics. The central difference
method can be applied to discretize the electromagnetic field components, the
atomic populations of different levels, and the induced polarizations in space
and time. Then time marching of electromagnetic waves in leapfrog fashion can
be constructed, and consequently the atomic evolutions can be monitored. The
electromagnetic waves are governed by Maxwell’s equation as(8)∇×E=−μ0∂H∂t,∇×H=ε0εr∂E∂t+∂P∂t+J. We briefly describe the discretization algorithm as shown in
Figure 2, at time (n+0.5)Δt,
we first update the magnetic field components and population density N0.
Then, we update the induced polarization based on (2) at time (n+1)Δt,
where the knowledge of two preceding steps is required. With the known magnetic
field and induced polarization, we can calculate the electric field components.
At time (n+1)Δt, we also update the remaining atomic population
densities: N3,N2, and N1. Since the population density at the highest
energy level, N3,
is related only to N0 which was updated at the half-time step, the
population density is computed in order from high to low levels to maintain
consistency and minimize the memory storage.Figure 2
Flow chart of time evolution of  population density functions and electromagnetic waves.To excite the system, a source,J,
must be introduced. Two different sources are considered, one to investigate spontaneous
emission and one to investigate stimulated emission. To characterize the
amplification (stimulated emission) of an active medium, we launch a probe
signal into the system. With the presence of a pumping signal, population inversion
takes place within the active medium. As a result, the probe signal is
amplified during propagation through the medium. To study the PL and lasing dynamics
in the active system, we introduce randomly distributed sources within the
system to simulate the spontaneous emission. Each source generates waves with a
Lorentzian spectrum distribution centered at ωs and amplitude dependent on the population
density of N2.
Since spontaneous emission is radiated incoherently, the phase associated with
the emission is chosen randomly.


## 4. Numerical Simulations of Active Nanocrystal Devices

The demonstration of photoluminescence in Si-ncs under room
temperature is a significant step towards the development of Si-based light
emitting materials. The indirect nature of the energy band gap is partially
resolved due to quantum confinement in lower-dimension silicon such as Si/SiO2 superlattices, porous silicon, Si-ncs embedded in SiO2
, and rare earth-doped Si-ncs.Based on the extensive experimental measurements of Si-ncs, the
principle structural, and optical parameters, such as emission life time,
absorption cross-section, emission cross-section, emission linewidth, and peak
wavelength can be deduced, as listed in the following table [10, 21].In simulations, the active medium is uniformly pumped with a
shorter wavelength source, that is, 457 nm, and the pump photon flux,Ip/ℏω,
varies from 1015~1022 cm−2s−1. Then,
the pump rate, Wp=δpIp/ℏω,
can be determined as 10~108 s−1. The effective index of silicon
nanocrystals is calculated using Bruggman’s effective medium approximation. In
this section, we investigate the gain characteristics of Si-ncs in both 1D
stacks [25] and 2D waveguiding structures
[26] by incorporating the optical
and structural properties above as simulation parameters in the coupled rate
and electromagnetic wave equations.


### 4.1. Light Amplification and Amplified Spontaneous Emission in 1D Nanocrystals

Based on the theoretical model developed in the previous
section, we first consider light amplification in Si-ncs by applying pump and
probe signal transmission simulation. The structure investigated is a
supperlattice of Si/SiO2 on the top of a quartz substrate, as shown
in Figure 3(a). The thickness of Si-nc is 250 nm. In the FDTD simulation, the
mesh size is 20 nm, and the time step is 2.25×10−17
seconds.(a) 1D Si-nc active device for stimulated
emission simulation study. (b) 1D Si-nc active device
for amplified stimulated emission
simulation study.
(a)(b)First, we consider a continuous plane wave with a wavelength of
750 nm illuminating the sample from free space. The steady-state results with
two different pump rates are shown in Figure4(a). The red lines indicate
interfaces between Si-ncs and their claddings. The solid and dash lines
represent the field distributions across the devices with and without a pump
rate of 106.
In the transmitted region (SiO2 substrate), an apparent
amplification can be identified. To
accurately characterize the light amplification of Si-ncs, we calculate the transmittance
based on the Poynting vector. Then, the light amplification factor is defined
by normalizing the transmitted powers with pump to without pump. In doing so,
the interface coupling loss can be subtracted, and consequently, net gain
contributed from stimulated emission can be extracted. We plot both the
amplification factor and population inversion in Figure 4(b) for pump rates
varying from 102 to 108.
As we see from the figure, both the amplification factor and normalized
population inversion density are nonlinearly dependent on the pump rate. While
the pump rate increases to 108,
the inversion population density starts to saturate and leads to a maximum
amplification factor of 1.046.(a) Amplitude distributions of
electric field across the device, where in the transmitted region, an
apparent amplification can be observed under the pump rate of106 compared with no pump case. (b)
Amplification of transmitted signal and population inversion ΔN inside the active medium verse the pump rate varying from 102 to 108.
(a)(b)In addition to the modeling of stimulated emission for the
probe signal amplification through active Si-ncs, we consider photoluminescence
and microcavity-based ASE from an Si-nc device, as shown in Figure3(b). Instead
of using an external plane wave, we introduce an artificial source to simulate the
spontaneous emission process in an active medium.The spontaneous emission occurs when electrons spontaneously
drop from an upper (N2) to a lower-energy level (N1)
while emitting electromagnetic radiation at the transition frequency. While
atoms emit this kind of fluorescence or spontaneous emission, each individual
atom acts exactly like a small random oscillating radiator. Each atom radiates
independently, with a temporal phase angle that is independent of all the other
radiating atoms. To model this process, we introduce many dipole sources
randomly distributed within the system in the FDTD simulation. Each source
generates waves in the form of a Lorentzian spectrum distribution centered at 750 nm, with its
amplitude dependent on the time varying population density of N2.
Since the spontaneous emission is incoherent, the phase associated with the
photon is random, and thus we code a random phase to each dipole source in the
FDTD modeling. To attain the PL characteristics of Si-nc, a few detectors are
placed to collect temporal photon luminescence both in free space and the
quartz substrate. The recorded temporal electromagnetic waves are Fourier
transformed to the frequency domain in order to obtain the PL spectrum.It is well known that the spontaneous emission is not only determined
by the active material, but it also relates to the optical structure. By introducing
a microcavity, we can control and manipulate the spontaneous emission. Based on
photonic crystal band gap materials, many microcavities have been designed to
enhance the spontaneous emission in active material systems [27]. The simplest configuration
consists of planar stacks of alternating high- and low-index thin films deposited
using conventional planar fabrication processing, such as PECVD [9]. To this end, we designed a
Fabry-Perot resonator consisting of one-dimensional photonic crystal structures
to confine the optical mode inside the active Si-nc material, as shown in Figure 3(b). The device consists of Si-ncs sandwiched with some variable number of
periods of a one-dimensional PhC lattice at both ends, that is, Nu and Nd. To form a band gap within the PL spectrum, Si3N4 and SiO2 are used as alternate high- and low-index materials with
their indices of 2.05 and 1.454 at the wavelength of 750 nm. The lattice period
is designed as 210 nm and the filling factor of Si3N4 is
chosen as 50%. Transmission characteristics have been performed to confirm that
a band gap opens between 638 and 879 nm. With a 250 nm thick Si-nc medium as a
defect, a strong single-cavity
mode located at a wavelength of 751 nm can be recognized within the band gap.
The Q factor is measured as 85 while the numbers of DBRs are Nu=1
and Nd=6.Applying the modeling tools developed in this work, we
performed numerical simulations to illustrate the enhancement effect of
spontaneous emission due to the introduction of a microcavity in an active Si-nc
system. During the time evolution of the population densities, the population inversion
is found to beΔN=5.4×1024 after the system reaches steady state. A
comparison of spontaneous emission spectra with and without DBR measured in the
free space is illustrated in Figure 5. All the spectra are normalized to the
peak PL intensity of Si-nc devices without DBR. As we can see from the figure, the
PL spectrum without DBR shows a typical broad emission band of 250 nm centered
at 750 nm. By introducing the microcavity, a sharp emission enhancement can be
observed. The resonant wavelength is slightly redshifted to a longer wavelength at 760.1 nm and the
linewidth becomes smaller (7.2 nm). The peak PL exhibits an enhancement factor
of 16.4.Figure 5
Enhanced amplified spontaneous emission (PL)
with microcavity consisting of 1D DBRs with the layers ofNu=4 and
Nd=6 compared with the one without DBR.Owing to the large reflectance of DBR mirrors, the optical
field at resonance is well confined inside the microcavity. As a consequence,
the resonant optical mode has a longer photon lifetime and thus longer
interaction with the gain medium resulting in the enhanced PL signal at the
resonant wavelength.In addition to the study of active characteristics in 1D Si-ncs,
we are also interested in edge emission from nanocrystals, in which a
two-dimensional (2D) system is required. In the following section, we will
investigate Si-nc waveguiding structures for this application.


### 4.2. Light Amplification and Amplified Spontaneous Emission in
Nanocrystal Waveguiding Structures

In two dimensions, we consider an active planar waveguide made
of higher index Si-nc in a SiO2 matrix on top of a lower index SiO2 substrate. Both TE and TM propagations are simulated to identify their
amplification and ASE characteristics. In cases where we study spontaneous
emission, dipole sources representing the spontaneous emission source have
freedom to emit in any direction. Gain guiding and total internal reflection
(TIR) combine to direct the spontaneous emission along the length of the Si-nc
region.The basic 2D geometry we consider consists of a slab waveguide
composed of a 200 nm Si-nc layer on a SiO2 substrate surrounded by
air. We now consider two devices based on this geometry as shown schematically
in Figure 6.(a) Configuration for waveguide
amplification study. A Gaussian pulse
in air is incident on a pumped Si-nc waveguide. A detector in air at the end of the
waveguide measures the amplification by normalizing to the output without
pumping. (b) Configuration for amplified spontaneous emission study. Optional DBRs create a microcavity to
enhance spontaneous emission of pumped Si-ncs.
(a)(b)The first device is a waveguide amplifier, shown in Figure6(a). In this case, a Gaussian pulse introduced in
air couples into the waveguide, propagates along the length of the waveguide,
interacts with the pumped active material, and arrives at the detector,
positioned in air at the opposite end. To study the waveguide's behavior as an amplifier,
we normalize the detected signal at various pump strengths to the signal when
the pumping and material losses are set to zero. This allows us to study
amplification without loss due to reflections at the interfaces. The normalized amplification for a 10μm
waveguide is shown in Figure 7 for the TE and TM cases at two pump rates. Both the TE and TM cases exhibit
amplification under sufficient pumping, although the TM case is clearly
stronger. This can be understood by considering the TE mode profile which
includes a longer evanescent tail, extending into the substrate. The broader
mode results in a reduced modal volume in the Si-nc region where amplification
is possible; consequently amplification of the TE mode is reduced relative to
the more confined TM mode. The oscillations in the spectral response of both
modes are due to Fabry-Perot effects. The high-index contrast Si-nc to air interface
introduces enough reflection to create a weak microcavity.Figure 7
Amplification
through a pumped Si-nc waveguide for TE and TM cases. Detected signal under
pumping is normalized to detected signal without pumping to eliminate loss
due to reflection at waveguide interfaces.The second device shown in Figure6(b) employs two DBRs to
enhance the Fabry-Perot microcavity which introduced oscillations in the
amplification spectrum studied above. The DBR microcavities are considered for
their role in an ASE device. Spontaneous
emission is modeled as dipole sources with random phase positioned throughout
the Si-nc region. The PL spectrum is detected in air to the right of the
waveguide. Each DBR consists of
alternating layers of air and Si-nc/SiO2 with a period of 300 nm,
optimized to open a band gap between 650 nm and 900 nm. The air holes are
“etched” through the Si-nc region and 200 nm into the SiO2 region to
create the DBR. The detected PL signal with DBR is normalized to the peak PL
signal without DBR. A 5μm Si-nc region
is bordered by 3 periods of DBR on either side. The resulting ASE for the TE
and TM cases is shown in Figure 8. The
steady-state response at the peak wavelength shows the high-optical confinement
within the microcavity. The inclusion of DBRs enhances the PL signal by a
factor of 4 in the TE case and nearly a factor of 8 in the TM case.DBRs enhance PL signal
of Si-ncs by ~4X in the TE case and ~8X in the TM case. (a) The steady-state
amplitude of theEz component is shown, the TM case corresponds to
the peak wavelength of ~770 nm while the TE case corresponds to its peak
wavelength of 677 nm. (b) The spectral response for TE and TM with 3 periods
of DBRs is shown and normalized to the peak PL emission for the no-DBR case.
(a)(b)


## 4.1. Light Amplification and Amplified Spontaneous Emission in 1D Nanocrystals

Based on the theoretical model developed in the previous
section, we first consider light amplification in Si-ncs by applying pump and
probe signal transmission simulation. The structure investigated is a
supperlattice of Si/SiO2 on the top of a quartz substrate, as shown
in Figure 3(a). The thickness of Si-nc is 250 nm. In the FDTD simulation, the
mesh size is 20 nm, and the time step is 2.25×10−17
seconds.(a) 1D Si-nc active device for stimulated
emission simulation study. (b) 1D Si-nc active device
for amplified stimulated emission
simulation study.
(a)(b)First, we consider a continuous plane wave with a wavelength of
750 nm illuminating the sample from free space. The steady-state results with
two different pump rates are shown in Figure4(a). The red lines indicate
interfaces between Si-ncs and their claddings. The solid and dash lines
represent the field distributions across the devices with and without a pump
rate of 106.
In the transmitted region (SiO2 substrate), an apparent
amplification can be identified. To
accurately characterize the light amplification of Si-ncs, we calculate the transmittance
based on the Poynting vector. Then, the light amplification factor is defined
by normalizing the transmitted powers with pump to without pump. In doing so,
the interface coupling loss can be subtracted, and consequently, net gain
contributed from stimulated emission can be extracted. We plot both the
amplification factor and population inversion in Figure 4(b) for pump rates
varying from 102 to 108.
As we see from the figure, both the amplification factor and normalized
population inversion density are nonlinearly dependent on the pump rate. While
the pump rate increases to 108,
the inversion population density starts to saturate and leads to a maximum
amplification factor of 1.046.(a) Amplitude distributions of
electric field across the device, where in the transmitted region, an
apparent amplification can be observed under the pump rate of106 compared with no pump case. (b)
Amplification of transmitted signal and population inversion ΔN inside the active medium verse the pump rate varying from 102 to 108.
(a)(b)In addition to the modeling of stimulated emission for the
probe signal amplification through active Si-ncs, we consider photoluminescence
and microcavity-based ASE from an Si-nc device, as shown in Figure3(b). Instead
of using an external plane wave, we introduce an artificial source to simulate the
spontaneous emission process in an active medium.The spontaneous emission occurs when electrons spontaneously
drop from an upper (N2) to a lower-energy level (N1)
while emitting electromagnetic radiation at the transition frequency. While
atoms emit this kind of fluorescence or spontaneous emission, each individual
atom acts exactly like a small random oscillating radiator. Each atom radiates
independently, with a temporal phase angle that is independent of all the other
radiating atoms. To model this process, we introduce many dipole sources
randomly distributed within the system in the FDTD simulation. Each source
generates waves in the form of a Lorentzian spectrum distribution centered at 750 nm, with its
amplitude dependent on the time varying population density of N2.
Since the spontaneous emission is incoherent, the phase associated with the
photon is random, and thus we code a random phase to each dipole source in the
FDTD modeling. To attain the PL characteristics of Si-nc, a few detectors are
placed to collect temporal photon luminescence both in free space and the
quartz substrate. The recorded temporal electromagnetic waves are Fourier
transformed to the frequency domain in order to obtain the PL spectrum.It is well known that the spontaneous emission is not only determined
by the active material, but it also relates to the optical structure. By introducing
a microcavity, we can control and manipulate the spontaneous emission. Based on
photonic crystal band gap materials, many microcavities have been designed to
enhance the spontaneous emission in active material systems [27]. The simplest configuration
consists of planar stacks of alternating high- and low-index thin films deposited
using conventional planar fabrication processing, such as PECVD [9]. To this end, we designed a
Fabry-Perot resonator consisting of one-dimensional photonic crystal structures
to confine the optical mode inside the active Si-nc material, as shown in Figure 3(b). The device consists of Si-ncs sandwiched with some variable number of
periods of a one-dimensional PhC lattice at both ends, that is, Nu and Nd. To form a band gap within the PL spectrum, Si3N4 and SiO2 are used as alternate high- and low-index materials with
their indices of 2.05 and 1.454 at the wavelength of 750 nm. The lattice period
is designed as 210 nm and the filling factor of Si3N4 is
chosen as 50%. Transmission characteristics have been performed to confirm that
a band gap opens between 638 and 879 nm. With a 250 nm thick Si-nc medium as a
defect, a strong single-cavity
mode located at a wavelength of 751 nm can be recognized within the band gap.
The Q factor is measured as 85 while the numbers of DBRs are Nu=1
and Nd=6.Applying the modeling tools developed in this work, we
performed numerical simulations to illustrate the enhancement effect of
spontaneous emission due to the introduction of a microcavity in an active Si-nc
system. During the time evolution of the population densities, the population inversion
is found to beΔN=5.4×1024 after the system reaches steady state. A
comparison of spontaneous emission spectra with and without DBR measured in the
free space is illustrated in Figure 5. All the spectra are normalized to the
peak PL intensity of Si-nc devices without DBR. As we can see from the figure, the
PL spectrum without DBR shows a typical broad emission band of 250 nm centered
at 750 nm. By introducing the microcavity, a sharp emission enhancement can be
observed. The resonant wavelength is slightly redshifted to a longer wavelength at 760.1 nm and the
linewidth becomes smaller (7.2 nm). The peak PL exhibits an enhancement factor
of 16.4.Figure 5
Enhanced amplified spontaneous emission (PL)
with microcavity consisting of 1D DBRs with the layers ofNu=4 and
Nd=6 compared with the one without DBR.Owing to the large reflectance of DBR mirrors, the optical
field at resonance is well confined inside the microcavity. As a consequence,
the resonant optical mode has a longer photon lifetime and thus longer
interaction with the gain medium resulting in the enhanced PL signal at the
resonant wavelength.In addition to the study of active characteristics in 1D Si-ncs,
we are also interested in edge emission from nanocrystals, in which a
two-dimensional (2D) system is required. In the following section, we will
investigate Si-nc waveguiding structures for this application.


## 4.2. Light Amplification and Amplified Spontaneous Emission in
Nanocrystal Waveguiding Structures

In two dimensions, we consider an active planar waveguide made
of higher index Si-nc in a SiO2 matrix on top of a lower index SiO2 substrate. Both TE and TM propagations are simulated to identify their
amplification and ASE characteristics. In cases where we study spontaneous
emission, dipole sources representing the spontaneous emission source have
freedom to emit in any direction. Gain guiding and total internal reflection
(TIR) combine to direct the spontaneous emission along the length of the Si-nc
region.The basic 2D geometry we consider consists of a slab waveguide
composed of a 200 nm Si-nc layer on a SiO2 substrate surrounded by
air. We now consider two devices based on this geometry as shown schematically
in Figure 6.(a) Configuration for waveguide
amplification study. A Gaussian pulse
in air is incident on a pumped Si-nc waveguide. A detector in air at the end of the
waveguide measures the amplification by normalizing to the output without
pumping. (b) Configuration for amplified spontaneous emission study. Optional DBRs create a microcavity to
enhance spontaneous emission of pumped Si-ncs.
(a)(b)The first device is a waveguide amplifier, shown in Figure6(a). In this case, a Gaussian pulse introduced in
air couples into the waveguide, propagates along the length of the waveguide,
interacts with the pumped active material, and arrives at the detector,
positioned in air at the opposite end. To study the waveguide's behavior as an amplifier,
we normalize the detected signal at various pump strengths to the signal when
the pumping and material losses are set to zero. This allows us to study
amplification without loss due to reflections at the interfaces. The normalized amplification for a 10μm
waveguide is shown in Figure 7 for the TE and TM cases at two pump rates. Both the TE and TM cases exhibit
amplification under sufficient pumping, although the TM case is clearly
stronger. This can be understood by considering the TE mode profile which
includes a longer evanescent tail, extending into the substrate. The broader
mode results in a reduced modal volume in the Si-nc region where amplification
is possible; consequently amplification of the TE mode is reduced relative to
the more confined TM mode. The oscillations in the spectral response of both
modes are due to Fabry-Perot effects. The high-index contrast Si-nc to air interface
introduces enough reflection to create a weak microcavity.Figure 7
Amplification
through a pumped Si-nc waveguide for TE and TM cases. Detected signal under
pumping is normalized to detected signal without pumping to eliminate loss
due to reflection at waveguide interfaces.The second device shown in Figure6(b) employs two DBRs to
enhance the Fabry-Perot microcavity which introduced oscillations in the
amplification spectrum studied above. The DBR microcavities are considered for
their role in an ASE device. Spontaneous
emission is modeled as dipole sources with random phase positioned throughout
the Si-nc region. The PL spectrum is detected in air to the right of the
waveguide. Each DBR consists of
alternating layers of air and Si-nc/SiO2 with a period of 300 nm,
optimized to open a band gap between 650 nm and 900 nm. The air holes are
“etched” through the Si-nc region and 200 nm into the SiO2 region to
create the DBR. The detected PL signal with DBR is normalized to the peak PL
signal without DBR. A 5μm Si-nc region
is bordered by 3 periods of DBR on either side. The resulting ASE for the TE
and TM cases is shown in Figure 8. The
steady-state response at the peak wavelength shows the high-optical confinement
within the microcavity. The inclusion of DBRs enhances the PL signal by a
factor of 4 in the TE case and nearly a factor of 8 in the TM case.DBRs enhance PL signal
of Si-ncs by ~4X in the TE case and ~8X in the TM case. (a) The steady-state
amplitude of theEz component is shown, the TM case corresponds to
the peak wavelength of ~770 nm while the TE case corresponds to its peak
wavelength of 677 nm. (b) The spectral response for TE and TM with 3 periods
of DBRs is shown and normalized to the peak PL emission for the no-DBR case.
(a)(b)


## 5. Lasing Dynamics in a Photonic Crystal-Based Silicon Microcavity

In this section, we employ the developed technique to
investigate the time evolution of population density functions and
electromagnetic dynamics of a novel photonic crystal-based microcavity. During
the lasing activity, the strong feedback of the local light intensity saturates
the amplification in the system through the nonlinearity of active material,
and eventually leads to a steady optical mode output.Light propagation in a PhC is most appropriately interpreted
through a dispersion diagram, which characterizes the relationship between the
frequency of the wave,ω, and its associated wave vector, k. Dispersion
surfaces provide the spatial variation of the spectral properties of a certain
band within the photonic crystal structure. An electromagnetic wave propagates
along the direction normal to the dispersion surface, which stems from the
relation of the group velocity vg=∇kω(k).
The ability to shape the equifrequency contours (EFC), and thereby engineer the
dispersion properties of the PhC, opens up a new paradigm for the design of
optical devices [28–30]. For the applications of
self-collimation, we desire a flat EFC, in which case the wave is only allowed
to propagate along those directions normal to the sides of the straight
curvatures. As such, it is possible to vary the incident wave vector over a
wide range of angles and yet maintain a narrow range of propagating angles
within the PhC.Based on the dispersion waveguiding property in photonic
crystals, a novel class of photonic-crystal-embedded microcavity (PCEM) coupled
with waveguide [30] has been theoretically
investigated using FDTD algorithm, in which both optical resonant mode and
quality factor are particularly considered. Figure 9(a) depicts the schematic
of our design. A photonic crystal cavity consists of a silicon square slab
perforated by an N×N array of square lattice. The air holes are
back-filled with the gain medium, that is, Er-doped glasses, as shown in the
inset of Figure 9(a). The hole has a radius of 0.3a , where a is the lattice
constant. The silicon and glass have refractive indices of 3.5 and 1.5,
respectively. After the light couples into the resonator from the waveguide, it
will propagate along the ΓM direction and then will be reflected
at the edge of the resonator. As such, with the assistance of the four clear
edges of silicon slab as mirrors, a traveling wave-based cavity can be formed
along the optical path indicated by an arrow loop in Figure 9(b), which is
similar to a whispering gallery mode in a microdisk resonator. Along one edge
of this microcavity, a conventional dielectric waveguide is used for the
in-/outcoupling between the cavity and waveguide. To realize the active
operation in such a microcavity, we fill the air holes with a certain gain
medium to achieve nonlinear gain dynamics. In our design, low-index highly
doped Erbium Ion
glass is considered as a gain material for the light amplification. The
proposed device has several advantages. (1) Since we are working at the first
band of the dispersion surface, the self-collimation mode is far below the
light cone, therefore the wave inside the resonator is well confined inside the
slab with minimal out-plane propagation loss even with the presence of a substrate.
(2) The self-collimation mode is very close to the band edge, so the slow group
velocity will enhance the nonlinear interaction between light and gain
material.(a) Schematic of silicon laser
based on dispersion-engineered
photonic crystals where the active material is introduced by backfilling the
air holes of the PhC. (b) The highest cavity mode below the band edge.
(a)(b)We simulate the proposed device using the FDTD method. A6×6 array of glass-filled cylinders is initially simulated; the
highest resonant mode below the band edge of the first dispersion band is shown
in Figure 9(b). We observe that most optical modes are well confined within the
low-index materials. This unique property may benefit the lasing mechanism and
lower the optical pumping threshold. The cavity properties are largely
dependent on the design parameters, such as gap size between the waveguide and
resonator, and array size. An extensive study has been performed to investigate
the passive cavity performance. The gap between the cavity and waveguide is a critical
design parameter to achieve optimal coupling and Q factor. An optimal gap size
of 160 nm is found in the following design. In this case, we measured a Q factor
of 520 and drop efficiency of 32.2% for a resonant mode at 1532 nm. We also
investigate the Purcell factor as an additional figure of merit for the
microcavity. The Purcell factor dictates the enhancement in the radiative
recombination rate due to optical confinement provided by the cavity. Based on
the mode profile at the resonant wavelength of 1532 nm, the PhC microcavity exhibits
an effective mode volume of 3.27*(λ/n)3 and a Purcell factor of 10.88.With the appropriate design of the microcavity based on the dispersion
engineering of photonic crystals, we further considered backfilling the air
holes in photonic crystals with a gain medium, that is, Er-doped glass, to achieve active operation,
as shown in the schematic view of the proposed novel silicon laser in Figure9(a). For the Er3+ Ion, the typical lifetime is on the order of 10−3~10−2 s.
Numerically, using a rigorous EM algorithm in the time domain to simulate the
lasing dynamics, this is far beyond the state-of-the-art computational
capability, particularly in a high-dimension system. To phenomenologically
investigate the lasing dynamics in the microcavity with the backfilled gain
medium, we scale the lifetime of electron population transition accordingly while
maintaining a reasonable timescale associated with different relaxation processes.
To this end, we shorten the lifetime of atomic transitions to reduce the
computation time necessary to achieve the steady state. The lifetimes are given
by τ10=10−12 s, τ21=10−10 s, and τ32=10−13 s. The
transition frequency associated with the energy levels N2 and N1 is chosen as 200 THz and linewidth is taken to be 6 THz. The pump rate into level
N3 is chosen as Wp=2×108/s. The initial state
of the simulation is included as follows: all the electrons are in the ground
state, so there is no field in the cavity and no spontaneous emission. In this
paper, we choose N0=3×1024/m3. After the
electrons are pumped, the system starts to evolve both in terms of population densities
and electromagnetic waves. To monitor the EM dynamics, a detector is placed in
the waveguide.In the FDTD simulation, a total time of 66 ps is simulated. The
time stepΔt is chosen as 3.3×10−7 s, so a total of 2 million FDTD
iterations are used. The evolution of electromagnetic waves starts from an
initial small noise current over the photonic crystal cavity. A detector is
placed in the waveguide to monitor the lasing dynamics, and the total field in
the output waveguide is illustrated in Figure 10(a). In the figure, we observe
the lasing performance. For clarity, we zoom in on a portion of the transient
plot near 60 ps as shown in Figure 10(b), which indicates a nearly single lasing
mode.Figure 10
(a) Lasing dynamics by monitoring output in the straight dielectric waveguide, (b) steady-state output of single optical mode as shown within the time window in (a).Figure11 shows the calculated time evolution of the electron
population inversion between energy levels 2 and 1 at the position of one of the
central holes in the cavity. The plotted population inversion is normalized to
total electron concentration. During the simulation, the ground state
population density remains relatively unchanged compared to higher-level
populations. In the beginning, the
population inversion linearly increases, leading to significant amplification
of electromagnetic waves in the cavity. In the meantime, the desired cavity
mode is gradually established. Eventually, a convergent population inversion,
−6.9×10−4, can be observed. The population dynamics are consistent
with the lasing output from the waveguide.Figure 11
Normalized population inversion at pump rate of2×108/s.In addition, snapshots of the 2D magnetic fields after the
simulation ran for 10, 30, and 50 ps are plotted in Figure12, where the lasing
dynamics of the electromagnetic field is gradually established within the
cavity. In the end, the high-intensity distribution of EM field can be observed
in the cavity area, particularly in the low-index material region and steady
results are output from both ends of the straight waveguide.Snapshots of 2D field
distribution at three time instants: 10, 30, and 50 ps.
(a)(b)(c)


## 6. Conclusion

In this paper, a time-domain analysis of active silicon-based
photonic devices is proposed. The simulation model couples Maxwell’s equations
with the rate equations to fully describe the nonlinearity of active medium,
and numerically solves them using an ADE-FDTD scheme. Both stimulated and
spontaneous emissions
are taken into account in the active medium system. Light amplification characteristics
due to stimulated emission are investigated under various pump rates in both 1D
and 2D waveguiding silicon nanocrystals. To achieve amplified spontaneous
emission, microcavities based on photonic band gap materials are carefully
designed.  These Fabry-Perot
microcavities are shown to significantly enhance the spontaneous emission
signal. At last, we propose a novel PhC-based microcavity, in which a low-index
gain medium is incorporated into the air holes in photonic crystals, to study
the lasing characteristics.Table 1
Optical properties and simulation parameters of Si-ncs.
Pump wavelength (nm)532Pump photon fluxIp/ℏω (cm-2 s-1)1015~1022Absorption cross-sectionσp (cm2)10-14Emission cross-sectionσs (cm2)3×10−16Si-nc concentrationNtot (m-3)6×1024Emission wavelengthλ (nm)750Emission spectrum linewidth (nm)200τ10 (s)10-15τ21 (s)10-5τ32 (s)10-15Refractive index of Si-ncs1.9+i0.0014Refractive index of SiO21.454



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*Source: 615393-2008-06-16.xml*