Advances in Optical Technologies

Volume 2008, Article ID 615393, 11 pages

http://dx.doi.org/10.1155/2008/615393

## Quantum Electrodynamic Modeling of Silicon-Based Active Devices

Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA

Received 25 February 2008; Accepted 2 May 2008

Academic Editor: Pavel Cheben

Copyright © 2008 Shouyuan Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 Figure 1, time-domain population dynamics can be expressed with a four-level rate equation formulism: where are the transition population densities for different atomic levels, which are related to the total concentration are the lifetimes associated with the transitions from energy to is the central frequency of radiation of the materials related to the atomic transition energy levels through and is the induced radiation rate or excitation rate depending on its sign. The pump rate is defined as , where is the absorption cross section, is the photon flux, and is the intensity of the pumping signal.

Based on the CEO model, the net macroscopic polarization induced in the presence of an applied electric field for an isotropic medium can be described by the following equations [22]: where is the instantaneous population density difference between energy levels 1 and 2 of an atomic transition, given by . The coupling coefficient, , 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 .

In addition, in the frequency domain, the polarization due to the gain or absorption materials can be derived from (2) as The material susceptibility can then be introduced as . Assuming the host material has a relative dielectric constant of , the complex dielectric constant can be expressed as 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 On the other hand, if we know the radiative cross-section, , of the gain medium at peak gain frequency , the amplification coefficient can be written as As a result, we can relate the coupling coefficient with the radiative cross-section through the amplification coefficient as 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 We briefly describe the discretization algorithm as shown in Figure 2, at time , we first update the magnetic field components and population density . Then, we update the induced polarization based on (2) at time , 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 , we also update the remaining atomic population densities: and Since the population density at the highest energy level, , is related only to 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.

To excite the system, a source, , 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 and amplitude dependent on the population density of . 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/SiO_{2} superlattices, porous silicon, Si-ncs embedded in SiO_{2}
, 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, ,
varies from 10^{15}~10^{22} cm^{−2}s^{−1}. Then,
the pump rate, ,
can be determined as 10~10^{8} 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/SiO_{2} 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
seconds.

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 Figure 4(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 .
In the transmitted region (SiO_{2} 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 to .
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 ,
the inversion population density starts to saturate and leads to a maximum
amplification factor of 1.046.

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 Figure 3(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 (*N _{2}*) to a lower-energy level (

*N*) 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

_{1}*N*. 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.

_{2}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, *N _{u}* and

*N*. To form a band gap within the PL spectrum, Si

_{d}_{3}N

_{4}and SiO

_{2}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 Si

_{3}N

_{4}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 and .

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 be 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.

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 SiO_{2} matrix on top of a lower index SiO_{2} 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 SiO_{2} substrate surrounded by
air. We now consider two devices based on this geometry as shown schematically
in Figure 6.

The first device is a waveguide amplifier, shown in Figure 6(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 10m 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.

The second device shown in Figure 6(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/SiO_{2} 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 SiO_{2} region to
create the DBR. The detected PL signal with DBR is normalized to the peak PL
signal without DBR. A 5m 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.

#### 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 .
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 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 0.3*a* , 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 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.

We simulate the proposed device using the FDTD method. A 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 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 Figure 9(a). For the Er^{3+} 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 s, s, and s. The
transition frequency associated with the energy levels *N _{2}* and

*N*is chosen as 200 THz and linewidth is taken to be 6 THz. The pump rate into level

_{1}*N*is chosen as /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 /m

_{3}^{3}. 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 step is chosen as 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 11 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, , can be observed. The population dynamics are consistent with the lasing output from the waveguide.

In addition, snapshots of the 2D magnetic fields after the simulation ran for 10, 30, and 50 ps are plotted in Figure 12, 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.

#### 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.

#### Acknowledgment

This work is partially sponsored by the research Contracts FA9550-06-1-0470 and 5710002024 from AFOSR.

#### References

- B. Jalali, M. Paniccia, and G. Reed, “Silicon photonics,”
*IEEE Microwave Magazine*, vol. 7, no. 3, pp. 58–68, 2006. View at Publisher · View at Google Scholar - L. Pavesi, “Will silicon be the photonic material of the third millenium?”
*Journal of Physics: Condensed Matter*, vol. 15, no. 26, pp. R1169–R1196, 2003. View at Publisher · View at Google Scholar - M. E. Castagna, S. Coffa, M. Monaco et al., “Si-based materials and devices for light emission in silicon,”
*Physica E*, vol. 16, no. 3-4, pp. 547–553, 2003. View at Publisher · View at Google Scholar - L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, “Optical gain in silicon nanocrystals,”
*Nature*, vol. 408, no. 6811, pp. 440–444, 2000. View at Publisher · View at Google Scholar - A. Fojtik, J. Valenta, I. Pelant, M. Kalal, and P. Fiala, “On the road to silicon-nanoparticle laser,”
*Journal of Materials Processing Technology*, vol. 181, no. 1–3, pp. 88–92, 2007. View at Publisher · View at Google Scholar - D. J. Lockwood, Z. H. Lu, and J.-M. Baribeau, “Quantum confined luminescence in Si/${\text{SiO}}_{2}$ superlattices,”
*Physical Review Letters*, vol. 76, no. 3, pp. 539–541, 1996. View at Publisher · View at Google Scholar - V. Ovchinnikov, A. Malinin, V. Sokolov, O. Kilpelä, and J. Sinkkonen, “Photo and electroluminescence from PECVD grown a-Si:H/${\text{SiO}}_{2}$ multilayers,”
*Optical Materials*, vol. 17, no. 1-2, pp. 103–106, 2001. View at Publisher · View at Google Scholar - J. Lee, J. H. Shin, and N. Park, “Optical gain at 1.5 $\mu $m in nanocrystal Si-sensitized Er-doped silica waveguide using top-pumping 470 nm LEDs,”
*Journal of Lightwave Technology*, vol. 23, no. 1, pp. 19–25, 2005. View at Publisher · View at Google Scholar - T. Creazzo, E. Marchena, B. Redding, T. Hodson, and D. W. Prather, “Fabrication and characterization of silicon/silicon dioxide super lattices for silicon based light emitting devices,” in
*Nanoengineering: Fabrication, Properties, Optics, and Devices IV*, vol. 6645 of*Proceedings of SPIE*, pp. 1–8, San Diego, Calif, USA, August 2007. View at Publisher · View at Google Scholar - L. Pavesi, S. Gaponenko, and L. Dal Negro,
*Towards the First Silicon Laser*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. - H. Rong, R. Jones, A. Liu et al., “A continuous-wave Raman silicon laser,”
*Nature*, vol. 433, no. 7027, pp. 725–728, 2005. View at Publisher · View at Google Scholar - O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,”
*Optics Express*, vol. 12, no. 21, pp. 5269–5273, 2004. View at Publisher · View at Google Scholar - A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,”
*IEEE Transactions on Antennas and Propagation*, vol. 46, no. 3, pp. 334–340, 1998. View at Publisher · View at Google Scholar - X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,”
*Physical Review Letters*, vol. 85, no. 1, pp. 70–73, 2000. View at Publisher · View at Google Scholar - P. Sebbah and C. Vanneste, “Random laser in the localized regime,”
*Physical Review B*, vol. 66, no. 14, Article ID 144202, 10 pages, 2002. View at Publisher · View at Google Scholar - S. Shi and D. W. Prather, “Lasing dynamics of a silicon photonic crystal microcavity,”
*Optics Express*, vol. 15, no. 16, pp. 10294–10302, 2007. View at Publisher · View at Google Scholar - S. Shi, G. Jin, and D. W. Prather, “Electromagnetic simulation of quantum well structures,”
*Optics Express*, vol. 14, no. 6, pp. 2459–2472, 2006. View at Publisher · View at Google Scholar - D. Biallo, A. D'Orazio, M. De Sario, V. Petruzzelli, and F. Prudenzano, “Time domain analysis of optical amplification in ${\text{Er}}^{3+}$ doped ${\text{SiO}}_{2}$-${\text{TiO}}_{2}$ planar waveguide,”
*Optics Express*, vol. 13, no. 12, pp. 4683–4692, 2005. View at Publisher · View at Google Scholar - S.-H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,”
*Optics Express*, vol. 12, no. 16, pp. 3827–3833, 2004. View at Publisher · View at Google Scholar - K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/${\text{SiO}}_{2}$ waveguide: experiments and model,”
*Applied Physics Letters*, vol. 77, no. 11, pp. 1617–1619, 2000. View at Publisher · View at Google Scholar - L. Dal Negro, M. Cazzanelli, N. Daldosso et al., “Stimulated emission in plasma-enhanced chemical vapour deposited silicon nanocrystals,”
*Physica E*, vol. 16, no. 3-4, pp. 297–308, 2003. View at Publisher · View at Google Scholar - A. E. Siegman,
*Lasers*, University Science Books, Mill Valley, Calif, USA, 1986. - J.-Y. Sung, A. Tewary, M. L. Brongersma, and J. H. Shin, “Cavity $Q$ measurements of silica microspheres with nanocluster silicon active layer,”
*IEEE Journal on Selected Topics in Quantum Electronics*, vol. 12, no. 6, pp. 1388–1392, 2006. View at Publisher · View at Google Scholar - L. Dal Negro, M. Cazzanelli, B. Danese et al., “Light amplification in silicon nanocrystals by pump and probe transmission measurements,”
*Journal of Applied Physics*, vol. 96, no. 10, pp. 5747–5755, 2004. View at Publisher · View at Google Scholar - S. Shi, T. Creazzo, B. Redding, and D. W. Prather, “Simulation of light amplification and enhanced spontaneous emission in silicon nanocrystals,” in review,
*Journal of Optics A: Pure and Applied Optics*, 2008. - B. Redding, S. Shi, and D. W. Prather, “Time analysis of active silicon nanocrystal waveguiding structures,”
*Optics Express,*accepted for publication, 2008. - A. Belarouci and F. Gourbilleau, “Microcavity enhanced spontaneous emission from silicon nanocrystals,”
*Journal of Applied Physics*, vol. 101, no. 7, Article ID 073108, 4 pages, 2007. View at Publisher · View at Google Scholar - D. W. Prather, S. Shi, D. M. Pustai et al., “Dispersion-based optical routing in photonic crystals,”
*Optics Letters*, vol. 29, no. 1, pp. 50–52, 2004. View at Publisher · View at Google Scholar - M. R. Newton, K. A. Morey, Y. Zhang et al., “Anisotropic diffusion in face-centered cubic opals,”
*Nano Letters*, vol. 4, no. 5, pp. 875–880, 2004. View at Publisher · View at Google Scholar - K. Tsia and A. Poon, “Dispersion-guided resonances in two-dimensional photonic-crystal-embedded microcavities,”
*Optics Express*, vol. 12, no. 23, pp. 5711–5722, 2004. View at Publisher · View at Google Scholar