## Abstract

In this paper we propose an electromagnetic analysis of active silicon nano-crystal (Si-nc) waveguide devices. To account for the nonlinearity in the active medium we introduce a four level rate equation model whose parameters are based on experimentally reported material properties. The electromagnetic polarization serves to couple the quantum mechanical and electromagnetic behavior within the ADE-FDTD scheme. The developed modeling tool is used to simulate waveguide amplifiers, enhanced spontaneous emission microcavities, and the temporal lasing dynamics of active Si-nc based devices.

© 2008 Optical Society of America

## 1. Introduction

Silicon, owing to its excellent electronic material properties, availability, and cheap, efficient processing, has played major roles in microelectronics during past decades and promises to be the key material in the 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 within a single chip, research into silicon microphotonics has attracted more and more attention in recent years [1, 2]. While a number of passive devices have been demonstrated, the development of active devices continues to be limited by silicon’s indirect band gap resulting in inefficient light emission. The ability for low dimensional silicon to mediate this problem, as suggested by experimental demonstration of luminescence and optical gain [3–8], may provide the key breakthrough in the development of active photonic devices on a silicon platform.

Active photonic devices, addressing such applications as laser systems, amplifiers, laser cooling of atoms, quantum computing, plasmons, polaritons, and enhanced spontaneous emissions in microcavities, will require a detailed understanding of the interaction of electromagnetic fields with quantum mechanics based material phenomena such as light emission and gain. This requirement has generated increased interest in device design based on quantum electrodynamics, where atoms in active systems are treated quantum mechanically while the electromagnetic wave is treated classically. In this way, light interaction with an active medium can be fully understood using a classical harmonic oscillator model and the rate equations of electron population density. To study the temporal quantum electrodynamics of these systems we employ the two-dimensional Finite-difference and time domain (2D FDTD) method in conjunction with a set of auxiliary differential equations (ADEs) which allow us to represent the material’s quantum behavior by a four-level rate equation model. This method has been used to study absorption, gain and lasing dynamics in various atomic systems [9–12]. In particular, this model has been successfully employed for the analysis of one-dimenisonal Si-ncs, [13].

In this paper we present a time domain simulation of the quantum electrodynamic behavior of active Si-nc waveguide devices. We investigate both stimulated and spontaneous emission in devices designed for the applications of light amplification, enhanced spontaneous emission, and lasing.

## 2. Coupled rate equation and electrodynamic model

As shown in Fig. 1, the Si-nc material’s gain and emission behavior is dictated by a four level atomic system. While this system ignores non-linear processes such as Auger recombination and free carrier absorption, it provides a phenomenological framework under which to develop devices dependant on this gain material. Furthermore, optical gain has been demonstrated experimentally [3] and thus we study devices whose performance is predicated on operation within this regime. The resulting system is governed by a set of rate equations [13, 14],

where *N _{i}*(

*i*=0,1,2,3) is the transition population density for different atomic levels.

*τ*are the lifetimes associated with the transitions from energy

_{ij}*E*and

_{i}*E*.

_{j}*W*is the pumping rate,

_{p}*ω*is the central frequency of radiation of the materials related to the atomic transition energy levels through

_{s}*ω*=(

_{s}*E*

_{1}-

*E*

_{2}/)

*ħ*and (

**E**(

*t*)/

*ħω*)·(

_{s}*d*

**p**(

*t*)/

*dt*) is the induced radiation rate or excitation rate depending on its sign.

The link from these rate equations to the electromagnetic fields in a device simulation is made through the classical electron oscillator model, which relies on the macroscopic polarization **P**(*t*) which, for an isotropic medium, can be described by the following equations,

where Δ*N*
_{12}(*t*) is the instantaneous population density difference between energy levels 1 and 2 in the above rate equation model. *κ* is the coupling coefficient that bridges the quantum world to the electromagnetic world. By deriving separate expressions for the amplification /absorption coefficient *α*, first in terms of the imaginary part of a wave vector describing an incident plane wave and then in relation to the experimentally measured emission cross section [13], we arrive at an expression for *κ* as:

This coupling coefficient is entirely determined by experimental observations of the stimulated emission cross section *σ _{s}*, the linewidth of the emission cross section Δ

*ω*, and the relative dielectric constant

_{s}*ε*. The polarization then factors into Maxwell’s equations as,

_{r}To excite the system we introduce a source, *J*, to represent an external Gaussian source when we investigate amplification via stimulated emission or to represent spontaneous emission when we study enhanced spontaneous emission and lasing. We use the FDTD method to numerically solve the system with coupled rate and Maxwell’s equations using the standard time marching scheme [15]. In this way, we monitor the temporal quantum electrodynamics for active Si-nc waveguide devices.

## 3. Numerical simulation of Si-nc waveguiding devices

The optical and structural properties of a Si-nc type material depend strongly on the specific processes followed in its fabrication [3]. In general, the model presented above can be applied to study devices incorporating a variety of Si-nc type materials for which sufficient experimental characterization is provided. In the current work we study a Si-nc material based on that reported in [14, 16] in which an SiO_{x} film was deposited by PECVD and subjected to a high temperature anneal at 1250°C.

From [14] we take the radiative emission lifetime, *τ*
_{21}, to be 10^{-5} s. The non-radiative decay lifetimes, *τ*
_{10} and *τ*
_{32}, are taken as 10^{-15} s. The absorption cross section, *σ _{p}*, is taken as 10

^{-14}cm

^{2}. The emission cross section,

*σ*, is taken as 3×10

_{s}^{-16}cm

^{2}. The peak emission wavelength is 750nm with an emission linewidth of 200nm.

The pumping rate which factors into the rate equations is defined as *W _{p}*=

*σ*/

_{p}I_{p}*ħω*where

_{p}*I*is the pump intensity at the frequency

_{p}*ω*corresponding to the pump wavelength of 457nm. If the photon flux varies from 10

_{p}^{15}to 10

^{22}cm

^{-2}s

^{-1}then the pumping rate ranges from 10 to 10

^{8}s

^{-1}.

Although the Si-nc material is composed of individual nanocrystals within an SiO_{2} matrix, we treat it as a single material with a single refractive index and propagation loss. To do this we introduce an effective index calculated using Bruggeman’s effective medium approximation [17]. The Si-nc material is composed of 25% nanocrystals with an index of 3.73+0.009i at the peak emission wavelength, embedded in an SiO2 matrix with an index of 1.454. The resulting effective index is 1.9+0.0014i. The imaginary component introduces a propagation loss, which manifests in our simulation as a material conductivity of 116.

We then incorporate this material into a series of two-dimensional device structures. The devices consist of active planar waveguides made of higher index Si-nc on top of a lower index SiO_{2} substrate with index of 1.454. Both TE and TM propagations are simulated to identify their amplification and amplified spontaneous emission (ASE) characteristics. In cases where we study spontaneous emission, randomly distributed 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 waveguide.

The basic 2D geometry we consider consists of a slab waveguide composed of a 200nm Si-nc layer on an SiO_{2} substrate. We consider two devices based on this geometry as shown schematically in Fig. 2. The first device relies on an external source passing through the pumped Si-nc waveguide and is used to study amplification. The second includes DBRs on the edge to create a microcavity and is used to study amplified spontaneous emission and lasing dynamics.

#### 3.1: Light amplification in Si-nc waveguide

Consider a waveguide amplifier, shown in Fig. 2 (a). The amplifier relies on stimulated emission which occurs when an incident electromagnetic wave induces an electron in an excited state to drop to a lower energy state and emit a new photon with direction and phase commensurate with that of the incident photon. This phenomenon clearly affects both the quantum mechanical system and the electromagnetic waves.

The effect of stimulated emission on the quantum mechanical system is manifest in the third term in Eq. (1)’s description of N_{2} and N_{1}, (**E**(*t*)/*ħω _{s}*)·(

*d*

**p**(

*t*)/

*dt*). The natural decay from N

_{1}to N

_{0}is very fast causing the population in N

_{1}to remain negligibly small. A sufficiently strong electric field will thus deplete the steady state population inversion. The small signal approximation holds when the incident electric field is too weak to affect the population inversion. Our waveguide amplifier study operates in the regime where this is true. Moving beyond the small signal approximation will allow us to study effects such as gain saturation and lasing dynamics. This regime will be addressed in section 3.3. On the other hand, the effect of stimulated emission on the electromagnetic behavior is described by the induced polarization and its effect on the electric field. The polarization increases with population inversion and electric field. In this way, as an electromagnetic (EM) field propagates through an active material with a population inversion (determined by the pumping rate), a polarization is generated which serves to amplify the EM field.

To study amplification we consider a 10µm long Si-nc waveguide on a SiO_{2} substrate. A Gaussian beam, introduced in air at the left edge of the computational region, couples into the waveguide, propagates along the length of the waveguide, interacting with the pumped active material, and arrives at the detector, positioned in air at the opposite end. The Si-nc waveguide, subject to a given pumping rate, is allowed to reach its steady state population inversion before the pulse is introduced. This pulse does not affect the population inversion due to the small signal approximation; however, it does induce local polarization which amplifies the EM field components. The signal which reaches the detector in air is recorded for various pumping rates and normalized to the detected signal for an identical pulse with a pumping rate of zero and the material loss set to zero. This normalization serves to eliminate the effect due to reflections at the air/Si-nc interfaces when considering the devices behavior as an amplifier. The normalized spectral response, shown in Fig. 3(a), is derived by taken the Fourier transform of the detected pulse. The amplification profile for the TE and TM cases are shown at two pumping rates: 10^{4} and 10^{8}. 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 active Si-nc region; 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 2D plots of amplitude of the steady state fields at the peak wavelength of 750nm are shown in Fig 3(b) for the TE and TM cases. Ignoring Fabry-Perot based oscillations, the amplification of the electromagnetic signal during propagation is evident.

#### 3.2: Enhanced spontaneous emission

In this section we address the luminescent behavior of Si-ncs and present device structures which tune and enhance the material emission. Experimental studies have demonstrated the photo and electro-luminescent properties of Si-ncs. Both of these phenomena rely on incoherent spontaneous emission, in which an electron in an excited state (N_{2}) decays to a lower state (N_{1}) as dictated by the radiative emission lifetime (*τ*
_{21}). The subsequent emission has random phase and direction while the frequency is commensurate with the difference in energy between the two states. In this work we model the spontaneous emission of Si-ncs as a collection of randomly oriented dipoles each with a randomly distributed phase.

The device structure is similar to that of the waveguide amplifier case and is shown in Fig 2.b. Here, instead of introducing an external source, we introduce a spontaneous emission source in the form of random dipoles. The Si-nc material is still subject to a given pumping rate, introducing a population inversion, and thus the same stimulated emission mechanisms exist. The subsequent amplification serves to gain-guide the initially random emissions, along with the waveguide’s TIR properties, directing the PL signal along the length of the waveguide. We can enhance this effect by incorporating a Distributed Bragg Reflector (DBR) on either side of the waveguide. Previous work has demonstrated enhanced spontaneous emission in planar DBR microcavities for top emission [13, 18]. Here we are considering a lateral resonator for enhanced edge emission or coupling to other in plane devices. As we can see from sec. 3.1, the air-bordered waveguide creates a weak Fabry-Perot resonance. Increasing the interface reflection by incorporating DBRs increases the microcavity Q. By thus increasing the cavity lifetime we increase the stimulated emission and effectively enhance the PL signal as detected in air at the end of the waveguide.

Each DBR consists of alternating layers of air and Si-nc/SiO_{2} with a period of 300nm, designed to open a bandgap between 650nm and 900nm. The air holes are “etched” through the Si-nc region and 200nm into the SiO_{2} region to create the DBR. The detected PL signal with DBRs is normalized to the peak PL signal without DBRs. A 5µm long Si-nc waveguide region is bordered by 3 periods of DBR on either side. The resulting ASE for the TE and TM case is shown in Fig. 4. The steady state response at the peak emission wavelength shows the high optical confinement within the microcavity. The inclusion of DBRs enhances the peak PL signal by a factor of 4 in the TE case and nearly a factor of 8 in the TM case.

#### 3.3: Lasing dynamics

In the previous two sections we dealt with relatively low amplitude EM fields which did not significantly affect the population inversion. In this section we consider the ability of our model to treat devices where the small signal approximation is no longer valid and the EM field is strong enough to deplete the population inversion.

To study these phenomena we optimize the DBR microcavity device introduced in sec. 3.2 to consist of an 8µm cavity with 6 periods of DBR on each side. Each DBR period in the optimized geometry is 250nm wide with 62.5nm air holes. The resulting microcavity supports several modes within the gain spectrum, including a central mode at 759nm with a Q of 525.

A spontaneous emission source is introduced as described in sec 3.2. In this case we are interested in monitoring both the evolution of the EM fields and the temporal population inversion and thus we begin our simulation with the entire population in the ground state. The temporal behavior of the Ez field at the detector (positioned in air outside the cavity) and the population inversion within the cavity are shown in Fig. 5(a) when the cavity is subjected to a pumping rate of 5×10^{11}s^{-1}. Initially the EM field is weak and there is no population inversion. As time progresses, the pumping rate induces a population inversion and the cavity exhibits round-trip gain. The EM field increases in amplitude until it becomes strong enough that it depletes the population inversion. As the population inversion declines, the round-trip gain in the cavity decreases until the cavity becomes lossy and the strength of the EM field declines. As the EM field decays, the high pumping rate begins to restore the population inversion and before the EM field dies out, the cavity again exhibits gain, amplifying the EM field once more. After a few oscillations of this kind, the system achieves steady state operation. In the steady state, the pumping rate and the EM field strength are balanced, resulting in a constant population inversion and constant optical output. At this point the system is acting as a laser in CW operation. The cavity supports multiple modes, as shown in Fig. 5(b). The strongest mode has a peak emission at 759.4nm with a linewidth of 0.4nm.

In order to identify the lasing threshold for this device, the lasing dynamics are studied at pumping rates ranging from 5×10^{11}s^{-1} to 10^{14}s^{-1}. We plot the peak optical output as a function of pump power in Fig. 6. As we approach the lasing threshold, the computational time required to reach a steady state condition increases due to the reduced gain in the system. By extrapolating a linear fit to the lasing cases simulated, we identify a lasing threshold at pump power of 4.3×10^{11}s^{-1}, corresponding to an optical output power of zero.

## 4. Conclusion

In this paper we have presented a time domain analysis of active Si-nc waveguiding devices. The analysis accounts explicitly for both the electromagnetic and quantum mechanical behavior in waveguiding devices with active components using the ADE-FDTD scheme. We study both stimulated and spontaneous emission based phenomena in the Si-nc material. Waveguide amplifier devices are presented under varying pumping rates for the TE and TM modes. Microcavities consisting of Si-nc waveguides bounded by DBRs are designed and shown to enhance the spontaneous emission. By optimizing the DBR microcavity and increasing the pumping rate we are able to identify steady state lasing conditions and analyze the coupled electromagnetic and energy level population dynamics which lead to this phenomena.

## Acknowledgment

This work is partially sponsored by the AFOSR.

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