## Abstract

In this paper we propose a novel silicon microcavity design based on the dispersion engineered photonic crystals (PhCs). With the unique self-collimation property of PhCs, we optimize the passive cavity by tuning the design parameters, such as coupling gap size and array size, to achieve higher Q factor and drop efficiency. Highest cavity mode below the band edge is of particular interest. The strong mode confinement in the low index active material offers an opportunity to realize a lasing mechanism. To investigate the lasing dynamics we introduce the rate equations of atomic system into the electromagnetic polarization to fully describe the nonlinearity of active medium. With these auxiliary differential equations we solve the time evolutions of the electromagnetic waves and atomic populations by using the FDTD method.

© 2007 Optical Society of America

## 1. Introduction

Silicon, the leading material in microelectronics during the last past decades, also promises to be the key material in the future. Silicon based microchip has been proven as an extremely powerful technology platform for a variety of applications of microelectronics, and now becomes the basis of complex microprocessor, larger memory circuits and other digital and analog electronics. The rapid progress of microprocessors may soon or later be severely limited by the transmission bandwidth capability of electronic connections. As a result, there is definitely a strong demand for fast communication links. To eliminate the bottleneck of electronic circuit and establish fast data communication between circuit boards, between chips on a board, or even with a single chip, the research of silicon photonics has attracted more and more attention due to its intrinsic high data transmission rate.

While a wide variety of passive silicon photonic devices were developed, recent activities have focused on the achieving active functionalities, mostly light amplification and generation, in silicon photonic devices. However, due to the fundamental limitation related to the indirect nature of the bulk Si bandgap the semiconductor has very low light emission efficiency. As a result, the development of silicon gain and laser becomes one of challenging goal in the silicon photonics and optoelectronics. Other driving force to stimulate a significant amount of researches is the need of low cost photonic devices to the applications of the future computing and communication. In the past recent, many different approaches have been taken to achieve this goal: silicon nanocrystals, Er doped Si-Ge, Si-Ge quantum dots, and Er doped silicon [1, 2]. More recently, stimulated Raman scattering effect has been used to demonstrate the light amplification and lasing in the silicon both in pulse and continuous-wave operation. [3, 4] In the content of the presented work, we attempt to design a novel silicon laser based on the gain material imbedded in the host silicon material.

To this end, we will design a novel optical cavity based on the dispersion engineered photonic crystals (PhCs). To observe the lasing dynamics, we will incorporate the rate equations of a four level atomic system into the device to simulate the gain and absorption of the active material. By solving the Maxwell’s equations with these auxiliary differential equations with the Finite-difference Time-domain (FDTD) method, we will track the time evolutions of the electromagnetic waves and atomic populations.

## 2. Study of dispersion based photonic crystal cavity

The 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 wavevector, **k**. Dispersion surfaces provide the spatial variation of the spectral properties of a certain band within the photonic crystal structure. Electromagnetic wave propagates along the direction normal to the dispersion surface as shown in Fig. 1, which stems from the relation of the group velocity **v**
_{g}=∇_{k}
*ω*(**k**). The equi-frequency contour (EFC), which can be obtained from the dispersion surface at certain frequency, can lead to beam divergence or convergence as shown in the figure. The ability to shape the EFCs, and thereby engineer the dispersion properties of the PhC, opens up a new paradigm for the design of optical devices [5–7]. 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 wavevector 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 [7] has been theoretically investigated by using FDTD algorithm, in which both optical resonant mode and quality factor are particularly considered.

Consider a silicon photonic crystal slab perforated by a square lattice with air holes back-filled by the gain medium, i.e. Er-doped glasses, as shown in the inset of Fig.2 (a). The hole has 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. The dispersion surface of first band diagram is plotted in Fig. 1(a). By carefully selecting the frequency, one can obtain a flat curvature within certain angular range at specified frequency, i.e. 0.18*c/a*, as depicted in Fig.1 (b), where the blue circle is the dispersion contour in free space with frequency of 0.18*c/a*. Such flatness of the curvature offers self-collimation along ΓM direction within a wide incident angular range.

Self-collimation effect has been widely used in the applications of optical routing, sensors, variable beam splitters, self-collimated optical emitters to enhance the emission from a light emitting diode [8]. In the presented paper we propose a novel design of silicon microcavity based on the self-collimation phenomena. Figure 2 depicts the schematic of our design. A photonic crystal cavity consists of silicon square slab perforated by an N×N array of square lattice. Based on the dispersion property that we discussed above the electromagnetic waves near the band edge of first band will propagate along ΓM direction. After the light couple into the resonator from the waveguide, it will propagate along ΓM direction and then will be reflected at the edge of 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, illustrated as arrow loop in Fig. 2(a), which is similar as whispering gallery mode in microdisk resonator. Along one edge of this microcavity a conventional dielectric waveguide is used for the in-/out-coupling between the cavity and waveguide. To realize the active operation in such microcavity, we will fill the air holes with certain gain medium to achieve nonlinear gain dynamics. In our design, low index highly doped Erbium Ion glass is considered as gain materials for the light amplification. The proposed device has several advantages: 1) since working at the first band of dispersion surface, the self-collimation mode is far below the light cone, therefore, the wave inside resonator is well confined inside the slab with minimal out-plane propagation loss even with the presence of substrate; 2) the self-collimation mode is very close to the band edge, the slow group velocity will enhance the nonlinear interaction between light and gain material.

We simulate the proposed device by using Finite-difference time-domain (FDTD) method. 6×6 array of glass filled cylinders is initially simulated; the highest resonant mode below the band edge of first dispersion band, is shown in Fig. 2(b). As we can observe that most optical mode are well confined within the low index materials. This unique property may benefit the lasing mechanism and lower the threshold optical pumping. The cavity property is largely dependent on the design parameters, such as gap size between the waveguide and resonator, and array size. Extensive study has been done to investigate the passive cavity performance. The gap between the cavity and waveguide is critical design parameter to achieve optimal coupling. First, we fixed the number of array and tune the gap size to study the Q factor and drop efficiency from the waveguide to the resonator. The gap is continually changed from the 150nm to 300nm. As we can see from Fig. 3 that the Q factor increases as the gap size increases due to reduction of the coupling loss, while the drop efficiency expectedly decreases. An optimal gap size of 160nm is chosen in the following design. In this case we measured Q factor of 520 and drop efficiency of 32.2%. In addition, we fix the gap size and increase the number of array size from 6 to 11 and find as the number of array increases the resonant frequency slightly shift to high end, and the Q-factor increase accordingly, which offer us another degree of freedom to tune the Q factor. In the case of array size of 10, the Q factor is 1900 and the drop efficiency remains as high as up to 56%.

With the appropriate design of the microcavity based on the dispersion engineering of photonic crystals, we will further consider to backfill the air holes in photonic crystals with gain medium to achieve active operation.

## 3. Rate equations of a four-level atomic system

To simulate a laser dynamics in an optical cavity associated with a gain or active material, a conventional rate equation model [9] can be employed for the simulations of the nonlinearity and dispersion properties, i.e. a uniform glass host containing a dispersion of highly doped erbium ions. Coupled with Maxwell’s equations, we are able to model the time evolution of the atomic energy level populations as well as the optical signal propagation, amplification and absorption in the devices. In this paper we proposed a simplified realistic four-level atomic system as sketched in Fig. 3, where the principal transitions induced by the presence of the pump and signal beams are included. The absorption of pump radiation at 980nm wavelength promotes the electrons from level 0 to level 3. After decaying to level 2, these electrons provide the signal gain at 1500nm via the transition to level 1. Here we describe the time domain population dynamics by using a four-level rate equation formulism:

where *Ni (i=0,1,2,3)* is the transition population density for different atomic levels. *W _{p}* is the fixed pumping rate to transfer the electron from the ground state level 3. The ground state population density

*N*is assumed to be very large compared to the population density of the high energy levels and is basically constant with time.

_{0}*τ*are the lifetime associated with the transitions from energy

_{ij}*E*and

_{i}*E*is the central frequency of radiation of the materials related to the atomic transition energy levels through $\frac{1}{\hslash {\omega}_{a}}\mathbf{E}\left(t\right)\xb7\frac{d\mathbf{P}\left(t\right)}{\mathrm{dt}}$ is induced radiation rate or excitation rate dependent on its sign.

_{j.}ω_{a}Based on classical electron oscillator model, the net macroscopic polarization **P**(*t*) induced with the presence of applied electric field **E**(*t*) for an isotropic medium can be described by the following equations,

where *ΔN(t)* is instantaneous population density difference between energy levels 1 and 2 of atomic transition, which is given by *ΔN(t)=N1(t)-N2(t)*. *κ=γ _{r}/γ_{c}*, where

*γ*=1/

_{r}*τ*is the radiative decay rate for this from

_{21}*E*and

_{2}*E*transition and

_{1}*γ*=

_{c}*e*

^{2}

*ω*

^{2}

_{a}/(

*m6πε*) is the classical rate. Δω

_{0}c^{3}_{a}is the total energy decay which describes the transition linewidth. It can be easily derived from the above equation that the amplification line shape Lorentizen and homogeneous 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 is introduced into system, where the source is homogeneously distributed across the device. The amplification takes place when the external pumping mechanism produces population inversion

*ΔN(t)*<

*0*.

By doing so, then the effects of the nonlinear and active medium on the propagation of electromagnetic waves are taken into account through the polarization response of the medium. Coupled with the Maxwell’s equation in our optical devices, we are able to simulate the proposed photonic crystal laser cavity and more appropriate to understand the time evolution of such system.

## 4. Dynamics of electromagnetic wave in cavity system

Based on the theory we developed, in this section we will introduce Finite-difference Time-domain (FDTD) method to solve the coupled equations of atomic system and Maxwell’s equations for the application of lasing dynamics in the microcavity system.

As is well known, the FDTD method [10] has become one of the more widely used numerical techniques for solving electromagnetic boundary value problems. In short, the FDTD method directly discretize Maxwell’s equations in both time and space by using central difference technique, and iteratively updates the electrical and magnetic field components at all points in the computational space using the leapfrog fashion. The complicated material properties can be also incorporated in computation. To this end, the method is able to easily deal with the interactions between electromagnetic fields and objects having complex shape and/or inhomogeneous, dispersive and passive/gain media.

Over the past several years, a number of different methods have been proposed to account for material dispersion with absorption and gain using the FDTD method. [11–13] One the most commonly used and memory efficient algorithm is auxiliary differential equation (ADE) method. With the assistance of the ADEs, i.e. the rate equations and induced polarization, we are able to numerically simulate our novel laser structure. The central difference method can be applied to discretize the electromagnetic field components, the atomic populations of different levels, the induced vectorial polarizations in space and time. Then time marching of electromagnetic waves in leapfrog fashion can be constructed and in the meanwhile the atomic evolutions can be monitored. In such way we are able to simulate the lasing action in the photonic cavity. We briefly describe the discretization algorithm: at time of (*n+0.5)Δt*, we first update the magnetic field components and population density *N _{0}*. Then we will update the induced polarization based on Eq. (2) at time of

*(n+1)Δt*, where the knowledge of previous and two-step before are required for the updating. With the known magnetic filed and induced polarization, we can calculate the electric field components. At time of

*(n+1)Δt*, we will also update the remaining atomic population density,

*N*and

_{3}, N_{2}*N*. Since the high level population density is related to lower level atomic transition, the population density is computed in the order from high to low levels to maintain the consistence and minimize the memory storage.

_{1}As shown in the schematic view of proposed novel silicon laser in Fig.2, the holes are back filled with an active material, i.e. Er doped Glass. The medium is uniformly pumped with an external source. For the Er^{3+} Ion, the typical lifetime is on the order of 10^{-3}~10^{-2}s. Numerically, with rigorous EM algorithm in time domain to simulate the lasing dynamics is far beyond the state of art computational capability, particularly in high dimension system. To phenomenally investigate the lasing dynamics in the microcavity with the backfilled gain medium, we scale the lifetime of electron population transition accordingly and yet maintain the reasonable timescale associated to different relaxation process. To this end we shorten the lifetime of atomic transition to reduce the computation time that needed to achieve the steady state. The lifetime are given by *τ _{10}*=10

^{-12}

_{s},

*τ*

_{21}=10

^{-10}

_{s}, and

*τ*=10

_{32}^{-13}s. The transition frequency associated with the energy levels

*E*and

_{2}*E*is chosen as 200THz and linewidth is taken to be 6THz. The pump rate into level

_{1}*E*is chosen W

_{3}_{p}= 2×10

^{8}/s. The initial state of the simulation is included as follow: all the electrons are on the ground state, so there is no field in the cavity and no spontaneous emission. In this paper we choose

*N*=3×10

_{0}^{24}/m

^{3}. After the electrons are pumped, the system start to evolve both in population density and electromagnetic waves. To monitor the EM dynamics a detector is placed in the waveguide.

In the FDTD simulation, a total time of 66ps is simulated. The time step *Δt* is chosen as 3.3×10^{-17}
*s*, so total 2 million FDTD iterations is used. The evolution of electromagnetic waves starts from an initial small noise over the photonic crystal cavity. A detector is placed in the waveguide to monitor the lasing dynamics, and the total field in the output straight waveguide is illustrated in Fig. 5(a). In the figure we can obviously lasing performance. For the clarity, we zoom the transient plot near 60ps as shown in Fig. 5(b), which indicates nearly single lasing model.

In addition, snapshots of 2D magnetic fields at three instant time of 10, 30 and 50*ps* are plotted in Fig. 6, where the lasing dynamics of electromagnetic field is gradually established within the cavity. At last, 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 outputted from both ends of straight waveguide.

Figure 7 shows the calculated time evolution of the electron population inversion between energy level 2 and 1 at the position of one of central holes in the cavity. The plotted population inversion is normalized to the initial population density of the ground state *N _{0}*. During the simulation of the population evolution, the ground state population density almost remains unchanged compared to higher-level populations. In the beginning the population inversion linearly increases, so it leads to significant amplification of electromagnetic wave in the cavity. In the meanwhile the desired cavity mode is established gradually. At last, a convergent population inversion, -6.9×10

^{-4}, can be observed. The population dynamics are well consistent with the lasing output from waveguide. In addition to this, we also calculate the optical output intensity in the waveguide by varying the pumping rate from 0.05×10

^{8}/s to 8×10

^{8}/s to study the lasing threshold. To obtain a stable lasing behavior, particularly near the threshold point, due to lower available gain a longer simulation time is required to actively excite the optical mode inside the microcavity. At the pumping rate of 0.05×10

^{8}/s, 24 million FDTD iterations are used. Figure 8 shows the optical output intensity as the function of the pumping rate. The lasing threshold can be predicted from the output optical intensity as the pumping rate reduces. As shown in the inset of Fig. 8, and we have measured the lasing threshold to be 0.07×10

^{8}/s for the system.

## 5. Conclusion

In this paper we proposed a novel silicon laser design based on the dispersion engineering of photonic crystals. By carefully designing the photonic crystals, we took advantages of the self-collimation property to suppress the optical cavity modes. The relative high concentration of optical mode in lower index gain material will offer benefit to improve the lasing performance. Detailed studies have been performed to investigate the passive cavity design in terms of the coupling gap and the array size of air holes to maximize Q factor while maintains the high drop efficiency. To simulate the lasing dynamics in such optical cavity, we incorporated the rate equations of a four-level atomic system into Maxwell’s with the assistance of the induced electric polarization contributed from the response of a collection of atoms. Finally, we numerically implemented time evolutions of electromagnetic waves and atomic populations by using FDTD method to the application of our engineered dispersion based photonic crystal laser.

## References and links

**1. **B. Jalali, M. Paniccia, and G. Reed, “Silicon photonics,” IEEE Microwave Magazine **7**, 58–68 (2006) [CrossRef]

**2. **L. Pavesi, “Will silicon bethe photonic material of the third millenium,” J. Phys. Condens. Matter **15**, R1169–R1196 (2003) [CrossRef]

**3. **H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continues-wave Raman silicon laser,” Nature **433**, 725–728 (2005) [CrossRef]

**4. **O. Boyraz and B. Jalali, “Demonstartion of a silicon Raman laser,” Opt. Express **12**, 5269–5273 (2004) [CrossRef]

**5. **D. W. Prather, S. Shi, D. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Routing Optical Waves Without Waveguides,” Opt. Lett. **29**, 50–52 (2004) [CrossRef]

**6. **M. R. Newton, K. A. Morey, Y. H. Zhang, R. J. Snow, M. Diwekar, J. Shi, and H. S. White, “Anisotropic diffusion in face-centered cubic opals,” Nano. Lett. **4**, 875–880 (2004) [CrossRef]

**7. **K. K. Tsia and A. W. Poon, “Dispersion-guided resonances in two dimensional photonic-crystal embedded microcavities,” Opt. Express **12**, 5711–5722 (2004) [CrossRef]

**8. **D.W. Prather, A. Sharkawy, and S. Shi, “Handbook of Nanoscience, Engineering and Technology,” 1,b (2002)

**9. **S. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express **12**, 3827–3833 (2004) [CrossRef]

**10. **A. Taflove, “Computational Electromagnetics: The Finite-Difference Time Domain Method,” (1995)

**11. **A. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. **46**, 334–340 (1998) [CrossRef]

**12. **X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. **85**, 70–73 (2000) [CrossRef]

**13. **P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B **66**, 144202 (2002) [CrossRef]