We present a method for performing time domain simulations of a microphotonic system containing a four level gain medium based on the finite element method. This method includes an approximation that involves expanding the pump and probe electromagnetic fields around their respective carrier frequencies, providing a dramatic speedup of the time evolution. Finally, we present a two dimensional example of this model, simulating a cylindrical spaser array consisting of a four level gain medium inside of a metal shell.
© 2012 Optical Society of America
Interest in microphotonic lasing systems has been increasing over the past few years. As a result, it has become more important to be able to numerically simulate these lasing systems. Several finite difference time domain (FDTD) simulations of a four level gain medium embedded in a microphotonic system have been presented previously [1–9], but these simulations all use structured (cubic) grids and consequently accurately model curved geometries. There have also been methods developed to model spherical gain geometries by expanding electromagnetic fields as sums of spherical Bessel functions [10, 11]. These methods overcome the limitations of structured grids for spherical geometries, but in turn are limited to only modelling spherical geometries. In principle it should be possible to model a microphotonic lasing system with an FDTD simulation utilizing unstructured grids, but to the best knowledge of the authors this has not been demonstrated. The finite element method (FEM) can utilize unstructured grids and as a result can model a wide variety of geometries. In this paper we present a FEM model of a microphotonic system with gain arising from a four level quantum system. In addition to developing a FEM microphotonic lasing model, an approximation is introduced whereby the pump and probe fields are solved for separately, with each field described by the slowly varying complex valued field amplitude of a constant frequency carrier wave. This approximation allows for much larger time steps and a considerable speedup in simulation time.
In the first section of this paper we will describe the dynamics of the microphotonic lasing system, the carrier wave approximation, and finally the finite element formulation of the problem. In the second section we present a two dimensional model of a one dimensional cylindrical spaser array as an example of this new simulation method.
2. FEM microphotonic lasing simulation
2.1. Field equations of a microphotonic lasing system
The simulation we present of a microphotonic lasing system requires the time domain modelling of several different fields and their mutual interactions. These fields include the electromagnetic field, the electric polarization field inside a metal with a Drude response, the electric polarization field of the gain medium, and the population density fields of the different energy levels of the gain medium. Each of these fields evolve according a particular differential equation that must be solved when simulating a microphotonic lasing system. The field equation for the electromagnetic field is
The gain medium is modelled as simple four level quantum system, described schematically in Fig. 1. The 1 → 2 transition is an electric dipole transition with a frequency of ωa. Similarly, the 0 → 3 transition is also an electric dipole transition with frequency ωb. Spontaneous decay between the i-th level to the j-th level occurs at the decay rate of 1/τij. These decay rates include both radiative (spontaneous photon emission) and non-radiative (spontaneous phonon emission) decay processes. In the case of spontaneous photon emission, our model does not produce a photon. Coupling of the gain medium to the electromagnetic field is only allowed for stimulated photon emission.
The electromagnetic response of the four level gain system is given by12].
Together, this system of equations (Eqs. (1), (3)–(5)) completely describes the dynamics of the microphotonic lasing system. The main disadvantage of solving this system of differential equations is the small time step required. In practice, ∼ 100 time steps per period of the pumping laser beam are required for an adequate simulation. A typical lasing simulation could require over 100,000 lasing periods, making the computational requirements of the simulation prohibitively large.
2.2. Period averaged approximation
There is a simple method for dramatically speeding up the simulation time. The electromagnetic field as well as the polarization fields oscillates at two frequencies. These two frequencies are approximately equal to the frequency of the 1 → 2 transition frequency ωa, and the 0 → 3 transition frequency ωb. Much of the computational effort required in this time domain simulation is spent on these simple, approximately harmonic oscillations. A good approximation is to assume these fields oscillate harmonically, with complex valued amplitudes that are slowly changing in time. We can ignore the fast oscillations and instead simulate the relatively slower time dependence of these amplitudes.
Since there are two frequencies, we divide our electromagnetic field into two separate fields
By inserting the above equation into Eq. (1), the field equation for A, we derive two new field equations
Finally, the new differential equations for the occupation number densities are
Finally, we mention that while we have developed the preceding approximation using a FEM model, this approximation is not limited to the FEM. It could potentially be used to speedup both the FDTD models of microphotonic lasing systems [1–9] as well as the time domain models of spherical lasing geometries utilizing spherical Bessel functions [10, 11].
2.3. Finite element formulation
Now that we have derived the period averaged field equations (Eqs. (7), (9)–(12)) for the microphotonic lasing system, we can convert these differential equations into weak forms that can be solved in a finite element simulation.
The weak forms for the field equations of the electromagnetic fields (Eq. (7)) are14, 15]. These weak forms enforce both the electromagnetic field equations as well as a natural boundary condition [14, 15]. The finite element method requires that the integral of the weak form over the simulation domain be set to zero. As an example, if we apply this requirement to the weak form FA1, we find that by integrating by parts we obtain a volume integral enforcing the electromagnetic field equation as well as a second boundary integral enforcing a boundary condition on the field, Eq. (14) forces the tangential component of the magnetic field H1 to zero. This perfect magnetic conductor boundary condition is not desirable for our simulation, so we will modify it to allow for a boundary that absorbs and emits plane waves at normal incidence to the boundary.
For a flat boundary at a large enough distance from the inclusions in the simulation domain that evanescent waves are negligibly small, if the remaining propagating fields are normal to this flat boundary then the vector potential can be represented as the sum of two vector potentials,Eq. (14) that is within the brackets and substitute Eq. (15) for A1 we get Eq. (16) by a test function Ã1 and integrating over the domain boundary gives us a new boundary weak term 15]) which allows for plane waves normal to the boundary to be absorbed and for the incident plane wave to be emitted into the domain normal to the boundary. A matched boundary condition for A2 can be enforced in the same manner.
The weak forms for the remaining field equations are simpler since these differential equations only involve derivatives with respect to time. The weak form for the polarization of Drude metal inclusions isEq. (12). Also, we can avoid solving for N0i by taking advantage of the fact that N0i = Nint − N1i − N2i − N3i where Nint is the initial value of N0i when N1i = N2i = N3i = 0.
3. Cylindrical spaser array
As an example of a microphotonic lasing system simulation we present a two dimensional model of a spaser (surface plasmon amplification by stimulated emission of radiation [16,17]). The time domain FEM simulation was performed using the commercial software COMSOL Multiphysics 3.5. For time stepping, the Generalized-α method was used with the damping parameter ρinf = 1. A copy of the model can be obtained by contacting the corresponding author by email.
The spaser is a one dimensional array of cylinders, each cylinder being infinite in extent in their axial direction. Each cylinder has a core consisting of a four level gain medium with a radius of r1 = 30nm and an outer shell composed of Ag with an outer radius of r2 = 35nm. A diagram of the simulation domain is provided in Fig. 2. The artificial gain medium is characterized by the lifetimes τ10 = 10−14s, τ21 = 10−11s, τ32 = 10−13s and τ30 = 10−12s. The coupling constants in Eq. (10) are σa = 10−4C2/kg and σb = 5 · 10−6C2/kg, and the linewidths of their corresponding transitions are Γa = 2 · 1013s−1 and Γb = 1/τ30 = 1012s−1. Finally, the initial population density parameter is Nint = 5 · 1023m−3. The population densities of the four level gain medium obeys the rate equations given in Eq. (11), and the gain medium interacts with the electromagnetic field through the gain polarization which obeys Eq. (10). The Ag layer interacts with the electromagnetic field through the Drude polarization which evolves according to Eq. (9).
Since the cylinder array is a single layer, it can be characterized as a metasurface . As a metasurface, the electromagnetic response is given by the surface polarizabilityEquation (21) is adapted from Ref. , modified to be consistent with SI units and taking for granted that the metasurface is embedded in vacuum. The surface polarizability α̂ is defined from the scattering matrix S. The S matrix is defined from the amplitude of the electric field of the scattered waves and is adjusted so that the effective thickness of the characterized array is zero . For a symmetric and reciprocal array, such as the cylindrical spaser array, the S matrix components S11 = S22 are the reflection amplitude of a scattered wave and S12 = S21 are the transmitted amplitude of the scattered wave.
The surface polarizability of the cylindrical array is plotted in Fig. 2. The reflection and transmission amplitudes used to calculate the surface polarizability were calculated from a frequency domain FEM simulation (COMSOL Multiphysics) where the Ag had a relative permittivity of and the gain medium is simply a dielectric with permittivity εG = 9. We see from the surface polarizabilities that there is an electric resonance near λ0 = 1220nm and a magnetic resonance near λ0 = 830nm. Figure 2 also show fields profiles for each of these resonances calculated using a FEM eigenfrequency simulation. Also shown are the wavelengths of the corresponding resonances λr = 2πc/Re(ωr), and a resonance quality factor Q = 2πRe(ωr)/Im(ωr), where ωr is a complex eigenfrequency returned by the same FEM eigenfrequency simulation.
We are interested in using both resonances to achieve lasing, one resonance for enhancing the pumping of the gain medium and the other resonance for enhancing the lasing transition. Therefore we choose the energy levels of the artificial four level gain medium so that the 0 → 3 transition approximately matches the higher frequency magnetic resonance ωb = 2πc/830nm, and the 1 → 2 transition approximately matches the lower frequency electric resonance ωa = 2πc/1221nm. The presence of a electronic transition will modify the spectrum of the cylindrical array for frequencies near that transition. Figure 3 plots the surface polarizability near the magnetic resonance at λ0 = 831nm for the cylindrical array where the gain medium now has the relative permittivity . Figure 3 also plots the total absorption of the cylindrical array, as well as separately plotting the absorption in the Ag and in the gain medium. Like Fig. 2, the data for these plots were calculated from a frequency domain FEM simulation.
We can see from Fig. 3 that the interaction of the electronic transition with the magnetic shape resonance shown in Fig. 2(c) causes these resonances to hybridize. As a result the response of the cylindrical array for frequencies near that transition is strongly modified. Instead of a single magnetic resonance we see now see multiple resonances, both electric and magnetic. Examining the absorption plotted in Fig. 3(c) we see that the gain medium strongly absorbs at the magnetic resonance near λ0 = 826nm. For our lasing simulations this will be the pump frequency. There is no way to know exactly what the lasing frequency will be without first running the time domain lasing simulation, except to say that it will be approximately equal to the frequency of the 1 → 2 transition ωa. A good initial guess is to set ω1 = ωa, but after running the lasing simulation this can be adjusted to better match true lasing frequency. In what follows, we have used ω1 = 2πc/1219.3nm.
Figure 4 shows data from a time domain simulation of the cylindrical spaser array using the parameters defined above. The initial state of the simulation is prepared with a previous simulation where the system is pumped with the field A2, with an intensity of 8W/mm2, while the incident probe field is set to A1 = 0. The pump beam is turned on slowly with A2 having the profileFig. 4 begins in this steady state population inversion. Shortly after t=0, a short probe pulse is emitted into the simulation domain. This excites the polarization field , which in turn begins the lasing process. The intensity of the resulting lasing field plotted in Fig. 4(a) spikes initially, but after about 30000 lasing periods it settles into steady state lasing. Figure 4 also plots the difference between the integral of the population densities N2 and N1 for the system beginning in population inversion.
The time step used for the simulations in Fig. 4 varies throughout the simulation. When the pump is initially turned on the time step must be less then the pump rise time τpu. Once the pump is at a maximum the time step can be increased while the gain system approaches steady state. When the time is reset and a probe pulse is introduced the time step must be made smaller than the width of the probe pulse, and must remain small to resolve the resulting oscillations of the interaction between the probe pulse and the resonators as well as the initial exponential growth of the lasing beam. As the laser approaches steady state the time step can again be increased. At all times the time step must be smaller than the inverse rate of change of any transient beams (pump, probe or lasing beams). If ω1 is not close to the resulting lasing frequency the phase of A1 will rapidly change and will require a correspondingly small time step. Once the system begins lasing, the actual lasing frequency can be inferred from this oscillation in the phase of A1, and ω1 can be changed in the middle of the simulation. This causes the phase of A1 to slowly change allowing for a larger time step.
There is a minimum pump intensity required for the light generated due to stimulated emission to overcome the internal losses in the cylindrical array. Figure 4(c) plots the steady state lasing intensity vs. the pump intensity. A linear fit to the lasing data points indicates a threshold pump intensity of 7.15W/mm2. This threshold intensity depends on a number of variables, including all of the parameters of the gain medium, as well as the cylinder plasmon resonances used to enhance both the pump and lasing transition (Figs. 2 and 3). These resonances in turn depend on the geometry and material parameters of the cylindrical array.
While there is a minimum threshold intensity for the array to exhibit lasing, we can observe interesting changes in the spectrum of the array at lower pump intensities. We saw by comparing Figs. 2 and 3 that the spectrum of the cylindrical array was changed by the presence of the 0 → 3 transition. As we pump the array at increasing intensities we observe a similar change in the spectrum due to the 1 → 2 transition. Figure 5 plots the surface polarizability (Eq. (21)) of the electric resonance for different pump intensities. These plots were created by pumping the cylindrical array with the field A2 for a long period of time (t ≫ τ21), and then injecting a Gaussian probe field A1 with a much weaker intensity. Applying a Fourier transform to the resulting time domain reflected and transmitted probe fields gives us the reflection and transmission amplitudes in the frequency domain, allowing us to calculate the surface polarizability according to Eq. (21).
From Fig. 5, we see that for increasing values of the pump intensity, the lineshape of resembles a LorentzianFig. 5(a) that as we increase the pump intensity, it is as if the positive valued scattering frequency γα is made smaller, narrowing the lineshape. We see in Fig. 2(b) that at even higher pump intensities, γα continues to shrink, passing through zero, and the imaginary part of changes sign, indicating gain. As the pump intensity continues to increase, γα continues to grow more negative and the lineshape begins to broaden.
Even though we have gain at the pump intensities in Fig. 5(b), we still do not have lasing because the gain is not large enough to overcome radiative losses.
We have presented a finite element method simulation for a microphotonic lasing system. We have shown how to achieve a massive speedup in the simulation by separating the various fields into fields that oscillate at the carrier frequencies ω1 or ω2, with slowly changing complex valued amplitudes. A demonstration of this simulation was provided with a two dimensional model of a one dimensional cylindrical spaser array as an example. The threshold pump intensity for this array was determined. Finally, we have shown how the linewidth of the lasing transition changes for various pump intensities.
Chris Fietz would like to acknowledge support from the IC Postdoctoral Research Fellowship Program. Work at Ames Laboratory was supported by the Department of Energy (Basic Energy Science, Division of Materials Sciences and Engineering) under contract no. DE-ACD2-07CH11358.
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