Surface Plasmon Hybridization of Whispering Gallery Mode Microdisk Laser

Oka Kurniawan, Iftikhar Ahmed, and Li Er PingInstitute of High Performance Computing, A*STAR

I Fusionopolis Way #16-16 ConnexisSingapore 138632

kurniawano@ihpc.a-star.edu.sg

Abstract-We study the effects of metal layers on the modes of microdisk lasers. It is found that metal layers hybridize the whispering gallery modes of microdisks with the surface plasmon modes. We observe a significant enhancement in the electric field normal to the metal layers. This hybridization effect can be utilized to design a surface plasmon source similar to laser for plasmonic applications.

I. INTRODUCTION

Many efforts have been done to bring the speed of light into the current electronic circuitry. Interests in the integration of photonics and electronic devices have been taken up by the major industry players. Recently, plasmonic offers a striking potential of both high speed and miniaturization of photonics devices [1]. Many works have been done on the passive components. Unlike these, our work focuses its attention on the active component which can be used as the source for a plasmonic circuit.

Plasmonic devices rely on the two basic phenomena. One is the localized surface plasmon. This phenomenon has been utilized in SERS (Surface Enhanced Raman Scattering) for imaging [2]. The other one is the propagating surface plasmon polariton. In this mode the electromagnetic wave travels at the boundary between a metal and a dielectric [3]. The electron cloud at the metal is modulated by the electric field at the boundary and enables a wave to propagate several micrometers away with a finite momentum ħksp.

It is well known that the surface plasmon mode momentum (ħksp) is greater than that of the free-space photon momentum (ħk0) [4]. This means that light from free-space requires an additional change of momentum to couple to a surface plasmon polariton mode. Several techniques have been devised to overcome this. The most common ones make use of prisms, scatterings, and gratings. It is obviously beneficial if one can design a surface plasmon source that couples directly to plasmonic waveguides.

Few works have been done on the surface plasmon source. Bergman and Stockman showed theoretically that a stimulated amplification of plasmon mode similar to laser can be achieved [5, 6]. They calculated the gain of the stimulated emission when a V-shape metal structure is embedded inside an active dielectric matrix. Few years later, Koller at el. successfully demonstrated a surface plasmon source based on organic light emitting diodes [7]. The light

emitting diode is attached on top of a metal layer where surface plasmon polariton wave can propagate. A year later, Walters et al. proposed a silicon-based electrical source for surface plasmon polariton [8]. The electric field between the metals excites electrons which in turn excite the surface plasmon. In this work, we study the effects of metal layers when attached to microdisk lasers. The cavity structure of a microdisk promises higher field intensity through its confinement.

It will be shown that the metal layers hybridize the whispering gallery mode of microdisk lasers with the surface plasmon mode. The electric field normal to the metal layers is enhanced significantly and become the dominant component.

II. THEORETICAL MODELS

The structure that we study is shown in Fig. 1. The dark regions indicate the metal layers while the bright one indicates dielectric or semiconductor. The microdisk laser is sandwiched between two metal layers on the top and the bottom. The microdisk is made of a III-V Semiconductor material which has a bandgap at 1.55 µm and a refractive index of 3.5. The metal layers are gold (Au).

Fig. 1. Microdisk laser sandwiched between two metal layers on top and bottom. We are studying the effect of the metal layers on the whispering

gallery mode of the microdisk laser.

The structure shown in Fig. 1 is studied using a full three dimensional dynamic thermal electron quantum medium finite difference time domain (DTEQM-FDTD) [9]. This code has been previously used to study the light extraction

from elliptical microdisk laser and the effect of external magnetic field on such extraction [9, 10]. A three dimensional simulation domain is required to take into account the thickness of the microdisk laser. The DTEQM-FDTD takes into account the electron dynamics based on multi-level multi-electron quantum system which is governed by Pauli Exclusion Principle, state filling, and dynamical Fermi-Dirac Thermalization [11]. In this way, we are able to treat various solid-state or atomic media. Previous studies consider the medium only as a dielectric and did not take into account the electron dynamics which are considered in this work.

Fig. 2. (left) Electron dynamics in 4-level model. (right) Multi-level multi-electron model semiconductor material in DTEQM-FDTD.

The left figure in Fig. 2. shows the 4-level 2-electron model where an electron is excited above the bandgap energy. This electron then undergoes intraband decay to the band edge through phonon, and finally recombines with the hole in the valence band. This recombination excites a photon with energy equal to the energy difference between the conduction and the valence band. To model the semiconductor medium more accurately, we use additional energy levels and electron dynamics. In this multi-level multi-electron model, the conduction and valence bands are divided into several groups which can be represented by a single dipole. In this work, five groups of energy level are used in each conduction and valence bands, and the energy width for each group is Δλ = 25 nm. In the simulation, the microdisk is electrically injected at a rate of 3×109 s-1. The temperature is set to room temperature at 300 K.

To model the electromagnetic wave, we solve the Maxwell’s equation using the finite difference time domain (FDTD) method. The difference with a conventional FDTD method is that we include the effect of the electron dynamics into the gain medium as mentioned in the previous paragraph. This is done by adding a macroscopic polarization density for the gain medium. Hence, our Maxwell equations can be written as

(1)

(2)

where E and H are the electric field and the magnetic field respectively, and ε0 and μ0 are the free-space permittivity and permeability. The refractive index n in the above equation is the effective refractive index value of the medium. The microscopic polarization density P is derived from the dipole transition of electrons taking into account the state filling and Pauli Exclusion Principle.

The second region that needs to be modeled is the metallic structures. For this region, we incorporate the Lorentz-Drude dispersive model. We use one Drude term and one Lorentz pole as follows:

(3)

The gold (Au) parameters used in the calculation follows that of Reference [12], i.e. ε∞ = 5.9573, ωD/2π = 2113.6 THz, γD/2π = 15.92 THz, ΩL/2π = 650.07 THz, ΓL/2π = 104.86 THz, and Δε = 1.09.

III. RESULTSWe first study the conventional microdisk laser without

any metal layers. The microdisk has a 1 µm radius and 120 nm thickness. The total electric field distribution is plotted in Fig. 3.

Fig. 3. Total electric field distribution of conventional microdisk laser. We observe the whispering gallery mode traveling at the near boundary of the

disk with M=8.

The plot shows a whispering gallery mode traveling inside the microdisk as predicted. The whispering gallery mode solution can be obtained by solving the Helmholtz equation of a dielectric disk. The solution of the wave function is given by ψ = JM(x)eiMθ, where JM are Bessel functions of the first kind, and x = neff(ω)r/c is the root of the Bessel

functions [13]. Putting in M=8 gives us a resonance wavelength at around 1.7 µm. This, however, assumes that the wave function is terminated at the radial boundary of the microdisk. It does not take into account the bending loss in the microdisk.

Our FDTD simulation gave a resonance wavelength at around 1.47 µm (Fig. 4). The simulation takes into account the bending loss due to refractive index difference at the radial boundary. A similar wavelength has been reported previously in [10].

Fig. 4 also shows a second peak at 1.37 µm. At the initial state, we observe the mode at 1.37 µm dominates. Our simulation model allows us to see the evolution of the electric field distribution over time. Hence, we were able to observe that several modes were competing, and how the mode at 1.47 µm overcame the mode at 1.37 µm after some time. The steady state dominant mode is at wavelength 1.47 µm as shown in the plot.

Fig. 4. Dominant electric field intensity of conventional microdisk. The dominant mode is at 1.47µm, while the second mode is at 1.37 µm.

Fig. 5. Total electric field of plasmonic microdisk laser. The whispering gallery mode is hybridized by the surface plasmon mode.

When the metal layers are attached at the top and bottom of the microdisk lasers, the total electric field distribution is

shown in Fig. 5. Both the fields at the inside and outside of the microdisk are affected. More modes can be observed in the plasmonic microdisk laser than the conventional microdisk laser. Observation of the field distribution over time shows that the whispering gallery mode can still be found. However, this mode is now mixed with the surface plasmon modes. Hence we observe there is a mixture of wave circling the microdisk as well as wave traveling back and forth in the radial direction. This wave that is traveling back and forth in the radial direction evolves to be the dominant field as it can be seen in the field distribution outside the microdisk which is now propagating in the radial direction.

In the conventional microdisk laser shown in Fig. 3, the mode can be considered to be almost TE (transverse electric). This means that the dominant electric field is transverse to the direction of propagation. Our calculation indicates that the dominant electric field for the conventional microdisk points in the radial direction. In fact the normal component, which is the z direction, is very small.

This mode observation agrees with the results by Perhaia et al. in [14]. They studied the mode when a single metal layer is attached to a microdisk laser. As they varied the radius of the metal layer, they found an increasing hybridization of the whispering gallery mode with the surface plasmon mode. In our case, the surface plasmon polariton at the top interface is coupled with the bottom interface and creates a stronger field enhancement.

The fast fourier transform of the electric field when the metals are attached to the microdisk laser is shown in Fig. 6. In this figure, we only plot the normal component of field, which is the z direction.

When metal layers are attached to the microdisk lasers, the normal component of the electric field is enhanced significantly. It is observed that the enhancement at wavelength 1.47 µm is about 20,000 times. This significant enhancement in the normal direction changes the dominant field from the radial direction to the normal direction. The electric field normal to the metal layers shows the characteristic of surface plasmon polariton mode.

Fig. 6, Enhancement of electric field in the normal direction due to the metal layers.

When one solves the Maxwell equation for a metal-dielectric-metal medium, one will obtain a TM (transverse magnetic) mode propagating at the surface. For example, when the mode propagates in the x direction, the non-zero field components are Ex, Ez, and Hy. Consider, then, the case when the wave is traveling at the bottom but inside the disk of Fig. 3. The propagation direction is in the x direction and the wave is moving to the right in anti-clock wise fashion. The dominant electric field was found in the negative y direction, or in other words, in the radial direction. But when the metal layers are attached to the microdisk, the dominant electric field now changes to the z direction, which is normal to the plane in Figs. 3 and 5.

Fig. 6 also shows us that the metal layers create more modes as shown in the increasing number of peaks. In the conventional microdisk we only find one dominant mode at 1.47 µm and a secondary mode at 1.37 µm. However, the plasmonic microdisk shows another dominant peak at the wavelength near 1.47 µm, and a smaller peak at wavelength 1.42 µm. The plasmonic microdisk, then, must be designed well in order to have a single mode output. This will be a challenging yet interesting work which we will pursue in the future.

IV. CONCLUSIONWe have studied the plasmonic effects of microdisk lasers

when two metal layers are attached. It has been shown that the whispering gallery mode is hybridized with the surface plasmon mode. The dominant electric field changes from pointing in the radial direction to the normal of the planes. This change can be utilized to extract the surface plasmon mode into a metal-insulator metal waveguide which has a similar plasmonic mode.

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