## Abstract

Semiconductor plasmonic lasers at submicron and nanometer scales exhibit many characteristics distinct from those of their conventional counterparts at micron scales. The differences originate from their small sizes and the presence of metal plasma surrounding the cavity. To design a laser of this type, features such as metal dispersion, optical energy confinement, and group velocity have to be taken into account properly. In this paper, we provide a comprehensive approach to the design and performance evaluation of plasmonic Fabry-Perot nanolasers. In particular, we show the proper procedure to obtain the key parameters, especially the quality factor and threshold gain, which are usually neglected in conventional semiconductor Fabry-Perot lasers but become important for nanolasers.

© 2010 Optical Society of America

## 1. Introduction

Ultrasmall semiconductor lasers with a low threshold current and decent output power are the key to high-density integration on a single photonic chip. Among them, the edge-emitting type, which includes the conventional Fabry-Perot (FP) and distributed-feedback (DFB) lasers with a waveguide geometry, is the most important category due to the ease of power delivery and coupling via optical waveguides. The intuitive approach to make these lasers smaller is the reduction of the cavity length and the waveguide core size. While the decrease of the cavity length is limited by the mirror loss, which may be compensated by the high-reflectivity coating or an appropriate feedback structure under a proper phase matching condition [1], the miniaturization of the waveguide core is usually infeasible in a dielectric surrounding. The main obstacle is the diffraction limit, which puts a lower bound on the transverse size of a dielectric-guided mode. With a small waveguide core, a significant portion of the modal profile spreads into the dielectric cladding, which decreases the overlap with gain medium and increases the chance of scattering loss. One of the potential candidates to eliminate the restriction is the plasmonic confinement from the metal cladding [2]. Lasing with the metal cladding has been successfully demonstrated experimentally [3, 4]. A substrate-free metal-cavity surface-emitting microlaser with continuous-wave operation at room temperature has also been achieved [5]. The negative real part of the metal permittivity at optical frequencies is efficient in reducing the field penetration into the metal cladding and can confine the mode in the waveguide core. However, the main concern is the significant material loss of metal, which requires an active medium capable of providing enough gain to the lasing mode. The issue of metal loss becomes more important for a smaller waveguide core. The realization of a plasmonic FP nanolaser is then the search of an operation condition under which the significant metal and mirror losses are balanced by the high material gain in a small waveguide cavity.

The differences between guided modes in plasmonic and dielectric waveguides also require attention in the design of plasmonic FP cavities. For example, while the modal gain and loss of guided modes in a dielectric waveguide are related to the material gain and loss via the power confinement factors, this connection in a plasmonic waveguide can be incorrect due to the Poynting’s vector flowing opposite to the propagation direction in the plasma metal region; see a rigorous formulation in Ref. [6]. Another difference is the facet reflectivity. For conventional semiconductor waveguides, the Fresnel formula, which uses the effective index of the guided mode at the semiconductor side and free-space refractive index at the output side, often works well. However, for small waveguides, the approximation breaks down [7]. The presence of metal cladding further complicates the situation. The facet reflectivity is related to the mirror loss and affects the threshold material gain. Also, in the conventional FP laser, the group velocity of the guided mode and material group velocity in semiconductor are seldom distinguished from each other. However, this condition does not hold in a plasmonic waveguide and can influence the estimation of the quality (*Q*) factor. All of these issues point to the fundamental working principle of a laser: whether there is enough gain to compensate the overall loss.

In this paper, we first present the rate equations for a multi-longitudinal-mode plasmonic FP nanolaser. The laser structure is a silver (Ag) coated circular FP waveguide cavity [8]. While the real laser device may be more complicated due to various technical concerns, this simple structure is a good example to address the issues mentioned earlier. We then explore various physical parameters including guided-mode properties, facet reflectivities, confinement factors, group velocities, quality factors, and threshold material gain of this plasmonic FP nanolaser as well as their differences from those of typical dielectric FP lasers. Finally, we utilize these parameters in the rate equations to investigate the operation characteristics of this plasmonic FP nanolaser.

## 2. Rate Equations

Figure 1(a) shows a silver-coated nanocavity with a short vertical post and subwavelength cross section. For devices with electrical current injection, a thin insulator layer is present between the cladding and core to avoid short circuit. This structure is the limiting case of the FP nanocavity in Fig. 1(b) when the dimension of multiple standing waves is shrunk to about half of an effective wavelength. Additional feedback structures are usually required to reduce the significant radiation loss in these structures [3]. For such a small cavity, only a few modes in the interested frequency range exist, and often only one mode (or a few if allowed by cavity symmetry) can reach the threshold condition for lasing. In this case, the rate equations with a single lasing mode are sufficient to understand the laser behavior. On the other hand, when the post length increases to micron size, modes correspond to multiple standing waves are present. Although their resonance frequencies may be quite separated, often the broad gain spectrum in semiconductors under high pumping does not mean the mode with lowest threshold material gain is the ultimate lasing mode. Under such circumstances, the interplay between semiconductor gains and parameters characterizing the modes becomes an important factor for the lasing behavior. The rate equations with multiple modes, which take these subtleties into account, are required to understand the lasing action of such a plasmonic FP nanolaser.

We use the general multimode rate equations to model the performance of a nanolaser:

where *n* is the carrier density in the active region; *η _{i}* is the injection efficiency;

*I*is the injection current from the top contact to the active region;

*q*is the electron charge;

*V*is the volume of the active region;

_{a}*R*

_{nr}(

*n*) is the nonradiative recombination rate modeled with a defect recombination coefficient

*A*and Auger coefficient

*C*;

*R*

_{sp,cont}(

*n*) and

*R*

_{sp,b}(

*n*) are the spontaneous emission rates directly into the continuum modes and a discrete mode

*b*, respectively;

*R*

_{st,b}(

*n*),

*S*,

_{b}*τ*

_{p,b}, and Γ

_{E,b}are the stimulated emission coefficient, photon density, photon lifetime, and energy confinement factor of mode

*b*;

*v*

_{g,a}(

*ω*) and

_{b}*g*(

*h̅ω*) are the material group velocity and material gain in the active region for mode

_{b},n*b*, respectively; and

*v*

_{g,a}(

*ω*) =

_{b}*c*/

*n*

_{g,a}(

*ω*), where

_{b}*n*

_{g,a}(

*ω*) is the material group index in the active region. The coefficient

_{b}*A*in Eq. (1c) is assumed to originate mainly from surface recombination, namely,

*A*=

*v*Ω

_{s}_{a}/

*V*, where

_{a}*v*is the surface recombination velocity, and Ω

_{s}_{a}is the total surface area of the active region. Also, in Eq. (1d), the spontaneous emission

*R*

_{sp,cont}(

*n*) into continuum modes is described by an effective photon lifetime

*τ*

_{rad}, and

*f*

_{c,k}and

*f*

_{v,k}are the occupation numbers at wave vector

**k**in conduction band c and valence band

*v*, respectively [9].

Various properties of the modes determine the corresponding parameters entering the above rate equations.We will explore the key physical quantities and their expressions in the following sections. The details of other quantities can be found in Ref. [6].

## 3. Guided Modes in the Plasmonic Circular Waveguide

As shown in Fig. 1(b), the propagation direction of the guided modes is denoted as the *z* direction. The cladding is assumed to be infinitely thick, which is a good approximation if the metal cladding is at least several skin depths thick. The material in the core is In_{0.53}Ga_{0.47}As bulk semiconductor, which functions as the gain medium. The diameter *D* of the core is 300 nm, and the cavity length *L* is 3 *µ*m. At the two ends of the circular cylinder, thin Ag films can be coated to reduce the mirror loss.

We adopt the Drude model fitted to the permittivity of silver in Ref. [10]. For simplicity, we neglect both the imaginary parts of the relative metal permittivity *ε*
_{p}(*ω*) and semiconductor permittivity *ε*
_{s}(*ω*) in the calculation of mode profiles. The modal loss or gain due to the imaginary parts Im[*ε*
_{p}(*ω*)] and Im[*ε*
_{s}(*ω*)] is estimated by the variational approach, which works well if Im[*ε*
_{p}(*ω*)] and Im[*ε*
_{s}(*ω*)] are at least one order of magnitude smaller than their respective real parts Re[*ε*
_{p}(*ω*)] and Re[*ε*
_{s}(*ω*)], and if the frequency of the guided mode is much above its cutoff frequency and below the plasmon resonance frequency.

There are two cases for the guided modes. If the propagation constant *k _{z}* <

*k*

_{0}(Re[

*ε*

_{s}(

*ω*)])

^{1/2}, where

*k*

_{0}is the vacuum wave vector, the guided modes are similar to those of a typical fiber waveguide. On the other hand, if

*k*>

_{z}*k*

_{0}(Re[

*ε*

_{s}(

*ω*)])

^{1/2}, the guided mode is surface-wave-like. In the case of

*k*<

_{z}*k*

_{0}(Re[

*ε*

_{s}(

*ω*)])

^{1/2}, the guided modes can be classified into transverse-electric modes TE

_{0n}and transverse-magnetic modes TM

_{0n}(azimuthal mode number

*m*= 0), as well as HEmn and EHmn modes (

*m*> 0). We follow the mode convention in Refs. [11] and [12]. The HE

_{mn}and EH

_{mn}modes approach TE

_{mn}and TM

_{mn}modes, respectively, as the silver-coated waveguide approaches a PEC waveguide, for example, when metal permittivity approaches negative infinity, or the thickness of an insulator layer between metal and semiconductor core approaches zero. With this convention, the HE

_{mn}modes are TE-like, and EH

_{mn}modes are TM-like in the regime

*k*<

_{z}*k*

_{0}(Re[

*ε*

_{s}(

*ω*)])

^{1/2}. For the other case

*k*>

_{z}*k*

_{0}(Re[

*ε*

_{s}(

*ω*)])

^{1/2}, there is exactly one mode for each azimuthal mode number

*m*. We denote these modes as surface-plasma-polariton modes (SPP

_{m}), which are TM-like.

Figure 2(a) shows the effective index *n*
_{eff} ≡ *k _{z}*/

*k*

_{0}of the guided modes in the lossless waveguide. In Fig. 2(b), we also show the counterparts of various guided modes with the same core but covered by the perfect electric conductor (PEC). For the PEC circular waveguide, the classifications of the TE

_{mn}and TM

_{mn}modes are exact. The comparison of the modes from two waveguides indicates the existence of the cutoff frequencies at which

*k*and

_{z}*n*

_{eff}approach zero. These waveguide cutoffs are different from those of the guided modes in a dielectric waveguide, at which

*k*and

_{z}*n*

_{eff}do not approach zero, nevertheless, with guided modes turned into radiation modes. From this viewpoint, plasmonic waveguides are more similar to PEC waveguides than to dielectric waveguides in nature. On the other hand, plasmonic and PEC waveguides are still different. The TM

_{01}and HE

_{m1}modes of the plasmonic waveguide gradually evolve into the SPP

_{m}modes, which are absent in the PEC waveguides. Except for TM

_{01}mode, the turning point of each HE

_{m1}mode into the corresponding SPP

_{m}mode marks the transition from TE-like into the TM-like behavior. This fact also explains why the TE

_{01}mode cannot evolve into a surface-plasma-polariton mode because it lacks the necessary field components 𝓗

_{z}(

*) and 𝓔*

**ρ**_{ρ}(

*) for a TM-like surface wave*

**ρ***at any frequency*. Also, the frequencies of the SPP

_{m}modes can never exceed the plasmon resonance frequency

*ω*

_{res}at which Re[

*ε*

_{p}(

*ω*

_{res}) +

*ε*

_{s}(

*ω*

_{res})] = 0. Since the SPP

_{m}modes (

*m*≥ 1) evolve from corresponding HE

_{m1}modes, it implies that each HE

_{m1}mode has to emerge below

*ω*

_{res}. Note that the plasmon resonance creates a very steep dispersion for SPP

_{m}modes nearby. In reality, this steepness will be smeared out by the imaginary parts Im[

*ε*

_{p}(

*ω*)] and Im[

*ε*

_{s}(

*ω*)] of permittivities, and is not easily observed experimentally.

The imaginary parts Im[*ε*
_{p}(*ω*)] and Im[*ε*
_{s}(*ω*)] of the permittivities make the propagation constant *k _{z}* complex. The real parts of the propagation constant and effective index are not affected much except near the cutoff and plasmon resonance. The imaginary parts Im[

*ε*

_{p}(

*ω*)] and Im[

*ε*

_{s}(

*ω*)] make Re[

*n*

_{eff}] slightly nonzero and thus convert the strict cutoff into a

*soft cutoff*. Figure 3(a) shows the effective index from two-dimensional (2D) finite-element method (FEM) for a few guided modes with Im[

*ε*

_{p}(

*ω*)] included but Im[

*ε*

_{s}(

*ω*)] still set to zero, corresponding to the condition of

*ideal cold cavity*. The counterparts in the lossless waveguide are also plotted for comparison. The agreement of the two calculations is in general good except near the cutoff and plasmon resonance.

The expression of modal loss *α*
_{i} = 2Im[*k _{z}*] is given as [6]

where *A* is the waveguide cross section; *P _{z}* is the time-averaged power flow of the mode;

*n*(

*,*

**ρ***ω*) and

*α*(

*,*

**ρ***h̅ω*) are the space-dependent real part of the material refractive index and material absorption, respectively; ${\eta}_{0}(=\sqrt{\frac{{\mu}_{0}}{{\epsilon}_{0}}})$ is the intrinsic impedance. If one substitutes the

*true mode profile in the presence of loss*into Eq. (2), the value of modal absorption loss α

_{i}will be exact. On the other hand, a variational estimation of the modal loss can be made by substituting the field profiles of the lossless waveguide into Eq. (2) [13]. Figure 3(b) shows such a comparison between the variational estimation and results from 2D FEM calculation. In most of the frequencies, the agreement between two approaches is good, except near the cutoff and plasmon resonance, where the variational approach indicates zero power flows (

*P*= 0) at the cutoff and plasmon resonance. In the real situation, a small power flowing into the waveguide is still present and gets dissipated by the material loss as manifested by Im[

_{z}*ε*

_{p}(

*ω*)] and Im[

*ε*

_{s}(

*ω*)].

In the rest part of the paper, we will use the results from the lossless waveguide and adopt the variational approach.

## 4. Facet Reflectivity

The facet reflectivity *R* is an important parameter related to the mirror loss, and it determines how short the cavity length could be before the mirror loss is too large to overcome. Here we use the idea of standing-wave ratio (SWR) from the transmission-line theory to obtain the facet reflection coefficient *r* and the corresponding reflectivity *R* = ∣*r*∣^{2} of a guided mode [14]. If we send the forward-propagating wave purely consisting of a particular guided mode toward the reflection facet, some incident power will be transmitted, scattered, or coupled to other structures nearby. The field which is coupled back to the original guided mode interferes with the incident wave and form a standing-wave pattern along the *z* axis. Let us first assume that the waveguide is lossless. After picking up a transverse coordinate * ρ* and moving along the axis which is parallel to the

*z*axis and passes through

*, we can obtain the maximum and minimum field strengths*

**ρ***E*

_{max}and

*E*

_{min}of the dominant transverse component of the guided mode. The standing-wave ratio SWR is defined as

and is related to the magnitude of the reflection coefficient ∣*r*∣ as follows:

Equation (5) provides a simple way to obtain the magnitude of the reflection coefficient. However, in many calculations, this simple formula is not practical due to (1) the presence of the modal loss which may wash out the standing-wave pattern, (2) the near field close to the reflection facet that may blur field profile nearby, and (3) the inability to resolve the extreme field strength *E*
_{max} and *E*
_{min} as a result of the coarse sampling grids. In that case, we obtain the reflection coefficient by fitting the magnitude of dominant transverse component *𝓔*(*z*) from the three-dimensional (3D) numerical calculation with the following standing-wave formula (assuming exp(−*iωt*) phasor convention):

$$=\mid {\mathcal{E}}_{\mathrm{inc}}\mid \sqrt{{e}^{-2\mathrm{Im}\left[{k}_{z}\right]\left(z-{z}_{\mathrm{facet}}\right)+{\mid r\mid}^{2}{e}^{2\mathrm{Im}\left[{k}_{z}\right]\left(z-{z}_{\mathrm{facet}}\right)}+2\mid r\mid \mathrm{cos}\{2\mathrm{Re}\left[{k}_{z}\right](z-{z}_{\mathrm{facet}})-{\theta}_{r}\}}},$$

where *𝓔*
_{inc} is the incident amplitude of dominant transverse component at the transverse coordinate ** ρ**,

*z*

_{facet}is the position of the reflection facet; and

*θ*is the phase angle of the complex reflection coefficient. In Eq. (6), the three quantities ∣

_{r}*𝓔*

_{inc}∣, ∣

*r*∣, and

*θ*are the fitting parameters to be extracted while Re[

_{r}*k*] and Im[

_{z}*k*] can be obtained from other calculations such as 2D FEM and are given as the input parameters. When using Eq. (7), the reflection components of other guided modes should be projected out. In our calculations, the standing-wave patterns show that the amplitudes of other excited and reflected guided modes are insignificant, and we will not consider this effect.

_{z}In Fig. 4, we present an example of reflectivity calculation. The incident wave consists of the HE_{11} mode. The junction between the waveguide and air plays the role of reflection facet, and we set the facet position *z*
_{facet} = 0. The wavelength is 1443.65 nm, which is one of the resonance wavelengths of this plasmonic circular cavity. In Fig. 4(a), we show the standing-wave pattern of the *x*-polarized field in the *yz* plane. This 3D calculation is performed using the commercial FEM software COMSOL Multiphysics [15]. To minimize the unwanted reflection from the computation domain, perfect matched layers (PMLs) are constructed at the boundaries of the free space. We can see that a clear standing-wave pattern of the HE_{11} mode is formed in the waveguide region, and part of the incident power has been transferred to the free space. From the normalized field strength ∣*𝓔 _{x}*(

*z*)∣ [max(∣

*𝓔*(

_{x}*z*)∣) = 1] at

*ρ*= 0 [Fig. 4(b)], we can obtain ∣

*r*∣ = 0.881 (

*R*= 0.777), and

*θ*= −0.165

_{r}*π*. When performing this kind of curve fitting, it is necessary to exclude the first one or two standing wave periods because they are often contaminated by the near field close to the facet, as can be seen from the slight deviation between the fitted curve and original data near

*z*=0. The effective index

*n*

_{eff}of this mode is 2.927+0.01007i. If we use the Fresnel formula with unity refractive index of air and effective index of the mode, we would obtain ∣

*r*∣ = 0.491 (

*R*= 0.241) and

*θ*= 8.47 × 10

_{r}^{−4}

*π*, which underestimates the reflectivity. Although the enhancement of the reflectivity for this small waveguide compared with that of the conventional waveguide seems promising, the much shorter waveguide length nevertheless enlarges the mirror loss significantly. The Ag coating, which can further decrease the mirror loss, will be considered in Section 6.

## 5. Quality Factor and Threshold Material Gain

The (cold-cavity) quality factor *Q* of a cavity mode is indicative of how long the stored energy of that mode remains in the cavity when *interband transitions are absent*. A higher *Q* factor usually means that the cavity mode has more chance to lase, though other factors such as how well the mode overlaps with the gain medium (confinement factor) and whether the mode resonance frequency lies within the gain spectrum also play equally important roles. The *Q* factor is related to the photon lifetime *τ*
_{p} which enters the rate equation [Eq. (1b)] via the resonance frequency *ω* of the mode: *Q* = *ωτ*
_{p}. For a FP mode, the inverses of *τ*
_{p} and *Q* are [6]

where *v*
_{g,z}(*ω*) ≃ (*∂*Re[*k _{z}*]/

*∂ω*)

^{−1}is the

*group velocity of the guided mode*;

*Q*

_{abs}and

*Q*

_{mir}are the quality-factor components due to modal absorption loss

*α*

_{i}and mirror loss ln(1/

*R*)/

*L*, respectively; and we have assumed that the two facets have the same reflectivity

*R*. In conventional semiconductor FP lasers,

*v*

_{g,z}(

*ω*) is close to the material group velocity in the semiconductor, and they are seldom distinguished from each other. However, the dispersion of the plasmonic waveguide with a small core is much more significant than that of the conventional semiconductor dielectric waveguide and may result in a considerable deviation in the estimations of

*Q*and

*τ*

_{p}. Experimental data indeed show that the group indices of the FP lasing modes in plasmonic waveguides are very different from those of conventional semiconductor waveguides [4, 16]. Denote the waveguide group index as

*n*

_{g,z}(

*ω*) =

*c*/

*v*

_{g,z}(

*ω*), where

*c*is speed of light in vacuum. In Fig. 5(a), we show the waveguide group indices of the HE

_{11}and SPP

_{1}modes in a lossless waveguide and compare them with the material group index

*n*

_{g,a}(

*ω*) of the active medium InGa

_{0.53}As

_{0.47}. The two waveguide group indices are always larger than that of InGa

_{0.53}As

_{0.47}, and the difference is non-negligible. The deviations become more significant near the waveguide cutoff of the HE

_{11}mode and the plasmon resonance of the SPP

_{1}mode, where the waveguide group velocity is close to zero. This indicates that the waveguide dispersion has to be included in the estimation of quality factor for the small plasmonic waveguides.

The threshold material gain gth is one of the critical factors on whether an active optical device can lase. It can be related to the photon lifetime *τ*
_{p} via two possible confinement factors [6]:

where Γ_{wg} is the waveguide confinement factor which characterizes the *modal gain*; and *v*
_{g,a}(*ω*) and Γ_{E} are the material group velocity of the active medium and energy confinement factor in the rate equations:

In Eq. (10b), *ε*
_{g}(* ρ,ω*) is the group permittivity, defined as

*∂*Re[

*ωε*(

*)]/*

**ρ**,ω*∂ω*, and the quantities with subscript “a” are physical parameters specific to the active gain medium (In

_{0.53}Ga

_{0.47}As). The two confinement factors in Eq. (9) with their respective group velocities will result in the same threshold material gain

*g*

_{th}. In addition, if the waveguide confinement factor Γ

_{wg}is used, Eqs. (9) and (10a) lead to the well-known gain-loss balance condition of a FP cavity. In Fig. 5(b), we show both Γ

_{wg}and Γ

_{E}for the HE

_{11}and SPP

_{1}modes. While Γ

_{wg}is always larger than unity and can become huge at the waveguide cutoff of HE

_{11}mode and plasmon resonance of SPP

_{1}mode, the energy confinement factor remains smaller than unity, which is physical because the sum of

*ε*

_{g}(

*) and Re[*

**ρ**,ω*ε*(

*)] is always larger than zero, even for metals [17]. In addition, in Eq. (9), while the effect of huge Γ*

**ρ**,ω_{wg}on the product

*v*

_{g,z}(

*ω*)Γ

_{wg}=

*c*Γ

_{wg}/

*n*

_{g,z}(

*ω*) will be eventually compensated by the waveguide group index

*n*

_{g,z}(

*ω*) that shows the same trend as that of Γ

_{wg}near the HE

_{11}waveguide cutoff and SPP

_{1}plasmon resonance, the energy confinement factor Γ

_{E}is more indicative of how well the mode overlaps with the gain medium. We will show that it is the energy confinement factor Γ

_{E}rather than

*Q*factor that predicts the right trend of threshold material gain.

To verify the expression of the *Q* factor in Eq. (8b), we use an alternative approach to obtain this quantity. If we use a source with a variable frequency to excite the modes in the cavity, the spectral response of the storage energy *U* in the cavity, defined as the energy-density integral in the active volume *V*
_{a} (In_{0.53}Ga_{0.47}As)

will show resonance peaks corresponding to different mode wavelengths [18]. In Eq. (11), the 3D electric-field profile 𝓔 (**r**) will automatically contain the resonance condition of standing wave. For a particular cavity mode whose resonance wavelength and full-width-at-half-maximum (FWHM) linewidth are *λ*
_{r} and Δ*λ*, respectively, its *Q* factor is estimated as

In Fig. 6, we show the spectrum of storage energy *U* as a function of the wavelength using a linear-polarized plane wave incident along the *z* axis as the source for the FP cavity with waveguide/air interfaces. The calculation is performed using 3D FEM. All the excited cavity modes have the transverse profile of the HE_{11} mode. We use Lorentzian fitting to each resonance peak to find the corresponding *λ*
_{r} and Δ*λ*, and the estimated *Q* factors are listed in Table 1 [shown as *Q* (3D)]. At each resonance wavelength, we also list *Q*
_{abs} and *Q*
_{mir} from the reflectivity calculation in the table. The *Q* factors form the FP formula in Eq. (8b) [*Q* (FP)] and the corresponding material threshold gain *g*
_{th,FP} are also presented. The two *Q* factors obtained from different methods are close in magnitude, ranging from 140 to 150. The deviation between them may come from the error in the reflectivity calculations due to the proximity of PML layers with the waveguide output. The error is smaller in the high-reflectivity cases presented in the next section because less power is transmitted into the free space. Also, from Table 1, although the longer-wavelength modes have lower quality factors, their required threshold material gains are also smaller due to their higher energy confinement factors Γ_{E}, as can be seen from Fig. 5(b). This indicates that in addition to the *Q* factor, other parameters should be taken into account when assessing the feasibility of a lasing mode.

In Table 1, the required material gains for the HE_{11} FP modes in this cavity with waveguide/air interfaces are all above 1000 cm^{−1}. It is necessary to reduce the threshold further. In the next section, metal coating will be considered to lower the mirror loss.

## 6. The Effect of High-Reflectivity Metal Coating

The metal coating at FP cavity facets can increase the reflectivity *R* and thus reduce the mirror loss. Since the material properties of metal cannot be altered much, a proper design of coating thickness should reduce the overall loss so that a reasonable and achievable material gain is able to support lasing action. For bulk semiconductors, the typical material gain under a reasonable carrier density is about a few hundreds at room temperature. It is thus necessary to lower the mirror loss of air/waveguide interfaces, which have a threshold of over 1000 cm^{−1} as shown in Table 1.

Figure 7(a) shows the resonance spectrum of the stored energy with a silver coating thickness of *t* = 10 nm at both the waveguide outputs. The FWHM inewidths of each cavity mode become narrower, reflecting the increase of the reflectivities of all the modes above 88 % (Table 2). The corresponding *Q* factors calculated from the FP formula are above 200, which can lower the threshold material gain to the range of 700–900 cm^{−1}. In Fig. 7(b), we show the 2D standing-wave pattern of the mode at *λ* =1414.63 nm. The 10 nm Ag coating on the waveguide facet at *z* = 0 significantly reduces the power transmitted into the air region, and sharpens the contrast of the field peaks and valleys, indicating the increase of the SWR. From Table 2, we also note that the relative deviations between *Q* factors obtained from the FP formula and 3D FEM calculations become smaller when compared with those of the bare waveguide/air interface. This indicates that apart from the uncertainty in the reflectivity calculations, the correct usage of the group velocity *v*
_{g,z} and modal absorption loss *α*
_{i} of the guided mode is important.

Figure 8 shows similar plots to Fig. 7 with a silver coating thickness *t* = 30 nm. From Fig. 8(b), the standing-wave pattern of the mode at *λ* = 1405.65 nm shows an even smaller portion of the output power than that of the case with a 10 nm silver coating. From Table 3, the *Q* factor of each mode exceeds 240 due to the high reflectivity (> 97%). The corresponding threshold material gains all drop below 670 cm^{−1}. At this thickness, due to the more accurate reflectivity calculations, the *Q* factors from the FP formula and 3D FEM calculations agree very well.

A drawback of the high-reflectivity coating is the poor extraction efficiency. An accompanying issue is the amount of output power under a high-reflectivity condition. Since the energy from the stimulated emissions inside the cavity will finally vanish, it is either dissipated by the material loss or escapes the cavity. Although reducing the amount of energy leaving the cavity can decrease the threshold, it implies that a significant percentage of the generated photons become metal heat rather than output, which reduces the extraction efficiency. Thus, a proper thickness should be designed so that the threshold material gain is achievable while a reasonable extraction efficiency and output power can still be maintained.

## 7. Numerical Results of the Plasmonic Fabry-Perot NanoLaser

The final issues are whether In_{0.53}Ga_{0.47}As has sufficient gain so that this plasmonic nanolaser can lase at room temperature, and which FP mode lases. In the following calculation, we focus on the case of 30 nm Ag coating on the waveguide facet. The photon densities corresponding to the four FP modes in Table 3 (each has a degeneracy of two for *m* = 1) are included in the rate equations. There may be other cavity modes corresponding to different transverse or standing-wave profiles. However, these modes usually have much higher thresholds, so we do not take them into account. The surface recombination velocity is set to 200 cm/s, corresponding to a coefficient *A* = 2.8 × 10^{9} s^{−1} [19]. The Auger coefficient is set to 4.67 × 10^{−29} cm^{6}s^{−1} [20]. The injection efficiency is assumed to be unity, and the radiative lifetime *τ*
_{rad} for the continuum modes is set to 5 *µ*s. In Fig. 9, we show the room-temperature gain spectra of bulk In_{0.53}Ga_{0.47}As under different carrier densities. From Table 3, the threshold material gains of these FP modes range from 610–670 cm^{−1}. The gain spectra indicate that the required threshold carrier density is between 3–4 × 10^{18} cm^{−3}, which is achievable for In_{0.53}Ga_{0.47}As at room temperature.

Figure 10(a) shows the stimulated emission rates of the four FP modes and nonradiative recombination rate. Due to the significant band filling to high-energy states at room temperature, it is the mode with the second longest wavelength (*λ* = 1529.59 nm) which reaches the largest modal gain, rather than the mode with the lowest threshold material gain (*λ* = 1597.88 nm), that lases. Above a current of 0.79 mA, the stimulated emission rate of the mode at λ =1529.59 nm surpasses the nonradiative recombination rates. Above this current density, all the other transition rates, including the spontaneous emission rates of the four FP modes and continuum modes shown in Fig. 10(b), are pinned at their fixed levels. The carrier density is also pinned at 3.13 × 10^{18} cm^{−3}. This indicates that the lasing action of the mode at *λ* =1529.59 nm begins to dominate. Also, from Fig. 10(b), the spontaneous emission rates of the four FP modes are close in magnitude even though only one mode can reach its threshold. Comparing Fig. 10 (a) and (b), we also find that the magnitudes of the four spontaneous emission rates are much smaller than their stimulated emission counterparts.

Figure 11 shows the output powers of the four modes, taking into account that about 30–37% of the power is absorbed when passing through the Ag-coated facets. The light output power *P _{b}* of mode

*b*is

where *T _{b}* and

*R*are the transmittance and reflectivity of the Ag-coated layers for mode

_{b}*b*(assuming two identical facets), which satisfy

*R*+

_{b}*A*+

_{b}*T*= 1 (

_{b}*A*is the absorbance of Ag layers). In the limit of high reflectivity (

_{b}*R*≈ 1), the square bracket in Eq. (13) approaches

_{b}*v*

_{g,z}(

*ω*)

_{b}*T*/

_{b}*L*, which is just the photon escape rate from the facets. The trends of the output powers are close to the stimulated emission rates once population inversion is reached.We note that the output power level is in the range of microwatts, which is much smaller than that of a typical vertical-cavity surface-emitting laser (one to a few milliwatts) at the same injection level. This is partly due to the scalability of power with the device size, and due to the silver coating on the device. Since most of the generated photons cannot escape away from the cavity and get dissipated easily by metal, a reasonable output power requires a high injection current, though a moderate injection current is enough for lasing.

When the cavity length *L* is short, the field normalization and energy confinement factor with the 3D field profile should be used, as formulated in Ref. [6]. The FP formulation assumes a field of the single-mode forward and backward wave propagations exp(±*ik _{z}z*) in a long cavity, and thus the mode mixing near the facets and field variation of standing wave are ignored. An ultrashort cavity can have significant mode mixing near waveguide boundaries and a profile with only a few standing waves. Under such circumstances, the direct 3D field solution should be used, and the energy confinement factor based on the 3D rather than 2D field in Eq. (10b) needs to be applied.

## 8. Conclusion

We have presented a rigorous rate-equation model including important physical parameters on the analysis of plasmonic Fabry-Perot nanolaser using an In_{0.53}Ga_{0.47}As circular waveguide with a silver cladding. Many approximations valid in conventional semiconductor Fabry-Perot lasers cannot be directly applied to the plasmonic Fabry-Perot laser in the submicron or nanometer scales. We show the importance of the reflectivity and *waveguide group velocity* when evaluating the quality factor of the cold cavity, where the approximations of the Fresnel formula and material group velocity may not be appropriate. These effects can influence the estimation of the required threshold material gain. Our theory accounts for metal plasma and waveguide dispersion (waveguide group index), and energy confinement factor. The theoretical formulation not only provides the design and operation condition of the plasmonic nanolasers but also can be used to predict the performance, such as the light output power as a function of the injection current. Future work on the high-speed modulation of plasmonic nanolasers is in progress.

## Acknowledgments

This work was sponsored by the DARPA NACHOS Program under Grant No. W911NF-07-1-0314.We thank many technical discussions with Chien-Yao Lu and Chi-Yu Ni at the University of Illinois at Urbana-Champaign.

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