Abstract

We develop a frequency-domain formulation in the form of generalized eigenvalue problems for reciprocal microlasers and nanolasers. While the goal is to explore the resonance properties of dispersive cavities, the starting point of our approach is the mode expansion of arbitrary current sources inside the active regions of lasers. Due to the Lorentz reciprocity, a mode orthogonality relation is present and serves as the basis to distinguish various cavity modes. This scheme can also incorporate the asymmetric Fano lineshape into the emission spectra of cavities. We show how to obtain the important parameters of laser cavities based on this formulation. The proposed approach could be an alternative to other computation schemes such as the finite-difference-time-domain method for reciprocal cavities.

© 2011 OSA

1. Introduction

There is significant progress in microlasers and nanolasers [110] as the top-down fabrication technology advances. The geometries and sizes of ultrasmall laser cavities can be well controlled, which makes the fine tuning of mode characteristics such as lasing wavelengths more accurate and flexible. Nevertheless, it often takes much effort and time to implement active photonic devices in the subwavelength regime. To save cost and time, the preliminary cavity calculation for performance estimations is often preferred before device fabrications. After characterization measurements, if the optimization of device structures is necessary, the more serious modeling has to be carried out. With these demands, a physical and efficient formulation for the modeling of microcavities and nanocavities is indispensable.

Two common approaches to the resonance modes of microcavities and nanocavities are the finite-difference-time-domain (FDTD) method [1114] and complex eigenfrequency method [15]. Both have their own advantages and disadvantages. The merits of the FDTD method, for instances, are the easy implementations on the Yee lattice [11], simultaneous spectral searches for resonance modes with broadband excitations, and the lower memory usage than those of other computation schemes. On the other hand, some drawbacks of the FDTD method may bring inconveniences into the cavity modeling. These shortcomings include (1) time-consuming computations for modes with high quality (Q) factors due to the stability issue from the size of time steps [12]; (2) poor mesh adaptivity to arbitrary cavity structures; (3) excitation-dependent outcomes of the mode patterns; and (4) few convenient ways to incorporate arbitrary frequency dispersions of materials into time-domain calculations. Albeit the drawbacks, the FDTD method has been applied in the modeling of active photonic devices [1620]. The scheme is particularly useful when the dynamic properties of lasers, which are not easily conceivable from other approaches, are of interest.

The complex eigenfrequency method resolves some of the issues in the FDTD method and is represented as a generalized eigenvalue problem from the source-free Maxwell’s equations:

××Ecav(r)=(ωcavc)2ɛ¯¯r(r,ωcav)Ecav(r),
where c is the speed of light in vacuum; ωcav and Ecav(r) are the complex eigenfrequency and electric field of the cavity mode; and ɛ̿r(r,ωcav) is the relative permittivity tensor. The implementation of this generalized eigenvalue problem with the finite-element method (FEM) [2123] removes the problem of mesh adaptivity in issue (2). In addition to the FEM, the eigenvalue problem can be also implemented using the integral-equation method [2429]. For issue (3), as an eigenvalue-type approach without sources, the cavity structure itself determines the mode profiles, and multimode excitations from arbitrary sources are avoided.

The complex eigenfrequency method is not flawless, however. For dispersive cavities, the eigenfrequency ωcav also comes into play in the relative permittivity tensor ɛ̿r(r,ωcav). Taking the dispersion into account implies that the eigenfrequency ωcav should be obtained iteratively and self-consistently, which is the penalty for issue (4). In addition, the relative permittivity tensor ɛ̿r(r,ω) needs to be extended to the complex frequency ω. It is not always clear how this generalization is made theoretically or empirically if the experimental data at real frequencies are the only reliable sources of dispersions. Also, even though the Q factor is easily calculated from the real and imaginary parts of ωcav [Q = −Re[ωcav]/2Im[ωcav], where Im[ωcav] < 0 for the time dependence exp(−cavt)], the complex eigenfrequency results in a nonphysical divergent far field due to the complex vacuum wave vector k0 = ωcav/c (Im[k0] < 0) and the outgoing-wave boundary condition [|exp(ik0r)|r→∞ = exp(−Im[k0]r)|r→∞ → ∞, assuming the cavity is surrounded by vacuum]. All these issues limit the applicability of the eigenfrequency method to the modeling of dispersive cavities.

Excitation sources are the origin of many critical issues mentioned above. At the level of classical electrodynamics, as long as the system loss, including the absorption and radiation ones, outnumbers the system gain (even slightly), the physically meaningful fields must be generated by some sorts of sources. For instance, a plane wave in the source-free region is actually generated by a current sheet somewhere else. The nonphysical divergent far field in the eigen-frequency method is also due to the absence of sources. In fact, the dispersion effect is easier to manage if carefully-constructed current sources sinusoidally oscillating at real frequencies are properly embedded into the cavity.

We present a full frequency-domain formulation for reciprocal microlasers and nanolasers and demonstrate how to obtain important parameters of laser cavities. The central part of this approach is a generalized eigenvalue problem which resembles that of self-supporting lasing modes [15, 3035] (a set of discrete real frequencies and corresponding threshold gains are searched in this case). However, unlike the frequency-domain schemes including the eigenfrequency method and approach of self-supporting modes, our motivation is the mode expansion of arbitrary sinusoidal current sources inside the active regions of lasers. The proposed formulation is more advantageous than the conventional schemes in the following respects: (1) no divergent far fields when compared with the eigenfrequency method because the source is present; (2) no multimode excitations when compared with the FDTD method because this approach is a generalized eigenvalue problem inherently; (3) the ability to directly include arbitrary frequency dispersions of materials when compared with all the schemes mentioned above because the frequency ω is real and given in the first place; (4) the straightforward mode expansion due to a natural orthogonality relation brought by the Lorentz reciprocity; and (5) the capability of modeling spectral properties of modes, including the asymmetric Fano lineshape of the emission spectra, which is nontrivial in the eigenfrequency method and approach of self-supporting modes. These advantages make the formulation an alternative to other computation schemes such as the FDTD method in the cavity modeling.

The rest of the paper is organized as follows. We first briefly review the reciprocity theorem and introduce the concept of reciprocal cavities and lasers (section 2). The construction of the formulation is then discussed in detail (section 3 and appendix). We also demonstrate how to obtain the important parameters of microlaser and nanolaser cavities such as resonance frequencies, quality factors, threshold gains, and spontaneous emission coupling factors (section 4). The application of this formulation to a one-dimensional (1D) Fabry-Perot (FP) cavity is illustrated as an example (section 5). Additional remarks on the implementation and variation of the formulation are addressed (section 6), and a conclusion is given at the end (section 7).

2. Reciprocity theorem and reciprocal cavities/lasers

In this section, we briefly introduce the concept of (Lorentz) reciprocity and reciprocal cavities/lasers. The Lorentz reciprocity plays an important role in the construction of the formulation. Many of the results in the following sections need modifying if the Lorentz reciprocity does not hold. We will address the applicability of the formulation.

The schematic diagram of a microlaser or nanolaser cavity is depicted in Fig. 1. The active region of the laser (might be unconnected) is denoted as Ωa and is embedded in the cavity region, which could be an open structure and need not have physical boundaries from the surrounding. Sa is the surface of Ωa (or union of the surfaces from unconnected parts). We consider a cavity which is characterized by the relative permittivity tensor ɛ̿r(r,ω) but a relative permeability of unity. We also construct a region Ω which contains Ωa but does not necessarily cover the cavity region. The surface of Ω is denoted as S.

 

Fig. 1 The schematic diagram of the laser cavity. The active region is denoted as Ωa, and Ω is an arbitrary region which contains Ωa. Sa and S are the surfaces corresponding to Ωa and Ω, respectively.

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For two current sources Js,1(r) and Js,2(r) confined in Ωa but vanishing elsewhere, the reciprocity theorem states that if responses of the material system to any electromagnetic fields are linear, and the relative permittivity tensor ɛ̿r(r,ω) in the cartesian basis is symmetric inside Ω:

[ɛr(r,ω)]αα=[ɛr(r,ω)]αα,α,α=x,y,zrΩ,
the fields generated by the respective sources satisfy the following integral identity [36, 37]:
Ωadr[E1(r)Js,2(r)E2(r)Js,1(r)]=Sda[E1(r)×H2(r)E2(r)×H1(r)],
where E1(r) and H1(r) are the electric and magnetic fields generated by Js,1(r) alone; and E2(r) and H2(r) are the counterparts generated by Js,2(r). Note that the symmetric form in Eq. (2) is not restricted to the cartesian basis. As long as Eq. (2) holds, ɛ̿r(r,ω) is symmetric in any real orthogonal local bases inside Ω. If ɛ̿r(r,ω) is nonsymmetric, an additional volume integral over with an integrand proportional to E1(r)[ɛ¯¯r(r,ω)ɛ¯¯rT(r,ω)]E2(r), where the superscript “T” means transpose, is present at the left-hand side of Eq. (3).

The surface S belongs to a region Ω containing Ωa and therefore encloses Ωa. Let us assume that far away from the cavity is the isotropic free space or absorptive media/structures. In this way, the surface integral at the right hand-side of Eq. (3) vanishes as a result of extending the surface S to infinity, at which the two cross products nearly cancel each other due to the plane-wave approximation of outgoing waves in the far-field regime, or the fields just turn exponentially small due to the absorption present at infinity. If other outer spaces can lead to a vanishing surface integral in Eq. (3), they can also be considered. Under such circumstances, the general reciprocity theorem in Eq. (3) turns into the Lorentz reciprocity theorem [38]:

Ωadr[E1(r)Js,2(r)E2(r)Js,1(r)]=0.
The Lorentz reciprocity theorem is more restrictive than the general reciprocity theorem because the relative permittivity tensor has to be symmetric everywhere (Ω is the full space now).

A reciprocal cavity is a cavity to which the Lorentz reciprocity in Eq. (4) is applicable for two arbitrary current sources confined in Ωa. Practical cavities which exhibit a symmetric permittivity tensor in Eq. (2) everywhere, for example, dielectric spheres, cleaved ridge wavegeuides, and microdisks (assuming that outside the substrate is the free space), all belong to this type. For lasers, the nonlinearity such as the optical feedback from the stimulated emission is impermissible in the reciprocity theorem. Therefore, we adopt a loosened definition for reciprocal lasers, by which the effective permittivity tensor dressed by the nonlinearity remains symmetric, and the integral form in Eq. (4) stays valid for two weak current sources (dropping the perturbation to gain dynamics) inside Ωa.

The Lorentz reciprocity in Eq. (4) is critical to the formulation because it introduces a natural orthogonality relation between modes and provides a way to distinguish them as well as extract their magnitudes from an arbitrary source distribution in Ωa (see section 3.2). Its failure leads to the inapplicability of the proposed formulation to some cavity and laser systems. If the (effective) permittivity tensor ɛ̿r(r,ω) inside or outside the cavity becomes nonsymmetric and breaks the Lorentz reciprocity, the formulation should not be applied, though some qualitative estimations can be still made when the asymmetry is small. The nonsymmetric permittivity tensor ɛ̿r(r,ω) can take place when the time-reversal symmetry is broken. Such examples include cavities with the magneto-optic effect [39,40] or an external magnetic field [4145]. Still, most of the microlasers and nanolasers nowadays remain reciprocal during operation, namely, the effective permittivity tensors remain symmetric. Thus, this approach can be useful in a wide range of active photonic devices in the subwavelength regime.

3. Formulation of reciprocal cavities

In the following subsections, we will describe the generalized eigenvalue problem, mode orthogonality relation, and dyadic Green’s function for the sources localized in Ωa. We note that unless fixed by the real device layout, the active region should have a higher (identical) symmetry group than (to) that of the cavity structure (if there is any) so that the symmetry is preserved.

3.1. Generalized eigenvalue problem

In the frequency domain, the Maxwell’s equations in the presence of sources are

×E(r)=iωμ0H(r),
×H(r)=iωɛ0ɛ¯¯r(r,ω)E(r)+Js(r),
where E(r) and H(r) are the electric and magnetic fields; ɛ0 and μ0 are the vacuum permittivity and permeability, respectively; and Js(r) is the current source. Both the effects of absorption (cold cavity, but inter-state dipole absorption might be excluded) and gain (warm cavity) can be incorporated into ɛ̿r(r,ω), depending on the operation condition of the cavity. Also, in Eq. (5a) and (5b), the real frequency ω is given and can be continuously varied. It is different from the so-called resonance frequencies of cavity modes, which are a set of discrete real frequencies to be sought with specific criteria.

After eliminating H(r) in Eq. (5a) and (5b), we obtain the wave equation for E(r)

××E(r)(ωc)2ɛ¯¯r(r,ω)E(r)=iωμ0Js(r).
We now consider the current sources Js(r) that are only present in Ωa, namely,
Js(r)={Avectorfield,rΩa,0otherwise.
The constraint in Eq. (7) is often physical because the spontaneous emission dipole moments which trigger the lasing field usually coexist with gain in the active region only. Our goal is to find a set {js,n(r,ω)} of current sources, which is present only in Ωa and labeled by index n, for the mode expansion of Js(r) in Eq. (7). Denote the set of electric fields generated by {js,n(r,ω)} as {fn(r,ω)}. We want to prevent the multi-mode excitations from carelessly-constructed sources. A solution to this issue is to excite the electric field with a source which is proportional to the electric field itself in Ωa but vanishes elsewhere, namely, a self-duplicated vector field in Ωa. Thus, we assign the following ansatz to js,n(r,ω):
js,n(r,ω)=iωɛ0Δɛr,n(ω)U(r)fn(r,ω),
U(r)={1,rΩa,0,otherwise,
where U(r) is the indicator function for Ωa; and Δɛr,n(ω) is a complex parameter. With Eq. (8a), the wave equation of fn(r,ω) is transformed into a generalized eigenvalue problem:
××fn(r,ω)(ωc)2ɛ¯¯r(r,ω)fn(r,ω)=iωμ0js,n(r,ω)=(ωc)2Δɛr,n(ω)U(r)fn(r,ω),
where (ω/c)2Δɛr,n(ω) acts as the eigenvalue; and mode quantization indicated by index n is justified from the source confinement in Ωa, cavity structure, and outgoing-wave boundary condition. Note that fn(r,ω) is not divergent in the far-field zone because it is effectively generated by a source js,n(r,ω). Once fn(r,ω) is obtained, the corresponding magnetic field gn(r,ω) is derived from Faraday’s law in Eq. (5a):
gn(r,ω)=1iωμ0×fn(r,ω).

The parameter Δɛr,n(ω) acquires its frequency dependence when Eq. (9) is solved for different ω’s. Its physical interpretation becomes clear if we rewrite Eq. (9) as

××fn(r,ω)(ωc)2[ɛ¯¯r(r,ω)+Δɛr,n(ω)U(r)I¯¯]fn(r,ω)=0,
where I̿ is the identity tensor. Equation (11) resembles the source-free wave equation of which the solution fn(r,ω) oscillates at the real frequency ω, namely, a self-supporting mode. As indicated in Fig. 2, the real part Re[Δɛr,n(ω)] can be regarded as the average permittivity variation in Ωa required to shift the original resonance frequency ωn of mode n to the given frequency ω, while the imaginary part Im[Δɛr,n(ω)] is related to the necessary gain which compensates the loss and converts the emission spectrum Pn(ω′) (ω′ is a dummy variable for the frequency) into the delta-function spectrum centered at ω [for details about ωn and Pn(ω′), see section 4.1]. The self-supporting mode is a bonus of this formulation though the original motivation is aimed at current sources. We also note that Im[Δɛr,n(ω)] has to be negative [in the convention exp(−iωt)] because it represents gain. The magnitude |Im[Δɛr,n(ω)]| is smaller if the more gain is incorporated into ɛ̿r(r,ω), but Im[Δɛr,n(ω)] never turns positive under physical circumstances. If too much gain is present initially, the stimulated emission would clamp it so that the steady-state gain still corresponds to a negative imaginary part Im[Δɛr,n(ω)].

 

Fig. 2 The effect of Δɛr,n(ω) on the emission spectrum Pn(ω′). The real part Re[Δɛr,n(ω)] shifts the resonance frequency ωn to ω, while the imaginary part Im[Δɛr,n(ω)] compensates the loss and converts Pn(ω′) into a delta function centered at ω.

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On the other hand, some current sources are not expandable with the set {js,n(r,ω)}. This incompleteness is more evident for a homogeneous and isotropic active region characterized by a scalar permittivity ɛr,a(ω). In this case, we take the divergence in Eq. (11) for r ∈ Ωa:

[ɛr,a(ω)+Δɛr,n(ω)]fn(r,ω)=0,rΩa.
Since Δɛr,n(ω) ≠ −ɛr,a(ω) [otherwise, the fact that ∇ × ∇ × fn(r,ω) ∝ ∇ × gn(r,ω) = 0 in Ωa may lead to discontinuous tangential magnetic fields across Sa, and thus fictitious surface currents], the divergence of the field fn(r,ω) vanishes in Ωa. Therefore, the corresponding current source js,n(r,ω) is also divergenceless (solenoidal) in Ωa:
js,n(r,ω)=iωɛ0Δɛr,n(ω)fn(r,ω)=0,rΩa.
From Eq. (13), any current sources confined in Ωa that contribute to volume charge densities oscillating at ω (nonzero divergence) cannot be fully expanded by the set {js,n(r,ω)} (however, js,n(r,ω) may result in the surface charge density on Sa). Although real-space charge densities oscillating around the resonance frequencies of cavity modes are uncommon in typical lasers, these charge densities, once induced, affect both the near-field and far-field profiles and should be taken into account. For general reciprocal permittivity tensors in Ωa, the situation becomes less transparent because the set {js,n(r,ω)} may not be divergenceless. However, from the lesson of homogeneous and isotropic active region, it is probable that {js,n(r,ω)} does not span all the source configurations. Therefore, we introduce an analogous set {is,m(r,ω)} of current sources, where m is the mode index, to complement {js,n(r,ω)}. Similar to js,n(r,ω), we require is,m(r,ω) to be confined in Ωa only. The corresponding sets of electric and magnetic fields generated by {is,m(r,ω)} are denoted as {um(r,ω)} and {wm(r,ω)}, respectively, and they also have to satisfy the outgoing-wave boundary condition. We summarize the construction of these additional sets in the appendix.

With the sets {js,n(r,ω)} and {is,m(r,ω)}, we expand an arbitrary current source Js(r) confined in Ωa and oscillating at ω as follows:

Js(r)=ncnjs,n(r,ω)+mdmis,m(r,ω),
where cn and dm are the expansion coefficients. From the superposition principle of linear systems, the corresponding electric field E(r) and magnetic field H(r) are expressed as
E(r)=ncnfn(r,ω)+mdmum(r,ω),
H(r)=ncngn(r,ω)+mdmwm(r,ω).
Our next goal is the extractions of expansion coefficients cn and dm. This step is straightforward if various modes are orthogonal to each other via a certain form of inner product. We will show that the Lorentz reciprocity provides such a handy relation.

3.2. Mode orthogonality

From the Lorentz reciprocity in Eq. (4), we are now in a position to show how a natural orthogonality relation between various modes can be derived. With the sets {js,n(r,ω)} and {fn(r,ω)} constructed in section 3.1, we make the following assignments:

[Js,1(r),E1(r)]=[js,n(r,ω),fn(r,ω)]=[iωɛ0Δɛr,n(ω)U(r)fn(r,ω),fn(r,ω)],
[Js,2(r),E2(r)]=[js,n(r,ω),fn(r,ω)]=[iωɛ0Δɛr,n(ω)U(r)fn(r,ω),fn(r,ω)],
and substitute them into the Lorentz reciprocity theorem in Eq. (4):
iωɛ0[Δɛr,n(ω)Δɛr,n(ω)]Ωadrfn(r,ω)fn(r,ω)=0.
In Eq. (16), if Δɛr,n (ω) ≠ Δɛr,n(ω), the volume integral of the dot product fn (r,ω) · fn(r,ω) must vanish. On the other hand, if a degeneracy exists such that Δɛr,n (ω) = Δɛr,n(ω), we can still utilize the volume integral of the dot product as the inner-product rule to orthogonalize the modes. Thus, we obtain a natural orthogonality relation for the set {fn(r,ω)} as follows:
Ωadrfn(r,ω)fn(r,ω)=δnnΛn(ω),
where δnn is the Kronecker’s delta; and Λn(ω) is the complex normalization constant of fn(r,ω). We also define an analogous orthogonality relation to Eq. (17) for the set {js,n(r,ω)}:
Ωadrjs,n(r,ω)js,n(r,ω)=δnnΘn(ω),
Θn(ω)=[ωɛ0Δɛr,n(ω)]2Λn(ω).
where Θn(ω) is the normalization constant of js,n(r,ω).

For the set {is,m(r,ω)}, we also demand a similar orthogonality relation based on the same inner-product rule:

Ωadris,m(r,ω)is,m(r,ω)=δmmΞm(ω),
where Ξm(ω) is the normalization constant of is,m(r,ω). In addition, two current sources, one from {js,n(r,ω)} and the other from {is,m(r,ω)}, always have to be orthogonal to each other:
Ωadris,m(r,ω)js,n(r,ω)=Ωadrjs,n(r,ω)is,m(r,ω)=0.
The conditions in Eqs. (18a), (19), and (20) necessitate only one single orthogonality relation for any basis vector functions from {js,n(r,ω)} and {is,m(r,ω)}. This property, however, stems from a specifically-constructed set {is,m(r,ω)}, which is summarized in the appendix.

With the orthogonality relations in Eqs. (18a), (19), and (20), we can extract the expansion coefficients cn and dm in Eq. (14a), (14b), and (14c). The expressions of these coefficients can help construct the dyadic Green’s function, which is useful in the calculations of spontaneous emission coupling factors.

3.3. Dyadic Green’s function

For an arbitrary current source Js(r) confined in Ωa, we need to find its mode expansion coefficients cn and dm in Eq. (14a) in order to reconstruct the electric field E(r) and magnetic field H(r) from Eq. (14b) and (14c), respectively. We first dot product both sides of Eq. (14a) with the current source js,n (r,ω) and integrate it over Ωa. With the orthogonality relations in Eqs. (18a), (19), and (20), only the term corresponding to cn at the right-hand side of Eq. (14a) remains, and we obtain the expression of cn as follows:

cn=1Θn(ω)Ωadrjs,n(r,ω)Js(r).
With the same procedure but adopting is,m (r,ω) rather than js,n (r,ω), we derive the analogous expression of dm:
dm=1Ξm(ω)Ωadris,m(r,ω)Js(r).
For the applications in optics, the spatial profile and far-field pattern of the electric field E(r) are important. With the expressions of expansion coefficients in Eq. (21a) and (21b), we substitute them into the expansion series of E(r) in Eq. (14b). After renaming the dummy index n′ (m′) into n (m) and interchanging the variable r with r′, we then link the electric field E(r) to the current source Js(r) through the dyadic Green function G̿ee(r, r, ω):
E(r)=ΩadrG¯¯ee(r,r,ω)[iωμ0Js(r)],
G¯¯ee(r,r,ω)=nfn(r,ω)js,nT(r,ω)iωμ0Θn(ω)+num(r,ω)is,mT(r,ω)iωμ0Ξm(ω)=nfn(r,ω)fnT(r,ω)U(r)(ωc)2Δɛr,n(ω)Λn(ω)+mum(r,ω)is,mT(r,ω)iωμ0Ξm(ω).
The matrix multiplications js,nT(r,ω)Js(r) and is,mT(r,ω)Js(r) in the tensor operation of G̿ee(r, r′, ω) on Js(r′) represent the dot products js,n(r′, ω) · Js(r′) and is,m(r′, ω) · Js(r′), respectively. On the other hand, not every current source can be substituted into Eq. (22a). The dyadic Green’s function G̿ee(r, r′, ω) in Eq. (22b) is not applicable to the sources which spread outside the active region Ωa.

We will utilize the expression of the dyadic Green’s function G̿ee(r, r′, ω) in Eq. (22b) to calculate the spontaneous emission coupling factor in section 4.2.

4. Characteristics of reciprocal cavities

In section 3, we constructed the sets of current sources {js,n(r,ω)}, electric fields {fn(r,ω)}, and magnetic fields {gn(r,ω)} at a given real frequency ω. To further investigate the spectral properties of these modes, we need to link the calculations at different ω’s together. Through this connection, we can then define the cavity resonance.

With the field fl(r,ω) of a nondegenerate mode l at ω, the formal way to associate it with its counterpart fl(r,ω + Δω) at ω + Δω, where Δω is a small frequency difference, is to track how the field profile evolves from ω to ω + Δω and make a one-to-one link from {fn(r,ω)} to {fn(r,ω + Δω)}. An easier approach is the connection through eigenvalue Δɛr,l(ω), assuming a continuous and smooth frequency variation. On the other hand, in degenerate cases due to the cavity symmetry, once a field profile fl(r,ω) in the set {fl (r,ω)|Δɛr,l (ω) = Δɛr,l(ω)} of degenerate modes at ω (excluding the accidental degeneracy) is chosen, its counterpart fl(r,ω + Δω) has to satisfy a condition similar to the orthogonality relation in Eq. (17):

Ωadrfl(r,ω)fl(r,ω+Δω)δll.
Unlike Eq. (17), however, Eq. (23) originates from the symmetry viewpoint even though two equations coincide with each other at Δω = 0. Note that in degenerate cases, the continuity and smoothness of Δɛr,l(ω) may only provide the group-to-group rather than one-to-one correspondence of modes and are insufficient to uniquely identify a particular mode.

In the following derivations, we will assume that the proper one-to-one connection has been established for mode l at different frequencies. In this way, we can identify one of these frequencies as its resonance frequency and obtain the mode information from there.

4.1. Resonance frequency, lineshape, quality factor, and threshold gain

With the identification of mode l at each frequency, we look into the frequency dependence of the power generated by a current source proportional to js,l(r,ω). In this way, the forms of the current source Js(r) and electric field E(r) are

Js(r)=a(ω)js,l(r,ω)=a(ω)(iω)ɛ0Δɛr,l(ω)U(r)fl(r,ω),
E(r)=a(ω)fl(r,ω),
where a(ω) is the source strength. For a fair comparison between the responses at different frequencies, we demand a frequency-independent volume integral of |Js(r)|2
Ωadr|Js(r)|2=|a(ω)|2(ɛ0ω)2|Δɛr,l(ω)|2Ωadr|fl(r,ω)|2𝒥2Va,
where 𝒥 is a real constant; and Va is the volume of the active region. The constraint in Eq. (25) is the white-noise condition [46] for mode l so that the frequency-dependent strength a(ω) of Js(r) does not interfere with the intrinsic nature of mode l on the power spectrum. In this way, the expressions of the square magnitude |a(ω)|2 and power Pl(ω) become
|a(ω)|2=𝒥2Va(ɛ0ω)2|Δɛr,l(ω)|2Ωadr|fl(r,ω)|2,
Pl(ω)=Ωadr12Re[Js*(r)E(r)]=|a(ω)|2ɛ0ωIm[Δɛr,l(ω)]2Ωadr|fl(r,ω)|2=𝒥2Va2ɛ0Im[1ωΔɛr,l(ω)].

Equation (26b) indicates that the lineshape of the white-noise power is determined by the term Im{[ωΔɛr,l(ω)]−1}. To understand its behavior, we define a frequency ωl such that the absolute value |ωlΔɛr,l(ωl)| is the minimum. As shown in Fig. 3(a), if we plot the locus of η(ω) ≡ ωΔɛr,l(ω) parameterized by ω on the complex η plane, the differential change δη due to a small frequency variation δω around ωωl, when viewed as a two-dimensional (2D) vector, has to be perpendicular to that of η(ωl). If these two vectors were not perpendicular, |η(ω)| would have further reduced its magnitude by either moving forward or backward on the curve. In terms of complex numbers, the illustration in Fig. 3(a) implies

δηη(ωl)=iδω(Δωl/2),
where Δωl is a parameter that must be real. The presence of imaginary number i in Eq. (27) indicates a ±π/2 phase change (depending on the sign of δω), namely, the 2D vectors of δη and η(ωl) on the complex η plane are perpendicular. From Eq. (27) and δηη′(ωl)δω, where η′(ω) is the frequency derivative of η(ω), we can express the parameter Δωl as
Δωl=2iη(ωl)η(ωl)=2i[ωlΔɛr,l(ωl)]{[ωΔɛr,l(ω)]ω|ω=ωl}1.

With Eq. (27), we approximate η(ω) near ωl with the linear expansion in ωωl and substitute it into the power spectrum Pl(ω) in Eq. (26b):

η(ω)=ωΔɛr,l(ω)ωlΔɛr,l(ωl)[1i(ωωl)(Δωl/2)],
Pl(ω)𝒥2Va2ɛ0(Δωl/2)|ωlΔɛr,l(ωl)|{Im[ωlΔɛr,l(ωl)]|ωlΔɛr,l(ωl)|(Δωl/2)(ωωl)2+(Δωl/2)2}+Re[ωlΔɛr,l(ωl)]|ωlΔɛr,l(ωl)|(ωωl)(ωωl)2+(Δωl/2)2}.
Equation (29b) indicates that Pl(ω) has a Fano lineshape near ωl (weighted sum of the Lorentzian centered at ωl and its Hilbert transformation). As shown in Fig. 3(b), this line-shape is asymmetric with respect to ωl. In contrast to the Lorentzian, which has a maximum at ωl, the maximum of Fano lineshape is blueshifted (Re[ωlΔɛr,l(ωl)] > 0) or redshifted (Re[ωlΔɛr,l(ωl)] < 0). Although the asymmetry on the power spectrum is derived based on the white-noise source, the phenomenon is generic to most frequency-dependent sources.

 

Fig. 3 (a) The locus of η(ω) = ωΔɛr,l(ω) parameterized by ω on the complex η plane. At resonance frequency ωl, η(ωl) is closest to the origin of the complex plane. (b) The comparison between the Lorentzian and Fano lineshapes. The Fano lineshape is asymmetric with respect to ωl, and the peak is shifted from that of the Lorentzian.

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If |Re[ωlΔɛr,l(ωl)]| ≪ |Im[ωlΔɛr,l(ωl)]| (valid for most cavities with high Q factors), the physical interpretations of ωl and Δωl are more evident. In this case, the frequency ωl is nearly the peak frequency of Pl(ω). Therefore, we may view ωl as the resonance frequency of mode l and generalize it to the cases of asymmetric lineshapes. The electric field fl(r,ωl) at ωl is then the field profile of mode l. Correspondingly, we can identify the parameter Δωl as the full-width-at-half-maximum (FWHM) linewidth of the approximate Lorentzian. The (cold- or warm-cavity) quality factor Ql of mode l is defined as the ratio between ωl and Δωl:

Ql=ωlΔωl=i2Δɛr,l(ωl)[ωΔɛr,l(ω)]ω|ω=ωl.
Recall that Δɛr,l(ωl) is the permittivity variation required for the self-supporting mode l at ωl. With a homogeneous and isotropic active region [ɛ̿r(r,ω)|r∈Ωa = ɛa(ω)I̿] and the cold-cavity condition [inter-state dipole absorption excluded from ɛa(ω)], the threshold gain gth,l is
gth,l=2(ωlc)Im[ɛa(ωl)+Δɛr,l(ωl)ɛa(ωl)](ωlc)Im[Δɛr,l(ωl)]Re[ɛa(ωl)].

4.2. Spontaneous emission coupling factor and Purcell effect

The spontaneous emission coupling factor βl is defined as the ratio of the spontaneous emission power into mode l over the total spontaneous emission power. Classically, various spontaneous emission events are generated by uncorrelated dipoles. Therefore, the spontaneous emission current source Jsp(r,ω) per unit square-root energy [46] at ω satisfies the following relation:

Jsp,α*(r,ω)Jsp,α(r,ω)=δ(rr)c,vDcv(r,ω)jsp,cv,α*(ω)jsp,cv,α(ω)¯,
where α,α′ = x,y,z; Dcv(r,ω) is the position-dependent emitter density per unit energy due to transitions between states (bands) c and v, which includes the information of state occupations and vanishes for r ∉ Ωa; jsp,cv(ω) is the microscopic source responsible for the spontaneous emission; 〈...〉 means the ensemble average; and overline means averages or expectations of other types such as crystal symmetries. For dipole transitions, the source jsp,cv(ω) is
jsp,cv(ω)=2iωqdcv(ω),
where qdcv(ω) is the dipole moment; and a factor of two in Eq. (33) is due to the consistent usages between the harmonic sum and phasor notation (dcv(ω)exp(−iωt) + c.c. = Re[2dcv(ω)exp(−iωt)]).

With Eq. (32), the spontaneous emission rate rsp(ω) per unit energy is expressed as

rsp(ω)=1h¯ωΩadr12Re[Jsp*(r,ω)E(r,ω)]=1h¯ωΩaΩadrdr12Re[Jsp*(r,ω)G¯¯ee(r,r,ω)iωμ0Jsp(r,ω)]=μ02h¯(α,α),(c,v)ΩadrDcv(r,ω)Im{[Gee(r,r,ω)]ααjsp,cv,α*(ω)jsp,cv,α(ω)¯},
where E(r,ω) is the electric field generated by Jsp(r,ω); and the factor of 1/(h̄ω) is to convert the power into rate. We then substitute the expansion of the dyadic Green’s function G̿ee(r, r, ω) in Eq. (22b) into Eq. (34) and obtain
rsp(ω)=nrsp,n(ω)+mr˜sp,m(ω),
rsp,n(ω)=12ɛ0(c,v)ΩadrDcv(r,ω)h¯ωIm{[jsp,cv*(ω)fn(r,ω)][jsp,cv(ω)fn(r,ω)]¯[ωΔɛr,n(ω)]Λn(ω)},
r˜sp,m(ω)=12(c,v)ΩadrDcv(r,ω)h¯ωIm{[jsp,cv*(ω)um(r,ω)][jsp,cv(ω)is,m(r,ω)]¯iΞm(ω)},
where rsp,n(ω) is the spontaneous emission rate per unit energy into mode n in {fn(r,ω)}; and sp,m(ω) is the counterpart into mode m in {um(r,ω)}.

The total spontaneous emission rate Rsp, spontaneous emission rate Rsp,n into mode n in {fn(r,ω)}, and the counterpart sp,m into mode m in {um(r,ω)}, are the integrations of rsp(ω), rsp,n(ω), and sp,m(ω) over photon energy h̄ω, respectively:

Rsp=0d(h¯ω)rsp(ω)=nRsp,n+mR˜sp,m,
Rsp,n=0d(h¯ω)rsp,n(ω),
R˜sp,m=0d(h¯ω)r˜sp,m(ω),
and the spontaneous emission coupling factor βl into mode l in {fn(r,ω)} is written as
βl=Rsp,lRsp.

For a single dipole [see Eq. (33)] due to a two-level system (the occupied excited state |c〉, empty ground state |v〉, and transition frequency ωcv) and located at position rs ∈ Ωa, namely,

Dcv(r,ω)=δ(rrs)δ(h¯ωh¯ωcv),rsΩa,
the spontaneous emission rate Rsp,l into mode l in Eq. (36b) becomes
Rsp,l=2ωcvɛ0h¯Im{[qdcv*fl(rs,ωcv)][qdcvfl(rs,ωcv)][ωcvΔɛr,l(ωcv)]Λl(ωcv)}.
Equation (39) is indicative of the Purcell effect on a single dipole. The spatial enhancement comes from the modal strength fl(rs,ωcv) at position rs, and spectral enhancement originates from the lineshape effect from 1/[ωΔɛr,l(ωcv)] if ωcv is sufficiently close to ωl. To compare it with the counterpart Wsp,l from Fermi’s Golden rule (Lorentzian density of states of mode l):
Wsp,l=2πh¯|qdcvE^l(rs)2|2(Δωl/2)πh¯1(ωcvωl)2+(Δωl/2)2,
where Êph,l(rs) is the single-photon field of mode l, we rewrite Rsp,l in Eq. (39) with a few simplifications. First, for ωcvωl, the fastest-varying part with respect to ωcvωl is the factor 1/[ωcvΔɛr,l(ωcv)]. Except for this factor, we replace ωcv with ωl in other parts of Rsp,l. Second, if we set Λl(ωl) to a positive real number and assume that the phase of fl(r,ωl) does not vary rapidly in Ωa (except for ±π jumps near nodes or nodal surfaces), fl(r,ωl) is close to a real vector field in Ωa, namely, fl*(r,ωl)fl(r,ωl) for r ∈ Ωa. Third, we assume that the asymmetry of the lineshape is minor and only keep the symmetric Lorentzian. With these simplifications, the form of Rsp,l is reduced to that of Wsp,l in Eq. (40) with the following connection between the single-photon field Êl(rs) and mode profile fl(rs,ωl):
E^l(rs)=2h¯ωlɛ0fl(rs,ωl)Im[Δɛr,l(ωl)]QlΛl(ωl).

5. One-dimensional Fabry-Perot cavity

To see if the proposed formulation leads to reasonable outcomes, we apply it to a one-dimensional Fabry-Perot cavity, of which many analytical or approximate properties are known. The structure of the FP cavity is shown in Fig. 4(a). The length Lc of the cavity is 5 μm, and the counterpart La of the active region is 2 μm. The cavity starts at z = −Lc/2 and ends at z = Lc/2, and is uniform in the xy plane. The active region is evenly distributed in the central part of the cavity, and outside it are two passive regions with identical lengths Lp = 1.5 μm. The relative permittivity of the whole cavity, including the active and passive regions, is ɛc (no frequency dispersion). We only consider modes which are uniform in the xy plane. In this way, the cavity modes can be classified based on whether their electric fields are even or odd along the z axis. The free space outside the cavity is filled with air, of which the relative permittivity is unity. We will focus on the spectral properties such as resonance frequencies and lineshapes of modes.

 

Fig. 4 (a) The layout of the 1D FP cavity. The active region has the same permittivity as that (ɛc) of the whole cavity. (b) The resonance lineshapes of various cavity modes with even or odd electric fields when ɛc = 12.25. The lineshape of each mode closely resembles a Lorentzian near its resonance frequency.

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The field profiles of the 1D cavity are composed of forward- and backward-propagating waves with (complex) propagation constants in the active region (symmetric or antisymmetric combination) and passive regions, and outgoing waves in the free space. After matching the boundary conditions at z = ±La/2 and z = ±Lc/2, the permittivity variation Δɛr,n(ω) of mode n satisfies the transcendental equation:

1=±eika,nLa[ra,p+rp,fse2ikpLp1+ra,prp,fse2ikpLp],
ka,n=(ωc)ɛc+Δɛr,n(ω),kp=(ωc)ɛc,
ra,p=ɛc+Δɛr,n(ω)ɛcɛc+Δɛr,n(ω)+ɛc,rp,fs=ɛc1ɛc+1,
where + (−) is the sign for even (odd) modes, ka,n is propagation constant of mode n in the active region while kp is that of the passive region; and ra,p and rp,fs are the reflection coefficients when a plane wave is normally incident from the active region to passive region, and from passive region to free space (fs) filled with air, respectively. The permittivity variation Δɛr,n(ω) is obtained by self-consistently solving the transcendental equation in Eq. (42a) at different ω’s. Setting the relative permittivity ɛc = 12.25, we show the white-noise lineshapes of various FP modes [a common current parameter 𝒥 for all modes in Eq. (26b)] in Fig. 4(b). Since the modes exhibit high enough quality factors, their lineshapes resemble symmetric Lorentzians near their resonance frequencies.

For this 1D cavity, the resonance frequency ωn, quality factor Qn, and the threshold gain gth,n of mode n can be estimated from the nature of a FP resonator and represented as [4749]

ωnmnπcɛcLc,
1QnvgωnLcln(1|rp,fs|2)=2mnπln(ɛc+1ɛc1),
gth,n1Γz,nLcln(1|rp,fs|2)=2[1±sinc(mnπLa/Lc)]Laln(ɛc+1ɛc1),
where mn is the number of standing waves of mode n in the cavity; vg is the group velocity of the wave in the cavity and is identical to the phase velocity ( c/ɛc) in this case; and Γz,n = (La/Lc)[1 ± sinc(ωnLa/c)] is the longitudinal confinement factor [+ (−) for even (odd) modes; and sinc(x) = sin(x)/x]. The quantization of ωn in Eq. (43a) is due to an integral number of standing waves in the cavity region while Qn in Eq. (43b) is estimated from the fractional loss of the energy due to the power leakage at the two outputs in a round-trip period [47]. The threshold gain gth,n in Eq. (43c) is obtained from the round-trip balance condition, taking into account the effects of LaLc and the standing-wave pattern, as described by the expression of the longitudinal confinement factor Γz,n [48]. In Table 1, we show the theoretical estimations of h̄ωn, Qn, and gth,n and their FP counterparts from Eq. (43a) to (43c). The spectral parameters h̄ωn, Qn, and gth,n obtained from these two different approaches agree well. Therefore, we believe that the proposed approach has grasped the essential points of the cavity modeling.

Tables Icon

Table 1. The Comparison of h̄ωn, Qn, and gth,n between the Theoretical and FP Estimations

Figure 5(a) shows the square field magnitude of the even mode at a resonance photon energy of about 1.42 eV (n = 3 in Table 1). In addition to the standing-wave pattern of typical FP modes, the effect of the permittivity variation Δɛr,3(ω) can be observed in the active region (|z| < 1 μm). The growing envelop of the mode profile toward both ends of the active region indicates that the amplification in the active region compensates the radiation loss at two outputs of the cavity. Figure 5(b) shows the locus of h̄η(ω) = h̄ωΔɛr,3(ω) on the complex h̄η plane. The permittivity variation Δɛr,3(ω3) at resonance is about 0.00256–0.286i. Although the real part of the permittivity variation is much lower than the imaginary part in magnitude, this small but positive number indicates that the white-noise lineshape of this FP mode is a little bit asymmetric, and its peak photon energy is slightly blueshifted from h̄ω3 = 1.42 eV.

 

Fig. 5 The spectral characteristics of the mode at h̄ω3 = 1.42 eV. (a) The square magnitude of the mode profile. The pattern in the active region shows the amplification due to Im[Δɛr,3(ω3)]. (b) The locus of h̄η(ω) = h̄ωΔɛr,3(ω) on the complex h̄η plane. The real part Re[Δɛr,3(ω3)] = 0.00256 indicates a slight blueshift from ω3.

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To make the phenomenon of asymmetry more significant, we lower the relative cavity permittivity to one quarter of its original value (ɛc = 3.0625) and show the lineshape of an even mode l (solid black) in Fig. 6(a) (h̄ωl = 1.281 eV). For comparison, we also show the line-shape of an even mode l′ (dashed red) with a close resonance photon energy when ɛc = 12.25 (h̄ωl = 1.278 eV). The square magnitudes of the mode profiles corresponding to the two cases are shown in Fig. 6(b). The lower relative cavity permittivity makes the radiation loss larger. Therefore, the quality factor of mode l is lower than that of mode l′ (Ql = 21.52 versus Ql = 96.34). The larger radiation loss of mode l also leads to the more rapid field growth in the active region, as shown in Fig. 6(b). Usually, the asymmetry on the lineshape is more significant in the more lossy or leaky cavities. From Fig. 6(a), the asymmetry on the line-shape of mode l is indeed more prominent than that of l′. This behavior is also reflected in the more significant real part of the permittivity variation for mode l at its resonance frequency [Δɛr,l(ωl) = −0.0733 – 0.359i versus Δɛr,l (ωl) = −0.00190 – 0.311i]. In addition, a plateau-like feature takes place between photon energies 1.4 to 1.45 eV, which even goes beyond the applicability of Fano lineshape in Eq. (29b).

 

Fig. 6 (a) The white-noise lineshapes of two even modes with ɛc = 3.0625 and ɛc = 12.25, respectively. The mode corresponding to ɛc = 3.0625 has the more asymmetric lineshape. (b) The mode profiles (square magnitudes) of the two even modes with ɛc = 3.0625 and ɛc = 12.25, respectively. The more significant field amplification in the active region with ɛc = 3.0625 indicates the more leaky cavity in this case.

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6. Additional remarks

A practical issue is how to numerically implement the computation of the mode profiles. We focus on the set {fn(r,ω)} and show a generic computation domain based on the FEM in Fig. 7. Since we require the outgoing-wave boundary condition for the modes, perfect-matched layers (PMLs) are inserted in the inner sides of the computation domain to avoid unnecessary reflections. With fields significantly attenuated inside PMLs, the boundary conditions corresponding to the perfect electric conductor (PEC) or perfect magnetic conductor (PMC) can be imposed at the outer boundaries of the computation domain. The generalized eigenvalue problem is then implemented in an analogous numerical scheme to the mode calculations of cavities with the outer PEC or PMC boundaries. The only difference is that it is the permittivity variation Δɛr,n(ω) rather than the eigenfrequency that is to be sought. The construction should be applicable to the modeling of most laser cavities as long as the Lorentz reciprocity holds.

 

Fig. 7 The generic computation domain for the numerical implementation of the formulation. PMLs are inserted in the inner sides of the computation domain while the outer boundaries of the computation domain are set to PECs or PMCs.

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The computation time is also critical in scientific computations. The proposed formulation is based on a generalized eigenvalue problem solved at each sampling frequency. One can speed up the calculations with (1) efficient solvers for generalized eigenvalue problems, and (2) fewer frequency samplings. While (1) depends on details of solver implementations, (2) can be often improved with suitable estimation schemes. For example, the search for the resonance frequency ωl of mode l is equivalent to the minimization of the goal function |ωΔɛr,l(ω)|2 with respect to frequency ω. The convergence to the target frequency can be effective and robust using proper schemes such as quasi-Newton methods or conjugate gradient method.

Another issue is the ansatz of the current source. We may alternatively write the source as js,n(r,ω) = −iωɛ0γnU(r)T̿(r,ω)fn(r,ω), where γn is related to the eigenvalue; and T̿(r,ω) is a symmetric tensor in the cartesian basis. The orthogonality relations take the form of volume integrals of fnT(r,ω)T¯¯(r,ω)fn(r,ω) and js,nT(r,ω)T¯¯1(r,ω)js,n(r,ω) in Ωa. As an example, we can use T̿(r, ω) = ɛ̿r(r,ω) and directly set is,m(r,ω)U(r)ɛ¯¯r1(r,ω)Φm(r,ω)[Φm(r,ω) is a scalar function vanishing at rSa] without the projection process (see appendix). However, in this case, the physical interpretation of γn related to the eigenvalue is less clear, and we should not pursue the related formulation here.

7. Conclusion

We have presented a frequency-domain approach to reciprocal microlasers and nanolasers. The motivation of the approach is the mode expansion of arbitrary current sources in the active region. The proposed formulation can take the frequency dispersion of materials into account and avoid the effect of multimode excitation. We have shown how to obtain the important spectral parameters of cavity modes using this approach, and demonstrated that the asymmetric Fano lineshape comes into play naturally in this formulation. With the constructed dyadic Green’s function, we also derive the spontaneous emission coupling factor and Purcell effect on a single dipole. This approach may be adopted as an alternative to other computation schemes such as the FDTD method.

Appendix: Construction of Complementary Source Set {is,m(r,ω)}

The set {is,m(r,ω)} is constructed to complement the set {js,n(r,ω)}. In a homogeneous and isotropic active region, we may set the source is,m(r,ω) curl-free [is,m(r,ω) ∝ U(r)∇ϕ(r), where ϕ(r) is a scalar function] in Ωa to complement the divergenceless source js,m(r,ω). For general reciprocal active regions, we generalize this idea and write is,m(r,ω) as

is,m(r,ω)iωɛ0ΩadrP¯¯(c)(r,r,ω)Φm(r,ω),
where Φm(r,ω) is a scalar function; and P̿(c)(r,r′,ω) is the projection operator which eliminates the component spanned by {js,n(r,ω)} from the current source present in Ωa only:
P¯¯(c)(r,r,ω)=δ(rr)U(r)I¯¯njs,n(r,ω)js,nT(r,ω)Θn(ω).
The ansatz in Eq. (44) makes {is,m(r,ω)} orthogonal to {js,n(r,ω)}, as required in Eq. (20).

The continuity equation relates is,m(r,ω) to corresponding charge density ρm(r,ω):

is,m(r,ω)=iωρm(r,ω).
To construct a generalized eigenvalue problem for Φm(r,ω), we write ρm(r,ω) as
ρm(r,ω)=ɛ0(ωc)2Δκr,m(ω)U(r)Φm(r,ω),
and substitute the expressions in Eqs. (44) and (47) into Eq. (46):
[ΩadrP¯¯(c)(r,r,ω)Φm(r,ω)]=(ωc)2Δκr,m(ω)U(r)Φm(r,ω),
where (ω/c)2Δκr,m(ω) is the eigenvalue to be obtained. With the extra requirement Φm(r,ω) = 0,rSa, one can show that the orthogonality between the two current sources is,m (r,ω) and is,m(r,ω) [see Eq. (19)] is indeed satisfied. Once is,m(r,ω) is obtained, the electric field um(r,ω) and magnetic field wm(r,ω) are then calculated through the wave equation and Faraday’s law:
××um(r,ω)(ωc)2ɛ¯¯r(r,ω)um(r,ω)=iωμ0is,m(r,ω),
wm(r,ω)=1iωμ0×um(r,ω).

A significant simplification can be made with a homogeneous and isotropic active region. In this case, it can be shown that the set {js,n(r,ω)} is automatically orthogonal to U(r)∇Φm(r) [Φm(r,ω) = 0 for rSa] through the inner product in Eq. (20). In this case, Eq. (48) turns into the form of Schrodinger’s equation with the infinite potential barrier outside Ωa

[U(r)Φm(r,ω)]=(ωc)2Δκr,m(ω)U(r)Φm(r,ω),Φm(r,ω)=0,rSa,
and the source is,m(r,ω) = −iωɛ0U(r)∇Φm(r,ω) becomes curl-free in Ωa.

Acknowledgments

The author would like to thank Professor Shun Lien Chuang at the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, for the discussion which initiated this formulation. This work is sponsored by the research project of Research Center for Applied Sciences, Academia Sinica, Taiwan, and grant support of National Science Council, Taiwan, under contract number NSC100-2112-M-001-002-MY2.

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25. J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003). [CrossRef]  

26. S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004). [CrossRef]  

27. M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004). [CrossRef]   [PubMed]  

28. T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005). [CrossRef]  

29. J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006). [CrossRef]  

30. A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

31. B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998). [CrossRef]   [PubMed]  

32. B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998). [CrossRef]  

33. E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004). [CrossRef]  

34. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005). [CrossRef]  

35. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006). [CrossRef]  

36. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Coninuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

37. C. A. Balanis, Advanced Engineering Electromagnetics, 1st ed. (Wiley and Sons, 1989).

38. H. A. Lorentz, “The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd. 4, 176–187 (1896).

39. N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004). [CrossRef]  

40. N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005). [CrossRef]   [PubMed]  

41. Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982). [CrossRef]  

42. H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986). [CrossRef]  

43. G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007). [CrossRef]  

44. A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009). [CrossRef]  

45. G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010). [CrossRef]  

46. S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

47. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford, 1996).

48. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, 1st ed. (Wiley and Sons, 1995).

49. S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic Fabry-Perot nanolasers,” Opt. Express 18, 15039–15053 (2010). [CrossRef]   [PubMed]  

References

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  1. M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
    [CrossRef]
  2. M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
    [CrossRef] [PubMed]
  3. M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
    [CrossRef]
  4. K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
    [CrossRef] [PubMed]
  5. S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
    [CrossRef]
  6. C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
    [CrossRef]
  7. M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).
  8. B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
    [CrossRef]
  9. M. W. Kim and P. C. Ku, “Semiconductor nanoring lasers,” Appl. Phys. Lett. 98, 201105 (2011).
  10. C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
    [CrossRef] [PubMed]
  11. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  12. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech.23, 623–630 (1975).
    [CrossRef]
  13. A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Electromagn. Compat. 22, 191–202 (1980).
    [CrossRef]
  14. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  15. A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
    [CrossRef]
  16. S. H. 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] [PubMed]
  17. Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express 14, 3569–3587 (2006).
    [CrossRef] [PubMed]
  18. S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
    [CrossRef]
  19. S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
    [CrossRef] [PubMed]
  20. S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
    [CrossRef]
  21. J. Jin, The Finite Element Method in Electromagnetics (Wiley and Sons, 2002).
  22. K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photonics Rev., (2011).
    [CrossRef]
  23. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
    [CrossRef]
  24. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
    [CrossRef]
  25. J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
    [CrossRef]
  26. S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
    [CrossRef]
  27. M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
    [CrossRef] [PubMed]
  28. T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
    [CrossRef]
  29. J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
    [CrossRef]
  30. A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).
  31. B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
    [CrossRef] [PubMed]
  32. B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
    [CrossRef]
  33. E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
    [CrossRef]
  34. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
    [CrossRef]
  35. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
    [CrossRef]
  36. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Coninuous Media, 2nd ed. (Butterworth-Heinemann, 1984).
  37. C. A. Balanis, Advanced Engineering Electromagnetics, 1st ed. (Wiley and Sons, 1989).
  38. H. A. Lorentz, “The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd. 4, 176–187 (1896).
  39. N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
    [CrossRef]
  40. N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
    [CrossRef] [PubMed]
  41. Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
    [CrossRef]
  42. H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986).
    [CrossRef]
  43. G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
    [CrossRef]
  44. A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
    [CrossRef]
  45. G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
    [CrossRef]
  46. S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).
  47. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford, 1996).
  48. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, 1st ed. (Wiley and Sons, 1995).
  49. S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic Fabry-Perot nanolasers,” Opt. Express 18, 15039–15053 (2010).
    [CrossRef] [PubMed]

2011

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
[CrossRef] [PubMed]

2010

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[CrossRef] [PubMed]

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic Fabry-Perot nanolasers,” Opt. Express 18, 15039–15053 (2010).
[CrossRef] [PubMed]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

2009

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[CrossRef]

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

2007

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
[CrossRef]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

2006

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[CrossRef]

Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express 14, 3569–3587 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

2005

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

2004

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
[CrossRef]

S. H. 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] [PubMed]

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
[CrossRef] [PubMed]

2003

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

1998

B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
[CrossRef] [PubMed]

B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

1986

H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986).
[CrossRef]

1982

Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
[CrossRef]

1980

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Electromagn. Compat. 22, 191–202 (1980).
[CrossRef]

1977

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

1966

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1896

H. A. Lorentz, “The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd. 4, 176–187 (1896).

Amanti, M. I.

G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

Aoki, H.

H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986).
[CrossRef]

Arakawa, Y.

Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
[CrossRef]

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

Baba, T.

S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 1st ed. (Wiley and Sons, 1989).

Beck, M.

G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

Beere, H.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Benson, R. M.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

Benson, T. M.

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C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

Bokor, J.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

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S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
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S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
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S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

Chuang, S. L.

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
[CrossRef] [PubMed]

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic Fabry-Perot nanolasers,” Opt. Express 18, 15039–15053 (2010).
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C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
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[CrossRef]

Germann, T. D.

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

Harris, J.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

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N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
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B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

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Hoyler, N.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Hu, Q.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

Huang, Y.

Iwamoto, S.

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

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M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
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S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
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M. W. Kim and P. C. Ku, “Semiconductor nanoring lasers,” Appl. Phys. Lett. 98, 201105 (2011).

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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

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S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

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B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
[CrossRef] [PubMed]

B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

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K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photonics Rev., (2011).
[CrossRef]

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S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[CrossRef]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

Ku, P. C.

M. W. Kim and P. C. Ku, “Semiconductor nanoring lasers,” Appl. Phys. Lett. 98, 201105 (2011).

Kumagai, N.

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

Kumar, S.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

Kurdoglyan, M. S.

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
[CrossRef] [PubMed]

Kwon, S.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

Lee, S. Y.

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
[CrossRef] [PubMed]

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Lee, Y. H.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Leong, E. S. P.

Liddle, A.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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Lloyd-Hughes, J.

G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

Lomakin, V.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

Lu, C. Y.

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
[CrossRef] [PubMed]

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

Marell, M.

Mayer, M. A.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

Mizrahi, A.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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Nezhad, M. P.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photonics Rev., (2011).
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T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
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M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

Nosich, A. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

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M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

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Pohl, U. W.

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

Qureshi, N.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
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B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

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A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

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S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
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M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
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B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

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N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
[CrossRef]

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A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
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E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
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B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
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M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

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M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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Smotrova, E. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
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E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
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E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
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E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
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G. Scalari, D. Turčinková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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Vlasov, A. G.

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

Vuckovic, J.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
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A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
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J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

Williams, B. S.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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Zhukovsky, S. V.

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
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S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
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S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
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Appl. Phys. Lett.

S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

M. W. Kim and P. C. Ku, “Semiconductor nanoring lasers,” Appl. Phys. Lett. 98, 201105 (2011).

B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
[CrossRef]

Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
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IEEE J. Sel. Top. Quantum Electron.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

IEEE Trans. Antennas Propag.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

IEEE Trans. Electromagn. Compat.

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Electromagn. Compat. 22, 191–202 (1980).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

Nano Lett.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

Nat. Photonics

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
[CrossRef]

Phys. Rev. A

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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Phys. Rev. B

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Phys. Rev. Lett.

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[CrossRef]

Phys. Stat. Solidi B

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
[CrossRef]

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

The schematic diagram of the laser cavity. The active region is denoted as Ωa, and Ω is an arbitrary region which contains Ωa. Sa and S are the surfaces corresponding to Ωa and Ω, respectively.

Fig. 2
Fig. 2

The effect of Δɛr,n(ω) on the emission spectrum Pn(ω′). The real part Re[Δɛr,n(ω)] shifts the resonance frequency ωn to ω, while the imaginary part Im[Δɛr,n(ω)] compensates the loss and converts Pn(ω′) into a delta function centered at ω.

Fig. 3
Fig. 3

(a) The locus of η(ω) = ωΔɛr,l(ω) parameterized by ω on the complex η plane. At resonance frequency ωl, η(ωl) is closest to the origin of the complex plane. (b) The comparison between the Lorentzian and Fano lineshapes. The Fano lineshape is asymmetric with respect to ωl, and the peak is shifted from that of the Lorentzian.

Fig. 4
Fig. 4

(a) The layout of the 1D FP cavity. The active region has the same permittivity as that (ɛc) of the whole cavity. (b) The resonance lineshapes of various cavity modes with even or odd electric fields when ɛc = 12.25. The lineshape of each mode closely resembles a Lorentzian near its resonance frequency.

Fig. 5
Fig. 5

The spectral characteristics of the mode at h̄ω3 = 1.42 eV. (a) The square magnitude of the mode profile. The pattern in the active region shows the amplification due to Im[Δɛr,3(ω3)]. (b) The locus of h̄η(ω) = h̄ωΔɛr,3(ω) on the complex h̄η plane. The real part Re[Δɛr,3(ω3)] = 0.00256 indicates a slight blueshift from ω3.

Fig. 6
Fig. 6

(a) The white-noise lineshapes of two even modes with ɛc = 3.0625 and ɛc = 12.25, respectively. The mode corresponding to ɛc = 3.0625 has the more asymmetric lineshape. (b) The mode profiles (square magnitudes) of the two even modes with ɛc = 3.0625 and ɛc = 12.25, respectively. The more significant field amplification in the active region with ɛc = 3.0625 indicates the more leaky cavity in this case.

Fig. 7
Fig. 7

The generic computation domain for the numerical implementation of the formulation. PMLs are inserted in the inner sides of the computation domain while the outer boundaries of the computation domain are set to PECs or PMCs.

Tables (1)

Tables Icon

Table 1 The Comparison of h̄ωn, Qn, and gth,n between the Theoretical and FP Estimations

Equations (70)

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× × E cav ( r ) = ( ω cav c ) 2 ɛ ¯ ¯ r ( r , ω cav ) E cav ( r ) ,
[ ɛ r ( r , ω ) ] α α = [ ɛ r ( r , ω ) ] α α , α , α = x , y , z r Ω ,
Ω a d r [ E 1 ( r ) J s , 2 ( r ) E 2 ( r ) J s , 1 ( r ) ] = S d a [ E 1 ( r ) × H 2 ( r ) E 2 ( r ) × H 1 ( r ) ] ,
Ω a d r [ E 1 ( r ) J s , 2 ( r ) E 2 ( r ) J s , 1 ( r ) ] = 0 .
× E ( r ) = i ω μ 0 H ( r ) ,
× H ( r ) = i ω ɛ 0 ɛ ¯ ¯ r ( r , ω ) E ( r ) + J s ( r ) ,
× × E ( r ) ( ω c ) 2 ɛ ¯ ¯ r ( r , ω ) E ( r ) = i ω μ 0 J s ( r ) .
J s ( r ) = { A vector field , r Ω a , 0 otherwise .
j s , n ( r , ω ) = i ω ɛ 0 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) ,
U ( r ) = { 1 , r Ω a , 0 , otherwise ,
× × f n ( r , ω ) ( ω c ) 2 ɛ ¯ ¯ r ( r , ω ) f n ( r , ω ) = i ω μ 0 j s , n ( r , ω ) = ( ω c ) 2 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) ,
g n ( r , ω ) = 1 i ω μ 0 × f n ( r , ω ) .
× × f n ( r , ω ) ( ω c ) 2 [ ɛ ¯ ¯ r ( r , ω ) + Δ ɛ r , n ( ω ) U ( r ) I ¯ ¯ ] f n ( r , ω ) = 0 ,
[ ɛ r , a ( ω ) + Δ ɛ r , n ( ω ) ] f n ( r , ω ) = 0 , r Ω a .
j s , n ( r , ω ) = i ω ɛ 0 Δ ɛ r , n ( ω ) f n ( r , ω ) = 0 , r Ω a .
J s ( r ) = n c n j s , n ( r , ω ) + m d m i s , m ( r , ω ) ,
E ( r ) = n c n f n ( r , ω ) + m d m u m ( r , ω ) ,
H ( r ) = n c n g n ( r , ω ) + m d m w m ( r , ω ) .
[ J s , 1 ( r ) , E 1 ( r ) ] = [ j s , n ( r , ω ) , f n ( r , ω ) ] = [ i ω ɛ 0 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) , f n ( r , ω ) ] ,
[ J s , 2 ( r ) , E 2 ( r ) ] = [ j s , n ( r , ω ) , f n ( r , ω ) ] = [ i ω ɛ 0 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) , f n ( r , ω ) ] ,
i ω ɛ 0 [ Δ ɛ r , n ( ω ) Δ ɛ r , n ( ω ) ] Ω a d r f n ( r , ω ) f n ( r , ω ) = 0 .
Ω a d r f n ( r , ω ) f n ( r , ω ) = δ n n Λ n ( ω ) ,
Ω a d r j s , n ( r , ω ) j s , n ( r , ω ) = δ n n Θ n ( ω ) ,
Θ n ( ω ) = [ ω ɛ 0 Δ ɛ r , n ( ω ) ] 2 Λ n ( ω ) .
Ω a d r i s , m ( r , ω ) i s , m ( r , ω ) = δ m m Ξ m ( ω ) ,
Ω a d r i s , m ( r , ω ) j s , n ( r , ω ) = Ω a d r j s , n ( r , ω ) i s , m ( r , ω ) = 0 .
c n = 1 Θ n ( ω ) Ω a d r j s , n ( r , ω ) J s ( r ) .
d m = 1 Ξ m ( ω ) Ω a d r i s , m ( r , ω ) J s ( r ) .
E ( r ) = Ω a d r G ¯ ¯ ee ( r , r , ω ) [ i ω μ 0 J s ( r ) ] ,
G ¯ ¯ ee ( r , r , ω ) = n f n ( r , ω ) j s , n T ( r , ω ) i ω μ 0 Θ n ( ω ) + n u m ( r , ω ) i s , m T ( r , ω ) i ω μ 0 Ξ m ( ω ) = n f n ( r , ω ) f n T ( r , ω ) U ( r ) ( ω c ) 2 Δ ɛ r , n ( ω ) Λ n ( ω ) + m u m ( r , ω ) i s , m T ( r , ω ) i ω μ 0 Ξ m ( ω ) .
Ω a d r f l ( r , ω ) f l ( r , ω + Δ ω ) δ l l .
J s ( r ) = a ( ω ) j s , l ( r , ω ) = a ( ω ) ( i ω ) ɛ 0 Δ ɛ r , l ( ω ) U ( r ) f l ( r , ω ) ,
E ( r ) = a ( ω ) f l ( r , ω ) ,
Ω a d r | J s ( r ) | 2 = | a ( ω ) | 2 ( ɛ 0 ω ) 2 | Δ ɛ r , l ( ω ) | 2 Ω a d r | f l ( r , ω ) | 2 𝒥 2 V a ,
| a ( ω ) | 2 = 𝒥 2 V a ( ɛ 0 ω ) 2 | Δ ɛ r , l ( ω ) | 2 Ω a d r | f l ( r , ω ) | 2 ,
P l ( ω ) = Ω a d r 1 2 Re [ J s * ( r ) E ( r ) ] = | a ( ω ) | 2 ɛ 0 ω Im [ Δ ɛ r , l ( ω ) ] 2 Ω a d r | f l ( r , ω ) | 2 = 𝒥 2 V a 2 ɛ 0 Im [ 1 ω Δ ɛ r , l ( ω ) ] .
δ η η ( ω l ) = i δ ω ( Δ ω l / 2 ) ,
Δ ω l = 2 i η ( ω l ) η ( ω l ) = 2 i [ ω l Δ ɛ r , l ( ω l ) ] { [ ω Δ ɛ r , l ( ω ) ] ω | ω = ω l } 1 .
η ( ω ) = ω Δ ɛ r , l ( ω ) ω l Δ ɛ r , l ( ω l ) [ 1 i ( ω ω l ) ( Δ ω l / 2 ) ] ,
P l ( ω ) 𝒥 2 V a 2 ɛ 0 ( Δ ω l / 2 ) | ω l Δ ɛ r , l ( ω l ) | { Im [ ω l Δ ɛ r , l ( ω l ) ] | ω l Δ ɛ r , l ( ω l ) | ( Δ ω l / 2 ) ( ω ω l ) 2 + ( Δ ω l / 2 ) 2 } + Re [ ω l Δ ɛ r , l ( ω l ) ] | ω l Δ ɛ r , l ( ω l ) | ( ω ω l ) ( ω ω l ) 2 + ( Δ ω l / 2 ) 2 } .
Q l = ω l Δ ω l = i 2 Δ ɛ r , l ( ω l ) [ ω Δ ɛ r , l ( ω ) ] ω | ω = ω l .
g th , l = 2 ( ω l c ) Im [ ɛ a ( ω l ) + Δ ɛ r , l ( ω l ) ɛ a ( ω l ) ] ( ω l c ) Im [ Δ ɛ r , l ( ω l ) ] Re [ ɛ a ( ω l ) ] .
J sp , α * ( r , ω ) J sp , α ( r , ω ) = δ ( r r ) c , v D c v ( r , ω ) j sp , c v , α * ( ω ) j sp , c v , α ( ω ) ¯ ,
j sp , c v ( ω ) = 2 i ω q d c v ( ω ) ,
r sp ( ω ) = 1 h ¯ ω Ω a d r 1 2 Re [ J sp * ( r , ω ) E ( r , ω ) ] = 1 h ¯ ω Ω a Ω a d r d r 1 2 Re [ J sp * ( r , ω ) G ¯ ¯ ee ( r , r , ω ) i ω μ 0 J sp ( r , ω ) ] = μ 0 2 h ¯ ( α , α ) , ( c , v ) Ω a d r D c v ( r , ω ) Im { [ G ee ( r , r , ω ) ] α α j sp , c v , α * ( ω ) j sp , c v , α ( ω ) ¯ } ,
r sp ( ω ) = n r sp , n ( ω ) + m r ˜ sp , m ( ω ) ,
r sp , n ( ω ) = 1 2 ɛ 0 ( c , v ) Ω a d r D c v ( r , ω ) h ¯ ω Im { [ j sp , c v * ( ω ) f n ( r , ω ) ] [ j sp , c v ( ω ) f n ( r , ω ) ] ¯ [ ω Δ ɛ r , n ( ω ) ] Λ n ( ω ) } ,
r ˜ sp , m ( ω ) = 1 2 ( c , v ) Ω a d r D c v ( r , ω ) h ¯ ω Im { [ j sp , c v * ( ω ) u m ( r , ω ) ] [ j sp , c v ( ω ) i s , m ( r , ω ) ] ¯ i Ξ m ( ω ) } ,
R sp = 0 d ( h ¯ ω ) r sp ( ω ) = n R sp , n + m R ˜ sp , m ,
R sp , n = 0 d ( h ¯ ω ) r sp , n ( ω ) ,
R ˜ sp , m = 0 d ( h ¯ ω ) r ˜ sp , m ( ω ) ,
β l = R sp , l R sp .
D c v ( r , ω ) = δ ( r r s ) δ ( h ¯ ω h ¯ ω c v ) , r s Ω a ,
R sp , l = 2 ω c v ɛ 0 h ¯ Im { [ q d c v * f l ( r s , ω c v ) ] [ q d c v f l ( r s , ω c v ) ] [ ω c v Δ ɛ r , l ( ω c v ) ] Λ l ( ω c v ) } .
W sp , l = 2 π h ¯ | q d c v E ^ l ( r s ) 2 | 2 ( Δ ω l / 2 ) π h ¯ 1 ( ω c v ω l ) 2 + ( Δ ω l / 2 ) 2 ,
E ^ l ( r s ) = 2 h ¯ ω l ɛ 0 f l ( r s , ω l ) Im [ Δ ɛ r , l ( ω l ) ] Q l Λ l ( ω l ) .
1 = ± e i k a , n L a [ r a , p + r p , fs e 2 i k p L p 1 + r a , p r p , fs e 2 i k p L p ] ,
k a , n = ( ω c ) ɛ c + Δ ɛ r , n ( ω ) , k p = ( ω c ) ɛ c ,
r a , p = ɛ c + Δ ɛ r , n ( ω ) ɛ c ɛ c + Δ ɛ r , n ( ω ) + ɛ c , r p , fs = ɛ c 1 ɛ c + 1 ,
ω n m n π c ɛ c L c ,
1 Q n v g ω n L c ln ( 1 | r p , fs | 2 ) = 2 m n π ln ( ɛ c + 1 ɛ c 1 ) ,
g th , n 1 Γ z , n L c ln ( 1 | r p , fs | 2 ) = 2 [ 1 ± sinc ( m n π L a / L c ) ] L a ln ( ɛ c + 1 ɛ c 1 ) ,
i s , m ( r , ω ) i ω ɛ 0 Ω a d r P ¯ ¯ ( c ) ( r , r , ω ) Φ m ( r , ω ) ,
P ¯ ¯ ( c ) ( r , r , ω ) = δ ( r r ) U ( r ) I ¯ ¯ n j s , n ( r , ω ) j s , n T ( r , ω ) Θ n ( ω ) .
i s , m ( r , ω ) = i ω ρ m ( r , ω ) .
ρ m ( r , ω ) = ɛ 0 ( ω c ) 2 Δ κ r , m ( ω ) U ( r ) Φ m ( r , ω ) ,
[ Ω a d r P ¯ ¯ ( c ) ( r , r , ω ) Φ m ( r , ω ) ] = ( ω c ) 2 Δ κ r , m ( ω ) U ( r ) Φ m ( r , ω ) ,
× × u m ( r , ω ) ( ω c ) 2 ɛ ¯ ¯ r ( r , ω ) u m ( r , ω ) = i ω μ 0 i s , m ( r , ω ) ,
w m ( r , ω ) = 1 i ω μ 0 × u m ( r , ω ) .
[ U ( r ) Φ m ( r , ω ) ] = ( ω c ) 2 Δ κ r , m ( ω ) U ( r ) Φ m ( r , ω ) , Φ m ( r , ω ) = 0 , r S a ,

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