## Full frequency-domain approach to reciprocal microlasers and nanolasers–perspective from Lorentz reciprocity |

Optics Express, Vol. 19, Issue 22, pp. 21116-21134 (2011)

http://dx.doi.org/10.1364/OE.19.021116

Acrobat PDF (1236 KB)

### 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

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]

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]

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]

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]

*Q*) factors due to the stability issue from the size of time steps [12

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]

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]

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

*c*is the speed of light in vacuum;

*ω*

_{cav}and

**E**

_{cav}(

**r**) are the complex eigenfrequency and electric field of the cavity mode; and

*ɛ*̿

_{r}(

**r**,

*ω*

_{cav}) is the relative permittivity tensor. The implementation of this generalized eigenvalue problem with the finite-element method (FEM) [21–23

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]

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

*ω*

_{cav}also comes into play in the relative permittivity tensor

*ɛ*̿

_{r}(

**r**,

*ω*

_{cav}). Taking the dispersion into account implies that the eigenfrequency

*ω*

_{cav}should be obtained iteratively and self-consistently, which is the penalty for issue (4). In addition, the relative permittivity tensor

*ɛ*̿

_{r}(

**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

*ω*

_{cav}[

*Q*= −Re[

*ω*

_{cav}]/2Im[

*ω*

_{cav}], where Im[

*ω*

_{cav}] < 0 for the time dependence exp(−

*iω*

_{cav}

*t*)], the complex eigenfrequency results in a nonphysical divergent far field due to the complex vacuum wave vector

*k*

_{0}=

*ω*

_{cav}/

*c*(Im[

*k*

_{0}] < 0) and the outgoing-wave boundary condition [|exp(

*ik*

_{0}

*r*)|

_{r}_{→∞}= exp(−Im[

*k*

_{0}]

*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.

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]

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]

*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.

## 2. Reciprocity theorem and reciprocal cavities/lasers

_{a}and is embedded in the cavity region, which could be an open structure and need not have physical boundaries from the surrounding.

*S*

_{a}is the surface of Ω

_{a}(or union of the surfaces from unconnected parts). We consider a cavity which is characterized by the relative permittivity tensor

*ɛ*̿

_{r}(

**r**,

*ω*) but a relative permeability of unity. We also construct a region Ω which contains Ω

_{a}but does not necessarily cover the cavity region. The surface of Ω is denoted as

*S*.

**J**

_{s,1}(

**r**) and

**J**

_{s,2}(

**r**) confined in Ω

_{a}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

*ɛ*̿

_{r}(

**r**,

*ω*) in the cartesian basis is

*symmetric*inside Ω: the fields generated by the respective sources satisfy the following integral identity [36, 37]: where

**E**

_{1}(

**r**) and

**H**

_{1}(

**r**) are the electric and magnetic fields generated by

**J**

_{s,1}(

**r**) alone; and

**E**

_{2}(

**r**) and

**H**

_{2}(

**r**) are the counterparts generated by

**J**

_{s,2}(

**r**). Note that the symmetric form in Eq. (2) is not restricted to the cartesian basis. As long as Eq. (2) holds,

*ɛ*̿

_{r}(

**r**,

*ω*) is symmetric in any

*real orthogonal*local bases inside Ω. If

*ɛ*̿

_{r}(

**r**,

*ω*) is nonsymmetric, an additional volume integral over with an integrand proportional to

*S*belongs to a region Ω containing Ω

_{a}and therefore encloses Ω

_{a}. 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]: 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}. 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 Ω

_{a}.

*orthogonality relation*between modes and provides a way to

*distinguish*them as well as

*extract their magnitudes*from an arbitrary source distribution in Ω

_{a}(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

*ɛ*̿

_{r}(

**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

*ɛ*̿

_{r}(

**r**,

*ω*) can take place when the time-reversal symmetry is broken. Such examples include cavities with the magneto-optic effect [39

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]

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]

## 3. Formulation of reciprocal cavities

_{a}. 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

**E**(

**r**) and

**H**(

**r**) are the electric and magnetic fields;

*ɛ*

_{0}and

*μ*

_{0}are the vacuum permittivity and permeability, respectively; and

**J**

_{s}(

**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

*ɛ*̿

_{r}(

**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.

**H**(

**r**) in Eq. (5a) and (5b), we obtain the wave equation for

**E**(

**r**) We now consider the current sources

**J**

_{s}(

**r**) that are only present in Ω

_{a}, namely, 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 {

**j**

_{s,n}(

**r**,

*ω*)} of current sources, which is present only in Ω

_{a}and labeled by index

*n*, for the mode expansion of

**J**

_{s}(

**r**) in Eq. (7). Denote the set of electric fields generated by {

**j**

_{s,n}(

**r**,

*ω*)} as {

**f**

*(*

_{n}**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 Ω

_{a}but vanishes elsewhere, namely, a self-duplicated vector field in Ω

_{a}. Thus, we assign the following ansatz to

**j**

_{s,n}(

**r**,

*ω*): where

*U*(

**r**) is the indicator function for Ω

_{a}; and Δ

*ɛ*

_{r,n}(

*ω*) is a complex parameter. With Eq. (8a), the wave equation of

**f**

*(*

_{n}**r**,

*ω*) is transformed into a generalized eigenvalue problem: where (

*ω*/

*c*)

^{2}Δ

*ɛ*

_{r,n}(

*ω*) acts as the eigenvalue; and mode quantization indicated by index

*n*is justified from the source confinement in Ω

*, cavity structure, and outgoing-wave boundary condition. Note that*

_{a}**f**

*(*

_{n}**r**,

*ω*) is not divergent in the far-field zone because it is effectively generated by a source

**j**

_{s,n}(

**r**,

*ω*). Once

**f**

*(*

_{n}**r**,

*ω*) is obtained, the corresponding magnetic field

**g**

*(*

_{n}**r**,

*ω*) is derived from Faraday’s law in Eq. (5a):

**j**

_{s,n}(

**r**,

*ω*)}. This incompleteness is more evident for a homogeneous and isotropic active region characterized by a scalar permittivity

*ɛ*

_{r,a}(

*ω*). In this case, we take the divergence in Eq. (11) for

**r**∈ Ω

_{a}: Since Δ

*ɛ*

_{r,n}(

*ω*) ≠ −

*ɛ*

_{r,a}(

*ω*) [otherwise, the fact that ∇ × ∇ ×

**f**

*(*

_{n}**r**,

*ω*) ∝ ∇ ×

**g**

*(*

_{n}**r**,

*ω*) =

**0**in Ω

_{a}may lead to discontinuous tangential magnetic fields across

*S*

_{a}, and thus fictitious surface currents], the divergence of the field

**f**

*(*

_{n}**r**,

*ω*) vanishes in Ω

_{a}. Therefore, the corresponding current source

**j**

_{s,n}(

**r**,

*ω*) is also divergenceless (solenoidal) in Ω

*: From Eq. (13), any current sources confined in Ω*

_{a}_{a}that contribute to volume charge densities oscillating at

*ω*(nonzero divergence) cannot be fully expanded by the set {

**j**

_{s,n}(

**r**,

*ω*)} (however,

**j**

_{s,n}(

**r**,

*ω*) may result in the surface charge density on

*S*

_{a}). 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 Ω

_{a}, the situation becomes less transparent because the set {

**j**

_{s,n}(

**r**,

*ω*)} may not be divergenceless. However, from the lesson of homogeneous and isotropic active region, it is probable that {

**j**

_{s,n}(

**r**,

*ω*)} does not span all the source configurations. Therefore, we introduce an analogous set {

**i**

_{s,m}(

**r**,

*ω*)} of current sources, where

*m*is the mode index, to complement {

**j**

_{s,n}(

**r**,

*ω*)}. Similar to

**j**

_{s,n}(

**r**,

*ω*), we require

**i**

_{s,m}(

**r**,

*ω*) to be confined in Ω

_{a}only. The corresponding sets of electric and magnetic fields generated by {

**i**

_{s,m}(

**r**,

*ω*)} are denoted as {

**u**

*(*

_{m}**r**,

*ω*)} and {

**w**

*(*

_{m}**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.

**j**

_{s,n}(

**r**,

*ω*)} and {

**i**

_{s,m}(

**r**,

*ω*)}, we expand an arbitrary current source

**J**

_{s}(

**r**) confined in Ω

_{a}and oscillating at

*ω*as follows: where

*c*and

_{n}*d*are the expansion coefficients. From the superposition principle of linear systems, the corresponding electric field

_{m}**E**(

**r**) and magnetic field

**H**(

**r**) are expressed as Our next goal is the extractions of expansion coefficients

*c*and

_{n}*d*. This step is straightforward if various modes are

_{m}*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

**j**

_{s,n}(

**r**,

*ω*)} and {

**f**

*(*

_{n}**r**,

*ω*)} constructed in section 3.1, we make the following assignments: and substitute them into the Lorentz reciprocity theorem in Eq. (4): In Eq. (16), if Δ

*ɛ*

_{r,n′}(

*ω*) ≠ Δ

*ɛ*

_{r,n}(

*ω*), the volume integral of the dot product

**f**

_{n}_{′}(

**r**,

*ω*)

*·*

**f**

*(*

_{n}**r**,

*ω*) must vanish. On the other hand, if a degeneracy exists such that Δ

*ɛ*

_{r,n′}(

*ω*) = Δ

*ɛ*

_{r,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 {

**f**

*(*

_{n}**r**,

*ω*)} as follows: where

*δ*

_{n′n}is the Kronecker’s delta; and Λ

*(*

_{n}*ω*) is the complex normalization constant of

**f**

*(*

_{n}**r**,

*ω*). We also define an analogous orthogonality relation to Eq. (17) for the set {

**j**

_{s,n}(

**r**,

*ω*)}: where Θ

*(*

_{n}*ω*) is the normalization constant of

**j**

_{s,n}(

**r**,

*ω*).

**i**

_{s,m}(

**r**,

*ω*)}, we also demand a similar orthogonality relation based on the same inner-product rule: where Ξ

*(*

_{m}*ω*) is the normalization constant of

**i**

_{s,m}(

**r**,

*ω*). In addition, two current sources, one from {

**j**

_{s,n}(

**r**,

*ω*)} and the other from {

**i**

_{s,m}(

**r**,

*ω*)}, always have to be orthogonal to each other: The conditions in Eqs. (18a), (19), and (20) necessitate only one single orthogonality relation for any basis vector functions from {

**j**

_{s,n}(

**r**,

*ω*)} and {

**i**

_{s,m}(

**r**,

*ω*)}. This property, however, stems from a specifically-constructed set {

**i**

_{s,m}(

**r**,

*ω*)}, which is summarized in the appendix.

*c*and

_{n}*d*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.

_{m}### 3.3. Dyadic Green’s function

**J**

_{s}(

**r**) confined in Ω

_{a}, we need to find its mode expansion coefficients

*c*and

_{n}*d*in Eq. (14a) in order to reconstruct the electric field

_{m}**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

**j**

_{s,n′}(

**r**,

*ω*) and integrate it over Ω

_{a}. With the orthogonality relations in Eqs. (18a), (19), and (20), only the term corresponding to

*c*

_{n′}at the right-hand side of Eq. (14a) remains, and we obtain the expression of

*c*

_{n′}as follows: With the same procedure but adopting

**i**

_{s,m′}(

**r**,

*ω*) rather than

**j**

_{s,n′}(

**r**,

*ω*), we derive the analogous expression of

*d*

_{m′}: 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

**J**

_{s}(

**r**) through the dyadic Green function

*G*̿

_{ee}(

**r**,

**r**

^{′},

*ω*):

*G̿*

_{ee}(

**r**,

**r**′,

*ω*) on

**J**

_{s}(

**r**′) represent the dot products

**j**

_{s,n}(

**r**′,

*ω*) ·

**J**

_{s}(

**r**′) and

**i**

_{s,m}(

**r**′,

*ω*) ·

**J**

_{s}(

**r**′), respectively. On the other hand, not every current source can be substituted into Eq. (22a). The dyadic Green’s function

*G̿*

_{ee}(

**r**,

**r**′,

*ω*) in Eq. (22b) is not applicable to the sources which spread outside the active region Ω

_{a}.

*G̿*

_{ee}(

**r**,

**r**′,

*ω*) in Eq. (22b) to calculate the spontaneous emission coupling factor in section 4.2.

## 4. Characteristics of reciprocal cavities

**j**

_{s,n}(

**r**,

*ω*)}, electric fields {

**f**

*(*

_{n}**r**,

*ω*)}, and magnetic fields {

**g**

*(*

_{n}**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*.

**f**

*(*

_{l}**r**,

*ω*) of a nondegenerate mode

*l*at

*ω*, the formal way to associate it with its counterpart

**f**

*(*

_{l}**r**,

*ω*+ Δ

*ω*) at

*ω*+ Δ

*ω*, where Δ

*ω*is a small frequency difference, is to track how the field profile evolves from

*ω*to

*ω*+ Δ

*ω*and make a one-to-one link from {

**f**

*(*

_{n}**r**,

*ω*)} to {

**f**

*(*

_{n}**r**,

*ω*+ Δ

*ω*)}. An easier approach is the connection through eigenvalue Δ

*ɛ*

_{r,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

**f**

*(*

_{l}**r**,

*ω*) in the set {

**f**

_{l′}(

**r**,

*ω*)|Δ

*ɛ*

_{r,l′}(

*ω*) = Δ

*ɛ*

_{r,l}(

*ω*)} of degenerate modes at

*ω*(excluding the accidental degeneracy) is chosen, its counterpart

**f**

*(*

_{l}**r**,

*ω*+ Δ

*ω*) has to satisfy a condition similar to the orthogonality relation in Eq. (17): Unlike Eq. (17), however, Eq. (23) originates from the symmetry viewpoint even though two equations coincide with each other at Δ

*ω*= 0. Note that in degenerate cases, the continuity and smoothness of Δ

*ɛ*

_{r,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.

*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

*l*at each frequency, we look into the frequency dependence of the power generated by a current source proportional to

**j**

_{s,l}(

**r**,

*ω*). In this way, the forms of the current source

**J**

_{s}(

**r**) and electric field

**E**(

**r**) are 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 |

**J**

_{s}(

**r**)|

^{2}where 𝒥 is a real constant; and

*V*

_{a}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

**J**

_{s}(

**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

*P*(

_{l}*ω*) become

*ω*Δ

*ɛ*

_{r,l}(

*ω*)]

^{−1}}. To understand its behavior, we define a frequency

*ω*such that the absolute value |

_{l}*ω*Δ

_{l}*ɛ*

_{r,l}(

*ω*)| is the minimum. As shown in Fig. 3(a), if we plot the locus of

_{l}*η*(

*ω*) ≡

*ω*Δ

*ɛ*

_{r,l}(

*ω*) parameterized by

*ω*on the complex

*η*plane, the differential change

*δη*due to a small frequency variation

*δω*around

*ω*≈

*ω*, when viewed as a two-dimensional (2D) vector, has to be perpendicular to that of

_{l}*η*(

*ω*). If these two vectors were not perpendicular, |

_{l}*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 where Δ

*ω*is a parameter that must be

_{l}*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*(

*ω*) on the complex

_{l}*η*plane are perpendicular. From Eq. (27) and

*δη*≃

*η*′(

*ω*)

_{l}*δω*, where

*η*′(

*ω*) is the frequency derivative of

*η*(

*ω*), we can express the parameter Δ

*ω*as

_{l}*η*(

*ω*) near

*ω*with the linear expansion in

_{l}*ω*−

*ω*and substitute it into the power spectrum

_{l}*P*(

_{l}*ω*) in Eq. (26b):

*P*(

_{l}*ω*) has a Fano lineshape near

*ω*(weighted sum of the Lorentzian centered at

_{l}*ω*and its Hilbert transformation). As shown in Fig. 3(b), this line-shape is asymmetric with respect to

_{l}*ω*. In contrast to the Lorentzian, which has a maximum at

_{l}*ω*, the maximum of Fano lineshape is blueshifted (Re[

_{l}*ω*Δ

_{l}*ɛ*

_{r,l}(

*ω*)] > 0) or redshifted (Re[

_{l}*ω*Δ

_{l}*ɛ*

_{r,l}(

*ω*)] < 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.

_{l}*ω*Δ

_{l}*ɛ*

_{r,l}(

*ω*)]| ≪ |Im[

_{l}*ω*Δ

_{l}*ɛ*

_{r,l}(

*ω*)]| (valid for most cavities with high

_{l}*Q*factors), the physical interpretations of

*ω*and Δ

_{l}*ω*are more evident. In this case, the frequency

_{l}*ω*is nearly the peak frequency of

_{l}*P*(

_{l}*ω*). Therefore, we may view

*ω*as the resonance frequency of mode

_{l}*l*and generalize it to the cases of asymmetric lineshapes. The electric field

**f**

*(*

_{l}**r**,

*ω*) at

_{l}*ω*is then the field profile of mode

_{l}*l*. Correspondingly, we can identify the parameter Δ

*ω*as the full-width-at-half-maximum (FWHM) linewidth of the approximate Lorentzian. The (cold- or warm-cavity) quality factor

_{l}*Q*of mode

_{l}*l*is defined as the ratio between

*ω*and Δ

_{l}*ω*: Recall that Δ

_{l}*ɛ*

_{r,l}(

*ω*) is the permittivity variation required for the self-supporting mode

_{l}*l*at

*ω*. With a homogeneous and isotropic active region [

_{l}*ɛ̿*

_{r}(

**r**,

*ω*)|

_{r∈Ωa}=

*ɛ*

_{a}(

*ω*)

*I̿*] and the cold-cavity condition [inter-state dipole absorption excluded from

*ɛ*

_{a}(

*ω*)], the threshold gain

*g*

_{th,l}is

### 4.2. Spontaneous emission coupling factor and Purcell effect

*r*

_{sp}(

*ω*) per unit energy is expressed as

**E**(

**r**,

*ω*) is the electric field generated by

**J**

_{sp}(

**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

*G̿*

_{ee}(

**r**,

**r**,

*ω*) in Eq. (22b) into Eq. (34) and obtain

*r*

_{sp,n}(

*ω*) is the spontaneous emission rate per unit energy into mode

*n*in {

**f**

*(*

_{n}**r**,

*ω*)}; and

*r̃*

_{sp,m}(

*ω*) is the counterpart into mode

*m*in {

**u**

*(*

_{m}**r**,

*ω*)}.

*R*

_{sp}, spontaneous emission rate

*R*

_{sp,n}into mode

*n*in {

**f**

*(*

_{n}**r**,

*ω*)}, and the counterpart

*R̃*

_{sp,m}into mode

*m*in {

**u**

*(*

_{m}**r**,

*ω*)}, are the integrations of

*r*

_{sp}(

*ω*),

*r*

_{sp,n}(

*ω*), and

*r̃*

_{sp,m}(

*ω*) over photon energy

*h̄ω*, respectively: and the spontaneous emission coupling factor

*β*into mode

_{l}*l*in {

**f**

*(*

_{n}**r**,

*ω*)} is written as

*c*〉, empty ground state |

*v*〉, and transition frequency

*ω*) and located at position

_{cv}**r**

_{s}∈ Ω

_{a}, namely, the spontaneous emission rate

*R*

_{sp,l}into mode

*l*in Eq. (36b) becomes Equation (39) is indicative of the Purcell effect on a single dipole. The spatial enhancement comes from the modal strength

**f**

*(*

_{l}**r**

_{s},

*ω*) at position

_{cv}**r**

*, and spectral enhancement originates from the lineshape effect from 1/[*

_{s}*ω*Δ

*ɛ*

_{r,l}(

*ω*)] if

_{cv}*ω*is sufficiently close to

_{cv}*ω*. To compare it with the counterpart

_{l}*W*

_{sp,l}from Fermi’s Golden rule (Lorentzian density of states of mode

*l*): where

**Ê**

_{ph,l}(

**r**

*) is the single-photon field of mode*

_{s}*l*, we rewrite

*R*

_{sp,l}in Eq. (39) with a few simplifications. First, for

*ω*≈

_{cv}*ω*, the fastest-varying part with respect to

_{l}*ω*−

_{cv}*ω*is the factor 1/[

_{l}*ω*Δ

_{cv}*ɛ*

_{r,l}(

*ω*)]. Except for this factor, we replace

_{cv}*ω*with

_{cv}*ω*in other parts of

_{l}*R*

_{sp,l}. Second, if we set Λ

*(*

_{l}*ω*) to a positive real number and assume that the phase of

_{l}**f**

*(*

_{l}**r**,

*ω*) does not vary rapidly in Ω

_{l}_{a}(except for ±

*π*jumps near nodes or nodal surfaces),

**f**

*(*

_{l}**r**,

*ω*) is close to a real vector field in Ω

_{l}_{a}, namely,

**r**∈ Ω

_{a}. Third, we assume that the asymmetry of the lineshape is minor and only keep the symmetric Lorentzian. With these simplifications, the form of

*R*

_{sp,l}is reduced to that of

*W*

_{sp,l}in Eq. (40) with the following connection between the single-photon field

**Ê**

*(*

_{l}**r**

*) and mode profile*

_{s}**f**

*(*

_{l}**r**

_{s},

*ω*):

_{l}## 5. One-dimensional Fabry-Perot cavity

*L*

_{c}of the cavity is 5

*μ*m, and the counterpart

*L*

_{a}of the active region is 2

*μ*m. The cavity starts at

*z*= −

*L*

_{c}/2 and ends at

*z*=

*L*

_{c}/2, and is uniform in the

*x*–

*y*plane. The active region is evenly distributed in the central part of the cavity, and outside it are two passive regions with identical lengths

*L*

_{p}= 1.5

*μ*m. The relative permittivity of the whole cavity, including the active and passive regions, is

*ɛ*

_{c}(no frequency dispersion). We only consider modes which are uniform in the

*x*–

*y*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.

*z*= ±

*L*

_{a}/2 and

*z*= ±

*L*

_{c}/2, the permittivity variation Δ

*ɛ*

_{r,n}(

*ω*) of mode

*n*satisfies the transcendental equation: where + (−) is the sign for even (odd) modes,

*k*

_{a,n}is propagation constant of mode

*n*in the active region while

*k*

_{p}is that of the passive region; and

*r*

_{a,p}and

*r*

_{p,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 Δ

*ɛ*

_{r,n}(

*ω*) is obtained by self-consistently solving the transcendental equation in Eq. (42a) at different

*ω*’s. Setting the relative permittivity

*ɛ*

_{c}= 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.

*ω*, quality factor

_{n}*Q*, and the threshold gain

_{n}*g*

_{th,n}of mode

*n*can be estimated from the nature of a FP resonator and represented as [47–49

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]

*m*is the number of standing waves of mode

_{n}*n*in the cavity;

*v*

_{g}is the group velocity of the wave in the cavity and is identical to the phase velocity (

*= (*

_{z,n}*L*

_{a}/

*L*

_{c})[1 ± sinc(

*ω*

_{n}L_{a}/

*c*)] is the longitudinal confinement factor [+ (−) for even (odd) modes; and sinc(

*x*) = sin(

*x*)/

*x*]. The quantization of

*ω*in Eq. (43a) is due to an integral number of standing waves in the cavity region while

_{n}*Q*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

_{n}*g*

_{th,n}in Eq. (43c) is obtained from the round-trip balance condition, taking into account the effects of

*L*

_{a}≠

*L*

_{c}and the standing-wave pattern, as described by the expression of the longitudinal confinement factor Γ

*[48]. In Table 1, we show the theoretical estimations of*

_{z,n}*h̄ω*,

_{n}*Q*, and

_{n}*g*

_{th,n}and their FP counterparts from Eq. (43a) to (43c). The spectral parameters

*h̄ω*,

_{n}*Q*, and

_{n}*g*

_{th,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.

*n*= 3 in Table 1). In addition to the standing-wave pattern of typical FP modes, the effect of the permittivity variation Δ

*ɛ*

_{r,3}(

*ω*) can be observed in the active region (|

*z*| < 1

*μ*m). 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*) =

*h̄ω*Δ

*ɛ*

_{r,3}(

*ω*) on the complex

*h̄η*plane. The permittivity variation Δ

*ɛ*

_{r,3}(

*ω*

_{3}) 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

*h̄ω*

_{3}= 1.42 eV.

*ɛ*

_{c}= 3.0625) and show the lineshape of an even mode

*l*(solid black) in Fig. 6(a) (

*h̄ω*= 1.281 eV). For comparison, we also show the line-shape of an even mode

_{l}*l*′ (dashed red) with a close resonance photon energy when

*ɛ*

_{c}= 12.25 (

*h̄ω*

_{l}_{′}= 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*′ (

*Q*= 21.52 versus

_{l}*Q*

_{l}_{′}= 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 [Δ

*ɛ*

_{r,l}(

*ω*) = −0.0733 – 0.359

_{l}*i*versus Δ

*ɛ*

_{r,l′}(

*ω*

_{l′}) = −0.00190 – 0.311

*i*]. 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).

## 6. Additional remarks

**f**

*(*

_{n}**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 Δ

*ɛ*

_{r,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.

*ω*of mode

_{l}*l*is equivalent to the minimization of the goal function |

*ω*Δ

*ɛ*

_{r,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.

## 7. Conclusion

## Appendix: Construction of Complementary Source Set {i_{s,m}(*r*,*ω*)}

**i**

_{s,m}(

**r**,

*ω*)} is constructed to complement the set {

**j**

_{s,n}(

**r**,

*ω*)}. In a homogeneous and isotropic active region, we may set the source

**i**

_{s,m}(

**r**,

*ω*) curl-free [

**i**

_{s,m}(

**r**,

*ω*) ∝

*U*(

**r**)∇

*ϕ*(

**r**), where

*ϕ*(

**r**) is a scalar function] in Ω

_{a}to complement the divergenceless source

**j**

_{s,m}(

**r**,

*ω*). For general reciprocal active regions, we generalize this idea and write

**i**

_{s,m}(

**r**,

*ω*) as where Φ

*(*

_{m}**r**,

*ω*) is a scalar function; and

*P̿*

^{(c)}(

**r**,

**r**′,

*ω*) is the projection operator which eliminates the component spanned by {

**j**

_{s,n}(

*r*,

*ω*)} from the current source present in Ω

_{a}only: The ansatz in Eq. (44) makes {

**i**

_{s,m}(

*r*,

*ω*)} orthogonal to {

**j**

_{s,n}(

**r**,

*ω*)}, as required in Eq. (20).

**i**

_{s,m}(

**r**,

*ω*) to corresponding charge density

*ρ*(

_{m}**r**,

*ω*): To construct a generalized eigenvalue problem for Φ

*(*

_{m}**r**,

*ω*), we write

*ρ*(

_{m}**r**,

*ω*) as and substitute the expressions in Eqs. (44) and (47) into Eq. (46): where (

*ω*/

*c*)

^{2}Δ

*κ*

_{r,m}(

*ω*) is the eigenvalue to be obtained. With the extra requirement Φ

*(*

_{m}**r**,

*ω*) = 0,

**r**∈

*S*

_{a}, one can show that the orthogonality between the two current sources

**i**

_{s,m′}(

**r**,

*ω*) and

**i**

_{s,m}(

**r**,

*ω*) [see Eq. (19)] is indeed satisfied. Once

**i**

_{s,m}(

**r**,

*ω*) is obtained, the electric field

**u**

*(*

_{m}**r**,

*ω*) and magnetic field

**w**

*(*

_{m}**r**,

*ω*) are then calculated through the wave equation and Faraday’s law:

**j**

_{s,n}(

**r**,

*ω*)} is automatically orthogonal to

*U*(

**r**)∇Φ

*(*

_{m}**r**) [Φ

*(*

_{m}**r**,

*ω*) = 0 for

**r**∈

*S*

_{a}] 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 Ω

_{a}and the source

**i**

_{s,m}(

**r**,

*ω*) = −

*iωɛ*

_{0}

*U*(

**r**)∇Φ

*(*

_{m}**r**,

*ω*) becomes curl-free in Ω

_{a}.

## Acknowledgments

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**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(140.3945) Lasers and laser optics : Microcavities

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 1, 2011

Revised Manuscript: September 18, 2011

Manuscript Accepted: September 27, 2011

Published: October 10, 2011

**Citation**

Shu-Wei Chang, "Full frequency-domain approach to reciprocal microlasers and nanolasers–perspective from Lorentz reciprocity," Opt. Express **19**, 21116-21134 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21116

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### References

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