## Abstract

We propose to use a combination of intersubband transitions in semiconductor quantum wells with a two dimensional photonic crystal cavity to obtain narrow, strong thermal radiation spectra. Single peak thermal radiation is obtained due to the Lorentzian shape absorption spectrum of the intersubband transition and the single mode cavity embedded within the photonic band gap. We present an analysis based on the quantum Langevin theory. It is shown that local radiance of the narrow emission peak can be maximized to ~80% of the radiation from the blackbody devices when the photon dissipation rates of the cavity mode due to the intersubband absorption and that due to the radiation to the free space modes are equal. Guidelines for concrete device design are introduced, and an example device structure is shown.

©2009 Optical Society of America

## 1. Introduction

Thermal radiation devices such as incandescent lamps have wide ranges of emission spectra which contain large amount of unusable spectral components. A variety of research has been carried out to develop thermal radiation devices that emit only the desired components in order to improve the energy efficiency. So far, most of this research has taken an approach based on the blackbody radiation model [1,2] which indicates that the thermal radiation is proportional to the photonic density of states in the limit of long interaction times. Wavelength size structures such as open cavities [3–8] and photonic crystals [9–11] have been used to modify the photonic density of states, thus modulating the thermal radiation spectra. Periodic microgrooves on metal or semiconductor surfaces [6,12–14] have also been utilized to couple out surface modes, which have large photonic density of states at specific frequency ranges, to free space modes to obtain similar effects. Alternatively, it is possible to take an approach based on a radiative transfer model which states that the radiation from a small volume element of a material is proportional to the blackbody radiation intensity ${I}_{b}$ multiplied by the absorption coefficient of the material *α*. Accordingly, the radiation intensity from the material having a characteristic (or interaction) length *L* is obtained from ${I}_{b}\left(1-\mathrm{exp}(-\alpha L)\right)$ when reflections at the boundaries can be ignored. This model indicates that the control of the wavelength dependence of the absorption coefficient is also effective for controlling the thermal radiation spectra [15,16] when the size of the structure (or interaction length) is comparable to or smaller than the inverse of the absorption coefficient.

We propose a new approach to establish control over the thermal radiation by modulating both the absorption coefficient of a material and the photonic states of a structure. This is a two-step approach: Firstly the radiation spectral ranges are roughly limited by controlling the absorption coefficient spectra of the material composing the device. Secondly more precise optimizations are carried out by controlling the photonic states of the device structure. Since the radiation range of the material is limited by the control of the absorption coefficient spectra by the first method, the wavelength range to be controlled by the second is narrow enough that precise control of the radiation can be expected by modulating the photonic states of the structure. In this paper, we discuss concrete methods to control the absorption coefficient spectra and photonic states. Based on the discussion, a device that utilizes intersubband transitions [17–19] and a two-dimensional photonic crystal slab cavity [20–22] is proposed. The proposed device is theoretically investigated based on the quantum Langevin theory [23] to obtain a strategy for device optimization and to study the expected performances.

## 2. Proposal of thermal radiation devices

For the control of absorption coefficient spectra, various methods and materials are available, for example: interband transition in semiconductors (no absorption below the bandgap), intersubband transition (ISB-Ts) in quantum wells (QWs) [17,18] or quantum dots (QDs) [19], optically active phonons [15], rare earth materials [16] (isolated absorption peaks), etc. We chose to investigate ISB-Ts in QWs because the transition wavelength and absorption magnitude can be easily tuned by design and material selection of the QWs. Control of the transition wavelength is important for obtaining thermal radiation at the desired wavelength. Control of the absorption magnitude is also important since the wavelength dependence vanishes when the inverse of the absorption coefficient is smaller than the interaction length. For control of the photonic states, we chose photonic crystals (PCs) [24,25] since they permit precise control. For example, point defects artificially introduced in PCs will act as photonic cavities [20–22,26,27] with properties that can be tuned by design. Alternatively, the photonic band edges can be used as a cavity with a large density of states [28]. In particular, the two-dimensional (2D) PC slab [20–22] is very promising for the following reasons: First, 2D-PC slabs are very compatible with QWs for ISB-Ts since they can be easily fabricated by making periodic arrays of air holes in QW layers and then removing the material underneath. Second, extensive research on point defect cavities in 2D-PC slab has been reported and their optical characteristics such as numbers of modes, modal volumes, *Q* factors, and radiation patterns can be controlled by their design.

Based on these points, we propose to utilize the combination of ISB-Ts in QWs with a 2D-PC slab for the control of the thermal radiation, where two potential device structures are schematically shown in Fig. 1 (a) and (b) . For the case where point defect cavities (Fig. 1 (a)) are used, it should be taken into account that the dipole moments of the ISB-Ts are perpendicular to the QW plane, so the PC should be designed to have photonic band gaps in the transverse magnetic (TM) mode. For example, triangular lattice PCs with triangular air holes [29] and triangular lattice PCs with three stacked layers (air-hole arrays / dielectric-rod arrays / air-hole arrays) [30] can support TM PBGs. When the bandedge modes are used as a cavity (Fig. 1 (b)), the PBG is not always necessary so that there is more freedom in the PC design. The former structure is expected to enable more complete control over the thermal radiation, however fabrication will be more difficult since a complicated structure is required to obtain TM PBG. On the other hand, the latter is easier to fabricate since no PBG is required while the control of the radiation properties is restricted by the multimode properties of the bandedge modes.

In this work, we will concentrate on the single mode, point defect cavity case since more complete control of the thermal radiation is expected. The multi mode case can be discussed as an extension of the single mode, however, such discussion is out of the scope of this work and will be discussed in a separate paper.

## 3. Quantum mechanical model and analysis

Figure 2
shows the model to analyze the proposed system. We approximated the ISB-Ts in the QWs as *N* two-level electron systems (TLSs). We also approximate a point defect cavity in 2D-PC slab as single mode cavity and various phonon modes of the material forming the QW as local Bosonic thermal baths. In reality, the electrons in the subbands have freedom to propagate in-plane direction. However, the energy separation between the upper and the lower subbands are almost same for all the in-plane k-vectors, and hence, each set of upper and lower sublevels defined by an in-plane k-vector and a spin can be approximated as TLS when an electron is included. Those sets that include two electrons or no electron are optically ignored, where the former case is almost ruled out for usual QW structures having a large ISB energy compared to the thermal energy. Therefore, *N* is almost equal to the numbers of electrons in the QW within the cavity. In this model the three forms of interaction are taken into account as follows: 1. Each TLS interacts with the local Bosonic thermal bath. 2. The TLSs also interact with a single mode cavity. 3. The single mode cavity also interacts with free space photon modes. Here, we ignored the interaction between the single mode cavity and in-plane motion of electrons (or free-carrier absorption) because interaction strength is much weaker compared to the case 2. Although, the free-carrier absorption could produce spectrally broad and weak background, the intensity is considered to be much weaker than the emission of the present interest.

In this framework, the model’s Hamiltonian is written as

*i*-th TLS, ${\widehat{a}}_{cav}^{+}$(${\widehat{a}}_{cav}^{}$) is the creation (annihilation) operator of the cavity mode, ${\widehat{b}}_{i\lambda}^{+}$(${\widehat{b}}_{i\lambda}^{}$) is the creation (annihilation) operator of the local phonon modes around

*i*-th TLS, ${\widehat{d}}_{\mu}^{+}$(${\widehat{d}}_{\mu}^{}$) is the creation (annihilation) operator of the free space photon modes, and ${\omega}_{2i}$, ${\omega}_{1i}$, ${\omega}_{cav}$, ${\omega}_{\lambda}$, ${\omega}_{\mu}$are the corresponding frequencies, respectively. The local phonon modes and free space photon modes are reservoirs in this model with their temperatures are fixed by the external conditions. ${g}_{cav,21i}$ is the coupling constant between

*i*-th TLS and the cavity mode, which differs depending on the position of TLSs relative to the cavity field. ${g}_{\mu ,cav}$ is the coupling constant between the free space modes and the cavity mode, and ${g}_{21i,i\lambda}$ is the coupling constant between

*i*-th TLS and the local thermal bath. The equations of motion for the operators ${\widehat{\sigma}}_{i}^{}$, ${\widehat{a}}_{cav}^{}$are obtained from the Heisenberg equation, and by eliminating the reservoir operators using the Markov approximation and moving to the interaction picture, the following Heisenberg-Langevin equations are obtained:

*i*-th TLS, respectively. ${\gamma}_{Si}^{}=\pi DO{S}_{thermal\_bath\_i}({\omega}_{21}){\overline{\left|{g}_{21i,i\lambda ({\omega}_{\lambda}={\omega}_{21i})}\right|}}^{2}$is the relaxation rate of the polarization due to the interaction with the local thermal bath, and ${\gamma}_{cav}^{}=\pi DO{S}_{free\_space}({\omega}_{cav}){\overline{\left|{g}_{\mu ({\omega}_{\mu}={\omega}_{cav}),cav}\right|}}^{2}$ is the relaxation rate of the cavity field due to the interaction with the free space photon modes, where $DO{S}_{thermal\_bath\_i}$ and $DO{S}_{free\_space}$ are the density of states of the local thermal bath and the free space photon modes, respectively. ${\overline{n}}_{TBi}^{}=\u3008{b}_{i\lambda ({\omega}_{\lambda}={\omega}_{21i})}^{+}{b}_{i\lambda ({\omega}_{\lambda}={\omega}_{21i})}^{}\u3009={\left\{\mathrm{exp}(\hslash {\omega}_{21i}/k{T}_{TBi}^{})-1\right\}}^{-1}$ is the expectation value of the population of the local thermal bath with a temperature of ${T}_{TBi}$ and an energy of $\hslash {\omega}_{21}$, where

*k*is the Boltzmann constant. The noise operators are expressed by${\widehat{F}}_{}$, which obey the usual Langevin correlations.

Before solving these equations, the parameters are simplified using the following boundary conditions that take into account the aim and the structure of the device under analysis. First, we use uniform QW materials, so the transition energies and damping constants are the same for all the TLS (${\omega}_{21i}={\omega}_{21}$and ${\gamma}_{Si}^{}={\gamma}_{S}^{}$ for all *i*). We also use the resonant condition (${\omega}_{cav}={\omega}_{21}$) since maximum emission is expected. We consider the situation where the broad areas of the QW materials were heated by current injection or some other method. Therefore, the temperature of the QW materials is assumed to be uniform (${T}_{TBi}^{}={T}_{TB}^{}$ and ${\overline{n}}_{TBi}^{}={\overline{n}}_{TB}^{}$ for all *i*). Simultaneously, we use the condition ${\overline{n}}_{TB}>>{\overline{n}}_{FS}$, which means the QW materials are heated up to a sufficiently high temperature to generate enough energy flow from the device to free space. Finally, the cavity linewidth is assumed to be much narrower than $2\hslash {\gamma}_{S}$ since we aim to control the bare radiation from the material by the cavity. In addition to the above boundary conditions, we have to take into account the fact that ${\gamma}_{S}$is much larger than $\left|{g}_{cav,21i}\right|$ so the weak coupling condition is satisfied [31]. We solve the equations under the above conditions to obtain the steady state solution since the intention is to use the device for continuous emission. Formal integration of (2a) yields

Substituting (5) into (2d) yields the field equation of motion:

We can ignore the fluctuation in the second term of (6) because of the weak coupling condition ($\left|{g}_{cav,21i}\right|/{\gamma}_{S}<<1$), and hence we replace the population operator by the steady state value of the reservoir average, namely

To evaluate (7), we substitute (5) into (2b) and use the steady state condition and the relation ship${N}_{{}_{1i}}^{}+{N}_{{}_{2i}}^{}=1$, which yields

The last term is obtained by considering the weak coupling condition and the fact that $\u3008{\widehat{A}}_{cav}^{+}(t){\widehat{A}}_{cav}(t)\u3009$ is always smaller than ${\overline{n}}_{TB}$ at steady state. This equation means the population differences can be considered to be determined by the temperature of the thermal bath and to be the same for all *i*. To express (6) more simply, we re-define the coefficient of the second term of (6) that represents the absorption ratio by TLSs as

Here we used approximation (7) and introduced the average coupling constant

Also, the third term of (6) is re-defined as a truncated noise term:

Here, using (4a), (8)and (9), the correlation for (11) is obtained as

Substituting (9) and (11) into (6) yields

Formally integrating (13) and using steady state condition and transforming to the Heisenberg picture yields,

Using (14) and the conjugate, and substituting (12), the spectrum of the cavity mode photons in the steady state condition $U(\omega )$can be obtained as

The boundary condition that ${\gamma}_{S}$is much larger than the cavity linewidth corresponds to ${\gamma}_{cav}+{\gamma}_{abs}<<{\gamma}_{S}$. Applying this condition to (15) yields

The output power spectrum $P(\omega )$ is obtained by multiplying cavity photon escape rate $2{\gamma}_{cav}$ and the photon energy $\hslash \omega $to (16), namely

Equation (17) shows that the spectral peak power of the radiation is determined by the ratio between${\gamma}_{cav}$ and ${\gamma}_{abs}$. Figure 3 shows the spectral peak power as a function of ${\gamma}_{cav}/{\gamma}_{abs}$, which is maximized when the condition

is satisfied, resulting in a maximum peak spectral power ${P}_{\mathrm{max}}$ ofIt is seen in Fig. 3 that the tolerance for the maximum peak power condition is relatively large; more than 75% of the maximum power is obtained when $1/3<{\gamma}_{cav}/{\gamma}_{abs}<3$. In addition, (16) indicates that the linewith *Γ* is determined by sum of the relaxation rate to the free space and the absorption rate by the TLSs, namely

From (18), (19) and (20) we determine that the linewidth can be changed while maintaining the same maximal peak power.

Now we compare the spectral power given by (17) to the spectral radiance of the black body radiation ${I}_{BB}(\omega )$at the same temperature as the thermal bath, which is given by

For direct comparison, the units of (17) should be converted from spectral power to spectral radiance which is the power per unit frequency per unit area per unit solid angle. For this purpose we assume that the surface area of the cavity is a circle with a radius of *r* and the radiation pattern obeys the propagation of a Gaussian beam having the same radius *r* at the focal point. Although it is possible to calculate the radiation pattern and surface area of the given photonic crystal cavity by using numerical simulation methods, here, we choose to analyze using the Gaussian model to obtain the overall picture. (A numerical example for a specific structure will be shown later.) With this assumption, the radiation half angle $\Delta \theta $is expressed by

*λ*is the wavelength of the radiated light, and the corresponding solid angle of the radiation is given as

By dividing (17) by a factor of 2 (which represents the fact that the light is radiated from both upper and lower surfaces of the 2D-PC slab cavity), the surface area of the cavity ${S}_{cav}=\pi {r}^{2}$ and the solid angle (23), the spectral radiance *I* is found to be

By dividing (24) by (21), we obtain the relative spectral radiance (emissivity) $\epsilon (\omega )$of the proposed device:

Figure 4 shows the calculated emissivity for various ${\gamma}_{cav}$ and ${\gamma}_{abs}$, in which ${\left\{\mathrm{exp}(\hslash \omega /k{T}_{TB})-1\right\}}^{-1}$ is approximated to be ${\overline{n}}_{TB}$ for simplicity. It is seen that the maximum emissivity, $\pi /4~0.8$ is obtained for various linewidths provided that (18) is satisfied.

## 4. Discussion

It is indicated from (25) that the peak spectral radiance of the proposed device can be as much as 80% of the black body radiation for the optimal case. Furthermore, the linewidth can be controlled according to (20) where (18) must be maintained to keep the maximum intensity. In addition, (24) indicates that the surface area of the cavity is independent from the spectral radiance. Therefore, the surface area can be chosen according to the required radiation angle using (22). However, it is usually difficult to keep the single mode condition for larger cavities, so there should be a limitation for the cavity size in using (24) and (25). Otherwise the influence of the multimode condition must be taken into account. For small single mode cavities, practical devices will be an array of cavities in order to obtain a sufficiently large radiation power. Therefore, the area that is needed to define a cavity, including the surrounding photonic crystal, should be taken into account, when calculating the averaged radiance of the device. Devices that radiate from the whole photonic crystal area cavity such as band edge modes will be more efficient from the view point of the averaged radiance, but the multimode characteristics of these large area cavities limit the controllability over thermal emission properties, such as linewidths, radiation angles, and etc.

Now, we discuss more concrete design conditions for the QWs and the single mode cavity. From (9), (18) and (20), the conditions required to obtain the target linewidth ${\Gamma}_{\text{target}}$and maximal spectral radiance simultaneously are rewritten in terms of the design parameters as

Therefore, from the first and second terms of (26) the required *Q* factor of the cavity is determined as

This is the primary condition required for the control of the photonic states. Another required condition is given by the first and third terms of (26), where the three parameters$\overline{g}$, *N*, ${\overline{n}}_{TB}$, and ${\gamma}_{S}$ can be used for the design. While ${\gamma}_{S}$ is difficult to control since it is determined by the interaction between TLSs and the environment, *N*can be controlled by the design of the QWs, ${\overline{n}}_{TB}$ can be controlled by the operating temperature, and $\overline{g}$ can be controlled by the design of QWs and the cavity. The detailed expression of the individual coupling constant ${g}_{cav,21i}$ in (10) is written as

*i*-th TLS and $\overrightarrow{M}$ is the dipole moment of TLS which can be assumed to be same for all TLSs. Substituting (28) into (10) yields,

Here, we introduce the spatial distribution function of TLSs $\rho (\overrightarrow{x})$, of which the spatial integral is normalized to be 1. By using this function, (29) is rewritten as

We introduce the coefficient *Λ* that represents the overlap between the cavity electric field and distribution of TLSs as

This overlap coefficient *Λ* ranges from 0 to 1, where 1 indicates that all the TLSs are located at the position where the cavity electric field is largest and the dipole moments of all the TLSs are aligned to the direction of the cavity electric field. By substituting (31) into (30), a simple form of the average coupling constant becomes

Substituting (32) into (26) yields the detailed design condition as

Here, *N* corresponds to the number of electrons in the QWs within the cavity which is determined by the doping density and the volume of the QWs within the cavity. For the conversion of parameters, we rewrite the cavity modal volume as ${V}_{cav}={S}_{cav}\times {t}_{cav}$where ${S}_{cav}$ is the effective surface area of the cavity mode as defined above, and ${t}_{cav}$ is the effective thickness of the cavity mode. Furthermore, the total thickness of the QWs is defined as ${t}_{QW}$and the effective doping density within the QWs is defined as ${n}_{dope}$, respectively. Using these definitions, *N*is rewritten as

*Θ*is the area filling factor of the QW plane which is defined as the ratio of the area of QW to ${S}_{cav}$, with the removal of QWs in the air hole locations of PC taken into account. Substituting (34) into (33) yields

From these discussions, one of the most feasible design guidelines can be expressed as follows:

- (A) Determine the target wavelength, linewidth, peak spectral radiance (or power), and radiation angle.
- (B) Determine the operation temperature according to the target peak spectral radiance using (24) and (18) or the target peak maximum spectral power using (19). ${\overline{n}}_{TB}$ is determined accordingly.
- (C) Determine the
*Q*factor required from the target line width using (27). Design the cavity that has ${\omega}_{cav}$ corresponding to the target wavelength and the determined*Q*factor. The radiation angle should also be taken into account. In this step, ${S}_{cav}$, ${t}_{cav}$, ${\overrightarrow{E}}_{cav}(\overrightarrow{x})$, ${\epsilon}_{r\mathrm{max}}$ and*Θ*are determined according to the cavity design. - (D) Design a QW that has ${\omega}_{21}$ corresponding to the target wavelength. In this step, $\left|\overrightarrow{M}\right|$ is determined according to the design of the QW.
- (E) Determine the positions of the QWs within the slab. In this step,
*Λ*is determined by the positions of QWs and ${\overrightarrow{E}}_{cav}(\overrightarrow{x})$ from (31). - (F) Determine ${t}_{\text{QW}}$ and ${n}_{dope}$using (35) and other parameters which were already determined in steps (B) to (E).

Here, we show an example of the design of the thermal radiation device, where the target wavelength, linewidth, and peak spectral radiance are 11μm, 0.11μm, and 80% of the black body radiation at 600 K (or ~80 W/m^{2}/str/μm), respectively. A target for the radiation angle is not set in this example. The operating temperature is determined to be 600K according to step (B). Following step (C), the target *Q* factor of the cavity is determined to be 200. We have designed, using three dimensional finite difference time domain method, a defect cavity shown in Fig. 5
, which can confine the light of TM-like mode [30]. This cavity has a *Q* factor of 200, and the lowest cavity mode has a normalized resonant frequency of 0.359 $c/a$, where *a* is the lattice constant of PC. From the target wavelength, *a* is determined to be 4.0 μm, and accordingly ${S}_{cav}$ and ${t}_{cav}$ are roughly estimated to be about 50 μm^{2} and 5 μm, respectively. Figure 6
shows the cross-sectional view of this cavity mode. The ISB-Ts can couple to this mode efficiently due to the *E*
_{z} component in the center of the slab (Fig. 6 (b)), while radiation in the direction normal to the slab becomes possible due to the *E*
_{y} component near the surfaces (Fig. 6 (a)). The radiation solid angle $\Delta \Omega $ has been roughly estimated to be ~0.75π. *Θ*is estimated to be ~0.5 from the cavity mode distribution.

Following step (D), the QW structure is designed (according to the method shown in Ref. 32) to be GaAs (29 mono-layers) / Al_{0.3}Ga_{0.7}As (60 mono-layers), where the transition wavelength and the dipole moment are 11 μm and 2.1 e × nm, respectively (e is the unit charge). Usually, ${\gamma}_{S}^{}$ is 10~20 ps^{−1} in such n-doped QW structures [33,34] so that we can roughly assume $2\hslash {\gamma}_{S}^{}$~10 meV. When we decide to locate this QW at the middle layer of the PC slab, *Λ* is estimated to be ~0.5 from the cavity mode distribution (step (E)). Finally, ${t}_{\text{QW}}$ and ${n}_{dope}$ are determined according to step (F), and the obtained result is ${n}_{dope}\times {t}_{QW}$ = 8.2 × 10^{16} m^{−2}. When we set the numbers of QWs to be 50 where ${t}_{QW}$ becomes ~410 nm, ${n}_{dope}$ is determined to be 2.0 × 10^{17} cm^{−3}. These parameters are summarized in Table 1
and are consistent with the current fabrication capabilities. The spectral radiance of the designed device at 600 K is calculated from (24), and the result is shown in Fig. 7
together with that of the blackbody radiation at the same temperature. It is seen from the figure that the peak spectral radiance of the device is about 80% of that of the blackbody and the radiation outside the target range is well suppressed.

The obtained spectral radiance of 80% is a local spectral radiance at the surface of the cavity and is useful for microscopic applications such as thermal emitters for FTIR microscopes. However, for macroscopic applications, practical devices will be arrays of cavities as mentioned before, and in this case, the macroscopic spectral radiance averaged over the whole device area would be a more appropriate figure of merit. According to our calculations, the cavities can be packed to a spacing of 4*a* in the densest case while still avoiding a mutual coupling effect. Therefore, the maximum fraction occupied by the radiation area among the whole photonic crystal area is 22.5%, and the maximum macroscopic radiance is evaluated to be about 18%. In order to increase the macroscopic spectral radiance, we must increase the cavity filling factor while avoiding mutual coupling or design a large area cavity that can avoid multimode effects.

## 5. Conclusion

We have proposed a new approach for controlling thermal radiation in which the modulations of both photonic states and electronic states are utilized. A concrete device structure based on an intersubband transition and a two dimensional photonic crystal slab cavity has been proposed. The performance of the proposed device has been analyzed based on the quantum mechanical model. It has been shown that the local spectral radiance can be increased to ~80% of the black body radiation and that an arbitrary linewidth can be obtained by appropriately setting the *Q* factor of the cavity and the absorption by the QWs according to the obtained equations. The possibility of controlling the radiation angle has also been shown. Following the obtained design strategy, an example device was successfully designed to emit 11 μm light with a linewith of 0.11 μm and with a local spectral radiance of 80 W/m^{2}/str/μm at 600K, which corresponds to the 80% of the black body radiation. As for a macroscopic spectral radiance, the value decreases to 18% due to the relatively low cavity filling factor in the structure employed here. We believe that these results will accelerate the development of highly efficient thermal radiation devices.

## Acknowledgements

We acknowledge the financial support from Research Programs (Grant-in-Aid, COE, and Special Coordination Fund) for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and also from the Core Research for Evolutional Science and Technology of the Japan Science and Technology Agency.

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**31. **$\left|{g}_{cav,21i}\right|={\left[{\omega}_{cav}/2\hslash {V}_{cav}{\epsilon}_{0}{\epsilon}_{r\mathrm{max}}\right]}^{1/2}\left|\overrightarrow{M}\right|$for the best position (see Eq. (28)). If we substitute ${\epsilon}_{r\mathrm{max}}$ = 3.4, ${V}_{cav}={\left({\lambda}_{cav}/2\right)}^{3}$,${\omega}_{cav}=2\pi c/{\lambda}_{cav}$ and use ${\lambda}_{cav}$ = 10μm and $\left|\overrightarrow{M}\right|$ = 21eÅ ^{32}, $\left|{g}_{cav,21i}\right|$is evaluated to be ~5ns^{−1}. In contrast, ${\gamma}_{S}$is reported to be of the order of 10~20 ps^{−1} even at 300 K and becomes larger for higher temperatures [33, 34]. Thus ${\gamma}_{S}>>\left|{g}_{cav,21i}\right|$ holds true for the devices under analysis.

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