# On the applicability of Kirchhoff's law to the lasing regime

Aurelian Loirette-Pelous, Jean-Jacques Greffet

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Aurelian Loirette-Pelous, Jean-Jacques Greffet. On the applicability of Kirchhoff's law to the lasing regime. Optica, 2024, 11 (12), pp.1621-1627. 10.1364/optica.531864. hal-05288295

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# On the applicability of Kirchhoff's law to the lasing regime

AURELIAN LOIRETTE-PELOUS<fnref n="1" />,<fnref n="2" /> ID AND JEAN-JACQUES GREFFET<fnref n="1" />,∗ ID

Received 3 June 2024; revised 19 October 2024; accepted 20 October 2024; published 26 November 2024

In 1860, Kirchhoff showed that the radiance emitted by a hot body is the product of its absorptivity by a universal function of temperature and frequency, which he defined as blackbody radiance. With the advent of semiconductors, it has been shown that Kirchhoff's law can be extended to account for electroluminescence and photoluminescence in a cavity. Hence, the question of the applicability of Kirchhoff's law in the lasing regime arises naturally. In this work, we show that Kirchhoff's law accounts for the lasing transition. It also enables the modeling of important features of the lasing regime, such as the frequency, directivity, and polarization of both the first lasing mode and the non-lasing background. Finally, we show that it enables to recover the Schawlow-Townes spectral linewidth. In summary, Kirchhoff's law appears to be a versatile tool to model light emission from the thermal to the lasing regime.

Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

https://doi.org/10.1364/OPTICA.531864

## 1. INTRODUCTION

Since its derivation in 1860 [1], Kirchhoff's law has been the basic tool to model thermal radiation from astrophysics to microscale [2]. It expresses in an elegant and simple way the radiance emitted by an isothermal body as the product of its absorptivity and the blackbody radiance. Later on, several generalizations of this law have been established. An important step is its derivation in an electromagnetic framework [3,4], while the original derivation was based on geometrical optics. This extends its use to bodies at equilibrium that cannot be described using geometrical optics such as subwavelength particles or metasurfaces [3,5,6]. This generalization beyond geometrical optics has thus largely participated to the growth of the field of coherent thermal emission [7] as a design tool for new thermal sources [8–18]. A Kirchhoff's law accounting for pumping has also been derived by Würfel to describe the electroluminescence or photoluminescence by a semiconductor slab [19]. It is obtained by replacing $\omega$ by $\omega - \mu/\hbar$ in Planck's function where $\mu$ is the difference between quasi-Fermi levels induced by pumping. $\mu$ is called photon chemical potential [19]. This generalization has been used to design optimized light emitting diodes [20,21], in the field of photovoltaics [22], and recently to model the emission by pumped media embedded in resonators or metasurfaces [23–27]. Interestingly, while it is usually thought that Kirchhoff's law is restricted to the modeling of spontaneous emission, it can also be used in the gain regime [19]. Amplified spontaneous emission can thus be accounted for [19,28]. Amplified spontaneous emission in a planar cavity has been numerically investigated, predicting the onset of large and narrowed emission lines [29].

Taking everything into account, a question arises naturally: is it possible to use Kirchhoff's law in the lasing regime?

In this article, we show that the generalization of Ref. [6] called local Kirchhoff's law is able to model the light emitted both in the lasing and the non-lasing regime by arbitrary cavities containing a gain medium in local thermodynamic equilibrium. More precisely, we limit our discussion to solid gain media such that (i) quasi-Fermi levels and (ii) a permittivity can be defined. The first condition requires that the electron-electron interaction time is much shorter than the electron-hole recombination time [30]; the second condition requires that the polarization dephasing time is much shorter than the electron-hole recombination time [31]. This results in the adiabatic elimination of the polarization from laser equations so that rate equations are sufficient to model the gain. Our starting point is the basic definition of lasing, namely, the macroscopic occupation of a leaky cavity mode that occurs when its gain compensates for its losses. Such a condition does not appear explicitly in Kirchhoff's law. Here, we exhibit this condition analyzing carefully the absorption cross section. We then apply this formalism to a planar VCSEL. We show that the local Kirchhoff's law enables to predict quantitatively the emitted radiance as a function of direction and polarization for both the lasing and the non-lasing parts. Finally, we show that in the lasing regime, the Schawlow-Townes linewidth is contained in Kirchhoff's law. We stress that Kirchhoff's law only predicts the emitted power once the photon chemical potential is known. As such, it is not a laser theory. Additional rate equations are needed to obtain the photon chemical potential as a function of pumping. This is analogous to the use of Kirchhoff's law to predict the emitted radiance by hot

<footnote n="1">¹Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127, Palaiseau, France</footnote>

<footnote n="2">²aurelianjoan.loirettepelous@ehu.eus</footnote>

<footnote>∗jean-jacques.greffet@institutoptique.fr</footnote>
bodies once the temperature is known. In this latter case, additional budget equations are required to model the body temperature, given the thermal fluxes.

## 2. MODEL: LOCAL KIRCHHOFF’S LAW

Let us introduce the local Kirchhoff’s law derived in Ref. [6] for homogeneous bodies in local thermodynamic equilibrium. For the sake of simplicity, we consider in what follows a semiconductor with conduction and valence bands. In local thermodynamic equilibrium, the electronic distribution of the conduction (resp. valence) band is described by a Fermi-Dirac distribution characterized by a temperature $T_c$ (resp. $T_v$) and a chemical potential $\mu_c$ (resp. $\mu_v$) called quasi-Fermi level. We further assume that $T_c = T_v$ and refer to the temperature by $T$ in the following. The emitted power $dP$ by a body in the far field at a frequency $\omega$, in a direction $\mathbf{u}$ and in a polarization state $p$ can then be cast into the form

$$ \frac{dP}{d\Omega}(\omega, -\mathbf{u}, p, T, \mu) = \sigma_{\text{abs}}(\omega, -\mathbf{u}, p, T, \mu) \\ \times \frac{\omega^2 \frac{\hbar\omega}{8\pi^3 c^2} \exp(\frac{\hbar\omega-\mu}{k_B T}) - 1}{1}, \quad (1) $$

where $\sigma_{\text{abs}}$ is the absorption cross section of the body, $c$ is the speed of light in vacuum, $\hbar$ is the reduced Planck’s constant, and $k_B$ is the Boltzmann constant. The parameter $\mu = \mu_c - \mu_v$ is called photon chemical potential [19]. In practice, $\mu$ quantifies the pumping strength and can be adjusted by tuning the injection current or the pump beam intensity. Note that it is uniquely related to the excited electron number (see, e.g., Section 2.C. of Supplement 1). Let us also note that at zero pumping ($\mu = 0$), all bands are at thermodynamic equilibrium so that the usual Kirchhoff’s law is recovered. Finally, we note that for a large planar source with surface $\Sigma$, the absorption cross section can be related to the absorptivity $A$ by $\sigma_{\text{abs}} = A \cos(\theta) \Sigma$ [where $\theta$ is defined from the surface normal; see Fig. 2(a)], and the emitted power can be related to the spectral radiance $L_e$ by $dP = L_e \cos(\theta) \Sigma d\Omega$.

We also recall the general definition of the absorption cross section of an isotropic and homogeneous active body of volume $V_{\text{act}}$ [33]:

$$ \sigma_{\text{abs}}(\omega, \mathbf{u}, p, T, \mu) = \frac{\omega}{c} \text{Im}[\varepsilon(\omega, T, \mu)] \int_{V_{\text{act}}} \times \frac{|\mathcal{E}(\mathbf{r}', \omega, \mathbf{u}, p, T, \mu)|^2}{|\mathcal{E}_{\text{inc}}(\omega, \mathbf{u}, p)|^2} d^3\mathbf{r}', \quad (2) $$

where $\varepsilon$ is the permittivity of the active body and $\mathcal{E}_{\text{inc}}(\omega, \mathbf{u}, p)$ is the electric field corresponding to an incident plane wave with frequency $\omega$, direction $\mathbf{u}$, and polarization state $p$. The electric field induced by this plane wave within the active medium is written $\mathcal{E}(\mathbf{r}', \omega, \mathbf{u}, p, T, \mu)$, where $\mathbf{r}'$ is the position in the medium. Hence, the ratio $|\mathcal{E}/\mathcal{E}_{\text{inc}}|^2$ describes the local intensity enhancement in the body.

## 3. UNVEILING THE LASING REGIME IN LOCAL KIRCHHOFF’S LAW

Equation (2) shows that the absorption cross section is composed of two terms: (i) $\text{Im}[\varepsilon]$, which accounts for losses (or gain above

the transparency threshold), and (ii) the intensity enhancement integral over the active medium, which depends on the cavity resonance. Importantly, in local thermodynamic equilibrium, both quantities depend on $\mu$ (i.e., on pumping). Hence, they are both likely to play a role in a transition to lasing. We elucidate this dependence in the following paragraphs and show that Eq. (1) enables to account for the lasing regime of a pumped medium described by quasi-Fermi levels and embedded in an arbitrary resonant environment.

### A. Divergence of the Bose-Einstein Distribution

It appears from Eq. (1) that Kirchhoff’s law does not exhibit explicitly the laser condition. Moreover, a striking feature complicates the analysis. The right hand side of Eq. (1) contains the Bose-Einstein (BE) distribution $1/[\exp(\frac{\hbar\omega-\mu}{k_B T}) - 1]$. When the photon chemical potential reaches the energy of a resonator mode, this distribution diverges. This peculiar behavior is reminiscent of Bose-Einstein condensation, a phenomenon also due to stimulated emission. Thus, it was suggested in Ref. [6] that Kirchhoff’s law could be useful to model photon Bose-Einstein condensation [34–36]. However, this point of view neglects the dependence of the absorption on the photon chemical potential. This issue has been tackled in Refs. [19,29]. We summarize here their results. To this end, let us focus on the case of a semiconductor-based active medium in local thermodynamic equilibrium, for which explicit and accurate models of the dependence of the imaginary part of the permittivity on pumping are available [37]:

$$ \text{Im}[\varepsilon(\omega, \mu_c, \mu_v)] \propto |d_{cv}|^2 \rho_J(\omega) [f_v(\omega, \mu_c) - f_c(\omega, \mu_v)], \quad (3) $$

where $d_{cv}$ is the transition matrix element between the conduction and valence bands and $\rho_J$ is the joint density of states. The electronic distribution in the conduction/valence band is noted $f_{c/v}(\omega, \mu_{c/v}) = 1/[\exp(\frac{E_{c/v}(\omega)-\mu_{c/v}}{k_B T}) + 1]$, where $E_{c/v}(\omega)$ is the conduction/valence electron energy associated to a vertical transition with energy $\hbar\omega$ [37,38], and $\mu_{c/v}$ is the conduction/valence band quasi-Fermi-level. Further, note that in intrinsic semiconductors, $\mu_c$ and $\mu_v$ are not independent (see, e.g., Section 2.C. of Supplement 1). Hence, pumping is fully characterized by the single parameter $\mu = \mu_c - \mu_v$, which we will use in the following. In Eq. (3), the difference $f_v - f_c$ results from the difference $f_v[1 - f_c] - f_c[1 - f_v]$, where $f_v[1 - f_c]$ describes the probability of an electron excitation process due to photon absorption, while $f_c[1 - f_v]$ describes the probability of stimulated emission. This term thus represents a net absorption probability.

Now, we analyze the net absorption probability $f_v - f_c$ in Eq. (3) together with the BE distribution in Eq. (1). In local thermodynamic equilibrium, simple algebra leads to

$$ \frac{f_v(\omega, \mu) - f_c(\omega, \mu)}{\exp(\frac{\hbar\omega-\mu}{k_B T}) - 1} = f_c(\omega, \mu)[1 - f_v(\omega, \mu)]. \quad (4) $$

The right hand side can be viewed as the probability of emission in a mode whereas the left hand side is the product of the absorption by Bose-Einstein distribution. This equation is actually a generalization to local equilibrium of the so-called Van Roosbroeck-Shockley relation [39], which connects the emission and absorption rates in a homogeneous medium.

Inserting Eq. (4) in the combination of Eqs. (1)–(3), the Bose-Einstein distribution in Kirchhoff’s law is replaced by
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Fabry-Perot cavity. The internal reflection factor of the mirrors is noted by $r_{22}$, and the length of the cavity by $L$. The cavity is homogeneously filled with a semiconductor-based active medium in local thermodynamic equilibrium (medium 2, orange-shaded area), with a non-dispersive refractive index $n_2(\omega) = n_R + in_I(\omega)$. The refractive index outside the cavity (medium 1) is $n_1 = 1$.</figcaption></figure>

$$ \frac{dP}{d\Omega} \propto f_c(\omega, \mu)[1 - f_v(\omega, \mu)]. \quad (5) $$

This equation shows that the emitted power does not diverge at $\omega = \mu$. Actually, Eq. (4) shows that the BE distribution diverges exactly at the active medium transparency frequency where the net absorption goes to zero, so that the divergence is regularized. Moreover, while above the transparency frequency the BE distribution becomes negative, Eq. (4) shows that the power emitted remains positive since the net absorption also becomes negative (i.e., absorption becomes gain). Hence, Eq. (5) shows that there is no inconsistency with using Kirchhoff's law in the gain regime [19,29].

## B. Lasing Regime of a Fabry-Perot Cavity

We now continue our search for lasing in local Kirchhoff's law focusing on the intensity enhancement integral in the absorption cross section [Eq. (2)]. The intensity is expected to be enhanced in the presence of a cavity. For the sake of simplicity, we start with a toy model, a planar Fabry-Perot cavity with gain described in Fig. 1. In such an infinite planar device, the emitted power in Eq. (1) must be replaced by the spectral radiance, and the absorption cross section by the absorptivity $A$. The latter can be conveniently expressed as $A = 1 - R - T$, where $T = |\mathcal{E}_{\text{trans}}|^2/|\mathcal{E}_{\text{inc}}|^2$ is the cavity transmittance and $R = |\mathcal{E}_{\text{ref}}|^2/|\mathcal{E}_{\text{inc}}|^2$ the cavity reflectance. The calculation of $T$ and $R$ is given in Section 1 of Supplement 1. The absorptivity can be cast in the form

$$ A = \frac{X(\omega)}{|1 - r_{22}^2 e^{2ikL}|^2}, \quad (6) $$

where $k = k_R + ik_I = (n_R + in_I)\omega/c$, $r_{22} = |r_{22}| \exp(i\phi)$ is the internal reflection factor in the cavity, which we assume independent of $\omega$ for the sake of simplicity, and the numerator $X(\omega)$ is a non-resonant real function of $\omega$. The denominator vanishes if $r_{22}^2 e^{2ikL} = 1$. This entails a phase condition that defines the cavity modes $2\phi + 2n_R\omega_I L/c = l \times 2\pi$, where $l$ is an integer, and an amplitude condition, $|r_{22}|e^{-k_I L} = 1$. Introducing the well-known short-hands that are the cavity photon round-trip time $\tau_{RT} = 2n_R L/c$, the mirror loss rate $\kappa = -(2c/n_R L) \ln(|r_{22}|)$, and the net gain rate $g(\omega, \mu) = -2\omega n_I(\omega, \mu)/n_R$, Eq. (6) can be cast into the alternative form

$$ A(\omega, \mu) = \frac{X(\omega)}{|1 - e^{i\tau_{RT}[\omega - \omega_I]} e^{-\tau_{RT}[\kappa - g(\omega, \mu)]/2}|^2}. \quad (7) $$

Close to the resonance frequency of the $l$-st mode, i.e., $[\omega - \omega_I]\tau_{RT} \ll 1$, and assuming a good quality factor, i.e., $[\kappa - g(\omega, \mu)]\tau_{RT} \ll 1$, Eq. (7) can be expanded to first order into

$$ A(\omega, \mu) \approx \frac{X(\omega_I)}{\tau_{RT}^2} \frac{1}{[\omega - \omega_I]^2 + \left[ \frac{\kappa - g(\mu)}{2} \right]^2}, \quad (8) $$

where $g_I(\mu) = g(\omega_I, \mu)$ is the net gain rate of mode $l$. This form shows that the absorptivity, and thus the emitted power, resonate when two conditions are simultaneously fulfilled: (i) $\omega$ coincides with the mode frequency $\omega_I$, and (ii) the gain of the mode is equal to its radiative losses, i.e., $g_I(\mu) = \kappa$. These conditions are the lasing conditions.

## C. Generalization to Arbitrary Resonators

In the previous section, we have provided insight on the use of local Kirchhoff's law in the lasing regime for a Fabry-Perot cavity homogeneously filled with a semiconductor gain medium in local thermodynamic equilibrium. Here, we generalize this result to the lasing regime of any device featuring a gain medium in local thermodynamic equilibrium in an arbitrary resonator. To this end, we will assume that the gain is homogeneous within the active region and that the lasing mode does not overlap spectrally with other modes.

We start by deriving an alternative expression for the absorption cross section Eq. (2). Let us note that the electric field $\mathcal{E}$ inside the emitting body can be expanded over the quasi-normal modes (QNMs) of the open resonator [40]:

$$ \mathcal{E}(\mathbf{r}', \omega, \mathbf{u}, p) = \sum_l \alpha_l(\omega, \mathbf{u}, p) \hat{\mathcal{E}}_l(\mathbf{r}'), \quad (9) $$

where $\hat{\mathcal{E}}_l$ is the electric field of the normalized $l$-est QNM, and where the amplitudes $\alpha_l(\omega, \mathbf{u}, p)$ are given by [40]

$$ \alpha_l(\omega, \mathbf{u}, p) = \frac{\omega}{\tilde{\omega}_l - \omega} \int_{V_{\text{res}}} [\varepsilon(\mathbf{r}') - 1] \mathcal{E}_{\text{inc}}(\omega, \mathbf{u}, p) \times \exp\left(i \frac{\omega}{c} \mathbf{u} \cdot \mathbf{r}'\right) \cdot \hat{\mathcal{E}}_l(\mathbf{r}') d^3\mathbf{r}'. \quad (10) $$

Note that the integral is taken over the volume of the resonator, which includes both the active medium and lossy media. This form shows explicitly the resonance of mode $l$ as $\omega$ approaches the QNM frequency $\tilde{\omega}_l = \omega_I + i\Gamma_I/2$, where $\omega_I$ and $\Gamma_I$ are both real valued [40]. Inserting this expansion in the absorption cross section Eq. (2), we find that the contribution of the resonant mode $l$ can be cast in the form

$$ \sigma_{\text{abs}}(\omega, \mathbf{u}, p) \approx \frac{\omega}{c} \text{Im}[\varepsilon(\omega)] \frac{Y(\omega)}{[\omega - \omega_I]^2 + [\Gamma_I/2]^2}, \quad (11) $$

where $Y(\omega)$ is given in terms of the overlap integral appearing in Eq. (10).

Equation (11) generalizes Eq. (8) obtained in the previous section to any resonator with well-separated modes. When the active medium is in local thermodynamic equilibrium, Eq. (11) can be inserted in Kirchhoff's law [Eq. (1)] to calculate the emitted power. The mode decay rate also exhibits the dependence $\Gamma_I(\mu)$ on the quasi-Fermi levels splitting of the active medium. Indeed, $\Gamma$ is a net decay rate that includes the gain supplied to a mode through its
overlap with the active medium. The mode decay rate can thus be split without loss of generality as $\Gamma_l(\mu) = \kappa_l^{\text{rad}} + \kappa_l^{n-\text{rad}} - g_l(\mu)$, where $\kappa_l^{\text{rad}}$ (resp. $\kappa_l^{n-\text{rad}}$) is the total radiative (resp. non-radiative) loss rate and $g_l(\mu)$ the total gain rate of mode $l$. Again, it appears that the power emitted increases resonantly when two conditions are simultaneously fulfilled: (i) the emission is resonant, i.e., $\omega = \omega_l$, and (ii) the gain of the resonant mode is equal to its total losses, i.e., $\Gamma_l = 0 \Leftrightarrow g_l(\mu) = \kappa_l^{\text{rad}} + \kappa_l^{n-\text{rad}}$. This proves that the lasing regime is included in local Kirchhoff's law in the general case of any device featuring a gain medium in local thermodynamic equilibrium in a resonator.

# 4. APPLICATION: MODELING A VCSEL USING LOCAL KIRCHHOFF'S LAW

In this section, we outline the advantages of using the local Kirchhoff's law to model emission by a laser. As a test case, we consider a large area VCSEL.

## A. Model

A VCSEL is schematically depicted in Fig. 2(a). In this type of device, two Bragg mirrors ensure a high quality factor Fabry-Perot cavity. Placing a few quantum wells closely packed at a field antinode (here at the center of the cavity) then supplies sufficient gain to start lasing. In addition, it is shown in Fig. 2(b) that the cavity modes dispersion (top panel, dashed cyan line) of this device is non-trivial. This is due to the presence of transverse modes that span a large emission angle range on a small energy domain. Small material properties or geometrical modification can then change the lasing mode and dramatically shift the emission angle in the lasing regime [41]. Since a prominent strength of Kirchhoff's law is its ability to predict the spectrum and the directivity of the thermal [8–18] and photoluminescent [26,27] emission, this type of device represents a good testbed for these properties in the lasing regime.

To run our simulations, we resort to two main hypotheses. (i) We assume that the VCSEL is infinite in directions perpendicular to the growth axis, which is a good approximation for a large area

device. Hence, the emitted power in Eq. (1) must be replaced by the spectral radiance $L_e$, and the absorption cross section by the absorptivity $A$ of the cavity. (ii) We assume that the quantum wells are sufficiently close to each other so that they operate in the same conditions, that is, the same value of $\mu$ can be used for all quantum wells. In addition, other model characteristics and the procedure to run the simulations are described in Section 2.C. of Supplement 1. We further use parameter values representative of the large area VCSEL studied in Ref. [32], which are given in Section 2 of Supplement 1.

## B. Emission Spectrum, Directivity, Polarization, and Minimization of the Lasing Condition

We show in Figs. 2(b)–2(d) the simulated spectral radiance $L_e(\hbar\omega, \theta, p = \text{TE})$ in TE polarization as a function of frequency and emission angle, for several values of $\mu$. The bottom plots show the corresponding spectral exitance, which is the spectral radiance integrated over all emission angles. First, we observe in Fig. 2(b) (top) that the cavity modes dispersion measured experimentally in Ref. [32] (dashed cyan line) is correctly reproduced in the simulation. Importantly, Kirchhoff's law further provides the power emitted through any of these cavity modes. For $\mu$ from zero up to 1.275 eV, a similar broad and slightly peaked at $\theta = 0$ emission is seen in Figs. 2(b) and 2(c) lower panels. This is the LED regime of the device, characterized by the exponential behavior $\exp(-\hbar\omega/k_B T)$ [39]. It can also be checked that the spatial existence scales as $\propto \exp(\mu/k_B T)$ compared to its equilibrium value. This is indeed expected for a LED, for which $\mu \approx eV$, where $V$ is the voltage applied to a single quantum well. In Fig. 2(c), the position of $\mu$ is indicated by a dashed white line. This shows the transparency of the active medium. Hence, the active medium is in population inversion at energies lower than $\mu$, so that the modes lying below the line experience gain. Still, here it is seen that gain is not large enough to reach the lasing regime. This threshold is passed in Fig. 2(d), where the value of $\mu$ is raised further so that a sharp peak becomes visible on the spectral exitance. The lasing mode is the lowest energy cavity mode that emits in the direction

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Simulation of the light emitted by a VCSEL using the local Kirchhoff's law. (a) Design of the VCSEL. TE/TM stand for transverse electric/magnetic polarization. (b)–(d) Simulations of the spectral radiance $L_e$ (top plots) in TE polarization against energy $\hbar\omega$ and emission angle $\theta$, for $\mu = 0$ eV in (b), $\mu = 1.275$ eV in (c), and $\mu = 1.290193$ eV in (d). The bottom plots show the corresponding spectral exitance, which is the spectral radiance integrated over all emission angles. The dashed cyan line in (b) reproduces the parabolic fit made in Ref. [32] to the experimental cavity mode dispersion. Fit parameters are: $\hbar\omega = \hbar\omega_{\theta=0} + (\hbar\omega_{\theta=0})^2 \sin(\theta)^2 / 2m_{\text{ph}}c^2$ with $\hbar\omega_{\theta=0} = 1.271$ eV and $m_{\text{ph}} = 2.2881 \times 10^{-35}$ kg [32]. The dashed white line in (c) stands for $\hbar\omega = \mu$. Linear color scale is used in (b) and (c), while logarithmic color scale is used in (d). Also note that a smaller energy and angle window is used in (d) compared to (b) and (c). All plots are normalized to maximum. The procedure and the parameters used for the simulation are detailed in Section 2 of Supplement 1.</figcaption></figure>
normal to the growth axis ($\theta = 0$). Interestingly, this result is consistent with the experimental observations of Ref. [32], from which the parameters used in the simulations have been taken.

Let us now use these results to highlight the opportunities offered by the local Kirchhoff's law. First, it is seen that Kirchhoff's law enables to model light emission continuously from the thermal to the LED and even the lasing regime, by simply increasing the photon chemical potential. Second, increasing gradually $\mu$ enables to identify the mode that is going to lase first. Here, this is better seen on the spatial exitance, but we stress that Kirchhoff's law provides the full energy/emission angle map for a given polarization [top plots in Figs. 2(b)–2(d)]. The frequency and direction in which lasing takes place is thus known without resorting to extra calculations. Hence, Kirchhoff's law provides a tool to design a laser with tailored frequency, directionality, and polarization of the emission.

The local Kirchhoff's law also gives access to the population inversion required for lasing. Indeed, increasing gradually $\mu$ in Eq. (1) enables to find the value for which the gain = loss condition is reached, which we note as $\mu_{g=l}$. For example, Fig. 2(d) shows that $\mu_{g=l} \gtrsim 1.290193$ eV for our VCSEL. Interestingly, given some targeted spectral, directional, and/or polarization properties, Kirchhoff's law enables to find a design minimizing the value of $\mu_{g=l}$. Hence, Kirchhoff's law also provides a tool to optimize the efficiency of a device to be designed.

Finally, we note that the transition to lasing in Figs. 2(b)–2(d) goes along with a drastic narrowing of the spectral linewidth of the modes. This behavior is indeed expected for a laser. This raises the question whether the local Kirchhoff's law enables a quantitative prediction of the first-order time coherence of the emitted light. We examine this issue more thoroughly in the following section.

### C. Spectral Linewidth in the Lasing Regime

The full width at half maximum (FWHM) of a laser line is expected to follow the so-called Schawlow-Townes linewidth [42,43] (see Section 3.A. of Supplement 1 for a discussion of this equation):

$$ \Delta\omega_{\text{ST}} = \frac{\hbar\omega_{l}\kappa^2}{P_l}, \qquad (12) $$

where $\omega_l$ is the frequency of the lasing mode $l$, $\kappa$ is the mirrors loss rate, which is the FWHM of the empty cavity spectrum, and $P_l$ is the power emitted through the mode $l$. Alternatively, the Schawlow-Townes linewidth can be cast in the form $\Delta\omega_{\text{ST}} = \kappa/N_l$, noting that the power emitted through the mode $l$ is related to the number $N_l$ of photons in the mode by $P_l = \hbar\omega_l\kappa N_l$.

In Fig. 3, red dots show the evolution of the spectral linewidth of the lasing mode (corresponding to $\theta = 0^\circ$) against the total power emitted in this mode. Both quantities are calculated for the VCSEL model using local Kirchhoff's law. While the spectral linewidth is nearly constant at small emitted power, we recall that large emitted power goes along with a drastic spectral narrowing. In particular, it is seen that in the large emitted power regime, the linewidth calculated with local Kirchhoff's law is in perfect agreement with the Schawlow-Townes linewidth (dashed black line). Hence, it seems that the local Kirchhoff's law quantitatively predicts the linewidth of the light emitted, even in the lasing regime.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Spectral linewidth (FWHM) against total emitted power for the $\theta = 0^\circ$ mode of the VCSEL analyzed in Fig. 2. Red dots show the linewidth extracted from a simulation using local Kirchhoff's law. The dashed black line shows the linewidth expected in the Schawlow-Townes theory Eq. (12), with a slight modification discussed in Section 3.A. of Supplement 1. The procedure to calculate $\Delta\hbar\omega_{\theta=0}$, $P_{\theta=0}$ as well as to extract $\kappa$ is exposed in Section 3.B. of Supplement 1. $P_{\theta=0}$ is normalized to its value at the lowest emitted power in the figure.</figcaption></figure>

We now provide a general proof of this statement. We start from Eq. (11), which states that the FWHM of the lasing mode $l$ as given by Kirchhoff's law reads $\Delta\omega_K = \kappa - g_l(\mu)$. We aim at connecting the quantities involved in this expression to the mode photon number $N_l$. To that end, let us write the steady-state balance between photon gain (through spontaneous and stimulated emission) and losses (through radiative + non-radiative losses and absorption) in the lasing mode $l$:

$$ R_l^{\text{em}}(1 + N_l) = (\kappa + R_l^{\text{abs}})N_l, \qquad (13) $$

where $R_l^{\text{abs}}$ is the absorption rate of one photon in the mode $l$ by the active medium. By definition, $R_l^{\text{abs}}$ and $R_l^{\text{em}}$ are related to the net gain rate $g_l(\mu)$ of the mode by $g_l(\mu) = R_l^{\text{em}} - R_l^{\text{abs}}$. Remarkably, using this definition and rearranging Eq. (13), a relation between the mode net losses and the mode photon number is obtained: $\kappa - g_l(\mu) = R_l^{\text{em}}/N_l$. This relation eventually enables to cast Kirchhoff's law linewidth in the form

$$ \Delta\omega_K = \frac{R_l^{\text{em}}}{N_l}. \qquad (14) $$

Further, Eq. (13) entails $R_l^{\text{em}} = (\kappa + R_l^{\text{abs}})N_l/(1 + N_l) \approx \kappa$ in the limit of large $N_l$ and low reabsorption. In addition, we assume negligible non-radiative losses, so that $\kappa$ is the empty cavity linewidth. Inserting these approximations into Eq. (14) eventually enables to show that Kirchhoff's law linewidth is equal to the Schawlow-Townes linewidth:

$$ \Delta\omega_K = \Delta\omega_{\text{ST}} = \frac{\kappa}{N_l}. \qquad (15) $$

Finally, we note that the spectral narrowing is due to the replacement of $\kappa$ by $\kappa - g$. In other words, it is due to the effective reduction of the cavity decay rate $\kappa$ by the introduction of the gain $g$. This semiclassical mechanism is the same that was considered by Schawlow and Townes [42–44]. In addition, we note that Eq. (15) gives the Schawlow-Townes linewidth in its original and most widely known form. Still, the derivation of Schawlow and Townes relies on the assumption of low reabsorption with respect to losses, which is not always justified. Hence, comparisons with the Schawlow-Townes linewidth in the form of Eq. (14) should be preferred, as done in Fig. 3.
# 5. CONCLUDING REMARKS

In the previous part, we evidenced several advantages of using the local Kirchhoff's law to model a laser, focusing on a large area VCSEL with semiconductor quantum wells as active media. However, the method can be applied to a large variety of light emitting devices beyond this example. First, let us stress that absorption can be computed with Maxwell solvers for any type of optical resonator that may combine gain, lossy regions, and field enhancement regions. This includes metasurfaces, nanoscale resonators such as micropillars or plasmonic nanostructures. Second, the technique can also be applied to a large variety of gain media. The derivation given in Section 3 is based on a pumped medium whose band populations are characterized by quasi-Fermi levels. Hence, Kirchhoff's law can deal with most 3D- [45], 2D- [22], or 1D- [46] semiconductor media, with possible generalization to account for additional bands. The formalism can also be applied to organic molecules provided that the so-called Kennard-Stepanov relation is valid [47–49]. This relation is sometimes also called the Neporent-McCumber relation (see Ref. [50] and references therein) and plays a role equivalent to the pivotal Van Roosbroeck-Shockley relation Eq. (4) valid for semiconductors. A recent study showed that Kirchhoff's law can be used to model colloidal quantum dots assemblies [26]. Third, we note that the local Kirchhoff's law presented in this article assumes spatially and temporally homogeneous $\mu$ and $T$. Actually, it has been proven that the local Kirchhoff's law can be also defined locally in time and space [6].

We now return to the photon Bose-Einstein condensation issue raised in Section 3.A. While the divergence of the $1/[\exp(h\nu/\hbar\mu) - 1]$ term is not related to condensation, we have shown that Eq. (1) accounts for lasing. In fact, it was shown in Ref. [51] that photon Bose-Einstein condensation is a particular case of lasing, when $R_l^{abs} \gg \kappa_l$ for all the modes (i.e., including non-lasing ones) in systems described by quasi-Fermi levels. We stress that Kirchhoff's law can be used in this regime. In particular, Kirchhoff's law can account both for lasing and non-lasing modes, as seen in Fig. 2. Hence, it is a valuable tool to study thermalization in these devices.

Finally, let us stress that Kirchhoff's law enables to compute the emitted power knowing $T$ and $\mu$. Connecting $\mu$ to the pumping conditions requires an additional modeling of the populations that depend on the system studied. For instance, coupled rate equations connecting the populations of excited carriers and photons need to be solved in class-B lasers [52,53]. Nevertheless, as we have discussed in Section 4.B, Kirchhoff's law enables to fully model and design the emission properties of a laser using $\mu$ and $T$ as free parameters.

In conclusion, we have shown that the local Kirchhoff's law Eq. (1) accounts for the lasing regime of a gain medium in local thermodynamic equilibrium in an arbitrary resonator. We have shown that it enables modeling with a high level of accuracy important features of the lasing regime, such as the frequency, directivity, polarization, and spectral linewidth of the first lasing mode. It also enables efficiency optimization. These results provide a new theoretical tool to design and analyze a large class of lasers. In particular, we believe that it will be of interest to model lasers based on unconventional cavities, such as spasers [54–60], or metasurface-based lasers [61–69]. It can also be used to model the photon Bose-Einstein condensation regime, enabling to compute both the condensed and uncondensed phases [34–36,51,70,71].

**Funding.** Agence Nationale de la Recherche (ANR-17-CE24-0046).

**Acknowledgment.** We are grateful to Stéphane Barland and Gian Luca Lippi for helpful discussions, and to Jean-Paul Hugonin for assistance with RETICOLO software [72].

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

**Supplemental document.** See Supplement 1 for supporting content.

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