]Equal contribution as first author.

]Equal contribution as first author.

]Equal contribution as first author.

# Generalization of Kirchhoff’s Law of Thermal Radiation:
The Inherent Relations Between Quantum Efficiency and Emissivity

M. Kurtulik  [  Russell Berrie Nanotechnology Institute, Technion – Israel
Institute of Technology. Haifa 3200003, Israel  M. Shimanovic  [  Department
of Mechanical Engineering, Technion – Israel Institute of Technology. Haifa
3200003, Israel  T. Bar Lev  [  Department of Mechanical Engineering, Technion
– Israel Institute of Technology. Haifa 3200003, Israel  R. Weill  Department
of Mechanical Engineering, Technion – Israel Institute of Technology. Haifa
3200003, Israel  A. Manor  Russell Berrie Nanotechnology Institute, Technion –
Israel Institute of Technology. Haifa 3200003, Israel  M. Shustov  Department
of Mechanical Engineering, Technion – Israel Institute of Technology. Haifa
3200003, Israel  C. Rotschild<EMAIL_ADDRESS>Russell Berrie
Nanotechnology Institute, Technion – Israel Institute of Technology. Haifa
3200003, Israel  Department of Mechanical Engineering, Technion – Israel
Institute of Technology. Haifa 3200003, Israel

###### Abstract

Planck’s law of thermal radiation depends on the temperature, $T$, and the
emissivity, $\epsilon$, of a body, where emissivity is the coupling of heat to
radiation that depends on both phonon-electron nonradiative interactions and
electron-photon radiative interactions. Another property of a body is
absorptivity, $\alpha$, which only depends on the electron-photon radiative
interactions. At thermodynamic equilibrium, nonradiative interactions are
balanced, resulting in Kirchhoff’s law of thermal radiation that equals these
two properties, i.e., $\epsilon=\alpha$. For non-equilibrium, quantum
efficiency ($QE$) describes the statistics of photon emission, which like
emissivity depends on both radiative and nonradiative interactions. Past
generalized Planck’s equation extends Kirchhoff’s law out of equilibrium by
scaling the emissivity with the pump-dependent chemical-potential $\mu$,
obscuring the relations between the body properties. Here we theoretically and
experimentally demonstrate a prime equation relating these properties in the
form of $\epsilon=\alpha(1-QE)$, which is in agreement with a recent universal
modal radiation law for all thermal emitters. At equilibrium, these relations
are reduced to Kirchhoff’s law. Our work was done on photoluminescence. Future
expansion to electro-luminescence and other quantum emission processes, will
uncover the shared fundamental principles of radiation.

††preprint: APS/123-QED

##  I Introduction

Planck thermal radiation is the emission of a body at non-zero temperature. It
is characterized by the temperature of the body $T$, and its ability to
thermally emit radiation, given by emissivity $\epsilon$, which is a material
and cavity property reflecting the contribution of both radiative
$\gamma_{r}$, and non-radiative $\gamma_{nr}$ rates. While $\gamma_{nr}$
couples heat to excite electrons, $\gamma_{r}$ couples the excited electrons
to photon emission. In the reverse process, these rates respectively govern
non-radiative recombination and photon absorption, denoted by $\alpha$.
Although absorptivity depends only on the radiative rates, while emissivity
depends on both radiative and non-radiative rates, at thermodynamic
equilibrium, thermal excitation is balanced by the non-radiative
recombination, leading to the equality between emissivity and absorptivity:

$\epsilon=\alpha$ (1)

known as Kirchhoff’s law of thermal radiation. Out of equilibrium, i.e., a
body emits radiation into an environment having a different temperature or
chemical potential, Kirchhoff’s law does not apply, since non-radiative
processes are not canceled out (thermal excitation $\neq$ non-radiative
recombination). This phenomenon can be seen in semiconductors, light-emitting
diodes, atoms and photoluminescence [1]. An extreme example of non-equilibrium
is when absorptivity becomes negative at high excitation rates while the
emissivity remains positive [2, 3, 4].

Photoluminescence (PL), as an example of non-equilibrium emission, involves
the absorption of photons followed by fast thermalization of the excited
electrons and emission of (typically low-energy, red-shifted) photons [5, 6,
7, 8, 9, 10, 11]. Since thermalization results in the energy difference
between the incoming and outgoing photon energy, it is defined as out of
equilibrium. Nevertheless, thermodynamics statistically analyzes PL at quasi-
equilibrium [12, 13, 14, 15], using the conventional thermodynamic variables
such as temperature, emissivity, and chemical potential $\mu$ [16, 17], where
the latter describes the excitation above thermal. The generalized Planck’s
law describes such non-equilibrium emission [17, 18] as:

$L\left(h\nu,T,\mu\right)=\epsilon(h\nu)\cdot\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu-\mu}{k_{B}T}}-1}\approx
L_{Th}e^{\frac{\mu}{k_{B}T}}$ (2)

where L is the spectral radiance (having units of Watts per frequency, per
solid angle, and per unit area); $\epsilon$ is the emissivity, which is a
material and cavity property that depends on the density of states (DoS) of
the radiative and non-radiative transitions; $T$ is the temperature; $h\nu$ is
the photon energy; $k_{B}$ is Boltzmann’s constant; and $L_{Th}$ is the
thermal radiance when the pump is off. The chemical potential $\mu$ is the
Gibbs free energy per emitted photon and is also the gap that is opened in
semiconductors between the quasi-Fermi levels under excitation. Eq. (2) is
relevant at a specific frequency band, where the chemical potential is a
constant. Without knowing the inherent relation of $\mu(T)$, however, Eq. (2)
can only be fitted to an observed PL, without predicting its evolution.
Previous studies on non-equilibrium emission by external pump excitation
assume the temperature of the surroundings to be the same as the PL body [1,
2, 3, 4]. In such cases, Eq. (2) can be seen as a generalization of
Kirchhoff’s law, demonstrating that the line shape of the non-thermal emission
is the same as the thermal emission enhanced by the (scalar) chemical
potential, $\epsilon(h\nu)e^{\frac{\mu}{k_{B}T}}$ [13].

Under pump excitation, at low temperature, Eq. (2) also defines the PL quantum
efficiency, as the ratio between the emitted and absorbed photon rates:

$QE=\frac{\text{\\# of emitted photons}}{\text{\\# of absorbed
photons}}=\frac{L(T\rightarrow 0,\mu)}{\alpha\cdot L_{pump}(T_{p})}$ (3)

where $L(T\rightarrow 0,\mu)$ is the rate of photons emitted by a pumped body
at $T\rightarrow 0K$, and $L_{pump}(T_{p})$ is the incoming photon rate into
the system with a brightness temperature $T_{p}$, which defines the radiance
of the pump, at a specific wavelength, as equal to black-body radiance at the
temperature $T_{p}$. $QE$ is a material and geometric property of the body
defined at low temperatures where thermal excitation is negligible and
similarly to emissivity, it reflects the competition between radiative and
nonradiative rates, where higher $\gamma_{r}$ and lower $\gamma_{nr}$ support
higher $QE$.

In the formalism of Eq. (2), in contrast to Planck’s formula, the emissivity
becomes pump-dependent, losing its original meaning as a material and cavity
property, and unable to provide insight into the temperature-dependent
evolution of the radiation. This state of affairs leaves many open questions.
For example, what is the inherent relation between emissivity and absorptivity
for a material that thermally emits $(T>0)$ towards an environment at a
different temperature? Is there an inherent relation between emissivity and
quantum efficiency ($QE$), since both depend on the same parameters? In
similarity to Kirchhoff’s law, what is the inherent evolution of non-
equilibrium radiation with temperature? The aim of this paper is to answer
these questions by extending Kirchhoff’s law to non-equilibrium and to define
the inherent relations between fundamental properties of radiation.

We begin with the rate equation for a 2-level system and demonstrate how
spontaneous and stimulated phonon rates lead to the prime equation. Next, we
interpret the solution and show how it aligns with previous studies on
photoluminescence at elevated temperatures. We continue with a numerical
solution for a 3-level system that generalizes the results obtained for the
2-level system. Finally, we complete the study with an experimental
demonstration that directly measures emissivity, absorptivity, and quantum
efficiency as predicted by the prime equation. We also explain why prior
experimental studies did not reveal the inherent relations between these
properties.

##  II methods

###  II.1 2-Level System Analysis

Observing of the rate equations of a 2-level system reveals the inherent
summation of quantum and thermal radiations. Following Siegman [21], a
photoluminescent body placed in an open cavity is described by the following
equations:

$\displaystyle\frac{dn_{2}}{dt}=$
$\displaystyle(n_{1}-n_{2})B_{r12}n_{ph12}-n_{2}\gamma_{r12}+$
$\displaystyle(n_{1}-n_{2})B_{nr12}n_{pn12}-n_{2}\gamma_{nr12}$ $\displaystyle
4\pi\cdot\Delta\nu\cdot\frac{dn_{ph12}}{dt}=$ $\displaystyle
n_{pump}\Gamma_{p}-n_{ph12}\Gamma_{12}-$
$\displaystyle(n_{1}-n_{2})B_{r12}n_{ph12}+n_{2}\gamma_{r12}$

where $n_{1},n_{2}$ are the electron population densities of the ground and
excited states ($N=n_{1}+n_{2}$); $\gamma_{r},\gamma_{nr}$ are the
corresponding radiative and non-radiative spontaneous rates
($\gamma_{r12},\gamma_{nr12}$);
$B_{r12}=\gamma_{r}/DoS_{ph},B_{nr12}=\gamma_{nr}/DoS_{pn}$ are the Einstein
coefficients for the stimulated absorption or emission rates for both
radiative and non-radiative processes [12]; $DoS_{ph},DoS_{pn}$ are the
density of states of the photons and phonons, respectively;
$\Gamma_{p},\Gamma_{out}$ are the coupling rates in (via pump) and out of the
cavity, respectively; $n_{ph},n_{pn},p_{pump}$ are the field densities for
photons, phonons and incoming photons induced by the pump, respectively. These
densities obey $n_{pump}=DoS_{ph}\cdot f(T_{p})$, $n_{ph}=DoS_{ph}\cdot f(T)$
and $n_{pn}=DoS_{pn}\cdot f(T)$, where $f(T),f(T_{p})$ are the distribution
functions obeying the Bose-Einstein statistics for black-bodies, according to
the temperature, $T$, and brightness temperature of the pump, $T_{p}$.
Conventional solutions for the rate equations assume thermal phononic
distribution. Our solution is general and considers both spontaneous and
stimulated phonon rates. At steady-state, Eq. (4) is solved for $n_{2}$ and
substituting it into Eq. (4) determines the rate of photons leaving the
cavity:

$n_{ph12}\Gamma_{out}=\Gamma_{out}\frac{n_{pump}\Gamma_{p}+NB_{nr12}n_{pn12}\frac{\gamma_{r}}{\gamma_{r}+\gamma_{nr}}}{\Gamma_{out}+NB_{r12}\left(1-\frac{\gamma_{r}}{\gamma_{r}+\gamma_{nr}}\right)}$
(5)

We distinguish between two mechanisms of emission:

  1. 1.

Quantum emission, which is pump dependent and accounts only for scatterings
and PL:

$L_{quantum}=\Gamma_{out}\frac{n_{pump}\Gamma_{p}}{\Gamma_{out}+NB_{r12}\left(1-\frac{\gamma_{r}}{\gamma_{r}+\gamma_{nr}}\right)}$
(6a)

  2. 2.

Thermal emission, which is temperature dependent and radiates according to
Planck’s law of thermal radiation:

$L_{thermal}=\Gamma_{out}\frac{NB_{nr12}n_{pn12}\frac{\gamma_{r}}{\gamma_{r}+\gamma_{nr}}}{\Gamma_{out}+NB_{r12}\left(1-\frac{\gamma_{r}}{\gamma_{r}+\gamma_{nr}}\right)}$
(6b)

In our model, we assume the fundamental material parameters — $\gamma_{r}$,
$\gamma_{nr}$, $DoS_{ph}$, and $DoS_{pn}$, to be temperature-independent.
Consequently, $\epsilon$, $QE$, $B_{r12}$, and $B_{nr12}$ are also
temperature-independent. Based on this assumption, the prime equation is
derived:

$\epsilon=\alpha\times(1-QE)$ (7)

This equation is reduced to the familiar Eq. (1) in equilibrium. The complete
derivation can be found in Section 3 in the Supplementary Material.

Figure 1:  Illustration of PL vs. temperature.

###  II.2 Support in Prior Experimental Results

Recent experimental work on PL at elevated temperatures [19], shows a quasi-
conserved photon emission rate in the low-temperature range, far above thermal
emission, which corresponds to the $QE$. At this temperature range, the PL
spectrum is blue-shifted with temperature rise. At higher temperatures, the PL
rate increases exponentially, approaching the thermal emission. Fig. 1
illustrates the PL rate evolution with temperature based on [19], excluding
the very high-temperature range (above 1500K), which will be discussed at the
end of the paper. The green line depicts the pump photon rate $L_{p}$, while
the red line depicts the absorbed photon rate $\alpha\times L_{p}$, where
$\alpha$ is the absorptivity. The blue curves are the PL (solid line) and
thermal emission (dashed line) for a specific material and geometry for pump-
on and pump-off respectively.

At low temperatures, the PL rate is defined by $\alpha\times L_{p}\times QE$.
As the temperature increases, the PL increases monotonically up to a
temperature at which the PL photon rate equals the absorbed photon rate
(marked with a star in Fig. 1), defined as the universal point. The rate of
electrons that are photonically excited and recombine non-radiatively at low
temperature is $(1-QE)\times\alpha\times L_{p}$, where at the universal point,
this portion is restored by thermal contribution. Therefore, this temperature-
dependent evolution equals the thermal emission of a material such that:

$\epsilon\times
L_{B.B}\left(h\nu,T_{0}\right)=\left(1-QE\right)\times\alpha\times
L_{p}(T_{p})$ (8)

where $L_{B.B}\left(h\nu,T_{0}\right)$ is the black-body emission at $T_{0}$,
which, together with $L_{p}$, are quantities independent of the specific body
properties, while $\epsilon$, $\alpha$ and the $QE$ are material and geometry
properties. Thus, we can deduct from this equality the following relation at
the universal point:

$L_{B.B}(h\nu,T_{0})=L_{p}(T_{p})$ (9)

By definition, $T_{0}$ in Eq. (9) equals the brightness temperature of the
pump, $T_{p}$, manifesting the zeroth law of thermodynamics at the universal
point. Inserting Eq. (9) in Eq. (8), results in the prime equation, Eq. (7).
By assuming the material properties to be independent of temperature, the
prime equation becomes valid at any temperature.

Intuitively, according to the prime equation, $\epsilon$ must be reduced,
compared to $\alpha$, by the $QE$ amount. This also agrees with the
understanding that high-$QE$ materials have low phonon-electron nonradiative
rates.

Recent theoretical work on a universal modal radiation law [20] supports a
very similar conclusion while analyzing a scattering system in thermodynamic
equilibrium. We may treat photoluminescence as part of the scattering
coefficient $|s_{p}|^{2}$ and reach our prime equation. The similarity between
the two approaches is explained in greater detail in the Supplementary
Material in section 1.

###  II.3 Agreement With Prior Theoretical Formalism

The separation between quantum and thermal emission described in (6a), (6b)
necessitates the interpretation of the chemical potential $\mu$, in accordance
with the generalized Planck’s law in Eq. (2), which can be expressed as:

$\epsilon\times L_{B.B}e^{\frac{\mu}{k_{B}T}}=\alpha\times QE\times
L_{p}+\alpha\times(1-QE)\times L_{B.B}$ (10)

and by applying Eq. (7), the chemical potential is reduced to:

$\displaystyle\mu$
$\displaystyle=k_{B}T\cdot\ln\left(\frac{L_{quantum}}{L_{thermal}}+1\right)$
(11) $\displaystyle=k_{B}T\cdot\ln\left(\frac{L}{L_{thermal}}\right)$

where $L_{quantum}=\alpha\times QE\times L_{p}$ and
$L_{thermal}=\epsilon\times L_{B.B}$ which is in agreement with [18].

###  II.4 3-level system numerical simulation

Following Siegman [21], we study the case of a 3-level system in a cavity
emitting toward a zero-K environment (describing, for example, the 830–900 nm
emission lines of $Nd^{+3}$ as depicted in [19]). Such a detailed balance
considers only photonic, electronic, and phononic transitions and omits other
processes associated, for example, with defects, which may cause additional
temperature-dependent quenching. As such, this model describes the upper limit
of temperature-dependent luminescence. Nevertheless, any additional factors
can be embedded in the radiative and non-radiative rates for a specific
solution.

Fig. 2(a) shows the considered energy levels having a ground state and a broad
excited level consisting of two closely spaced levels, with fast non-radiative
thermalization between them $(\gamma_{nr23})$. This drives towards a Boltzmann
distribution of the excited electron populations between the two levels,
$n_{2}$ and $n_{3}$ [22] (detailed solution in Section 2 of the Supplementary
Material). Also, here, the brightness temperature $T_{p}$ defines the pump
rate within the angular and spectral coupling window $\Gamma_{p}$. The system
is described by:

$\displaystyle\frac{dn_{2}}{dt}=$
$\displaystyle(n_{1}-n_{2})B_{r12}n_{ph12}-n_{2}\gamma_{r12}+$
$\displaystyle(n_{1}-n_{2})B_{nr12}n_{pn12}-n_{2}\gamma_{nr12}+$
$\displaystyle(n_{3}-n_{2})B_{nr23}n_{pn23}+n_{3}\gamma_{nr23}$
$\displaystyle\frac{dn_{3}}{dt}=$
$\displaystyle(n_{1}-n_{3})B_{r13}n_{ph13}-n_{3}\gamma_{r13}+$
$\displaystyle(n_{1}-n_{3})B_{nr13}n_{pn13}-n_{3}\gamma_{nr13}-$
$\displaystyle(n_{3}-n_{2})B_{nr23}n_{pn23}-n_{3}\gamma_{nr23}$ $\displaystyle
4\pi\cdot\Delta\nu\cdot\frac{dn_{ph12}}{dt}=$ $\displaystyle
n_{pump}\Gamma_{p}-n_{ph12}\Gamma_{12}-$
$\displaystyle(n_{1}-n_{2})B_{r12}n_{ph12}+n_{2}\gamma_{r12}$ $\displaystyle
4\pi\cdot\Delta\nu\cdot\frac{dn_{ph13}}{dt}=$ $\displaystyle
n_{pump}\Gamma_{p}-n_{ph13}\Gamma_{13}-$
$\displaystyle(n_{1}-n_{3})B_{r13}n_{ph12}+n_{2}\gamma_{r13}$

where all the parameters are identical to those introduced in Eq. (4) and Eq.
(4), modified to describe a 3-level system. Solving the equations for
different regimes reveals different observables. For simplicity, we plot the
solution for various values of $\gamma_{r}$ and $\gamma_{nr}$ as temperature-
independent. For the temperature-dependent case, the solution can be depicted
as crossing between different curves at different temperatures.

(a)

(b)

(c)

Figure 2: A 3-level system with fast non-radiative thermalization between
upper energy levels. (b) PL emission for three different QE: $QE=0,\;0.5$ and
1 systems (black, blue and red lines, respectively), optically excited by a
pump at brightness temperature $T_{p}=1000K$. The green and purple lines show
individual emissions from lower and higher excited energy levels for the
$QE=0.5$ case. (c) Above the critical temperature $T>T_{c}$, the PL emission
(blue line) is bounded by a thermal body when $QE=0$ (black line) and by the
thermal emission for the specific $QE$ ($QE=0.5$, red line). The dashed green
line shows the constant difference between the PL emission and the thermal
emission for $QE=0.5$. Figure 3: Generalization of Kirchhoff’s law for non-
equilibrium radiation. Absorptivity of different $QE$ samples, under optical
excitation, measured at room temperature (yellow) and multiplied by $(1-QE)$
of each sample (red). The blue line is the measured emissivity of each sample.
The inset presents the independently measured $QE$ as a function of Er
concentration. Figure 4: Difference between PL and thermal emission rates in
the measured temperature range.

The solutions for the photon rate (radiative transitions to the ground state)
are given by $n_{ph12}\Gamma_{12}$ and $n_{ph13}\Gamma_{13}$, for equally
incoming and output coupling rates
$\Gamma_{12}=\Gamma_{13}=\Gamma_{p}=\Gamma$. We also assume an optical
excitation $n_{pump12}\Gamma_{p}$ and $n_{pump13}\Gamma_{p}$ at $T_{p}=1000K$
for various QEs (0, 0.5, and 1). The results are depicted in Fig. 2(b). These
three cases are all chosen to have the same absorptivity, depending on
$\gamma_{r}$, and different losses due to different $\gamma_{nr}$ rates. The
red line describes both the absorbed pump rate and the $QE=1$ case. The black
line in Fig. 2(b) describes the thermal emission, when setting $QE=0$ (while
keeping $\gamma_{r}$ constant), describing the case where
$\gamma_{nr12},\gamma_{nr13}\gg\gamma_{r12},\gamma_{r13}$.

Under this regime, nearly all the absorbed photons recombine nonradiatively,
which results in thermal emission that increases exponentially until it
reaches the $QE=1$ line at the universal point $T=T_{p}$. This point is the
intersection between the three lines ($QE=0$, $QE=1$, and $QE=0.5$). A body
with the $QE\cong 0$, emits the same amount of radiation to the environment
regardless of the optical excitation. It is in the local equilibrium set by
the dominant nonradiative rates, and obeys Kirchhoff’s law. For the PL
emission of $QE=0.5$ (The blue line), at $0K$, half of the absorbed photons
are lost due to nonradiative recombination $\gamma_{nr}$. Agreeing with the
experimental observations [19], in the low (non-zero) temperature range, the
total photon rate (the sum of $n_{ph12}\Gamma_{12}$ and $n_{ph13}\Gamma_{13}$)
is quasi-conserved, accompanied by a blue-shift of the spectrum-emitted photon
rate. That is, $n_{ph12}\Gamma_{12}$ decreases while $n_{ph13}\Gamma_{13}$
increases, as the temperature increases. The ratio between these emissions is
given by the Boltzmann distribution as long as
$\gamma_{nr23}\gg\gamma_{r12},\gamma_{r13},\gamma_{nr12},\gamma_{nr13}$ [22].
A further increase in temperature leads to an increase in the photon rate
until it reaches the intersection between the $QE=1$ case (red line) and the
$QE=0$ case (black line) at the universal point $T=T_{p}$. The photon rate
continues, thereafter, to rise above it.

In the absence of an optical pump ($n_{pump}=0$), one observes the thermal
emission for the body for any $QE$ emitting towards a zero-K environment. Fig.
2(c) shows the PL line for $QE=0.5$ (blue line) in addition to its
corresponding thermal emission (red line). For comparison, we also show the
thermal emission for the case of $QE=0$ (black line), all have the same
$\gamma_{r}$ values. The difference between the PL and thermal lines for
$QE=0.5$ is a constant, invariant of temperature (depicted by the green dashed
line).

This general solution is, as far as we know, the first explanation for the
experimentally observed transition from the rate-conservation region
accompanied by a blue shift to thermal emission, where the photon rate
increases at any wavelength, as shown in [19]. Evidently, the thermal emission
for $QE>0$ is reduced, compared to the thermal emission of $QE=0$, due to a
lower $\gamma_{nr}$ value compared to $\gamma_{r}$. In addition, the PL
emission beyond the universal point is restricted to remaining between the
$QE=0$ line and the thermal line for the same $QE$.

We find from figure Fig. 2(c) that the ratio between the thermal emission of a
body (for any $QE>0$, in the absence of a pump) and the emission curve of a
thermal body having the same absorptivity ($QE=0$ and
$\epsilon_{(QE=0)}=\alpha$) is a temperature-independent and $QE$-dependent
constant we name $\epsilon_{(QE>0)}$, which results in
$\epsilon_{(QE>0)}=1-QE$. It is evident, that the thermal contribution leads
to recovery of the universal point and balancing nonradiative recombination
which are proportional to $\alpha\times(1-QE)$. As suggested by Eq. (7), the
total emissivity can be expressed as:

$\epsilon=\epsilon_{(QE=0)}\times\epsilon_{(QE>0)}=\alpha\times(1-QE)$ (13)

The total thermal emission of a body toward a zero-K environment (in absence
of a pump) is:

$L\left(h\nu,T\right)=\Gamma\times\alpha\times(1-QE)\times\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{k_{B}T}}-1}$
(14)

For the case where $\gamma_{r}$ and $\gamma_{nr}$ are temperature dependent,
as for Planck’s radiation, the emission curve crosses from one $QE$ line to
the other, yet still reaching the universal point. For this reason, Eq. (13)
is valid also in the form $\epsilon(T)=\alpha(T)\times(1-QE(T))$.

In addition, the equations show that in the case where the coupling rate
$\Gamma\to 0$, any material approach $QE=0$. This means that the emission from
any material enclosed in an optical cavity becomes black-body emission. The
cavity reduces the photons escaping the system thereby effectively reducing
$\gamma_{r}$, while unchanging $\gamma_{nr}$ which is a material property. The
relation $\gamma_{nr}\gg\ \gamma_{r}$, together with multiple self-absorption
events in the cavity that effectively enhance absorptivity results in a black-
body.

##  III Experiment

Conventional absorptivity measurements are conducted in specific directions,
while emissivity and $QE$ experiments are performed in an integration sphere,
which measures hemispherical radiation. Such measurements do not align with
Eq. (7). Instead, it is necessary to measure hemispherical absorptivity, which
is the absorptivity at all directions. This method is elaborated in Section 6
of the Supplementary Materials.

For measuring the emissivity of materials with different $QE$s, we choose
Erbium-doped YAG (Yttrium Aluminum Garnet, $Y_{3}Al_{5}O_{12}$) crystals with
four different $Er^{+3}$ ion concentrations having mass ratios of 0.2%, 2%, 3%
and 5% [23]. The different concentrations affect the $QE$ through self-
quenching, with minimal change in the band gap. The dimensions of the 0.2% and
3% $Er^{+3}$ ion concentration crystals are 10×10×4 $mm^{3}$ (Roditi), while
the 2% and 5% Er:YAG crystals are 11×11×5 $mm^{3}$ (Crytur). As illustrated in
Fig. 5(a), each sample is placed on a silver plate, which is heated by a
ceramic heater (Thorlabs HT19R). Due to the high thermal conductivity of
silver, along with its relatively low emissivity, it is possible to uniformly
heat the sample, as well as perform temperature and thermal emission
measurements. The temperature of the silver plate is measured with a k-type
thermocouple and controlled with a PID controller (Eurotherm 2416). The
thermocouple is placed inside a blind hole, with its tips at the center of the
silver plate. The sample is covered with an additional silver plate, having an
additional thermocouple to monitor and minimize the temperature gradient over
the sample. The power spectrum is acquired using multimode fiber, coupled to a
calibrated spectrometer (Cornerstore 260) connected to an InGaAs detector
(Andor iDus), placed in the vicinity of the sample face. The acquired power
spectrum is normalized by the emission area and solid angle. As an example,
the thermal emission of the 5% Er:YAG crystal at the temperature range of
296K-803K is presented in Fig. 5(b). The thermal emissions of the other three
samples are presented in the Supplementary Material, section 4. A detailed
description of the measured emission normalization in the free-space system is
given in the Supplementary Material, section 5.

(a)

(b)

Figure 5: Experimental setup and measured power spectra of the thermal
emission. (a) The experimental setup. Each Er:YAG sample is placed on a silver
plate that is positioned on a heater. The lower silver plate is grooved to
align the sample (shown in the inset). The sample is heated by a PID-
controlled heater. The temperature of the lower plate is measured by a
thermocouple placed in a blind hole, drilled in the silver plate. The emitted
thermal radiation of the sample is collected by a fiber and the power spectrum
is measured by a calibrated spectrometer. To increase emission into the
peripheral directions, the sample is covered on top with a second (non-
heated)silver plate. (b) The power spectrum of the thermal emission of the 5%
Er:YAG crystal.

Fig. 3 depicts the obtained emissivity and absorptivity, proving
experimentally the generalization of Kirchhoff’s law as presented by Eq. (7).
We note that the measured emissivity, after normalization is the hemispherical
emissivity. This value is compared to the hemispherical absorptivity measured
inside an integration sphere (see Supplementary Material, section 6). The
yellow line depicts the average hemispherical absorptivity at the spectral
range (1450–1650 nm) of Erbium samples having different $QE$s. The $QE$s are
measured at room temperature [24] and presented in the inset as a function of
Erbium concentration. The blue line presents the hemispherical emissivity of
each sample measured at the same spectral band as the absorptivity. We present
the results obtained for 803K, while in Section 7 of the Supplementary
Material we show the emissivity vs. temperature, which in our case is
constant. Had the $QE$ of the samples been zero, the blue and yellow lines
would coincide, representing $\epsilon=\alpha$. From Fig. 3 it is clear that
the measured emissivity (blue line) coincides with the absorptivity multiplied
by $(1-QE)$ of each sample (red line).

Finally, we experimentally measure our hypothesis that the PL emission is the
sum of a constant ($QE$) contribution at zero-Kelvin and a temperature-
dependent thermal emission. Fig. 4 shows the measured difference between the
PL and emissivity of each sample vs. the temperature at ranges where the
thermal emission can be obtained accurately (300K–750K). Clearly, all
measurements show a constant $QE$ contribution over thermal emission while
both increase (500K–750K). The temperature gradient is noticeable only at high
temperatures where it reaches the maximal gradient value of 50K. The error
bars represent the resulting uncertainty in the photon rate. We note that in
[19] the PL merges with thermal emission at temperatures above 1600K. This can
be interpreted as quenching of the $QE$ at high temperatures or as a result of
the uncertainty in temperature that leads to high uncertainty in the flux.

##  IV Conclusion

In this paper, we theoretically and experimentally demonstrate the prime
equation for emission out of equilibrium as a material property of the form of
$\epsilon=\alpha(1-QE)$, relating emissivity, absorptivity, and $QE$. At
equilibrium, this relation is reduced to Kirchhoff’s law due to the balance of
non-radiative (thermal) processes. Starting from the theoretical model
describing a 2-level system through the detailed balance approach, we derive
this relation analytically and simulate it for a 3-level system.
Experimentally, we demonstrate the prime equation through direct measurement
of its properties. We observe at high $QE$ that the emissivity is three-fold
lower than absorptivity, with an excellent fit to the prime equation. Our work
was focused on photoluminescence. As we look ahead, expanding into areas like
electro-luminescence and other quantum emission processes will help uncover
the underlying shared principles of radiation.

Funding

European Union’s Seventh Framework Program (H2020/2014-2020]) under grant
agreement n° 638133-ERC-ThforPV.

Israel Science Foundation (Grant No. 1230/21).

Disclosures

The authors declare no conflicts of interest.

Supplemental document

See Supplementary Material for supporting content.

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