# Non-Planck thermal emission from two-level media

Igor A. Nechepurenko  Moscow Institute of Physics and Technology, Dolgoprudny
141700, Russia  Dukhov Research Institute of Automatics, Moscow 127055, Russia
Denis G. Baranov<EMAIL_ADDRESS>Center for Photonics and 2D
Materials, Moscow Institute of Physics and Technology, Dolgoprudny 141700,
Russia

###### Abstract

Thermal emission is a universal phenomenon of stochastic electromagnetic
emission from an object composed of arbitrary materials at elevated
temperatures. A defining feature of this emission is the monotonic and rapid
growth of its intensity with the object’s temperature for most known
materials. This growth originates from the Bose-Einstein statistics of the
thermal photonic field. The fact that the material’s ability to absorb and
emit light may change with temperature, however, is often ignored. Here, we
carry out a theoretical study of thermal emission from structures
incorporating ensembles of two-level media. We investigate this effect in a
range of geometries including thin films and compact nanoparticles, and
establish the general dependencies in the evolution of thermal emission from
systems including two-level media. Thermal emission from such structures is
essentially Non-Planckian and exhibits a universal asymptotic behavior in the
limit of high temperatures. These results might have important implications
for the design of thermal energy harvesting and thermal vision systems.

##  I Introduction

Any absorbing object radiates a broadband spectrum of electromagnetic
radiation into the environment according to the principles of statistical
mechanics. The class of idealized black bodies – objects that perfectly absorb
any incident electromagnetic radiation – emit broadband and incoherent thermal
radiation, whose spectral radiance (spectral power density per unit area per
unit solid angle radiated along the observation angle $(\theta,\varphi)$ is
described by the Planck’s law Planck (1901):

$L_{BB}=\frac{\omega^{2}}{4\pi^{3}c^{2}}\frac{\hbar\omega}{e^{\hbar\omega/k_{B}T}-1}.$

Upon integration over the whole spectral and angular range it yields the
Stefan-Boltzmann law describing the integral black-body radiation intensity
Stefan (1879):

$I_{BB}=\sigma T^{4},$

where $\sigma$ is the Stefan-Boltzmann constant. As this law suggests, the
integral intensity of emission from a black-body quickly grows with
temperature. This rapid $T^{4}$ growth originates from the Bose-Einstein
statistics of the population of the surrounding photonic field Boyd (1983).

This dependence is modified for materials and nanostructures with selective
absorption spectra, for which the spectral radiance $L$ is defined by the
structure’s absorptivity $\alpha(\omega)$. This correspondence between the
emissivity and the absorptivity, known as the Kirchhoff’s law Kirchhoff
(1860); Greffet  _et al._ (2018), has enabled a wide range of thermal emitters
with narrowband Celanovic  _et al._ (2005); Liu  _et al._ (2011); Inoue  _et
al._ (2015), directional Greffet  _et al._ (2002); Laroche  _et al._ (2005);
Chalabi  _et al._ (2016), and tunable Inoue  _et al._ (2014); Du  _et al._
(2017); Kats  _et al._ (2013) emission spectra that have found numerous
practical applications for thermal energy management Byrnes  _et al._ (2014);
Davids  _et al._ (2020) and radiative cooling Gentle and Smith (2010); Raman
_et al._ (2014); Mandal  _et al._ (2018); Zhai  _et al._ (2017), (see Baranov
_et al._ (2019) for a review). Although the Kirchoff’s law enables a slower
growth of thermally emitted intensity with temperature from narrowband
emitters, a more dramatic evolution of emitted intensity with temperature is
needed in certain situations. Finding a material or a structure exhibiting
strongly non-Planckian thermal emission is of great importance for many
practical applications dealing with thermal light, such as energy recycling
Byrnes  _et al._ (2014) and infrared imaging Qu  _et al._ (2018). To achieve
this, the emissivity of the material itself must vary with temperature.

Media and nanostructures whose ability to emit light (i.e., their emissivity)
varies with temperature, form the set of thermochromic materials. There has
been a considerable progress in studies on non-Planckian thermal emission from
nanostructures made of thermochromic crystals such as GST Du  _et al._ (2017);
Qu  _et al._ (2017), VO2 Kats  _et al._ (2013), or SmNiO3 Shahsafi  _et al._
(2019). An interesting case of thermochromics is presented by so called
negative differential emission, wherein the total thermal emission intensity
does not grow, but in contrast, reduces upon heating in a certain interval of
temperatures Kats  _et al._ (2013). An extreme regime is the zero-
differential, or invariant, thermal emission - the regime wherein the total
intensity of emission from an object remains (nearly) constant in a range of
temperatures Shahsafi  _et al._ (2019). When viewed through a thermal camera,
such an object would appear to maintain its temperature even when heated or
cooled within that range of temperatures.

In this paper, we theoretically investigate thermal emission from structures
incorporating ensembles of two-level emitters. Due to the statistical behavior
of the two-level medium population, its emissivity quickly falls with
temperature, thus rendering the resulting thermal emission non-Planckian. We
investigate thermal emission in a range of common geometries including thin
films and compact nanoparticles, and find the general dependencies in the
evolution of thermal emission from systems incorporating two-level media.
Furthermore, we demonstrate that within the framework of a two-level medium,
the intensity of emitted thermal radiation universally approaches a constant
value in the limit of high temperatures for any finite structure. We expect
these results to be valuable for the development of energy recycling systems
and thermal camouflage.

Figure 1: (a) Sketch of a thermal emitter incorporating a medium formed by
two-level emitters. The system exchanges electromagnetic energy with the
environment at temperature $T$. (b) Temperature-dependent equilibrium
population inversion $\delta n$ of a two-level medium.

##  II Results

Ensemble of two-level emitters forming a continuous medium is a commonly used
generic system in studies of thermal emission. Fig. 1(a) illustrates the
process of thermal emission from such a system: a generic cavity enclosing a
two-level medium exchanges energy with the surrounding photonic field at
temperature $T$. As a result of this exchange, the structure emits a
continuous spectrum of thermal radiation. The integrated thermal emission
intensity in the far field from a structure characterized by an absorptivity
$\alpha$ has the form:

$I=\int_{0}^{\infty}{\alpha(\omega)L_{BB}d\omega},$ (1)

where $L_{BB}$ is the black-body radiance and $\alpha$ is the structure
absorptivity spectrum.

Dielectric properties of a two-level medium can be described by the Lorentzian
permittivity:

$\varepsilon(\omega)=\varepsilon_{\infty}-f\delta
n\frac{\omega_{0}^{2}}{\omega_{0}^{2}-\omega^{2}-i\gamma\omega},$ (2)

where $\varepsilon_{\infty}$ is the non-resonant permittivity of the medium,
$\omega_{0}$ and $\gamma$ are the resonance frequency and linewidth, $f$ is
the dimensionless bulk oscillator strength, and $\delta n=n_{e}-n_{g}$ is the
ensemble average population inversion of the TLS. One often overlooked detail
is that the population inversion of the two-level medium is a function of
temperature itself. At a zero temperature all emitters are in their ground
state and $\delta n=-1$. Applying the Gibbs distribution to an ensemble of
two-level systems at temperature $T$, we find the thermal population:

$\delta
n=\frac{e^{-\hbar\omega_{0}/k_{B}T}-1}{1+e^{-\hbar\omega_{0}/k_{B}T}}.$ (3)

The population inversion becomes less negative with temperature, Fig. 1(b),
thus effectively making the material less absorbing, hence suppressing its
thermal emittance. This dependence readily allows us to calculate the
absoprtivity and integrated thermal emission intensity of from an arbitrary
two-level medium at a temperature $T$.

To begin our analysis of thermal emission from structures incorporating two-
level media, we consider thermal emission from a homogeneous film described by
permittivity of the form Eq. (2) with $\varepsilon_{\infty}=6$, $f=0.4$,
$\gamma/\omega_{0}=0.05$ in air. Fig. 2(a) shows the resulting emissivity
$\epsilon(\omega)$ of the film for a series of temperatures $T/\omega_{0}$ (we
use dimensionless temperature $k_{B}T/\hbar$ in the plots). These spectra
clearly indicate that the emissivity of the film made of a two-level medium
quickly drops with temperature as a result populating the excited level of the
emitters, Fig. 1(b).

Figure 2: (a) Temperature-dependent emissivity of a thin film of thickness
$d=\lambda_{0}/10$ made of a two-level medium described by the Lorentz
permittivity $\varepsilon(\omega,T)$, Eq. (2), in vacuum. (b) Temperature-
dependent integral intensity of thermal radiation $I$ emitted in the normal
direction from a thin film made of a two-level medium for a series of
thickness values $d$. The black line is the blackbody emission intensity;
dashed lines show the corresponding emission intensity from a harmonic Lorentz
film of the same thickness.

Fig. 2(b) presents integral total emission intensity $I$ (radiated normally to
the interface) as a function of temperature for a film of thickness $d$ in
free space in comparison to the intensity emitted by a black-body $I_{BB}$. It
readily shows that the emission intensity radiated by a film of two-level
medium saturates at temperatures $k_{B}T\approx\hbar\omega_{0}$ irrespective
of the film thickness and approaches a constant in the limit of high
temperatures. Correspondingly, due to this saturation it starts to lag behind
the black-body emission $I_{BB}$ at temperatures above the characteristic
temperature of the two-level medium.

Interestingly, the intensity of thermal emission from two-level films features
a super-Planckian growth at low temperatures, $k_{B}T<0.3\hbar\omega_{0}$,
Fig. 2(b). This is the result of the blackdoby emission peak approaching the
resonant frequency of the two-level medium: the population inversion is nearly
stationary at low temperatures, but the magnitude of the Planck’s blackbody
factor $L_{BB}$ grows rapidly in the vicinity of $\omega_{0}$ (with the
blackdoby emission peak approaching the resonance from the low-energy side).
Since most of the emitted intensity originates from energies close to
$\omega_{0}$, this leads to super-Planckian growth of the integral intensity.
This observation might be interesting in the context of super-resolution based
on the super-linearity of thermal radiation Graciani and Amblard (2019).

It is instructive to compare the resulting thermal emission from a two-level
medium film with the case of a harmonic Lorentz medium, i.e., a medium formed
by an ensemble of resonant harmonic oscillators, rather than two-level
transitions. Because of a different statistical behavior of the harmonic
oscillator, its absorptivity (and emissivity, respectively) does not get
saturated at elevated temperature and remains constant: $\delta n=-1$. The
resulting integral emission intensity of harmonic Lorentz films grows slower
than the blackbody intensity, but does not remain constant at high
temperatures (dashed curves in Fig. 2(b)). Fitting the resulting intensity in
the double logarithmic scale yields linear dependence at high temperatures,
$I\propto T$. This scaling can be understood by noting that the emissivity of
a harmonic film $\epsilon(\omega)$ does not vary with temperature; at the same
time, at high temperatures $L_{BB}\propto T$ at a fixed frequency, resulting
in the linear growth of the integral emission intensity.

As another representative example of emitting structure, we consider a thin
film of a two-level medium of thickness $L$ placed a distance $h$ from a
lossless metallic mirror. The resulting dependence of the integral thermal
emission radiated at a normal from the two-level medium film exhibits a
behavior very similar to that observed with a thin film in free space, Fig.
3(a).

Figure 3: (a) Temperature-dependent integral intensity of thermal radiation
$I$ emitted in the normal direction from a two-level medium film of thickness
$d$ placed on top of a perfect electric conductor. The dashed line is the
corresponding blackbody emission intensity; (b) Temperature-dependent
intensity of thermal emission from a two-level medium film of thickness $d$
placed between a mirror with 90% reflectance and a perfect electric conductor.
The black line is the blackbody emission intensity; dashed lines show the
corresponding emission intensity from a harmonic Lorentz film of the same
thickness in the same geometry.

Next, we consider a similar two-level medium film, but enclosed in a Fabry-
Perot cavity formed by another semi-transparent mirror on top (described by a
lossless model $\varepsilon_{D}=1-\omega_{p}^{2}/\omega^{2}$ with
$\omega_{p}=12\omega_{0}$ chosen for the demonstration). The cavity can
selectively enhance thermal emission at its resonant wavelengths, thus
decreasing the bandwidth of thermal emission from a resonant medium even
further Celanovic  _et al._ (2005). The resulting dependencies of integral
emission intensity shown in Fig. 3(b) demonstrate a behavior qualitatively
similar to other thin film geometries examined above - the integral intensity
approaches a constant at high temperatures, whereas emission intensity from a
harmonic Lorentz film grows as $I\propto T$. Notably, thicker cavities filled
with the two-level medium can exhibit lower emission intensities, as we can
infer from comparing the curves corresponding to $d=0.17\lambda_{0}$ and
$d=0.45\lambda_{0}$. This behavior originates from the critical coupling of
cavities of certain thickness, resulting in maximized absorption and
emissivity.

Finally, we examine non-Planck thermal emission from compact objects
incorporating two-level absorbing media. As a tutorial example we consider a
solid sphere of radius $r$ made of a two-level material described by the
permittivity Eq. (2). For the case of a compact object, the integral power of
thermal radiation emitted in the observation direction $(\theta,\varphi)$ per
unit area of the object is determined by the corresponding absorption cross-
section $\sigma_{(\theta,\varphi)}$ Biehs and Ben-Abdallah (2016); Sakat  _et
al._ (2018):

$I=\frac{1}{S}\int_{0}^{\infty}{\sigma_{(\theta,\varphi)}(\omega)L_{BB}d\omega},$
(4)

where $S$ is the geometrical cross-section. Fig. 4 presents the resulting
thermal power per unit area emitted by a sphere for two illustrative cases: a
subwavelength sphere with $\varepsilon_{\infty}=1$, whose response is
dominated by the Frolich (plasmon-like) resonances occurring at negative
permittivities, and a wavelength-scale sphere with $\varepsilon_{\infty}=6$,
whose response is dominated by dielectric Mie resonances coupled with the
Lorentz resonance of the two-level medium. Despite the difference, both
situations feature qualitatively similar dependencies of thermal emission
intensity, saturating near $k_{B}T\approx\hbar\omega_{0}$.

Figure 4: (a) Temperature-dependent integral intensity of thermal radiation
$I$ emitted from a subwavelength sphere of radius $r$ in air made of two-level
medium. The dashed lines show the corresponding emission intensity from a
sphere made of the harmonic Lorentz material; (b) The same as (a) for a
wavelength-scale sphere described by the same permittivity with
$\varepsilon=6$.

It is possible to build an analytical argument showing that in the limit of
high temperatures the integral emission intensity from an arbitrary finite
structure incorporating only a perfect two-level medium as a source of thermal
fluctuations approaches an asymptote. Absorptivity $\alpha$ (or alternatively
scattering cross-section $\sigma$ for a compact object) of the system under
excitation with a plane wave
$\mathbf{E}_{inc}=\mathbf{E}_{0}e^{i\mathbf{k}_{inc}\mathbf{r}}$ from a
certain direction $(\theta,\varphi)$ can be found as work performed by the
total field $\mathbf{E}_{tot}$ on the induced electric current density
$\mathbf{j}(\mathbf{r})$:

$\alpha(\omega,T)\propto\frac{1}{I_{inc}}\int_{V}\frac{1}{2}\operatorname{Re}(\mathbf{E}_{tot}(\mathbf{r})\cdot\mathbf{j}^{*}(\mathbf{r}))d^{3}\mathbf{r}=\\\
\frac{\omega}{2I_{inc}}\int_{V}\varepsilon_{0}\
\operatorname{Im}\varepsilon(\mathbf{r},\omega,T)|\mathbf{E}_{tot}(\mathbf{r},\omega,T)|^{2}d^{3}\mathbf{r}.$
(5)

where $I_{inc}$ in the incident energy flux. Expanding the imaginary part of
the permittivity in the series over $1/T$, we obtain

$\operatorname{Im}\varepsilon(\omega,T)=A\frac{1}{T}+o(1/T),$ (6)

where $A$ is a constant. The total field can be expanded in a similar way. In
the limit of high temperatures the population inversion $\delta n$ approaches
zero, and the material becomes non-resonant. Assuming the limiting structure
($\delta n=0$) does not supports bound states in the continuum, the induced
electric field across the structure $\mathbf{E}_{tot}(\mathbf{r},\omega)$ can
be expanded as

$\mathbf{E}_{tot}(\mathbf{r},\omega,T)=\mathbf{E}_{\infty}(\mathbf{r},\omega)+\delta\mathbf{E}(\mathbf{r},\omega,T)O(1/T),$
(7)

where $\mathbf{E}_{\infty}(\mathbf{r},\omega)$ is the limiting field
distribution determined at every wavelength and illumination direction
$(\theta,\varphi)$ solely by the geometry and $\varepsilon_{\infty}$.
Substituting this into Eq. (5) we obtain $\alpha(\omega,T)\propto 1/T$ in the
limit of high temperatures. Finally, combining this absorptivity with the
black-body radiance $L_{BB}$, integrating within the linewidth of the emitter
$\gamma$ and keeping only the leading term in the expansion, we find the
asymptotic behavior of the total thermal emission:

$I_{TLS}=\int_{0}^{\infty}{\alpha(\omega,T)L_{BB}(\omega,T)d\omega}=C+O(1/T),$
(8)

where $C$ is a constant.

This is a universal result that holds for any ensemble of truly two-level
systems. Unfortunately, this saturation behavior is quite slow and becomes
pronounced only at temperatures above the transition frequency of the resonant
medium, $k_{B}T>\hbar\omega_{0}$, which makes this effect not suitable for
realization of temperature-independent thermal emission Shahsafi  _et al._
(2019). However, we expect that combining this behavior with other
thermochormic materials, for example, with multi-layered structures
incorporating dielectric layers with temperature-dependent refractive indices,
could shift the temperature-independent emission regime to lower temperatures.

We realize that a two-level medium is an idealization, and any real material
inevitably will contain higher-energy vibrational or electronic transitions
associated to respective resonances of the material’s permittivity. However,
even if one accounts for these transitions, they will contribute to spectral
radiance only in the frequency range near the respective resonance. Therefore,
if the quantity of interest is the emission intensity integrated within a
certain wavelength range (which is the case, for example, for most of thermal
imaging devices), the resulting behavior of this quantity will exhibit a
saturation similar to the ones demonstrated above.

##  III Conclusion

To conclude, we have studied temperature dependencies of thermal emission from
structures incorporating resonant two-level media. We have examined a range of
practically common geometries such as thin films, cavities, and spherical
nanoparticles, and found that thermal emission from all these systems shares
one common feature: its intensity is non-Planckian, and approaches a constant
value (dependent on the structure) in the limit of high temperature. This
asymptotic behavior stems from the trade-off between the diminishing
emissivity of the two-level medium, and the growing population of the photonic
field. These results might have far reaching implications for the design of
thermal energy harvesting and thermal vision systems.

##  IV Acknowledgments

The work was supported by the Russian Science Foundation (21-12-00316). D.G.B.
acknowledges support from the Russian Federation President Grant
(MK-1211.2021.1.2).

## References

  * Planck (1901) M. Planck, Annalen der Physik 4 (1901).
  * Stefan (1879) J. Stefan, Akademie der Wissenschaften 79, 391 (1879).
  * Boyd (1983) R. W. Boyd,  _Radiometry and the detection of optical radiation_ (John Wiley and Sons, New York, NY, USA, 1983).
  * Kirchhoff (1860) G. Kirchhoff, Annalen der Physik und Chemie 109, 275 (1860).
  * Greffet  _et al._ (2018) J.-J. Greffet, P. Bouchon, G. Brucoli, and F. Marquier, Physical Review X 8, 021008 (2018).
  * Celanovic  _et al._ (2005) I. Celanovic, D. Perreault, and J. Kassakian, Phys. Rev. B 72, 2 (2005).
  * Liu  _et al._ (2011) X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, Phys. Rev. Lett. 107, 045901 (2011).
  * Inoue  _et al._ (2015) T. Inoue, M. D. Zoysa, T. Asano, and S. Noda, Optica 2, 27 (2015).
  * Greffet  _et al._ (2002) J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, Nature 416, 61 (2002).
  * Laroche  _et al._ (2005) M. Laroche, C. Arnold, F. Marquier, R. Carminati, J. J. Greffet, S. Collin, N. Bardou, and J. L. Pelouard, Opt. Lett. 30, 2623 (2005).
  * Chalabi  _et al._ (2016) H. Chalabi, A. Alù, and M. L. Brongersma, Phys. Rev. B 94, 094307 (2016).
  * Inoue  _et al._ (2014) T. Inoue, M. De Zoysa, T. Asano, and S. Noda, Nature materials 13, 928 (2014).
  * Du  _et al._ (2017) K. Du, Q. Li, Y. Lyu, J. Ding, Y. Lu, and Z. Cheng, Light Sci. Appl. 6, e16194 (2017).
  * Kats  _et al._ (2013) M. A. Kats, R. Blanchard, S. Zhang, P. Genevet, C. Ko, S. Ramanathan, and F. Capasso, Phys. Rev. X 3, 041004 (2013).
  * Byrnes  _et al._ (2014) S. J. Byrnes, R. Blanchard, and F. Capasso, Proc. Natl. Acad. Sci. U.S.A 111, 3927 (2014).
  * Davids  _et al._ (2020) P. S. Davids, J. Kirsch, A. Starbuck, R. Jarecki, J. Shank, and D. Peters, Science 367, 1341 (2020).
  * Gentle and Smith (2010) A. R. Gentle and G. B. Smith, Nano letters 10, 373 (2010).
  * Raman  _et al._ (2014) A. P. Raman, M. A. Anoma, L. Zhu, E. Rephaeli, and S. Fan, Nature 515, 540 (2014).
  * Mandal  _et al._ (2018) J. Mandal, Y. Fu, A. C. Overvig, M. Jia, K. Sun, N. N. Shi, H. Zhou, X. Xiao, N. Yu, and Y. Yang, Science 362, 315 (2018).
  * Zhai  _et al._ (2017) Y. Zhai, Y. Ma, S. N. David, D. Zhao, R. Lou, G. Tan, R. Yang, and X. Yin, Science 355, 1062 (2017).
  * Baranov  _et al._ (2019) D. G. Baranov, Y. Xiao, I. A. Nechepurenko, A. Krasnok, A. Alù, and M. A. Kats, Nature materials 18, 920 (2019).
  * Qu  _et al._ (2018) Y. Qu, Q. Li, L. Cai, M. Pan, P. Ghosh, K. Du, and M. Qiu, Light: Science & Applications 7, 26 (2018).
  * Qu  _et al._ (2017) Y. Qu, Q. Li, K. Du, L. Cai, J. Lu, and M. Qiu, Laser Photon. Rev. 11, 1700091 (2017).
  * Shahsafi  _et al._ (2019) A. Shahsafi, P. Roney, Y. Zhou, Z. Zhang, Y. Xiao, C. Wan, R. Wambold, J. Salman, Z. Yu, J. Li, _et al._ , Proceedings of the National Academy of Sciences 116, 26402 (2019).
  * Graciani and Amblard (2019) G. Graciani and F. Amblard, Nature Communications 10, 5761 (2019).
  * Biehs and Ben-Abdallah (2016) S.-A. Biehs and P. Ben-Abdallah, Phys. Rev. B 93, 165405 (2016).
  * Sakat  _et al._ (2018) E. Sakat, L. Wojszvzyk, J.-P. Hugonin, M. Besbes, C. Sauvan, and J.-J. Greffet, Optica 5, 175 (2018).