# Dipole model for far-field thermal emission of a nanoparticle above a planar
substrate

Florian Herz  Institut für Physik, Carl von Ossietzky Universität, D-26111
Oldenburg, Germany  Svend-Age Biehs<EMAIL_ADDRESS>Institut für
Physik, Carl von Ossietzky Universität, D-26111 Oldenburg, Germany

(August 27, 2024)

###### Abstract

We develop a dipole model describing the thermal far-field radiation of a
nanoparticle in close vicinity to a substrate. By including in our description
the contribution of eddy currents and the possibility to choose different
temperatures for the nanoparticle, the substrate, and the background, we
generalize the existing models. We discuss the impact of the different
temperatures, particle size, emission angle, and the distance dependence for
all four combinations of gold and SiC nanoparticles or substrates.

##  I Introduction

The theoretical problem of thermal emission of a nanoparticle in the vicinity
of a planar substrate within the dipole model together with the framework of
fluctuational electrodynamics has a relatively long history. In the first
works it was used to determine the exchange of thermal radiation between a
spherical nanoparticle and a planar substrate when bringing the nanoparticle
into the near-field of the substrate Dorofeyev97 ; Dorofeyev98 ; JPendry99 ;
MuletEtAl01 . In subsequent works, the contribution due to multiple
interactions between the nanoparticle and the surface VolokitinPersson07 , the
contribution of eddy currents DedkovKyasov07 ; ChapuisEtAl08b , and mixed
magnetic and dielectric contributions Joulain2  where added to the
description, as well as the generalization to ellipsoidal nanoparticles
HuthEtAl2010 , the inclusion of surface roughness for the substrate Biehs2010
, and multipolar moments for the nanoparticle Doro2008 . In a similar manner
also the scattering of the thermal near-field by a nanoparticle into the far-
field Joulain2 ; Jarzembski ; Herz  and the direct thermal emission of a
nanoparticle into the far-field Joulain2 ; Asheichyk2017 ; Herz  has been
studied.

This interest into thermal emission is triggered by several infrared near-
field thermal imaging setups where typically a sharp tip of an atomic force
microscope (AFM) cantilever, scattering optical microscope (SNOM), or scanning
tunneling microscope (STM) is brought close to a surface DeWilde ; Kittel2008
; WorbesEtAl2013 ; Huth2011 ; Jones ; WengEtAl2018 . For example, in the so-
called thermal radiation scanning tunneling microscope (TRSTM) DeWilde ,
thermal radiation of a heated surface is scattered by a sharp tip into the
far-field where its spectrum can be measured Babuty . A similar experiment can
be made with the thermal infrared near-field spectroscope (TINS) Jones ;
O'Callahan  which is based on an AFM probe and allows for heating of the
surface and the tip with respect to the environment. The so-called scanning
noise microscope (SNoiM) is a highly improved TRSTM which does not need to
heat the tip or the substrate to get a signal like for TRSTM or TINS Lin ;
WengEtAl2018 ; Komiyama  due to an ultra-sensitive single-photon detector
working at $4.2$ K. On the other hand, the SNoiM is measuring the heat flux in
a narrow wavelength band around $14.2\,\mu{\rm m}$ and in its present state it
cannot measure broad spectra like the TINS or TRSTM. A simple approach to
model such experimental setups is to approximate the foremost part of the
probe as a nanoparticle Joulain2 ; Jarzembski ; Herz . Then, in lowest order
one can attribute the main influence on the signal due to the local density of
states (LDOS) at the tip position DeWilde ; Jones ; O'Callahan . Although a
comparison to the LDOS can explain many properties of the measured spectra,
this model, for example, cannot explain the red-shift of the surface phonon
polariton (SPhP) frequency for increasing distances between a Si tip and a SiC
substrate Jones . Furthermore, the modeling of these experiments by the
scattering of the near-field by a nanoparticle neglects at least one of the
following two facts: First, apart from the induced electric dipole moment,
eddy currents can emerge, too. Those eddy currents are particularly important
for metallic particles DedkovKyasov07 ; ChapuisEtAl08b ; Dong  and were
included in the dipole model in Ref. Joulain2 . Second, in the experiments the
tip of the probes, the substrate, and, additionally, the background can have
different temperatures. Here, we provide a generalization of the model
Joulain2 ; Herz  to take these different temperatures into account.
Furthermore, instead of neglecting divergent terms of the Green function
appearing in the dipole model, we conduct a renormalization procedure as
described for the coupled dipole model in Ref. Lakhtakia .

Our work is organized as follows. In Sec. II we introduce the renormalization
method and the dressed polarizabilities. In Sec. III we determine the
analytical expressions for the power emitted by the nanoparticle, the surface
and the mixed terms. In Sec. IV we discuss the numerical example of a Au or
SiC nanoparticle above a Au or SiC substrate. Finally, in Sec. V we summarize
our results and give a conclusion.

##  II Fluctuational fields and dressed polarizability

Consider the following setup as sketched in Fig. 1: A nanoparticle with radius
$R$ and temperature $T_{\text{p}}$ is placed at distance $d$ to a semi-
infinite planar substrate at $z\leq 0$ having a temperature $T_{\text{s}}$.
The materials of the nanoparticle and substrate are assumed to be non-
magnetic, homogeneous, and isotropic. They are embedded in vacuum which is
filled with a thermal fluctuating electromagnetic field at temperature
$T_{\text{b}}$ due to the thermal photons in the vacuum part. Hence, we assume
that the substrate, the nanoparticle, and the backround can be considered to
be in local thermal equilibrium at the given temperatures. Furthermore, we
will assume that the fields due to the thermal sources in the substrate and
nanoparticle and the background fields are statistically independent. We want
to determine the spectral power of heat radiation emitted into the far-field
through a plane parallel to the substrate at distance $z>d+R$, i.e. above the
nanoparticle (see detection plane in Fig. 1). Due to the translational
symmetry in x-y direction or rotational symmetry with respect to the z-axis,
only the z-component of the mean Poynting vector of the total thermal
fluctuational fields will be of interest.

Figure 1: Sketch of the considered configuration of a nanoparticle with radius
$R$ at temperature $T_{\text{p}}$ and distance $d$ above a planar substrate at
temperature $T_{\text{s}}$ immersed in a background with temperature
$T_{\text{b}}$. The whole setup is considered to be underneath a detection
plane through which the heat flux is measured.

To determine the mean Poynting vector in this configuration, the expressions
of the total electric and magnetic fields are required. It is possible to
obtain these expressions by solving the volume-integral equation Hecht

$\displaystyle\underline{\mathbf{F}}_{\text{tot}}(\mathbf{r})$
$\displaystyle=\underline{\mathbf{F}}_{\text{env}}(\mathbf{r})+\text{i}\mu_{0}\omega\int_{V_{\text{p}}}\\!\\!\rm{d}^{3}r^{\prime}\underline{\underline{\mathds{G}}}(\mathbf{r},\mathbf{r}^{\prime})\underline{\mathbf{J}}(\mathbf{r}^{\prime})$
(1)

introducing angular frequency $\omega$, vacuum permeability $\mu_{0}$, and
imaginary unit i. The total field $\underline{\mathbf{F}}_{\text{tot}}$
consists of the field of the environment $\underline{\mathbf{F}}_{\text{env}}$
and the field generated by the current densities
$\underline{\mathbf{J}}(\mathbf{r}^{\prime})$ inside the nanoparticle. Hence,
the volume integral is carried out over the volume $V_{\text{p}}$ of the
nanoparticle. Here, we take the electric and magnetic source currents into
account in order to describe dielectric and magnetic nanoparticles
DedkovKyasov07 ; ChapuisEtAl08b ; HuthEtAl2010  so that the above quantities
are given by the block vectors

$\displaystyle\underline{\mathbf{F}}(\mathbf{r})$
$\displaystyle=\left(\mathbf{E}(\mathbf{r}),\mathbf{H}(\mathbf{r})\right)^{t},$
(2) $\displaystyle\underline{\mathbf{J}}(\mathbf{r})$
$\displaystyle=\left(\mathbf{J}_{\text{E}}(\mathbf{r}),\mathbf{J}_{\text{H}}(\mathbf{r})\right)^{t},$
(3)

and block matrix

$\underline{\underline{\mathds{G}}}(\mathbf{r},\mathbf{r}^{\prime})=\begin{pmatrix}\mathds{G}_{\text{EE}}(\mathbf{r},\mathbf{r}^{\prime})&\mathds{G}_{\text{EH}}(\mathbf{r},\mathbf{r}^{\prime})\\\
\mathds{G}_{\text{HE}}(\mathbf{r},\mathbf{r}^{\prime})&\mathds{G}_{\text{HH}}(\mathbf{r},\mathbf{r}^{\prime})\end{pmatrix}.$
(4)

Note that the environment fields with index “env” include both the fields of
the background and the substrate.

The volume-integral equation is typically simplified by using the dipole
approximation which is implemented by replacing the current densities
$\underline{\mathbf{J}}$ by dipole moments $\underline{\mathbf{d}}$ located at
the position $\mathbf{r}_{\text{p}}$ of the nanoparticle’s center via the
relation
$\underline{\mathbf{J}}(\mathbf{r})=-\text{i}\omega\underline{\mathbf{d}}\delta(\mathbf{r}-\mathbf{r}_{\text{p}})$.
The disadvantage of this implementation is that it leads to the necessity for
evaluating the Green’s functions
$\underline{\underline{\mathds{G}}}(\mathbf{r}_{\text{p}},\mathbf{r}_{\text{p}})$
which can be decomposed into a vacuum part and a scattering part
$\underline{\underline{\mathds{G}}}=\underline{\underline{\mathds{G}}}_{\rm
vac}+\underline{\underline{\mathds{G}}}_{\rm sc}$. It is well known that the
real part of the vacuum Green’s function diverges in such cases Yaghjian  and
is, therefore, very often neglected. One simply reimplements this part by
renormalizing the polarizability of the nanoparticle leading to a “dressed”
polarizability as detailed in Ref. Messina . Here, we will follow another
route of employing the dipole or long wavelength approximation (LWA) as
discussed in detail in Refs. Lakhtakia ; Albaladejo . To this end, we first
derive the fields $\underline{\mathbf{F}}_{\text{in}}$ inside the nanoparticle
by consider the nanoparticle as a finite sized subwavelength sphere, i.e. for
propagating waves the sphere must be much smaller than the wavelength and much
smaller than the decay length of the field for evanescent modes. In this case,
it is reasonable to assume that the current density is approximately constant
within this very small finite volume Lakhtakia ; Albaladejo  and can,
therefore, be expressed as

$\underline{\mathbf{J}}(\mathbf{r})=-\text{i}\frac{\omega}{V_{p}}\underline{\mathbf{d}}$
(5)

where $\underline{\mathbf{d}}=(\mathbf{p},\mathbf{m})^{t}$ contains the
electric and magnetic dipole moments $\mathbf{p}$ and $\mathbf{m}$
corresponding to the entries $\mathbf{J}_{\text{E}}$ and
$\mathbf{J}_{\text{H}}$. Therefore, for $\mathbf{r}\in V_{\text{p}}$ we can
take the current density in the second term in Eq. (1) out of the volume
integral so that only the Green’s function is integrated. This integral can be
understood as a volume average of the function defined by

$\displaystyle\langle\underline{\underline{\mathds{G}}}(\mathbf{r})\rangle$
$\displaystyle=\frac{1}{V_{\rm p}}\int_{V_{\rm
p}}\\!\\!\rm{d}^{3}r^{\prime}\underline{\underline{\mathds{G}}}(\mathbf{r},\mathbf{r}^{\prime}).$
(6)

Hence, the volume-integral equation for $\mathbf{r}\in V_{\text{p}}$ is

$\underline{\mathbf{F}}_{\text{in}}(\mathbf{r})=\underline{\mathbf{F}}_{\text{env}}(\mathbf{r})+\mu_{0}\omega^{2}\langle\underline{\underline{\mathds{G}}}(\mathbf{r})\rangle\underline{\mathbf{d}}.$
(7)

Now, the dipole moment can be separated into a fluctuating and induced
contribution due to the fields inside the nanoparticle volume by

$\underline{\mathbf{d}}=\underline{\mathbf{d}}_{\text{fl}}+\varepsilon_{0}V\underline{\underline{\tilde{\boldsymbol{\chi}}}}\underline{\mathbf{F}}_{\text{in}}(\mathbf{r})$
(8)

with the vacuum permittivity $\varepsilon_{0}$ and

$\underline{\underline{\tilde{\boldsymbol{\chi}}}}=\begin{pmatrix}\chi_{\text{E}}&0\\\
0&\chi_{\text{H}}\end{pmatrix}.$ (9)

Here, $\chi_{\text{E/H}}$ are the electric material response to the
electric/magnetic field. In conclusion, we can express the fields inside the
nanoparticle by the closed formula (omitting the arguments for convenience)

$\displaystyle\underline{\mathbf{F}}_{\text{in}}$
$\displaystyle=\left[\mathds{1}-k_{0}^{2}V\underline{\underline{\tilde{\boldsymbol{\chi}}}}\langle\underline{\underline{\mathds{G}}}\rangle\right]^{-1}\left[\underline{\mathbf{F}}_{\text{env}}+\mu_{0}\omega^{2}\langle\underline{\underline{\mathds{G}}}\rangle\underline{\mathbf{d}}_{\text{fl}}\right]$
(10)

defining the vacuum wave number $k_{0}=\omega/c$. Inserting Eq. (10) into Eq.
(8), we get

$\displaystyle\underline{\mathbf{d}}$
$\displaystyle=\left[\mathds{1}+k_{0}^{2}\underline{\underline{\boldsymbol{\alpha}}}\langle\underline{\underline{\mathds{G}}}_{\text{EE}}(\mathbf{r}_{\text{p}})\rangle\right]\underline{\mathbf{d}}_{\text{fl}}+\varepsilon_{0}\underline{\underline{\boldsymbol{\alpha}}}\,\underline{\mathbf{F}}_{\text{env}}(\mathbf{r}_{\text{p}}).$
(11)

where we have introduced the polarizability block matrix

$\underline{\underline{\boldsymbol{\alpha}}}=V\underline{\underline{\tilde{\boldsymbol{\chi}}}}\left[\mathds{1}-k_{0}^{2}V\underline{\underline{\tilde{\boldsymbol{\chi}}}}\langle\underline{\underline{\mathds{G}}}(\mathbf{r}_{\text{p}})\rangle\right]^{-1}\equiv\begin{pmatrix}\boldsymbol{\alpha}_{\text{EE}}&\boldsymbol{\alpha}_{\text{EH}}\\\
\boldsymbol{\alpha}_{\text{HE}}&\boldsymbol{\alpha}_{\text{HH}}\end{pmatrix}.$
(12)

The components of this polarizability block matrix are

$\displaystyle\boldsymbol{\alpha}_{\text{EE}}$
$\displaystyle=\frac{V\chi_{\text{E}}\mathbf{e}_{\perp}\otimes\mathbf{e}_{\perp}}{1-k_{0}^{2}V\chi_{\text{E}}\langle
G_{\text{EE},\perp}(\mathbf{r}_{\text{p}})\rangle+\frac{k_{0}^{4}V^{2}\chi_{\text{E}}\chi_{\text{H}}\langle
G_{\text{HE}}(\mathbf{r}_{\text{p}})\rangle^{2}}{1-k_{0}^{2}V\chi_{\text{H}}\langle
G_{\text{HH},\perp}(\mathbf{r}_{\text{p}})\rangle}}$
$\displaystyle\quad+\frac{V\chi_{\text{E}}\mathbf{e}_{z}\otimes\mathbf{e}_{z}}{1-k_{0}^{2}V\chi_{\text{E}}\langle
G_{\text{EE,z}}(\mathbf{r}_{\text{p}})\rangle},$ (13)
$\displaystyle\boldsymbol{\alpha}_{\text{EH}}$
$\displaystyle=\frac{k_{0}^{2}V\chi_{\text{H/E}}\alpha_{\text{EE/HH},\perp}\langle
G_{\text{HE}}(\mathbf{r}_{\text{p}})\rangle}{1-k_{0}^{2}V\chi_{\text{H/E}}\langle
G_{\text{HH/EE},\perp}(\mathbf{r}_{\text{p}})\rangle}\mathds{X}$ (14)

with
$\mathbf{e}_{\perp}\otimes\mathbf{e}_{\perp}=\mathbf{e}_{x}\otimes\mathbf{e}_{x}+\mathbf{e}_{y}\otimes\mathbf{e}_{y}$
and
$\mathds{X}=\mathbf{e}_{x}\otimes\mathbf{e}_{y}-\mathbf{e}_{y}\otimes\mathbf{e}_{x}$.

Furthermore, $\boldsymbol{\alpha}_{\text{HE}}=\boldsymbol{\alpha}_{\text{EH}}$
holds and $\boldsymbol{\alpha}_{\text{HH}}$ follows from
$\boldsymbol{\alpha}_{\text{EE}}$ by interchanging the field indices. Note
that $\boldsymbol{\alpha}_{\rm EE}^{t}=\boldsymbol{\alpha}_{\rm EE}$,
$\boldsymbol{\alpha}_{\rm HH}^{t}=\boldsymbol{\alpha}_{\rm HH}$, and
$\boldsymbol{\alpha}_{\rm EH}^{t}=-\boldsymbol{\alpha}_{\rm EH}$. The explicit
expressions for the volume averaged Green’s functions can be found in appendix
B. These four polarizabilities are the general “dressed polarizabilities”
because they are not the polarizabilities of the isolated nanoparticle but
also contain information about the interaction with the substrate material by
means of the scattering part of the Green’s function. In its structure the
purely electric part resembles the results in recent works Messina ;
Albaladejo . By the magnetic contribution we retrieved the results for
recently derived dressed polarizabilities Joulain2  apart from the missing
vacuum contribution which is included in our approach.

With the expression of the dipole moment in terms of the dressed
polarizability we can now derive the analytical expression for the field
outside the nanoparticle. For $\mathbf{r}\notin V_{\text{p}}$ the volume
average of the Green’s function is simply
$\langle\underline{\underline{\mathds{G}}}(\mathbf{r})\rangle=\underline{\underline{\mathds{G}}}(\mathbf{r},\mathbf{r}_{\text{p}})$
Albaladejo  within the long-wavelength approximation. Hence, the field outside
the nanoparticle Eq. (1) can be written as

$\underline{\mathbf{F}}_{\text{out}}(\mathbf{r})=\underline{\mathbf{F}}_{\text{env}}(\mathbf{r})+\mu_{0}\omega^{2}\underline{\underline{\mathds{G}}}(\mathbf{r},\mathbf{r}_{\text{p}})\underline{\mathbf{d}}.$
(15)

Inserting Eq. (11), we obtain

$\begin{split}\underline{\mathbf{F}}_{\text{out}}(\mathbf{r})&=\mu_{0}\omega^{2}\underline{\underline{\mathds{G}}}(\mathbf{r},\mathbf{r}_{\text{p}})\left[\mathds{1}+k_{0}^{2}\underline{\underline{\boldsymbol{\alpha}}}\langle\underline{\underline{\mathds{G}}}(\mathbf{r}_{\text{p}})\rangle\right]\underline{\mathbf{d}}_{\text{fl}}\\\
&\quad+k_{0}^{2}\underline{\underline{\mathds{G}}}(\mathbf{r},\mathbf{r}_{\text{p}})\underline{\underline{\boldsymbol{\alpha}}}\,\underline{\mathbf{F}}_{\text{env}}(\mathbf{r}_{\text{p}})+\underline{\mathbf{F}}_{\text{env}}(\mathbf{r}).\end{split}$
(16)

This is the final result for the fields outside the nanoparticle expressing it
in terms of the fluctuational environmental field and the fluctuating dipole
moments. The first term describes the directly emitted field of the
nanoparticle. The scattering of the environmental field at the nanoparticle is
included in the second contribution. The last term solely corresponds to the
environmental field
$\underline{\mathbf{F}}_{\text{env}}(\mathbf{r})=\underline{\mathbf{F}}_{\text{s}}(\mathbf{r})+\underline{\mathbf{F}}_{\text{b}}(\mathbf{r})$
incorporating the substrate and background field evaluated at position
$\mathbf{r}$.

##  III Power emitted into the far field

The power emitted into the far-field in the configuration in Fig. 1 is given
by the integration of the z-component of the Poynting vector over a x-y plane
above the nanoparticle. Formally, the spectral power is given by (Einstein
notation)

$\begin{split}P_{\omega}^{\rm tot}&=\int{\rm
d}^{2}x\,\big{\langle}\\!\\!\big{\langle}\mathbf{S}^{\text{out}}\big{\rangle}\\!\\!\big{\rangle}^{\omega}\cdot\mathbf{e}_{z}\\\
&=2\text{Re}\int{\rm
d}^{2}x\,\epsilon_{\emph{ijz}}\big{\langle}\\!\\!\big{\langle}\mathbf{E}_{\text{out}}(\mathbf{r})\otimes\mathbf{H}_{\text{out}}^{\dagger}(\mathbf{r})\big{\rangle}\\!\\!\big{\rangle}^{\omega}_{ij}\end{split}$
(17)

where $\epsilon_{ijk}$ denotes the Levi-Civita symbol, $\,\,{\rm d}^{2}x$ is
an infinitesimal area element of the x-y detection plane which is indicated in
Fig. 1, and
$\big{\langle}\\!\\!\big{\langle}\cdot\big{\rangle}\\!\\!\big{\rangle}^{\omega}$
defines the stochastic average in frequency space. By inserting the fields
from Eq. (16) and assuming that the fluctuational fields of the substrate,
background and nanoparticle are statistically independent, we obtain

$P_{\omega}^{\rm tot}=P_{\omega}^{s}+P_{\omega}^{\rm env}+P_{\omega}^{\rm
np}.$ (18)

Here,

$P_{\omega}^{s}=2\text{Re}\int{\rm
d}^{2}x\,\epsilon_{\emph{ijz}}\big{\langle}\\!\\!\big{\langle}\mathbf{E}_{\text{env}}(\mathbf{r})\otimes\mathbf{H}_{\text{env}}^{\dagger}(\mathbf{r})\big{\rangle}\\!\\!\big{\rangle}^{\omega}_{ij}$
(19)

is the thermal radiation of the substrate at temperature $T_{\text{s}}$ into
its environment at temperature $T_{\text{b}}$ without the nanoparticle and its
analytical expression is well known Polder . On the other hand
$P_{\omega}^{\rm np}$, including all the terms with correlation functions of
the fluctuational dipole moments $\underline{\mathbf{d}}_{\rm fl}$, describes
the thermal emission of the nanoparticle in the presence of the substrate.
Finally, $P_{\omega}^{\rm env}$ contains all the other terms including
correlation functions of the environment fields and incorporates, therefore,
corrections to the thermal emission. For example, it includes the power
emitted by the substrate which is absorbed by the nanoparticle. Note that the
thermal emission of the nanoparticle into the substrate, as measured in near-
field imaging experiments in Refs. Kittel2008 ; WorbesEtAl2013 , can be
evaluated in a similar manner by integrating the mean Poynting vector over a
plane between the particle and the substrate. However, here we focus on the
far-field emission, only.

In the following we will evaluate all contributions to the total power. To
this end, it is necessary to deal with the field correlation functions of the
environment fields
$\mathds{C}_{\text{env}}^{\text{FF'}}=\big{\langle}\\!\\!\big{\langle}\mathbf{F}\otimes\mathbf{F}^{\prime}\big{\rangle}\\!\\!\big{\rangle}$
where F and F’ symbolize the electric or magnetic fields. Due to the
statistical independence of the substrate and background fields, we can
separate the correlation function into a substrate and background part

$\displaystyle\mathds{C}_{\text{env}}^{\text{FF'}}(\mathbf{r},\mathbf{r}^{\prime},\omega)$
$\displaystyle=\mathds{C}_{\text{s}}^{\text{FF'}}(\mathbf{r},\mathbf{r}^{\prime},\omega)+\mathds{C}_{\text{b}}^{\text{FF'}}(\mathbf{r},\mathbf{r}^{\prime},\omega).$
(20)

The fluctuational fields of the substrate are generated by thermal sources in
the substrate volume $V_{\text{s}}$. Hence, we can assume local equilibrium
for the substrate at temperature $T_{\text{s}}$ and evaluate these correlation
function using Rytov’s formulae givingEckhardt

$\displaystyle\mathds{C}^{\text{FF'}}_{\text{s,leq}}(\mathbf{r},\mathbf{r}^{\prime})$
$\displaystyle=2\Theta_{\text{s}}k_{0}^{2}\mu_{0}\omega\text{Im}(\varepsilon_{\text{s}})$
$\displaystyle\quad\times\int_{V_{\text{s}}}\\!\\!\rm{d}^{3}r^{\prime\prime}\mathds{G}_{\text{FE}}^{\text{s}}(\mathbf{r},\mathbf{r}^{\prime\prime})\mathds{G}_{\text{F'E}}^{\text{s}\dagger}(\mathbf{r}^{\prime},\mathbf{r}^{\prime\prime}).$
(21)

We employ the definition
$\Theta_{\gamma}=\hbar\omega\bigl{(}n_{\gamma}+\frac{1}{2}\bigr{)}$ where
$n_{\gamma}$ is the Bose-Einstein occupation number ($\gamma={\rm s,p,b}$)

$\displaystyle n_{\gamma}$
$\displaystyle=\left(e^{\frac{\hbar\omega}{k_{B}T_{\gamma}}}-1\right)^{-1}$
(22)

with the Boltzmann constant $k_{\rm B}$ and the reduced Planck constant
$\hbar$. The Green’s functions used here are those with source points within
the substrate and observation points outside the substrate as explicitely
given in appendix A.

To evaluate the correlation functions for the background fields, we recall the
expressions for global thermal equilibrium Agarwal ; Eckhardt

$\displaystyle\mathds{C}^{\text{EE/HH}}_{\text{env,eq}}(\mathbf{r},\mathbf{r}^{\prime})$
$\displaystyle=2\Theta\mu_{0}\omega\text{Im}\left[\mathds{G}_{\text{EE/HH}}(\mathbf{r},\mathbf{r}^{\prime})\right],$
(23)
$\displaystyle\mathds{C}^{\text{EH}}_{\text{env,eq}}(\mathbf{r},\mathbf{r}^{\prime})$
$\displaystyle=-2\text{i}\Theta\mu_{0}\omega\text{Re}\left[\mathds{G}_{\text{EH}}(\mathbf{r},\mathbf{r}^{\prime})\right].$
(24)

To obtain an expression for the background contributions in local thermal
equilibrium, we exploit the fact that, when
$T_{\text{s}}=T_{\text{p}}=T_{\text{b}}$, Eq. (20) must fulfill these
expressions for global equilibrium. Then, the desired background correlation
can be evaluated from Eqs. (21)-(24). Thus, we obtain, for example, for the
correlation function with two electric fields

$\displaystyle\mathds{C}^{\text{EE}}_{\text{b,leq}}(\mathbf{r},\mathbf{r}^{\prime})$
$\displaystyle=2\Theta_{\text{b}}\mu_{0}\omega\Bigl{[}\text{Im}\left[\mathds{G}_{\text{EE}}(\mathbf{r},\mathbf{r}^{\prime})\right]$
$\displaystyle\quad-
k_{0}^{2}\text{Im}(\varepsilon_{\text{s}})\int_{V_{\text{s}}}\\!\\!\rm{d}^{3}r^{\prime\prime}\mathds{G}_{\text{EE}}^{\text{s}}(\mathbf{r},\mathbf{r}^{\prime\prime})\mathds{G}_{\text{EE}}^{\text{s}\dagger}(\mathbf{r}^{\prime},\mathbf{r}^{\prime\prime})\Bigr{]}.$
(25)

Combining both the substrate and background correlation functions, we
exemplary end up with

$\begin{split}\mathds{C}^{\text{EE}}_{\text{env}}(\mathbf{r},\mathbf{r}^{\prime})&=2\Theta_{\text{b}}\mu_{0}\omega\text{Im}\left[\mathds{G}_{\text{EE}}(\mathbf{r},\mathbf{r}^{\prime})\right]\\\
&\quad+2\left(\Theta_{\text{s}}-\Theta_{\text{b}}\right)k_{0}^{2}\mu_{0}\omega\text{Im}(\varepsilon_{\text{s}})\\\
&\quad\times\int_{V_{\text{s}}}\\!\\!\rm{d}^{3}r^{\prime\prime}\mathds{G}_{\text{EE}}^{\text{s}}(\mathbf{r},\mathbf{r}^{\prime\prime})\mathds{G}_{\text{EE}}^{\text{s}\dagger}(\mathbf{r}^{\prime},\mathbf{r}^{\prime\prime}).\end{split}$
(26)

Similar expressions hold for the other correlation functions. Hence, the
correlation functions $\mathds{C}^{\text{FF'}}_{\text{env}}$ always consist of
a global equilibrium term and a local equilibrium correction vanishing for
$T_{\text{s}}=T_{\text{b}}$. Therefore, we can also separate the emitted power
into exactly these two contributions

$P_{\omega}^{\rm env}=P_{\omega}^{\text{eq}}+P_{\omega}^{\text{leq}}.$ (27)

We will refer to $P_{\omega}^{\text{eq}}$ by “equilibrium contribution” (EQC)
and to $P_{\omega}^{\text{leq}}$ by “local equilibrium contribution” (LEQC).

In the following, we provide the explicit expressions for the different terms.
The special cases of these expressions for a perfect metallic or black body
substrate are given in appendices D and E

###  III.1 Contribution of the substrate

First, we recall the known expression for the thermal radiation of the planar
substrate into the background. It reads Polder

$P_{\omega}^{\text{s}}=A\bigl{(}\Theta_{\text{s}}-\Theta_{\text{b}}\bigr{)}\sum_{k=\text{E,H}}\int_{0}^{k_{0}}\\!\\!\frac{{\rm
d}k_{\perp}}{2\pi}k_{\perp}(1-|r_{k}|^{2})$ (28)

where $r_{\text{E}}$ and $r_{\text{H}}$ are the Fresnel coefficients for p-
and s-polarized light as defined in appendix A and $A$ is the area of a plane
above the substrate through which the heat flux is flowing.

###  III.2 EQC

Using the expressions for the correlation functions, we find

$\begin{split}P_{\omega}^{\text{eq}}&=-4\Theta_{\text{b}}\mu_{0}\omega\epsilon_{ijz}\int{\rm
d}^{2}x\,\\\
&\quad\times\text{Re}\Bigl{[}\mathds{G}_{\text{E}k}(\mathbf{r},\mathbf{r}_{\text{p}})\Bigl{[}\boldsymbol{\chi}^{k}+\mathds{F}_{k}\boldsymbol{\chi}^{\bar{k}}\mathds{F}_{k}^{\dagger}\Bigr{]}\mathds{G}_{\text{H}k}^{\dagger}(\mathbf{r},\mathbf{r}_{\text{p}})\\\
&\quad+\mathds{G}_{\text{E}k}(\mathbf{r},\mathbf{r}_{\text{p}})\Bigl{[}\boldsymbol{\chi}^{k}\mathds{F}_{\bar{k}}^{\dagger}+\mathds{F}_{k}\boldsymbol{\chi}^{\bar{k}}\Bigr{]}\mathds{G}_{\text{H}\bar{k}}^{\dagger}(\mathbf{r},\mathbf{r}_{\text{p}})\Bigr{]}_{ij}\end{split}$
(29)

where we introduced the generalized susceptibility block matrix (Einstein
summation over $k=\text{E,H}$)

$\begin{split}\boldsymbol{\chi}^{k}&=k_{0}^{2}\text{Im}\left(\boldsymbol{\alpha}_{kk}\right)-k_{0}^{4}\boldsymbol{\alpha}_{kk}\text{Im}\Bigl{(}\langle\mathds{G}_{kk}(\mathbf{r}_{\text{p}})\rangle\\\
&\quad+\langle\mathds{G}_{\text{HE}}(\mathbf{r}_{\text{p}})\rangle\mathds{F}_{\bar{k}}\Bigr{)}\boldsymbol{\alpha}_{kk}^{\dagger},\end{split}$
(30)

analogue to the one introduced for the purely electric case Krueger ; Herz2 ,
and

$\mathds{F}_{k}=k_{0}^{2}V\chi_{k}\left[\mathds{1}-k_{0}^{2}V\chi_{k}\langle\mathds{G}_{kk}(\mathbf{r}_{\text{p}})\rangle\right]^{-1}\langle\mathds{G}_{\text{HE}}(\mathbf{r}_{\text{p}})\rangle.$
(31)

The bar notes the “inversion” $\bar{\text{E}}=\text{H}$ and
$\bar{\text{H}}=\text{E}$.

Finally, by inserting the Green’s functions from appendix A and a lengthy
calculation we find

$\begin{split}P_{\omega}^{\text{eq}}&=-k_{0}\Theta_{\text{b}}\Bigl{[}\left(\chi_{\text{E},\perp}+|F_{\text{E}}|^{2}\chi_{\text{H},\perp}\right)I_{\text{E},\perp}^{\text{pr}}+\chi_{\text{E},z}I_{\text{E},z}^{\text{pr}}\\\
&\quad+\frac{\varepsilon_{0}}{\mu_{0}}\left[\left(\chi_{\text{H},\perp}+|F_{\text{H}}|^{2}\chi_{\text{E},\perp}\right)I_{\text{H},\perp}^{\text{pr}}+\chi_{\text{H},z}I_{\text{H},z}^{\text{pr}}\right]\\\
&\quad+2\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\text{Re}\Bigl{[}\left(\chi_{\text{H},\perp}F_{\text{E}}-\chi_{\text{E},\perp}F_{\text{H}}^{*}\right)I_{\text{c}}^{\text{pr}}\Bigr{]}\Bigr{]}\end{split}$
(32)

with

$\displaystyle I_{k,\perp}^{\text{pr}}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}}{k_{z}k_{0}}\biggl{(}\frac{k_{z}^{2}}{k_{0}^{2}}\big{|}1-r_{k}e^{2\text{i}k_{z}d}\big{|}^{2}$
$\displaystyle\quad+\big{|}1+r_{\bar{k}}e^{2\text{i}k_{z}d}\big{|}^{2}\biggr{)},$
(33) $\displaystyle I_{k,z}^{\text{pr}}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}^{3}}{k_{0}^{3}k_{z}}\big{|}1+r_{k}e^{2\text{i}k_{z}d}\big{|}^{2}$
(34)

and

$\begin{split}I_{\text{c}}^{\text{pr}}&=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}}{k_{0}^{2}}\bigl{[}\left(1+r_{\text{H}}e^{2\text{i}k_{z}d}\right)\left(1-r_{\text{H}}e^{2\text{i}k_{z}d}\right)^{*}\\\
&\quad+\left(1-r_{\text{E}}e^{2\text{i}k_{z}d}\right)\left(1+r_{\text{E}}e^{2\text{i}k_{z}d}\right)^{*}\bigr{]}.\end{split}$
(35)

###  III.3 Nanoparticle contribution

By employing the fluctuation-dissipation theorem for the electric and magnetic
dipole moments

$\displaystyle\big{\langle}\\!\\!\big{\langle}\mathbf{p}_{\text{fl}}\otimes\mathbf{p}_{\text{fl}}^{\dagger}\big{\rangle}\\!\\!\big{\rangle}^{\omega}$
$\displaystyle=\frac{2}{\mu_{0}\omega^{3}}\Theta_{\text{p}}\mathds{A}_{\text{EE}}^{-1}\boldsymbol{\chi}^{\text{E}}\mathds{A}_{\text{EE}}^{-1\dagger},$
(36)
$\displaystyle\big{\langle}\\!\\!\big{\langle}\mathbf{m}_{\text{fl}}\otimes\mathbf{m}_{\text{fl}}^{\dagger}\big{\rangle}\\!\\!\big{\rangle}^{\omega}$
$\displaystyle=\frac{2}{\mu_{0}\omega^{3}}\Theta_{\text{p}}\mathds{A}_{\text{HH}}^{-1}\boldsymbol{\chi}^{\text{H}}\mathds{A}_{\text{HH}}^{-1\dagger},$
(37)

we obtain

$\displaystyle P_{\omega}^{\text{np}}$
$\displaystyle=4\Theta_{\text{p}}\mu_{0}\omega\epsilon_{ijz}\int{\rm
d}^{2}x\,$
$\displaystyle\quad\times\text{Re}\Bigl{[}\mathds{G}_{\text{E}k}(\mathbf{r},\mathbf{r}_{\text{p}})\Bigl{[}\boldsymbol{\chi}^{k}+\mathds{F}_{k}\boldsymbol{\chi}^{\bar{k}}\mathds{F}_{k}^{\dagger}\Bigr{]}\mathds{G}_{\text{H}k}^{\dagger}(\mathbf{r},\mathbf{r}_{\text{p}})$
$\displaystyle\quad+\mathds{G}_{\text{E}k}(\mathbf{r},\mathbf{r}_{\text{p}})\Bigl{[}\boldsymbol{\chi}^{k}\mathds{F}_{\bar{k}}^{\dagger}+\mathds{F}_{k}\boldsymbol{\chi}^{\bar{k}}\Bigr{]}\mathds{G}_{\text{H}\bar{k}}^{\dagger}(\mathbf{r},\mathbf{r}_{\text{p}})\Bigr{]}_{ij}.$
(38)

Hence, we can write

$P_{\omega}^{\text{np}}=-P_{\omega}^{\text{eq}}|_{T_{\text{s}}=T_{\text{p}}}.$
(39)

This is a logical result because in global thermal equilibrium
$P_{\omega}^{\rm s}=P_{\omega}^{\rm leq}=0$ holds so that
$P_{\omega}^{\text{np}}=-P_{\omega}^{\text{eq}}$ must be fulfilled to have
$P_{\omega}^{\rm tot}=0$. In appendix C we compare the thermal emission
$P_{\omega}^{\rm tot}=P_{\omega}^{\text{np}}+P_{\omega}^{\text{eq}}$ for the
purely electric case and $T_{\text{s}}=T_{\text{p}}$ with previously obtained
results.

###  III.4 LEQC

Similarly, we obtain for the LEQC

$P_{\omega}^{\text{leq}}=P_{\omega,1}^{\text{leq}}+P_{\omega,2}^{\text{leq}}$
(40)

with

$\begin{split}P_{\omega,1}^{\text{leq}}&=k_{0}^{3}\left(\Theta_{\text{b}}-\Theta_{\text{s}}\right)\text{Im}\biggl{[}2\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\alpha_{\text{HE}}R_{c}^{\text{pr}}\\\
&\quad+\sum_{j\in\\{\perp,z\\}}\left(\alpha_{\text{EE},j}R_{\text{E},j}^{\text{pr}}+\frac{\varepsilon_{0}}{\mu_{0}}\alpha_{\text{HH},j}R_{\text{H},j}^{\text{pr}}\right)\biggr{]}\end{split}$
(41)

and the integrals ($k=\text{H,E}$)

$\displaystyle R_{k,\perp}^{\text{pr}}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}}{k_{0}k_{z}}\Bigl{[}\frac{k_{z}^{2}}{k_{0}^{2}}(1-|r_{k}|^{2})\left[1-r_{k}e^{2\text{i}k_{z}d}\right]$
$\displaystyle\quad+(1-|r_{\bar{k}}|^{2})\left[1+r_{\bar{k}}e^{2\text{i}k_{z}d}\right]\Bigr{]},$
(42) $\displaystyle R_{k,z}^{\text{pr}}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}^{3}}{k_{0}^{3}k_{z}}(1-|r_{k}|^{2})\left[1+r_{k}e^{2\text{i}k_{z}d}\right],$
(43) $\displaystyle R_{c}^{\text{pr}}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}}{k_{0}^{2}}\Bigl{[}(1-|r_{\text{H}}|^{2})r_{\text{H}}-(1-|r_{\text{E}}|^{2})r_{\text{E}}\Bigr{]}e^{2\text{i}k_{z}d}$
(44)

as well as

$\displaystyle P_{\omega,2}^{\text{leq}}$
$\displaystyle=\frac{k_{0}^{6}}{8}\left(\Theta_{\text{s}}-\Theta_{\text{b}}\right)\text{Re}\biggl{[}\Bigl{(}\Gamma_{\text{E},\perp}|\alpha_{\text{EE},\perp}|^{2}+\frac{\varepsilon_{0}}{\mu_{0}}\Gamma_{\text{H},\perp}|\alpha_{\text{HE}}|^{2}$
$\displaystyle\quad+2\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\Gamma_{c}\alpha_{\text{EE},\perp}\alpha_{\text{HE}}^{*}\Bigr{)}I_{\text{E},\perp}^{\text{pr}}+2\Gamma_{\text{E},z}|\alpha_{\text{EE},z}|^{2}I_{\text{E},z}^{\text{pr}}$
$\displaystyle\quad+\frac{\varepsilon_{0}^{2}}{\mu_{0}^{2}}\Bigl{[}\Bigl{(}\Gamma_{\text{H},\perp}|\alpha_{\text{HH},\perp}|^{2}-2\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}\Gamma_{c}^{*}\alpha_{\text{HH},\perp}\alpha_{\text{HE}}^{*}$
$\displaystyle\quad+\frac{\mu_{0}}{\varepsilon_{0}}\Gamma_{\text{E},\perp}|\alpha_{\text{HE}}|^{2}\Bigr{)}I_{\text{H},\perp}^{\text{pr}}+2\Gamma_{\text{E},z}|\alpha_{\text{HH},z}|^{2}I_{\text{H},z}^{\text{pr}}\Bigr{]}$
$\displaystyle\quad+2\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\Bigl{[}-\Gamma_{\text{E},\perp}\alpha_{\text{EE},\perp}\alpha_{\text{HE}}^{*}+\frac{\varepsilon_{0}}{\mu_{0}}\Gamma_{\text{H},\perp}^{*}\alpha_{\text{HH},\perp}^{*}\alpha_{\text{HE}}$
$\displaystyle\quad+\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\Bigl{(}\Gamma_{c}\alpha_{\text{EE},\perp}\alpha_{\text{HH},\perp}^{*}-\Gamma_{c}^{*}|\alpha_{\text{HE}}|^{2}\Bigr{)}\Bigr{]}I_{\text{c}}^{\text{pr}}\biggr{]}$
(45)

with the integrals ($k=\text{H,E}$)

$\displaystyle\Gamma_{k,\perp}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}}{k_{0}k_{z}}\left(\frac{k_{z}^{2}}{k_{0}^{2}}\left(1-|r_{k}|^{2}\right)+1-|r_{\bar{k}}|^{2}\right)$
$\displaystyle\quad+\int_{k_{0}}^{\infty}\frac{\text{d}k_{\perp}}{\pi}\frac{k_{\perp}e^{-2|k_{z}|d}}{k_{0}|k_{z}|}\text{Im}\left(\frac{|k_{z}|^{2}}{k_{0}^{2}}r_{k}+r_{\bar{k}}\right),$
(46) $\displaystyle\Gamma_{k,z}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}^{3}}{k_{0}^{2}k_{z}}\left(1-|r_{k}|^{2}\right)$
$\displaystyle\quad+\int_{k_{0}}^{\infty}\frac{\text{d}k_{\perp}}{\pi}\frac{k_{\perp}^{3}}{k_{0}^{2}|k_{z}|}e^{-2|k_{z}|d}\text{Im}(r_{k}),$
(47) $\displaystyle\Gamma_{c}$
$\displaystyle=\int_{0}^{k_{0}}\frac{\text{d}k_{\perp}}{2\pi}\frac{k_{\perp}}{k_{0}^{2}}(2-|r_{\text{H}}|^{2}-|r_{\text{E}}|^{2})$
$\displaystyle\quad+\int_{k_{0}}^{\infty}\frac{\text{d}k_{\perp}}{\text{i}\pi}\frac{k_{\perp}}{k_{0}^{2}}e^{-2|k_{z}|d}\left(\text{Im}(r_{\text{H}})-\text{Im}(r_{\text{E}})\right).$
(48)

One can expect the first part $P^{\rm leq}_{\omega,1}$ to be negative for
$T_{\text{b}}<T_{\text{s}}$. Since it is proportional to the imaginary parts
of the polarizabilities, it can be interpreted as that fraction of the thermal
emission of the substrate which is absorbed or shielded by the nanoparticle.
We have checked that by adding, for example, the power $P_{\omega}^{\rm s}$
from Eq. (28) emitted by the substrate for a surface area $A$ which equals the
cross section of the nanoparticle, then $P_{\omega}^{\rm s}+P_{\omega,1}^{\rm
leq}$ would be positive for all frequencies. The second part $P^{\rm
leq}_{\omega,2}$ which is proportional to the square of the polarizabilities
describes, in contrast to $P_{\omega,1}^{\text{leq}}$, the power of the
substrate scattered by the nanoparticle into the far field. The
$\Gamma$-integrals are well known from the heat flux received by a small
sphere from a semi-infinite substrate when neglecting multiple reflections
Krueger ; Herz2 .

##  IV Numerical evaluation

Figure 2: Semi-logarithmic plot of the z-component of spectral power signals
for $R=10$ nm, $d=40$ nm, $T_{\text{p}}=700$ K, $T_{\text{s}}=500$ K, and
$T_{\text{b}}=300$ K for all configurations of gold and SiC as materials of
nanoparticle and substrate. The vertical dashed lines indicate the SPhP and
TOP wavenumber. Panel (a) shows the spectrum of the total signal, panel (b)
the EQC and $P^{\text{np}}_{\omega}$, and panel (c,d) the two LEQCs.

Alltogether the total spectral power emitted into the far-field is given by

$P_{\omega}^{\rm tot}=P_{\omega}^{\rm s}+(P_{\omega}^{\rm eq}+P_{\omega}^{\rm
np})+P_{\omega,1}^{\rm leq}+P_{\omega,2}^{\rm leq}$ (49)

As already mentioned $P_{\omega}^{\rm s}$ does not depend on the presence of
the nanoparticle and is therefore not of interest in the following discussion.
To evaluate these formulas for the emitted power, we choose to compare four
different configurations where nanoparticle and substrate can consist of the
dielectric material SiC or the metal gold. The permittivity of SiC is modelled
by a Lorentz oscillator Palik

$\varepsilon_{\text{SiC}}(\omega)=\varepsilon_{\infty}\frac{\omega_{l}^{2}-\omega^{2}-\text{i}\Gamma\omega}{\omega_{t}^{2}-\omega^{2}-\text{i}\Gamma\omega}$
(50)

with $\varepsilon_{\infty}=6.7$, $\omega_{l}=1.827\times 10^{14}\text{
rad}\text{ s}^{-1}$, $\omega_{t}=1.495\times 10^{14}\text{ rad}\text{
s}^{-1}$, and $\Gamma=0.9\times 10^{12}\text{ rad}\text{s}^{-1}$. For gold we
employ the Drude model Ordal

$\varepsilon_{\text{Au}}(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2}+\text{i}\Gamma\omega}$
(51)

with $\varepsilon_{\infty}=8.344$, $\omega_{p}=1.372\times 10^{16}\text{
rad}\text{ s}^{-1}$, and $\Gamma=4.059\times 10^{13}\text{ rad}\text{
s}^{-1}$. Both are necessary for $\chi_{k}$. For isotropic spherical particles
those material constants are given by DedkovKyasov07 ; Dong

$\displaystyle\chi_{\text{E}}$
$\displaystyle=\frac{9\text{i}}{2k_{0}^{3}}\frac{\varepsilon_{\rm
p}j_{1}(y)[xj_{1}(x)]^{\prime}-j_{1}(x)[yj_{1}(y)]^{\prime}}{\varepsilon_{\rm
p}j_{1}(y)[xh_{1}^{(1)}(x)]^{\prime}-h_{1}^{(1)}(x)[yj_{1}(y)]^{\prime}},$
(52) $\displaystyle\chi_{\text{H}}$
$\displaystyle=\frac{9\mu_{0}\text{i}}{2\varepsilon_{0}k_{0}^{3}}\frac{j_{1}(y)[xj_{1}(x)]^{\prime}-j_{1}(x)[yj_{1}(y)]^{\prime}}{j_{1}(y)[xh_{1}^{(1)}(x)]^{\prime}-h_{1}^{(1)}(x)[yj_{1}(y)]^{\prime}}$
(53)

with $x=k_{0}R$ and $y=\sqrt{\varepsilon_{\text{p}}}x$. $j_{1}$ and
$h_{1}^{(1)}$ are the spherical Bessel and Hankel functions of the first kind,
respectively. The prime notation indicates the derivation with respect to the
considered argument. For small radii $R$, meaning $x,y<\\!\\!\\!<1$, these two
expressions can be simplified within the LWA by

$\displaystyle\chi_{\text{E}}$
$\displaystyle=3\frac{\varepsilon_{\text{p}}-1}{\varepsilon_{\text{p}}+2},$
(54) $\displaystyle\chi_{\text{H}}$
$\displaystyle=\frac{\mu_{0}}{\varepsilon_{0}}\frac{(k_{0}R)^{2}}{10}(\varepsilon_{\text{p}}-1).$
(55)

Note, that $\text{Im}(\chi_{\text{H}})$ describes the absorptivity of a small
nanoparticle due to the induction of eddy currents as discussed in detail in
Refs. HuthEtAl2010 ; Tomchuk . Therefore, our general approach introducing the
magnetic response $\chi_{\text{H}}$ is by virtue of the correct form of
$\chi_{\text{H}}$ including the contributions of eddy currents which are
important for metallic nanoparticles.

###  IV.1 Spectral signatures

To ensure that the dipole approximation is applicable, the nanoparticle with
$R=10$ nm is placed above the substrate at distance $d=40$ nm. All three
components have different temperatures: $T_{\text{p}}=700$ K,
$T_{\text{s}}=500$ K, and $T_{\text{b}}=300$ K. In the plots we refer to the
nanoparticle by “np” and to the substrate by “sub”. Additionally, we highlight
the surface mode resonance (SPhP) wavenumber and the transversal optical
phonon (TOP) wavenumber of the SiC substrate by vertical dashed lines. To
employ semi-logarithmic plots even for negative values, as necessary for
$P_{\omega,1}^{\rm leq}$, we adopt the method outlined by Webber Webber .

In Fig. 2(a) we show the total spectral power without the power emitted by the
semi-infinite material $P_{\omega}^{\rm s}$ from Eq. (28), i.e. we show
$P_{\omega}=P_{\omega}^{\rm tot}-P_{\omega}^{\rm s}=P^{\rm np}_{\omega}+P^{\rm
eq}_{\omega}+P^{\rm leq}_{\omega,1}+P^{\rm leq}_{\omega,2}$. It can be seen
that $P_{\omega}$ is negative for SiC substrates at the TOP wavenumber and,
partly, for a SiC nanoparticle above a gold substrate outside the reststrahlen
band. For SiC nanoparticles the SPhP mode resonance dominates the spectrum. To
see where these effects are coming from, we show the contributions
$P_{\omega}^{\rm eq}+P_{\omega}^{\rm np}$, $P_{\omega,1}^{\rm leq}$, and
$P_{\omega,2}^{\rm leq}$ in Fig. 2(b)-(d), separately.

Figure 3: Semi-logarithmic plot of the z-component of the electric (left) and
magnetic (right) spectral power. (a,b) show the EQC and
$P_{\omega}^{\text{np}}$, (c,d) and (e,f) show the two parts of the LEQC.
Parameters are the same as in Fig. 2.

The quantity $P_{\omega}^{\rm eq}+P_{\omega}^{\rm np}$ is plotted in Fig. 2(b)
corresponding to the total spectral power emitted in the case where
$T_{\text{s}}=T_{\text{b}}$. This power is positive and because of
$T_{\text{p}}\gg T_{\text{b}}$ it is mainly emitted by the nanoparticle. The
spectra are showing resonances for the SiC nanoparticle close to the TOP and
SPhP wavenumbers. The pronounced peaks near the SPhP wavenumber are
blueshifted due to the strong coupling between the nanoparticle and the
substrate. Large shifts of the SPhP resonance frequency – but towards lower
frequencies because of different material compositions – were already
encountered theoretically Joulain2 ; Edalatpour2 ; Herz  and observed
experimentally Babuty ; Jones . For the Au nanoparticle above the Au substrate
there are, of course, no resonances in the infrared. Nonetheless, even for Au
nanoparticles the TOP and SPhP resonances of the SiC substrate have, due to
the interaction between the nanoparticle and the substrate, some impact on the
emitted power. Since the metal substrate is more reflective than the SiC
substrate, the signals emitted by the nanoparticles into the far field are
larger for the metal substrates than for SiC substrates, because for Au the
part radiated towards the substrate is reflected into the far-field whereas
for SiC this radiation is partially absorbed by the substrate. This explains
also the dip in the power spectrum at the SPhP wavenumber for the Au
nanoparticle above the SiC substrate.

Now, Fig. 2(c) and (d) show $P_{\omega,1}^{\rm leq}$ and $P_{\omega,2}^{\rm
leq}$ which would be the power emitted in the case $T_{\text{p}}=T_{\text{b}}$
when neglecting, again, the directly emitted power by the substrate
$P^{\text{s}}_{\omega}$. As discussed above, $P_{\omega,1}^{\rm leq}$ can be
understood as that power of the substrate emission $P_{\omega}^{\rm s}$ which
is absorbed in the nanoparticle due to the multiple interactions between the
particle and the substrate. As can be seen in Fig. 2(c) this contribution can
be negative which is as mentioned in Sec. III.D due to the omisson of the
substrate contribution. Therefore, it is not astonishing that
$P_{\omega,1}^{\rm leq}$ has a resonance with negative value at the SPhP
wavenumber for the SiC nanoparticle where the thermal radiation coming from
the substrate is strongly absorbed. For the SiC substrate there is also a
resonance with negative value at the TOP wavenumber which simply means that
also a part of the power emitted by the surface is reflected by the
nanoparticle and then reabsorbed by the substrate. The scattered power
$P_{\omega,2}^{\rm leq}$, on the other hand, is always positive and the
scattering is particluarly strong around the SPhP wavenumber if the particle
or the substrate are made of SiC.

Let us conclude the investigation of the spectra by ascertaining the impact of
the electric and magnetic part for each configuration and the different
contributions $P_{\omega}^{\rm eq}+P_{\omega}^{\rm np}$, $P_{\omega,1}^{\rm
leq}$, and $P_{\omega,2}^{\rm leq}$ shown in Fig. 3. We do not show the impact
of the mixed terms because they are negligibly small compared to the other
purely electric or magnetic ones. It is interesting that the resonant thermal
emission and scattering near the SPhP mode stem from the electric part due to
resonance in $r_{\text{E}}$ and the peak corresponding to the TOP mode stems
from the magnetic part [compare Fig. 3 (a) and (b) and Fig. 3 (e) and (f)].
Because of the higher magnitudes of the electric part, the SPhP mode dominates
the spectrum in Fig. 2a for SiC nanoparticles. Compared to the electric
contribution, the magnetic one is by two orders of magnitudes smaller for the
SiC nanoparticle and about one order of magnitude larger for the gold
nanoparticle. Thus, as could be expected, the magnetic part has a bigger
impact on the signal for metallic than for dielectric nanoparticles. This can
be in particularly observed in the magnetic emitted power $P_{\omega}^{\rm
eq}+P_{\omega}^{\rm np}$ in Fig. 3(b) and in the magnetic scattered power
$P_{\omega,2}^{\rm leq}$ shown in Fig. 3(f).

###  IV.2 Temperature and size dependence

Loosely speaking we have the proportionalities

$\displaystyle P_{\omega}^{\rm eq}+P_{\omega}^{\rm np}$
$\displaystyle\propto(T_{\rm p}-T_{\rm b})\text{Im}(\alpha),$ (56)
$\displaystyle P_{\omega,1}^{\rm leq}$ $\displaystyle\propto(T_{\rm b}-T_{\rm
s})\text{Im}(\alpha),$ (57) $\displaystyle P_{\omega,2}^{\rm leq}$
$\displaystyle\propto(T_{\rm b}-T_{\rm s})|\alpha|^{2},$ (58)

where depending on whether the magnetic or electric part dominates one has to
consider $\alpha_{\rm EE}$ or $\alpha_{\rm HH}$. The cross terms can typically
be neglected. That means that the spectra are highly dependent on the
temperature and the radius of the nanoparticle. For instance, by switching the
temperatures from $T_{p}=700\,{\rm K}$ and $T_{\rm s}=500\,{\rm K}$ as in Fig.
2 to $T_{p}=500\,{\rm K}$ and $T_{\rm s}=700\,{\rm K}$ one obtains the results
shown in Fig. 4. It can be clearly seen that for all material combinations the
spectra highly depend on the choice of temperature. It is also clear, that the
signs of the different contributions $P_{\omega}^{\rm eq}+P_{\omega}^{\rm
np}$, $P_{\omega,1}^{\rm leq}$, and $P_{\omega,2}^{\rm leq}$ highly depend on
the relative value of the different temperatures in the system. In addition,
in an experimental setup being able to control $T_{\rm s}$, $T_{\rm b}$, and
$T_{\rm p}$ separately it would be possible to measure only $P_{\omega,1}^{\rm
leq}$ and $P_{\omega,2}^{\rm leq}$ by choosing $T_{\rm p}=T_{\rm b}$ or only
$P_{\omega}^{\rm eq}+P_{\omega}^{\rm np}$ by choosing $T_{\rm s}=T_{\rm b}$.

Figure 4: Semi-logarithmic plot of the z-component of spectral power for
$R=10$ nm, $d=40$ nm, $T_{\text{p}}=700$ K, $T_{\text{s}}=500$ K, and
$T_{\text{b}}=300$ K ($T_{\rm p}>T_{\rm s}$, solid lines) for all
configurations of gold and SiC as materials of nanoparticle and substrate as
in Fig. 2 and with the switched temperatures $T_{\text{s}}=700$ K, and
$T_{\text{p}}=500$ K ($T_{\rm s}>T_{\rm p}$, dashed lines).

Since for small radii the polarizabilites fulfill $\alpha_{\rm EE}\propto
R^{3}$ and $\alpha_{\rm HH}\propto R^{5}$ it is clear from the above
proportionalities that the electric part of $P_{\omega}^{\rm
eq}+P_{\omega}^{\rm np},P_{\omega,1}^{\rm leq}$ is $\propto R^{3}$ and the
magnetic part is $\propto R^{5}$ whereas the electric part of
$P_{\omega,2}^{\rm leq}$ is $\propto R^{6}$ and the magnetic part is $\propto
R^{10}$ suggesting that for large enough radii always the scattering term will
dominate as long as $T_{\rm s}\neq T_{\rm b}$ Herz . In Fig. 5 we explicitely
show the radius dependence for small radii when $d=100\,{\rm nm}$. It can be
seen that for radii smaller than $10\,{\rm nm}$ the emitted power scales like
$R^{3}$ for the dielectric and magnetic nanoparticle whereas for larger radii
the magnetic contribution starts to dominate for the metallic nanoparticle
leading to a $R^{5}$ dependence. Of course these dependencies highly depend on
the material parameters and the particular dominance of either
$P_{\omega}^{\rm eq}+P_{\omega}^{\rm np},P_{\omega,1}^{\rm leq}$ or
$P_{\omega,2}^{\rm leq}$.

Figure 5: Semi-logarithmic plot of the z-component of spectral power as
function of the radius for $d=100$ nm, $T_{\text{p}}=700$ K,
$T_{\text{s}}=500$ K, and $T_{\text{b}}=300$ K for all configurations of gold
and SiC as materials of nanoparticle and substrate as in Fig. 2 at the surface
mode wavenumber. The dashed line indicates an $R^{3}$ and the dotted line an
$R^{5}$ power law.

###  IV.3 Angle dependence

Similar to the approach in Ref. Joulain2  our general expressions obtained by
integration of the mean Poynting vector over the whole x-y plane can still be
used to obtain also the emission into a certain solid angle. That is because
the integrals over $\mathbf{k}_{\perp}$ are taking the emission into all
possible directions in the half-space above the nanoparticle into account. To
get the angle-dependent expressions, the $k_{\perp}$ integrals for the
quantitities $P_{\omega}^{\rm s},I^{\rm pr}_{k,\perp},I^{\rm pr}_{k,z},I^{\rm
pr}_{c},R^{\rm pr}_{k,\perp},R^{\rm pr}_{k,z},R^{\rm pr}_{c}$ must be
converted following the scheme

$\begin{split}\int_{0}^{k_{0}}\\!\\!\frac{{\rm
d}^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}f(\mathbf{k}_{\perp})&=\int_{0}^{k_{0}}\\!\\!\frac{{\rm
d}k_{\perp}}{2\pi}\,k_{\perp}f(k_{\perp})\\\
&=\int_{0}^{\pi/2}\\!\\!\frac{{\rm
d}\theta}{2\pi}\,\cos(\theta)\sin(\theta)k_{0}^{2}f(k_{0}\sin(\theta))\end{split}$
(59)

where $f$ denotes the integrand of the different quantities and $\theta$ is
the angle between $\mathbf{k}_{\perp}$ and the surface normal as indicated in
Fig. 6. Note, that in our calculations we have already used the rotational
symmetry to simplify the two-dimensional integrals to a one-dimensional
integral. Therefore, when it is necessary to consider only the thermal
emission into a certain solid angle range $\Delta\theta$ and $\Delta\phi$ it
is inevitable to make the replacements

$\begin{split}\int_{0}^{k_{0}}\\!&\frac{{\rm
d}k_{\perp}}{2\pi}k_{\perp}f(k_{\perp})\\\
&\rightarrow\frac{\Delta\phi}{2\pi}\int_{\Delta\theta}\frac{{\rm
d}\theta}{2\pi}\,\cos(\theta)\sin(\theta)k_{0}^{2}f(k_{0}\sin(\theta))\end{split}$
(60)

in the terms $P_{\omega}^{\rm s},I^{\rm pr}_{k,\perp},I^{\rm pr}_{k,z},I^{\rm
pr}_{c},R^{\rm pr}_{k,\perp},R^{\rm pr}_{k,z},R^{\rm pr}_{c}$.

To illustrate the angle dependence of the spectra we show in Fig. 6 the
dependence of the emitted power spectrum on the opening angle $\theta$ as
sketched in the inset for a SiC nanoparticle above a SiC substrate using Eq.
(59). It is obvious that the spectrum is highly depending on the opening
angle. It is interesting that for small angles a resonance appears close to
TOP wavenumber which is not present for large angles, whereas for an angle
$\theta=60^{\circ}$ the spectrum around the surface mode wavenumber
corresponds already quite well to the full spectrum. Hence, for a modeling of
experiments the inclusion of the correct opening angle over which the
scattered radiation is collected is very important.

Figure 6: Semi-logarithmic plot of the z-component of the spectral power as
function of the opening angle $\theta$ for a SiC nanoparticle above a SiC
substrate. Inset: Definition of the angle $\theta$. Parameters are the same as
in Fig. 2

###  IV.4 Distance dependence

Finally, we want to discuss the distance dependence of the total power without
the power emitted by the substrate

$P=\int_{0}^{\infty}\frac{{\rm d}\omega}{2\pi}(P_{\omega}^{\rm
tot}-P_{\omega}^{\rm s}).$ (61)

This quantity is plotted over the distance normalized to the particle radius
$R$ in Fig. 7 for the four possible material configurations. In Fig. 7 also
the different contributions are shown separately. In principle, three
different regimes are visible. The extreme near-field ($d\sim R$) where the
coupling between the particle and the evanescent waves is dominating. In this
regime the scattering contribution $P^{\rm leq}_{2}$ gives a strong
increasement of the heat flux when approaching the medium. On the other hand,
such an enhancment cannot be seen in the total heat flux with our set of
parameters, since in this case the contribution of $P^{\rm eq}+P^{\rm np}$
dominates for the SiC nanoparticle and $P^{\rm leq}_{1}$ dominates for the
gold nanoparticle. However, by increasing the nanoparticle radius or the
substrate temperature $P^{\rm leq}_{2}$ will eventually dominate the total
power emission. The transition region towards the far-field where due to
reflections there will be an interference pattern is starting at distances
$d\approx 300R$ which correspond to the thermal wavelength for
$T_{\text{p}}=700$ K and $R=10$ nm. The oscillations are exponentially damped
so that a constant value is reached in the far-field. Finally, there is an
intermediate regime for $2R<d<300R$. Due to the high reflectivity, the emitted
power (without the direct thermal emission of the substrate) for the Au
substrates is larger than for the SiC substrate in the far-field. It is
interesting that the oscillations most pronounced for SiC nanoparticles. In
the extreme near-field ($d\sim R$) the power emission for the SiC nanoparticle
is the largest for $R=10\,{\rm nm}$. For other choices of parameters and in
particular the radius the power emitted by the Au nanoparticle can be larger
than that of the SiC nanoparticle for certain configurations as can be seen in
Fig. 5 for radii close to 100 nm. It should, of course, be kept in mind that
in the extreme near-field regime also higher multipole orders need to be
included Naraynaswamy2008 ; Otey ; Becerril  and phonon tunneling might also
play a role Pendry ; GangChen .

Figure 7:  Logarithmic plot of the total power $P$ without the direct surface
emitted power from Eq. (61) as function of distance together with the
different contributions $P^{\rm eq}+P^{\rm np}$, $P_{1}^{\rm leq}$, and
$P_{2}^{\rm leq}$. All other parameters are the same as in Fig. 2.

Since scanning thermal microscopes like the TRSTM, TINS, and SNoiM are capable
of measuring either spectra or the distance dependence in a narrow frequency
band around a single frequency we also show in Fig. 8 some results for the
distance dependence of the emitted spectral power at the surface mode
frequency $\omega_{\rm SPhP}$ and a frequency slightly larger than the
longitudinal optical phonon frequency $\omega_{l}$ of SiC. First of all, it
can be seen that the oscillations in the far-field regime are present in all
material combinations for single frequencies and therefore are in the
integrated power $P$ shown in Fig. 7 albeit smeared out for gold
nanoparticles. Furthermore, in the near-field regime an increased or decreased
power emission can be seen for the SiC nanoparticle. This effect is
particularly strong at the surface mode frequency and survives after
integration in the total emitted power for the gold substrate in Fig. 7. For
the Au nanoparticle there can also be an increased power emission when
approaching the material depending on the substrate and frequency. Furthermore
far away from the surface mode frequency the power emission of the Au
nanoparticle can be larger than that of the SiC nanoparticle which, of course,
is particularly strong close to the surface mode frequency.

Figure 8:  Logarithmic plot of the spectral power $P_{\omega}$ as function of
distance at the surface mode frequency and the wavenumber $1100\,{\rm
cm}^{-1}$. All other parameters are the same as in Fig. 2.

##  V Conclusion

We have presented a generalized dipole model within the framework of
fluctuational electrodynamics and with a rigurous treatment of the divergence
in the Green’s functions. Our model is taking eddy currents in the
nanoparticle into account as well as the possibility to have three different
temperatures for the nanoparticle, substrate, and background. We derived an
expression for the spectral power emitted into far-field and discussed
numerical emission spectra and the integrated power for all four combinations
of the materials SiC and gold for the nanoparticles and the substrate. We have
shown that the impact of the three different temperatures can alter the
results tremendously. For example by considering the specific cases
$T_{\text{s}}=T_{\text{b}}$ and $T_{\text{p}}=T_{\text{b}}$ either direct
thermal emission of the nanoparticle or scattering dominates. The relative
values of the three temperatures have a strong impact on the sign of the
different contributions. We have discussed the impact of the magnetic field
contribution due to the eddy currents in metals. Finally, we discussed the
distance dependence of the total power emission without the direct thermal
emission of the substrate. A strong near-field increasement can always be seen
when the scattering term dominates, whereas in the regime where the direct
thermal emission of the nanoparticle dominates there is not necessarily a
near-field increasement in thermal emission. Which of both terms dominates the
power emission highly depends on the parameters, but one can see from the
expressions that for large enough radii scattering always dominates.

In principle, our dipol model can be applied to thermal imaging experiments
like TRSTM, TINS, or SNoiM but several further extensions are necessary in
upcoming works. To treat, e.g. anisotropic or non-reciprocal materials, the
evaluation and usage of the Green’s functions has to be adapted as well as the
definition of the expressions for $\chi_{k}$ in the Lippmann-Schwinger
equation in Eq. (1). To have a better model for thermal imaging experiments,
one could include surface roughness of the surface by perturbation theory
Biehs2010 ; Chen  or a specific surface profile by combining our model with a
numerical method which evaluates the Green’s function. Additionally, the exact
near-field behavior for $d<3R$ requires taking different multipole orders into
account instead of relying on the dipole approximation alone. A simple way to
improve the modeling of the tip shape used in the near-field imaging
experiments could be to extend our work to the case of many dipoles in a
discrete dipole approximation PBAmanybody ; Edalatpour2 ; EkerothEtAl2017 .

###### Acknowledgements.

S.-A. B. acknowledges support from Heisenberg Programme of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under the project No.
404073166.

##  Appendix A Green’s functions

The Green’s functions for $z>z^{\prime}$ above a planar, isotropic, and semi-
infinite substrate at $z<0$ are within the Weyl representation Hecht ; Sipe

$\mathds{G}_{\text{FF'}}(\mathbf{r},\mathbf{r}^{\prime})=\int\frac{\text{d}^{2}k_{\perp}}{(2\pi)^{2}}e^{\text{i}\mathbf{k}_{\perp}(\mathbf{x}-\mathbf{x}^{\prime})}\mathds{G}_{\text{FF'}}(k_{\perp},z,z^{\prime})$
(62)

with $\mathbf{x}=(x,y)^{t}$,
$\mathbf{x}^{\prime}=(x^{\prime},y^{\prime})^{t}$,
$\mathbf{k}_{\perp}=(k_{x},k_{y})^{t}$, and
$\text{F},\text{F'}=\text{H},\text{E}$. For example, the electric Green’s
function for electric sources including the vacuum and scattering contribution
is given by Hecht ; Sipe

$\begin{split}\mathds{G}_{\text{EE}}(k_{\perp},z,z^{\prime})&=\frac{\text{i}{\rm
e}^{\text{i}k_{z}(z-z^{\prime})}}{2k_{z}}\sum_{k=\text{E,H}}\Bigl{(}\mathbf{a}_{k}^{+}(k_{0})\otimes\mathbf{a}_{k}^{+}(k_{0})\\\
&\qquad+e^{2\text{i}k_{z}z^{\prime}}r_{k}\mathbf{a}_{k}^{+}(k_{0})\otimes\mathbf{a}_{k}^{-}(k_{0})\Bigr{)}\end{split}$
(63)

introducing the polarization vectors Hecht ; Sipe

$\displaystyle\mathbf{a}_{\text{H}}^{\pm}(k_{\mu})$
$\displaystyle=\frac{1}{k_{\perp}}\left(k_{y},-k_{x},0\right)^{t},$ (64)
$\displaystyle\mathbf{a}_{\text{E}}^{\pm}(k_{\mu})$
$\displaystyle=\frac{1}{k_{\mu}k_{\perp}}\left(\mp k_{x}k_{\mu,z},\mp
k_{y}k_{\mu,z},k_{\mu,z}^{2}\right)^{t}$ (65)

with $k_{\mu}^{2}=\epsilon_{\mu}\omega^{2}/c^{2}$ and
$k_{\mu,z}^{2}=k_{0}^{2}\epsilon_{\mu}-\kappa^{2}$. Here, $r_{\text{H/E}}$ are
the Fresnel amplitude reflection coefficients

$r_{\text{H}}=\frac{k_{z}-k_{\text{s},z}}{k_{z}-k_{\text{s},z}},\quad
r_{\text{E}}=\frac{\varepsilon_{\text{s}}k_{z}-k_{\text{s},z}}{\varepsilon_{\text{s}}k_{z}-k_{\text{s},z}}$
(66)

with $k_{\text{s},z}=\sqrt{\varepsilon_{\text{s}}k_{0}^{2}-k_{\perp}^{2}}$.
$\varepsilon_{\text{s}}$ denotes the substrate permittivity. To obtain
$\mathds{G}_{\text{HH}}$ one can simply interchange
$r_{\text{E}}\leftrightarrow r_{\text{H}}$ and multiply by
$\varepsilon/\mu_{0}$. To obtain $\mathds{G}_{\text{HE}}$ replace the left-
handed vector $\mathbf{a}_{\text{E}}^{+}$ by
$\mathbf{a}_{\text{H}}^{+}/\omega\mu_{0}$ and $\mathbf{a}_{\text{H}}^{+}$ by
$-\mathbf{a}_{\text{E}}^{+}/\omega\mu_{0}$. One obtains
$\mathds{G}_{\text{EH}}$ from $\mathds{G}_{\text{HE}}$ by interchanging
$r_{\text{E}}\leftrightarrow r_{\text{H}}$ and switching the sign.

The electric Green’s function for the transmitted part of the fields with
electric sources inside the semi-infinite medium ($z^{\prime}<0$) is given by
the transmitted or scattered part, only, for $z>0$ and reads

$\begin{split}\mathds{G}_{\text{EE}}^{\text{s}}(k_{\perp},z,z^{\prime})&=\frac{\text{i}{\rm
e}^{\text{i}(k_{z}z-k_{\text{s},z}z^{\prime})}}{2k_{\text{s},z}}\sum_{k=\text{E,H}}t_{k}\mathbf{a}_{k}^{+}(k_{0})\otimes\mathbf{a}_{k}^{+}(k_{\text{s}})\end{split}$
(67)

introducing the amplitude transmission coefficients

$t_{\text{H}}=\frac{2k_{\text{s},z}}{k_{\text{s},z}+k_{z}},\quad
t_{\text{E}}=\frac{k_{\text{s}}}{k_{0}}\frac{2k_{\text{s},z}}{k_{\text{s},z}+k_{z}}.$
(68)

To obtain the other Green’s functions, one can perform the same replacements
as above.

##  Appendix B Volume average of the Green’s functions

The volume averaged functions can be separated into a vacuum contribution and
a scattered one due to the reflecting substrate. The vacuum contribution for
the fully electric Green’s function is Albaladejo ; Fikioris ; Lakhtakia ;
Yaghjian

$\displaystyle V\langle\mathds{G}_{\text{EE,vac}}(\mathbf{r})\rangle$
$\displaystyle=\frac{1}{k_{0}^{2}}\left[\frac{2}{3}\left(1-\text{i}k_{0}R\right)e^{\text{i}k_{0}R}-1\right]\mathds{1}.$
(69)

To obtain the magnetic counterpart, multiply Eq. (69) with
$\varepsilon_{0}/\mu_{0}$. The vacuum part of the function
$\mathds{G}_{\text{EH/HE}}$ vanishes. For the considered media the scattered
part of the Green’s function is well known Sipe . See also appendix A. Its
volume average contains no singularities rendering it possible to approximate
the integral as product of the nanoparticle volume and the integrand evaluated
at $\mathbf{r}^{\prime}=\mathbf{r}=d\mathbf{e}_{z}$ within the long wavelength
approximation (LWA). Thus, we obtain

$\displaystyle\langle\mathds{G}_{\text{EE/HH,sc}}\rangle$
$\displaystyle=\text{i}\int_{0}^{\infty}\frac{\text{d}k_{\perp}}{8\pi}\frac{k_{\perp}e^{2\text{i}k_{z}d}}{k_{z}}\biggl{[}\biggl{(}r_{\text{H/E}}-r_{\text{E/H}}\frac{k_{z}^{2}}{k_{0}^{2}}\biggr{)}$
$\displaystyle\quad\times\mathbf{e}_{\perp}\otimes\mathbf{e}_{\perp}+2\frac{k_{\perp}^{2}}{k_{0}^{2}}r_{\text{E/H}}\mathbf{e}_{z}\otimes\mathbf{e}_{z}\biggr{]},$
(70) $\displaystyle\langle\mathds{G}_{\text{EH/HE,sc}}\rangle$
$\displaystyle=-\frac{\text{i}}{k_{0}}\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\int_{0}^{\infty}\frac{\text{d}k_{\perp}}{8\pi}k_{\perp}e^{2\text{i}k_{z}d}\left(r_{\text{H}}-r_{\text{E}}\right)\mathds{X}.$
(71)

##  Appendix C Comparision with previous results for electrical case and
$T_{\text{s}}=T_{\text{b}}$

If we would restrict ourselves to the purely electric contribution, we have to
set $\chi_{\text{H}}=0$. Then, the polarizability matrices
$\boldsymbol{\alpha}_{\text{HH/HE}}$ contain no non-zero entries anymore and
the matrices $\boldsymbol{\alpha}_{\text{EE}}$ and
$\boldsymbol{\chi}^{\text{E}}$ simplify to

$\displaystyle\boldsymbol{\alpha}_{\text{EE}}$
$\displaystyle=\sum_{j\in\\{\perp,z\\}}\frac{V\chi_{\text{E}}\mathbf{e}_{j}\otimes\mathbf{e}_{j}}{1-k_{0}^{2}V\chi_{\text{E}}\langle
G_{\text{EE},j}(\mathbf{r}_{\text{p}})\rangle},$ (72)
$\displaystyle\chi^{\text{E}}_{\perp/z}$
$\displaystyle=\frac{k_{0}^{2}V\text{Im}(\chi_{\text{E}})}{\Big{|}1-k_{0}^{2}V\chi_{\text{E}}\langle
G_{\text{EE},\perp/z}(\mathbf{r}_{\text{p}})\rangle\Big{|}^{2}}.$ (73)

Hence, for $T_{\text{s}}=T_{\text{b}}$ we would end up with

$\displaystyle P_{z}^{\rm tot}$
$\displaystyle=\sum_{j\in\\{\perp,z\\}}\frac{k_{0}^{3}V\left(\Theta_{\text{p}}-\Theta_{\text{b}}\right)\text{Im}(\chi_{\text{E}})I_{\text{E},j}^{\text{pr}}}{\big{|}1-k_{0}^{2}V\chi_{\text{E}}\langle
G_{\text{EE},j}(\mathbf{r}_{\text{p}})\rangle\big{|}^{2}}.$ (74)

This result resembles the one in Eq. (62) in Ref. Herz  but with
$V\chi_{\text{e}}$ instead of the “naked” electric polarizability used there
and a different vacuum contribution of the Green’s function. Within the range
of validity of the dipole model, our new result and that in Ref. Herz  give
the same numerical values if we neglect the vacuum correction term
$\mathds{G}_{\text{vac}}$.

##  Appendix D Substrate as perfect metal

If the substrate would be a perfect metal ($r_{\text{E/H}}=\pm 1$), the
nanoparticle directly emits

$\displaystyle P_{\omega}^{\text{np}}+P_{\omega}^{\text{eq}}$
$\displaystyle=\frac{2k_{0}}{3\gamma^{3}\pi}(\Theta_{\text{p}}-\Theta_{\text{b}})$
$\displaystyle\quad\times\Bigl{[}\left(\chi_{\text{E},\perp}+|F_{\text{E}}|^{2}\chi_{\text{H},\perp}\right)I_{\perp}^{-}+\chi_{\text{E},z}I_{z}^{-}$
$\displaystyle\quad+\frac{\varepsilon_{0}}{\mu_{0}}\left[\left(\chi_{\text{H},\perp}+|F_{\text{H}}|^{2}\chi_{\text{E},\perp}\right)I_{\perp}^{+}+\chi_{\text{H},z}I_{z}^{+}\right]$
$\displaystyle\quad+6\gamma\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\bigl{[}\chi_{\text{H},\perp}\text{Im}(F_{\text{E}})+\chi_{\text{E},\perp}\text{Im}(F_{\text{H}})\bigr{]}I\Bigr{]}$
(75)

with

$\displaystyle I_{\perp}^{\mp}$ $\displaystyle=2\gamma^{3}\mp
3\left[\gamma\cos(\gamma)+\left(\gamma^{2}-1\right)\sin(\gamma)\right],$ (76)
$\displaystyle I_{z}^{\mp}$ $\displaystyle=\gamma^{3}\mp
3\left[\gamma\cos(\gamma)-\sin(\gamma)\right],$ (77) $\displaystyle I$
$\displaystyle=\sin(\gamma)-\gamma\cos(\gamma).$ (78)

Here, we used $\gamma=2k_{0}d$. In this case the LEQC and
$P_{\omega}^{\text{s}}$ vanish.

##  Appendix E Substrate as black body

By assuming the substrate to be a black body, we set $r_{\text{E/H}}=0$. Then,
we get for the directly emitted power of the nanoparticle

$\displaystyle P_{\omega}^{\text{np}}+P_{\omega}^{\text{eq}}$
$\displaystyle=\frac{k_{0}}{3\pi}(\Theta_{\text{p}}-\Theta_{\text{b}})\left[2\chi_{\perp}^{\text{E}}+\chi_{z}^{\text{E}}+\frac{\varepsilon_{0}}{\mu_{0}}\left(2\chi_{\perp}^{\text{H}}+\chi_{z}^{\text{H}}\right)\right].$
(79)

The LEQCs are

$\displaystyle P_{\omega,1}^{\text{leq}}$
$\displaystyle=\frac{k_{0}^{3}}{3\pi}\left(\Theta_{\text{b}}-\Theta_{\text{s}}\right)\text{Im}\Bigl{[}2\alpha_{\text{EE},\perp}+\alpha_{\text{EE},z}$
$\displaystyle\quad+\frac{\varepsilon_{0}}{\mu_{0}}\left(2\alpha_{\text{HH},\perp}+\alpha_{\text{HH},z}\right)\Bigr{]}$
(80)

and

$\displaystyle P_{\omega,2}^{\text{leq}}$
$\displaystyle=\frac{k_{0}^{6}}{16\pi^{2}}\left(\Theta_{\text{s}}-\Theta_{\text{b}}\right)\biggl{[}\frac{\varepsilon_{0}}{\mu_{0}}\text{Re}\left(\alpha_{\text{EE},\perp}\alpha_{\text{HH},\perp}^{*}\right)$
$\displaystyle\quad+\frac{8}{9}|\alpha_{\text{EE},\perp}|^{2}+\frac{4}{9}|\alpha_{\text{EE},z}|^{2}$
$\displaystyle\quad+\frac{\varepsilon_{0}^{2}}{\mu_{0}^{2}}\left(\frac{8}{9}|\alpha_{\text{HH},\perp}|^{2}+\frac{4}{9}|\alpha_{\text{HH},z}|^{2}\right)\biggr{]}.$
(81)

In this case $P_{\omega}^{\text{s}}$ is simply the well-known power exchanged
between two blackbodies at temperatures $T_{\text{s}}$ and $T_{\text{b}}$
through an area $A$.

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