# Effective refractive index of a silicon dioxide with
implanted Ag nanoparticles and Er3+ ions

Igor Kuzmenko1,2, Y. Avishai1,3, Y. B. Band1,2,4 1Department of Physics, Ben-
Gurion University of the Negev, Beer-Sheva 84105, Israel
2Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva
84105, Israel
3Yukawa Institute for Theoretical Physics, Kyoto, Japan
4The Ilse Katz Center for Nano-Science, Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel

###### Abstract

We consider light propagation in a silicon dioxide substrate with implanted
${\mathrm{Er}}^{3+}$ ions and silver nanoparticles that are randomly and
homogeneously distributed in the substrate. When their densities are large
enough, the medium can have a negative refractive index over a certain range
of frequencies, within which the following exotic property ensues: increasing
the electric and magnetic plasma frequencies, the medium transparency is
augmented.

##  I  Introduction

Negative refraction (NR) is a phenomenon in which electromagnetic waves are
refracted at an interface of a material with a refraction angle that is
negative relative to commonly observed refraction angles [1, 2, 3, 4, 5]. It
was believed that, in order for NR to occur at a given frequency, the real
part of the (electric) permittivity ($\varepsilon$) and real part of the
(magnetic) permeability ($\mu$) must both be negative at this particular
frequency [1, 2, 3, 4, 5, 6, 7, 8, 9]. Such materials are sometimes called
“double negative” materials. NR meta-materials, i.e., specially designed NR
materials made from assemblies of multiple elements fashioned from composite
materials have been developed [3, 6, 7, 8, 9]. Moreover, it has often been
stated that light at frequencies such that
${\mathrm{Re}}[\varepsilon(\omega)]{\mathrm{Re}}[\mu(\omega)]<0$ {i.e.,
${\mathrm{Re}}[\varepsilon(\omega)]>0$ and ${\mathrm{Re}}[\mu(\omega)]<0$ or
{${\mathrm{Re}}[\varepsilon(\omega)]<0$ and ${\mathrm{Re}}[\mu(\omega)]>0$} is
not able to propagate in materials [1, 2, 3, 4, 5, 6, 7, 8, 9]; these are
simplistic interpretations based upon the assumption that the permittivity and
permittivity are real functions of frequency. Reference [10] is an exception;
it used the amplitude-phase representation of the permittivity, permeability
and refractive index. It considers two resultant complex refractive indices,
$n(\omega)=\pm\sqrt{\varepsilon(\omega)\mu(\omega)/(\varepsilon_{0}\mu_{0})}$.
While the resultant complex refractive index $n(\omega)$ with the minus sign
is not a physical solution, because it gives negative absorption (i.e., gain),
the refractive index $n(\omega)$ with the plus sign has the right properties.
Recently, Ref. [11] considered negative refraction in achiral materials using
the Drude model, and chiral materials using the Drude-Born-Fedorov model. That
paper shows that the time-averaged Poynting vector always points along the
wave vector, the time-averaged energy flux density is always positive, and the
time-averaged energy density is positive (negative) when the refractive index
is positive (negative). The phase velocity is negative when the real part of
the refractive index is negative, and the group velocity generally changes
sign several times as a function of frequency near resonance.

Here we study the optical properties a silicon dioxide substrate doped with
silver nanoparticles and Er3+ ions. The electric permittivity of silver is
negative over a wide frequency range, and Er3+ ions have a magnetic dipole
optical transition from the ground state to the first excited state with a
transition frequency of $\omega_{0}=2\pi\times 194$ THz [12]. Therefore, SiO2
doped with silver and erbium can exhibit a negative refractive index in a
frequency range just above $\omega_{0}$, and the width of the range depends on
a number density of silver nanoparticles and a number density of erbium ions.
Outside this frequency range, the imaginary part of the complex refractive
index is much larger than the real part, and light cannot propagate in the
medium. We also study the propagation of light from vacuum into the medium
through an interface. When the refractive index is negative, the medium is
transparent and the reflection coefficient is small. If the refractive index
is positive, the reflection coefficient is close to unity, and the light is
reflected from the interface.

NR materials are man-made meta-materials. NR meta-materials have led to
significant technological advancements [4, 5, 6, 7, 8, 9] including: (1)
superlensing, i.e., overcoming the diffraction limit of conventional lenses,
allowing for sub-wavelength imaging for high-resolution microscopy [13, 14],
(2) cloaking using devices that can manipulate the flow of light around an
object, rendering it invisible to observers [4, 15, 16], (3) terahertz
imaging, spectroscopy, and communication systems, enabling non-invasive
inspections in biomedical imaging and security screening [17], (4) antennas
incorporating NR meta-materials that can enhance the radiated power of the
antenna NR by focusing electromagnetic radiation by a flat lens versus
dispersion [18, 19, 20]. Note that there has been one report of a naturally
occurring NR material, the Dirac semi-metal Cd3As2 [21].

The manuscript is organized as follows. Section II describes a medium
consisting of a SiO2 substrate doped with Ag nanoparticles and Er3+ ions. In
Sec. III, we calculate the complex permittivity
$\varepsilon_{\mathrm{eff}}(\omega)$ of SiO2 with implanted silver
nanoparticles, and show that the real part
$\varepsilon^{\prime}_{\mathrm{eff}}(\omega)$ is negative over a wide
frequency interval. In Sec. IV, we calculate the complex permeability
$\mu_{\mathrm{Er}}(\omega)$ of Er3+ ions implanted into the SiO2 substrate,
and show that there is a frequency interval wherein its real part
$\mu^{\prime}_{\mathrm{Er}}(\omega)<0$. In Sec. V, we calculate the complex
refractive index $n_{\mathrm{eff}}(\omega)$ of the SiO2 substrate with Ag
nanoparticles and Er3+ ions, and show that $n^{\prime}_{\mathrm{eff}}(\omega)$
can be positive or negative depending on the sign of
$\mu^{\prime}_{\mathrm{Er}}(\omega)$. In Sec. VI we discuss the position-
dependence of the electric and magnetic fields inside the sample for
$\omega_{\pm}=\omega_{0}\pm 2\pi\times 50~{}\text{kHz}$ and show that for
$\omega=\omega_{+}$, $n^{\prime}_{\mathrm{eff}}(\omega_{+})<0$, and the
imaginary part, $n^{\prime\prime}_{\mathrm{eff}}(\omega)$ is small,
$n^{\prime\prime}_{\mathrm{eff}}(\omega_{+})\ll|n^{\prime}_{\mathrm{eff}}(\omega_{+})|$.
Therefore the electromagnetic fields propagate with very small absorption (the
medium is almost transparent for light with the frequency $\omega_{+}$), but
for $\omega=\omega_{-}$, $n^{\prime}_{\mathrm{eff}}(\omega_{-})$ is positive
and $n^{\prime}_{\mathrm{eff}}(\omega_{-})\ll
n^{\prime\prime}_{\mathrm{eff}}(\omega)$, hence the electromagnetic fields are
strongly reflected and decrease exponentially near the edge of the medium. In
Sec. VII, we consider light propagating in a vacuum and impinging on the
interface separating the vacuum and the medium, and show that the medium is
transparent for light with $\omega=\omega_{+}$ where the real part of the
refractive index is negative. However, for $\omega_{-}<\omega_{0}$,
$n^{\prime}_{\mathrm{eff}}(\omega)>0$, and the light is reflected from the
interface.

##  II SiO2 substrate doped with $\text{Er}^{3+}$ ions and Silver
nanoparticles

Doping silicon dioxide (SiO2) with impurities can change its optical
properties. References [22, 23] consider a silicon dioxide substrate doped
with silver nanoparticles. A variety of shapes of the silver nanoparticles are
possible, such as triangular, pentagonal, prolate, spheres, rods, cubes and
prisms [22]. The size of the nanoparticles can range from one nanometer to
hundreds of nanometers [22]. Reference [12] studies Er3+ ions implanted in a
SiO2 substrate. Er3+ doped zinc phosphate glasses containing spherical Ag
nanoparticles with diameter in the range of 20-40 nm were synthesized in Ref.
[24].

Here we theoretically analyze the optical properties of a SiO2 substrate with
implanted silver nanoparticles and Er3+ ions, as schematically shown in Fig.
1, when it is subject to a ray of light of wavelenght $\lambda$. The
nanoparticles, assumed to have the same size and shape are randomly and
homogeneously distributed in the substrate, as are the ions [25]. The
nanoparticle size, the averaged distance between the neighboring
nanoparticles, and between the ions are much smaller than $\lambda$ so we can
parametrize the medium by an effective complex permittivity and permeability.
We will focus our attention on light frequencies just above the resonance
frequency $\omega_{0}=2\pi\times 194$ THz of the magnetic dipole transition in
Er3+ ions [12]. This reference showed that when the density of the Er3+ ions
is high enough, the real part of the permeability is negative over a certain
frequency region, and when the density of the Ag nanoparticles is high enough,
the real part of the permittivity can be negative. Hence, the medium can have
a negative refractive index.

Figure 1: Silicon dioxide substrate (aqua) with implanted Er3+ ions (purple)
and silver nanoparticles (Ag, orange) localized at random locations within the
substrate. The Cartesian basis vectors are $\hat{\mathbf{x}}$,
$\hat{\mathbf{y}}$ and $\hat{\mathbf{z}}$. Note that
$N_{\mathrm{Er}^{+}}>N_{\mathrm{Ag}}$.

##  III Effective permittivity of silver nanoparticles

In Gaussian units, the permittivity of silver is calculated using the Drude
model [26],

$\varepsilon_{\mathrm{Ag}}(\omega)=\varepsilon_{\infty}-\frac{\omega_{p,e}^{2}}{\omega^{2}+i\omega\gamma_{\mathrm{Ag}}},$
(1)

where $\varepsilon_{\infty}=5\pm 2$ is the background permittivity,
$\gamma_{\mathrm{Ag}}=(60\pm 8)\times 10^{12}~{}\text{s}^{-1}$ is the width
(the inverse mean free lifetime) of the permittivity, and
$\omega_{p,e}=(13.5\pm 0.3)\times 10^{15}~{}\text{s}^{-1}$ is the plasma
frequency [26]. The silver and silicon dioxide permeabilities are
$\mu_{\mathrm{Ag}}=\mu_{\rm SiO_{2}}=1$. We calculate the effective
permittivity of silver nanoparticles doped into SiO2 substrate assuming the
silver nanoparticles have spherical shape of radius $r_{0}=40$ nm.

When a nanoparticle of radius $r_{0}$ is placed in an external electric field
$\bm{\mathcal{E}}(t)=\bm{\mathcal{E}}_{0}\,e^{-i\omega t}$, the induced dipole
momentum is

$\mathbf{p}(t)=\frac{4\pi}{3}\,r_{0}^{3}\,\big{(}\varepsilon_{\mathrm{Ag}}(\omega)-1\big{)}\,\bm{\mathcal{E}}(t)\equiv\alpha(\omega)\bm{\mathcal{E}}(t).$
(2)

The polarizability $\alpha(\omega)$ of a single nanoparticle is related to the
effective permittivity $\varepsilon_{\mathrm{eff}}(\omega)$ of a set of
nanoparticles with number density $N_{\mathrm{Ag}}$ by the Lorentz-Lorenz
formula,

$\frac{\varepsilon_{\mathrm{eff}}(\omega)-\varepsilon_{\rm
SiO_{2}}(\omega)}{\varepsilon_{\mathrm{eff}}(\omega)+2\varepsilon_{\rm
SiO_{2}}(\omega)}=\frac{4\pi N_{\mathrm{Ag}}\alpha(\omega)}{3},$ (3)

which can also be written as the Clausius-Mossotti relation,

$\varepsilon_{\mathrm{eff}}(\omega)=\varepsilon_{\rm
SiO_{2}}(\omega)\,\frac{\displaystyle
1+2\,\Big{(}\frac{4\pi}{3}\Big{)}^{2}\,N_{\mathrm{Ag}}r_{0}^{3}\,\big{[}\varepsilon_{\mathrm{Ag}}(\omega)-1\big{]}}{\displaystyle
1-\Big{(}\frac{4\pi}{3}\Big{)}^{2}\,N_{\mathrm{Ag}}r_{0}^{3}\,\big{[}\varepsilon_{\mathrm{Ag}}(\omega)-1\big{]}}.$
(4)

When the nanoparticle density vanishes, $N_{\mathrm{Ag}}r_{0}^{3}\to 0$,
$\varepsilon_{\mathrm{eff}}(\omega)$ reduces to the SiO2 permittivity
$\varepsilon_{\rm SiO_{2}}(\omega)$. For $\omega_{+}=\omega_{0}+2\pi\times 50$
kHz, where $\omega_{0}=2\pi\times 194$ THz, $\varepsilon_{\rm
SiO_{2}}(\omega_{+})=2.37$ is real and positive. When
$N_{\mathrm{Ag}}r_{0}^{3}\neq 0$, $\varepsilon_{\mathrm{eff}}(\omega)$ is
complex, and the imaginary part is positive and proportional to
$\mathrm{Im}[\varepsilon_{\mathrm{Ag}}(\omega)]$. There is a value of the
dimensionless parameter $\mathcal{N}=N_{\mathrm{Ag}}r_{0}^{3}$,
$\mathcal{N}_{c}=0.0002405$, such that for $\mathcal{N}<\mathcal{N}_{c}$,
$\mathrm{Re}[\varepsilon_{\mathrm{eff}}(\omega_{+})]>0$, and for
$\mathcal{N}>\mathcal{N}_{c}$,
$\mathrm{Re}[\varepsilon_{\mathrm{eff}}(\omega_{+})]<0$. Note that for
$N_{\mathrm{Ag}}r_{0}^{3}=\mathcal{N}_{c}$, the average distance between the
neighboring nanoparticles, $N_{\mathrm{Ag}}^{-1/3}=16.08\,r_{0}$ is much
larger than $r_{0}$. Here we take $\mathcal{N}=8\mathcal{N}_{c}$, for which

$\varepsilon_{\mathrm{eff}}(\omega_{+})=-3.322+0.058\,i.$ (5)

Figure 2 shows the real and imaginary parts of the permittivity,
$\varepsilon^{\prime}_{\mathrm{eff}}(\omega)$ and
$\varepsilon^{\prime\prime}_{\mathrm{eff}}(\omega)$, versus $\omega$. There is
a density dependent threshold frequency $\omega_{\mathrm{th}}(\mathcal{N})$
such that $\varepsilon^{\prime}_{\mathrm{eff}}(\omega)<0$ for
$\omega<\omega_{\mathrm{th}}(\mathcal{N})$, and
$\varepsilon^{\prime}_{\mathrm{eff}}(\omega)>0$ for
$\omega>\omega_{\mathrm{th}}(\mathcal{N})$. In Fig. 2(a),
$\omega_{\mathrm{th}}(\mathcal{N})$ is the frequency where
$\varepsilon^{\prime}_{\mathrm{eff}}(\omega_{\mathrm{th}}(\mathcal{N}))=0$.
Note that $\varepsilon^{\prime\prime}_{\mathrm{eff}}(\omega)\geq 0$ for all
$\omega\geq 0$.

Figure 2: Real and imaginary parts,
$\varepsilon^{\prime}_{\mathrm{eff}}(\omega)$ and
$\varepsilon^{\prime\prime}_{\mathrm{eff}}(\omega)$, of the permittivity of
SiO2 doped with silver nanoparticles versus $\omega$ for a several values of
$\mathcal{N}$.

##  IV Relative permeability of the ions

The lowest-energy states of Er3+ ions, ordered from the lowest up, have $L=6$,
$S=\frac{3}{2}$ and $J=\frac{15}{2},\frac{13}{2},\frac{11}{2},\frac{9}{2}$
[12]. The magnetic dipole transition from the ground state $J=15/2$ to the
excited state $J^{\prime}=13/2$ has frequency $\omega_{0}=2\pi\times 194$ THz.

The magnetic susceptibility can be calculated using a Drude formula (see
Appendix B for details):

$\chi_{\mathrm{Er}}(\omega)=-\frac{N_{\mathrm{Er}}\mu_{B}^{2}\omega_{0}}{5\hbar}\,\frac{1}{\omega^{2}-\omega_{0}^{2}+i\gamma_{\mathrm{Er}}\omega},$
(6)

where $N_{\mathrm{Er}}$ is the number density of ions, and
$\gamma_{\mathrm{Er}}$ is the line-width of the
${}^{4}I_{15/2}$–${}^{4}I_{13/2}$ magnetic dipole transition. The nomenclature
for spectral terms ${}^{4}I_{J}$ for Er3+ ions is clarified in Appendix A. The
lifetime of Er3+ ions implanted in a bulk SiO2 substrate is
$\gamma_{\mathrm{Er}}^{-1}=(14.3\pm 0.3)^{-1}$ ms [12], hence
$\gamma_{\mathrm{Er}}=(70\pm 1)~{}\text{s}^{-1}$. Equation (6) is derived in
Appendix B.

The permeability of Er3+,
$\mu_{\mathrm{Er}}(\omega)=1+4\pi\chi_{\mathrm{Er}}(\omega)$ is

$\mu_{\mathrm{Er}}(\omega)=1-\frac{\omega_{p,m}^{2}}{\omega^{2}-\omega_{0}^{2}+i\gamma_{\mathrm{Er}}\omega},$
(7)

where the magnetic plasma frequency is

$\omega_{p,m}^{2}=\frac{4\pi N_{\mathrm{Er}}\mu_{B}^{2}\,\omega_{0}}{5\hbar}.$
(8)

The real part of the permeability, $\mu^{\prime}_{\mathrm{Er}}(\omega)$, has a
minimum at $\omega_{\min}=\sqrt{\omega_{0}(\omega_{0}+\gamma_{\mathrm{Er}})}$,

$\mu^{\prime}_{\mathrm{Er}}(\omega_{\min})=1-\frac{\omega_{p,m}^{2}}{\gamma_{\mathrm{Er}}(2\omega_{0}+\gamma_{\mathrm{Er}})}.$

For
$\omega_{p,m}>\sqrt{\gamma_{\mathrm{Er}}(2\omega_{0}+\gamma_{\mathrm{Er}})}=2\pi\times
6.56\times 10^{7}$ Hz, $\mu^{\prime}_{\mathrm{Er}}(\omega_{\min})$ is
negative. Hereafter, we take $N_{\mathrm{Er}}=10^{19}~{}\text{cm}^{-3}$, and
$\omega_{p,m}=2\pi\times 7.96\times 10^{9}$ Hz. The real part of the
permeability is negative for
$\omega_{0}<\omega<\sqrt{\omega_{0}^{2}+\omega_{p,m}^{2}}$, where
$\sqrt{\omega_{0}^{2}+\omega_{p,m}^{2}}-\omega_{0}=2\pi\times 163$ kHz. For
$\omega=\omega_{+}$,

$\mu_{\mathrm{Er}}(\omega_{+})=-2.26+0.00036\,i.$ (9)

Figure 3 shows the real part, $\mu^{\prime}_{\mathrm{Er}}(\omega)$, and the
imaginary part, $\mu^{\prime\prime}_{\mathrm{Er}}(\omega)$, of the
permeability versus $\omega$. For $\omega_{0}<\omega<\omega_{0}+2\pi\times
163$ kHz, $\mu^{\prime}_{\mathrm{Er}}(\omega)<0$, otherwise
$\mu^{\prime}_{\mathrm{Er}}(\omega)>0$. Figure 3(b) shows the permeability
near the $M_{1}$ resonance frequency $\omega_{0}$.

Figure 3: (a) Real (solid red) and imaginary (dashed blue) parts of the
permeability, $\mu^{\prime}_{\mathrm{Er}}(\omega)$ and
$\mu^{\prime\prime}_{\mathrm{Er}}(\omega)$, of SiO2 doped with Er3+ ions,
versus $\omega$ for $N_{\mathrm{Er}}=1\times 10^{19}~{}\text{cm}^{-3}$. (b)
Zoom of the frequency region near $\omega_{0}$.

##  V Refractive index

The refractive index $n_{\mathrm{eff}}(\omega)$ of a SiO2 substrate implanted
with Ag nanoparticles and Er3+ ions is given by

$n_{\mathrm{eff}}(\omega)=\sqrt{\varepsilon_{\mathrm{eff}}(\omega)\,\mu_{\mathrm{Er}}(\omega)}\,.$
(10)

In Eq. (10), it is convenient to write $\varepsilon_{\mathrm{eff}}(\omega)$
[see Eq. (4)] and $\mu_{\mathrm{Er}}(\omega)$ [see Eq. (7)] as

$\displaystyle\varepsilon_{\mathrm{eff}}(\omega)$ $\displaystyle=$
$\displaystyle|\varepsilon_{\mathrm{eff}}(\omega)|\,e^{i\theta_{\varepsilon}(\omega)},$
$\displaystyle\mu_{\mathrm{Er}}(\omega)$ $\displaystyle=$
$\displaystyle|\mu_{\mathrm{Er}}(\omega)|\,e^{i\theta_{\mu}(\omega)}.$

For $\omega=\omega_{+}$, $|\varepsilon_{\mathrm{eff}}(\omega_{+})|=3.322$,
$|\mu_{\mathrm{Er}}(\omega_{+})|=2.262$,
$\theta_{\varepsilon}(\omega_{+})=\pi-0.017$ and
$\theta_{\mu}(\omega_{+})=\pi-0.00016$, hence $n_{\mathrm{eff}}(\omega_{+})$
is

$\displaystyle n_{\mathrm{eff}}(\omega_{+})$ $\displaystyle=$
$\displaystyle\sqrt{|\varepsilon_{\mathrm{eff}}(\omega_{+})|\,|\mu_{\mathrm{Er}}(\omega_{+})|}\,e^{i[\theta_{\varepsilon}(\omega_{+})+\theta_{\mu}(\omega_{+})]/2}$
(11) $\displaystyle\approx$
$\displaystyle\sqrt{|\varepsilon_{\mathrm{eff}}(\omega_{+})|\,|\mu_{\mathrm{Er}}(\omega_{+})|}\,\times$
$\displaystyle\bigg{\\{}-1+\frac{i[2\pi-\theta_{\varepsilon}(\omega_{+})+\theta_{\mu}(\omega_{+})]}{2}\bigg{\\}}.\quad$

Numerically, $n_{\mathrm{eff}}(\omega_{+})=-2.741+0.024\,i$. Note that the
real part $n^{\prime}_{\mathrm{eff}}(\omega_{+})$ of the refractive index is
negative. The imaginary part $n^{\prime\prime}_{\mathrm{eff}}(\omega_{+})$ is
very small, hence the medium is optically transparent.

Figure 4: For $N_{\mathrm{Ag}}=3.002\times 10^{13}~{}\text{cm}^{-3}$ and
$N_{\mathrm{Er}}=1.0\times 10^{19}~{}\text{cm}^{-3}$, the figure shows the
real part (solid red) and the imaginary part (dashed blue) of the refractive
index in Eq. (10). The purple dot is for $n_{\mathrm{eff}}(\omega_{-})$ and
the green dot is for $n_{\mathrm{eff}}(\omega_{+})$.

The complex refractive index plotted as a function of frequency is shown in
Fig. 4. For $\omega<\omega_{0}$,
$\mathrm{Im}[n_{\mathrm{eff}}(\omega)]\gg\mathrm{Re}[n_{\mathrm{eff}}(\omega)]$
and the medium is not transparent. For
$\omega_{0}<\omega<\omega_{1}=\omega_{0}+2\pi\times 163.1~{}\text{kHz}$,
$\mathrm{Re}[n_{\mathrm{eff}}(\omega)]<0$ and
$\mathrm{Im}[n_{\mathrm{eff}}(\omega)]\ll 1$, hence the medium is transparent
for this frequency interval. For $\omega>\omega_{1}$, the medium is not
transparent.

We consider here two additional cases:

  * •

$N_{\mathrm{Ag}}\neq 0$, and $N_{\mathrm{Er}}=0$;

  * •

$N_{\mathrm{Ag}}=0$, and $N_{\mathrm{Er}}\neq 0$.

For $N_{\mathrm{Er}}=0$, the permeability is $\mu=1$, and the refractive index
$n_{\varepsilon}(\omega_{+})$ is

$\displaystyle n_{\varepsilon}(\omega_{+})$ $\displaystyle=$
$\displaystyle\sqrt{|\varepsilon_{\mathrm{eff}}(\omega_{+})|}\,e^{\frac{i}{2}\theta_{\varepsilon}(\omega_{+})}$
(12) $\displaystyle=$ $\displaystyle 0.016+1.823\,i.$

Note that the imaginary part of $n_{\varepsilon}(\omega_{+})$ is large, hence
the SiO2 substrate with implanted Ag nanoparticles is not transparent.

For $N_{\mathrm{Ag}}=0$, the permittivity $\varepsilon_{\rm
SiO_{2}}(\omega_{+})$ is real and positive, and the refractive index
$n_{\mu}(\omega_{+})$ is

$\displaystyle n_{\mu}(\omega_{+})$ $\displaystyle=$
$\displaystyle\sqrt{\varepsilon_{\rm
SiO_{2}}(\omega_{+})\,|\mu_{\mathrm{Er}}(\omega_{+})|}\,e^{\frac{i}{2}\,\theta_{\mu}(\omega_{+})}$
(13) $\displaystyle=$ $\displaystyle 0.00019+2.316\,i.$

As in the former case, the imaginary part of $n_{\mu}(\omega_{+})$ is large,
hence the SiO2 substrate with implanted Er3+ ions is not transparent. But the
imaginary part of $n_{\mathrm{eff}}(\omega)$ is very small. How is it possible
that upon implanting Ag nanoparticles and the Er3+ ions into the SiO2
substrate, the medium is optically transparent? We will answer this question
in Sec. VII, but first we consider the spatial dependence of the electric,
magnetic, polarization and magnetization fields inside the sample.

##  VI  Space and time dependence of the fields

In this section we describe the dependence on $\mathbf{r}$ and $t$ of the
electric and displacement fields $\mathbf{E}(\mathbf{r},t)$,
$\mathbf{D}(\mathbf{r},t)$, as well as the magnetic, magnetic induction,
magnetization and polarization fields, $\mathbf{H}(\mathbf{r},t)$,
$\mathbf{B}(\mathbf{r},t)$, $\mathbf{M}(\mathbf{r},t)$,
$\mathbf{P}(\mathbf{r},t)$. Specifically, we consider the cases where the real
parts of the permittivity and permeability are positive and negative,
respectively.

Consider light with frequency $\omega$ propagating along the $z$-axis and
polarized along the $x$-axis. The light wave number is
$k\,n_{\mathrm{eff}}(\omega)$, where $k=\omega/c$ is the wave number in
vacuum. The complex electric field of the light is

$\bm{\mathcal{E}}(z,t)=E_{0}\,e^{i[n_{\mathrm{eff}}(\omega)kz-\omega
t]}\,\hat{\mathbf{x}},$ (14a) where $E_{0}$ is assumed to be real and
positive. The complex displacement and polarization fields are
$\displaystyle\bm{\mathcal{D}}(z,t)$ $\displaystyle=$
$\displaystyle\varepsilon_{\mathrm{eff}}(\omega)\,E_{0}\,e^{i[n_{\mathrm{eff}}(\omega)kz-\omega
t]}\,\hat{\mathbf{x}},$ (14b) $\displaystyle\bm{\mathcal{P}}(z,t)$
$\displaystyle=$
$\displaystyle\frac{\varepsilon_{\mathrm{eff}}(\omega)-1}{4\pi}\,E_{0}\,e^{i[n_{\mathrm{eff}}(\omega)kz-\omega
t]}\,\hat{\mathbf{x}}.$ (14c)

Applying Faraday’s induction law, we find the complex induction magnetic
field,

$\bm{\mathcal{B}}(z,t)=n_{\mathrm{eff}}(\omega)\,E_{0}\,e^{i[n_{\mathrm{eff}}(\omega)kz-\omega
t]}\,\hat{\mathbf{y}}.$ (14d)

The complex magnetic and magnetization fields are

$\displaystyle\bm{\mathcal{H}}(z,t)=\eta(\omega)\,E_{0}\,e^{i[n_{\mathrm{eff}}(\omega)kz-\omega
t]}\,\hat{\mathbf{y}},$ (14e)
$\displaystyle\bm{\mathcal{M}}(z,t)=\frac{\mu_{\mathrm{Er}}(\omega)-1}{4\pi}\,\eta(\omega)\,E_{0}\,e^{i[n_{\mathrm{eff}}(\omega)kz-\omega
t]}\,\hat{\mathbf{y}}.\quad$ (14f)

Here we write the permittivity and permeability as
$\varepsilon_{\mathrm{eff}}(\omega)=|\varepsilon_{\mathrm{eff}}(\omega)|\,e^{i\theta_{\varepsilon}}$
and
$\mu_{\mathrm{Er}}(\omega)=|\mu_{\mathrm{Er}}(\omega)|\,e^{i\theta_{\mu}}$,
and define $\eta(\omega)$ as

$\eta(\omega)=\sqrt{\frac{|\varepsilon_{\mathrm{eff}}(\omega)|}{|\mu_{\mathrm{Er}}(\omega)|}}\,e^{i\theta_{\eta}},$
(15)

where $\theta_{\eta}=\frac{1}{2}\,[\theta_{\varepsilon}-\theta_{\mu}]$. The
physical fields $\mathbf{E}(z,t)$, $\mathbf{D}(z,t)$, $\mathbf{P}(z,t)$,
$\mathbf{B}(z,t)$, $\mathbf{H}(z,t)$ and $\mathbf{M}(z,t)$ are the real parts
of the complex fields specified in Eq. (14).

Figure 5: (a) The spatial dependence of the fields $\mathbf{E}(z,0)$ [red],
$\mathbf{D}(z,0)$ [green] and $4\pi\mathbf{P}(z,0)$ [blue], and (b) the fields
$\mathbf{B}(z,0)$ [red], $\mathbf{H}(z,0)$ [green] and
${\color[rgb]{0,0,1}4\pi}\mathbf{M}(z,0)$ [blue] for $\omega=\omega_{+}$ where
the real parts of the permittivity and permeability are negative.

The spatial dependence of the fields shown in Fig. 5 are for
$\omega=\omega_{+}$, where the real parts of the permittivity and permeability
are negative. Figure 5 shows that $\mathbf{E}(z,0)$ [red] is opposite to
$\mathbf{D}(z,0)$ [green] and $\mathbf{P}(z,0)$ [blue]. This is because
$\mathrm{Re}[\varepsilon_{\mathrm{eff}}(\omega)]<-1$, and
$\mathrm{Im}[\varepsilon_{\mathrm{eff}}(\omega)]$ is small. Figure 5 shows
that $\mathbf{H}(z,0)$ [green] is opposite to $\mathbf{B}(z,0)$ [red] and
$\mathbf{M}(z,0)$ [blue], since $\mathrm{Re}[\mu_{\mathrm{Er}}(\omega)]<-1$
and $\mathrm{Im}[\mu_{\mathrm{Er}}(\omega)]$ is small.
$\mathrm{Im}[\mu_{\mathrm{Er}}(\omega)]$ increases when the frequency
approaches the resonance frequency $\omega_{0}$, and $\mathbf{H}(z,0)$ and
$\mathbf{M}(z,0)$ get a phase shift due to the imaginary part of the
permittivity.

Figure 6: (a) The fields $\mathbf{E}(z,0)$ [red], $\mathbf{D}(z,0)$ [green]
and $4\pi\mathbf{P}(z,0)$ [blue]. (b) the fields $\mathbf{B}(z,T/4)$ [red],
$\mathbf{H}(z,T/4)$ [green] and ${\color[rgb]{0,0,1}4\pi}\mathbf{M}(z,T/4)$
[blue] for $\omega=\omega_{-}$ where the real parts of the permittivity is
negative and real part of permeability is positive. Here $T=2\pi/\omega$.

The fields are shown in Fig. 6 for $\omega_{-}=\omega_{0}-2\pi\times
50~{}\text{kHz}$, where the real part of the permittivity is negative and the
real part of the permeability is positive. Figure 6 shows that
$\mathbf{E}(z,0)$ [red] is opposite to $\mathbf{D}(z,0)$ [green] and
$\mathbf{P}(z,0)$ [blue], since
$\mathrm{Re}[\varepsilon_{\mathrm{eff}}(\omega)]<-1$, and
$\mathrm{Im}[\varepsilon_{\mathrm{eff}}(\omega)]$ is small. The real part of
the refractive index is much smaller than the imaginary part, hence the fields
exponentially vanish with $z$. Figure 6 shows the fields $\mathbf{B}(z,T/4)$,
$\mathbf{H}(z,T/4)$ and $4\pi\mathbf{M}(z,T/4)$, where $T=2\pi/\omega$. Since
$\mathrm{Im}[n_{\mathrm{eff}}(\omega)]\gg\mathrm{Re}[n_{\mathrm{eff}}(\omega)]$,
the fields are very small at $t=0$. The fields decrease with $z$ and vanish
for $z\to\infty$. Therefore the light with $\omega<\omega_{0}$ cannot
propagate in the medium. When we have an interface separating the medium and
the vacuum, the light is reflected from the interface.

##  VII Light reflection from the interface between the vacuum and medium

Consider light with frequency $\omega$ propagating in vacuum and impinges on a
planar interface (in the $x-y$ plane) between vacuum and the medium with
complex permittivity $\varepsilon_{\mathrm{eff}}(\omega)$, permeability
$\mu_{\mathrm{Er}}(\omega)$ and refractive index $n_{\mathrm{eff}}(\omega)$.
Let the light propagate along the $z$-direction which is perpendicular to the
interface and polarized along the $x$-axis. The electric and magnetic fields
in vacuum are

$\displaystyle\bm{\mathcal{E}}_{\mathrm{vac}}(z,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\Big{[}\mathcal{E}_{0}\,e^{ikz-i\omega
t}+\mathcal{E}_{2}\,e^{-ikz-i\omega t}\Big{]}\hat{\mathbf{x}},$ (16a)
$\displaystyle\bm{\mathcal{H}}_{\mathrm{vac}}(z,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\Big{[}\mathcal{E}_{0}\,e^{ikz-i\omega
t}-\mathcal{E}_{2}\,e^{-ikz-i\omega t}\Big{]}\hat{\mathbf{y}},$ (16b)

where $k=\omega/c$ is the wave-number of the light in vacuum,
$\mathcal{E}_{0}$ is the real amplitude of the incident light and
$\mathcal{E}_{2}$ is the complex amplitude of the reflected light. The
electric and magnetic fields of the light in he medium are

$\displaystyle\bm{\mathcal{E}}_{\mathrm{med}}(z,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\mathcal{E}_{1}\,e^{in_{\mathrm{eff}}(\omega)\,kz-i\omega
t}\,\hat{\mathbf{x}},$ (17a)
$\displaystyle\bm{\mathcal{H}}_{\mathrm{med}}(z,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\frac{n_{\mathrm{eff}}(\omega)}{\mu_{\mathrm{Er}}(\omega)}\,\mathcal{E}_{1}\,e^{in_{\mathrm{eff}}(\omega)\,kz-i\omega
t}\,\hat{\mathbf{y}},$ (17b)

where $\mathcal{E}_{1}$ is the complex amplitude of the transmitted light. The
electric and magnetic fields in Eqs. (16) and (17) satisfy the following
boundary conditions at the interface $z=0$:

$\displaystyle\bm{\mathcal{E}}_{\mathrm{vac}}(0,t)$ $\displaystyle=$
$\displaystyle\bm{\mathcal{E}}_{\mathrm{med}}(0,t),$ (18a)
$\displaystyle\bm{\mathcal{H}}_{\mathrm{vac}}(0,t)$ $\displaystyle=$
$\displaystyle\bm{\mathcal{H}}_{\mathrm{med}}(0,t),$ (18b)

which imply

$\displaystyle\mathcal{E}_{0}+\mathcal{E}_{2}$ $\displaystyle=$
$\displaystyle\mathcal{E}_{1},$ (19a)
$\displaystyle\mathcal{E}_{0}-\mathcal{E}_{2}$ $\displaystyle=$
$\displaystyle\eta(\omega)\,\mathcal{E}_{1}.$ (19b)

Equations (19) imply

$\displaystyle\mathcal{E}_{1}$ $\displaystyle=$
$\displaystyle\frac{2\mathcal{E}_{0}}{1+\eta(\omega)},$ (20a)
$\displaystyle\mathcal{E}_{2}$ $\displaystyle=$
$\displaystyle\mathcal{E}_{0}-\frac{2\eta(\omega)\,\mathcal{E}_{0}}{1+\eta(\omega)}.$
(20b)

The energy flux (Poynting) vector of the electromagnetic field is
$\mathbf{S}=\frac{c}{4\pi}\overline{\mathbf{E}\times\mathbf{H}}$, where the
over-line means the temporal average,
$\mathbf{E}=\tfrac{\bm{\mathcal{E}}+\bm{\mathcal{E}}^{*}}{2}$ and
$\mathbf{H}=\tfrac{\bm{\mathcal{H}}+\bm{\mathcal{H}}^{*}}{2}$. The Poynting
vector of the incident, transmitted and reflected light at $z=0$ are

$\displaystyle S_{0}(0)$ $\displaystyle=$
$\displaystyle\frac{c}{8\pi}\,\mathcal{E}_{0}^{2},$ (21a) $\displaystyle
S_{1}(0)$ $\displaystyle=$
$\displaystyle\frac{c}{4\pi}\,\mathrm{Re}\big{[}\eta(\omega)\big{]}\,\big{|}\mathcal{E}_{1}\big{|}^{2},$
(21b) $\displaystyle S_{2}(0)$ $\displaystyle=$
$\displaystyle\frac{c}{8\pi}\,\big{|}\mathcal{E}_{2}\big{|}^{2}.$ (21c)

The reflection coefficient $\mathcal{R}(\omega)$ and the transmission
coefficient $\mathcal{T}(\omega)$ are defined as

$\mathcal{T}(\omega)=\frac{S_{1}(0)}{S_{0}(0)},\quad\mathcal{R}(\omega)=\frac{S_{2}(0)}{S_{0}(0)}.$

Using Eqs. (21), we get

$\displaystyle\mathcal{T}(\omega)$ $\displaystyle=$
$\displaystyle\frac{4\,\mathrm{Re}[\eta(\omega)]}{1+|\eta(\omega)|^{2}+2\mathrm{Re}[\eta(\omega)]},$
(22) $\displaystyle\mathcal{R}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1+|\eta(\omega)|^{2}-2\mathrm{Re}[\eta(\omega)]}{1+|\eta(\omega)|^{2}+2\mathrm{Re}[\eta(\omega)]}.$
(23)

Figure 7: The reflection coefficient (solid red curve) and the transmission
coefficient (dashed blue curve) versus frequency for
$N_{\mathrm{Ag}}=3.002\times 10^{13}~{}\text{cm}^{-3}$ and
$N_{\mathrm{Er}}=1\times 10^{19}~{}\text{cm}^{-3}$.

The reflection and transmission coefficients versus $\omega$ are shown in Fig.
7. For $\omega<\omega_{0}$, $\mathcal{R}(\omega)\approx 1$ and
$\mathcal{T}(\omega)\ll 1$, hence almost all the light is reflected from the
interface. For $\omega>\omega_{0}$, $\mathcal{R}(\omega)$ decreases and
reaches its minimum, $\mathcal{R}(\omega_{\min})=1.855\times 10^{-5}$, at
$\omega_{\min}=\omega_{0}+2\pi\times 37.74~{}\text{kHz}$.
$\mathcal{R}(\omega)$ increases for $\omega>\omega_{\mathrm{min}}$. For
$\omega=\omega_{+}$, $\mathcal{R}(\omega_{+})=0.009189$ is small.

##  VIII Summary and Conclusion

We calculated electric permittivity and magnetic permeability of SiO2 with
implanted Ag nanoparticles and Er3+ ions. We showed that when the number
density of the nanoparticles and the number density of ions are larger than
critical values, there is a frequency interval just above the $M_{1}$
transition frequency of Er3+ where the real part $n^{\prime}(\omega)$ of the
complex refractive index $n(\omega)$ is negative, and the imaginary part
$n^{\prime\prime}(\omega)$ is very small, see Fig. 4. Outside of this
interval, $n^{\prime}(\omega)$ is much smaller than
$n^{\prime\prime}(\omega)$. We also studied transmission and reflection of
light from the interface separating vacuum and the medium. When the refractive
index is negative, the medium is transparent and the reflection coefficient is
small. When the refractive index is positive, the reflection coefficient is
close to 1, and the light almost completely reflected from the interface.
Hence the medium can be used as an infrared filter to selectively transmit
light. We plan to generalize the theory to treat anisotropic crystals such as
quartz and isotropic materials subject to a DC electric field.

##  Appendix A Spectral terms of atoms

The notation to denote the atomic energy levels (i.e., the spectral terms of
the atoms or ions) is clarified in Ref. [28]: states with different values of
the total angular momentum $L$ are denoted by capital Latin letters, as
follows:

$\begin{array}[]{cccccccccccccc}L&=&0&1&2&3&4&5&6&7&8&9&10&\ldots\\\
&&S&P&D&F&G&H&I&K&L&M&N&\ldots\end{array}$

Above and to the left of this letter is placed the number $2S+1$, called the
multiplicity of the term, where $S$ is the total spin. When $L\geq S$, the
number of fine-structure components is $2S+1$. Below and to the right of the
letter is placed the value of the total orbital angular momentum $J$. Thus the
symbols ${}^{4}I_{15/2}$ and ${}^{4}I_{13/2}$ denote the levels with $L=6$,
$S=3/2$, $J=15/2$ and $13/2$.

##  Appendix B Derivation of the magnetic susceptibility in Eq. (6)

Interaction of an Er3+ ion with an alternating magnetic field,
$\mathbf{B}(t)=\frac{1}{2}\mathbf{B}_{0}\,e^{-i\omega t}+\mathrm{c.c.}$, is
given by the Hamiltonian

$H_{\mathrm{dm}}=g_{L}\mu_{B}\,\mathbf{L}\cdot\mathbf{B}(t)+g_{S}\mu_{B}\,\mathbf{S}\cdot\mathbf{B}(t),$
(24)

where $\mu_{B}$ is the Bohr magneton, $g_{L}\approx 1$ and $g_{S}\approx 2$
are the electron $g$-factors. $\mathbf{B}_{0}$ is assumed to be real and
directed in the $z$ direction. For the frequency $\omega$ far detuned from the
resonance frequency $\omega_{0}$, $H_{\mathrm{dm}}$ can be treated as a weak
perturbation. The second order correction $\epsilon_{2}(m_{J},B_{0})$ to the
ground-state energy due to the time-dependent interaction $H_{\mathrm{dm}}$ is

$\displaystyle\epsilon_{2}(m_{J},B_{0})$ $\displaystyle=$
$\displaystyle-\frac{\mu_{B}^{2}B_{0}^{2}}{4\hbar}\,\bigg{|}\Big{\langle}\frac{15}{2},m_{J}\Big{|}g_{L}L_{z}+g_{S}S_{z}\Big{|}\frac{13}{2},m_{J}\Big{\rangle}\bigg{|}^{2}$
(25)
$\displaystyle\times\bigg{[}\frac{1}{\omega_{0}-\omega-\frac{1}{2}\,i\gamma}+\frac{1}{\omega_{0}+\omega+\frac{1}{2}\,i\gamma}\bigg{]},$

where $m_{J}$ is the projection of $\mathbf{J}$ on the $z$ axis, and $\gamma$
is the resonance width of the $J=\frac{15}{2}$ to $J^{\prime}=\frac{13}{2}$
quantum transition. Taking into account the equalities,

$\displaystyle\Big{\langle}\frac{15}{2},m_{J}\Big{|}g_{L}L_{z}+g_{S}S_{z}\Big{|}\frac{13}{2},m_{J}\Big{\rangle}$
$\displaystyle\qquad\qquad=\big{(}g_{S}-g_{L}\big{)}\,\Big{\langle}\frac{15}{2},m_{J}\Big{|}S_{z}\Big{|}\frac{13}{2},m_{J}\Big{\rangle}$
$\displaystyle\qquad\qquad\approx\Big{\langle}\frac{15}{2},m_{J}\Big{|}S_{z}\Big{|}\frac{13}{2},m_{J}\Big{\rangle},$
$\displaystyle\frac{1}{\omega_{0}-\omega-\frac{1}{2}\,i\gamma}+\frac{1}{\omega_{0}+\omega+\frac{1}{2}\,i\gamma}\approx\frac{2\omega_{0}}{\omega_{0}^{2}-\omega^{2}-i\omega\gamma},$

we can rewrite Eq. (25) as

$\displaystyle\epsilon_{2}(m_{J},B_{0})$ $\displaystyle\approx$
$\displaystyle-\frac{\mu_{B}^{2}B_{0}^{2}}{2\hbar}\,\frac{\omega_{0}}{\omega_{0}^{2}-\omega^{2}-i\omega\gamma}$
(26)
$\displaystyle\times\bigg{|}\Big{\langle}\frac{15}{2},m_{J}\Big{|}S_{z}\Big{|}\frac{13}{2},m_{J}\Big{\rangle}\bigg{|}^{2}.$

The energy density $W_{2}(B_{0})$ of an ensemble of the Er3+ ions is

$W_{2}(B_{0})=N_{\mathrm{Er}}\sum_{m_{J}}p_{m_{J}}\,\epsilon_{2}(m_{J},B_{0}),$
(27)

where $N_{\mathrm{Er}}$ is the number density of the Er3+ ions, $p_{m_{J}}$ is
the probability to find an ion is in the ground state with the magnetic number
$m_{J}$, and

$\sum_{m_{J}}p_{m_{J}}=1.$

For equilibrium, $p_{m_{J}}=1/(2J+1)$ and we have

$\frac{1}{2J+1}\sum_{m_{J}}\big{|}\langle
L,S,J^{\prime},m_{J}|S_{z}|L,S,J,m_{J}\rangle\big{|}^{2}=\frac{2}{5},$

where $J=\frac{15}{2}$ and $J^{\prime}=\frac{13}{2}$ on the left hand side of
the equation.

Magnetic susceptibility $\chi_{\mathrm{Er}}(\omega)$ of the ensemble of the
Er3+ ions is

$\chi_{\mathrm{Er}}(\omega)=-\frac{\partial^{2}W_{2}(B_{0})}{\partial
B_{0}^{2}}.$ (28)

Substituting Eq. (27) into Eq. (28), we get

$\displaystyle\chi_{\mathrm{Er}}(\omega)$ $\displaystyle=$
$\displaystyle-\frac{N\mu_{B}^{2}\omega_{0}}{2\hbar(\omega^{2}-\omega_{0}^{2}+i\gamma\omega)}\sum_{m_{J}}p_{m_{J}}$
(29) $\displaystyle\times\big{|}\langle
L,S,J^{\prime},m_{J}|S_{z}|L,S,J,m_{J}\rangle\big{|}^{2},$

where $|L,S,J,m_{J}\rangle$ is the ground-state wave function with $L=6$,
$S=3/2$ $J=15/2$, and the projection $m_{J}$ of the orbital angular momentum
$\mathbf{J}$ on the $z$ axis,, $|L,S,J,m_{J}\rangle$ is the excited-state wave
function with $J^{\prime}=13/2$, and $p_{m_{J}}$ is the probability to find an
ion in the state $|L,S,J,m_{J}\rangle$. For equilibrium, Eq. (29) reduces to
Eq. (6).

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