# Lighting of a monochromatic scatterer with virtual gain

R. Ali<EMAIL_ADDRESS>Applied Physics Department, Gleb Wataghin
Physics Institute, University of Campinas, Campinas 13083-859, SP, Brazil

###### Abstract

In this work, we discuss the scattering features of a dipolar particle made of
large refractive index material by employing the concept of virtual gain and
virtual loss. The virtual gain and loss can be achieved in a lossless passive
nanostructure by shaping the temporal waveform of incident signals in the
complex frequency plane. We show that an appropriate tuning of excitation time
of the impinging field allows to capture and release the electromagnetic
energy on-demand for an arbitrary time scale in a lossless nanosphere. Thus,
the nanosphere obliges to emit monochromatic magnetic light which can be tuned
throughout the whole visible spectrum by varying the size of the nanosphere.
This proposal may find fruitful applications in lab-on-a-chip technologies and
the realization of monochromatic sectoral multipole light source with a large
quality factor at nanoscale level.

##  I Introduction

Scattering of light by nanosphere is a core problem in electromagnetism which
is thoroughly discussed in many applications such as sensing Kabashin2009 ;
Ali2020 , light manipulating devices Kuznetsov2016 ; Almeida2004 ;
Taghinejad2019 , optical manipulation Ashkin1986 ; Ali2020c ; Ali2021  and
light emitting devices Kuznetsov2012 . Particularly, the nano-optical devices
made of large refractive index nano-structures utilize the optical scattering
for routing the optical fields at nanoscales level which can be efficiently
controlled via electric and magnetic resonances Kuznetsov2012 ; Ali2020b .

Remarkably, nanosphere made of semiconductors ubiquitous unique optical
scattering properties due to strong magnetic response appears owing to sharp
magnetic dipolar resonances with comparatively low losses in visible and near-
infrared frequency-domain Evlyukhin2012 ; Dominguez2019 . The theoretical and
experimental results predict that interaction of the impinging light with a
nanosphere of large refractive index $({\rm m_{s}}>3$ e.g., of silicon,
germanium, tin, gallium arsenide, molybdenum diselenide, lead selenide,
mercury sulfide and so on) rotates the displacement currents inside the sphere
and yielding a magnetic moment oscillating through the center of the sphere
Person2013 ; fu2013 ; Etxarri2011 ; geffrin2012 ; alaee2015 ; Medina2011 .
Thus, a strong magnetic response has appeared in a nonmagnetic sphere in the
visible spectral range Evlyukhin2012 . Therefore, they are considered as basic
elements for the designing of complex metamaterials with desired optical
properties such as optical clocking Farhat2012 , single particles Mie
nanolaser Mylnikov2020 , and Huygen source Kuznetsov2012 ; Ali2020b .

One of the important applications of such resonators is an all-dielectric Mie
nano-laser which utilizes a gain functionalized dielectric nanostructure with
a single lasing mode at room temperature Mylnikov2020 ; Tiguntseva2020 ;
Hoang2020 . A typical Mie-nanolaser starts lasing a coherent monochromatic
light at the certain threshold gain at which the enhancement of stimulated
emission is achieved. It is well established that magnetic resonance supported
lasing mode can be accomplished at a lower lasing threshold and large quality
Q factor Kuznetsov2012 ; Zhang2018 . Therefore, a small optical gain is enough
to overcome the radiative losses and increase its Q factor several times as
compared to a regular Mie mode. Since the lasing modes are associated with Mie
resonances of the resonator, therefore, they can be tuned by changing the
dimensions of the particle, structural properties and amount of optical gain.

In this article, we provide an alternative method to design an all-passive and
all-dielectric monochromatic lighting source without employing optical gain.
In this context, we consider Silicon nano-sphere illuminated by non-
monochromatic incident signal oscillating in complex frequency plane. Under
the time-harmonic convention $e^{-i\omega t},$ the scattering zeros and
scattering poles are located in the upper and lower frequency half plane,
respectively. For the case of monochromatic incidence, a lossless cavity
cannot admit the scattering zeros and scattering poles but they can be reached
by adding appropriate material losses and gain, respectively, to the cavity.
On the contrary, for non-monochromatic incident signal oscillating in the
complex plane, the cavity admits the scattering zeros and poles which can be
pushed towards the real axis by tailoring the temporal profile of the incident
field. This temporal modulation of incident signal allows matching the
exponentially diverging perfectly absorbing mode corresponding to the
scattering zero over a finite time interval. To this end, the scattering
signal has to be zero. Since the cavity is lossless (Ohmic losses are zero),
thus, the energy associated with the incident signal will be virtually
absorbed or stored in the cavity.

On the other hand, this stored energy can be released on demand by cutting the
exponential growth of the impinging signal. Thus, the cavity is obliged to
give a strong scattering signal owing to release of the stored energy in the
absence of gain. In a nutshell, the transient behavior of the incident signal
dictates the lossless cavity to captures the incident signals for a finite
time scale and release them on demand. This concept of the capturing and
releasing of the incident signal are known as virtual absorption and virtual
gain, respectively, which has been recently proposed and experimentally
demonstrated by Andrea Alú′s group Baranov2017a ; Lepeshov2020 ; Radi2020 ;
Li2020 ; Baranov2017 .

Virtual loss and virtual gain in a lossless cavity can be achieved in the
laboratory by exciting the cavity with multiple incident signals with
controllable relative intensities and relative phases in real time. The
spatiotemporal modulation of the input waveforms is then used to store and
release the electromagnetic energy at will Baranov2017 . In this way, the
cavity can engage the scattering zeros in the upper complex frequency half-
plane when the incident signal grows faster than decay. On the contrary, the
virtual gain can be achieved by engaging the scattering poles positioned in
the lower complex frequency half plane, where the incident signal grows weaker
than its decay rate. Thus, the cavity is obliged to store energy in the upper
and release energy in the lower frequency plane. This counterintuitive
phenomenon plays crucial role in many applications, i. e. coherent perfect
virtual absorption Baranov2017 ; Longhi2018 , critical coupling of the signals
to a high-Q lossless resonators Radi2020  and tuning the direction of
radiation pressure Lepeshov2020 .

In this work, we use the well established concepts and findings displayed in
the Refs. Kuznetsov2012 ; Person2013 ; fu2013 ; Etxarri2011 ; geffrin2012 ;
alaee2015 ; Medina2011 ; Baranov2017a ; Lepeshov2020 ; Radi2020 ; Li2020 ;
Baranov2017  to achieve, for the first time, all passive monochromatic nano-
emitter by pairing the Mie resonances with the scattering poles in a balanced
way, which allows to emit a three orders magnitudes large monochromatic signal
at a nanoscale level.

The rest of the paper is organized as follows. Section II is devoted to the
brief description of methodology, where the scattering cross sections are
discussed using the Mie scattering theory. In Sec. III, we present numerical
analysis and results are discussed. Finally, we summarize our results and
conclude the findings in Sec. IV.

##  II Methodology

We start by considering a plane wave scattering by a nanoparticle of radius
$R$, refractive index ${\rm m_{s}}$ suspended in a nonabsorbing medium with
refractive index ${\rm n_{m}}$. The incident time-harmonic optical field of
angular frequency $\omega$ is propagating along the z-axis, polarization along
the x-axis and impinging on the nanosphere placed at the origin. The
electromagnetic self-oscillation of the nanoparticle can be determined by
following the standard approach presented in Ref. Bohren . Here, the incident
electromagnetic fields are expressed in spherical coordinates in terms of
vector spherical harmonics and the corresponding incident field ${\bf
E_{in}}$, scattered field ${\bf E}_{s}$ and field inside ${\bf E}_{int}$ the
sphere are given as (see Ch. 4 of the Ref. Bohren )

${\bf
E}_{in}=E_{0}\sum^{\infty}_{\ell=1}i^{\ell}\frac{2\ell+1}{\ell(\ell+1)}(M^{(1)}_{o1\ell}-iN^{(1)}_{e1\ell}),$
(1) ${\bf
E}_{s}=E_{0}\sum^{\infty}_{\ell=1}i^{\ell}\frac{2\ell+1}{\ell(\ell+1)}(ia_{\ell}N^{(3)}_{e1\ell}-b_{\ell}M^{(3)}_{o1\ell}),$
(2) ${\bf
E}_{int}=E_{0}\sum^{\infty}_{\ell=1}i^{\ell}\frac{2\ell+1}{\ell(\ell+1)}(c_{\ell}M^{(1)}_{o1\ell}-id_{\ell}M^{(1)}_{e1\ell}),$
(3)

where, ${\bf M}_{e1\ell}$, ${\bf M}_{o1\ell}$, ${\bf N}_{e1\ell}$ and ${\bf
N}_{o1\ell}$ are the vector spherical harmonics, and the explicit expression
of these vector functions are present in Ref. Bohren . In addition,
$a_{\ell}$, $b_{\ell}$ and $c_{\ell}$, $d_{\ell}$ are the sets of the Mie
coefficients representing the scattered electromagnetic field amplitudes
inside the surrounding medium and field amplitudes inside the spherical
particle, respectively. These Mie coefficients can be determined by applying
the subsidiary boundary conditions at the surface of the sphere $(r=R)$. For
instance, the electric and magnetic fields must obey the following boundary
conditions at the boundary between the sphere and the surrounding medium
Bohren

$({\bf E}_{in}+{\bf E}_{s}-{\bf E}_{int})\times\hat{n}=0,$ (4) $({\bf
H}_{in}+{\bf H}_{s}-{\bf H}_{int})\times\hat{n}=0.$ (5)

Finally, the quantification of energy received and scattered by the sphere is
defined in terms of dimensionless quantities and can be expressed as
extinction efficiency $Q_{ext}$, scattering efficiency $Q_{s}$, and absorption
efficiency $Q_{abs}$ Ali2020c ; Bohren ; Ali2018  as

Figure 1:  (a) Normalized extinction efficiency (red) by a sphere of radius
${\rm R}$ as a function of normalized real frequency. The scattering
contribution of a magnetic dipole (dashed), electric dipole (black), magnetic
quadruple (dotted), and electric quadruple (green). (b) Real (red) and
imaginary (black) part of denominator ${\rm D}$ of magnetic dipolar Mie-
coefficient as a function of normalized imaginary frequency for fixed real
frequency at 0.75.

$Q_{ext}=\frac{2}{x^{2}}\sum^{\infty}_{\ell=1}(2\ell+1)Re(a_{\ell}+b_{\ell}),$
(6)

$Q_{s}=\frac{2}{x^{2}}\sum^{\infty}_{\ell=1}(2\ell+1)(|a_{\ell}|^{2}+|b_{\ell}|^{2}),$
(7)

$Q_{abs}=Q_{ext}-Q_{s}.$ (8)

where, $x=ka$ ($k=2\pi/\lambda$) is the size parameter, and $\lambda$ is the
incident wavelength. The scattering Mie coefficients $a_{\ell}$ and $b_{\ell}$
are representing the scattering amplitudes of the electric and magnetic field,
respectively, which can easily be determined by simplification of the Eqs. 4 &
5 and one can read as Bohren ; Ali2018

$\displaystyle
a_{\ell}=\frac{m\psi_{\ell}(mx)\psi^{\prime}_{\ell}(x)-\mu\psi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}{m\psi_{\ell}(mx)\xi^{\prime}_{\ell}(x)-\mu\xi_{\ell}(x)\psi^{\prime}_{\ell}(mx)},$
(9) $\displaystyle
b_{\ell}=\frac{\mu\psi_{\ell}(mx)\psi^{\prime}_{\ell}(x)-m\psi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}{\mu\psi_{\ell}(mx)\xi^{\prime}_{\ell}(x)-m\xi_{\ell}(x)\psi^{\prime}_{\ell}(mx)},$
(10)

where, $\psi_{\ell}$, $\xi_{\ell}$ are Riccati-Bessel functions DLMF25.12 ,
${\rm m=m_{s}/n_{m}}$ and ${\rm\mu=\mu_{s}/\mu_{m}}$ are the relative
refractive index and relative permeability, respectively. The index $\ell$
denotes the $\ell^{th}$ order of spherical harmonics. It is worthy to
emphasize that for monochromatic incident light, a losses cavity presents
$Q_{abs}=0$ and all energy received by the sphere is being scattered and
allowing $Q_{ext}=Q_{s}$. However, for the case of non-monochromatic
incidence, the situation dramatically changes and a passive cavity presents
nonzero $Q_{abs}$ without heat losses.

Figure 2:  Normalized (a) extinction efficiency and (b) absorption efficiency
by a sphere of radius ${\rm R}$ illuminated by non-monochromatic incident
field as a function of normalized imaginary frequency for fixed real frequency
at at fixed ${\rm Re(\omega R/c=0.75)}$

##  III Results and discussion

In the following discussion, we consider two-port non-monochromatic light
source $E^{in}(t)\propto E_{0}e^{-i\omega t}$ incident upon a nanoparticle of
radius $R$ and refractive index ${\rm m_{s}=4}$, where
$\omega=\omega^{\prime}+i\omega^{\prime\prime}$, $\omega^{\prime}$ and
$\omega^{\prime\prime}$ are the real and imaginary frequencies, respectively.
Under this time convention, a passive sphere can engage the scattering zero
for $\omega^{\prime\prime}>0$ and scattering poles for
$\omega^{\prime\prime}<0$ in the complex frequency plane. As the incident
signal is oscillating in the complex frequency plane, the position of the
scattering zero and scattering poles can be conveniently tailored by tuning
the time evolution of complex excitation of the incident field, instead of
using optical losses or optical gain. Thus, tailoring of the temporal
wavefront of incident signal one can push the scattering zeros and scattering
poles close to the real frequency axis over a finite time interval. For
instance, in the upper complex frequency plane, the incident signal is growing
exponentially $E^{in}(t)\propto E_{0}e^{|\omega^{\prime\prime}|t}$, as the
complex frequency increases the lossless cavity is obliged to exhibit
$Q_{ext}>Q_{s}$ and at some points the scattering output approaches to zero.
Hence, the scattering zero has been reached and all incident energy is being
stored in the lossless cavity in terms of so-called virtual loss (without
Ohmic losses) in the resonator. On the contrary, in the lower complex
frequency plane incident signal $E^{in}(t)\propto
E_{0}e^{-|\omega^{\prime\prime}|t}$ exponentially attenuates, where the signal
decays faster than its growth rate, and hence the particle scatters more
energy than it receives $Q_{ext}<Q_{s}$. Since the particle has no gain
therefore the additional scattered energy is coming from the stored energy
Trainiti2019 . In this context, the scattering signal will be much stronger,
than the ordinary Mie scattering mode calculated at the real frequency axis,
and monochromatic close to the complex poles due to virtual gain, which is the
main goal of this work.

In order to make discussion independent of the incident frequency range, we
calculate the dimensionless scattering cross sections as a function of
dimensionless frequency ${\rm\omega R/c}$, where c is the speed of light.
Figure 1(a) demonstrates the normalized scattering efficiency as a function of
normalized monochromatic frequency (at the real frequency axis). It is
worthwhile to note that the sphere is nonmagnetic and having large refractive
index, the incident field rotates the displacement currents and providing a
strong ring-shaped electric field inside the sphere. Consequently, a strong
transversely oscillating magnetic field in the center of the ring appears at
$\lambda_{0}/m_{s}\approx 2R$ and providing an oscillating magnetic dipole
(MD) at ${\rm Re(\omega R/c)=0.75}$ Kuznetsov2016 . It is important to
emphasize that the sizes of the nanosphere ($R\approx\lambda_{0}/2m_{s}$)
required to excite the dipolar resonance are sufficiently large and hence the
nonlocal and quantum effects can be ignored Ruppin1973 ; Gubbin2020 . As
frequency grows other resonances corresponding to electric dipoles (ED),
magnetic quadruple (MQ) and electric quadrupole (EQ occur at ${\rm Re(\omega
R/c)=1}$, ${\rm Re(\omega R/c)=1.1}$, and ${\rm Re(\omega R/c)=1.4}$,
respectively, as expected fu2013 ; Etxarri2011 ; geffrin2012 ; Medina2011 . It
is clearly shown that in the long-wavelength regime the fundamental resonance
MD has larger Q factor than ED. Thus, MD should be a good candidate for the
scattering of magnetic light for longer time scale. It is worthy to stress
that sphere is lossless and hence $Q_{s}=Q_{ext}$

Figure 3:  (a) The squared absolute value of the dipolar magnetic Mie
coefficient $|b_{1}|^{2}$ as a function of normalized real frequency. (b) The
scattered frequency shift in the visible region, where the scattering response
of $|b_{1}|^{2}$ is calculated by considering a sphere of radii ${\rm 70\,nm}$
(orange), ${\rm 80\,nm}$ (red) and ${\rm 90\,nm}$ (dashed) as a function of
wavelength. The normalized imaginary frequency is fixed for all calculations
at ${\rm Im(\omega R/c)=-0.02}$.

On the other hand, if we shin a non-monochromatic field the situation
drastically changes and the lossless cavity leaves nonzero absorption
footprint in the complex frequency plane. The theoretical study predicts that
the real frequencies corresponding to the resonances should engage scattering
poles in the lower frequency plane. The position of such poles can be
analytically calculated by solving the denominator of corresponding scattering
amplitudes. As we are interested in a magnetic monochromatic dipolar light
source, therefore, we numerically solve denominator of magnetic dipolar
($\ell=1$) Mie coefficient
$D={\mu\psi_{\ell}(mx)\xi^{\prime}_{\ell}(x)-m\xi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}$
along the complex frequency axis for ${\rm Re(\omega R/c)=0.75}$ and shown in
Fig. 1(b). It is straightforward to notice that ${\rm Re(D)}$ (red) and ${\rm
Im(D)}$ (black) are vanishing at ${\rm Im(\omega R/c)=-0.02}$, and it is a
clear indication that a pole of $b_{1}$ in complex frequency plane has been
reached. In order to discuss the scattering behavior in complex frequency
plane close to the pole, we calculate the extinction efficiency Fig. 2(a) and
abortion efficiency Fig. 2(b) as a function of imaginary frequency for ${\rm
Re(\omega R/c)=0.75}$.

It is important to stress that a resonator engages corresponding complex zero
in the upper complex frequency plane and captures energy (virtual loss) during
the transient excitation. This stored energy can be paired with poles and
released on will as depicted in Fig. 2, where the particle has to decay
incident signal faster than it grows and hence scattering efficiency should be
different than the extinction efficiency (for $\omega^{\prime\prime}\neq 0$).
It is clearly shown in Fig. 2(a) that when magnetic dipolar resonance is
aligned with scattering pole positioned at ${\rm Im(\omega R/c)=-0.02}$ and
the resonator scatters more energy than it receives, thus negative extinction
efficiency is appearing. Hence it is immediate to tell that the rest part of
the scattering energy is coming from the virtual gain which is the consequence
of the negative absorption for $\omega^{\prime\prime}<0$ as shown in Fig.
2(b). This figure shows that the absorption is zero at the real frequency axis
while $Q_{abs}>0$ and $Q_{abs}<0$ in the upper and lower frequency plane,
respectively, as one can refer to the inset of Fig. 2(b). It is worth
mentioning that the phenomenon of perfect energy capture and release at will
has been demonstrated experimentally Baranov2017 ; Trainiti2019 .

Figure 4: Density plot showing the scattering pattern of light by a sphere of
radius R illuminated by a non-monochromatic incident signal as a function of
real and imaginary frequency.

Figure 3(a) demonstrates the squared absolute value of the magnetic dipole as
a function of ${\rm Re(\omega R/c)}$ for ${\rm Im=-0.02}$. It is shown that
the magnetic dipolar scatterer behaves like a monochromatic magnetic light
source of large output signal. The scattering signal is about three order
magnitude stronger than the value achieved for monochromatic incidence as
shown in Fig. 1(a). Such type of scattering phenomenon can also be achieved by
employing an optical gain medium Mylnikov2020 . The scattered magnetic light
by the nanoparticle is so strong and it can be easily seen under a dark-field
optical microscope. It is important to emphasize that the wavelength of the
monochromatic scattered magnetic light signal can be varied throughout the
whole visible spectral range from violet to red by just changing by changing
the cavity size and refractive index. For concreteness, we validate our
results by considering the realistic material and physical parameters, and the
corresponding scattering signal is demonstrated in Fig. 3(b). Here we present
squared absolute value of the magnetic dipole $|b_{1}|^{2}$ of a Si nanosphere
as a function of wavelength for different sizes: ${\rm R=70\,nm}$ (orange),
${\rm R=80\,nm}$ (red) and ${\rm R=90\,nm}$ (dashed). It is clearly shown that
for a small increment in the size of the nanoparticle the resonance peak moves
from ${\rm 480\,nm}$ to ${\rm 700\,nm}$. Finally, we expand this concept to
parametric space to elucidate the role of other resonances by plotting the
scattering amplitudes as a function of normalized real and imaginary frequency
in Fig. 4. It is clearly shown that there are four scattering peaks
corresponding to MD, ED, EQ, and MQ. Since we are interested in a smallest
scatterer, therefore, we are considering the MD dipolar resonance for dipolar
magnetic light. On the other hand, the ED signal appears with a low Q factor
and the MD and ED would not be important in the long wavelength or dipolar
limit.

##  IV Summary

In this paper, we have revisited the scattering response of non-monochromatic
incident light, oscillating with the complex frequency, by a large refractive
index nanosphere using the Mie scattering theory. Our numerical analysis shows
that one can achieve monochromatic magnetic light with a large quality factor
than an ordinary Mie mode without employing optical gain and population
inversion at room temperature. To this end, the monochromatic scattering
signal is achieved by spatiotemporal modulation of the exponentially growing
incident signal. Thus, fine tailoring of the excitation in the complex
frequency plane provides an optimal control to tune the location of the
scattering poles and move them close to the real frequency axis, instead of
using optical gain which was considered to be an essential requirement. Our
findings have shown that the efficient tuning of the incident signal’s
excitation in complex frequency plane allows to pair the scattering poles and
scattering zero in a balanced way. Thus, providing an opportunity to realize
monochromatic dipolar electric, magnetic light Trigo2019 . This smallest Mie-
resonant is able to emit single-valued wavelength light which can be tuned
throughout the visible region by varying the size of the nanosphere. Finally,
we confident that this novel scattering paradigm can pave the way to design
the multipolar light source, multifunctional photonic designs for routing
light at the nanoscale level.

### Acknowledgment

We thank S. Iqbal, G. Wiederhecker, F. A. Pinheiro, F. S. S. Rosa and P. A. M.
Neto for inspiring discussions. This work is partially supported by Fundacão
de Amparo a Pesquisa do Estado de São Paulo (FAPESP) (2020/03131-2). The
author declares no conflicts of interest.

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