# Inverse Design in the Complex Plane: Manipulating Quasi–Normal Modes

J. R. Capers<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Exeter, Stocker Road, Exeter, EX4 4QL  D. A. Patient
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Exeter,
Stocker Road, Exeter, EX4 4QL  S. A. R. Horsley  Department of Physics and
Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL

###### Abstract

Utilising the fact that the frequency response of a material can be decomposed
into the quasi–normal modes supported by the system, we present two methods to
directly manipulate the complex frequencies of quasi–normal modes in the
complex plane. We first consider an ‘eigen–permittivity’ approach that allows
one to find how to shift the permittivity of the structure everywhere in order
to place a single quasi–normal mode at a desired complex frequency. Secondly,
we then use perturbation theory for quasi–normal modes to iteratively change
the structure until a given selection of quasi–normal modes occur at desired
complex frequencies.

##  I Introduction

Quasi–normal modes (QNMs) are the complex frequency bound states of a system.
They were first used in quantum mechanics to describe alpha decay Gamow1928 ;
Bethe1937 , and have since found utility in modelling radiation in many
different systems, from black holes Chandrasekhar1975 , photonic resonators
Kristensen2020  to leaky waveguides Ghatak1985 ; Hu2009 . QNMs correspond to
the poles of the scattering matrix in the complex frequency plane
Alpeggiani2017 ; Tikhodeev2017 , where the waves at the boundary of the system
are purely out–going. The effect of a structured environment can, for example,
be analysed by decomposing the Purcell factor in terms of these QNMs
Zschiedrich2018 , and through calculating how small changes in the system
perturb the QNMs, deeper insight into sensing has been developed Yang2015 ;
Both2022 . Here, motivated by the connection between the location of poles in
the complex plane and physical properties such as transmission, we combine
ideas from inverse design with the QNM approach to modelling resonator systems
to design materials that have poles at specific complex frequencies. Perhaps
the simplest example of a system supporting QNMs is a homogeneous dielectric
slab (refractive index $n_{R}$ in some background index $n_{B}$). For this
simple case, the complex frequencies of the QNMs can be found analytically
Chandrasekhar1975 ; Kristensen2020  as

$k_{m}L=\frac{2\pi
m+i\ln\left[\left(n_{R}-n_{B}\right)^{2}/\left(n_{R}+n_{B}\right)^{2}\right]}{2n_{R}},$
(1)

where $m$ is an integer and $L$ is the width of the slab. Figs. 1(a-c)
demonstrate that poles in the reflection coefficient as a function of complex
$k$ correspond to QNMs, which are in turn associated with peaks in
transmission. Examining the field, shown in Fig. 1(c), at a complex $k$ value
associated with a QNM shows the characteristic exponential growth in space.

Figure 1: The quasi–normal modes of a dielectric slab (a-c) and the absorbing
stack (d-e). The reflection coefficient in the complex plane (a) and
transmission ($\sqrt{1-|r|^{2}}$) along the real frequency axis (b). The red
crosses represent the analytic solution to Eq. (1) for $n_{b}=1$, $n_{r}=\pi$,
$L=1$. The real component of the QNMs are associated with the peaks in
transmission. The real (blue), imaginary (orange dashed) and absolute (black)
field distribution (c) of the $m=3$ mode is shown to have the characteristic
exponential growth in space. For the (near) perfect absorber (depicted in
inset, green layers are Germanium, yellow are Silicon Oxide, with a Tungsten
substrate), the complex reflection coefficient (d) shows a single QNM. The
absorption spectrum (e) shows a large resonance in the mid-IR, with a resonant
frequency ($\lambda_{0}$) and linewidth ($\Gamma$) directly associated with
the QNM, which can be understood in terms of the poles of the associated
Lorentzian (red dashed line).

More complicated systems can also be understood in terms of QNMs. For example,
multilayer dielectric absorbers (e.g. the mid–infrared absorber presented in
Sakurai2019 ) can be understood this way. The absorption of the structure
given in Sakurai2019 , along with the reflection coefficient in the complex
wavelength plane is shown in Figs. 1(d-e). Fitting the Lorentzian

$\mathcal{L}(\lambda)=\frac{\Gamma}{(\lambda-\lambda_{0})^{2}+\Gamma^{2}}$ (2)

to the absorption peak, we find the peak wavelength is $\lambda_{0}=5.15\mu$m
and the linewidth $\Gamma=0.0138\mu$m. This corresponds to a pole of the
reflection coefficient in the complex plane at $\lambda_{0}+i\Gamma$, as shown
in Fig. 1(e).

While QNMs provide a valuable framework to understand resonators, the ability
to _design_ the spectral response of materials is key to e.g. more efficient
photovoltaic cells Zoysa2012  and sensors Liu2010 . For sensing applications,
narrow resonances at particular wavelengths are desirable Landy2008 ; Luo2016
; Lochbaum2017 , while energy harvesting requires large absorption over a
broad band Aydin2011 ; Pala2009 ; Zhou2021 ; Ding2016 .

When designing spectral features, one can employ the physical insight provided
by QNMs to greatly simplify the problem. For example, one way to approach the
inverse design problem for absorbers is to try to move the QNM to a desired
complex frequency Grigoriev2013 . In this way, one can tailor scattering
effects Wu2020 , design absorbers Ming2019  and manipulate exceptional points
Yan2020  with minimal numerical complexity. To date, however, these approaches
address the forwards problem, finding how the pole moves if the resonator
geometry is changed. We instead solve the inverse design problem of designing
materials with poles at specific complex frequencies, using only simple
techniques.

We present two methods for placing QNM poles at arbitrary complex frequencies.
Firstly, we re–formulate the eigenvalue problem of the Helmholtz equation to
find a complex constant value by which the permittivity of a structure should
be shifted to place a pole in the desired location. Secondly, we employ QNM
perturbation theory to find how to change the spatial distribution of material
to move around several poles in the complex frequency plane. These methods
enable the simultaneous control of resonance wavelength _and_ linewidth, for
the design of absorbers and sensors.

##  II Eigen–Permittivities

One way to _find_ the locations of quasi–normal modes (QNMs) is to formulate
the Helmholtz equation for the out–of–plane electric field $\phi$, as an
eigenvalue problem for complex wave–numbers $k$

$-\frac{1}{\varepsilon(x)}\derivative[2]{\phi}{x}=k^{2}\phi.$ (3)

However, to find the QNMs the correct boundary condition must be imposed on
$\phi$. Originally derived by Sommerfeld Sommerfeld_pde , but since used to
model black hole radiation Zerilli1970 ; Kapur1938 , the appropriate boundary
condition is that the wave is purely outgoing. For example, on the either side
of a planar medium,

$\derivative{\phi(x)}{x}=\pm ik\phi(x),$ (4)

as $x\rightarrow\pm\infty$. To numerically find the QNMs of our system, we
imposed this boundary condition within a finite difference approximation,
adapting the elements of the Laplacian at the boundaries e.g. for $N$ points
the value of the field at the final point on the right of the system is fixed
to be $\phi_{N+1}=\phi_{N}+ik\Delta x\phi_{N}$, giving

$\derivative[2]{\phi}{x}\approx\frac{1}{(\Delta
x)^{2}}\begin{pmatrix}(ik\Delta x-1)&1&0&0\\\ 1&-2&1&0\\\ 0&1&-2&1\\\
0&0&1&(ik\Delta x-1)\end{pmatrix}\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\\\ \phi_{4}\end{pmatrix}.$ (5)

It is now evident that solving the eigenvalue problem required to find the
QNMs is challenging Lalanne2019  as the eigenvalue $k^{2}$ also appears in the
boundary condition. To avoid solving this non–linear problem, it has recently
been noted by Chen et al. Chen2019  that the analysis of QNMs can be
simplified by working in terms of real wave–numbers but extending the
_permittivity_ into the complex plane. Despite the utility of the normal mode
framework of Chen et. al. Chen2019  for employing modal expansions we are
trying to engineer the resonance location (related to ${\rm Re}[k]$) and
linewidth (given by ${\rm Im}[k]$). The location of the QNM frequency
trivially encodes these features we are trying to engineer. Employing the
insight of Chen et. al., we write the permittivity as a spatial variation plus
a constant background $\varepsilon(x)=\varepsilon_{s}(x)+\varepsilon_{b}$
allows us to recast the Helmholtz equation as an eigenvalue problem for the
permittivity

$-\frac{1}{k^{2}}\left(\frac{d^{2}}{dx^{2}}+k^{2}\varepsilon_{s}(x)\right)\phi(x)=\varepsilon_{b}\phi(x).$
(6)

Rather than using this to find the QNMs of a system, we show that this can be
used to design the complex frequencies of the QNMs.

To do this, we take a known spatially varying permittivity, such as the
dielectric step or absorber stack e.g. from Sakurai2019 , and choose a
$k\in\mathbb{C}$ at which we would like a QNM to occur. We then numerically
solve the eigenvalue problem Eq. (6) using the finite difference method Eq.
(5), along with standard matrix libraries, to find a complex
eigen–permittivity that allows us to form a structure with
$\varepsilon(x)=\varepsilon_{s}(x)+\varepsilon_{b}$ with a pole at the chosen
complex frequency.

We first apply the method to the homogeneous slab. In Fig. 2 we design the new
structure to support a QNM at the frequency $k=1.5-0.05i$. For the $N\times N$
Laplacian matrix, there are $N$ possible values for $\varepsilon_{b}$ that
will satisfy this condition. Taking the lowest absolute valued background (to
minimise numerical error) permittivity $\varepsilon_{b}=-4.99-2.32i$, we find
that the new structure now supports a QNM at our chosen $k$. This is shown in
Fig. 2(a). The transmission, Fig. 2(b), shows a large peak at the real
frequency associated with the QNM and has values $|t|>1$ due to the gain that
has been added to the system. Although the location of the pole can be
manipulated solely by changing the height of the barrier, in order to
manipulate the real and imaginary parts independently, control over both the
real and imaginary permittivity is required. As might be anticipated, in order
to move a pole closer to the real frequency axis, without changing the
resonant frequency, gain is required. Conversely, loss is required to move the
pole further away from the real axis. The field profile, shown in Fig. 2(c),
still has the exponential growth characteristic of QNMs.

Figure 2: A background permittivity $\varepsilon_{b}=-4.99-2.32i$ is found as
a solution to Eq. (5) which, when combined with the original structure
$\varepsilon_{s}(|x|<L/2)=\pi^{2}$ will contain a pole at the desired complex
frequency of $k=1.5-0.05i$. The reflection coefficient of the new structure is
plotted in the complex plane (a). The transmission along the dashed white
line, where $\rm{Im}[k]=0$ is plotted (b), alongside the field distribution
plotted at the complex frequency $k$ (c). Overlaid on the transmission
calculations are results found using COMSOL Multiphysics COMSOL .

Next, we apply the same eigen–permittivity method to the absorbing stack shown
in Fig. 1(e). For this structure, we must take care that the correct boundary
conditions are imposed. The opaque metal substrate requires the Dirichlet
boundary condition $\phi=0$, while the outgoing wave boundary condition must
be imposed at the top of the stack. Choosing two target bandwidths, for the
same resonance wavelength, $\lambda_{1}=(6.5+0.03i)\mu$m and
$\lambda_{2}=(6.5+0.15i)\mu$m, we obtain background permittivities of
$\varepsilon_{b,1}=3.27-0.01i$ and $\varepsilon_{b,2}=3.28+0.29i$. The effect
of the background shift on the pole locations is shown in Fig. 3(a-b).
Accordingly, the poles are found at the expected complex frequencies. The
absorption, shown in Fig. 3(c) plotted along the white dashed line
($\rm{Im}[\lambda]=0$) is also provided, with a fitted Lorentzian to extract
the properties of the resonances and verify that it corresponds to the QNM
frequencies.

Figure 3: The original absorbing stack, shown in Fig. 1(e) has been modified
into two structures that contain a QNM at $\lambda_{1}=(6.5+0.03i)\mu{\rm m}$
and $\lambda_{2}=(6.5+0.15i)\mu{\rm m}$ respectively. The former is close to
the real axis, corresponding to a narrow bandwidth, while the latter has a
broader bandwidth. Plotted on (a) and (b) respectively are the reflection
coefficients in the complex plane, showing that a QNM is indeed located at the
chosen complex frequency. The absorption spectra of the two structures are
plotted as a function of real wavelength (c). Fitted Lorentzians in dashed red
(blue) correspond to fitting to the narrow (broad) resonance, verifying the
complex frequencies of the QNMs. For the broadband case, we must fit a sum of
3 Lorentzians to accurately model the spectral profile, and obtain the correct
fitting parameters.

We can also apply this design procedure to impose the condition of coherent
perfect absorption (CPA) at a given complex frequency. This can be understood
as the time reverse of QNMs Chong2010  were the wave is purely incoming rather
than outgoing. The wavelengths at which a structure behaves as a perfect
absorber are related to the locations of zeros on the real axis, rather than
poles. With our eigen-permittivity formulation, we can find the background
permittivity value required to make the device a perfect absorber at a
frequency of choice. To do this, we simply take the outgoing boundary
condition Eq. (4) and replace it with the incoming boundary condition

$\frac{d\phi(x)}{dx}=\mp ik\phi(x)$ (7)

as $x\rightarrow\pm\infty$. This changes the boundary elements in the
Laplacian Eq. (5) from $ik\Delta x-1$ to $-ik\Delta x-1$.

Applying the above changes to the Laplacian matrix, we can take e.g. a slab of
dielectric, and rather than choose a complex frequency, pick a real frequency
that we wish CPA to occur at. We take a dielectric slab of length $L=1$ and
initial permittivity $\pi^{2}$ and choose the arbitrary CPA frequency to be
125 MHz. The resulting background permittivity required is
$\epsilon_{b}=-9.55+i0.63$. To verify that there is coherent perfect
absorption at the chosen frequency, we construct the scattering matrix for the
slab under incidence from the left and right side

$\begin{pmatrix}\phi_{L}^{\rm scattered}\\\ \phi_{R}^{\rm
scattered}\end{pmatrix}=\begin{pmatrix}r_{L}&t_{R}\\\
t_{L}&r_{R}\end{pmatrix}\begin{pmatrix}\phi_{L}^{\rm in}\\\ \phi_{R}^{\rm
in}\end{pmatrix},$ (8)

noting that CPA occurs when an eigenvalue of the scattering matrix goes to
zero Chong2010 . The scattering matrix can be constructed analytically from
the transfer matrix or found numerically in full–wave solvers such as COMSOL
COMSOL . In Fig. 4 we plot the smallest eigenvalue of the scattering matrix of
the slab as a function of frequency. A clear dip is seen at the desired
frequency. We also show field profiles both under incidence from only one side
and from both sides at different frequencies. Under incidence from only the
left side, one can see the usual interference between reflected and incident
field to the left of the slab and the constant transmitted field. Under
excitation from both sides, but away from the target CPA frequency one can see
reflection from both sides. At the target CPA frequency of 125 MHz, an almost
constant field amplitude is observed, indicating perfect absorption.

Figure 4: An example of using our eigen–permittivity method to design a
structure that exhibits coherent perfect absorption, shown schematically in
a). Under incidence from one direction, the structure scatters in the usual
way, but under incidence from both sides reflection vanishes. We design a
permittivity step of length $L=1$ of permittivity
$\varepsilon=\pi^{2}+(-3.98+i\ 1.59)$ that exhibits this behaviour at the
desired frequency of 125 MHz. To verify this, we show b) the smallest
eigenvalues of the scattering matrix of the structure. Vanishing eigenvalue
indicates coherent perfect absorption. This can be clearly observed at the
target frequency of 125 MHz. The fields c), also indicate coherent perfect
absorption. Under excitation from one side or off of the target frequency,
reflections are observed. At the design frequency, there is a standing wave.

So far, all of the examples provided have been in 1D. However our method is
straightforwardly extended to higher dimensions. To illustrate this we
consider a 2D square dielectric resonator, shown in Fig. 5(a). The resonator
is a silicon cross inside a gallium arsenide square. To find how to change the
permittivity to place a pole at a particular complex frequency, we must solve
the eigenvalue problem Eq. (6) in 2D. To do this, we use COMSOL’s coefficient
form PDE interface, which allows one to solve problems of the form

$\lambda^{2}e_{a}\phi-\lambda
d_{a}\phi+\nabla\cdot\left(-c\nabla\phi-\alpha\phi+\gamma\right)+\beta\nabla\phi+a\phi=f,$
(9)

where $\lambda$ is the eigenvalue. Choosing the coefficients to be
$e_{a}=0,c=1,d_{a}=1,a=-k^{2}\varepsilon$, this becomes exactly the eigenvlaue
problem we would like to solve

$\nabla^{2}\phi+k^{2}\varepsilon\phi=-\lambda\phi,$ (10)

where $\lambda=k^{2}\varepsilon_{b}$. The outgoing wave boundary condition can
be applied to the outside edge of the resonator using the ‘flux/source’
boundary condition. Generally, this boundary condition is

$-\boldsymbol{n}\cdot\left(-c\nabla\phi+\alpha\phi+\gamma\right)=g-\phi u,$
(11)

where $\boldsymbol{n}$ is a unit–vector normal to the surface of the resonator
at a given point. It is not necessary for $\boldsymbol{n}$ to be normal to the
surface, it only needs to point outwards. With our parameter choices this
becomes

$\boldsymbol{n}\cdot\nabla\phi=-q\phi.$ (12)

Setting $q=ik$ gives the correct out–going boundary condition. Solving this
eigenvalue problem for the 2D geometry shown in Fig. 5(a), and choosing the
location of the pole to be $f=500+1i$ THz, we find a background permittivity
of $\varepsilon_{b}=-1.37+i0.88$. To verify that a QNM is now located at the
correct complex frequency, we excite the resonator with a nearby point dipole
and examine the total scattered power before and after the permittivity shift
is applied. This is shown in Fig. 5(b). Once the shift is applied, there is a
clear peak in scattered power at the desired wavelength. Additionally, the
fields when the resonator is driven at 500 THz are shown in Fig. 5(c-d). Once
the permittivty of the resonator is shifted, scattering at the desired
frequency is greatly enhanced by the presence of the QNM.

Figure 5: An example of using our eigen–permittivity framework to place the
quasi-normal modes of a 2D resonator. The resonator, shown inset in a), is
made of two different permittivities, $\varepsilon_{1}$ (silicon at 550 nm)
and $\varepsilon_{2}$ (gallium arsenide at 550 nm). We apply our framework to
find a permittivity offset to move a pole to the complex frequency (500 + 1i)
THz. The background is $\varepsilon_{b}=-1.37+i0.88$. To verify the location
of the pole, we excite the resonator with a point electric dipole, located at
(-1100 nm, 0), and calculate the total scattered power, shown in b). A clear
peak is present in the spectrum of the shifted structure at the desired
frequency of 500 THz, which is not present in the spectrum of the un–shifted
structure. Examining the fields of the resonator driven by a nearby dipole at
a frequency of 500 THz, c) and d), we see that the excitation of the mode in
the sifted structure greatly increase the scattering.

Although simple to implement, this eigen–permittivity method only allows you
to choose the complex frequency of a single QNM. We now explore the
possibility of applying an iterative method to move one or more QNMs to
desired complex frequencies, by changing the spatial variation of the
permittivity profile.

##  III Optimisation Approach to Moving Poles

The second method we present to move quasi–normal modes (QNMs) to desired
complex frequencies is to use an iterative procedure, based on perturbation
theory. Standard Rayleigh–Schrödinger perturbation theory LL3  of Hermitian
quantum mechanics connects a change in the potential $\delta V$ to a change in
the $n^{\rm th}$ energy level $E_{n}$ through the matrix element

$\delta E_{n}=\langle\phi_{n}|\delta V|\phi_{n}\rangle,$ (13)

where the states are normalised so that
$\langle\phi_{n}|\phi_{m}\rangle=\delta_{nm}$. Usually the perturbation to the
potential is known and the energy level shifts are calculated (e.g. in the
textbook analysis of the Stark effect LL3  §76). Being able to analytically
connect structure and function is the key to inverse design, allowing one to
find derivatives of a quantity of interest (here, the energy) in terms of
derivatives of the structure (the potential). With this observation, it is
possible to use perturbation theory backwards to find how one should change
the potential to get a particular energy level. This idea can be extended to
move the complex frequency of a QNM of an electromagnetic resonator. Instead
of a potential, we seek to design a permittivity profile $\varepsilon(x)$ that
has a QNM, $k_{n}$, at a particular complex frequency. However, as QNMs grow
in space, they cannot be normalised. The expressions that connect a change in
the permittivity profile to a change in the complex wave–number $k_{n}$
requires some modification. Regularisation techniques have been used to
develop a perturbation theory for QNMs in both quantum mechanics ZelDovich1961
; Leung1998  and electromagnetism Muljarov2010 . Perturbation theory can be
used to connect a change in the permittivity $\delta\varepsilon(x)$ to a
change in the complex frequency of the QNM Perelomov1998  through

$\delta
k_{n}=\frac{1}{2k_{n}}\frac{\int_{-L/2}^{L/2}\phi_{n}^{2}(x)\delta\varepsilon(x)dx}{\langle\phi_{n}|\phi_{n}\rangle},$
(14)

where $k=k^{\prime}+ik^{\prime\prime}$ and the inner product is now Leung1998

$\langle\phi_{n}|\phi_{n}\rangle=\int_{-L/2}^{L/2}\phi_{n}^{2}(x)dx+i\left[\phi_{n}^{2}(-L/2)+\phi_{n}^{2}(L/2)\right].$
(15)

If we change the permittivity by a small amount $\Delta\varepsilon$ at a
particular location $x_{i}$ so that
$\delta\varepsilon(x)=\Delta\varepsilon\delta(x-x_{i})$, we find that

$\delta
k_{n}=\frac{1}{2k_{n}}\frac{\phi_{n}^{2}(x_{i})\Delta\varepsilon}{\langle\phi_{n}|\phi_{n}\rangle}.$
(16)

As this is true for all $x_{i}$, we can divide by the small change in
permittivity to find the gradient of the wave–number with respect to the
permittivity

$\partialderivative{k_{n}}{\varepsilon}=\frac{\phi_{n}^{2}(x)}{2k_{n}\langle\phi_{n}|\phi_{n}\rangle}.$
(17)

Importantly, this gives a continuous function for the derivative of the
complex frequency of the QNM with respect to the spatial structure of the
permittivity. For example, say we would like to move mode $k_{n}$ to the
complex frequency $k_{\star}$. We can write a suitable figure of merit and
it’s derivative as

$\displaystyle\mathcal{F}$ $\displaystyle=(k_{n}-k_{\star})^{2},$ (18)
$\displaystyle\partialderivative{\mathcal{F}}{\varepsilon}$
$\displaystyle=2(k_{n}-k_{\star})\partialderivative{k_{n}}{\varepsilon}.$ (19)

Updating the permittivity from iteration $i$ to $i+1$ is done according to

$\varepsilon^{(i+1)}(x)=\varepsilon^{(i)}(x)+\gamma\partialderivative{\mathcal{F}}{\varepsilon},$
(20)

where $\gamma$ is the step size. This makes the evaluation of the figure of
merit gradients extremely efficient, similar to the adjoint method Lalau-
Keraly2013 . Combining this with gradient descent optimisation Press2007 , we
have found how to update the permittivty distribution in order to arbitrarily
change the complex frequencies of the QNMs.

Figure 6: An example of using our iterative method to move a pole to a desired
location. Beginning from a) a step of dielectric which supports a QNM at
$k=1-0.1i$, our iterative method designs the permittivity distribution shown
in b), which supports a QNM at the desired frequency $k_{\star}=0.8-0.01i$.
The resulting transmission of the structure is shown in c), and compared to a
full–wave solver. Fitting a Lorentzian to the transmission peak associated
with $k_{\star}$, we extract find that the peak is at $k_{0}=0.799$ with width
$\Gamma=0.0109$. The path of the pole over the optimisation is shown in d).

An example of this procedure is shown in Fig. 6. We begin by selecting a QNM
of the system: the frequency of which we want to modify. The complex
wave–number of this mode can be found by root–finding in the complex plane,
using i.e. Newton’s method. Specifying a target frequency of the pole
$k_{\star}$, then using Eqns. (17, 19, 20) to iteratively update the
permittivity distribution allows the pole to be moved to the desired complex
frequency. At every iteration, $\phi_{n}$ and $k_{n}$ must be re–calculated.
In the example of Fig. 6 we move the pole originally at $k=1-0.1i$ to
$k_{\star}=0.8-0.01i$, and show that yields a structure with a peak in
transmission at the designed frequency with the designed width. It should be
noted that while we can move the pole to an arbitrary complex frequency,
complete control of both the real and imaginary parts of the permittivity is
required.

As another example of this method, we consider trying to move several poles
simultaneously. In Fig. 7 we take the poles originally at $k=1-0.1i$, $2-0.1i$
and $3-0.1i$ and move them to three different values $k_{1},k_{2}$ and
$k_{3}$. Interestingly, due to the presence of other nearby poles, the
transmission profile of the resulting structure becomes more complex, however
a clear narrow transmission peaks associated with $k_{1},k_{2}$ and $k_{3}$
are evident. If one controls all poles of interest over a given range of $k$
values, almost complete control over the transmission profile can be obtained.

Figure 7: An example of using the iterative method we present to move 3 poles
to desired complex frequencies at the same time. Beginning from a permittivity
step shown in a), the real poles associated with ${\rm Re}[k]=1,2,3$ are moved
to the targets: $k_{1}=0.8-0.007i$, $k_{2}=3.5-0.008i$ and $k_{3}=1.8-0.009i$.
The resulting permittivity profile is shown in b) and its transmission
coefficient in c). Clear peaks are seen at the three target values of $k$. The
path of the poles over the optimisation is shown in d).

##  IV Conclusions and Outlook

In this work we address the inverse design problem: ‘how should one change a
photonic system to ensure a quasi–normal mode appears at a pre-determined
complex frequency?’. We propose two approaches to answer this question. The
first is to re-express the permittivity of a system as the original
permittivity profile, plus some global background shift
$\varepsilon_{s}(x)+\varepsilon_{b}$. This allows us to write the Helmholtz
equation as an eigenvalue problem for the background permittivity
$\varepsilon_{b}$. By choosing a target complex frequency, we can find a
(complex) background permittivity that can be added to the structure so that a
QNM occurs at the desired complex frequency with the desired linewidth. This
method could be used to modify existing structures to control the frequency
and bandwidth of a resonant system. We also show that we can apply this method
in order to construct materials that, at a single frequency of operation, act
as coherent perfect absorbers.

The second approach we develop is an iterative procedure based on perturbation
theory: a small change in permittivity can be connected to the shift in
complex frequency of a QNM. By defining a suitable figure of merit, and
combining with gradient descent optimisation, we can iteratively change the
spatial permittivity profile to move a QNM closer to a target frequency. This
procedure can also be used to move multiple QNMs to different target
frequencies. This iterative approach can be further modified in many ways. For
example by restricting the search space of $\delta\varepsilon$ to only allow
loss rather than gain, or to ensure $\varepsilon(x)>1$. Also, rather than
manipulating the full spatial form of $\varepsilon$, we could seek to change
only a few free parameters such as width and height of the dielectric step.

The approaches we have developed open up several avenues of exploration to
design, for example, broadband absorbers for solar cells and thermal emitters.
Since the framework also applies to leaky waveguides, our methods could also
be used to design leaky–wave antennas. Rather than manually changing
structural parameters until a QNM appears at the correct complex frequency,
the methods we present leverage the benefits of inverse design to rapidly
design materials that have the desired properties. Importantly, as our methods
allow QNMs to be placed exactly, both resonance frequency and linewidth can be
tuned with a high degree of accuracy.

## Acknowledgements

The authors would like to thank Josh Glasbey and Ian Hooper for many
illuminating discussions and Jake Binsley for his assistance with Blender.

We acknowledge financial support from the Engineering and Physical Sciences
Research Council (EPSRC) of the United Kingdom, via the EPSRC Centre for
Doctoral Training in Metamaterials (Grant No. EP/L015331/1). J.R.C also wishes
to acknowledge financial support from Defence Science Technology Laboratory
(DSTL). S.A.R.H acknowledges financial support from the Royal Society
(URF\R\211033). All data and code created during this research are openly
available from the corresponding authors, upon reasonable request.

## References

  * [1] G. Gamow.  Zur quantentheorie des atomkernes.  Zeitschrift für Physik, 51(204-212), 1928.
  * [2] H. A. Bethe.  Nuclear physics b: Nuclear dynamics, theoretical.  Rev. Mod. Phys, 9(69), 1937.
  * [3] S. Chandrasekhar and S. Detweiler.  The quasi-normal modes of the schwarzschild black hole.  Proc. R. Soc. Lond. A, 344(441-452), 1975.
  * [4] P. T. Kristensen, K. Herrmann, F. Intravaia, and K. Busch.  Modeling electromagnetic resonators using quasinormal modes.  Adv. Opt. Photonics, 12(612-708), 2020.
  * [5] A. K. Ghatak.  Leaky modes in optical waveguides.  Opt. Quantum Electron., 17(311-321), 1985.
  * [6] J. Hu and C. R. Menyuk.  Understanding leaky modes: slab waveguide revisited.  Adv. Opt. Photonics, 1(58-106), 2009.
  * [7] Filippo Alpeggiani, Nikhil Parappurath, Ewold Verhagen, and L. Kuipers.  Quasinormal-mode expansion of the scattering matrix.  Phs. Rev. X, 7(021035), 2017.
  * [8] S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara.  Quasiguided modes and optical properties of photonic crystal slabs.  Phs. Rev. B, 66(045102), 2002.
  * [9] Lin Zschiedrich, Felix Binkowski, Niko Nikolay, Oliver Benson, Günter Kewes, and Sven Burger.  Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators.  Phys. Rev. A, 98:043806, Oct 2018.
  * [10] J. Yang, H. Giessen, and P. Lalanne.  Simple analytical expression for the peak-frequency shifts of plasmonic resonances for sensing.  Nano Lett., 15(3439-3444), 2015.
  * [11] S. Both, M. Schäferling, F. Sterl, E. A. Muljarov, H. Giessen, and T. Weiss.  Nanophotonic chiral sensing: How does it actually work?  ACS Nano., 16(2822-2832), 2022.
  * [12] A Sakurai, K Yada, T Simomura, S Ju, M Kashiwagi, H Okada, T Nagao, K Tsuda, and J Shiomi.  Ultranarrow-Band Wavelength-Selective Thermal Emission with Aperiodic Multilayered Metamaterials Designed by Bayesian Optimization.  ACS Cent. Sci., 5(2):319–326, feb 2019.
  * [13] M. De Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and Susumu Noda.  Conversion of broadband to narrowband thermal emission through energy recycling.  Nature Photon., 6(535-539), 2021.
  * [14] N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen.  Infrared perfect absorber and its application as plasmonic sensor.  Nano. Lett., 10(2342-2348), 2010.
  * [15] N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla1.  Perfect metamaterial absorber.  Phys. Rev. Lett., 100(207402), 2008.
  * [16] S. Luo, J. Zhao, D. Zuo, and X. Wang.  Perfect narrow band absorber for sensing applications.  Opt. Express, 24(9288-9294), 2016.
  * [17] A. Lochbaum, Y. Fedoryshyn, A. Dorodnyy, U. Koch, C. Hafner, and J. Leuthold.  On-chip narrowband thermal emitter for mid-ir optical gas sensing.  ACS Photonics, 4(1371-1380), 2017.
  * [18] K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater.  Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers.  Nat. Commun., 2(517), 2011.
  * [19] R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma.  Design of plasmonic thin-film solar cells withbroadband absorption enhancements.  Adv. Mater., 21(3504-3509), 2009.
  * [20] Y. Zhou, Z. Qin, Z. Liang, D. Meng, H. Xu, D. R. Smith, and Y. Liu.  Ultra-broadband metamaterial absorbers from long to very long infrared regime.  Light Sci Appl, 10(138), 2021.
  * [21] F. Ding, J. Dai, Y. Chen, J. Zhu, Y. Jin, and S. I. Bozhevolnyi.  Broadband near-infrared metamaterial absorbers utilizing highly lossy metals.  Scientific Reports, 6(39445), 2016.
  * [22] V. Grigoriev, A. Tahri, S. Varault, B. Rolly, B. Stout, J. Wenger, and N. Bonod.  Optimization of resonant effects in nanostructures via weierstrass factorization.  Phys. Rev. A, 88(011803), 2013.
  * [23] T. Wu, A. Baron, P. Lalanne, and K. Vynck.  Intrinsic multipolar contents of nanoresonators for tailored scattering.  Phys. Rev. A, 101(011803), 2020.
  * [24] X. Ming and L. Sun.  Optimization of broadband perfect absorber by weierstrass factorization.  IEEE Photonics J., 11(IEEE Photonics J.), 2019.
  * [25] Wei Yan, Philippe Lalanne, and Min Qiu.  Shape deformation of nanoresonator: A quasinormal-mode perturbation theory.  Phys. Rev. Lett., 125:013901, Jul 2020.
  * [26] A. Sommerfeld.  Partial Differential Equations in Physics.  Academic Press, 1949.
  * [27] F. J. Zerilli.  Effective potential for even-parity regge-wheeler gravitational perturbation equations.  Phys. Rev. Lett., 24(737-738), 1970.
  * [28] P. L. Kapur and R. Peierls.  The dispersion formula for nuclear reactions.  Proc. R. Soc. Lond. A, 166(277-295), 1938.
  * [29] P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss.  Quasinormal mode solvers for resonators with dispersive materials.  J. Opt. Soc. Am. A, 36(686-704), 2019.
  * [30] P. Y. Chen, D. J. Bergman, and Y. Sivan.  Generalizing normal mode expansion of electromagnetic green’s tensor to open systems.  Phys. Rev. Appl., 11(044018), 2019.
  * [31] COMSOL Multiphysics v. 6.0, 2022.
  * [32] Y. D. Chong, Li Ge, Hui Cao, and A. D. Stone.  Coherent perfect absorbers: Time-reversed lasers.  Phys. Rev. Lett., 105(053901), 2010.
  * [33] L. D. Landau and E. M. Lifshitz.  Quantum Mechanics: Non-relativistic Theory.  Pergamon Press, 2rd edition, 1965.
  * [34] Y. B. Zel’Dovich.  On the theory of unstable states.  Soviet Physics JETP, 12(3), 1961.
  * [35] P. T. Leung, Y. T. Liu, W. M. Suen, C. Y. Tam, and K. Young.  Logarithmic perturbation theory for quasinormal modes.  J. Phys. A, 31(3271), 1998.
  * [36] E. A. Muljarov, W. Langbein, and R. Zimmermann.  Brillouin-wigner perturbation theory in open electromagnetic systems.  EPL, 92(50010), 2010.
  * [37] A. M. Perelomov and Y. B. Zel’Dovich.  Quantum Mechanics: Selected Topics.  World Scientific, 1998.
  * [38] C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch.  Adjoint shape optimization applied to electromagnetic design.  Opt. Express, 21:21693–21701, 2013.
  * [39] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery.  Numerical Recipes: The Art of Scientific Computing (3rd Ed.).  Cambridge University Press, 2007.