# Plasmon localization and giant fields in holographic metasurface for SERS
sensors

Andrey K. Sarychev<EMAIL_ADDRESS>Andrey Ivanov  Andrey N.
Lagarkov  Ilya Ryzhikov  Konstantin Afanasev  Institute for Theoretical and
Applied Electrodynamics, Russian Academy of Sciences, 125412, Moscow, Russia
Igor Bykov  Institute for Theoretical and Applied Electrodynamics, Russian
Academy of Sciences, 125412, Moscow, Russia;
Laboratory of Nano-Bioengineering, National Research Nuclear University MEPhI
(Moscow Engineering Physics Institute), 31 Kashirskoe shosse, Moscow 115521,
Russia  Grégory Barbillon  EPF-Ecole d’Ingénieurs, 3 bis rue Lakanal, 92330
Sceaux, France
Nikita Bakholdin  Mikhail Mikhailov  National Research University “Moscow
Power Engineering Institute”, Krasnokazarmennaya 14, Moscow 111250, Russia
Alexander Smyk  Alexander Shurygin  James River Branch llc, 8, Tvardovsky st.,
Strogino Industrial Park Moscow 123458, Russia  Alexander Shalygin  Lomonosov
Moscow State University, Faculty of Physics, GSP-1, 1-2 Leninskiye Gory,
Moscow 119991, Russia

###### Abstract

We present SERS-active metal holographic metasurfaces fabricated from metal
periodical nanograting deposited on a dielectric substrate. The metasurface
consists of a modulated dielectric, which is covered by a thin silver layer.
The metasurface operates as an open plasmon resonator. The theory of plasmons
excited in the open resonator formed by a metal nanograting is presented. The
large local electromagnetic field is predicted for optical frequencies. The
excitation of plasmons is experimentally demonstrated in the metasurface
designed on a 4-inch Si wafer. The enhancement of the local electric field
results in surface-enhanced Raman scattering (SERS). To investigate the SERS
effect, the metasurfaces are covered by molecules of 4-mercaptophenylboronic
acid, which form covalent bonds with the silver nanolayer and serve as a
proof-of-concept. Finally, we obtain a detection limit of 230 nM for molecules
of 4-mercaptophenylboronic acid.

††preprint: APS/123-QED

##  I Introduction

The design of a nanoresonator with a high enhancement of the local electric
field is an important point for the application to the surface-enhanced Raman
scattering (SERS) and other plasmon-enhanced spectroscopies. It is important
to increase the selectivity and sensitivity of the biological and chemical
sensing [1, 2, 3, 4]. The novel fabrication techniques allow to produce
sophisticated resonant substrates with the controllable shape and arrangement
of the resonators [5]. These techniques include focused ion-beam lithography
[6], electron-beam lithography [7, 8, 9, 10], X-ray, UV, plasmonic cavity lens
(PCL) and interference lithographs [11, 12, 13, 14, 15, 16], nanoimprint
Lithography (NIL) [17, 18, 19], nanosphere lithography (NSL) [20, 21, 22, 23],
laser-induced transfer metal nanodroplets [24]. Some of them allow to realize
reproducible SERS substrates with a lower cost and a larger active surface
area, for example, interference lithography. In references [11, 15], the gold
and hybrid gold-silver nanodisk array fabricated via X-ray interference
lithography is proposed. The effectiveness of the structures has been proven
by the detection of a low concentration of Rhodamine 6G (R6G) molecules via
SERS with enhancement factor $10^{5}-10^{6}$. The authors admitted high
sensitivity, reproducibility and stability of such structures. In reference
[16], silver nanohole arrays are proposed. A photoresist layer sandwiched with
two Ag layers increases Q-factor of nanoresonators and the sensitivity. The
SERS substrates exhibited enhancement factors (EFs) of $10^{7}$ that are
capable of monolayer detection of R6G molecules. It should be noted that
optically mirror underlayer, which is designed to increase Q-factor, was
theoretically predicted and experimentally investigated by the authors [4, 26,
25, 27]. Theoretical predictions and experimental studies have shown giant
electromagnetic field fluctuations in the case of almost touched plasmonic
nanoparticles [29, 28, 30, 31, 32, 33, 34, 35, 36]. It was shown that
plasmonic nanocavities confining the light to unprecedentedly small volumes,
support multiple types of modes. Different nature of these modes leads to the
mode beating within the nanocavity and the Rabi oscillations, which alter the
spatio-temporal dynamics of the hybrid system [37]. By intermixing plasmonic
excitation in nanoparticle arrays with excitons in a $WS_{2}$ monolayer inside
a resonant metal microcavity, the hierarchical system was fabricated with the
collective microcavity-plasmon-exciton Rabi splitting exceeding $0.5\,eV$ at
room temperature. Gap-surface plasmon metasurfaces, which consist of a
subwavelength thin dielectric spacer sandwiched between an optically thick
film of metal and arrays of metal subwavelength elements arranged in a
strictly or quasiperiodic fashion, have a possibility to fully control the
amplitude, phase, and polarization of the reflected light [30, 38].

Open resonators can very effectively accumulate the electromagnetic energy
(see, e.g., [39]). In the optical frequency range, a corrugated metal surface
can operate as an open resonator and generate large local electric field [40,
41, 42, 43]. The method of the transformation optics was developed to
calculate optical properties of metamaterials [44, 45]. This method was used
to investigate reflection from the metal gratings in recent papers [46, 47,
48, 49]. The authors develop an original conformal mapping that transforms
one-side metal grating into rectangular metal slab that optical properties can
be found. For instance, to calculate the reflection of an electromagnetic
wave, the metal grating is replaced by thin plane metal film. The developed
method was used to theoretically investigate broadband THz absorption in
graphene metasurfaces [50], optical properties of singular metal surfaces [48,
51], non-local effects [52], and calculating of the energy loss of an electron
flying over the metal modulation [53].

Strong motivation for the investigation of open plasmon resonators is the SERS
effect, which is mainly due to the local electric field enhancement. The SERS
effect enables identification of trace molecules captured by the corrugated
metal surface. The SERS is extremely important for medical diagnostics, for
instance, for cancer detection, imaging and therapy, drug delivery,
quantitative control of biomarkers including glycated proteins and
cardiovascular biomarkers [55, 56, 57, 58, 59, 60].

In this paper, we present the analytical theory as well as experimental
observation of the plasmons excited in open resonators formed by a periodic
metal grating. The resonance condition of the open plasmon resonator is found,
when it is possible to achieve large electromagnetic field enhancement both
inside and in the vicinity of the plasmonic grating. This will increase the
sensitivity of SERS-probing and other surface-enhanced spectroscopies. The
modulated silver film is produced by four-ray holographic process and its
optical properties are measured. The silver metasurfaces are covered by
molecules of 4-mercaptophenylboronic acid to investigate the SERS in
metasurfaces made by optical interference lithography. Holographic SERS
substrates are easy-made and low-cost for mass production, and they have
allowed to achieve a detection limit of 230 nM for molecules of
4-mercaptophenylboronic acid (4-mPBA).

##  II Analytical theory

We consider periodic metal gratings that support localized plasmons whose size
can reach few hundred nanometers. The metal gratings with two-side modulation
efficiently concentrate and store electromagnetic energy. The local electric
field is resonantly enhanced. The localized plasmons are analytically
calculated using a quasistatic approximation for the nanogratings whose
dimensions (thickness and modulation) are much smaller than the wavelength
$\lambda$.

###  II.1 Plasmon resonance in metal nanotube

To understand how the metasurface works, let us solve a simple two-dimensional
plasmonic system, which is interesting by itself since it has a non-trivial
analytical solution. We investigate the plasmon resonance and the
corresponding electromagnetic field enhancement in the metal nanotube that
radius and thickness are much smaller than the wavelength $\lambda$. The
length of the metal nanotube is much larger than the radius. The vector of the
external electric field lies in the $u-v$ plane perpendicular to tube axis.
Since the transversal dimensions of the nanotube is much smaller than the
wavelength the problem is reduced to the two dimensional problem of the
excitation of the infinite metal nanotube by the $E=\\{E_{u},E_{v}\\}$
electric field, which is normal to the tube axis. For further simplification,
we neglect retardation effects and use the quasistatic approximation when the
plasmonic electric field is found from the electric potential $\varphi(u,v)$.
The two dimensional problem can be solved by considering potential, electric
field, and charge in $u-v$ plane using theory of functions of a complex
variable. All the distances are measured in terms of the internal radius of
the nanotube.

Thus, we calculate the plasmon resonance and the field enhancement in a metal
ring with radius $r_{0}$, the thickness $d_{0}$, and the metal permittivity
$\varepsilon_{m}$, shown in Fig. 1 by red color. To find the field in the
ring, we introduce the Descartes coordinates $u$, $v$, and the complex
variable $w=u+iv$. The ring has the center at $u=v=0$. It is surrounded by a
dielectric host $\varepsilon_{d}$, blue color in Fig. 1; the medium inside the
ring has permittivity $\varepsilon_{1}$ (light green in Fig. 1). The
electromagnetic field is excited by the point ”magnetic” monopole, which
generates in the surrounding space the electric potential
$\varphi_{s}=\Re\varphi_{sing}$, where the complex potential equals to

$\varphi_{sing}=iE_{0}\log(w-x_{0}),$ (1)

where $u=x_{0}$ and $v=0$ are coordinates of the monopole (horizontal blue
line in Fig. 1). The exact nature of the monopole is irrelevant for the
material presented below.

Figure 1: Metal disk in plane $u-v$ with permittivity of internal space
$\varepsilon_{1}$, metal permittivity $\varepsilon_{m}$, permittivity of outer
space $\varepsilon_{d}$, and a ”magnetic” monopole placed at $u=x_{0},v=0$.

The monopole oscillates with the frequency $\omega$, and the amplitude $E_{0}$
is assumed, for simplicity, to be positive $E_{0}>0$. Recall the quasistatic
approximation is used, when the field retardation is neglected. Then, the
electric potential $\varphi$ is the solution of the Laplace equation
$\triangle\varphi=0$ that holds in all the space but the monopole center. As
usual in two-dimensional (2D) problem, a complex electric field is introduced
$\tilde{E}=E_{u}-iE_{v}=-d\varphi/dw$. The field is expanded into radial
$E_{r}=\Re(w\tilde{E})/r$ $\equiv(E_{u}u+E_{v}v)/r$ and angular
$E_{\phi}=-\Im(w\tilde{E})/r$ $\equiv(-E_{u}v+E_{v}u)/r$ components, where
$r=\left|w\right|=\sqrt{u^{2}+v^{2}}$. The electric charges, induced by the
monopole, are uniformly distributed in the metal ring when the monopole is
placed in the center. Therefore, angular component of the electric field
remains to be the same inside and outside the ring. For the central monopole
$(x_{0}=0)$ with amplitude $E_{0}>0$, the interior electric field
$E^{(i)}(r<1)$, the field $E^{(m)}(1\leq r\leq 1+d_{0})$ in the ring, and the
outer field $E^{(e)}(r>1+d_{0})$ have only angular components. The interior,
ring, and outer fields are all equal to
$E^{(i)}_{r}=E^{(m)}_{r}=E^{(e)}_{r}=0$ and
$E^{(i)}_{\varphi}=E^{(m)}_{\varphi}=E^{(e)}_{\varphi}=E_{0}/r$. As we see,
the central monopole does not excite the plasmon regardless of the value of
the metal permittivity $\varepsilon_{m}$. Yet, this is an unstable solution.
Plasmons are excited when monopole is shifted from the center. Suppose that
the monopole is set at $u=x_{0}<1,v=0$. The interior complex potential equals
to

$\displaystyle\varphi^{(i)}(w)=\varphi_{sing}(w)+\varphi^{(i)}_{reg}(w)=$
$\displaystyle iE_{0}[\log(w-x_{0})+\varphi^{(i)}_{reg}(w)],\;\;r\leq 1$ (2)

where $\varphi^{(i)}_{reg}(w)$ is a regular part of the interior potential
that can be expressed in series of $w$ that only contains positive powers. To
simplify the consideration, we neglect for a moment the loss in the metal
assuming that permittivity $\varepsilon_{m}$ has zero imaginary part. The
complex interior field equals to

$\displaystyle\tilde{E}^{(i)}(w)=-\dfrac{d\,\varphi(w)}{d\,w}=$ (3)
$\displaystyle-
iE_{0}\left[\frac{1}{w-x_{0}}+\sum_{n=1}^{\infty}A_{n}w^{n-1}x_{0}^{n}\right],\;\;r\leq
1,$

where $0<x_{0}<1$ and it is still assumed that $E_{0}$ is real. The first term
in Eq. (3) is expanded in series of $(x_{0}/w)^{n}$ for
$x_{0}<\left|w\right|<1$, and we obtain

$\displaystyle\tilde{E}^{(i)}(w)=-iE_{0}\left[w^{-1}+\right.$ (4)
$\displaystyle\left.\sum_{n=1}^{\infty}x_{0}^{n}\left(A_{n}w^{n-1}+w^{-n-1}\right)\right],\;\;r\leq
1$

where coefficients $A_{n}$ take real values. The electric field in the metal
ring is expanded in Laurent series:

$\displaystyle\tilde{E}^{(m)}(w)=-iE_{0}\left[w^{-1}+\right.$ (5)
$\displaystyle\left.\sum_{n=1}^{\infty}x_{0}^{n}\left(B_{n}w^{-n-1}+C_{n}w^{n-1}\right)\right],\;\;1<r<1+d$

and the outer complex field decreases when the radius $r\equiv\left|w\right|$
goes to infinity:

$\displaystyle\tilde{E}^{(e)}(w)=-iE_{0}\left[w^{-1}+\sum_{n=1}^{\infty}x_{0}^{n}D_{n}w^{-n-1}\right],$
(6) $\displaystyle r\geq 1+d$

These complex fields can be split in radial and angular parts, namely,

$E=\left\\{E_{r},E_{\phi}\right\\}=\left\\{\Re\left[\tilde{E}\frac{w}{r}\right],-\Im\left[\tilde{E}\frac{w}{r}\right]\right\\}.$

For example, the internal field $E^{(i)}$ has the following radial and angular
components

$\left\\{E^{(i)}_{r},E^{(i)}_{\phi}\right\\}=E_{0}\left\\{0,r^{-1}\right\\}\\\
+E_{0}\sum_{n=1}^{\infty}r^{-n-1}x_{0}^{n}\left\\{\left(A_{n}r^{2n}-1\right)\sin(n\phi),\left(A_{n}r^{2n}+1\right)\cos(n\phi)\right\\},\;r\leq
1$ (7)

The $\left\\{E_{r},E_{\phi}\right\\}$ expansion of the internal $E^{(i)}$,
ring $E^{(m)}$, and outer $E^{(e)}$ fields holds even if the coefficients
$A_{n},B_{n},C_{n},D_{n}$ are complex values. The coefficients ascribe complex
values since the metal permittivity has imaginary part
$\varepsilon_{m}=\varepsilon_{m1}+i\varepsilon_{m2}$ due to ohmic loss.

The important value of the intensity
$I^{(i)}=\left|E^{(i)}\right|^{2}/\left|E_{0}\right|^{2}$ of the internal
field, which is averaged over the internal surface of the metal ring,
($\left|w\right|\equiv r=1$) equals to

$\left\langle
I^{(i)}\right\rangle=\dfrac{1}{2\pi}\int_{0}^{2\pi}\left|E^{(i)}(r=1,\phi)\right|^{2}\,d\phi=1+\frac{1}{2}\sum_{n=1}^{\infty}x_{0}^{2n}\left(\left|A_{n}\right|^{2}+1\right)\equiv
1+\sum_{n=1}^{\infty}I_{n}$ (8)

where the coefficients $A_{n}$ are given below by Eq. (12). The coefficients
$A_{n},B_{n},C_{n},D_{n}$ in Eqs. (3) – (8) are found from the boundary
conditions for the electric fields at internal surface $(\left|w\right|=1)$
and outer surface $(\left|w\right|=1+d_{0})$ of the ring:

$\displaystyle\varepsilon_{1}E^{(i)}_{r}=\varepsilon_{m}E^{(m)}_{r},\;E^{(i)}_{\varphi}=E^{(m)}_{\varphi},\;\;\left|w\right|=1,$
(9)
$\displaystyle\varepsilon_{m}E^{(m)}_{r}=\varepsilon_{d}E^{(e)}_{r},\;E^{(m)}_{\varphi}=E^{(e)}_{\varphi},\;\;\left|w\right|=1+d_{0}$
(10)

where $\varepsilon_{1}$, $\varepsilon_{m}$ and, $\varepsilon_{d}$ are the
permittivities of the internal space $(r\leq 1)$, metal ring $(1<r<1+d_{0})$
and the outer space $(r\geq 1+d_{0})$, respectively. We substitute the series
(3) – (7) in the above boundary equations and equate the coefficients at the
same power of $x_{0}$. It is easy to verify that the coefficients at
$x_{0}^{n}$ in all series (3)-(7) have the same angular dependence. Thus, the
following basic equations are obtained:

$\displaystyle
A_{n}+1=B_{n}+C_{n},\;\;\varepsilon_{1}\left(A_{n}-1\right)=\varepsilon_{m}\left(C_{n}-B_{n}\right),$
$\displaystyle
B_{n}+C_{n}d_{2n}=D_{n},\;\;\varepsilon_{m}\left(B_{n}-C_{n}d_{2n}\right)=\varepsilon_{d}D_{n}$
(11)

where $d_{2n}=(1+d_{0})^{2n}$. Solution of these equations gives the
coefficients

$\displaystyle
A_{n}=\frac{d_{2n}\left(\varepsilon_{1}-\varepsilon_{m}\right)\left(\varepsilon_{d}+\varepsilon_{m}\right)-\left(\varepsilon_{m}+\varepsilon_{1}\right)\left(\varepsilon_{d}-\varepsilon_{m}\right)}{det_{n}},$
(12) $\displaystyle
B_{n}=\frac{2d_{2n}\left(\varepsilon_{d}+\varepsilon_{m}\right)}{det_{n}},\;C_{n}=\frac{2\left(\varepsilon_{m}-\varepsilon_{d}\right)}{det_{n}},\;D_{n}=\frac{4d_{2n}\varepsilon_{m}}{det_{n}}$

that completely determine the electric field inside, in, and out of the
plasmonic ring. The determinant $det_{n}$ of equations Eq. (II.1) that is the
denominator in Eqs. (12) equals to

$\displaystyle
det_{n}=d_{2n}\left(\varepsilon_{m}+\varepsilon_{1}\right)\left(\varepsilon_{m}+\varepsilon_{2}\right)+$
$\displaystyle\left(\varepsilon_{1}-\varepsilon_{m}\right)\left(\varepsilon_{m}-\varepsilon_{d}\right)$
(13)

Zeroing of the determinant $det_{n}=0$ gives the condition for the $n$-th
resonance exciting in the plasmonic ring. For the given permittivities of the
metal ring $\varepsilon_{m}$ and outer space $\varepsilon_{d}$, the ring
resonates when its thickness takes the following values:

$\displaystyle
d_{0res,n}=r_{0}\,\Re\left[\left(\frac{\left(\varepsilon_{1}-\varepsilon_{m}\right)\left(\varepsilon_{d}-\varepsilon_{m}\right)}{\left(\varepsilon_{m}+\varepsilon_{1}\right)\left(\varepsilon_{d}+\varepsilon_{m}\right)}\right)^{\frac{1}{2n}}-1\right]\simeq$
$\displaystyle
r_{0}\left[\left(\frac{\left(\varepsilon_{1}-\varepsilon_{m1}\right)\left(\varepsilon_{d}-\varepsilon_{m1}\right)}{\left(\varepsilon_{m1}+\varepsilon_{1}\right)\left(\varepsilon_{d}+\varepsilon_{m1}\right)}\right)^{\frac{1}{2n}}-1\right]$
(14)

where we restore natural dimensions and insert the radius $r_{0}$ of the ring.
$n=1,2,3,\ldots$ is the resonance number, i.e., $2n$ is the number of nodes in
the ring field $E^{(m)}$ (see Fig. 2). In transition to the second estimation
in Eq. (II.1), it is taken into account that the real part
$\varepsilon_{m1}=\Re[\varepsilon_{m}]$ of the metal permittivity is much
larger in absolute value than the imaginary part
$\varepsilon_{m2}=\Im[\varepsilon_{m}]\ll\left|\varepsilon_{m1}\right|$ for
the ”good” optical metals like silver or gold. We obtain an expected result:
the ring resonates if and only if the real part of the metal permittivity is
negative. For the red and infrared parts of the optical spectrum, the real
part $\varepsilon_{m1}$ of the metal permittivity is negative and large in
absolute value. Then, the dispersion equation Eq. (II.1) simplifies to

$d_{0res,n}\simeq
r_{0}\frac{\varepsilon_{d}+\varepsilon_{1}}{n|\varepsilon_{m}|}$ (15)

where $r_{0}$ is the radius of the ring. We obtain that the resonance
thickness of the metal layer is inverse proportional to the absolute value of
the metal permittivity.

The dimensionless intensity $\left\langle I^{(i)}\right\rangle$ of the
resonance electric field averaged over the internal surface of the ring with
thickness $d_{0res,n}$ estimates as $\left\langle
I^{(i)}(d=d_{0res,n})\right\rangle=I_{0\max,n}\simeq I_{n}$ from Eq. (8):

$\displaystyle I_{0\max,n}\simeq
2\left[\frac{\epsilon_{1}\epsilon_{m1}\left(\epsilon_{d}^{2}-\epsilon_{m1}^{2}\right)}{\epsilon_{m2}\left(\epsilon_{d}+\epsilon_{1}\right)\left(\epsilon_{m1}^{2}-\epsilon_{1}\epsilon_{d}\right)}\right]^{2}\left(\frac{x_{0}}{r_{0}}\right)^{2n}$
(16)

where $\varepsilon_{m1}=\Re\varepsilon_{m}$ and
$\varepsilon_{m2}=\Im\varepsilon_{m}$. Excitation of various resonances in the
silver ring is illustrated in Fig. 2 for the wavelength $\lambda=785\,nm$. The
silver permittivity for this wavelength is $\varepsilon_{m}\simeq-30+0.4i$
[61], and the permittivity of the outer space is chosen to be
$\varepsilon_{d}=2$, which corresponds to the permittivity of the photoresist
in our experiments. The first three resonances are displayed in Figure 2. The
electric field in a metal ring has two, four and six nodes for the first,
second and third resonances, respectively. Note the first dipole resonance
electric field, shown in Figure 2a, fills all the internal space of the ring.
This result corresponds to the electric field distribution in the metal
cylinder excited by an uniform external electric field [62].

Figure 2:  Electric field in the silver ring for the plasmon resonances
$n=1,2,3$ at an excitation wavelength of $\lambda=785\,nm$. Silver
permittivity is $\varepsilon_{m}\approx-30+0.4i$, internal permittivity
$\varepsilon_{1}=1$, outer permittivity $\varepsilon_{d}=2$, ring thickness
$d_{0res,n}=0.11,\,0.052,\,0.034$, all dimensions are normalized to the ring
radius $r_{0}$. The white spot corresponds to the monopole, which is placed at
$u=x_{0}=0.9,\,v=0$.

A plasmon excited in a metal tube can be experimentally investigated. Suppose
that the nanochannels with radius $r_{0}$ are formed in a dielectric material.
Nanochannels can be produced, for instance, in the silicon by the
electrochemical etching (see [63] and reference therein). The surface of the
nanochannels is covered by a silver layer with the thickness $d_{res,1}\simeq
r_{0}(\varepsilon_{d}+1)/(\left|\varepsilon_{m}\right|)$ given by Eq. (15).
Suppose further that the Raman-active analyte is injected in these channels.
Then, the Raman signal from the molecules in the nanochannels is much enhanced
since the scattering is incoherent and the Raman intensities are summed up.
The Raman enhancement factor (SERS) estimates as $G\sim I^{2}_{max,n}$ (see,
e.g., [64]), where $I_{max,n}\gg 1$ is given by Eq. (16) and the parameter
$x_{0}\sim r_{0}$. For the typical Raman spectroscopy wavelength of
$\lambda=785\,nm$, the silver permittivity is about
$\varepsilon_{m}\simeq-30+0.4i$ and the enhancement factor estimates as
$G\sim\left(\varepsilon_{m1}/\varepsilon_{m2}\right)^{4}\sim 10^{7}$. This
esteem holds for any argument of the complex amplitude $E_{0}$. For instance,
in the case $E_{0}=-2iQ,\,Q>0$ the monopole potential (1) is the potential of
the usual electric charge, which density equals to
$Q\delta(u-x_{0})\delta(v)$. Any other field sources such as a dipole or
quadrupole, can be combined from electric monopoles.

###  II.2 Giant electric field in metal grating

We consider a modulated metal film. The metal grating is placed at the plane
$X=0$. The electromagnetic wave propagates along X-axis being
${}^{\prime\prime}y^{\prime\prime}$-polarized, i.e.,
$E\sim\\{0,E_{0},0\\}\exp(ikX)$, where $k=\omega/c$ is the wave-vector. The
wave impinges on the front surface of the grating, where the plasmon is
excited due to the modulation (see Figure 3a). In this section devoted to the
analytical consideration, we assume that the film modulation $h$ as well as
its period $L$ are much smaller than the wavelength $\lambda=2\pi/k$. In this
case, we can approximate $E\sim\\{0,E_{0},0\\}\exp(ikX)\simeq\\{0,E_{0},0\\}$
that is the plasmon grating is excited by uniform field $E_{0}$, which is
$y$-directed and oscillates with frequency $\omega$. It is convenient to
introduce the complex variable $Z=X+iY$. We use the quasistatic approach,
therefore, the incident field is described by the complex potential
$\varphi(Z)=iZE_{0},$ where we assume that the field $E_{0}$ takes real
values. Then, $E_{x}=-\Re[d\varphi(Z)/dZ]=0$,
$E_{y}=\Im[d\varphi(Z)/dZ]=E_{0}$ (see Fig.3). It is convenient to shift from
the real coordinates X and Y to the dimensionless coordinates x and y that are
defined by using the period of the film modulation $L$, namely, $x=2\pi X/L$
and $y=2\pi Y/L$. That is the periodically modulated metal film has ”natural”
period of $2\pi$ being considered in $x$ and $y$ frame as shown in Figure 3a.
Recall we use the cuasistatic approximation, since the amplitude of metal
modulation is assumed to be much smaller than the wavelength of the incident
light. The potential of the local electric field does not change after re-
scaling of the coordinates, since there is no characteristic length in the
Laplace equation:

$\varphi(z)=izE_{0},$ (17)

where $z=x+iy$. The shape of the modulated metal is chosen in such way that
the front surface $\\{x_{f},y_{f}\\}$ of the modulated metal film (left red
surface in Fig. 3a) is given by the parametric equation, which it is
convenient to write in a complex form

$z^{(f)}(\phi)=x^{(f)}(\phi)+iy^{{(f)}}(\phi)=\log[\exp(i\phi)-x_{0}]$ (18)

where the parameter $x_{0}$ is in the interval $0<x_{0}<1$, the variable
$\phi$ changes in the limits $-\infty<\phi<\infty$; the logarithm is the full
analytical function defined in all Riemann surfaces so that $y$ also varies
within the limits of $-\infty<y<\infty$. The back surface
$x_{b}(\phi)>x_{f}(\phi)$ of the metal film (right red surface in Fig. 3a) is
given by the parametric equation:

$\displaystyle z^{(b)}(\phi)=x^{(b)}(\phi)+iy^{(b)}(\phi)=$
$\displaystyle\log[(1+d_{0})\exp(i\phi)-x_{0}]$ (19)

where the parameter $\phi$ still changes in the limits $-\infty<\phi<\infty$.
The shape of the metal film is fully defined by its period $L$ the amplitude
of the modulation $h=(x^{(f)}_{max}-x^{(f)}_{min})\frac{L}{2\pi}$ and the
largest film thickness $d=(x^{(b)}_{min}-x^{(f)}_{min})\frac{L}{2\pi}$. The
parameters $d_{0}$ and $x_{0}$ in the Eqs. (18) and (II.2) are expressed in
terms of the film modulation $h$ and the film thickness $d$, namely,

$x_{0}=\tanh\left(\dfrac{h_{1}}{2}\right),\;\;d_{0}=2\dfrac{\exp(d_{1})-1}{\exp(h_{1})+1}$
(20)

where $h_{1}=h(\frac{2\pi}{L})$ and $d_{1}=d(\frac{2\pi}{L})$. Therefore, the
period $L$, the modulation $h$, and thickness $d$ completely determine the
shape of the modulated metal film indeed. Note that the film defined by Eqs.
(II.2) and (18) mimics a metal film obtained, for example, by the metal
evaporation on the modulated dielectric substrate. During the deposition
process, the metal concentrates on tops of the film (see Fig. 3a and 7).

Figure 3: (a) The design of plasmonic grating based on a metal film with a
permittivity $\varepsilon_{m}$ deposited on the modulated dielectric substrate
(shown in blue) with a permittivity $\varepsilon_{d}$, metal film has
thickness $d$, ”y”-polarized light is incident from left; star ”$\ast$”
denotes the point with coordinates $x=x^{(f)}_{max}$ and $y=\pi$, where
surface electric field has maximum; small open square ”$\square$” denotes the
point with coordinates $x=x^{(f)}_{min}$ and $y=0$, where surface electric
field has minimum, the film modulation
$h=\left(x^{(f)}_{max}-x^{(f)}_{min}\right)(L/2\pi)$. (b)  Intensity of local
electric field at surface of silver film, which is modulated with period L;
the field is shown in the point $X_{max}=x^{(f)}_{max}(L/2\pi)$ and
$Y_{max}=\pi(L/2\pi)$ [star ”$\ast$” in  (a)] as function of the film
thickness $d$, where the period is equal to $L=50\,nm$, amplitude of the
modulation is equal to $h=11\,nm$, silver permittivity equals to
$\varepsilon_{m}\approx-30+0.4i$ at 785 $nm$, permittivity of dielectric
substrate equals to $\varepsilon_{d}=2$, permittivity of front medium
$\varepsilon_{1}=1$. Vertical color lines correspond to plasmon resonances
given by Eq. (23).

To find the optical electric field in and around a modulated metal film,
illuminated by the incident light, we use new coordinates $u$ and $v$ that
convenient to introduce in the complex form:

$w=u+iv=\exp(z)+x_{0},$ (21)

where $z=x+iy$. In the transition from $x,y$ frame to $u,v$ frame, the whole
metal film transforms to the ring $1\leq\left|w\right|\leq 1+d_{0}$. The front
and back surfaces of the grating are transformed to the internal
$\left|w\right|=1$ and outer $\left|w\right|=1+d_{0}$ surface of the ring. All
the space in front of the grating (left from the grating in Fig. 3a)
transforms in the space inside the ring $\left|w\right|<1$ in Fig. 1a, the
space behind the grating (right from the grating in Fig. 3a) transforms into
outer space of the ring $\left|w\right|>1+d_{0}$. That is the metal film in
Fig. 3a is rolled into the metal tube in Fig. 1. The approach is similar to
the map used in references [30, 46]. The electric potential given by Eq. (17)
transforms in $\\{u,v\\}$ frame to the potential $\varphi_{sing}$ in Eq. (1).
Therefore, the complex electric potential $\varphi(z)$ for a modulated metal
film equals to $\varphi(w(z))$, where $\varphi(w)$ is the potential, which is
found in the previous subsection for the metal ring. The complex electric
field inside and around the modulated metal film is obtained from the
following equation
$\tilde{E}(z)=E_{x}-iE_{y}=-d\varphi(w(z))/dz=[-d\varphi(w)/dw][dw/dz]=[dw/dz]\tilde{E}(w)$,
where the complex field $\tilde{E}(w)$ is given by Eqs. (3) – (7) and (12).
The equation for the electric field ${E_{x},E_{y}}$ can be rewritten in the
matrix form as follows:

$\begin{bmatrix}E_{x}(x,y)\\\
E_{y}(x,y)\end{bmatrix}=\left[\begin{array}[]{cc}r-\frac{u{x_{0}}}{r}&\frac{v{x_{0}}}{r}\\\
-\frac{v{x_{0}}}{r}&r-\frac{u{x_{0}}}{r}\\\
\end{array}\right]\begin{bmatrix}E_{r}(u,v)\\\ E_{\phi}(u,v)\end{bmatrix},$
(22)

where $\\{E_{x},E_{y}\\}$ is the field in the metal film (see Figure 3 a),
$\\{E_{r},E_{\phi}\\}$ are radial and angular components of the field are
given by Eq. (7), which is found for the plasmon resonance in the metal
nanoring (see Figure 2), $r\equiv\left|w\right|=\sqrt{u^{2}+v^{2}}$ is the
radius in $\left\\{u,v\right\\}$ frame. The values of $u=\exp(x)\cos(y)+x_{0}$
and $v=\exp(x)\sin(y)$ are obtained from Eq. (21).

Below, we suppose that $E_{0}=1$ and measure all fields in terms of the
applied field. The local electric field at the modulated metal surface in the
point with coordinates

$\displaystyle X_{max}=(L/2\pi)x^{(f)}_{max}=(L/2\pi)\log(1+x_{0}),$
$\displaystyle Y_{max}=L/2+mL\,(m=0,\pm 1,\pm 2,\ldots)$

of the modulated silver film is shown in Figure 3b as a function of the film
thickness $d$. The wavelength of the incident light is $\lambda=785\,nm$. The
film is deposited on the dielectric with permittivity $\varepsilon_{d}$, and
the permittivity of the front medium $\varepsilon_{1}=1$. The surface electric
field is much enhanced for the film thickness corresponding to the excitation
of plasmon resonances. The resonance thickness $d_{res,n}$ is obtained from
Eqs. (II.1 ) and (20):

$d_{res}=\frac{L}{2\pi}\log\left[\frac{\left(\frac{\left(\varepsilon_{1}-\varepsilon_{m1}\right)\left(\varepsilon_{d}-\varepsilon_{m1}\right)}{\left(\varepsilon_{1}+\varepsilon_{m1}\right)\left(\varepsilon_{d}+\varepsilon_{m1}\right)}\right)^{\frac{1}{2n}}-\tanh\frac{h_{1}}{2}}{1-\tanh\frac{h_{1}}{2}}\right]$
(23)

where $L$ is the period and $h_{1}=h(2\pi/L)$ is the dimensionless film
modulation. The resonances are indicated by vertical lines in Figures 3b and
5a. The distributions of the local electric field for the three resonances are
displayed in Figure 4. The local electric field is much enhanced and exceeds
the incident field $E_{0}$ by two-three orders of magnitude. The enhanced
field spreads over the grating achieving its maxima in the depressions of the
front surface. The amplitude of the maximum field in the point with
coordinates $X_{max},Y_{max}$ at the front surface of the metal film (star
$"\ast"$ in Fig. 3a) is given by the following equation obtained from Eq. (7)
by substitution there $r=1$ and $f=\pi$ and multiplying the result by matrix
(22)

$\displaystyle
E_{x}(X_{max},Y_{max})=E_{0}(1-x_{0})\left[\frac{1}{1+x_{0}}+\sum_{n=1}^{\infty}\left(-x_{0}\right)^{n}\frac{\left(\varepsilon_{1}+\varepsilon_{m}\right)\left(\varepsilon_{m}-\varepsilon_{d}\right)+\left(d_{0}+1\right)^{2n}\left(\varepsilon_{1}-\varepsilon_{m}\right)\left(\varepsilon_{d}+\varepsilon_{m}\right)}{\left(\varepsilon_{1}-\varepsilon_{m}\right)\left(\varepsilon_{m}-\varepsilon_{d}\right)+\left(d_{0}+1\right){}^{2n}\left(\varepsilon_{1}+\varepsilon_{m}\right)\left(\varepsilon_{d}+\varepsilon_{m}\right)}\right],$
$\displaystyle E_{y}(X_{max},Y_{max})=0$ (24)

where the parameters $x_{0}$ and $d_{0}$ are given by Eqs. (20). The important
characteristic is the amplitude of the resonance field averaged over the
surface of the metal film, which has resonance thickness $d_{res,n}$ given by
Eq. (23). The amplitude is still given by Eq. (16), where the ratio
$x_{0}/r_{0}$ is replaced by $\tanh\frac{h_{1}}{2}$. For the practically
important case $\left|\varepsilon_{m}\right|\gg\varepsilon_{d}$, the averaged
intensity at the $n$-$th$ resonance is estimated as follows:

$I_{\max,n}\sim\left[\frac{\epsilon_{1}\epsilon_{m1}}{\epsilon_{m2}\left(\epsilon_{d}+\epsilon_{1}\right)}\tanh^{n}\frac{h_{1}}{2}\right]^{2}$
(25)

where $h_{1}=h(\frac{2\pi}{L})$ and we skip numerical coefficient on the order
one.

Figure 4: Intensity of local electric field in a modulated silver film for the
blue, dark yellow, and green resonances displayed in Figure 3b, where the
period is $L=50\,nm$, modulation amplitude is $h=11\,nm$, silver permittivity
at $\lambda=785\,nm$ is $\varepsilon_{m}=-30+0.4i$, permittivity of dielectric
substrate is $\varepsilon_{d}=2$, and permittivity of front space
$\varepsilon_{1}=1$.

The analytical results are compared with results of computer simulations.
Computer simulations are performed for the electromagnetic field distribution
in a weakly profiled silver film deposited on a photoresist. The dimensions of
the computer model are the same than those described in Figures 3b and 4
($\varepsilon_{1}=1$, $\varepsilon_{d}=2$, i.e., the refractive index of the
photoresist is $n=1.41$). The simulations are done in COMSOL environment. The
Maxwell equations are solved by the finite element method (FEM).

Figure 5: (a) Electric field intensity for different hicknesses d of the
silver film (illumination under normal incidence, with the following
parameters: L = 50 nm and h = 11 nm). The red vertical lines indicate
resonance thicknesses obtained from analytical theory. (b) Distributions of
the electric field intensity corresponding to two resonances ($R_{1}$) and
($R_{2}$) displayed in figure 5a.

We demonstrate the enhancement of the electric field $|E/E_{0}|^{2}$ in the
modulated thin metal film with period $L=50\;nm$ and amplitude of the
modulation $h=11\;nm$ (see Figure 5). It can be seen that positions of the
analytical and numerical resonances are in a good agreement. The distribution
of the electric field for two resonances shown in Figure 5a are displayed in
Figure 5b. Computer simulations are in agreement with analytical theory (see
the blue and dark yellow resonances in figure 4). We also simulate the
resonant enhancement of the electric field for a modulated metal film whose
parameters correspond to the experiment discussed below (see Figure 6). Figure
6a,b show the distribution of the electric field intensity calculated with the
result of Eq. (24) of the analytical theory, and computer simulations,
respectively. Resonances are due to the excitation of dipolar and multipolar
plasmons in the modulated metal film. The electric field distribution in
dipolar and multipolar plasmons is in qualitative agreement with analytical
theory [see the resonances: (I), (II), and (III) displayed in figure 6a,b].
The field distribution in the resonance (IV) in Figure 6b,c differs from all
other resonances. The electric field expands from the metal surface on the
distance comparable with the wavelength and period of the modulation. We
speculate that in this case the plasmon resonance hybridizes with plasmon
propagating over the modulated metal surface (see, e.g., [40]). The
hybridization could be the subject for the further consideration.

Figure 6:  Intensity of electric field for different thicknesses of silver
film d with the following parameters: period L = 750 nm and amplitude of the
modulation h = 100 nm (illumination at $\lambda$ = 785 nm, under normal
incidence), calculated by using (a) the result of Eq. (24) and (b) computer
simulations. (c) Computer simulation of distribution of local electric field
corresponding to resonances: (I), (II), (III) and (IV) displayed in figure
Fig. 6b.

In summary, we present the analytical theory of plasmons excited in a
modulated metal film deposited on a dielectric substrate. Computer simulations
and results given by the analytical theory are in a good agreement. The
resonance conditions are found as a function of the metal permittivity, the
modulation and the thickness of the film. When the grating is excited by the
incident light, the surface electric field is much enhanced. The quasistatic
theory qualitatively described the field enhancement even in the case
$kL\simeq 1$, since the field is concentrated in the nanograting whose
thickness and modulation are much smaller than $\lambda$. The size of a
plasmon maxima $l_{p}$ is on the order of the film modulation $l_{p}\approx h$
(see Figs.4,5,6). The radiation loss, which is propotional to $l_{p}dk^{2}$
remains to be less than one. For example, for silver film with parameters
$h=100\,nm,d=20\,nm$ and $\lambda=785\,nm$ (see Fig.6), $l_{p}dk^{2}\approx
0.1<1$. The positions of the plasmon resonances from computer simulations are
almost coincide with results of Eq. (23). The resonance enhancement of the
electric field obtained in computer simulations is very large. Yet, the
computer results are few times smaller than results of the quasistatic Eq.
(24). The discrepancy is due to the radiation loss that can be taken into
account by using a perturbation theory. Yet, we do not expect qualitative
difference between the quasistatic approximation and exact result while
plasmons are localized like in resonances (I) - (III) (Figure 6). The
developed analytical theory can be used for the design of new SERS substrates
and other optical sensors. The theory can be extended to multilayered metal-
dielectric films.

##  III Experimental investigation of holographic metasurfaces

###  III.1 Fabrication process

The metasurfaces are fabricated by the following steps. As a first step
$1.5-1.9\;\mu m$ recording layer of positive photoresist (Microposit S1800) is
deposited by spin-coating on the silicon wafer. Then, fifteen areas of two-
dimensional modulated films ($1.5\times 1.5\;cm^{2}$) are formed by
holographic lithography technique. The principle scheme of recording setup
employs a phase-modulator HED-6001 Monochrome LCOS, and an ultraviolet laser
($\lambda=405\;nm$) (see Figure 7a). Exposure dose is varied from 4. 1 to 8. 5
$\mu J/cm^{2}$. The angle between interfering light waves is around 49∘. Next,
the resist is developed in Microposit 303A developer. Afterwards, two
successive depositions of a 80-nm $SiO_{2}$ layer and a 60-nm silver layer are
realized by electron beam evaporation. The metasurfaces with period from
$720\,nm$ to $770\,nm$ and modulation amplitude from 8 nm to 30 nm are
fabricated (see Figure 7b). The principle scheme of the metasurface morphology
is shown in Figure 7c. The morphology of the fabricated metasurfaces is
measured with a scanning electron microscopy (SEM, JEOL), and SEM images of
these metasurfaces are shown in Figure 7d,e. Finally, modulation amplitude
measurements are made with an atomic force microscopy (AFM; Solver Pro NT-MDT)
for different exposure doses (see Figure 7f).

Figure 7: (a) Principle scheme of the recording setup. (b) Photo of 4-inch
wafer where holographic metasurfaces are fabricated. (c) Principle scheme of
the metasurface design with the following parameters: period is $L=720-770$
$nm$, modulation amplitude is $h=8-30$ $nm$, silver layer thickness is $d=60$
$nm$, thickness of $SiO_{2}$ is $h_{Si}$ = 80 $nm$, photoresist thickness is
$h_{Ph}=1.5-1.9$ $\mu m$. SEM images of the metasurfaces (Period $L\approx$
760 $nm$, and film modulation $h\approx$ 30 $nm$) with a scale bar of 2 $\mu
m$ for (d) and 500 $nm$ for (e). (f) Hole depth versus exposure dose (depth
measured by AFM).

###  III.2 Deposition of 4-MercaptophenylboronicAcid for SERS

In order to study the sensitivity of the holographic metasurfaces, we employed
molecules of 4-mercaptophenylboronic acid (4-mPBA), which are small molecules
(thickness of a 4-mPBA monolayer is around 0.8 $nm$ [65]). We have chosen the
boronic acid, which is a specific substance because the boron group covalently
binds diols and forms a boron ester. Therefore, boronic acid is considered as
a specific agent for sugars and their fragments. There is a strong covalent
fixation of this acid on the surface of silver due to the thiol group (–SH).
The analyzed molecules are covalently bound to the silver, thereby increasing
the probability of the location of molecules in the region of generation of
giant electromagnetic fields. In addition, we prepared solutions of 4-mPBA in
ethanol with concentrations varying from 2. 3 $\times$ 10-8 M to 2. 3 $\times$
10-3 M. Then, for SERS experiments, an aliquot of each 4-mPBA solution was
applied onto holographic metasurface and then dried at room temperature.

###  III.3 Optical Characterization

For characterizing the optical properties of metasurfaces, the reflectance
spectra have been recorded by using a spectroscopic ellipsometer SE 850 DUV
(Sentech, Germany). For the SERS measurements, a Raman spectrometer InnoRam
(BWTek, USA) with an excitation wavelength of 785 $nm$ and a spectral
resolution of 4 $cm^{-1}$ was employed. The acquisition time and the laser
power have been set at 1 s and 3. 84 mW, respectively. A microscope objective
($\times$50, NA = 0.8) was used in order to focus the laser beam on the
metasurface, and the Raman signal of the 4-mPBA molecules immobilized on
metasurface was detected by this same objective in a backscattering
configuration.

##  IV Results and discussion

###  IV.1 Optical properties of holographicmetasurfaces

To determine the macroscopic optical properties of metasurfaces, the angular
reflectance spectrum is measured by ellipsometry in the range from
$20^{\circ}$ to $70^{\circ}$. The reflectance spectrum is measured for two
polarizations: $Rp$ for $p$-polarization of the incident light and $Rs$ for
$s$-polarization of the incident light. The measured reflectance spectra for
$Rp$ and $Rs$ show deep gaps with a complex behavior versus period and angle
(see Figure 8).

Figure 8: Experimental angular reflectance spectra from the metasurface for
different angles of the incident light and with the following structure
parameters: (a) $L$ = 720 $nm$, depth $h$ $\approx$ 30 $nm$; (b) $L$ = 720
$nm$, depth $h$ $\approx$ 20 $nm$; (c) $L$ = 770 $nm$, depth $h$ $\approx$ 18
$nm$.

Raman instruments are usually attached to microscopes, where the excitation
and the inelastically scattered light are delivered. The light is collected
from the sample through the same objective in a backscattering geometry. The
objective is characterized by a numerical aperture (NA) equal to 0.8 in our
case. This means that the cone angle is equal to
$2\alpha=\arcsin{(NA)}=106^{\circ}$. So, we have to take into account all the
rays of the incident light at angles $\alpha$ from $0^{\circ}$ to $53^{\circ}$
degrees. The focal length $f$ is related to the lens diameter by the ratio
$(D/2f)=\tan\alpha$ ($D$ \- lens diameter). We find the average reflection in
the approximation of the geometric optics by integrating the reflection over
the basis of the cone of the incidence. The principle scheme of the polarized
light collected by the lens is displayed in Figure 9a. The average reflection
is expressed as follows:

$R_{average}=2(\frac{1}{NA^{2}}-1)\int_{0}^{\alpha_{max}}\frac{\sin\alpha}{\cos\alpha^{3}}R(\alpha)d\alpha$
(26)

where $\alpha_{max}=\arcsin{(NA)}$. In addition, each ray has a transverse and
longitudinal field component, so we average $Rp$ and $Rs$. The resulting
average reflection is displayed in Figure 9b.

Figure 9: (a) Schematic illustration of the polarized light impinged to the
sample by the lens. The red and green arrows show the directions of the wave-
vector and the electric field, respectively. (b) Average experimental
reflectance spectra from modulated silver film (metasurface), the reflectance
is averaged over the light cone, which is focused by the Raman microscope on
the sample.

We observe minima on the spectral range from $400\;nm$ to $800\;nm$ and more
specifically close to the excitation wavelength of $785\;nm$. We assume that
these minima in the reflectance spectrum correspond to plasmon resonances.
These plasmonic resonances (close to the excitation wavelength of $785\;nm$)
will permit the enhancement the Raman signal of 4-mPBA molecules.

###  IV.2 SERS detection of 4-mercaptophenylboronic acid molecules

For all the SERS experiments, a modulated silver metasurface with a period
$L=760\;nm,h=30\;nm$ is used. This metasurface has the maximum available
modulation depth $h$ = 30 $nm$. With a higher modulation depth, the number of
electromagnetic modes increases. Thus, this modulation allows obtaining deep
minima in the reflectance spectrum close to the excitation wavelength of
$785\;nm$ (see figure 9b). Then, to evaluate the sensitivity of these
metasurfaces, 4-mPBA molecules are grafted onto them by using the
functionalization protocol described in section 3.2. Next, the SERS spectra
have been recorded at the excitation wavelength of 785 $nm$. Figure 10 shows
the SERS spectra of 4-mPBA molecules on metasurfaces obtained for each
concentration varying from 2. 3 $\times$ 10-8 M to 2. 3 $\times$ 10-3 M. From
the SERS spectrum obtained for the concentration of 2. 3 mM depicted in Figure
10, seven characteristic Raman peaks of 4-mPBA molecules are well-
distinguished [59, 66, 67]. The Raman peaks at 420 $cm^{-1}$, 693 $cm^{-1}$
and 1072 $cm^{-1}$ correspond to the C–C–C in-plane bending mode coupled with
C–S stretching mode (called respectively: $\beta_{CCC}$ and $\nu_{CS}$). Then,
the peak at 480 $cm^{-1}$ corresponds to the C–C–C out-of-plane bending mode
coupled to O–B–O in-plane bending mode (called respectively: $\gamma_{CCC}$
and $\beta_{OBO}$). The Raman peaks at 1000 $cm^{-1}$ and 1030 $cm^{-1}$
correspond, respectively, to the C–C–C in-plane bending mode (called
$\beta_{CCC}$) and the C–H in-plane bending mode (called $\beta_{CH}$).
Finally, the Raman peak at $1580\;cm^{-1}$ corresponds to the C–C stretching
vibrational modes (called $\nu_{CC}$). For determining the detection limit, we
have chosen the most intense Raman peak, which is located at 1072 $cm^{-1}$,
and a detection limit of 230 nM was achieved for the sensing of 4-mPBA
molecules (see Figure 10 where the blue zone corresponds to the zone of the 3
$\times$ noise level for the last curve) with a signal-to-noise ratio $>3$.

Figure 10:  SERS spectra of 4-mPBA molecules recorded on the metasurface for
the concentrations of 2. 3 $\times$ 10-3 M, 2. 3 $\times$ 10-5 M, 2\. 3
$\times$ 10-7 M and 2. 3 $\times$ 10-8 M. At the bottom of the figure, graph
of SERS intensity versus 4-mPBA concentration, where the blue zone corresponds
to the zone of the 3 $\times$ noise level.

In addition, the distribution of Raman hotspots generated by periodic holes
(grid) repeats the morphology of the surface, which is shown on the optical
image displayed in Figure 7. Therefore, we can see the SERS effect distributed
uniformly over the entire metasurface. The pattern of the Raman signal in
Figure 11 repeats the periodic structure of the modulated film. Therefore, the
maxima of the Raman signal repeat the maxima of the plasmon field $I(x,y)$.

Figure 11: Optical image of the intensity distribution for the Raman bands
1060–1100 $cm^{-1}$.

One could suggest that the SERS is a local effect, namely, $G(x,y)\propto
I(x,y)^{2}$. Investigation of the modulated metal film with progressively
smaller period $L$ can confirm locality of SERS effect.

##  V Conclusion

In this paper, we presented the low-cost fabrication and the study of a
resonant metasurface consisting of a periodically modulated metal film that
exhibits an anomalous optical response due to the excitation of localized
surface plasmons. The distribution of the enhanced electric field generated by
the film modulation has repeated the surface morphology, and this distribution
was uniform over this entire surface. The holographic metasurface allowed
achieving a detection limit of 230 nM for the 4-mPBA molecules serving here as
a proof-of-concept of SERS sensing. Thus, the low-cost and easy-made
holographic SERS metasurfaces modified with 4-mPBA molecules can be used for
quantitative biosensing of glycated proteins and sugars. Moreover, the
analytical theory can allow the design and the optimization of various
plasmonic sensors.

##  VI Funding

This work was supported by the Russian Foundation for Basic Research (Grant
No. 20-21-00080). The Raman measurments was supported by the Russian Science
Foundation (Grant No. 21-79-30048).

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