# Some results on the Turán number of $k_{1}P_{\ell}\cup k_{2}S_{\ell-1}$
111This work was supported by the National Nature Science Foundation of China
(Nos.11871040, 12271337)

Tao Fang<EMAIL_ADDRESS>Xiying Yuan  Department of Mathematics, Shanghai
University, Shanghai 200444, P.R. China

###### Abstract

The Turán number of a graph $H$, denoted by $ex(n,H)$, is the maximum number
of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let
$P_{\ell}$ denote the path on $\ell$ vertices, $S_{\ell-1}$ denote the star on
$\ell$ vertices and $k_{1}P_{\ell}\cup k_{2}S_{\ell-1}$ denote the path-star
forest with disjoint union of $k_{1}$ copies of $P_{\ell}$ and $k_{2}$ copies
of $S_{\ell-1}$. In 2013, Lidický et al. first considered the Turán number of
$k_{1}P_{4}\cup k_{2}S_{3}$ for sufficiently large $n$. In 2022, Zhang and
Wang raised a conjecture about the Turán number of $k_{1}P_{2\ell}\cup
k_{2}S_{2\ell-1}$. In this paper, we determine the Turán numbers of
$P_{\ell}\cup kS_{\ell-1}$, $k_{1}P_{2\ell}\cup k_{2}S_{2\ell-1}$, $2P_{5}\cup
kS_{4}$ for $n$ appropriately large, which implies the conjecture of Zhang and
Wang. The corresponding extremal graphs are also completely characterized.

###### keywords:

Turán number, path-star forest, extremal graph

###### MSC:

[2010] 05C05 05C35

##  1 Introduction

In this paper, all graphs considered are undirected, finite and contain
neither loops nor multiple edges. The vertex set of a graph $G$ is denoted by
$V(G)$, the edge set of $G$ by $E(G)$, the number of the vertices in $G$ by
$v(G)$ and the number of edges in $G$ by $e(G)$. Let $K_{n}$, $P_{n}$,
$S_{n-1}$ denote the complete graph, path and star on $n$ vertices,
respectively. For a vertex $v\in V(G)$, let $N_{G}(v)$ denote the set of
vertices in $G$ which are adjacent to $v$ and $d_{G}(v)$ denote the degree of
a vertex $v$, i.e., $d_{G}(v)=\left|N_{G}(v)\right|$. Given two vertex-
disjoint graphs $G$ and $H$, let $G\cup H$ denote the disjoint union of graphs
$G$ and $H$, $kG$ the disjoint union of $k$ copies of $G$, and $G\vee H$ the
graph obtained from $G\cup H$ by joining all vertices of $G$ to all vertices
of $H$. We use $\overline{G}$ to denote the complement of the graph $G$. For
any set $S\subseteq V(G)$, let $G[S]$ denote the subgraph of $G$ induced by
$S$, $\left|S\right|$ denote the cardinality of $S$. For a graph $G$ and its
subgraph $H$, let $G-H$ denote the subgraph induced by $V(G)\backslash V(H)$.

The Turán number of a graph $H$, $ex(n,H)$, is the maximum number of edges in
$G$ of order $n$ that does not contain a copy of $H$. Denote by
$\mathbb{EX}(n,H)$ the set of graphs on $n$ vertices with $ex(n,H)$ edges
containing no $H$ as a subgraph and call the graph from $\mathbb{EX}(n,H)$ the
extremal graph for $H$ or $H$-extremal graph. If $\mathbb{EX}(n,H)$ contains
only one graph, we may simply use $\mbox{\rm{EX}}(n,H)$ instead.

The study of Turán numbers of forests began with the famous result of Erdös
and Gallai [4] in 1956. Then in 1975, Faudree and Schelp [5] gave an
improvement of the extremal graph for $P_{k}$.

######  Theorem 1.1.

[4] Let $n=d(\ell-1)\geqslant 2$, where $d\geqslant 1$. Then

$ex(n,P_{\ell})=\frac{(\ell-2)n}{2}.$

Furthermore,

$\mbox{\rm{EX}}(n,P_{\ell})=dK_{\ell-1}.$

The following two symbols are defined in [12]. Let $n\geqslant
m\geqslant\ell\geqslant 2$ be three positive integers and
$n=(m-1)+d(\ell-1)+r$ with $d\geqslant 0$ and $0\leqslant r<\ell-1$. Define

$[n,m,\ell]=\binom{m-1}{2}+d\binom{\ell-1}{2}+\binom{r}{2}.$

Let $n$ and $s$ be two positive integers and $n\geqslant s$. Define

$[n,s]=\binom{s-1}{2}+(s-1)(n-s+1).$

######  Theorem 1.2.

[5] Let $n=d(\ell-1)+r$, where $d\geqslant 1$ and $0\leqslant r<\ell-1$. Then

$ex(n,P_{\ell})=[n,\ell,\ell].$

Furthermore, if $\ell$ is even, $r={\ell}/2$ or $(\ell-2)/2$, then

$\mathbb{EX}(n,P_{\ell})=\left\\{dK_{\ell-1}\cup
K_{r},\left(\left(d-s-1\right)K_{\ell-1}\right)\cup\left(K_{\frac{\ell-2}{2}}\vee\overline{K}_{\frac{\ell}{2}+s\left(\ell-1\right)+r}\right),s=0,1,\cdots,d-1\right\\};$

if otherwise, then

$\mbox{\rm{EX}}(n,P_{\ell})=dK_{\ell-1}\cup K_{r}.$

We follow the notation and terminology of [10]. A linear forest is a forest
whose connected components are paths. A star forest is a forest whose
connected components are stars. A path-star forest is a forest whose connected
components are paths and stars. In 2011, Bushaw and Kettle [3] determined the
Turán numbers of $kP_{\ell}$ for sufficiently large $n$, which was extended by
Lidiciký et al. [10]. Yuan and Zhang [11, 12] determined the Turán numbers of
linear forests containing at most one odd path for all $n$. For special linear
forest, Bielak and Kieliszek [2] and Yuan and Zhang [12] independently
determined $ex(n,2P_{5})$ for all $n$ and characterized all extremal graphs.

######  Lemma 1.1.

[2, 12] Let $n\geqslant 10$. Then

$ex(n,2P_{5})=\mbox{\rm{max}}\big{\\{}[n,10,5],3n-5\big{\\}}.$

The extremal graphs are $K_{9}\cup\mbox{\rm{EX}}(n,P_{5})$ and
$K_{3}\vee\left(K_{2}\cup\overline{K}_{n-5}\right)$.

By calculations, when $n\geqslant 38$, $[n,10,5]<3n-5$ holds. Hence, we may
get the following result from Lemma 1.1.

######  Lemma 1.2.

When $n\geqslant 38$, we have

$ex(n,2P_{5})=3n-5.$

The extremal graph is $K_{3}\vee\left(K_{2}\cup\overline{K}_{n-5}\right)$.

The following lemma is based on Theorem 1.7 of [12].

######  Lemma 1.3.

[12] Let $k\geqslant 2$ be a positive integer, $\ell$ be an even number and
$n\geqslant\ell k$. Then

$ex(n,kP_{\ell})=\mbox{\rm{max}$\big{\\{}[n,\ell k,\ell],[n,\ell
k/2]\big{\\}}$}.$

The extremal graphs are $\mbox{\rm{EX}}(n-\ell k+1,P_{\ell})\cup K_{k\ell-1}$
and $K_{\ell k/2-1}\vee\overline{K}_{n-\ell k/2+1}$.

By calculations, when $k\geqslant 2$ and $n\geqslant(2{\ell}^{2}+3\ell-4)k+3$,
$[n,\ell k,\ell]<[n,\ell k/2]$ holds. Hence, we may get the following result
from Lemma 1.3.

######  Lemma 1.4.

Suppose $k\geqslant 2$, $\ell$ are positive integers and
$n\geqslant(2{\ell}^{2}+3\ell-4)k+3$. Then

$ex(n,kP_{2\ell})=\binom{\ell k-1}{2}+(\ell k-1)(n-\ell k+1).$

The extremal graph is $K_{\ell k-1}\vee\overline{K}_{n-\ell k+1}$.

For sufficiently large $n$, Lidický et al. [10] determined the Turán number of
stars forests. Later, Lan et al. [7] determined the Turán number of
$kS_{\ell}$ for $n$ appropriately large related to $k$ and $\ell$.
Furthermore, Li et al. [9] determined the Turán number of $kS_{\ell}$, where
$k\geqslant 2$ and $\ell\geqslant 3$, for all $n$.

######  Lemma 1.5.

[7] If $\ell\geqslant 3$ and $n\geqslant\ell+1$, then

$ex(n,S_{\ell})\leqslant\left\lfloor\frac{(\ell-1)n}{2}\right\rfloor,$

with one extremal graph is the $\left(\ell-1\right)$-regular graph on $n$
vertices.

######  Theorem 1.3.

[9] If $k\geqslant 2$ and $\ell\geqslant 3$, then

$ex(n,kS_{\ell})=\begin{cases}\binom{n}{2},&\text{if }n<k(\ell+1),\\\
\binom{k\ell+k-1}{2}+\binom{n-k\ell-k+1}{2},&\text{if }k(\ell+1)\leqslant
n\leqslant(k+1)\ell+k-1,\\\
\binom{k\ell+k-1}{2}+\left\lfloor\frac{(\ell-1)(n-k\ell-k+1)}{2}\right\rfloor,&\text{if
}(k+1)\ell+k\leqslant n<\frac{k{\ell}^{2}+2k\ell+2k-2}{2},\\\
\binom{k-1}{2}+(n-k+1)(k-1)+\left\lfloor\frac{(\ell-1)(n-k+1)}{2}\right\rfloor,&\text{if
}n\geqslant\frac{k{\ell}^{2}+2k\ell+2k-2}{2}.\end{cases}$

In this paper, we mainly consider the Turán numbers of some kinds of path-star
forests. The Turán numbers and the extremal graphs for $P_{\ell}\cup
kS_{\ell-1}$, $k_{1}P_{2\ell}\cup k_{2}S_{2\ell-1}$ and $2P_{5}\cup kS_{4}$
will be presented in Section 2, and their proofs will be provided in Section
3.

##  2 Main results

Now, we introduce the following three kinds of graphs to state the main
results. Set

$G_{1}(n,k,\ell)=K_{k}\vee(dK_{\ell-1}\cup K_{r}),\mbox{ where
$n=k+d(\ell-1)+r$, $0\leqslant r<\ell-1$,}$
$G_{2}(n,k_{1},k_{2},2\ell)=K_{\ell k_{1}+k_{2}-1}\vee\overline{K}_{n-\ell
k_{1}-k_{2}+1},$
$G_{3}(n,k)=K_{k+3}\vee\left(K_{2}\cup\overline{K}_{n-k-5}\right).$

By calculations, we have the following facts.

$e(G_{1}(n,k,\ell))=\left(k+\frac{\ell}{2}-1\right)n-\frac{k^{2}+(\ell-1)(k+r)-r^{2}}{2},$
(2.1) $e(G_{2}(n,k_{1},k_{2},2\ell))=(\ell k_{1}+k_{2}-1)n-\frac{(\ell
k_{1}+k_{2})(\ell k_{1}+k_{2}-1)}{2},$ (2.2)
$e(G_{3}(n,k))=(k+3)n-\frac{k^{2}+7k+10}{2}.$ (2.3)

Denote a kind of path-star forest by $F(k_{1},k_{2};\ell)=k_{1}P_{\ell}\cup
k_{2}S_{\ell-1}$. Lidický et al. [10] first investigated the Turán number of
$F(k_{1},k_{2};4)$ for sufficiently large $n$. Lan et al. [7] considered the
Turán number of $F(k_{1},k_{2};4)$ for $n\geqslant 10k_{1}+13k_{2}+3$. Later,
Zhang and Wang [13] considered the Turán number of $F(k_{1},k_{2};6)$ for
$n\geqslant 23k_{1}+31k_{2}+3$ and proposed Conjecture 2.1.

######  Theorem 2.1.

[7] Suppose $n=k_{2}+3d+r\geqslant 10k_{1}+13k_{2}+3$, where $k_{1},k_{2},d,r$
are positive integers and $r\leqslant 2$. Then

$ex(n,F(k_{1},k_{2};4))=\mbox{\rm{max}}\big{\\{}e(G_{1}(n,k_{2},4)),e(G_{2}(n,k_{1},k_{2},4))\big{\\}}.$

Furthermore, the extremal graph is $G_{1}(n,k_{2},4)$ when $k_{1}=1$ and
$G_{2}(n,k_{1},k_{2},4)$ when $k_{1}>1$. In particular,
$G_{2}(n,k_{1},k_{2},4)$ is also an extremal graph when $k_{1}=1$ and $r=1$ or
$r=2$.

######  Theorem 2.2.

[13] Suppose $n=k_{2}+5d+r\geqslant 23k_{1}+31k_{2}+3$, where
$k_{1},k_{2},d,r$ are positive integers and $r\leqslant 4$. Then

$ex(n,F(k_{1},k_{2};6))=\mbox{\rm{max}}\big{\\{}e(G_{1}(n,k_{2},6)),e(G_{2}(n,k_{1},k_{2},6))\big{\\}}.$

Furthermore, the extremal graph is $G_{1}(n,k_{2},6)$ when $k_{1}=1$ and
$G_{2}(n,k_{1},k_{2},6)$ when $k_{1}>1$.

######  Conjecture 2.1.

[13] Suppose $k_{1}\geqslant 1$, $k_{2}$ and $\ell\geqslant 2$ are integers
and $n=k_{2}+d(2\ell-1)+r$, where $0\leqslant r<2\ell-1$. Then

$ex(n,F(k_{1},k_{2};2\ell))=\mbox{\rm{max}$\big{\\{}e(G_{1}(n,k_{2},2\ell)),e(G_{2}(n,k_{1},k_{2},2\ell))\big{\\}}$}.$

We may point out that when $k_{1}=1$ and $r=2$ or $r=3$,
$G_{2}(n,k_{1},k_{2},6)$ is also an extremal graph of $F(k_{1},k_{2};6)$. This
fact was ignored in Theorem 2.2. Our results are given in the next three
theorems, which determine the Turán numbers and the extremal graphs for
$F(1,k;\ell)$ (see Theorem 2.3), $F(k_{1},k_{2};2\ell)$ (see Theorem 2.4) and
$F(2,k;5)$ (see Theorem 2.5), respectively. The results of Theorem 2.3 and
Theorem 2.4 imply Conjecture 2.1.

######  Theorem 2.3.

Suppose $n=k+d(\ell-1)+r\geqslant(\ell^{2}-\ell+1)k+(\ell^{2}+3\ell-2)/2$,
where $\ell\geqslant 4,0\leqslant r<\ell-1$. Then

$ex(n,F(1,k;\ell))=e(G_{1}(n,k,\ell)).$

Moreover,

$\mathbb{EX}(n,F(1,k;\ell))=\begin{cases}\left\\{G_{1}(n,k,\ell),G_{2}(n,1,k,\ell)\right\\},&\mbox{if
$\ell$ is even, and $r=\frac{\ell}{2}$ or $r=\frac{\ell-2}{2}$,}\\\
\left\\{G_{1}(n,k,\ell)\right\\},&\mbox{otherwise}.\end{cases}$

######  Theorem 2.4.

Suppose $n\geqslant(2{\ell}^{2}+3\ell-4)k_{1}+(4{\ell}^{2}-2\ell+1)k_{2}+3$,
where $k_{1}\geqslant 2,\ell\geqslant 2$. Then

$ex(n,F(k_{1},k_{2};2\ell))=e(G_{2}(n,k_{1},k_{2},2\ell)).$

Moreover,

$\mbox{\rm{EX}}(n,F(k_{1},k_{2};2\ell))=G_{2}(n,k_{1},k_{2},2\ell).$

######  Theorem 2.5.

Suppose $n\geqslant 21k+38$. Then

$ex(n,F(2,k;5))=e(G_{3}(n,k)).$

Moreover,

$\mbox{\rm{EX}}(n,F(2,k;5))=G_{3}(n,k).$

##  3 Proofs of the main results

###  3.1 The Turán number and the extremal graphs for $F(1,k;\ell)$

Write $n=k+d(\ell-1)+r$, where $0\leqslant r<\ell-1$, and

$H=K_{k}\vee\left(\left(\left(d-s-1\right)K_{\ell-1}\right)\cup\left(K_{\frac{\ell-2}{2}}\vee\overline{K}_{\frac{\ell}{2}+s\left(\ell-1\right)+r}\right)\right),$

where $\ell$ is an even integer. Recall that $F(1,k;\ell)=P_{\ell}\cup
kS_{\ell-1}$. We first present the following lemma which help us to determine
the extremal graphs for $F(1,k;\ell)$.

######  Lemma 3.1.

If $n\geqslant\ell k+\ell$ and $s\in\\{0,1,\cdots,d-2\\}$, then $H$ contains a
copy of $F(1,k;\ell)$.

###### Proof.

If $s\in\\{0,1,\cdots,d-2\\}$, we have $d-s-1\geqslant 1$. In $H$, let

$V(K_{k})=\\{u_{1},u_{2},\cdots,u_{k}\\},$
$V\left((d-s-1)K_{\ell-1}\right)=\\{v_{1},v_{2},\cdots,v_{\ell-1}\\}\cup
V_{1},$
$V\left(K_{\frac{\ell-2}{2}}\vee\overline{K}_{\frac{\ell}{2}+s\left(\ell-1\right)+r}\right)=\\{w_{1},w_{2},\cdots,w_{\ell}\\}\cup
V_{2},$

where $v_{1},v_{2},\cdots,v_{\ell-1}$ are the vertices of an induced subgraph
$K_{\ell-1}$ of $(d-s-1)K_{\ell-1}$, and $w_{1}\in
V\left(K_{\frac{\ell-2}{2}}\right)$. We may check that
$H[\\{u_{1},v_{1},v_{2},\cdots,v_{\ell-1}\\}]$ is a path on $\ell$ vertices,
and $H[\\{w_{1},w_{2},\cdots,w_{\ell}\\}]$ is a star on $\ell$ vertices with
center vertex $w_{1}$. We may find another $(k-1)$ copies of $S_{\ell-1}$ with
$(k-1)$ center vertices in $\\{u_{2},u_{3},\cdots,u_{k}\\}$ and
$\left(k-1\right)\left(\ell-1\right)$ leaves vertices in $V_{1}\cup V_{2}$.
Hence, we have $F(1,k;\ell)\subseteq H$. ∎

###### Proof of Theorem 2.3.

We suppose $n\geqslant(\ell^{2}-\ell+1)k+(\ell^{2}+3\ell-2)/2$ in this
subsection. Recall that

$G_{1}(n,k,\ell)=K_{k}\vee\left(dK_{\ell-1}\cup K_{r}\right)$

and

$G_{2}(n,1,k,\ell)=K_{k+\frac{\ell}{2}-1}\vee\overline{K}_{n-k-\frac{\ell}{2}+1}.$

First we prove that both $G_{1}(n,k,\ell)$ and $G_{2}(n,1,k,\ell)$ are
$F(1,k;\ell)$-free. If $G_{1}(n,k,\ell)$ contains a copy of $F(1,k;\ell)$,
then each $S_{\ell-1}$ contains at least one vertex of $K_{k}$, and the
$P_{\ell}$ contains at least one vertex of $K_{k}$, which is a contradiction.
If $\ell$ is even and $G_{2}(n,1,k,\ell)$ contains a copy of $F(1,k;\ell)$,
then each $S_{\ell-1}$ contains at least one vertex of
$K_{k+\frac{\ell}{2}-1}$, and the $P_{\ell}$ contains at least $\ell/2$
vertices of $K_{k+\frac{\ell}{2}-1}$, which is a contradiction. Hence,
$G_{2}(n,1,k,\ell)$ is $F(1,k;\ell)$-free. Furthermore, by (2.1) and (2.2), we
have

$e(G_{1}(n,k,\ell))-e(G_{2}(n,1,k,\ell))=\frac{(\ell-2r)(\ell-2r-2)}{8}\geqslant
0,$

with equality if and only if $r=\ell/2$ or $r=(\ell-2)/2$. Thus we have

$ex(n,F(1,k;\ell))\geqslant\mbox{\rm{max}}\big{\\{}e(G_{1}(n,k,\ell)),e(G_{2}(n,1,k,\ell))\big{\\}}=e(G_{1}(n,k,\ell)).$
(3.1)

Now we will show the inequality

$ex(n,F(1,k;\ell))\leqslant e(G_{1}(n,k,\ell))$ (3.2)

by induction on $k$. For $k=0$, $n=d(\ell-1)+r$, we have
$G_{1}(n,0,\ell)=dK_{\ell-1}\cup K_{r}$ and $F(1,0;\ell)=P_{\ell}$. Then by
Theorem 1.2, $ex(n,F(1,0;\ell))=[n,\ell,\ell]=e(G_{1}(n,0,\ell))$ holds.
Suppose that $k\geqslant 1$ and (3.2) holds for all $k^{\prime}<k$. Let $G$ be
an $n$-vertex $F(1,k;\ell)$-free graph with $e(G)=ex(n,F(1,k;\ell))$. By (3.1)
and (2.1), we have

$\displaystyle e(G)$ $\displaystyle\geqslant e(G_{1}(n,k,\ell))$
$\displaystyle=\left(k+\frac{\ell}{2}-1\right)n-\frac{k^{2}+(\ell-1)(k+r)-r^{2}}{2}$
$\displaystyle>\left(k+\frac{\ell}{2}-2\right)n-\frac{(k-1)(k+\ell-2)}{2}$
$\displaystyle=(k-1)\left(n-\frac{k}{2}\right)+\frac{(\ell-2)(n-k+1)}{2}$
$\displaystyle\geqslant(k-1)\left(n-\frac{k}{2}\right)+\left\lfloor\frac{(\ell-2)(n-k+1)}{2}\right\rfloor$
$\displaystyle\geqslant ex(n,kS_{\ell-1}),$

which implies $G$ contains $k$ copies $S_{\ell-1}$ by Theorem 1.3 and Lemma
1.5. By induction hypothesis,

$ex(n-\ell,F(1,k-1;\ell))\leqslant e(G_{1}(n-\ell,k-1,\ell)).$

Since $G$ is $F(1,k;\ell)$-free, $G-S_{\ell-1}$ is $F(1,k-1;\ell)$-free.
Hence,

$e(G-S_{\ell-1})\leqslant ex(n-\ell,F(1,k-1;\ell))\leqslant
e(G_{1}(n-\ell,k-1,\ell)).$ (3.3)

Let $m_{0}$ be the number of edges incident with the vertices of $S_{\ell-1}$
in $G$, that is $m_{0}=e(G)-e(G-S_{\ell-1})$. Noting that
$n\geqslant(\ell^{2}-\ell+1)k+(\ell^{2}+3\ell-2)/2$, by (3.1) and (3.3), we
have

$\displaystyle m_{0}$ $\displaystyle=e(G)-e(G-S_{\ell-1})$
$\displaystyle\geqslant e(G_{1}(n,k,\ell))-e(G_{1}(n-\ell,k-1,\ell))$
$\displaystyle=n+(\ell-1)k+\frac{\ell^{2}-5\ell+2}{2}$
$\displaystyle\geqslant\ell(\ell k+\ell-1).$

That is, each copy of $S_{\ell-1}$ in $G$ contains a vertex with degree at
least $\ell k+\ell-1$. Let $U\subseteq V(G)$ be a set of vertices with degree
at least $\ell k+\ell-1$ and each vertex in $U$ belongs to distinct
$S_{\ell-1}$. Then $\left|U\right|=k$. Let $\overline{U}=V(G)\backslash U$.
Then $\left|\overline{U}\right|=n-k$. Set $N\left(U\right)=\bigcup_{u\in
U}N\left(u\right)$ and $W_{0}=N\left(U\right)\cap\overline{U}$. Then
$\left|W_{0}\right|\geqslant(\ell-1)k+\ell$. If $G[\overline{U}]$ contains a
copy of $P_{\ell}$, we set $W_{1}=W_{0}\backslash V(P_{\ell})$, then we have
$\left|W_{1}\right|\geqslant\left|W_{0}\right|-\ell\geqslant\left(\ell-1\right)k$.
For any $u\in U$, we have

$d_{G[W_{1}]}\left(u\right)\geqslant(\ell k+\ell-1)-(k-1)-\ell=(\ell-1)k.$

We may find $k$ copies of $S_{\ell-1}$ in $G-P_{\ell}$ with $k$ center
vertices in $U$ and $(\ell-1)k$ leaves vertices in $W_{1}$. Hence
$F(1,k;\ell)\subseteq G$, which is a contradiction. Therefore,
$G[\overline{U}]$ is $P_{\ell}$-free. Recall that
$\left|\overline{U}\right|=n-k=d(\ell-1)+r$. By Theorem 1.2, we have

$e\left(G[\overline{U}]\right)\leqslant ex(n-k,P_{\ell})=[n-k,\ell,\ell].$
(3.4)

Hence, by (3.4) and (2.1), we have

$\displaystyle e(G)$
$\displaystyle=e(G[U])+e\left(U,\overline{U}\right)+e\left(G[\overline{U}]\right)$
$\displaystyle\leqslant\binom{k}{2}+k(n-k)+[n-k,\ell,\ell]$
$\displaystyle=\left(k+\frac{\ell}{2}-1\right)n-\frac{k^{2}+(\ell-1)(k+r)-r^{2}}{2}$
$\displaystyle=e(G_{1}(n,k,\ell)).$

Thus (3.2) holds, and therefore, $e(G)=ex(n,F(1,k;\ell))=e(G_{1}(n,k,\ell))$
holds.

Now we determine the extremal graphs for $F(1,k;\ell)$. If
$e(G)=e(G_{1}(n,k,\ell))$, then the equality case of (3.4) holds, and
$G=K_{k}\vee G[\overline{U}]$. By Theorem 1.2, we consider the following two
cases. (a) $G[\overline{U}]=dK_{\ell-1}\cup K_{r}$, where $0\leqslant
r<\ell-1$. Then

$G=K_{k}\vee\left(dK_{\ell-1}\cup K_{r}\right)=G_{1}(n,k,\ell).$

(b) $\ell$ is even, $r={\ell}/2$ or $r=(\ell-2)/2$, and
$G[\overline{U}]=\left(\left(d-s-1\right)K_{\ell-1}\right)\cup\left(K_{\frac{\ell-2}{2}}\vee\overline{K}_{\frac{\ell}{2}+s\left(\ell-1\right)+r}\right)$,
where $s=0,1,\cdots,d-1$. Noting that $G$ is $F(1,k;\ell)$-free, we have
$s=d-1$ by Lemma 3.1, and then
$G=K_{k+\frac{\ell}{2}-1}\vee\overline{K}_{n-k-\frac{\ell}{2}+1}=G_{2}(n,1,k,\ell)$.

Hence, the extremal graph for $F(1,k;\ell)$ is $G_{1}(n,k,\ell)$, or
$G_{2}(n,1,k,\ell)$ if $\ell$ is even, $r=\ell/2$ or $r=(\ell-2)/2$. The proof
is completed. ∎

###  3.2 The Turán number and the extremal graph for $F(k_{1},k_{2};2\ell)$

###### Proof of Theorem 2.4.

We suppose $n\geqslant(2{\ell}^{2}+3\ell-4)k_{1}+(4{\ell}^{2}-2\ell+1)k_{2}+3$
in this subsection. Recall that

$G_{2}(n,k_{1},k_{2},2\ell)=K_{\ell k_{1}+k_{2}-1}\vee\overline{K}_{n-\ell
k_{1}-k_{2}+1}$

and

$F(k_{1},k_{2};2\ell)=k_{1}P_{2\ell}\cup k_{2}S_{2\ell-1}.$

If $G_{2}(n,k_{1},k_{2},2\ell)$ contains a copy of $F(k_{1},k_{2};2\ell)$,
then each $S_{2\ell-1}$ contains at least one vertex of $K_{\ell
k_{1}+k_{2}-1}$ and each $P_{2\ell}$ contains at least $\ell$ vertices of
$K_{\ell k_{1}+k_{2}-1}$. This is a contradiction. Hence
$G_{2}(n,k_{1},k_{2},2\ell)$ is $F(k_{1},k_{2};2\ell)$-free and

$ex(n,F(k_{1},k_{2};2\ell))\geqslant e(G_{2}(n,k_{1},k_{2},2\ell)).$ (3.5)

Now we prove Theorem 2.4 by induction on $k_{2}$. For $k_{2}=0$,
$n\geqslant(2{\ell}^{2}+3\ell-4)k_{1}+3$, $G_{2}(n,k_{1},0,2\ell)=K_{\ell
k_{1}-1}\vee\overline{K}_{n-\ell k_{1}+1}$ and
$F(k_{1},0;2\ell)=k_{1}P_{2\ell}$ hold, and the results follow from Lemma 1.4.
Suppose that $k_{2}\geqslant 1$ and Theorem 2.4 holds for all
$k_{2}^{\prime}<k_{2}$. Suppose $G$ is an $F(k_{1},k_{2};2\ell)$-free graph
with $e(G)=ex(n,F(k_{1},k_{2};2\ell))$. Hence, by (3.5) and (2.2), we have

$\displaystyle e(G)$ $\displaystyle\geqslant e(G_{2}(n,k_{1},k_{2},2\ell))$
$\displaystyle=(\ell k_{1}+k_{2}-1)n-\frac{(\ell k_{1}+k_{2})(\ell
k_{1}+k_{2}-1)}{2}$
$\displaystyle>(\ell+k_{2}-2)n-\frac{(k_{2}-1)(k_{2}+2\ell-2)}{2}$
$\displaystyle=\frac{(k_{2}-1)(k_{2}-2)}{2}+(k_{2}+\ell-2)(n-k_{2}+1)$
$\displaystyle\geqslant ex(n,k_{2}S_{2\ell-1}),$

which implies $G$ contains $k_{2}$ copies $S_{2\ell-1}$ by Theorem 1.3 and
Lemma 1.5. By induction hypothesis,

$ex(n-2\ell,F(k_{1},k_{2}-1;2\ell))=e(G_{2}(n-2\ell,k_{1},k_{2}-1,2\ell)).$

Since $G$ is $F(k_{1},k_{2};2\ell)$-free, $G-S_{2\ell-1}$ is
$F(k_{1},k_{2}-1;2\ell)$-free. Hence,

$e(G-S_{2\ell-1})\leqslant
ex(n-2\ell,F(k_{1},k_{2}-1;2\ell))=e(G_{2}(n-2\ell,k_{1},k_{2}-1,2\ell)).$
(3.6)

Let $m_{0}$ be the number of edges incident with the vertices of $S_{2\ell-1}$
in $G$. Noting that $\ell\geqslant 2$, $k_{1}\geqslant 2$ and
$n\geqslant(2{\ell}^{2}+3\ell-4)k_{1}+(4{\ell}^{2}-2\ell+1)k_{2}+3$, by (3.5)
and (3.6), we have

$\displaystyle m_{0}$ $\displaystyle=e(G)-e(G-S_{2\ell-1})$
$\displaystyle\geqslant
e(G_{2}(n,k_{1},k_{2},2\ell))-e(G_{2}(n-2\ell,k_{1},k_{2}-1,2\ell))$
$\displaystyle=n+(2{\ell}^{2}-\ell)k_{1}+(2\ell-1)k_{2}-4\ell+1$
$\displaystyle\geqslant 2\ell(2\ell(k_{1}+k_{2})-1)+(2\ell-4)(k_{1}-1)$
$\displaystyle\geqslant 2\ell(2\ell(k_{1}+k_{2})-1).$

Then we may construct a vertex subset $U\subseteq V(G)$ of order $k_{2}$ whose
each vertex has degree at least $2\ell(k_{1}+k_{2})-1$. Write
$\overline{U}=V(G)\backslash U$. Then $\left|\overline{U}\right|=n-k_{2}$. By
(3.5), we have

$e\left(G[\overline{U}]\right)=e(G)-e(G[U])-e\left(U,\overline{U}\right)\geqslant
e(G_{2}(n,k_{1},k_{2},2\ell))-e(G[U])-e\left(U,\overline{U}\right).$

We consider the following two cases.

Case 1. $G[U]$ is a clique, and each vertex in $U$ is adjacent to each vertex
in $\overline{U}$.

In this case, $e(G[U])=k_{2}(k_{2}-1)/2$ and
$e\left(U,\overline{U}\right)=k_{2}(n-k_{2})$. Then by Lemma 1.4, we have

$\displaystyle e\left(G[\overline{U}]\right)$ $\displaystyle\geqslant
e(G_{2}(n,k_{1},k_{2},2\ell))-e(G[U])-e\left(U,\overline{U}\right)$
$\displaystyle=(\ell k_{1}-1)\left(n-\frac{\ell k_{1}}{2}-k_{2}\right)$
$\displaystyle=ex(n-k_{2},k_{1}P_{2\ell}).$

If $e\left(G[\overline{U}]\right)>ex(n-k_{2},k_{1}P_{2\ell})$, then we have
$k_{1}P_{2\ell}\subseteq G[\overline{U}]$. Set $W=\overline{U}\backslash
V(k_{1}P_{2\ell})$. Note that

$\left|W\right|=\left|\overline{U}\backslash
V(k_{1}P_{2\ell})\right|=n-k_{2}-2\ell
k_{1}\geqslant(2{\ell}^{2}+\ell-4)k_{1}+(4{\ell}^{2}-2\ell)k_{2}+3>(2\ell-1)k_{2},$

and each vertex in $U$ is adjacent to each vertex in $W$. So there are $k_{2}$
copies of $S_{2\ell-1}$ in $G[V(G)\backslash V(k_{1}P_{2\ell})]$ with $k_{2}$
center vertices in $U$ and $(2\ell-1)k_{2}$ leaves vertices in $W$. Hence, we
have $k_{1}P_{2\ell}\cup k_{2}S_{2\ell-1}\subseteq G$, which is a
contradiction.

Hence $e\left(G[\overline{U}]\right)=ex(n-k_{2},k_{1}P_{2\ell})$ and
$G[\overline{U}]$ does not contain $k_{1}$ copies of $P_{2\ell}$. By Lemma 1.4
again,

$G[\overline{U}]=\mbox{\rm{EX}}(n-k_{2},k_{1}P_{2\ell})=K_{\ell
k_{1}-1}\vee\overline{K}_{n-\ell k_{1}-k_{2}+1}.$

Hence,

$G=K_{k_{2}}\vee\left(K_{\ell k_{1}-1}\vee\overline{K}_{n-\ell
k_{1}-k_{2}+1}\right)=G_{2}(n,k_{1},k_{2},2\ell).$

Case 2. $G[U]$ is not a clique, or some vertex in $U$ is not adjacent to some
vertex in $\overline{U}$.

In this case, either $e(G[U])<k_{2}(k_{2}-1)/2$ or
$e\left(U,\overline{U}\right)<k_{2}(n-k_{2})$ holds. Then by Lemma 1.4, we
have

$\displaystyle e\left(G[\overline{U}]\right)$ $\displaystyle\geqslant
e(G_{2}(n,k_{1},k_{2},2\ell))-e(G[U])-e\left(U,\overline{U}\right)$
$\displaystyle>(\ell k_{1}+k_{2}-1)n-\frac{(\ell k_{1}+k_{2})(\ell
k_{1}+k_{2}-1)}{2}-\frac{k_{2}(k_{2}-1)}{2}-k_{2}(n-k_{2})$
$\displaystyle=(\ell k_{1}-1)\left(n-\frac{\ell k_{1}}{2}-k_{2}\right)$
$\displaystyle=ex(n-k_{2},k_{1}P_{2\ell}),$

which implies $k_{1}P_{2\ell}\subseteq G[\overline{U}]$. Set
$W=\overline{U}\backslash V(k_{1}P_{2\ell})$. For any vertex $u\in U$,

$d_{G[W]}(u)\geqslant(2\ell(k_{1}+k_{2})-1)-(k_{2}-1)-2\ell
k_{1}=(2\ell-1)k_{2}.$

Hence, we can find $k_{2}$ copies of $S_{2\ell-1}$ in $G[V(G)\backslash
V(k_{1}P_{2\ell})]$ with $k_{2}$ center vertices in $U$ and $(2\ell-1)k_{2}$
leaves vertices in $W$. Hence, there is a copy of $k_{1}P_{2\ell}\cup
k_{2}S_{2\ell-1}$ in $G$, which is a contradiction. The proof is completed. ∎

###  3.3 The Turán number and the extremal graph for $F(2,k;5)$

###### Proof of Theorem 2.5.

We suppose $n\geqslant 21k+38$ in this subsection. Recall that

$G_{3}(n,k)=K_{k+3}\vee\left(K_{2}\cup\overline{K}_{n-k-5}\right)$

and

$F(2,k;5)=2P_{5}\cup kS_{4}.$

If $G_{3}(n,k)$ contains a copy of $F(2,k;5)$, then each $S_{4}$ contains at
least one vertex of $K_{k+3}$ and each $P_{5}$ contains at least two vertices
of $K_{k+3}$. This is a contradiction. Hence $G_{3}(n,k)$ is $F(2,k;5)$-free
and

$ex(n,F(2,k;5))\geqslant e(G_{3}(n,k)).$ (3.7)

Now we prove Theorem 2.5 by induction on $k$. For $k=0$, $n\geqslant 38$,
$G_{3}(n,0)=K_{3}\vee\left(K_{2}\cup\overline{K}_{n-5}\right)$ and
$F(2,0;5)=2P_{5}$ hold. Hence the results follow from Lemma 1.2. Suppose that
$k\geqslant 1$ and the results hold for all $k^{\prime}<k$. Suppose $G$ is an
$F(2,k;5)$-free graph with $e(G)=ex(n,F(2,k;5))$. Then by (3.7) and (2.3), we
have

$\displaystyle e(G)$ $\displaystyle\geqslant e(G_{3}\left(n,k\right))$
$\displaystyle=(k+3)n-\frac{k^{2}+7k+10}{2}$
$\displaystyle>\left(k+\frac{1}{2}\right)n-\frac{k^{2}+2k-3}{2}$
$\displaystyle=\left(k-1\right)\left(n-\frac{k}{2}\right)+\frac{3(n-k+1)}{2}$
$\displaystyle\geqslant\left(k-1\right)\left(n-\frac{k}{2}\right)+\left\lfloor\frac{3(n-k+1)}{2}\right\rfloor$
$\displaystyle\geqslant ex(n,kS_{4}),$

which implies $G$ contains $k$ copies $S_{4}$ by Theorem 1.3 and Lemma 1.5. By
induction hypothesis,

$ex(n-5,F(2,k-1;5))=e(G_{3}(n-5,k-1)).$

Since $G$ is $F(2,k;5)$-free, $G-S_{4}$ is $F(2,k-1;5)$-free. Hence,

$e(G-S_{4})\leqslant ex(n-5,F(2,k-1;5))=e(G_{3}(n-5,k-1)).$ (3.8)

Let $m_{0}$ be the number of edges incident with the vertices of $S_{4}$ in
$G$. Noting that $n\geqslant 21k+38$, by (3.7) and (3.8), we have

$\displaystyle m_{0}$ $\displaystyle=e(G)-e(G-S_{4})$ $\displaystyle\geqslant
e(G_{3}(n,k))-e(G_{3}(n-5,k-1))$ $\displaystyle=n+4k+7$
$\displaystyle\geqslant 5(5k+9).$

Then we can construct a vertex subset $U\subseteq V(G)$ of order $k$ whose
each vertex has degree at least $5k+9$. Let $\overline{U}=V(G)\backslash U$.
Then $\left|\overline{U}\right|=n-k$. Note that

$e\left(G[\overline{U}]\right)=e(G)-e(G[U])-e\left(U,\overline{U}\right)\geqslant
e(G_{3}(n,k))-e(G[U])-e\left(U,\overline{U}\right).$

We consider the following two cases.

Case 1. $G[U]$ is a clique and each vertex in $U$ is adjacent to each vertex
in $\overline{U}$.

In this case, $e(G[U])=k(k-1)/2$ and $e\left(U,\overline{U}\right)=k(n-k)$.
Then by Lemma 1.2, we have

$\displaystyle e(G[\overline{U}])$ $\displaystyle\geqslant
e(G_{3}(n,k))-e(G[U])-e\left(U,\overline{U}]\right)$ $\displaystyle=3(n-k)-5$
$\displaystyle=ex(n-k,2P_{5}).$

If $e\left(G[\overline{U}]\right)>ex(n-k,2P_{5})$, then $2P_{5}\subseteq
G[\overline{U}]$. Set $W=\overline{U}\backslash V(2P_{5})$. Note that

$\left|W\right|=\left|\overline{U}\backslash V(2P_{5})\right|=n-k-10\geqslant
21k+38-k-10>4k$

and each vertex in $U$ is adjacent to each vertex in $W$. Hence, there are $k$
copies of $S_{4}$ in $G[V(G)\backslash V(2P_{5})]$ with $k$ center vertices in
$U$ and $4k$ leaves vertices in $W$, and then $2P_{5}\cup kS_{4}\subseteq G$,
which is a contradiction.

Hence $e\left(G[\overline{U}]\right)=ex(n-k,2P_{5})$ and $G[\overline{U}]$
does not contain $2$ copies of $P_{5}$. By Lemma 1.2 again,

$G[\overline{U}]=\mbox{\rm{EX}}(n-k,2P_{5})=K_{3}\vee\left(K_{2}\cup\overline{K}_{n-k-5}\right),$

and then

$G=K_{k+3}\vee\left(K_{2}\cup\overline{K}_{n-k-5}\right)=G_{3}(n,k).$

Case 2. $G[U]$ is not a clique or some vertex in $U$ is not adjacent to some
vertex in $\overline{U}$.

In this case, either $e(G[U])<k(k-1)/2$ or
$e\left(U,\overline{U}\right)<k(n-k)$ holds. Then by Lemma 1.2, we have

$\displaystyle e\left(G[\overline{U}]\right)$ $\displaystyle\geqslant
e(G_{3}(n,k))-e(G[U])-e\left(U,\overline{U}\right)$
$\displaystyle>(k+3)n-\frac{k^{2}+7k+10}{2}-\frac{k(k-1)}{2}-k(n-k)$
$\displaystyle=3(n-k)-5$ $\displaystyle=ex(n-k,2P_{5}),$

which implies $2P_{5}\subseteq G[\overline{U}]$. Set $W=\overline{U}\backslash
V(2P_{5})$. For any vertex $u\in U$,

$d_{G[W]}(u)\geqslant(5k+9)-(k-1)-10=4k.$

Hence, we can find $k$ copies of $S_{4}$ in $G[V(G)\backslash V(2P_{5})]$ with
$k$ center vertices in $U$ and $4k$ leaves vertices in $W$. Hence, there is a
copy of $2P_{5}\cup kS_{4}$ in $G$, which is a contradiction. The proof is
completed. ∎

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###### Abstract

A simple metamaterial absorber suitable for fabrication over large areas
associated with a Fano-like resonance is proposed. The proposed designed of
the metamaterial absorber consists of photoresist disk arrays on a silicon
substrate followed by the deposition of a tri-layer of Au/ZnS/Au on the top of
the structure. Due to the tri-layer, there is formation of a cavity between
the substrate and gold layer on the top of photoresist disk and a waveguide in
the ZnS layer in between the gold layers on the substrate and the ZnS layer
itself. The Fano-like resonance arises due to the interference of the cavity
mode and the guided-mode resonance as well as the Wood’s anomaly. The cavity
mode resonance works as continuum mode whereas the guided mode resonance and
Wood’s anomaly work as discrete modes. The spectral position of the resonance
can be tuned by just controlling the thickness of the tri-layer instead of the
structural size and shape modification of the micro/nano-structures as usually
done in conventional metamaterial absorbers. This design can be easily scaled
for fabrication over large areas as it separate the structuring and deposition
processes, and makes them sequential thereby avoiding expensive and complex
lift-off or etching processes which are usually required for conventional
metamaterial processing.

A simple metamaterial absorber associated with Fano-like resonance

Raghwendra Kumara,b and S. Anantha Ramakrishnaa

##  1 Introduction

Electromagnetic (EM) metamaterials are artificial nano/micro-structured
composites of sub-wavelength metallic and dielectric components that exhibit
exotic electromagnetic responses such as complete absorption of
electromagnetic waves [1, 2, 3, 4, 5], negative refraction [6], invisibility
cloaking [7], sub-diffraction limited imaging using super lenses [8], etc,
which are not available in natural materials [9]. These composites usually
consist of identical structured unit cells, arranged typically in a periodic
fashion, and can be considered as bulk homogeneous media described as an
effective medium through effective medium parameters [10]. Among these
responses, the complete/perfect absorption of EM radiations as metamaterial
absorbers have received considerable attention due their potential
applications for radar stealth at microwave frequencies, sensitive infrared
detectors and micro-bolometer arrays [11], controllable thermal emitters for
manipulating the black body emittance [12] and thermophotovoltaics [13] at
infrared frequencies, etc. A perfect metamaterial absorber with nearly perfect
absorption caused by simultaneous excitation of electric and magnetic
resonances to create an impedance match of a resonant medium with the
surrounding medium was first proposed by Landy et al. in 2008 [1]. After that,
a lot of absorbers based on different physical mechanisms have been
demonstrated theoretically and experimentally in a wide spectral range [14,
15, 16]. It has been possible to easily adapt metamaterial for different
frequencies due to the scale invariance of Maxwell’s equations. The main
purpose of metamaterial absorber is to reduce undesired reflections and to
increase absorption within the given spectral range.

Metamaterial absorbers can be realized using different physical mechanisms
such as anti-reflection coating, resonant/impedance matching [17, 18, 19, 20],
localized surface plasmons resonance, Ohmic losses in the visible regime and
lossy dielectric in the microwave regime, cavity resonance [21], guided mode
resonance [22], etc. A typical metamaterial absorber has a tri-layered
structure, with a structured top conducting layer separated by a continuous
dielectric layer from a bottom conducting layer. The excitation depends on the
electric resonance of the top layer and the impedance matching arises from the
magnetic resonance created by the sheet current distributions on the top
resonating structure and the bottom conducting layer (image charges and
currents), which are anti-parallel. The thickness of the dielectric layer
determines the capacitance for the LC resonance and the optimal impedance
matching for coupling maximum radiation into the resonant structure.
Generally, metamaterial absorbers show symmetric (Gaussian) absorption
profiles, however the asymmetric absorption profiles have been also absorbed
due to the interference of different modes arises in the system [23, 24].

Usually, the Fano resonance [25, 26] is used to describe asymmetric resonance,
which arises due to the constructive and destructive interference of discrete
resonance states with broad band continuum states. This phenomenon and the
underlying mechanism, being common and appearing in many domains of physical
sciences, are found in a variety of micro/nano-structures such as plasmonics
and metamaterials [27, 28], photonic crystals [29, 30, 31], quantum dots [32],
etc. The asymmetric Fano resonance profile promises a variety of applications
in a wide range of photonic devices, such as sensors [33, 34], optical filters
[35], switches [36], detectors [37, 38], non-linear devices [39] and slow-
light devices [40], etc. In the last few years, with advancements in
micro/nano fabrication techniques and the development of integration
techniques, Fano resonance has been widely investigated in micro/nano
fabricated structures [27, 28, 31, 41, 42, 43].

The metamaterial perfect absorbers based on impedance matched resonances
typically involve a complex design, and the fabrication tolerances are quite
severe. However, for many infra-red applications, such as solar thermal
devices or thermal camouflage, control of the emissivity and the absorptivity
is required over very large area surfaces ranging from few square meters to
even square kilo meters. It is a great manufacturing challenge to produce
metamaterials that have complex micrometric or even sub-micrometric structures
containing metallic as well as dielectric constituents over these large areas
with good fidelity and over realistic time scales. In this work, we present
numerical studies on a novel design of metamaterial absorber for controlling
the emissivity/absorptivity at infra-red frequencies. The metamaterial
consists of a tri-layer of Au/ZnS/Au thin films deposited on a structured
photoresist disk arrays, and depends on a cavity and guided mode resonance
(GMR) of the structure to produce high absorptivity. The simplified design of
the metamaterial absorber can be easily implemented for sequential patterning
and deposition processes without lift-off or etching, making the process
applicable for large area metamaterial surfaces [44].

##  2 Design of the metamaterial by electromagnetic simulations

The unit cell of the metamaterial absorber is shown in Fig. 1. It consists of
a photoresist disk of height ($h$) 500 nm and diameter ($d$) 1.68 $\mu$m on a
silicon substrate with consecutive layers of bottom gold of thickness $t_{1}$
(160 nm), ZnS of thickness $t_{2}$ (300 nm) and top gold of thickness $t_{3}$
(30 nm) on the top of both the disk as well as on the other parts of the
silicon substrate not covered by the disk. In essence, the structure consists
of three disks, out of two are gold disks and one is ZnS disk placed on the
photoresist disk, with continuous thin films of Au/ZnS/Au containing holes
filled with the photoresist at a lower level on the substrate. Due to the tri-
layer deposition, a cavity is formed between the bottom layer of gold on the
top of photoresist disk and silicon substrate, because the refractive index of
the photoresist is lower than that of silicon, and gold is itself highly
reflecting. There is formation of waveguide in the ZnS layer between the gold
layers on the silicon substrate and the ZnS itself.

The numerical simulation is performed by radio frequency module of COMSOL
Multiphysics software [45] based on the finite element method. Periodic
boundary conditions were used along the transverse x and y directions to
simulate an infinite array of unit cells to reduce the complexity of the
simulation. Perfectly matched layers (PML) were used along the propagation
direction (z - direction) to prevent the reflection of the waves from the top
and bottom domain boundaries with sufficient distance from the structure. The
dispersive refractive indices (real and imaginary parts) of evaporated gold,
ZnS and silicon were taken from Refs [46, 47] and [48], respectively. The
refractive index of the photoresist in simulation is considered as 1.58. The
transmittance ($T$) and reflectance ($R$) of the structure is calculated by
using port conditions at a plane above the structure in air and below the
structure in the silicon substrate. The absorbance is calculated using energy
conservation as $A=1-R-T$ at each wavelength.

Figure 1: Schematic diagram of the unit cell of the metamaterial absorber with
illustration of different existing modes. $t_{1}$, $t_{2}$ and $t_{3}$ are the
thickness of the bottom gold layer, ZnS layer and top gold layer,
respectively. $h$ and $d$ are the height and diameter of the photoresist disk.
$g$ is the gap between gold layers on the top of photoresist disk and the ZnS
layer on the silicon substrate.

The simulated reflectance, transmittance and absorbance spectra of the
metamaterial are shown in Fig. 2(a). The transmittance spectrum consists of
two peaks at wavelengths 9.8 $\mu$m and 9 $\mu$m. The reflectance spectrum
consists of three minima at wavelengths of 6.4 $\mu$m, 9.1 $\mu$m and
9.85$\mu$m. The absorbance spectrum consists of three maxima at wavelengths of
6.4 $\mu$m, 9.1 $\mu$m and 10 $\mu$m. Color maps of the normalized magnetic
field at different resonant wavelengths 9.1 $\mu$m, 6.4 $\mu$m, 9.6 $\mu$m and
10 $\mu$m are shown in Figs. 2(b), (c), (d) and (e), respectively. It is clear
from Fig. 2(b) that the fields are strongly confined in the waveguide as well
as in the cavity at a wavelength of 9.1 $\mu$m. The field is highly localized
in the ZnS layer between the two layers of gold on the top of photoresist disk
at wavelength of 6.4 $\mu$m. Figs. 2(d) and (e) show the normalized magnetic
field maps at the peak and dip of the corresponding transmittance spectrum,
respectively. It is clear from Fig. 2(d) that the fields are distributed along
the Au/Si interface and weakly within the cavity. The fields are distributed
along the Au/Si interface and not localized within the cavity at 10 $\mu$m
wavelength (see Fig. 2(e)). It is clear that there is no contribution of the
resonance at wavelength 6.4 $\mu$m in the resultant interfering resonances at
9.1 $\mu$m wavelength. In order to understand the contribution of the
different resonances in the interference at 9.1 $\mu$m, we note that the
resonance at 6.4 $\mu$m, which is not close to 9.1 $\mu$m can be eliminated by
removing the top layer gold and ZnS from the photoresist disk.

Figure 2: (a) simulated reflectance, transmittance and absorbance spectra
corresponding to the metamaterial absorber, (b) color plot of the normalized
magnetic field at resonant wavelength 9.1 $\mu$m, (c) color plot of the
normalized magnetic field at resonant wavelength 6.4 $\mu$m, (d) color plot of
the normalized magnetic field at the resonant wavelength 9.6 $\mu$m, (e) color
plot of the normalized magnetic field at the resonant wavelength 10.0 $\mu$m.
The parameters used for simulations: periodicity of the disk array, disk
height, disk diameter, ZnS thickness, bottom layer Au thickness, top layer Au
thickness are 2.8 $\mu$m, 0.5 $\mu$m, 1.68 $\mu$m, 300 nm and 30 nm,
respectively.

The simulated reflectance, transmittance and absorbance spectra of the
metamaterial absorber without the ZnS and top Au layers on the photoresist
disk are shown in Fig. 3(a), where the corresponding unit cell shown in the
inset. Fig. 3(b) shows the normalized time average power flow in color map,
and to understand the power flow in the system, an arrow plot of the time-
averaged power is plotted in the same figure. It is clear that the power flows
into the cavity and the waveguide as well as along the interface of gold and
silicon. The color maps of the normalized magnetic fields at wavelengths of
9.1 $\mu$m and 10 $\mu$m are shown in Figs. 3(c) and (d), respectively.

Figure 3: (a) simulated reflectance, transmittance and absorbance spectra
corresponding to the metamaterial absorber without top Au and ZnS layers on
the disk, (b) Color plot of the normalized time average power with Poynting
vector field lines at resonant wavelength 9.1 $\mu$m, (d) color plot of the
normalized magnetic field at the resonant wavelength 9.1 $\mu$m, (e) color
plot of the normalized magnetic field at the resonant wavelength 10.0 $\mu$m.
The parameters used for simulations: periodicity of the disk array, disk
height, disk diameter, ZnS thickness, bottom gold layer thickness and top gold
layer thickness are 2.8 $\mu$m, 0.5 $\mu$m, 1.68 $\mu$m, 300 nm, 160 nm and 30
nm, respectively.

Fig. 4 shows the transmittance, reflectance and absorbance spectra for
different periodicities of the photoresist disk array. The other parameters
like the thickness of the bottom and top gold layer, ZnS layer and height of
the photoresist disk are kept constant. With increasing the periodicity of the
disk array, an extra resonance appears. The extra resonance position shifts
towards longer wavelengths with increasing the periodicity of photoresist disk
array. But, the resonance corresponding to the cavity mode remains almost
independent of the periodicity of the disk array. There is an asymmetric
resonance for a fixed value of periodicity 2.8 $\mu$m, which arises due to the
interference of the fixed resonance (cavity resonance) and the resonance is
caused by changing the periodicity (Guided-mode resonance) of the disk array
[Fig. 4(b)]. The interfering resonances corresponding to the different modes
separate when the period of the disk array becomes 4 $\mu$m (see in Fig.
4(c)). At the periodicity of 2 $\mu$m, only one resonance appears that arises
due to the confinement of fields into the cavity, whereas at 5 $\mu$m
periodicity an extra resonance appears around 12.4 $\mu$m.

Figure 4: Simulated reflectance, transmittance and absorbance spectra at
normal incidence for different periodicities of the photoresist disk array.
The parameters used for simulations: disk height, disk diameter, bottom Au
layer thickness, ZnS layer thickness and top Au layer thickness are 0.5
$\mu$m, 1.68 $\mu$m, 160 nm, 300 nm and 30 nm, respectively.

Fig. 5 shows the color-maps of the normalized magnetic field and arrow volume
plot of the normalized time average power flow at different resonance
wavelengths for 4 $\mu$m periodicity of the disk array. Fig. 5(a) shows the
color plot of the normalized magnetic field at 8.8 $\mu$m wavelength, and at
this resonance wavelength field is only localized in the cavity formed between
gold layers on the photoresist disk and Si substrate. The corresponding color
map of the normalized time average power and its arrow volume plot is shown in
the figure 5(d). Fig. 5(b) shows the color map of the normalized magnetic
field at 13.7 $\mu$m wavelength, and at this resonance wavelength field is
only extended into the silicon substrate from the gold layer side. The
corresponding color map of the normalized time average power and its arrow
volume plot are shown in the figure. 5(e). The arrow power flow indicates the
power flow in the substrates along the gold and silicon interface. Fig. 5(c)
shows the color map of the normalized magnetic field at 10.3 $\mu$m wavelength
and at this resonance wavelength field is localized in the sandwiched ZnS
layer between the bottom gold layer on the substrate and the top gold layer on
ZnS. The corresponding color map of the normalized time average power and its
arrow volume plot is shown in Fig. 5(f). It is clear that most of the power is
flowing into the ZnS sandwiched layer.

Figure 5: Color plots of the normalized magnetic field at resonance wavelength
(a) $\lambda$ = 8.8 $\mu$m, (b) $\lambda$ = 10.3 $\mu$m and (c) $\lambda$ =
13.7 $\mu$m. The arrow volume plots of the time average power flow at
resonance wavelength (d) $\lambda$ = 8.8 $\mu$m, (e) $\lambda$ = 10.3 $\mu$m
and (f) $\lambda$ = 13.7 $\mu$m.

Figs. 6(a) and (b) show the simulated reflectance spectra at different
incident angles for the TE and TM polarizations, respectively. There is a
slight decrement in the minimum of reflectance spectra around the 9 $\mu$m
wavelength with increasing the angle of incidence in the case of TE
polarization, whereas it remains almost constant with a slight redshift in the
case of TM polarization. The minimum at around 10 $\mu$m wavelength in the
case of TE polarization slightly decreases in amplitude with increasing the
angle of incidence whereas it slightly decreases in amplitude with redshift in
the case of TM polarization. An extra minimum appears in the reflectance
spectra around 7.5 $\mu$m wavelength in the case of TM polarization, and it
shifts towards longer wavelengths with increasing the angle of incidence. The
minimum in the reflectance spectra around 6.5 $\mu$m wavelength remains almost
constant with increasing the angle of incidence in both polarizations. The
minimum in the reflectance spectrum around 4.5 $\mu$m wavelength slightly
increases in amplitude with increasing the angle of incidence for both
polarizations but faster in the case of TM polarization compared to that of TE
polarization. Splitting appears in the minimum at 4.5 $\mu$m wavelength with
increasing the angle of incidence in the case of TM polarization, whereas
there is no such splitting in the case of TE polarization.

Figure 6: The simulated reflectance spectra of the fabricated sample for the
(a) TE polarization, (b) TM polarization at different incident angles. The
parameters used for simulations: disk height, disk diameter, bottom Au layer
thickness, ZnS layer thickness and top Au layer thickness are 0.5 $\mu$m, 1.68
$\mu$m, 155 nm, 300 nm and 30 nm, respectively.

Fig. 7 shows the simulated electromagnetic response of the tri-layered
metamaterial absorber for different values of the $g$ parameter at normal
incidence. The value of $g$ was changed by changing the thickness of the
bottom gold layer, and the other parameters are kept constant. In the
reflectance spectra [see Fig. 7(a)], the minimum at 9.1 $\mu$m wavelength
shifts towards shorter wavelengths, and the minimum at 9.8 $\mu$m wavelength
remains almost constant with increasing the value of $g$. While the minimum at
6.4 $\mu$m wavelength increases with increasing the gap. In the transmittance
spectra [see Fig. 7(b)], the fundamental order transmission amplitude at 9.8
$\mu$m wavelength decreases whereas the higher order transmission amplitude at
6.4 $\mu$m wavelength increases with increasing the gap. The transmission peak
at 9.1 $\mu$m wavelength shifted towards shorter wavelengths with increasing
the gap. In the absorbance spectra [see Fig. 7(c)], the absorption peak around
the 10 $\mu$m wavelength decreases with a minimal shift towards shorter
wavelengths with increasing the gap. There is a shift in the absorption peak
around 9 $\mu$m towards shorter wavelengths with increasing the gap whereas
the absorption peak around 6.4 $\mu$m wavelength increases in amplitude with
increasing the gap.

Figure 7: The simulated (a) reflectance, (b) transmittance and (c) absorbance
spectra of the tri-layered metamaterial for different values of $g$ as 10 nm,
15 nm, 20 nm, 25 nm and 30 nm at normal incidence. The other parameters such
as disk height, disk diameter, ZnS layer thickness and top Au layer thickness
are 0.5 $\mu$m, 1.68 $\mu$m, 300 nm and 30 nm, respectively.

Fig. 8(a) shows the simulated reflectance spectrum (black line) at normal
incidence of the metamaterial absorber with the Fano fit (red line) using the
coupled mode theory of one port and two modes. There are two minima at 9.1
$\mu$m and 9.8 $\mu$m wavelengths in the reflectance spectrum, which
corresponds to the cavity mode resonance and guided mode resonance,
respectively. Fig. 8(b) shows a fit of the Fano line shape combining the
individual resonance. This was plotted using the same fitting parameters as
used to fit the asymmetric reflectance spectrum in Fig. 8(a). The individual
response was plotted by switching off the other corresponding resonance. For
example for CM resonance, the value of $\omega_{2}=0$, and similarly for GMR,
the value of $\omega_{1}=0$ [see Eq. 1].

Figure 8: (a) Simulated reflectance spectrum (black line) with Fano fit (red
line), (b) Fano fitted reflectance spectra of the corresponding individual
resonance. The parameters used for simulations: periodicity of the disk array,
disk height, disk diameter, bottom layer Au thickness, ZnS layer thickness and
top layer Au thickness are 2.8 $\mu$m, 0.5 $\mu$m, 1.68 $\mu$m, 160 nm, 300 nm
and 30 nm, respectively. The fitting parameters; $\omega_{1}=65.6$ THz,
$\omega_{2}=60.73$ THz, $\gamma_{1}=6.42$ THz, $\gamma_{2}=1.86$ THz,
$\gamma^{{}^{\prime}}_{1}=6.4$ THz, $\gamma^{{}^{\prime}}_{2}=3.0$ THz and
$\omega_{0}=6.53$ THz.

##  3 Discussion

The mechanism of absorption of EM waves can be understood from the
illustration of different excited modes, shown in the schematic diagram of the
unit cell of the metamaterial absorber [see Fig. 1]. Formation of a cavity
between the bottom layer gold on the top of photoresist disk and silicon
substrate occurs because of the lower refractive index of the photoresist than
that of silicon. The diffracted wave from the disk gets multiply reflected
between the gold layer and the silicon substrate and causes the trapping of
the electromagnetic wave into the photoresist medium [49, 50]. There is a
formation of a waveguide in the ZnS layer sandwiched between the gold layer
the on the silicon substrate and ZnS layer itself. Waves diffracted from the
different disks enter through the gap between the bottom layer of gold on the
photoresist disk and top layer gold on the ZnS layer on uncovered portions of
the substrate by the photoresist disk. The diffracted light, which enters
through the gap couples into the cavity mode as well as to the waveguide mode
[51], and some parts of it just propagate at the interface of silicon and gold
layers. The same happens by coupling through the gap with other disks. Thus,
diffracted coupled waves travel in forward and backward directions in the
waveguide and cause the formation of standing waves. The diffracted waves
propagating along the plane at the interface between gold and silicon get
absorbed, and do not contribute to the transmittance, and the corresponding
phenomenon is known as Wood’s anomaly [52, 53].

Fig. 2(a) shows the simulated electromagnetic response of the metamaterial
absorber. The asymmetric electromagnetic response is caused by simultaneous
excitation and interaction of the cavity mode, guided mode and Wood’s anomaly
resonances, which results in a Fano-like resonance. This Fano-like resonance
arises due to the interference of the continuum mode (cavity mode resonance)
and the dark mode (guided mode resonance and Wood’s anomaly) resonances. The
simulated absorption spectrum consists of three peaks at wavelengths 6.4
$\mu$m, 9.1 $\mu$m and 10 $\mu$m. The peak at a wavelength of 6.4 $\mu$m
corresponds to the metamaterial absorption peak due to the presence of the
tri-layer (Au/ZnS/Au) on the top of the photoresist disk [1, 54, 3]. There is
no contribution of this resonance in the Fano-like resonance, so it was
neglected in the further simulations by not including the top gold and ZnS
layer on the photoresist disk in the simulations.

Fig. 3 shows the simulated electromagnetic response without considering the
top gold and ZnS layer on the photoresist disk. However, there is still an
appearance of a peak in the absorption spectrum at 6.4 $\mu$m which is due to
the localized surface plasmons at the corner of the gold disk placed on the
top of the photoresist disk. There is no effect of the removal of the top gold
and ZnS layers from the photoresist disk on the other resonances, which is
seen in Fig. 3(a). The arrow plot of the normalized time average power flow
with its color map is shown in Fig. 3(b). The simultaneous excitation of
cavity mode, guided mode and Wood’s anomaly can be seen through the coupling
of power flow into each mode. The simultaneous excitation of all these modes
can be seen in the color map of the normalized magnetic fields in Fig. 3(c).
The propagation of the diffracted wave at the interface of gold and silicon
can be seen in Fig. 3(d), which results in almost zero transmittance.

To understand the Fano-like resonance better and to elucidate the contribution
of different modes in the Fano resonance, the simulations were carried out for
different periodicities of the photoresist disk array which are shown in Fig.
4. Only the cavity mode resonance exists in the case of photoresist disk array
with 2 $\mu$m periodicity. The existence of this mode is independent of the
periodicity of the disk array and is always associated with the unit cell, and
hence regarded as continuum mode, whereas the existence of the guided mode
resonance and Wood’s anomaly is conditional and hence regarded as the discrete
mode. Interference of the continuum and discrete modes is conditional, and it
only happens at 2.8 $\mu$m periodicity (Fig. 4b). For periodicity of 4 $\mu$m,
these interfering modes get separated in frequency (Fig. 4c), and the modes
corresponding to the resonance wavelengths of 8.8 $\mu$m, 10.3 $\mu$m and 13.7
$\mu$m are the cavity mode, guided mode and Wood’s anomaly, respectively. The
existence of the cavity mode can be seen even at 5 $\mu$m periodicity (Fig.
4d).

The color maps of the normalized magnetic field and normalized time average
power flow with its arrow volume plot at different resonant wavelengths
corresponding to 4 $\mu$m periodicity of disk array are shown in Fig. 5.
Excitation of cavity mode resonance can be understood by the confinement of
the magnetic field in the cavity and arrow volume plot of the time average
power from Figs. 5(a) and (d), respectively. Excitation of the guided mode
resonance can be understood by the confinement of the magnetic field in the
waveguide and arrow volume plot of the time average power from Figs. 5(c) and
(f), respectively. Similarly, excitation of the Wood’s anomaly resonance can
be understood from Figs. 5(d) and (e).

The effect of spoof SPPs can be understood from the angle dependent
reflectance spectra of the metamaterial absorber for the TE and TM
polarizations [see Figs. 6(a) and (b)]. In case of TE polarization, magnitude
of electric remains constant so there is only slight variation in the
magnitude of deeps and peaks in the reflectance spectra at different incident
angles. In case of TM polarization, magnitude of electric field changes due to
which coupling condition for spoof SPPs changes, and hence there is finite
variation in the magnitude of deeps and peaks as well as spectral shift in the
resonance [55].

The minimum in the reflectance spectra as well as the corresponding peak in
absorbance and transmittance spectra around 9 $\mu$m wavelength shift towards
shorter wavelengths with increasing the value of $g$ [ Fig. 7]. This can be
understood from the circuit model theory of metamaterial absorber. The
resonance frequency can be defined as $\omega=1/\sqrt{L}C$ on the basis of the
circuit model, where $L$ and $C$ are effective inductance and capacitance
corresponding to the metamaterial absorber. In the proposed metamterial
absorber with increasing the value of $g$ there is decreased in the effective
capacitance which corresponds to increase in the resonance frequency and hence
shorter wavelengths shift in resonance.

The coupled mode theory (CMT) has also been used to describe the Fano
asymmetric line profile which arises due to the interference of continuum and
discrete mode resonances in the metamaterial absorbers [41, 56]. Since our
system contains two main resonances; one is cavity resonance and other is GMR,
so the CMT with the one-port and two modes resonator model can be used to
describe the asymmetric reflectance spectra of the metamaterial absorber [57,
31]. The reflectance r corresponding our metamaterial absorber is

$r=-1+\frac{2(W_{1}\gamma_{2}+W_{2}\gamma_{1}-2\gamma_{0}\sqrt{\gamma_{1}\gamma_{2}})}{(W_{1}W_{2}-|\gamma_{0}|^{2})}$
(1)

where
$W_{i}=j(\omega-\omega_{i}+\gamma_{i}+\gamma^{{}^{\prime}}_{i}),(i=1,2)$,
$\omega_{i}$ is resonant frequency corresponding to the $i^{th}$ mode,
$\gamma_{i}$ is the radiation loss rate due to the interaction between the
$i^{th}$ mode and the outgoing wave, $\gamma^{{}^{\prime}}_{i}$ is the
intrinsic loss rate associated with the $i^{th}$ mode, and $\gamma_{0}$ is the
coupling between the modes. The first two terms of the numerator of the second
part of Eq. 1 describe the responses of two modes independently after coupling
modulation, whereas the last term $\sqrt{\gamma_{1}\gamma_{2}}$ describes the
responses of the cross-coupling between the two modes after coupling
modulation. For the metamaterial absorber, the radiation losses for the two
modes are $\gamma_{1}=6.42$ THz and $\gamma_{2}=1.86$ THz. The intrinsic loss
rates of the two modes are $\gamma^{{}^{\prime}}_{1}=6.4$ THz and
$\gamma^{{}^{\prime}}_{2}=3.0$ THz. The coupling between modes occurs at
$\omega_{0}=6.53$ THz. The resonant frequencies of the two modes are
$\omega_{1}=65.6$ THz and $\omega_{2}=60.73$ THz.

The fitting parameters match well with the expected parameters such as the
spectral position of resonances, bandwidth of the individual resonance, etc.
Now, it is clear that the asymmetric profile is caused by the interference of
mainly the CM resonance and the GM resonance. The response of the individual
resonance using the same fitting parameters as for the fitting of the
reflectance spectrum [see in Fig. 8(a)] is plotted in Fig. 8(b).

##  4 Conclusions

We have proposed a very simple design of a metamaterial absorber with band
selective absorptivity/emissivity that is suitable for large area fabrication.
The origin of absorption as well as the asymmetric nature of the absorption
profile has been understood. The absorption is caused due to the simultaneous
excitation of the cavity and guided mode resonances in the structures whereas
the asymmetric profile is caused due to the interference of the continuum
mode, discrete modes (guided mode and Wood’s anomalies). The advantage of this
design is that it can be easily implemented for fabrication over large areas
by separating the structuring and deposition processes and making them
sequential, as well as avoiding expensive and complex lift-off or etching
processes. Our approach can be easily adopted for rapid productions of large
areas fabrication on the industrial level. The spectral position of the
resonance can be tuned without structural size and shape modification just by
changing the gap between the gold layers on the top of photoresist pillar and
the ZnS layer at the substrate.

##  5 Acknowledgement

RK thanks Council of Scientific $\&$ Industrial Research (CSIR), India for
fellowship.

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