\section{Introduction}

Stencil computations are widely implemented in many numerical algorithms that involve structured grids. In acoustic field simulation based on two- or three-dimensional transient wave equations, the finite difference time domain method is a standard approach leading to stencil operations \cite{Hamilton2015}. In an explicit time-stepping finite difference scheme, a solution value at each point in a time-space grid is calculated using a linear combination of values at its spatial neighbors from previous time steps. Among such schemes the main attention in the literature has been given to two-step schemes (which operate over three time levels $t_{k+1}=(k+1)\tau$, $t_{k}=k\tau$ and $t_{k-1}=(k-1)\tau$ where $\tau$ is a fixed time increment). They have been intensively studied and reviewed in many articles and books (see, e.g., $^{2-7}$).

Explicit two-step numerical schemes for the scalar wave equation can also be devised based on spherical means representations as it is done in \cite{Alpert2000} where an integral evolution formula with three time levels is derived. For the 2D case, the evolution formula has been implemented in \cite{Li2004} and \cite{Hagstrom2015} using piecewise polynomial interpolation in 2D mesh cells and numerical integration.

The two-step schemes pose some challenges when imposing the initial conditions. To calculate the value of a sought solution $u$ at the first time level $t_1$, one needs values of $u$ from time levels $t_0$ and $t_{-1}$. The initial condition for $u$ provides the required values at $t_0$. However, $u({\bf x},t_{-1})$ must be inferred from the other initial condition. So, there is a need for transforming the two-step scheme at the first time step to a one-step form which has been discussed, e.g., in \cite{Strikwerda2004} and \cite{Langtangen2017}. A conventional approach for the 2D case (described, e.g., in \cite{Langtangen2017}) uses the central difference in time for approximating the initial condition for $v=\partial u / \partial t$.  For any point ${\bf x}$ in $R^2$, the value of $u({\bf x},t_{-1})$ is inferred as $u({\bf x},t_1)-2 \tau v({\bf x},0)$. Even though this approach is attractive due to its simplicity, it is worthwhile to consider the numerical schemes for both the first time step and the next steps derived using a unified approach.

The present work focuses on building explicit time-stepping stencil computation schemes for the transient 2D acoustic (scalar) wave equation using spherical means formulas including Poisson's formula \cite{Evans1998} and a similar integral formula involving three time levels \cite{Alpert2000} that are described in section 2. A general form of the implemented time-stepping algorithm is presented in section 3 where two different integral expressions are given for the first time-step and for the next steps. Sections 4 and 5 consider some known results regarding polynomial interpolation on stencils and exact integration needed for deriving explicit stencil computation schemes from these integral expressions. Particular numerical schemes for five-, nine- and 13-point stencils are obtained in section 6. For each scheme, two separate expressions are derived: 1) a one-step expression for the first time-step through the stencil interpolated initial conditions; 2) a two-step expression for the second and next time-steps. All the obtained stencil expressions for the first time-step have not been previously presented in the literature. A derived two-step expression for the nine-point stencil is also new. In contrast, the obtained two-step expressions for five- and 13-point stencils coincide with the corresponding finite-difference stencil expressions. It is shown by simulation that the derived numerical schemes can significantly improve accuracy of stencil computations in comparison with conventional approaches.

\section{Representation Formulas}
Consider a Cauchy problem for the transient 2D scalar wave equation
\begin{gather}  \label{we1}
\frac{\partial^2 u}{\partial t} -
c^2 \left( \frac{\partial^2 u}{\partial x_1^2}
+ \frac{\partial^2 u}{\partial x_2^2}
\right)=0,
\quad
u=u({\bf x},t), \quad {\bf x}=(x_1,x_2) \mbox{ in } \mathbb{R}^2,
\\
u|_{t=0}=u_0({\bf x}), \quad \frac{\partial u}{\partial t}|_{t=0} =v_0({\bf x}).
\end{gather}
Its solution is given by the representation formula which is often named Poisson's formula for the 2D wave  equation \cite{Evans1998}:
\begin{equation}
u({\bf x},t)=\frac{1}{2\pi c t^2} \int_{|{\bf y}-{\bf x}|^2<t^2}
\frac{t u_0({\bf y})+t \ \nabla u_0({\bf y}) \cdot ({\bf y} - {\bf x}) +t^2 v_0({\bf y}) }{\sqrt{c^2 t^2-|{\bf y}-{\bf x}|^2}} d y_1 d y_2
\label{RF1}
\end{equation}
where $\nabla$ denotes the gradient operator and ${\bf y}=(y_1,y_2) \in \mathbb{R}^2$ is a variable of integration.

Rewriting the previous integral on the unit disk, we get
\begin{equation}  \label{RF2}
u({\bf x},t)=\frac{1}{2\pi} \int_{|{\bf z}|<1}
\frac{u_0({\bf x}+c t {\bf z})+c t \nabla u_0({\bf x}+c t {\bf z}) \cdot {\bf z} }{\sqrt{1-|{\bf z}|^2}} d z_1 d z_2
+\frac{t}{2\pi} \int_{|{\bf z}|<1}
\frac{v_0({\bf x}+c t {\bf z})}{\sqrt{1-|{\bf z}|^2}} d z_1 d z_2
\end{equation}
where ${\bf z}=(z_1,z_2) \in \mathbb{R}^2$ is a new variable of integration.

Formulas \eqref{RF1} and \eqref{RF2} are also valid for negative $t$ which can be proved based on the time reversal property of the wave equation.
By substituting $-t$ for $t$ in formula \eqref{RF2} and changing ${\bf z}$ to $-{\bf z}$ inside its integrals, we obtain the following analog of that formula for negative time:
\begin{equation}  \label{RF3}
u({\bf x},-t)=\frac{1}{2\pi} \int_{|{\bf z}|<1}
\frac{u_0({\bf x}+c t {\bf z})+c t \nabla u_0({\bf x}+c t {\bf z}) \cdot {\bf z} }{\sqrt{1-|{\bf z}|^2}} d z_1 d z_2  \\
-\frac{t}{2\pi} \int_{|{\bf z}|<1}
\frac{v_0({\bf x}+c t {\bf z})}{\sqrt{1-|{\bf z}|^2}} d z_1 d z_2.
\end{equation}
The only difference between the right-hand sides of \eqref{RF2} and \eqref{RF3} is the opposite signs of the second term. So, one can eliminate this term by summing \eqref{RF2} and \eqref{RF3}. Shifting the initial moment in the resulting formula from $t=0$ to $t=t_*$, we obtain the following expression involving three time points with a time increment $\tau$:
\begin{equation}
u({\bf x},t_* + \tau)+u({\bf x},t_* - \tau)=\frac{1}{\pi} \int_{|{\bf z}|<1}
\frac{u({\bf x}+c t {\bf z},t_*)+c t \nabla u({\bf x}+c t {\bf z},t_*) \cdot {\bf z} }{\sqrt{1-|{\bf z}|^2}} d z_1 d z_2
\label{RF4}
\end{equation}
where the right-hand side does not include the time derivative $v$. The same representation formula (in a different form) has been derived in \cite{Alpert2000} without using Poisson's formula.

Both formulas \eqref{RF2} and \eqref{RF4} will be used below to build a time-marching stencil computation algorithm for the wave equation.
\section{An Integral Time-Stepping Algorithm}

Consider a uniform time grid $\{t_0=0, t_1=\tau,\ldots, t_k=k\tau,\ldots\}$ where $\tau$ is a fixed time-step. Denote by $u_k({\bf x})$ the restriction of $u({\bf x},t)$ to a moment $t=t_k$.
Next, denote by $A({\bf x},\tau)$ and $B({\bf x},\tau)$ the following integral operators acting on continuous functions defined in $\mathbb{R}^2$:
\begin{equation}
A({\bf x},\tau)f(\cdot)=\frac{1}{2 \pi} \int_{|{\bf z}|<1}
\frac{f({\bf x}+c \tau {\bf z})+c \tau \nabla f({\bf x}+c \tau {\bf z}) \cdot {\bf z} }{\sqrt{1-|{\bf z}|^2}} d z_1 d z_2,
\label{Op1}
\end{equation}
\begin{equation}
B({\bf x},\tau)f (\cdot)=\frac{\tau}{2 \pi} \int_{|{\bf z}|<1}
\frac{f({\bf x}+c \tau {\bf z})}{\sqrt{1-|{\bf z}|^2}} d z_1 d z_2
\label{Op2}
\end{equation}

The time-stepping algorithm proposed here consists of two procedures based on the representation formulas \eqref{RF2} and \eqref{RF4}:

1) \textit{The procedure for the first time-step} which, according to \eqref{RF2}, calculates $u_1({\bf x})$ as
\begin{equation}
u_1({\bf x})=A({\bf x},\tau)u_0(\cdot)+B({\bf x},\tau)v_0(\cdot), {\bf x} \in \mathbb{R}^2
\label{A1}
\end{equation}

2) \textit{The procedure for the second and next time-steps} which, using \eqref{RF4} for $t_*=t_k$, calculates $u_{k+1}({\bf x})$  as
\begin{equation}
u_{k+1}({\bf x})=2 A({\bf x},\tau)u_k(\cdot)-u_{k-1}({\bf x}), {\bf x} \in \mathbb{R}^2, k=1,2,...
\label{A2}
\end{equation}

While formula \eqref{A2} involves three time levels (two time-steps), formula \eqref{A1} for the first time-step incorporates only two time levels (one time-step) without using any finite difference approximation of the time derivative.

\section{Using Polynomial Interpolation on Stencils}

Consider a two-dimensional uniform Cartesian grid $\{(i_1h,i_2h)\}$ where $i_1$ and $i_2$ are integers, and $h$ is the grid spacing in both directions $x_1$ and $x_2$. Suppose that the evaluation point ${\bf x}$ in formulas \eqref{A1} and \eqref{A2} is a grid point ${\bf x}_{ij}=(ih,jh)$. Our intention is to choose a stencil in the Cartesian grid  and reduce the integral formulas \eqref{A1} and \eqref{A2} to linear combinations of the stencil node values of $u_k({\bf x})\, (k=0,1,2,\ldots)$ and $v_0({\bf x})$. Such a reduction will be done by using polynomial interpolation.

Assume that a particular stencil with $m$ nodes is chosen for polynomial interpolation. The corresponding index set $\{(q_1,q_2)\}$ is denoted by $Q_m$. The stencil index components $q_1$ and $q_2$ are numbered relative to the referencing point located at the evaluation point ${\bf x}_{ij}$. So, polynomial interpolation in a neighborhood of the evaluation point ${\bf x}_{ij}$ will be carried out using interpolation points
\begin{equation}
{\bf x}_{i,j}+{\bf x}_{q_1,q_2}={\bf x}_{i+q_1,j+q_2}, \quad (q_1,q_2) \in Q_m.
\end{equation}

Following \cite{McKinney1972} we associate with $Q_m$ a set of $m$ distinct bivariate monomials
\begin{equation} \label{McKinney}
\mathcal{M}_m= \{x_1^{\alpha(q_1)} x_2^{\alpha(q_2)}, (q_1,q_2) \in Q_m \}
\end{equation}
where
\begin{equation}  \label{alpha}
\alpha(q)=
\begin{dcases}
2|q|-1  & \text{if } q < 0  \\
2q & \text{if } q\geq 0.
\end{dcases}
\end{equation}
The above function has a unique inverse function
\begin{equation}  \label{invMcKinney}
q(\alpha)=(-1)^\alpha \left[\frac{\alpha+1}{2}\right]
\end{equation}
where $[\cdot]$ is the whole part function.
So, for each index value $(q_1,q_2)) \in Q_m$ there exists a unique monomial from $\mathcal{M}_m$ and vice versa.

Consider a polynomial space $\mathcal{P}_m$ spanned by $\mathcal{M}_m$. We will use only those stencils for which the Lagrange interpolation problem is unisolvent in $\mathcal{P}_m$ (see \cite{Gasca2012}). In this case, there exists a Lagrange basis for $\mathcal{P}_m$ that can be built as described below.

Suppose that there is an ordering imposed on the monomials in $\mathcal{M}_m$
\begin{equation} \label{monomials}
\{\mu_1({\bf x}), \ldots, \mu_m({\bf x})\}.
\end{equation}
The corresponding stencil nodes are numbered accordingly using \eqref{McKinney}-\eqref{invMcKinney}:
\begin{equation}     \label{nodes}
\{{\bf x}_{(1)}, ..., {\bf x}_{(m)}\}.
\end{equation}
Thus, one can compute the following matrix  \cite{Gasca2012}:
\begin{equation} \label{matrix1}
D=\left[\mu_s({\bf x}_{(r)})\right]_{m \times m}
\end{equation}
If this matrix is non-singular, i.e., $\det(D) \ne 0$, which means that the Lagrange basis exists, then the inverse matrix
\begin{equation} \label{matrix2}
C=\left[c_{sr}\right]_{m \times m}=D^{-1}
\end{equation}
can be calculated. Its components are instrumental in expressing the Lagrange basis functions through the chosen set of monomials:
\begin{equation} \label{Lagrange}
L_s({\bf x})=\sum_{r=1}^m c_{sr} \mu_r ({\bf x}), \quad s=1,...,m.
\end{equation}
Once the Lagrange basis is obtained, we again need two indexes to denote the Lagrange basis functions in accordance with the two index notation for grid points. The sequence \eqref{monomials} defines a relationship $s=g(\alpha_1,\alpha_2)$ between the ordinal number $s$ and monomial exponents $(\alpha_1,\alpha_2)$. Then, the resulting relationship between an index pair $(q_1,q_2)$ and the corresponding ordinal number $s$ is given by the following expression:
\begin{equation} \label{Lagrange3}
s=\gamma(q_1, q_2)=g(\alpha(q_1),\alpha(q_2)).
\end{equation}
where $\alpha(\cdot)$ is specified in \eqref{alpha}.
Therefore, by introducing a new (two index) notation for the Lagrange basis functions
\begin{equation} \label{Lagrange4}
\phi_{q_1,q_2}({\bf x})=L_{\gamma(q_1,q_2)}({\bf x}),
\end{equation}
we get the interpolation formula for a continuous function $f({\bf x})$ in the form
\begin{equation} \label{Lagrange5}
\tilde{f}({\bf x})=\sum_{(q_1, \, q_2) \in Q_m} f_{i+q_1,j+q_2} \phi_{q_1 q_2} ({\bf x})
\end{equation}

Even though different sets of monomials and their sequences can be employed for building Lagrange bases, we will use a particular method of monomial ordering that is described below.

Denote by $\mathcal{M}$ the set of all monomials $x_1^{\alpha_1} x_2^{\alpha_2}$ where $\alpha_1$ and  $\alpha_2$  are natural numbers including $0$. For each monomial $x_1^{\alpha_1} x_2^{\alpha_2}$, the corresponding ordinal number $s$ will be assigned using the following function:
\begin{equation}  \label{g}
s=g(\alpha_1,\alpha_2 )=\frac{(\alpha_1+\alpha_2)(\alpha_1+\alpha_2+1)}{2}
+\begin{dcases}
\alpha_1 - \alpha_2 & \text{if } \alpha_2 < \alpha_1, \\
\alpha_2 - \alpha_1 + 1   & \text{if } \alpha_2 \geq \alpha_1.
\end{dcases}
\end{equation}
It is easy to prove that the function \eqref{g} provides a one-to-one correspondence between $\mathcal{M}$ and the set of all positive natural numbers with the usual ordering. The order induced by \eqref{g} uses the total degree as the first sorting parameter (similarly to the more common graded lexicographic order) while the difference between individual degrees with the same total degree is used as the next sorting variable.

 The set $\mathcal{M}$ endowed with the order induced by \eqref{g} will be denoted by $\mathcal{M}^*$. The initial segment of $\mathcal{M}^*$ with $m$ members is denoted below by $\mathcal{M}_{\le m}^*$.  It will be shown in section 6 that such ordered monomial sets  play a useful role in building particular numerical schemes.

\section{Calculating the Integrals}
Let ${\bf x}={\bf x}_{ij}$ in the integral operators \eqref{Op1} and \eqref{Op2}. Without loss of generality, assume that the origin of the 2D coordinate system is located at ${\bf x}_{ij}$ which can be achieved by a parallel translation of coordinates. Let $f({\bf y})$ appearing in \eqref{Op1} and \eqref{Op2}  be a monomial in local scaled variables $y_1/h$ and $y_2/h$:
\begin{equation} \label{monomial2}
\mu({\bf y})=\left(\frac{y_1}{h}\right)^{\alpha_1} \left(\frac{y_2}{h}\right)^{\alpha_2} , \quad \alpha_1,\alpha_2 = 0,1, \ldots.
\end{equation}
Then the integrals in \eqref{Op1} and \eqref{Op2}  can be exactly calculated and expressed through the Courant number
\begin{equation} \label{Courant}
\lambda=\frac{c\tau}{h}
\end{equation}
Indeed, using the table of integrals of Gradshteyn \& Ryzhik \cite{Gradshteyn1965}, expressions \eqref{Op1} and \eqref{Op2} are reduced to the following exact values:
\begin{equation} \label{exact1}
A({\bf x}_{ij},\tau)\mu(\cdot)=0, \, B({\bf x}_{ij},\tau) \mu(\cdot)=0 \, \mbox{  if $\alpha_1$ or $\alpha_2$ are non-negative odd integers},
\end{equation}
\begin{equation} \label{exact2}
\begin{aligned}
A({\bf x}_{ij},\tau) \mu(\cdot)=\frac{(\alpha_1-1)!! (\alpha_2-1)!!}{(\alpha_1+\alpha_2-1)!!} \lambda^{\alpha_1+\alpha_2}, \\
B({\bf x}_{ij},\tau) \mu(\cdot)=\frac{\tau}{\alpha_1+\alpha_2+1} A({\bf x}_{ij},\tau) \mu(\cdot) \\
\mbox{if $\alpha_1$ and $\alpha_2$ are both non-negative even integers},
\end{aligned}
\end{equation}
where $(\cdot)!!$ is the double factorial. It is assumed that $(-1)!!=1,0!!=1$.

The above formulas allow one to exactly calculate integrals \eqref{Op1} and \eqref{Op2} when $f(\cdot)$ is a polynomial from the Lagrange basis (see section 4).
\section{Particular Explicit Two-Step Schemes}
Now we can start building some numerical schemes by transforming the procedures \eqref{A1} and \eqref{A2} into algebraic expressions. All the functions $u_0({\bf x}), v_0({\bf x})$ and $u_k({\bf x}), k=1,2,...$ included in these procedures will be interpolated in a stencil's center neighborhood using the same stencil nodes. The following standard notations for grid values of the solution and initial conditions will be used in the computation schemes:
\begin{equation} \label{grid0}
u_{ij}^0=u({\bf x}_{ij},0),v_{ij}^0 =v({\bf x}_{ij},0),
\end{equation}
\begin{equation} \label{grid1}
u_{ij}^k = u({\bf x}_{ij},k\tau), k=1,2,\ldots.
\end{equation}
\subsection{The five-point stencil}
Consider building the Lagrange basis for the space of complete second degree polynomials. In this case, the monomial basis sequence ordered according to \eqref{g} is as follows:
\begin{equation} \label{mono6}
\mathcal{M}_{\le 6}^*=\lbrace 1,x_1,x_2,x_1 x_2,x_1^2,x_2^2 \rbrace
\end{equation}
The corresponding interpolation stencil node sequence will be written according to \eqref{invMcKinney} as
\begin{equation*} \label{index6}
\lbrace(0,0),(-h,0),(0,-h),(-h,-h),(h,0),(0,h)\rbrace
\end{equation*}
The matrix $D$ for this stencil is non-singular with $\det(D)=4 h^8$. As a result, we get the Lagrange basis as
\begin{equation} \label{LB6}
\begin{split}
\phi_{0,0}=L_1 &= 1+\frac{x_1 x_2}{h^2}-\frac{x_2^2}{h^2}-\frac{x_1^2}{h^2}, \quad
\phi_{-1,0}=L_2=-\frac{x_1}{2h}-\frac{x_1 x_2}{h^2}+\frac{x_1^2}{2h^2}, \\
\phi_{0,-1}&=L_3 =-\frac{x_2}{2h}-\frac{x_1 x_2}{h^2}+\frac{x_2^2}{2h^2},  \quad
\phi_{-1,-1}=L_4 = \frac{x_1 x_2}{h^2}, \\
&\phi_{1,0}=L_5 =\frac{x_1}{2 h}+\frac{x_1^2}{2 h^2}, \quad
\phi_{0,1}=L_6 =\frac{x_2}{2 h}+\frac{x_2^2}{2 h^2}.
\end{split}
\end{equation}
One can see that the Lagrange basis \eqref{LB6} includes monomials in the scaled variables $x_1/h$ and $x_2/h$ with coefficients independent of $h$.

Using formulas from section 5 and notation \eqref{Lagrange4} we get
\begin{eqnarray} \label{A6}
A({\bf x}_{ij},\tau)\phi_{00}(\cdot)=1-2 \lambda ^2,  \quad A({\bf x}_{ij},\tau)\phi_{-1,-1}(\cdot)=0, \nonumber \\
A({\bf x}_{ij},\tau)\phi_{\pm 1,0}(\cdot)=A({\bf x}_{ij},\tau)\phi_{0, \pm 1}(\cdot)=\frac{1}{2} \lambda ^2,
\end{eqnarray}
\begin{eqnarray} \label{B6}
B({\bf x}_{ij},\tau)\phi_{0,0}(\cdot)=\tau (1-\frac{2}{3}\lambda^2),  \quad B({\bf x}_{ij},\tau)\phi_{-1,-1}(\cdot)=0, \nonumber \\
B({\bf x}_{ij},\tau)\phi_{\pm 1,0}(\cdot)=B({\bf x}_{ij},\tau)\phi_{0, \pm 1}(\cdot)=\frac{\tau}{6} \lambda ^2,
\end{eqnarray}
So, all coefficients for node $(-h,-h)$ disappear and the corresponding computational scheme contains only 5 spatial points which is shown in Figure 1.
\begin{figure}[ht]
	\centering
	\begin{tikzpicture}
	\begin{axis}[xmin=-2,   xmax=2,
	ymin=-2,   ymax=2,
	extra x ticks={-2,-1,0,1,2},
	extra y ticks={-2,-1,0,1,2},
	extra tick style={grid=major}]
	\addplot [only marks,mark=*, mark size=3pt]
	coordinates
	{(0,0) (-1,0) (1,0) (0,-1) (0,1)};
	\end{axis}
	\end{tikzpicture}
	\caption{Index set for the five-point numerical scheme}
	\label{fig:Q5}
\end{figure}
\subsubsection{A new first time-step expression}
As a result, the proposed five-point numerical scheme is as follows:

1) \textit{for the first time-step}

\begin{multline} \label{comp5.1}
u_{ij}^1 = u_{ij}^0+\tau v_{ij}^0
+\frac{\lambda^2}{2} \left( u_{i-1,j}^0+u_{i+1,j}^0+u_{i,j-1}^0+u_{i,j+1}^0-4 u_{ij}^0 \right) \\
+\frac{\tau\lambda^2}{6} \left(v_{i-1,j}^0+v_{i+1,j}^0+v_{i,j-1}^0
+v_{i,j+1}^0-4 v_{ij}^0 \right);
\end{multline}
2) \textit{for the second and next time-steps}
\begin{equation} \label{comp5.2}
u_{ij}^{k+1} = 2 u_{ij}^k - u_{ij}^{k-1}
+\lambda^2 \left( u_{i-1,j}^k+u_{i+1,j}^k+u_{i,j-1}^k+u_{i,j+1}^k-4 u_{ij}^k \right) ,\quad k=1,2,\ldots.
\end{equation}

A conventional approach for the five-point stencil (see, e.g., \cite{Langtangen2017}) uses the same procedure \eqref{comp5.2} for the second and next time-steps. However, for the first time-step,  this approach uses another formula (rather than \eqref{comp5.1}) based on the central difference in time for approximating the initial condition for $v$ combined with \eqref{comp5.2} for $k=0$:
\begin{equation} \label{comp5.1a}
u_{ij}^1 = u_{ij}^0+\tau v_{ij}^0
+\frac{\lambda^2}{2} \left( u_{i-1,j}^0+u_{i+1,j}^0+u_{i,j-1}^0+u_{i,j+1}^0-4 u_{ij}^0 \right).
\end{equation}
Comparing \eqref{comp5.1} and \eqref{comp5.1a} one can see that the first three terms of the right-hand parts of these equations coincide, but the forth term present in the right-hand side of \eqref{comp5.1} is absent in \eqref{comp5.1a}. Thus, the difference between these two first time-step expressions depends on properties of $v_0({\bf x})$. If, for example, $v_0({\bf x})$ is a linear function of spatial coordinate, then there is no difference between  \eqref{comp5.1} and \eqref{comp5.1a}. However, for more general cases encountered in practice, the difference may exist and can be influential from the accuracy point of view which is shown in subsection 6.1.2.

The stability condition for numerical schemes utilizing \eqref{comp5.2} is well known from von Neumann stability analysis \cite{Strikwerda2004}:
\begin{equation} \label{stability5}
\lambda \le \lambda_{\max}=\frac{\sqrt 2}{2}.
\end{equation}
\subsubsection{Numerical simulation}
To make a numerical comparison of both approaches, we use the initial and boundary conditions corresponding to a standing wave exact solution:
\begin{equation} \label{exact sol}
u_e(x_1,x_2,t)=\sin(2 \pi x_1) \sin(2 \pi x_2) \sin(2\sqrt2 \pi c t).
\end{equation}
This solution creates the following pair of initial conditions for the numerical simulation:
\begin{equation} \label{Exact IC}
u_0(x_1,x_2)=0, \quad v_0(x_1,x_2)=2\sqrt2\pi c \ \sin(2 \pi x_1) \sin(2 \pi x_2).
\end{equation}
We consider the unit square $\Omega=[0,1]^2$ as a space region for the numerical solution and apply boundary condition $u=0$ on $\partial\Omega$ generated by \eqref{exact sol}. In addition, we assume that $c=1$. Let $n_t$ be the number of time-steps and $n \geq 2$ be the space discretization number related to $h$ by equation $nh=1$. The above numerical schemes have been employed over a spatial index set $\{(i,j), 0<i,j<n\}$
using grid boundary conditions
\begin{equation*}
u_{0,j}^k=u_{n,j}^k=u_{i,0}^k=u_{i,n}^k=0, \quad 0 \leq i,j \leq n,
\quad k \geq 0.
\end{equation*}
The accuracy of both numerical schemes have been estimated using the relative $L^2$ error defined as
\begin{equation} \label{error}
E(n,n_t)=\left(\frac{\sum_{k=1}^{n_t}\sum_{i=0}^{n}\sum_{j=0}^{n} \left(u_{ij}^k-u_e(ih,jh,k\tau)\right)^2}{\sum_{k=1}^{n_t}\sum_{i=0}^{n}\sum_{j=0}^{n} \left(u_e(ih,jh,k\tau)\right)^2}\right)^{1/2}
\end{equation}

The relative $L^2$ errors for the proposed new five-point scheme \eqref{comp5.1}$\&$\eqref{comp5.2} (based on Poisson's formula) and the conventional one \eqref{comp5.1a}$\&$\eqref{comp5.2} are denoted by $E_{P5}(n,n_t)$ and $E_{C5}(n,n_t)$, respectively. Calculated values of $E_{P5}(n,n_t)$, $E_{C5}(n,n_t)$ and their ratio for different combinations of $n$, $n_t$  and $\lambda=0.707$ are presented in table 1 below. For every value of $n$ used, three values of $n_t$ are considered: $n_t=1$ (to check errors after the first step), $n_t=n$ and $n_t=2n$ (to demonstrate the error accumulation process).
\begin{table}[h]
	\centering
	\ra{1.3}
	\caption{Wave \eqref{exact sol} simulations using the five-point stencil.}
	\begin{tabular}{@{}cccccc@{}}\toprule
		$n \quad$ & $n_t \quad$ & $\lambda$ & $E_{P5} \quad$ & $E_{C5}$  \\   \midrule
		10 &  1 & $0.707$ & $9.0843\cdot10^{-4}$   & $6.8938\cdot10^{-2}$   \\
		10 & 10 & $0.707$ & $9.1540\cdot10^{-4}$   & $6.8945\cdot10^{-2}$   \\
		10 & 20 & $0.707$ & $9.1604\cdot10^{-4}$   & $6.8945\cdot10^{-2}$   \\ \midrule
		20 &  1 & $0.707$ & $5.4767\cdot10^{-5}$   & $1.6636\cdot10^{-2}$   \\
		20 & 20 & $0.707$ & $5.6800\cdot10^{-5}$   & $1.6638\cdot10^{-2}$   \\
		20 & 40 & $0.707$ & $5.7372\cdot10^{-5}$   & $1.6638\cdot10^{-2}$   \\ \midrule
		40 &  1 & $0.707$ & $3.3924\cdot10^{-6}$   & $4.1230\cdot10^{-3}$   \\
		40 & 40 & $0.707$ & $4.0331\cdot10^{-6}$   & $4.1234\cdot10^{-3}$   \\
		40 & 80 & $0.707$ & $4.4928\cdot10^{-6}$   & $4.1234\cdot10^{-3}$   \\ \midrule
		80 &  1 & $0.707$ & $2.1158\cdot10^{-7}$   & $1.0285\cdot10^{-3}$   \\
		80 & 80 & $0.707$ & $4.3820\cdot10^{-7}$   & $1.0286\cdot10^{-3}$   \\
		80 &160 & $0.707$ & $6.5824\cdot10^{-7}$   & $1.0286\cdot10^{-3}$   \\ \bottomrule
	\end{tabular}
\end{table}
Table 1 demonstrated a much higher accuracy of the new scheme in comparison with the conventional one (with the error ratio $E_{C5}/E_{P5}$ exceeding $10^3$ for more dense grids).  Observed increases of the relative $L^2$ errors from $n_t=n$ to $n_t=2n$ are negligible (less than $10^{-6}$).
\subsection{A nine-point square stencil}
Consider the initial segment $\mathcal{M}_{\le 11}^*$ of $\mathcal{M}^*$ with the last member equal to $x_1^2 x_2^2$:
\begin{equation} \label{monomials11}
\{1,x_1,x_2,x_1 x_2,x_1^2,x_2^2,x_1^2 x_2,x_1 x_2^2,x_1^3, x_2^3,x_1^2 x_2^2\}
\end{equation}
This set includes all the monomials of total degree $\le 3$ and one monomial of total degree $=4$, and is the minimal initial segment of $\mathcal{M}^*$ that includes the bivariate tensor-product of the monomial bases for the second degree polynomials in each spatial coordinate. The corresponding index set $Q_{11}$ is shown in Figure 2 where both solid dots and empty circles denote interpolation points.
\begin{figure}[b!]
	\centering
	\begin{tikzpicture}
	\begin{axis}[xmin=-3,   xmax=2,
	ymin=-3,   ymax=2,
	extra x ticks={-3,-2,-1,0,1,2},
	extra y ticks={-3,-2,-1,0,1,2},
	extra tick style={grid=major}]
	\addplot [only marks,mark=*, mark size=3pt]
	coordinates {(0,0) (-1,0) (0,-1) (-1,-1) (1,0) (0,1) (1,-1) (-1,1) (1,1)};
	\addplot [only marks,mark=o, mark size=3pt]
	coordinates {(-2,0) (0,-2)};
	\end{axis}
	\end{tikzpicture}
	\caption{Index set for the 11-point interpolation stencil}
	\label{fig:Q9}
\end{figure}
The corresponding Lagrange basis can be easily calculated using matrix \eqref{matrix1} and presented similar to \eqref{LB6}. However, it is preferable to avoid presenting long expressions that include 11 monomials and display only terms that will be used later.  Denote by $\phi^e_{q_1,q_2}$ a part of $\phi_{q_1,q_2}$ that includes all the monomial terms with even exponents in both coordinates. Then we get
\begin{equation} \label{LB11}
\begin{split}
\phi^e_{0,0}=1-\frac{x_1^2}{h^2}-\frac{x_2^2}{h^2}+\frac{x_1^2 x_2^2}{h^4}, \quad &\phi^e_{-2,0}=\phi^e_{0,-2}=0, \\
\phi^e_{\pm 1,0}=\frac{1}{2}\left(\frac{x_1^2}{h^2}- \frac{x_1^2 x_2^2}{h^4} \right), \quad
\phi^e_{0,\pm 1}=\frac{1}{2}&\left(\frac{x_2^2}{h^2}- \frac{x_1^2 x_2^2}{h^4} \right), \quad \phi^e_{\pm 1,\pm 1}=\frac{1}{4}\frac{x_1^2 x_2^2}{h^4}.
\end{split}
\end{equation}
\subsubsection{A new nine-point time-stepping scheme}
It follows from the results \eqref{exact1} and \eqref{exact2} of section 5 that functions $\phi^e_{q_1,q_2}$ rather than the complete Lagrange basis will be used in building numerical schemes. Therefore, two points (-2,0) and (0.-2) (presented by empty circles in Figure 2) will disappear in the corresponding numerical scheme. The remaining nodes (solid circles in Figure 2) create the nine-point square-shaped computational stencil. Using \eqref{exact1}-\eqref{exact2} we obtain:
\begin{equation} \label{A11}
\begin{split}
A({\bf x}_{ij},\tau)\phi_{00}(\cdot)&=1-2 \lambda ^2+\frac{\lambda ^4}{3},  \quad A({\bf x}_{ij},\tau)\phi_{\pm 1,\pm 1}(\cdot)= \frac{\lambda ^4}{12}, \\
A({\bf x}_{ij},\tau)&\phi_{\pm 1,0}(\cdot)= A({\bf x}_{ij},\tau)\phi_{0,\pm 1}(\cdot)=\frac{\lambda ^2}{2}-\frac{\lambda ^4}{6},
\end{split}
\end{equation}
\begin{equation} \label{B11}
\begin{split}
B({\bf x}_{ij},\tau)\phi_{00}(\cdot)&=\tau\left(1-\frac{2\lambda ^2}{3}+\frac{\lambda ^4}{15}\right),  \quad B({\bf x}_{ij},\tau)\phi_{\pm 1,\pm 1}(\cdot)= \frac{\tau\lambda ^4}{60}, \\
B({\bf x}_{ij},\tau)&\phi_{\pm 1,0}(\cdot)= B({\bf x}_{ij},\tau)\phi_{0,\pm 1}(\cdot)=\frac{\tau\lambda ^2}{6} \left(1-\frac{\lambda ^2}{5}\right).
\end{split}
\end{equation}
Now we can use \eqref{A1} and \eqref{A2} to build a new nine-point numerical scheme similar to  \eqref{comp5.1}-\eqref{comp5.2}. However, to avoid writing long expressions, some additional notations will be needed:
\begin{equation} \label{notations2}
\begin{split}
\delta^k_{ij}(q_1,q_2)=u^k_{i+q_1,j+q_2}+u^k_{i-q_2,j+q_1}
+u^k_{i1-q_1,j-q_2}
& +u^k_{i+q_2,j-q_1}-4 u^k_{i,j},  \\
\epsilon^0_{ij}(q_1,q_2)=v^0_{i+q_1,j+q_2}+v^0_{i-q_2,j+q_1}
+v^0_{i-q_1,j-q_2}
& +v^0_{i+q_2,j-q_1}-4 v^0_{i,j}, \\
&k=0,1,2,\ldots.
\end{split}
\end{equation}
Thus, the following time-stepping numerical scheme is derived:

1) \textit{for the first time-step}
\begin{equation} \label{comp9.1}
\begin{split}
u_{ij}^1 = u_{ij}^0+\tau v_{ij}^0+\frac{\lambda^2}{2} \Bigg[ \left(1-\frac{\lambda^2}{3}\right) \delta^0_{ij}(1,0)+\frac{\lambda^2}{6}\delta^0_{ij}(1,1) \Bigg] \\
+\frac{\tau\lambda^2}{6}\Bigg[\left(1-\frac{\lambda^2}{5}\right) \epsilon^0_{ij}(1,0)+\frac{\lambda^2}{10}\epsilon^0_{ij}(1,1)
\Bigg];
\end{split}
\end{equation}

2) \textit{for the second and next time-steps}
\begin{multline} \label{comp9.2}
u_{ij}^{k+1} = 2 u_{ij}^k - u_{ij}^{k-1}+\lambda^2 \Bigg[ \left(1-\frac{\lambda^2}{3}\right) \delta^k_{ij}(1,0)+\frac{\lambda^2}{6}\delta^k_{ij}(1,1)\Bigg],\quad k=1,2, \ldots.
\end{multline}
Using the von Neumann stability analysis method (see, e.g., \cite{Strikwerda2004}), we obtain the stability condition for the above numerical scheme as
\begin{equation} \label{stability9}
\lambda \le \lambda_{\max}=\frac{\sqrt {3-\sqrt 3}}{2} \approx 0.796.
\end{equation}

To our knowledge this numerical scheme has not been presented previously in the literature. A conventional explicit nine-point square-shaped scheme (dubbed the isotropic scheme (see, e.g., \cite{Trefethen1982}) has a different form obtained using a nine-point finite difference approximation of the Laplace operator in the two-dimensional space \cite{Kantorovich1958}:
\begin{equation} \label{comp9.2b}
u_{ij}^{k+1} = 2 u_{ij}^k - u_{ij}^{k-1}+\lambda^2\Bigg[\frac{2}{3} \delta^k_{ij}(1,0) +\frac{1}{6}\delta^k_{ij}(1,1)\Bigg],\quad k=1,2, \ldots.
\end{equation}

Comparing the coefficients in \eqref{comp9.2} and \eqref{comp9.2b} it is easy to see that the term $\delta^k_{ij}(1,1)$ is less influential in the new nine-point scheme than in the conventional nine-point scheme (with a coefficient ratio equal to $\lambda^2$). The stability condition for the conventional scheme is less restrictive than for the new one: $\lambda \le \lambda_{\max}=\sqrt {3}/{2} \approx 0.866$. On the other hand, the new scheme has some accuracy advantages in comparison with the conventional nine-point approach for $\lambda \le 0.796$ which is shown in the next subsection.
\subsubsection{Numerical simulation}
We consider simulation results for the proposed nine-point scheme \eqref{comp9.1}-\eqref{comp9.2} using initial and boundary conditions generated by the exact solutions of the previous subsection. Let $c=1$. We assume that the unit square $\Omega=[0,1]^2$ is used as a space grid region, the time grid interval is $[0,\lambda]$ with $n_t=n$. A comparison is made with simulated results on the same grids for the conventional scheme \eqref{comp9.2b}.
Even though no expression for the first time-step corresponding to the conventional nine-point scheme is presented in \cite{Trefethen1982},  the usual approach based on the central difference for approximating the initial condition for $v$ combined with \eqref{comp9.2b} provides the corresponding expression
\begin{equation} \label{comp9.1b}
u_{ij}^1 = u_{ij}^0+\tau v_{ij}^0 +\frac{\lambda^2}{2}\Bigg[ \frac{2}{3}\delta^0_{ij}(1,0) +\frac{1}{6} \delta^0_{ij}(1,1) \Bigg]
\end{equation}
which is used for numerical simulation.

The simulation results are presented in the next table for standing wave simulations. We have used two values of $\lambda$ for the simulation: $\lambda=0.707$, as in subsection 6.1, and $\lambda=0.796$, according to the stability condition \eqref{stability9}. Relative $L^2$ errors for the schemes \eqref{comp9.1}$\&$\eqref{comp9.2} and  \eqref{comp9.1b}$\&$\eqref{comp9.2b} are denoted by $E_{P9}$ and $E_{C9}$, respectively. According to the simulated data, the new scheme \eqref{comp9.1}$\&$\eqref{comp9.2} appears to be more accurate than the other scheme for both $\lambda=0.707$ and $\lambda=0.796$.
\begin{table}[h]
	\centering
	\ra{1.3}
	\caption{Wave \eqref{exact sol} simulations using nine-point stencils}
	\begin{tabular}{@{}ccccc@{}}\toprule
		$ n $ & $ n_t $ & $\lambda$ & $E_{P9}$  & $E_{C9}$ \\   \midrule
		10 & 10 & $0.707$    & $3.7058\cdot10^{-2}$
		 &$1.1741\cdot10^{-1}$ \\
		10 & 10 & $0.796$    & $2.9587\cdot10^{-2}$ &  $1.1241\cdot10^{-1}$ \\	 \midrule
		20 & 20 & $0.707$    & $8.9333\cdot10^{-3}$ &  $2.8002\cdot10^{-2}$ \\
		20 & 20 & $0.796$    & $8.0697\cdot10^{-3}$ &  $2.7523\cdot10^{-2}$ \\  \midrule
		40 & 40 & $0.707$    & $2.3723\cdot10^{-3}$ &  $6.8821\cdot10^{-3}$ \\
		40 & 40 & $0.796$    & $2.5737\cdot10^{-3}$ &  $6.8668\cdot10^{-3}$ \\   \midrule
		80 & 80 & $0.707$    & $7.5573\cdot10^{-4}$ &  $1.7084\cdot10^{-3}$ \\
		80 & 80 & $0.796$    & $1.0274\cdot10^{-3}$ &  $1.7187\cdot10^{-3}$ \\ \bottomrule
	\end{tabular}
\end{table}

 The obtained results show a loss in accuracy of the both nine-point schemes for the simulated problem in comparison with the new five-point scheme results presented in Table 1.
\subsection{A 13-point stencil}
The next numerical stencil is based on complete bivariate interpolation polynomials of the fourth degree. Consider the initial segment $\mathcal{M}_{\le 15}^*$ of $\mathcal{M}^*$:
\begin{equation} \label{monomials15}
\{1,x_1,x_2,x_1 x_2,x_1^2,x_2^2,x_1^2 x_2,x_1 x_2^2,x_1^3, x_2^3,x_1^2 x_2^2, x_1^3 x_2, x_1 x_2^3, x_1^4,x_2^4 \}
\end{equation}
This set includes all the monomials of total degree $\le 4$.
The corresponding index set $Q_{15}$ is presented in Figure 3 where both thirteen solid dots and two empty circles denote interpolation points.
\begin{figure}[ht]
	\centering
	\begin{tikzpicture}
	\begin{axis}[xmin=-3,   xmax=3,
	ymin=-3,   ymax=3  ,
	extra x ticks={-3,-2,-1,0,1,2,3},
	extra y ticks={-3,-2,-1,0,1,2,3},
	extra tick style={grid=major}]
	\addplot [only marks,mark=*, mark size=3pt]
	coordinates {(0,0) (-1,0) (0,-1) (-1,-1) (1,0) (0,1) (1,-1) (-1,1) (1,1) (-2,0) (2,0) (0,-2) (0,2)};
	\addplot [only marks,mark=o, mark size=3pt]
	coordinates {(-2,-1) (-1,-2)};
	\end{axis}
	\end{tikzpicture}
	\caption{Index set for the 15-point interpolation stencil}
	\label{fig:Q13}
\end{figure}

After calculating the corresponding Lagrange basis (see section 4) and using \eqref{exact1}-\eqref{exact2}, one can determine that all the coefficients in expressions \eqref{A1}-\eqref{A2} related to the two empty circles will disappear. As a result, we get a numerical scheme that involves the 13-point stencil (the solid dots in Figure 3). The notations \eqref{notations2} will be used in presenting the scheme to make the expressions more compact:

1) \textit{for the first time-step}
\begin{equation} \label{comp13.1}
\begin{split}
u_{ij}^1 = u_{ij}^0+\tau v_{ij}^0+\frac{\lambda^2}{2} \Bigg[ \frac{4-2\lambda^2}{3} \delta^0_{ij}(1,0)
+\frac{\lambda^2}{6}\delta^0_{ij}(1,1)
+\frac{\lambda^2-1}{12}\delta^0_{ij}(2,0) \Bigg] \\
+\frac{\tau\lambda^2}{6}\Bigg[ \left(\frac{4}{3}-\frac{2\lambda^2}{5}\right) \epsilon^0_{ij}(1,0)
+\frac{\lambda^2}{10}\epsilon^0_{ij}(1,1)
+\left(\frac{\lambda^2}{20}-\frac{1}{12}\right)\epsilon^0_{ij}(2,0)
\Bigg];
\end{split}
\end{equation}

2) \textit{for the second and next time-steps}
\begin{equation} \label{comp13.2}
\begin{split}
u_{ij}^{k+1} = 2 u_{ij}^k - u_{ij}^{k-1}
+\lambda^2 \Bigg[\frac{4-2\lambda^2}{3} \delta^0_{ij}(1,0)
+\frac{\lambda^2}{6}\delta^0_{ij}(1,1)
+\frac{\lambda^2-1}{12}\delta^0_{ij}(2,0) \Bigg],  \\
k=1,2, \ldots.
\end{split}
\end{equation}
Formula \eqref{comp13.2} for the second and next time-steps completely coincides with that obtained previously \cite{Cohen1987,Cohen1996} using the finite-difference method.

However, expression \eqref{comp13.1} for the first time-step  has not been presented in the literature before. An advantage of using this expression rather than the conventional one (based on the central difference for approximating the initial condition for $v$) is shown in Table 3 where simulation results for the standing wave \eqref{exact sol} solution are presented. The relative $L^2$ errors for the conventional and new (Poisson's) approaches are denoted by $E_{C13}$ and $E_{P13}$, respectively.  For simplicity sake, the periodic boundary conditions have been incorporated in the simulation taking into account that the solution \eqref{exact sol} is periodic in both spatial directions.
Since the maximal Courant number needed for stability of this scheme is $1/\sqrt{2}$, a value of $\lambda=0.707$ was used.
\begin{table}[h!]
	\centering
	\ra{1.3}
	\caption{Wave \eqref{exact sol} simulations using the 13-point stencil.}
	\begin{tabular}{@{}cccccc@{}}\toprule
		$n \quad$ & $n_t \quad$ & $\lambda$ & $E_{P13} \quad$ & $E_{C13}$   \\   \midrule
		10 & 10 & $0.707$ & $4.2146\cdot10^{-5}$   & $6.8938\cdot10^{-2}$   \\	\midrule
		20 & 20 & $0.707$ & $6.6004\cdot10^{-7}$   & $1.6636\cdot10^{-2}$   \\  \midrule
		40 & 40 & $0.707$ & $1.1471\cdot10^{-8}$   & $4.1230\cdot10^{-3}$   \\ \midrule
		80 & 80 & $0.707$ & $2.8884\cdot10^{-10}$   & $1.0285\cdot10^{-3}$   \\ \bottomrule
	\end{tabular}
\end{table}\textit{}

A large advantage in accuracy for the new scheme demonstrated in Table 3 can be attributed to a higher accuracy of the new first time-step expression \eqref{comp13.1} in comparison with the conventional one. On the other hand,  it is worth to notice that the relative $L^2$ error $E_{C13}$ (corresponding to the conventional first time-step approach) in Table 3 has almost the same values as the error $E_{C5}$ in Table 1 despite using a higher degree interpolation stencil in Table 3. That is, an error introduced at the first time-step probably suppresses advantages of using a higher degree interpolation at later time-steps.

\section{Summary}
A new method is implemented to build explicit time-stepping stencil computation schemes for the transient 2D acoustic wave equation. It is based on using Poisson's formula and a similar three time level expression combined with polynomial stencil interpolation of the solution at each time-step and exact integration. As a result, for each chosen 2D stencil and a set of monomials, a unified time-marching scheme is created that includes two explicit computation procedures: for the first time-step and for the next steps.

Particular explicit stencil computation schemes (with five, nine and 13 space points) are derived. All of the obtained first time-step computation expressions are different from those used in conventional finite-difference methods. The obtained two-step stencil expressions for the five- and 13-point stencils (where the complete interpolation polynomials have been used)  coincide with the corresponding finite difference schemes. The obtained two-step expression for the nine-point stencil is new. Its stability region is determined by the von Neumann analysis.

Simulation comparison results are presented for a benchmark problem with an exact solution. It is demonstrated by simulation that the proposed stencil computation approach maintains an accuracy advantage in comparison with conventional finite difference schemes which is mostly attributed to the new first time-step computation expressions.


\section*{Supplementary Material}

In this Supplementary Material (SM), we include some miscellaneous extra material, which is not central to the description of our results in the main text, but can provide some extra information on the nature of the calculations performed and the scope of the results. The numbers of the equations and figures in this material contain the prefix ``SM'' to distinguish the usual numbers of the equations and figures in the main text.

\subsection*{Robustness of scheme to collision angle}
In the main text, we show in Fig.~3 the angular distribution of high-energy ($s > 0.11$ corresponding to $>1.1\,\trm{GeV}$) $E$-polarised photons emitted by a pump electron bunch with an angular divergence \mbox{$\Theta=0.2~\textrm{mrad}$}. Because the bunch angular divergence is several times broader than the field-induced angular spread, \mbox{$2\xi/\gamma_p$}, the harmonic structure in the polarised-photon angular distribution is smoothed out.

Integrating the angular distributions over a particular detector acceptance angle $\theta$ and multiplying the number of the electrons in the bunch, we can obtain the total number of photons accepted by the detector. In the main text, as an example we present a calculation of the photon source brilliance from an electron bunch incident at an angle $\vartheta_{i}=100\,$mrad with divergence $\Theta=0.2\,$mrad.

In Fig.~\ref{Fig_Brilliance}, we present the brilliance obtained for the high-energy $E$-polarised photon source for a broad range of bunch incident angles and different bunch angular divergence. As shown, the brilliance decreases less than $5\%$ as the bunch incident angle $\vartheta_i$ is increased from $0$ to $200\,\textrm{mrad}$, and further increase of the incident angle to $500\,\textrm{mrad}$ would result in a faster decay of the brilliance.

\subsection*{Increased Brilliance of Highly Polarised Source}
One way to improve the brilliance of the polarised source, would be to reduce the electron bunch's angular divergence. In Fig.~\ref{Fig_Brilliance}, we show example results for an angular divergence of $\Theta=0.05\,\textrm{mrad}$.  It follows that the angular divergence of the photon beam is determined by the properties of the electron bunch and we find that the brilliance of the photon source is inversely proportional to the electron bunch divergence, $\propto\Theta^{-2}$.

\begin{figure}[t!!]
 \center{\includegraphics[width=0.44\textwidth]{figureSM1}}
\caption{Brilliance of the $E$-polarised high-energy photons ($s>0.11$ corresponding to $>1.1~\textrm{GeV}$) for different incident angle $\vartheta_i$ and electron beam angular divergence $\Theta=0.05~\textrm{mrad}$ and $0.2~\textrm{mrad}$. The acceptance angle of the detector is set to $\theta=\Theta$.
The field parameters are the same as in the main text (for a laser pulse with $\xi=0.5$, $\omega_l=1.24~\textrm{eV}$ and duration $26.7\,\trm{fs}$ ($8$ cycles)). The electron bunch has a mean energy of  $10\,$GeV and a root-mean-square energy spread of $6\%$. The photon detector is set downstream along the bunch propagation direction.}
\label{Fig_Brilliance}
\end{figure}

\subsection*{Calculation of the angular spectrum}
As mentioned in the main text, to obtain a brilliant photon source, a high-energy electron bunch is required. The calculation we performed for the electron bunch, assumed a normalised momentum distribution of the form:
\bea
&\rho(\bm{p})=\frac{1}{\sqrt{2\pi^3}\sigma_{\parallel}\sigma^2_{\perp} m^3}\exp\left[-\frac{(\bm{p}\cdot\bm{n}-\tilde{p})^2}{2\sigma^2_{\parallel}m^2}-\frac{\left|\bm{p}-\bm{n}(\bm{p}\cdot\bm{n})\right|^2}{\sigma^2_{\perp} m^2}\right]\,,\nonumber
\eea
with modulus average momentum $\tilde{p}$ and rms momentum spread in the longitudinal, $\sigma_{\parallel} m$, and transverse, $\sigma_{\perp} m $ directions, where $\bm{n}$ is the incident direction of the electron bunch.

The differential probability of emitting a photon in the polarisation state $\eps_{j}$ with momentum $k$ is straightforwardly acquired from Eq.~(1) of the main text:
\begin{align}
\frac{d^3\tsf{P}_{j}}{ds\,d^2\bm{r}} &= \frac{\alpha}{(2\pi\eta)^{2}}\int^{\infty}_{s} \frac{d\lambda}{\lambda^2}\int d^2 p^{\LCperp} p^{0}\rho(\bm{p}) \nonumber \\
& \times \frac{s}{t}\int d\phi\,d\phi'~\tsf{T}_{j}~\e^{i\int^{\phi}_{\phi'}\frac{k\cdot \pi_{p}(\vphi)}{m^2t\eta}d\vphi}\,,\label{eqn:sfi1}
\end{align}
where $\lambda=\vkap\cdot p/m^2\eta$ is the electron light-front momentum normalised by the interaction energy parameter $\eta$ used in the manuscript, $s=\vkap\cdot k/m^2\eta$, $t=\lambda-s$. The expression of $\tsf{T}_{j}$ ($j=1,2$ for the linear case, and $j=+,-$ for the circular case) is same as in the manuscript except the dependency on $s$ and $t$ is replaced with $s/\lambda$ and $t/\lambda$, respectively.

From Eq.~(\ref{eqn:sfi1}), the angular distribution of the polarised photon can be acquired:
\begin{align}
\frac{d^2\tsf{P}_{j}}{dr_x\,dr_y} =\frac{\alpha}{(2\pi\eta)^{2}} \int^{\infty}_{s_d} \frac{d\lambda}{\lambda^2}\int d^2 p^{\LCperp}\, p^{0}\rho(\bm{p})\,g\big(\lambda, \bm{r}-\frac{p^{\LCperp}}{m\lambda}\big)
\label{eqn:sfi2}
\end{align}
in which we use the shorthand
\begin{align}
g\left(\lambda, \bm{r}-\frac{p^{\LCperp}}{m\lambda}\right)=\int^{\lambda}_{s_d}ds\frac{s}{t}\int d\phi\,d\phi'~\tsf{T}_{j}~\e^{i\int^{\phi}_{\phi'}\frac{k\cdot \pi_{p}(\vphi)}{m^2 t \eta}d\vphi}\,.
\end{align}

\begin{figure}[t!!]
 \center{\includegraphics[width=0.48\textwidth]{figureSM2}}
\caption{Angular distribution $d^2\tsf{P}/d\theta_xd\theta_y$ of the polarised photons ($s_d=0$) in a circularly-polarised background (upper panels) and a linearly-polarised background (bottom panels). Left column: (a) and (c) angular distribution of $E$-polarised photons. Right column: (b) and (d) angular distribution of $B$-polarised photons. The field parameters are same as in the main text. The parameters of electron bunch are listed: angular divergence $\Theta=0.05\,$mrad, incident angle $\vartheta_i=100\,$mrad, average energy $10$~GeV and root-mean-square energy spread $6\%$, and. The photon detector is set downstream along the bunch propagation direction.}
\label{Fig_bunch_angle_dist_005}
\end{figure}
To view the angular harmonic structure shown in Fig.~1 from the single-particle results, the bunch angular divergence has to be reduced to the same level of the angular spread induced by field, thus to be $\Theta \approx 2\xi/\gamma_p$. Fig.~\ref{Fig_bunch_angle_dist_005} shows the angular distribution of polarised photons emitted by a pump electron bunch of average energy $\tilde{p}=\gamma_p m$ ($10\,$GeV),
$\sigma_{\parallel}=0.03\gamma_p$ and $\sigma_{\perp}=2.5\times 10^{-5} \gamma_p$ corresponding to a $6\%$ root-mean-square energy spread and an angular divergence $\Theta=0.05\,\textrm{mrad}$, incident in the direction $\bm{n}=[\sin\vartheta_{i},0,-\cos\vartheta_{i}]$ at an angle $\vartheta_{i}=100\,\textrm{mrad}$. As shown in the figure, well-defined harmonic structures in the angular distribution can be clearly observed.

\bibliographystyle{apsrev}

 \providecommand{\noopsort}[1]{}


\section{\label{Intro}Introduction}

In quantum chromodynamics (QCD), parton distribution and fragmentation functions (PDFs and FFs) are used to describe the long distance bound state nature of hadrons. Historically PDFs and FFs have been taken to only be dependent on the longitudinal momentum fraction $x$ and $z$, respectively. In the last several decades this has been extended to include transverse momentum dependence so that the ``unintegrated" transverse-momentum-dependent (TMD) PDFs and FFs are written explicitly as dependent on both the longitudinal momentum and transverse momentum of the partons. Since the framework explicitly includes small transverse momenta, the reevaluation of important principles of QCD like factorization and universality has been necessary. \par

In particular, the role of color interactions due to soft gluon exchanges between participants of the hard scattering and remnants of the interaction in collisions involving a hadron have been found to have profound effects regarding these principles. For example, the Sivers function, a particular TMD PDF that correlates the transverse spin of the proton with the orbital angular momentum of the parton, was predicted to be the same magnitude but opposite in sign when measured in semi-inclusive deep-inelastic scattering (SIDIS) and Drell-Yan (DY)~\cite{collins_signchange}. This prediction arises from the different color flows that are possible between these two interactions due to the possibility of soft gluon exchanges in the initial-state and final-state for DY and SIDIS, respectively. Factorization of the nonperturbative functions is still predicted to hold in both SIDIS and DY.\par

In $p$$+$$p$ collisions where two nearly back-to-back hadrons are measured, soft gluon exchanges are possible in both the initial and final state since there is color present in both the initial and final states. In this process, factorization breaking was predicted in a TMD framework~\cite{trog_fact,collins_qiu_fact}. For processes where factorization breaking is predicted, individual TMD PDFs and TMD FFs become correlated with one another and cannot be written as a convolution of nonperturbative functions. The ideas behind the predicted sign change of certain TMD PDFs and factorization breaking result from the same physical process of soft gluons being exchanged between participants of the hard scattering and remnants of the collision. These predictions represent major qualitative departures from purely perturbative approaches which do not consider the remnants of the collision at all. \par

In calculations of TMD processes where factorization is predicted to hold, the Collins-Soper (CS) equation is known to govern the evolution of nonperturbative functions in a TMD framework with the hard scale of the interaction $Q^2$~\cite{cs1,cs2}. Contrary to the purely perturbative collinear DGLAP evolution equations, the CS evolution equation for TMD processes involves the Collins-Soper-Sterman (CSS) soft factor~\cite{css_soft}, which contains nonperturbative contributions. The theoretical expectation from CSS evolution is that momentum widths sensitive to nonperturbative transverse momentum should increase with the hard scale of the interaction. This can intuitively be thought of as due to an increased phase space for hard gluon radiation, and thus a broader transverse momentum distribution. This behavior has been studied and confirmed in several phenomenlogical analyses of DY and Z boson data (see e.g.~\cite{dy1,dy2,sidisdy}) as well as phenemonological analyses of SIDIS data (see e.g.~\cite{sidisdy,sidis1,sidis2}). Since the CS evolution equation comes directly out of the derivation of TMD factorization~\cite{css_tmd}, it then follows that a promising avenue to potentially observe factorization breaking effects is to study possible deviations from CSS evolution in processes where factorization breaking is predicted such as dihadron and direct photon-hadron angular correlations in $p$$+$$p$ collisions. \par

\section{Dihadron and Direct Photon-Hadron Angular Correlations}

Dihadron and direct photon-hadron angular correlations are both predicted to be sensitive to factorization breaking effects because there are hadrons in both the initial and final states, thus the potential for soft gluon exchanges in both the initial and final states exists. Additionally these processes can be treated in a TMD framework when the two particles are nearly back-to-back and have large $p_T$; a hard scale is defined with the large $p_T$ of the particles and the process is also sensitive to the convolution of initial-state and final-state small transverse momentum scales $k_T$ and $j_T$. Here $k_T$ refers to the initial-state partonic transverse momentum due to the confined nature of the partons and soft or hard gluon radiation, and $j_T$ refers to the final-state fragmentation transverse momentum due to soft or hard gluon radiation in the hadronization process. \par

 \begin{figure}[thb]
 \includegraphics[width=1.0\linewidth]{kTkinematics.pdf} \\
 \includegraphics[width=1.0\linewidth]{dp_kt_kinematics.pdf}
 \caption{\label{fig:ktkinematics}
 A schematic diagram showing event hard-scattering kinematics for (a) dihadron and (b) direct photon-hadron processes in the transverse plane. Two hard scattered partons with transverse momenta $\hat{p}_T^{\rm trig}$ and $\hat{p}_T^{\rm assoc}$ are acoplanar due to the initial-state partonic $k_T$, given by $|\vec{k}_T^1+\vec{k}_T^2|$. The scattered partons result in two fragmented hadrons that are additionally acoplanar due to the final-state transverse momentum perpendicular to the jet axis $j_{T_y}$. For (b) direct photon-hadrons only one away-side jet fragment is produced since the direct photon is colorless. The transverse momentum component perpendicular to the trigger particle's axis is labeled as $\mbox{$p_{\rm out}$}\xspace$.
 }
 \end{figure}

Figure~\ref{fig:ktkinematics} shows a kinematic diagram in the transverse plane for both dihadron (top) and direct photon-hadron (bottom) events. At leading order the hard scattered partons are exactly back-to-back, but due to initial-state $k_T$ the partons are acoplanar by some amount $|\vec{k}_T^1+\vec{k}_T^2|$. The fragmentation process introduces an additional transverse momentum component $j_{T_y}$ which is assumed to be Gaussian distributed about the parton axes such that $\sqrt{\langle j_T^2\rangle}=\sqrt{2\langle j_{T_y}^2\rangle}$. The transverse momentum component perpendicular to the trigger particle's axis is labeled as $\mbox{$p_{\rm out}$}\xspace$ and is sensitive to initial and final state $k_T$ and $j_T$, where the trigger particle refers collectively to the direct photon or near-side hadron. Measuring the azimuthal angular separation between the near-side trigger particle and away-side associated particle allows calculating \mbox{$p_{\rm out}$}\xspace with the following equation:
\begin{equation}
\mbox{$p_{\rm out}$}\xspace = \mbox{$p_{\rm T}^{\rm assoc}$}\xspace\sin\mbox{$\Delta\phi$}\xspace
\end{equation}

The results presented were measured by the PHENIX collaboration at the Relativistic Heavy Ion Collider at Brookhaven National Lab. The PHENIX detector covers a pseudorapidity interval of $|\eta|<0.35$, and has two arms which in total span an azimuthal region of $\Delta\phi\sim\pi$. Lead scintillator and lead glass electromagnetic calorimeters provide measurements of isolated direct photons and neutral pions via their two photon decay. To measure away-side particles, the PHENIX detector employs a drift chamber and pad chamber tracking system that measures nonidentified charged hadrons. These results were measured from $p$$+$$p$ collisions at a center-of-mass energy of $\sqrt{s}=510$ GeV, and were recently submitted to the arXiv~\cite{ppg195}.

\section{Results}
\begin{figure*}[thb]
\includegraphics[width=1.0\linewidth]{dphi_correlations_rmspoutfits_nearsidefits.pdf}
\caption{\label{fig:dphis}
	Per trigger yields of charged hadrons as a function of $\mbox{$\Delta\phi$}\xspace$ are shown in several \mbox{$p_{\rm T}^{\rm trig}$}\xspace and \mbox{$p_{\rm T}^{\rm assoc}$}\xspace bins for both \mbox{$\pi^{0}$}\xspace-hadron and direct photon-hadron correlations. The near side yield of the isolated direct photon-hadron correlations is not shown due to the presence of an isolation cut. The blue and red solid lines shown on the away-sides of the distributions are fits to extract the quantity \mbox{$\sqrt{\langle p_{\rm out}^{2}\rangle}$}\xspace.
	}
\end{figure*}

Figure~\ref{fig:dphis} shows the measured per-trigger yields as a function of \mbox{$\Delta\phi$}\xspace for both \mbox{$\pi^{0}$}\xspace-hadron and direct photon-hadron correlations. The \mbox{$\pi^{0}$}\xspace-hadron correlations show the expected two jet structure, with two peaks that are nearly back-to-back around \mbox{$\Delta\phi$}\xspace$\sim0$ and \mbox{$\Delta\phi$}\xspace$\sim\pi$. The near side of the direct photon-hadron correlations is omitted due to the presence of an isolation cut on the direct photons; thus the near side is not physically interpretable. Additionally, the away-side jets are sensitive to the effects of $k_T$ broadening, so they are the yields of interest. The direct photon-hadron yields have a smaller yield than the \mbox{$\pi^{0}$}\xspace-hadron yields due to the direct photon emerging from the hard scattering at leading order; thus the direct photon-hadron correlations probe a smaller jet energy than the \mbox{$\pi^{0}$}\xspace-hadron correlations. \par

The measured $\mbox{$p_{\rm out}$}\xspace$ per-trigger yields are shown in Fig.~\ref{fig:pouts} for both $\mbox{$\pi^{0}$}\xspace$-$\mbox{\text{h}$^\pm$}\xspace$ and direct photon-$\mbox{\text{h}$^\pm$}\xspace$ angular correlations. The open points show the distributions for $\mbox{$\pi^{0}$}\xspace$-hadron correlations while the filled points show the distributions for the direct photon-hadron correlations. The distributions are constructed for only away-side charged hadrons that are sensitive to initial-state $k_T$ and final-state $j_T$. The $\mbox{$p_{\rm out}$}\xspace$ distributions show two distinct regions; at small $\mbox{$p_{\rm out}$}\xspace\sim0$ where the particles are nearly back-to-back a Gaussian shape can be seen, while at larger $\mbox{$p_{\rm out}$}\xspace$ a power law shape is clear. These two shapes indicate a transition from nonperturbatively generated transverse momentum due to soft gluon emission in the Gaussian region to perturbatively generated transverse momentum due to hard gluon emission in the power law region. The distributions are fit with a Gaussian function at small $\mbox{$p_{\rm out}$}\xspace$ and a Kaplan function over the entire distribution. The Gaussian function clearly fails at $\mbox{$p_{\rm out}$}\xspace\sim$1.3 while the Kaplan function accurately describes the transition from Gaussian behavior to power law behavior. \par

\begin{figure*}[thb]
\includegraphics[width=1.0\linewidth]{pouts_allfits.pdf}
\caption{\label{fig:pouts}
Per trigger yields of charged hadrons as a function of $\mbox{$p_{\rm out}$}\xspace$ are shown in several $\mbox{$p_{\rm T}^{\rm trig}$}\xspace$ bins for both $\mbox{$\pi^{0}$}\xspace$-$\mbox{\text{h}$^\pm$}\xspace$ and direct photon-$\mbox{\text{h}$^\pm$}\xspace$ correlations. The distributions are fit with a Gaussian function at small $\mbox{$p_{\rm out}$}\xspace$ and a Kaplan function over the entire range. The Gaussian fit clearly fails after $\sim$1.3 GeV/c, indicating a transition from nonperturbatively generated $k_T$ and $j_T$ to perturbatively generated $k_T$ and $j_T$.
}
\end{figure*}

To search for effects from factorization breaking, a comparison to the expectation from CSS evolution must be made using momentum widths which are sensitive to nonperturbative transverse momentum. To make a comparison, two different momentum widths were extracted from the measured correlations. Figure~\ref{fig:rmspouts} shows the root mean square of \mbox{$p_{\rm out}$}\xspace as a function of $\mbox{$p_{\rm T}^{\rm trig}$}\xspace$, which is extracted from fits to the entire away-side jet region in Fig.~\ref{fig:dphis}. The away-side fits can be seen in Fig.~\ref{fig:dphis} as red and blue solid lines around $\mbox{$\Delta\phi$}\xspace\sim\pi$. The values of \mbox{$\sqrt{\langle p_{\rm out}^{2}\rangle}$}\xspace clearly decrease for both the \mbox{$\pi^{0}$}\xspace-hadron correlations and direct photon-hadron correlations, which is the opposite of what is predicted from CSS evolution. The direct photon-hadron correlations show a stronger dependence than the \mbox{$\pi^{0}$}\xspace-hadron correlations in the same region of \mbox{$p_{\rm T}^{\rm trig}$}\xspace. Since the values of \mbox{$\sqrt{\langle p_{\rm out}^{2}\rangle}$}\xspace are extracted from the entire away-side jet region, \mbox{$\sqrt{\langle p_{\rm out}^{2}\rangle}$}\xspace is sensitive to both perturbatively and nonperturbatively generated $k_T$ and $j_T$. While it is dominated by nonperturbatively generated transverse momentum since the majority of charged hadrons are in the nearly back-to-back region $\mbox{$p_{\rm out}$}\xspace\sim0$ or $\mbox{$\Delta\phi$}\xspace\sim\pi$, an observable that is sensitive to only nonperturbatively generated transverse momentum is the most ideal for comparisons to CSS evolution.\par

\begin{figure}[thb]
\includegraphics[width=\linewidth]{rmspout_vspttrig.pdf}
\caption{\label{fig:rmspouts}
The extracted values of \mbox{$\sqrt{\langle p_{\rm out}^{2}\rangle}$}\xspace are shown as a function of the interaction hard scale \mbox{$p_{\rm T}^{\rm trig}$}\xspace for both \mbox{$\pi^{0}$}\xspace-hadron and direct photon-hadron correlations. The momentum widths decrease with the interaction hard scale, which is opposite the prediction from CSS evolution.
}
\end{figure}

Since the Gaussian fits to the \mbox{$p_{\rm out}$}\xspace distributions are taken in only the nearly back-to-back region, the widths of the Gaussian functions are momentum widths that are sensitive to only nonperturbative transverse momentum. Figure~\ref{fig:widths} shows the measured Gaussian widths as a function of the interaction hard scale \mbox{$p_{\rm T}^{\rm trig}$}\xspace for both direct photon-hadron and \mbox{$\pi^{0}$}\xspace-hadron correlations. Similarly to the values of \mbox{$\sqrt{\langle p_{\rm out}^{2}\rangle}$}\xspace from Fig.~\ref{fig:rmspouts}, the Gaussian widths decrease as a function of the hard scale, which is the opposite of the prediction from CSS evolution. The direct photon-hadron widths show a stronger dependence on \mbox{$p_{\rm T}^{\rm trig}$}\xspace than the \mbox{$\pi^{0}$}\xspace-hadron widths, similar to the values of \mbox{$\sqrt{\langle p_{\rm out}^{2}\rangle}$}\xspace. \par

\begin{figure}[thb]
\includegraphics[width=\linewidth]{gausswidths_withpythia.pdf}
\caption{\label{fig:widths}
The measured and {\sc pythia} simulated Gaussian widths from the \mbox{$p_{\rm out}$}\xspace distributions are shown as a function of the hard scale \mbox{$p_{\rm T}^{\rm trig}$}\xspace. The widths, sensitive to only nonperturbative transverse momentum, decrease with \mbox{$p_{\rm T}^{\rm trig}$}\xspace. Surprisingly, {\sc pythia} nearly replicates the evolution behavior for both the direct photon-hadron and \mbox{$\pi^{0}$}\xspace-hadron correlations despite a $\sim$15\% difference in the magnitude of the {\sc pythia} widths.
}
\end{figure}

Since theoretical calculations were not available, {\sc pythia}~\cite{pythia} was used to simulate direct photon-hadron and \mbox{$\pi^{0}$}\xspace-hadron correlations and study the evolution of momentum widths with the hard scale of the interaction. The Perugia0~\cite{perugia} tune was used since it was tuned to Tevatron Z boson data at small \mbox{$p_T$}\xspace; therefore it should adequately reproduce events with a small amount of total \mbox{$p_T$}\xspace. {\sc pythia} direct photon and dijet events were produced and the \mbox{$p_{\rm out}$}\xspace distributions were constructed directly from the simulation in exactly the same way that they were measured in data. Although the magnitude of the widths from the simulation is roughly 15\% different in each bin, the simulation remarkably nearly reproduces the measured evolution of the Gaussian widths as seen in Fig.~\ref{fig:widths}. It is plausible that {\sc pythia} could be sensitive to the effects from factorization breaking due to the way a \mbox{$p$+$p$}\xspace event is processed. Unlike a standard perturbative QCD calculation, {\sc pythia} forces all particles to color neutralize in the event. This includes allowing initial and final-state gluon interactions, which are the necessary physical mechanism for factorization breaking and are additionally necessary to color neutralize all of the objects in the event. \par

\section{Conclusion}

In hadronic collisions where at least one final-state hadron is measured, factorization breaking has been predicted in a TMD framework. When color is present in both the initial and final states, soft gluon exchanges between participants in the hard scattering and the remnants of the collision are possible, leading to novel color flows throughout the entire scattering process. Nearly back-to-back dihadron and direct photon-hadron angular correlations in \mbox{$p$+$p$}\xspace collisions at \mbox{$\sqrt{s}$}\xspace=510 GeV from the PHENIX experiment at the Relativistic Heavy Ion Collider were measured to probe possible effects from factorization breaking~\cite{ppg195}. The transverse momentum component perpendicular to the near-side trigger particle, \mbox{$p_{\rm out}$}\xspace, was used to compare predictions from CSS evolution. CSS evolution, which comes directly out of the derivation of TMD factorization~\cite{css_tmd}, predicts that momentum widths sensitive to nonperturbative transverse momentum should increase with increasing hard scale due to the broadened phase space for perturbatively generated gluon radiation. This dependence has been observed in phenomenological fits to both DY and SIDIS data~\cite{dy1,dy2,sidisdy,sidis1,sidis2}. \par

The measured correlations at PHENIX show the opposite dependence from the prediction of  CSS evolution; momentum widths in hadronic collisions where at least one final-state hadron is measured decrease with the interaction hard scale \mbox{$p_{\rm T}^{\rm trig}$}\xspace~\cite{ppg195}. Remarkably, {\sc pythia} replicates this behavior in both direct photon-hadron correlations and \mbox{$\pi^{0}$}\xspace-hadron correlations. While {\sc pythia} certainly does not consider effects from factorization breaking as it relies on collinear factorization, the necessary physical mechanism that results in the predicted factorization breaking is present in {\sc pythia}; gluon exchanges with the remnants are possible in a {\sc pythia} simulated event since all colored objects are forced to color neutralize in any given event, unlike a standard perturbative QCD calculation. \par


\section{Introduction}
The discovery of halo nuclei (neutron-halo and proton-halo) in the neighborhood of drip lines is one of the major achievements of the advancements of the radioactive ion-beam facilities. Halo structure is characterized by a relatively stable and denser core surrounded by weakly bound one or more valence nucleon(s) giving rise to long extended tail in the density distribution. This low-density tail is supposed to be the consequence of quantum mechanical tunneling of the last nucleon(s) through a-shallow barrier following an attractive well that appears due to the short-range nuclear interaction, at energies smaller than the height of the barrier. In halo nuclei, one seldom finds any excited bound states because of the utmost support one bound state at energies less than 1 MeV. Halo nuclei have high scientific significance as they exhibit one or more resonance state(s) just above the binding threshold. The observed halo nuclei-$^{17}$B, $^{19}$C show one-neutron halo; $^6$He, $^{11}$Li, $^{11, 14}$Be show two-neutron halo; $^8$B, $^{26}$P show one proton-halo; $^{17}$Ne, $^{27}$S show two proton-halo and $^{14}$Be, $^{19}$B show four neutron-halo structure respectively \cite{tanihata-1985, kobayashi-2012, hwash-2017, jensen-2000, schwab-2000, tanaka-2010}. Halo-nuclei is characterized by their unusually large r.m.s. matter radii (larger than the liquid-drop model prediction of $R_A\propto A^{1/3}$) \cite{audi-2003, acharya-2013} and sufficiently small two-nucleon separation energies (typically less than 1 MeV). Tanaka et al. 2010. \cite{tanaka-2010} observed of a large reaction cross-section in the drip-line nucleus $^{22}$C, Kobayashi et al. 2012 \cite{kobayashi-2012}, conducted research on one- and two-neutron removal reactions from the most neutron-rich carbon isotopes, Gaudefroy et al 2012 \cite{gaudefroy-2012} carried a direct mass measurements of $^{19}$B, $^{22}$C, $^{29}$F, $^{31}$Ne, $^{34}$Na and some other light exotic nuclei. Togano et al, 2016 \cite{togano-2016} studied interaction cross-section of the two-neutron halo nucleus $^{22}$C. \newline Nuclear matter distribution profile of such nuclei has an extended low-density tail forming a halo around the more localized dense nuclear core. Thus, in addition to bound state properties, continuum spectra is another significant parameter that is highly involved in the investigation of structure and interparticle interactions in the exotic few-body systems like the halo nuclei. It is worth stating here that the study of resonances is of particular interest in many branches of physics involving weakly bound systems in which only few bound states are possible.\newline In the literature survey, we found three main theoretical approaches that were used to explore the structure of 2n-halo nuclei. The first one is the microscopic model approach in which the valence neutrons are supposed to move around the conglomerate of other nucleons (protons and neutrons) without having any stable core. The second one is the three-body cluster model in which the valence nucleons are assumed to move around the structureless inert core. And the third one is the microscopic cluster model in which the valence nucleons move around the deformed excited core \cite{saaf-2014, nesterov-2010, korennov-2004}. There are several theoretical approaches which are employed for computation of resonant states. Some of those are the positive energy solution of the Faddeev equation \cite{cobis-1997}, complex coordinate rotation (CCR) \cite{csoto-1993, ayoma-1995}, the analytic computation of bound state energies \cite{tanaka-1997}, the algebraic version of resonating group method (RGM) \cite{vasilevsky-2001}, continuum-discretized coupled-channels (CDCC) method clubbed to the cluster-orbital shell model (COSM) \cite{ogata-2013}, hyperspherical harmonics method (HHM) for scattering states \cite{danilin-1997}, etc. In most of the theoretical approaches, Jacobi coordinates are used to derive the relative coordinates separating the center of mass motion.
\newline One of the most challenging obstacles that are involved in the calculation of resonances in any weakly bound nucleus is the large degree of computational error. In our case, we overcome this obstacle by adopting a novel theoretical approach by interfacing the algebra of supersymmetric quantum mechanics with the algebra involved in the hyperspherical harmonics expansion method. In this scheme, one can handle the ground state as well as the resonant states on the same footing. The technique is based on the fact that, for any arbitrarily given potential (say, {$U$}), one can construct a family of isospectral potentials ($\hat{U}$), in which the latter depends on an adjustable parameter ($\lambda$). And when the original potential has a significantly low and excessively wide barrier (poorly supporting the resonant state), $\lambda$ can be chosen judiciously to enhance the depth of the well together with the height of the barrier in $\hat{U}$. This enhanced well-barrier combination in $\hat{U}$ facilitates trapping of the particle which in turn facilitates the computation of resonant state more accurately at the same energy, as that in the case of {$U$}. This is because, {$U$} and $\hat{U}$ are {\bf strictly isospectral}.
\newline To test the effectiveness of the scheme we apply the scheme to the first $0^+$ resonant states of the carbon isotopes $^A$C, for A equal to 18 and 20 respectively. We chose three-body (2n+$^{\rm A-2}$C) cluster model for each of the above isotopes, where outer core neutrons move around the relatively heavier core $^{A-2}$C. The lowest eigen potential derived for the three-body systems has a shallow well following a skinny and sufficiently wide barrier. This skinny-wide barrier gives rise to a large resonance width. One can, in principle, find quasi-bound states in such a shallow potential, but that poses a difficult numerical task. For a finite height of the barrier, a particle can temporarily be trapped in the shallow well when its energy is close to the resonance energy. However, there is a finite possibility that the particle may creep in and tunnel out through the barrier. Thus, a more accurate calculation of resonance energy is easily masked by the large resonance width resulting from a large tunneling probability due to a low barrier height. Hence, a straightforward calculation of the resonance energies of such systems fails to yield accurate results.
\newline We adopt the hyperspherical harmonics expansion method (HHEM) \cite{ballot-1982} to solve the three-body Schr\"{o}dinger equation in relative coordinates. In HHEM, three-body relative wavefunction is expanded in a complete set of hyperspherical harmonics. The substitution of the wavefunction in the Schr\"{o}dinger equation and use of orthonormality of HH gives rise to an infinite set of coupled differential equations (CDE). The method is an essentially exact one, involving no other approximation except an eventual truncation of the expansion basis subject to the desired precision in the energy snd the capacity of available computer. However, hyperspherical convergence theorem \cite{schneider-1972} permits extrapolation of the data computed for the finite size of the expansion basis, to estimate those for even larger expansion bases. However, the convergence of HH expansion being significantly slow one needs to solve a large number of CDE's to achieve desired precision causing another limitation, hence we used the hyperspherical adiabatic approximation (HAA) \cite{ballot1-1982}to construct single differential equation (SDE) to be solved for the lowest eigen potential, ${U}_0(\rho)$) to get the ground state energy $E_0$ and the corresponding wavefunction $\psi_0(\rho)$ \cite{das-1982}.
\newline We next derive the isospectral potential $\hat{U}(\lambda,\rho)$ following algebra of the SSQM \cite{cooper-1995, khare-1989, nieto-1984}. Finally, we solve the SDE for $\hat{U}(\lambda,\rho)$ for various positive energies to get the wavefunction. We then compute the probability density corresponding to the wavefunction for finding the particle within the deep-sharp well following the enhanced barrier. A plot of probability density as a function of energy shows a sharp peak at the resonance energy. The actual width of resonance can be obtained by back-transforming the wave function $\hat{\psi}(\lambda, \rho)$ corresponding to $\hat{U}(\lambda,\rho)$ to $\psi(\rho)$ of $U(\rho)$.
\newline The paper is organized as follows. In sections 2, we briefly review the HHE method. In section 3, we present a precise description of the SSQM algebra to construct the one-parameter family of isospectral potential $\hat{U}(\lambda,\rho)$. The results of our calculation are presented in section 4 while conclusions are drawn in section 5.

\section{Hyperspherical Harmonics Expansion Method}
For a the three-body model of the nuclei $^{A-2}$C+n+n, the relatively heavy core $^{A-2}$C is labeled as particle 1, and two valence neutrons are labelled as particle 2 and 3 respectively. Thus there are three possibile partitions for the choice of Jacobi coordinates. In any chosen partition, say the $i^{th}$ partition, particle labelled $i$ plays the role of spectator while remaining two paricles form the interacting pair. In this partition the Jacobi coordinates are defined as
\begin{equation}
\vec{x_{i}} = a_i(\vec{r_{j}} - \vec{r_{k}}); \vec{y_{i}} = \frac{1}{a_i} \left(\vec{r_{i}} -\frac{m_{j}\vec{r_{j}} + m_{k} \vec{r_{k}}}{ m_{j} + m_{k}} \right); \vec{R}=  \frac{\sum_{i=1}^3 m_{i}\vec{r_{i}}}{M}      \end{equation}
where $i,j, k$ form a cyclic permutation of 1,2,3. The parameter $a_i= \left[\frac{m_{j} m_{k}M}{m_{i}(m_{j}+m_{k})^{2}} \right]^{\frac{1}{4}}$; $m_{i}, \vec{r_{i}}$ are the mass and position of the $i^{th}$ particle and $M(=\sum_{i=1}^3m_{i})$, $\vec{R}$ are those of the centre of mass (CM) of the system. Then in terms of Jacobi coordinates, the relative motion of the three-body system can be described by the equation \begin{equation}
\left\{ - \frac{\hbar^{2}}{2\mu} \nabla_{x_{i}}^{2}- \frac{\hbar^{2}}{2\mu} \nabla_{y_{i}}^{2}+
V_{jk} (\vec{x_{i}})
 +V_{ki} (\vec{x_{i}}, \vec{y_{i}} ) + V_{ij} (\vec{x_{i}}, \vec{y_{i}})-E
\:  \right\} \Psi (\vec{x_{i}}, \vec{y_{i}}) = 0
\end{equation}
where ${\mu = \left[ \frac{m_{i} m_{j} m_{k}}{M} \right]^{\frac{1}{2}}}\rightarrow$ is the reduced mass of the system, $V_{ij}$ represents the interaction potential between the particles $i$ and $j$, $x_{i} = \rho \cos \phi_{i}$; $y_{i}= \rho \sin \phi_{i}$; $\phi_i=\tan^{-1}(\frac{y_i}{x_i})$; $\rho=\sqrt{x_i^2+y_i^2}$. The hyperradius $\rho$ together with five angular variables $\Omega_{i} \rightarrow \{\phi_{i}, \theta_{x_{i}}, {\cal \phi}_{x_{i}}, \theta_{y_{i}}, {\cal \phi}_{y_{i}} \}$ constitute hyperspherical coordinates of the system. The Schr\"{o}dinger equation in hyperspherical variables $(\rho, \Omega_{i})$ becomes
\begin{equation}
\left\{ - \frac{\hbar^{2}}{2\mu}\frac{1}{\rho^5} \frac{\partial^2}{\partial\rho^2} - \frac{\hbar^{2}}{2\mu}\frac{4}{\rho}\frac{\partial}{\partial\rho}+
\frac{\hbar^{2}}{2\mu}\frac{\hat{{\cal K}}^{2}(\Omega_{i})}{\rho^{2}} + V(\rho, \Omega_{i}
) - E \right\} \Psi(\rho, \Omega_{i})= 0.
\end{equation}
In Eq.(3) $V(\rho, \Omega_{i}) = V_{jk} + V_{ki} + V_{ij}$ is the total interaction potential in the $i^{th}$ partition and $\sl{\hat{{\cal K}}^{2}}(\Omega_{i})$ is the square of the hyperangular momentum operator satisfying the eigenvalue equation
\begin{equation}
\hat{{\cal K}}^{2}(\Omega_{i}) {\cal Y}_{K \alpha_{i}}(\Omega_{i}) = K (K + 4) {\cal Y}_{K \alpha_{i}}(\Omega_{i}) \end{equation}
$K$ is the hyperangular momentum quantum number and $\alpha_{i}$ $\equiv\{l_{x_{i}}, l_{y_{i}}, L, M \}$, ${\cal Y}_{K\alpha_{i}}(\Omega_{i})$ are the hyperspherical harmonics (HH) for which a closed analytic expressions can be found in ref. \cite{cobis-1997}.
\newline In the HHEM, $\Psi(\rho, \Omega_{i})$ is expanded in the complete set of HH corresponding to the partition "$i$" as
\begin{equation}
\Psi(\rho, \Omega_{i}) = \sum_{K\alpha_{i}}\frac{\psi_{K\alpha_{i}} (\rho)}{\rho^{5/2}} {\cal Y}_{K\alpha_{i}}(\Omega_{i}) \end{equation}
Use of Eq. (5), in Eq. (3) and application of the orthonormality of HH leads to a set of coupled differential equations (CDE) in $\rho$
\begin{equation}
\left\{-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{d\rho^{2}}
+\frac{\hbar^{2}}{2\mu}\frac{(K+3/2)(K+5/2)}{\rho^2}-E\right\}
\psi_{K\alpha_{i}}(\rho)
+\sum_{K^{\prime}\alpha_{i}^{\prime}} {\cal M}_{K\alpha_{i}}^{K^{\prime}\alpha_{i}^{\prime}}\psi_{K^{\prime}\alpha_{i}^{\prime}}(\rho)=0.
\end{equation}
where \begin{equation}
{\cal M}_{K\alpha_{i}}^{K^{\prime}\alpha_{i}^{\prime}} = \int {\cal Y}_{K\alpha_{i}}^{*}(\Omega_{i}) V(\rho, \Omega_{i}) {\cal Y}_{K^{\prime} \alpha_{i}^{~\prime}}(\Omega_{i}) d\Omega_{i}.
\end{equation}
The infinite set of CDE's represented by Eq. (6) is truncated to a finite set by retaining all K values up to a maximum of $K_{max}$ in the expansion (5). For a given $K$, all allowed values of $\alpha_{i}$ are included. The size of the basis states is further restricted by symmetry requirements and associated conserved quantum numbers. The reduced set of CDE's are then solved by adopting hyperspherical adiabatic approximation (HAA) \cite{ballot1-1982}. In HAA, the CDE's are approximated by a single differential equation assuming that the hyperradial motion is much slower compared to hyperangular motion. For this reason, the angular part is first solved for a fixed value of $\rho$. This involves diagonalization of the potential matrix (including the hyper centrifugal repulsion term) for each $\rho$-mesh point and choosing the lowest eigenvalue $U_0(\rho)$ as the lowest eigen potential \cite{das-1982}. Then the energy of the system is obtained by solving the hyperradial motion for the chosen lowest eigen potential ($U_0(\rho)$), which is the effective potential for the hyperradial motion \begin{equation}
\left\{-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{d\rho^{2}} + U_0(\rho) - E \right\} \psi_{0}(\rho) = 0 \end{equation}
Renormalized Numerov algorithm subject to appropriate boundary
conditions in the limit $\rho \rightarrow 0$ and $\rho\rightarrow\infty$
is then applied to solve Eq. (8) for E ($\leq E_0$). The hyper-partial wave $\psi_{K\alpha_{i}}(\rho)$ is given by
\begin{equation}
\psi_{K \alpha_{i}}(\rho) = \psi_{0}(\rho) \chi_{K \alpha_{i},0}(\rho)
\end{equation}
where $\chi_{K \alpha_{i},0}(\rho)$ is the ${(K\alpha_{i})}^{th}$ element of the eigenvector, corresponding to the lowest eigen potential $U_0(\rho)$.

\section{Construction of Isospectral Potential}
In this section we present a bird's eye view of the scheme of construction of one parameter family of isospectral potentials. We have from Eq. (8) \begin{equation}
U_0(\rho) = E_0 + \frac{\hbar^{2}}{2\mu}\frac{\psi_0^{\prime\prime}(\rho)}{\psi_0(\rho)} \end{equation}
In 1-D supersymmetric quantum mechanics, one defines a superpotential for a system in terms of its ground state wave function ($\psi_{0}$) \cite{cooper-1995} as
\begin{equation}
W(\rho)= -\frac{\hbar}{\sqrt{2m}}\frac{\psi_{0}^{\prime}(\rho)}{\psi_{0}(\rho)}.
\end{equation}
The energy scale is next shifted by the ground state energy $(E_{0})$ of the
potential $U_0(\rho)$, so that in this shifted energy scale the new potential become
\begin{equation}
U_1(\rho) = U_0(\rho) - E_{0}=\frac{\hbar^{2}}{2\mu}\frac{\psi_0^{\prime\prime}(\rho)}{\psi_0(\rho)}\end{equation}
having its ground state at zero energy. One can then easily verify that $U_1(\rho)$ is expressible in terms of the superpotential via the Riccati equation
\begin{equation}
U_1(\rho) = W^{2}(\rho) - \frac{\hbar}{\sqrt{2m}}W^{\prime }(\rho).
\end{equation}
By introducing the operator pairs
\begin{equation}
\left. \begin{array}{lcl}
A^{\dag}&  =&  -\frac{\hbar}{\sqrt{2m}}\frac{d}{d\rho}+W(\rho)\\
A  &=&  \frac{\hbar}{\sqrt{2m}}\frac{d}{d\rho}+W(\rho)
\end{array} \right\}
\end{equation}
the Hamiltonian for $U_1$ becomes
\begin{equation}
H_1 = -\frac{\hbar^{2}}{2m} \frac{d^{2}}{d\rho^{2}} + U_1(\rho) = A^{\dag}A.
\end{equation}
The pair of opertors $A^{\dag}, A$ serve the purpose of creation and annihilation of nodes in the wave function. Next we introduce a partner Hamiltonian $H_{2}$, corresponding to the SUSY partner potential $U_2$ of $U_1$ as
\begin{equation}
H_{2}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{d\rho^{2}}+U_2(\rho)=AA^{\dag}
\end{equation}
where
\begin{equation}
U_2(\rho)=W^{2}(\rho)+\frac{\hbar}{\sqrt{2m}}W^{\prime}(\rho).
\end{equation}
Energy eigen values and wavefunctons corresponding to the SUSY partner Hamiltonians $H_1$ and $H_2$ are connected via the relations
\begin{equation}
\left. \begin{array}{lcl}
E_n^{(2)} & = & E_{n+1}^{(1)}, E_0^{(1)}=0 \; (n=0, 1, 2, 3,...),\\
\psi_n^{(n)}&=&\sqrt{E_{n+1}^{(1)}}A\psi_{n+1}^{(1)}\\
\psi_{n+1}^{(1)}&=&\sqrt{E_n^{(2)}}A^{\dagger}\psi_{n}^{(2)}\\
\end{array} \right\}\\
\end{equation}
where $E_{n}^{(i)}$ represents the energy of the $n^{th}$ excited state of $H_{i}$ (i=1, 2). Thus $H_{1}$ and $H_{2}$ have identical spectra, except the fact that the partner state of $H_{2}$ corresponding to the ground state of $H_{1}$ is absent in the spectrum of $H_{2}$ \cite{cooper-1995}. Hence the potentials $U_1$ and $U_2$ are {\bf not strictly isospectral}.
\newline However, one can construct, a one parameter family of {\bf strictly isospectral} potentials $\hat{U_1}(\lambda, \rho)$, explointing the fact that for a given $U_1(\rho)$, $U_0(\rho)$ and $W(\rho)$ are not unique (see Eqs. (12) \& (13)), since the Riccati equation is a nonlinear one. Following \cite{cooper-1995, khare-1989, darboux-1982}, it can be shown that the most general superpotential satisfying Riccati equation for $U_1(\rho)$ (Eq. (16)) is given by
\begin{equation}
\hat{W}(\rho)=W(\rho)+\frac{\hbar}{\sqrt{2m}}\frac{d}{d\rho}\log [I_{0}(\rho) +
\lambda]
\end{equation}
where $\lambda$ is a constant of integration, and $I_0$ is given by
\begin{equation}
I_{0}(\rho)={\displaystyle\int}_{\rho^{\prime}=0}^{\rho} {[\psi_{0}(\rho^{\prime})]}^{2} d\rho^{\prime},
\end{equation}
in which $\psi_0(\rho)$ is the normalized ground state wave function of $U_0(\rho)$. The potential
\begin{equation}
\hat{U_1}(\lambda, \rho)= \hat{W}^{2}(\rho)-\frac{\hbar}{\sqrt{2m}}
\hat{W}^{\prime}(\rho)
= U_1(\rho) - 2\frac{\hbar^{2}}{2m}
\frac{d^{2}}{d\rho^{2}}\log [I_0(\rho) + \lambda],
\end{equation}
has the same SUSY partner $U_2(\rho)$. $\hat{U_1}(\lambda, \rho)$ has its ground state at zero energy with the corresponding wavefunction given by
\begin{equation}
\hat{\psi_1}(\lambda, \rho)= \frac{\psi_1}{I_0+\lambda}.
\end{equation}
Hence, potentials $\hat{U_1}(\lambda, \rho)$ and $U_1(\rho)$ are {\bf strictly isospectral}. The parameter $\lambda$ is arbitrary in the intervals $-\infty<\lambda<-1$ and $0<\lambda<\infty$. $I_{0}(\rho)$ lies between 0 and 1, so the interval $-1\leq \lambda\leq 0 $ is forbidden, in order to bypass singularities in $\hat{U_1}(\lambda, \rho)$. For $\lambda \rightarrow \pm \infty$, $\hat{U_1} \rightarrow U_1$ and for $\lambda \rightarrow 0+$, $\hat{U_1}$ develops a narrow and deep attractive well in the viscinity of the origin. This well-barrier combination effectively traps the particle giving rise to a sharp resonance. This method has been
tested successfully for 3D finite square well potential \cite{das-2001} choosing parameters capabe of supporting one or more resonance state(s) in addition to one bound state. Nuclei $^{18, 20}$C have in their ground states $T=1, J^{\pi}=0^+$ and there exists a resonance state of the same $J^{\pi}$. Thus, the forgoing procedure starting from the ground state of $^{18, 20}$C will give $T=1, J^{\pi}=0^+$ resonance(s). In an attempt to  search for the correct resonance energy, we compute the probability of finding the system within the well region of the potential $\hat{U_1}(\lambda, \rho)$ corresponding to the energy $E$ ($>0$) by integrating the probability density up to the top of the barrier:
\begin{equation}
G(E)=\int_{\rho^{\prime}=0}^{\rho_B} |\hat{\psi_E}(\rho^{\prime},\lambda)|^2d\rho^{\prime}
\end{equation}
where $\rho_B$ indicates position of the top of the barrier component of the potential $\hat{U_1}(\lambda, \rho)$ for a chosen $\lambda$. Here $\hat{\psi_E}(\lambda, \rho)$ that represents the solution of the potential $\hat{U_1}(\lambda, \rho)$, corresponding to a positive energy $E$, is normalized to have a constant amplitude in the assymptotic region. Plot of the quantity $G(E)$ against increasing $E$ ($E > 0$) shows a peak at the resonance energy $E=E_R$. Choice of $\lambda$ has to be made judiciously to avoid numerical errors entering in the wavefunction in the extremely narrow well for $\lambda\rightarrow 0+$. The width of resonance can be obtained from the mean life of the state using the energy-time uncertainty relation. The mean life is reciprocal to the decay constant. And the decay constant is the product of the number of hit per unit time on the barrier and the corresponding probability of tunneling through the barrier.

\section{Results and discussions}
Eq.(6) is solved for the GPT n-n potential [31] and core-n SBB potential [32]. The range parameter for the core-n potential $b_{cn}$ is slightly adjusted to match the experimetal ground state spectra. The calculated two-neutron separation energies ($S_{2n}$), the relative convergence in energy (=$\frac{E(K_{max}+4)-E(K_{max})}{E(K_{max}+4)}$) and the rms matter radii (R$_A$) for gradualy increasing $K_{max}$ are listed in Table 1 for both of $^{18}$C and $^{20}$C. Although the computed results indicate a clear convergence trend with increasing $K_{max}$, it is far away from full convergence even at $K_{max}=24$. For this reason, we used an extrapolation technique succesfully used for atomic systems \cite{das-1994, khan-2012} as well as for nuclar system \cite{khan-2001}, to get the converged value of about 4.91 MeV for $^{18}$C and 3.51 MeV for $^{20}$C as shown in columns 2 and 4 of Table 2. Partial contribution of the different partial waves to the two-neutron separation enegies corresponding to $l_x = 0, 1, 2, 3, 4$ are presented in Table 3. Variation of the two-neutron separation as a function of $K_{max}$ is shown in Figure 1 for both the nuclei $^{18}$C and $^{20}$C. In Figure 2 we have shown the relative convergence trend in enrgies as a function of $K_{max}$. In Figures 3 and 4 we have presented a 3D view of the correlation density profile of the halo nuclei $^{18}$C and $^{20}$C. While in Figures 5 and 6 we have shown the 2D projection of the 3D probability density distribution. The figures clearly indicate the halo structure of the nuclei comprising a dense core surrounded by low density tail.
\newline After getting the ground state energy and wavefunctions we constructed the isospectral potential invoking principles of SSQM to investigate the resonant states. The lowest eigen potential obtained for the ground states of $^{18}$C and $^{20}$C as shown in yellow lines in Figures 7 and 8 exhibits shallow well followed by a broad and low barrier. This low well-barrier combination may indicate resonant states. However, since the well is very shallow and the barrier is not sufficiently high, the resonance width is very large and a numerical calculation of the resonant state is quite challenging. Hence, we constructed the one-parameter family of isospectral potentials $\hat{U}(\lambda, \rho)$ following Eq.(19) by appropriate selection $\lambda$ parameter values, such that a narrow and sufficiently deep well followed by a high barrier is obtained which are also shown in Figures 7 and 8 in representative cases. The enhanced well-barrier combination effectively traps the particles to form a strong resonant state. Calculated parameters of the isospectral potential for some $\lambda $ values, along with the original lowest eigen potential $U_0(\rho)$ (which corresponds to $\lambda\rightarrow\infty$) are presented in Table 4. One can, for example, note from Table 4 under $^{18}$C that, when $\lambda$ changes from = 0.1 to 0.0001, the depth of the well increases from -24.2 MeV at 2.7 fm to -247.6 MeV at 1.3 fm while the height of the barrier increases from 5.4 MeV at 5.1 fm to 121.5 MeV at 1.9 fm. The same trend is observed for $^{20}$C also. Thus the application of SSQM produces a {\bf dramatic effect} in the isospectral potential $\hat{U_1}(\lambda, \rho)$ as $\lambda$ approaches 0+. Further smaller positive values of $\lambda$ are not desirable since that will make the well too narrow to compute the wave functions accurately by a standard numerical technique.
The probability of trapping, $G(E)$, of the particle within the enhanced well-barrier combination as a function of the particle energies E shown in Figures 9 and 10 exhibit resonance peak at the energies $E_R\simeq 1.89$ MeV for $^{18}$C and at energy $E_R\simeq 3.735$ MeV for $^{20}$C respectively. It is interesting to see that the resonance energy is independent of the $\lambda$ parameter. The enhancement of accuracy in the determination of $E_R$ is the principal advantage of using Supersymmetric formalism. Since $\hat{U_1}(\lambda;\rho)$ is strictly isospectral with $U(\rho)$, any value of $\lambda$ is admissible in principle. However, a judicious choice of $\lambda$ is necessary for accurate determination of the resonance energy. The calculated two-neutron separation energies are in excellent agreement with the observed values $4.910\pm 0.030$ MeV for $^{18}$C and $3.510\pm 0.240$ MeV for $^{20}$C \cite{audi1-2003} and also with results of Yamaguchi et al
\cite{yamaguchi-2011} as presented in Table 5. The calculated RMS matter radii also agree fairly with the experimental values \cite{ozawa1-2001}.

\section{Summary and conclusions}
In this communication we have investigated the structure of $^{18, 20}$C using hyperspherical harmonics expansion method assuming $^{16,18}$C$+n+n$ three-body model. Standard GPT \cite{gogny-1970} potential is chosen for the $n-n$ pair while a three-term Gaussian SBB potential \cite{sack-1954} with adustable range parameter is used to compute the ground state energy and wavefunction. The one parameter familly of isospectral potentials constructed using the ground state wavefunctions succesfully explains the resonance states in both the systems. The method is a robust one and can be applied for any weakly bound system even if the system lacks any bound ground state.

\ack
\hspace*{1cm} This work has been supported by computational facility at Aliah University, India.
\section*{References}