rearranging a scattering matrix
$begingroup$
Please I need some assistance. I am formulating a scattering matrix problem for some multilayers.
In my formulation, I ended up with the following matrix equation
$$
begin{bmatrix}
c^{ ′+}_2 \
c^{ ′-}_2 \
end{bmatrix} =
begin{bmatrix}
I & J & \
M & N & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_1 \
end{bmatrix}
$$
where I, J, M and N are matrices themselves. Now, the problem is rearranging this equation (up) to take the form of a scattering matrix as shown below
$$
begin{bmatrix}
c^{ ′-}_1 \
c^{ ′+}_2 \
end{bmatrix} =
begin{bmatrix}
S_{11} & S_{1} & \
S_{21} & S_{22} & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_2 \
end{bmatrix}
$$
The farthest I could go on my own was by decomposing it into a two sets of linear equations (not even sure if this was the way to go about it) and from there, I had no clue what else to do. I could not find any resources/clues on this problem out there. I would very much appreciate it if you could give me any reference,hint or clue on how to proceed with this. Thanks.
linear-algebra matrices matrix-equations
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
$endgroup$
add a comment |
$begingroup$
Please I need some assistance. I am formulating a scattering matrix problem for some multilayers.
In my formulation, I ended up with the following matrix equation
$$
begin{bmatrix}
c^{ ′+}_2 \
c^{ ′-}_2 \
end{bmatrix} =
begin{bmatrix}
I & J & \
M & N & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_1 \
end{bmatrix}
$$
where I, J, M and N are matrices themselves. Now, the problem is rearranging this equation (up) to take the form of a scattering matrix as shown below
$$
begin{bmatrix}
c^{ ′-}_1 \
c^{ ′+}_2 \
end{bmatrix} =
begin{bmatrix}
S_{11} & S_{1} & \
S_{21} & S_{22} & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_2 \
end{bmatrix}
$$
The farthest I could go on my own was by decomposing it into a two sets of linear equations (not even sure if this was the way to go about it) and from there, I had no clue what else to do. I could not find any resources/clues on this problem out there. I would very much appreciate it if you could give me any reference,hint or clue on how to proceed with this. Thanks.
linear-algebra matrices matrix-equations
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
$endgroup$
add a comment |
$begingroup$
Please I need some assistance. I am formulating a scattering matrix problem for some multilayers.
In my formulation, I ended up with the following matrix equation
$$
begin{bmatrix}
c^{ ′+}_2 \
c^{ ′-}_2 \
end{bmatrix} =
begin{bmatrix}
I & J & \
M & N & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_1 \
end{bmatrix}
$$
where I, J, M and N are matrices themselves. Now, the problem is rearranging this equation (up) to take the form of a scattering matrix as shown below
$$
begin{bmatrix}
c^{ ′-}_1 \
c^{ ′+}_2 \
end{bmatrix} =
begin{bmatrix}
S_{11} & S_{1} & \
S_{21} & S_{22} & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_2 \
end{bmatrix}
$$
The farthest I could go on my own was by decomposing it into a two sets of linear equations (not even sure if this was the way to go about it) and from there, I had no clue what else to do. I could not find any resources/clues on this problem out there. I would very much appreciate it if you could give me any reference,hint or clue on how to proceed with this. Thanks.
linear-algebra matrices matrix-equations
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
$endgroup$
Please I need some assistance. I am formulating a scattering matrix problem for some multilayers.
In my formulation, I ended up with the following matrix equation
$$
begin{bmatrix}
c^{ ′+}_2 \
c^{ ′-}_2 \
end{bmatrix} =
begin{bmatrix}
I & J & \
M & N & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_1 \
end{bmatrix}
$$
where I, J, M and N are matrices themselves. Now, the problem is rearranging this equation (up) to take the form of a scattering matrix as shown below
$$
begin{bmatrix}
c^{ ′-}_1 \
c^{ ′+}_2 \
end{bmatrix} =
begin{bmatrix}
S_{11} & S_{1} & \
S_{21} & S_{22} & \
end{bmatrix} begin{bmatrix}
c^{ ′+}_1 \
c^{ ′-}_2 \
end{bmatrix}
$$
The farthest I could go on my own was by decomposing it into a two sets of linear equations (not even sure if this was the way to go about it) and from there, I had no clue what else to do. I could not find any resources/clues on this problem out there. I would very much appreciate it if you could give me any reference,hint or clue on how to proceed with this. Thanks.
linear-algebra matrices matrix-equations
linear-algebra matrices matrix-equations
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
asked Jan 3 at 0:41
LEWIS ASILEVILEWIS ASILEVI
11
11
add a comment |
add a comment |
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