How do I use the matrix elements, all of which are functions of lambda, to obtain R(lambda)? I get an error when I try to use the matrix elements to get R(lambda).)

If we let M be a 2x2 matrix with matrix elements M1, M2, M3, and M4, all of which are dependent on lambda, I cannot seem to use the matrix elements to plot R(lambda) vs lambda. M1, M2, M3, and M4 do get displayed in the command window, but I get an error when I try to use them to obtain R(lambda).

Soure code:

lambda_0 = 0.980;                                                          % Cavity length of DBR mirror in micron
d_G = 0.06975562993;                                                       % Thickness of GaAs layer in micron
d_A = 0.08148984639;                                                       % Thickness of AlAs layer in micron
n_G0 = 3.512261308;                                                        % GaAs index of refraction at 980 nm
n_A0 = 3.006509533;                                                        % AlAs index of refraction at 980 nm
lambda = 0.9:0.01:1.1;                                                     % Wavelength range in micron
n_G = @(lambda) sqrt(10.906+(0.97501./(lambda.^2-(0.52886)^2))-...         % Index of refraction of GaAs as a function of wavelength
    0.002467.*lambda.^2);
n_A = @(lambda) sqrt(7.986+(0.97501./(lambda.^2-(0.19886)^2))-...          % Index of refraction of AlAs as a function of wavelength
    0.0059457.*lambda.^2);
M_G = @(lambda) [cos((2*pi*d_G.*n_G(lambda))./lambda),...                  % Characteristic 2x2 Matrix of GaAs
    (-i./n_G(lambda)).*sin((2*pi*d_G.*n_G(lambda))./lambda);...
    -(i.*n_G(lambda)).*sin((2*pi*d_G.*n_G(lambda))./lambda),...
    cos((2*pi*d_G.*n_G(lambda))./lambda)];
M_A = @(lambda) [cos((2*pi*d_A.*n_A(lambda))./lambda),...                  % Characteristic 2x2 Matrix of AlAs
    (-i./n_A(lambda)).*sin((2*pi*d_A.*n_A(lambda))./lambda);...
    -(i.*n_A(lambda)).*sin((2*pi*d_A.*n_A(lambda))./lambda),...
    cos((2*pi*d_A.*n_A(lambda))./lambda)];

syms lambda
M = @(lambda) (M_G(lambda)*M_A(lambda))^32;                                % Characteristic matrix of DBR mirror consisting of 
% 32 pairs of AlAs/GaAs layers
tmp=num2cell(M(lambda));
[M1,M2,M3,M4]=deal(tmp{:})

syms lambda
M1 = @(lambda) M1;
M2 = @(lambda) M2;
M3 = @(lambda) M3;
M4 = @(lambda) M4;

R = @(lambda) (M2(lambda)./M1(lambda)).^2;                                 % Reflection coefficient of DBR mirror as a function of
% wavelength


figure(1)
plot(lambda,R(lambda),'k'),xlabel('Wavelength (um)'),ylabel('Reflection Coefficient')

Answer 1

[Karan Singh]

Hi Craig,

From what I understand, the goal is plot the parameters as “M1”, “M2”, “M3”, and “M4”, as they do get displayed in the command window, but you get an error when I try to use them to obtain "R(lambda)”.

The error you're encountering occurs because the “plot” function expects numeric data for the x and y axes. In your case, “lambda” is defined as a symbolic variable, and “R(lambda)” is a symbolic expression. To resolve this issue, you need to convert the symbolic expressions to numeric values before plotting.

Here's an updated version of your code that should work correctly:

lambda_0 = 0.980; 
d_G = 0.06975562993; 
d_A = 0.08148984639; 
n_G0 = 3.512261308; 
n_A0 = 3.006509533; 
lambda = 0.9:0.01:1.1; 

n_G = @(lambda) sqrt(10.906+(0.97501./(lambda.^2-(0.52886)^2))-0.002467.*lambda.^2); 
n_A = @(lambda) sqrt(7.986+(0.97501./(lambda.^2-(0.19886)^2))-0.0059457.*lambda.^2); 

M_G = @(lambda) [cos((2*pi*d_G.*n_G(lambda))./lambda), (-1i./n_G(lambda)).*sin((2*pi*d_G.*n_G(lambda))./lambda);... 
    -(1i.*n_G(lambda)).*sin((2*pi*d_G.*n_G(lambda))./lambda), cos((2*pi*d_G.*n_G(lambda))./lambda)]; 
M_A = @(lambda) [cos((2*pi*d_A.*n_A(lambda))./lambda), (-1i./n_A(lambda)).*sin((2*pi*d_A.*n_A(lambda))./lambda);... 
    -(1i.*n_A(lambda)).*sin((2*pi*d_A.*n_A(lambda))./lambda), cos((2*pi*d_A.*n_A(lambda))./lambda)]; 

M = @(lambda) M_G(lambda)*M_A(lambda); 
M_values = arrayfun(@(l) M(l), lambda, 'UniformOutput', false); 
M1 = cellfun(@(m) m(1, 1), M_values); 
M2 = cellfun(@(m) m(1, 2), M_values); 

R = @(lambda) (M2./M1).^2; 

figure(1) 
plot(lambda, R(lambda), 'k') 
xlabel('Wavelength (um)') 
ylabel('Reflection Coefficient') 

In this updated code, I've used “arrayfun” to evaluate the symbolic expressions “M(l)” for each value in the “lambda” array. Then, I used “cellfun” to extract the specific elements “M1” and “M2” from the cell array “M_values”. Finally, the “R(lambda)” calculation and plotting should work without errors.

Attached below are some documentation links that you may find helpful:

• Apply function to each element of array - MATLAB arrayfun (mathworks.com)
• Apply function to each cell in cell array - MATLAB cellfun (mathworks.com)
• Plot graph nodes and edges - MATLAB plot (mathworks.com)
• Anonymous Functions - MATLAB & Simulink (mathworks.com)

Hope this helps!

Karan Singh Khati