This section discusses how to obtain the H field components of far field projections.
Calculating \( \vert H \vert^2\) in the far field
To calculate \(\mid H\mid^{2}\) in the far field, you can simply use the relation \(\mid H\mid^{2}=n^{2}\frac{\epsilon_0}{\mu_0}\mid E\mid^{2}\)
Calculating H in the intermediate and far field
We use the farfieldexact script commands to calculate the components of the electric field at any specific point in the farfield. In this example (based on the nanoslit device shown in Wang et. al.), we can calculate the E field intensity on top of the device to analyze the focusing properties of this nanoslit structure. This is done using the following script commands:
# define desired region of x and y
x = linspace(-5e-6,5e-6,200);
y = linspace(1e-6,50e-6,500);
# do far field projection
E2 = farfieldexact2d('monitor1',x,y);
E2 = sum(abs(E2)^2,3); # calculate E2 from Ex, Ey, Ez
# plot E field intensity profile as a function of x,y
image(x*1e6,y*1e6,E2,'x (microns)','y (microns)','E field intensity');
Next, we can calculate the H fields numerically from the E fields using Maxwell's equation:
$$\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}=i\omega \mu_0H_x$$
$$\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}=i\omega \mu_0H_y$$
$$\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial z}=i\omega \mu_0H_z$$
For example, in the attached script, Hz is calculated using the following script commands:
delta = 1e-9; # used to calculate the numerical derivative;
f = getdata("T","f");
Ey1 = pinch(farfieldexact2d("T",x-delta,y),3,2);
Ey2 = pinch(farfieldexact2d("T",x+delta,y),3,2);
Ex1 = pinch(farfieldexact2d("T",x,y-delta),3,1);
Ex2 = pinch(farfieldexact2d("T",x,y+delta),3,1);
dEy_dx = (Ey2-Ey1)/(2*delta);
dEx_dy = (Ex2-Ex1)/(2*delta);
Hz = (dEy_dx - dEx_dy) / (1i * 2* pi * f * mu0);
# plot H field intensity profile as a function of x,y
image(x*1e6,y*1e6,abs(Hz)^2,"x (um)","y (um)","Hz");
Related publications
B. Wang and G. P. Wang, "Directional beaming of light from a nanoslit surrounded by metallic heterostructures," Appl. Phys. Lett. 88, 013114 (2006)