The cardinal sine function ($sinc x= \frac{\sin x}{x}$) admits a maximum at 0, and becomes zero for $x$ multiple of $\pi$.
%pylab inline
Populating the interactive namespace from numpy and matplotlib
x = linspace(-5*pi,5*pi, 500)
y = sin(x)/x
plt.plot(x, y,"--r", linewidth=2,label=r"$sinc \ x$")
plt.legend()
<matplotlib.legend.Legend at 0x7f67ea8cdc10>
The expression for the electric field is written as follows: \begin{equation} E(X, Y) = E_0 \ sinc (\frac{\pi X b}{\lambda D}) \ sinc (\frac{\pi Y h}{\lambda D}) \end{equation}
The Intesity is given by $ I(X,Y) = K E(X, Y) E(X, Y)^*$, where $K$ is a constant of proporsonality and $*$ is symbol for complex conjugate. Therefor, with $I_0 = K(E_0)^2$ and $k = \frac{2 \pi}{\lambda}$ is the wavenumber:
\begin{equation} I(X,Y) = I_0 sinc^2 (\frac{ k b}{2 D}X) \ sinc^2 (\frac{ k h}{2 D}Y) \end{equation}Let: $B = \frac{k b}{2 D}$ and $H = \frac{k h}{2 D}$ \begin{equation} I(M) = I_0 \ sinc^2 (B X) \ sinc^2 (H Y) \end{equation}
This function is the product of two cardinal sine functions, one is a function of $X$ and the other is a function of $Y$. The intensity distribution will have the shape of two perpendicular lines of light each formed by a succession of more or less intense spots.
%run "scripts/rect_apertureV3.py"
from ipywidgets import interactive
from IPython.display import clear_output, display, HTML
w=interactive(rect_apert, lamda=(400, 800),
b=(0.01*1.E-3,0.3*1.E-3,0.01*1.E-3),
h=(0.01*1.E-3,0.3*1.E-3,0.01*1.E-3),
D=(0.5,3))
display(w)
%run "scripts/Diff_1slit_1D_V3.py"
w=interactive(Diff_1S, lamda=(300, 800),
b=(0.01*1.E-3,0.3*1.E-3,0.01*1.E-3),
D=(0.5,3))
display(w)
%run "scripts/Diff_1slit_2D_V3.py"
w=interactive(Diff_1S, lamda=(400, 800),
b=(0.02*1.E-3,0.3*1.E-3,0.01*1.E-3),
D=(0.5,2))
display(w)
where $J_1$ is the Bessel function of the first kind and $A1(M)$ and $A2(M)$ are given by:
$A_1(M)=k r \frac{r'_1(M)}{D}$ where $r'_1(M)=\sqrt{(X_1(M)-d)^2+Y_1^2(M)}$
$A_2(M)=k r \frac{r'_2(M)}{D}$ where $r'_2(M)=\sqrt{(X_2(M)+d)^2+Y_2^2(M)}$
$k$ is the wavenumber, $r$ is the radius of the circular aperture and $D$ is the distance between the screen and the aperture.
%run "scripts/Rayleigh_1D_V3.py"
w=interactive(Rayleigh_1d, lamda=(400, 800),
r=(1.E-5,5*1.E-4,1.E-5),
d=(1*1.E-3,20*1.E-3,1*1.E-3),
D=(0.5,3))
display(w)
%run "scripts/Rayleigh_2D_V3.py"
w=interactive(Rayleigh_2d, lamda=(400, 800),
r=(1.E-5,5*1.E-4,1.E-5),
d=(1*1.E-3,20*1.E-3,1*1.E-3),
D=(0.5,3))
display(w)