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Shadows (Ray Optics)
Developed by Ernest Behringer  Published July 16, 2016
This set of exercises guides the student in exploring computationally the behavior of light patterns and shadows generated by simple light sources together with apertures in thin, opaque barriers. It requires the student to generate, and describe the results of simulating, light patterns and shadows. Diffraction is ignored. The numerical approach used is summing over a twodimensional spatial grid while applying a logical mask ('transparency function'). Please note that this set of computational exercises can be affordably coupled to simple experiments with small light bulbs and apertures cut into (or barriers cut out of) opaque paper sheets. A possible extension is to compare the predicted light patterns to experimental measurements. This set of exercises could be incorporated as an initial activity in an intermediate optics laboratory.
Subject Area  Waves & Optics 

Levels  First Year and Beyond the First Year 
Available Implementations  Python and Easy Java Simulations 
Learning Objectives 
Students who complete this set of exercises will be able to
* predict and visually represent the irradiance distribution at a screen generated by a point light source and an aperture (**Exercise 1**);
* predict and visually represent the irradiance distribution at a screen generated by multiple point light sources and an aperture (**Exercise 2**);
* predict and visually represent the irradiance distribution at a screen generated by multiple point light sources and a complex aperture (**Exercise 3**);
* predict and visually represent the irradiance distribution at a screen generated by a twodimensional array of point light sources and an aperture (**Exercise 4**);, and
* predict and visually represent the irradiance distribution at a screen generated by a twodimensional array of point light sources and an opaque barrier (**Exercise 5**).

Time to Complete  120 min 
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
###Exercise 1: Irradiance at a screen due to a point source
We imagine that a point source of light at $P_s=(x_s,y_s,z_s)$ illuminates an aperture in an opaque barrier located at $z=0$ as shown below.
![Alt Figure](images/Shadows/Single_source_perspective_view.png "")
The irradiance of the light appearing at point $P=(x,y,z)$ on the viewing screen located at $z=z_{sc}$ is then given by
$$
I(x,y,z) = {{A}\over{r^2}}T(x,y)
$$
where $A$ is a constant, $r$ is the distance from $P_S$ (the source) to $P$ (the screen point), and $T(x,y)$ is a function we call the 'transparency function'. If the line from the source at $P_s$ to the screen point $P$ passes through the aperture, then $T(x,y) = 1$; otherwise, $T(x,y)=0$. In the case where the aperture is a rectangle of width $w$ in the $x$direction and height $h$ in the $y$direction, show that the value of the transparency function is given byAlt Figure
$$
\begin{equation}
T(x,y)=\begin{cases}
1, & \text{if $\Bigglx_S + {{xx_S}\over{zz_S}}\biglz_S\bigr\biggr < {{w}\over{2}}$ and $\Biggly_S + {{yy_S}\over{zz_S}}\biglz_S\bigr\biggr < {{h}\over{2}}$}\\
0, & \text{otherwise}.
\end{cases}
\end{equation}
$$
Compute the irradiance on an array of uniformly spaced screen points on the viewing screen for an aperture with $w=4.0$ cm and $h=3.0$ cm in the following cases:
$x_S$ [cm]  $y_S$ [cm]  $z_S$ [cm]  $z_{sc}$ [cm]
::  ::  ::  ::
0.0  0.0  20.0  40.0
5.0  0.0  20.0  40.0
0.0  5.0  20.0  40.0
0.0  0.0  10.0  40.0
0.0  0.0  40.0  40.0
0.0  0.0  20.0  60.0
0.0  0.0  20.0  20.0
Does the computed light pattern change as you expect when you change the source and screen locations? Are the edges of the light pattern sharp or fuzzy? (How do you define "sharp" and "fuzzy"?) Why?
###Exercise 2: Irradiance due to $N$ point sources with a rectangular aperture
If instead of one point source, suppose $N$ point sources are uniformly spaced and also arranged along a line segment of total length $L$ that is parallel to the $x$axis, as shown below.
![Alt Figure](images/Shadows/NSources_top_view.png "")
Calculate the irradiance at the screen for the aperture of Exercise 1 if one source is on the symmetry axis, $L= 4.0$ cm, and: (a) $N=3$; (b) $N=11$; (c) and $N=101$. How many distinct shadow regions (areas characterized by different irradiance) appear in each case?
###Exercise 3: Irradiance due to $N$ point sources and an Lshaped aperture
Now imagine that you have $N$ point sources, as in Exercise 2, but now the aperture is an Lshaped aperture as shown below.
![Alt Figure](images/Shadows/Laperture.png "")
What do you expect the irradiance pattern at the screen to look like? Why? Now calculate the irradiance pattern at the screen for the case $N=101$ and $w=2.0$ cm. Does the computed pattern resemble your prediction?
###Exercise 4: Irradiance due to $N \times N$ point sources and a rectangular aperture
Imagine you have $N\times N$ point sources arranged uniformly on a square grid, with side length $L$, and illuminating the rectangular aperture of Exercise 1. What do you expect the irradiance pattern to look like when the aperture is much smaller than the point source array? When the point source array is much smaller than the aperture? Use the aperture from Exercise 1, with $z_s = 20$ cm and $z_{sc} = 40$ cm and the following parameters to calculate the irradiance at the screen: $N=11$ and (a) $L = 3.0$ cm, (b) $L = 0.3$ cm, and (c) $L = 30.0$ cm.
###Exercise 5: Irradiance due to a model circular extended source and a rectangular barrier
Imagine you have $N\times N$ point sources arranged on a square grid of side length $L$. Now allow *only* the sources within a circle of radius $L/2$ to illuminate a rectangular barrier. What do you expect the irradiance pattern at the screen to look like when the barrier is much smaller than the array of point sources? Calculate the irradiance pattern when $L=3.0$ cm and $N = 11$, with a *barrier* of the same dimensions as the aperture of Exercise 1, and with $z_s = 20$ cm and $z_{sc} = 40$ cm. Make sure to 'extinguish' sources in the $N\times N$ grid that lie outside a radius of $L/2$.
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Credits and Licensing
Ernest Behringer, "Shadows (Ray Optics)," Published in the PICUP Collection, July 2016.
The instructor materials are ©2016 Ernest Behringer.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license