EXERCISE 1: IRRADIANCE AT A SCREEN DUE TO A POINT SOURCE

We imagine that a point source of light at Ps=(xs,ys,zs) illuminates an aperture in an opaque barrier located at z=0 as shown below.

The irradiance of the light appearing at point P=(x,y,z) on the viewing screen located at z=zsc is then given by

(2)I(x,y,z)=Ar2T(x,y)

where A is a constant, r is the distance from PS (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 Ps 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

(3)T(x,y)={1,if |xS+xxSzzS|zS||<w2 and |yS+yySzzS|zS||<h20,otherwise.

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:

xS [cm] yS [cm] zS [cm] zsc [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?