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Shadows (Ray Optics)

Developed by E. Behringer

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 two-dimensional 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 Implementation Python
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 two-dimensional array of point light sources and an aperture (Exercise 4);, and
  • predict and visually represent the irradiance distribution at a screen generated by a two-dimensional array of point light sources and an opaque barrier (Exercise 5).
Time to Complete 120 min
Exercise 3

EXERCISE 3: IRRADIANCE DUE TO Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> POINT SOURCES AND AN L-SHAPED APERTURE

Now imagine that you have Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> point sources, as in Exercise 2, but now the aperture is an L-shaped aperture as shown below.

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 Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">101style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> and wstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">2.0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> cm. Does the computed pattern resemble your prediction?

Shadows_Exercise_3.py

#

# Shadows_Exercise_3.py

#

# A linear array of N point sources located a specified distance from

# an L-shaped aperture composed of two rectangular apertures,

# one of which is centered on the origin.

# A screen is located a specified distance from

# the aperture.

#

# This file will generate a filled contour plot of

# the irradiance of light reaching the screen

# versus lateral coordinates (x,y)

#

# Written by:

#

# Ernest R. Behringer

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

# (734) 487-8799

# This email address is being protected from spambots. You need JavaScript enabled to view it.

#

# 20160112-13 by ERB one aperture

# 20160119 by ERB compound aperture

# 20160609 clean up by ERB

#

# import the commands needed to make the plot

from pylab import axis,xlim,ylim,xlabel,ylabel,show,contourf,colorbar,figure,title

from matplotlib import cm

# import the command needed to make a 1D array

from numpy import meshgrid,absolute,where,zeros,linspace

# inputs

zs = -20.0 # distance between the source and aperture [cm]

zsc = 40.0 # distance between the screen and aperture [cm]

nso = 101 # number of point sources

length_so = 4.0 # length of the array of point sources [cm]

ap1_width = 2.0 # aperture width [cm]

ap1_height = 1.0 # aperture height [cm]

ap2_width = 1.0 # second aperture width [cm]

ap2_height = 3.0 # second aperture height [cm]

ap2_x = -0.5 # second aperture center x-coordinate [cm]

ap2_y = 2.0 # second aperture center y coordinate [cm]

screen_width = 60.0 # screen width [cm]

screen_height = 60.0 # wcreen height [cm]

nw = 240 # Number of screen width intervals

nh = 240 # Number of screen height intervals

# initialize needed arrays to zero

rso = zeros((nso,2)) # (x,y) location of the source [cm,cm]

screen_x = zeros((nw+1,nh+1)) # x-coordinates of screen points

screen_y = zeros((nw+1,nh+1)) # y coordinates of screen points

rsq = zeros((nw+1,nh+1)) # r squared values

irradiance = zeros((nw+1,nh+1)) # irradiance values

x0 = zeros((nw+1,nh+1)) # x coordinate of intersection at aperture plane

y0 = zeros((nw+1,nh+1)) # y coordinate of intersection at aperture plane

# create 1D arrays to create a meshgrid for contour plotting

screen_xx = linspace(-0.5*screen_width,0.5*screen_width,nw+1)

screen_yy = linspace(-0.5*screen_height,0.5*screen_height,nh+1)

# generate the meshgrid

screen_xx, screen_yy = meshgrid(screen_xx,screen_yy)

# Set up point source coordinates

for i in range (0,nso):

for j in range (0,2):

if j==0: # the x-coordinate is

rso[i,j] = -0.5*length_so + i*length_so/(nso-1)

else: # j = 1 and the y-coordinate is

rso[i,j] = 0.0

# Calculate grid increments

deltaw = screen_width/nw # grid increment, width [cm]

deltah = screen_height/nh # grid increment, height [cm]

# Define the array of screen x and screen y values

for i in range (0,nw+1):

for j in range (0,nh+1):

screen_x[i,j] = -0.5*screen_width + deltaw*i

screen_y[i,j] = -0.5*screen_height + deltah*j

# Calculate the irradiance at each screen point

for k in range (0,nso):

for i in range (0,nw+1):

for j in range (0,nh+1):

# First calculate square of distance from source to screen

rsq[i,j] = (screen_x[i,j]-rso[k,0])**2 + (screen_y[i,j]-rso[k,1])**2 + (zsc-zs)**2

# Calculate x and y coordinates at the aperture

x0[i,j] = rso[k,0] + abs(zs)*(screen_x[i,j] - rso[k,0])/(zsc - zs)

y0[i,j] = rso[k,1] + abs(zs)*(screen_y[i,j] - rso[k,1])/(zsc - zs)

# Check if the ray coordinates at the aperture fall within the apertures

maskx1 = where(absolute(x0) < 0.5*ap1_width,1.0,0.0)

masky1 = where(absolute(y0) < 0.5*ap1_height,1.0,0.0)

maskx2 = where((x0 < ap2_x + 0.5*ap2_width)&(x0 > ap2_x - 0.5*ap2_width),1.0,0.0)

masky2 = where((y0 < ap2_y + 0.5*ap2_height)&(y0 > ap2_y - 0.5*ap2_height),1.0,0.0)

# Calculate the irradiance (note that we are accumulating irradiance)

irradiance = irradiance + (maskx1*masky1 + maskx2*masky2)/rsq

# make a filled contour plot of the period vs overlap and length ratios

figure()

contourf(screen_yy,screen_xx,irradiance,100,cmap=cm.bone)

title('Illumination pattern: \(w = \)%s cm, \(h = \)%s; \(N = \)%d'%(ap1_width,ap1_height,nso))

axis('equal')

xlim(-0.5*screen_width,0.5*screen_width)

ylim(-0.5*screen_height,0.5*screen_height)

xlabel("\(x\) [cm]")

ylabel("\(y\) [cm]")

colorbar().set_label(label='Irradiance [arb. units]',size=16)

show()

EXERCISE 3: IRRADIANCE DUE TO N N  POINT SOURCES AND AN L-SHAPED APERTURE
 

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Software Requirements

SoftwareRequirements

Android iOS Windows MacOS
with best with Chrome Chrome Chrome Chrome
support fullscreen? Yes. Chrome/Opera No. Firefox/ Sumsung Internet Not yet Yes Yes
cannot work on some mobile browser that don't understand JavaScript such as..... cannot work on Internet Explorer 9 and below

 

Credits

Fremont Teng; Loo Kang Wee

end faq

 Version:

  1. https://www.compadre.org/PICUP/exercises/Exercise.cfm?A=Shadows&S=6
  2. http://weelookang.blogspot.com/2018/06/shadows-ray-optics.html 

end faq

 

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