About
Developed by E. Behringer
This set of exercises guides the student to model the results of an experiment to determine the profile of a laser beam using a knife-edge technique. It requires the development of the model of the knife-edge profile, and fitting of the model profile to experimental data. Here, the computational tasks are handled by built-in functions of the computational tool being used to complete these exercises.
Subject Area | Waves & Optics |
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Level | Beyond the First Year |
Available Implementation | Python |
Learning Objectives |
Students who complete this set of exercises will be able to
|
Time to Complete | 120 min |
EXERCISE 1: THE TEM MODE: “LINE CUTS”
The irradiance (power per unit area) of a laser oscillating in the TEM mode can be expressed as
where is the maximum irradiance of the beam and is a parameter that describes the width (i.e., the spatial extent) of the beam. You can show that when , the locations at which are . Show that when and , the irradiance is and show that the above expression for the irradance is equivalent to
where the “scaled variables” and are defined as . We can relate this expression to the total power transmitted by the beam:
(a) Assume mW and mm to generate a line plot of the irradiance on the vertical axis versus on the horizontal axis for for different values of and .
(b) Repeat (a) but now plot the irradiance divided by the maximum irradiance for that particular value of (in other words, plot ) versus on the horizontal axis. Comment on the resulting plot.
#
# Beam_Profile_Exercise_1.py
#
# This file will generate a plot of
# the irradiance I(x,y) of the TEM00 Gaussian Mode of a laser
# as a function of scaled x for several different values of scaled y.
#
# This file will also generate a plot of
# the scaled irradiance I(x,y) of the TEM00 Gaussian Mode of a laser
# as a function of scaled x for several different values of scaled 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.
#
# 20160628 ERB
#
from pylab import xlim,xlabel,ylim,ylabel,show,plot,figure,grid,legend
from numpy import linspace,exp,pi,zeros
# Generate scaled x and y values
max_xsc = 2.0 # maximum value of x position [mm]
min_xsc = -2.0 # minimum value of x position [mm]
Npts = 81
xsc = linspace(min_xsc,max_xsc,Npts) # array of scaled x values
ysc = [0.0,0.2,0.4,0.6,0.8,1.0]
# other inputs
w0 = 1.00 # beam width [mm]
P_0 = 1.00 # laser power [mW]
I_0 = 2.00*P_0/(pi*w0*w0) # maximum irradiance [mW/mm2]
# Set up array of irradiances
I = zeros((6,Npts))
for i in range(0,6):
I[i] = I_0 * exp(-2.0*(xsc*xsc + ysc[i]*ysc[i]))
# Generate the line plot of irradiance versus scaled x values
# for different scaled y values
figure()
# Here we plot unscaled I versus the scaled x for different scaled y values,
plot(xsc,I[0],'mo-',label='\(\\tilde y = 0.0\)',markevery=8)
plot(xsc,I[1],'bD-',label='\(\\tilde y = 0.2\)',markevery=4)
plot(xsc,I[2],'g>-',label='\(\\tilde y = 0.4\)',markevery=2)
plot(xsc,I[3],'ys-',label='\(\\tilde y = 0.6\)',markevery=5)
plot(xsc,I[4],color='#ffaa00',marker='^',linestyle='-',label='\(\\tilde y = 0.8\)',markevery=4)
plot(xsc,I[5],'r*-',label='\(\\tilde y = 1.0\)',markevery=3)
# Define the limits of the horizontal axis
xlim(min(xsc),max(xsc))
# Label the horizontal axis, with units
xlabel("\(\\tilde x\)", size = 16)
# Define the limits of the vertical axis
ylim(min(I[0]),max(I[0]))
# Label the vertical axis, with units
ylabel("\(I\) [mW/mm\(^2\)]", size = 16)
grid('on')
legend()
show()
# Generate the line plot of scaled I versus scaled x
figure()
# Here we plot scaled I versus scaled x for different scaled y values,
plot(xsc,I[0]/max(I[0]),'mo-',label='\(\\tilde y = 0.0\)',markevery=8)
plot(xsc,I[1]/max(I[1]),'bD-',label='\(\\tilde y = 0.2\)',markevery=4)
plot(xsc,I[2]/max(I[2]),'g>-',label='\(\\tilde y = 0.4\)',markevery=2)
plot(xsc,I[3]/max(I[3]),'ys-',label='\(\\tilde y = 0.6\)',markevery=5)
plot(xsc,I[4]/max(I[4]),color='#ffaa00',marker='^',linestyle='-',label='\(\\tilde y = 0.8\)',markevery=4)
plot(xsc,I[5]/max(I[5]),'r*-',label='\(\\tilde y = 1.0\)',markevery=3)
# Define the limits of the horizontal axis
xlim(min(xsc),max(xsc))
# Label the horizontal axis, with units
xlabel("\(\\tilde x\)", size = 16)
# Define the limits of the vertical axis
ylim(0.0,1.0)
# Label the vertical axis, with units
ylabel("\(I/I_{max,\\tilde{y}}\)", size = 16)
grid('on')
legend()
show()
Translations
Code | Language | Translator | Run | |
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Credits
Fremont Teng; Loo Kang Wee
Sample Learning Goals
[text]
For Teachers
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Research
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Video
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Version:
- https://www.compadre.org/PICUP/exercises/exercise.cfm?I=134&A=laser_beam_profile
- http://weelookang.blogspot.com/2018/06/laser-beam-profile-exercise-1-tem00.html
Other Resources
[text]
end faq
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