About

Subject Area	Waves & Optics
Level	Beyond the First Year
Available Implementation	Python
Learning Objectives	Students who complete this set of exercises will be able to express an equation predicting the profile of a laser oscillating in the TEM00 mode in terms of dimensionless (“scaled”) variables suitable for coding (Exercise 1); produce both line plots and contour plots of the (scaled) irradiance of the beam versus (scaled) position(s) (Exercises 1 and 2); develop a model of, and plot, the knife-edge profile of the laser beam (Exercise 3); and fit the model, i.e., of the irradiance versus knife-edge position, to experimental data (Exercise 4)
Time to Complete	120 min

EXERCISE 1: THE TEM00 MODE: “LINE CUTS”

The irradiance I (power per unit area) of a laser oscillating in the TEM00 mode can be expressed as

where I0 is the maximum irradiance of the beam and w0 is a parameter that describes the width (i.e., the spatial extent) of the beam. You can show that when y=0, the locations x1/2 at which I=I0/2 are x1/2=±(0.5ln2)1/2w0=±0.59w0. Show that when y=0 and x=w0, the irradiance is I=I0exp(−2)=0.135 and show that the above expression for the irradance is equivalent to

where the “scaled variables” x~ and y~ are defined as x~≡x/w0. We can relate this expression to the total power P0 transmitted by the beam:

(a) Assume P0=1.0 mW and w0=1.0 mm to generate a line plot of the irradiance I on the vertical axis versus x~ on the horizontal axis for −2≤x~≤2 for different values of y~=0.0,0.2,0.4,0.6,0.8 and 1.0.

(b) Repeat (a) but now plot the irradiance divided by the maximum irradiance for that particular value of y~ (in other words, plot I/Imax,y~) versus x~ 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

Credits

Fremont Teng; Loo Kang Wee

Sample Learning Goals

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For Teachers

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Research

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Video

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 Version:

1.  https://www.compadre.org/PICUP/exercises/exercise.cfm?I=134&A=laser_beam_profile
2. http://weelookang.blogspot.com/2018/06/laser-beam-profile-exercise-1-tem00.html

Other Resources

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