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Laser Beam Profile

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
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

Beam_Profile_Exercise_2.py

# Beam_Profile_Exercise_2.py

# This file will generate a filled contour plot of

# the irradiance I(x,y) of the TEM00 Gaussian Mode of a laser

# versus x and y.

# This file will also generate a filled contour plot of

# the scaled irradiance I(x,y)/I_max of the TEM00 Gaussian Mode of a laser

# versus scaled x and 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.

# with contributions from graduate student

# Najwa Sulaiman

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

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

# 20160529 ERB

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

from matplotlib.colors import LinearSegmentedColormap

from numpy import linspace,meshgrid,exp,sqrt,pi,amax

# Define the colormaps for these contour plots.

# For a HeNe laser beam profile, we might want red to be bright

# and black to be dark. This is the color map labeled

# black_red1.

# If instead we want white to be dark,

# then we use the color map labeled white_red1.

# The first digit in each 3-tuple is the scaled intensity

# (0 for lowest value, 1 for highest value).

# The second two digits are identical and represent the

# R-, G-, or B-value at that scaled intensity. Here,the

# R-value increases linearly with intensity while the

# G- and B-values are always zero.

cdict1 = {'red': ((0.0, 0.0, 0.0),

(1.0, 1.0, 1.0)),

'green': ((0.0, 0.0, 0.0),

(1.0, 0.0, 0.0)),

'blue': ((0.0, 0.0, 0.0),

(1.0, 0.0, 0.0))}

black_red1 = LinearSegmentedColormap('Black_Red1',cdict1)

cdict2 = {'red': ((0.0, 1.0, 1.0),

(1.0, 1.0, 1.0)),

'green': ((0.0, 1.0, 1.0),

(1.0, 0.0, 0.0)),

'blue': ((0.0, 1.0, 1.0),

(1.0, 0.0, 0.0))}

white_red1 = LinearSegmentedColormap('White_Red1',cdict2)

#x and y array values to be used later

max_x = 2.0 # maximum value of x position [mm]

min_x = -2.0 # minimum value of x position [mm]

max_y = max_x # maximum value of y position [mm]

min_y = min_x # minimum value of y position [mm]

Npts = 81

x = linspace(min_x,max_x,Npts) #x data point array

y = linspace(min_y,max_y,Npts) #y data point array

# create mesh of points on which irradiance is evaluated

X, Y = meshgrid(x,y)

# Now scale the x and y values by the beam width w0

w0 = 0.5 # beam width w0 [mm]

scaled_x = x/w0

scaled_y = y/w0

#creating grid table for function to be set on

Xsc, Ysc = meshgrid(scaled_x,scaled_y)

# other inputs

P_0 = 1.00 # laser power [mW]

I_0 = 2.00*P_0/(pi*w0*w0) # maximum irradiance [mW/mm2]

I = I_0 * exp(-2.0*(X*X + Y*Y))

# define the scaled irradiance

Imax = amax(I)

Isc = I/Imax

# Generate the contour plot of irradiance

# versus x and y with black as dark and red as bright

figure()

# Use the colormap black_red1

contourf(X,Y,I,100,cmap=black_red1)

# Define the limits of the horizontal axis

xlim(min_x,max_x)

# Label the horizontal axis, with units

xlabel("\(x\) [mm]", size = 16)

# Define the limits of the vertical axis

ylim(min_y,max_y)

# Label the vertical axis, with units

ylabel("\(y\) [mm]", size = 16)

# Plot title

title('\(w_0 = \)%s mm, \(P_0 = \)%s mW'%(w0,P_0))

# Show the colorbar

colorbar(label='Irradiance [mW/mm\(^2\)]')

show()

# Generate the contour plot of scaled irradiance

# versus scaled x and scaled y with black as dark and red as bright

figure()

# Use the colormap black_red1

contourf(Xsc,Ysc,Isc,100,cmap=black_red1)

# Define the limits of the horizontal axis

xlim(min_x,max_x)

# Label the horizontal axis, with units

xlabel("\(\\tilde{x}\equiv x/w_0\)", size = 16)

# Define the limits of the vertical axis

ylim(min_y,max_y)

# Label the vertical axis, with units

ylabel("\(\\tilde{y}\equiv y/w_0\)", size = 16)

# Plot title

title('\(w_0 = \)%s mm, \(P_0 = \)%s mW'%(w0,P_0))

# Show the colorbar

colorbar(label='Scaled irradiance')

show()

Exercise 2

EXERCISE 2: THE TEM MODE: A CONTOUR PLOT

(a) A filled contour plot is a plot that indicates value by color as a function of two variables. Make a plot of the irradiance versus the variables and for , with mm and mW.

(b) Using the same values for and , make a filled contour plot of the scaled irradiance versus the scaled variables and for .

Translations

Code	Language		Translator	Run

Credits

Fremont Teng; Loo Kang Wee

students in modeling and analyzing the profile of a laser beam using the knife-edge technique. Exercise 2 specifically focuses on generating contour plots of the TEM00 Gaussian mode of a laser beam's irradiance.

Main Themes and Important Ideas/Facts:

Objective: The primary goal of this exercise is for students to learn how to generate and interpret contour plots of the irradiance of a laser beam operating in the TEM00 mode. This includes understanding both the actual irradiance and the scaled irradiance in relation to spatial coordinates. The exercise builds upon the foundational understanding developed in Exercise 1, which likely covers the equation for the TEM00 mode.
TEM00 Gaussian Mode: The exercise explicitly deals with the "TEM00 Gaussian Mode" of a laser. This is the fundamental transverse electromagnetic mode of a laser resonator, characterized by a Gaussian intensity profile.
Contour Plots: The exercise emphasizes the creation of "filled contour plots," which visually represent the irradiance value as a function of two spatial variables (x and y) using color. This allows for a clear visualization of the laser beam's intensity distribution.
Scaled Variables: A key aspect of the exercise involves using "dimensionless ('scaled') variables suitable for coding." The provided Python code demonstrates the scaling of the x and y positions by the beam width (w_0) ((\tilde{x}\equiv x/w_0) and (\tilde{y}\equiv y/w_0)) and the scaling of the irradiance by the maximum irradiance (I_{max}) ((Isc = I/Imax), which is also related to (2 P_0 / (\pi w_0^2))). The learning objective explicitly states that students will be able to "produce both line plots and contour plots of the (scaled) irradiance of the beam versus (scaled) position(s) (Exercises 1 and 2)."
Computational Implementation: The exercise is designed to be implemented using a computational tool, with Python being explicitly mentioned as an "Available Implementation." The provided Python code (Beam_Profile_Exercise_2.py) serves as a template or example of how to generate these contour plots using libraries like pylab and matplotlib.
Knife-Edge Technique: Although Exercise 2 focuses on contour plots, the "About" section indicates that the broader set of exercises aims to model an experiment using the "knife-edge technique" to determine the laser beam profile. Exercise 2 provides the foundation for understanding the theoretical profile that would be measured in such an experiment.
Irradiance Equation: While the exact equation isn't explicitly stated in the text excerpt, the Python code provides it: I = I_0 * exp(-2.0*(X*X + Y*Y)), where (I_0) is the maximum irradiance. This equation describes the Gaussian intensity profile of the TEM00 mode. The scaled irradiance is then (Isc = exp(-2.0*(XscXsc + YscYsc))).
Color Mapping: The code defines two colormaps (black_red1 and white_red1) to control how the irradiance values are represented by color in the contour plots. The comments explain the logic behind these colormaps, allowing students to visualize the intensity distribution with brighter colors indicating higher intensity. For a HeNe laser, "we might want red to be bright and black to be dark," which corresponds to black_red1.
Specific Exercise Tasks: Exercise 2 explicitly asks students to:

(a) "Make a plot of the irradiance versus the variables (x) and (y) for (− 2 w 0 ≤ x , y ≤ 2 w 0), with (w 0 = 0.5) mm and (P 0 = 1.0) mW."
(b) "Using the same values for (w 0) and (P 0), make a filled contour plot of the scaled irradiance (I ( x , y ) / ( 2 P 0 / π w 2 0 )) versus the scaled variables (\tilde{x}) and (\tilde{y}) for (− 2 ≤ \tilde{x} , \tilde{y} ≤ 2)."

Quotes from the Source:

"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."
"Students who complete this set of exercises will be able to... produce both line plots and contour plots of the (scaled) irradiance of the beam versus (scaled) position(s) (Exercises 1 and 2)."
"This file will generate a filled contour plot of the irradiance I(x,y) of the TEM00 Gaussian Mode of a laser versus x and y."
"This file will also generate a filled contour plot of the scaled irradiance I(x,y)/I_max of the TEM00 Gaussian Mode of a laser versus scaled x and scaled y."
"For a HeNe laser beam profile, we might want red to be bright and black to be dark. This is the color map labeled black_red1."
"A filled contour plot is a plot that indicates value by color as a function of two variables."
"Make a plot of the irradiance versus the variables (x) and (y) for (− 2 w 0 ≤ x , y ≤ 2 w 0), with (w 0 = 0.5) mm and (P 0 = 1.0) mW."
"Using the same values for (w 0) and (P 0), make a filled contour plot of the scaled irradiance (I ( x , y ) / ( 2 P 0 / π w 2 0 )) versus the scaled variables (\tilde{x}) and (\tilde{y}) for (− 2 ≤ \tilde{x} , \tilde{y} ≤ 2)."

Learning Objectives for Students:

Upon completion of this set of exercises, students will be able to:

Express the equation for the TEM00 laser mode in scaled variables.
Generate line plots and contour plots of both irradiance and scaled irradiance.
Develop a model and plot the knife-edge profile of a laser beam.
Fit the model of irradiance versus knife-edge position to experimental data.

Time to Complete: The estimated time to complete the entire set of exercises is 120 minutes.

Developed By: E. Behringer, with contributions from Najwa Sulaiman (Eastern Michigan University).

Subject Area: Waves & Optics.

Level: Beyond the First Year.

Available Implementation: Python.

Conclusion:

"PICUP Laser Beam Profile EXERCISE 2: THE TEM00 MODE: A CONTOUR PLOT" provides a valuable hands-on learning experience for students studying waves and optics, particularly lasers. It combines theoretical understanding of the TEM00 Gaussian mode with practical computational skills in Python to visualize the intensity profile of a laser beam through contour plots. The exercise is a step towards modeling a real-world experimental technique (knife-edge method) for characterizing laser beams. The use of scaled variables makes the model more general and suitable for coding and analysis.

Study Guide: TEM00 Laser Beam Profile and Contour Plots

Key Concepts

TEM00 Mode: The fundamental transverse electromagnetic mode of a laser beam, characterized by a Gaussian intensity profile.
Irradiance (I(x,y)): The power per unit area of the laser beam at a given point (x, y) in the transverse plane.
Gaussian Profile: The bell-shaped distribution of irradiance in the TEM00 mode, where the intensity is maximum at the center and decreases exponentially with radial distance.
Beam Width (w0): A parameter characterizing the size of the laser beam, defined as the radius at which the irradiance drops to 1/e² (approximately 13.5%) of its maximum value.
Contour Plot: A graphical representation of a function of two variables (here, irradiance as a function of x and y), where lines or colored regions connect points of equal value.
Scaled Variables (x̃, ỹ):
Dimensionless variables obtained by dividing the spatial coordinates (x, y) by the beam width (w0), simplifying the equation and allowing for a universal representation of the beam profile.
Scaled Irradiance (Isc): The irradiance normalized by its maximum value (Imax), providing a dimensionless measure of the relative intensity across the beam profile.
Knife-Edge Technique: An experimental method used to determine the beam profile by measuring the power transmitted past a translating knife edge.
Computational Modeling: Using computational tools (like Python) to simulate and visualize physical phenomena, such as the laser beam profile.
Fitting: The process of adjusting the parameters of a theoretical model to best match experimental data.

Quiz

Describe the intensity distribution of a laser beam operating in the TEM00 mode. What mathematical function characterizes this profile?
Define the term "beam width" (w0) in the context of a Gaussian laser beam. What is the significance of this parameter?
What is a contour plot, and how does it visually represent the irradiance of a laser beam? Explain the information conveyed by the colors or lines in such a plot.
Explain the purpose of using scaled variables (x̃ and ỹ) when describing the TEM00 laser beam profile. What advantage do they offer for analysis and coding?
What is scaled irradiance (Isc), and how is it calculated from the irradiance I(x,y)? What information does the scaled irradiance plot provide?
According to the learning objectives, what are students expected to be able to do after completing these exercises related to the laser beam profile?
What is the knife-edge technique mentioned in the "About" section, and what is its purpose in the context of laser beam profiling?
What programming language is indicated as an "Available Implementation" for these exercises? Briefly describe the role of code in these exercises.
Identify the parameters P0 and w0 in the context of the laser beam. How do these parameters influence the irradiance profile?
What do the provided Python code snippets (e.g., the cdict definitions) suggest about how the contour plots are being generated and customized?

Quiz Answer Key

A TEM00 laser beam has a Gaussian intensity distribution, meaning the intensity is highest at the center and decreases exponentially in a bell-shaped curve as the distance from the center increases. This profile is mathematically described by a Gaussian function.
The beam width (w0) is the radius at which the irradiance of the laser beam drops to 1/e² (approximately 13.5%) of its peak value. It serves as a measure of the beam's size and how rapidly its intensity decreases with distance from the center.
A contour plot uses colors or lines to represent different levels of a value across a two-dimensional area. In the context of laser irradiance, it shows how the intensity of the beam varies with position (x, y), with each color or contour line indicating a specific irradiance value.
Scaled variables (x̃ = x/w0, ỹ = y/w0) make the mathematical expression for the beam profile dimensionless and independent of the specific beam width. This allows for a universal representation of the TEM00 mode and simplifies coding and comparisons across different laser beams.
Scaled irradiance (Isc = I/Imax) is the irradiance at a point (x, y) divided by the maximum irradiance of the beam. Its plot shows the relative intensity distribution, ranging from 0 to 1, across the scaled coordinates, highlighting the shape of the Gaussian profile regardless of the absolute power.
Students completing these exercises will be able to express the TEM00 mode equation in scaled variables, produce line and contour plots of irradiance (scaled and unscaled), develop a model of the knife-edge profile, and fit this model to experimental data.
The knife-edge technique is an experimental method where a sharp edge is moved across the laser beam, and the transmitted power is measured as a function of the knife-edge position. This allows for the inference of the beam's intensity profile by analyzing how the power is blocked.
Python is the indicated "Available Implementation" for these exercises. Code is used as a computational tool to generate the contour plots of the laser beam irradiance and scaled irradiance based on the mathematical model of the TEM00 mode.
P0 represents the total power of the laser beam, while w0 is the beam width. P0 scales the overall irradiance of the beam (higher power means higher intensity), and w0 determines the spatial extent and how rapidly the intensity decreases from the center (larger w0 means a wider beam with a slower decrease in intensity).
The cdict definitions in the Python code specify custom colormaps (black_red1 and white_red1) for the contour plots. This indicates that the colors used to represent irradiance levels are defined by how the red, green, and blue components of the color vary with the scaled intensity, allowing for visualization with bright regions corresponding to high irradiance and dark regions to low irradiance.

Essay Format Questions

Discuss the significance of the TEM00 mode in laser applications. How does its Gaussian intensity profile contribute to the unique properties and uses of lasers compared to other light sources?
Explain the process of scaling variables in the context of the TEM00 laser beam profile. What are the theoretical and practical advantages of using scaled irradiance and scaled spatial coordinates?
Describe how a contour plot effectively visualizes a two-dimensional function like the irradiance of a laser beam. What information can be readily obtained from a contour plot of the TEM00 mode, both in terms of absolute and scaled irradiance?
The exercises mention a "knife-edge technique" for experimentally determining the laser beam profile. Discuss how computational modeling and fitting (as suggested in the exercises) can be used in conjunction with such experimental data to characterize a laser beam.
Analyze the learning objectives provided for this set of exercises. How do these objectives build upon each other to provide a comprehensive understanding of the TEM00 laser beam profile and its characterization?

Glossary of Key Terms

Irradiance:
Power per unit area, typically measured in Watts per square meter (W/m²) or in these materials, milliwatts per square millimeter (mW/mm²). It describes the intensity of electromagnetic radiation at a given point.
TEM00 Mode: The fundamental or lowest-order transverse electromagnetic mode of a laser resonator. Its intensity profile in the plane perpendicular to the beam propagation direction is Gaussian.
Gaussian Beam: A beam of electromagnetic radiation whose transverse electric and magnetic field amplitude profiles are well approximated by Gaussian functions. Laser beams in the TEM00 mode exhibit this profile.
Beam Waist (w0): The minimum radius of a Gaussian beam, occurring at the point where the beam is most tightly focused. In these materials, it is referred to as the beam width.
Contour Plot: A graphical technique for representing a 3D surface by plotting constant elevation curves on a 2D plane. In this context, the "elevation" is the irradiance value.
Scaled Variable: A dimensionless variable obtained by dividing a physical quantity by a characteristic value of that quantity. For position, it's typically scaled by the beam width. For irradiance, it's often scaled by the maximum irradiance.
Maximum Irradiance (Imax): The highest value of irradiance in the laser beam profile, occurring at the center of the TEM00 mode.
Knife-Edge Technique: An experimental method used to measure the spatial profile of a beam by scanning a sharp edge across it and measuring the transmitted power as a function of the edge's position.
Computational Tool: Software or programming language (like Python) used to perform calculations, simulations, and data visualization.
Fitting (Model Fitting): The process of finding the parameters of a mathematical model that best describe a set of data, often by minimizing the difference between the model's predictions and the observed data.

Sample Learning Goals

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Research

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{accordionfaq faqid=accordion4 faqclass="lightnessfaq defaulticon headerbackground headerborder contentbackground contentborder round5"}

Frequently Asked Questions: Laser Beam Profiling (TEM00 Mode)

1. What is the main goal of the "PICUP Laser Beam Profile" exercises?

The primary objective of these exercises is to guide students through the process of modeling and analyzing the spatial profile of a laser beam operating in the TEM00 mode. This involves understanding the mathematical description of the beam's irradiance, visualizing it through plots, modeling how a knife-edge would interact with this profile, and ultimately fitting this model to experimental data obtained using the knife-edge technique.

2. What is the TEM00 mode of a laser?

The TEM00 mode, or Transverse ElectroMagnetic 00 mode, is the fundamental Gaussian mode of a laser beam. It represents the lowest-order transverse electromagnetic mode within the laser resonator and is characterized by a Gaussian-shaped intensity profile in the plane perpendicular to the beam's direction of propagation. This means the intensity is highest at the center of the beam and decreases radially according to a Gaussian function.

3. What is irradiance in the context of a laser beam?

Irradiance (often referred to as intensity) describes the power per unit area carried by the laser beam at a given point in space. It is a measure of how much laser power is incident on a surface of a specific area. In the context of the TEM00 mode, the irradiance varies spatially according to a Gaussian distribution.

4. What is a contour plot and why is it used in these exercises?

A contour plot is a graphical representation of a function of two variables, where lines or colored regions connect points of equal value. In these exercises, contour plots are used to visualize the irradiance of the laser beam as a function of the x and y positions in a plane perpendicular to the beam's propagation. The color or the contour lines indicate areas of equal irradiance, allowing for a clear understanding of the beam's spatial intensity distribution.

5. What are scaled variables and why are they useful for modeling the laser beam profile?

Scaled variables are dimensionless quantities obtained by dividing physical variables by a characteristic parameter of the system. In this case, the x and y positions are scaled by the beam width, (w_0), resulting in dimensionless coordinates (\tilde{x} = x/w_0) and (\tilde{y} = y/w_0). Similarly, the irradiance is scaled by the maximum irradiance, (I_{max}). Using scaled variables simplifies the mathematical expressions, makes them suitable for coding and numerical analysis, and allows for a more general representation of the beam profile that is independent of specific laser parameters.

6. What is the knife-edge technique mentioned in the "About" section?

The knife-edge technique is an experimental method used to measure the spatial profile of a laser beam. It involves scanning a sharp edge (the knife-edge) across the beam and measuring the power transmitted past the edge as a function of the knife-edge's position. By analyzing how the transmitted power changes with the knife-edge's location, the intensity profile of the beam can be reconstructed.

7. What software or programming language is primarily used in these exercises?

The exercises mention that Python is an available implementation, and a Python script (Beam_Profile_Exercise_2.py) is provided which generates the contour plots. This suggests that Python, along with libraries like pylab and numpy, is the primary computational tool used for modeling and visualizing the laser beam profile in these exercises.

8. What are some of the learning objectives for students completing these exercises?

Upon completion of these exercises, students should be able to:

Express the equation for the TEM00 laser beam profile using scaled variables.
Generate line plots and contour plots of the scaled irradiance as a function of scaled position.
Develop a model of the knife-edge profile of the laser beam.
Fit this model to experimental data obtained from a knife-edge scan to determine the beam profile.

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Details: Written by Loo Kang Wee; Parent Category: 06 Modern Physics; Category: 04 Laser and Semi-conductors; Published: 30 April 2018; Created: 30 April 2018; Last Updated: 22 March 2025; Hits: 7402

Breadcrumbs

🔦PICUP Laser Beam Profile EXERCISE 2: THE TEM00 MODE: A CONTOUR PLOT

About

EXERCISE 2: THE TEM00 MODE: A CONTOUR PLOT

Translations

Credits

Study Guide: TEM00 Laser Beam Profile and Contour Plots

Key Concepts

Quiz

Quiz Answer Key

Essay Format Questions

Glossary of Key Terms

Sample Learning Goals

For Teachers

Video

Version:

Other Resources

end faq

Frequently Asked Questions: Laser Beam Profiling (TEM00 Mode)

1. What is the main goal of the "PICUP Laser Beam Profile" exercises?

2. What is the TEM00 mode of a laser?

3. What is irradiance in the context of a laser beam?

4. What is a contour plot and why is it used in these exercises?

5. What are scaled variables and why are they useful for modeling the laser beam profile?

6. What is the knife-edge technique mentioned in the "About" section?

7. What software or programming language is primarily used in these exercises?

8. What are some of the learning objectives for students completing these exercises?

EXERCISE 2: THE TEM MODE: A CONTOUR PLOT