TEM00 MODE: “LINE CUTS” JavaScript Simulation Applet HTML5" resource. This resource is designed as an exercise for students to model and understand the intensity profile of a laser beam operating in the fundamental transverse electromagnetic mode (TEM00). It utilizes a JavaScript simulation applet and focuses on the "line cuts" technique to analyze the beam's irradiance distribution. The exercise aims to help students connect theoretical models with experimental observations.

Main Themes and Important Ideas/Facts:

1. Objective of the Exercise: The primary goal of this set of exercises is to guide students in modeling an experiment that determines the profile of a laser beam using a knife-edge technique. Exercise 1 specifically focuses on understanding the irradiance profile of a TEM00 laser beam.
2. TEM00 Mode Irradiance Equation: The exercise introduces the fundamental equation describing the irradiance (I(x, y)) of a laser oscillating in the TEM00 mode:
3. (I ( x , y ) = I 0 exp [ − 2 ( x 2 + y 2 ) w 2 0 ]) (Equation 2) where (I_0) is the maximum irradiance, and (w_0) is the beam width parameter.
4. Beam Width Parameter ((w_0)): The parameter (w_0) is defined as describing the "width (i.e., the spatial extent) of the beam." The exercise further clarifies that at (y = 0), the locations where the irradiance drops to half its maximum value ((I = I_0 / 2)), denoted as (x_{1/2}), are approximately (\pm 0.59 w_0).
5. Scaled Variables: The exercise introduces the concept of "scaled variables" (\tilde{x} \equiv x / w_0) and (\tilde{y} \equiv y / w_0). Using these scaled variables, the irradiance equation can be simplified to a dimensionless form:
6. (I ( x , y ) = I 0 exp [ − 2 ( x ~ 2 + y ~ 2 ) ]) (Equation 3) This scaling makes the equation more convenient for computational modeling.
7. Relationship to Total Power: The irradiance is also related to the total power (P_0) transmitted by the beam:
8. (I ( x , y ) = 2 P 0 π w 2 0 exp [ − 2 ( x ~ 2 + y ~ 2 ) ]) (Equation 4) This equation connects the irradiance profile to a measurable quantity, the total power of the laser beam.
9. Computational Tasks: Exercise 1 involves computational tasks that are handled by built-in functions of the computational tool (presumably Python, as indicated in the "Available Implementation"). Students are expected to:

• Express the TEM00 mode profile equation in terms of scaled variables suitable for coding.
• Generate line plots of the irradiance (I) versus scaled position (\tilde{x}) for different values of scaled position (\tilde{y}).
• Generate line plots of the scaled irradiance (I / I_{max, \tilde{y}}) versus scaled position (\tilde{x}) for different values of scaled position (\tilde{y}).

1. JavaScript Simulation Applet: The resource includes an embedded JavaScript simulation applet, allowing for interactive exploration of the laser beam profile. The provided iframe suggests that the simulation visualizes the TEM00 mode and likely facilitates the understanding of "line cuts."
2. Python Implementation: The provided Python script ("Beam_Profile_Exercise_1.py") demonstrates how to generate the plots requested in Exercise 1. It uses libraries like pylab and numpy to define scaled x and y values, calculate irradiance based on the TEM00 mode equation, and generate line plots of both unscaled and scaled irradiance.
3. Learning Objectives: Upon completing the set of exercises (including Exercise 1), students will be able to:

• Express the TEM00 mode equation using scaled variables.
• Produce line plots and contour plots of the scaled irradiance.
• Develop a model of the knife-edge profile of the laser beam.
• Fit the model to experimental data.

1. Target Audience and Level: The exercise is designed for students "Beyond the First Year" in a "Waves & Optics" subject area, suggesting an undergraduate physics or engineering level.
2. Time to Complete: Exercise 1 is estimated to take approximately 120 minutes to complete.

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."
• "The irradiance (I) (power per unit area) of a laser oscillating in the TEM 00 mode can be expressed as (I ( x , y ) = I 0 exp [ − 2 ( x 2 + y 2 ) w 2 0 ])"
• "where the “scaled variables” (\tilde{x}) and (\tilde{y}) are defined as (\tilde{x} \equiv x / w_0)."
• "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);"
• "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);"

Additional Information:

• The resource is developed by E. Behringer.
• It is part of the PICUP (Partnership for Integration of Computation into Undergraduate Physics) project.
• The content is licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License.
• The webpage provides links to different versions of the exercise and other related resources, including numerous other physics and mathematics simulations developed under the Open Educational Resources / Open Source Physics @ Singapore initiative.

Conclusion:

The PICUP Laser Beam Profile Exercise 1 provides a comprehensive and computationally focused approach for students to understand the fundamental TEM00 mode of a laser beam. By introducing scaled variables and requiring students to generate plots using computational tools, the exercise encourages a deeper understanding of the mathematical model and its physical representation. The inclusion of a JavaScript simulation applet offers an interactive way to visualize the beam profile, complementing the analytical and computational aspects of the exercise. This resource appears valuable for undergraduate physics and engineering students studying optics and lasers.

Frequently Asked Questions: PICUP Laser Beam Profile Exercise

• What is the primary goal of the "PICUP Laser Beam Profile EXERCISE 1"? This exercise aims to guide students in understanding and modeling the irradiance profile of a laser beam operating in the TEM00 mode. Specifically, it involves expressing the mathematical equation for this profile using scaled variables suitable for computational modeling, generating plots of the irradiance, and laying the groundwork for more advanced exercises involving knife-edge profiling and fitting to experimental data.
• What is the mathematical expression for the irradiance of a TEM00 laser beam, and what do the terms represent? The irradiance (I(x, y)) of a TEM00 laser beam is given by the equation (I(x, y) = I_0 \exp \left[ -2 \frac{(x^2 + y^2)}{w_0^2} \right]). In this equation, (I_0) represents the maximum irradiance (power per unit area) at the center of the beam, (x) and (y) are the spatial coordinates perpendicular to the direction of beam propagation, and (w_0) is the beam width parameter that characterizes the spatial extent of the beam.
• What are "scaled variables" in the context of this exercise, and why are they useful? Scaled variables, denoted as (\tilde{x}) and (\tilde{y}), are dimensionless quantities defined by dividing the spatial coordinates (x) and (y) by the beam width parameter (w_0) (i.e., (\tilde{x} \equiv x / w_0) and (\tilde{y} \equiv y / w_0)). Using scaled variables simplifies the expression for irradiance to (I(\tilde{x}, \tilde{y}) = I_0 \exp [-2 (\tilde{x}^2 + \tilde{y}^2)]), making it more general and convenient for coding and analysis as it removes the dependence on the specific value of the beam width.
• How is the total power of the laser beam related to its irradiance profile? The total power (P_0) transmitted by the laser beam is related to the irradiance (I(x, y)) by the equation (I(x, y) = \frac{2 P_0}{\pi w_0^2} \exp [-2 (\tilde{x}^2 + \tilde{y}^2)]). This equation shows that the irradiance is proportional to the total power and inversely proportional to the square of the beam width parameter, indicating that a wider beam or a lower power will result in lower irradiance at any given scaled position.
• What does plotting the irradiance (I) versus the scaled position (\tilde{x}) for different values of (\tilde{y}) reveal about the laser beam profile? Generating these line plots allows visualization of how the irradiance of the laser beam changes as a function of the scaled horizontal position ((\tilde{x})) at different scaled vertical positions ((\tilde{y})). These plots demonstrate the Gaussian shape of the TEM00 mode, showing that the irradiance is maximum at the center ((\tilde{x} = 0, \tilde{y} = 0)) and decreases exponentially as the distance from the center increases in either the (x) or (y) direction.
• What is the significance of plotting the scaled irradiance (I / I_{max, \tilde{y}}) versus the scaled position (\tilde{x})? Plotting the irradiance normalized by its maximum value for a specific (\tilde{y}) allows for a direct comparison of the beam's spatial profile at different vertical positions, independent of the absolute irradiance values. The resulting plots for different (\tilde{y}) values will overlap, indicating that the shape of the irradiance profile along the (x) direction remains Gaussian, and only the peak irradiance changes as (\tilde{y}) varies. This highlights the separability of the (x) and (y) dependence in the Gaussian profile.
• What computational tools are suggested or used in this exercise? The exercise utilizes a JavaScript simulation applet for interactive exploration. Additionally, the provided Python script (Beam_Profile_Exercise_1.py) demonstrates how to generate the described plots using the pylab and numpy libraries, indicating that Python is a suitable tool for completing such computational tasks.
• Beyond Exercise 1, what further investigations into laser beam profiles are mentioned? The "About" section indicates that this exercise is the first in a set of exercises. Subsequent exercises involve producing contour plots of the irradiance (Exercise 2), developing a model of the knife-edge profile of the laser beam (Exercise 3), and fitting this model to experimental data obtained using a knife-edge technique (Exercise 4). These further steps aim to provide a comprehensive understanding of experimentally determining a laser beam's profile.