This briefing document summarizes the main themes and important ideas presented in the PICUP (Partnership for Integration of Computation into Undergraduate Physics) Laser Beam Profile Exercise 3. This exercise focuses on using the knife-edge technique to experimentally determine the width of a laser beam oscillating in the TEM00 mode. The exercise guides students to develop a model of the knife-edge profile and to understand the relationship between the knife-edge position and the power detected.

Main Themes and Important Ideas

1. Experimental Determination of Laser Beam Width: The primary goal of this exercise is to teach students how to experimentally determine the beam width ($w_0$) of a laser. The document states: "You can experimentally determine the value of the parameter $w_0$ that describes the width of the beam by measuring the profile of the laser beam using a knife-edge mounted on a linear translation stage together with an optical detector connected to a power meter."
2. The Knife-Edge Technique: The exercise details the knife-edge technique, where a knife-edge is translated across the laser beam, and the transmitted power is measured by an optical detector. The principle is that as the knife-edge progressively blocks the beam, the power reading decreases from the total power to zero. "The idea is that you allow the laser beam to enter the detector while you translate the knife-edge across the beam and monitor the power meter reading. Initially, when the beam is not blocked, the reading gives the total power of the beam; finally, the beam is completely blocked from entering the detector by the knife-edge and the reading would be zero (in the absence of background light)."
3. Mathematical Model of the Knife-Edge Profile: The exercise provides a mathematical model that describes the power received by the detector as a function of the knife-edge position ($x$). The initial integral form is given as:
4. P ( x ) = 2 P 0 π w 2 0 ∫ + ∞ − ∞ d y ∫ + ∞ x exp [ − 2 ( x ~ 2 + y ~ 2 ) ] d x (5)
5. where $P_0$ is the total power of the laser beam.
6. Relationship to the Error Function: A key theoretical step in the exercise is showing that the integral expression for the power is equivalent to an expression involving the error function ($erf(u) = ∫ + ∞ u exp ( − t 2 ) d t$):
7. P ( x ) = P 0 2 [ 1 − e r f ( 2 x w 0 ) ] (6)
8. This equation provides a direct relationship between the knife-edge position, the beam width, the total power, and the detected power.
9. Computational Implementation: The exercise includes a Python code snippet (Beam_Profile_Exercise_3.py) that demonstrates how to calculate and plot the predicted knife-edge beam profile using the derived equation. The code defines parameters like the number of points, the beam width ($w_0 = 0.5$ mm), and the range of the knife-edge position ($− 3 w 0 ≤ x ≤ 3 w 0$). It utilizes the scipy.special.erf function to calculate the error function and pylab for plotting.
10. Scaled Variables: Exercise 1 (mentioned in the "About" section) focuses on expressing the laser beam profile equation in terms of dimensionless ("scaled") variables suitable for coding. This suggests that the overall set of exercises emphasizes the importance of scaling for computational modeling.
11. Learning Objectives: Students completing this exercise will be able to:

• "develop a model of, and plot, the knife-edge profile of the laser beam"
• (As indicated by the broader set of objectives) understand how the knife-edge technique can be used to determine laser beam parameters.

Key Facts

• The exercise focuses on a laser oscillating in the TEM00 mode, which has a Gaussian intensity profile.
• The knife-edge is assumed to travel in the $+x$-direction.
• $w_0$ is the parameter describing the width of the laser beam.
• $P_0$ represents the total power of the laser beam.
• The error function, $erf(u)$, is crucial in expressing the power received by the detector as a function of the knife-edge position.
• The provided Python code calculates and plots the normalized transmitted power ($P(x)/P_0$) versus the knife-edge position $x$.

Quotes

• "You can experimentally determine the value of the parameter $w_0$ that describes the width of the beam by measuring the profile of the laser beam using a knife-edge mounted on a linear translation stage together with an optical detector connected to a power meter."
• "Imagine that the knife-edge travels in the $+ x -direction, and that the position of the knife-edge is$x$. Then the power received by the detector is$P ( x ) = 2 P 0 π w 2 0 ∫ + ∞ − ∞ d y ∫ + ∞ x exp [ − 2 ( x ~ 2 + y ~ 2 ) ] d x\( (5)"
• "Show that this expression is equivalent to \)P ( x ) = P 0 2 [ 1 − e r f ( 2 x w 0 ) ]\( (6)"

Conclusion

Exercise 3 of the PICUP Laser Beam Profile series provides a hands-on approach to understanding laser beam profiles and experimental techniques for characterizing them. By deriving and implementing the mathematical model of the knife-edge profile, students gain practical experience in linking theoretical concepts with experimental measurements. The use of computational tools like Python is integral to the exercise, reflecting the PICUP initiative's goal of integrating computation into undergraduate physics education. The exercise builds upon previous exercises that establish the fundamental description of a TEM00 laser beam.

Frequently Asked Questions: Knife-Edge Laser Beam Profiling

1. What is the purpose of the knife-edge technique in the context of laser beams?

The knife-edge technique is an experimental method used to determine the spatial profile, specifically the width (\)w_0\(), of a laser beam. By translating a sharp edge across the beam and measuring the transmitted power with an optical detector, the beam's intensity distribution can be inferred.

2. How does the experimental setup for the knife-edge technique work?

The setup involves a laser beam, a knife-edge mounted on a linear translation stage, and an optical detector connected to a power meter. The laser beam passes towards the detector, and the knife-edge is moved incrementally across the beam's path. At each position of the knife-edge (\)x\(), the power meter records the portion of the beam's power that is not blocked by the edge and reaches the detector.

3. What does the power meter reading tell us during a knife-edge scan?

When the knife-edge is far from the beam, the power meter reads the total power (\)P_0\() of the laser. As the knife-edge begins to block the beam, the power reading decreases. When the knife-edge completely blocks the beam, the power reading becomes zero (ideally, neglecting background light). The change in power as a function of the knife-edge position reveals the beam's profile.

4. What is the mathematical expression that describes the power received by the detector as a function of the knife-edge position?

The power \)P(x)$received by the detector when the knife-edge is at position$x$(moving in the +x direction) is given by the integral:$P ( x ) = \frac{2 P_0}{\pi w_0^2} \int_{-\infty}^{+\infty} d y \int_{x}^{+\infty} \exp \left[ - 2 \left( \frac{x'^2 + y'^2}{w_0^2} \right) \right] d x'$This integral represents the portion of the laser beam's intensity distribution that is not blocked by the knife-edge at position$x\(.

5. How is the integral expression for \)P(x)\( simplified?

The integral expression simplifies to a form involving the error function (\)erf$):$P ( x ) = \frac{P_0}{2} \left[ 1 - erf \left( \frac{\sqrt{2} x}{w_0} \right) \right]$where$erf(u) = \frac{2}{\sqrt{\pi}} \int_{0}^{u} \exp(-t^2) dt$. The provided source uses a slightly different definition of the error function,$erf(u) = \int_{u}^{+\infty} \exp(-t^2) dt$, leading to$P ( x ) = \frac{P_0}{2} \left[ 1 - erf \left( \frac{\sqrt{2} x}{w_0} \right) \right]\(.

6. What is \)w_0\( and why is it an important parameter in this experiment?

\)w_0$is the beam waist radius, representing the width of the laser beam at its narrowest point. It is a crucial parameter that characterizes the laser beam's spatial profile and divergence. The knife-edge technique allows for the experimental determination of this$w_0$value by fitting the theoretical model of$P(x)\( to the measured power data.

7. How can the experimental data obtained from the knife-edge scan be used to find \)w_0\(?

By plotting the measured transmitted power \)P(x)$as a function of the knife-edge position$x$, and then fitting the theoretical model$P ( x ) = \frac{P_0}{2} \left[ 1 - erf \left( \frac{\sqrt{2} x}{w_0} \right) \right]$to this data, the value of$w_0$can be extracted. This fitting process involves adjusting the parameter$w_0\( in the model until the curve best matches the experimental data points.

8. What are some learning objectives associated with performing this knife-edge laser beam profiling exercise?

Students who complete this exercise are expected to learn how to:

• Model the profile of a TEM00 laser beam using a mathematical equation.
• Express this equation in terms of scaled, dimensionless variables suitable for coding.
• Generate line plots and contour plots of the laser beam's irradiance.
• Develop a model for the knife-edge profile of the laser beam.
• Plot the predicted knife-edge profile.
• Fit this model to experimental data to determine the beam width (\)w_0$).