EJS layout to Exploring Optical Resolution: From Circular Apertures to Single Slits

https://sg.iwant2study.org/ospsg/index.php/1260
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Using the GPT trigonometry:

https://sg.iwant2study.org/ospsg/index.php/1260
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Thanks for the clarification! If you want to create a second arrow that, together with the first one, forms a straight line, you'll need to ensure that the second arrow’s direction and size match the first arrow in such a way that their angles and lengths line up perfectly.

You already have an arrow starting at (calculateFlashlightXPosititon(), calculateFlashlightYPosititon().y1) with the following lengths:

SizeX = (0.2 - calculateFlashlightXPosititon())

SizeY = (0 - calculateFlashlightYPosititon().y1)

To ensure the second arrow aligns with the first arrow (creating the appearance of one continuous straight line), the second arrow's direction should follow the same slope as the first arrow.

Here's how to compute the SizeX and SizeY for the second arrow:

Find the slope of the first arrow: The slope m of the first arrow can be calculated using the coordinates of the arrow’s start and end points:

Start: (calculateFlashlightXPosititon(), calculateFlashlightYPosititon().y1)

End: (0.2, 0)

The slope is given by:

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const m = (0 - calculateFlashlightYPosititon().y1) / (0.2 - calculateFlashlightXPosititon());

This gives you the angle of the line that connects the two points.

Use the same slope for the second arrow: To make the second arrow appear as a continuation of the first, you need the second arrow’s slope to be the same. The second arrow starts at (0.2, 0), so its SizeX and SizeY need to follow the same slope m:

The relationship between SizeX and SizeY is:

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SizeY / SizeX = m

Choose the length of the second arrow: You can decide the length of the second arrow, which we can call L2. Then, you can calculate SizeX and SizeY using the slope m:

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const SizeX2 = L2 / Math.sqrt(1 + m * m);

const SizeY2 = m * SizeX2;

This ensures that the second arrow follows the same slope as the first arrow, and the two arrows will appear to be connected in a straight line.

Summary:

Calculate the slope m of the first arrow using its start and end points.

Apply the same slope to the second arrow.

Set the length L2 of the second arrow and calculate its SizeX2 and SizeY2 using the formula.

Would you like a specific code example based on your current setup? Let me know if this approach aligns with your objective! 

Exploring Optical Resolution: From Circular Apertures to Single Slits

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Introduction

Optical resolution, the ability of an optical system to distinguish between two closely spaced objects, plays a critical role in both scientific research and practical applications. Traditionally, circular apertures are used to demonstrate this phenomenon, resulting in the well-known Airy disk pattern and Rayleigh criterion. However, by turning our attention to single-slit apertures, we open up a new dimension in understanding optical resolution.

In this post, we’ll dive into a simulation I’ve developed, which explores optical resolution through a single-slit aperture, comparing it with the circular aperture model. Let’s explore the physics of single-slit diffraction, its mathematical basis, and how users can apply the Rayleigh criterion to resolve two light sources.

How the Design Layout Supports Better Sense-Making: A World View and Diffraction Pattern Comparison

One unique aspect of this optical resolution simulation is the thoughtfully designed layout, which utilizes the left and right panel format. The left panel presents a "world view" of the physical setup, showing light sources, the slit, and how they interact with the surrounding environment. Meanwhile, the right panel provides a visualization of the resulting diffraction pattern on the observation screen. This side-by-side representation offers a significant improvement in understanding compared to traditional diffraction pattern simulations that often provide only the output pattern without the context of the experimental setup.

This enhanced layout is made possible by EJS (Easy JavaScript Simulations), which allows for the creation of custom simulations that combine visual elements with dynamic interactivity. By integrating the setup and the result visually, learners are more effectively supported in understanding the link between cause and effect.

Integrating Context with Results

In traditional diffraction simulations, learners are often presented with a view of the resulting diffraction pattern alone, with limited visual context about how this pattern is generated. This can make it challenging to connect the observed effects with the specific properties of the setup, such as slit width, wavelength, and source separation. By offering both the "world view" and the diffraction pattern side by side, users can simultaneously see the cause—light from two sources passing through a slit—and its effect—the intensity pattern on the screen. This integrated approach fosters a deeper sense of how changes in the physical configuration lead directly to changes in the observed pattern.

For example, when users adjust the source separation or the slit width in the left panel, they immediately see how this affects the diffraction pattern in the right panel. Such immediate feedback is key for developing an intuitive understanding of the relationships between physical parameters and their outcomes.

Engagement Through Interactivity

The EJS framework allows users to interact directly with the simulation in real-time. By dragging sliders or inputting values, learners can modify parameters such as wavelength, slit width, or source separation, and see the resulting changes to both the physical setup and the diffraction pattern simultaneously. This interactivity creates a hands-on learning experience, where students are actively engaged in experimentation rather than passively observing results.

The combination of these interactive elements with a clear visual layout helps learners to internalize the concept of diffraction more effectively. Rather than needing to imagine how an adjustment to the physical setup might impact the pattern, they can experiment and see the impact immediately, which accelerates the learning process.

Making Abstract Concepts Tangible

Diffraction and interference are inherently abstract concepts that can be challenging to grasp. The world view on the left panel provides a tangible, visual representation of the elements involved in generating the diffraction pattern. Users can see the light sources, their paths, and the slit, which helps contextualize the diffraction process. This spatial representation is particularly helpful for visual learners who benefit from seeing physical arrangements instead of relying purely on mathematical descriptions or isolated intensity graphs.

The right panel, showing the diffraction pattern, is not merely a static output but is dynamically linked to the setup in the left panel. The evolving intensity curves and interference fringes directly correlate to the physical parameters, providing a richer understanding of how light behaves under different conditions. The visual feedback on whether the sources are resolvable according to the Rayleigh criterion further adds an element of clarity that makes the learning process more intuitive.

Advantages Over Traditional Simulations

Traditional diffraction simulations typically fall short by focusing solely on the final diffraction pattern, lacking any visual connection to the physical setup. This separation often forces learners to engage in mental gymnastics, trying to imagine the physical causes that might produce the observed effects. In contrast, the EJS-based layout merges these two critical aspects of the experiment into one cohesive view, minimizing cognitive load and supporting better understanding.

The side-by-side layout not only reduces the abstractness of the phenomenon but also encourages exploration and hypothesis testing. Students can form predictions about how a particular change—such as decreasing the slit width—will affect the diffraction pattern and then immediately test these predictions within the simulation. Such active engagement transforms the learning process from a theoretical exercise into an experiential one, thereby deepening comprehension and retention.

Conclusion

The left panel world view and right panel diffraction pattern layout represent a thoughtful design that bridges the gap between physical setup and observed phenomenon. By integrating these two views, learners can better understand the connection between input parameters and their effects on diffraction, fostering a deeper grasp of the principles of wave optics. This is a powerful demonstration of how EJS enables a more interactive and context-rich learning experience, making abstract optical concepts more accessible and intuitive.

Understanding Single-Slit Diffraction

When monochromatic light passes through a narrow slit, it diffracts and forms an interference pattern on a screen. This diffraction pattern consists of a central bright maximum, surrounded by alternating minima and secondary maxima. The distribution of light intensity depends on the slit’s width and the wavelength of light.

The mathematical formula for the intensity, $I(\theta)$, as a function of angle $\theta$ from the central axis is given by:

$$I(\theta) = I_0 \left(\frac{\sin(\alpha)}{\alpha}\right)^2$$

Where:

• $I_0$ is the peak intensity at $\theta = 0$
  
• $\alpha = \frac{\pi a \sin(\theta)}{\lambda}$,
• $a$ is the width of the slit,
• $\lambda$is the wavelength of light.

The simulation you’ll interact with models two light sources diffracting through a single slit, allowing you to manipulate parameters such as wavelength, slit width, and source separation.

The Rayleigh Criterion for Single Slits

The Rayleigh criterion defines the minimum angular separation, $\theta_{\text{min}}$, at which two point sources can still be resolved. For single-slit diffraction, this is expressed as:

$$\theta_{\text{min}} = \frac{\lambda}{a}$$

This differs from the circular aperture’s Rayleigh criterion, which has an additional factor of 1.22 due to the geometry. With single-slit diffraction, this factor is absent, simplifying the criterion and relating resolution directly to the slit width and wavelength.

Simulation Overview

The simulation demonstrates two point sources passing through a single-slit aperture, projecting diffraction patterns onto a screen. By adjusting parameters, users can explore the relationship between:

• Wavelength (λ): Influences the diffraction color and energy.
• Slit Width (a): Affects the diffraction pattern's spread.
• Source Separation (d): The distance between two light sources.
• Distance to Screen (L): The distance where the diffraction pattern is observed.

These parameters control the visibility of the diffraction pattern and the resolvability of the sources.

Calculating Angular Separation

In the simulation, the angular separation $\theta$ between the two sources is essential to determining resolvability:

$$\theta = \frac{2d}{L}$$

Where $d$ is half the distance between the two sources and $L$ is the distance to the screen. This formula assumes small angles, valid when the source separation is small relative to the screen distance.

Comparing to the Rayleigh Criterion

To determine resolvability, the calculated angular separation $\theta$ is compared with $\theta_{\text{min}}$:

• Resolvable: If $\theta \geq \theta_{\text{min}}$, the sources are distinguishable.
• Not Resolvable: If $\theta < \theta_{\text{min}}$, the diffraction patterns overlap, and the sources appear as one.

Visualizing the Diffraction Pattern

The simulation provides a real-time visual display of the intensity distribution. The main features include:

• Central Maximum: The bright central peak.
• First Minima: Points where intensity drops to zero.
• Pattern Overlap: As the sources move closer, their diffraction patterns merge, illustrating the limit of resolvability.

Interactivity and Educational Value

Users can adjust the parameters to observe how changing wavelength, slit width, and source separation affects diffraction and resolution. This interactive exploration makes abstract concepts tangible and highlights the wave nature of light.

From Circular Apertures to Single Slits

While circular apertures are commonly used in telescopes and cameras, single slits are fundamental in experimental physics, providing a simpler mathematical treatment and unique insights into diffraction.

Key modifications in this simulation include:

• Rayleigh Criterion: Adjusted to match single-slit diffraction.
• Intensity Calculation: Adapted to use the sinc function, relevant to single-slit apertures.
• Angular Separation: Simplified to reflect slit geometry.

Technical Implementation

This JavaScript-based simulation ensures broad accessibility. Key calculations include:

• Defining the Sinc Function: To avoid division by zero when $\alpha = 0$.

javascript
function sinc(x) {
  return x === 0 ? 1 : Math.sin(x) / x;
}

• Calculating Intensity: Using the sinc function to compute intensity values on the screen.

javascript
let alpha = (Math.PI * a * x) / (lambda * L);
let intensity = Math.pow(sinc(alpha), 2);

• Resolvability Check: Comparing $\theta$ and $\theta_{\text{min}}$ to determine if the sources are resolvable.

Conclusion

This simulation offers a hands-on tool for understanding optical resolution through single-slit diffraction. By manipulating variables like wavelength, slit width, and source separation, users can directly observe how diffraction limits resolution.

The shift from circular apertures to single slits not only simplifies the mathematical approach but also opens up new educational opportunities, allowing deeper exploration of light’s wave properties.

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Try the Simulation

https://sg.iwant2study.org/ospsg/index.php/interactive-resources/physics/04-waves/01-superposition/1260-single-slit-and-2-light-source-optical-resolution-and-rayleigh-criterion

Experience the simulation and discover how light's wave nature influences what we can observe!

References

• Hecht, E. (2002). Optics (4th ed.). Addison-Wesley.
• Young, H. D., & Freedman, R. A. (2012). University Physics with Modern Physics (13th ed.). Pearson.
• Jenkins, F. A., & White, H. E. (1957). Fundamentals of Optics (3rd ed.). McGraw-Hill.