Exploring Optical Resolution: From Circular Apertures to Single Slits using Rayleigh criterion = θ min = λ / a

 Exploring Optical Resolution: From Circular Apertures to Single Slits

Introduction

Optical resolution is a fundamental concept in physics and engineering that describes the ability of an optical system to distinguish between two closely spaced objects. Traditionally, this concept is often demonstrated using circular apertures, leading to the well-known Airy disk pattern and the Rayleigh criterion with a factor of 1.22. However, by shifting our focus to single-slit apertures, we can explore optical resolution from a different perspective, using the Rayleigh criterion appropriate for slit diffraction.

This blog post delves into a newly developed simulation that models optical resolution through a single-slit aperture. We'll discuss the physics behind single-slit diffraction, the modifications made from the circular aperture model, and how the Rayleigh criterion applies in this context.

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Understanding Single-Slit Diffraction

When monochromatic light passes through a single narrow slit, it spreads out or diffracts, creating an interference pattern on a screen placed behind the slit. This pattern consists of a central maximum flanked by successive minima and secondary maxima. The intensity distribution of this pattern is governed by the slit width and the wavelength of the light.

The mathematical expression for the intensity $I$ at an 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 maximum intensity at $\theta = 0$,
• $\alpha = \frac{\pi a \sin(\theta)}{\lambda}$,
• $a$ is the slit width,
• $\lambda$ is the wavelength of the light.

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The Rayleigh Criterion for Single Slits

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

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

This formula differs from the circular aperture case, which includes a factor of 1.22 due to the geometry of the circular aperture leading to the Airy disk pattern. In the single-slit scenario, the absence of this factor simplifies the criterion and directly relates the minimum resolvable angle to the ratio of the wavelength and slit width.

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Simulation Overview

The simulation models two point sources of light passing through a single-slit aperture and projecting onto a screen. By adjusting parameters such as:

• Wavelength ($\lambda$): The color or energy of the light.
• Slit Width ($a$): The width of the single slit through which light diffracts.
• Source Separation ($d$): The distance between the two point sources.
• Distance to Screen ($L$): How far the observation screen is from the slit.

Users can observe how these factors influence the diffraction patterns and the ability to resolve the two sources.

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Calculating Angular Separation

The angular separation $\theta$ between the two sources is essential for applying the Rayleigh criterion. In the simulation, this is calculated using the geometry of the setup:

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

where:

• $d$ is half the distance between the two sources (since they are positioned symmetrically about the center),
• $L$ is the distance from the slit to the screen.

This formula assumes small angles, which is valid when $d$ is much smaller than $L$.

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Comparing to the Rayleigh Criterion

By comparing the calculated angular separation $\theta$ to $\theta_{\text{min}}$, users can determine if the two sources are resolvable:

• Resolvable: If $\theta \geq \theta_{\text{min}}$, the two sources can be distinguished on the screen.
• Not Resolvable: If $\theta < \theta_{\text{min}}$, the diffraction patterns overlap significantly, and the sources appear as one.

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Visualizing the Diffraction Pattern

The simulation provides a visual representation of the intensity distribution on the screen. The key features include:

• Central Maximum: The brightest spot directly opposite the slit.
• First Minima: Points where the intensity drops to zero, located at angles satisfying $\alpha = \pm \pi$
  
• Overlap of Patterns: When the sources are close together, their diffraction patterns overlap, affecting resolvability.

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Interactivity and Educational Value

By manipulating the parameters, users can explore various scenarios:

• Changing Wavelength ($\lambda$): Observing how different colors of light affect the diffraction pattern.
• Adjusting Slit Width ($a$): Seeing the impact of a wider or narrower slit on resolution.
• Varying Source Separation ($d$): Understanding how increasing or decreasing the distance between sources affects their resolvability.

This hands-on approach enhances comprehension of diffraction and interference phenomena, making abstract concepts more tangible.

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From Circular Apertures to Single Slits

The transition from modeling circular apertures to single slits in the simulation was motivated by the desire to explore different aspects of optical diffraction and resolution. While circular apertures are common in telescopes and cameras, single slits are fundamental in experimental physics and offer a more straightforward mathematical treatment.

Key Modifications in the Simulation:

• Rayleigh Criterion Adjustment: Updated to $\theta_{\text{min}} = \lambda / a$to reflect single-slit diffraction.
• Intensity Calculations: Modified the intensity formula to use the sinc function appropriate for a single slit.
• Angular Calculations: Simplified angular separation computations based on slit geometry.

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Technical Implementation

The simulation was developed using JavaScript for broad accessibility. Key computational steps include:

• Defining the Sinc Function: To handle calculations where $\alpha = 0$ and avoid division by zero.

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

• Calculating Intensity: Using the sinc function to compute the intensity at each point on the screen.

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

• Determining Resolvability: Comparing $\theta$ and $\theta_{\text{min}}$ within the code to output whether the sources are resolvable.

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Conclusion

This simulation offers a valuable tool for visualizing and understanding the principles of optical resolution through single-slit diffraction. By adjusting fundamental parameters, users gain insights into how light behaves when it encounters obstacles, a concept that is central to many areas of physics and engineering.

The shift from circular apertures to single slits not only simplifies the mathematical treatment but also broadens the educational scope, allowing exploration of fundamental wave phenomena in optics.

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Try the Simulation

link

Experience the simulation yourself and explore the fascinating world of diffraction and optical resolution. Adjust the wavelength, slit width, source separation, and observe how light's wave nature influences what we see.

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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.

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About the Author

lookang is a physics enthusiast and developer passionate about making complex scientific concepts accessible through interactive simulations. With a background in optics and computational physics, lookang enjoys creating educational tools that bridge theory and visualization.

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Feedback

Have thoughts or questions about the simulation? Feel free to leave a comment or reach out via weelookang@gmail.com