You want to photograph a circular diffraction pattern whose central maximum has a diameter of 1.0 cm. You have a helium-neon laser (λ=633 nm) and a 0.12-mm-diameter pinhole. How far behind the pinhole should you place the screen that's to be photographed?

Verified step by step guidance

1. Determine the relationship between the diameter of the central maximum and the distance to the screen using the formula for the angular position of the first minimum in a circular diffraction pattern: \( \sin \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of the light and \( D \) is the diameter of the pinhole.
2. Approximate \( \sin \theta \) as \( \tan \theta \) for small angles, which allows us to relate the diameter of the central maximum \( d \) to the distance to the screen \( L \): \( \tan \theta = \frac{d/2}{L} \).
3. Combine the two equations: \( \frac{d/2}{L} = 1.22 \frac{\lambda}{D} \). Rearrange to solve for \( L \): \( L = \frac{d}{2 \cdot 1.22 \cdot \lambda / D} \).
4. Substitute the given values into the equation: \( d = 1.0 \, \text{cm} = 0.01 \, \text{m} \), \( \lambda = 633 \, \text{nm} = 633 \times 10^{-9} \, \text{m} \), and \( D = 0.12 \, \text{mm} = 0.12 \times 10^{-3} \, \text{m} \).
5. Simplify the expression to calculate \( L \), ensuring all units are consistent. This will give the distance from the pinhole to the screen where the central maximum has the specified diameter.

Key Concepts

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through small openings. In the context of light, diffraction patterns are created when coherent light, such as that from a laser, passes through a pinhole, resulting in a series of bright and dark fringes. The central maximum is the brightest part of the pattern, and its size is influenced by the wavelength of the light and the size of the aperture.

Rayleigh Criterion

The Rayleigh criterion is a formula used to determine the minimum resolvable detail in optical systems, particularly in diffraction patterns. It states that two point sources are resolvable when the center of one diffraction pattern coincides with the first minimum of another. This criterion helps in understanding how the size of the central maximum relates to the wavelength of light and the aperture size, which is crucial for calculating the distance to the screen.

Geometric Optics and Projection

Geometric optics deals with the propagation of light in terms of rays, which can be used to analyze how light travels from the pinhole to the screen. The distance from the pinhole to the screen can be calculated using the geometry of the diffraction pattern, where the size of the central maximum and the wavelength of light are key factors. This understanding allows for the determination of the optimal placement of the screen to capture the desired diffraction pattern.