A scientist needs to focus a helium-neon laser beam (⋋ = 633 nm) to a 10-μm-diameter spot 8.0 cm behind a lens. What minimum diameter must the lens have?

Verified step by step guidance

Understand the problem: The goal is to determine the minimum diameter of the lens required to focus a laser beam to a specific spot size. This involves diffraction and the lens's aperture. The key formula to use is the diffraction-limited spot size equation.

Use the formula for the diffraction-limited spot size: \( d = \frac{2.44 \lambda f}{D} \), where \( d \) is the spot diameter, \( \lambda \) is the wavelength of the laser, \( f \) is the focal length of the lens, and \( D \) is the diameter of the lens. Rearrange this formula to solve for \( D \): \( D = \frac{2.44 \lambda f}{d} \).

Substitute the given values into the formula: \( \lambda = 633 \text{ nm} = 633 \times 10^{-9} \text{ m} \), \( f = 8.0 \text{ cm} = 0.08 \text{ m} \), and \( d = 10 \text{ μm} = 10 \times 10^{-6} \text{ m} \).

Perform the substitution: \( D = \frac{2.44 \times (633 \times 10^{-9}) \times 0.08}{10 \times 10^{-6}} \).

Simplify the expression to find the minimum diameter \( D \). Ensure the units are consistent throughout the calculation to avoid errors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Diffraction Limit

The diffraction limit refers to the fundamental limit on the resolution of optical systems due to the wave nature of light. It is determined by the wavelength of light and the aperture size of the lens. A smaller spot size can be achieved with larger apertures, but diffraction effects can cause the beam to spread, limiting the minimum achievable spot size.

Lens Diameter and Focusing

The diameter of a lens affects its ability to focus light. A larger lens can collect more light and reduce the effects of diffraction, allowing for a smaller focused spot size. The relationship between lens diameter, focal length, and the size of the focused spot is crucial for determining the minimum diameter required to achieve the desired focus.

Rayleigh Criterion

The Rayleigh criterion is a formula used to determine the minimum resolvable detail in an optical system. It states that two point sources are resolvable when the central maximum of one diffraction pattern coincides with the first minimum of another. This criterion is essential for calculating the minimum lens diameter needed to focus a laser beam to a specific spot size.

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