A helium-neon laser (λ = 633 nm) is built with a glass tube of inside diameter 1.0 mm, as shown in FIGURE P33.62. One mirror is partially transmitting to allow the laser beam out. An electrical discharge in the tube causes it to glow like a neon light. From an optical perspective, the laser beam is a light wave that diffracts out through a 1.0-mm-diameter circular opening. What is the diameter (in mm) of the laser beam after it travels 3.0 m? Note that the wave model is appropriate because the spreading, at this distance, is significantly larger than the size of the opening.

검증된 단계별 안내

Step 1: Understand the problem. The laser beam diffracts as it exits the circular opening of diameter 1.0 mm. The spreading of the beam is governed by the principles of diffraction, specifically the single-slit diffraction formula. The goal is to calculate the diameter of the beam after it travels 3.0 m.

Step 2: Recall the formula for the angular width of the central maximum in single-slit diffraction: θ = 1.22 * (λ / D), where λ is the wavelength of the light (633 nm = 633 × 10⁻⁹ m), and D is the diameter of the circular opening (1.0 mm = 1.0 × 10⁻³ m). This formula gives the angular width in radians.

Step 3: Calculate the linear width of the beam at a distance of 3.0 m using the relationship: w = 2 * L * tan(θ), where L is the distance the beam travels (3.0 m), and θ is the angular width. Since θ is small, tan(θ) ≈ θ, so the formula simplifies to w ≈ 2 * L * θ.

Step 4: Substitute the values into the simplified formula: θ = 1.22 * (λ / D), and then w ≈ 2 * L * θ. This will give the diameter of the beam at the given distance.

Step 5: Convert the final result into millimeters (mm) for the diameter of the laser beam after traveling 3.0 m. Ensure all units are consistent throughout the calculation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Diffraction

Diffraction is the bending of waves around obstacles and openings. In the context of light waves, when a beam passes through a small aperture, such as the 1.0 mm opening in the laser tube, it spreads out rather than traveling in a straight line. This phenomenon is crucial for understanding how the laser beam expands as it travels, especially over longer distances.

Wave Model of Light

The wave model of light describes light as a wave that exhibits properties such as interference and diffraction. This model is essential for analyzing the behavior of laser beams, particularly when they pass through small openings. It allows us to predict how the beam will spread and change in diameter as it propagates through space.

Beam Divergence

Beam divergence refers to the angle at which a laser beam spreads as it moves away from its source. It is influenced by the diameter of the aperture and the wavelength of the light. Understanding beam divergence is key to calculating the diameter of the laser beam after it has traveled a certain distance, as it determines how much the beam will expand over that distance.