Ch 33: Wave Optics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 33: Wave Optics

Problem 62c

Chapter 33, Problem 62c

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.

Verified step by step guidance

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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Key Concepts

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

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.

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Related Practice

Textbook Question

Light of wavelength 600 nm passes though two slits separated by 0.20 mm and is observed on a screen 1.0 m behind the slits. The location of the central maximum is marked on the screen and labeled y = 0. A very thin piece of glass is then placed in one slit. Because light travels slower in glass than in air, the wave passing through the glass is delayed by 5.0×10⁻¹⁶ s in comparison to the wave going through the other slit. What fraction of the period of the light wave is this delay?

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Textbook Question

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. Can a laser beam be perfectly parallel, with no spreading? Why or why not?

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Textbook Question

A radar for tracking aircraft broadcasts a 12 GHz microwave beam from a 2.0-m-diameter circular radar antenna. From a wave perspective, the antenna is a circular aperture through which the microwaves diffract. If the antenna emits 100 kW of power, what is the average microwave intensity at 30 km?

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Textbook Question

Use your expression from part a to find an expression for the separation Δy on the screen of two fringes that differ in wavelength by Δλ.

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Textbook Question

A diffraction grating has slit spacing d. Fringes are viewed on a screen at distance L. Find an expression for the wavelength of light that produces a first-order fringe on the viewing screen at distance L from the center of the screen.

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