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The beam from an argon laser (of wavelength \(515 \mathrm{~nm}\) ) has a diameter \(d\) of \(3.00 \mathrm{~mm}\) and a continuous energy output rate of \(4.00 \mathrm{~W}\). The beam is focused onto a diffuse surface by a lens whose focal length \(f\) is \(3.50 \mathrm{~cm}\). A diffraction pattern such as that of Fig. \(36-10\) is formed, the radius of the central disk being given by $$ R=\frac{1.22 f \lambda}{d} $$ (see Eq. 36-12 and Fig. 36-14). The central disk can be shown to contain \(84 \%\) of the incident power. (a) What is the radius of the central disk? (b) What is the average intensity (power per unit area) in the incident beam? (c) What is the average intensity in the central disk?

Step by step solution

01

Calculate the Radius of the Central Disk

To find the radius \( R \) of the central disk, we use the formula \( R = \frac{1.22 f \lambda}{d} \). The given values are \( \lambda = 515 \text{ nm} = 515 \times 10^{-9} \text{ m} \), \( f = 3.50 \text{ cm} = 0.035 \text{ m} \), and \( d = 3.00 \text{ mm} = 0.003 \text{ m} \). Plugging in these values gives:\[R = \frac{1.22 \times 0.035 \times 515 \times 10^{-9}}{0.003} = 7.401 \times 10^{-6} \text{ m} = 7.401 \mu\text{m}\]

02

Calculate Average Intensity of the Incident Beam

The average intensity \( I \) of the incident beam is given by the formula \( I = \frac{P}{A} \), where \( P = 4.00 \text{ W} \) is the power and \( A \) is the cross-sectional area of the beam. Since the beam is circular, \( A = \pi \left(\frac{d}{2}\right)^2 \). Substituting the diameter \( d = 3.00 \text{ mm} = 0.003 \text{ m} \), we find:\[A = \pi \left(\frac{0.003}{2}\right)^2 = 7.069 \times 10^{-6} \text{ m}^2\]Thus, the intensity is:\[I = \frac{4.00}{7.069 \times 10^{-6}} = 5.66 \times 10^5 \text{ W/m}^2\]

03

Calculate Average Intensity in the Central Disk

The central disk contains \( 84\% \) of the incident power. The intensity in the central disk \( I_c \) can be calculated using \( I_c = \frac{0.84 P}{A_c} \), where \( A_c = \pi R^2 \) is the area of the central disk. Given \( R = 7.401 \mu\text{m} = 7.401 \times 10^{-6} \text{ m} \), we calculate:\[A_c = \pi (7.401 \times 10^{-6})^2 = 1.72 \times 10^{-10} \text{ m}^2\]Thus, the intensity is:\[I_c = \frac{0.84 \times 4.00}{1.72 \times 10^{-10}} = 1.95 \times 10^{10} \text{ W/m}^2\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diffraction Pattern

When light passes through a small aperture or around an edge, it tends to spread out. This spreading effect is called diffraction and creates a distinct pattern on a surface. The pattern typically consists of a series of light and dark fringes. For a circular aperture, as in the case of the argon laser beam being focused by the lens, the central region of the diffraction pattern is a bright spot called the central disk, surrounded by concentric dark and light rings.

The radius of the central disk can be calculated using the formula: \[ R = \frac{1.22 f \lambda}{d} \] where:

• \(R\) is the radius of the central disk,
• \(f\) is the focal length of the lens,
• \(\lambda\) is the wavelength of the laser light,
• \(d\) is the diameter of the beam.

This pattern and the formula are crucial in optics, impacting how light is focused and visualized using lenses.

Average Intensity

Intensity in the context of optics refers to the power per unit area carried by a wave. For laser beams, knowing the average intensity is important for applications where precision and safety are key. The intensity can be computed using the formula: \[ I = \frac{P}{A} \] where:

• \(I\) is the intensity,
• \(P\) is the power of the laser,
• \(A\) is the area over which the power is distributed.

To find the area for a circular laser beam, use: \[ A = \pi \left(\frac{d}{2}\right)^2 \] For the central disk, which receives 84% of the beam's power, the average intensity can be found similarly, adjusting the power to 84% of the total. This ensures precise control over how the laser interacts with the surface it hits.

Argon Laser

An argon laser is a type of gas laser that employs argon ions to produce light. It is commonly used in scientific research, medical procedures, and various industrial applications. The laser light emitted by an argon laser is typically in the visible spectrum, usually blue or green, with a specific wavelength; in this exercise, it is 515 nm. Argon lasers are valued for their

• Continuous and high-output power, suitable for precise applications
• Stability and high-intensity beams, which make them ideal for optical experiments
• The ability to focus tightly due to their small beam diameter

Understanding how an argon laser works, including its power and output characteristics, is key to effectively using it in optical configurations, like producing diffraction patterns.

Focal Length

The focal length of a lens, denoted as \(f\), is the distance from the lens to the point where the beam comes to focus. It plays a significant role in determining how light behaves as it passes through the lens. For the argon laser in the problem, the focal length is 3.50 cm. A shorter focal length means a lens can focus light to a point more quickly, resulting in a more pronounced diffraction pattern, leading to a smaller central disk. Conversely, a longer focal length results in a larger disk.

• Affecting the size and intensity of the projected beam
• More pronounced effects at shorter focal lengths

Understanding the focal length helps in designing systems that require specific beam shapes or patterns, crucial in fields like microscopy and astronomy.