To determine the required dimensions of the rectangular aperture that will allow the central maximum of the laser light to fill a rectangle measuring 110 mm wide and 6 mm high on a wall 4.5 m away, we can apply principles from optics, particularly those related to diffraction and the behavior of light passing through apertures.
Understanding the Problem
We have a laser beam with a wavelength of 632.8 nm (nanometers) and a circular cross-section of 2 mm in diameter. The light travels to a wall 4.5 m away, where we want the central maximum of the diffraction pattern to fit within a rectangle of 110 mm by 6 mm. The goal is to find the dimensions of the rectangular aperture that will create this effect.
Key Concepts
- Diffraction: When light passes through an aperture, it spreads out, creating a diffraction pattern. The central maximum is the brightest part of this pattern.
- Angular Width of the Central Maximum: The width of the central maximum can be related to the size of the aperture and the wavelength of the light.
Calculating the Required Aperture Dimensions
To find the dimensions of the aperture, we can use the formula for the angular width of the central maximum in a single-slit diffraction pattern:
θ = λ / w
Where:
- θ is the angular width in radians.
- λ is the wavelength of the light (632.8 nm = 632.8 x 10-9 m).
- w is the width of the aperture.
Next, we can relate the angular width to the physical width on the wall using the distance to the wall (L = 4.5 m):
Width on the wall = L * θ
Since we want the width of the central maximum to be 110 mm (0.11 m), we can set up the equation:
0.11 = 4.5 * (λ / w)
Solving for Width of the Aperture
Rearranging the equation gives:
w = (4.5 * λ) / 0.11
Substituting the values:
w = (4.5 * 632.8 x 10-9) / 0.11
Calculating this:
w ≈ 2.57 x 10-5 m = 25.7 μm
Height of the Aperture
Now, we need to find the height of the aperture. The height of the central maximum is given as 6 mm (0.006 m). We can use a similar approach as before, but we will consider the height instead of the width. The formula remains the same:
h = (4.5 * λ) / H
Where H is the height of the central maximum on the wall. Setting up the equation:
0.006 = 4.5 * (λ / h)
Rearranging gives:
h = (4.5 * λ) / 0.006
Substituting the values:
h = (4.5 * 632.8 x 10-9) / 0.006
Calculating this:
h ≈ 0.0477 m = 47.7 mm
Final Dimensions of the Aperture
In summary, the required dimensions of the rectangular aperture are approximately:
- Width: 25.7 μm
- Height: 47.7 mm
These dimensions will ensure that the central maximum of the laser light fills the specified rectangle on the wall 4.5 m away. Understanding these principles of diffraction and how they relate to aperture size is crucial in optics and various applications, from laser technology to imaging systems.
