Question

Question 9 (6 points)

A laser beam is aimed through a circular aperture of diameter 1 mm.
a. If the laser beam is red with a wavelength of 632.8 nm, what is the angle from the center of the Airy disk to the first dark ring? (2 points)




b. If the screen you are projecting the Airy disk onto is 2 m from the aperture, what is the distance between the center of the disk and the first dark ring? (2 points)


c. How does the distance between the center of the disk and the first dark ring change as you move the screen closer to the aperture? (1 point)



d. You change to a green laser with a wavelength of 532 nm. How does this affect the separation of the rings in the Airy disk? (1 point)

Answer 1

Answer: a. The angle from the center of the Airy disk to the first dark ring can be calculated using the formula:

θ = 1.22 λ/D

where θ is the angle in radians, λ is the wavelength of the laser beam, and D is the diameter of the circular aperture. Substituting the values given, we get:

θ = 1.22 (632.8 x 10^-9 m) / (1 x 10^-3 m) = 7.89 x 10^-4 radians

b. The distance between the center of the disk and the first dark ring can be calculated using the formula:

r = 1.22 λ L / D

where r is the distance in meters, λ is the wavelength of the laser beam, L is the distance between the aperture and the screen, and D is the diameter of the circular aperture. Substituting the values given, we get:

r = 1.22 (632.8 x 10^-9 m) (2 m) / (1 x 10^-3 m) = 1.59 x 10^-3 m

c. As the screen is moved closer to the aperture, the distance between the center of the disk and the first dark ring will decrease, since the formula for r includes the distance L as a factor.

d. The separation of the rings in the Airy disk depends on the wavelength of the laser beam, according to the formula:

Δr = λL / D

where Δr is the separation between adjacent bright fringes, λ is the wavelength of the laser beam, L is the distance between the aperture and the screen, and D is the diameter of the circular aperture. Since the green laser has a shorter wavelength (532 nm) than the red laser (632.8 nm), the separation between the rings in the Airy disk will be smaller for the green laser. Specifically, we have:

Δr_green = (532 x 10^-9 m) (2 m) / (1 x 10^-3 m) = 1.06 x 10^-3 m

Δr_red = (632.8 x 10^-9 m) (2 m) / (1 x 10^-3 m) = 1.27 x 10^-3 m

So the separation between the rings in the Airy disk is smaller for the green laser.

Explanation: :)