Question

34.35 Monochromatic blue light \((\lambda=449 \mathrm{nm})\) is beamed into a Michelson interferometer. How many fringes move by the screen when the movable mirror is moved a distance \(d=\) \(0.381 \mathrm{~mm} ?\)

Short answer

Answer: Approximately 1696 fringes move across the screen.

Step-by-step solution

Convert the given wavelength and distance to the same units

We need to make sure that the values for the wavelength and distance are in the same units. The given wavelength is 449 nm, and the distance the mirror is moved is 0.381 mm. Convert the distance to nanometers, as follows: \(d = 0.381\,\text{mm} \times \frac{1000\,\text{nm}}{1\,\text{mm}} = 381000\,\text{nm}\) Now, both wavelength and distance values are in nanometers.

Calculate the path length difference between the interferometer's arms

Remember that the path length difference between the two arms of the interferometer is given by \(2d\). So, we can calculate it as follows: \(\Delta L = 2d = 2 \times 381000\,\text{nm} = 762000\,\text{nm}\)

Calculate the number of fringes that move across the screen

To find the number of fringes that pass the screen, divide the path length difference by the wavelength of the light. Then, round the result to the nearest whole number to get the number of fringes: \(N = \frac{\Delta L}{\lambda} = \frac{762000\,\text{nm}}{449\,\text{nm}} \approx 1696\) So, when the movable mirror is moved a distance \(d = 0.381\,\text{mm}\), approximately 1696 fringes move across the screen in the Michelson interferometer.

Key concepts

Interference Fringes

In a Michelson interferometer, one can observe fascinating patterns known as interference fringes. These fringes result from the interaction of light waves as they overlap and combine. As light waves from different paths meet, they either amplify each other or cancel each other out, depending on their phases.
Consequently, some areas appear bright (constructive interference), while others appear dark (destructive interference). These alternating light and dark bands are what we call interference fringes.

• Constructive Interference: When two waves align perfectly, their amplitudes add up, creating a brighter fringe.
• Destructive Interference: When waves meet out of phase, they cancel each other out, resulting in a darker fringe.

Understanding interference fringes is crucial when studying optical systems since they provide insight into light wave behaviors and properties.

Wavelength Conversion

Converting wavelengths and distances to the same units is an important step in calculations involving light. In the context of the Michelson interferometer, the wavelength is often given in nanometers (nm).
For accurate calculations, other measurements, such as the distance a mirror is moved, should also be in the same units. This avoids confusion and errors in the calculations. Converting from millimeters to nanometers can be achieved as follows:

• Conversion Factor: 1 millimeter is equal to 1000 nanometers.

By using this conversion factor, we can ensure that all measurements are consistent, making the calculations straightforward and reliable.

Path Length Difference

In an interferometer, the path length difference is key to determining the behavior and outcome of interference patterns. It refers to the difference in the path lengths that two beams of light travel in an interferometer.
The path length difference is influenced by how much one of the mirrors is moved. In a Michelson interferometer, when a mirror is moved a distance of \( d \), the path length difference is calculated as \( 2d \).

• The factor of 2 arises because the beam travels to the mirror and back.

Path length difference is crucial as it dictates where the interference fringes will appear. When the path length difference is an integer multiple of the wavelength, constructive interference occurs. Conversely, a half-integer multiple leads to destructive interference.

Optical Interference

Optical interference arises when two or more light waves overlap and combine, forming a new wave pattern. This fundamental concept of optics is at the core of the functioning of a Michelson interferometer.
In this device, beams are split into two paths. When recombined, the resulting interference pattern depends on the path length difference of these split beams. Optical interference is not only critical for understanding the behavior of interferometers, but it also has applications across multiple fields.

• Applications:
  • Precision measurement tools like interferometers for distance and wavelength measurements.
  • Creating holograms.
  • Improving the resolution of telescopes and microscopes.

Understanding optical interference allows scientists and engineers to manipulate light in precise and innovative ways, advancing technology and scientific discoveries.