Question

The Michelson interferometer is used in a class of commercially available optical instruments called wavelength meters. In a wavelength meter, the interferometer is illuminated simultaneously with the parallel beam of a reference laser of known wavelength and that of an unknown laser. The movable mirror of the interferometer is then displaced by a distance \(\Delta d,\) and the number of fringes produced by each laser and passing by a reference point (a photo detector) is counted. In a given wavelength meter, a red He-Ne laser \(\left(\lambda_{\mathrm{Red}}=632.8 \mathrm{nm}\right)\) is used as a reference laser. When the movable mirror of the interferometer is displaced by a distance \(\Delta d\), a number \(\Delta N_{\text {Red }}=6.000 \cdot 10^{4}\) red fringes and \(\Delta N_{\text {unknown }}=7.780 \cdot 10^{4}\) fringes pass by the reference photodiode. a) Calculate the wavelength of the unknown laser. b) Calculate the displacement, \(\Delta d\), of the movable mirror.

Short answer

\(\Delta d = \frac{1000 * 632.8 * 10^{-9}}{2}\) Calculating, we get: \(\Delta d = 316.4 * 10^{-6}\) meters Now, we can use this calculated displacement \(\Delta d\) and the number of fringes produced by the unknown laser to find its wavelength. Rearrange the formula to solve for the unknown laser's wavelength \(\lambda_{Unknown}\): \(\lambda_{Unknown} = \frac{2 \Delta d}{\Delta N_{Unknown}}\) Plug in the values to calculate \(\lambda_{Unknown}\): \(\lambda_{Unknown} = \frac{2 * 316.4 * 10^{-6}}{950}\) Calculating, we get: \(\lambda_{Unknown} = 667.2 * 10^{-9}\) meters So, the wavelength of the unknown laser is approximately \(667.2\) nm.

Step-by-step solution

a) Calculate the wavelength of the unknown laser.

First, let's use the formula for the number of fringes formed in an interferometer, which is given by: Number of fringes, \(\Delta N = \frac{2 \Delta d}{\lambda}\) Here, \(\Delta d\) is the distance the movable mirror is displaced, and \(\lambda\) is the wavelength of the laser. We know the reference laser's wavelength and the number of fringes produced, so let's rearrange this formula and find the displacement \(\Delta d\) for the reference laser: \(\Delta d = \frac{\Delta N_{Red} * \lambda_{Red}}{2}\) Plug the given values to calculate \(\Delta d\):

Key concepts

Wavelength Measurement

Wavelength measurement refers to determining the distance between two consecutive peaks of a wave, such as light. This characteristic is crucial for understanding how different types of light behave and interact with materials. In the context of a Michelson interferometer, wavelength measurement becomes essential for identifying unknown laser sources.
The procedure uses the principle of interference, where two light waves overlap to form a pattern of dark and bright spots known as fringes. To measure the wavelength of an unknown laser, the interferometer compares it with a reference laser of known wavelength.

This involves:

• Displacing a mirror by a known distance.
• Counting the fringes produced by both the reference and unknown laser.
• Using the fringe count to calculate the unknown laser's wavelength, based on its proportionality to the displacement and known wavelength.

This method allows for precise wavelength determination, making it indispensable in various scientific and technological applications.

Optical Instruments

Optical instruments are devices designed to collect and manipulate light. These instruments, often using lenses and mirrors, serve a variety of crucial functions, from magnification to measurement. In the case of the Michelson interferometer, it acts as an important optical tool for wavelength measurement.
The interferometer exploits the interference effect to measure wavelengths with unparalleled accuracy. It's composed of a beam splitter, mirrors, and a photodetector, allowing it to generate and count fringes.
Some key points about optical interferometers include:

• They enable precise distance and wavelength measurements.
• Utilize interference to analyze light properties.
• Applied in various fields such as astronomy, engineering, and physics.

These devices help scientists gain insights into light properties, aiding in tasks ranging from creating holograms to laser calibration.

Laser Fringes

Laser fringes are the pattern of light and dark bands seen when two coherent light beams overlap, forming interference. This pattern is a signature phenomenon observed when lasers, such as those in an interferometer, are used.
In the Michelson interferometer, these fringes arise due to the slight path differences introduced by the movable mirror. The dark and bright spots correspond to constructive and destructive interference, respectively.
The process of counting these fringes plays a vital role in determining the wavelength. To elaborate:

• Each fringe represents one complete cycle of wave interference, caused by a full wavelength difference in path length.
• The number of fringes counted is directly linked to the displacement of the mirror and the wavelength of the light used.
• This count allows for the precise calculation of unknown wavelengths, using known reference light sources.

Fringes are fundamental in precision measurements, highlighting subtle changes in distance or wavelength.

He-Ne Laser

The He-Ne laser stands for Helium-Neon laser, a popular type of gas laser that emits light typically at a wavelength of 632.8 nm. It is well-known for its bright, coherent red light, making it an ideal reference source in optical instruments.
The He-Ne laser is crucial in the operation of devices like the Michelson interferometer. Its consistent wavelength forms a benchmark for measuring unknown lasers' wavelengths by comparison.
Some notable properties and uses of the He-Ne laser include:

• High stability and low cost, making it accessible for various lab applications.
• Used in barcode readers, optical research, and educational demonstrations.
• Provides a stable and precise reference point due to its well-defined wavelength.

The He-Ne laser's reliable performance ensures it remains a staple in both scientific research and practical applications, such as wavelength meters.