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Michelson interferometer

This applet lets you study the Michelson interferometer and see the evolution of light rings as the parameters of the system are changed. The case of a point source, which corresponds to the Twyman interferometer, is also analyzed.

Michelson interferometer

The Michelson interferometer consists of an extended source located at the focal plane of a lens (so the emerging rays are parallel), a semireflecting plate that divides the beam into two, and two mirrors. The beams, after reflection at the mirrors, are re-combined at the plate exit and interfere. The interferences are observed at the focal plane of a collimator lens. As the vertical beam traverses the plate three times while the horizontal beam passes through it only once, a compensating plate of the same material and thickness as the semireflecting plate is added in the horizontal arm.

If the two beams that interfere have the same intensity, the resulting intensity will be:

where λ is the wavelength of light (in the vacuum) and Δ is the optical path difference between the two light beams, which is due to the different length of the interferometer's arms (d). If we consider a beam that enters and exits the system making an angle θ with the axis, it is easy to demonstrate that:

n is the refraction index of the medium where the interferometer is placed. A constructive or destructive interference will take place whenever this path length difference is an integer multiple of the wavelength or an odd integer multiple of half a wavelength, respectively:

From these conditions we can see that all the rays that form the same angle with the axis will be in the same interference state. When observing the interference pattern on the focal plane of a lens of focal length f', rings of radii

will appear. We may observe that, as the angle or radii becomes greater, the corresponding order of interference gets smaller. On the other hand, in the center of the image (θ=0), the optical path difference is 2nd, therefore there is only a maximum or minimum of the intensity when this number is an integer multiple of half or one wavelength.

By modifying the system parameters (wavelength in the vacuum λ, distance d between mirrors, refraction index), you can see the evolution of the interference pattern, which is shown in a color that corresponds to the wavelength. The display area width changes with the lens focal length, as the rings' radii are directly proportional to it. By pressing the "Graphic" button you can see the intensity profile, whereas by pressing the "Image" button you go back to the interference pattern image, which is shown by default at the beginning of the program. We should point out that the separation between maxima gets smaller (rings get closer) as they are further away from the center.

If the "Diagram" button is pressed, the program shows a diagram to calculate the optical path difference between the two light beams. By pressing the same button, which is now labeled as "Setup", the Michelson interferometer schema is shown again, as by default.

This program can be used to measure the wavelength of the light source, by counting the number of bright or dark rings that appear from or disappear into the center of the pattern (θ=0) when the distance between the mirrors is changed between two points that give constructive or destructive interference, respectively. If the number of rings that appear or disappear is N and d₁ and d₂ is the initial and final separation of the mirrors, the wavelength can be calculated as:

While rings appear from the center as the separation of the mirrors is increased, they disappear into it when it is decreased.

Twyman interferometer

The Twyman interferometer has the same arrangement as the Michelson interferometer except that it is illuminated by a point source. Therefore, all beams will be in the same interference state and the interference pattern will have a constant intensity. Its value will depend on the optical path difference 2nd, so a variation by λ/4n of the distance between the mirrors is required to change from a maximum to a minimum intensity value.