Deriving the Michelson Interferometer Equation: d_m = (m[lambda])/2

[NovaKing]

does anyone know how to derive this equation regarding the Michelson interferometer:

d_m = (m[lambda])/2

where d_m is the physical distance of a micrometer division, m is the number of fringes that crosses a screen given some d_m and lambda is the wavelength of the laser used to create the interference pattern.

I understand that the path difference divided by the wavelength is responsible for the number of fringes that pass by a certain mark, but I don't understand where the 2 comes from. Can someone help me please?

 

[NovaKing]

oh hehe never mind.


as it turns out because the path difference divided by the wavelength is number of fringes that pass I can set up a diagram to evaluate the problem.

Quite simply, the change in distance for the interferometer which has an arm that changes length is just 2L - 2(L-d_m) which gets 2d_m as the change in distance. Thus:

2d_m = m[lambda]

or

d_m = (m[lambda])/2


how silly of me

 

[Entropia]

The Michelson interferometer is a device used to measure small changes in the distance between two points. It works by splitting a beam of light into two paths, reflecting them back and recombining them, creating an interference pattern. The resulting pattern is dependent on the path difference between the two beams, which can be altered by changing the distance between the mirrors.

To derive the equation d_m = (m[lambda])/2, we need to understand the concept of path difference and how it relates to the number of fringes observed. The path difference is the difference in the distance traveled by the two beams before they are recombined. In the Michelson interferometer, this distance is equal to the physical distance between the two mirrors, denoted as d_m.

When a beam of light is split into two paths, it undergoes a phase shift due to the different distances traveled. This phase shift is given by 2πd_m/λ, where λ is the wavelength of the light. This means that for every full wavelength of light, there will be a phase difference of 2π. As the mirrors are moved closer or further apart, the number of wavelengths of light that fit between them will change, resulting in a change in the interference pattern.

Now, to calculate the number of fringes that cross a screen, we divide the path difference by the wavelength. This gives us the equation m = 2πd_m/λ. Solving for d_m, we get d_m = (mλ)/2π. However, in most cases, we are interested in the physical distance between the fringes, not the phase difference. This is where the 2 comes in.

The 2 in the equation represents the distance between two consecutive fringes, which is equal to one full wavelength. Therefore, to get the physical distance between fringes, we need to divide by 2, giving us the final equation d_m = (m[lambda])/2.

In summary, the 2 in the equation represents the distance between two consecutive fringes and is necessary to convert the phase difference into a physical distance. I hope this explanation helps in understanding the derivation of the Michelson interferometer equation.