# Interferometer problem about periodic fringe patterns [closed]

Please,kindly help me; my teacher assigned me problem set, and she never went over anything about interferometers in class. Also, there is almost no information about about how to solve interferometer problems in my book. I have no idea how to set this problem up, and I have no intuition concerning how I should solve it... Can someone help me set this problem up and explain it? I know if this example is solved, I will be able to solve the rest of the problems:

Sodium D lines, of wavelength(a)=589.0 nanometers and wavelength(b)=589.6 nanometers are used on a Michelson interferometer. When the first mirror moves, these lines periodically appear and disappear. Explain, in detail, this phenomenon and write an equation that would allow you to know how much the mirror must move to make the lines reappear and disappear one time.

• Did you try to google "Michelson interferometer" to get a basic understanding of how they work? – Floris Aug 7 '14 at 22:33
• Yes, I did. I need to know how to solve problems though, and form an intuition. – Mitch Aug 7 '14 at 23:12
• I actually did spent a little while calculating this question. I understand that it can be hard to figure out how to solve a certain kind of problem that you have no experience with. I do not believe that this question deserves an up arrow, but I gave it one because it had -1, and I didn't consider his approach to be $\textbf{totally}$ unreasonable. If he had just listed a problem with no explanation, or goal of learning, I would agree with the down arrow. But in my opinion (which clearly doesn't hold as much value as an experience member) I think neutral is fine here. – Gödel Aug 7 '14 at 23:23

Your situation is unfortunate, but this will become very typical as you get further in school. I will do my best to explain this problem fully:

To begin with, in this problem, two interference patterns are formed, each pattern unique to one of the wavelengths provided to you. It is important to understand here that the fringe patterns might overlap, but do $\textbf{not}$ $\textit{interfere}$ with one another (in terms of wave front interference). Consequently, it can be concluded that the bright fringes of one wavelength will eventually share locations with the dark fringes of the other wavelength.

When this occurs,no fringes will be visible, as there will be no dark bands to differentiate a between a single fringe, and the fringes adjacent to it.

In order to make the transition between $\textbf{periods of fringe absence / appearance}$ the mirror's change in position must produce an $\textbf{integer number}$ of fringe shifts for each wavelength, and the number of shifts for the shorter wavelength must be one more than the number for the longer wavelength.

$\textbf{This next bit is imperative for understanding the Michelson Interferometer:}$

The light travels the length of the apparatus $\textbf{twice}$, so the change in the position of the mirror must also be accounted for $\textbf{twice}$.

Thus, even though a fringe shift possesses a wavelength value of $\lambda$ , we will denote it as $\cfrac{\lambda}{2}$ and the change in position of the mirror $(\bigtriangleup X)$ will become $2(\bigtriangleup X)$

To wrap this problem up, let's make $\epsilon_n$ the number of fringes produced by a given wavelength.

$\epsilon_1$ = $\cfrac{2(\bigtriangleup X)}{\lambda_a}$ and $\epsilon_2$ = $\cfrac{2(\bigtriangleup X)}{\lambda_b}$, therefore $\epsilon_2$ = ($\epsilon_1 + 1$)

From here we can see that $\cfrac{2(\bigtriangleup X)}{\lambda_b}$ = $\cfrac{2(\bigtriangleup X)}{\lambda_a} + 1$ , meaning that $\bigtriangleup X = \cfrac{\lambda_a\lambda_b}{2(\lambda_a-\lambda_b)}$

I will let you plug in your values, you should get a number much less than a meter, but much greater than a nanometer. This was not an easy problem for what I am assuming is an A level physics course. Let me know if you have questions. Good luck.

• Thank you, this gave me the numerical value in the back of the book, and it was very helpful. I think I am getting the hang of solving these problems. – Mitch Aug 7 '14 at 23:14
• Glad I could help. Approach other problems with a similar method (they will all have their "trick"). – Gödel Aug 7 '14 at 23:25