# How to know what m value to plug into thin film interference equations

• carrotcake123
In summary, the value of "m" in the thin film interference equations depends on the question being asked and can be either 0 or 1 for the thinnest film, while any other value of "m" will result in a thicker film. Constructive interference occurs when the path difference is an integer number of wavelengths, while destructive interference occurs when the path difference is an integer + 1/2 number of wavelengths. The key to understanding thin film interference lies in understanding the 180 degree phase shift that occurs at interfaces between materials with different indices of refraction.
carrotcake123
How do I work out what m value (0, 1/2, 1 etc) to put in the thin film interference equations like 2nt = (m + 1/2)*lambda? Does it depend if it's constructive or destructive? Could someone help explain, thanks!

No, it's always constructive. The question is what path difference causes constructive interference.

Forget about reflections and thin films for a minute. Suppose I had two sources that were emitting waves exactly in phase. Different places in space will be closer to one point than the other, so there's a path difference. Where will I get constructive interference?

Answer: When the path difference is an integer number of wavelengths.

Now, what if one of my sources was 180 degrees out of phase with the other one? If they travel an equal distance, they cancel out. So where do I have constructive interference?

Answer: When the path difference is an integer + 1/2 number of wavelengths. If the path difference is 1/2 wavelength, then that adds a 180 degree phase shift, which added to the original 180 degree phase difference puts them in phase.

And that's the key to thin films. Look at the two interfaces. You get a 180 degree phase shift when the interface is going from a lower to higher index of refraction, for instance air (n = 1.0) to water (n = 1.33) or water to oil with n = 1.50.

If you get a phase shift from one surface but not the other, then it's going to take an path difference that is an integer + 1/2 number of wavelengths to get them back in phase.

If there's no phase shift at either interface, or there's a phase shift at both interfaces, then to get the two waves in phase means the path difference is an integer number of wavelengths.

Clear?

berkeman
RPinPA said:
No, it's always constructive. The question is what path difference causes constructive interference.

Wait, this is not right. There are destructive interference in thin-film interference. That's the whole point of anti-reflective coating!

To the OP: Here's a page out of my class lecture notes that may help:

Here, "t" is the film thickness, and "n" is the index of refraction of the film itself. The rest should be self-explanatory.

And to answer your question, the value of "m" that you should use depends on the question being asked. Often, you will be asked to find the thinnest film that will cause such-and-such. In that case, you want the smallest "t", meaning that you choose m=0 or 1. Any other value of m will produce larger t. For the top equation, using m=0 makes no sense, because it means that there is no film at all.

Zz.

vanhees71
RPinPA said:
No, it's always constructive. The question is what path difference causes constructive interference.

Forget about reflections and thin films for a minute. Suppose I had two sources that were emitting waves exactly in phase. Different places in space will be closer to one point than the other, so there's a path difference. Where will I get constructive interference?

Answer: When the path difference is an integer number of wavelengths.

Now, what if one of my sources was 180 degrees out of phase with the other one? If they travel an equal distance, they cancel out. So where do I have constructive interference?

Answer: When the path difference is an integer + 1/2 number of wavelengths. If the path difference is 1/2 wavelength, then that adds a 180 degree phase shift, which added to the original 180 degree phase difference puts them in phase.

And that's the key to thin films. Look at the two interfaces. You get a 180 degree phase shift when the interface is going from a lower to higher index of refraction, for instance air (n = 1.0) to water (n = 1.33) or water to oil with n = 1.50.

If you get a phase shift from one surface but not the other, then it's going to take an path difference that is an integer + 1/2 number of wavelengths to get them back in phase.

If there's no phase shift at either interface, or there's a phase shift at both interfaces, then to get the two waves in phase means the path difference is an integer number of wavelengths.

Clear?
I cannot express how grateful I am for this 17 sentence paragraph and how it explained what countless videos, lectures, and textbooks have been somehow unable to convey. Tysm.

## 1. What is the "m" value in thin film interference equations?

The "m" value in thin film interference equations represents the order of interference, which is determined by the number of times the light wave reflects off the top and bottom surfaces of the thin film.

## 2. How do I calculate the "m" value for thin film interference?

The "m" value can be calculated by dividing the thickness of the thin film by the wavelength of the incident light. This will give you the number of times the light wave reflects off the top and bottom surfaces of the thin film.

## 3. Can the "m" value be a decimal or a negative number?

No, the "m" value must be a positive integer. If the result of the calculation in question 2 is a decimal or a negative number, it means that there is no interference occurring in the thin film.

## 4. How does the "m" value affect the interference pattern in thin films?

The "m" value determines the number of bright and dark fringes in the interference pattern. A higher "m" value will result in more fringes, while a lower "m" value will result in fewer fringes.

## 5. Can the "m" value be changed after the interference pattern has been formed?

No, the "m" value is determined by the properties of the thin film and the incident light, and cannot be changed once the interference pattern has been formed. However, changing the thickness or the wavelength of the light can result in a different "m" value and therefore a different interference pattern.

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