# Interference due to a thin film

## Homework Statement

A thin film with a refractive index of n=1.50 is surrounded by air. Wavelengths of light 360, 450, and 602 nm are the only visible wavelengths missing from the reflected light. What is the thickness of the film.

## The Attempt at a Solution

One of the rays will be 180 degrees out of phase from the other so destructive interference should occur when:

2t = mλ'

So I tried:

2t = mλ'
2t = m(λ/n)
t = (1)(360/1.5)
t = 120 nm

But then I tried the same thing for another one of the given wavelengths:

2t = mλ'
2t = m(λ/n)
t = (1)(450/1.5)
t = 150 nm

For what is a seemingly simple problem I cannot see what I am doing incorrectly.
(the answer is 600 nm I just don't understand how to find it)

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First off it seems like you made a typo. 602 nm light will not experience destructive interference from a thin film of 600 nm. It seems like you meant to type 600 nm light.

Now to the problem. Clearly you are only allowed to have one thickness. There will be one thickness that satisfies destructive interference for all those wavelengths (it doesn't have to be the same order though). So what you should do is make a table of all the thicknesses that cause destructive interference (try the first 5 orders) and see what matches up.

Okay, but why doesn't it work out for all orders?

Okay, but why doesn't it work out for all orders?
It works for exactly one order (or none at all) for any given wavelength.
However, it works for a different order for different wavelengths.

You'll notice these are only the wavelengths of visible light where the destructive interference is observable...if infrared light were considered, you'd notice a few more, and if ultraviolet light were used you'd notice many, many more.

I'm still struggling to understand why.

For every integer value of m won't the two rays be exactly out of phase? And therefore won't there be destructive interference for any integer value of m?

I'm still struggling to understand why.

For every integer value of m won't the two rays be exactly out of phase? And therefore won't there be destructive interference for any integer value of m?
Yes that formula will show all possible thicknesses that will cause destructive interference. However, you are definitely only allowed one thickness that causes destructive interference for all those wavelengths. Think about it, there's no way that the film would change in thickness for different wavelengths of light.

For enlightenment, try plotting wavenumber vs. m for the known solutions.

What is wavenumber?

1/wavelength

I will try to reword what I am struggling with.

I have 3 wavelengths of light for which destructive interference occurs 360, 450, and 602 nm (602 nm is what is in the question).

Destructive interference will occur at:

For λ =360 nm

t = 120m

For λ = 450 nm

t = 150m

For λ = 602 nm

t = 200m

Okay, so now I have 3 equations and m may vary between them. It is obvious that with the restriction that m must be an integer that t=600 nm with m=5,4,3 respectively.

I don't have a good understanding of what m is so I am confused as to why it varies between different wavelengths.

m has no units.

$${m{\lambda}\over{n}}=2t$$
$$m{{\lambda}\over{2}}=nt$$
m is the number of half-wavelengths between the front and back of the film. (corrected for index of refraction, of course)

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I will try to reword what I am struggling with.

I have 3 wavelengths of light for which destructive interference occurs 360, 450, and 602 nm (602 nm is what is in the question).

Destructive interference will occur at:

For λ =360 nm

t = 120m

For λ = 450 nm

t = 150m

For λ = 602 nm

t = 200m

Okay, so now I have 3 equations and m may vary between them. It is obvious that with the restriction that m must be an integer that t=600 nm with m=5,4,3 respectively.

I don't have a good understanding of what m is so I am confused as to why it varies between different wavelengths.
I don't think it should be 602 nm. If you plug that number into the equation (602/1.5)m = 2t

You end up with t = 200 and 2/3...so something is clearly wrong.

Like PhaseShifter said, m tells you the number of wavelengths (or half-wavelengths depending on the equation) in the path length difference (in this case 2t). Regardless of m the result is the same, destructive interference. So your job is to find the path length difference that will cause ALL of those wavelengths to destructively interfere without changing the thickness.