# Thin-film interference: transmission & reflection of colors

#### Beth N

1. Homework Statement
This is a 2 part question. I have a follow up question from the problem

1. A sheet of glass having an index of refraction of 1.40 is to be coated with a 187-nm thick film of material having a refractive index of 1.55 such that yellow light with a wavelength of 580nm (in vacuum) incident normally is preferentially transmitted. Are other parts of the visible light spectrum also preferentially transmitted?

2. A sheet of glass having an index of refraction of 1.40 is to be coated with a 187-nm thick film of material having a refractive index of 1.55 such that yellow light with a wavelength of 580nm (in vacuum) incident normally is preferentially transmitted. Will the transmission of any color be sharply reduced? If so, which color?

2. Homework Equations

Constructive interference:

$2t= \frac {m \lambda_{air}} {n_{film}}$ where m is number of reflective phase change = 0,2,4...

$2t= \frac {(m+ {\frac {1} {2}}) \lambda_{air}} {n_{film}}$ where m= 1,3, 5....

3. The Attempt at a Solution

The answer is given in the picture, which shows that yellow is preferentially transmitted, and violet is preferentially reflected. My question is, what about other colors within the visible spectrum? Can I say that for yellow and violet, the effect is 100%, while all other colors (like red, green...) has some component that are reflected and some are transmitted?
I guess I'm confused on the wording "preferential". Does preferential mean the effect is 100%?

Thank you so much!

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Homework Helper
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In general, the preferential transmittance that occurs is not 100%, and the preferential reflectance that occurs is not 100%. A complete treatment of this topic would take you well beyond what the scope of your present course is trying to cover. For normal incidence, the Fresnel reflection coefficient (for the electric field) is $\rho=\frac{n_1-n_2}{n_1+n_2}$. It takes some detailed calculations to work out the precise intensities for this Fabry-Perot type interference. Let's see of I can find you a "link" on the Fabry-Perot effect. In general, to answer your question, wavelengths close to the wavelength that gets optimal preferential reflection or transmission will also see similar effects. $\\$ Edit: No luck with a good "link". The Optics book by Hecht and Zajac treats this case in full detail.

#### Beth N

In general, the preferential transmittance that occurs is not 100%, and the preferential reflectance that occurs is not 100%. A complete treatment of this topic would take you well beyond what the scope of your present course is trying to cover. For normal incidence, the Fresnel reflection coefficient (for the electric field) is $\rho=\frac{n_1-n_2}{n_1+n_2}$. It takes some detailed calculations to work out the precise intensities for this Fabry-Perot type interference. Let's see of I can find you a "link" on the Fabry-Perot effect. In general, to answer your question, wavelengths close to the wavelength that gets optimal preferential reflection or transmission will also see similar effects.
Hi Charles,

Thank you for your answer. Unfortunately my course have not covered the Fabry Perot effect so this may not be useful in solving the problem in the context of my class (although would be interesting to know). I was just wondering, in general, where the other wavelengths "go", because the problem only mention yellow and violet.

Homework Helper
Gold Member
2018 Award
Hi Charles,

Thank you for your answer. Unfortunately my course have not covered the Fabry Perot effect so this may not be useful in solving the problem in the context of my class (although would be interesting to know). I was just wondering, in general, where the other wavelengths "go", because the problem only mention yellow and violet.
There is always a partial reflection and partial transmission for this problem with a thin-film layer. It normally is not 100%, even for wavelengths that are found from the formulas to be completely preferential, i.e. where the interference is optimized by the extra path difference by the part of the beam that undergoes multiple reflections.

#### Beth N

There is always a partial reflection and partial transmission. It normally is not 100%, even in the case of completely preferential, where the interference is optimized by the extra path difference by the part of the beam that undergoes multiple reflections.
So most (but not all) yellow light are transmitted (m) , and most violet are reflected (m+ 0.5), while red or green lies somewhere in between (m+0.2 etc)?

Homework Helper
Gold Member
2018 Award
When the wavelength is found to be preferential for a given type of interference (constructive or destructive), you will find that the amount of light appearing at those wavelengths will be altered the most by the coating. $\\$ For the uncoated glass, the transmission will generally be about 96% with 4% reflected, independent of wavelength. A thin film coating will affect those numbers, with an amount that is wavelength dependent. For the most preferred wavelengths for transmission, (constructive interference of the multiple reflections with the original incident beam that goes straight through), you might change those numbers to 99% and 1%, and the least preferred, (where the constructive interference occurs with the multiple reflections along with the reflection of the original incident beam), might be 92% and 8%. $\\$ At the present level of instruction, they are not expecting you to quantify these results. They just want you to get the general idea of the kind of interference that is occurring. For in-between wavelengths, those numbers would tend to stay more near 96% and 4%. $\\$ When constructive interference occurs upon transmission of the multiple reflections, destructive interference occurs upon reflection, and visa-versa)

#### Beth N

When the wavelength is found to be preferential for a given type of interference (constructive or destructive), you will find that the amount of light appearing at those wavelengths will be altered the most by the coating. $\\$ For the uncoated glass, the transmission will generally be about 96% with 4% reflected, independent of wavelength. A thin film coating will affect those numbers, with an amount that is wavelength dependent. For the most preferred wavelengths for transmission, (constructive interference of the multiple reflections with the original incident beam that goes straight through), you might change those numbers to 99% and 1%, and the least preferred might be 92% and 8%. $\\$ At the present level of instruction, they are not expecting you to quantify these results. They just want you to get the general idea of the kind of interference that is occurring. For in-between wavelengths, those numbers would tend to stay more near 96% and 4%.
I think it makes more sense now, thank you!

"Thin-film interference: transmission & reflection of colors"

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