What is the equation for constructive interference in thin film interference?

• fedecolo
In summary, a question was asked about finding the wavelengths of light transmitted and reflected when white light is incident on a thin film of water between two glass surfaces. The solution involves setting equations for constructive interference and assuming no light is absorbed by the material. It is important to note that there is always interference in both the transmitted and reflected waves, and standard textbooks provide examples of this.
fedecolo

Homework Statement

Between two pieces of glass (##n_1=1.70##), there is a thin film of water (##n_2=1.33## and width ##d=1 \mu m##). If there is normal-incidence of white light on the water surface, find:
(a) which wavelenghts can be seen in the light transmitted (answer: 667 nm,533 nm,444 nm, 381 nm)
(b) which wavelenghts can be seen in the light reflected (answer: 593 nm,484 nm, 410 nm, (355 nm) )

The Attempt at a Solution

(a) I don't know how to deal with light transmitted
(b) I set the ##2d= \left(m-\frac{1}{2} \right) \cdot \frac{\lambda}{n_2}## to find the lambdas for ##m=1,2...## but I obtain only ##\lambda_{m=1}=355 nm## (below the light that can be seen).

Any help?

fedecolo said:
(a) I don't know how to deal with light transmitted
You can assume that none of the light is absorbed by the material. So, the total amount of incoming light energy must be conserved. As less light is reflected, more light must be transmitted. And vice versa.

(b) I set the ##2d= \left(m-\frac{1}{2} \right) \cdot \frac{\lambda}{n_2}## to find the lambdas for ##m=1,2...## but I obtain only ##\lambda_{m=1}=355 nm## (below the light that can be seen).
Your equation looks OK. But, in order to get a wavelength of 355 nm, I have to let m = 8. You will need to choose values of m that yield visible wavelengths.

TSny said:
You can assume that none of the light is absorbed by the material. So, the total amount of incoming light energy must be conserved. As less light is reflected, more light must be transmitted. And vice versa.

Thanks. But how can I set the equation for the constructive interference in the first case? There is no interference since the light is transmitted(?)

fedecolo said:
Thanks. But how can I set the equation for the constructive interference in the first case? There is no interference since the light is transmitted(?)
There is always interference (constructive, destructive, or something in between) in the transmitted waves and in the reflected waves. Standard textbooks show how you get two reflected rays that interfere. See if you can show how to get two transmitted waves that interfere. Hint: One of the transmitted rays has no reflections. The other transmitted wave has more than one reflection.

What is thin film interference?

Thin film interference is a phenomenon that occurs when a beam of light is reflected off of a thin film, such as a soap bubble or a layer of oil on water. This results in interference patterns that are visible as different colors.

What causes thin film interference?

Thin film interference is caused by the interaction of light waves with the different layers of the thin film. When light passes through the film, some of it is reflected off the top layer and some is transmitted through the film. The reflected and transmitted waves then interact with each other, leading to interference patterns.

How does the thickness of the film affect the interference pattern?

The thickness of the film is a crucial factor in determining the interference pattern. When the film is very thin, the reflected and transmitted waves are nearly in phase, resulting in bright interference fringes. As the film gets thicker, the waves become more out of phase, leading to dark fringes.

Can the color of the interference pattern be changed?

Yes, the color of the interference pattern can be changed by altering the thickness of the film or the wavelength of the incident light. Thicker films will produce different colors, and using different colors of light will also result in different interference patterns.

What are some real-life applications of thin film interference?

Thin film interference is used in many everyday items, such as anti-reflective coatings on glasses and camera lenses, anti-glare coatings on computer and phone screens, and in the production of color-changing pigments for cosmetics and paints. It is also used in more advanced technologies, such as thin film solar cells and optical filters.

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