Thin Film Interference in a Soap Bubble

[calvert11]

Homework Statement

Question:

A soap bubble of index of refraction 1.48 strongly reflects both the red and the green
components of white light. What film thickness allows this to happen? (In air, the wavelength of red light is 716 nm, of green light 511.429 nm.)

n = 1.48
$$\lambda$$(r) = 716 nm
$$\lambda$$(g) = 511.429 nm

Homework Equations

2L = (m + 1/2)*(λ/n)

The Attempt at a Solution

I used the equation for thin film interference (maxima): 2L = (m + 1/2)*(λ/n)

Since the light strongly reflects, I assume m = 0.

I end up with 2 equations giving thickness: one for red light, one for green light:

L(r) = 716e-9/(4*1.48)
L(g) = 511.429e-9/(4*1.48)

But I don't know how to proceed from there.

 

[jbsteele]

How do I combine these two equations to get the film thickness that allows both red and green light to strongly reflect?

 

[tomcue]

Thin film interference in soap bubbles is a fascinating phenomenon that occurs due to the interaction between light and the thin layer of soap film. The interference occurs because of the difference in path lengths of light waves reflected from the top and bottom surfaces of the film.

In this case, the soap bubble has an index of refraction of 1.48, which means that the speed of light is slower in the bubble than in air. This causes a phase shift in the reflected light, resulting in constructive interference for certain wavelengths.

To determine the film thickness that allows strong reflection of both red and green light, we can use the equation for thin film interference:

2L = (m + 1/2)*(λ/n)

Where L is the film thickness, m is the order of the interference (in this case, m = 0), λ is the wavelength of light, and n is the index of refraction.

Substituting the values given in the question, we get:

L(r) = (0.5)*(716e-9)/(1.48) = 242.57 nm

L(g) = (0.5)*(511.429e-9)/(1.48) = 173.13 nm

Therefore, a film thickness of 242.57 nm allows for strong reflection of red light, and a film thickness of 173.13 nm allows for strong reflection of green light. This means that the film thickness that allows for strong reflection of both red and green light is the average of these two values, which is approximately 207.85 nm.

It is important to note that this calculation assumes ideal conditions and does not take into account factors such as the curvature of the bubble or the presence of impurities in the film. Nevertheless, it provides a good estimate of the film thickness that allows for strong reflection of both red and green light in a soap bubble of index of refraction 1.48.