Question

A thin film having an index of refraction of 1.50 is surrounded by air. It is illuminated normally by white light. Analysis of the reflected light shows that the wavelengths 360, 450, and 600 nm are the only missing wavelengths in the visible portion of the spectrum. That is, for those wavelengths, there is destructive interference.

(a) What is the thickness of the film?
nm

(b) What wavelengths in the visible portion of the spectrum are brightest in the reflected interference pattern? (Enter your answers from smallest to largest.)

? (nm)

(c) If this film were resting on glass that has an index of refraction of 1.60, what wavelengths in the visible spectrum would be missing from the reflected light? (Enter your answers from smallest to largest.)
....... nm
....... nm
....... nm

Answer 1

Part A)

For thin film interference of this type, the formula that applies is ...

2nt = m(wavelength)

2(1.5)(t) = m(450 X 10^-9) for the first wavelength, then...

2(1.5)(t) = (m + 1)(360 X 10^-9) for the second wavelength

Lets start from there...

From the first equation...

3t = 450 X 10^-9m

m = 3t/(450 X 10^-9)

Sub into the second...

3t = (m+1)(360 X 10^-9)

3t = (3t/(450X 10^-9) + 1)(360 X 10^-9)

3t = 2.4t + 360 X 10^-9

.6t = 360 X 10^-9

t = 6 X 10^-7 m (600 nm)

Part B)

In this case, the formula switches to 2nt = (m + .5)(wavelength)

We need to find values in the visible spectrum...

Start with m = 0 and move up to find appropriate values

When m = 0...

2(1.5)(6 X 10^-7) = (0 +.5)(wavelength)

wavelength = 3.6 X 10^-6 m (Not a valid answer)

When m = 1

2(1.5)(6 X 10^-7) = (1 +.5)(wavelength)

wavelength = 1.2 X 10^-6 m (Not a valid answer)

When m = 2

2(1.5)(6 X 10^-7) = (2 +.5)(wavelength)

wavelength = 7.2 X 10^-7 m Depending on your book, this could be a valid answer. 720 nm is close to the visible spectrum

Now try m = 3

2(1.5)(6 X 10^-7) = (3 +.5)(wavelength)

wavelength = 5.14 X 10^-7 m (514 nm) This is a valid answer

Now try m = 4

2(1.5)(6 X 10^-7) = (4 +.5)(wavelength)

wavelength = 4.0 X 10^-7 m (400 nm) This is a valid answer and the smallest answer so anything beyond will not work.

Thus the valid answer are 400 nm, 514 nm, and 720 nm (If your book accepts 720 because its right on the fringe of visible)

Part C)

In this case, the formula that applies is 2nt = (m + .5)(wavelength)

Again we will do trial and error to find values in the visible specturm...

When m = 0 and 1 the answers will not be valid

When m = 2...

2(1.5)(6 X 10^-7) = (2.5)(wavelength)

wavelength = 7.2 X 10^-7 m (720 nm)

When m = 3...

2(1.5)(6 X 10^-7) = (3.5)(wavelength)

wavelength = 5.143 X 10^-7 m (514.3 nm)

When m = 4...

2(1.5)(6 X 10^-7) = (4.5)(wavelength)

wavelength = 4.0 X 10^-7 m (400 nm)

Anything beyond is outside the visible spectrum

So the valid answers are 400 nm, 514.3 nm and 720 nm (For the 720 nm, it is right on the fringe of the visible spectrum so your answer key may not accept that one)