Thin-film interference question

[aznshark4]

Homework Statement

The table lists the range of wavelengths in vacuum corresponding to a given color. If one looks through a film which has a refractive index of 1.333 and thickness of 340 nm (nanometers), which color will be 100% transmitted through the film?

Table (Color/Wavelength):

• red/780nm-622nm
• orange/622nm-597nm
• yellow/597nm-577nm
• green/577nm-492nm
• blue/492nm-455nm
• violet/455nm-390nm

A) red
B) yellow
C) violet
D) green
E) white

Homework Equations

refractive index of air is 1, so n(air)<n(film)>n(air) condition is met. relevant equations will be:
$$t(min)=\frac{\lambda}{2*n(film)}$$

where n(film) is the refractive index of the film and t(min) is minimum film thickness

The Attempt at a Solution

I tried this problem, making n(film)=1.333 and t(min)=340 nm, solving for $$\lambda$$.
I got $$\lambda$$=2*n(film)*t(min)=2*1.333*340=906nm
However, this exceeds the wavelength for any of the colors, and the answer should be (C). What did I do wrong?

 

[ehild]

You get maximum transmittance through the film if its thickness is any integer multiple of lambda/(2n)



ehild

 

[aznshark4]

yeah, but how do you come up with the correct wavelength?

 

[ehild]

Find the possible wavelengths which are between 780nm and 390 nm. All of them are correct.

ehild

 

[Ravian]

how this relationship between thickness, refractive index and wavelength is obtained? is there any relationship between complex refractive index with film thickness?

 

[aznshark4]

Uhh, I don't think so. The refractive index depends on the material the medium is made of, and that's it. What I was asking here was an interference question. When the medium is at a certain thickness, the light reflected from one end of the medium interferes with that from the other end to cancel each other out.