Find the Optimal Film Thickness for Yellow and Purple Light Reflection

[CINA]

Homework Statement

A client has asked you to produce a sheet of material that will appear yellow for transmitted light and purple for reflected light. You decide that "yellow light" is 600 nm and that "purple light" is 400 nm. If you use MgF2 (n2= 1.39), how think should the sheet be? (Assume the film is in air, the measured wavelength are in air, and the light is normally incident on the film.)

Homework Equations

reflected light:
delta(L)=2*t (distance the light travels in the film)
phase change due to reflection = pi
n18lambda1=n2*lambda2

The Attempt at a Solution

2*t=(1/2 + m)*lambda2. Lambda1 is 600 nm since that's the color we want to see in the air, so I can just plud that value into lambda 2. I'm solving for t but my question is, what is m? can it be any value? I'm assuming that what I have so far is actually right, if it is not, please tell me.

 

[lippyka]

2*t=(1/2 + m)*lambda2 2*t= (1/2 + m)*400 nm 2*t=200nm + 400nm*m 2*t-200nm = 400nm*m 2*t/400nm - 1/2 = mt/200nm - 1/4 = m So, the thickness of the sheet should be 200 nm - 1/4 multiplied by the refractive index of MgF2 (1.39).