What is the minimum thickness that would minimize reflection of light

[msk172]

Homework Statement

A transparent film (n = 1.5) is deposited on a glass lens (n = 1.65) to form a nonreflective coating. What is the minimum thickness that would minimize reflection of light with wavelength 500 nm in air?

Homework Equations

2nt = mλ
n = refractive index of film
t = thickness of film
m = 1, 2, 3, …
λ = light wavelength in vacuum (air)

The Attempt at a Solution

I am understanding how to find values for similar questions using the above equation, but get lost when asked about finding the "minimum thickness". I do not see any "tmin" type equations. Any help greatly appreciated. Thanks in advance!

 

[Doc Al]

2nt = mλ

You have the equation that allows you to determine the various film thicknesses that will produce destructive interference. What's the minimum thickness given by that equation? (Which value of m gives the smallest value of t?)

 

[msk172]

Doc,

Thanks for the quick reply. I am still trying to work this problem out. Starting on the left side, using nfilm of 1.5 and multiplying it by 2, I get 3t on the left side. The opposite side, I have mλ. λ is specified as 500nm (in air). I am not sure what you are getting at when you ask which value of m gives the smallest value of t? (m obviously being 1, 3, 3, etc.. but how high can this realistically go?).

 

[msk172]

Perhaps I am getting hung up with not working on coming up with a new λ based on the provided n values. I think the second n value is messing with me (nglass). Does this have any bearing on the solution? Initially I thought yes, however based on your reply, I am starting to feel as though it is not relevant.

 

[Doc Al]

I am not sure what you are getting at when you ask which value of m gives the smallest value of t? (m obviously being 1, 3, 3, etc.. but how high can this realistically go?).

Who cares how high it can go? You want the smallest thickness.

Solve for t as a function of m.

(Hint: When you realize what you need to do, you'll probably smack yourself.)

 

[msk172]

Who cares how high it can go? You want the smallest thickness.

Solve for t as a function of m.

(Hint: When you realize what you need to do, you'll probably smack yourself.)

I know, right :-) I always over-think the easy ones and make them the hardest. I'll keep looking at it.

 

[msk172]

So.. 3t=m500. for tmin, you'd want m to equal 1, no? Thus, 500/3? It doesn't like that answer...

 

[Doc Al]

I think the second n value is messing with me (nglass). Does this have any bearing on the solution? Initially I thought yes, however based on your reply, I am starting to feel as though it is not relevant.

The only relevance of the second index value (for the glass) is how it compares to the index value of the film, since there relative size determines the phase change--or lack thereof--upon reflection at the boundary.

 

[Doc Al]

So.. 3t=m500. for tmin, you'd want m to equal 1, no? Thus, 500/3?

Sounds right to me.

It doesn't like that answer...

What units does it want? That answer is in nm.

Is this Mastering Physics, by any chance? Often it's picky about rounding off.

 

[msk172]

Heh.. Glad my logic sounds right to you.. I was getting ready to jump off the balcony.. Haha. This is CHIP at Purdue. I'm not sure how rounding could be an issue. Plugged 166.67 in. Affirm on it wanting nm.

 

[Doc Al]

2nt = mλ

D'oh! That's the formula for constructive interference, not destructive. That's the problem.

Derive (or look up) the correct formula.

(Sorry about that! )

 

[msk172]

Heh, got it. M=0 obviously, +.5. Thanks for the assistance!

 

[Doc Al]

You're welcome. (Sorry for not spotting that earlier. )

 

[msk172]

You're welcome. (Sorry for not spotting that earlier. )

Pshh. My fault, actually. No clue how the word "minimize" somehow translated to "constructive" interference in my mind. Hah. Thanks so much for your help.