Reflectivity coefficient of a composition of layers

[LCSphysicist]

Homework Statement

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Relevant Equations

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> A Ti02 film of index 2.5 is placed on glass of index 1.5 to increase
the reflection in the visible. Choosing a suitable value for wavelength, how
thick a layer in microns would you want, and what reflectivity would this
give you?

My question is about the last question presented above. Precisely, i am having some trouble to understand what it means by "reflectivity", i guessed that it is the coefficiente R that points the intensity obtained from a reflection, in this case normal, so generally $r_{12}=(\frac{n_1 - n_2}{n_1 + n_2})^2$. In this case, i thought that, since we want to increase the reflection and at the same time decrease the intensity, the reflected wave need to be out of phase by pi and the reflectivity would be $|r_{10}-r_{12}|$, where 0 is the air, one is the film and two is the glass.

But, the answer is $r=(\frac{n_0 - n_1²/n_2}{n_0 + n_1²/n_2})^2$, i have no idea how does the author got this answer. Any help?

 

[haruspex]

The answer to the first part is any multiple of half a wavelength, right?

For the second part, I have trouble believing the given answer. It says that if $n_1^2=n_0n_2$ then there's no reflection.
You don't explain how you got your answer. Did you take into account a reflected ray can undergo any odd number of traversals of the film?
I get $1-\frac{4n_0n_1n_2}{(n_0+n_2)(n_0n_2+n_1^2)}$, but not confidently.
(Without this comment, Latex seems to lose a subscript 1 at the end there. With this comment it’s fine!)