Deriving the antireflective coating formula n1=sqrt(n0*ns)

[reb30]

Hello

I'm trying to solve a problem where I'm asked to find the optimum index of refraction for an antireflective coating on a piece of glass. In trying to derive an equation for it I keep coming across this in the internet (where 1,0, and s should be subscripts)

n1=sqrt(n0ns)

where n1 is the index of refraction of the coating, n0 is that of the air, and ns is the glass (see here - http://en.wikipedia.org/wiki/Antireflection_coating) . I can't work out how this was derived for the life of me and can't find it anywhere in books or the internet. Does anyone have any ideas?

Thanks a lot

Robert

 

[ijustlost]

I also have found this near impossible, despite spending hours working on it. My books quote that the relation given above should give zero reflection, but the only way I can make that happen is if n0 = ns.

I understand theoretically how and why this should work, I just can't get it to work mathematically..

 

[ijustlost]

@reb30:

I don't know if you own a copy of the textbook "Optics" by Hecht's but there's a derivation on page 375. Unfortuanatly it's very complicated and is done for general angle of incidence, rather than normal incidence. I couldn't make it work for simplified start conditions..