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A transparent film (n=1.3) is deposited on a glass lens (n=1.5) to form a reflective coating. What is the minimum thickness that would maximize reflection of light with a wavelength of 500.0 nm in air?

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  • 4 weeks ago

    Assume the light is normally incident on the surface of the film.  We want the light that enters the film to hit the glass, reflect due to the index mismatch and arrive back at the surface of the film in phase with the light reflected from the surface of the film.  The light reflected from the surface of the film is shifted 180 deg in phase relative to the incoming light.

    The light reflected at the film - glass interface is shifted 180 deg in phase as well.  there is an additional phase shift due the to optical path through the film.  Call the film thickness t, the wavelength of light wl and the index of the film nf.  We want 2t to be a integer number of wavelengths so the light reflected from the glass arrives at the surface of the film in phase with light reflected from the film's surface.

    Now the wavelength of light in the film

     wl_f = wl/n

    (can find this by noting speed of light in a medium with index n is u = c/n and d = f*wl --> u = c/n = f*wl/n  and since the frequency is not changed by the medium, wl_medium = wl/n).

    so we want 2t = m*wl/nf  where m = 1, 2, 3 ...

    t = m*wl/(2nf)  --> min. thickness occurs when m = 1 so

    t = w/(2nf) = 192.3 nm

    Note this is a little different than the oil film on water example often used to illustrate thin film interference.  In that example, water has a lower index than oil so there is an additional 180 deg phase shift when the light reflects off the water that has to be accounted for.

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