Thickness of thin films?
A jewelry maker has asked a glass studio to produce a sheet of glass that will appear blue (wavelength = 480nm) for reflected light and red (wavelength = 649nm) for transmitted light.
a. If the glass studio uses a coating of refractive index MgF(2) coating refractive index of n = 1.39 what should the minimum thickness of the coating be?
b. If the glass studio makes the MgF(2) film thicker, there will be certain thicknesses for which the film is again strongly reflecting for blue light (wavelength 480nm). If the studio uses the third such thickness (counting the minimum thickness as #1), what other wavelengths of the visible spectrum besides blue will be strongly reflected?
This sort of scheme is usually used for antireflection coatings using phase cancellation, but can be used for frequency-specific reinforcement as well. For reinforcement, the phase shift of the ray reflected from the MgF-glass boundary should be equal to (or an integer multiple of 2pi different from) that of the ray reflected from the air-MgF boundary. Both reflections, being of the type where the ray travels through a lower-N medium to a boundary with a higher-N medium, produce a phase shift of pi. For the two reflected rays to be in phase, the additional two-way path length of the coating, 2T, must be an integer mult. of λ, thus T must be an integer mult. of λ/2.
A. With N = 1.39, λ in MgF = 480/1.39 nm. Then T must be at minimum 0.5*480/1.39 nm.
B. If a thickness of 3*0.5*480/1.39 nm is used, any other wavelength for which this thickness is an integer mult. of λ/2 will be reinforced; that is (wavelengths in air) 3*480/(1,2,3,4…) = 1440, 720, 480, 320, etc. nm. In the visible spectrum, this means 720 nm is added if you stretch the visible definition a bit.
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