Thin-Film Interference

Press About to burst. The top of the film is now 10 nm of water, and it is black. One water surface on its own reflects 2.055% of 550 nm light, and ray optics, adding the film's two surfaces, expects 4.027%. The film reflects 0.197%, under a tenth of a single surface. The two reflections are almost the same size, but the one from the top, where light goes from air into water, comes back turned over (r = −0.1434), and the one from the bottom, water into air, does not (+0.1434). In a film much thinner than the wavelength they return almost in step and cancel. Thicken the film and the light from the bottom falls behind by δ = 4πn₁d/λ; at a quarter wave, 103.0 nm, the two add and the film reflects 7.892%, nearly twice the ghost. Just before black the film is a dim blue-grey, because the shortest wavelengths, with the largest δ, are the last to cancel. Newton watched it happen on a soap bubble under a glass cover: a black spot grew at the top where, he wrote, "At first I thought there had been no Light reflected from the Water in that place", and then, looking closer, saw a candle faintly reflected in it.

Press A 3 µm film. At 550 nm this film reflects 7.635%, almost the most any thickness can. Yet in white light it reflects 4.013%, against 4.025% for the ghost, and it looks the same grey. The spectrum shows why: 11 reflection peaks between 380 and 780 nm, so each of the three colour-matching functions the page integrates spans several bright fringes and dark ones, and the sum comes out as the ghost. Now press Green cancelled. At 206 nm there is a single trough in the visible, centred on 550 nm, which removes green and leaves a violet 0.256 from white. The most colourful film anywhere on the scale is within a few nanometres of this one. Colour needs a gap in the spectrum as wide as a colour, which only a thin film makes, and from about 2 µm up to the 5 µm the page reaches, the film never again comes more than 0.005 from white, against 0.256 for this violet. The second book of Newton's Opticks opens with it, as something others had already seen: thin plates show colours "according to their various thinness, altho' at a greater thickness they appear very clear and colourless". Ray optics gets a thick film right only because white light averages the fringes away. Light of a single wavelength never does.

Press Coated lens. 100 nm of magnesium fluoride on crown glass cuts the reflection at 550 nm from 4.239% to 1.247%. Some of that is the gentler steps in index: adding the two surfaces as ray optics does gives 2.754%. The rest is cancellation. Both reflections turn the wave over, since each goes into a denser material, and a quarter-wave layer delays the second by half a turn (δ = 0.5013 turn), so they arrive out of step. Now press Coating twice as thick. At 200 nm the delay is a whole turn, the reflections are back in step, and the face reflects 4.239% at 550 nm, the same as bare glass; in white light it reflects 4.082%, against 4.24% bare. More coating is no better; the right thickness is. Because a quarter wave fits only one wavelength, the coated face reflects 1.634% at 450 nm and 1.418% at 650 nm, more at both ends of the spectrum than in the middle, which is why the coated face in the picture is faintly violet. Full cancellation needs a layer of index √1.5185 = 1.2323: type it as the film index, set d to the quarter wave the results give, 111.6 nm, and the 550 nm reflection falls to 0.000%. Carl Zeiss applied on 1 November 1935 for a patent on a layer evaporated onto the glass in a vacuum. It names as its forerunner a British patent of Taylor's from 1904, prompted by old lenses whose weathered patches looked more transparent than the polished glass beside them.