Lloyd's Mirror Experiment

Following Young's success with double slit interference, other scientists tried to replicate an interference pattern using separate light sources. This approach proved to be fruitless because separate light sources are not in phase; that is, the light produced by two random sources will never be in step to produce bright or dark bars. In 1834, Humphrey Lloyd produced an interference pattern of bright and dark bars with an ingenious arrangement illustrated at left.The top ray of light travels directly to the screen. Another ray travels to a mirror and reflects from same. It then proceeds to the screen as though it were coming from a phantom slit below the mirror. Lloyd was delighted to see an interference pattern of bright and dark bars. The bad news was that the bars were not where they belonged. Where there were supposed to be a bright bar, it was dark, and conversely.Lloyd reasoned that the only difference between his experiment and a traditional double slit setup was a change in phase for the ray of light incident on the mirror. Furthermore, this change of phase was exactly 1/2 wavelength. Finally, this change of phase occurred only when light was in a low index material and reflected off a higher index material.

Derivation of Thin Film Interference

Consider light incident on a thin slab of glass  whose index is nag. The slab is situated in vacuo.The wave equation yields line 1 outside the glass	1
Inside the glass the wave equation yields line 2.	2
Divide line 1 by line 2 to get line 3.	3
The left hand side of line 3 is the definition of index of refraction for glass.	4
Line 5 shows that the wavelength of light is different in glass than in vacuo.	5
Two factors contribute to interference--path length difference and the Lloyd's mirror effect. We get constructive interference if the PLD is one half wavelength and the thickness is 1/4 wavelength	6
constructive interference occurs for other thicknesses, namely, odd multiples of quarter wavelengths. We generate these values as shown	7	m = 0, 1, 2...
Line 7 shows the wavelength in glass. We need the wavelength in vacuo. We substitute from line 5.	8	m = 0, 1, 2...
Rearranging terms, line 9 describes constructive interference	9	m = 0, 1, 2...
Destructive interference comes when the thickness is even multiples of quarter wavelengths	10	m = 0, 1, 2...
Finally, lines 9 & 10 work when the LMPC applies to just one surface of the film. If the LMPC occurs at both surfaces, then the equations at lines 9 & 10 are REVERSED