## Interference

Interference and coherent sources:  Monochromatic light is light with a single frequency.  Coherence is a definite, unchanging phase relationship between two waves.  The overlap of waves from two coherent sources of monochromatic light forms an interference pattern.  The principle of superposition states that the total wave disturbance at any point is the sum of the disturbances from the separate waves.

Two-source interference of light:  When two sources are in phase, constructive interference occurs where the difference in path length from the two sources is zero or an integer number of wavelengths; destructive interference occurs where path difference is a half-integer number of wavelengths.  If two sources separated by a distance $d$ are both very far from a point $P$, and the line from the sources make an angle $\theta$ with the line perpendicular to the line of the sources, then the condition for constructive interference at $P$ is $d\sin\theta=m\lambda, (m=0,\pm 1,\pm 2,\ldots)$.  The condition for destructive interference is $d\sin\theta=(m+\frac12)\lambda, (m=0,\pm1,\pm2,\ldots)$.  When $\theta$ is very small, the position $y_m$ of the $m$th bright fringe on a screen located at distance $R$ from the sources is: $y_m=R\frac{m\lambda}d, (m=0,\pm1,\pm2,\ldots)$.

Intensity in interference pattern:  When two sinusoidal waves with equal amplitude $E$ and phase difference $\phi$ are superimposed, the resultant amplitude $E_P$ and intensity $I$ are as follows: $E_P=2E|\cos\frac{\phi}2|$, $I=I_0\cos^2\frac{\phi}2$.  If the two sources emit in phase, the phase difference $\phi$ at a point $P$ (located a distance $r_1$ from source 1 and a distance $r_2$ from source 2) is directly proportional to the path difference $r_2-r_1$: $\phi=\frac{2\pi}{\lambda}(r_2-r_1)=k(r_2-r_1)$.

Interference in thin films:  When light is reflected from both sides of a thin film of thickness $t$ and no phase shift occurs at either surface, constructive interference between the reflected waves occurs when $2t$ is equal to an integral number of wavelengths.  If a half-cycle phase shift occurs at one surface, this is the condition for destructive interference.  A half-cycle phase shift occurs during reflection whenever the index of refraction in the second material is greater than in the first.

Michelson interferometer:  The Michelson interferometer uses a monochromatic light source and can be used for high-precision measurements of wavelengths.  Its original purpose was to detect motion of the earth relative to a hypothetical ether, the supposed medium for electromagnetic waves.  The ether has never been detected, and the concept has been abandoned; the speed of light is the same relative to all observers.  This is part of the foundation of the special theory of relativity.