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Interference in Space

Figure:   A replica of Thomas Young's original drawing (1803) showing the interference pattern created by two similar waves being emitted "in phase" (going up and down simultaneously) from two sources separated by a small distance. The arrows point along lines of constructive interference (crests on top of crests and troughs underneath troughs) and the dotted lines indicate "lines of nodes" where the crests and troughs cancel.

Suppose spherical waves emanate from two point sources oscillating in phase (one goes "up" at the same time as the other goes "up") at the same frequency, so that the two wave-generators are like synchronized swimmers in water ballet.^14.24 Each will produce outgoing spherical waves that will interfere wherever they meet.

The qualitative situation is pictured in Fig.14.13, which shows a "snapshot" of two outgoing spherical^14.25waves and the "rays" ($\Vec{k}$ directions) along which their peaks and valleys (or crests and troughs, whatever) coincide, giving constructive interference. This diagram accompanied an experimental observation by Young of "interference fringes"" (a pattern of intensity maxima and minima on a screen some distance from the two sources) that is generally regarded as the final proof of the wave nature of light.^14.26

Figure: Diagram showing the condition for constructive interference of two "rays" of the same frequency and wavelength $\lambda$ emitted in phase from two sources separated by a distance $d$. At angles for which the difference in path length $\Delta \ell$ is an integer number ($m$) of wavelengths, $m \lambda$, the two rays arrive at a distant detector in phase so that their amplitudes add constructively, maximizing the intensity. The case shown is for $m = 1$.

If we want to precisely locate the angles at which constructive interference occurs ("interference maxima") then it is most convenient to think in terms of "rays" ($\Vec{k}$ vectors) as pictured in Fig. 14.14.

The mathematical criterion for constructive interference is simply a statement that the difference in path length, $\Delta \ell = d \, \sin \vartheta_m$, for the two "rays" is an integer number $m$ of wavelengths $\lambda$, where the $_m$ subscript on $\vartheta_m$ is a reminder that this will be a different angle for each value of $m$:

$$\fbox{ \rule[-0.5\baselineskip]{0pt}{1.5\baselineskip} \hb . . . . . . aystyle d \, \sin \vartheta_m \; = \; m \, \lambda $} }\; .$$ (14.46)

$$\hbox{\rm (criterion for {\sc Constructive Interference})}$$

Conversely, if the path length difference is a half-integer number of wavelengths, the two waves will arrive at the distant detector exactly out of phase and cancel each other out. The angles at which this happens are given by

$$\fbox{ \rule[-0.5\baselineskip]{0pt}{1.5\baselineskip} \hb . . . . . . r} \; = \; \left( m + {1\over2} \right) \, \lambda $} }\; .$$ (14.47)

$$\hbox{\rm (criterion for {\sc Destructive Interference})}$$

Footnotes

. . . ballet.^14.24

This notion of being "in phase" or "out of phase" is one of the most archetypal metaphors in Physics. It is so compelling that most Physicists incorporate it into their thinking about virtually everything. A Physicist at a cocktail party may be heard to say, "Yeah, we were 90$^\circ$ out of phase on everything. Eventually we called it quits." This is slightly more subtle than, " . . . we were 180$^\circ$ out of phase . . . " meaning diametrically opposed, opposite, cancelling each other, destructively interfering. To be "90$^\circ$ out of phase" means to be moving at top speed when the other is sitting still (in $SHM$, this would mean to have all your energy in kinetic energy when the other has it all in potential energy) and vice versa. The $\Vec{E}$ and $\Vec{B}$ fields in a linearly polarized $EM$ wave are 90$^\circ$ out of phase, as are the "push" and the "swing" when a resonance is being driven (like pushing a kid on a swing) at maximum effect, so in the right circumstances "90$^\circ$ out of phase" can be productive . . . . Just remember, "in phase" at the point of interest means constructive interference (maximum amplitude) and "180$^\circ$ out of phase" at the point of interest means destructive interference (minimum amplitude - zero, in fact, if the two waves have equal amplitude).

. . . spherical^14.25

OK, they are circular waves, not spherical waves. You try drawing a picture of spherical waves!

. . . light.^14.26

Young's classic experiment is in fact the archetype for all subsequent demonstrations of wave properties, as shall be seen in the Chapter(s) on QUANTUM MECHANICS.