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Interference

We discussed how waves differ from particles and how one goes about characterizing them. But to truely understand how peculiar waves are one has to become familiar with the concept of wave interference. Unlike particles waves are spatially distributed, and unlike particles two different waves can "share the same location" and by that I mean that two different waves can occupy the same space. A question then comes to mind - if we look at a certain spot that "contains" two different waves, how then do we tell one from the other? The unswer is - we dont. There is no notion of two different waves, since when two different waves overlap spatially and temporaly (at the same place and time) their amplitudes add up to produce a third wave, a new wave.

The figure above illustrated two (one-dimensional) waves that ocupy the same position in space. The two are ofthe same wavelength and differ by a certain phase factor (they are shifted - their peaks and troughs don't overlap). The blue wave at the top is the result of the interference of the two waves. If one carefully examines how the amplitudes are combined it becomes apparent that two peaks (one positive, the other negative both of the same magnitude) can cancel each other in what is commonly referred to as destructive interference. In the case that two peaks of the same sign and magnitude overlap, they produce a larger peak in a constructive interference.


Intensity of Interference

We wish to understand what happens to the intensity of two waves that undergo interference. We restrict ourselves to electromagnetic waves, and we consider their intensity since it is what we percieve with our eyes.

Where

To be more precise, we see the time-averaged intensity. Taking the time average of the intensity and assuming both amplitudes are the same

We get that the time-averaged intensity is

Where

By looking at the equation for I3, it is easy to see that a phase difference that corresponds to a half a wavelength results in a zero intensity which is the destructive interference. And phase differences that correspond to a full wavelength result in four times the intensity - the constructive interference.


Coherence

Two beams of light are said to be coherent if their phase difference remains constant as they propagate through space, and as time passes. The path along two beams remain coherent is termed coherence length and the time during which two beams remain coherent is called coherence time. For the effects of interference to be noticable, it is important that the two waves are coherent. White light, which has a very short coherence length - meaning that after a very short distance it is no longer coherent, does not produce noticable interference effects.

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