Path Difference and Phase

Two waves leaving in step can arrive out of step if they travel different distances. That extra distance is the path difference. What matters is not the raw distance but how it compares to one wavelength. A path difference of exactly one wavelength means the second wave has gone through one full cycle more than the first, so it arrives back in step: crest meets crest. That is constructive interference, a bright fringe. A path difference of half a wavelength means one wave is at a crest exactly when the other is at a trough. They are 180 degrees out of phase, and if their amplitudes match, they cancel completely: a dark spot.

The conversion is direct and proportional. Every full wavelength of path difference equals 360 degrees of phase difference, so phase difference equals 360 degrees times the path difference divided by the wavelength. A path difference of 0, 1, 2, or any whole number of wavelengths gives 0 degrees phase difference effectively, meaning the waves line up and reinforce. A path difference of 0.5, 1.5, 2.5 wavelengths gives 180 degrees, pure cancellation. Anything in between gives partial reinforcement or partial cancellation, which is why fringe patterns fade smoothly between bright and dark rather than switching instantly.

This idea is the engine behind every interference and diffraction pattern in this unit. Double slit bright fringes sit where path difference equals a whole number of wavelengths; dark fringes sit at half-integers. The same logic explains thin film colors and diffraction grating spacing. Path difference is the one quantity you calculate from geometry, and phase difference is what that geometry means physically for the waves meeting at a screen.