Condition for the constructive interference of waves from a crystal film

Constructive interference of waves

In order for two waves to simultaneously strenghen each other (that is, constructively interfere), they must be in phase. After reflection from a thin crystal grating with spacing d, two waves are in the same phase only if the additional distance $l$ that one reflected wave must travel is an integer multiple of the wavelength λ longer than the distance that a second reflected wave must travel.
The wavelength difference $l$ between the two waves depends on the incoming wave's angle of incidence θ and the spacing d of the grating planes. Geometrical considerations give the result$$l=2\cdot d\cdot \sin(\theta)$$ where the factor of 2 arises because the overall additional path includes the distance traveled by both the incoming and outgoing waves. The so called Bragg condition for constructive interference is given by $$\text n\cdot \lambda=2\cdot d\cdot \sin(\theta)$$where n ∈ N_{0} is the order of the interference maximum. (In the graphic, the wave picture for n=2 is depicted. The travelling distance difference of the waves is two wavelengths to create the second order interference maximum.)