We have had so much fun with two slits. Naturally, we want to add another slit to make it 3-slit interference, right?

Firstly, let me emphasize that we are talking about 3 slits which are equally spaced out. In other words, consecutive slits have the same slit separation *d*. Hence waves from consecutive slits arrive at (each point on) the screen with the same path difference .

This has the important implication that when the 3 light rays arrive at a point on the screen, ray B will lag ray A by the same phase difference as ray C lags ray B.

Let’s consider what happens when .

Ray C will arrive exactly one cycle after ray B, which arrives exactly one cycle after ray A. So all 3 sinusoidal waves arrive in phase at P, superposing constructively to form a resultant wave of amplitude 3*A* (assuming the amplitude of each wave is *A*).

Let’s now consider what happens when .

Ray C will arrive one-third of a cycle after ray B, which arrives one-third of a cycle after ray A. It is like we have 3 sinusoidal waves spaced apart with 120° between (A and B), (B and C) and (C and A). Do you realize that these 3 sinusoids are going to sum up to zero? The 3 rays interfere destructively to cancel one another out!

Amazingly, D.I. also occurs among these 3 waves when .

With some thought, we can deduce other values of at which C.I. and D.I. will occur.

C.I. when | D.I. when |

0 | , |

λ | , |

2λ | , |

Alright. I guess we are now confident enough to add even more slits!

If we have 4 slits (and 4 rays), the first D.I. occurs when the 4 waves are spaced apart by . This happens when the path difference is .

If we have 10 slits (and 10 rays), the first D.I. occurs when the 10 waves are spaced apart by . This happens when the path difference is .

By extension, we can deduce that for *N* slits:

Bright fringes are formed at angles *θ* where ,

Dark fringes are formed at angles *θ* where , and

The graphs below show how the interference pattern’s intensity profile changes as we add more and more slits (while maintaining the same slit separation *d*).

Note that as *N* increases the number of bright fringes do not change. Neither do their positions. However, they do become brighter, since they have amplitude *NA* and intensity (assuming each ray arrives on the screen with amplitude *A* and intensity *I*_{0}). In the graphs above, the intensity has been “equalised” across the different *N* values because we have chosen to plot the normalized intensity , where is .

Note also that as *N* increases, the number of dark fringes increases. In fact, there are dark fringes between two consecutive bright fringes. When *N* is a large number, we see dark fringes everywhere, except at those locations where there are very narrow and bright fringes.

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**Video Explanation **