# Diffraction Grating interference pattern

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## Homework Statement

This is the interference pattern on a viewing screen behind two slits.

(image of 7 bright fringes with black fringes between every pair of bright fringes)

How would the pattern change if the two slits were replaced by 20 slits having the same spacing d between adjacent slits? Would the number of fringes on the screen increase, decrease, or stay the same?

## Homework Equations

dsin(theta)=m(lambda)

## The Attempt at a Solution

The formula above says that for constructive interference, the path-length difference between successive slits needs to be an integer multiple of wavelength (lambda). However, I don't see anything in my book about the number of fringes going up or down with N (the number of slits) on the grating. Any advice?

I know that the fringe spacing increases, the width of each fringe decreases, and the brigntness of each fringe increases, but I don't know about the *number* of fringes and whether it goes up or down.

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ehild
Homework Helper
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Two-thirds of the way down, I see: * This pattern has maxima where all cosine terms are either 1, or all are -1 .
o All cosines are +1, when π (d/λ) sinθ = 2 m π
o All cosines are -1, when π (d/λ) sinθ = (2 m + 1)π
* Hence, any integer number of π will do, and the directions of intensity maxima in pattern from diffraction grating are
sinθm = m λ/d
i.e. exactly the same as for two-slit interference.
* The value of m is called the order of the maximum .

This tells me that maxima can be found at angles of θm=sin-1(mλ/d) but it doesn't tell me anything about the number of fringes for an N = 2 interference pattern versus an N = 20 interference pattern.

Also, I assume there is a point where the bright fringes dissipate as the order of maximum gets larger and larger (m increases).

ehild
Homework Helper
" and the directions of intensity maxima in pattern from diffraction grating are
sinθm = m λ/d
i.e. exactly the same as for two-slit interference. "

If the directions are the same as in case of two slits, the bright strips are at the same positions independent on the number of slits. If there are seven bright strips in between the minima of the diffraction pattern of a single slit, it will stay the same if you have a grating with many slits of the same width and spacing, only the bright strips will be narrower and brighter.

ehild

Gold Member
Thanks for the clarification ehild!