Would an infinitely (or very long) diffraction grating produce a diffraction pattern?

Question

Take a typical science lab diffraction grating which is producing a pattern on a screen.

Let's consider the location of say the first maximum on the right side.

Let's draw straight lines from the slits to the first maximum and think in terms of ''rays'' travelling from the slits to the maximum on the screen.

These rays all converge at the maximum and although they are close to being parallel we know that they are not perfectly parallel (because they converge).

Therefore the differences between the distances that neighboring rays have to travel will not be exactly the same.

i.e. If one ray has to travel one lambda more than its neighboring ray, then the following ray will have to travel slightly more or less than its neighboring ray.

If there are many slits very close together I can see that if the variance is small compared to the wavelength then we can ignore these differences and assume that each ray travels exactly one wavelength more/less than its neighboring ray.

But if the grating is long enough, these differences will eventually build up until they are significant enough to cause destructive interference.

My understanding is that a very long grating will have destructive interference at all locations and will not produce a pattern.

Is this true ?

What are peoples thoughts on this ?

Answer 1

Would a very long diffraction grating produce a diffraction pattern?

Yes. You might need to set the screen further back to ensure that the far-field condition is still satisfied, but given that the screen is sufficiently far away (for the length of the grating), arbitrarily large gratings can be used.

(Alternatively, you can drop the idealized introductory-textbook configuration and use a realistic design, which includes things like a curved screen and focusing optics before the grating.)

Would an infinitely long diffraction grating produce a diffraction pattern?

This is equivalent to asking "can one get infinitely far away from an infinitely long grating?". This kind of ill-defined double limit is not worth considering.

Answer 2

All secondary peaks of the pattern are going to become dimmer and dimmer until they disappear completely as the number of slits is increased. But the central peak will become more and more bright and sharp, since light from each slit interferes constructively at the center of the pattern