Young's double slit interference

[ppy]

Homework Statement

In a young's double slit experiment the slits have the same width and are separated by a distance a. an observation screen is placed at a distance L=1m from the slits at a point on the screen a distance y from the optical axis the optical path difference between the 2 waves from the slits is

r1-r2=(ay/L)

the double slits are illuminated with light with longitudinal coherence l=10x10^-6m if the interference pattern is observed on the screen around the position y=1cm what is the max value for a for which interference fringes are observed?

I am confused how can interference fringes be observed. For interference coherent light is needed but the longitudinal coherence is less than the distance from the slits to the screen so surely by the time the waves are at the screen the waves aren't coherent anymore so no interference pattern is observed?

Much help needed. Do I use that the path length difference must be greater than zero for interference ?

Thanks

 

[Ibix]

A not entirely accurate picture of this is to imagine an army on the march. Every man marches at the same pace, and every man in each regiment is in phase, but not necessarily in phase with the men in the regiment in front or behind. The longitudinal coherence length is about half the length of a regiment - the distance over which you can be reasonably confident of predicting the phase of a man's pace given the phase of the man passing in front of you.

So the answer is that as long as the path difference doesn't exceed the longitudinal coherence length then the wavefronts from the two slits should have a reliable phase relationship. Thus we simply require $a\leq lL/y$ where $l$ is the longitudinal coherence length. That gives us $a\leq 10^{-3}$m.