Fraunhofer diffraction and lenses Suppose we have the diffraction pattern for a single slit in the Fraunhofer approximation. In order to see this diffraction patter at finite distance we locate a converging lens. 
Is the diffraction pattern always centered on the slit? what happens if I move the lens up and down?

 A: The lens in this setup performs an optical Fourier transform. On the screen in the rear focal plane you observe the (intensity distribution of the) Fourier transform of the light field in the front focal plane – which is basically the slit here. The "zero" frequency (and thus the center of the pattern in this case) is given by the intersection of the optical axis of the lens and the screen – which by the way should be perpendicular to this axis.
Shifting the lens up and down to a certain extent is the same as shifting the slit down and up. Thus the pattern on the screen will be the Fourier transform of the shifted slit. Since a transverse shift in "real" space adds a simple linear phase gradient to the Fourier transform and only its intensity is observed on the screen, the observed pattern is unchanged.
In summary: If you move the slit, the pattern stays the same. If you shift the lens, the unchanged pattern follows this shift.
A: The work of the lens is accumulating the parallel beam at its focus, when the lens is at the center, like in the image the most parallel beam reach to the lens and the lens focus the most beam at the center,but if you move the lens up or down  the lens can not gather intensity or parallel beam as befor ,and we see the corner of the diffraction pattern,, so because the most parallel beam exist near the coardinate of y=0 in the image the diffraction pattern always has the highest intensity at the center,and its not related to moving the lens up or down.and moving the lens up or down cause that we just see the corner of the diffraction better.
