I see that slotted waveguide antennas are usually built from elements with diagonal offset (as shown in attached figure).
The longitudinal offset is understood to me (phase difference), however the horizontal offset is not clear.
If the waveguide is excited by TE 10 fundamental mode, the E field keeps his sign along X axis.
Can you please help me figuring out? Does the horizontal offset also has something to do with the accumulated phase?
Spacing the slots at ½λg intervals in the waveguide is an electrical spacing of 180° —
each slot is exactly out of phase with its neighbors, so their radiation will cancel each
other. However, slots on opposite sides of the centerline of the guide will be out of phase
(180°), so we can alternate the slot displacement around the centerline and have a total
phase difference of 360° between slots, putting them back in phase.
Slotted waveguide antennas can often be very useful. The basic idea is that in moving the slots around the amplitude and phase distribution of an array can be controlled. The trick is finding the appropriate placement and then figuring out how to manufacture the thing. Modern simulation tools and numeric controlled machining makes these much easier to realize. Slotted waveguides appear often in shipboard and airborne radars.
I note that there seem to be threads on the same subject here. You might fish through some of them to see if any are helpful.