(Solved): 2. 4f system [60 pt] Consider a 4f imaging system of unit magnification (i.e. ...

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2. $$4f$$ system [60 pt] Consider a $$4f$$ imaging system of unit magnification (i.e. both lenses of focal length $$f$$ ), with an unobstructed circular aperture of radius $$a$$. This imaging system has an amplitude point spread function $$H(?)$$. (1) Derive $$H(?)$$ in the case where an obstructing disk of radius $$b<a$$ is inserted into the aperture. (2) Derive $$H(?)$$ in the case where the disk in (1) is fully transmitting but produces a phase shift of $$90^{?}$$ (3) Derive $$H(?)$$ in the case where the disk in (1) is fully transmitting but produces a phase shift of $$180^{?}$$. (4) Consider imaging an on-axis point source of light at the object plane with the above systems in (1), (2) or (3). Compared to the unobstructed aperture system, is it possible to obtain an increase in the image intensity on axis (i.e. at the point $$?=0$$ in the image plane)? If so, under what conditions? Is it possible to obtain a null in the image intensity on axis? If so, under what conditions?