I) A di'raetive beamsplitter can be made with a material that acts as a half wave plate whose axis of rotation changes across the plate. In other words, the plate can be described by a Jones matrix of the form: we [[1, 3,] seem. (1) where 9(33) 2 or, and RM) is the rotation matrix. Show that this splitter separetes a beam into its circular components in the following way: Do this by supposing an xpolarized beam is incident, and calculate the xdependence of the :r and 3,: components of the eld after passing through the plate. Then take the Fourier transform with respect to :r: to express the eld in terms of its kw components. Calculate the angle ,8 in terms of a and either the wavelength or frequency of the incident light. (1}) Another way to make use of a positiondependent half wave plate rotation is to convert input linear polarization to radial polarization. In this version the rotation of the local halfwave plates varies with the azimuthal (polar) angle (r. It is possible to accomplish this by making a dense array of half wave plates, with a controllable angle of the fast axis to the input polarization. 'We want the input linear polarization to be converted to radial, which can be written as: :er l 132 ,. ,. 2' (323: + ya) (2) ()[ it!\" The following image shows schematically the input and the output polarization states. For the radially polarized beam, the eld points outwards at one point in time, then a half cycle later it points inwards. Notice that the eld goes to zero at the origin [it must, since the polarization state would be undened there). Design the converter by specifying the halfwave plate rotation function that will produce radial polarization with linear polarization input. (c) Now we take the output beam from part (b) and direct into the splitter described in part (a). Calculate expressions for two diffracted beams that come out, including polarization state and any additional phase functions.