 its thickness is precisely calculated in order for this net phase delay to be AQ = /2 (corresponding to a quarter wavelength, hence, the name). Intuitively, we may already infer that the QWP transforms the input polarizations from linear to circular and the converse, and this inference is absolutely correct! Let us prove such a property now. Assuming a linearly polarized incident E-field, as defined in Eq. (25.8), and the QWP axes oriented at 45° (cw) from the vertical axis, the incident E-field is projected along the QWP fast and slow axes according to the definition: (25.16) After traversing the QWP, the output E-field has become (25.17) Substituting E1 = 1, E2 = 0 (incident ray in the orientation, or aligned with the fast axis, or, conventionally, |0)), we obtain: Econ== (25.18) which is immediately identified as the left-circular polarization state, or n, or 1+). Clearly, the case E1 = 0, E2 = 1, corresponding to an incident ray in the orientation 25.8 Electromagnetic waves, polarization states, photons, and quantum measurements aligned with the slow axis, or, conventionally, |1)), yields the right-circular polarization state, or U, or |->. Thus, the QWP converts linear polarization states into circular polarization states, according to the transformations |0) |+) and |1) Let us show next the converse operation. Assume an incident ray that is circularly polarized, according to the base n or U (equivalently, |+) or |-)). It suffices it to substitute E1 = 1 and E2 = i in Eqs. (25.16) and (25.17) to obtain for the output E-field: (25.19) As expressed in the vertical or horizontal basis (i.e., after rotating the reference axes by 45° ccw), the output E-field is, finally, (25.20) which (within arbitrary phase factors) corresponds to the linearly polarized states I (top) and (bottom), respectively. Thus, the QWP converts circular polarization states into linear polarization states, according to the transformations |+) -> |0) and |-) |1), as previously announced. We may summarize all of the above conclusions by rec