Power spectrum equalized scalar representation of wide-angle optical field propagation
It may be desirable to represent optical fields using scalar approximations, due to its simplicity. Since the optical field is an electromagnetic wave, in order to implement an optical setup, a mapping from such a scalar field to the vector electromagnetic field is needed. In the conventional scalar-to-vector field mapping, a large error in power spectrum arises in wide-angle fields due to the neglected large longitudinal component of the electric field. This error could be severe in wide-angle or off-axis imaging setups. In order to find another scalar-to-vector field mapping that compensates for this large magnitude error, first, a general constraint on monochromatic electromagnetic fields to appropriately represent them by a scalar wave in free space is developed. The development of the general constraint begins by formulating the computations of the components of the magnetic field as the outputs of linear-shift invariant (LSI) systems, where the inputs to the LSI systems are the transverse components of the electric field. Furthermore, if one of the transverse components of the electric field can be computed from the other one using a LSI operation, a scalar field, which is related to the transverse components through another LSI system, can be used to fully describe the electromagnetic field. Under this constraint, the required condition on the filters which relates the scalar field to the electric field is presented by taking into consideration the longitudinal component, such that the power spectra of the scalar field and the corresponding electric field are equal. The filters are specified for the electric fields with zero longitudinal component and simple polarization features, as well. Moreover, for the electric fields with simple polarization features, some discrete simulations are performed to compare the scalar field intensity pattern and the corresponding electric field intensity patterns for the conventional and proposed mapping cases. The simulation results show that the excessive amplification of the large frequency components is compensated by the proposed filters, and hence, the undesired effects of the filters used in the computation of the longitudinal component disappear. In this respect, if equality of the power spectra of the scalar field and the corresponding electric field is of concern in an application, the proposed scalar-to-vector wave field mapping should be used.