Form and structure factors for impedance and reflection from periodic layers

In an exact treatment of the Maxwell equations, we derive form and structure factors for reflection from periodic layers, and we show that these factors are significantly different from their analogs in kinematic x-ray diffraction. Quite generally, we show that reflection and impedance can be written precisely as the sum of an additive form factor and the product of a structure factor and a second form factor. This additive form factor does not have an analog in kinematic x-ray diffraction. It is demonstrated that the form factors are found by analytic continuation to an arbitrary wavelength of expressions for the impedance both at long wavelengths and at quarter wavelengths. A correction to the Bragg law relating fringe spacing to the total structure thickness is derived. We go beyond previous numerical work by deriving simple analytic exact expressions for reflection and impedance of periodic layers for all frequencies within the reflection passband, and for an arbitrary number of periods in the structure, an arbitrary index profile within each period, arbitrary layer thicknesses (not just quarter-wave layers), and for arbitrary sizes of the refractive index differences.