Christiansen filter using cylindrical step lens of odd symmetry

Abstract

A type of Christiansen filter that uses a cylindrical step lens of odd symmetry to achieve wavelength discrimination is proposed. This kind of Christiansen filter has pass band narrower than that of a traditional Christiansen filter, with more means of control for its performance. It could approximate arbitrary functions over a frequency range of interest. The filter's response is analyzed against the step parameters of the lens, and a systematic design method of inverse scattering is developed for the synthesis of such filter. The design technique uses Fourier transformation to correlate the lens' parameters with the filter's performance. With this method the lens shape can be derived from the characteristic pattern of its scattering, which is set to the prescribed filter response. This enables the synthesis of such filter from a desired, target performance. The synthesis of a desired, prescribed response by properly configuring the steps of the lens of the filter is established. It is derived that, due to the limitation of the step lens itself, only certain filter responses can be synthesized and realized physically in Christiansen filter form. Several popular filtering functions are discussed in detail. Two Christiansen filters centered at 545 nm with a full width at half maximum of 2 nm are designed with step lenses, and their selectivity is compared. When a response function cannot be realised in the Christiansen filter form, the underlying difficulty is discussed.