# K-series approximation of vectorial optical fields for designing diffractive optical elements with subwavelength feature sizes

## TL;DR

This paper introduces a wave-vector series approximation method for vectorial optical fields in subwavelength diffractive optical elements, enabling efficient and accurate design by addressing computational challenges of vectorial simulations.

## Contribution

It proposes a k-series approximation algorithm based on RCWA for efficient vectorial field simulation in subwavelength DOEs, outperforming scalar models in accuracy.

## Key findings

- Predicted intensity profiles match FDTD simulations with fractional error.
- Algorithm achieves 10-100 times better accuracy than scalar models.
- Requires reasonable computational resources for complex designs.

## Abstract

Diffractive optical elements (DOEs) are widely applied as compact solutions for desired light manipulations via wavefront shaping. Recent advanced chip applications further require their feature sizes to move down to the subwavelength, which inevitably brings forth vectorial effects of optical fields and makes the typical scalar-based theory invalid. However, simulating and optimizing their vectorial fields, which are associated with billions of adjustable parameters in the optical element, are difficult to do, because of the issues of numerical stability and the highly-demanding computational cost. To address this problem, this research proposes an applicable algorithm by means of a wave-vector (k) series approximation of vectorial optical fields. On the basis of the semi-analytical rigorous coupled wave analysis (RCWA), an adequate selection scheme on k-series enables computationally efficient yet still predictive calculations for DOEs. The performance estimations for exemplary designs by the finite difference time domain (FDTD) method show that the predicted intensity profiles by the proposed algorithm agree with the target by just a fractional error. Together with optimizing the geometrical degrees of freedom (e.g., DOE depth h) as compensation for errors from the truncation of k-series, the algorithm demonstrates its outperformance by one or two orders of magnitude in accuracy versus the scalar-based model, and demands only a reasonable computational resource.