# Shaping the longitudinal intensity pattern of Cartesian beams in lossless and lossy media

## TL;DR

This paper introduces a novel method to control the longitudinal intensity pattern of Cartesian beams, enabling advanced beam shaping for applications like optical tweezers, demonstrated with Gaussian and Airy beams.

## Contribution

The authors develop a technique to modulate the longitudinal intensity of Cartesian beams, filling a gap in beam shaping methods analogous to Frozen Waves in cylindrical coordinates.

## Key findings

- Method successfully modulates intensity patterns in Cartesian beams.
- Experimental demonstrations confirm the theoretical predictions.
- Tunable spot sizes achieved in Gaussian and Airy beam examples.

## Abstract

Several applications, such as optical tweezers and atom guiding, benefit from techniques that allow the engineering of optical fields' spatial profiles, in particular their longitudinal intensity patterns. In cylindrical coordinates, methods such as Frozen Waves allow an advanced control of beams' characteristics, but in Cartesian coordinates there is no analogous technique. Since Cartesian beams may also be useful for applications, we develop here a method to modulate on-demand the longitudinal intensity pattern of any (initially) unidimensional Cartesian beam with concentrated wavevector spectrum, thus encompassing all paraxial unidimensional beams. To this end, we write the total beam as a product of two unidimensional beams and explore the degree of freedom provided by the additional Cartesian coordinate. While in the plane where this coordinate is zero the chosen unidimensional beam keeps its structure with the additional desired intensity modulation, a sinusoidal-like oscillation appears in the direction of this variable and creates a spot whose size is tunable. Examples with Gaussian and Airy beams are presented and their corresponding experimental demonstrations are performed to show the validity of the method.