# Tailored Magnetic Linear Birefringence in Wedge-Shaped Co Nanocluster Assemblies
## 1 Introduction  
Since its discovery in calcite crystals, birefringence is a very well known effect and largely studied in optical physics. In anisotropic crystalline structures, the transmitted light propagates with two different refractive indexes, as its polarization plane is perpendicular or parallel to the optical axis of the birefringent material. For the case of uniaxial crystals, those are the ordinary and extraordinary transmitted rays, respectively. This effect has countless applications in optical devices, e.g., polarization rotators, waveplates, light modulators, or filters. For isotropic or disordered media, the necessary existence of an anisotropy axis to yield birefringence is to be induced externally somehow, e.g., stress or form birefringence in polymeric tapes. More recently, the use of pulsed lasers to irradiate different substrates has shown to be a powerful tool to obtain high-resolution bidimensional nanostructures with a defined-form anisotropy [1, 2]. These laser-induced periodic surface structures (LIPSS) consist of a two-dimensional (2D) ripple array, where the transmitted light travels with two different velocities as its polarization plane is parallel or perpendicular to the ripple direction. Such approach to induce form birefringence has been extended to polymeric substrates [3], showing a retardation factor comparable or higher than LIPSS in glass systems [4, 5]. Thus, LIPSS seem to be promising candidates for optical devices in flat optics, and are much more flexible and affordable than crystalline materials. Magnetic linear birefringence (MLB) is a much more subtle magneto-optical effect, and notwithstanding, not less interesting both for fundamental research and potential applications in magnetic sensors [6]. In the case of magnetized media with some kind of magnetic anisotropy, the magnetization reversal is accomplished with two different mechanisms as the external magnetic field, \(H\), is applied parallel or perpendicular to that anisotropy axis. Consequently, the electrical field, E, of the polarized light transmitting across those systems interacts differently with the magnetization vector, depending on their relative orientation. That leads the light to propagate with two different refractive indexes as E aligns parallel or perpendicular to \(H\), i.e., magnetic linear birefringence [7]. The first experimental evidence of this nonlinear magneto-optical effect was observed by [PERSON] et al. [8] in Co, Fe, and Ni-Fe thin films prepared by vacuum deposition in a magnetic field, \(H_{0}\). Thus, the surface magnetization, M, was partially induced to follow a preferential alignment (easy magnetic axis along \(H_{0}\) direction), and show a hard magnetic perpendicular axis. Recently, other alternative methods have been tested to increase the magnitude and, above all, the accuracy of MLB in thin films, i.e., towards the _ideal_ magnetic linear birefringence [8]. This could be the case of Co films grown on glass or silicon substrates, and nanopatterned later to obtain a defined and regular ripple nanostructure, showing easy and hard magnetic axes, parallel and perpendicular to the ripple direction, respectively [9; 10]. The interesting applicability of that magneto-optical effect to flexible flat optics has been also studied in LIPSS on polymeric substrates and coated with a thin film of permalloy, showing the coexistence of form and magnetic linear birefringence [3]. For all the above, a deep insight of the relevant sources in the magnetic anisotropy is crucial to control the processes for magnetization reversal in 2D systems, where MLB is required later. During the last decades, wide research has been driven to the interplay between volume and interface contributions ruling the magnetization anisotropy in epitaxial thin films and multilayers [11; 12; 13; 14]. In the case of ultrathin 2D magnetic systems, a dominant interface contribution can force the magnetization vector to orientate perpendicular to the film plane, overcoming the shape effect.