Propagation Properties of Partially Coherent Vortex Cosine-Hyperbolic-Gaussian Beams in Uniaxial Crystals

Optics Physics
partially coherent beams uniaxial crystals vortex beams cosh-gaussian beams light propagation

Propagation Properties of Partially Coherent Vortex Cosine-Hyperbolic-Gaussian Beams in Uniaxial Crystals

Research Background and Problem Statement

In the field of optics, the propagation characteristics of laser beams in anisotropic media (such as uniaxial crystals) have been a research hotspot. This research not only helps to understand the fundamental physical mechanisms of light-matter interactions but also provides theoretical support for designing wave plates, polarizers, compensators, and optical modulation devices. In recent years, with the deepening study of complex beams (such as vortex beams and partially coherent beams), scientists have begun to focus on the behavior of these novel beams in anisotropic media. However, there has been no systematic study on the propagation properties of partially coherent vortex cosine-hyperbolic-Gaussian beams (PCVCHGB) in uniaxial crystals.

To fill this gap, M. Lazrek et al. conducted relevant research. They aimed to reveal the propagation laws of PCVCHGB in uniaxial crystals through theoretical derivation and numerical simulation, and explore key issues such as intensity distribution, coherence changes, and beam shape evolution. This research not only enriches the theory of beam propagation but also provides important references for the practical application of partially coherent beams in anisotropic media.

Source of the Paper

This paper was co-authored by M. Lazrek, M. Yaalou, Z. Hricha, and A. Belafhal, all from the Laser Physics Group of the LPNAMME Laboratory, Department of Physics, Chouaïb Doukkali University, Morocco. The paper was published in the journal Optical and Quantum Electronics in 2025, with the article number 57:151, and DOI: 10.1007/s11082-025-08047-w.

Research Content and Methods

a) Research Process and Experimental Methods

This study mainly includes the following steps:

  1. Theoretical Derivation
    Based on the Huygens-Fresnel diffraction integral and ABCD matrix optics, the authors derived an analytical formula for the propagation of PCVCHGB in uniaxial crystals. This formula considers the initial beam parameters (such as decentered parameter (b), topological charge (m), coherence length (\sigma_0)) and the refractive index ratio (n_e/n_0) of the uniaxial crystal.
    During the derivation process, the authors utilized the definition of the hyperbolic cosine function, the addition formula of Hermite polynomials, and some special integral formulas. Ultimately, they obtained the expression describing the cross-spectral density of the beam (Equation (10)), which is the core theoretical achievement of the entire study.

  2. Numerical Simulation
    Based on the above theoretical formula, the authors conducted extensive numerical simulations to analyze the intensity distribution and coherence changes of the beam at different propagation distances. The simulation parameters included the initial beam waist radius (\omega_0=5\mu m), wavelength (\lambda=632nm), ordinary refractive index of the crystal (n_0=2.616) (using rutile crystal as an example), and Rayleigh distance (z_r=324.58\mu m).
    The simulations primarily examined the effects of the following variables:

    • Decentered parameter (b): Small values (e.g., (b=0.1)) correspond to hollow vortex Gaussian beams, while large values (e.g., (b=4)) correspond to four-petal vortex Gaussian beams.
    • Refractive index ratio (n_e/n_0): Used to evaluate the influence of crystal anisotropy on beam propagation.
    • Coherence length (\sigma_0): Used to study the modulation effect of partial coherence on beam shape evolution.
    • Topological charge (m): Used to analyze the influence of vortex structure on beam distribution.

  3. Algorithms and Data Analysis
    During data analysis, the authors employed various mathematical tools, including recursive relations of Hermite polynomials, complex number operations, and two-dimensional integral calculations. Additionally, the authors plotted numerous three-dimensional intensity distribution graphs and contour plots to visually demonstrate the evolution process of the beam at different propagation distances.

b) Main Results

  1. Short-Distance Propagation Characteristics
    Numerical simulations show that at short-distance propagation ((z_r)), PCVCHGB can maintain its initial shape. For small (b) values, the beam appears as a hollow vortex Gaussian beam, with a dark central region surrounded by a bright ring; for large (b) values, the beam appears as a four-petal vortex Gaussian beam. This indicates that the decentered parameter (b) significantly regulates the initial shape of the beam.

  2. Long-Distance Propagation Characteristics
    As the propagation distance increases, the beam gradually loses its initial shape. For small (b) values, the bright ring gradually evolves into an elliptical shape; for large (b) values, the four-petal structure gradually expands and overlaps, eventually forming a star-shaped pattern. This phenomenon is closely related to the anisotropy of the uniaxial crystal, where the larger the refractive index ratio (n_e/n_0), the faster the beam diffuses along the (x)-direction, while diffusion along the (y)-direction is slower.

  3. Coherence Changes
    The authors further studied the spatial coherence changes of the beam. The results show that the coherence length (\sigma_0) has a significant impact on the beam shape evolution. When (\sigma_0) is small, the beam can maintain a better shape in the far field; when (\sigma_0) is large, the beam easily loses its shape in the far field. Additionally, increasing the topological charge (m) enlarges the central dark region and alters the far-field distribution pattern.

c) Conclusions and Significance

Through theoretical derivation and numerical simulation, this study systematically reveals the propagation characteristics of PCVCHGB in uniaxial crystals. The research shows: - PCVCHGB can maintain its initial shape during short-distance propagation, but gradually evolves into an elliptical Gaussian-like beam due to crystal anisotropy during long-distance propagation. - The decentered parameter (b), coherence length (\sigma_0), and topological charge (m) significantly affect the beam shape evolution. - The refractive index ratio (n_e/n_0) of the uniaxial crystal determines the difference in diffusion speed of the beam along the (x) and (y) directions.

This study not only deepens the understanding of the propagation behavior of partially coherent beams in anisotropic media but also provides theoretical guidance for designing new optical devices and modulating beam shapes. For example, by adjusting the initial parameters of the beam or selecting appropriate crystal materials, precise control over beam shape and coherence can be achieved.

d) Highlights of the Study

  1. Innovative Theoretical Derivation
    The authors first derived the analytical formula for the propagation of PCVCHGB in uniaxial crystals, laying a theoretical foundation for subsequent research.

  2. Comprehensive Parameter Analysis
    The study covers multiple key parameters such as decentered parameter (b), coherence length (\sigma_0), topological charge (m), and refractive index ratio (n_e/n_0), revealing their comprehensive impact on beam propagation characteristics.

  3. Rich Numerical Simulations
    Numerous three-dimensional intensity distribution graphs and contour plots vividly demonstrate the evolution process of the beam at different propagation distances, providing readers with a clear physical picture.

e) Other Valuable Information

In addition to theoretical derivation and numerical simulation, the authors also discussed the propagation characteristics of PCVCHGB in other media (such as turbulent atmosphere and oceanic turbulence). These supplementary studies further expand the application scope of this beam.

Significance and Value of the Research

This study not only fills the gap in the research on the propagation properties of PCVCHGB in uniaxial crystals but also provides important references for the practical application of partially coherent beams. For example, in optical communication, beam parameters can be adjusted to improve the stability and anti-interference capability of signal transmission; in optical imaging, the special shape of the beam can be utilized to achieve high-resolution imaging. In summary, this research makes significant contributions to the development of beam propagation theory and the advancement of practical applications.