The project that produced these data involved the analysis and modeling of atmospheric Asian dust particles and background marine air particles collected in Hawaii, USA at the Mauna Loa Observatory (MLO) of the National Oceanic and Atmospheric Administration. Individual heterogeneous dust particles were analyzed with focused ion-beam scanning electron microscopy and energy-dispersive X-ray spectroscopy. Three-dimensional spatial and composition models of the particles from focused ion-beam tomography were used in the calculation of optical properties by the discrete dipole approximation method with incident light at 589 nm. Optical properties included the phase function, extent of linear polarization, scattering, backscattering, and extinction cross sections, and the backscatter fraction. In addition, the project involved optical property calculations for particles as geometric shapes with volumes equivalent to those of the MLO particles. Shapes modeled for each particle included spheres, spheroids, ellipsoids, cubes, square prisms, rectangular prisms, tetrahedra, and triangular pyramids. The representation of particles as geometric shapes is an important component of algorithms used in remote sensing (satellite-based and ground-based) for determining how atmospheric dust affects climate. The dataset consists of Microsoft Excel and Word files, text files, and bitmap image files. The Asian dust particles in the dataset are labeled: CaMg 1D, CaMg 2N, CaMg 3D, CaMg 4N1, Ca-rich 1D, Ca-rich 2N, Ca-rich 3D, Ca-rich 4N1, Ca-rich 4N2. The background marine air particles are labeled: Ca-S 1D, Ca-S 2N, Ca-S 3D, Ca-S 4N. Data are contained in seven folders arranged by the following topics: 1 -- Particle compositions by FIB-SEM-EDX and volumes of material phases within particles (folder: 1_Particle_Compositions_Volumes); 2 -- Spatial and optical parameters for optical modeling of particles and geometric shapes (folder: 2_Particles_Shapes_Spatial_Optical_Parameters (and subfolders)); 3 -- Complex refractive indices for particles and shapes based on Maxwell Garnett average dielectric function (folder: 3_Complex_RIs_Maxwell_Garnett (and subfolders)); 4 -- Results from discrete dipole approximation modeling software DDSCAT ver. 7.3 (folder: 4_DDSCAT_Scattering_Output (and subfolders)); 5 -- Mueller scattering matrix elements (folder: 5_Matrix_Elements (and subfolders)); 6 -- Root-mean-square calculations for phase function and degree of linear polarization (folder: 6_PhaseFunction_LinearPolarization_RMS); 7 -- Calculations for the backscatter fraction (folder: 7_Backscatter_Fraction)