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Spatially resolved (sub)millimeter polarization has been detected by ALMA in an increasing number of disks around young stellar objects. The majority of the observations show polarization patterns that are consistent with self-scattering, especially at ALMA Band 7. The inferred sizes of the grains are typically of order 100 microns, which is very different from the millimeter size commonly inferred from the dust opacity index beta. In an effort to resolve this discrepancy, we first introduced the so-called "Coplanar Isotropic Radiation Field" approximation, which enables the computation of the (signed) polarization fraction analytically. With an oft-adopted dust composition (used, e.g., by DSHARP), we find that models with big dust grains produce very small polarization with reversed orientation, which hasn't been observed. The semi-analytic results are validated through Monte Carlo radiative transfer simulations. In these models, the "correct" polarization orientation and the small beta index are mutually exclusive. To resolve this tension, we explore a wide range of dust models, parameterized by their complex refractive index. We find that both the fraction of the polarization and whether it is reversed or not depend on the refractive index in a complex way, and this dependence is mapped out on an n-k plane for a representative dust size distribution (MRN with amax=3mm) and wavelength of 870 microns. In particular, refractory organics dust grains produce reversed polarization, whereas grains of the same size but made of absorptive carbonaceous materials can produce a percent-level polarization that is not reversed; the latter may alleviate the tension between the grain sizes inferred from the scattering-induced polarization and the opacity index beta. We conclude that scattering-induced polarization has the potential to probe not only the sizes of the grains but also their composition.

The Effects of Dust Optical Properties on the Scattering-induced Disk Polarization by Millimeter-sized Grains