Extended Source Effects in Substructure Lensing

We investigate the extended source size effects on gravitational lensing in which a lens consists of a smooth potential and small mass clumps ("substructure lensing"). We first consider a lens model that consists of a clump modeled as a singular isothermal sphere (SIS) and a primary lens modeled as an external background shear and convergence. For this simple model, we derive analytic formulae for (de)magnification of circularly symmetric top-hat sources with three types of parity for their lensed images, namely, positive, negative, and doubly negative parities. Provided that the source size is sufficiently larger than the Einstein radius of the SIS, we find that in the positive (doubly negative) parity case, an extended source is always magnified (demagnified) in comparison with the unperturbed macrolens system, whereas in the negative parity case, the (de)magnification effect, which depends on the sign of convergence minus unity, is weaker than those in other parities. It is shown that a measurement of the distortion pattern in a multiply lensed image enables us to break the degeneracy between the lensing effects of clump mass and those of clump distance if the lensing parameters of the relevant macrolens model are determined from the position and flux of multiple images. We also show that an actual density profile of a clump can be directly measured by analyzing the "fine structure" in a multiply lensed image within the Einstein radius of the clump.