Mutual Interference And Structural Properties of Object Images in the Vicinity of the Gravitational Lens Cusp Point

Gravitational lens system could form multiple images of the same radiating surface of a cosmic object. The radiation fluxes of these images interfere to one another (Refsdal, 1964). The practical importance of this effect is seems to be as follows: (i) it is the only direct test of microlensing (Schneider et. al. 1985); (ii) this effect may say about fine structure of a lensed object; (iii) it must be allowed for simulations of microlensing process. The maximum effect is expected in the case when a cusp point of the microlensing caustic is projected on the lensed object. To investigate the interference effect in this case, the exact solution of the lens equation in the vicinity of the cusp point was obtained (Mandzhos,1993): $${x_{1,2}} = 2|A|.\cos \{{\Phi _0} + \left[{1 \pm sign\left(\xi \right)} \right]\frac{\pi}{3}.sign\left(\xi \right),$$ $${{x}_{3}} = 2|A| \cdot \cos \{ {{\Phi }_{0}} + \frac{4}{3}\pi \} \cdot sign\left( \xi \right),y = \frac{1}{2}\eta - \frac{1}{4}v{{x}^{2}}$$ (1) were ξη are dimensionless coordinates of a radiating element on the object plane; x,y are coordinates of the point at which the lens plane is intersected by light beam; $${{\Phi }_{0}} = \frac{1}{3}Arc\tan \sqrt {{h\frac{{{{\eta }^{3}}}}{{{{\xi }^{2}}}} - 1,|A| = k{{\eta }^{{1/2}}}}}$$ furthermore: $$h = \frac{2}{9}\frac{{{v^3}}}{{3{v^2} - 2u}},k = \sqrt {\left| {\left. {\frac{{2v}}{{3{v^2} - 2u}}} \right|} \right.}.$$ (2) Where \(u = {\partial ^4}\psi /\partial {x^4};v = {\partial ^3}\psi /\partial {x^2}\partial y,\psi\) is the lens scalar potential.