In this paper we have studied one of the main tests of GR, the Gravitational Lensing: massive objects can modify the structure of spacetime, with the consequence that photons will not follow straight paths. This effect has a remarkable consequence: we will detect multiple images of lensed light-source, which will not be synchronized due to the different paths followed by light. In

Section 2 we have divided this delay in two contributions, the Shapiro, or potential, delay and the geometric delay, which we calculated following the standard analysis [

19], obtaining an approximate expression, (

53), known in the Literature [

19]. This formula is important because it is directly related to the value of the Hubble constant

${H}_{0}$, so we can obtain a direct measurement of its value studying the time delay of lensed images. However, the results of the H0LiCOW collaboration [

15] are not compatible with the measurement obtained by the PLANCK collaboration [

10]; this tension is a strong motivation to improve the expression of time delay (

53). In

Section 3 we studied two slightly different approaches: we developed a more rigorous treatment for the Shapiro delay and a more precise value for the geometric delay, obtaining the time delay formula (

65) involving higher orders in the angles

${\alpha}_{1,2}$, which identify the images of the source

S. The crucial fact to notice is that it can be traced back to the Taylor series of the cosine, hence it goes like even powers of the angles. Now, it has been possible to give a preliminary estimate of the second order correction of the time delay formula (

65), applied to a typical source like the twin quasar Q0957+561. For this lensing phenomenon, the angular separations are of the order of one arcsecond, i.e.,

${10}^{-5}$$rad$. Using the lens parameters, the coefficient

${c}_{2}$ in (

65) is of the order of unity. Hence, the second order correction is of the order

${10}^{-10}$ which is far too small to be detected with the lenses at our disposal. For lenses with bigger angular separation (around 22 arcseconds), the second order correction reaches

${10}^{-8}$, which is still too little. The important conclusion is that, at least for the lenses appearing in the CASTLES catalogue [

20], the standard formula (

57) for the time delay seems to be acceptable within the actual instrumental capabilities. This even more motivates the search for an alternative formula for time delay, which goes beyond the simple expansion in powers of the angles.

In

Section 4 we proposed a new approach: in analogy with the first Born-Oppenheimer approximation for the scattering amplitude in non-relativistic Quantum Mechanics, we considered the lens as a kind of cosmological scattering target, and consequently we divided the space in two regions: one where the gravitational potential originated by the lens is negligible, and another one, closer to the lens, where the gravitational potential is different from zero. This led to consider a more complicated geometry, which gave us the possibility to calculate the total delay in a single shot. We believe that our result represent an important improvement, because it allows to avoid the inaccuracies of the standard analysis. We also checked that the expression we have obtained for the time delay (

97) can be reduced to, hence includes, the known result (

53).