3.1 Lens equation

The basic setup for such a simplified gravitational lens scenario involving a point source and a point lens is displayed in Figure 2View Image. The three ingredients in such a lensing situation are the source S, the lens L, and the observer O. Light rays emitted from the source are deflected by the lens. For a point-like lens, there will always be (at least) two images S1 and S2 of the source. With external shear – due to the tidal field of objects outside but near the light bundles – there can be more images. The observer sees the images in directions corresponding to the tangents to the real incoming light paths.
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Figure 2: Setup of a gravitational lens situation: The lens L located between source S and observer O produces two images S1 and S2 of the background source.

In Figure 3View Image the corresponding angles and angular diameter distances DL, DS, DLS are indicated3. In the thin-lens approximation, the hyperbolic paths are approximated by their asymptotes. In the circular-symmetric case the deflection angle is given as

&tidle;α(ξ) = 4GM---(ξ-)1, (4 ) c2 ξ
where M (ξ) is the mass inside a radius ξ. In this depiction the origin is chosen at the observer. From the diagram it can be seen that the following relation holds:
𝜃DS = βDS + α&tidle;DLS (5 )
(for 𝜃, β, &tidle;α ≪ 1; this condition is fulfilled in practically all astrophysically relevant situations). With the definition of the reduced deflection angle as α(𝜃) = (DLS ∕DS )α&tidle;(𝜃), this can be expressed as:
β = 𝜃 − α(𝜃). (6 )
This relation between the positions of images and source can easily be derived for a non-symmetric mass distribution as well. In that case, all angles are vector-valued. The two-dimensional lens equation then reads:
⃗ ⃗ ⃗ β = 𝜃 − ⃗α(𝜃). (7 )
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Figure 3: The relation between the various angles and distances involved in the lensing setup can be derived for the case &tidle;α ≪ 1 and formulated in the lens equation (6View Equation).

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