## Online material

### Appendix A: MmD with an absorbed extended continuum

We consider a line profile where absorption is present. We write where *F*_{ce} represents the flux from an extended source of continuum, which is only macrolensed as the emission line flux *F*_{ee}, *F*_{cc} ≠ 0 is the flux from the source of continuum compact enough to be microlensed, and *τ*_{cc} (*τ*_{ce}) is the optical depth of the absorber in front of *cc* (*ce*). Using these relations in Eqs. (3) and (4) with *A* = *Mμ*, we obtain In these equations, *A*′ and *M*′ are parameters tuned to satisfy constraints on *F*_{M} and *F*_{Mμ}. The constraint *F*_{Mμ} = 0 at wavelengths where *F*_{cc}*e*^{−τcc} = 0 and *F*_{ee} ≠ 0 can only be satisfied with *M*′ = *M*. If *M* is correctly estimated, *F*_{Mμ} = *F*_{cc}*e*^{−τcc} ≥ 0 independently of the existence of a nonmicrolensed extended continuum.

In the absence of a nonmicrolensed extended continuum (i.e., *F*_{ce} = 0), making *F*_{M} = 0 at wavelengths where *F*_{ee} = 0 and *τ*_{cc} = *τ*_{ce} = 0 (i.e., in the unabsorbed continuum) gives *A*′ = *A*. In the presence of a nonmicrolensed extended continuum, making *F*_{M} = 0 in the unabsorbed continuum gives *A*′ ≠ *A*. In this case, *F*_{M} = *F*_{ce} (*e*^{−τce} − *e*^{−τcc}) at wavelengths where the continuum is absorbed and *F*_{ee} = 0, which can result in either positive or negative values for *F*_{M} whether *τ*_{ce}<*τ*_{cc} or *τ*_{ce } > *τ*_{cc}, respectively. Making *F*_{M} ≥ 0 allows the detection of the extended continuum in the latter case (*τ*_{ce } > *τ*_{cc}), but *A*′ = *A* is derived only if *e*^{−τce} = 0 at some wavelengths where *F*_{ee} = 0 and *e*^{−τcc} ≠ 0. Without absorption (i.e., *τ*_{ce} = *τ*_{cc} = 0), such a nonmicrolensed extended continuum would remain unnoticed.

Note that *μ*′ = *A*′/*M*, determined by making *F*_{M} = 0 in the unabsorbed continuum, can be seen as the average microamplification of the whole continuum source, while *μ* = *A*/*M* refers to the microamplification of the compact continuum source. They are related by (A.5)

### Appendix B: Equivalent width in the absorption trough

In order to understand how microlensing affects the EW, we can write Eq. (5), with more explicit expression of *F*_{λ} and of the continuum flux ℱ. We assume that the latter is constant over the emission line and the absorption trough and is well approximated by the continuum flux measured in the vicinity of the line. We consider a broad emission line characterized by a flux *F*_{λ,E}, and a continuum originating from two regions, a compact region *cc*, compact enough to be microlensed, and a more extended region *ce*, which is not microlensed. We can specialize Eq. (5) in the presence of absorption. For a lensed image *i*, we obtain (B.1)where *μ*_{i} is the amplitude of micromagnification of the compact continuum *cc*, and *τ*_{cc}, *τ*_{ce}, and *τ*_{E} are the optical depths of the gas absorbing the compact continuum, the extended continuum, and the broad emission line, respectively. Contrary to Appendix A, we make explicit the fact that the emission line can be absorbed (i.e., *F*_{ee} = *e*^{−τE}*F*_{λ,E}).

Equation (B.1) clearly shows that if *F*_{λ,ce} = *F*_{λ,E} = 0 (i.e., no extended continuum or emission line), and if the continuum does not vary over the range [*λ*_{1},*λ*_{2}] (i.e., *F*_{λ,cc} ≈ *F*_{cc}), then the EW is independent of the amplitude of microlensing *μ*_{i}. In the following, we compare the difference of equivalent widths Δ*EW* = *E**W*_{D} − *E**W*_{A}, between image *D*, characterized by microlensing *μ*, and image *A*, unaffected by microlensing, in the presence of flux from the emission line and/or from an extended continuum region.

#### Appendix B.1: Compact continuum+emission line

We first consider the situation where there is only the compact source of continuum *F*_{cc} and the emission line over the wavelength range [*λ*_{1},*λ*_{2}]. We then have *F*_{λ,ce} = 0 in (B.1). Assuming that the continuum flux does not vary over the range [*λ*_{1},*λ*_{2}] such that *F*_{λ,cc} = *F*_{cc}, we can rewrite (B.1): The difference of equivalent width, Δ*EW* = *E**W*_{D} − *E**W*_{A}, can be derived from the above equation specialized for the pair of images, i.e., (B.4)It results from this equation that when *μ* > 1, one may only observe *E**W*_{D } > *E**W*_{A}.

#### Appendix B.2: Compact + extended continua

We now consider that there is no flux from the emission line (*F*_{λ,E} = 0) over the wavelength range [*λ*_{1},*λ*_{2}], but two sources of continuum: a compact continuum microlensed by a factor *μ*_{i} and an extended continuum *F*_{ce} which is not microlensed. Assuming that the continuum flux is constant over the range [*λ*_{1},*λ*_{2}] such that *F*_{λ,cc} = *F*_{cc} and *F*_{λ,ce} = *F*_{ce}, we obtain where we defined ℱ_{i} = *μ*_{i}*F*_{cc} + *F*_{ce}, the flux in the unabsorbed continuum.

The difference of equivalent width Δ*EW* = *E**W*_{D} − *E**W*_{A}, is then (B.7)We clearly see that a difference of equivalent width is only detected if there is a difference of opacity between the absorbers covering the extended and the compact continua, i.e., *τ*_{cc} ≠ *τ*_{ce}. If *μ* > 1, we find Δ*EW* > 0 (*E**W*_{D } > *E**W*_{A}), and if *τ*_{cc } > *τ*_{ce}, and conversely Δ*EW* < 0 (*E**W*_{A } > *E**W*_{D}) if *τ*_{cc}<*τ*_{ce}.

#### Appendix B.3: Compact + extended continua + emission line

The situation where the source of light under the absorption is the sum of a compact and an extended continuum and of an emission line, directly results from the two previous cases. If we define the flux of the unabsorbed continuum of image *A* and *D*, ℱ_{A} = *F*_{cc} + *F*_{ce}, and ℱ_{D} = *μ**F*_{cc} + *F*_{ce}, then we can write the following difference of equivalent width Δ*EW* = *E**W*_{D} − *E**W*_{A} between images *D* and *A*:(B.8)

From this equation, we see that when *μ* > 1 and *τ*_{cc } > *τ*_{ce} (i.e., when the compact continuum is more absorbed than the extended continuum), Δ*EW* > 0. Conversely, Δ*EW* < 0 can only be obtained if *τ*_{cc}<*τ*_{ce}, i.e., when the compact continuum is less absorbed than the extended continuum. Positive value of Δ*EW* may also occur depending on the absorption rate of the different components, and on the relative contribution of the extended continuum and of the emission line.

*© ESO, 2015*