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It is well known that multiply imaged quasars are likely to be a ected by microlensing. Quadruply imaged systems are especially useful laboratories for studying microlensing because their macrolens models are relatively well constrained. We begin with analytical results for a simple family of galaxy models. These results can be used to estimate the magni cations and time delays for the quadruple systems. We compute expected brightness uctuations due to microlensing in several such systems for a range of source sizes. Among these we treat for the rst time the limiting case of a completely unresolved source. For the case of MG0414+0534 we interpret the discrepant optical-to-radio ux ratios for the di erent components as the result of microlensing. On the assumption that the lensing galaxy is roughly isothermal this gives a 95% con dence upper limit on the optical source size of 10 16 cm (hMi=0:1M) 1=2 where hMi is the average stellar mass of the lensing galaxy. For more centrally concentrated macromodels the upper limit is greater. Su ciently long strings of photometric observations can in principle constrain the degree of concentration.

We study gravitational lens time delays for a general family of lensing potentials, which includes the popular singular isothermal elliptical potential and singular isothermal elliptical density distribution but allows general angular structure. Using a novel approach, we show that the time delay can be cast in a very simple form, depending only on the observed image positions. Including an external shear changes the time delay proportional to the shear strength, and varying the radial profile of the potential changes the time delay approximately linearly. These analytic results can be used to obtain simple estimates of the time delay and the Hubble constant in observed gravitational lenses. The naive estimates for four of five time delay lenses show surprising agreement with each other and with local measurements of H 0 ; the complicated Q 0957+561 system is the only outlier. The agreement suggests that it is reasonable to use simple isothermal lens models to infer H 0 , although it is still important to check this conclusion by examining detailed models and by measuring more lensing time delays.

The observables in a strong gravitational lens are usually just the image positions and sometimes the flux ratios. We develop a new and simple algorithm which allows a set of models to be fitted exactly to the observations. Taking our cue from the strong body of evidence that early‐type galaxies are close to isothermal, we assume that the lens is scale‐free with a flat rotation curve. External shear can be easily included. Our algorithm allows full flexibility regarding the angular structure of the lensing potential. Importantly, all the free parameters enter linearly into the model and so the lens and flux ratio equations can always be solved by straightforward matrix inversion. The models are only restricted by the fact that the surface mass density must be positive. We use this new algorithm to examine some of the claims made for anomalous flux ratios. It has been argued that such anomalies betray the presence of substantial amounts of substructure in the lensing galaxy. We demonstrate by explicit construction that some of the lens systems for which substructure has been claimed can be well fitted by smooth lens models. This is especially the case when the systematic errors in the flux ratios (caused by microlensing or differential extinction) are taken into account. However, there is certainly one system (B 1422+231) for which the existing smooth models are definitely inadequate and for which substructure may be implicated. Within a few tens of kpc of the lensing galaxy centre, dynamical friction and tidal disruption are known to be very efficient at dissolving any substructure. Very little substructure is projected within the Einstein radius. The numbers of strong lenses for which substructure is currently being claimed may be so large that this contradicts rather than supports cold dark matter theories.

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Recent observations indicate that many if not all galaxies host massive central black holes. In this paper we explore the influence of black holes on the lensing properties. We model the lens as an isothermal ellipsoid with a finite core radius plus a central black hole. We show that the presence of the black hole substantially changes the critical curves and caustics. If the black hole mass is above a critical value, then it will completely suppress the central images for all source positions. Realistic central black holes are likely to have masses below this critical value. Even in such subcritical cases, the black hole can suppress the central image when the source is inside a zone of influence, which depends on the core radius and black hole mass. In the subcritical cases, an additional image may be created by the black hole in some regions, which for some radio lenses may be detectable with high‐resolution and large dynamic range VLBI maps. The presence of central black holes should also be taken into account when one constrains the core radius from the lack of central images in gravitational lenses.

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We study a general elliptical potential of the form ψ(x 2 + y 2 /q 2 ) (0 < q ≤ 1) plus an additional shear (with an arbitrary direction) as models for the observed quadruple lenses. It is shown that a minimum additional shear is needed even just to reproduce the observed positions alone. We also obtain the dependence of the axial ratio, q, on the orientation of the major axis of potential. A general relation also exists between the shear, the position angle and axial ratio of the lensing galaxy. The relation shows a generic degeneracy in modelling quadruple lenses. In particular, it shows that only the ratio of the ellipticity, ǫ ≡ (1−q 2 )/(1+q 2 ), to the magnitude of shear, γ can be determined.All these results are valid regardless of the radial profile of the potential. Our formalism applies when the galaxy position is observed, which is the case for seven of the eight known quadruple lenses. Application to these seven cases reveals two quadruple lenses CLASS 1608+656 and HST 12531-2914, requiring highly significant shear with magnitude ≈ 0.2. For HST 12531-2914, there must be a misalignment between the major axis of light and the major axis of potential (mass). We conclude that detailed modelling of quadruple lenses can yield valuable quantitative information about the shape of lensing galaxies and their dark matter halos.