We present the geometrical optics for refraction of a distant background radio source by an interstellar plasma lens , with specific application to a lens with a Gaussian profile of free electron column density .
The refractive properties of the lens are specified completely by a dimensionless parameter , \alpha , which is a function of the wavelength of observation , the free electron column density through the lens , the lens-observer distance , and the diameter of the lens transverse to the line of sight .
A lens passing between the observer and a background source , due to the relative motions of the observer , lens , and source , produces modulations in the light curve of the background source .
Because plasma lenses are diverging , the light curve displays a minimum in the background source ’ s flux density , formed when the lens is on-axis , surrounded by enhancements above the nominal ( unlensed ) flux density .
The exact form of the light curve depends only upon the parameter \alpha and the relative angular sizes of the source and lens as seen by the observer .
Other effects due to lensing include the formation of caustic surfaces , upon which the apparent brightness of the background source becomes very large ; the possible creation of multiple images of the background source ; and angular position wander of the background source .
If caustics are formed , the separation of the outer caustics can be used to constrain \alpha , while the separation of the inner caustics can constrain the size of the lens .
We apply our analysis to two sources which have undergone extreme scattering events : 0954 + 654 , a source for which we can identify multiple caustics in its light curve , and 1741 - 038 , for which polarization observations were obtained during and after the scattering event .
We find general agreement between modelled and observed light curves at 2.25 GHz , but poor agreement at 8.1 GHz .
The discrepancies between the modelled and observed light curves may result from some combination of substructure within the lens , an anisotropic lens shape , a lens which only grazes the source rather than passing completely over it , or unresolved substructure within the extragalactic sources .
Our analysis also allows us to place constraints on the physical characteristics of the lens .
The inferred properties of the lens responsible for the scattering event toward 0954 + 654 ( 1741 - 038 ) are that it was 0.38 AU ( 0.065 AU ) in diameter , with a peak column density of 0.24 pc cm ^ { -3 } ( 10 ^ { -4 } pc cm ^ { -3 } ) , an electron density within the lens of 10 ^ { 5 } cm ^ { -3 } ( 300 cm ^ { -3 } ) , and a mass of 6.5 \times 10 ^ { -14 } M _ { \sun } ( 10 ^ { -18 } M _ { \sun } ) .
The angular position wander caused by the lens was 250 mas ( 0.4 mas ) at 2.25 GHz .
In the case of 1741 - 038 , we can place an upper limit of only 100 mG on the magnetic field within the lens .