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GENERAL LENSING BACKGROUND
Outline


As is done in physical optics, lensing systems are often divided into planes: source plane, lensing plane and observer plane (Figure GL1). We then assume that light travels without deflection between the planes and experiences a sharp deflection, α, at the lensing plane. This approximation is widely used when studying the lensing effects of compact objects, such as clusters and galaxies. Obviously, a more sophisticated approach is needed when studying the lensing effects of extended objects, such as largescale structure (LSS). For now, however, we shall continue to use the thin lens approximation.
Fisher GL1: Diagram illustrating the typical geometry of a lensing system. In the thin lens approximation the system is divided into three planes; (1) the observer plane, where the observer sits, (2) the image/lensing plane which is where the lens sits. This is also thought of at as the plane that defines the angular position of the images on the sky. Finally (3) The source plane is the plane on which the sources sit.
The simple geometry illustrated in Figure GL1 shows us that the position of the source on the source plane, η, can be described in terms of the deflection angle, α, and the distances between the various planes in the problem,
GL1
This can also be expressed in terms of angular coordinates, which gives us the lens equation,
GL2
noting that the lengths used here are in angular diameter distances. This equation is linear from image to source, which means that one image will come from one source and is nonlinear from source to image, i.e. one source can have many images. Whether a lensing system produces many images or one image, tends to mark the boundary between the strong lensing regime (multiple images) and weak lensing.
The 2D potential
Gravitational lensing can be studied using a 2D lensing potential, ψ, where the deflection angle is given by the gradient of this potential. The lens equation, hence, becomes
GL3
The deflection potential, ψ, is the 2D analogue of the Newtonian gravitational potential and can be such that
GL4
this way we can calculate the deflection angels in terms of the surface density of mass, Σ, which is the total mass along the line of sight, Σ = ∫ ρdl. Often it is convenient to express this surface density in terms of the dimensionless quantity κ which is known as the convergence. To do this we have to define the critical surface density, . It can be shown that if a mass distribution has k > 1 then multiple images can be formed. Therefore, a surface density Σ > Σ_{c} is sometimes seen as the boundary between weak and strong gravitational lensing.
Distortion matrix
The lens equation holds for all points on the source place. An extended source, therefore, can be thought of as a collection of point sources (or a bundle of light rays). Since the light from different points on the source will experience varying degrees of deflection depending on their trajectory through the lens, it is conceivable that the image produced will be distorted. The distortion of the images is found by differentiating the lensing equation, which leads to the Jacobian
GL5
or related to the Hessian matrix ψ_{ij} of ψ using the equations (GL3) and (GL4)
. GL6
Where equation (GL4) can now be rewritten
GL7
There is another set of derivatives of ψ which of a paramount importance. The shear γ is defined is a complex quantity γ=γ_{1}+iγ_{2}, and is related to the deflection potential as
GL8
and
GL9
It is now clear that equation (GL6) be simply written in term of the convergence and the shear
GL10
or
GL11
Where we have used the complex notation γ_{1}=γ cos(2π) and γ_{2} = γ sin(2π) and introduced the reduced shear
GL12
From equations (GL10) and (GL11) it can be seen that
 The convergence, κ, causes an isotropic focusing of light raysvand an isotropic magnification of the source. Convergence alone does not alter the shape of the source galaxy.
 The shear, γ, introduces an anisotropic mapping from the source to the image plane causing the image to become stretched. γ the magnitude of the distortion and φ is the direction of distortion.
 γ_{1} causes images to be strectched in the x (+ve γ_{1}) or y (ve γ_{1}) directions.
 γ_{2} causes images to be streched along the 45 degree directions "southwest to northeast" (+ve γ_{2}) and "northwest to southeast" (ve γ_{2}).
The equations above describe the mapping of a circular source to an elliptic image with the ratio of the semimajor axes of the ellipse to the radius of the original, circular source being 1κ+γ.
Magnification
Liouville's theorem and the absence of absorption or emission of photons during gravitational lensing imply that the surface brightness of a lensed galaxy must be conserved. The flux observed from the image and the unlensed source are integrals over the respective surface brightness. The ratio of the image and source flux is the magnification; since the surface brightness is conserved the magnification is also simply the ratio of the area of the unlensed galaxy to the lensed image.
We can therefore calculate the magnification, μ, of a lensed object by simply considering the geometric change in the image shape described by the distortion matrix (equations GL10 and GL11). The magnification is obtained by taking the inverse of the determinant of A, or one can defined a magnification matrix M
GL13
for a circularly symmetric lens the ratio of the surface areas dθ^{2} to dβ^{2} is simply μ=(θdθ\βdβ). Note that the magnification observable is the magnitude of the magnification μ.
Five Easy Steps
 Exercise 1
 Exercise 2
 Exercise 3
 Exercise 4
 Exercise 5
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