= Point-mass lens deflection and focusing
{title2=$f(b)\simeq c^2b^2/(4GM)$}
Deflection changes one <light ray>'s direction; focusing concerns relative directions of neighboring <light rays> and their bundle's cross section. To see the difference for a <point-mass gravitational lens>, send parallel incident <light rays> past the lens. A ray at <impact parameter> $b$ meets the <optical axis> approximately $f(b)=b/\widehat\alpha=c^2b^2/(4GM)$ beyond the lens. Doubling $b$ quadruples this crossing distance. Thus the lens does not have a common <focal length> for all rays, although it can produce <lensing magnification> and an <Einstein ring> for a specified source-observer geometry.
The <point-mass gravitational lens> also illustrates why focusing must not be identified just with positive <lensing convergence>. Away from the central <mass>, differentiate the reduced deflection $\boldsymbol\alpha=\theta_E^2\boldsymbol\theta/r^2$: its two-dimensional <divergence> is zero, so $\kappa=0$ there. Nevertheless its radial and tangential <derivatives> differ, giving <lensing shear> of magnitude $\theta_E^2/r^2$ and $\det A=1-\theta_E^4/r^4$. A vacuum ray bundle can be distorted and magnified. In the same ideal transparent <geometrical optics> model, <surface brightness> is preserved; the larger apparent <solid angle> supplies the increased received light.
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