
A lens is an optical instrument that focuses or disperses light beams through refraction to form images or manipulate light paths.A lens is typically a transparent body with two spherical surfaces. Its curvature together with the refractive index (how much the material slows light) determines the focal length—the distance from the lens to the camera sensor when focused at infinity. A camera objective contains several lenses made of different glasses, because a single lens produces rainbow fringes and blurriness at the edges. The human eye is an amazing example: its crystalline lens changes thickness, focusing without a single moving part.
The earliest lenses made of ground rock crystal date back to 700 BCE, Assyria. The scientific approach to lens systems began with Newton's invention of the reflecting telescope (to avoid the rainbow fringe) and the subsequent development of achromatic doublets by Chester Moore Hall in the 1730s. In 1665, Robert Hooke, using a compound microscope assembled from several lenses, first revealed the cellular structure of living matter in his 'Micrographia'.
The law of refraction (Snell's law) n1 sinθ1 = n2 sinθ2, when light hits a curved boundary, gives different ray slopes at different points. For a thin lens, the lens formula works: 1/f = 1/u + 1/v, where f is the focal length, u is the object distance, v is the image distance. In the wave picture, the lens introduces a spatial phase delay proportional to the square of the distance from the axis, and thereby uses diffraction to turn a diverging wavefront into a converging one.
Lenses in phone cameras and in telescopes catching distant galaxies. Tiny lenses also live in the eye, focusing the world onto the retina.
The largest objective lens ever made—the Yerkes Observatory telescope (1897)—has a diameter of 102 centimeters. It is so heavy that the glass sags under its own weight, which is why modern telescopes use mirrors.
| y | variable | Source position on the sky — angular position radian |
| x | variable | Image position on the sky — angular position radian |
| α | variable | Deflection angle of light — deflection angle radian |
| kappa | variable | weak lensing convergence — convergence |
| theta | variable | angular position on the sky — angular position radian |
| chi | variable | comoving distance to the source — comoving distance metre |
| chi' | variable | comoving distance along the line of sight — comoving distance metre |
| q | variable | lensing efficiency function — lensing efficiency |
| a | variable | scale factor — scale factor |
| f_K | variable | comoving angular diameter distance — comoving angular diameter distance metre |
| delta | variable | matter density contrast — density contrast |
| H_0 | constant | current expansion rate of the universe — hubble constant = 2.2e-18 per second |
| Omega_m | constant | ratio of matter density to critical density today — matter density parameter = 0.315 |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| ∫ | operator | summation over infinitesimal elements — integral |
| d_{ls} | variable | angular diameter distance between lens and source — angular diameter distance metre |
| d_s | variable | angular diameter distance from observer to source — angular diameter distance metre |
| d_l | variable | angular diameter distance from observer to lens — angular diameter distance metre |
| \Omega_K | variable | curvature density parameter — curvature density parameter |
















