light

electromagnetic radiation & optics - a historical account

telescopes

photographic astigmatism and field curvature

the angle of incidence = the angle of reflection

the incident ray, the reflected ray & the normal to the surface lie in the same plane

principal focus is the convergence point for rays parallel to & close to the principal axis of the mirror, and is located halfway between the mirror & its centre of curvature, ie. focal length = radius/2

magnification = q/p;

absolute index of refraction of a medium = speed in vacuum / speed in medium (this is dependent on wavelength!)

relative index of refraction of 2nd medium relative to first = speed in medium 1 / speed in medium 2 = abs. index medium 2 / abs. index medium 1

on still sunny days, there may be a layer of hot, expanded air in contact with the heated ground in which light travels faster, thus light entering it obliquely will be refracted upwards resulting in observer seeing inverted images of distant objects suggestive of reflections in a smooth pool of water. These are often seen on highways.

when the air near the ground is cooler than the upper strata (eg. over snow or over water), rays of light are deviated downward, thus one may see an image of a ship above the ship itself, or the curvature may allow one to see objects below the horizon.

 

if p = distance of object from mirror of focal length f, then q = distance of image from mirror

abs.index medium 1/p + abs.index medium 2/q = (abs.index medium 2 - abs.index medium 1)/radius

by convention, radius is positive if measured from surface to centre in direction of light leaving the surface.

if p = distance of object from lens of focal length f, then q = distance of image from lens

1/p + 1/q = 1/f and magnification = q/p

the amount by which it can change the curvature of a wave (in diopters) = 1/focal length in metres, thus a +2 is a convex lens with focal length of 500mm

when using multiple lenses, just sum the diopter values to get the total diopter value

positive diopter lenses are similar in shape to the convex lenses found in common magnifying glasses. 

negative diopter lenses may use concave elements, but this shape may be masked by multiple glass elements.

when lenses are used in combination, each magnifies the image from the preceding lens, thus resultant magnification is the product of individual magnifications.

when thin lenses are in contact, total power is sum of the powers, ie. 1/f = 1/f₁ + 1/f₂

rays that enter the lens near its edge are brought to focus closer to the lens than are the central rays resulting in spherical aberration

this can be minimised by using a diaphragm in front of the lens to decrease its effective aperture, although the sharper image will be at cost of less light.

for a more detailed explanation see http://toothwalker.org/optics/spherical.html

correcting spherical aberration of a spherical lens surface - the Wasserman-Wolf problem appears to have been solved by Héctor A. Chaparro-Romo and Rafael G. González-Acuña in 2018 ¹⁾ and also solved the Levi-Civita Problem ²⁾

another form of spherical aberration occurs for object points that are laterally displaced from the principal axis

this can be minimised by using a compound lens having several surfaces, or more simply by an aperture stop that eliminates rays which are not near the principal axis

a lens defect whereby horizontal & vertical lines in an object are brought to a focus in different planes

arises from a lack of symmetry of a lens or lens system about the line from the centre of the lens to an object.

see also: photographic astigmatism and field curvature

caused by fact that magnification varies at different parts of the image and may produce barrel-shaped or pin-cushion distortion.

as a result of optical dispersion, short wavelengths are refracted more than long wavelengths, resulting in fringes of colour around an object

its presence resulted in Newton inventing the reflecting telescope which does not have this aberration

it can be corrected in lens by combining a strong positive lens made of glass with a low dispersion, with a weak negative lens of highly dispersive glass. Each component has a difference of power for two colours & if these are equal in magnitude but of opposite sign, then they will cancel with a net power of zero. Such a lens is called an achromatic lens.

new types of glasses such as flourite and ED invented in the 1990's are even better at minimising chromatic aberration and these are called apochromatic (APO).