the incident ray, the reflected ray & the normal to the surface lie in the same plane
spherical mirrors:
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
the mirror equation:
if p = distance of object from mirror of focal length f, then q = distance of image from mirror
1/p + 1/q = 1/f
magnification = q/p;
refraction
speed of light depends on the medium in which it travels (water = 3/4 that in air)
refraction (change of direction) occurs when light passes from a medium of one density to a medium of another density
a material of lesser speed is called “optically more dense”
refractivity of a medium:
absolute index of refraction:
absolute index of refraction of a medium = speed in vacuum / speed in medium (this is dependent on wavelength!)
examples for wavelengths of 5893A:
air = 1.0003; crown glass = 1.517; diamond = 2.419; quartz = 1.4585; ice at -8degC = 1.31; water at 20degC = 1.333;
relative index of refraction:
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
optical dispersion:
the variation of the index of refraction with the wavelength of light is called the optical dispersion of a substance
short wavelengths are refracted more than long wavelengths
substances with high optical dispersion, usually also have a high index of refraction (eg. diamond)
Snell's law:
for a given wavelength & a pair of substances, the ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction ® is constant
abs.index medium 1 * sin i = abs.index medium 2 * sin r
examples:
underwater viewing:
the apparent depth of an object under water when viewed from above the water, is less than its actual depth below the surface
real depth = 1.333 x apparent depth (1.333 is the rel.index of water to air)
mirages:
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.
looming:
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.
total internal reflection:
for light passing from a medium to a less dense medium, there is an incident angle whereby refraction will no longer occur & the light is internally reflected instead, this occurs as the angle of refraction approaches 90deg. (when sin r = 1)
thus critical angle of incidence (ic) can be determined from sin ic = abs.index medium 2 / abs.index medium 1
therefore, for water/air interface when viewed from under water, the critical angle is 48deg.
spherical surfaces:
for rays with small angles to perpendicular of surface:
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.
thin lenses:
if the object is at infinity, the image is at the principal focus
thin lens equation:
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
power of a lens:
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.
photographers will most often encounter diopter lenses as a kit of +1, +2, +3, and +4 diopter lenses for macro-photography.
for a camera lens focussed at infinity, adding a closeup positive diopter lens will change the focus to (1/diopter value) in meters. So a +4 diopter lens will be 1/4 meter, a +10 diopter lens will be 1/10th meter (4 inches), and a +20 diopter lens will be only 1/20th of a meter or about 5 cm. (or 2 inches).
the new effective focal length in metres = camera lens focal length in meters / (1+ diopter value*focal length in metres)
the effective f/ratio will be altered as the aperture stays the same but you now have a new focal length
combinations of lens:
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
aberrations:
spherical aberration:
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.
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)
coma:
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
astigmatism:
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.
caused by fact that magnification varies at different parts of the image and may produce barrel-shaped or pin-cushion distortion.
chromatic aberration:
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).
field curvature
anastigmatic lens:
a lens corrected for such defects as spherical aberration, astigmatism, distortion & chromatic aberration.