The ability to manipulate light dates back to ancient times. The understanding of the nature of light, which involved debates over whether it is composed of waves or particles, began in the 1500s and 1600s. The discovery of and the elaboration of principles needed to design optics with confidence began in the early 1800s.
It seems quite likely that reflections in calm lakes and ponds were seen and wondered at since the dawn of human existence, perhaps millions of years ago, but no trace remains. The earliest optical devices we have found are stone and obsidian mirrors from the Bronze Age in Europe and the Middle East. It is likely that at about the same time people noticed their reflections in the blades of metal swords, axes and armor if they were highly polished. Flat mirrors reflect light at the same angle as it is incident at, and the formation of an image takes place in the eye of the beholder. A mirror can concentrate light if its surface is made concave. There are Greek accounts of Archimedes’ “burning mirrors” being used to ignite the sails of Roman ships in battle, using concentrated solar energy. These mirrors were made of polished metal. A helmet or breastplate, being convex on its outside, spreads light and, if of high enough quality, forms a reduced-scale, wide angle image, rather than a concentration of light or a magnified image.
Transparent materials transmit light and can also manipulate it. We find glass jewelry which spreads light into colors, a property called dispersion, from the Bronze Age on, and this was undoubtedly long predated by the discovery that natural crystals such as quartz, calcite, amethyst, and emeralds, created colorful dispersal and multiple reflections of light. And again going back to early human times our ancestors saw and must have wondered at the colors of mother-of-pearl, rainbows, sun-dogs, lunar halos and other natural phenomena.
Small apertures in opaque surfaces when illuminated from one side will form images on a surface placed on the opposite side. This phenomenon was remarked in classical Greek times, dating back to 250 BC. It allows one to safely view a partial solar eclipse in shadows and images cast on the ground by a leafy tree.
All these phenomena can be explained based on the nature of light and its behavior in different materials and at material boundaries. If a material is internally uniform in density and transparency, then light travels through it in straight lines and only changes direction, or refracts, at the boundary to another medium. At each boundary the path direction changes, and the angle of change (with respect to a perpendicular to the surface at the penetration point) is called the angle of refraction. If the surface is flat, the light entering it at a given angle is refracted the same amount everywhere on the surface. If the surface is curved, then the angle of refraction varies over the surface. If the shape is part of a sphere, the piece is said to be a lens and it can focus or defocus light, depending on whether the surfaces are convex or concave. (It should also be noted that at each boundary between different materials a fraction of the light is reflected: for glass and air it amounts to about 4%, and can give rise to multiple internal reflections inside lenses or rear-surface mirrors.)
The angular amount of refraction of a material is measured by a number unique to each material, called the index of refraction, the ratio of bending in a material to that of empty space, which is set equal to one. Air and other low-density gases have indices of refraction just a bit larger than one. Water at sea level has an index of refraction of 1.33, natural crystals, glass and plastics generally fall in the range 1.25 to 1.8. Some relatively exotic composition glasses have indices of refraction well over 2.5, which means lenses made out of them can be of thinner material and still bend light as much as thicker lenses made of lower index glass. (An aside: Einstein's relativity shows that, in the presence of matter, space itself curves and thus light's path in space is curved proportionate to the distance and density of that matter. It shows up in astronomy in curved arc images of distant galaxies seen around massive foreground galaxies. This effect is ignored in classical optics and so far has found no application in conventional photography!)
Lens making became a profession in the late renaissance in Europe. The Dutch optician, Snell (1580-1626), discovered and published a short mathematical relation for the bending of light in 1626, by sending light beams through glass surfaces of varying shapes and indices of refraction and carefully measuring the angle of incidence of the light as it struck the glass, and then the angle it was deviated to inside. He also measured the angles at the light ray's emergence on the far surface, into the air.
Snell's law is: n1 sin θ1 = n2 sin θ2.
In words that is: the index of refraction of light in the first medium times the sine of the angle of incidence at the boundary is equal to the index of the light in the second medium times the sine of the angle of refraction in the second material. The unprimed numbers are the numbers in the first medium, the primed numbers refer to the second medium. The Greek letter theta (θ) refers to the angle in degrees. Recall that the sine of an angle is a trigonometric property and can be found in mathematical reference tables or calculated by hand or computer. Since by its definition, the sine of any angle between zero degrees and 90 degrees is somewhere between 0 and 1, and since the index of any ordinary transparent material is greater than one, a light ray entering glass from air will be bent to a smaller angle than the angle of incidence it had at entry. When a ray leaves the denser material, passing from glass to air, the ray will bend back to a larger angle.
Snell's law was empirically derived. It can be derived theoretically using the electromagnetic wave theory of light founded by James Clerk Maxwell in the 1860s, and also by Fermat's principle of least time of travel. The latter idea relies on the fact that the index of refraction is not only a measure of bending strength but also the ratio of the speed of light in a material to that in empty space.
Snell's law and a little further work led to a useful formula, still applicable with certain restrictions, for describing practical optical systems. It is called the lens maker's formula:
1/fl = (1/OD) + (1/ID).
The OD is the distance from the object in question to the center of the lens, the ID is the distance from the lens center to the focused image, and fl is the focal length, the distance from the lens center to the focused image when the object is at infinity. The focal length is also, by definition, one half the radius of curvature of the lens surface. It should be noted that this formula can be repeatedly applied to follow light through a series of lenses. With proper observation of positive and negative sign conventions it can also be used to study mirrors. The restrictions are that the lens have a shallow curvature, or equivalently that it has a very long focal length compared to its diameter. And light rays are assumed to travel in straight lines except when they cross material boundaries. This is called the Thin Lens Approximation. The lens maker's formula is a shortcut of use in deciding where an image will focus for an object at a given distance using a lens of known focal length, or for a quick design of a simple optical system. This formula also can be used to get the image magnification, which is the ratio: M= ID/OD.
Aside: Fresnel (1788-1827) in the early 1800s garnered a large prize fund from the French Academy of Science by designing a very thin lens, for use in lighthouse lights. These were used successfully for that purpose but now find much larger use as lenses for theatrical spotlights, in camera viewfinder focusers, and also as novelty wide-angle lenses for automobile rear windows. Think of taking a set of nested concentric circular cookie-cutters and slicing a lens into a series of rings. Then shave off the bulk of the glass in the rear, leaving only the front curvature. Cement the resulting rings to a thin flat sheet, and the result is like a Fresnel lens (which is actually molded). This is a really thin lens!
Using the Thin Lens Approximation and simple geometrical rules, it is possible to quickly draw the principle light paths in a lens system. For each lens, a light ray down the system axis (a perpendicular through the lens center) travels straight on. A ray parallel to the axis, emerging from an off-axis point of the object as if it came from infinity, is bent by the lens to pass through the rear focal point, and a ray passing from the same object point through the front focal point (at an angle) emerges from the lens parallel to the axis (note the symmetry!). Where the latter two rays cross is the point where the original point of the object is imaged. In cases where the image rays converge to a focus, the image is termed real.
If, in leaving the lens, the rays only diverge, then an image can only be seen by the use of an additional lens, say that in your eye, and the image is called virtual. Convex lenses and concave mirrors can give real or virtual images, depending on whether the object is closer or farther than one focal length away from the optic. Concave lenses and convex mirrors yield only virtual images.
Snell's Law, which describes the refraction of light, along with the law of reflection (the angle of incidence equals the angle of reflection), can be applied at each point of a boundary surface to predict the path of light rays as they pass through. By doing this step by step at all points (or a sample) of a surface of known shape, one can follow a ray of light through a system of any complexity, and in reasonable detail. This process is called ray tracing, and until the advent of computers was carried out by hand. There some complications to this process, however.
First, the index of refraction of any real transparent material varies with the color (wavelength or frequency) of light. This effect is called dispersion, and explains why prisms are able to spread white light out into the visible spectrum. In general the index is greatest in the blue and diminishes continuously into the red and infrared. Mirrors do not suffer from this problem. This effect causes any lens to send blue rays to a different focus point than green or red ones. The result is called chromatic aberration, and results in color fringes surrounding images of objects with sharp edges, such as the Moon. It was discovered in the middle 1700s that this problem could be removed by sandwiching together two lenses of different indices of refraction, one convex and the other concave, carefully choosing their focal lengths, to make the dispersion of the second approximately cancel that of the first. These pairs are called achromatic lenses. All modern lenses for cameras and telescopes are achromats.
Second, every lens or mirror has a natural limit to resolution caused by the wave nature of light, called the diffraction limit. Waves bend around any edge, straight, curved or jagged. This is termed diffraction. Light does this and this results in fuzzy shadow edges (visible under careful examination) and also fuzziness in the whole image. The larger the opening of the lens or mirror, the less important this is and the sharper the image. The longer the wavelength of light (the redder it is) the worse the effect. This diffraction limit can not be evaded in conventional optics.
The diffraction effect can actually be put to use, however. A simple round pinhole carefully made in an opaque sheet will give a real image on almost any size surface behind, at almost any distance. What you gain in areal coverage and depth of focus you sacrifice in speed and exposure time. In conventional optics (lenses or mirrors)the aperture is wide and concentrates a lot of light. In pinhole cameras, just a tiny amount of light passes through. A conventional “fast” lens might have a focal length to diameter ratio (“f/stop” or “speed”) of f/1.2 or f/1.8. The speed of a pinhole is usually f/150 or more.
Aside: A more efficient way than a pinhole to use diffraction to manipulate light and form an image is a second invention of Fresnel: the zone plate. Fresnel found an exact formula to compute the widths of transparent gaps between and widths of concentric opaque rings to form a lens based on diffraction. A zone plate looks just like a bullseye, but it is a mathematical construct.
There are also distortions of image shape due to lens or mirror shape. If the surfaces are spherical then off-axis rays do not focus at the same distance as on-axis ones. Spherical aberration, along with comatic aberration (images at the edge of the field of view spread out into fan “tails” like comets), barrel distortion and pincushion distortion (rectangular objects have “swollen” or “collapsed” images, respectively) along with chromatic aberration all have to be reduced to make an optical system produce high quality images. With the use of ray tracing in modern computers all of these problems can be solved.
Compound lenses have been designed for many different purposes. Perhaps the two greatest challenges are to find excellent wide angle lenses, and to find zoom lenses which maintain focus and image quality, along with maximum speed at every magnification. Again computers have allowed many different solutions to these problems.
If one surface of a compound optical system can be made to change shape microscopically and the image sharpness can be sensed instant by instant, you can use computer control to integrate this information and “sculpt” the flexible surface to make ultra sharp images in real time. Such star “de-twinkling” systems (“rubber” mirrors) are now available for astronomers and can make earth-based telescopic images almost as sharp as those made from Earth orbit. But they are not yet available for ordinary commercial cameras. However, overall motion compensation (“de-jiggle”) is here in binoculars and some digital cameras. It may not be long before it appears in film cameras too.
Holography, the making of 3-dimensional images using laser light, can use or dispense with lenses and mirrors to achieve focused images. The images are formed by preserving the distance, brightness and color information carried by light waves, in the form of microscopic interference patterns, which are recorded on super-high resolution emulsions. Only laser light has the color purity and wave orderliness to form stable interference patterns, and all motion must stop for the duration of the exposure, down to a millionth of an inch, to avoid blurring out the interference pattern.
Holograms can, however, be made of small moving objects if the laser can emit a very intense short flash. Holographic large-scene snapshots are not yet on the horizon.
See also: Lenses: 1. 1830s-1850s.
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