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         We'll start our discussion of optics with an observation so obvious that it took centuries before anybody quantified it and wrote it down: Objects twice as far away appear half as large.
         The somebody in question was Leonardo da Vinci, who left us a great many of our first truly scientific observations on the subjects of optics, vision, and perspective.  It is one of his drawings on perspective that graces the "Tech Talk" button on the home page of this website.
         For centuries men such as Newton and Huygens argued over wave versus "corpuscular" theories of light.  (And it wasn't until the early part of the 20th century that physics finally settled the debate by declaring that light has both particle and wave properties.)
         Geometric optics is essentially a science that treats light as a stream of photons (and thus adheres more to the corpuscular theory).  These photons have paths that can be described by rays.  A few geometric formulae, a small amount of physics, and a whole lot of common sense will take you far in geometric optics.
         The first useful concept about light is that it does have a finite speed, and that speed can change depending upon the medium through which it passes.  It moves at its maximum speed in a vacuum.  But other materials such as glass or air will slow it down a bit.  And every medium through which light can pass has a characteristic "index of refraction," whose values are related to the speed(s) of light in that medium.
         As light passes from one medium to another (from air to glass, for example) it bends at an angle described by Snell's Law:
 

na sin a = nb sin b

where na and nbare the indices of refraction of media A and B, and sin a and sin b are the angles (measured from the normal to the surface) at which the light travels through the media.
         Below is a table of the indices of refraction for yellow sodium light (wavelength = 589nm) for common materials.
 

Medium

Index of Refraction

Ice (H­2­O)

1.309

Methyl Alcohol (CH3OH)

1.329

Water (H2O)

1.333

Ethyl Alcohol (CH3OH)

1.362

Flourite (CaF2)

1.434

Carbon Tetrachloride (CCl4)

1.461

Turpentine

1.472

Glycerin

1.473

Benzene

1.501

Rock Salt (NaCl)

1.544

Quartz (SiO2)

1.544

Carbon Disulfide (CS2)

1.628

Zircon (ZrO2 – SiO2)

1.923

Fabulite (SrTiO3)

2.409

Diamond (C)

2.417

Rutile (TiO2)

2.616

2.903
 
         Note that we specify a particular color of light (in this case, the yellow light emitted by a sodium lamp) when we describe the index of refraction of a material.  This is because shorter wavelength light (bluer light) has more energy than longer wavelength like (redder light).  And in any medium other than a vacuum, light with slightly higher energy will move slightly faster than light with slower energy.  As a result, the bluer light will have a slightly different index of refraction than the redder light.  It's on this principle that prisms (which split light into the colors of the rainbow) are based.  And this difference in indices of refraction of different wavelengths of light in a material is the cause of "chromatic aberration" in a lens.  We will discuss this shortcoming in lens designs later.
         Because the indices of refraction of air and glass are different, it becomes possible to create a lens by forming a piece of glass where the surfaces at which light enters and light leaves are not parallel to each other.

         The more common lens types(plano-convex, bi-convex, plano-concave, bi-concave, and convex-concave) are shown below.

         Lenses are commonly defined by their focal lengths.  A "focal length" is the distance behind a lens at which an object an infinite distance away will image to a single point.  The concept is best explained with an illustration, which is presented below.  Note that objects at distances other than "infinity" will appear as smaller versions of themselves.

         Ideally, the surfaces of a lens should be ground as paraboloids (or be made perfectly flat), but in practice lens makers find it a great deal easier to cast and grind glass in spherical shapes -- which are often close enough to paraboloids in shape to make decent lenses.
         The focal length of a lens in relation to the radii of curvature of its surfaces is defined by the lens maker’s equation:
 
1/f = (n-1) (1/R1 - 1/R2)

 

Where "f" is the focal length of the lens, n is the index of refraction of the material of which the lens is made, and R1 and R2 are the radii of curvature of the lens' two surfaces.  (Here, as a practical matter, we're assuming that we're dealing with a lens with a spherical surface.)
         Note that the index of refraction, as mentioned before, is subtly different for different wavelengths (colors) of light.  This fact should be enough to tell you that different colors of light are not going to focus to the same point behind the lens.  And whereas this property of differing indices of refraction is sometimes exploited to decompose white light into its constituent colors, in the case of a lens, the difference in focal length that is dependent upon wavelength is generally called "chromatic aberration" and is considered to be undesirable.
         To combat chromatic aberration, designers of photographic lenses usually incorporate multiple pieces of glass (of differing indices of refraction) in a lens so that the otherwise ill effects in another element can reverse the ill effects of chromatic aberration in one element.  (This is one of those rare cases where two wrongs can make a right.)
         One of the more common lens designs (the Zeiss "Tessar") is shown below.  Note that the Tessar lens consists of a plano-convex element followed by a bi-concave element and a convex-concave doublet (formed by cementing a bi-concave element of one index of refraction to a bi-convex element of a different index).  The Tessar is thus commonly thought of as being a "four element" lens.

         Other aberrations (other than chromatic aberrations) include spherical aberrations and astigmatism.
         Spherical aberrations (recently made famous by the Hubble Space Telescope) cause point sources of light on the lens axis to fail to be rendered as point images.  In fact, in the case of spherical aberrations, they converge to some circle of a minimum radius known as the "circle of confusion."  If a point source passes off axis through a lens with spherical aberration, the result is a comet-shaped image, and the effect is known as "coma."
         Astigmatism in a lens causes off-axis point sources to image as lines.  This is due to the radius of curvature of the lens being different in different axes.
 
RATING LENSES
 
         For photographers, lenses are commonly classified by a combination of three numbers.  The first is the focal length of the lens, usually expressed in millimeters for small- and medium-format lenses and in inches for those lenses intended for use with large format cameras.
         For any given film format, there is a "natural" or "normal" focal length at which the image is neither enlarged (as it would be with a “telephoto” lens) nor reduced (as it would be for a "wide angle" lens).  This natural focal length is generally taken to be the diagonal measure of the image.  For example, on 35mm film, the actual height of the image is 24mm (5.5mm either side of the image is reserved for the film's sprocket holes), and the width of the image is 36mm.  The diagonal measure of the image is therefore (using Pythagoras' theorem) the square root of the sum of the squares of 24 and 36, or about 43mm.  In practice, lenses around 50mm are therefore said to be "normal" lenses for 35mm.

         To illustrate the fact that differing film formats have different natural (and by extension telephoto and wide-angle) focal lengths, I'll take a case from recent personal experience.  I own lenses of 50mm and 135mm lengths for my 35mm cameras, but I don't currently have a good 80mm lens.  I do, however, own a 6cm x 7cm medium format camera, and I wanted another lens for it -- and one that'd produce the same effect in 6cm x 7cm format as an 80mm lens would in 35mm.  So I calculated that the "normal" focal length for a 60mm x 70mm image is about 92mm.  I then calculated that in the 35mm film format an 80mm lens is about 1.85 times the normal focal length.  So to get an equivalent lens in 6cm x 7cm format, I'd have to get a lens of 1.85 x 92mm = 170mm.  In practice, the closest thing I could find at a reasonable price was 150mm, so that's what I ended up getting.  (There were 165mm lenses available, and this would have been closer to the desired focal length; but the prices were a bit high for my liking...)
 
F-STOP  
         The second number that defines a lens is its light gathering power, or minimum f-stop (field stop).  Because as focal lengths increase, the field of view of a lens decreases, the amount of light the lens gathers also decreases.  F-stops are a convenient expression of the size of the lens opening relative to the lens focal length; and in this combination of numbers, the relative changes in light gathering power can be normalized out of equations required for calculation of exposure.
         The formula for the f-stop of a lens is the focal length divided by its diameter.  Thus, lower f-stop numbers (corresponding to either shorter focal lengths or larger lens diameters) represent greater light gathering power.
         Commonly f-stops are numbered according to a sequence such that each increasing number represents a halving of the amount of light that passes through the lens.  For example, in the normal sequence of f-stops, the number "f2" immediately follows "f1.4."  The reason is that the actual area of the opening in a lens (its aperture) is found by the formula:
 
area = p x (radius2)

 
         The area of the opening of an f1.4 lens, relative to focal length (and therefore normalized for light gathering power) would therefore be 1/(1.42 x pi), or approximately 0.16.  The opening of a lens set to f2 would be 1/(22 x p), or approximately 0.08.  Note that the second number is roughly half the first number.  A lens set to f2 is therefore said to be "one stop down" from a lens set at f1.4.
         The normal sequence of f-numbers is:
 
1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, 90, 128, 180, 256, 360, and finally 512.
 
         In practice only the best lenses will reach down as low (numerically) as f1.2 or f1.4, and most fixed-focal-length lenses will begin their scales at either f2 or f2.8.  And most common lenses will top out at f-stop settings of f16 or f22.  Large format lenses may begin their scales at f8 and run up to f128 or f180; but these lenses are usually of extremely long focal length, since the "natural" or "normal" focal length of an 8" x 10" sheet film camera is 12.8 inches (325mm).  In order to make an f1.4 lens at the focal length, the outermost element of the lens would have to be just over nine inches in diameter, and in practice grinding a lens that large without aberrations is an expensive and tricky proposition.  Further, there are also mechanical considerations related to the physical problem of supporting that weight of glass.  And this is just the first of what could be from four to seven elements.
         Instead, at least in the case of a large format lens, the designer grinds smaller discs of glass and sets the lower f-stop number to something a little large, numerically speaking.  For the large format photographer, this means using slower shutter speeds to gather enough light to make a proper exposure.  But since an 8" x 10" large format view camera weighs a freaking ton, it's almost certainly going to end up sitting on a tripod, anyway.  Slower shutter speeds are therefore probably considerably less of an issue.
         Zoom lenses, which are expected to cover a variety of focal lengths, rarely reach down very low in f-stop number either, largely because as the elements of the lens move in relation to each other (in order to achieve the "zoom" effect), they also tend to block incoming light or otherwise throw it away. Also, they have enough elements in them that their overall cost becomes a consideration, and the constraints imposed by price (particularly for "consumer grade" equipment) force the compromise of grinding lenses at smaller radii.  Finally, the average consumer (and by and large, it's the amateur photographers and not the pro's who are using zoom lenses) demands that his lenses be lightweight.  With 9 to 12 elements in the average zoom lens (as opposed to maybe four to six in the average fixed-focal-length lens), each element has to weigh about half as much, and this severely restricts the amount of glass that can go into each element.  Zoom lenses therefore generally only open up to about f5.6 or so.
         Zoom lenses are unpopular with professionals for another reason than just lack of light gathering power:  Every time you add an element to a lens, you add one more piece of glass that can be dirty, misaligned, or poorly ground.  Beyond that, there are some limits imposed by the fundamental physics of light that say that every time you add another element in an optical system, no matter how well ground it is, it's going to degrade your image at least somewhat.  So for a serious lens designer the objective is to correct inherent aberrations introduced by differing indices of refraction for air and various types of glass by using the *minimum* number of elements.  This is one of those places where the "KISS" principle applies with a vengeance.
 
MECHANICAL CONSIDERATIONS  
         The third number that defines a lens is the radius of the outer filter ring on it (which determines what size filters it will accept.  This is purely a mechanical consideration, but it can become significant. Just ask anyone who shelled out the $130 required to pick up a decent IR filter as big as, say, 62mm.
         Putting the three defining numbers together, we see the kinds of lens descriptions that commonly appear in photography catalogs.  For example:
 
50mm (52) f1.4 Nikon AIS E-

 
would identify the lens for sale as a used Nikon 50mm lens with maximum opening of f1.4 and a filter ring size of 52mm.  The "AIS" designation would identify the family of the lens (and indirectly tell you what camera bodies it'd likely fit).  And finally, "E-" would identify the lens as being in "Excellent Minus" condition,, meaning it'd probably show very minimal external wear and would most likely not have any pitting or damage to the glass at all.  A lens like this one would fit nicely on something like one of my Nikon F3HP's and with a little bargain shopping should cost not much more than $100.  (I actually own two of these lenses, and I've made countless photos with them.  As I recall I paid $65 for one in "average" condition and $85 for one in "Excellent Minus" condition.)
 
PINHOLE PHOTOGRAPHY
 
Some optical systems contain no glass elements at all.  In extremis, a pinhole will form an image; and the first "camera obscura" (literally "dark box") probably had a pinhole for a lens.  The focal length of a "pinhole lens" may be calculated from the formula:
 
where "D" is the diameter of the pinhole in inches and "f" is the focal length in inches.  In practice, a #10 sewing needle will poke a 1/50" hole in a piece of cardboard and produce a "lens" with about an 8" focal length.  Using our formula for "f-stop" above, we see that such a lens has an intrinsic f-stop of f400.  Looking at our progression of f-numbers, we see that f400 falls about 9 1/2 stops down from f16.  Using the "sunny 16" rule (see the "exposure" section of this website), for an ISO 400 film (a reasonably "fast" film), our exposure time for a brightly lit scene would be between half a second and a full second with this camera-lens combination.  Making a picture on an overcast day with this pinhole camera would probably be a ten to fifteen second affair.  Bring your tripod.
         For those of you inspired to make a pinhole camera like the one described, feel free to download a few drawings (and instructions) here.
         In practice there are lower limits on just how small you probably want to make the pinhole, since as the pinhole gets smaller, secondary effects of diffraction (a function of the wave-like properties of light) begin to creep in and adversely affect the image.
         Any lens at a particular aperture will have a "depth of field."  Depth of field is considered to be a range of distances around the point of focus of a lens for which objects will be in "acceptable" focus.  Since the definition of "acceptable" focus is somewhat subjective, most lens manufacturers by convention have chosen to define the limits of focus by "circle of confusion."  These days, most lens manufacturers...
 
FOURIER THEORY  
         Study of the phenomenon of "depth of field" tends to lead naturally into a discussion of Fourier theory.  Without going into the nastier details of the theory, in effect it says that in the case of lenses, the more light you try to let into them, the less their depth of field is going to be.  As a way to understand that, consider the case of buying stocks.  If you bought 10 share of a particular stock on one day, you'd have ten shares.  If you bought 10 shares every day the market was open in a year, you might end up with more like 2500 shares (since there are about 250 market days in a year).  But if I were to ask you information about what you paid for a stock on March 15th, if you kept track of your purchases on a day-to-day basis, you could say, for example, "nine dollars and fifty cents."  On the other hand, if at the end of the year you knew you'd bought 2500 shares and paid $25,000 for them all, the most you could probably tell me is that on average you'd paid $10/share.  You would have destroyed the information about what happened on a specific day, and you couldn't tell me that on March 15th you'd got yourself a bargain.  March 15th would be out of focus to any fiducial lens you put on it.
         Optical lenses work much the same way.  As you let more light in, any one individual bundle of light is less and less discernable in the crowd.  Information gets lost, and the further you move away from the lens' actual point of focus the more information gets lost.  It's sort of like my focusing in on the average price of a stock in March (and maybe setting the point of focus at March 15th) and then asking you about the price in April or October.  I'm willing to bet that if the point of focus is somewhere in March, your guess of the April price is going to be a whole lot better than any guess you make about the average October price.
         By the same token, if I take an average of the prices of the stock in March and ask you about the March 15th price, you're probably going to give me a pretty good guess.  And in the period of interest there were probably only about 20-25 market days, so you bought maybe, at most, 250 shares.
Similarly, if I open my lens up to include the purchase of something more like 700 shares (let's say I look at all your stock purchases for February, March, and April), you're going to have a more difficult job pinpointing what the individual price might have been on any given day in the period.
         Again, by analogy, opening up an optical lens to include more bits of light will make it harder to focus on any one point.  This is what happens when you open a lens aperture and your depth of field falls off.
         Looking back at our pinhole, we see that it's a very small aperture and therefore should produce an image with exceedingly high depth of field.  And in practice it does, right up until the size of the hole gets small enough that the effects of diffraction at the edges of the hole begin to corrupt the image.