Saturday, September 15, 2012

The Science of Photography -- Part Three

In my second installment of “The Science of Photography” I introduced the water bucket analogy and described in detail the shutter speeds available in modern cameras. The key to the explanation was that each shutter speed is either twice as fast or half as fast as the predecessor or successor value. That is the key point. This doubling or halving with each click of the shutter speed control will be combined with a similar double or halving of the light admitted by the lens with each click of the f/stop control.

In this article we will dig deep into the mystery of the f/stop. What it is, how it is calculated, and probably more math than you’ve experienced in the last week. There is a lot to say about f/stops. As authors like to say, the plot thickens.

Ultimately we will lean how f/stop is used in conjunction with the shutter speed control to adjust the total light that is admitted to the film or sensor. With proper adjustment of the variables, the amount of light will be just enough to “fill the bucket” and provide a correct exposure. But first, let me introduce you to, the value you’ve known for all these years, the f/stop.

F/Stops

An f/stop is a ratio. It is the ratio between the diameter of the aperture in the lens and the focal length of the lens. Therefore, f/stop is a characteristic of a given lens. Since most (good) lenses are made in either Japan or somewhere in Europe, the focal length is generally measured in millimeters, so that is the measurement I’ll use.

A very common lens has a focal length of 50 mm. Assuming a 50mm lens, then f/2 indicates that the diameter of the aperture is 25mm. That is, the ratio is 50/25 = 2 or f/stop = 2. So that seems pretty simple, although how it is useful for calculating the correct exposure may not be so clear yet.

Very simple lenses in very inexpensive cameras and in most cameras in cell phones are fixed aperture. That is, the aperture is not adjustable. But better cameras, that is more expensive cameras, have adjustable aperture.  The way it is made adjustable is by incorporating a set of between five and fifteen blades that can be adjusted inward and outward. The hole in the middle of the set of blades is roughly circular. We will use that fact in a few paragraphs, along with the formula for the area of a circle -- you know, the one with π, to calculate area. So stand by for some calculations.

But first, let’s look at some actual f/stop values:

1.4    2.0    2.8    4    5.6    8    11    16    22

This is the standard sequence of f/stops from f/1.4 to f/22. To begin with, you must realize that the f/1.4 setting lets in the most light while the f/22 setting lets in the least. The f/1.4 setting is when the aperture blades are retracted to their fullest extent and the f/22 is when the aperture blades are closed down to minimize the area the most. We say that the lens is “stopped down.”

Since the f/stop is part of a ratio or fraction, and it is really the denominator or bottom of the fraction, then, as the f/stop number gets larger, the ratio or fraction gets smaller. Compare 1/2 to 1/4. The larger four in the denominator makes the fraction smaller. That is why the larger the f/stop, the smaller the diameter of the aperture and the less light that comes in. Therefore, f/22 admits the least light into the lens.

Also, even though the sequence of numbers may seem arbitrary, each of these f/stops has precisely the same halving/doubling relationship as the shutter speed sequence. And therein lies the “magic.”

We can derive these numbers if we perform a little geometric mathematics. The aperture blades create a “hole” shaped as a polygon For example, with eight blades you get an octagon (looks like a stop sign). With a fifteen blade aperture, you get a pentadecagon, also called a pentakaidecon -- ain’t math fun!

However, it is OK to treat the aperture opening as a simple circle. Therefore the area of the opening is equal to π times the radius squared. That is, take the radius in millimeters, multiply it by itself, and then multiply that by approximately 3.1416. (Those wishing more accuracy can use 3.14159265 or even a longer approximation. Best would be to just press the π key on the calculator.)

Recall that our 50mm lens stopped down to f/2.0 meant that the aperture had a diameter of 25 mm. But, since we are interested in the amount of light admitted, we must calculate the area. If the diameter is 25mm, then the radius is half that for 12.5mm. Square the radius and we get 156.25. Multiply the radius squared by π gives 156.25 X 3.1416 =  490.9 square mm.

So the area of the 50mm lens aperture when stopped down to f/2.0 is about 500 mm squared.

Look at my list of f/stops above and you’ll see the f/stop to the right of 2.0 is 2.8. Let’s do some calculating on that value.

Because the f/stop is a ratio of the focal length to diameter, our 50mm lens at f/2.8 would have an aperture  diameter of 50/2.8 = 17.86mm. Remember, we have to divide that by 2 to get the radius of 8.93mm, so the area of the circle would be π X 8.93 squared, or 250.5 square mm. Rounding off a bit, that's about 250 sq. mm at f/2.8 and 500 sq. mm at f/2, a double/half relationship.

So the odd sequence of numbers isn’t so odd after all if you do all the converting from diameter to radius and squaring that number and then multiplying by three something. No wonder f/stops are so confusing.

But, on the other hand, they are simplicity indeed. Each larger number step to the right means half the light and each small number step to the left means twice the light up to the limit of the lens itself.

Now you may ask yourself, “why don’t they just call it the area of the aperture instead of these flakey f/stop numbers? Well, as confusing as f/stop numbers may be, think about using the diameter. “I took this picture with my 50 mm lens at 1/250th of a second and an aperture of 63 square millimeters.” Wait, you say. “That isn’t so bad. I like that better then f/5.6.” But, and here is the great beauty of f/stops, what if it was not a 50mm lens, but a 70mm lens. Then the actual area would be different -- BUT THE F/STOP WOULD BE THE SAME!

You see, all this f/stop business came from an era before cameras had built in light meters. In the old days -- also known as “when I was young” or “the time of the dinosaurs" -- we used hand held light meters and they would calculate shutter and f/stop settings independent of the focal length of the lens in the camera.

Knowing only the area of the aperture requires also knowing the length of the lens to understand the amount of light coming through the lens. The f/stop figure incorporates both of these in one useful, if initially confusing measure, and the lens length is immaterial. The f/stop is basically a shorthand notation for this important characteristic. When you set a lens to f/8, you mean for the focal length of this lens, open the aperture of the lens to a diameter that results in a circle that has a diameter that is one-eighth the value of the focal length. Fortunately for us, the lens makers figure out all these things and just mark the f/stops on the lens. So all those weird numbers are really for our benefit.

There is so much more to say about f/stop, but I’ll stop for now. In my next installment I’ll continue the thrilling story of the f/stop ... yes gentle readers, there is so much more story to tell. Ultimately, I’ll explain how to combine f/stop and shutter speed and control both the exposure and the effect you wish to capture. But you will have to wait for the next installment or two to reach that climax.

http://mickey-cheatham.blogspot.com/2012/09/the-science-of-photography-part-four.html