How does aperture affect my photographs?
Why should I care about the aperture with which a photo was taken?
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The aperture is the opening through which light goes to reach the camera's sensor.
The size of the aperture affects not only the amount of time required to take a photo, but also the depth of field within it.
With a wide aperture (so a low number, like f/1.8) gives a shallow depth of field - sometimes less than a millimetre with a macro lens. Because a lot of light is reaching the sensor (be it film or digital), this allows for fast shutter speeds
With a narrow aperture (so a high number like f/22), the depth of field is much greater, which is useful for things like landscape photography - it will limit the amount of light reaching your sensor, so you will get slower shutter speeds, which makes a tripod handy.
Aperture simply means "opening", and in photography we use the term to refer to the diameter of the opening in a special adjustable diaphragm within each lens. When this diaphragm is constricted, there's less physical space for light to get in, so naturally the exposure is darker, and a more-open aperture allows more light and results in a lighter exposure.
Aperture has another important property. When the aperture is very small, the admitted light is highly "collimated", which is a fancy way of saying "all the rays are nicely parallel to each other". This results in a sharp focus for all the light that comes in. When the aperture is more open, only the rays which closely match the focus point are collimated — which means that whatever you've focused on is sharp, but farther or closer parts of the scene will be increasingly blurry.
Lens aperture is usually given as an f-number, which is a ratio of the focal length of the lens to the effective size of the aperture. This is more convenient than using physical diameter, because it works out that the actual amount of light gathered for a given physical aperture size depends on the focal length — so, if one uses the ratios, the exposure is the same regardless of lens length. (The counter-intuitive side-effect of this scheme is that smaller f-numbers let in more light.)
These f-numbers are used in photography in a sequence which may seem irrational: f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, and so on. However, these are chosen for a simple reason: each one has twice the area of the previous, letting in twice as much light. (There's no mystery to the numbers — the area of a circle is π × radius squared, and you can quickly work out that to double the area, one must simply increase the diameter by a factor of √2.)
Each step in the sequence is called "one stop", presumably because on manual lenses there's a physical catch which makes the dial which controls aperture nicely stop at each of these pre-defined points. However, the term "one stop" is generally also used by analogy for shutter speed and sensor ISO to mean "the amount by which this factor must be adjusted to double or half the recorded exposure".
An important thing to keep in mind is that while a small aperture excludes non-parallel light rays, as the size of the aperture approaches the wavelength of the light being captured, another effect comes into play: diffraction, which is a bending and scattering of waves as they pass through the aperture. In practical terms, this begins to affect APS-C form-factor cameras at around f/8, and so stopping down much beyond may increase depth of field at the expense of decreased sharpness in the in-focus areas. At some point, the effect of diffaction becomes strong enough that stopping down the aperture further doesn't provide any benefit at all.
Aperture is how we describe the size of the opening that determines how much light can come into your lens (like an iris on your eye). Assuming two lenses have the same size opening, a wider aperture value (smaller f stop #) will let more light in than the same size lens with a larger minimum aperture value.
A wide aperture will need a shorter shutter speed, which means you can capture high speed motion quicker. More light means a lower ISO setting is needed, which means less graininess in low light. A wider aperture will allow you to take photos in lower levels of light.
A wide aperture will result in a shallower depth of focus (the in focus part of the image will be smaller in the direction directly outward from the camera)
The aperture of a lens is the opening in the diaphragm that controls how much light is allowed to reach your focal plane (film or sensor). The aperture is usually rated as a fraction of the focal length, hence the f/x.y nomenclature. The wider an aperture is, the more light it allows through, and the smaller the number under f/ is. A common wide aperture is f/2.8, and is called a "fast" lens. A common tight aperture is f/5.6, and is called a "slow" lens. Tighter apertures allow less light through, which requires an decrease in shutter speed to maintain the same exposure.
For a much more detail answer about aperture, including the basic mathematics behind how they affect light, you can see my reply to this question:
Some other frequent questions about aperture:
Is there such a thing as a maximum aperture that a lens can be open to?
Yes, it's contained in the name and marketing materials of, and is one of the most prominent markings on, any camera lens.
If you're asking about lens design in general, the maximum aperture is limited by the size and weight of its lens elements and of certain limitations of the lens design. Also, with bigger, heavier elements and a larger lens body comes greater cost.
What about a minimum aperture that it can be closed to?
This is a less important specification and is not usually provided everywhere the maximum aperture is provided. Unlike the maximum aperture, there is rarely a good reason to use a minimum aperture of any lens. It would be normal to assume that the minimum aperture of a DSLR lens is going to be f/22 or smaller, and yet it would be rare to have a reason to go smaller than, say, f/11.
When you use smaller apertures there is a point at which the diffraction from the aperture blades start to make the image blurrier than any benefit that the small aperture has.
Is there a lens with the narrowest aperture in the world?
Again, it's not a very important statistic so there's no real reason to strive to it.
However, you may be interested in looking into what a pinhole camera is. It is a camera which has a tiny opening for light to pass through. Effectively it is an aperture so small that no lens is needed to focus the light.
Is there one with the widest?
Lenses get bigger, heavier and more expensive with increasing aperture. There are physical limits to maximum aperture. Lenses with a f/0.95 maximum aperture do exist, which means that the effective aperture diameter is a little larger than the focal distance.
See also this discussion thread. There are various interpretations of what "widest aperture" means.
Aperture is often confused and considered backwards - larger hole, smaller number - but this isn't really the case. Aperture is measured in a ratio, so its really a ratio of the diameter of the aperture in comparison to the focal length of the lens. This is why it seems backwards, because a smaller number in the denominator is really a larger number (e.g. 1/ 2.8 > 1/5.6). As said in other answers, the size of the aperture affects the amount of light let in, so changing it forces you to compensate with shutter speed and/or ISO and/or ambient light changes.
Aperture is how open your lens is (lack of a better phrase). The lower the number, the more light you allow in, but the less depth of field. The higher the number, the more depth of field and the less light being let in. Example: Taking a shot of something up close might need a lower aperture to capture ONLY that item, person, etc, and leave the background out of focus, while leaving the subject well lit (depending on your other settings). While a larger shot, might require a higher aperture, in order to keep everything in focus.
One thing to consider is that although some lenses do have low apertures, there can be aberrations at the lower f numbers. This is due to the lenses not being perfect. So to get the best sharpness across an image a more middle-range aperture may be ideal, as it cuts out light that has passed through the slightly imperfect glass at the centre of the lens. It does depend on your subject if this matters to you.
The best answer to your question is a rather a dumb down answer; as I was searching for the same when I was starting out in photography, because everyone was talking about aperture in manual mode.
Hope that help.