Crop Factor: Does Sensor Size Affect Focal Length, Aperture, and Light Gathering?
Crop Factor is a comparative (imaginary) number based on the diagonal length of a full frame (FF) sensor divided by the diagonal length of another given sensor, to be able to calculate an "equivalent" focal length and aperture that will capture the same "frame" with the same field of view (FOV) and depth of field (DOF) relative to what a full frame sensor at a given focal length and aperture would.
This article assumes you have the basic knowledge of how a camera works and about what depth of field and field of view is. Also knowing what aperture is and how it effects exposure is good to know. Even though I will describe things in its most basic forms it helps to know your stuff so that you do not miss any important information.
First to better understand crop factor one must know how and where the calculation for crop factor comes from. To find the crop factor of any given sensor first add the squared length and width of the sensor and then square root it. This calculates the diagonal length of the sensor. Do this for a Full frame sensor which yields 43.27 mm, and place this over the other diagonal sensor length and then divide. (Don't worry, this is the most math that will be done in this article.)
So on an APS-c sized sensor with a 26.68 mm diagonal length the crop factor would be
43.27 mm / 28.85 mm = 1.5 crop factor
Smaller sensor = larger crop factor.
Larger than FF sensor size = smaller crop factor
Lets say you saw an image taken on a full frame camera using a lens of fixed focal length shot at a certain aperture: what the crop factor will do is provide a focal length and aperture size for any given sensor to recreate the exact same frame (taking the same picture of something at the same distance away) and yield the same DOF and FOV (crop factor only has to do with FOV and DOF and not exposure, but more on that later). To achieve this you would multiple the crop factor to your cameras focal length and aperture to find the same "equivalent" focal length and aperture as that full frame camera to get the same frame or picture.
The focal length and aperture are not really changing. When you buy a 50mm lens for a APS-C camera it is truly a 50mm lens but will take the same picture as a 75mm lens on a full frame camera. When applying crop factor one is just translating it into an 'equivalent' number for more accurate comparison, but let me explain this in more detail.
Why are we doing this to find equivalent focal lengths and apertures to compare with full frame cameras? The only reason is so photographers can be speaking the same language no matter what sensor sized camera they have. If we all know what FOV a 50 mm equivalent focal length will give then the person with a micro four thirds camera can tell the person with a medium format camera what equivalent focal length was used to capture a specific image. Then the medium format sensor sized camera guy or gal can go buy the appropriate lens to take a similar image.
As to the reason why FF and not another size is due to the movie industry. 35mm film was popular to use during analog times, and became kind of a standard (but not the only sized used) for motion pictures. 35mm film is about the same size as a full frame sensor.
Focal length and Field of View
I will not talk about compression or how different focal lengths will change the perceived distance of objects. That is not important to crop factor and more important to focal length.
Focal length is the distance from the lens element to the sensor, or more specifically where the light rays converge to the image sensor along the focal plane. (I will have an illustration)
Why does FOV change for same focal length lens on a different sensor size? This is due to the smaller sensor (smaller then FF) using a narrower area of light. Opposite applies to sensors larger than FF. This is better shown with an illustration.
As you can see even though the same focal length was maintained across two cameras with different sensor sizes the field of view was narrowed because less range of light will hit the smaller sensor.
We can also change or get the same FOV by changing the focal length of the lens. This can be illustrated by increasing the focal length of the FF camera lens to get a narrower FOV to be the same as the APS-c camera or vice versa.
Example: If you wanted to take an image at a point in a room with a FF 50mm lens and wanted to get that same image and field of view (FOV) at the same point in the room on an APS-c sensor camera with a 1.5 crop factor, you would need to use a 33.3mm lens. (Of course the lens will be rounded and sold as a 33mm.)
33.3 mm x 1.5 = 50 mm
Aperture and Depth of Field
The aperture f-stop number (or f-number) is a calculated number which gives the ratio of the focal length to diameter of the entrance pupil (aperture opening). Aperture in a camera is adjusted by changing the entrance pupil diameter only (on a prime lens), but the calculation to get the f-stop number also involves the focal length. Because the f-number changes with the aperture opening and closing, f-number can be used to describe the change in light due to the change in aperture size.
Aperture is calculated taking the focal length over the diameter of the entrance pupil and dividing them.
Focal length / diameter of entrance pupil = aperture in an f-stop number
F-stop number does not change depending on your sensor size since its a ratio based on physical properties of the lens, and instead will (more or less) scale with the change of focal length and size of lens due to change in sensor size. This is convenient because the change in f-stop number, used to identify change in how much light is coming into the camera through the lens, can be universally translated to any lens on any camera. The change of one stop of light will be the same change f-stop number if using a full frame sensor, other sensor size, or even if using different focal lengths. F-stop is another universal language for talking about exposure.
This number helps gauge the amount of light coming into the camera, so even though changing from an f/2.8 to a f/1.4 on a full frame is a visually larger and higher diameter of change then going from f/2.8 to f/1.4 on a micro four thirds camera it is still the same amount of stops of light or same proportional change in light coming into the camera. It is the same proportional change in amount of light but different sized openings. If using the same focal length lens then the change in diameter will be the same.
Edit: This means that exposure from aperture is not affected by sensor size. The 'speed' of a lens does not change compared to a FF camera. A 1.2 f lens is very fast and the same speed if it is on a FF camera or micro four thirds camera.
There are three factors that effect DOF; focal length, aperture size, and camera to subject distance. Lets focus on aperture and how it affects depth of field. As you open an aperture you produce a shallower (or smaller) depth of field or 'less stuff in focus'. DOF is the depth or length of stuff in focus, behind and in front of the main or center focus point.
Focusing and DOF can get kind of confusing, but i will try to simply explain as best as possible.
First we must understand how a subject or just a point can be in focus. For simplicity we will say two rays of light come off of a point at a certain distance and hit the edges of lens opening at opposite ends. Those rays then start to converge. Again to simplify i will say that if these rays of light do not converge at or very close to a plane called the 'plane of focus' then the point where those light rays came from will be out of focus.
To expand on this idea, if we say infinite rays come off that point at all directions then the rays converge inside the lens and create a cone. If that cone does not converge at the plane of focus (converging behind or in front) then it will be sliced at the plane of focus creating a circle cross section called the 'circle of confusion'. If this circle of confusion is larger then the 'permissible circle of confusion' then that point will be out of focus.
We then can have a depth of field because we have a range of what can be in focus. This is a distance where the circle of confusions can be smaller then the permissible circle of confusion and in turn in focus. By increasing the aperture then the angles of the cone become smaller creating a stubbier cone. A stubbier cone will have a larger change in the circle of confusion when traveling along the optical axis (simply put is the axis which goes through center of lens and is perpendicular to the plane of focus) which is why a larger aperture will yield a shallower DOF.
When the aperture is closed as small as possible then light rays entering will be at similar angles or have less angle variation between them yielding much larger area in focus as the cone will be long and skinny.
This topic can go further, but for basic understanding of DOF for a photographer, this is good enough. I will like to say that aperture will not change the image itself. You are still getting the same image but with a different DOF and exposure, and FOV stays the same.
Now that we have aperture out of the way let us see how the other two factors change DOF. Depth of field also changes depending on the distance of the object and the focal length, so there is a lot more going on besides aperture.
Holding aperture and focal length constant we can see that depending on the distance the object is away from the camera will change if it is in focus.
Lets take a real life example where we don't change the aperture size, but instead we change the focus point.
A great example is using the best camera you have. Your eyes. Place an object ( or you can use your hand as well) at eye level with plenty a view behind it. When you stand a step back from that object, everything in the background is blurry, but you can still make out some detail. If you then approach the object to the point where your face is almost touching it everything behind it becomes much more blurry. You have shifted the focal point bringing it closer to the lens (your eye).
Objects closer to the camera will have a smaller DOF then objects farther away. You can see this in the picture below. I kept all exposure setting constant (same aperature!) and just moved the object.
Two things might change when using 'equivalent' lenses (ex. 50mm on FF and 33mm on APS-c): That lens has an actual focal length and opening diameter different from the 'equivalent' lens on a FF sensor camera.
Equivalent focal length is imaginary, or for comparative purposes, and actual focal length is the length light adheres to. Light is not biding by the crop factor and will only consider the actual focal length it needs to travel. So even if you are using an 'equivalent' focal length your DOF will change because your actual focal length changed.
As mentioned before the f-stop number calculated using focal length and opening diameter used to tell how much light is coming into the camera, because it is a ratio you can use a smaller diameter and focal length and still achieve the same f-stop number. Even though you get the same f-stop number (same proportional light entering) the physical dimensions of the lens are still different. In the real world light is going to abide by those physical dimensions meaning your DOF will change.
So because of these two factors change with sensor size, no matter if you are using the equivalent lens or standing the same distance away, DOF will change. So to get the same image frame at the same spot you need to get an 'equivalent' focal length lens and shoot at an 'equivalent' aperture by applying the crop factor.
However, one can use the same real focal length with the same real aperture on different sensor sizes and take a picture at the same distance away (even though the image will be cropped or have different FOV among other things) to achieve a picture with the same DOF. If you were to crop all those pictures to get the same FOV you will have the same image (more or less) with the same DOF. I think this is an important paragraph as it shows how imaginary crop factor is. You could go your whole life not knowing about applying crop factor to aperture and it would never bother you, but this also means that when you buy equivalent lenses, the lens for the larger sensor size will have a shallower depth of field for the same aperture size and vice versa.
To get the same DOF for the same image or frame (again at the same location with same FOV using an equivalent focal length) we apply the crop factor to the aperture. Unlike focal length when applying crop factor to aperture you will need to divide.
Example: If I wanted to stand in the same place but achieve the same DOF that a FF f/1.8 lens provides using a APS-c sized sensor camera (1.5 crop factor) then I would need to open up the aperture to f/1.2.
The problem with this is DOF might change to achieve the same 'equivalent' aperture, but the aperture and the amount of light coming into the lens is also changing. By increasing that aperture you also increased the amount of light coming into the camera. You would have to compensate for that by adjusting the other two exposure triangle points; shutter speed and iso.
Edit: I do not know if I made this clear but the crop factor only applies to the DOF of the aperture and not its ability to let light into the camera. The 'speed' of a lens stays the same. You can take a picture with the same exposure settings (same iso, shutter speed, and f-number) on any sized sensor and get the same exposure.
Light and Sensor Size
One misconception is that sensor size corresponds to iso performance or that a smaller sensor will capture less light needing to compensate to capture a similarly exposed image as a FF camera would.
Let me start by saying crop factor has nothing to do with the next two sections. I am switching gears a little just to roll out some of the sensor size misconceptions. Some of the next information could dive straight into some hard science such as quantum mechanics, but ill keep it simple. You don't need all that fancy-fancy to understand this.
So, now I can tell you that smaller sensors, compared to FF, receive less light, but on a smaller area, but because light intensity entering the camera is the same, the same light per unit area will be captured. So smaller sensors capture the same light per unit area as larger ones. That is the key phrase here, "same light per unit area".
This means that a larger sensor and a smaller sensor (lets say they are identical sensors except one is smaller than the other) will capture the same light per unit area so that the images between them will have the same exposure. Sensor size does not change the light gather capability.
ISO and Sensor Size
Iso performance is a different story. Three main factors effect ISO performance; sensor quality, image processor quality, and size of pixels. (New technologies will also affect iso performance, but I am disregarding this for the explanation). The first two don't depend on sensor size. The third doesn't either. The third is related to how large each pixel is on the sensor. This relates to how many pixels per area or pixel density and then the size of the sensor compared to that density. Smaller pixels mean they will gather less light then larger pixels, and in turn less information per pixel and more noise. Noise is the camera trying to compensate for lack of information at a given pixel.
Smaller sensor does not mean smaller pixels. Reduce the pixel density on a smaller sensor and the pixels grow. Sensor size does not determine iso performance.
This means you can not have two (identical) sensors (with identical image processors), one being 24 megapixel full frame sized and the other 24 megapixel APS-c sized and expect them to get the same performance. Smaller pixels have smaller area so they pick up less light information (simple version).
This also means we should all switch to large format. Might just buy me one for shits and giggles.
The conclusion is who gives a f**k about sensor size. If your using anything in between large format and 8mm frame then you are set to take pictures. Crop factor is imaginary and is just a translation key of focal lengths and apertures for one universal photography language. Use whatever focal length and aperture you think is visually pleasing.
A more useful and scientific conclusion is crop factor is an imaginary tool to allow photographers using different sensor sized cameras to communicate with each other easily. A useful focal length is 50 mm, but I wont list all the equivalent focal lengths for every sensor size, the crop factor calculation deals with that. It all depends on how you want to take a picture and finding the correct FOV and DOF to get the desired results.
I know when I first started photography I used a crop sensor, but all the available information was based in actual focal lengths, so at first I was confused why I needed to be so far away or close up to take similar pictures as some of my ideals. Then I discovered the idea of crop factor. As my lens lineup grew I stopped thinking about equivalent lengths and started thinking about the images I wanted to capture and how I would capture them. Besides all the crop factor stuff I hope what people really learn is YOU get to choose how and what you want to take a picture of. Do not let this imaginary person know as crop factor tell you how to take a picture or how to have fun doing it.
Comment below if you disagree with something, find a mistake, or just want to add to the discussion. Here are two nice videos that are for and against applying crop factor to aperture. They both illustrate why you would or would not but this article tells