Doug Kerr
Well-known member
In recent times, we see the metric "equivalent focal length" reported for various camera-lens combinations in reviews and such. In this note I will discuss what that is all about.
Full-frame 35-mm equivalent focal length
We are all familiar with the concept of full-frame 35-mm equivalent focal length (even though the blue part of its name is not usually mentioned).
The concept is this. Assume we have a camera whose sensor is (in linear dimension) 2/3 the size of the sensor in a full-frame 35-mm camera (usually just called "full frame").
And imagine we use on that camera a lens with a focal length of 50 mm (fixed focal length or a zoom lens set at the moment to that focal length).
Then, the field of view afforded by that lens is the same as would be afforded on a camera with a full-frame 35-mm size sensor by lens of focal length 75 mm [75/(2/3) or 75•1.5].
We say that, "In place on our camera, the lens in use has a full-frame 35-mm equivalent focal length of 75 mm."
The factor 1.5 is called the "[full-frame 35-mm] equivalent focal length factor. It is also spoken of by other, repugnant, names (such as "crop factor").
[Full-frame 35-mm] equivalent f-number
In recent times, we hear of the equivalent f-number (accurately called the full-frame 35-mm equivalent f-number). In recent camera reviews in dpr, for example, we see this metric, in the case of a zoom lens, plotted as a function of the focal length setting (or maybe its the full-frame 35-mm equivalent focal length of the focal length setting, I don't remember just now). What is that metric, and what does it signify?
What is it?
According to the dpr tutorial on this matter, that metric is calculated by taking the actual f-number of interest and multiplying it by multiplying it by the "crop factor"; that is, by the full-frame 36-mm equivalent focal length factor.
What does it tell us?
It is said that the equivalent f-number (I will use its "short name" from now on for conciseness) can be used to help us with the following matters.
Depth of field and out-of-focus blur performance
These are two distinct, but related, properties of a camera with a certain lens aboard. The latter is of importance in estimating the "diameter" of the blur figures produced for out-of-focus objects, usually in connection with the nature of the "bokeh" that is produced.
Suppose we think in terms of "our" camera, with a certain sensor size, and a lens aboard with a certain focal length, set to a certain focal distance. The most reasonable way to compare depth-of-field performance between two cameras of different sensor size is to assume:
• Focal lengths providing the same field of view on both cameras.
• Cameras focused at the same distance.
• Choice of the circle of confusion diameter limit (COCDL) that is the same, as a fraction of the sensor size, on the two cameras.
If we adopt those conditions, then the depth of field performance of "our" camera is, to a first approximation, would be the same as would be attained on a full-frame 35-mm camera with a lens aboard with a focal length that produced the same field of view, set to the same focal distance, whose f-number is the equivalent focal length factor times the actual f-number. That is, a lens with the same f-number as the "equivalent f-number" for our lens.
The same is true for out-of-focus blur performance (except there the matter of the choice of a COCDL does not enter in).
So, using as an example our hypothetical camera with a sensor size 2/3 the size of the sensor on a full-frame 35-mm camera, consider a lens with a focal length of 50 mm and an aperture setting of f/2.0, focused at a distance of 10 m.
The same field of view would be attained on a full-frame 35-mm camera with a lens of focal length 75 mm. And, with such a lens aboard, using (for depth of field calculations) a COCDL 3/2 that we used for "our" camera, we would get approximately the same depth of field, and the same out-of-focus blur performance, on that camera with the lens aperture set to f/3.
Thus they say that the "equivalent f-number" of an f/2.0 lens on our "smaller-sensor" camera is f/3.0.
Light gathering ability
The equivalent f-number of a lens can be thought of as indicative of its "light-gathering ability". Before I discuss that further, let me give some background on the significance of the f-number of a lens itself.
The f/number of a lens is the ratio of the focal length to the diameter of the entrance pupil (we can simplistically think of the "diameter of the aperture"). This metric tells us, for a certain scene luminance, the illuminance upon the sensor, one of the factors in the photometric exposure, which determines the response of the sensor to the light upon it. (The other factor is the exposure time, or "shutter speed".) And this is independent of the focal length of the lens.
How does this happen? Well, firstly, the total amount of light "gathered" from part of the scene that is in the field of view of the camera is proportional to the area of the entrance pupil, which is proportional to the square of the actual diameter of the aperture.
But the illuminance upon the sensor depends on both the total light that is gathered (from the part of the scene that is in the field of view) and the area over which it is spread on the sensor (that will of course be the area of the sensor).
But, if we consider a certain small part of the scene, the area over which the light gathered from it is spread on the sensor depends on the inverse of the distance from the (exit pupil of) the lens to the sensor.
And if the camera is focused at infinity (and the significance of the f-number only strictly applies to that situation), that distance is exactly the focal length of the lens.
If we do the algebra, we then find out that, for a given scene luminance, the illuminance on the sensor is proportional to the reciprocal of the square of the f-number.
And that is independent of the focal length of the lens or the size of the sensor.
And all our photographic exposure reckonings are based on that principle - the determining factor is the f-number.
Now it is often said that "the f-number of the lens tells us its light-gathering ability". And from the above we can see that this is not true. In fact, the amount of light gathered by the lens (from a certain field of view on a scene of a certain luminance) is proportional to the square of its aperture diameter, not the f-number (or its square, or the inverse of its square).
Now I've shown above that with regard to matters of photographic exposure, it is the f-number, not the "total light gathered from the visible part of the scene", in which we are interested.
But with regard to another area of concern, we are interested in that "total light gathered".
Information theory teaches us that in the matter of signal-to-noise ratio it is the energy in our "signal" that is pivotal. The implications of this on a digital camera sensor is that for a certain exposure time, the total energy on the sensor is proportional to the total amount of light falling on the sensor.
So, following that whole trail, we might find that (for any given exposure time, and for some scene of arbitrary luminance) the overall noise performance of the camera is largely dictated, among other things, by the diameter of the lens aperture.
And the equivalent f-number tells us that diameter in a relative way that can be related to the familiar notion of the f-number and our poster boy for sensor size, the full-frame 35-mm sensor size. Lets see how that works.
Image that on our camera, with a sensor of dimensions 2/3 those of the sensor on a full-frame 35-mm camera, we have a 50 mm f/2.0 lens. The diameter of its aperture is 25 mm.
Now, if we go to a full-frame 35-mm sensor size camera, with a lens giving the same field of view (that would be 75 mm), and aspire to have that same aperture diameter (so "the same amount of light will be gathered"), we find that would be an f/3.0 lens.
Which is of course the "equivalent f-number" for our lens when on our camera.
************
So that's what that is all about, and how it works.
Best regards,
Doug
Full-frame 35-mm equivalent focal length
We are all familiar with the concept of full-frame 35-mm equivalent focal length (even though the blue part of its name is not usually mentioned).
The concept is this. Assume we have a camera whose sensor is (in linear dimension) 2/3 the size of the sensor in a full-frame 35-mm camera (usually just called "full frame").
And imagine we use on that camera a lens with a focal length of 50 mm (fixed focal length or a zoom lens set at the moment to that focal length).
Then, the field of view afforded by that lens is the same as would be afforded on a camera with a full-frame 35-mm size sensor by lens of focal length 75 mm [75/(2/3) or 75•1.5].
We say that, "In place on our camera, the lens in use has a full-frame 35-mm equivalent focal length of 75 mm."
The factor 1.5 is called the "[full-frame 35-mm] equivalent focal length factor. It is also spoken of by other, repugnant, names (such as "crop factor").
[Full-frame 35-mm] equivalent f-number
In recent times, we hear of the equivalent f-number (accurately called the full-frame 35-mm equivalent f-number). In recent camera reviews in dpr, for example, we see this metric, in the case of a zoom lens, plotted as a function of the focal length setting (or maybe its the full-frame 35-mm equivalent focal length of the focal length setting, I don't remember just now). What is that metric, and what does it signify?
What is it?
According to the dpr tutorial on this matter, that metric is calculated by taking the actual f-number of interest and multiplying it by multiplying it by the "crop factor"; that is, by the full-frame 36-mm equivalent focal length factor.
What does it tell us?
It is said that the equivalent f-number (I will use its "short name" from now on for conciseness) can be used to help us with the following matters.
Depth of field and out-of-focus blur performance
These are two distinct, but related, properties of a camera with a certain lens aboard. The latter is of importance in estimating the "diameter" of the blur figures produced for out-of-focus objects, usually in connection with the nature of the "bokeh" that is produced.
Suppose we think in terms of "our" camera, with a certain sensor size, and a lens aboard with a certain focal length, set to a certain focal distance. The most reasonable way to compare depth-of-field performance between two cameras of different sensor size is to assume:
• Focal lengths providing the same field of view on both cameras.
• Cameras focused at the same distance.
• Choice of the circle of confusion diameter limit (COCDL) that is the same, as a fraction of the sensor size, on the two cameras.
Those who believe that the COCDL should be chosen based on sensor sensel pitch or measured resolution will not find this discussion, nor the entire notion of "equivalent f-number" with regard to DoF considerations, comforting.
If we adopt those conditions, then the depth of field performance of "our" camera is, to a first approximation, would be the same as would be attained on a full-frame 35-mm camera with a lens aboard with a focal length that produced the same field of view, set to the same focal distance, whose f-number is the equivalent focal length factor times the actual f-number. That is, a lens with the same f-number as the "equivalent f-number" for our lens.
The same is true for out-of-focus blur performance (except there the matter of the choice of a COCDL does not enter in).
So, using as an example our hypothetical camera with a sensor size 2/3 the size of the sensor on a full-frame 35-mm camera, consider a lens with a focal length of 50 mm and an aperture setting of f/2.0, focused at a distance of 10 m.
The same field of view would be attained on a full-frame 35-mm camera with a lens of focal length 75 mm. And, with such a lens aboard, using (for depth of field calculations) a COCDL 3/2 that we used for "our" camera, we would get approximately the same depth of field, and the same out-of-focus blur performance, on that camera with the lens aperture set to f/3.
Thus they say that the "equivalent f-number" of an f/2.0 lens on our "smaller-sensor" camera is f/3.0.
Light gathering ability
The equivalent f-number of a lens can be thought of as indicative of its "light-gathering ability". Before I discuss that further, let me give some background on the significance of the f-number of a lens itself.
The f/number of a lens is the ratio of the focal length to the diameter of the entrance pupil (we can simplistically think of the "diameter of the aperture"). This metric tells us, for a certain scene luminance, the illuminance upon the sensor, one of the factors in the photometric exposure, which determines the response of the sensor to the light upon it. (The other factor is the exposure time, or "shutter speed".) And this is independent of the focal length of the lens.
How does this happen? Well, firstly, the total amount of light "gathered" from part of the scene that is in the field of view of the camera is proportional to the area of the entrance pupil, which is proportional to the square of the actual diameter of the aperture.
But the illuminance upon the sensor depends on both the total light that is gathered (from the part of the scene that is in the field of view) and the area over which it is spread on the sensor (that will of course be the area of the sensor).
But, if we consider a certain small part of the scene, the area over which the light gathered from it is spread on the sensor depends on the inverse of the distance from the (exit pupil of) the lens to the sensor.
And if the camera is focused at infinity (and the significance of the f-number only strictly applies to that situation), that distance is exactly the focal length of the lens.
If we do the algebra, we then find out that, for a given scene luminance, the illuminance on the sensor is proportional to the reciprocal of the square of the f-number.
And that is independent of the focal length of the lens or the size of the sensor.
And all our photographic exposure reckonings are based on that principle - the determining factor is the f-number.
Now it is often said that "the f-number of the lens tells us its light-gathering ability". And from the above we can see that this is not true. In fact, the amount of light gathered by the lens (from a certain field of view on a scene of a certain luminance) is proportional to the square of its aperture diameter, not the f-number (or its square, or the inverse of its square).
Now I've shown above that with regard to matters of photographic exposure, it is the f-number, not the "total light gathered from the visible part of the scene", in which we are interested.
But with regard to another area of concern, we are interested in that "total light gathered".
Information theory teaches us that in the matter of signal-to-noise ratio it is the energy in our "signal" that is pivotal. The implications of this on a digital camera sensor is that for a certain exposure time, the total energy on the sensor is proportional to the total amount of light falling on the sensor.
So, following that whole trail, we might find that (for any given exposure time, and for some scene of arbitrary luminance) the overall noise performance of the camera is largely dictated, among other things, by the diameter of the lens aperture.
And the equivalent f-number tells us that diameter in a relative way that can be related to the familiar notion of the f-number and our poster boy for sensor size, the full-frame 35-mm sensor size. Lets see how that works.
Image that on our camera, with a sensor of dimensions 2/3 those of the sensor on a full-frame 35-mm camera, we have a 50 mm f/2.0 lens. The diameter of its aperture is 25 mm.
Now, if we go to a full-frame 35-mm sensor size camera, with a lens giving the same field of view (that would be 75 mm), and aspire to have that same aperture diameter (so "the same amount of light will be gathered"), we find that would be an f/3.0 lens.
Which is of course the "equivalent f-number" for our lens when on our camera.
************
So that's what that is all about, and how it works.
Best regards,
Doug
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