#### 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.

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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#### Michael Nagel

##### Active member
Doug,

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.

This subject has been addressed quite often the last few years. Should be nothing new for those among us who work with Kleinbild, MF and LF on a regular base.
Both cover the entire aspect of equivalence, f-number being part of it.

Best regards,
Michael

#### Doug Kerr

##### Well-known member
Hi, Michael

Doug,

This subject has been addressed quite often the last few years. Should be nothing new for those among us who work with Kleinbild, MF and LF on a regular base.
Both cover the entire aspect of equivalence, f-number being part of it.

Thanks for that observation and those links.

Best regards,

Doug

#### Tom dinning

##### Registrant*
If "we" do the algebra?
You do it. You're the expert.
Just give me a short answer I can understand at the end, in monosyllables and nothing exceeding 2 decimal points.
I'm still figuring out which way to put the battery in.

xxx

#### Doug Kerr

##### Well-known member
Hi, Tom,

If "we" do the algebra?
You do it. You're the expert.
I did it.

Just give me a short answer I can understand at the end, in monosyllables and nothing exceeding 2 decimal points.
Yes. Perhaps you mean 2 decimal places [please excuse the use of polysyllabic words here]. Then, 1.76 . Or, with 2 decimal points, 8.3.6 .
I'm still figuring out which way to put the battery in.
Big end first, I think.

Best regards,

Doug

#### Doug Kerr

##### Well-known member
Hi, Michael,

Doug,

Both cover the entire aspect of equivalence, f-number being part of it.

I finally had a chance to read both of those articles ("lightly", more so for the second one).

They are quite good. They illuminate some aspects of, in particular, sensor size that are not often recognized. I especially enjoyed the discussion in the Johnson article (Luminous Landscape) of how sensor size, through a chain of relationships, can influence the potential efficacy of "phase comparison" automatic focus systems.

Perhaps familiar to many, but not to me, was the relationship, based on diffraction considerations in concert with other factors, that suggests that the maximum usable f-number for a camera is linearly related to the diagonal size of the sensor.

This is all fun stuff.

Thanks again.

Best regards,

Doug

#### Asher Kelman

##### OPF Owner/Editor-in-Chief
Doug Kerr said:
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.

Doug,

Do you intend to mean, "in absolute terms, the same amount of light gathered" by a sensor of a different size to the "full frame" 35mm standard sensor size? Or could it be that you are considering the same amount of light incident and received "per unit surface area"?

If the former is your frame of reference, where would be the utility? What might be the interest in matching "total" light received to a small sensor to use to make a picture with the same "total light" on a much larger sensor of the same class of sensitivity, (i.e. Film, v for example, 10 year old digital technology versus current 2016 sensors)?

Asher

#### Doug Kerr

##### Well-known member
Hi, Asher,

Doug,

Do you intend to mean, "in absolute terms, the same amount of light gathered" by a sensor of a different size to the "full frame" 35mm standard sensor size? Or could it be that you are considering the same amount of light incident and received "per unit surface area"?

Well, in different parts of my presentation, I speak of both.

But in the passage you quote, I am referring to the total amount of light that is gathered (and delivered onto the sensor).

If the former is your frame of reference, where would be the utility? What might be the interest in matching "total" light received to a small sensor to use to make a picture with the same "total light" on a much larger sensor of the same class of sensitivity, (i.e. Film, v for example, 10 year old digital technology versus current 2016 sensors)?

The relevance is that from a standpoint of information theory, the total energy involved in conveying the information is a pivotal factor. The various tradeoffs we encounter (for example, when choosing an aperture, the tradeoffs between photometric exposure and depth of field) can all be (at least in a general way) traced to that concept.
I remember exactly where I was standing (in about 1965) when that came to me. I was at the time pondering the theoretical background behind depth of field calculations in connection with an article on that topic I was writing for some photographic magazine.​

Of course when we introduce into the comparison dramatically different technology (e.g., film vs. CFA digital sensors) other variables add to the complexity of the overall situation.

But ultimately, the telling limiting property is total energy involved. As we improve our technology, we are able to more nearly attain the theoretical limit of overall performance vs. available energy (just as is true of improved encoding for information transmission).

Recall that in the usual discussion's of Claude Shannon's seminal finding regarding information throughput and error rate as a function of the energy and bandwidth available, it is often said, "This formula tells the absolute limit of performance but gives no hint as to how it could actually be attained."
Of course my work in telecommunications led me to the understanding that "energy is the food of life".

Best regards,

Doug

#### Doug Kerr

##### Well-known member
Hi, Asher,

To expand a little less abstractly:

Suppose we have two cameras A with a smaller sensor than B. Both have the same sensel count, and so as a first order should deliver the same overall resolution (assuming in each case a lens that is able to "do its part").

We will assume that both sensors are operated at the same ISO sensitivity, and that we will use the same shutter speed for an exposure on each camera (perhaps dictated by subject motion considerations).

Thus (to shortcut various well-known relationships) we would be led to the use of the same f-number for both exposures.

The result will be the same illuminance on the sensor in each case (luminous flux per unit area). Given the consistent exposure time, that means the same luminous energy per unit area in the two cases.

As a first-order approximation, the noise performance of individual sensel photodetectors (when the sensor is operated at a certain ISO sensitivity) is related to the photometric energy they receive during the exposure.

In camera B, with the larger sensor and teh same sensel count, the area of the individual sensel photodetectors is greater than in camera A. Thus, considering that the total luminous energy per unit area is the same for both cases, we find that in camera B the photometric energy received by a sensel photodetector is greater than for camera A.

Thus we can expect better noise performance in camera B. And (given the various assumptions I have made) this is relateable to its greater sensor size.

Now of course if we depart from any of those assumptions, the conclusion would be different.

Best regards,

Doug

#### Doug Kerr

##### Well-known member

We have become familiar with the concept of the full-frame 35-mm equivalent focal length* of a lens when it is used on a camera with a sensor size other than "full-frame 35-mm").
*I use here its "entire" name, which we rarely hear, just to make most clear what I speak of.​

Of course, this metric allows us to speak of the field of view attained by "our" lens on "our" smaller-sensor camera in terms of the field of view that would be attained by a lens of that "equivalent focal length" on a camera with a full-frame 35-mm size sensor.

And it it a very useful convention, which I rely on all the time (notwithstanding that, a few years ago, I used to regularly rail out against it on intellectual grounds).

And, over the years, "we" have just about exterminated the notion that this quantity is the focal length of "our" lens in place on "our" camera.

I will note before I move on that the ff35 equivalent focal length (to use a somewhat shortened form of its full name) is the focal length of the lens of interest (some would feel compelled to say "actual focal length") multiplied by the ratio of the size of an ff35 sensor to the size of "our" sensor.

In recent years, another quantity has come into use: the full-frame 35-mm equivalent f-number of a lens when it is used on a camera with a sensor size other than "full-frame 35-mm"). This is calculated as the f-number of the lens of interest multiplied by the ratio of the size of an ff35 sensor to the size of "our" sensor.

The significance of this metric is that it allows us to visualize the depth-of-field performance, or the out-of-focus blur performance, of our lens in terms of what f-number would be needed on a lens on a camera with a ff35 size sensor to get that same performance.

And again, this convention can be a useful one.

But sadly, just as in the case of the ff35 equivalent focal length, there weaves though the usage of this quantity the notion that this somehow is "the f-number" of the lens of interest, on our smaller-sensor camera, with regard to the main thing we use the f-number to indicates: one of the two aspects of photographic exposure.

I saw this the other day in a review that spoke of lens with a maximum aperture of, say, f/4.0 used on a camera with an "APS-C" size sensor. (This was a Canon EF-M lens, only suited for such a camera.) The author said that its "equivalent" f-number was f/6.4 (true enough), and then commented that thus this lens was really "pretty slow". The intimation was that the exposure performance of the lens, on the camera for which it was intended, was that of an f/6.4 lens.

Another example is in a chart in a dpr review showing the "equivalent f-number" of the integrated lenses on various smaller-sensor cameras (including the Panasonic FZ1000 and FZ2000/2500 and the Sony RX10 III), where part of the commentary said:
When the FZ2500 reaches its maximum telephoto position it's equivalent aperture is about 1/3-stop slower than the Sony.

Now we are used to using the term "slower" to refer to a lens with a smaller aperture than some other lens (traditionally, because such a lens requires a longer exposure for the same photometric exposure), but it hardly seems apt when applied to a number that does not tell us the lens aperture.

What is being referred to is that, under the conditions of the comparison, the "equivalent" aperture of the FZ2500 is f/14 and that of the Sony f/11. But recall that these quantities are intended to allow us to compare the depth-of-field and out-of-focus blur performance of the lens; they are not actual f/numbers in that they do not tell us about the effect of that lens on exposure.

So (assuming that I got involved in this notion at all), I might have written:

When the FZ2500 reaches its maximum telephoto position its equivalent f-number is about 1/3-stop higher than the Sony.​

[Yes, I would have not incorrectly used the apostrophe in "its".]​

or even:

When the FZ2500 reaches its maximum telephoto position its equivalent aperture is about 1/3-stop smaller than the Sony.​

So, just be careful what you think - or maybe even say - is the meaning of "equivalent f-number".

Best regards,

Doug

#### fahim mohammed

##### Well-known member
Gives us the DOF. Good enough for me.

Blur. NO. How the acceptable in-focus parts dissolve into the unfocussed parts is determined by the construction of the lens glass elements. E
e.g. Fast, sharp, smooth, rendition...etc is a different matter. This rendition of out-of-focus elements is
a matter of constant debate between aficionados of various lens construction parameters. The result is usually very subjective.

To repeat, as far as DOF calcs go...you are generally correct. Blur etc. is a different matter.

And, as is often said but not adhered to, a picture is worth a lot of words.

The DOF of this lens can be determined by calculations. The ' Blur ' , the way the in-focus areas dissolve into the out-of-focus areas..is a different issue. The demarkation can be ugly, smooth, fast, gradual etc.

Once again, in the illustration below the DOF is the same. It is the same lens. The Change into out-of-focus areas is an aesthetic determination ( and based on lens construction..how many aperture blades, for example ).

No maths is going to describe the visual experience. It is subjective and personal. Partially, that is the reason for the difference in price of various lenses ( amongst other construction factors ).

p.s. Camera, subject and bg distances also play a part. Examples above are at different distances.

#### Doug Kerr

##### Well-known member
Hi, Fahim,

Gives us the DOF. Good enough for me.
<snip>

All well said.

Thanks.

Best regards,

Doug