Is f/4 always f/4, even on a Different Sensor Size?

Short Answer:

Yes and No.

Long Answer:

What is "f/4"?

A typical camera lens has a variable size, roughly circular, hole called the "Aperture". When the Aperture is wider, more light is let in. When the Aperture is smaller, less light is let in.

The notation "f/4" means that the diameter of the Aperture hole is the focal length ("f") divided by 4. On a 50mm lens f/4 is a 12.5mm Aperture diameter. On a 100mm lens, f/4 is a 25mm Aperture Diameter.

This notation is often called the "f/stop" or the "relative aperture".

Why We Use Relative Aperture

The reason photographers use relative Aperture is because at the same relative aperture, you get the same light per unit area on the film/sensor, independent of focal length.

If you swap out your 50mm lens with a 100mm lens, but keep the same aperture diameter, you will end up 1/4 the light per unit area hitting your sensor. If you were shooting film, this would be a problem, as you pretty much need to hit the same exposure, no matter what lens you have mounted. However, if you keep the same relative aperture, you maintain the same exposure.

Replace your 50mm 12.5mm diameter aperture lens with a 100mm 25mm aperture diameter lens, and the light per unit area remains the same. Note that a 50mm lens with a 12.5mm Aperture has the same relative aperture (f/4) as a 100mm lens with a 25mm Aperture.

Keeping the same exposure for every shot on a roll is extraordinarily helpful when shooting film.

Implementation Details vs. Results

Focal lengths, sensor sizes, light per unit area, are all implementation details. They are steps along the path to creating our final image. As with many journeys, there is more than one reasonable path to the destination.

At the same Angle of View, same subject, same Aperture Diameter, and same shutter speed, you get essentially the same results (same framing, same motion blur, same Depth of Field, same diffraction blurring, and same overall image noise). The choice of focal length, aperture diameter and sensor size are implementation details along the path.

Many photographers care more about the results than the implementation details.

Consider a full frame camera and a 2X crop body. The full frame has a 50mm lens at f/4, and the crop body has a 25mm lens. Both combinations result in the same 46° Angle of View. Assume the same subject and shutter speed for both cameras.

In terms of implementation details, if the crop body lens is also set to f/4, we get the same light per unit area on the sensor (this would be critical if we were shooting film). From that implementation detail perspective, f/4 on the full frame is the same as f/4 on the crop body.

On the other hand, the f/4 crop body image will have deeper depth of field, more diffraction, and more image noise. In terms of the results, f/4 on a full frame yields a different result than f/4 on a 2X crop body.

Now, if we were to set the 25mm lens on the crop body to f/2, we get four times the light per unit area on the sensor (which happens to have 1/4 the area). If we were shooting film, this would be a problem. If we are shooting digital, we can reduce the ISO setting by two stops to compensate.

By opening up to f/2, the 25mm crop body lens has the same 12.5mm Aperture diameter as the 50mm f/4 lens on the full frame. They also have the same Angle of View. This means the resulting images will have the same Depth of Field, same overall image noise, same framing, same diffraction issues, etc. In other words, you will get essentially the same image from a full frame 50mm f/4, as a 2X crop body with a 25mm f/2. From the perspective of the results, f/2 on a 2X crop body, has an "equivalent f/stop" of f/4.

It's equivalent in the same way as a 25mm lens on a 2X crop has an equivalent focal length of 50mm. The 25mm lens is still a 25mm lens, but the results match what you would expect from a full frame with a 50mm lens. Put a 25mm f/2 on a 2X crop body, and it is still f/2, but the results match what you would expect from f/4 on a full frame.

Conclusion

So is f/4 always f/4? The answer is "Yes, but the results won't be what you would get from f/4 on a full frame".

True enough. And 25mm is always 25mm, but the results won't be what you would get from 25mm on a full frame.

www.josephjamesphotography.com/equivalence/

Actually, in the context it is usually asked, we fave f_1/4 and f_2/4 with f_1 and f_2 different, depending on the formats, so they are not the same.

Sounds like a philosophic question, and you give the typical answer "Yes and No".

Maybe fix the typo in the title :-)

Thank you for pointing out the typo.

As this is the beginners forum, I'll try my hand at a stripped down simplified, non-techie, absolute beginners version of an answer. Feel free to suggest edits:

Lenses contain a part called an iris diaphragm. It works a bit like the iris of the eye. Think of it as a throttle for light - in bright light the iris shrinks down to a pin-prick to restrict the amount of light, in low light is opens up as much as possible to let in as much light as it can. If you didn't have a variable iris, you'd be blinded by bright light and stumbling around in darkness in low light.

The camera lens iris is a circular blind consisting of multiple overlapping blades that combine to create a roughly circular hole. The hole in the iris is how the light gets in and it can be varied in size (by adjusting what is called the "aperture" control) to increase or reduce the amount of light the lens lets in.

Historically we have a convention for labeling the settings for the size of the opening. It goes like this: f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22 and so on. On lenses which feature aperture rings, you'll usually see it marked like this (most lenses won't use all of the sequence for various reasons).

At this point, you do not need to know how this scale is derived or why the numbers look so odd. The important thing is what the numbers tell us. And what they tell us is that as you turn the aperture ring or roll the aperture control wheel on your camera to scroll through these settings, the amount of light let in by the iris halves with each successive step.

eg If you start at f/2.8 and change the setting to f/4 (one step), the amount of light the iris allows through the lens halves. Or if you go the other direction from f/4 to f/2.8, the amount of light the iris lets in doubles. This relationship is true for any pair of numbers in the scale: moving from f/8 to f/11 halves the amount of light the iris allows through the lens, moving from f/5.6 to f/4 doubles the amount of light. If you change the aperture setting by more than one step (eg from f/2.8 to f/5.6 - 2 steps) the amount of light is halved, then halved again. And so on for every extra step.

Basically, the f numbers are used for controlling the amount of light the lens lets in a standardised fashion from lens to lens. This halving and doubling relationship is the same for the f numbers on every lens. And yes, in terms of throttling the amount of light going through the lens, f/4 means the same for every sensor size.

So good, so far, but things are a bit more complicated (of course they are!) than this, because the iris has other effects beyond controlling the amount of light the lens lets in (for example, it is important in controlling "depth of field" and the effects of this can vary across sensor sizes) but what we have considered so far is a good place to start. Consult other articles in this series for a deeper dive when you are comfortable.

A useful introduction to the concept of lens aperture. But that's a separate issue from what I was addressing in my original post, which was on the interaction of sensor size and aperture.

I was trying to address the question: If I beginner has a smartphone with an f/2.0 aperture and a 50 mm equivalent focal length, will that give him the same results as a full frame camera with a 50mm lens at f/2.0?

My thinking was to write things as a series of linked posts starting off as simply as possible then getting more sophisticated. I did answer the question in a hand wavy way at the end with:

Basically, the f numbers are used for controlling the amount of light the lens lets in a standardised fashion from lens to lens. This halving and doubling relationship is the same for the f numbers on every lens. And yes, in terms of throttling the amount of light going through the lens, f/4 means the same for every sensor size

followed by a segue to the next part that doesn't exist yet

So good, so far, but things are a bit more complicated (of course they are!) than this, because the iris has other effects beyond controlling the amount of light the lens lets in (for example, it is important in controlling "depth of field" and the effects of this can vary across sensor sizes) but what we have considered so far is a good place to start. Consult other articles in this series for a deeper dive when you are comfortable.

But there are many ways to approach this. I was using the question more as a vehicle for considering the style of beginners Q&A than the specifics of the question.

You imply that the physical hole (diaphragm/iris) is for example 25mm in your example but it is the entrance pupil (the virtual image of the diaphragm as view from the front of the lens) not the diaphragm that is 25mm

Of course you're right, Bill. For a lot of calculations, I find the thin lens model sufficient. But, as you've demonstrated on your website, sometimes you need to peel back more of that onion.

In what way would the results differ?

At same print size, diffraction, DOF, noise...

While Jim Kasson asnwered your question directly just above, I would like to answer your question with a question: why do you care about the aperture? Most beginners probably shoot in Auto or P mode, so the camera chooses the f-number for them. A little more advanced, it's common for people to shoot in A mode (Av for Canon). So, presumably, these people care about the f-number.

Again, why? Why care what the f-number is?

The f-number affects two primary attributes (it also affects diffraction and sun-stars, although sun-stars are also strongly dependent on lens design):

1) The amount of light projected onto the sensor
2) DOF

Usually, people think of (1) in terms of exposure, which is the amount of light per area that is projected on the sensor while the shutter is open. This is all well and good when using the camera in hand.

However, this raises another question: why care about exposure?

The only reason one would care about exposure is noise and blown highlights -- lower exposures result in more noisy photos (because less light results in a more noisy photo) and higher exposures [may] result in more of the photo being blown out. Let's concentrate on the noise end of exposure.

Putting it all together, the reason we care about the f-number is:

1) Noise
2) DOF

So, we use lower f-numbers to get less noisy photos, but this necessarily results in a more shallow DOF. We use higher f-numbers to get a deeper DOF, but this either results in a more noisy photo (if we use the higher f-number with the same exposure time) or increases the risk/severity of motion blur (when we use a longer exposure time to compensate for the higher f-number to maintain the exposure).

As it turns out, assuming sensors that record the same proportion of light projected on them and sensors that add in the same amount of additional electronic noise, it's the total amount of light making up the photo (or portion thereof) that dictates the noise, not the amount of light/area projected on the sensor. Thus, the same exposure on a sensor twice the size (4x the area) will result in two stops more (4x as much) light making up the photo (or any portion thereof), and hence a photo that is half as noisy.

If, instead, we used twice the f-number on the sensor that is twice the size, with the same exposure time, the exposure will be two stops less (1/4 as much), but since the sensor area is 4x larger, will result in the same total amount of light making up the photo (or portion thereof), and thus equally noisy photos.

Now for DOF. It turns out that for a given perspective (subject-camera distance) and framing, the DOF is determined by the diameter of the aperture (entrance pupil), not the f-number, for given viewing conditions (same display size, same display medium, same viewing distance, same visual acuity, etc.). So, if the smaller sensor used, say, 25 mm f/4, the aperture diameter would be 25 mm / 4 = 6.25 mm. A sensor twice the size would use 25 mm x 2 = 50 mm for the same framing and thus use f/8 for the same aperture diameter (50 mm / 8 = 6.25 mm) if they wanted the same DOF.

Thus, we would say that f/4 on a smaller sensor is "equivalent to" f/8 on a sensor twice the size, since it results in the same aperture diameter, which results in the same DOF (for the same perspective and framing), same motion blur (for the same exposure time), and same total amount of light projected on the sensor, hence the same noisiness.

Alternatively, if we used f/4 on both systems, the sensor twice the size would have half the DOF and produce a photo half as noisy, again, for the same scene, perspective, framing, and exposure time.

Hope that makes sense!

That's the point of the f/4 not always looking like f/4 when you change formats. Isn't that what this thread is about?

The title of the thread is: Is f/4 always f/4, even on a Different Sensor Size?

That doesn't mean that the effects of diffraction are nearly impossible to measure, as you claimed.

I compare Velvia-50 transparencies, I don't have an optical bench.
Edit: I find out of focus image degradation to be more objectionable than diffraction degradation.

How do they compare to transparencies shot without diffraction?

I agree we should avoid discussing film (and car) topics in this thread.