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Equations for F stops and lens size?
Hello. I am puzzled by what seems to be an inversion of the equation for lens size and maximum aperture..

That is, it is often said that the F-stop rating of a lens is simply the maximum aperture divided by the focal length..

That seems to be at least partly correct, and it is easy to verify with long lenses. A 300mm F5 lens has a larger piece of glass in front than a 300mm F8. And a 300mm F2.8 is really big..

But with smaller focal lengths the equations seem to reverse. Below about 35mm or so, the lenses seem to be growing larger again. And very wide-angle low-light lenses are really large. My 55mm F1.4 lens is large, and apparently, a 24mm F1.2 would be huge, if you could get one. But even a 20mm F1.0 should only be 20mm in diameter, if the above equation were true. And 20mm in diameter isn't very large at all..

So what gives? What are the real equations that determine the size of the chunk of glass in front when they are designing wide-angle low-light lenses?.

By the way, I think there has to be another term in the equation because it seems that you cannot go below F1.0. I've never heard of a lens like 24mm F0.33....

TIA..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comments (26)

Terrance13 wrote:.

... it is often said that the F-stop rating of a lens is simplythe maximum aperture divided by the focal length..

Oops. Reverse that. I meant Focal length divided by maximum aperture..

I need this coffee to kick in...

Comment #1

The aperture is not the diameter of the first piece of glass, it is the entrance pupil. Take the covers of a lens, hold it against the sky and you will see it. If you do that with a wideangle and twist it sideways and up and down you will see why the front element needs to be big..

Shoot...

Comment #2

Nokem wrote:.

The aperture is not the diameter of the first piece of glass, it isthe entrance pupil. Take the covers of a lens, hold it against thesky and you will see it. If you do that with a wideangle and twist itsideways and up and down you will see why the front element needs tobe big..

Shoot..

Ah, okay, thank you. That clarifies a lot. We still don't have any equations, but I'm getting the idea. As we go lower and lower in focal length, the angle of view gets wider and wider until the lens turns into a fisheye, which means that the front piece of glass has to be increasingly larger, and it also has to curve more and more until it bulges like a hemishere..

Still, is there some factor in there that becomes close to zero as the F-stop size approaches 1.0? I smell a divide-by-zero error. It seems that it is impossible to make a lense with an F-stop rating less than 1.0. I've never heard of such a thing..

In fact, I only saw one instance of a guy selling huge old pieces of glass on eBay who claimed that they were F/1.1 and F/1.0 lenses. I don't know whether his estimates of the properties of the lenses were correct..

And I'm curious. Thanks again..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #3

The f number is the focal length of the lens divided by the diameter of the entrance pupil usually at the aperture or stop..

The bigger the diameter of the aperture, the smaller the f number. If the diameter of the aperture is larger than the focal length, the the f number will be less than 1..

The problem with very fast l[ = small f numbers ] enses seems to be with the correction of residual abberations and/or the need for fancy glass or aspherics..

There have certainly been lenses with f numbers below 1.0, but they usually suffered from softness at full aperture/abberations..

Leica make an f1 lens for the M series the Noctilux..

Very fast lenses are not so necessary nowadays film emulsions are much better at high ISOs, as are DSLR sensors I started with the original Kodachrome with an ISO/ASA of 10 digital cameras don't go anything like so low nowadays..

Bertie..

Comment #4

Still, is there some factor in there that becomes close to zero asthe F-stop size approaches 1.0? I smell a divide-by-zero error. Itseems that it is impossible to make a lense with an F-stop ratingless than 1.0. I've never heard of such a thing..

Do a search of Joseph Wisniewski's recent posts. I think he discussed this to some degree with a few other people. What I recall is that f/0.5 is as fast as a lens can be given it is made from spherical glass. But that doesn't mean it will be very good. Just that it is possible and that there is some mathematical reason why you can't go faster when using spherical surfaces..

Jay Turbervillehttp://www.jayandwanda.com..

Comment #5

Sherwoodpete wrote:.

Http://www.canon.com/.../camera-museum/camera/lens/s/data/50-85/s_50_095.html.

Http://www.pbase.com/elliot/image/45262012.

Thanks for the tip. Now that's one beautiful piece of glass..

F 0.95 And obviously no divide-by-zero error where we all vanish into a black hole... .

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #6

Sherwoodpete wrote:.

Carl Zeiss 50mm f/0.70.

Http://forums.dpreview.com/...forums/read.asp?forum=1037&message=20496240.

It says:"A few were built for NASA but cant find any info about it or pictures. ".

Okay, thank you..

I'll keep my eyes open and see if I can find a used on at the thrift store... I wish....

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #7

Rc53 wrote:.

Very fast lenses are not so necessary nowadays film emulsions aremuch better at high ISOs, as are DSLR sensors I started with theoriginal Kodachrome with an ISO/ASA of 10 digital cameras don't goanything like so low nowadays..

Bertie.

Ah, now we are coming to the crux of the matter.What got me started wondering about this question was shooting at night.I like to go out and photograph in the afternoon, and evening, and whenit gets too dark, I have to switch to my old manual focus Sears 55mm F1.4lens, which works when every other lens I have is useless. (On an OIy E-510.).

But it still has problems with *really* dim light, so I find myself thinking about F1.2 or F1.0, and wondering what a really great lens for shooting in the dark would be like..

I mean, if I can still see, then a camera should be able to too, wouldn't you think?If you had a high enough ISO and a fat enough light-catcher....

And of course I can't help but notice that lenses faster than F1.2 are as common as hen's teeth..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #8

Jay Turberville wrote:.

Still, is there some factor in there that becomes close to zero asthe F-stop size approaches 1.0? I smell a divide-by-zero error. Itseems that it is impossible to make a lense with an F-stop ratingless than 1.0. I've never heard of such a thing..

Do a search of Joseph Wisniewski's recent posts. I think hediscussed this to some degree with a few other people. What I recallis that f/0.5 is as fast as a lens can be given it is made fromspherical glass. But that doesn't mean it will be very good. Justthat it is possible and that there is some mathematical reason whyyou can't go faster when using spherical surfaces..

Jay Turbervillehttp://www.jayandwanda.com.

Okay, thanks for the post. I'll track them down.Have a good day..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #9

Jay Turberville wrote:.

Do a search of Joseph Wisniewski's recent posts. I think hediscussed this to some degree with a few other people. What I recallis that f/0.5 is as fast as a lens can be given it is made fromspherical glass..

Actually, no assumption about the construction of the lens is needed, because the f/0.5 barrier isn't really a lens limitation as such. It is the limit imposed by the finite size (in the appropriate sense) of the bundle of light rays incident on the sensor..

Based on simple geometry, an f/0.5 lens might be expected to have a marginal ray angle of arctan(1) = 45 degrees. But the simple geometry is not correct at low f-numbers, and the marginal ray angle is actually arcsin(1) = 90 degrees. This means that all possible incidence angles have been used up, and there is simply no place to put any more light to make a faster lens..

Here's one of my older posts with more details and some relevant links:http://forums.dpreview.com/...forums/read.asp?forum=1032&message=20801807.

Incidentally, I believe there is an f/1.0 limit for EXIF information - it stores an APEX aperture value as an unsigned rational..

Alan Martin..

Comment #10

Terrance13 wrote:.

Hello. I am puzzled by what seems to be an inversion of the equationfor lens size and maximum aperture..

That is, it is often said that the F-stop rating of a lens is simplythe maximum aperture divided by the focal length..

Invert your above equation and it works better..

That seems to be at least partly correct....

No, it doesn't. .

Charlie DavisNikon 5700 & Sony R1HomePage: http://www.1derful.infoBridge Blog: http://www.here-ugo.com/BridgeBlog/..

Comment #11

Unless I'm forgetting something from my old math classes in high school, you'd only have a "divide-by-zero" error if you try to make an f/0.0 lens. Nothing breaks down mathematically with f-stops less than 1 (such as the f/.95, and f/.7 lenses mentioned).

You should check out some scenes in Kubrick's "Barry Lyndon". I believe he used a 50mm f/.7 lens. They had to get the actors to sit REALLY still as the DOF was so shallow at such a low f-stop..

Terrance13 wrote:.

Sherwoodpete wrote:.

Http://www.canon.com/.../camera-museum/camera/lens/s/data/50-85/s_50_095.html.

Http://www.pbase.com/elliot/image/45262012.

Thanks for the tip. Now that's one beautiful piece of glass.F 0.95 And obviously no divide-by-zero error where we all vanishinto a black hole... .

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #12

Http://www.visual-memory.co.uk/sk/ac/len/page1.htm.

What fantastic eye-candy Barry Lyndon is! Every scene looks like it was painted onto the film. The story's not bad either .

GordonGordon SolomonAssistant technical writer, dpreview.com..

Comment #13

Alan Martin wrote:.

This means that all possibleincidence angles have been used up, and there is simply no place toput any more light to make a faster lens..

Thanks for the input. Good stuff..

And another way to think of it is to realize that a really huge lens would have to bend incoming rays of light at almost a 90 degree angle to get them into the center of the lens. Alas, the laws of physics are unforgiving when you try to break them..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #14

Chuxter wrote:.

Invert your above equation and it works better..

Yes, I realized the error about 5 minutes after I posted it, and corrected it in the second post. The morning coffee wasn't working yet... .

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #15

100mm lens Max F/stop = f/2...........Aperture size = 50mm or 1/2 of the focal length..

The F/stop is really a fraction...F= the focal length....in this case 100mm.

2 = the factor divided into the focal length.....Hence F/2 = 100mm divided by 2 = the aperture diameter..in this case it is 50mm..

So f/2 for a 100mm is 50mm.

Now to confuse you more.........

F/2 for a 50mm = 25mm Diameter BUT LETS IN THE SAME (OR CLOSE) LIGHT AS THE F/2 ON THE 100mm lens AT F/2 that is a 50mm Diameter.

SO.....

F/4 on a 100mm = a 25mm Diameter or 1/4 the focal length 100/4F/8 on a 100mm = 12.5 Diameter or 1/8 the focal length...100/8and so on..

It is a bit more complex....but for simple understanding.....this works fine..

The Front element group is use mostly to gather light for the other other elements that do more of the focusing of that light..So it can be much larger that the max f/stop diameter. Again...the front element group is more complex in design and function..But for illustrative examples...it works for this thread..

Peter .

Image control:Zoom outZoom 100%Zoom inExpand AllOpen in new window.

Enjoy your photography images, even if your wife doesn't ! ;-(http://laurence-photography.com/http://www.pbase.com/peterarbib/Cameras in profile...

Comment #16

Terrance13 wrote:.

And another way to think of it is to realize that a really huge lenswould have to bend incoming rays of light at almost a 90 degree angleto get them into the center of the lens..

Sort of, but it's even worse. As the diameter of the aperture opening (the "entrance pupil") approaches twice the focal length (f/0.5), not only do the light rays at the edge have to be bent more to reach their target on the sensor, but the bending has to take place *closer* to the sensor plane..

To reach the limit, the bending would have to take place *at* the sensor plane, at a 90 degree angle. Obviously it is not possible to achieve this, let alone exceed it. And the lens wouldn't have to be really huge - an 8mm f/0.5 normal lens for a 1/2.5" sensor, if it could be made, would only have a 16 mm entrance pupil..

This is an idealization of course, since each light ray is actually bent numerous times, once at each lens element surface. But it's a reasonably good model of the "black box" behaviour of a lens; the place where the light rays are notionally bent is called the "rear principal plane". (Except that as J. Brian Caldwell said, it's actually a sphere, whose diameter of twice the focal length is responsible for the f/0.5 limit.).

Alas, the laws of physics are unforgiving when you try to break them..

Well, people have already found a way to "break" this one! It turns out that if the space between the lens and the sensor is filled with a liquid having a refractive index of n > 1, then the f/0.5 limit becomes 0.5 / n..

Lenses as fast as f/0.35 have been made for special applications such as optical microscopy and photolithography... but don't expect to see one on a digital camera anytime soon.  .

Alan Martin..

Comment #17

When I first started out about a year ago.. I couldnt figure out the significance of the standard F-stop sizes.. F4.0, F2.8, F1.4 etc,, then a friend of mine explained to me that the F-stop is simply ratio of the diameter of the aperature to the distance between the front and rear lens.. Using this knowledge I figured out why the f-stop numbers weren't integers or whole numbers... Here's how I figured it out:.

For an given lens, the length is fixed.. we'll call that Length L..

Using the above explanation for f-stop, for a f4.0 means:.

4d = L (where d is the diameter of the aperature).

Therefore:.

D= L/4.

We know that the Area of a circle is calculated by:.

A = pi * r ^2.

A = pi * (d/2)^2 of the aperature.

Therefore the area of the aperature opening at f4.0 can be calculated as:.

A' = pi * (L/8)^2A' = pi * (L^2)/64.

Now, Using the same equations, for an f2.8 on the same lens, L remains constant..

L = 2.8 * dd = L/2.8.

A'= pi * (d/2)^2A' = pi * (L/2.8)^2A' = pi * (L/5.6)^2A' = pi * (L^2)/32.

Now compare the Areas for the two f-stops:.

For f4.0 > A' = pi * (L^2)/64.

For f2.8 > A' = pi * (L^2)/32.

If you divide both equations by (pi * L^2), you see that.

A' = 1/64A' = 1/32.

Which means that for a given fixed lens, the aperature area for f4.0 is half the size as it is when it's set to f2.8. therefore, for a given fixed length lens, shot with the same shutter, speed, only half the amount of light gets through the lens, when set at f4.0, vice f2.8...

Comment #18

Thanks for a really illuminating post.No pun intended..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #19

Correct me if I'm wrong reg. SLR lenses and compact lenses:.

To get 35 mm full-frame equivalent on an APS-C or a compact your real focal length would be:APS-C (crop 1.6): 22 mmCompact: 7.5 mm.

If both cameras boast f/2.8 the absolute aperture (the diameter of the hole in mm) for both lenses would be:APS-C: 7.9 mmCOmpact: 2.7 mm.

Obviously (?) more light enters to the APS-C sensor, but is this light somehow in optic physics disappearing because of the longer focal length???.

Hope the question makes sense...

Comment #20

The APS sensor will get more light. The number of photons per unit of area will be the same, but the APS sensor is bigger and will collect more light..

Shoot...

Comment #21

What good is a wide aperture if your DOF is razor thin? you have to consider this when you're out shooting stuff.

Having such shallow DOF requires proper setup of your objects, a friend standing sideways will have only part of his/her body in focus if your taking a picture of them lets say 10 feet away or so at 1.2 or 1.4.

You will be forced to run smaller apertures just to get the correct DOF, otherwise you're wasting time...

Comment #22

Gooshin wrote:.

What good is a wide aperture if your DOF is razor thin? you have toconsider this when you're out shooting stuff.

Oh, I know. I mostly do long shots at night, riverscapes and cityscapes, at infinity focus or very close to it..

Trying to focus close yields pictures with only the middle of something in focus.That's no fun..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #23

Shejken wrote:.

Correct me if I'm wrong reg. SLR lenses and compact lenses:To get 35 mm full-frame equivalent on an APS-C or a compact your realfocal length would be:APS-C (crop 1.6): 22 mmCompact: 7.5 mm.

Obviously (?) more light enters to the APS-C sensor, but is this light somehowin optic physics disappearing because of the longer focal length???.

Hope the question makes sense..

Yes, the question makes sense. But there is some confusion there. I was confused just the same way at first, too..

The focal length of the lens never changes. You can put a 300mm lens (assume full frame, maybe a 35mm film camera legacy lens) on a full-frame Canon or on an APS-C camera or on a 4/3 Olympus, and it's still a 300mm lens..

Only the field of vision changes. The Oly 4/3 camera uses only the center part of the image circle that the lens creates, and enlarges it to "full screen", so that the lens appears to have more magnification than it really does. But the focal length is still a constant..

Where the confusion creeps in is where some people say that a 300mm lens on a full frame camera is equivalent to a 600mm lens on an Oly 4/3 camera. No, it isn't. It's still a 300mm lens. Only the Field Of View (FOV) changes. So a better way to describe the situation would be to say that "A 300mm full-frame lens, when mounted on a 4/3 camera, gives an FOV equivalent to a 600mm lens on a full-frame camera.".

Now you can argue that the Oly camera is wasting a lot of light because it isn't using the light in the part of the picture that doesn't land on the sensor. True, but the legacy lens was larger to start with (compared to a Zuiko Digital lens), and gathered more light, and distributed it in a larger circle. The extra glass and extra photons are pretty irrelevant. They neither increase nor decrease your light intensity on the 4/3 sensor..

Now the previous poster is correct when he says that the APS sensor will gather more light. And a full-frame sensor will capture even more light. Obviously, if you make your sensor four times as large, thus making each pixel on it four times as large, each pixel will get more photons..

** 'Now I know what it's like to be high on life.** It isn't as good, but my driving has improved.'** == Nina, on 'Just Shoot Me', 13 Jan 2006...

Comment #24

Shejken wrote:.

Correct me if I'm wrong reg. SLR lenses and compact lenses:To get 35 mm full-frame equivalent on an APS-C or a compact your realfocal length would be:APS-C (crop 1.6): 22 mmCompact: 7.5 mm.

If both cameras boast f/2.8 the absolute aperture (the diameter ofthe hole in mm) for both lenses would be:APS-C: 7.9 mmCOmpact: 2.7 mm.

Obviously (?) more light enters to the APS-C sensor, but is thislight somehow in optic physics disappearing because of the longerfocal length???.

There's nothing mysterious going on, it's just that when you say that a larger opening gathers more light, you mean light flux in lumens (for a given field of view) - while the common claim that "f/2.8 gives the same amount of light on any camera" is only true for light *intensity* in lux (ignoring transmission losses)..

Light intensity (illuminance) is light flux per unit area (1 lx = 1 lm/m), so it takes more total light flux to illuminate a larger sensor to the same intensity. That means more photons per second, assuming the same spectrum - and indeed, much of the image-quality advantage of a larger sensor comes down to "more photons"..

(The light flux through an aperture is also proportional to the field of view from which the light is accepted, and this cancels out the difference in absolute aperture when different focal lengths are used on the *same* camera.).

Alan Martin..

Comment #25

Terrance13 wrote (with one small correction, see below):.

Where the confusion creeps in is where some people say that a 300mmlens on an Oly 4/3 camera is equivalent to a 600mm lens on a full framecamera. No, it isn't. It's still a 300mm lens. Only the Field Of View (FOV)changes. So a better way to describe the situation would be to say that"A 300mm full-frame lens, when mounted on a 4/3 camera, gives an FOVequivalent to a 600mm lens on a full-frame camera.".

The former way of saying it is fine too, if it is mentioned or understood that the equivalence is only for FOV. What causes confusion is if that part is left out, or worse, if "equivalent" is misinterpreted to mean that a 300mm lens on a 4/3 camera somehow "is" 600mm. (BTW, you had it reversed, but I corrected it in the quote above.).

Also, your alternative is more complicated than it needs to be. It doesn't have to be a "full-frame lens" - the equivalence applies equally to a genuine 4/3 system lens (it's the sensor size that makes the difference). And there is no need to say that the FOV is "equivalent" when it is actually equal (on the diagonal)..

But the real value of equivalence is that it applies to many more things besides FOV - including DOF / background blur, diffraction, and total light captured (in lumen-seconds or photons) - *if* the 35mm format camera is operated at twice the f-number and four times the ISO of the 4/3 camera..

Alan Martin..

Comment #26

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This question was taken from a support group/message board and re-posted here so others can learn from it.

 

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