# Why smaller AoV APS-C has same aperture as larger AoV FF lens?

I am quite confused,

FF focal length 50mm f/2.0 has aperture diameter 25mm

APS-C lens 50mm f/2.0 has aperture diameter 25mm

Then both lenses have same size front glass.

However Angle of View for lens with focal length 50mm of on FF camera = 40° APS-C camera = 27°

If we talking about manual focus lens from FF camera i can use it on APS-C camera then i will get different field of view so about 33% of outer side of the lens will not be used?

Then why aperture for APS-C has to be 25mm, it can as well be 33% less = 16.5mm?

Perhaps i lack understanding how image is sent trough the different glass in lens, and only difference between APS-C lens and lens for FF is different circle of projection is provides?

But its a fact APS-C lenses are cheaper than FF lenses, so they must have less glass somewhere, if not front glass then what glass is smaller on 50mm APS-C lens versus 50mm FF lens?

EDIT: To clarify what i am asking is what physical difference is between construction (glass configuration) of the APS-C lens vs FF/35mm lens. Because i know from testing APS-C lens projects smaller Circle of Projection (CoP) than FF/35mm lens. It must have either less diameter of glass or glass curved differently to provide smaller CoP?

The difference in angle of view is due to the smaller image sensor size, not the lens or it's aperture.

APS-C lenses can be made cheaper and smaller because their image circle only has to cover the smaller image sensor on APS-C cameras.

• I already know they project different circle of projection, wanted to know what is actual physical difference in construction. Thanks though. Commented Jan 20, 2022 at 10:01

Angular Field and Angle of View Lens makers use the term “angular field”, which is the angle as measured from the center (axis) of the lens. Thus, a lens with an angular field of 20° to the photographer will be an angle of view of 40°.

TV sets are sold by their corner-to-corner (diagonal) measure because this is the largest dimension of the rectangular screen size. Likely the diagonal is less useful than the length measurement, however the diagonal being larger is the more impressive value.

If we consider the 35mm full frame format, a 50mm lens delivers 27° vertical, 40° horizonal and 46° diagonal. It’s the larger diagonal angle of view that gets published. These are important facts because they tell us the span of the field that will be included in the image. The published angle of view is only valid when the subject is at infinity; when imaging near objects, the field is smaller.

As you have been told, the diameter of the iris (aperture) has only a small effect on the angle of view. Now all lenses project an image of the outside world onto the surface of film or digital sensor. As an explement, you can demonstrate this for yourself. Hold most any lens near a white piece of paper, and with a little finagling, you can adjust paper-to-lens distance and create a projected image of the outside world.

You will discover that this projected image is circular. Close examination will reveal that only the central portion of this image is useful. This image is brightest at the center, dimming and becoming fuzzy the further from center you look. The photographically useful inside portion is called the “circle of good definition”. The falloff is gradual; thus camera makers hide the edges with a rectangular mask.

The size of this squared-off opening is called the format size. For the full frame, the rectangle measures 24 by 36mm with a diagonal of 43.3mm. For the APS-C the format measurements are 16 by 24mm with a diagonal of 28.8mm. Mount a 50mm lens on an APS-C and the angle of views are 18° vertical 27° horizontal 32° diagonal.

Changing the diameter of the iris (aperture) does slightly alter the angle of view. This is because the image projected by the lens is sharpest and brightest at its center. As you stop down the iris (aperture) the circle of good definition expands slightly because the fringes of the camera lens have the greatest curve (figure), thus likely guilty to project substandard light rays.

Continued stopping down to tiny aperture diameters induces degraded light rays. This is because a higher percentage of the arriving rays are brushed by the iris blades. These near misses induce interference and diffraction -- twin plagues that degrade the image. To avoid, the camera maker masks off the circle of good definition forming the format dimensions. Thus, the mask focuses the camera to see only the central portion of the projected image (the circle of good definition) is thus circumscribed, and the peripheral rays are discarded.

• Thus, a lens with an angular field of 20° to the photographer will be an angle of view of 40°. I don't get that, guess i am more of a visual learner. Wish you provided a diagram like this. Thank you for explanation of how lenses work in general, but what i wanted to know is what actual physical difference in glass configuration between lenses for 35mm (full frame) & APS-C formats that is causing APS-C lenses to reflect circle of projection that is smaller in diameter. Commented Jan 20, 2022 at 10:25
• It's simple geometry! From the rear nodal, a triangle is constructed. Its height is the focal length. The hypotenuse is the side from rear nodal to edge of film / sensor. The base therefore 1/2 of the film / sensor measurement. A right triangle is the result. It is easy to work a right triangle using trigonometry. After all sides and angles are discovered, you multiply the angles by 2. Don't forget, all this math was before computers and calculators. They used trig tables and slide rules to ease the drudgery of the math. Tables and slide rule and pen and pencil all they had. Commented Jan 20, 2022 at 15:10
• Angle of view (in degrees) = 2 ArcTan( sensor width / (2 X focal length)) * (180/π) Note: If your calculator is working in radians, you need the (180/π) if in degrees omit, For those with a modern scientific calculator with ArcTan function. Rember all these terms and math 100 + years old -- only pen and pencil and math tables look up trig tables you will see the degree of difficulty. If we can see further than other men, it's because we stand on the shoulders of giants. Commented Jan 20, 2022 at 16:43

The lens' angle of view (AoV) is the same regardless of what camera it is on. The system's AoV is the combination of the lens' AoV and the sensor's AoV.

"If we talking about manual focus lens from FF camera i can use it on APS-C camera then i will get different field of view so about 33% of outer side of the lens will not be used?"

That part is not correct, all of the same objective area is used for the same exposure (f#) in both cases. All of the light required to make a complete image exists at all points on the objective element... and every part of the objective element used contributes light to all areas of the image at the image plane (as restricted by aperture).

Aperture area/size is about exposure/luminous density, it is not about AoV... even the smallest aperture records the same scene. It is easier to understand aperture area as the stacking of multiple images/exposures, from different areas of the objective lens, onto the image plane... the size of the image plane is irrelevant in this aspect.

If you take the FF/50mm image and crop it to APS size in post the remaining AoV changes, but the exposure doesn't... that's exactly the same as what happens when you crop the lens' image circle by using an APS sensor instead.

The size of the projected image circle can be changed in a number of ways. When designing a lens for a specific image circle size the distance from the image plane can be varied. This actually changes the lens' focal length, and reduces the resulting exposure density (<f#) due to the inverse square law (spread of light; this is also called bellows factor).

When the lens is designed to create a larger image circle at (nearly) the same distance it's angle of view is altered... this is why a longer lens is considered "normal" on larger formats. In this case the smaller format is cropping the larger image circle; and it is essentially what is happening in your example.

But, in order to eliminate/absorb unwanted light (because it will not be used), internal or external baffles can be used to restrict the projected image circle size and shape. This is common as it allows a lens to be sharper nearer the corners; and because stray light is not good for IQ.

Note that aperture/objective area(size) has no effect in any of these drawings.

This is an example of an external baffle. But the "baffle" could actually just be the lens barrel with a recessed element, or the last "lens" may actually just be protective glass with a baffle behind/around it.

• The lens' AoV doesn't need to be as wide for an APS-C lens. The entrance pupil need only be visible from the front of the lens at the angles needed to provide the smaller image circle. Light from wider angles needed to provide a larger image circle do not need to be collected by the front of the lens. f-number is determined by the size of the e.p. when viewed from along the lens' center optical axis. It has nothing to do with whether or not the e.p. is visible from angles outside the designed AoV projected by the exit pupil. Commented Jan 19, 2022 at 19:02
• @MichaelC, light from wider angles are not required to create a larger image circle. The lens' AoV and FL together dictate the size of the image circle... At its' largest possible size, the entrance pupil is the objective element at the front of the lens. And varying (restricting) the size of the entrance pupil ultimately dictates the exposure of said image circle. Commented Jan 19, 2022 at 20:33
• If the focal length is the same, a larger image circle means light from a wider angle of view needs to be collected. Commented Jan 19, 2022 at 22:20
• @Steven Kersting Thanks for clarification could you provide link to read about it with some visual examples, provided its not a optical physics textbook :) Also please see updated question in my original post. Commented Jan 20, 2022 at 10:47
• @LilBro, I've edited the answer for the updated question. Commented Jan 20, 2022 at 16:19

A lot of long, detailed, technically correct answers already, so I'll try to add a simple explanation.

TL;DR The sizes of the lens elements have some influence on the vield of view, but not the way you describe in your question.

The lens opening diameter does not determine the field of view. Imagine a (horrible-quality) single-element lens with 5mm diameter and a 50mm focal lens. This lens will happily project a ray coming from 45 degrees onto the image plane 50mm behind the lens, at a spot 50mm from the sensor center. This is just basic optics, that rays directed to the center of a single element will pass through it in a straight line.

For this to happen, it doesn't matter whether the element has a 1mm, 5mm, or 50mm diameter. This will happen even for insanely high angles, so the lens has a virtually unlimited vield of view. What limits the field of view is how much of the (unlimited) projected image you catch with your sensor. A bigger sensor catches more, so it results in a bigger field of view.

Then, what is it that differentiates an APS-C lens from an FF lens?

Real-world lenses aren't single-element ones, but long, complex combinations of multiple elements, so typically our 45 degree incoming ray won't make it through all the elements. It will hit the lens housing somewhere before reaching the rear element. With smaller angles, only parts of the incoming rays will pass through, resulting in a darker image ("vignetting"), or they won't perfectly focus on one spot ("blurring"). So, every lens has a maximum angle up to which it works as expected. Trying to get a larger field of view out of the lens will give disappointing results like dark and/or blurry edges (don't use an APS-C lens with a full frame body).

An FF lens is one that supports a larger field of view for a given focal length than the same-focal-length APS-C lens. To achieve that, most of the elements have to be bigger than in the APS-C one, to allow for rays with a bigger angle to pass through instead of hitting the wall. (And, having to cover a wider range of angles with good quality is an additional challenge.)

One element in the lens is designed to be the limiting one, deciding which rays to collect and which to reject, and all the others have to be big enough not to clip any of these rays. (The limiting element also is the one that decides about the lens speed.)

Let's assume the front element is the limiting one (e.g., a 200mm f/4.0 lens will need a 50mm front diameter). Then the other elements need to be larger if you want to allow for a wider range of angled rays (= field of view), and the biggest size increase is needed for the elements furthest away from the limiting element, being the rear one in our case.

## Short version

For an APS-C sensor, the equivalent of a full-frame lens of focal length F and maximum aperture N has a focal length F/1.5 and a maximal aperture N/1.5 (ie the equivalent of a FF 50mm f2.8 would be an APS-C 33mm f1.8).

A comparison, taking into account this equivalence, of existing objectives shows that the results are not as clear as one might think. However, this type of comparison is difficult.

## Long version

The area to be covered by the image projected in the plane of the sensitive surface is smaller in the case of an APSC-C sensor than in the case of a 24mm x 36mm sensor. As a result, APS-C lenses designed to cover smaller sensor than full-frame ones can therefore have a smaller size. But when a same lens is used on a 24mm x 36mm sensor and a APS-C one, the resulting image are different.

Thus, the first step is to determine what is to be compared. The second step will be how to compare. The last step will be to make some comparisons.

## I. What to compare

### Angle of view

The angle of view is the angle between the lateral limits of the field embraced by the view on the sensor. When the size of the sensor decreases the angle decreases. It doesn't matter if part of the image projected by the lens is unused and located outside the sensor.

Therefore, for an APS-C lens to cover the same angle of view as a 24mmx36mm lens its focal length must be different. It is easy to show that the equivalent focal length is F/1.5, 1.5 being the ratio between the diagonal lengths of the two types of sensors.

In the following we will assume that this focal length equivalence is taken into account

### Aperture (illumination)

The aperture of a lens is related to the illumination of the surface on which the image is projected: the photographic film or the sensor.

The aperture, the "aperture diameter" and the focal length are characteristics of the lens regardless of the size of the sensor of the camera.

Instead of "aperture diameter", we should speak of entrance pupil which is the image of the aperture diaphragm through the upstream part of the optical system.

These three values are linked by the definition of the aperture number given by the formula aperture number = diameter of the entrance pupil / focal length.

### Confusion circle / depth of field

#### Confusion circle

A photograph is not seen by (or on) the sensor[*] but usually on a print or a screen. Moreover the human eye has limits in acuity.

From the minimum angle that allows the eye to distinguish two points close together on the print, it is possible to determine the size below which a circular spot on the sensitive surface will be perceived as a point (the diameter of this spot is the circle of confusion). It must be the smaller the magnification factor of the sensitive surface to the print is large.

To have the same print size between an APS-C sensor and a full frame sensor, the magnification factor must be multiplied by 1.5. As a result the circle of confusion for an APS-C sensor is 1.5 smaller than for a full-frame sensor.

#### Depth of field

The depth of field corresponds to the range of distance in which a point of the scene will give on the sensitive surface a spot with a diameter smaller than the circle of confusion.

From the above, it can be shown for an equivalent focal length (F/1.5), the equivalent aperture to obtain the same depth of field with an APS-C lens is equal to N/1.5, N being the aperture of the equivalent full frame lens.

In the following we will assume that this aperture equivalence is taken into account

#### Other points

Photographic lenses come with different characteristics, the most obvious are the presence or not of a stabilization mechanism, or of an autofocus mechanism. These two mechanisms usually involve dedicated optical elements.

Moreover, the lenses can have other optical elements to correct the inevitable optical aberrations.

Finally, the types of lenses used can be different, even talking about rarer solutions such as elements taking the form of Fresnel lenses.

## II. How to compare

(to be eventually completed) It would undoubtedly be desirable to compare the lenses according to their optical element. However such a comparison, in addition to the important work that it supposes, finds its limit indeed the optical formulas are never completely identical. This is why the comparison retained at this stage is very rough and based only on the weight of the lenses. A good part of this weight is that of the optical elements indeed

## III. Comparison

(to be eventually completed)

In the following, the equivalences introduced above are taken into account for each pair of objectives.

Canon EF 70-200mm F4L IS II USM: Weight 780 g
Fujifilm XF 50-140mm F2.8 R LM OIS WR: Weight 995 g

Canon EF 85mm F1.8 USM: Weight 425 g
Fujifilm XF 56mm F1.2 R: Weight 405 g

Voigtländer 50 mm / 1:2,0 APO-Lanthar aspherical VM: Weight 288 g
Voigtländer Nokton 35mm F1.2 X mount: Weight 196g


## Conclusion

Taking into account the equivalences, data of existing lenses shows that the results are not as clear as one might think.

[*]: There is a different approach to the one mentioned above for the sharpness. In this approach what is sought is not the perception of the image by the observer taking into account the magnification and the visual acuity but more simply the production of an image file as sharp as possible. In this case the limiting factor is no longer the visual acuity of the observer but the size of the sensor pixels. Some cameras of the Fujifilm X range allow to define the range of sharpness either from the photographic approach or from this last approach.

Then why aperture for APS-C has to be 25mm?

Because the f-number is a measure of focal length divided by the diameter of the entrance pupil. That is, the focal length of the lens divided by how wide the aperture looks when viewed from the subject's position in front of the lens.

But its a fact APS-C lenses are cheaper than FF lenses, so they must have less glass somewhere, if not front glass then what glass is smaller on 50mm APS-C lens versus 50mm FF lens?

The lenses that determine the exit pupil. That is, the lenses near the rear of the lens that project the image that can be seen from the rear of the lens.

Because I know from testing APS-C lens projects smaller Circle of Projection (CoP) than FF/35mm lens. It must have either less diameter of glass or glass curved differently to provide smaller CoP?

It must be the latter and is usually also the former.

If we only change the refractive index of the last lens element and leave everything else the same, then we also change the focal length of the entire lens. If we have spread the same angle of view out over a larger image circle we've increased magnification and thus increased the focal length. If we have concentrated the same angle of view into a smaller image circle we've reduced magnification and thus decreased the focal length. Of course, if we change the focal length but have the same entrance pupil size at the front of the lens, we've also changed the f-number because we're dividing a different focal length by the same entrance pupil diameter.

So the entire lens system, from front to back, has to be designed so that the size of the image circle projected also results in the desired focal length and f-number.

Beyond that, though, APS-C lenses can get away with fewer corrective lenses for the classic optical aberrations. Aberrations tend to be worse as one moves further from the center of the image circle projected by a lens. Since APS-C lenses project a smaller image circle than full frame lenses, they don't need the same level of corrective optics to make the edges and corners perform at a certain level compared to the corrective optics needed for the same edge and corner performance with the larger image circle projected by a full frame lens.

These aberrations are not caused by defective manufacturing of lens elements that need to be corrected. They're caused by the nature of light and how different wavelengths are refracted by different amounts when passing through the same refractive medium.

Compare the block diagram of the Canon EF-S 55-250mm f/4-5.6 IS STM:

to the block diagram of the Canon EF 70-300mm f/4-5.6 IS II:

The front and middle groups are similar, but the elements in the rear groups of the full frame lens are larger, relative to the size of the front elements, to allow projecting a larger image circle. Notice also the difference between the last element of the APS-C lens and the FF lens. Both are meniscus lenses, but the one in the APS-C lens is a positive meniscus turned one way, which narrows the cone of light it projects while the last element of the FF lens is a negative meniscus turned the other way which enlarges the cone of light it projects.

Both of these lenses are what one would consider "consumer grade", and they each perform about the same optically when either is mounted on the same APS-C camera.

Now compare both of those to the more corrected, and more expensive, EF 70-300mm f/4-5.6 L IS:

Notice how many more corrective elements there are in the rear group of the "luxury" lens?

Each lens has the following number of lens elements and groups:

EF-S 55-250mm f/4-5.6 IS STM - 15 elements in 12 groups
EF 70-300mm f/4-5.6 IS II - 17 elements in 12 groups
EF 70-300mm f/4-5.6 L IS - 19 elements in 14 groups

All three lenses extend/retract the front barrel to zoom and to focus and share the same basic designs including the locations of each's respective IS group and focusing group.

Note: The block diagrams are not drawn to the same scale. The EF-S lens has 58mm filter threads, while the other two lenses have 67mm filter threads.

• Great answer wish you provided more details though regarding the optical configuration differences between lenses for APS-C and FF/35mm formats. Commented Jan 20, 2022 at 10:49