Is there a theoretical limit to aperture?

If you want to know about theoretical possibilities, it is instructive to look at microscope lenses. Resolution limits been very well worked out for these lenses, and there is probably a lot less misinformation than for camera lenses. They also operate much closer to theoretical limits than camera lenses do.

The theoretical resolution limit with microscopes is 0.61 * lambda/NA at the object plane, where NA is the index of refraction times the sine of the half angle. That certainly applies to a flat field and a flat image. The front of a very-high-NA lens has to be flat or nearly as well, for practical reasons. A camera lens is like an infinity-corrected microscope lens, except that the object and image are reversed. The resolution limit applies to the microscope at the object plane and to the camera at the image plane, so the same formula still applies.

A well-corrected 40x objective with NA=0.85 is not uncommon. That's f/0.3. Some 60x objectives have NA= 0.95. That's f/0.16. The limit in air is 1.00, because that's the maximum value of the sine. That corresponds to f/0. As JACS pointed out, that corresponds to collection of all the light over a hemisphere. You cannot collect any more of the light unless the objective surrounds the subject, and that's the geometrical reason for the limit. Water- and oil-immersion lenses can have higher NA's. NA of 1.40 is common for oil immersion lenses.

These lenses are small and highly corrected for with open aperture at one distance only, with a specific cover glass thickness. Conventional camera lenses are large, less highly corrected, and they must operate over wide variations in distance and apertures. That's why lenses of f/0.95 are pretty fuzzy, and lenses faster than that are extremely rare. I think I remember aspherical condenser lenses with about f/0.5, but they give extremely poor images. With optical techniques using new concepts, perhaps extremely fast lenses could be made, however.

ThrillaMozilla said:

If you want to know about theoretical possibilities, it is instructive to look at microscope lenses. Resolution limits been very well worked out for these lenses, and there is probably a lot less misinformation than for camera lenses. They also operate much closer to theoretical limits than camera lenses do.

The theoretical resolution limit with microscopes is 0.61 * lambda/NA at the object plane, where NA is the index of refraction times the sine of the half angle. That certainly applies to a flat field and a flat image. The front of a very-high-NA lens has to be flat or nearly as well, for practical reasons. A camera lens is like an infinity-corrected microscope lens, except that the object and image are reversed. The resolution limit applies to the microscope at the object plane and to the camera at the image plane, so the same formula still applies.

A well-corrected 40x objective with NA=0.85 is not uncommon. That's f/0.3. Some 60x objectives have NA= 0.95. That's f/0.16. The limit in air is 1.00, because that's the maximum value of the sine. That corresponds to f/0. As JACS pointed out, that corresponds to collection of all the light over a hemisphere. You cannot collect any more of the light unless the objective surrounds the subject, and that's the geometrical reason for the limit. Water- and oil-immersion lenses can have higher NA's. NA of 1.40 is common for oil immersion lenses.

These lenses are small and highly corrected for with open aperture at one distance only, with a specific cover glass thickness. Conventional camera lenses are large, less highly corrected, and they must operate over wide variations in distance and apertures. That's why lenses of f/0.95 are pretty fuzzy, and lenses faster than that are extremely rare. I think I remember aspherical condenser lenses with about f/0.5, but they give extremely poor images. With optical techniques using new concepts, perhaps extremely fast lenses could be made, however.

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The problem with microscope lenses is that the object plane and the image one are switched compared to photo ones. In the object plane, rays focus close to the focal point and in the image one, they are almost parallel (compared to the rays on the other side). With photo lenses at normal distances, it is exactly the opposite.

I believe that the explanation provided by cpw and alanr0 is right, and I want to thank them for their patience.

ThrillaMozilla said:

The theoretical resolution limit with microscopes is 0.61 * lambda/NA at the object plane, where NA is the index of refraction times the sine of the half angle.

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Hi Thrilla,

Yes, that's the well known Rayleigh diffraction limit, which occurs when two points are separated by the distance to the first zero of the Airy disc. In photography we write it 1.22*lambda*N, with N the f-number.

ThrillaMozilla said:

A well-corrected 40x objective with NA=0.85 is not uncommon. That's f/0.3. Some 60x objectives have NA= 0.95. That's f/0.16. The limit in air is 1.00, because that's the maximum value of the sine. That corresponds to f/0.

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Notation of the above for photography would be f-number N = 0.59, 0.53 resp. and 0.5 for the limit.

ThrillaMozilla said:

These lenses are small and highly corrected for with open aperture at one distance only, with a specific cover glass thickness. Conventional camera lenses are large, less highly corrected, and they must operate over wide variations in distance and apertures. That's why lenses of f/0.95 are pretty fuzzy, and lenses faster than that are extremely rare. I think I remember aspherical condenser lenses with about f/0.5, but they give extremely poor images. With optical techniques using new concepts, perhaps extremely fast lenses could be made, however.

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Yes, but no faster than N = 0.5 in air, just as the limit for NA is 1.

Jack

J A C S said:

The problem with microscope lenses is that the object plane and the image one are switched compared to photo ones. In the object plane, rays focus close to the focal point and in the image one, they are almost parallel (compared to the rays on the other side). With photo lenses at normal distances, it is exactly the opposite.

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Why is that a problem? The same principles apply.

J A C S said:

I believe that the explanation provided by cpw and alanr0 is right, and I want to thank them for their patience.

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Indeed.

Jack Hogan said:

Yes, that's the well known Rayleigh diffraction limit, which occurs when two points are separated by the distance to the first zero of the Airy disc. In photography we write it 1.22*lambda*N, with N the f-number.

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Yes, I've seen that, but I've usually seen the f number defined as f/lens diameter. Is the latter just a demented approximation? Sorry if you have answered this already. I haven't read and understood everything in this thread.

Ah, never mind, I see, it's covered earlier by a quotation from a textbook. One more piece of misinformation to unlearn.

Incidentally, the light-gathering capability of a fast lens is limited somewhat by reflections at shallow angles, but in fact the performance can be quite good. In polarized light microscopy it is common to observe the back focal plane of the objective, and there you can see that light transmission is quite good all the way out to the edge of the objective.

J A C S said:

The problem with microscope lenses is that the object plane and the image one are switched compared to photo ones.

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Not a problem. In a microscope the Rayleigh diffraction limit applies to the object plane; in a camera the limit applies to the image plane.

Jack Hogan said:

J A C S said:

The problem with microscope lenses is that the object plane and the image one are switched compared to photo ones. In the object plane, rays focus close to the focal point and in the image one, they are almost parallel (compared to the rays on the other side). With photo lenses at normal distances, it is exactly the opposite.

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Why is that a problem? The same principles apply.

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With a 50mm lens, for example, with you main subject at 10m, you collect a tiny portion of the light coming. Increase the physical aperture 10x, and you collect (almost) 100x the light.

With a 50mm lens focused at 50mm, you already collect a significant portion of the light coming from the microscopic object. Increase the aperture 10x, you will get a few times more light but not anywhere close to 100x (compare the solid angles).

ThrillaMozilla said:

J A C S said:

The problem with microscope lenses is that the object plane and the image one are switched compared to photo ones.

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Not a problem. In a microscope the Rayleigh diffraction limit applies to the object plane; in a camera the limit applies to the image plane.

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I was not talking about the diffraction limit.

J A C S said:

With a 50mm lens, for example, with you main subject at 10m, you collect a tiny portion of the light coming. Increase the physical aperture 10x, and you collect (almost) 100x the light.

With a 50mm lens focused at 50mm, you already collect a significant portion of the light coming from the microscopic object. Increase the aperture 10x, you will get a few times more light but not anywhere close to 100x (compare the solid angles).

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Are you sure about that? I'm not, because of the shallow angle at which off-axis rays strike the detector. But to be honest, I don't want to puzzle it out to be able to argue the point.

ThrillaMozilla said:

J A C S said:

With a 50mm lens, for example, with you main subject at 10m, you collect a tiny portion of the light coming. Increase the physical aperture 10x, and you collect (almost) 100x the light.

With a 50mm lens focused at 50mm, you already collect a significant portion of the light coming from the microscopic object. Increase the aperture 10x, you will get a few times more light but not anywhere close to 100x (compare the solid angles).

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Are you sure about that? I'm not, because of the shallow angle at which off-axis rays strike the detector. But to be honest, I don't want to puzzle it out to be able to argue the point.

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Me neither (maybe Alan?) but I thought reciprocity ruled. We may not like the result but...

ThrillaMozilla said:

J A C S said:

With a 50mm lens, for example, with you main subject at 10m, you collect a tiny portion of the light coming. Increase the physical aperture 10x, and you collect (almost) 100x the light.

With a 50mm lens focused at 50mm, you already collect a significant portion of the light coming from the microscopic object. Increase the aperture 10x, you will get a few times more light but not anywhere close to 100x (compare the solid angles).

Click to expand...

Are you sure about that? I'm not, because of the shallow angle at which off-axis rays strike the detector. But to be honest, I don't want to puzzle it out to be able to argue the point.

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I am not even talking about the detector. Think about the light hitting the front element of the lens.

BTW, in the first scenario, a 50cm front element would still have almost perpendicular rays hitting the front element. In the second one, not. If you want to think about the angles at the lens, fine, and this in fact confirms what I said in the microscope scenario but I prefer to think about the solid angle. Just a different way to measure the same thing.

J A C S said:

I am not even talking about the detector. Think about the light hitting the front element of the lens.

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Well I'm talking about the detector, and I suspect that blows away the difference.

And as for the light hitting the front element, it doesn't matter. If it's not hitting the front element at a shallow angle, it's leaving the back element at a shallow angle. And the angle of incidence on the surface is actually shallower for the microscope configuration, so if there is a difference, it's worse for the microscope. But I already mentioned that you can observe the back focal plane directly, and the transmission through microscope lenses at these low angles is visually very good. Otherwise a 60x/0.95 lens would not be possible.

ThrillaMozilla said:

J A C S said:

I am not even talking about the detector. Think about the light hitting the front element of the lens.

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Well I'm talking about the detector, and I suspect that blows away the difference.

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But that is not under discussion here. How much can the detector detect of that light was never asked. When the angle is very small, the light is spread over a larger area, so nothing is lost unless the detector cannot detect it well. There is energy conservation after all (see also below) unless you do not want to have it.

ThrillaMozilla said:

And as for the light hitting the front element, it doesn't matter. If it's not hitting the front element at a shallow angle, it's leaving the back element at a shallow angle.

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That does not matter, energy conservation. The phase volume is preserved. Small angle, but large cross section (parallel to the sensor).

ThrillaMozilla said:

And the angle of incidence on the surface is actually shallower for the microscope configuration, so if there is a difference, it's worse for the microscope.

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Yes, that is what I am saying, for the microscope is worse.

ThrillaMozilla said:

But I already mentioned that you can observe the back focal plane directly, and the transmission through microscope lenses at these low angles is visually very good. Otherwise a 60x/0.95 lens would not be possible.

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So there is no problem with small angles after all?

With small angles, there would be greater loss of light due to reflection (which is supposed to be minimized by the coating) and the fact that the light does not propagate along rays only. But for the purpose of this discussion, let us stay away from it.

Since I'm no longer certain what the question is, I don't want to conjecture more about it.

I do note, however, that the equations for reflections from a surface are symmetrical, so it doesn't matter which direction the light is going. The same principles apply to a microscope lens as to a camera lens, and that equivalence also applies to light transmission.

Interesting observation you made about the solid angles. As I said, I suspect it makes no difference, but I can't argue the case definitively.

ThrillaMozilla said:

Since I'm no longer certain what the question is, I don't want to conjecture more about it.

I do note, however, that the equations for reflections from a surface are symmetrical, so it doesn't matter which direction the light is going. The same principles apply to a microscope lens as to a camera lens, and that equivalence also applies to light transmission.

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Light coming from a point source in microscopy, after it passes through the lens, creates a parallel (almost) family of rays along the axis with intensity decreasing going away from the axis. It fact, that intensity density is integrable and the total integral for a hypothetical (and impossible to exist) infinitely fast flat lens is half of the total light from the source.

Light coming from a very distant object hits the lens almost perpendicularly and has almost constant intensity over a large area (say, 50cm) that is still much smaller compared to the distance to the object (say, 10m). On the image plane, a 50cm aperture with FL=50mm (such a lens cannot exist) would have almost half of the solid angle.

So the difference is that in the two cases, the intensities are quite different.

Detail Man said:

cpw said:

J A C S said:

Jack Hogan said:

Knoxis said:

...there is technically no theoretical limit to how big aperture can get. You could get an F0.01 lens if the money and resources were available.

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Joe is correct, the physical lower limit on f-number (N) is 0.5. The reason is that f/D is only an approximation valid when the opening angle theta' is small. The actual definition of f-number in air is

N = 1/[2sin(theta')]

from which it becomes obvious that N can never be less than 0.5, as better explained by Nakamura .

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This does not take the index of refraction into account and does not explain why we cannot have a lens faster than 1/2 in f/D sense. Like a 50mm (single element) lens with 1m diameter. I believe the answer is that such a lens would not be able to focus rays in an acceptable way but I have not seen a good exposition. For a single lens element that should be doable but I am not sure about a multiple element one.

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Hi, Jack, Nakamura correctly writes down the formula for F#, but as JACS points out doesn't say where it comes from. It comes from a well corrected lens satisfying the Abbe sine condition, thereby making the principal surface of the lens curved (spherical).

Most of the time, I see lenses being drawn with flat principal surfaces, but as you go faster, the curvature of the principal surface begins to show up (brian used to write about this back in the day). So you can easily draw the situation as follows: draw a horizontal line as your optical axis, and mark the image focus point on it on the right. Then take a compass, set it to your focal length f, and centered on the focus point, sweep out an arc above and below the axis. Now, draw a line parallel ...

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Did you possibly intend to (directly above) write "draw a line perpendicular ..." ?

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Chris, thanks for your reply in your post here: http://www.dpreview.com/forums/post/57371258

Detail Man said:

Detail Man said:

... to the axis and a distance r away from the axis. r is = d/2, the radius of your entrance pupil. Now, mark the point where this line intersects the arc. So, the arc from the axis to this point is half of the principal surface. From this point, draw a ray to the focus point, which is the marginal ray and defines theta`, it is also of length f because it is on the circle. From this layout you can immediately see that r/f = sin(theta`).

Now, how big can this curve get? Well, if you keep increasing r, you see that the maximum size it can go is a forward facing hemisphere with r = f, and so f/d, (which still holds by the way) has its minimum value of F# = f/d = f/(2f) = 0.5!

I wrote about this more here (with a reference on where you can see a curved principal surface):

http://www.dpreview.com/forums/post/39973191

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Chris,

I think (with the perpendicular/parallel substitution asked about above performed) that I follow you above.

In my attempts to leave behind a simplistic (image-side) single-lens analysis [about Numerical Aperture (NA)], as shown in this diagram on the .Wikipedia web-page. including the (in my case misleading) identity:

Source: https://upload.wikimedia.org/wikipe...g/568px-Numerical_aperture_for_a_lens.svg.png

... am pondering the MIT diagrams displayed below. A simplifying assumption that the lens-system is focused at "infinity" (unaffected by non-unity values of Image, and Pupil, Magnification factors) is made.

It seems (set me straight if I am off-base in my interpretations) that (in the case of a multi-element lens-system), a region of (common analytical) interest exists relative to the Entrance Pupil (as analytically considered from the "object-side" of the lens-system), which is (sometimes) more often referred-to.

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While the depicted Numerical Aperture (in the 1st image below) is directly related to the size of the Entrance Pupil (using Marginal Rays originating in the center of the Entrance Window) and is a function of the size of the Aperture Stop), period, ...

... the Field of View (in the 1st image below) is not directly related to the size of the Entrance Pupil, and is defined by the size of Field Stop (using Chief Rays originating from the outer points of the Entrance Window), which itself determines both the Entrance Window size as well as the projected size of the Exit Window [with a maximum (relevant) value corresponding to maximum numerical physical size of the linear dimensions of a (however-shaped) image-sensor photo-active-area].

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As we know, when remaining in radial coordinates when deriving Field of View, "focal length" is unnecessary.

Detail Man said:

Lens-system "focal length" (of which two separate values exist for a thick-lens-system's image-side and object-side) seems not to be an implicitly required known quantity in order to determine the Numerical Aperture [and thus, (at least, in "paraxial" regions within an image-frame), the image-plane Exposure].

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It seems that assumptions of "constancy" (of transduced photon-flux irradiating an image-frame defined by an image-sensor's utilized photo-active-area) depends upon the extent/scope/efficacy of multi-element lens-system optical corrections implemented in the particular lens-system being considered in an analysis.

Detail Man said:

The particular physical location of the Front Nodal Point (from which Focus Distance is derived for use in precise calculations involving Hyperfocal Distances and Depths of Field based upon some "deemed COC" diameter), ...

... as well as the particular physical location of the Principle Planes, appear to also not be implicitly required known quantities in order to determine the Field of View, (as well as the non-specified physical location of Panoramic Pivot Point located at the center of the lens-system Entrance Pupil).

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Optically corrected Principle Surfaces are as I am now appreciating not (when "off-axis") "Principle Planes".

Detail Man said:

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Source: Pages 1-12: http://ocw.mit.edu/courses/mechanic...ndows-single-lens-camera/MIT2_71S09_lec06.pdf

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Source: Pages 3-4: http://ocw.mit.edu/courses/mechanic...ndows-single-lens-camera/MIT2_71S09_lec06.pdf

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Question: I make an assumption that the (effective, optical, from the perspective of the image-sensor) apparent maximum user-adjustable size of the Aperture Stop (in the multi-element lens-system diagram displayed above) cannot exceed the non-user-adjustable size of any Field Stop(s) existing with a system - or else the user would unable to fully "open up" an adjustable Aperture Stop.

(If) at that widest above-described user-adjustable Aperture Stop, some additional larger (effective, optical, from the perspective of the image-sensor) designed size "margin" exists where it comes to any linear/radial dimensions of Field Stop(s), then all seems to make sense - without any "fuss or muss".

(However), if an (effective, optical, from the perspective of the image-sensor) fully opened Aperture Stop coincided with, or exceeded any Field Stop(s) [in any analytically relevant (effective, optical, apparent) dimension(s)], then it seems (to me) that the system would have not one, but two (or potentially more), intra-system locations at which diffraction-patterns would be generated (?) ...

Such (it seems) would be "a real mess" ! I presume that such a situation is never the (designed) case ?

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Answer:

Due to Marginal Ray paths (but not for Chief Ray paths) it seems that the (effective, optical, projected) size of internal Field Stop(s) cannot (thus it is assumed does not) in lens-system designs be made smaller in size than = > the (effective, optical, projected) size of the internal Aperture Stop:


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This brief "YouTub" video provides (for me) an interesting and informative presentation regarding the subject of Aperture Stops (affecting the effective physically utilized proportion of some particular lens-element within a multi-element system), as opposed to Field Stops (affecting the effective physically utilized proportion of some particular Entrance/Exit Window existing outside of a multi-element system, such as an Exit Window defined by the utilized photo-active-area of an image-sensor assembly) - including their specific contributions as coefficients in the terms describing each of the 5 Seidel aberrations:

URL for the video (starting from the beginning):


Chris, could you possibly (at your leisure, if you may have time) check out this short video ? His latter descriptive associations are perplexing me, relative to my interpretations of the initial set of statements.

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Another longer (at ~53 Min) but more comprehensive MIT lecture discusses the matter of "Principle Surfaces", as well as just about everything else that (I myself) have been pondering on this thread:


Note: Instructor George Barbastathis utilizes the same (MIT) doc/diagrams that are linked-to above.

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(For a not fully, ideally optically corrected lens-system), the following information is a bit "mind bending" (?):

... the center of perspective and the no-parallax point are located at the apparent position of the aperture, called the “entrance pupil”.

Contrary to intuition, this point can be moved by modifying just the aperture, while leaving all refracting lens elements and the sensor in the same place.

Likewise, the angle of view must be measured around the no-parallax point and thus the angle of view is affected by the aperture location, not just by the lens and sensor as is commonly believed.

Physical vignetting may cause the entrance pupil to move, depending on angle, ...

From: http://www.janrik.net/PanoPostings/NoParallaxPoint/TheoryOfTheNoParallaxPoint.pdf

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DM

Yeah, you might be right.