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This is something I just realised I don't entirely know the answer to, so I'm going to ask it here as I think it's interesting.

In most non-scientific writing on depth of field, diagrams generally show camera and subjects as being perfectly parallel, e.g.

enter image description here

However, is this a more accurate representation of the focal plane?

enter image description here

Are there ways to optically alter the shape of the focal plane?

Note: Obviously these diagrams are two dimensional, but I'm assuming in the second diagram the shape would be spherical with the sensor at the center.

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You could, in theory, build a lens with a curved focal plane - I don't think such lenses exist, however. see : physics.stackexchange.com/q/81349/44080 As for altering the focal plane, tilt-shift lenses are one common way this is manipulated. The focal plane remains a flat plane, but tilt-shift lens allows you to incline and rotate the focus plane with respect to the sensor. en.wikipedia.org/wiki/Tilt%E2%80%93shift_photography – J... Feb 12 at 14:06
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@J... this is getting into the Scheimpflug principle. Also related are the papers by Leonard Evens. View Camera Focus and Depth of Field is the one I remember reading when learning how to focus a view camera. – user13451 Feb 12 at 14:38
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@J...: First, most lenses (especially wide angles) do exhibit at least a little curvature of field (but generally a lot less than shown in the diagram above). Although they're pretty rare there are a few lenses with intentionally curved fields. Perhaps the most unusual in this respect is the Minolta 24/2.8 VFC, which allowed the user to control its curvature of field. – Jerry Coffin Feb 12 at 16:21
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Curve the sensor ! whatdigitalcamera.com/technology_guides/… – Olivier Feb 12 at 18:43
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@J... Uncorrected all lenses would demonstrate field curvature. – Michael Clark Feb 12 at 19:20
up vote 18 down vote accepted

The effect is called field curvature. A good discussion comes from Nikon. It is a lens aberration that can reduce the resolution of the lens when coupled with a flat sensor. In the old days, the film could be bent a little to try to follow the image plane and reduce the effect, but our sensors today are rigid. It can be reduced with lens design.

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This is the correct answer. I would add that any ideal single element lens will show such a behavior, unless the effect is compensated for by another lens or a non-homogenous optical formula. This is actually a complicating factor in the design of good lenses and there are hopes that by making controllable flexible sensors we can use much simpler and much more compact lenses in the future. – retrography Feb 12 at 16:12
    
Film is also thick compared to the wells of a sensor. – user13451 Feb 12 at 20:27
    
So, in summary, lenses "naturally" have a curved field, and we add extra lens elements to try to flatten it? – MathematicalOrchid Feb 13 at 11:22
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@MathematicalOrchid Yes. That is basically correct – Michael Clark Feb 14 at 3:03
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And since the correction is likely to be perfect only at one focal distance, that distance may be optimized differently for different kinds of lenses. For example, a dedicated macro lens might be designed to have the flattest focal plane at its minimum focal distance. – Kevin Krumwiede Feb 14 at 4:54

A single converging lens with real thickness has a curved focal plane. Most lenses offered by manufacturers include corrective elements to flatten the focal plane to one degree or another. There are some well known and highly desired lenses known for flattening the focal plane particularly well: The Zeiss Planar series for example. There are also lenses known and desired for not correcting some or all of their field curvature and the "look" photos taken using those lenses exhibit. The Canon EF 85mm f/1.2 L II is one such lens.

What is the shape of the focal plane?

A single element lens with normal surfaces using mathematically simple optical formulas will demonstrate field curvature. When projected on a flat sensor/film, the varying distances from the center of the lens to the middle vs. corners of the sensor will cause loss of focus on the edges and in the corners if the center is properly in focus. If a film or sensor could be constructed so that all parts were equidistant from the optical center of the lens, everything would be in equal focus. Such a sensor would cover the same portion (expressed in angular degrees) of an arc of a sphere as the amount of arc covered by the lens in the camera's field of view. The radius of curvature would vary by the refractive index of the lens.

In modern practice, there are few, if any, simple single element lenses being offered by manufacturers and used for photography as defined within the scope of photo.stackexchange.com. The shape of the focal plane is entirely dependent upon the design of the lens. Spherical aberration/field curvature can be left totally uncorrected or can be highly corrected depending upon the decisions made by the lens designers and the effectiveness of their design.

When discussing cardinal point optics it must be kept in mind that zero thickness lenses do not actually exist. They are theoretical. From the wikipedia article for Cardinal point (optics):

The only ideal system that has been achieved in practice is the plane mirror.

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With an optically perfect lens, the focal plane is parallel to your sensor, and it has the same shape as it, i.e. it is actually a plane. With a real-life lens, I guess you can get a bit of distortion of the plane, but it will essentially remain a plane. It has to be so for landscape photography where you want the whole image to be focused at infinity at the same time, and it is the case for any decent lens.

Your first diagram is more correct than the second. In the second diagram, you are neglecting the fact that the corners of your sensor are farther to the optical center of your lens than the center of the sensor.

This is something to take into account when using "focus then recompose" usual technique: by rotating your camera to recompose, you are moving the focal plane without changing the distance to subject, and you can indeed get the subject out of focus. This is especially true with a wide-angle lens at a wide aperture.

See e.g. How to shoot moving subject with Panasonic FZ 70/72 for more details.

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Thanks. So when measuring dof, where in the camera is the measurement to? The sensor? – Pedr Feb 12 at 12:31
    
I don't understand the question. The DoF is not "measured" by the camera. It is the optical result of the aperture, focal length, ... – Matthieu Moy Feb 12 at 12:55
    
+1 for the implications for "focus then recompose". That's a subtle consequence that is not often realized. – scottbb Feb 12 at 13:48
    
Sorry. I mean where exactly on the camera are measurements relating to focus taken - is it a point on the lens, the aperture, or the sensor? – Pedr Feb 12 at 14:11
    
@Pedr I'm not exactly sure what measurement you're asking for, but this question may cover it: photo.stackexchange.com/questions/21668/… – recognizer Feb 12 at 16:21

I think that a missing element to the obviously-otherwise-correct answers given is connecting to the wrong intuition in the question.

The intuition in the question is coming (I believe) not from some question about lens abberation, but from a wrong sense that the focal plane is based on the distance from the lens.

This question could perhaps be paraphrased as

"are the things that are sharply in focus are all at a specific distance from the lens - are they all situated on a curve with the radius of the focal length?"

The answer is "no, that is not how focus works". As stated by Matthew Moy, the focal plane for a perfect lens is a parallel to the sensor.

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Only if the lens has been corrected for field curvature. Most modern lenses have been correct, but a single element thin lens with regular surfaces has not. – Michael Clark Feb 14 at 2:47
    
Does that mean that a "single element thin lens with regular surfaces" does in face have a spherical focal plane as a result of the focus being at a fixed radius from the lens, or it is simply that such a lens has some field curvature due to "imperfections"? – GreenAsJade Feb 14 at 3:12
    
Fixed radius both front and back. The only difference is the size if the radius on either side which is due to the refractive index of the lens. If the convex lens is shaped to have have a constant refractive index from center to edge it will demonstrate field curvature. As the focus distance approaches infinity the DoF will probably become so great that the curvature will no longer be noticeable. – Michael Clark Feb 14 at 4:56

The shape of the focal plane is dependent of the optical formula. In particular the Zeiss Planar was named after its particularly flat focal plane which made it good for photography of books, but in general it looks more like your second drawing.

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Thanks for replying. It would be great if you could expand your answer a bit, perhaps with references and examples. – Pedr Feb 12 at 11:33

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