Principal plane of a thin lens

[HastiM]

Hello,

in geometric optics people usually say that the refraction of light beams due to a thin lens can be handled by introducing a hypothetical plane (the so called principal plane) at which all the refraction can be considered to happen. But most books only picture the situation for symmetric biconvex or biconcave thin lenses, in which case the principal plane is positioned such that it cuts the lens in half and is perpendicular to the optical axis.
The principal plane is important in other respects as well. Namely some properties of the thin lens are related to the principal plane, e.g. the focal length of the thin lens is measured w.r.t. the principal plane etc.

I am wondering how to place the principal plane appropriately if the thin lens is not symmetric anymore. How to do it for example for those thin lenses named by wikipedia?: https://upload.wikimedia.org/wikipedia/commons/a/a7/Lenses_en.svg

If I would have to guess I would take the midpoint between the "endpoints" of the lens, i.e. those two points where the surface of the lens intersects the optical axis. However, for some lenses the principal plane lies outside the lens itself so that the picture looks awkward to me (some light rays are refracted before they hit the lens).

I would very appreciate any help!

Best wishes

 

[Baluncore]

There are limits to how far an approximation used in teaching can be taken.
The shape of a lens is usually dependent on the manufacturing process.
For asymmetric lenses you might be able to consider different radii as different lenses, like the rings of a Fresnel lens.

 

[Drakkith]

I am wondering how to place the principal plane appropriately if the thin lens is not symmetric anymore.

You can calculate the position of the principal planes using Gaussian or Newtonian imaging equations if you know the curvature or power of each surface of the lens (though technically a thin lens should have its principal planes directly on top of itself since it has zero thickness).

 

[jtbell]

(though technically a thin lens should have its principal planes directly on top of itself since it has zero thickness)

Exactly. I've never seen an example where the refraction of a thin lens isn't assumed to occur at the plane of the lens itself. I've only ever seen the concept of principal planes used with thick lenses.

 

[Dr Transport]

https://en.wikipedia.org/wiki/Cardinal_point_(optics)

this explains it more fully...

 

[HastiM]

Thank you all for the help!

@ Dr. Transport: It is written in the article on wikipedia: "For a thin lens in air, the principal planes both lie at the location of the lens."

As it is seen in the pictures linked in my question above, thin lenses usually are not drawn as infinitesimal small but with some thickness. So I am confused by the definition "location of the lens". My question is: where is this "location of a lens"? Suppose we would have to point at that spot on the pictures. Where would we point at?

@jtbell: What do you mean by "plane of the lens"? Can you please tell me where that plane lies?

 

[Ibix]

The thin lens approximation treats the lens as a single plane. So the shape of the lens doesn't come into it - both front and back surface are in the same place. If the lens has significant thickness, then it make sense to ask where the two surfaces are located. But then the lens isn't thin.

 

[Dr Transport]

Exactly. I've never seen an example where the refraction of a thin lens isn't assumed to occur at the plane of the lens itself. I've only ever seen the concept of principal planes used with thick lenses.

The thin lens approximation treats the lens as a single plane. So the shape of the lens doesn't come into it - both front and back surface are in the same place. If the lens has significant thickness, then it make sense to ask where the two surfaces are located. But then the lens isn't thin.

exactly, the concept of principle planes is pretty much meaningless in thin lenses because of the infinitesimal approximation of the thickness.

 

[jtbell]

A "thin lens" is by definition one whose thickness is negligible for purposes of analysis (thin-lens formula etc.), drawing a ray diagram, etc. It's an idealization, an approximation to a real lens which is useful under certain circumstances, e.g. the thickness of the lens is negligible compared to the distances between the lens and the object and image.

If you're using the thin-lens equations to analyze a real lens, it doesn't make much difference where, exactly, you place the plane of refraction with respect to the real lens, so long as it's somewhere within the thickness of the lens. I simply place it halfway between the front and back surfaces of the lens. This introduces an error or uncertainty in the analysis, but that's part of the nature of approximations.

If it makes a practical difference where you place the plane of refraction within the thickness of the lens, then you shouldn't be using the thin-lens equations in the first place!

When I draw a thin-lens ray diagram, if I want to apply the ray-tracing rules precisely, I don't draw the lens as having a thickness, but instead like this:

On the left is a converging (convex) lens, on the right is a diverging (concave) lens.