# What lens shape gives perfect focus?

Assuming ray optics model is valid, what lens profile focuses light to a point? It must not be spherical, otherwise there would not be the term 'spherical abberation'.

The problem with your reasoning is that that ray optics model has a limited application. There is a "best form" to a biconvex lens, but the spherical aberration is not zero. Use of aspherics can reduce aberrations, but a singlet will only have zero spherical aberration for an extremely limited set of illumination conditions. For example, parabolic reflectors have zero aberration, but only for on-axis points.

In fact, simply reversing the orientation of a planoconvex lens will result in radically different amounts of spherical aberration.

lzkelley
segment of an ellipse

The problem with your reasoning is that that ray optics model has a limited application. There is a "best form" to a biconvex lens, but the spherical aberration is not zero. Use of aspherics can reduce aberrations, but a singlet will only have zero spherical aberration for an extremely limited set of illumination conditions. For example, parabolic reflectors have zero aberration, but only for on-axis points.

In fact, simply reversing the orientation of a planoconvex lens will result in radically different amounts of spherical aberration.

Yes that's a good point. In practice the ray model is not good enough, and you can't realize the zero aberration ideal. I guess I'm asking more of a mathematical question then, but it is optics-related so I posted it here. I was thinking that the specific term 'spherical aberration' was included in the theory of ray optics, so that's why I reasoned that the ideal lens shape in the ray model must not be spherical. I am imagining a plano-convex lens with light coming from infinity, being focused to a point on the axis of the lens.

lzkelley said:
segment of an ellipse
I was wondering if it was a conic section. I tried to work this out when I was in high school but gave up after a while.

lzkelley
It wouldn't be easy, but it should be doable to show that any incident line would reflect to the foci of an ellipse by finding the incident angle relative to the slope of the surface etc etc.

Yes that's a good point. In practice the ray model is not good enough, and you can't realize the zero aberration ideal. I guess I'm asking more of a mathematical question then, but it is optics-related so I posted it here. I was thinking that the specific term 'spherical aberration' was included in the theory of ray optics, so that's why I reasoned that the ideal lens shape in the ray model must not be spherical. I am imagining a plano-convex lens with light coming from infinity, being focused to a point on the axis of the lens.

Yes, the term 'spherical aberration' was introduced by recognizing that spherical refracting surfaces all have a particular aberration- as the height of a ray (travelling parallel to the optic axis) increases, the distance between the lens and the location where the ray crosses the optical axis changes. That is, focus changes with aperture.

The way aberrations are discussed in ray optics is very artifical, IMO. Ray tracing involves linear and higher-order approximations to the sine function- linear optics has no aberrations, but there are 5 aberrations in 3rd order optics (7 actually, but 2 of them- piston and tilt- do not affect the PSF) and more for 5th order optics with strange names you have not heard of, etc. etc.

So, you can see how aberrations form in optics- as the linear approximation to a sine function breaks down (say the numerical aperture of a lens increases), higher order terms are required for accuracy, and aberrations come along for the ride as a result.

The way aberrations are discussed in ray optics is very artifical, IMO. Ray tracing involves linear and higher-order approximations to the sine function- linear optics has no aberrations, but there are 5 aberrations in 3rd order optics (7 actually, but 2 of them- piston and tilt- do not affect the PSF) and more for 5th order optics with strange names you have not heard of, etc. etc.

So, you can see how aberrations form in optics- as the linear approximation to a sine function breaks down (say the numerical aperture of a lens increases), higher order terms are required for accuracy, and aberrations come along for the ride as a result.

Yes, I looked up 'spherical aberration' on wikipedia and was immediately struck by how arcane all the terminology was for treating aberrations as tack-ons to the the ray model. I couldn't really follow it. Too much of "To find the correction for this particular effect, use third order so-and-so's equation" or something similar. But I guess when you need to actually design a lens system, and are not just interested in mathematical curiosities, those things would be useful.

You seem to be pretty knowledgeable about this stuff. Do you do research in an optics-related field?

I'm tempted to say the answer to my mathematical question is a hyperbola. I did a google search for 'hyperbolic lens' and found http://www.physics.umd.edu/lecdem/services/demos/demosl6/l6-03.htm [Broken] which purports to be a spherical lens (left) compared to a hyperbolic lens (right). Of course, it might just be a picture of a crappy spherical lens next to a picture of a good spherical lens for all I know.

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My graduate school training emphasized optics and fluid mechanics; I've since used optics in my research- microscopy, spectroscopy, laser tweezers, sensor systems, light scattering, etc.

There are several optical designs out there you may be interested in- the Ritchey-Chretien has two hyperboloids but only corrects coma off-axis, IIRC

http://ourworld.compuserve.com/homepages/David_Ratledge/tm9.htm [Broken]

Another is the Maksutov-Cassegrain design, which has no spherical aberration but has a restricted field of view and low f/#:

http://en.wikipedia.org/wiki/Maksutov_telescope

Here's a site showing how a decent optical designer thinks:

http://members.cox.net/rmscott/lh_scope/lh_design_article/lh_design.html

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