Calculating focal length for aspherical lenses

[CaptainBarbosa]

How is it done? I've done a little research on Wikipedia and found the lensmaker's equation, but one of the values needed to compute focal length is the radius of the lens curvature. Strictly speaking, an aspherical curvature wouldn't have a set radius, would it? The reason I'm asking is because I'm trying to make a computer model of a lens and its fresnel equivalent. I can make a lens with an arbitrary curvature and convert this to a fresnel, but I would like to be able to model the curvature for specific focal points. (Given a fixed lens diameter.)

[Edit] For a plano-convex lens, I take it the flat side has infinite radius? Or am I totally screwed up in my thinking?

[Edit2] Also, what is the unit for focal length in the equation? Meters? Millimeters?

 

[Doc Al]

For a plano-convex lens, I take it the flat side has infinite radius? Or am I totally screwed up in my thinking?

You are correct.

Also, what is the unit for focal length in the equation? Meters? Millimeters?

Doesn't matter, as long as you use the same units for the lens radii. A common unit of lens power (which is 1/f) is diopters; to get power in diopters you must measure focal length in meters.

 

[CaptainBarbosa]

Thanks for that. Anybody know an answer to my aspherical lens question?

[Edit] Here's the results of a test I did with my lens that I made just eyeballing the curve...

 

[CaptainBarbosa]

Um, hello? Anybody?

 

[Captain Barbosa]

Due to the forum e-mail problem I had been using the other account, but I'll use this one from now on.

 

[ice109]

Thanks for that. Anybody know an answer to my aspherical lens question?

[Edit] Here's the results of a test I did with my lens that I made just eyeballing the curve...

what am i looking at here? a picture ? cgi?

 

[Captain Barbosa]

Yep, CGI. Here is a rendering of the lens I used in the foggy room test.

 

[fromzitu]

How is it done? I've done a little research on Wikipedia and found the lensmaker's equation, but one of the values needed to compute focal length is the radius of the lens curvature. Strictly speaking, an aspherical curvature wouldn't have a set radius, would it? The reason I'm asking is because I'm trying to make a computer model of a lens and its fresnel equivalent. I can make a lens with an arbitrary curvature and convert this to a fresnel, but I would like to be able to model the curvature for specific focal points. (Given a fixed lens diameter.)

[Edit] For a plano-convex lens, I take it the flat side has infinite radius? Or am I totally screwed up in my thinking?

[Edit2] Also, what is the unit for focal length in the equation? Meters? Millimeters?

Could u show me the lensmaker's equation? Thanks

 

[Doc Al]

Could u show me the lensmaker's equation?

http://en.wikipedia.org/wiki/Lens_(optics)#Lensmaker.27s_equation"

 

[fromzitu]

http://en.wikipedia.org/wiki/Lens_(optics)#Lensmaker.27s_equation"

O,I got it, just the same as spheerical lens. BTW,could you show me an example about how to get the curvature R1 and R2 for aspherical lens by sending a email to [COLOR="Blue"]<personal email removed>[/COLOR]? This confuse me for quite a long time.Many thanks! Your help will be appreciated.

 

[schip666!]

Wiki has a beginning stab at your answer: http://en.wikipedia.org/wiki/Radius_of_curvature_(optics )

In general I think aspheric lenses are used to correct for aberations so you might be able to come up with a close-fit spheric approximation to simplify things...

 

[Doc Al]

O,I got it, just the same as spheerical lens.

No, the lens maker's formula that I linked is meant for spherical lenses. I don't know of any simple formula for dealing with aspherical lenses. (Of course, you can treat it approximately as a spherical lens.)

 

[virgil elings]

For a plano-convex lens try:
r^2=(n^2-1)z^2 + 2f(n-1)z where r is radial and z is axial.
At z<<r, this gives a spherical shape with f=R/(n-1) where R is the radius of the sphere.
At r>>f, the shape is a straight line with r=sqrt(n^2-1)z
You can derive all of this by remembering that the idea of the lens is to produce constructive interference at the focal point, so all paths of an incoming parallel beam on the plano side should have the same number of wavelengths to the focal point independent of their position r.

 

[nasu]

How do you know that your a-spherical surface has a focal point, in the first place?
And how do you define it? The rays may converge in different points (if they do at all) for different incident directions.
I know that there are non-spherical shapes with focal points, but is this a general property, to expect for any shape?

 

[virgil elings]

nasu
Sorry, I forgot to mention that f is the distance to the focal point measured from the on-axis surface of the convex side. This formula was derived for a beam parallel to the axis. I would guess that the focal point gets smeared a little for incoming beams not parallel to the axis. Focusing is not a general property of a shape and in fact is only an approximation for a spherical surface (spherical aberration) but they are easy to make and understand. This aspherical lens has no spherical aberration.

 

[Gordianus]

Even though aspheric lenses have a varying radius of curvature, the lensmaker equation can be used, with fair accuracy, by using the lens radius at the center.

 

[Andy Resnick]

How do you know that your a-spherical surface has a focal point, in the first place?
And how do you define it? The rays may converge in different points (if they do at all) for different incident directions.
I know that there are non-spherical shapes with focal points, but is this a general property, to expect for any shape?

Prior to decent molded plastic optics, aspheres were rarely encountered. I don't have much information about them, but I can say the following:

Aspheres are typically characterized by their deviation from a reference sphere. If the sag for a sphere is given by

$$z =\frac{c\rho^{2}}{1+\sqrt{1-c^{2}\rho^{2}}}$$

where c is the radius of curvature and $\rho$ the radial coordinate. The sag of an asphere is simply:

$$z =\frac{c\rho^{2}}{1+\sqrt{1-(1+k)c^{2}\rho^{2}}}$$

where 'k' is the conic constant (k=0 for a sphere, k = -1 for a parabola, etc)

The most simple apsheric surface is a 'corrector plate', but large projector condensors can be aspheres. Aspheric elements are often used in telescope mirrors and conformal optics as well. The only relevant reference I have are two pages in O'Shea's 'Elements of Modern Optical Design', and he points to Chapter 3 of 'Applied optics and optical engineering' (R. Shannon and J.C. Wyant, eds)

Note that while use of aspheric elements can completely correct aberrations on axis, they fail off axis:coma and astigmatism are the most common.

 

[nasu]

Thank you for these details. Now I understand that by "a-spherical" you mean something more restricted that I did. I though that a-spherical means everything that is not a sphere, including an irregular kind of shape or with a periodic profile, or things like this. Or even something with a nice analytical profile which is not a conical section.
I suppose the OP had in mind something along the lines of your post above.

Maybe there is a difference in technical meaning between a-spherical and non-spherical?