Can spherical aberration be avoided in lenses with non-spherical curves?

[itszach]

I've been doing some calculations on lenses, and I'm a little confused about some of the diagrams I've seen. Looking at a ray diagrams, I see the light passing through the lens, at one angle, and then converging on a focal point from there. Using snells law to calculate refraction through the materials, the concept of focal point is confusing me. If light enters one side of the lens, and refracts, wouldn't it refract in the opposite direction upon leaving the material(because it's not going from an area of lower wave velocity to higher)? Also, based on that, wouldn't there be some sort of lens configuration that allowed for a no focal point to exist, and light to continue at the same angle it entered, just more concentrated? My knowledge of optics is limited, so I apologize I'm asking an obvious question. I've searched for an answer all over the place and can't seem to find one. Thanks in advance.

 

[Drakkith]

I think the shape of the lense comes into play here. On a convex lens the trailing edge is opposite in angle than the leading edge, so the light exiting the lens refracts in the opposite direction. That's mostly a guess though.

To concentrate any light you MUST have some sort of focal point. Otherwise no light get concentrated!

 

[itszach]

In a standard biconvex lens, yes, but not in a plano-convex or a negative meniscus lens, which still have a focal point that seems to disregard both refractions. And I don't believe it's true that you must have a focal point to concentrate light. You would need a focal line(which could probably be created by some variation of a parabolic equation for the lens).

 

[Drakkith]

In a standard biconvex lens, yes, but not in a plano-convex or a negative meniscus lens, which still have a focal point that seems to disregard both refractions. And I don't believe it's true that you must have a focal point to concentrate light. You would need a focal line(which could probably be created by some variation of a parabolic equation for the lens).

Wouldn't that still be a focal point, just not as concentrated or focused as it typically is?

 

[Drakkith]

In a standard biconvex lens, yes, but not in a plano-convex or a negative meniscus lens, which still have a focal point that seems to disregard both refractions. And I don't believe it's true that you must have a focal point to concentrate light. You would need a focal line(which could probably be created by some variation of a parabolic equation for the lens).

Also, a plano-convex or negative meniscus lens has a focal point, as it is the point at which the image appears to be coming from. See here: http://en.wikipedia.org/wiki/Focus_(optics )

I don't believe it ignores refractions, as that is how the light is bent in the first place.

 

[itszach]

No, because light would not need to hit at a specific point for the concentration to occur, the light would enter at the same angle it left at, meaning it would be concentrated, but moving linearly. And doesn't the point in which the image appears to be coming from apply mainly to diverging lens, as apposed to converging? I'm pretty sure they'll still focus light to a point.

 

[RedX]

In a standard biconvex lens, yes, but not in a plano-convex or a negative meniscus lens, which still have a focal point that seems to disregard both refractions. And I don't believe it's true that you must have a focal point to concentrate light. You would need a focal line(which could probably be created by some variation of a parabolic equation for the lens).

This is something I never really understood. Isn't the point of a lens to have a small focal point? Because that's what you want to do, to bend light?

So it seems to achieve the lowest focal point, you would choose a biconvex or biconcave lens. What's the point of the rest?

 

[Drakkith]

No, because light would not need to hit at a specific point for the concentration to occur, the light would enter at the same angle it left at, meaning it would be concentrated, but moving linearly. And doesn't the point in which the image appears to be coming from apply mainly to diverging lens, as apposed to converging? I'm pretty sure they'll still focus light to a point.

I don't understand. If all the light left at the same angle, none of it would be concentrated. Yes, a diverging lens causes the image to appear to come from its focal point. It doesn not concentrate it. I misunderstood you.

 

[Drakkith]

Hold on...are you sure this is correct?

In a standard biconvex lens, yes, but not in a plano-convex or a negative meniscus lens, which still have a focal point that seems to disregard both refractions.

Depending on the angle of each side and the thickness of the lens the entering light is refracted differently. I think I see what you mean when you say that it isn't focused, but still concentrated if the light enteres at 90 degrees and leaves at 90 degrees, but the light IS bent inside the lens. The focal point isn't really a point, as the entire beam of light is "focused" to a degree.

 

[itszach]

Right, that's exactly what I'm saying. My main question though, is why does that not happen to a degree in one of the standard basic lenses, because in a biconvex lens, the light does not leave the lens at an angle somewhat unrelated to the angle it entered at, because the light would be bent, and leave at a different angle than it entered, depending on the curvature of the lens. It follows from that that light would be refracted differently by those lenses. Any ray diagram I've seen has not included this second refraction, and it's not making sense to me.

 

[Drakkith]

Right, that's exactly what I'm saying. My main question though, is why does that not happen to a degree in one of the standard basic lenses, because in a biconvex lens, the light does not leave the lens at an angle somewhat unrelated to the angle it entered at, because the light would be bent, and leave at a different angle than it entered, depending on the curvature of the lens. It follows from that that light would be refracted differently by those lenses. Any ray diagram I've seen has not included this second refraction, and it's not making sense to me.

I'm having a hard time understanding you. Are you asking why the light doesn't exit the lens at the same angle it entered at?

 

[itszach]

Not quite, I'm asking why, in lens diagrams, the rays are only seeming to refract once, and not straighten out some after the light exits the lens.

 

[Drakkith]

Not quite, I'm asking why, in lens diagrams, the rays are only seeming to refract once, and not straighten out some after the light exits the lens.

Why would the light straighten out after it exits the lense?

Edit: Look at this picture: http://en.wikipedia.org/wiki/File:Lens1.svg
The light refracts upon entering the lense and upon exiting. Though it is simplified and only shows it refracting at the thickets point instead of right as it exits the lens.

 

[itszach]

Well in that diagram it shows the light refracting twice, but the second refraction refracts the light inward, instead of outward.

 

[Drakkith]

Well in that diagram it shows the light refracting twice, but the second refraction refracts the light inward, instead of outward.

Yeah. Like I said in my first post I think that has to do with the angle of the lens upon exit, but I'm not sure.

 

[itszach]

Then in the case of a plano convex lense, wouldn't the light refract differently upon exit than upon exit, causing a very blurred focal point?

 

[RedX]

If light enters one side of the lens, and refracts, wouldn't it refract in the opposite direction upon leaving the material(because it's not going from an area of lower wave velocity to higher)?

Yes, if it's biconvex or biconcave. Say light starts at the left and travels rightward where it goes through the lens, and then leaves the lens. Say the light ray has an initial angle that's upwards. When the light hits the convex surface on the left, then it is bent down a little bit. If it then hits a concave surface to exit the lens, it is bent down again. So a biconvex lens (where the left side is convex, and the right side is concave when viewed from left to right) always bends the light towards the axis. If the right side of the lens is convex (as viewed from left to right), then as the light exits the lens, it gets bent back upwards. So the 2nd surface of the lens "undoes" the first surface of the lens. This is why I questioned what's the point of having meniscus lenses? Why would you want to undo an effect?

Also, based on that, wouldn't there be some sort of lens configuration that allowed for a no focal point to exist, and light to continue at the same angle it entered, just more concentrated?

Yes, but it has to be a meniscus lens, with the same curvature on the left as on the right.

edit: I've attached an image of such a lens. I created it by drawing a circle, cutting it in half, and then copying the half circle and shifting it parallel to the right. Then I joined the two surfaces at the top and bottom with a red line for clarity. As you can see the line enters and leaves parallel. I'm assuming that the lens is very thin, so that the 1st surface is almost right next to the second surface. Of course in the picture the lens looks thick, but I'm not that good with paint. One way to see that the ray will remain parallel before and after the lens is to note that the surfaces of the lens are parallel, and if the thickness approaches zero, you just get light entering and exiting a planar slab of material, and light enters and leaves a planar slab of material parallel.

 

[Drakkith]

Then in the case of a plano convex lense, wouldn't the light refract differently upon exit than upon exit, causing a very blurred focal point?

Yes, which is why it wouldn't be a focal *point* really. More like a spot of focused light that doesn't converge beyond the lens size.

 

[Drakkith]

RedX, those lenses are useful in complex lense arrangements to avoid optical errors, such as coma or sphereical aberration.

 

[jtbell]

I suggest that you carefully draw a large-scale diagram of the lens on a sheet of paper, using a compass to get good circular (spherical) surfaces. Then trace a few rays by hand through the lens using a ruler and protractor, and applying Snell's Law at the surfaces.

The simple geometrical optics formulas for thin lenses, etc. are derived using the paraxial approximation in which the rays don't make very large angles with respect to the axis of the system. Therefore, keep all angles smaller than 10 or 15 degrees. If some rays have significantly larger angles, they do not all come together at a single image point; this is called spherical aberration.

[added] The example diagrams that you see in textbooks or on Web sites often have angles that are much larger than this, just because a true-to-scale diagram would be too narrow to read, with all the rays nearly horizontal and bunched up together. A lens like the one RedX diagrammed in post #17 is unrealistic because it would have far too much spherical aberration to be useful.

 

[itszach]

Yes, which is why it wouldn't be a focal *point* really. More like a spot of focused light that doesn't converge beyond the lens size.

But a plano-convex lens does have a focal point, which is where I'm confused.

 

[Drakkith]

But a plano-convex lens does have a focal point, which is where I'm confused.

Ohhh, I'm sorry, I was thinking of the negative miniscus lens still. The image here shows 2 refractions: http://www.glassperfection.com/images/plano-covexImage1.jpg

 

[itszach]

I suggest that you carefully draw a large-scale diagram of the lens on a sheet of paper, using a compass to get good circular (spherical) surfaces. Then trace a few rays by hand through the lens using a ruler and protractor, and applying Snell's Law at the surfaces.

The simple geometrical optics formulas for thin lenses, etc. are derived using the paraxial approximation in which the rays don't make very large angles with respect to the axis of the system. Therefore, keep all angles smaller than 10 or 15 degrees. If some rays have significantly larger angles, they do not all come together at a single image point; this is called spherical aberration.

[added] The example diagrams that you see in textbooks or on Web sites often have angles that are much larger than this, just because a true-to-scale diagram would be too narrow to read, with all the rays nearly horizontal and bunched up together. A lens like the one RedX diagrammed in post #17 is unrealistic because it would have far too much spherical aberration to be useful.

In my understanding of it, spherical aberration is caused by the spherical curvature, and non-spherical curves in lenses don't show it. Even without the spherical aberration though, it still seems like the focal point of the lens would be skewed by the variously angled rays of light refracting at various angles relative to the flat side of the lens. I'll take your suggestion and draw the picture though before I confuse myself further. Thanks