# Spherical aberration in Biconvex lens

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• VVS2000
In summary: Consider a collimated ray (##\theta## = 0) incident on the first spherical surface of the lens at a displacement ##h## off the optical axis.2) Calculate the surface normal vector ##\hat{n}_1## to the first spherical surface at displacement ##h##. Now calculate the angle ##\theta_{inc,1}## between the incident ray and ##\hat{n}_1##. Use Snell's law to find the new orientation of the ray inside the lens.3) Trace the path of the ray inside the lens to see where it intersects the second spherical surface of the lens, and apply Snell's law once more.4
VVS2000
I was recently looking for proven relations between focal length, radius of curvature, refractive index etc of a convex lens as I was working on an experiment, I did Find a relation, between Height from principal axis and focal length, and it was a huge relation!I did the experiment to verify it, and it holds good. But I still don't know how to even derive such a huge relation. the image is attached.

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What is your goal here? To get some intuition? To get an idea of where it comes from in the math? To get actual predictions or improve an optical design? It's a broad question.

Twigg said:
What is your goal here? To get some intuition? To get an idea of where it comes from in the math? To get actual predictions or improve an optical design? It's a broad question.
to get an idea of where it comes in from the math. Like whether it's a geometric derived result or some kind of solution obtained out of brute force numerical method

VVS2000 said:
to get an idea of where it comes in from the math. Like whether it's a geometric derived result or some kind of solution obtained out of brute force numerical method
Numerical meethods are very unlikely to come up with explicit expressions like in the sheet you posted (source?)

The short version is that you get this expression when you use Snell's law and solving for the focal point of an incident ray where ##\theta## is the angle between the incident ray and the normal axis to the biconvex lens at the point at distance h off the optical axis. Naturally, you have to use Snell's again where the ray exits the lens on the second convex surface.

In practice, virtually no one uses this formula. You can get the Taylor coefficients by using some neat ray tracing tricks, so we tend to use ray tracing software to do aberration analysis. It's far more computationally efficient and general. My point is: don't feel like you need to know the above expression.

VVS2000, vanhees71, BvU and 1 other person
BvU said:
Numerical meethods are very unlikely to come up with explicit expressions like in the sheet you posted (source?)
thomas k gaylord, georgia tech optical engineering notes

BvU
Twigg said:
The short version is that you get this expression when you use Snell's law and solving for the focal point of an incident ray where ##\theta## is the angle between the incident ray and the normal axis to the biconvex lens at the point at distance h off the optical axis. Naturally, you have to use Snell's again where the ray exits the lens on the second convex surface.

In practice, virtually no one uses this formula. You can get the Taylor coefficients by using some neat ray tracing tricks, so we tend to use ray tracing software to do aberration analysis. It's far more computationally efficient and general. My point is: don't feel like you need to know the above expression.
That's the thing, How would one apply snell's law and get such a result, if you have any hint to get me started on the right direction I can make an attempt to solve it. I just want to know the math behind it all, like how one would approach such a complex situation

BvU
Sorry for the slow reply. Here's a quick outline of steps that would get you started.

1) Consider a collimated ray (##\theta## = 0) incident on the first spherical surface of the lens at a displacement ##h## off the optical axis. (See the image you attached in the OP for reference.)

2) Calculate the surface normal vector ##\hat{n}_1## to the first spherical surface at displacement ##h##. Now calculate the angle ##\theta_{inc,1}## between the incident ray and ##\hat{n}_1##. Use Snell's law to find the new orientation of the ray inside the lens.

3) Trace the path of the ray inside the lens to see where it intersects the second spherical surface of the lens, and apply Snell's law once more.

4) Trace the path of the ray in free space (on the image side), and see where the ray focuses (i.e. where it intersects the optical axis). This focal length should be given by the formula you first quoted.

Alternatively, maybe just use geometry and trig instead for your mechanical drawings. A lot easier.

## What is spherical aberration in a biconvex lens?

Spherical aberration is an optical imperfection that occurs in biconvex lenses, where the light rays passing through the edges of the lens focus at a different point than the rays passing through the center.

## What causes spherical aberration in a biconvex lens?

Spherical aberration is caused by the spherical shape of the lens, which causes the light rays to bend more at the edges than at the center, resulting in a blurred or distorted image.

## How does spherical aberration affect the quality of an image?

Spherical aberration can cause blurring, distortion, and a decrease in image resolution, resulting in a lower quality image. It can also cause color fringing and reduce contrast in the image.

## Can spherical aberration be corrected in a biconvex lens?

Yes, spherical aberration can be corrected by using a combination of lenses with different curvatures or by using specialized lens designs, such as aspheric lenses.

## How can spherical aberration be minimized in a biconvex lens?

Spherical aberration can be minimized by using a lens with a smaller aperture, as this reduces the amount of light passing through the edges of the lens. Additionally, using higher quality lenses with more precise curvatures can also help reduce spherical aberration.

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