Modeling a lense using the intersection of two spheres

In summary, the conversation discusses modeling lens shapes using two circles of different radii and finding the point of intersection to correct spherical aberration. It is noted that multiple lenses can help with correction, but perfect correction may not always be possible. It is suggested to break down the process into smaller steps and use mathematical software for automation.
  • #1
Unit
182
0
Hi everybody!

I am trying to model a lense shape using two circles of radii R and r, with one at the origin and the other offset upwards vertically by distance D. D must be less than R + r, but it must be greater than the larger of R or r.

Thus, I have two equations, and the region of their intersection is a lense-shape!

[tex]x^2 + y^2 = R^2[/tex] called the "origin sphere"

[tex]x^2 + (y - D)^2 = r^2[/tex] called the "offset sphere"

Anyway, for the numerical analysis I'm doing, I have R = r = 5 (the size of my protractor) and D = 8, which works out nicely because the two circles intersect at (-3, 4) and (3, 4).

What I do:
-start off with incident light, represented by vertical rays, like x = 2
-find the point at which it collides with the offset sphere (xray, yray)
-draw a tangent (using the derivative)
-find the necessary angles (using the arctangent of slope)
-use Snell's law once ([itex]n_{air} \sin{\theta_1} = n_{glass} \sin{\theta_2}[/itex])
-determine an equation of a line y that has this equivalent angle and passes through the collision point: y = m(x - xray) + yray
-find there this new line collides with the origin sphere (the top curve of the lense)
-draw a tangent
-Snell's law again
-determine an equation of the final line, the twice-refracted line
-the y-intercept of that line is the "focal" point (judging by symmetry)

This is a lengthy process, each of these steps involving not-very-simplifyable expressions, so a "master equation" that does all this for me would be unwieldy.

What I have found, though (to my great disappointment), is that the "focus" is actually a region of intersections and not simply one point. So, I'm trying to think, would adding a second "lense" remove all spherical aberration? Is it even possible to remove all spherical aberration? By "lense", I mean any extra, defineable shape that will correct the aberration.

Thanks! :smile:

-Unit
 
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  • #2


Hello! It sounds like you have a very interesting problem to solve. Modeling lens shapes can be difficult, but it's important for understanding how light behaves in different materials and how we can manipulate it for various applications.

To answer your question, yes, adding a second lens can help correct spherical aberration. In fact, many optical systems use multiple lenses to achieve the desired effect. However, it's important to note that perfect correction of spherical aberration is not always possible. It depends on the specific parameters of your system and the materials used.

In terms of finding a "master equation" for your process, it may be helpful to break it down into smaller steps and use mathematical software or programming to automate some of the calculations. This can save you time and effort in the long run. Additionally, there may be existing equations or models that can help you with your specific problem.

As for removing all spherical aberration, it may not be possible in all cases. However, by using multiple lenses and optimizing their positions and shapes, you can minimize the effect of spherical aberration and achieve a more precise focus.

Good luck with your research and let us know if you have any further questions!
 
  • #3


Hi Unit,

Your approach to modeling a lense using the intersection of two spheres is definitely an interesting one. It seems like you have put a lot of thought and effort into your numerical analysis. However, as you have discovered, the "focus" of your lense is actually a region of intersections rather than a single point. This is due to spherical aberration, which is caused by the fact that spherical surfaces do not focus all incoming light rays to a single point.

To answer your question about removing spherical aberration, it is possible to minimize it by using multiple lenses or by using aspheric lenses (non-spherical surfaces). In fact, many modern lenses are designed with aspheric surfaces to minimize spherical aberration and improve image quality.

Adding a second "lense" may help to correct the spherical aberration, but it will also introduce other optical effects such as coma and astigmatism. Therefore, the design of a lense system involves finding a balance between these different aberrations.

In summary, while your approach to modeling a lense using two spheres is interesting, it may not accurately represent the behavior of a real lense due to the presence of spherical aberration. To better understand the behavior of lenses, it may be helpful to study the principles of geometric optics and aberration theory. Keep up the scientific thinking and exploration!
 

1. How is a lens modeled using the intersection of two spheres?

The two intersecting spheres represent the front and back surfaces of the lens. By finding the intersection points of these spheres, the shape and curvature of the lens can be determined.

2. What factors affect the modeling of a lens using two spheres?

The radius and position of the spheres relative to each other can greatly impact the resulting lens shape. Additionally, the refractive index of the lens material and the desired focal length also play a role in the modeling process.

3. Can a lens be accurately modeled using only two spheres?

In most cases, a lens can be approximated using two spheres. However, for complex lenses with varying curvature or asymmetrical designs, additional spheres or mathematical equations may be necessary for a more accurate model.

4. How does the modeling of a lens using spheres compare to other methods?

Modeling a lens using spheres is a simplified method that is commonly used in optical design software. Other methods, such as ray tracing or wavefront analysis, may provide more accurate results but can be more complex and time-consuming.

5. What applications use the intersection of two spheres for lens modeling?

The intersection of two spheres is commonly used in the design of simple lenses, such as those found in eyeglasses, cameras, and telescopes. It is also used in more complex optical systems, including microscopes and telescopes, as a starting point for further refinement.

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