# Modified plano-convex lens doubt

## Main Question or Discussion Point

I was thinking about the situation given my text about a plano convex lens which was produced with a manufacturing defect. It's plane surface is tilted outwards by a small angle 'z'. In the text its written that when a parallel light beam enters the lens parallel to x axis , it will still be able to focus it at:
$(R/(u-1) , -zR)$ where u is refractive index of lens and R is the radius of curvature of curved surface of lens and lens is at origin with rays along x axis starting from negative infinity. How does this happen?

My analysis: I think the tilted surface is just for bending the ray(refraction) by a small angle 'r' and now we have Plano convex lens with parallel beam of light at an angle 'r'.

## Answers and Replies

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Drakkith
Staff Emeritus
The plane surface of the lens is still flat, right? If so, what is -zR?

The plane surface of the lens is still flat, right? If so, what is -zR?
No. Plane surface is tilted at an angle z.

Andy Resnick
Think about it this way- your lens is an ideal lens plus a wedge prism. does that help?

Drakkith
Staff Emeritus
No. Plane surface is tilted at an angle z.
It's tilted, but it's not curved, right?

It's tilted, but it's not curved, right?
Yes. Its not curved.

Think about it this way- your lens is an ideal lens plus a wedge prism. does that help?
Igot the x-coordinate part. But how do you get the zR?

Drakkith
Staff Emeritus
Hmmm, I think your original analysis is pretty much correct. The tilted plane surface should just change the angle of the converging beam. I'm guessing -zR is the distance the new focal point is from the original focal point, but I'm not sure.

Andy Resnick
Igot the x-coordinate part. But how do you get the zR?
How about using the small angle approximation sin z = z?