Plano-convex lens with tilted plane surface focus point

[AdityaDev]

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'.

 

[Drakkith]

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

 

[AdityaDev]

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]

No. Plane surface is tilted at an angle z.

It's tilted, but it's not curved, right?

 

[AdityaDev]

It's tilted, but it's not curved, right?

Yes. Its not curved.

 

[AdityaDev]

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]

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?