Calculated focus of an asymmetric bi-convex lens

Question

So I've been working on modeling an optical system using ray tracing. I've finally gotten to the point where I am testing things however I've come across something I'm not entirely sure of. I'm casting a set of rays into an asymmetric bi-convex lens and applying Snell's law to each ray to trace them through the lens. I then calculate the expected focus using the following equation:

$$\frac{1}{f} = (n-1)\left[\frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{nR_1R_2}\right]$$

I've plotted this point, as well as the rays exiting the lens, and I'm seeing that they don't match up all the time. My rays are being cast perpendicular to a sensor that lies on the optical axis (as shown below) so they travel in parallel lines until hitting the lens. It is my understanding then that the rays should then refract through the lens, and all converge at the focus. Is that not the case? Am I misunderstanding something here, or is there some other equation for the focal point that I should be using?

I'm also confused by the fact that different ray diagrams seem to show different things, such as the following images where one shows that parallel light converges to the focal point, while another shows something completely different:

Any help would be greatly appreciated!

My Ray Traced Diagram

Answer 1

You can see that the top line in the second image is parallel to the lines in the first image and it does also go through the focal point. It just it shown continuing on to the image location. Only a collimated beam normal to the center of the lens where all lines are parallel and perpendicular to the lens plane will converge at the focal point. And conversely light eminating from a focal point will cumulate in a collimated beam, i.e. parallel lines.

Light from otherly placed objects will not converge at the focal point nor create a collimated beam.

It's hard to say why the beams in the ray tracing example don't converge at the focal point without seeing the calculations. And that equation only works if they are spherical lenses. And if it's a thick lens what you call the focal length depends on where you are measuring from.