I am trying to make a very basic ray-tracing simulation of lens.

  • A single ray origin trigger light photons of a supposed single frequency in all directions. (I do not consider chromatic aberration).
  • Some lens emulated as a couple of spherical superficies with refraction index.

After detecting an intersection between a ray and the closest lens, I get the normal at this point to the lens and apply the following refraction formula wikipedia:

$$\frac{sin(\theta_1)}{sin(\theta_2)} = \frac{n_2}{n_1}$$

The formula is applied twice: a first time when entering the lens with (e.g. index $2.0$) and a second time when exiting the lens with the inverse index (e.g: $0.5$).

I checked the results numerically, but the rays do not focus:

Graphical representation with refraction Index = 2.0

As you may observe, the focus "point" is closer for rays crossing the outer part of the lens, while is further for rays crossing the lens in the middle.

Some suppositions:

  • The formula is too simplistic for achieving a correct focus point
  • Lens are not exactly spherical
  • I do some mistake somewhere

**Why are those rays not converging to the same focus point? **

Example of calculation:

  • Ray: Origin=$(400,300)$ Direction=$(0,-1) $
  • Lens: Circle center=$(400-50\sqrt{2}, 100)$ Radius=$100$
  • Intersection: Position=$(400, 170.7107..)$ Normal=$(\sqrt{2}, \sqrt{2})$
  • Refracted ray inside the lens: Origin=$(400,170.7107...)$ Direction=$(-0.41144, -0.91144)$ $\theta_1=-45$ $\theta_2=-20.7048$ $\frac{n_1}{n_2}=2$
  • Intersection2: Position=$(323.027, 0.19627)$ Normal=$(0.062622, 0.99804)$
  • Refracted ray outside the lens: Origin$(323.027, 0.19627)$, Direction=$(-0.75, -0.66144)$ $\theta_1=20.7048$ $\theta_2=45$ $\frac{n_1}{n_2}=0.5$

You have found the defect of a perfect spherical lens called spherical aberration which manifests itself when dealing with rays which are far from the principal axis.
It is precisely the fact that the lens is spherical which causes this lens defect.

The simple lens formula is only an approximation for rays which are near to the principal axis.

enter image description here


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