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Recently, I did discover that it might be possible to print fully solid, air free and thus optical homogenous, even with PLA.

How can I design a proper focusing lens (or rather: single sided lens with a flat bottom) with this, ignoring the post-processing needed to get the surface smooth?

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  • $\begingroup$ I guess I'll ask, if not about post processing or print orientation...how does this apply to 3D printing? $\endgroup$
    – tjb1
    Jan 8, 2019 at 20:25

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To calculate the focal length of an optical element, the two main factors are the refraction index and the shape of the lens.

For a cylindrical lens with one optic active side (that is, one domed or bowled side), we can ignore the whole bottom cylinder and just take into account the top dome. The shape of the dome is determined by the radius of the circle that created it.

Thin, single sided lenses

For a thin, single-sided lens the rather complicated Gullstrand’s formula to calculate the focal length of lenses becomes rather simple:

$f = \frac {r}{(n-1)}$ for the bend facing the object

$f = \frac {r}{(1-n)}= \frac {-r}{(n-1)}$ for mounting it in reverse.

A Polymer database did give a refraction index of PLA as $n=1.465$.

Thick, single sided lenses

For a thick lens with a total thickness of $d$ and one active side, we solve first for the one active side, and then insert: $$f_1=\frac{r}{n} \land \frac {-r}{(1-n)} ; f_2=\infty ; P_i = \frac 1 {f_i}$$ $$P=P_1+P_2 -P_1P_2\frac d n ; P_2\to0$$ $$f=\frac{1}{P_1}=f_1$$ As long as one side of our lense stays flat, the thickness of the lense is mathematically not relevant (save for increasing dispersion).

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    $\begingroup$ That last line is not strictly true. The thickness doesn't affect the paraxial trace a lot, but does affect higher-order distortion terms. In addition, since your material is almost certainly dispersive ( index varies with wavelength), the thickness will lead to spectral distortion. BTW, your "mounting in reverse" formula is identical to the first formula: just multiply by $\frac{-1}{-1}$ $\endgroup$
    – Carl Witthoft
    Jan 8, 2019 at 15:34
  • $\begingroup$ @CarlWitthoft duh! missed a minus sign... $\endgroup$
    – Trish
    Jan 8, 2019 at 15:35
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    $\begingroup$ aha! Tho' I will say that us "Optickers" usually leave the sign of the radius of curvature as positive for a convex surface with the understanding that the focal length is always positive for a real image and negative for a virtual image. But so long as your notation is self-consistent, no big deal. $\endgroup$
    – Carl Witthoft
    Jan 8, 2019 at 16:31

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