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Qs: Doesn't a curved side lens produce less aberration than a flat lens in general because the overall angle of deviation is smaller in the curved lens?


Consider the following experimental results for the image resolution due to each geometry: enter image description here


It follows that the resolution $L_{curve}>L_{flat}$.

However this was unexpected because for spherical/parabolic lenses, spherical aberration is related to the angle of deviation via the approximate relation: $$abberation \approx \delta ^3$$ (derived from the fact that $sin(x)=x-x^3/3!+....$ and $\delta=angle_{in}-angle_{out}$ and deviation from Snell's law).

Implying that

enter image description here

for (a)(flat) & (b)(curve) $$abberation_{flat} \approx \delta^3$$

$$abberation_{curve} \approx 2\left(\frac{\delta}{2}\right)^3=\frac{\delta^3}{4} $$


Thus, assuming the resolution decreases with aberration, a curveside lens is expected to yield a higher resolution $L$, which contradicts with my experimental results (and I'm fairly confident that the results were correct as repeated readings were used).

The same conclusion was reached for the doublet lenses which were used to reduce chromatic aberration.

A resolution test chart & CMOS camera were used to determine the experimental resolution

enter image description here

What am I missing? why did the experimental results not agree with the theory?


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You have done a nice investigation. For the first two cases that you show with the single element lens, your reasoning is correct. For the doublet, the situation is different. A doublet is typically designed to work so one conjugate is at infinity. You can't necessarily make assumptions based on the surface curvatures of the outer elements. The lens is usually designed so that the positive element of the doublet faces infinity, and the negative element is to the short conjugate. This would correspond with the 3rd figure in your first picture, where you state the resolution is 114 lp/mm. From your observations with the doublet, I would expect the result to be as observed: the resolution is higher when the negative element faces the short conjugate.

To learn more about aberrations of lenses, the book "Modern Optical Engineering", by WJ Smith, McGraw-Hill, is highly recommended.

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    $\begingroup$ Yes - spherical aberration is reduced when the rays undergo less deviation at a given surface. Then when the contributions for all surfaces are added up, it is better to split the deviation between two surfaces. This is what Figure 1 (a) shows. If all the refraction occurs at one surface (Figure 1 (b)) then there will be more spherical aberration. Some vendor web sites (like Edmund Optics) will usually have an explanation of how to orient lenses to reduce aberrations. $\endgroup$
    – JB2
    Mar 25, 2021 at 16:54

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