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Is it possible to reduce the airy disk size produced by one lens with another lens placed after the previous one? For example, parallel ray incident on first lens L1 (without aberration), then there is Airy disk on the back focal plane of L1, if image this pattern by another lens, how does the resulting Airy disk size change?

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Airy disk intensity is described by $$I(r) = \left( \frac{ 2 \cdot J_1(v) }{v} \right)^2 \qquad \text{ with } v= \frac{2\pi}{\lambda}r\cdot NA $$ The Bessel function $J_1(v)$ of first kind has a first minimum at $$J_1(3.832) = 0$$ $$ \Rightarrow r_{\text{min}}=0.61\cdot \frac{\lambda}{NA}$$

At a given wavelength, the first minimum of the airy disk $r_{\text{min}}$ only depends on the numerical aperture NA. Imagine this NA as a capability to collect light.

Your proposal is to use a second lens to change the size of the Airy disk. A well designed second lens (diameter, focal length and position) is able to reduce the size of the Airy disk pattern.

In short terms: You can shink the size of the Airy disk, if your two-lens-system has a higher NA.

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In information terms* every time light passes through an aperture information that 'doesn't fit' through that aperture is permanently lost. Therefore the second lens has less information to play with and you will have a fuzzier image.

*I refer to information here the higher order fourier terms that are not sampled by the first lens.

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