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Numerical aperture of microscope

I understand that it's tempting to look at an equation such as NA = n sin θ, and wonder what meaning can be inferred if n or θ were negative.

Numerical aperture formula

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No, there's really not such a thing as a negative numerical aperture. Understand that NA is about an image forming lens, bringing the incoming light (in the case you mentioned, collimated) to a single focus point. Meaning, a net-convex system.