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The one practical disclaimer is: If the lens is not well corrected, but has significant aberrations, then decreasing the aperture can actually decrease the spot size. The reason is that light passing through the edge of the lens, which forms aberrations that increase the spot size, is now blocked by the smaller aperture. The resulting spot might have lower overall power, but also have a lower spot size. To know if this is true you must know about the details of the lens.

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I guess fundamentally it's just confusing because a coherent beam is usually implied to only have a NA related to the beam diameter, but after passing through an aperture it diffracts so the whole diameter of the lens is relevant in picking up and focusing the higher diffraction orders.

But if the aperture becomes an infinitesimal pinhole, then it's the same as a point source emitting light. And the equation for how well resolved that point can be is defined by the airy disk using the aperture size of the lens diameter.

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I would look at it like this. Aperture cleans up the mode of the light. Provided the intensity of light is unifrom over the aperture, I cannot see the problem with putting a tube lens and an objective after it and focusing it into the diffraction limited spot.

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Basically, your argument is that we are messing with the numerical aperture of the light beam, and limit the ammount of k-vectors available to us. I would say that a microscope arrangement can effectively shift the energy between different k-vectors (same frequency of course), so it really does not matter. What does matter however is how clean your mode is, since odd modes tend to occupy more space (cannot be focused into such small spots)

So as the aperture is shrunk down, do the two relationships describe different things? For example, does the NA associated with the aperture of the laser beam describe how small of a Gaussian spot can be formed, and the NA associated with the lens itself describe the size of the airy disk that all the light coming through the aperture can be condensed into?

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Aside from aberrations, the spot size that can be formed by a laser beam going through a lens system is determined by the last aperture that restricts the beam. Note that if an aperture is wider than the beam, it is not restricting the beam -- so it is necessary to look upstream for the "last" aperture to restrict the beam.

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If you put an aperture in a laser beam to block some of it, I would imagine that the spot it can be focused to becomes larger due to diffraction. The numerical aperture of the system is limited by that truncation of the beam. For a regular laser focusing without an aperture, the beam diameter determines the numerical aperture, not the lens diameter.

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For a well corrected lens, one with a small amount of aberration, putting an aperture will increase the size of the focused spot. The basic formula for the diameter of the first dark ring is (from Smith, Modern Optical Engineering, page 453) $$B = \frac{(2.44 \lambda f)}{D}$$ where B is the diameter of the dark ring, $\lambda$ is the wavelength, $f$ is the focal length, and $D$ is the aperture diameter. If you put an aperture that decreases D, you will increase the diameter of the dark ring. With Gaussian beams, the formulas are different, but the effect is the same: putting an aperture on something causes the light to diffract more, and that spreads out the light at the focal plane.

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If you had a perfect laser (which is impossible) all beams are parallel and with an aspheric lens the spot is very small, limited by Airy disk of lens. Adding any aperture increases diffraction at the aperture, thus spot size increases because the rays are no longer parallel, there is no perfect focus any more. The aperture of the lens, usually much larger that an aperture in the beam, does not have as great an effect. I.e., its effect is much less than the beam aperture.

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