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The transverse optical resolution of a lens, as determined by its point spread function, can be approximated as $w$ ~ λ/(2$NA$), where $NA$ is the numerical aperture and λ is the wavelength of the light. The $NA$ is ~ $D$/(2$f$), where $D$ is the lens diameter and $f$ is the focal length. However, what if I have a laser beam illuminating an object, and the beam diameter is less than $D$? Would I have to change $D$ in this equation to approximate the resolution?

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    $\begingroup$ Is your laser beam supposedly TEM00 gaussian? For a perfectly coherent beam, you should replace the aperture size with laser beam size. For real beams there are additional parameters such as beam quality you must take into account. If noone comes up with exact expressions, I will elaborate in a day or so. $\endgroup$ – Arturs C. Aug 25 '18 at 23:05
  • $\begingroup$ Interesting. I assume my beam is TEM00. So my numerical aperture indeed depends on the mode size at the input? $\endgroup$ – Sean Daley Aug 25 '18 at 23:20
  • $\begingroup$ Basically yes. A coherent gaussian beam of size D should perform as a diffraction limited beam through aperture D $\endgroup$ – Arturs C. Aug 25 '18 at 23:54
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    $\begingroup$ @ArtursC. Perhaps you can write this up as an answer. $\endgroup$ – flippiefanus Aug 27 '18 at 4:17
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1-minor point: laser illumination will create speckle patterns and ruin your image

2-if the laser illuminates the object then you'll get diffuse reflection and also transmission if the object is thin

3-diffuse reflection will shoot light in all directions so the effective NA of your lens wont change

4-transmitted light will have a much lower NA (because it's parallel) so if you were in a transmitted light imaging setup (for ex laser illumination vs kohler illumination on a microscope) then yes NA will change.

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When you are limited by the size width of the laser beam, i.e. you are not limited by the diffraction of the lens you are going throught, the relation between the waist of the "beam image" $w_0^{"}$ and the waist the "beam object" $w_0$ is given by the relation:

$$\frac{w_0^{"}}{w_0}= \frac{\lambda f}{\pi w_0}$$

You can prove this easily with the lens formula for gaussian beam from this article (S.A. Self, Focusing of spherical Gaussian beams and taking the case where the beamimage waist is at the focal point of the lens)

So to answer your question. It's depend on which aperture is limiting your spot size: the lens or the beam width

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