# Spot size based on a numerical aperture (NA)

When an optics vendor provides a spec for an objective lens, they often provide a numerical aperture value ($$NA$$). Based on the $$NA$$, the optical system's magnification will change. Physically, this is related to the spot size formed by a lens given a focal length $$f$$ and lens diameter $$D$$ (all of these effects are captured by the $$NA$$ quantity). The spot size can be approximated as:

$$w_0 = \lambda/(2NA)$$

1. Question 1: For a given optic, is the input beam typically assumed to be a plane wave?

For common laser sources the cross-sectional profile is Gaussian unless the beam is made to be very large, in which case it approximates a plane wave.

A more fundamental way of framing the question follows. We can recast the spot size equation in terms of $$f$$ and $$D$$:

$$w_0 = f\lambda/D$$

1. Question 2: Does this simple approximation assume that the input beam intensity is uniform across the lens diameter $$D$$ (i.e., it is a plane wave)?
2. Question 3: If not, then for Gaussian beams is it correct to expect a larger spot size?
• spot size in demanding applications is also influenced by the quality of the incoming beam .... there is always an amount of angular dispersion .... i.e. even a laser is not perfectly collimated .... and the collimation will differ slightly in x y dimension across the beam. Thus no such thing as a perfect small spot at the focal point. Also spherical aberration plays a role. Aug 31, 2023 at 11:59

Question 1: For a given optic, is the input beam typically assumed to be a plane wave?

From my experience usually in real-world optical devices you assume the input is a gaussian beam (the usual case for most lasers). For the most part (and based on what Nikon claims), the numerical aperture is more closely related to the refractive index of the medium and the "one-half angular aperture", or the half angle that forms from the optical axis to the edge of the objective.

https://www.microscopyu.com/microscopy-basics/numerical-aperture

While it is true that these depend on the optics inside, you are better off going with the more standard airy disc approximation:

https://www.edinst.com/de/blog/laser-spot-size-in-a-microscope/

$$D_{airy} = 1.22\frac{\lambda}{\text{NA}}$$

Now why it is important to do this comes up in your next question:

Question 2: Does this simple approximation assume that the input beam intensity is uniform across the lens diameter D (i.e., it is a plane wave)?

Not neccessarily. Here what you should also consider is that you will face several of the wave-like effects of laser propagation such as diffraction, which is why the airy disc approximation is a good rule of thumb when deciding the objective numerical aperture (as well as immersion medium).

Hence a) I would not consider it a plane wave, if you have to do a rigorous analysis I would already consider the beam to be gaussian.

If not, then for Gaussian beams is it correct to expect a larger spot size?

Here I think that the diffraction effects will dominate that of the wavefront but I cannot be too sure as I only used these diffraction-limited systems for two-photon microscopy kind of experiments and while the wavefront may affect the spot size, there are other more limiting factors. I would definitely recommend to start with the Airy disc approximation and if you need to do more rigorous calculations, then start already with a gaussian beam distribution and move on from there when designing your system.