# How to calculate spot size of a laser focused through a microscope objective?

I am trying to figure out how to approximate the spot size of a laser beam that is focused using a microscope objective where the size of the laser beam is smaller than the diameter of the objective. I have tried googling and reading various other questions online but still haven't been able to cobble a coherent plan together. From this excellent response I've been able to cobble together an understanding that the working distance and the focal length of a lens are different. But I'm still at sea as to how I convert the information I'm given about an objective, e.g. magnification, NA and focal length to spot size. Would it be sufficient to simply treat an objective as a simple optic with the focal length given by the manufacturer and then use something like this spot size calculator to figure out the focused size? If not, would someone mind walking me through the steps to approximate (doesn't need to be exact, back of an envelope is great) spot size for a $$5 \;\text{mm}$$ well-collimated beam that is focused using this objective?

## 2 Answers

The spot size calculator is just what you need. Assuming your laser has a TEM$$00$$ Gaussian beam (the most common and desirable beam cross section), the math behind it is very simple. $$w = 4\lambda f/\pi D$$.

The calculator gives the spot diameter assuming your optics are perfect. A lens as expensive as the one you picked should be pretty good. You might investigate cheaper alternatives. You only need one wavelength. Is that lens optimized over the visible wavelengths? You don't need to worry about off axis aberrations. Spherical aberration is your primary concern.

See Cross-sectional area of a Gaussian beam of particles for more about the nature of Gaussian beams. There is a discussion of lasers.

"From Fundamentals of Photonics" by Saleh and Teich

We have an (ideal) lens of focal length $$f$$ and we are sending in a Gaussian beam with waist $$W_0$$ (Rayleigh range $$z_0$$) which is focused distance $$z$$ in front of the lens. The lens refocuses the beam distance $$z'$$ from the lens with waist size $$W_0'$$.

The textbook derives the following:

\begin{align} W_0' =& M W_0\\ (z'-f) =& M^2(z-f)\\ M =& \frac{M_r}{(1+r^2)^{\frac{1}{2}}}\\ M_r =& \left|\frac{f}{z-f}\right|\\ r =& \frac{z_0}{z-f} \end{align}

These expressions are not very illuminating in my opinion but we can consider two limiting cases. The first is that $$z-f \gg z_0$$ which says that the focus of the beam is like a point source and the equipotential lines of the gaussian beam can be treated as rays. In this case $$r\ll 1$$ so $$M=M_r$$ and this is in fact the usual ray-optics result for magnification.

The other limit, which is perhaps of more interest to you, is the opposite case in which the lens is in the Rayleigh range of the incident beam and the Rayleigh range is longer than the focal length, that is, $$z_0 \gg f, z$$. This is what we might call a "collimated" Gaussian beam because for the entire important space having to do with the lens the Gaussian beam keeps essentially the same radius. In this case, $$r\gg 1$$ so that

$$M \rightarrow \frac{f}{z_0}$$

Ignoring negative signs which leads to the well known and useful formula:

\begin{align} W_0' =& \frac{\lambda f}{\pi W_0}\\ z' =& f \end{align}

That is if you have a collimated beam of waist $$W_0$$ impinging on a lens with focal length $$f$$ you can easily calculate the resultant waist using the formula just above.

Regarding some of your other questions, if your beam is collimated then the formula I've given will work if you use only the given focal length for the lens (note that the spot size depends on the size of the beam you send in). The focus will appear distance $$f$$ away from the front focal plane of the lens.

• Note that the front focal plane is not necessarily located at the frontmost surface of the objective. This explains the difference between focal length and working distance.

• The magnification of the objective is a pretty useless parameter when thinking about the objective as a standalone optic. The magnification is defined as follows. Suppose you use the objective to collimate the light from a sample. Then you place a tube lens downstream of the objective to refocus the light and create an image. This image will have a magnification given by the ratio of the focal lengths of the objective and the tube lens. This means the magnification varies depending on the tube lens focal length.. so what tube lens focal length should you use? Different suppliers use different conventional tube lengths in their microscopes thus determining magnification when certain objectives are used. See https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=10764. Apparently, 200 mm is a common focal length. So if your objective has a focal length of $$f=4mm$$ it would be described as a $$50x$$ objective. You could of course change the magnification by changing the tube lens. For this reason (as someone who builds imaging systems from scratch more often than using commercial microscopes) I feel the magnification is a pretty silly parameter for objectives.

• Finally Numerical Aperature. NA will not help you with this problem. NA is a subtle specification for an objective in my opinion. If you have an object distance $$d$$ away from an objective which has a radius $$R$$ you might guess that the NA is given by $$NA \approx \sin\left(\arctan\left(\frac{r}{d}\right)\right)$$. But then the NA would depend on the distance between the object and the lens and you could make it arbitrarily close to unity (maximal in free space) by moving the object closer and closer. Do you get better resolution in this way by increasing the NA? No! This is not how NA is specified at all for optics. Rather, you should think of NA not as the maximal collection angle for an objective but rather the maximal non-aberrated collection angle. You can always decrease the NA of an optic by placing an aperture stop. When the NA of an objective is specified this means that if the objective is used as specified (meaning with proper working distance, for example) then if you use the NA corresponding to the specced NA then aberrations will be negligible, the objective will be diffraction limited. You can reduce the NA by hand and worsen your resolution. If you can increase the NA (some objectives may have an aperture stop built into them to avoid this) you will not get an increase in resolution because you will introduce aberrations into the system due to the introduction of the ray with very large angles that are hard to pass through the imaging system without aberration.