# Calculate the diameter of of Gaussian beam at FWHM (or at 1/e2)

I need to collimate a gaussian laser beam out of a single mode fiber (NA0.15) with a single lens so that the full-width half maximum (FWHM) of collimated beam is 10mm.

This question may sound naive, and I understand that when lens is set at it focal length from the fiber, the diameter of the beam is easy to calculate: diameter = 2*FL/((sqrt(1-NA^2))/NA). So for instance with FL=40mm lens, the diameter will be 12.14mm.

But the term "diameter" is this case seems to me a bit vague and to be honest I can't figure out how much is the diameter of FWHM if the "diameter" is said to be 12.14mm. I could try, measure and live with that, but I am sure there should be a simple relation to calculate the diameter at FWHM (or at 1/e2) asssuming that the beam is perfectly gaussian, and knowing the NA of the fiber and the FL of the lens.

Correct me if I'm wrong, but in these calculations, the intensity doesn't play a role, because the signal comes from a single mode fiber.

Can you please suggest a solution for this question?

As you use a single mode fiber, the beam is nearly gaussian (for a single mode fiber: $$M^2=1.1$$). So you may use the gaussian formula to calculate the collimated beam radius.

You can have a look here: https://www.rp-photonics.com/fiber_collimators.html

For calculations, the simpler case is that of a single-mode fiber. Here, the beam radius can be calculated with fairly good accuracy using the following equation:

$$w_{collimated}=f\theta_{fiber}\simeq \frac{f\lambda}{\pi w_{fiber}}$$

This assumes that the beam profile of the fiber mode has an approximately Gaussian shape, so that we can apply the corresponding formula for the beam divergence half-angle $$\theta_{fiber}$$.

It is also assumed that the distance between fiber end and lens is close to the focal length f of the lens. If the distance is too small, the beam will diverge, and for too large distances it converges to a focus at some distance.

The beam radius in the fiber $$w_{fiber}$$ is the mode field diameter (MFD) given in the specification of the fiber or can be easily calculated.

The beam radius for a gaussian beam is $$w_{collimated}$$ is defined at $$1/e^2$$. The intensity does not play a role in the waist measurment, because it is the percentage of energy in the $$1/e^2$$ part that count. When intensity increases, the percentage stays the same, and so does the beam radius.

• Hi Aliceo,thanks a lot for your answer - that was exactly what I needed for my calculations. The NA of my fiber is 0.12 according to specs, so I simply used it as divergence half-angle term in the equation. Commented Jun 9, 2017 at 16:06