Gaussian Beam Optics Experiment: Measuring Beam Waist & Rayleigh Range

[n0_3sc]

I am doing a Gaussian Beam Optics Experiment - where I observe how lasers have a gaussian intensity profile.
I've measured the beam radius at several points from the Laser in the far-field (ie. position 'z' >> Rayleigh Range).

I want to know why is there a beam waist inside the laser.
How do I calculate the beam waist from the measurements I have.
And what is the significance of the Rayleigh Range?

Any other important points would help.

 

[marlon]

The reason that there is a beam waist is just the fact that a Gaussian beam has certain properties:

1) These beams are a solution of the scalar Helmholtz wave equation. This implies that such beams contain a complex exponential, hence such beams will undergo damping at some point in space !

2) In the paraxial approximation, such beams give the same results as predicted by the Fresnel equations. This means that gaussian beams are a useful concept in optics.

3) Waves all exhibit the property of diffraction. This means that a beam cannot be focussed to one single point. This is actually the answer to your first question.

Now, if you look at the expression of a spherical Gaussian wave $$e^{ - \frac{x^2 + y^2}{w^2}}$$, the w defines the latitude of the spot (image). This w is calculated at that specific distance (starting from the position of the wave-source) where the wave's amplitude is reduced from it's original value to 1/e and the intensity is reduced to (1/e)². Now, if you take a plain gaussian wave (R = 0 and z = 0). You will see that this w reaches a minimal value which is called the waist. This is not just one single point because we are working with Gaussian waves here that exhibit the diffraction property !

To calculate the waist, you need to know at what angle the waves diverges starting from the waist. This angle theta is equal to $$\theta = \frac{ \lambda}{ \pi w'}$$. The w' is the magnitude of the waist. This answers your second question.

More generally :
If you know the curvature radius R of the beam at a certain position and the spot magnitude w1 than you can calculate both the position and the magnitude of the waist with these formula's:

position
$$z = \frac {R}{1 + ( \frac{\lambda R}{\pi w1^2})^2}$$

magnitude
$$w' = \frac {w1}{ \sqrt{1 + ( \frac{\pi w1^2}{\lambda R})^2}}$$

marlon

 

[marlon]

To continue a little bit...

In the case of Gaussian laser beams one defines the length w of the waist w' as $$w = \sqrt{2} w'$$.

The connection between the Rayleigh range z' and the waist w' is $$2z' = \frac{2 \pi w'^2}{\lambda}$$. This answers your third question

In far field (z >> z') the connection between waist w' and spotmagnitude w1 is still $$w1 = \frac{\lambda z}{\pi w'} = z \theta$$

This last formula is the same as the one i gave to calculate the waist (it's magnitude) in my first post.

The concept of waist is very important because it is used to study the stability of gaussian laser beams in resonators. So people that work in photonics industry need these concepts every day.

regards again

marlon