Laser Beam Propagation: Calculating Power & Radius

[MiddleVision]

Hi all,

I am interested in finding the power of a laser diode as function of the distance z, assuming a free space propagation. I think to have enough information to work it out but I am stacked at the moment.
Given:

• the initial power, P₀ = 1mW
• beam width clip (e^-2 = 13.5%), w_x = 3μm, w_y = 1μm
• wavelength, λ = 650nm
• divergence beam width (full angular), θ = 1mrad
• M² = 5

What would be the power P(z) and the radius w(z) of the beam at a distance z?

Thanks in advance for any help!

 

[blue_leaf77]

The power is constant everywhere. There is no way the power will change unless the energy inside the laser changes, e.g. due to battery depletion.

 

[MiddleVision]

Thanks for your prompt reply. About the radius of the beam, how large would be the laser spot after, let's say, 1m or 50m?

 

[blue_leaf77]

The information about the angular divergence which you already have there should be sufficient to determine the width at a given distance, assuming this distance is far enough from the beam waist.

 

[MiddleVision]

I have used the following equation (2.8) to simulate the width at a given distance, using the beam width clip as w₀. How do I include the angular divergence in the simulation of w(z)?
Thanks

Source: Sun, Haiyin. Laser diode beam basics, manipulations and characterizations. Springer Science & Business Media, 2012.

 

[blue_leaf77]

How do I include the angular divergence in the simulation of w(z)?

Isn't the beam divergence given already in equation (2.11)? You can plug in this into equation (2.8).

 

[MiddleVision]

I thought that the 2.11 is valid only far away from the beam waist. After which value do you think it's "sensible" to use the far field approximation?
I have plotted the beam radii propagation for the X and Y axis and after a distance of about 4 meter the two curves are overlapping each other, i.e. same radius in the plane and the spot is not elliptical anymore. Is that an expected result for such a laser diode?

 

[blue_leaf77]

$\theta$ is indeed derived by taking the limit of $\frac{d w(z)}{dz}$ for $z\rightarrow \infty$, but it turns out that $\theta$ has such a form given in (2.11). Therefore you can regard this equation as an identity and you use it to replace any factor containing the RHS of (2.11) with $\theta$.

I have plotted the beam radii propagation for the X and Y axis and after a distance of about 4 meter the two curves are overlapping each other, i.e. same radius in the plane and the spot is not elliptical anymore. Is that an expected result for such a laser diode?

Well at least for a perfect Gaussian beam, the smaller beamwaist you have, the bigger the angular divergence is. So, the behavior of the beam you observed makes quite some sense.

 

[MiddleVision]

I had to refresh my memory about this topic, now it makes sense.

Thanks a lot for your replies :)