Laser Beam Diameter which Minimizes Volume?

In summary, the question asks for an expression to minimize the sampled volume of a 10 meter long gas column when interrogated with a HeNe laser with a wavelength of 633nm. The solution involves using the Gaussian beam equation, Rayleigh range, and beam waist to derive an expression for the beam diameter that will minimize the sampled volume. The volume can be expressed as a function of the beam diameter and can be minimized by finding the optimal beam diameter that converges at the center and then diverges again.
  • #1
LucidDragon
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Homework Statement


Consider a 10 meter long gas column. We interrogate the gas molecules with a HeNe laser (lambda=633nm) at the minimum possible gas volume. If we focus the beam tightly, it will eventually diverge and the sampled gas volume will increase. Consider a minimum beam diameter D. Derive an expression for the beam diameter D that minimizes the sampled volume.

Homework Equations


Gaussian Beam: Theta=ω/z=λ/(π*w0)
Rayleigh Range: z0=[π*(w0)^2 ] / λ
Beam Waist = 2*w0
2*w0=(4*λ*f) / (π * D)

The Attempt at a Solution


Assuming that gas column refers to a cylindrical column, then the Volume would be: V=L*π*r^2=10*π*r^2. However I'm not sure where to go from here. I tried rewriting volume as: V=10*π*w0^2, then taking the derivative and setting it to zero to minimize volume, but that leads nowhere and is likely wrong. Any help or even hints on where to start would be greatly appreciated. Thank you!
 
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  • #2
I would expect the ideal beam to converge up to the middle, then diverge again. The width at the center is a free parameter - a smaller width leads to a smaller volume close to the center but a larger volume at the edges as divergence increases. Express the volume as function of this parameter (probably via an integral) and minimize.
 

1. How is the laser beam diameter related to the volume it minimizes?

The laser beam diameter directly affects the volume it minimizes. A smaller diameter will result in a smaller volume, while a larger diameter will result in a larger volume.

2. What factors influence the optimal laser beam diameter for minimizing volume?

The optimal laser beam diameter for minimizing volume is influenced by factors such as the material being used, the distance between the laser and the material, and the power of the laser.

3. Is there a specific formula for calculating the optimal laser beam diameter for minimizing volume?

Yes, there is a formula that can be used to calculate the optimal laser beam diameter for minimizing volume. It takes into account the material properties, laser power, and distance between the laser and material.

4. Are there any limitations to using a smaller laser beam diameter to minimize volume?

Yes, there are limitations to using a smaller laser beam diameter. A smaller diameter may result in a weaker laser beam, which can affect the precision and accuracy of the cutting or shaping process.

5. Are there any alternative methods for minimizing volume besides adjusting the laser beam diameter?

Yes, there are alternative methods for minimizing volume, such as using different laser settings, changing the material properties, or adjusting the distance between the laser and material. However, the laser beam diameter is a crucial factor in minimizing volume and should be carefully considered in the overall process.

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