Gaussian Beam Focusing: Find A(0) at Beam Waist

[Lemenks]

A Gaussian beam has an intensity I(r,z), if the beam area at position Z is given by A(Z), then the beam gets focused by a lens of focal length f, what will the area of the beam be at the beam waist A(0) be?

So I have been trying to figure this out for ages, I had to replicate an experiment in which a student simply assumed the beam focused like a cone, in which case you say the beam is 0 at the beam waist and the beam is "close" to the beam waist and choose/measure a distance. This seemed like pretty poor experimental work to me as you can make the intensity become infinitely large by "choosing/measuring" a value closer to the beam waist. In this scenario I have taken position z from the beam waist to be the focal distance f:

A(f)/A(0) = (pi*w(f)^2)/(pi*w(0)^2) = (w(f)/(w(0))^2 = (f/x)^2

x is the distance from the beam waist that a person "chooses" or "measures". To see the above equation, I found it constructive to draw out a cone and put in the values.

I read several Gaussian optics manuals and a better expression seems to be

A(f)/A(0) = (f/ZR)

Where ZR is the Rayleigh length and is given by

ZR = pi*w(0)^2/gamma

However the problem with this is that I don't know w(0). I read a limit for w(0)>/= 2*gamma/pi, however this seems to result is nonsensical answers.

If anyone here knows or works with lasers, perhaps you could help explain it to me?

 

[Andy Resnick]

A Gaussian beam has an intensity I(r,z), if the beam area at position Z is given by A(Z), then the beam gets focused by a lens of focal length f, what will the area of the beam be at the beam waist A(0) be?

So I have been trying to figure this out for ages, I had to replicate an experiment in which a student simply assumed the beam focused like a cone, in which case you say the beam is 0 at the beam waist and the beam is "close" to the beam waist and choose/measure a distance. This seemed like pretty poor experimental work to me as you can make the intensity become infinitely large by "choosing/measuring" a value closer to the beam waist. In this scenario I have taken position z from the beam waist to be the focal distance f:

A(f)/A(0) = (pi*w(f)^2)/(pi*w(0)^2) = (w(f)/(w(0))^2 = (f/x)^2

x is the distance from the beam waist that a person "chooses" or "measures". To see the above equation, I found it constructive to draw out a cone and put in the values.

I read several Gaussian optics manuals and a better expression seems to be

A(f)/A(0) = (f/ZR)

Where ZR is the Rayleigh length and is given by

ZR = pi*w(0)^2/gamma

However the problem with this is that I don't know w(0). I read a limit for w(0)>/= 2*gamma/pi, however this seems to result is nonsensical answers.

If anyone here knows or works with lasers, perhaps you could help explain it to me?

I'm a little unclear about your geometry: you seem to have a well-specified gaussian beam (do you know I(r,z) or not?) that is focused by a lens placed at a particular z = Z, and then you want to know the properties of the refracted beam?

 

[Lemenks]

I'm a little unclear about your geometry: you seem to have a well-specified gaussian beam (do you know I(r,z) or not?) that is focused by a lens placed at a particular z = Z, and then you want to know the properties of the refracted beam?

Yes I know the intensity at I(r,z) and need to calculate it at I(r,0) - sorry if I made it unclear. I was just a little confused as to the proper way of making this calculation, the beam focusing as a cone shape seemed too approximate and also to contain really large errors so I was trying to see if there was a batter method.

 

[Andy Resnick]

Yes I know the intensity at I(r,z) and need to calculate it at I(r,0) - sorry if I made it unclear. I was just a little confused as to the proper way of making this calculation, the beam focusing as a cone shape seemed too approximate and also to contain really large errors so I was trying to see if there was a batter method.

Not sure what you have tried already- have you used these resources:

http://www.newport.com/Gaussian-Beam-Optics/144899/1033/content.aspx
http://uigelz.eecs.umich.edu/classes/pub/ece355/handouts/example_gaussian_beam_focused_lens.pdf
http://nicadd.niu.edu/~piot/phys_630/Lesson3.pdf