What is the significance of r in the intensity formula for a Gaussian beam?

[Carlos Criollo]

Hi!.

I´m modeling a gaussiam beam. I found that considering the electric field distribution of Gaussian laser pulses along the axis of propagation, we can write the intensity as:

Where y = I_l
R is the power is the power reflection coefficient
y = w_0 is the beam waist radius at y = z=0
t_l is the laser pulse duration

But, I don´t now what represents r in the formula. I find that r is the distance from the center axis of the beam, but i don´t understand this explanation.

Thank you very much.

 

[UltrafastPED]

The Gaussian decay is symmetric about the axis of propagation ... the further from the axis, the weaker the Gaussian beam. Thus r is the radial distance from the axis of propagation. Cylindrical coordinates, with z the axis of propagation, is a convenient coordinate system.

 

[Carlos Criollo]

Thank you!

I am implementing this equation with numerical values, so, is r a number or is a variable in the equation?

 

[Redbelly98]

r is a variable, representing (transverse) distance from the beam's central axis. So, the intensity has a maximum on the axis (r = 0), and becomes weaker at greater distances from the central axis.

 

[Carlos Criollo]

Therefore, r is the distance between the axis of propagation and what the variable is?

 

[Redbelly98]

No, it's the distance between the axis of propagation and the location where you are trying to calculate the intensity.

I don't know what you mean by "distance between the a.o.p. and what the variable is", since "what the variable is" does not describe an object or a location in space (as far as I can tell).

EDIT: maybe these images will help your understanding? (Don't worry too much about the captions included with the figures, I'm posting these mainly for illustration and have added some explanatory notes of my own)

The dotted line is the central axis of the beam.
"r" represents the distance above or below that line.

Here "x" is what we are calling "r".
The circular-shaped figure represents the beam intensity, with the beam directed out of the plane of the page.​

 

[Carlos Criollo]

Sorry, I´m from Colombia, so I have to improve my English. It is more clear for me. Therefore. I am trying to evaluate the Intensity of the Gauss Beam at the beam waist in (z=0), so what is the correct value of r?

 

[Carlos Criollo]

Additionaly, The beam waist is in z=0, so what is the correct value of r in z=0?

 

[Redbelly98]

The intensity at the beam waist is when z=0, and is given by the equation you originally posted.
The maximum intensity occurs for r=0 and t=0, again in the equation you originally posted.

So, if you want the intensity as a function of r, there is no "correct" value of r. It is a variable that can be any value between 0 and infinity.

But if you want to know the maximum value of the intensity, then use r=0. And t=0 also.

I hope that helps.

 

[Carlos Criollo]

It is more clear for me, but if I want to calculate the intensity at the beam waist, then, is there a specific value of r?

 

[Redbelly98]

You probably want to calculate the maximum intensity, in which case use r=0.