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

In summary: But the intensity is also valid for any other value of r, including negative values (since it is a squared term).In summary, the intensity of a Gaussian beam can be calculated using the equation I = I_l * (1 - R) * exp((-2 * r^2) / w_0^2) * exp((-2 * t^2) / t_l^2), where r is the distance from the central axis of the beam and t is the laser pulse duration. The beam intensity is strongest on the central axis and decreases as the distance from the axis increases. When evaluating the intensity at the beam waist (z=0), r=0 should be used to calculate the maximum intensity. However, r can take on
  • #1
Carlos Criollo
9
0
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:

1.png

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

But, I don´t now what represents [itex]r[/itex] 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.
 

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  • #2
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.
 
  • #3
Thank you!

I am implementing this equation with numerical values, so, is r a number or is a variable in the equation?
 
  • #4
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.
 
  • #5
Therefore, r is the distance between the axis of propagation and what the variable is?
 
  • #6
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)

microscopelasersfigure2.jpg

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

220px-Laser_gaussian_profile.svg.png

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.​
 
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  • #7
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?
 
  • #8
Additionaly, The beam waist is in z=0, so what is the correct value of r in z=0?
 
  • #9
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.
 
  • #10
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?
 
  • #11
You probably want to calculate the maximum intensity, in which case use r=0.
 

What is the physical meaning of intensity in a Gaussian beam?

The intensity of a Gaussian beam refers to the amount of power per unit area of the beam. It represents the concentration of energy in the beam and can be thought of as the brightness of the beam at a particular point.

How is the intensity of a Gaussian beam affected by beam parameters?

The intensity of a Gaussian beam is affected by the beam's waist size, divergence angle, and wavelength. A larger waist size and smaller divergence angle will result in a higher intensity, while a shorter wavelength will also increase the intensity.

What is the formula for calculating the intensity of a Gaussian beam?

The formula for calculating the intensity of a Gaussian beam is I = P/(πw0^2), where I is the intensity, P is the power of the beam, and w0 is the beam waist size.

How does the intensity of a Gaussian beam change as it propagates through space?

The intensity of a Gaussian beam will decrease as it propagates through space due to beam divergence. This means that the beam will spread out and become less concentrated, resulting in a decrease in intensity.

Can the intensity of a Gaussian beam be adjusted?

Yes, the intensity of a Gaussian beam can be adjusted by changing the beam parameters, such as the waist size and divergence angle. It can also be adjusted by using optical components such as lenses or apertures to manipulate the beam's size and shape.

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