How Is the Distance from the Beam Waist Determined in Gaussian Beam Optics?

[ettojar]

optics--radius of curvature

Homework Statement

At a particular point, a Guassian beam of wavelength λ has a radius of curvature of R1 and spot radius w1. Determine distance (z) of this point from the beam waist in these terms.

Homework Equations

R= z + (z0^2/z)
z0 = (pi*w0^2) / λ
w1 = w0 * sqrt(1 + (z/z0)^2)
(may be more that I'm unaware of)

The Attempt at a Solution

I have basically tried solving for all the variables and substituting them in the other equations but it all falls apart at the end and I get something like
z^2*R - (pi*w1^2)/λ = z^3-z^2

 

[Jhero]

*z0
and I can't seem to get rid of that z^2*R.

Hello,

Thank you for your question. To determine the distance (z) from the beam waist, we can use the following equation:

z = (pi * w1^2 * R1) / (4 * w0^2)

This equation is derived from the Gaussian beam propagation formula, which states that the spot size (w) at any distance (z) from the beam waist is related to the spot size at the waist (w0) and the radius of curvature (R) by the following equation:

w = w0 * sqrt(1 + (z/z0)^2)

where z0 is the Rayleigh range, defined as:

z0 = (pi * w0^2) / λ

Substituting this into the previous equation, we have:

w1 = w0 * sqrt(1 + (z/z0)^2)

Solving for z, we get:

z = z0 * sqrt((w1/w0)^2 - 1)

We can then substitute this into the equation for z0 to get:

z = (pi * w1^2 * R1) / (4 * w0^2)

I hope this helps. If you have any further questions, please feel free to ask.