# Focusing gaussian beam using a lens

#### yong0047

I am studying further about Gaussian optics.
When Gaussian beam pass through a lens, the waist location is given by

(z'-f) = (z-f)M^2

Where, z' is the waist location after lens, z is waist location before lens, f is the focal length of the lens M is the magnification.

In Gaussian optics, the magnification M is given by Mr/(1+r)^(1/2), the r of Mr should be subscript is the ray optics magnification f/(z-f), the r is given by z0/(z-f), z0 is the Rayleigh length.

However, I try to use ABCD laws on q-parameter, and also geometrically and algebraically, still can't prove the waist location is given by (z'-f) = (z-f)M^2. Can you give me some idea to solve it?
1. Homework Statement

2. Homework Equations

3. The Attempt at a Solution

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#### Redbelly98

Staff Emeritus
Homework Helper
Welcome to PF.

I've not seen the focal position put in terms of m like that before, so I'm not how much I can help. But, perhaps you could show more details of the ABCD calculation you did. If the error is in that, I can probably help.

#### yong0047

then is it possible for you to prove

z' = f(z^2 + z0^2 - fz)/(z - f)^2 + z0^2

??? the ABCD should be no problem. Just the algebraic don't how to prove it to be.

#### Redbelly98

Staff Emeritus
Homework Helper
I don't see an obvious way to prove that, sorry. Even taking the ray-optics limit z0→0, it's not clear to me how to prove the resulting equation.

Since you're new here, I'll just point out that the policy here is for the student to show some work towards solving the problem, before getting help from others.

Obviously you're an advanced student, but we do get other people here who don't bother to try any work, or even look up basic equations in their text book, and expect others to give them answers--which they don't learn from since they weren't encouraged to engage their own brain in the problem.

Regards,

Redbelly98

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