# How to find rayleigh range and beam waist if I took several measurements of beam radius at some fixed distance from the lens?

Figure 1.

Figure 2.

Set up is like in the figure 1. Collimated beam of 633 nm is passing through a biconvex (focusing) lens.

Figure 2 shows Rayleigh range, it's same scenario as figure 1, light just goes in opposite direction.

Goal is to find beam waist (the focus point) and Rayleigh range, $w_o$ and $z_o$ using the formula of the radius of beam at a position along z:

$2 w(z) = 2 w_o \sqrt{1+ (\frac{z}{z_o})^2}$

I think (correct me if I'm wrong, as this is only a guess) by using knife-edge method I can get beam radiuses at some distances from the beam waist.

In my measurement, the distance between the lens and power detector was fixed at 5cm, however distance between knife-edge and lens was being changed the following way: 15 mm, 30 mm and 45 mm.

So I got three different beam radius values

radius when knife edge is at 15 mm from lens = 1.53 mm

radius when knife edge is at 30 mm from lens = 2.67 mm

radius when knife edge is at 45 mm from lens = 3.69 mm

Now my problem is, how do I interpolate these three values to find $w_o$ and $z_o$ using the formula of the radius of beam at a position along z:

$2 w(z) = 2 w_o \sqrt{1+ (\frac{z}{z_o})^2}$

????

Especially considering that my z values (15mm, 30mm, 45mm) were from lens to knife-edge, and not from beam waist to knife-edge.

Also, what's the relationship between Rayleigh range and focal length?

Are they related?

If I subtract $\frac{d}{2}$ from $f$ in figure 1, would I get $z_o$ (Rayleigh range)?

So I think I should assume (correct me if I'm wrong, as I'm not sure at all) that my values when knife edge was at different distances from the lens, even though power detector was fixed at 5cm, as if I was somehow measuring beam radius at 15 mm, 30 mm and 45 mm from the lens.

However, I need to know the distance between focal point (beam waist) to where I measured beam radiuses.

I don't know that. In fact, I'm supposed to find distance from the lens to the beam waist/focus point.

So I guess I can do linear trendline (y = mx + b) in excel to fnd $w_o$ (beam waist), and this is what I get. I subtracted 2.75 mm (approximate value of focal length) from z values. So that z values would correspond to distance between focus point and knife edge.

Not sure what to do next. 0.668 mm seems to be when x = 0, or when z = 0, so right at the supposed beam waist... However, relationship between the radius of the beam between focus point and some distance z is not y = mx +b, it's given by the above formula. I don't know how to "fit" that formula to the curve...

Or should I set up three equations with two unknowns ($w_o$ and $z_o$) ??? But I don't know how, and how to solve such a system... any help?

You have three measurements on the curve: $$2\omega(z-z') = 2 \omega_0 \sqrt{1 + \left ( \frac{z-z'}{z_0}\right )^2}$$

where $z$ is your z value, $z'$ is the translation from your origin (at the lens) to real $z$ and $\omega(z-z')$ is your measured beam, $\omega_0$ the beam waist and $z_0$ is the Rayleigh range.

Now you have 3 equations and 3 unknowns: $$2(1.53\times10^{-3}) = 2\omega_0 \sqrt { 1+ \left(\frac{15\times10^{-3}-z'}{z_0} \right )^2}$$ $$2(2.67\times10^{-3}) = 2\omega_0 \sqrt { 1+\left(\frac{30\times10^{-3}-z'}{z_0}\right )^2 }$$ $$2(3.69\times10^{-3}) = 2\omega_0 \sqrt{ 1+\left(\frac{45\times10^{-3}-z'}{z_0}\right )^2 }$$

$z'$ is likely to be where the linear fit of your measurements meet the $x$ axis.

• "$z'$ is the translation from your origin (at the lens) to real $z$" what? I don't get that? I already know that focal length is approximately 2.75mm, why can't I just subtract it from $z_1$ value to get 15mm-2.75mm= 12.25mm?? I think by z' you meant distance between lens and focal point, right? By linear fit, do you mean how in my excel graph? But you didn't seem to subtract 2.75 mm in your calculations... So is it 0.668 mm? Also, I tried to put the 3 equations into wolfram, and wolfram only found complex solutions... i.imgur.com/5jQwNJU.jpg I need a real solution...
– Jack
Aug 12, 2018 at 19:05
• Your $z$ origin could be anywhere. You need to arrange things such that the waist of the beam is at the $z = 0$ That's what $z'$ is for. Aug 13, 2018 at 0:08
• You want to have $z'$ work for you. Somewhere that is easy to measure from rather than somewhere virtual like the focus of a lens. A lens mount perhaps. Aug 13, 2018 at 2:00
• "rather than somewhere virtual like the focus of a lens" but the z in the formula is supposed to be measured from the beam waist not from the lens. And beam waist happens to occur at the focus point/focus of a lens.
– Jack
Aug 13, 2018 at 3:44
• @Jack That's the function of $z'$ which translates your unhandy point (the waist of the beam) to a handy point (some optical mount or some mechanical reference). If your reference point is actually on the waist of the beam then $z'$ is 0. Your data suggests that the waist is -6.5 cm from where you've been measuring, but you'll a few more measurements to convince yourself that they have been taken outside the Rayleigh range. Aug 13, 2018 at 12:27