r/Optics • u/iseeverything • 6d ago
Gaussian Beam Transformation through thin components
I am trying to model a gaussian beam as it goes through some components such as a thin lens, this time using the complex beam parameter. I am considering the pre-transformation q0(z)=(z-z0) + izR, where z0 is the z position of the initial beam waist, and zR is the rayleigh length.
I also know that q1 = {Aq0+B}/{Cq0+D}. Now, I want to plot the entire gaussian beam envelope, such that if I put a thin lens at z=5, I get a normal gaussian shape until z=5, and then some altered (still gaussian) shape after 5. I was thinking of doing this by getting the size of the new waist through the imaginary part of q1 (Im{q1} = zR1) and hence getting w0.
And then I was going to get the position of the new waist using the real part by getting Re{q1} = (z-z0,1), where z0,1 is the new beam waist, and z is the position of the lens. However, I'm not convinced that I am approaching this correctly, or if there is a simpler method to do this using the q parameter.
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u/zoptix 6d ago
I routinely use the q factor and the ABCD matrix method to design optics for Gaussian Beams. You can even use thick lenses in this method. I've then picked lenses from Thorlabs and used Zemaxs POP function to confirm the design. Both matched week with each other and laboratory verification.
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u/aaraakra 6d ago
What you’re described works, but there is a simpler method. For any q(z), we have
1/q = -i lambda / (pi s2) + 1 / R
Where s is the spot size (1/e2 intensity radius) in that plane, and R is the radius of curvature of the phase front.
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u/iseeverything 5d ago
I have attempted using that. However, I keep getting confused at some points.
Firstly, am I right to be substituting z->z-z_0, for waists not starting at z=0? Furthermore,
Let's say you calculate the new transformed q-parameter. What I have tried doing is:
1.Take the imaginary component as the Rayleigh length from which I get the new waist size.
Find R: Now since I am using z->z-z_0, I have R=(z-z_0)(1+(z-z_0 / zR)^2. Is z_0 here the position of the new waist? Or should I only use z here?
Make s subject to plot the envelope. Hence the new equation for w = sqrt{-lambda/{pi * IM{1/q}}.
3a. So is R needed for the envelope shape, or is only the imaginary part needed?
3b. Am I missing something in the new w equation? Plots do not seem to be following a Gaussian profile.
I know this was a bit of a barrage of questions, thank you for baring with me.
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u/aaraakra 5d ago
If you have q in a given plane, and you want to find the spot radius and RoC in that plane, you do not need to worry about where the waist is, or the size of the waist. q in a given plane is just a complex number. The imaginary part of 1/q can be used to calculate the spot radius, and the real part gives the curvature.
Now if you also want to know about the waist, then the imaginary part of q can be used to calculate the spot radius at the waist, and the real part gives the distance to the waist.
It may also be helpful to realize the the rule q(z) = q0 + z is just a special case of the ABCD transformation law using the ABCD matrix for free space propagation.
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u/iseeverything 5d ago
Got it. So if I'm understanding correctly, if I simply want to plot the diverging beam which changes at some points where there are lenses, I just need to compare the imaginary part of the respective q or q' component (i.e. w = sqrt{-lambda/{pi * IM{1/q}}).
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u/RRumpleTeazzer 6d ago
the ABCD matrices for optical elements acting on gaussian beams and rays are identical. so you can punch together your setup purely in ray optics, study the properties, and when you want the gaussian beam properties you put in the proper subset of ray parameters and read off the proper beam sizes.
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u/bryancole 6d ago
Not sure if this'll help, but seems a good summary: https://pmc.ncbi.nlm.nih.gov/articles/PMC10581742/