Achieving an input Gaussian beam waist different from the output LG beam waist

Question

The Gaussian beam is converted to a Laguerre Gaussian beam using a spiral phase plate (SPP). For this operation, I want the incident Gaussian beam to have a beam waist that is different from the beam waist of the resulting LG beam. Is there any feasible solution to realize that idea?

Answer 1

Here you can see the situation as far as I understand: You want to have different beam waists for each beam.

As pointed out in the comment, you can use a 4f system (a system of lenses which will change the beam properties)

Here you can see the simplest 4f system consisting of two lenses, L1 and L2, with focal lengths f1 and f2, which you would place in the optical path of the LG beam. Typically plano-convex lenses are suggested in these setups in order to reduce aberrations but it will depend entirely on your budget, goals, etc. For simplicity, the lenses depicted are only convex lenses.

Magnification will be given by $-f2/f1$ so you can simply use the beam waist of your beam and scale it by this ratio using this simple system.

There are some small implications of using this particular system:

• As noted, aberrations can occur by adding more optical elements in the path of the beam. You should be wary that it is important to characterize your beam before and after in order to correctly assess if the system needs a different lens system in order to decrease optical aberrations.
• Here you may notice that I used the paraxial approximation for my explanation and it is important to note that: Yes, this is not explicitly intended for gaussian beams as it would be more formally treated through wave optics, but for a large number of applications it is valid to start with the paraxial approximation and move from there in order to achieve the intended goal.
• If there are specific parameters which are entirely dependent on the wave-like behavior of the gaussian beam, it is still valid to use typical optical elements e.g. lenses, etc.
• The aforementioned anaylisis is more from the engineering perspective of how to scale (e.g. "magnify" although the ratio can be <1 as I showed in the graphic example) your beam but depending on your aim you may need to use different techniques to analyze the problem e.g. ABCD matrix.

EDIT: From your question, I could not determine whether your intention is to change the beam waist of the input gaussian beam and have the LG beam keep the properties of the source gaussian beam, or if only the LG beam should have a different beam waist. Typically you apply modifications on your optical system on the output rather than on the input as there is where most of the effects of your system will be observed. Therefore I assumed the typical use case.