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# Maths behind optimal beam expansion

#### Eracoy

##### Active member
LPF Site Supporter
I am curious about the maths involved in dot size and energy reaching the terminal beam point if a beam expander is involved. The expander I have is a LaserGlow X10DR 10x expander that I have adapted to be a universal fit. It has two adjustment points by twisting either the expander's lens or the expander base. So, the effect is that I can adjust both the beam's focus AND the expansion factor. Also, because I have such a long lens holder, it can go well beyond 10x expansion factor. I'd say easily 20x. The problem is, then, that the beam is clipped on the outside, because the exit aperture has a finite diameter. Does anyone have any ideas about how to calculate the way to get the most beam intensity (W/m^2) at the final (large) dot? I think it is possible to define the optimal beam expansion factor in terms of only the distance the beam will travel.

My question specifically is this:
Given a laser with divergence θ, and a beam expander with N-times magnification, I can work out that the spot size is proportional to θ/N.
However, at expansions above N=10x, the wings of the beam are clipped by the lens diameter and the housing of the expander. Below is a diagram of my situation.

How can I describe the energy per unit square meter reaching the target dot based on only N (expansion factor) and L (distance to target)? How would this be with a single mode diode which clips all around the dot versus a multimode where it only clips the width of the beam?

Thanks.

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#### steve001

##### Well-known member
I am curious about the maths involved in dot size and energy reaching the terminal beam point if a beam expander is involved. The expander I have is a LaserGlow X10DR 10x expander that I have adapted to be a universal fit. It has two adjustment points by twisting either the expander's lens or the expander base. So, the effect is that I can adjust both the beam's focus AND the expansion factor. Also, because I have such a long lens holder, it can go well beyond 10x expansion factor. I'd say easily 20x. The problem is, then, that the beam is clipped on the outside, because the exit aperture has a finite diameter. Does anyone have any ideas about how to calculate the way to get the most beam intensity (W/m^2) at the final (large) dot? I think it is possible to define the optimal beam expansion factor in terms of only the distance the beam will travel.

My question specifically is this:
Given a laser with divergence θ, and a beam expander with N-times magnification, I can work out that the spot size is proportional to θ/N.
However, at expansions above N=10x, the wings of the beam are clipped by the lens diameter and the housing of the expander. Below is a diagram of my situation.

How can I describe the energy per unit square meter reaching the target dot based on only N (expansion factor) and L (distance to target)? How would this be with a single mode diode which clips all around the dot versus a multimode where it only clips the width of the beam?

Thanks.
Your limited by the specifications of the expander.
Optionally
Try larger diameter lenses. Also be sure the lenses are AR coated for that wavelength. Try ivroptical.c om for various laser apps.

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##### Well-known member
LPF Site Supporter
If you are looking for the power within the spot, research power density. Look at my all too long signature for helpful links to online calculators.

#### Eracoy

##### Active member
LPF Site Supporter
Now that I think about it, I'm not sure what of the system is adjustable. Right now, I have a raw diode outputting into a G2 lens at an adjustable distance. The output of that is then immediately fed into a beam expander. The adjustable part is the focus going in to the beam expander. My thought was that once it came out of the BE aperture it would be adjusted to be collimated at an adjustable beam expansion ratio. The other possibility is that I am misunderstanding the optics going on between the diode and the beam exit. If this were the case, I would have a constant 10x expansion ratio and would just be adjusting the input width that results in an adjusting output width. So, what do you think? I'm working on the math behind the two scenarios, but I was wondering which was actually going on in my case.

##### Well-known member
LPF Site Supporter
Can you take a picture of your been expander or share a link to its specifications?

#### steve001

##### Well-known member
Now that I think about it, I'm not sure what of the system is adjustable. Right now, I have a raw diode outputting into a G2 lens at an adjustable distance. The output of that is then immediately fed into a beam expander. The adjustable part is the focus going in to the beam expander. My thought was that once it came out of the BE aperture it would be adjusted to be collimated at an adjustable beam expansion ratio. The other possibility is that I am misunderstanding the optics going on between the diode and the beam exit. If this were the case, I would have a constant 10x expansion ratio and would just be adjusting the input width that results in an adjusting output width. So, what do you think? I'm working on the math behind the two scenarios, but I was wondering which was actually going on in my case.

So you have one beam expander and one g2 lens? I'm betting the expander has a fixed expansion power of 10x. Is that correct? The beam expander has no movable parts?

#### Eracoy

##### Active member
LPF Site Supporter
The beam expander is an X10DR as seen here. The expander itself has an adjustable focus by twisting the exit aperture relative to the base. The laser itself is attached via an adapting focus ring which contains a g2 lens:

The effect of this setup is that I have an adjustable focus in and out of the beam expander. By combining the two adjustments in different amounts, you can change the beam expansion factor to something other than the default of 10, which assumes a collimated beam imput.

For small input beam diameters, such as a 532 or single mode 405, I can increase the expansion factor to nearly 40x before the edges get clipped by housing. As a note, to focus this requires unscrewing the front all the way and holding it a couple centimeters apart.

My question is about finding a balance between two constraints. On the one hand, a high expansion factor is good for power-arriving-at-dot because it lowers divergence. On the other, the beam expander clips off more and more power from the edges of the beam as the expansion factor goes up.

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#### steve001

##### Well-known member
The beam expander is an X10DR as seen here. The expander itself has an adjustable focus by twisting the exit aperture relative to the base. The laser itself is attached via an adapting focus ring which contains a g2 lens:

The effect of this setup is that I have an adjustable focus in and out of the beam expander. By combining the two adjustments in different amounts, you can change the beam expansion factor to something other than the default of 10, which assumes a collimated beam imput.

For small input beam diameters, such as a 532 or single mode 405, I can increase the expansion factor to nearly 40x before the edges get clipped by housing. As a note, to focus this requires unscrewing the front all the way and holding it a couple centimeters apart.

My question is about finding a balance between two constraints. On the one hand, a high expansion factor is good for power-arriving-at-dot because it lowers divergence. On the other, the beam expander clips off more and more power from the edges of the beam as the expansion factor goes up.

I don't think math is going to help you in this case because you are constrained by the physical structures of the parts. Math would if you were designing a custom system to be built to accomplish the objective. In this case you'll just have to manually make adjustments to achieve what you want.

##### Well-known member
LPF Site Supporter
If the beam has a consistent power distribution, when 50% of the beam diameter is cut off that means only 25% of the power is getting through. If just 10% of the diameter is cut off that might not make much difference, or be a wash compared to the smaller spot, or higher beam intensity within that spot which is gained by having a lower divergence.

Being a math moron, I've been googling for an online calculator where I could enter different beam diameters to determine the power lost from having a 10-20% smaller diameter beam due to being cut off inside the tube, but haven't found one yet. I do know that a 50% reduction of diameter, if the power is consistent throughout the beam, means only 25% of the power gets through. However, because there is normally lower power at the edges of the beam I wouldn't worry about loosing 10% of the diameter very much. Although another caveat; this depends upon your specific diode and its power distribution throughout the beam. I doubt you would have a Gaussian distribution of power if using multimode diodes, but some single mode diodes might have close to that kind of beam power distribution to where the last 10% of the diameter doesn't contain much power compared to the core.

Gaussian power distribution​

I suppose knowing a 50% blockage of circumference means only 25% the power, the rest could be figured out. It just comes down to the area of a circle, humm.. I didn't google that term. For satellite antennas, what I work with in my job, doubling the diameter means 6 dB more gain which is 4 times more power received from or delivered to the satellite. That's what I am basing this off of.

My signature has links to some online calculators you can use to determine how much the divergence affects the spot size at X distances.

Since I am still learning about lasers, if a pro can answer this better or find something wrong with my statements, please chime in, I'd like to know if I am in error. Planters, you around?

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#### Eracoy

##### Active member
LPF Site Supporter
I am not so much interested in using this formula to find the optimal setting for my expander, but most want to see the shape of the curve of optimal ratios for a purely academic reason. So, here is what I have worked out so far as far as math:

Here, we view a cross section of the beam along the dot's radius. Note that before it hits the expander, it is perfectly gaussian (I assume this). However, afterwards it is gaussian but clipped to a certain radius from the center of the expanded beam. We are mostly interested in the area between the lengths d2 (expander aperture size) and d3 (dot size). We can look at some equations related to this:

I made some assumptions here, and all I'm looking for is the dimensions to add up and get a reasonably proportionate idea of what the optimal points are.

This would leave our final power/area equation at Power-Of-Dot / Area-Of-Dot in W/m^2. This is preliminary to get my thoughts down on what I'm looking for, but I'd appreciate any feedback.

##### Well-known member
LPF Site Supporter
The limit of my working with math when it comes to beam expanders is a simple formula, to determine the amount of expansion divide the negative focal length of the expander lens with the positive FL of the collimation lens, the result is the X factor of expansion. i.e. -25mm FL expander lens with a 250mm FL PCX lens is 10X, or -10mm FL expander lens into a 100mm positive FL PCX lens is 10X.

i.e. Introduction to Laser Spatial Filtering

Beam Expander without Spatial Filter

For using a laser beam expander devoid of a spatial filter, the Galilean telescope type can be selected. The Galilean type lacks an internal focus, an issue for higher power pulsed lasers. Figure 5 delineates a 20X Beam Expander with basic characteristics identical to those of the design illustrated in Figure 4, but with improved performance at a lower cost due to its simplified form.

The focal length of the negative input lens is -8mm. The use of a 160mm focal length output lens provides the overall beam expander with magnification of 20X. This factor is applicable to the increase of output beam diameter, and the reduction in beam divergence.