how can a beam at a focal point be larger than the original!? (head scratcher..)

[opcom]

This 'question' is about laser beam spot sizes and plano-convex (or positive meniscus) lens focal lengths. I got the formulas off the www and got some pretty weird answers, so I have tried to think and answer the first part and provide examples to show my work as it were. The last part, I cannot answer. Anyone know optics well?

This is long, but it makes sense and the math was not bad.

I used two examples, a HeNe laser and CO2, because the big difference in wavelengths seems to have a real effect on spot size when the focal lengths get long. Anyone care to comment?

A laser beam - due to diffraction and beam divergence which in turn are related to dimensions being finite - will focus to a spot size that is directly proportional to the focal length of the lens and inversely proportional to the diameter of the laser beam at the point it meets the lens. The formula is:

spot dia. = 1.27 x f x wavelength x M^2 / D

where
f is lens focal length,
D is the beam diameter at lens position (generally smaller than the lens diameter),
and
M^2 is a number indicating the beam diffraction and divergence, I am assuming "1" that is to assume a perfect Gaussian beam profile.

In the CO2 laser, the wavelength is 10.6 micron (0.0106mm), so the above relation:
spot dia. = 1.27 x 0.0106 x M^2 x f / D
becomes;
spot dia. (in mm) = 0.013 x M2 x f / D

It could be any wavelength of laser, green, red, does not matter..) All units in mm.

With a f = 100 mm (4 inch) focal length lens and a beam diameter of 6 mm we get (assume M^2=1 in every case)

spot size (in mm) = .013 x 100/6 = 0.2 mm = 200 micron
for beam dia 2mm we get:
spot size (in mm) = .013 x 100/2 = 0.5 mm = 500 micron

The smaller beam diameter made the bigger spot!
now how about this:

I use a lens with a 2000mm focal length - I theoretically want to focus the 2mm beam about 7 ft away from the laser.

spot dia. (in mm) = 0.013 x M^2 x f / D
spot dia. (in mm) = 0.013 x f / D
spot dia. (in mm) = 0.013 x 2000 / 2
spot dia. (in mm) = 0.013 x 1000 = 13mm ??

How did the 2mm diameter pass through a plano-convex focusing lens and get focused to a huge 13mm spot at the focal point?

That did not make sense to me so I considered another wavelength, 635nm of the HeNe laser..



-->OK run the numbers for 635nm helium neon with a 2mm beam diameter. I know the beam is usually smaller, like 0.2mm, but for the sake of the argument.

spot dia. = 1.27 x f x wavelength x M^2 / D

In the 635nm laser, the wavelength is 0.635 micron (0.000635mm), so the above relation:
spot dia. = 1.27 x 0.000635 x M^2 x f / D
becomes;
spot dia. (in mm) = 0.00080645 x M2 x f / D

With a f = 100 mm (4 inch) focal length lens and a beam diameter of 2 mm we get (assume M^2=1 in every case)

spot size (in mm) = 0.00080645 x 100/2 = 0.040325 mm = 40 micron

(In the CO2 example with a 2mm beam and a 100mm lens, the spot diameter was 500 microns)
now how about this:

I use a lens with a 2000mm focal length - right I theoretically want to focus the 2mm beam about 7 ft away from the laser.

spot dia. (in mm) = 0.00080645 x M^2 x f / D
spot dia. (in mm) = 0.00080645 x f / D
spot dia. (in mm) = 0.00080645 x 2000 / 2
spot dia. (in mm) = 0.00080645 x 1000 = 0.8mm ??

So, with this long 2 meter focal length, the 2mm beam is only reduced to 0.8mm. It is still much smaller than with the same lens and beam diameter applied to a CO2 laser, the difference in the final spot size being related to the wavelength.

Can we use some kind of obscene lens that will ruin (not-focus) the HeNe beam diameter just like with the CO2 laser?

a 20 meter focal length for the same HeNe laser:
Lens is not likely to exist but:
spot dia. (in mm) = 0.00080645 x 20000 / 2 = 8.0645
--> yes a 2mm HeNe beam becomes an 8mm beam at the 20-meter focal point of a convex lens.


Experts: Have I done the math right? Is this a sad truth about trying to be cheap on optics when focusing lasers at long distances?



Next - a 10x collimator.

Back to the CO2 laser and its huge wavelength. If a collimator is used to spread the beam to 20mm, and then the 2000mm focusing lens is used:

spot dia. (in mm) = 0.013 x M^2 x f / D
spot dia. (in mm) = 0.013 x f / D
spot dia. (in mm) = 0.013 x 2000 / 20
spot dia. (in mm) = 0.013 x 100 = 0.13mm
nice.. good for the engraver.

Apply it to the HeNe laser:
spot dia. (in mm) = 0.00080645 x M^2 x f / D
spot dia. (in mm) = 0.00080645 x f / D
spot dia. (in mm) = 0.00080645 x 2000 / 20
spot dia. (in mm) = 0.00080645 x 100 = 0.08mm

It is easy to see that the 20000mm lens would still provide a 0.8mm spot for the HeNe laser.

This also explains why the YAG laser here has only been able to be focused to about 1.5mm with a 4000mm plano-convex lens.

i still don't get just how a beam can be made larger at a focal point than it is going in.. Is there a fault in the formula?

-----

Next question, and I have not tried to answer this one because this is still confusing to me:

What is the relationship of how far from the collimator's 20mm output the 2000mm focal length lens is placed?

The reason I ask about the collimator is because all I see is blowing up the beam, only to turn around and reduce it, and I've been told you can't get a free lunch.

[planters]

The amount of documentation you provide is intimidating, but I'll have a go at the general principals. First the longer the wavelength the larger the theoretical (based on perfect lenses) spot size. The relationship is linear and proportional ie. double the wavelength-double the spot size. Second, the longer the focal length of the collimator the larger the spot size. Same rule, double the FL-double the spot size. These rules often are overwhelmed by the low quality of the laser and to a smaller extent by less than perfect lenses. This is why a 10.6 um co2 is used to cut out parts. In addition to the high power. the beam is usually of pretty good quality while a lamp pumped YAG often suffers with poor beam quality which undermines its short wavelength advantage. Multi-mode diodes often produce atrocious beam quality, but they're cheap and easily incorporated into projectors. Finally, if you take divergence into accoun then the beam is always growing and the spot size is equal to the beam diamete. A very long FL lens will not focus to a spot smaller than the emitter (only microns across in some diode lasers and even smaller in vertical cavity lasers). I probably missed something here, but its a start.

[opcom]

I agree with you, I see your point.

I still do not understand how the spot at the focal length could be many times larger than larger than the spot close to the laser. According to the formulas provided anyway.

For example, a 2mm beam from a 1-meter long laser (even a so-so Chinese CO2 engraving laser beam) will stay about 2mm over a distance of 10 feet. When this beam passes though a lens (right at the laser) with a specific focal length, it should at some point become 'focused', even if no smaller than the original.

Like this progression:
2mm beam through an optical flat, at 10 FT is still a 2mm beam.
(I suggest that an optical flat is the same as a lens with an infinite focal length.)
2mm beam through a lens with a 100mm focal point, at 100mm is a quite small point of 0.5mm.

So, somewhere between infinity and 100mm, lies 2mm and 0.5mm, so a focal length between infinity and 100mm should produce a spot between 2mm and 0.5mm. as common sense says.

Yet the formulas say that a focal point of 2000mm, which is certainly between 100mm and infinity, yields a spot size of 13mm. How wierd is that? Common sense says it ought to be somewhere between 2mm and 0.5mm.

This makes me question the formulas. Maybe I do not understand how lenses really work. I wonder if there is a free graphical program for playing with virtual lasers and lenses.

The collimator is a second challenge, as I understand it, it makes the spot larger and reduces its divergence. Collimators for CO2 are horribly expensive.

p.s..below the SSY-1 YAG, TEM02 visible on the right image.

[danielbriggs]

I wonder if there is a free graphical program for playing with virtual lasers and lenses.

Give Psst! a try...
http://www.st-andrews.ac.uk/~psst/

I think it will be able to help you answer most of your questions.

[planters]

I agree the formula yields a goofy result. Looking at your examples again I think there is no inconsistency, just an error based on a simplification. The spot size, based on the limitless growth of the FL grow to infinity as you approach a plain parallel window. But your statement (correct) that such a window will not change the diameter of the beam is not supported by your equalities. Think, infinite FL. and divide.