Why does a beam expander allow a smaller focused spot size?

[Nathan McCorkle]

I've read some equations and searched google, but I'm still not
understanding WHY expanding the beam and shortening the focal length
allows light to be focused into a smaller spot. I know Feynman would
probably tell me I shouldn't ask WHY... but I still wonder

The equations here are intelligible, but I'm not finding any
explanations.... to me it seems like you'd have more play with a
longer focal length (i.e. minute variations past the focus would be
less impact as the divergence is slower)
http://www.kellerstudio.de/repairfaq/sam/laserioi.htm#ioicdf

http://specialoptics.com/pdf/wp_laser_beam_expander_theory.pdf

Simon, care to physicize me?

--
Nathan McCorkle
Rochester Institute of Technology
College of Science, Biotechnology/Bioinformatics

[Jelmer Cnossen]

My first guess: Is it not the diffraction limit that controls the minimum spot size? 

IAW: d = lambda / (2 * NA), and since an expanded beam can focus again from a wider angle it results in a larger NA (= n sin max_angle).

http://en.wikipedia.org/wiki/Diffraction-limited_system  is about optical imaging system, but the same applies for a laser I would say.

So the main answer: diffraction

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[John Griessen]

On 03/29/2012 10:25 PM, Nathan McCorkle wrote:
> I've read some equations and searched google, but I'm still not
> understanding WHY expanding the beam and shortening the focal length
> allows light to be focused into a smaller spot.

Could be due to the way a coherent beam goes a certain distance with no
measurable spreading out, and then does spread out. As though coherence
causes some interaction of the particles/waves that
are shoulder to shoulder "so to speak".

Gaussian beam defines it. from http://en.wikipedia.org/wiki/Gaussian_beam
"Gaussian beam model is valid only for beams with waists larger than about 2λ/π."

They mention "radius of curvature of the wavefronts". That could be as close to
a why as I've found. I.e. -- if a wavefront is not perfectly flat in a plane,
how could it stay the same as it propagates? The main theories we use all assume
"no action at a distance", (or knowledge at a a distance either), so a part
of a curved wavefront not in line with the beam axis is going to wander off
in a different direction.

So, apart from "why a Gaussian beam is the best you can do", taking an imperfect point source
of coherent light and expanding the beam **while getting the path lengths through the lens system
to create a planar wavefront** gets you the best collimation. Then you could collimate it down
to get a beam width you wanted.

For the practical laser focusing in a CO2 laser, the light is never collimated to a planar
or even perfect gaussian beam -- it just takes advantage of the lower divergence in the
waist zone, the Rayleigh range zR by having the waist width be big--> the distance it rules the
system divergence is greater. If you get that distance to be same or more than your operating distances
you've optimized it. Next to use it as a small spot, focus with a short focal length lens.

The CO2 lasers start with a large width of tube so they don't even have the expense of
the expanding lens system -- that width gets set by the end mirrors.

[Simon Quellen Field]

You are getting close.

:-)

Diffraction is not the correct answer, although diffraction does cause the spot to be larger

than it would otherwise be.

Any beam of light has a divergence -- it spreads out as you get farther away.

There is no magic that makes the divergence zero until some distance.

The angle between parallel and the outside of the beam is called the divergence angle.

There is a fundamental law of of optics called the optical invariant: The product of the image

size and the ray angle is a constant.

If the beam has a radius R, and a divergence angle A, and we focus it with a lens that has a

focal length F, then the cone of light will have an angle R/F, and the spot size at the vertex of

that angle will be A times F.

" http://www.newport.com/Optics-Fundamentals/604533/1033/content.aspx "

" http://www.newport.com/Focusing-and-Collimating/141191/1033/content.aspx "

The shorter the focal length, the smaller the spot.

This is how magnifying glasses magnify.

The larger the diameter of the lens (for a given focal length) the smaller the spot.

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2012/3/30 John Griessen <jo...@industromatic.com>

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[John Griessen]

On 03/30/2012 10:43 AM, Simon Quellen Field wrote:
> There is a fundamental law of of optics called the optical invariant: The product of the image
> size and the ray angle is a constant.
>

Thanks, I'll study that.

John

[Nathan McCorkle]

On Fri, Mar 30, 2012 at 11:43 AM, Simon Quellen Field
<sfi...@scitoys.com> wrote:
> You are getting close.
> :-)
>
> Diffraction is not the correct answer, although diffraction does cause the
> spot to be larger
> than it would otherwise be.
>
> Any beam of light has a divergence -- it spreads out as you get farther
> away.

I'm still not quite there... does the smaller focal length allow less
overall divergence to occur? As in, if the focal length were longer,
the rays would have a longer time to diverge?

[John Griessen]

On 03/30/2012 11:19 AM, Nathan McCorkle wrote:
> does the smaller focal length allow less
> overall divergence to occur? As in, if the focal length were longer,
> the rays would have a longer time to diverge?

Yes, but you think of the focal length as negligibly short.

You take care of whatever divergence has happened by unmagnifying it
to a small spot with a short focal length lens system.

For the original question, You choose a fatter beam to get the near in
effects where divergence varies less, see http://en.wikipedia.org/wiki/File:GaussianBeamWaist.svg

You use the fat beam in the part of the sketch called Zr

"where the origin of the z-axis is defined, without loss of generality, to coincide with the beam waist, and where

http://upload.wikimedia.org/wikipedia/en/math/e/5/8/e58a707d1ccf4ed6cefac660633038a8.png

is called the Rayleigh range." http://en.wikipedia.org/wiki/Gaussian_beam

Then, for some light source, it might make sense to expand it to a collimated beam of a certain width,
then focus down to a spot. The advantage would be that over the different distances,
(as in the distances a laser cutter mirror moves to),
the spot size would be changing less because it is in the Rayleigh range.

[Nathan McCorkle]

On Fri, Mar 30, 2012 at 4:54 PM, John Griessen <jo...@industromatic.com> wrote:
> On 03/30/2012 11:19 AM, Nathan McCorkle wrote:
>>
>> does the smaller focal length allow less
>> overall divergence to occur? As in, if the focal length were longer,
>> the rays would have a longer time to diverge?
>
>
> Yes, but you think of the focal length as negligibly short.

Why? Its not negligibly short, the distance is measurable and matters.

>
> You take care of whatever divergence has happened by unmagnifying it
> to a small spot with a short focal length lens system.
>

I meant divergence after the final lens, the length between the lens
and the focus, not divergence before the lens (as those rays would
merely pass the lens)

> For the original question, You choose a fatter beam to get the near in
> effects where divergence varies less, see
> http://en.wikipedia.org/wiki/File:GaussianBeamWaist.svg
>
> You use the fat beam in the part of the sketch called Zr
>

I only posed 1 question. Zr isn't the fat part, its the thinnest part
of the beam, that diagram looks like rays coming to a focus, from left
to right, which pass through the focus.

> "where the origin of the z-axis is defined, without loss of generality, to
> coincide with the beam waist, and where
>
>  http://upload.wikimedia.org/wikipedia/en/math/e/5/8/e58a707d1ccf4ed6cefac660633038a8.png
>
> is called the Rayleigh range."
> http://en.wikipedia.org/wiki/Gaussian_beam
>
> Then, for some light source, it might make sense to expand it to a
> collimated beam of a certain width,
> then focus down to a spot.   The advantage would be that over the different
> distances,
> (as in the distances a laser cutter mirror moves to),
> the spot size would be changing less because it is in the Rayleigh range.
>

None of this really answers the 'why', which is really the main part
of my question. Describing the behavior isn't what I was looking for,
I was looking for the root/cause of the behavior.

That said, if I'm correct about
(http://en.wikipedia.org/wiki/File:GaussianBeamWaist.svg) being rays
coming to a focus and going through a focus... it seems that the
reason a wider beam with a shorter focal length produces a smaller
spot size is due to less interference as the rays converge, because
they start out in low concentration (beam gets expanded), then rapidly
head toward the focus. In the other case the rays are more
concentrate, and have a longer focus so more time to 'rub shoulders'

"
Because of this property, a Gaussian laser beam that is focused to a
small spot spreads out rapidly as it propagates away from that spot.
To keep a laser beam very well collimated, it must have a large
diameter. This relationship between beam width and divergence is due
to diffraction.
"

"
Richard Feynman[4] said that
"no-one has ever been able to define the difference between
interference and diffraction satisfactorily. It is just a question of
usage, and there is no specific, important physical difference between
them."
He suggested that when there are only a few sources, say two, we call
it interference, as in Young's slits, but with a large number of
sources, the process be labelled diffraction.

[Simon Quellen Field]

The beam expander reduces the divergence.

Since the spot diameter is the divergence times the focal length.

In the beam expander, the ratio of the input beam divergence to the output beam divergence is

the ratio of the output beam diameter to the input beam diameter.

If the output beam diameter is 10 times the diameter of the input beam, then the output divergence

will be 1/10th that of the input divergence.

For the same focal length, the spot size will then be 1/10th what it would have been using the

original beam.

I once built a megawatt UV laser with an output beam diameter of about an inch.

We used a 12 inch astronomical telescope as a beam expander to limit the divergence, so we

could try to hit the laser reflector that Apollo 11 left on the moon. Counting the photons we

received about 3 seconds later, we got better than 80% confidence that they were our

reflected photons.

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[Nathan McCorkle]

On Fri, Mar 30, 2012 at 7:16 PM, Simon Quellen Field <sfi...@scitoys.com> wrote:
> I once built a megawatt UV laser with an output beam diameter of about an
> inch.
> We used a 12 inch astronomical telescope as a beam expander to limit the
> divergence, so we
> could try to hit the laser reflector that Apollo 11 left on the moon.
> Counting the photons we
> received about 3 seconds later, we got better than 80% confidence that they
> were our
> reflected photons.

You're now officially a badass (in a good way)... "now I know what
you're thinking Nathan, did I fire 6 photons, or only 5?"
http://www.youtube.com/watch?v=u0-oinyjsk0

(but again, the WHY was more of what I was trying to uncover... I
don't doubt all the equations that you're recanting to me work and are
true, I want to know WHY they're true) (also some equations are only
'true' to engineers, but they aren't 'true' to scientists... e.g.
approximations)

[Simon Quellen Field]

It's just geometry (as is most of classical optics).

Think of the lens as a lever, or some other impedance matching device.

You trade beam diameter for beam divergence.

Another way of looking at it is that divergence is a type of error that gets magnified

by the lens as the diameter of the beam is reduced.

A small angle in a large beam gets magnified into a larger angle in the reduced beam.

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