We're going to learn that light has a shape.
Rather than by starting with Maxwell's equations and doing some new derivations,
we're going to start with
probably the most important and common example
that you're going to encounter in optical systems design,
and that's the Gaussian beam.
Most gas lasers affect lots and lots of lasers but gas lasers in particular.
Light coming out of optical fibers,
a lot of single mode coherent systems,
again particularly lasers, emit Gaussian beams.
And so this will be a nice way to get a feel for the math and
that kind of shape that light has
and then we'll go back and do some more careful derivations.
So, a Gaussian beam is just a nice smooth spot.
The transverse profile of the beam is a Gaussian,
E to the minus sum coordinate over a distance squared.
And it turns out that if you launch this in
the paraxial version of the wave equation you derive from Maxwell's equations,
you come up with a solution that looks like this.
At the focus for example we would have this profile of
the Gaussian beam and then it goes away diffracts away from that waist.
And you see that it evolves into what looked like
spherical waves with these blue lines that must be the rays,
those rays are as always perpendicular to the face fronts.
But notice that it comes to the focus and it doesn't have an infinitely small size.
Instead, and this is really interesting,
these blue lines which are the rays they are normal to the face fronts bend.
And we already said that rays go in straight lines and free space,
so that might bother you.
But we derived rays under the assumption that the amplitude was slowly
varying and noticed that in this region right here that's definitely not true.
So, that's why that understanding of straight lines goes away right there.
So, the light would come in towards the focus and rather than
the rays all crossing in an infinitely small point they bend and it defracts out again.
So, this is the solution to the problem we
showed in the last lecture of blowing up the universe.
There are three or four quantities here which are
interrelated and they're really important to understand.
And all beams no matter what their shape,
whether it's Gaussian or a square top hat sort of beam,
all these generally three shapes three numbers and we want to understand how they relate?
So, there's some number describing how big is this waist?
How small is the focus?
In the case of a Gaussian beam we describe that by w_0,
the beam waist at the focus,
you see that's the number right there,
that would be the E to the minus one amplitude point across the beam in radius.
We also have a quantity called the Raleigh range for more broadly that look at
these things and that describes the depth of focus of the beam,
and we'll come back to that in full mathematical expression.
And then finally, we have something which describes as we get away from the focus.
What's the angular extent of the beam?
So theta not, so w_0, z_0 and theta not.
The relationships down here are worth remembering or at
least being able to look up in a hurry because what they tell you is that,
once you pick the vacuum wavelength and the index of refraction of the medium you're in.
So that picks your wavelength in
the medium that once you've picked one of these three quantities,
the other two are fixed.
You cannot choose them independently.
And this is one of
the fundamental concepts of this whole course again light having a shape,
is it the waist of a beam,
it's depth of focus, its angular extent,
or fundamentally and type-B related by
very fundamental physics and you can't choose them independently.
So, notice for example that the size of the beam,
it's waist is inverse with this angle.
The depth of focus is actually related to the waist size squared.
Those are concepts you'll find in all of physics of
waves and though there's different constants out front.
The depth of focus really being related to the beam size squared
divided by the wavelength is a universal thing for all beam shapes.
So, these are questions that you'll need and be using
a lot when we design with Gaussian beams.
So, let's actually write down that solution that we get for the Gaussian beam.
And this looks a little bit awful.
But as a matter of fact most of the terms are pretty simple and
most of the time we don't actually need most of the terms so let's walk through it.
We have an electric field that at the waist that is if z = zero.
That's this point right here, z = zero.
Most of these other terms matter of fact go away and we're
left with just that Gaussian profile.
And that's what we have before.
But now as we move away from the focus,
we find a couple of things.
First, we find that a term that looks like a plane wave.
This is just linear phase accumulating as we go down the axis.
So, that doesn't seem too unreasonable.
It's a propagating wave.
This term here actually is the curvature of the beam.
That's what gives the beam these radius of curvature,
the face fronts here.
And then this last term is rarely
important but every once in a while is what's called the Gouy phase.
And it just expresses the fact that the wave gets
a little bit larger wavelength as it goes right through the focus,
and that has to do again with the fact that when you
confined a beam to a small area it changes a little bit.
Worth knowing it's there,
most of the time it's not important.
So, we have terms in here which are the beam radius as a function of position,
the radius of curvature of the beam as a function of positions z down the axis.
And this extra Gouy phase term.
The first two are the ones you may need,
as a matter of fact it's usually just the waist you care about.
We can actually plot those three terms which is just a good thing to
do to understand them and what you see is that
the waist is smallest at
the origin and goes up and that
blue line is actually just the exact same as these blue lines here.
It actually is the ray.
But with a minimum at the origin and then it goes up and eventually this
turns into lines because this is a problem.
The radius of curvature is actually
zero at sorry infinity at the origin that's this curve here.
Okay, check the radius of curvature is infinite at the origin.
Of course as you infinitely far away from
the focus you'd expect the radius of curvature to go back to infinity.
And it turns out that the radius of curvature is minimum right at z_0.
Which is it seems like a reasonable guess.
So, this distance, this characteristic distance in z is important,
it tells you where the radius of curvatures a minimum.
And it turns out that the waist has gone up by a square of
two in an electric field amplitude right about at z_0.
So that's sort of tells you again why we think of z_0 as the depth of focus.
Now this is the electric field.
It's an expression worth having,
sometimes you need that.
But most of the time you're going to be detecting light on
a piece of film or a detector and it's the intensity you care about.
And that's why it kind settle up that a lot of these terms are
important because when I take the absolute magnitude
squared of this electric field to get intensity all these space terms drop out.
And so this is what a Gaussian beam looks like in intensity.
And I've done some two important cases here.
So, we have nothing more in the expression than our Gaussian beam before.
There's an extra two here because this is the absolute magnitude squared of
the electric field and we have the term that was leading out front.
This of course helps us conserve energy.
And I've actually integrated this thing transversely that gives a factor of pie over two,
so that this actually has a unity integral transversely.
So, if you want to do sort of normalized energy this has an energy of one.
So, what if we make the waist relatively large?
Then we get a solution like this.
What you see here is you have something that looks not too different than we've drawn
before of parallel rays describing what we call it collimated beam.
So, now what we learn is there is no such thing as a purely perfectly collimated beam;
rays that are traveling parallel forever,
because that violates this shape idea.
Once you tell me the beam has a certain radius,
I can tell you it has to have a certain depth of focus, z_0.
And so if I go far enough I will see the beam diverge.
So it's worth putting in your head whenever
you say the word collimated beam, you put it in quotes.
Nothing is truly collimated it's simply if you stay well
within plus or minus the z_0 then it's roughly collimated.
Notice this axis down here z/ z_0 and I've stayed between plus and minus 0.5.
Okay. It looks fairly collimated.
But if I went out to plus or minus one,
I would see significant divergence.
The amplitude will get bigger by-
the diameter of the beam we get bigger by about a factor of two.
The other case of course that's interesting is the focusing beam.
So, now let's make at z = 0 relatively small waist.
And let's go forward or backward.
I've looked simply contorts the waist here and look at
profiles of this is w squared the radius of the Gaussian beam intensity.
And these again these are equivalent to the rays before we would have
drawn these rays and they would have crossed right at the origin.
Now, instead they come to a finite sized focus.
And the point is if you have a radius measured in something like w_0,
the Gaussian waist radius,
you know immediately z_0.
And if you go back look at this about z_0 at one
here within z_0 the beam's sort of unchanging.
But as you go backwards multiples z_0s the amplitudes dropped a lot.
So much it doesn't even show on this color scale and it's spread out.
The profile is still Gaussian here.
This is still a Gaussian but the size of that Gaussian beam
is larger and the amplitude is smaller. To conserve energy.
