por lo que esta conferencia vamos a ser
hablando de la línea natural de la ampliación por lo
Recuerdas volver a Creo que fue nuestra
primera conferencia que uno de los muy
propiedades útiles de los láseres era que
son monocromática sentido de que si
trazar la intensidad de la luz emitida
en comparación con la frecuencia de la luz que se
estaban muy cerca de una sola frecuencia
Ahora, en un mundo ideal sería
absolutamente una frecuencia que el láser
emite, pero por desgracia en el mundo real
mundo el mayor número de procesos que significa que
no podemos conseguir este caso ideal y todo
láseres emiten un rango de frecuencias y
este intervalo o banda de frecuencias que
que emiten generalmente se llama la una de
el láser ahora hay muchos mecanismos
para aumentar esta bro ancho de línea como
Se les conoce como los mecanismos de la ampliación
one we're going to be talking about
today is natural line broadening which
is the fundamental limit of what we can
get with the line width for a laser so
what causes this limit and you'll
remember back to our second lecture that
we talked about something called
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emission
which was some event that where we had
two energy states that an electron in
the higher energy state would randomly
drop down to a lower energy state
emitting some photon with an energy
equal to the difference in these energy
states which would relate to the
flatlands frequency by 2 pi the photon
frequency and Planck's constant now we
said that this was some random event
which it would which it is and of course
in there being random this photon also
has some random phase and direction that
it's emitted at but what actually
triggers this event I mean that's a very
deep question that we should ask what
causes this electron to just randomly
jump down like that have a mind of his
own and just go when it wants to you
remember we also talked about some time
constant that the population in this
level or the rate of change of
population of electrons in this level
was proportional to the actual
population itself and the constant of
proportionality we gave it some time
constants so it was an idea that
electrons have some lifetime in this
upper energy state now what actually
causes them to go down to this planet a
lower energy state and what determines
is time constant or rate of change is
very fun that's something very
fundamental in physics it's something
probably familiar with
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so there's many forms of this but one of
them uncertainty in energy multiplied by
the uncertainty in time has to be
greater or equal to this fundamental
limit Planck's constant now I'm not
going to prove this it's a well-known
formula or principle in physics but what
this essentially means is that if we
think about it if we have a time that
has some energy yeah then using
Heisenberg's uncertainty principle we
can get that idea that this photon can
exist or there's some uncertainty in its
time which is related to the photons
frequency so you can think about in the
vacuum is not actually empty the vacuum
contains an infinite set of modes which
all just happened to be on average err
in the grande state of being unoccupied
but in reality there's always virtual
particles and photons popping in and out
of existence all the time and this
relates to this Heisenberg's uncertainty
principle that we could have a really
really energetic particle popping out
but in order to obtain this you could
only exist for a very short period of
time so here we've written out that if
we have a photon that's admitted with
some frequency V or W then this implies
they can exist for some
some some time period which related or
has this inequality we relating to its
frequency yeah now this all sounds
really crazy at the moment but as it
turns out this is actually measurable so
one really cool thing is that at the
Australian National University they have
a experimental setup which actually
measures these vacuum fluctuations and
because they're random in nature they
can use this to generate perfectly or
absolutely random numbers which is
really useful for a lot of programming
applications also there's this crazy
thing called the cash Casimir force so
another idea is that these particles of
photons they brought into existence they
have to satisfy some boundary conditions
for example if we have some cavity set
up there only photons that can pop into
existence here are the modes of this
cavity because it has some boundary
conditions so for example white waves
that can create standing waves within
this cavity yeah now if we think about
this outside this cavity we have free
space or a vacuum so essentially there's
infinitely more modes of light that can
be brought or pop out of the vacuum
outside then can be created within this
cavity now photons carry some momentum
so there's something known as photon
pressure that would be pressing against
these plates and so the photon pressure
pressure on the inside of this cavity is
going to be a lot less than the photon
pressure pressure from the outside of
the cavity because we're gonna have
being created and here in
in the cavity and this actually exhort
exerts a measurable force on these
cavity walls so the name it is yes so
what we haven't have now is that when we
have this electron in an excited state
now some photon can just randomly pop
into existence satisfying this
inequality that we had before this
photon is current just happens to pop
into existence at this energy with an
energy akin to this difference and then
that's gonna cause some in this case you
can see there's a resemblance of
stimulated emission spontaneous emission
is stimulated emissions from photons
popping out of the vacuum and so this is
gonna create another photon with the
same phase and direction of this random
vacuum photon but then after this time
constant here this photon is gonna
disappear and we're only gonna be left
with that one that popped out so you can
imagine now you can see how this relates
to some fundamental time constant all
day if we have
some population up here and these
photons are gonna be popping in and out
of existence at random timing intervals
but it's gonna create some time constant
for how long these can be or stay in
this higher energy state one to go down
in meaning and then the other one be
read yeah
so if we now think about this case about
the electric field if we consider this
scenario where we have all these
electrons in the higher energy state and
they're stable up there and we just have
these vacuum fluctuations happening and
that's causing this emission of extra
photons and then the vacuum photon
disappears yeah we're gonna get some if
we think about the electrical field a
man this decaying exponential where the
decay rate is equal to this time
constant which is that time constant
that we were talking about from the
spontaneous emission in the second
lecture and we've just included a factor
two here to account that the intensity
of the electric field so that just makes
our life a little bit more easy later
and then
Phil oscillating at this frequency which
is related to the energy difference
between the two states okay now you can
imagine that if we what we want to do
now is look at this in frequency space
so how do we do that we do that by
taking the Fourier transform and you can
imagine that if we ignore this
exponential decaying constant we just
have this closed term here in frequency
space that's very equal which is going
to have a single peak at this frequency
Omega naught so and of course that
relates to it you know I
our ideal idea of a laser but in reality
we have this decaying time constant so
if we continued frequency space and
frequency which we can also write this
if we remember trig identities
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yes we have this so looking up standard
tables for Fourier transform we can find
a Fourier transform for these
expressions where we have both
oscillating frequency and some decay
rate we evaluate this integral using
those tables is this expression we know
that their intensity which is the
passivity of free space and sorry I
think before I incorrectly stated that
the intensity was equal to the square of
this
electric field where in fact it's just a
proportion emotional to it with the
constant of proportionality daintiness
okay so when we evaluate this what we
end up with is a Lorenzen frequency
space where this is the intensity of the
light which is going to determine the
peak of this function when we have
defined this Delta mu nu to be equal to
this time constant of spontaneous yeah
and so what this is as I mentioned and
if you plot this density this frequency
we get it where the one with this and
it's really interesting lead line with a
measured by something called full width
half maximum which is used because
interestingly if you try to evaluate the
standard deviation of a Lorenzen
function it turns out to be undefined or
infinity so the idea of standard
deviations for Lorenzi and functions
undefined so instead they had to
improvise and use
idea for with half maximum so it's
essentially the width of this profile
half the maximum so 0.5 where this is I
up here the maximum intensity okay so
that comes to the end of this lecture so
we can see that this fundamental time
constant for the spontaneous emission
fundamentally determines the line width
for lasers yeah but in reality there's a
lot of other mechanisms that broaden
this lines such as Doppler line broaden
broadening and other types that we're
going to talk about in future lectures
yeah until next time
