Hello everyone, welcome back ah in the last
lecture we were talking about the the scientific problem in nano photonics. So, when we
have a distribution of ah permittivities in certain region of space how do we solve for the
electromagnetic response? And I mentioned that this essentially it essentially involves solving
Maxwell's equations, but how do we solve that?
So, to do that we have to essentially solve the
wave equation. So, if you look at the Maxwell's equation the curl equations here you have two
of these curl equations curl of H and curl of E and when you combine these two you get
what is known as a wave equation this is a wave equation for ah free space ok. So, how do
we get this equation? So, since this is a central part of nano photonics I would like to take
a little bit of time try to derive this.
So, you might have already done this, but I
just want to go over this one more time. So, that you are familiar with the assumptions that we
are making and what is the role of each term ok. So, ah all of you definitely
are familiar with wave equation which is of this form this is essentially a
second order partial differential equation. So, you have the 2nd derivative in time
this is a 2nd derivative in time and.
Sorry. I have my students here. So, if they are asking
me question. So, let me yeah what is this?
Student: Sir it should be close to
that H by (Refer Time: 01: 47).
Well why why should it be H? It can be electric
field as well you can have the electric ok. So, yeah the wave equation has two forms; one is the
wave equation in terms of electric field and the other one is ah the wave equation for the magnetic
field. So, you can have either of these two things we can derive them. So, it depends yeah.
So, what I will do today is I will derive the wave equation for the electric field and I leave it as
an exercise for you to derive the wave equation of the magnetic field you can do that ok and it will
be similar in form. So, that is a nice exercise yeah thank you so much Sahan yeah. So, ah yeah we
have the second derivative in time on the right and we have the second derivative in the space.
So, this is a Laplacian operator; Laplacian operator which is essentially dou
square by dou x square plus dou square by dou y square plus dou square by dou z square this is
a Laplacian operator in the Cartesian coordinates. So, the wave equation essentially relates the
2nd order 2nd derivative in space and ah and time. So, through the velocity term.
So, I mean I leave it as a small exercise for you to check that the dimensions match
out. So, the coefficient of here is going to be the velocity term ok this is the wave
equation. So, how do we get this wave equation? So, we will start with the Curl equations
given by Maxwell and we will also make an assumption that the magnetization is
essentially 0. So, we will only take ah B is going to be mu H mu naught H ok
this is the assumption we are making.
So, with that I have curl of E which is
given by second ah the partial derivative of B with respect to time and that will be
minus the ah the partial derivative of the magnetic field with respect to time in the second
derivations second equation is simple enough. So, what we will do now is we will take the curl of
the first equation. So, let me take curl of a. So, what I want to do is take curl. So, I assume that
you have gone through a course on vector calculus and you know what curl divergence all mean.
If not I would encourage you to take it because these equations this vector calculus is very
fundamental to a lot of areas of engineering. So, you should take I mean it will be just
a you know month long course on vector calculus you should take it there will be lot of
resources that are available in that context.
So, let us consider curl of the
first equation. So, I have curl of E. So, I will take a curl of that and that on the
left hand side. So, that will give you on the right hand side minus mu naught I will take the
derivative out and I will say curl of H, right.
So, I know what is curl of H from the second
equation the B here. So, that will be equal to mu naught ah since the this is ok let us
take the 2nd derivative. So, so mu naught 2nd derivative of ah D with respect to time plus
the minus mu naught yeah the 1st derivative of ah J with respect to time. So, this is a this should
be instead of a plus it should be a minus here as well. So, this is what you get right.
And you also know that from the constitutive relations we know that D is equal to
epsilon naught E plus polarization P. So, we can substitute and expand, so what
you will end up getting is minus mu naught epsilon naught dou square E by dou t square minus mu naught dou square P by dou t square
minus mu naught dou J by dou t.
So, you already see that the left hand side
we are already seeing that you know second derivative in terms of the time. Second derivative
electric field in terms of time that we see on the right hand side of the wave equation ok,
in addition we have the few other terms ok.
So, how do we look at this? So, we will put
it back together instead of in my handwriting its I have type set it here. So, essentially
curl curl of E is this term ok and to simplify the left hand side we will use a vector identity
which is essentially given by this. So, the curl of vector field will be given by ah the first
term it consisting of divergence of the field and the second term consisting of Laplacian.
So, it turns out that if you are talking about free space or ah region of space where there is
no free charges. So, this rho f means free charge if there are no free charges then rho f is 0 and
also if the epsilon does not vary significantly over the wavelength ah scale, then you can
actually consider divergence of E to be 0 because of that we will be able to simplify
the equation to this form on the bottom here.
So, you see that you have the familiar ah part
which is the Laplacian on the left hand side and the first 2nd derivative of electric field in
terms of time, but there are a couple of other terms ok. If you consider let us say free space by
which we mean there are basically no there is no polarization and also the there are no free
currents J is 0 are 0 in free free space.
If you consider this scenario the equation
reduces to Laplacian of E is equal to mu naught epsilon naught dou square E by dou t square
ok. And the solutions to these equation this equation are essentially what are known as plane
waves, we will talk about them in a little while, but ah figuratively the solution
is going to be like ah this.
Now, you have the electric field which is confined
to a plane and it oscillates as a function of time and magnetic field will be perpendicular to that.
So, you have electric field and magnetic field which are perpendicular to each other and they
are both simultaneously perpendicular to the propagation direction. So, this is what we call as
a plane wave which is a solution of electromagnet the wave equation. And we will see
that we will study a little bit more detail in the ah coming sections ah.
But so what is the speed of this wave right, how fast does it propagate? So, for that we have
to compare the coefficient of the time derivative. So, in the original case we had ah one over
v square right velocity square and in the in the equation that we solved
we got mu naught epsilon naught.
So, if you can define the velocity
of a wave in terms of velocity of the wave or of the E M wave is going
to be 1 over root of epsilon naught mu naught ok. And if you look at the terms
you know we mentioned what is epsilon naught and what is mu naught in the earlier lecture.
You will see that if you compute these numbers you will end up getting equal to constant C
which is 3 into 10 power 8 the speed of light.
So, whenever you have electromagnetic waves
propagating in vacuum we have ah that the speed with which they propagate is known as the speed of
light which is dependent only on the fundamental constants permittivity and permeability of
free space right. So, this is a speed of light. So, this is how electromagnetic wave propagates
ok, this is the solution of a wave equation.
Now, what happens to let us say regular
medium? Let us say I take a piece of glass how does the electromagnetic wave propagate
in that? So, if you go back and look at the original equation here what would change when you
consider glass? Ok. Will glass have any currents? No because glass is insulator it cannot conduct
any current, so we will put the J term to be 0.
What will happen to the P, the polarization will
there be a polarization when you consider glass? And if you look back you will recall that
glass can actually have some polarization right we said the polarization P is given by ah
certain susceptibility epsilon naught chi times E right that is what we did in the last lecture. So,
we have to consider polarization we cannot make it 0 now. What is the impact of this? Ok.
So, if I substitute for P I will get mu naught epsilon naught chi dou square E by
dou t square ok. So, now what I can do is I can take the right hand side and I can take common
terms out. So, right hand side of the equation I will take let us say epsilon naught mu naught
and this is 1 and the second term has a chi. So, I will take one plus chi dou
square E by dou t square.
And this particular term we have already
introduced which we call as a relative permittivity ok. So, what this is telling you is
the wave equation has a similar form to that of vacuum. Now, if you take a wave equation in let us
say any dielectric media it has the same form only addition is that you have this additional term
of relative permittivity. So, what does it imply? What is the presence of permittivity show
us? Well we said that the coefficient is related to the speed of the wave.
So, in the previous case we said the coefficient was you know 1 over mu naught epsilon
naught, but if you have a wave in a medium in a dielectric medium what happens? Well the velocity
now is modified, its no longer going to be C, but the velocity is going to be C by root of
epsilon r ok and more commonly this is known as root of r is denoted by refractive index n ok
So, this is a refractive index. So, when you have electromagnetic wave propagating
in a medium any dielectric medium these are simple dielectrics ok which ah let us assume that
the they do not have any losses we will get into that in the next week. When you have a simple
dielectric the wave essentially slows down and the ratio of the speeds is a refractive index ok this
is a simple first approximation of what happens.
And I must also emphasize that we have made
some assumptions when I said P is going to be epsilon naught chi times E this is a simplifying
assumption the reason is there are some materials which are anisotropic; that means, the way a
wave propagates in the one direction let us say x direction the speed with which it propagates
is going to be different from the speed with which it propagates in the y direction.
There are some examples like calcite ok where this they have an anisotropic response,
if you have an anisotropic response then we have to actually consider polarization in a slightly
more complicated form. So, now, in this case the polarization ah the susceptibility is not going to
be a scalar previous case the susceptibility was a number, but when you have a anisotropic medium
we will have to consider what is known as tensor.
So, you have scalar which is simple number
a vector which is a array of numbers and then the higher ah form of that is what is
tensor which is basically two dimensional its matrix ok that is why I represent ij. So, its
a matrix of susceptibility numbers in various directions. So, we call it a tensor and then
we have to compute ok its much more involved we will not get into it right now except
you know when I talk about meta materials I will talk about the anisotropic response ok.
Ah The other assumption is I am making is that we are dealing with linear medium. What do I mean
by linear? Well it turns out that the P which is a function of E direct linearly dependent on
E is an assumption you can have higher order terms the polarization will also be dependent on
the square of the electric field and the cube of the electric field, it can happen and so on
it can have higher order responses and these are what are known as non-linear responses ok.
And this leads to a lot of interesting effects. So, for example, here this
is a 2nd order susceptibility and this is a 3rd order susceptibility and
so on and the presence of the 2nd and 3rd order susceptibilities can actually lead to very
interesting effects ah like harmonic generation and so on there is an entire field of non-linear
optics which I think probably had I think a few Nobel prizes in that domain which has which
is based on this higher order responses, right. So, in this course we will not really
get into the non-linear ah phenomena.
So, what we will do is ah let us look at that
you know the dielectric constant right or rather the effective index in a little bit more of
detail. So, in the previous slide I said its a the ratio of the speed, it seems like ok we are
all familiar with refractive index of glass being 1.5 silicon being 3.1 and so on. But it turns out
that the refractive index is not a simple number, but it exhibits a dispersion the technical
term is dispersion which essentially means that we have the refractive index which
is a function of wavelengths ok.
So, on the x axis I am showing you function or
rather x axis you have the wavelength on the y axis you have refractive index. So, what
you are seeing here is various materials in their corresponding refractive indices. And
immediately you see that the refractive index is a is a kind of you know it shows a variation
ok, it depends on the material there are some materials where its more or less constant and
then some others which is not. Now, for example, on the top here the blue is germanium ok and
germanium has a refractive index of about 0.8 eV.
The green is gallium arsenide which has a sorry
band gap band gap its not refractive index I am sorry germanium has a band gap of 0.8 ev and
the gallium arsenide has a band gap of 1.42 eV and similarly if I come down if I look at gallium
nitride which this has a which has a band gap of about 3.4 eV and finally, you have ah silicon
dioxide which has a band gap of ah Eg of GaN this is this is eg of SiO 2 ok which is about 9 eV.
And what you see is the band gap changes as a function of sorry the
refractive index is invested related to the band gap ok. So, this is a interesting observation
and why does this happen and so on we can talk about it in the subsequent lectures ok.
So, I just want to leave you with a thought. So, ah generally when I talk of optics or
photonics or electromagnetic we think of it as a separate domain and its sometimes scary
because there is a lot of mathematics involved.
But I want to underline the the unifying or the
similarity between various you know domains for example, optics or electromagnetics is really
very closely related to the electronics that many of you are familiar with. If you look back
at how the electromagnetic waves are generated you must have heard about antennas which
generate electromagnetic waves, we will consider that in a little bit of we will show a
few things about that in the ah down the line.
But right now if you have electric ah charges
that are moving they produce electric fields. So, essentially motion of charges electrons
give you light the light generation part. And analogously when you have an electric field
incident on a material it is going to make the charges move the the the electric field
in the light or electromagnetic wave will make the charges move and that is light detection.
So, you see that the electrons and the photons are very intimately related they are not completely
different domains they are ah related by some very fundamental physics. Here I am listing out a few
parameters to just emphasize this you know ah you must have heard about the wave particle duality,
right. So, you can treat electrons and waves as electrons and photons as waves or particles ok
both are valid and there are certain regimes where particle picture is useful and certain other
regimes where the wave picture is useful.
So, what you see is here let us say energy the
tabulated a few things for electrons and photons and you see that the energy of electron is
essentially you know given by ah p square by 2m. So, it is related to square of momentum and
this is called the parabolic dispersion if you are familiar with electronics you must have heard
about the parabolic dispersion for electrons. Whereas, if you come to photons the dispersion
is slightly different, its a linear dispersion the energy is linearly related to the momentum ok
And similarly yeah momentum you have mv mass into velocity this is a classical picture
if you look at the quantum picture it is h cross k and similarly for photons you can talk of
momentum in the quantum picture which is h cross k or you can also define in terms of de Broglie
wavelength I mean rather than h by lambda which is essentially what de Broglie relation is
and the wavelength of course, you can rearrange.
So, what I want to emphasize is that ah the
behavior of electrons and photons is quite similar ok, electrons are governed by Schrodinger
equation which is essentially a second order partial differential equation wherein you have the
2nd derivative in space and 1st derivative in time that is the Schrodinger equation.
In photons or electromagnetic waves we deal with the wave equation which is again a
second order differential equation, but this time it is a second second order in both space and
time ok that is a slight difference, alright. So, ah this is where I will stop,
I will take a few questions yeah.
Student: How does the wavelength
or other parameters compared for both electrons and photons?
Yeah that was a very interesting question ah one of the student was asking about
how does these parameters compare ok for let us say electrons and photons are they similar
in range or not ok. Well, electrons have ah much higher I mean you can accelerate electrons
to much higher energies compared to photons, if you take a photon the typical let us say
energy in the visible range is going to be about ah few eV let us say 2 to 3 eV that is in the
visible domain ok or even I would say 1 to 3 eV ok that is the energy of a photon ok ah.
Just remember I am talking in eV units its much more easier to deal with energy in this
way, if you talk of joules we will have to talk of 10 power minus 19. So, I would encourage
you to actually try to figure out how to ah convert between wavelength you know
various units let us say nanometers to you know eV to centimeter inverse and so on the
various units and you should be comfortable ok that is easier it makes your life easier
So, the photon energy is typically 1 to 3 ev whereas, electrons can be accelerated
for example, you know I mentioned an SEM. So, where in the energy of an
electron can be up to 100 kv ok it can be very high, it depends on how much you
accelerate to or you can talk of the wavelength the wavelength of photon in the visible range
is about ah let us say the center wavelength is 500 nanometers whereas, ah ah the wavelength
of an electron can be much much smaller ok.
And this has some implications we will talk about
when we are talking about the diffraction limit, alright. So, yeah so, there are
the numbers are quite different, but the underlying phenomena has many similarities
ok, similar to you might have when you studied quantum mechanics you might have studied
about tunneling of electrons. You can talk of a similar concept of photons where you know
tunneling of electromagnetic waves and there is ah forbidden regions and so on ok. So, yeah they
are related yeah. Anything else? If not yeah.
Thank you so much for your attention and I will
see you in the next lectures, take care bye.