Optics (Lec - 14) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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right everyone we're going to start now
so my name's Colleen Shepherd and I'm
going to be giving the lecture today
George is here to keep me in order and I
think probably he'll come up with some
comments at times maybe I hope anyway so
he started off by saying that there's
there's no real announcements except
that he's changed he's uploaded some
revised notes apparently I think in the
notes so I have fixed them and the
current version should be the you know
the one that's in the website now is the
corrected ones so minor changes I think
so ok so I'm gonna try and take off from
where George left it which was with
Maxwell's equations and the derivation
of the wave equation and so I think that
it's actually it's quite nice to see
this derivation of the wave equation
starting from Maxwell's equations
because it brings everything together
and allows you to see you know how the
these different areas of physics are
interrelated but but normally of course
we in optics we don't usually go to the
electromagnetic type theory so we
usually get as far as the wave equation
we show the existence of simple forms
like plane waves and spherical waves and
then we carry on then using those plane
waves and spherical waves in a
simplified form so but anyway back to
Maxwell's equations these were the
Maxwell's equations George before and
explained these and try to I'm not going
to go through the the meaning of them
again because I think you've got that
but
in the differential form you've just got
these four equations connecting the
electric field and the magnetic field
right so you'll see that there's two two
E's there oh sorry there's a there's
three e's and then there's three B's and
then there's a charge row and a current
density J and so how we get the wave
equation from that is quite simple it's
just the case of doing a bit of vector
manipulation so we've got that currently
you can see is minus DV DT from the
Maxwell's equations so if you take the
curl of both sides of that you get this
equation and comb Cobie is another of
the Maxwell's equations so you can
substitute in from the Maxwell's
equation into this one and then you'll
get another equation the what you get
after that
George hasn't actually written here but
basically what you'd obviously get then
this is curl of this so it's going to be
curl of curl and so so this is what you
actually end up with is is curl court
curly is is going to be and then the
other term is going to be something a
second derivative in the e and so
actually the equation maybe I'll write
down it's not actually written there but
after you've just done that substitution
you'll get something like is this thing
working or yeah it is you get something
like curl of curl e does that there we
are now see and these these things is
out I usually a lot of people right
across product like that as I'm sort of
upside down thee
Cole Carly plus mu naught epsilon naught
dick D to e DT squared equals nor is I
think what you get after you've just
substituted that in there and and then
finally so this is this has still got
this thing curler curl e which is quite
complicated if you try and work out what
that is in spherical polar or something
especially if you're expressing this E
is something which is spatially varying
on whatever this could be quite a
complicated thing to have to work out
but anyway but the you can get it into a
slightly more useable form often we do
this by using this identity curl curl e
is equal to grad of Divi minus del
squared e now this this like this is it
calls it here an identity really it's
actually the definition of what this
thing means and you know this is not
obvious what this means actually and
this but these other things
okay they're well-defined what they mean
this thing is actually just what it says
in Cartesian coordinates so you know
you've got L is equal to DDX in the I
Direction plus D dy in the J direction
plus DD Zed in the K direction so del
squared is going to be this dot this
which is going to be d 2 del squared
equals d 2 DX squared plus D so dy
squared plus d 2 DZ squared and you'll
notice that that's exactly the same form
as the normal laplacian right so the
more laplacian you've had before del
squared of a scalar but this is
different because this is still squared
of a vector
and so what we're saying is that for
Cartesian coordinates if we define del
squared of a vector by this thing we
find that it's got exactly the same form
as del squared of a scalar
now that doesn't actually work with any
other coordinate system so that's a bit
of a warning if you're going to
spiracles or cylindrical coordinates
you'll find that this del squared of a
vector is not the same as del squared
that you would use for a scalar but
anyway so but nevertheless let's do that
we'd stick in this as being that and
then we look at this term here and
mostly we're going to be interested in
for example regions where there's no
charges and very often also we're going
to be interested in anisotropic media if
these things are true you know that div
D equals Rho so div D is equal to naught
if there's no charges which means that
DV is equal to null if there's no
charges and it's also isotropic so I'm
saying this all out in in length because
you have to be very careful about this
sometimes this is only true DV equals
not is only true if you've got no
charges and it's anisotropic media so if
you've got for example let's say a
diffraction grating where you've got
some variations in permittivity then in
general you can't actually assume this
because excellence changing and
therefore without Deek was no div
excellent doesn't equal not all right
but if you're in free space or some
constant isotropic medium this is true
so this this term goes and so this is
just equal to minus del squared and so
you can see then this tells del squared
then it's just replaced by this and we
got a sign change there all right so
is the final expression and you'll
notice then that this is exactly the
same form as the normal wave equation
that we're very used to in the scalar
form except that the scalar is replaced
by vector so that's really very nice
okay so yeah so comparing with that of
wave equation this is the the normal
wave equation we've had the scalar wave
equation before and you can see they're
exactly the same except that the scalar
is replaced by a vector and the one over
C squared where C is the velocity of the
wave is is replaced by this thing so we
can say then that 1 over C squared is
equal to MU naught epsilon naught and
that gives us then C is 1 over the
square root of U naught Epsilon's L
naught and so here there's some figures
put in and we get the expression for the
speed of light in vacuum so that's very
neat I think I guess Maxwell must have
been really amazed when he did this some
I don't know you know you don't know of
course that the actual history of the
thing but he played around with these
equations didn't he and he came up with
this this form which looked nicely
symmetrical to him and then he comes out
with an expression comes out with a
value for the speed of light that it
predicts which is what they already knew
was true so he must have felt really as
though he was going to get in a Nobel
Prize or something but probably I don't
know maybe he was before the Nobel Prize
quite interesting and actually Naveen
was telling me that he'd been looking at
the original paper by Maxwell Maxwell's
equations if you look in the original
are horrible because he doesn't you know
the this terminology for vectors wasn't
invented then so it's done all in terms
of
components horribly complicated and I
think no vain said it was was Heaviside
who actually really derived them the
Maxwell's equations in the in the form
that we know and love so well nowadays
Heaviside you might remember these
famous for a few things but one of them
is a thing called the Heaviside function
which is basically a step function and
the other thing is the Heaviside layer
which is a an ionic layer in the
ionosphere that the reflects radio waves
from okay so so that's all about free
space so how can we now deal with matter
well it turns out actually there's lots
of different ways you can deal with lots
of levels you can deal with propagation
of electromagnetic waves in matter you
can for example treat it really from a
proper atomic point of view you can
think of the material being made up of
atoms which have got nuclei and
electrons clouds and so on or you can
think of it in terms of you know matter
just being like anisotropic material
where you don't really go into the
microscopic view of it but just think of
it in in a macroscopic way and actually
to some degree i think that very often
that the second of those is just as well
because normally we're not really
interested in in the actual atomic
nature of where these properties come
from but this is going back to how you
can think of it really in terms of atoms
so if you know atoms are made up of a
nucleus with an electron cloud and the
the nucleus is relatively fixed of
course because it's heavy but the
electron cloud because the electrons are
much lighter can be moved around
by the field that supplied so you get
the distortion of the electron cloud so
this is what this is showing here it's
showing here the nucleus here which is
pretty well fixed because it's so heavy
and then they said this is this electron
cloud you can think of it like being
joined to the nucleus by a lot of
Springs and then the electrostatic the
the forces on that electric cloud caused
by the electric field will tend so for
example displace that the the electron
cloud relatives and the nucleus so it
gets sort of distorted and the way
that's described is in terms of property
called the polarization then and so this
is showing how you can see that if you
move the negative charge is relative to
the positive charges you'll set up a
sort of dipole moment separate the
charges you've got a plus and a minus
they're separated in distance so that
acts like a dipole so that acts like a
gives you a dipole moment and then if
you sum over all those dipole moments
you'll get the total polarization
effects of applying that field and so so
what that does then is introduces these
charge variations which which distort
the the the charge that you really have
there which is caused by the bound
charges so this is what we get you can
you can say that D equals Rho so Divya
is is equal to Rho over epsilon all
assuming the yeah well this is epsilon
naught of course is really a constant
and we're breaking this up into its
bound and free components and we're
saying that the that the the free part
oh we've got something wrong with us
again and we
sorry that should say they should say
free there shouldn't it yeah
the free part and the bound part is this
part and and so this is the final result
then we can now take the epsilon not the
other side take this the other side and
we get that div of epsilon naught E Plus
P is equal to the free charges right and
then this thing in brackets here is what
we call D the displacement in the medium
so it's so if you've got this medium div
D equals Rho is is what we know is the
the the normal expression that we write
from from from Maxwell's equations and
now we've derived that that D in the
medium can be written in terms of the
polarization of the molecules that make
up the medium so how do these things
connect this shows us a we do this then
so we got D equals epsilon naught E Plus
P now in general of course we don't
really know what this does at the moment
and in general it can be very
complicated actually
it can be a nonlinear relationship with
the electric field but if I go back
again to the diagram you'll see that
this is a bit like you know you've got
Springs here Hookes law tells us in
mechanics if a system of Bayes Hookes
law then the extension is proportional
to the tension is what is it the
extension is proportional to the tension
so sorry I'm not a mechanical engineer
so I don't know these things
so you'd expect the same would be true
could be the same could be true here if
you're only applying a very weak field
on here then you might think that this
relationship might be linear you'll get
the these Springs behaving like Hookes
law so you'll get a
linear relationship between P and D now
also in analogy with the mechanical
situation yeah so if we're in a matter
like in a material why is it the
permittivity of free space and not just
the permittivity of the material like
why is it epsilon not and not epsilon
right here that you can write this this
is either you can think of as epsilon
naught epsilon epsilon E or you can
think of it as epsilon naught E Plus P
so these are two alternative ways of
looking at the same thing right so we've
got D equals epsilon naught E Plus P and
you can think of this as being epsilon e
so we were just about to carry onto this
on the next slide actually so you can
see tells us there's a relationship now
between epsilon and P right and we're
going to we're going to look what that
relationship is but in general it's
actually quite complicated because we
don't know the relationship between P
and E yet right so what I'm trying to
establish at the moment is that if this
thing if the fields are very weak then
you'd expect maybe a linear relationship
P will be proportional to e so this term
here will also be something times e and
so you'll be able to take out a as a
factor from this right now this is only
good true if this electric field is weak
compared with the you know the if the
force due to the electric field is weak
compared with the the forces involved in
the binding the electrons to the nucleus
and I suppose it was probably not until
the invention of the laser that probably
even people
thought you could ever possibly get to a
case where that might not be true but
now of course we know if you've got a
laser you can make really big fields you
can get very easily to this regime where
this polarization changes alright and so
you'll get nonlinear type terms come e
that's not a you're not going to do this
any at all in the course so but there is
this whole area of course of nonlinear
optics which is when this linear
approximation breaks down so yeah so
here we are so so this is back to this
what I said D equals epsilon e ok so
that's really the the answer to your
question and the D equals epsilon E but
D equals this other expression epsilon
naught E Plus P and now we say that P is
proportional to e this Chi is called the
electric susceptibility which i think is
a confusing term because it's often in
practice just called susceptibility
people just neglect the electric bit but
of course there is also a magnetic
susceptibilities which is very important
when you're doing magnetic materials and
unfortunately people use Chi for that
too so that really gets people confused
if you're not careful so we're not going
to be doing anything with magnetism so
maybe it's no problem for us but anyway
this is the electric susceptibility so
we can put in our P as Chi E and
therefore D equals epsilon naught 1 plus
Chi times e and this is equal to epsilon
e as we've said and you can see then we
have the epsilon must equal this thing
epsilon equals epsilon naught 1 plus Chi
and and yet and this is a another
expression we've had before in terms of
the refractive index so you can write
the
permittivity the relative permittivity
in terms of the refractive index right
so so that so this is the refractive
index coming in here so and then the
refractive index then is the square root
of 1 plus KY and so you see how all
these things are connected
you could also say one more of course
we've got where are we yeah we could say
that and we've got N squared equals 1
plus KY so KY is equal to N squared
minus 1 so this is another sort of form
you might come across at times the
electric sucks you can see then we know
in free space free space n is 1 so this
thing is zero
so there's no polarization of free space
which is what we know because there's
nothing in it so there's nothing to
polarize ok and so yeah bit this bottom
bit it just says that you can do the
same for magnetism we you can do
virtually the same sort of expansions
for magnetism but we're not going to
deal with this because we're not going
to deal with magnetic materials at all
but there are some quite interesting you
know magneto-optic materials and so on
which are quite important actually
nowadays in terms of things like data
optical data storage and so on so we're
going to we're just going to be assuming
that the from the magnetic point of view
the material just behaves exactly the
same Spri space so B equals mu naught H
yeah the electric polarization subjects
the field out of the original the
magnetization sorry today I should add
to the field magnetization subtracts the
field yeah I think this is the Charlene
has asked this question of why this is a
plus and this is a minus and the the
answer
I think it's all to do with all these
horrible things you learn about in
magnetism diamagnetism and paramagnetism
and sometimes you know it's whether
whether they oppose the change to which
they were due and all that sort of stuff
so I think it's just a case of whether
if you define magnetization in the way
that it's normally done then it turns
out there's a minus sign but yeah I
think you could equally well if you find
it as being the opposite side actually
yeah I I don't really know much about
magnetism so don't ask me too much about
that well sort of thing has to do with
the part that stuff that beam is
actually the conjugate of the electric
displacement so it includes the effect
of magnetization so another way to put
it is if you could please go back one if
you solve this if you take em to the
other side and multiply by mu naught
then you get exactly the equation that's
right yeah if you just expand this out
you get B equals mu no H plus M so it
would be exactly the same formula yeah
so the field is actually H naught to be
the a conjugate of the electric field E
is the magnetic field H naught being B
is the induction okay so okay so so now
then let's write down those Maxwell's
equations for these various different
cases so so this is how you can write
them in vacuum I said we first came
across them and if you've got vacuum and
in addition you've got no charges and no
currents then of course this term this
will be 0 and this jail will also be 0
and now you can see that they look
nicely symmetrical these two are both
equal to 0 and these things are both
equal to a first time derivative but
there there is a difference of a sign
but apart from that
nice for nicely symmetrical and then the
middle middle row here says if you've
got some tip it's a matter and it's
still allowed to have free charges and
currents then we can write write these
things so this is that what we just said
a minute ago if of course if you've got
no free charges then div D is zero and
and then if you've got a material matter
without the free charges then again we
put the rose and the J's zero and we end
up with these quite nice simple and
symmetric or sort of relationships and
in any of these cases of course you
could derive the the wave equation and
write it in that same form we had before
okay
and the as we said that the this term
here is is related to the speed of light
in the in the medium and what we find is
that the speed of light in the medium is
equal to the speed of light if free
space divided by the refractive index
all of those things are things we've had
before so I'll just stress again one
thing that I did say earlier this was
originally if you remember curl curly
and then when we got rid of the Coker
Lee we actually had to get rid of a what
was it grad of div E and we assumed that
Greg div E is zero
I'm just reminding you again that is not
always true so beware and you probably
won't come across anything in this
course but in the general wide world
it's not always true and the second
thing to remember is that this thing del
squared of a vector that del squared is
not equal to the same as the laplacian
except if you're in Cartesian
coordinates
ok so now we've got the the wave
equation we go back just like we did for
scalar waves we can look at some
different solutions of that wave
equation and the simplest possible one
is the equation of a plane wave and so
this is this is a picture of what it
looks like
I presume these pictures are taken from
Hecht are they but to me it seems very
perverse to have the wave propagating in
the X direction he's a bit strange there
but anyway and virtually every book has
the wave propagating in the Z direction
but never mind we won't worry too much
it obviously doesn't make any difference
but they this is the important thing is
we get the electric field vector and the
magnetic field vector E&B he's drawing
here it could equally well be in because
the two there B and H are going to be
proportional to each other all right
right angles to each other I'm just
telling you the results first I haven't
proved this yet and the wave is
propagating in a direction which is at
right angles to both of those and so
these three former a triad and it's a
right-handed triad such that e cross B
is in the direction the wave goes so
that's the way to always remember it
well that's the way I remember it is
that he cross B is in the direction the
field travels which means of course if
you've got a right hand coordinate
system normally we would say e in the X
direction and B in the Y direction and
then the wave moves in the Z direction
but he's taken it with E in the Y
direction like this in order to get it
so that it's still a right-handed
coordinate system and so how do we get
that well we first of all of course we
can say it's quite straightforward for
the e we can just solve this thing
for the as just as iets a scaler and get
an expression for e so e e is going to
be then polarized in a particular
direction we've then got to work out how
B is polarized for this particular value
of e and well Jill just said something
here which is I guess it's correct but
it's quite sort of condensed and I went
through well I would really derive it so
perhaps I'll say a bit more about that
and so what we've got then is well this
first part is all right we've got curl E
is minus DB DT and we know that the the
time dependence is this e to the minus I
Omega T so DDT of e to the minus I Omega
T is very straightforward that's just
motive the minus I Omega comes out the
front so so that's what we get curl e is
equal to minus DB DT is equal to I Omega
B and so we can say then that B is equal
to minus I over Omega curl e right so if
we if we know what E is we can work out
what B is from that and so we we take E
as being this thing right so in general
this this might not be traveling in the
X direction or any other particular
direction and it might be just pointing
in some direction in space so really I
guess you ought to sort of do it for
that general case and which means that
effectively what you've got to do is to
work out what curl of the e is and E is
effectively as you can see a scalar
quantity well let's forget about the
time dependence we're only looking at
the space now there's this scalar
quantity here
times a constant vector so we then have
to use another of our identities which
is the the curl of a scalar times a
vector so let's write it like that and
you might remember that that that is
equal to and I had to look it up because
I never know these things five times
curl a plus grad Phi cross a right so so
that is a general vector identity which
allows you to work out the curl of this
products of two things and in our case
this is a plane wave so this direction
of the e vector is constant so our a is
a constant so curl of a it's going to be
zero because it's constant so that goes
and we're then left with that the the
curl of Phi times I is is grad Phi cross
a and Phi is equal to Phi equals e to
the IKr
and a is equal to this direction of the
e vector right so if Phi equals e to the
IKr then grad Phi is equal to I K times
e to the IKr and so we stick that into
here
and after we've done that we can derive
this expression here we find that the
finally that the Magnetic vector is
equal to the cross product of these two
so that's why of course all these three
things have to be a right angles to each
other all right so this is the next
thing this is a picture of it moving
along then and we'll stress very
strongly something and that is that
these two components the electric and
magnetic field are in phase with each
other
I don't know what it is but I've often
found the students don't seem to pick
this up and they come up with these
funny ideas about them being in
quadrature and I think they get it mixed
up with some other things but for a
plane polarized wave the electric field
and the magnetic field you can see are
in phase with each other when the e is a
maximum the B is a maximum and then they
they will have this wave form this here
this is showing the wave as a function
of distance this is like a snapshot of
what happens at a particular time but
you can equally well get a very similar
sort of thing by looking at a particular
position and seeing how it changes with
time you'd also get sine waves like this
and you would also see that this these
two the electric and magnetic fields
have to be in phase with each other for
that case so okay this is expressing tag
showing you again this right-hand rule
or whatever it is for I don't know how
you do these things but anyway I always
like the I can never do these partly
because I'm left-handed I think it ruins
things but but this corkscrew rule is
the one that I always know that one and
my students will know it's because I'm
very handy with a corkscrew
so so this is showing then how this
electric field varies with space at a
fixed time and the distance then between
these Maxima in the electric field is
the is the wavelength right so so this
is going to propagate yes it's a
traveling wave so this this structure
moves bodily doesn't it in time along
the along this direction of propagation
ah and now we get on to this thing that
I was talking with George about on the
way the pointing vector a
I guess this I don't think you actually
you really sort of prove it though you
what it really does but basically the
pointing vector it turns out is a
measure of the energy flow so it's a way
of looking at the energy that's carried
by a wave and this is the definition of
it so s is equal to 1 over mu naught e
cross B actually you can see B equals
actually this should really be I think
this should really be motion it not mune
off really if it was in a general media
and V equals mu age of course so this is
really equal H so so so that's the way
it's sometimes derived something defined
but anyway and so but if you're in free
space then you can write it with mu
naught here like that and then using our
expression for the the what the velocity
of light then we can write it like this
but I think this is also true for free
space only I think that I think that's
right this has taken that presumably
this is from heads again ok so this is
what our waves look like Egypt each of
electric and magnetic fields is a cosine
type wave so this represents then a wave
which is moving in this direction
alright so so this thing in here
basically is the phase isn't it you've
got cause of a phase term so you can see
that as Omega T varies so that shape of
the wave is going to sort of move in a
particular direction and you'll notice
that I with this this is a vector this
is a vector but this is a constant
vector so this is a vector which tells
you the amplitude of that wave in
magnitude and direction and you'll
notice that these cosine bits are
exactly the same
that's because as we said the E and the
B are in phase with each other and so
that's very simple and so what we can
say is that if we represent the the
fields like this we can say according to
our definition of the pointing vector we
can write it like this and so we end up
laying cos x cos is cos squared so
notice notice that although cos of
course can go negative it is got this
periodic isn't it so it's negative half
as half as long as it's positive cos
squared is always positive so the way
this is defined the pointing vector is
always going in the same direction it's
always when you've got a plane wave the
energy is going in the direction of the
direction of propagation of the wave and
ok so this shows us our that's how you
we how we get all of this again the
selected that's sorry let me this shows
us
can I go back oh yeah no I'll come back
to funny here we are this shows us how
this pointing vector varies in space and
time right so this is pointing vector
has also got a sort of wave sight nature
but normally what we're interested in is
not the how it varies in space and time
but what the time averaged form of that
is going to look like and so we can do a
time average of it and so that's what
this this thing here is this s with the
lines on the other side is a is the time
average of s and so we've got B is K - K
times e over Omega and so therefore
we've got here a cross B and we know
what B is so you can put that
and we're gonna get now a squared right
so finally what we get is that the
pointing vector is proportional to the
electric the average the time average of
the electric field squared sorry are we
haven't done any averaging sorry but
this is the modulus so it's the one of
the two lines mean okay it's the
magnitude of the vector it's still a
function of time okay we'll take it as
that yeah and yeah okay and I think
really I think really in if it's in a
medium this is absolutely not absolutely
not
is that right okay so now we're going to
do the time averaging so this e e is a
time varying quantity e varies with both
space and time and so this is the this
is the modulus square of of it of the
vector so we put that in and we get then
that s is equal to something we call
square did so it's not much different
from what we had before okay and then
the next thing we do is to do the time
averaging so this is something which is
positive but oscillating time varying
and the speed of these variations is
very very fast we've said before the the
frequency of light is around 10 to the
15 Hertz which is faster than any any
detector that we have to measure the
fields directly right so you can't
actually measure the electric field or
the power of an electric of an optical
wave because the frequencies
hi so the detector it normal detector
like a photodiode or whatever would be
measuring some time average of this
because there's no way it can respond at
this sort of frequency so we do the time
average and so this is how we can do the
time average right this is the squares
the square things mean time average the
caret signs and the time average you
integrate this over a long time and
divide by the time to get the time
average and so this is called well
usually called intensity I think that
intensity is now frowned upon by purists
and they come up with new names every
every now and then so irradiance is it
is a word that you often read nowadays
in the literature and yeah so here it's
measured in watts per square meter which
is obviously a unit of power density
right there's one other thing that's
worth mentioning here and that is
actually that normal detectors actually
really don't measure the the pointing
vector they they actually usually
measure the electric energy density and
there's a thing called pointings theorem
which if you study electromagnetism will
show that the power flow out of a volume
is equal to the rate of change of the
stored energy and the stored energy is
made up of electric and magnetic energy
so all these things are related and it
turns out that for all practical
purposes normally it doesn't really
matter whether you're measuring the
pointing vector or the the the electric
energy density or whatever it's going to
give the same sort of answer but there
are cases where you have to be careful
okay and then finally we've got to do
this time average of course
add and the the time average of cos
squared is is just a half you remember
how you do that you go in so cos squared
you go into double angles and then you
get a cosign that cancels out and you
just left with the constant bit and then
finally then you put that half in and we
now got an expression for the intensity
defined as being the time average of the
pointing vector as being equal to the
electric field the amplitude of the
electric field squared multiplied by
some constant which in probably 99 cases
out of a hundred we don't even bother to
even think what that constant is let
alone thinking of whether there's a half
there so you'll see loads of books or
papers where they just say that the
intensity is equal to e squared which is
strictly not true I could guess but
people do say so with how do we how are
we going we we got one out yeah okay
so that's how you can do how you can
calculate the power flow for a plane
wave i guess that the method is not only
going to be true for plane waves but you
can apply it for other sorts of ways so
but the next that the next thing is you
might think back to what we were doing
with phasers and there are a few things
that are important to realize here what
you can do and what you can't do you
remember this is what we said from our
phasers we said that out what we what we
actually measure in the real world is a
wave that's like this it's a cosine
dependence in space and time and we say
that we use this this complex
representative where we say that this is
the real part of some complex
exponential and if you remember what the
reason for doing that is that these
Exponential's are much easier to
manipulate it means that you don't have
to remember all those horrible cause of
a plus B things and so on so it's much
easier to do the algebra using these so
what you don't do normally as you
remember is you do your all your algebra
with your e to the I some things and
then right at the very end normally you
find the real part so find what you've
got what the real field is in real real
space as we use so but here now you've
got to be very careful because you saw
that the pointing vector actually was
cos squared right so if this is a
nonlinear we performed a nonlinear
operation and if you if you do nonlinear
operations the same would be true for
nonlinear optics as well actually you
can't do it
in the this complex notation directly
because you just get the wrong answer if
you if you squared this you get an e to
the 2i KZ minus 5i wouldn't you which is
which is and now the real part of that
is not what we want so so the what
you've really got to do is so it's the
work through I guess in the whether
these the the surest way of doing it is
to work through in terms of the cosines
and so on rather than to go into the
each of the eyes so so this is saying a
bit more about this so so what we've got
there is this is showing how you can do
it in terms of the cosines so you can
see that the that for example here we've
got two fields and let's say that you
see the point is that although the
fields are additive the the the power of
those two fields
those two waves is not going to be equal
to the sum of the powers of the
two ways right so we can say that the
the EES will add coherently and then we
can work out the power flow of the sum
of these two fields and you can see of
course expanding this square you're
going to have the the power of the two
but you're also going to get this cross
term in the cross product term which
depends you can see here on the relative
phase of the two waves if they're out of
phase by 90 degrees then that will go to
zero but if they're in phase this is of
course it's going to be strong and so
that's that's one way of doing it so now
but in terms of phases then if you you
you can do it in terms of phases I'm
just saying you have to be careful how
you do it and so you can add you have to
add the phases first and then find the
modulus square you can't actually do the
squaring and then though the real part
alright so you add these two phases
together which gives the total field and
and then the intensity is going to be
the modular square of this and you
expand it and you can see we've got
exactly the same answer so doesn't
matter which way you do it as long as
you do it one of those ways what you
don't want to do is to try try squaring
the e to the I things and then taking
the real part because you'll get
something different then and that is the
very last slide and that is the dead on
time so so we've got a couple of minutes
for some questions okay any questions
yes Sharon well I'm quite curious about
the pointing vector and you mentioned
that it's actually a cross term and it's
similar like property propagation
direction like weird energy flows but
what about the peculiar case in
evanescent waves like does it mean
energy is no idea oh yes even essent
waves this is you're getting on to
something really controversial here I
think actually sorry I mean if the
pointing vector is the direction of the
energy flow then then it seems like
there's no propagation energy in in the
evanescent wave then well it is normally
said that there is no energy propagation
and an even s of wave but you know in a
way there is this there's sort of
transverse propagation of energy but
there's no energy flow across the if
you're doing total internal reflection
and you're looking at the stage say
there's no energy flow across the
boundary I think it's true to say is
that true yeah yes anything a bit
controversial yes yes you can
if you have the interface if you have
the interface of the TR if you compute
the pointing vector in the vertical
direction it turns out to be zero the
time average is zero so you don't have
energy flow but you do have energy
density yeah yeah so and the other thing
is you know the question of if you got
this is your total internal reflection
so here you get this even s and wave now
of course you know the whole of all
these questions of these sort of
innocent waves and so on they all rely
on you know assuming infinite playing
waves and infinite surfaces and things
like that which of course none of these
things can really be strictly true in
practice but if we do this then we'd
expect to get something which is
actually travelling in this direction
and so in some sense I think that there
is some sort of energy flow in that in
the transverse direction but I'm just
having a big argument about this we're
not for this case but to do with some
other project we're doing at the moment
we're
where where is to do with that when when
the H vector is zero and so the question
is is there energy flow well I guess
there isn't I think there also this
question of whether pointing vector
really always means the energy flow in
fact I remember having a conversation
with Emma wolf once about this and he
said he said well he said let me give
you an example imagine you have a DC
planar electric field in this direction
and you have a DC planar magnetic field
in this direction is there any energy
flow no that but you can work out what
across ages and so you know so I think
that he said he said he put this forward
as a as an example which makes you
wonder whether it necessarily means that
the really are always means energy flow
this the way we're talking about this in
the lab today because people have come
up with these solutions where for
example energy flow sometimes does this
so you might get the lines of the the
pointing vector might do something like
this and here you can even get so here
you can see that there has to be some
sort of zero round here because this is
going this way and this is going this
way and here you can actually get these
sort of closed edges of pointing vector
that's going around in circles now
whether this really represents something
that's physical or not I don't really
know I you know in water flow I guess
you could get this there's no reason why
you couldn't get eddies of water going
in circles but you know so this is what
you get that this is what you get if
you've just got a very simple example of
a lens
focusing the light you get this sort of
pattern if I look at this in some detail
bit you get this sort of behavior in
this region here and they've been papers
that have described this but what it
really means in practice well I guess
maybe you know you you have to I guess
nothing really means anything unless you
can measure it so the question is can
you come up with a detector that
measures pointing vector and I don't
know the answer to that as I said
earlier I think normally detectors
measure electric energy density so
people have both looked at what happens
in this focused region of a lens and
there are various ways you can look at
that people have used near field optics
you know a tapered fiber to look at it
or you can do some sort of tomography
type experiments or with a with a
detector that you rotate and things like
that but to actually measure the
pointing vector alone right any
questions from over there