Optics (Lec - 11) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

the following content is provided under
a Creative Commons license your support
will help MIT OpenCourseWare continue to
offer high quality educational resources
for free
to make a donation or to view additional
materials from hundreds of MIT courses
visit MIT opencourseware at ocw.mit.edu
okay so you guys worked in just in time
so today's topic is slightly different
but because before we start you ask the
question last time right and tell fish
health - okay again so cal pass my
friends here ask me a question and I
think you asked it in the lecture before
the quiz right so I answered it I guess
he was not convinced so he asked me
again just before the quiz started the
told him well we're gonna wait until
actually we the classic convinced so we
cannot discuss it together so was
thinking how to answer his question and
then I decided it was a really
interesting question so made some
additional slides to discuss the topic
that he asked and the slides are not on
the website so you know don't look for
them you cannot find them even though I
will probably post them anyway but for
now you just have to watch and
participate in the discussion and see
what happens okay so confess after
following when we started actually the
class I claimed that optical rays have
to satisfy form as principal so
therefore they must always go where the
path is minimal so in the case of a
green well use the notation here where
the grey area in the slide corresponds
to high index of refraction the sort of
white said is lower index so this is
like a typical green you can imagine
this profile may be quadratic or
something like that and and what we said
in the class and you all believed me
except for CalPERS again what I said in
the class is that if you have a ray that
enters this medium sort of off axis like
shown here the Ray will actually bend
inwards so if all the Rays will come to
a focus somewhere around there but
kalpesh you ask
you know well why doesn't that I do this
I mean these days here seems to actually
it seems to do the wrong thing doesn't
it it seems to go out towards the high
in this area whereas the minimum path
principle would suggest that that I
should go outwards like this towards the
lower index area so the question is what
will when I do will do this or that well
we know from experience I mean we know
from experience from experimental should
say that being optics do focus or those
who know that physics actually behaves
like this but the question is how is
that consistent with our description of
the physical system what is going on
here ok so a difficult passage question
and what I said when he asked it in
class more or less I cannot remember
exactly what I said but they said well
to properly contemplate the answer we
must consider not just this way if you
just have this way we don't really know
where they will go but we also have
another way the one that is one axis now
these two rays they actually originated
at the same point where did they
originate well at infinity right some
were far out to the left over here and
what I said is that basically these rays
because this ray goes through a longer
optical path because it is entering in a
high index medium this leg has to match
its optical path so this ray actually
enters a lower index so therefore it
starts with a sort of predicament that
points to a smaller optical path so in
order to catch sharp in order to match
the longer optical path of this ray this
way the sort of off axis ray must follow
along a geometrical distance so that the
longer geometrical distance times the
lower index kind of will balance will
compensate ok so this is what this is
where we left it but then
kalpesh pink came back to me next you
know it's a quiz and he told me well I
don't believe it you know why it's all
right you want to rephrase your question
okay
so I represented your objection well
okay okay so let's print so first of all
does anybody else from here from Boston
have some additional insight about this
I can go back to the original we'll call
it the kalpesh question yes we didn't
hear the question
is your button push yeah you have to
make sure that the red light is also on
okay yeah in the standard case of an
interface we have a wave wave
propagating from left to right and we
have an interface which is top to bottom
but in this case the interface could be
considered to be left to right currently
because the refractive index changes
from top to bottom so the interface I
mean I'm just thinking that if we
consider the interface to be from left
to right then the normal will be from
top to bottom and still the Snell's law
will lead us to the correct answer
correct actually by the time you reach
here where the Ray has already been you
can make the target here right yes now
today is really just in a little bit
yeah but the question is why should
there a start to individuals to begin
with why should it not Bend this way
yeah and you bend in this way that you
don't like this right there's something
something amiss here here okay
so there's no doubt as I said there's no
doubt that the correct physics is this
one the question is is our model correct
you know is a claim all the stuff we
developed about optical paths and so on
does it really agree with the physics
okay so I don't get any any other
comments or objections about this or
okay okay so let's go with a little bit
more care through this argument where
are what is happening let's take out the
medium so there's nothing now just free
space so the range started at infinity
they're heading towards infinity
everybody's happy with a straight path
right because clearly the straight line
in uniform space is the shortest between
two points and clearly these two rays
they never really meet but we say that
me that infinity at to the left and they
also meet again at infinity to the right
and the wind have traversed the same
optical path by the same by the time
they meet again at infinity
no matter you might answer we might ask
well what does it really mean I mean
after all the path itself is infinite
what does it mean they travel equal
infinite paths well I can get actually
around this problem pretty easily
because I can actually put some
boundaries to the left and to the right
and I can keep this boundaries at a
finite distance right this then it can
definitely measure the optical path then
you can start moving the boundaries
further away to the left and further
away to the right the optical path still
remains the same as I move the
boundaries right and I can continue this
process until the boundaries are really
really verified this is mathematically
really what we mean by infinity right so
I can get around this difficulty I can
certainly talk about equal parts even if
the paths are actually infinite okay
this is sort of a technicality but
nevertheless an important one so in
order that were on solid ground okay
so hope I've convinced you that Farah is
happy in this case of free space because
there is indeed propagate through the
minimum path and indeed they're all
equal in terms of path okay okay now
this is really not anything new this is
what I said before and basically this
says that the longer geometrical path
actually gets to be multiplied by a
lower index so therefore by the time the
Rays meet again here they still have
obviously equal optical paths because
the path to the left is equal they both
came from infinity then the two
trajectories here if you sort of
integrate the optical path all along you
would get equal answer okay are you
happy
maybe push the button yet instead of a
will coming towards whole cross-section
of the green lens suppose if you have a
small sport coming towards only towards
the top so will it go towards up or in
that case right so the question is
suppose I take out this rail right will
vary
will this thing also focus what do you
think
I think we have only a single rate
towards the top which is coming towards
the lower refractive index then it
should go towards the lower reflective
index because it doesn't need to match
the with the same phase as the central
wave okay let's let's not quite argue
about a single array because there's no
such thing as a single array if you
luminate a very real infinitesimal
optical pot over here as we will learn
in wave optics what you will get is not
a sort of stray light stray straight
line of life but you'll get a very
strong diffraction right so what will
happen if you illuminate a single spot
here is the light will spread violently
into the green and then what will happen
is a kind of a complicated issue right
so so so the way essentially by claiming
diffraction I got around the problem
that you're posing because as long as I
have a finite pencil of rays come in
here then the same thing will happen
then because I can always compare two
rays that enter at the slightly
different indices than the same argument
follows so this pencil you will also
kind of focus in the sense so basically
I can remove parts of that ray diagram
and the same thing will still happen
according to geometrical optics
diffraction which says differently but
according to the medical optics I can
remove portions of this race and still
have focusing there's another objection
that I haven't heard yet I am not
finished I I am not satisfied that I
answered kalpesh this question yet and
the cynical path is also skeptical and I
hope someone I have another slide to
bring up the objection but I'm wondering
if someone will bring it up or maybe
another objection is that there is an
objection that I thought oh but they may
be answered
questionable I prefer yeah why is it
that the Ray on top bends to slow down
to meet the Ray that's traveling to the
denser medium versus both of them
traveling to the less dense optical
medium because that objection that's
exactly the objection so the objection
is what who says that the central ray
should go straight right I sort of
silently took it for granted that the
central very here goes straight and
therefore this other Ray has to raise in
order to meet it right why don't the two
days go like this is this your question
I mean yeah why don't the two days what
do they both do like this they seem to
be doing the correct thing they seem to
be both going toward the lower index
okay we know again we can argue that
this would happen but we know from
physics that this does not happen so
what is the physical fact that cannot
possibly happen by symmetry okay thank
you
actually it is not a question of
symmetry because indeed in this case I
assume that the refractive index is
symmetric right however now let me see
if I can do this without revealing my
next slide
however I can make an asymmetric index
profile okay so so this is a simulation
of the green with the symmetric
quadratic index here I made the profile
asymmetric so basically the in this is
the equation here I should simply draw
it ah
except I don't have a marker anyway
basically here the index of refraction
is a parabola that has a different slope
above and different slope below so you
see that even though this is a symmetric
stubbornly the central a still goes they
explain therefore it is not symmetry
which makes the previous case impossible
but well I said symmetry but if you go
to your next next one that Ray that's
going straight along the center is it
gone yeah like that ray goes straight
doesn't it yep and that's because
there's no transverse gradient in the
Vetri in refractive index where where is
propagating okay I guess okay
yes that is true there's no moment to
push it right yeah and that's agree with
however bacteria steam does not answer
the good question because the reason
moment there is momentum work so there
isn't we apply Snell's law for example
is because we claim Pat minimization
right so I suppose you could argue this
way this is a valid argument actually
why why this ray must go straight
actually I don't know if it must go
straight but I have another argument why
the why basically this is impossible why
this is physically incorrect
if you do that isn't it true that the
bottom raised path is still longer than
the top raised path because of the the
index of refraction is low and then it's
just low all the way and yeah exact
exactly if you do that if you do that
there is no way ever that this ray will
catch up right because this ray now not
only is it going through a higher index
of refraction it is also following a
longer geometrical trajectory
so therefore if these two rays ever meet
again okay they might meet at a finite
distance the way I drew them here at
some point they will meet again right
well they might meet at infinity they
might somehow become parallel and meet
again at infinity but no matter what
this ray over here has travelled there
longer optical path and that now that is
a clear violation of Verma so basically
arrived at an impossibility so this is
physically then incorrect that is the
reason why the base have to know the
base don't have to Rend the Rays have to
bend inwards and so the proper very
interpretation of hermas principle and
actually the book has an interesting
discussion about this we can claim yes
that firm and minimizes the optical path
but perhaps or at least linguistically
more precise definition is that firma
requires that the path of the race is
stationary
now stationary in mathematics what it
means that if you perturb it in some way
you will get a bigger answer if you
perturb it in any way so remember I wish
I had the marker I think I can just use
a pencil
okay let's try with a pen if you tell
you guys tell me Boston if you can see
the memory which for math we did this
this analogy we said you can see it
right we we said I'm sorry I should have
said remember with Snell's law Snell's
law says that they're a must do
something like this in order to connect
this point to this point so
mathematically what we derive is that
any other possibility this this this
this and also why not this these are all
different possible Rea trajectories
there is a mathematical pool it's called
calculus of variation which does in
molecular thing it somehow
compares all of these possible
trajectories and then it arrives at the
minimum trajectory and it tells us no no
no no no no matter what these are
infinite possibility that you can
imagine but this pool calculus of
variations actually tells us eventually
that all of these give us a longer total
optical path and they're only true one
if you wish the one that minimizes the
stationary one is the one that they draw
kind of bolder here okay so this is the
proper way to interpret from a it is not
as a single ray will actually you know
look for the lower index of refraction
it is actually that when you have a
bundle of Rays that share a common
beginning and a common and then the
trajectory that this rays will follow
will be such that they're all equal and
the distal object right now is
stationary in the sense that if you were
to perturb it in any way if you were to
move some of this raise up or down or
band them and whatever you would end up
getting a longer trajectory
okay that's the that's the proper
interpretation of firma so this is
actually very interesting very
interesting topic and then and I confess
that when I was a student I was also
bothered by this right I mean you know
their eyes seem to be doing
after all it seems to be going really
where it shouldn't go why well the
reason is as we just described because
the path has to be stationary the bottom
line of this which will actually be
useful in our ensuing discussion of
Hamiltonian optics is that the index of
refraction is acting a little bit like a
gravity actually exactly like gravity
more than you would think the index of
refraction seems to be attracting the
Rays which is again is a little bit
counterintuitive if you interpret for my
as a minimum path but after that after
we went through this explanation it
makes sense right the the index of
refraction pretty much collects rays
Miri yeah Judah I think if if we
consider the interfaces left to right
the geometry works out and so if we
assume any ray coming at said like this
going from left bottom to top right then
it will it is going from higher index to
lower index which means it will tend to
increase its angle with respect to
normal so it will keep bending like this
and at some point there will be total
internal reflection and it will bind
towards so maybe I mean that's just
another way of looking at the problem
that if you consider the interface to be
normal to the change in the refractive
index then yeah law also justifies this
attraction right I mean the the point
that is really hard to swallow is why
this ray starts bending involves at this
interface after you started then it has
to fall right because the Snell says
that it should fall yeah so that is
consistent so really these arts you can
think of a mechanical analogy which
would be like a
imagine that you take a shoe and you cut
it in half right now you have this
hollow sphere and then you launch a
billiard ball into that surface what
will the billiard ball do well it will
spin right it will go down and then up
again and then down again and spin and
spin and spin so basically the surface
of this in this sort of mechanical
analogy the equivalent of the surface
shape is the index of refraction here
and of course the billiard ball is
subject to its own gravity and the
surface how do you call it the hats
contact from the from the surface and so
on and so forth but the bottom line is
that this is a
michalak a mechanical analog I'm sorry
this isn't whatever the mechanical
mechanical analogy for this is gravity
attraction okay yes provided you define
the gravitational potential
appropriately in terms of the index of
refraction and this will actually come
up near the end of the Hamiltonian
lecture so we'll show you exactly the
formula did everybody hear his question
no okay yeah I was asking is the analogy
between the gravitational force and the
reflective it takes complete yeah and
you have there okay okay now that I have
a marker
let me also answer so in this case that
what I did is on the left hand side we
have the familiar a quadratic index
profile so this is n 0 minus some
coefficient times X square on the right
hand side what I did is a chosen index
that is somehow like this ok I'm
exaggerating but so basically it is
still a parabola on the left and awry
but it has a different character on the
left and the right and the point here is
that the central ray still goes that of
course this is a terrible
right that's something really horrible
back here does not focus properly and so
on and so forth but the only reason I
did this is to show that that that that
the array here the gold straight not
because the index of refraction is
symmetric even though Collins point is
well-taken I mean the reason it goes
straight here again you can think of it
in terms of Snell's law it does not want
to go this way because it would you know
it would Bend to to a longer optical
path than there a next to it it does not
go want to go down because again it
would end up with a longer optical path
with respect to the Ray below it so
therefore it has to go straight right
can you imagine the sequel a situation
huh is it always that can I actually
bend this ray what would they have to do
to make the central ribbon actually
Colin answered it already
yeah they cannot see you can you yeah
curve the whole profile you have to put
a gradient along Z on the index of
refraction then if there is a tilted
interface which basically means a
gradient of the index of refraction here
it will Bend right because again Snell
tells it to bend so but since there's no
seams here the index has no Z dependency
then the Ray must go straight okay
okay so first one thanks thanks to
kalpesh for asking this question because
it turned out to be a really really
helpful one and in addition to that
there were kind of two more things to do
so I'm usually finished the Hamiltonian
optics discussion and then we'll start
talking about wave optics so we'll see
how far we'll go and I have to make some
announcements near the end so so I will
try to save about five minutes near the
end for my announcements okay
so last time we were discussing gradient
index optics and just to briefly remind
you we kind of went this is actually
what prompted the Cal passes question we
went through this derivation where the
gradient index element I will stop
saying gradient index it gives me
actually it's difficult for me to
pronounce a green red so when the green
element and we did this simple
derivation which basically said that
this green element is acting like a lens
and again with our newfound wisdom it is
acting like a lens by virtue of the
attraction of the Rays by the region of
high index near the center of this
element and also when we also went
through this other case of axial green
and we also said that this this is now
let's understand if you only have Zig
gradient then it is still not a lens you
know I'll let you think about this by
yourself in order to make it a lens you
have to have an X gradient in the index
of refraction so this case is somewhere
in between where you have a cigarette
so the index is higher in the front and
lower in the back but in order to get
optical power that is to get focusing
power in this case you actually grind in
addition you blind it to a sphere and we
said that the reason you do this axial
grin here is because you want to correct
for spherical aberration okay so if you
compare the paths now of this way to
this way okay in this case this array
goes through a longer optical path
because the index is uniformly high here
their fault
it burns faster in this case by lowering
the index of refraction that this ray
goes you actually force it to go to a
longer geometrical trajectory
therefore you are pushing it further to
the back okay this is another way to
interpret the way this element collects
you can interpret it with Snell's law
this is what we did last time but now
with our newfound wisdom on optical
paths and formal who can also explain it
this way
I lowered the optical path by virtue of
lowering the index here therefore the
Ray has to go outwards in order to
increase its travel distance okay and
still maintain the same path as the
central ray okay so the net effect is
that you're correcting the badly bad
quality focus here that was caused by
the spherical aberration and I believe
will rule as it last time is we're asked
this question well whatever I have an
arbitrary index of refraction as a
function of position what if you know
this is not quadratic this is not actual
this is some general function of our the
Cartesian coordinate what then what is
the what is the a lot material yeah what
is the path of that is what is the
trajectory of the rays that go through
this variable index medium and I don't
remember how far I went here but we
basically took a turn here and we
started talking about mechanics
- - probably to some of your surprise
so the know you remind me how far did
they go do they actually finish this
example don't know yeah I think so right
yeah we cover these yes okay so don't
feel like going it again anyway because
this is sort of good all the physics 101
kind of thing what what might be a
little bit a little bit unfamiliar is
this development here which some of you
may have seen some of you may not have
seen so this is really not doing
anything spectacular this is a simple
mathematical manipulation so we start
with the equation of motion for the
mechanical oscillator so the question of
motion for the mechanical or I'm sorry
not a question of motion but the energy
for the mechanical oscillator so the
energy consists of course of two terms
one of them is that if you wish the
elastic energy the potential that is due
to the stone restoring force of the
spring and the other is the kinetic
energy okay and V is the velocity so V
is basically X dot okay the the first
time derivative of the position okay so
this is one thing that we know this is
the energy of the system and since in
our model there is no dissipation the
energy must be concerned I can also
write an equation of motion for this
system and since it is a simple spring
first kind of system they expect my
equation of motion to be what to be
second order so to be something of this
sort and X double dot plus KX equal the
force right the excitation okay so let's
let's say that the system is free
there's no force so if it is moving it
is moving because of a prior excitation
right which if I started to
we'll go on forever because there's not
dissipation there's no place for the
energy to go if you get this thing
moving it will go on forever but now
there's no force so basically I can
write down the question of motion like
this as a homogeneous equation okay so
there is a trick that those of you who
use MATLAB to solve differential
equations are very familiar and the
trick is the following you break this
higher order differential equation you
break it down into as many as you need
first order differential equations since
this is a second-order equation you want
it in two first order equations and the
way you do it is basically you write an
equation that looks kind of stupid the
question says that X dot equals V this
is one equation and then the other
question comes from this one since X dot
equals V it means that X double dot
equals V dot this is the acceleration so
the second equation now comes from this
that is V dot equals minus whatever K
over m times X okay
now I can define the vector X V and what
this set of equation here has done it
has actually connected the first
derivative of this vector with the
vector itself via matrix and here is the
vector in its first derivative here is
the vector itself and what I have to do
put here in order to make this look
correct is I have to put one here and
then minus K over m here and then zeros
and then I've done so this is again if
you ever used MATLAB to solve an
ordinary differential equation you're
familiar with this this is MATLAB
actually forces you to rewrite your
equation this way but it turns out that
this kind of method actually preceded
lab MATLAB by by about a century because
this is a standard way of writing the
equation in actually believed it was too
incorrect who actually first came up
with this in a standard way
derive the dynamics of a dynamic /
overhaul of the system and this is not
of course I'm not going I don't want to
spend too much time on this but over
there what this slide basically points
are to you is the following if you take
this expression here which we'll call
the energy but now we'll call it it is
of course you cannot see it because it's
given okay now you can see it so this
expression hmm it's already called the
energy but today I will call it the
Hamiltonian and it will become apparent
in a second why but if you take this
expression this is now a function of the
two dynamical variables the same two
variables that appeared in this
expression over here they actually get
into this equation X comma V right it's
a bit of a funny way to think about it
but all right it certainly you see X
here you see V here and actually I don't
like V very much because if I keep using
V I will not come up with a correct set
of equations I will replace V with the
momentum so that's a my not sitting
instead of V I will replace it with P so
P the momentum is simply n times V right
so I haven't cheated in any major way I
just you know I just did the simple
multiplication so if you replace P here
then the question will look like this
nothing changes here this part will look
like 1/2 p square over N ok oh sorry
ok ok so nothing major here the magic
now it's not really magic there's a good
reason for it but you know it looks like
magic is that if I do the following if I
take that moon derivatives of this
expression with respect to X and P so if
I take the H the X and if I take the h
DP in six again
okay what I get if I get the HD X what
is that it is K x-ray and if I take the
HDP I will get B over m okay so if I
look carefully here this is P over m
right that's the velocity and this is
minus K P isn't it wait a minute I'm
sure I have of course I know why I have
V dot here instead of the dot let me
rewrite it in terms of P dot if I time
if I do in terms of P dot I multiplied
by M both sides of the question I will
get minus K X okay I don't have to do
anything else
all right so what do I get well I get
that V HDB which equals P over m is this
term in this equation and if I get the
HDX which is K X is actually this term
in this equation so basically by taking
the partial derivatives of this
expression the Hamiltonian with respect
to my two dynamical variables position
and momentum I came up with the equation
of motion be composed in the dynamical
variables MATLAB style or actually pork
arrays probably shift in in his grave
one calais type okay
in the way we write dynamical systems
okay so this is really what I say in the
top of the slide of course I said in a
slightly more formal way and then the
rest what I do is I simply solve this
equation but I will not do it here this
is a simple harmonic oscillator all
right the mechanical system I saw before
is a car that will move forever
eternally in a sinusoidal fashion and
it's kind of interesting I guess that
the
energy alternates between 100% kinetic
which happens when the cart the particle
moves to the rest position because then
it is the maximum velocity and has no
potential it is the last position of the
spring and it converts to completely
elastic when it goes to the edge of the
range of motion right because over there
it has to reverse therefore it will go
to velocity zero momentarily and the
kinetic energy has vanished at this
point of has become all potential okay
this probably boring because the stuff
is very well known what I really wanna
do is I wanna establish a connection
between this very nice very well known
mechanical stuff with with optics so
basically what I will do now is I will
derive I will not really derive but I
will show you a set of Hamiltonian
equations that actually describe write
trajectories and and what is really
spectacular is that as you pointed out
earlier they will turn out to have an
exact mechanical analogy if you use a
certain equation for the index of
refraction so we'll get to that so in
order to so there's two ways to derive
the Hamiltonian equations
does anybody know in the field of
mechanics does anybody know how we get
the Hamiltonian equations what is the
sort of the process well I sort of just
described it over here but but I guess
what I'm trying to get to is how did
this come about of course it came about
from my physical intuition right because
I know that they have kinetic you know
someone told me I don't know a thousand
years ago when I was a student in school
this is the case right but it's the
invariant property of the system it's
the total energy we put in at the start
okay yes so it is the invariance of the
system and also mechanical systems they
also have a firm a principle of their
own
if you recall mechanical systems also
try to minimize their potential as the
growth trajectory that they follow try
tries to minimize their potential okay
so a good example of that is what is
actually forget that this is optics
imagine that what you see here in this
sort of fancy looking looking diagram
this is not not the index of refraction
but this is a star imagine that the
center of this have a massive object a
massive gravitational attraction right
like the actually yeah the Sun right and
imagine that the way that it's not a ray
anymore but imagine that the red
trajectory is the path of the satellite
right
this convention isn't it if you if the
grayscale if you allow the grayscale to
denote gravity now not index of
refraction but potential due to the
gravitational field exit was well
exerted by the planet then if I launch a
spacecraft or satellite or whatever in
the vicinity of this gravitational field
as you know you maybe if you had a sort
of amateur astronomer or amateur how do
you call it
aspiring after that if you if you know
if you are then you know that depending
on how you launch your spacecraft
sometimes it will actually sort of
settle once and then escape that's very
bad right because it means that you are
less if you are in an astronaut you are
gone for two the space forever but also
you know that if you launch it properly
then you can actually get into orbit day
and orbit means that the dynamics here
is such that you will actually start
circling this object okay in fact we
never launch spacecraft to start
typically we launch them to planets
nearby but then all right okay so so it
turns out that the trajectory is that
this is a spacecraft follow actually
they also minimize their what
optical paths where you know there's no
index of refraction then it is just
gravity but they actually do follow the
minimum path around the star and in fact
this has been known for more than a
century I mean Copernicus even hinted at
that in his work right when you computed
the the the the orbits of the stars all
of their of the planets in the solar
system okay
it was very well known stuff okay so
what I will do now then is I will
actually drive you without really
proving anything but I will drive you to
a set of hamiltonian equations that
describe the same analogy but for optics
now so in order to derive our
Hamiltonian equations we actually need
two postulates now what does it mean
postulate do you know that word yeah but
I have a doubt regarding definition of
Hamiltonian why can't it be some other
invariant quantity like angular momentum
or linear momentum and energy in this
case okay so actually you're forcing me
to say what I didn't say before okay how
did this come about this comes about and
I have to apologize now because I will
throw a lot of jargon at you this comes
about from something called the
Lagrangian principle so the Lagrangian
is very actually if you write the
Lagrangian it looks very similar to this
one very similar to this the Lagrangian
okay the question you ask in the
mechanical system is what is the
trajectory that the system will follow
right so we have a particular set of
particles and you ask what is the
trajectory the Lagrangian actually looks
very similar to this but with a minus
sign
and I never quite remember if it is if
you put a minus sign here or here but
anyway one of those has a minus sign
okay and so what you do then is you put
this Lagrangian into this integral and
you say that that the trajectory that
the particle or follow actually
minimizes this integral now this has a
very good physical justification because
this is actually Newton's law the way
you get this Lagrangian is you start
with good old Newton which says that
then you multiply
okay haven't cheated the multiplied in
with the quantity that is hopefully
nonzero and then you can write it down
as one and
actually not quite zero anymore but
constant okay now confuse my mind for
science and well
well anyway
that's a good plan because because this
is a restoring force right so it yeah
well anyway happy may actually derives
the Hamiltonian so so I don't know how
to happen but but this is if you think
about it this is now the Hamiltonian
isn't it so by a simple trick one
Newton's law I derive that's not quite
what I was expecting but it happened
anyway
okay so answered your questions simpler
than I was expecting it to be okay so
there is a Hamiltonian right now why
that then why not the momentum that
really have to do with the Lagrangian
but cleanliness okay it's clear what I
did here it is just a simple
manipulation that that I learned from
professional Jerry with a metrical thank
you
okay I think I skipped this line hello
okay so we were at this word postulate
what is the possible what does postulate
mean but we accept something without we
say something and we accept something
that's right yeah so the postulate is a
good way to develop a scientific theory
in the following sense first of only
partly postulates cannot be very
complicated right so it is something
simple that we accept with two caveats
one is that we accept it because it
justify it is justified by observation
so postulate had practice it the
assumption that has never been violated
to our knowledge for example the Sun
rises every day as far as humanity has
recorded history this has happened every
day except of course the north
above the Arctic right where sometimes
the Sun does not rise okay fine so so
that's one and the second is that the
theory that follows from the postulate
has to be to be self-consistent right so
it has to be mathematically consistent
so typically the postulates are things
that I don't quite feel like proving but
they're good assumptions like a
something that cannot be very easily
challenged okay so the assumptions here
one of them is that razor continues okay
that really cannot be very easily
challenged because I mean if I were to
jump that we really strange right and I
can quote a number of physical reasons
why this would be really strange for one
is that relativity you know it means
that somehow a photon if we believe that
photons follow the trajectories it means
that somehow light we you know within
zero time actually travels the finite
distance that is impossible so that is
one good reason okay
so if you accept them that their eyes
must be continuous and actually it is
only continuity I'm not saying that
their eyes are differentiable so the
Rays can bend like this so here actually
obviously it is non differentiable but
it's still continuously so the postulate
actually talks about continuity not
differentiability piecewise
differentiable yes it is because it is
certainly piecewise differentiable here
piecewise differentiable there this is
actually technicality functions that are
not piecewise differentiable earlier
really nasty beasts like Weierstrass
functions and stuff like that which
normally would not deal with an
engineering so basically the second
condition here is is a mathematical
technicality so we don't have to dwell
on that okay if now they hear something
that is that probably as engineers we
got turned off by when we let it in
calculus but it's actually a useful
thing it is called the mean value
theorem and the mean value theorem says
the following says the textual if you
are given a continuous piecewise
differentiable function so here it is
okay so this is X this is f of X so this
curve here is f of X this is some
function I have X 1 X 2 f of X 1 f of X
2 ok the mean value theorem says that
according to the conditions that are
stated there is a point somewhere in
this interval satisfying the following
property that f of X 2 minus X 1 over x2
minus x1 equals F prime of X so so
basically this ratio I can pick a point
C somewhere in this interval such the
value such that the value of the
derivative of a function at this point
equals this ratio
and of course you can realize
immediately that if you make the
interval very very tiny then of course
you get a definition of the derivative
itself but the mini value theorem is a
little bit more general and it is the
mean value theorem that they use here in
order to write this equation this is
basically the mean value theorem where
the quantity of interest that I applied
is there a trajectory let me back up for
a second I should have started with
something else what am I really plotting
here so the red curve is the right
trajectory so I described the right
trajectory as a veteran truth so the
vector is a position vector right this
is a position this is the origin of the
coordinates and Q is the position vector
and then I parameterize it with a
variable S so basically s is an index as
I increase as I step through as I
basically walk along this trajectory
right okay so you can think of s as time
right as time goes forward the light
partitive zebra which are moving along
this curve okay and at each point on the
curve since it is continuous and
piecewise differentiable at each point I
can define a tangent except of course
the points worth is not where it is not
differentiable but let's skip those for
a moment if the point is differentiable
I can define a tangent and this tangent
is the vector P okay so this vector P is
actually the momentum of the Ray okay
very good so since this since the
tangent is associated with the
derivative in fact in high dimensional
space it is even more obvious because
the tangent really points towards the
well it is the tangent later you know
then this really is an expression of the
mean value theorem for the continuous
and pitch wise differentiable ray and of
course I can write it down as a
differential equation by taking the
limit and there is one element that is
sort of missing here does the
normalization
did they pick to normalize by the value
of the tangent well let's leave that
alone for now
let's let's assume that it is correct
and see if you to lead this to an
acceptable physical result right but so
what I'm trying to say is that the
denominator here is arbitrary I don't
really need it I can throw it out
it is a normalization parameter and I
put it there because with hindsight it
would lead it will lead me to a
physically acceptable result okay
the second postulate is a little bit
easier to swallow
thanks to Cal passes question because
the second postulate says that they're a
very similar to ascribe to a spacecraft
approach the planet array actually is
attracted by a high index of refraction
and the way we quantify this assumption
is we say that if the a enters the
region where the index of refraction is
variable that is it has a gradient so
that's the gradient in case you have
forgotten I hope not
so even the gradient in the index of
refraction the Ray will receive a kick
and the kick of the rain what a kick
medium really means is that its momentum
will change right therefore the
trajectory will Bend and the kick that
is the change in momentum will be
proportional to the gradient that is the
postman and I will not do it here but if
you it is a little bit of a long
derivation I have to warn you I went
through it and I decided actually I went
through it and then I decided not to put
it in the note because it it is a little
bit involved but this actually turns out
to be equivalent to Snell's law okay so
if you want I can give you a copy of my
notes where I derived it but if you
don't feel like going through two pages
of of calculus of variations then please
take my word for it this is actually
Snell's law but written in the family in
a funny way okay
so that's then the sort of the
variational version and we can write it
down as a differential equation and we
get this okay so now we're kind of going
backwards now
we're two differential equations yes why
the position variables are different on
the both sides s and s - oh in their
prime yeah s prime so the S prime is
actually the C from the mean value
theorem oh when you write occasion for s
prime can be anywhere in here I don't
know exactly what it is and very similar
to here actually again here I invoke the
mean value theorem and inside yeah but
of course when you do the differential
form it once read because the interval
collapses yeah yeah okay so now what I
have is actually two differential
equations one of them contains a
trajectory the other contains a momentum
so this is basically the two dimensional
analog to the dynamical system that
we're out here right have an equation
for the position this is it
I never difference in the question of or
the momentum this is it
this is the question for the position
and this is a question for the momentum
I use true for the position here because
that's a convention in geometrical
optics people use to not X but it is two
is the position vector okay notice fine
the question is is there anything that
is conserved here
did they derive any sort of Hamiltonian
well it turns out and again this
requires a bit of hindsight but I can
pick the Hamiltonian to be of this form
to be equal to the magnitude of the
momentum minus the index of refraction
okay let's write down the two equations
again okay so this is the questions
and the Hamiltonian that I picked is H
equals P minus n of 2 okay you can see
now that if I if I do the gradient now
in the high dimensional case instead of
that derivatives are having told me I
have to take the gradients right so if I
do the gradient of the Hamiltonian with
respect to led to the first one with
respect to n I'm sorry with respect to 2
this is really what I'm going to get
right
okay never do the Hamiltonian with
respect to P ok what is P P
but so for example if you to the age apx
you will get the x over the square root
okay
and then you can follow the age will be
y and so on
so therefore the gradient which is
defined as a vector
okay
so you can basically say from this
simple derivation here that by this
choice of Hamiltonian will tell me that
I wrote here I can actually derive
consistently the set of Hamiltonian
equation so I basically find that the QD
s equals the derivative or the gradient
of the Hamiltonian with respect to the
conjugate variable the momentum and the
other go around for DP D H so this is a
question then is the Hamiltonian it is
not the only one by the way having
bunions they're not unique clearly for
example I can add then arbitrary
constants here I can get another
Hamiltonian so so and we know that from
energy also and it is not unique I
cannot consume to it and it is still the
main channel so you need a reference
basically to to decide which way to come
well anyway this is one possible
Hamiltonian okay how how do we explain
it physically when the momentum is a
vector right and what we hamiltonian
says that the difference between the
magnitude of the vector and at the given
position the index of refraction the
difference has to be constant I can take
this constant to be zero since it is
arbitrary let me pick it to be zero if
that is the case what I'm really saying
is that the momentum where the vector of
the momentum is constrained to lie on a
sphere that actually is known as that
the cart sphere but it was not really
the cart was not the first one to make
this observation twas an arab scientist
whose name is i believe al hazen who
actually make this observation for the
first time and actually it has an also
derived Snell's law for the first time
so so basically what the Hamiltonian to
derive self is that the momentum of the
Ray is constrained to lie on the sphere
okay now this is I did not go through
the slide when we did Snell's law but it
is in your in the in your notes and you
may have noticed it if you were diligent
when you were studying the notes because
I actually showed this slide there's
another way to derive the cards law so
what the way what is happening here is I
have the familiar situation of Snell's
law we have them index or I'm sorry I've
a medium of index and to the left in the
medium of index and prime to the right
and in each medium I associate a sphere
with a radius equal to what the card
predicts so the sphere has radius M on
the left and another sphere with radius
and Prime on the right in this case when
the prime is bigger than n so this fear
is bigger on the right hand side so
Snell's law here is derived by applying
momentum conservation which actually
professor separ mentioned in the
beginning when we were discussing help
ashes question momentum conservation
says the following that that in the in
the horizontal sorry in the vertical
direction
up and down clearly this geometry is
invariant does not change
so therefore the momentum must be
conserved the momentum on the left hand
side which denote this be vertical in
the momentum of the right hand side that
pre-primary can must be equal and since
the momenta controls the carts
calculation equal to this then basically
these vertical components are obtained
by projecting the radius of the sphere
onto the vertical axis and of course if
you do this now you immediately get the
Snell's law because the you know this
angle is Theta so therefore this length
equals and sine theta on the left hand
side
and prime sine theta Prime with an
anchored side so that is yet another way
to obtain Snell's law and of course you
might wonder what happens to the
momentum on the bank on the longitudinal
dimension well that's okay you can have
a change in momentum there because there
is a vast universe a change in in media
right so the changing media actually
corresponds to stay to exchange of
momentum and momentum is conserved
fourth actually means that the medium
here if you actually use elect proper
electrodynamics there is actually a
force applied in this medium here due to
the change in momentum of the light but
I will this is beyond the scope of the
class I will not discuss it in any
little detail so this kind of picture
here within the card sphere it also
applies to a green medium so here is a
medium whose index of refraction is
variable so you go from low to high
again the convention is gray means
higher index and you can break it down
like we did again at the beginning of
the class you can break it down into
slices of progressively higher index and
you can apply Snell's law using if there
is no exam part of a success treatment
or the succession of the card sphere you
can apply it as their egg Goes Down and
then even because it has to match the
momentum all because it has to satisfy
smells law the to acquiesce to
attachment statements are exactly
equivalent their aim must bend its path
so this is yet another way to justify
why today is bending well it is bending
because it must conserve the value of
this communion
it must lie on the the card sphere but
of course in the grain medium you have
to be a little bit careful because the
sphere has a variable radius as you can
see in this diagram over here okay when
has come to to the other question that
was that is there an analogy between
mechanics and optics and because when
I'm a little bit later I don't want to
spend too
this analogy is a very interesting one
and let you start it and if you like we
can also discuss later but basically the
knowledge that comes from this we go
back and realize the Hamiltonian the
question of the mechanical system as a
sort of unassailable kinetic energy term
the first term over here is clearly the
kinetic energy determinate looks like
momentum squared over plan then the
potential without the potential before
in the case of the elastic spring the
last potential was V of 2 equals 1/2 k2
okay that's one special case of a
restoring force of a of an elastic
potential but it can be more generally
so we're added this V of Q and then the
rest of the definitions are basically
straight forward the momentum equals M
times the time derivative of the
trajectory anyway like I said I want to
go through this in great detail but
basically if if as the starting force we
use the index of refraction squared
divided by 2 m where m is like a
fictitious mass that we have to assign
to the photon then we basically get a
one to one another this is very
interesting because it is really acting
like a spring or like gravity but for
gravity is not exactly the same because
gravity is like 1 over R so you have to
make an approximation and blah blah blah
but but if you think of it as a as a
spring then you get basically a
one-to-one analogy then this kind of
energy okay then the last thing I want
to say because this sort of it has a
practical significance is that once we
write down the Hamiltonian I'm sorry
what's going down they have important
equations then it is very convenient in
addition 12 this nice physics and
intuition and so on it is very
convenient because we can just plug them
in MATLAB
solve them right so if someone is giving
us a gradient index medium now whether
it is quadratic or cubic or 1 through
whatever twenty-seventh polynomial or
anything for that matter we can just
solve it using the this set of equations
and here what I did is I did an example
of a quadratic index medium on axis off
axis you can see that on ax is actually
pretty good it is not completely free of
spherical but pretty close to free you
can see it does come to very nice
geometrical focus of course of axes it
has a terrible aberration that we called
what how do you call this particular
aberration that you see here okay yes
yeah can you can you use the index of
refraction to correct for coma yes
project in my group which says that
suppose you wanted to define an
arbitrary index distribution here in
order to correct for a given set of
aberration so someone is giving you
specifications they say I won't
eliminate comma for this angle of
incidence I won't eliminate other
aberrations for example as the
astigmatism and so on then in principle
you can come up with some index of
refraction and of R which if properly
chosen will eliminate this specified
aberration now of course the that's
pretty easy actually that doesn't need
research it's kind of like a calculus
computational problem but how do you
implement an arbitrary arbitrary index
of refraction that's difficult so we're
actually working on that in my group but
yeah in principle you can in practice
with commercial grins it is a little bit
difficult actually you would probably
have to use something more than the
grain as far as I know
on the previous slide he said can you
press the button please yeah
and the master button he bet it's broken
oh yeah on the previous slides you said
something about physical n has to be
greater than zero
yeah I said that n has to be bigger than
one rather than one yeah the question is
can n be less than 1 and less than zero
okay
from the point of view of geometrical
optics n is the speed of life in a bulk
medium and in ordinary you know bulk
media that we use in geometrical optics
life is always slower than vacuum so n
is always bigger than one in a few Naxos
when we do proper electromagnetics also
let me say that so far has been a
phenomenological quantity this is
another funny world like postulate what
does it mean phenomenological okay it
remains a quantity that they have not
justified its physical origin and just
pulled it out of a hat and I said hey
you know what the index of refraction in
the medium I'm sorry the speed of light
in the medium is less than the speed in
vacuum and so therefore energy together
now when we do the proper physical
origin so then we will see that indeed
and can be less than one it can even be
negative and that's another hot topic in
in result optics nowadays so it is not
necessary that's true but if you take it
into the mechanical analogy interesting
linner
if you allow them to go into this sort
of strange modern sort of regions where
it is less than 1 and even negative then
of the mechanical analogy you could end
up with a negative energy which is kind
of interesting it doesn't mean anything
this is just an analogy right you know
it doesn't invalidate anything but it is
kind of elusive
and let me say up front and becoming
smaller than one does not mean that I
violate relativity it does not mean that
the that something moves faster than the
speed of light and as we will see in
just today ok and is what we call the
phase velocity and the phase velocity of
a wave can be much greater than the
speed of light because the phase can is
no information that
the classical way to justify so so I can
allow actually and to be less than 1 and
then perfectly fine so it is not
necessary but so I should I should put a
qualification here this I don't know
not physically allowable but how you say
everyday in tuition allowable perhaps
but of course physicists can do much
better than that with the intuition
okay so let's resume for another 20
minutes or so and we'll continue on
Monday and so what I would like to do
today is start the description of wave
optics so basically there is a we we
followed geometrical optics and it was
good enough to explain a number of
interesting things for us imaging
systems gradient indices and so on but
the medical optics also has some serious
failures the most important of those is
that when light interacts with very
small features the medical optics face
so for example I used it when I answered
Cal patch this question if you limit
light to a single array that is to an
infinitesimally small spot then the
medical optics patently failed so the
the pictures that the gauges are
completely wrong in order to deal with
that we have to deal with light as a
wave phenomenon and there's an
interesting history there Newton was a
big proponent of geometrical or particle
description of life some other
contemporaries of his like Huygens plan
and sort of the Continental school there
were proponents of the of the wave
theory and for many years the question
remained and settled until Maxwell in
the end I think 1899 or 1896 or
something like that James Clerk Maxwell
actually professor of fluid mechanics
came up with the set of equations that
proved very convincingly that the light
is a wave and then 10 years later
quantum mechanics proved that light is
also a party so both were correct both
Newton and the Continental School will
connect correct so anyway so so you can
think now that were basically now moving
from England to the continental Europe
right because we're abandoning the
geometrical optics theory and we're
moving wave of it
the waves of course are a very very
pervasive phenomenon that we see that we
see around us and they manifest
themselves in many different forms so
we're all familiar with c/y waves in sea
water or a river and so on we're
familiar with sand wedge this is
actually a Singapore's founder they lick
when you are given a speech so we can
communicate by sound waves that's
another form of a material wave so like
water wave sound waves are material
where you set the air particles in
motion when you speak we're also
familiar with with electromagnetic or
anyways not because we can see them
these are actually not something that
you can touch but all of us will use
cell phones all fashion radios and so on
we use them this is actually an old
fashioned antenna
it's not that well fashioned if you go
to places like Europe like grace for
example many homes still have those
antennas in their roof so this is the
way people used to receive television
signals before cable TV and finally
these are the waves that are sort of not
quite obviously the wedge this is a very
interesting geological phenomenon in the
Philippines is called the Chocolate
Hills there's a number it's actually
hundreds of very small hills not very
tall you might say well what kind of a
wave is that the hills are not moving
well actually they're not moving in our
time scale but in general what they call
it geological terms they actually move
and sometimes this motion as you know
the motion of the earth it can actually
happen in time scales that are perceived
by humans and that is called an
earthquake it's a very nasty phenomenon
so so waves the point is here that waves
manifest themselves in many different
forms of course I omitted another
important manifestation which is quantum
mechanics quantum mechanics says that
everything is actually wave including
yourselves and
me and every particle is also described
this way
so electrons I can be thought of as
waves and so on and so forth anyway so
the waves are in fact so pervasive that
that there's no proper definition and to
paraphrase something that an artist said
one's about art he said that art cannot
be defined but you know it when you see
it okay that's actually a very good way
to put it at least as far as art is
concerned but it's also to about waves
there's no single because there is a
great very variation in the way waves
occur there's no really way there's no
proper definition what we attempted here
is the simplest possible all-inclusive
definition as a traveling disturbance
and as far as I can tell it is certainly
inclusive of all waves of course it
includes other things that may not be
waves like a car moving down the road is
also traveling from disturbance is
clearly not a wave but anyway another
way to perhaps describe waves is if you
if there's some evidences of wave
behavior and a very prominent one is
what you call interference so what you
see here is we will see these are called
spherical waves and we'll see two point
sources and meet in these spherical
waves and as they propagate out you see
this very characteristic out nation
between bright and dark spots I have not
even really said what this thing is is
it a sound wave a water wave an
electromagnetic wave in fact it could be
all of the above in all of these cases
if I represent as bright as the high
value of the disturbance and as black
the low value of the disturbance then
this sort of not quite periodic but
periodic like looking behavior is called
interference and when you observe this
sort of interference then it is a pretty
good indication that you're actually
new wave phenomenon in fact the reason
for an L and the Continental school
insisted the plight is a while is
because they actually observed in light
interference in experiments they
observed like Luke doing something like
that in experiment so they said well way
there it must be waived
so another phenomenon which is actually
it is related to interference one will
cover it it is quite a different name is
called diffraction but then when we
cover it we will see that really it is
not a new phenomenon it is interference
in this guys and the fraction basically
says that shadows the geometrical
definition of shadows is incorrect so if
you what you see here is a wave that is
impinging on the block so it is blocked
above and below and the small portion of
it is allowed to pass through the
metrical optics would have you believe
that you would basically get a
continuation unperturbed continuation of
the wave in the geometrical shadow of
this slip you can think of it as a slit
that the wave goes through but the
fraction surfers not again as you can
see there's a rather more complicated
pattern that emerges here and and that
is another evidence of the wave
phenomenon that that we will describe
quantitatively in great detail later ok
so let's take a look a little bit of in
some more waves so for example the first
one well it meets my definition is a
traveling disturbance but it's not a
very interesting one because it is a
stood say moving the wonder that is on
the right is perhaps more interesting
because now you see sort of it is more
believable that it is a wave right it
has it has some oscillation in addition
to the travel in distance by the way or
Silla Tory behavior is also
characteristic of wave right but it is
not necessarily periodical in fact but
you can have a periodic wave but that's
not very interesting it is the non
period
periodic waves that are the most
interesting okay so now if you let me go
back and play this movie once again okay
that's all very interesting but what you
will see here is you will see that it's
actually the same envelope that you saw
moving on this diagram you see it is the
same envelope moving over here but
inside you have some Wiggles that are
also moving and if you observe it very
carefully you'll see that the envelope
is moving with a different velocity and
there than the Wiggles the witness have
a name where they were the called
friends can you see so the readings are
actually moving faster than the envelope
so this brings us to the definition of
what they used once a professorship or
this question about an earlier use the
term phase velocity so what you saw in
the previous movie in the one that was
on the right hand side was an envelope
that was moving so here's the blue one
was the envelope and inside there was a
sinusoid the sinusoid is multiplied by
the envelope a wave that is composed of
a sinusoid multiplied by an envelope
it's said to have a carrier so the
carrier is a sinusoid itself right okay
the carrier can move at its own velocity
which surprisingly is usually different
than the velocity that the envelope is
moving whereas I surprisingly well some
simple sort of intuition might have said
well they better move at the same
velocity shouldn't they when in fact the
truth is the opposite more often than
not in fact we have to try very hard to
make them move with the same velocity
but more often than not they move at
different velocity so the velocity at
which the envelope is moving is called
the group velocity the velocity at which
the carrier is moving is called the
phase velocity and right now we will
just leave those like this later
we will actually derive them properly
and we will see how they come about and
whether how they are defined but I can
say for now
proof that the what we wrote in
geometrical optics see over n that is
the phase velocity of the light okay if
you want to put an envelope on the light
somehow which is done in special lasers
called board blocked lasers you can have
a laugh light wave that looks like this
it has an envelope as it propagates in
those yeah the group velocity is
generally different than the phase
velocity of the light okay so of course
I said earlier that periodic waves are
not very interesting nevertheless
for educational and analysis purposes
they're simple so therefore we start
with periodic waves when we analyze
waves and periodical also known as
harmonic waves and of course in a
periodic wave there's no envelope to
speak on and therefore has no group
velocity to speak of because the
envelope is flat there's no profile
going anywhere
the only thing you have is a sinusoid
that disturbance moving with the phase
velocity right here you only have the
carrier so the only velocity to speak
off is the phase velocity that's a
characteristic of the harmonic wave and
I will start put in some notation here
so the time you can see the time is a
run in index here and the horizontal
axis I will call it Z okay so we will
stick to Z as the axis of wave
propagation for a while right so now
we'll do some math and since since this
is a harmonic wave I can expect to
describe it as a sinusoid and the
sinusoid in order to characterize it and
in this a number of quantities which
have certain buzzwords we better start
defining our buzzer
now so we don't get confused when we
refer to them later in the class so the
constant goes in front of the sinusoid
it's called the amplitude the reason I
use E as a symbol for the constant is
because electromagnetic waves are
usually electric fields who see that
later after we do Maxwell's equations
for now this could be any symbol
actually in fact very quickly it
wouldn't affect when different symbol
but in any case this quantity that
appears here is called the amplitude
then the other quantity that we need to
describe a sinusoid is the frequency but
now the frequency were to be a little
bit careful because there's two
frequencies going on you can see very
clearly you can think of this as a
snapshot over the wave if I take this
picture and I freeze it I will get
something looks like this
so in this snapshot the axis is space
coordinate it is Z so the period that
you observe here the period of you
observe in the spatial axis it is called
the wave length okay we saw that before
we did a definition of the wave length
in I believe lecture number one okay now
suppose I do like this suppose I fix my
position somewhere on this axis and I
observe the value of the wave as it is
passing by me so now I'm fixed at a
given Z and observe the wave what do I
believe if I stand over here we have
another sensor denied because if I'm
sitting here the wire that will say will
be oscillating it will go in sinusoidal
fashion it will oscillate from a
positive value to a negative value so
that actually associates with the
temporal frequency of the wave for which
we simply use the term frequency so we
say frequency we mean the temporal
frequency and some people use F I prefer
to use the symbol nu for that quantity
finally I have not defined my tie
axis yet so when I'm looking at the
oscillation if I'm fixed and I look at
the oscillation of the wave at time 0
the wave might be absolute positive in
that case it means the wave is a cosine
well it may be zero in that case it
means it is a sine and how do I correct
for this discrepancy by adding a phase
delay right by adding for example in
this case PI over 2 so this additional
phase that an it word in order to fix
the original time is called the phase
delay and for a while I will use the
symbol field for the phase delay okay so
we define basically one of the
quantities that we need except nu is not
anywhere to be seen with this equation
what everything instead is this
expression Z minus CT and what is the
meaning of this expression
but traveling wave is a traveling wave
or another way to put it is a traveling
coordinate system if I right
we're not about doubt what this is
that's a sinusoid
now let's assign the soil in an axis of
coordinate C okay if I now say C equals
Z minus CT where it is time what I have
done is I've taken this axis and I've
started moving it with respect to its
rest position at T equals zero so this
is what I've done in the expression for
the way I've put this travellin
coordinate system into the sinusoid and
I don't even have the frequency in here
but you can see that time the time
variable is an expression for the
sinusoid so I can get the frequency very
easily and the way I get it is by
observing that since the wave is
periodic in the time domain it will
repeat one period later so the period is
the inverse of the frequency so if I
look at this expression over here let me
rewrite the way
okay
I can also write it haven't done
anything just expanded the paralysis so
when this quantity becomes 2 pi or any
integral multiple there on the wave will
repeat wanted so so that will happen
when the time equals the period of the
wave or another way to put it is that 2
PI over lambda times C over nu whether
the frequency renders over the period
you college to PI of course the PI goes
away and we derive something that we've
seen before C equals lambda nu and that
is what we used to call the dispersion
relation okay so this is how we connect
the phase velocity of the wave with the
wavelength and the frequency and okay
the phase delay is actually a very
important property of waves but it's not
really rickety and this is what I was
saying before just yet why is it called
dispersion relation yes there's no
discussion to be seen here isn't it
isn't there however if I put for example
an index of refraction then the index of
refraction is a function of lambda right
so then you get dispersion and of course
another case is this is the dispersion
relation in free space if you put
boundary conditions of the wave for
example if you confine it in a waveguide
the dispersion relation changes right I
will not do this in this class but but
that's another way where the wave
becomes dispersive wave okay so that's
explanation the phase delay I don't want
to spend too much time if it equals zero
you see this this means the wave is a
cosine the phase delay is zero and then
depending on what you see at T equals
zero you may have different phase delays
now the reason the phase
there is an important quantity is
because I want you to look at it very
carefully and tell me what you see so in
the anybody wanna volunteer what what
has happened here I'm going to play this
once again
how are the two different on the left
another right
okay on the left there again look at it
very carefully on the left the two
spherical waves they're always bright
together or dark put together so they
are see light as we say in phase when
they write the two waves are out of
phase so if you look at it carefully you
will see that when one is bright the
other is dark and vice versa I'll play
it again so you can see it okay and
because of this again I'm going to play
it once again and now don't look at the
Centers of the waves anymore but look at
the general patterns you would see that
the general patterns are actually
different they have the same general
shape but whether the pattern is dark or
bright at any given instant actually
depends on how the two waves started so
whether when we have with this quantit
superposition because I have waves
originating from two sources
simultaneously when I have a
superposition the structure of the
overall wave that the Oh Blair depends
very strongly on the relative phase
between the two the two sources in this
case okay so with that I think I will
stop and we will continue on Monday and
I will not post any new notes or Monday
because what I think we're far enough
behind that we should cuts up on Monday
but I would like to say is that it is
about time to start the projects for the
graduate class for 2010 so soon but not
not basically by by Thursday morning
Boston time and earlier I will post in
the website and list of possible
projects and some instructions on what
to do and then basically what you what I
would like you guys to do is spend the
next week
selecting your project and also
contacting your mentors
it's project will be mentored by one of
my graduate students and some son buy me
some possibly I don't know if calling
you might be interested in mentoring one
but anyway we'll figure out maybe Sam
from Singapore maybe not but but anyway
so we will assign mentors to the
projects and then we'll get our thirties
in May May 6th of May 7 but anyway the
sooner we start the better