Optics (Lec - 15) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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right so you can see it's me again so
George George can have a holiday again
but it's he's not really having a real
holiday because he still he's actually
sitting in and I guess but one reason
why we decided that he had to do that
was because uh I have to use his
computer because he's written these
talks in this strange application that
doesn't run on my computer okay so now
we get it
today we get we're just about getting
into interesting stuff now seems rather
a long time since the beginning of this
course before we get into the real wave
optics which I'm sure everyone would
agree is the most interesting part so
this this this lecture is about
interferometry and and then the next
lecture I don't know how how much I'm
going to get through today but the next
lecture is about diffraction so these
are two techniques to two phenomena that
they're obviously very closely related
to each other so in this talk really two
types of interferometer the Michelson
interferometer and the magazine der
interferometer okay so it took me a few
minutes to understand this picture but
anyway it looks a bit sort of messy but
in the end I George drew this obviously
otherwise I wouldn't be saying that but
what this is showing then is trying to
show how a wave vet varies both in
distance and in time so the waves
propagating along this Z direction so if
you look along this Zed axis you will
see this amplitude variation given by
the black line so a cosine type
variation and then at any point any
value of Z you can see plotted in red
how that how the amplitude will change
with time along this
here okay so you can see that you get a
sinusoidal variation in both in time and
in space and and reminder here then how
you can express this wave this is this
is a wave a sinusoidal shape wave which
is moving in the positive z direction
and you can either express it in this
form which is what you would actually
measure if you could really measure it
except that as we've said before you
can't really measure it because it's
moving too quickly but normally we
manipulate do the manipulations on the
expressions using this complex notation
because it makes it easier to do the
thumbs and yeah so this is showing then
oh this is a particular value of
distance then and showing how it changes
here the value of the phase of this
cosine wave at the time T equals null
according so the value of Z I guess
that's one that's trying to show okay
and then let's think now what happens if
we've got some wave a plane wave
propagating along at an angle with the
axis so here's this wave coming along
which is it's going along a an axis
aligned the direction of propagation of
the wave is along this direction which
is an angle theta with the axis and so
you can see that when this wave reaches
this planes Z equals fixed here so this
is some plane where we might put a
screen or detector or something you can
see that the different parts of this
face front are going to strike this
plane at different times or
you you can say that if you look at this
plane here this way you will experience
exhibit different phase at different
points along this line and so this is
what we're looking at here then as it
says here the the path delay increases
linearly with X so there'll be somewhere
where this arrives at a phase of zero I
guess is this point here and then on one
side the phase is going to be positive
and negative R according to which where
you go alright so mathematically this is
how we express this so this is our wave
the same as the previous slide but now
the wave is moving at this angle theta
to the axis so we get this term this is
our you remember we express this before
as K dot R for a wave moving in general
in a direction R and so so this this so
we can write this then this this is what
is is in terms of our phasor
representation right so we as before we
write this as an e to the I right thing
and then we also this e to the minus I
Omega T we just missed that out for
clarity just because we don't want to
have to write that every time so we
there's an e to the minus I Omega T
understood that we don't write and K of
course is 2 PI over lambda right so K is
related inversely to the the the
wavelength of the light and so this
thing here this complex number here you
can write this as e to the I Phi like
this where Phi is the phase of the wave
as a function which is a function of
position and and it can be written like
this all right so if we look at a
particular value
zedd then said is going to be fixed so
so this this this part e to the I to PI
over lambda is n cos theta is going to
be fixed and the phase variation is only
going to be because of this other part
here so this is another example and so
this shows spherical waves or rather
distorted spherical waves they've become
because of the advances in modern
technology that love to distort
everything you draw and make it so that
it's not the right shape anymore but
anyway this is a point source which is
giving out spherical waves and you can
see that here we put our screen again or
whatever it is that Z equals Z naught so
you can see that these waves as they
travel out here they're traveling a
different distance before they get to
the screen right so you can see that's
going to do two things one is of course
the amplitude is going to decay as 1
over R so the amplitude of the wave when
it reaches out here is going to be
smaller than the amplitude here but much
more important is the phase effect
because you can see the phase actually
might change tremendously because you
see the because the wavelength is so
small compared with these distances it
means that you only only have to go
quite a small distance before you get a
big change in the phase right so that's
what we want to be able to quantify and
so so here we're riding our spherical
wave just like we did before so this is
George's pep this the same as he's said
before you'll notice that it's got this
funny PI over 2 in there which he
introduced that before he has the
effective my
like a sign of course something minus PI
by 2 is like sign it's it's like this is
actually like a sync type function and
the rip that's the reason why I that PI
over 2 is there it's so that it doesn't
blow up when you look at what happens x
equals y equals n equals T equals naught
if you didn't have the PI over 2 there
you'd have horrible infinities so you
avoid those by putting this PI over 2
there and yeah so this is the radius R
is just the square root of x squared
plus y squared plus Z squared and we get
a so this is kr minus Omega T and this
is our R at the bottom and now we now we
make an approximation with the
approximation we're going to make is
that we're not actually really looking
at this geometry as shown in the picture
but we're looking at the geometry when
actually we're only looking in a small
region near to the axis what that really
in practice in physical practice that
means is that this distance Z Z naught
becomes very large we're a long way away
from the source and so we can think of
this this value Zed being very large
sorry you understand my funny English
language don't you when I took about
Zed's what you call Z's I've just
suddenly realized that anyway so that's
so we've already had this before in
earlier lectures as well this is a
square root that we can expand by the
binomial and assume that Zed is lot much
larger than x and y so you take out a
factor Zed and the square root becomes 1
becomes 1 plus 1/2 something and so
eventually you get this sort of thing so
that's what we do it that inside this
phase term
but the thing at the bottom we say we
don't have to be so careful with that
because it's not so sensitive to this
this amplitude because all it does that
it all it does is just changes the the
magnitude of the of the wave a small
amount right because we've said it x and
y's small but he up here you have to be
more careful because we're talking about
phases all right so anyway at the bottom
we can just neglect the x and y so
that's where the Zech has come from
there and when we do this binomial thing
on the top we get this expression here
then so the the dependence on x and y
you can see just becomes a quadratic
type term x squared plus y squared over
2 Z and so this is what this pictures
drawing here the path delay increases
quadratically with X right so if as you
go further this point here corresponds
to the center of the screen and then as
you go further away this increases with
some parabolic variation like this okay
so that's right that's doing it in terms
of the cosines if we go in terms of the
complex complex Exponential's much the
same sort of approach you can just
expand again this square root inside the
exponential now exactly the same really
and you end up getting this expression
here and so this is an example now you
can see of the of the power of this
complex exponential form though because
because this complex exponential of here
two terms or it might be more terms in
general you can just expand them can't
you they just be you can just say that
multiplied together whereas if it was
close
of course you'd have to remember all
these things cause of a plus B plus C
horrible I don't know why I'm making
funny noises but maybe I shouldn't keep
getting too near the speakers okay so
yeah so with the complex exponentials we
can break it up very simply and this
inside the braces then is our fire
before Wi-Fi so we can say this is e
naught e to the I Phi where the phase
Phi has just got this parabolic term
plus some constant term which
corresponds to this which corresponds of
course to this distance the phase that
you get through traveling in that
distance there okay so that's how you
get a faith you like I've got a question
yeah what happened to the PI over two in
the phasor well I am to the PI over two
in the phasor it's so we've got good
it's there isn't it and then it's
disappeared are the I saw out there it
is it's that one yeah okay so it's an
eye it's an eye outside and yeah that's
that's that's an interesting I think
George said something about that before
because it effectively it makes it makes
this wave 90 degrees out of phase and
you know I like to think of that as
being related to some sort of resonance
type phenomenon you know if you excite
something that resonates then it's going
to resonate the 90 degrees out of phase
isn't it so if you if you've got a point
source and you excite it with a wave
then the the scattered wave is going to
be 90 degrees out of phase
with with the the driving force just
because it's like a resonance and I
guess that's also true in the Huygens
thing that we we've come on
in the next lecture but yeah okay thanks
for pointing that out I've forgotten
that okay so all this other than that is
it's it's fine isn't it I hope so we're
now going on to discuss something about
the significance that yeah so the point
is I mentioned that the the there is
this modulus term intensity type term
but this is a very weak change so you
can imagine if you're trying to do an
experiment to actually measure the
properties of that wave field by looking
at the intensity would be quite tough
because we've said that these two terms
are very much smaller than that one
right so looking at the phase of the
signal is a much more sensitive way to
look at that that way from and as it
says the the phase can tell us all sorts
of things about the wave like whether
where the wave has come from how far
it's traveled the materials it's gone
through and the well and generally the
optical path right which is you remember
we had optical path before integral n DZ
so that will change the phase and so if
we can measure the phase we can find out
information about those things the
problem now as we've said before is that
the phase is not something you can
actually see directly because it's so
fast so 10 to the minus 10 to the 15
Hertz is typical very very fat much
faster than you can measure with any
electronics so how do we do this we do
it by some form of interferometry which
is basically mixing it with a reference
beam and then you look at the in
density of the of the interference
between the two ways so this is how we
do it so I said that we're going to look
at two to two types of interferometer
the first one is the Michelson
interferometer so this is how you make a
mikrosil interferometer you get light
from some source and split it into two
parts with a beam splitter and one of
those one of those beams is what we call
the signal arm this is what we're going
to try and measure so here this shows
this light going through some objects
and then being reflected by a mirror and
then coming back to a detector and then
the other arm is the reference arm so
this is a going to produce a reference
signal so effective really we're going
to be comparing the phrases in those two
arms so the reference beam comes down
also
so these add together so we can add them
as phasers we've got this is for the
reference beam ar e to the I Phi R for
this for the signal beam we've got a s e
to the I Phi s we add the amplitudes
because we're assuming it's coherent
maybe I ought to stress that a bit of
course and so this is assuming that this
is that this is a coherent these two
beams are coherent with respect to each
other right so if you had light here
from a laser that would be that would be
true okay and we'll probably come on to
cases where it might not be true later
on and once we got the amplitude then we
just have to find the modular square to
find the intensity and if you remember
there's actually a proportional sign
here there's some funny constants with
epsilon 1/2 and things which we just
don't usually worry about okay so these
these phases tell us the optical path
that the light has gone through
in these two arms in order to get to the
detector and so what we're measuring is
the difference in the optical path in
those two arms and so this shows how
that can work so this is an example
where these two waves arrive in phase
with each other so then they'll add
constructively and you'll get a big
signal like that on the other hand you
could imagine that they might not arrive
like that they might arrive in
anti-phase like this in which case you
can see that the two ways now are going
to cancel out and you're not going to
see now any interference in practice of
course it could be anything in between
here all right so as you change you can
change the relative phase of this light
by changing the position of this mirror
or the the size or refractive index of
this object that's in there or the
wavelength of the light or whatever it's
going to depend on all of these things
and so example measuring distance so
we've said that effectively what we're
doing is measuring the optical path so
optical path depends on a few things
doesn't it optical path depends on the
distance it depends on the wavelength
and it depends on the refractive index
so these are all things that you can
measure using this other system so here
is talking as an example there's no
object in this one anymore so this is
just free space and so in this case then
what you're what you can measure is the
difference in the optical path in this
arm and this arm and the optical path
what the refractive index is now the
same in each the wavelengths the same in
each so you could measure the distance
right so so this is a an interferometer
for measuring distance so you might what
where could you use this well imagine
this this might not actually
be a flat mirror it might actually have
some structure there in which case you
could actually measure what that
structure is right or you could use if
it was a mirror you could actually use
this to accurately measure the distance
of this mirror how far away is from from
the instrument so to speak right so the
interesting thing another interesting
thing this is the Michelson
interferometer and the Michelson
interferometer was as the name says was
invented by microsomal I think he
probably invented it
he certainly has his name attached to it
but he mainly used it for measuring
wavelength so we've come back to that
after I've gone through in the maths a
bit actually I think it's probably best
anyway so what we said is this if we add
together these two phases find the
modulus square and then we've got to
expand that so this is a square of a sum
of two things so we're going to get the
square of this the square of this and
then we're going to get some cross cross
terms and this is what it comes out to
be the two cross terms one is the
complex conjugate of the other and which
means that when you add them together
the final result is real that has to be
true of course because this is an
expression for intensity and intensity
has to be real and so we've written it
in this nice form a R squared plus a s
squared plus 2 AR is cos of something
and this this thing here the something
is the difference between the phases of
the two beams right and so this next
line puts it into a sort of general term
general form it's of the forms a
constant plus a cosine and and then
these quantities I naught m and Delta
Phi you can also write in terms of these
other
parameters we had earlier right so as it
says when both paths are clear then
these phases are just directly
proportional to the distance LS and L R
and so the phase difference is just ke
times the difference in the two paths
right so therefore if you measure the
signal you can if you know what the
wavelength is you can measure what the
difference in those two distances is
there's a bit of a proviso there which
is a quite a big proviso that you can
only do this of course modulo 2 pi right
so because this is a cosine that's going
to keep repeating and you won't know
which of the cycles you're on but so it
means interferometry is usually very
sensitive you can measure you can
measure very very small distances either
these much smaller than the wavelength
and people sometimes talk about
measuring hundredths of an angstrom and
amazing small quantities using
interferometry the LIGO projects I don't
know what they're claiming 10 to the
minus 15 meters or something isn't it
they're they're trying to measure to
detect these gravitational waves and
anyway yeah so so you only measure it
modulo two pi so that means then if
you're you're trying to use this method
to measure for example I'm looking at
the door at the back of the lecture
theater if I'm trying to use this to
measure the distance of that door I can
measure the distance so an accuracy
maybe of something smaller than a micron
but I wouldn't really know whether it
had added to that 10 meters or whatever
so that actually has been
a lot of people have put effort into
trying to develop interferometric
methods that you can get that extra
information so that you can actually do
what we call unwrapping because this
this is like they sometimes call this
phase wrapping the the phase wraps
because of the cosine and so you don't
know which of these lobes you're on yeah
bro going back to Michelson so Michelson
did this the other way around
he knows his distance and uses this
expression to to find the wavelength or
the ke value or whatever that's of
course what you're doing in an
interferometer isn't it if you do a
Fourier transform interferometer you're
using that as a as a way of measuring
the the wavelength of the light right in
the case of Michelson he was looking he
looked at various things he didn't look
at lasers cuz he there weren't any
lasers they but he looked at light from
discharge tubes and various different
types of lamp and so on and he looked at
how the signal would vary as you move
this mirror and it turns out that if you
do move the mirror and you know
accurately how you're moving the mirror
you can actually work out from that what
the spectral distribution of the source
is right that also suggests a way around
this this wrapping problem the way
around the rattle a way around the
wrapping problem is to use not just one
wavelength but many wavelengths so even
two wavelengths is a big help actually
yeah it's quite interesting that
Michelson I think I've spoken about this
with some of my students before that
Michelson
he wrote quite a lot about measuring
wavelengths using that interferometer
and then many many years later people
came up with this bright idea of using
what they call the Loco hearin sin to
Freud or the white light into fright er
to measure distances but as far as I
know Michelson never never suggested
this I've never found it in any of his
paper right anyway so this is another
application here we're going to look at
a real object that we put in there and
try and find out something about it so
we put in this lump of material a length
length L a certain refractive index and
you can see now then that the the fades
of this light when it gets there is
going to be changed by the gut by the
light traveling through that material
and we taken into account George that
it's gone through twice here that the or
even on the previous verb they thought
that there ought to be a tool in there
there ought to be a 2 in here on there
all right so this the phase difference
is actually twice this because this it
makes here has this amount of phase
difference on the way in and on the way
out
so I didn't notice that before either so
but it applies to this next one as well
so so this this one the light goes
through this object here and so this is
extra phased delay that comes because of
this object which is this bit here so
again there ought to be a factor of two
for all all of these terms and so so
there we are though we've got this extra
optical path and so now
we can measure the optical path of that
object again modulo 2 pi so if this is
if this is many wavelengths you might
have trouble to know exactly how long it
is of course because of the the the
wrapping of the phase again so so it's
so if to this modulo 2 pi we can
actually measure the this is the optical
path difference n minus 1 times L the
minus 1 of course is because the other
but if the reference beam has gone
through air when this has gone through
this material right so what we measure
is n minus 1 times L if we knew what n
is we can know what L is if we knew what
L is we could know what n is but as it
says there but if you don't know either
then you can't find either you already
know is what n minus 1 l is anyone got
any ideas about how you might what what
might you do to get both of those if you
if you wanted to know both of those any
ideas how you could do that based on
what though you'd have to have two
measurements president Worthen please
yeah someone suggested that used to lens
to set up simultaneous equations you're
not getting full credit for that sure
yeah yeah anyway someone got it so yes
so if you measured this for two
different wavelengths then you could
have a pair of simultaneous equations
and you could solve solve for both of
those that of course assuming this that
there's no dispersion if that if the
event changes with wavelength then
you're still back to square one unless
you do some more wavelets I guess but
anyway yes
it takes it just shows you that you can
do a lot of these experiments very
easily so now let's sort of these are
used this Michelson interferometer is
used a lot in metrology industrial sort
of metrology and you know all countries
have their well you in the US it used to
be called the National Bureau of
Standards didn't it but I don't know
that that exists NIST NIST that's right
it changed I don't know why it changed
its name but anyway so all countries
have got this they they have this
standard measures like there are
procedures for calibrating distances and
so on and the idea is that you know if a
company like General Motors is making
parts for a car that they will those
those parts will fit as a spare for
another car that's made by another
factory right so if they know if you say
you make some shaft that's got there's
supposed to be one inch in diameter and
you you you want it to be accurately one
inch diameter otherwise the bit from one
place wouldn't fit a bit from the other
there is actually a true story about
that a thing called the Vickers inch
Vickers is an engineering company in the
UK and this this Vickers inch was a big
thing that came up during the Second
World War where lots of factories all
over the UK were converted to make
munitions and some somewhere died down
the line people realized that Vickers
had a different inch from everyone else
so you know when they when they made a
shell and tried putting it in the barrel
of the gun he didn't quite fit so
eventually that was sorted out they they
they worked out what the problem was and
I guess it's that sort of
problem that makes it so that the
company is now obviously you know the
country countries that keep these
standards and so on so the sort of thing
you might do is measure the height of
this let's say you you you put in that
you'd like they have these things called
gauge blocks a lot of you and mechanical
engineers so you must know about gauge
blocks so these are you can buy the
standard lumps of metal that have got a
certain thickness and so you put this
lump of metal in here and measure the
thickness of it using an interferometer
and and then so that you know these
standards of course they have these what
they call primary standards and then
they're handed down the actual I'm not
quite sure how the thing that the length
works anymore but it's already defined
in terms of the wavelength of something
isn't it and and then all the countries
around the world
they all have to compare their meter to
make sure that they've all got the same
meter and then within each country so in
the US NIST would then be measuring
these gauge blocks that will be sent to
them from other companies to make sure
that they've got the right size and then
in the companies then they would compare
those gauge blocks against other other
gauge blocks and other things so that
this is what you use one of the things
you uses okay so now fringe visibility
yeah so we've said that you get these
fringes perhaps I were to go back and
explain that a bit more back to this one
so you can see here we've got a constant
term plus a sinusoidal type term or
cosine you sort of tighter and the you
can see that what this term M is called
the contrast or fringe visibility so if
M is 1 then you can see that they're
going to be for particular values of
Delta Phi this intensity
is going to go to zero right so that's
when you get you've got very high fringe
visibility and you can see that this
visibility depends on this AR to a Ras
over IR square plus a s squared and you
can show quite easily that that is going
to reach a maximum gonna it's going to
be equal to one when I R equals is so if
you ensure that a R equals a s then
you're going to get your visibility of
one I guess it can't be bigger than one
because that would mean that the
intensity to go negative which is not
possible right so this this plots what
it is what what you would see as a
function of Delta Phi and for a
particular value of n so there's some
background sometimes called the bias and
then you've got some Wiggles on top of
that and so this plot is plotted against
Delta Phi Delta Phi could change by
another number of different ways of
course one way it could change as if you
actually move the mirror another way it
could change is if you tilted the mirror
if you tilted the mirror then there'd be
different Delta Phi's at different
transverse positions on the mirror and
if we take a are equal a s then you've
got perfect contrast M equals 1 you see
it goes right down to 0 and that's the
the highest visibility of the fringes
that you can get if on the other hand
you've you find that if you take let's
say the reference beam to be much
stronger than the signal beam then you
can see that M is going to tend to zero
the contrast is going to be become very
weak actually sometimes people don't do
that contrast is not necessarily the
most important thing because if you
notice what we had before
you see that the the the let's go back
to the printer yeah let's go back to
this one here you can see that you've
got these three terms so what we're
saying is if you make a are much bigger
than is then you you you you make the
contrast is going to get smaller and
smaller because obviously this term is
going to become much bigger than this
one
but nevertheless that's actually make
this term bigger so if actually what
you're trying to do is measure this
signal in some background of noise it's
sometimes it's sometimes a good thing to
make a are bigger than is because if as
you increase IR it increases the
strength of this of this interference
term and that's that's the principle
that's used in heterodyne interferometry
so not always true that you try and make
it the contrast as big as possible yeah
there is one more factor of 2 missing
here at on the left hand side bottom the
swing of the fringes should be 2m times
I naught n times here this is the factor
2 missing right that's quite unusual
three slides in a row they're all
missing if the factor of two hands
laughs yes okay so that's the Michelson
interferometer so the micro-sim
interferometer is basically you can see
a sort of reflection interferometer and
we're now going to go onto the max at
design der interferometer which is
basically a transmission interferometer
so it's the principle is very similar
you split the light into two parts the
two parts two light beams go through a
different path and then you combine them
onto it onto a detector but so so this
works in transmission
mode here you've got you're looking
through an object so now you're not
going to get that factor of two of
course because the light is only going
through it once so the light goes
through here
you notice that the way this has been
drawn it's been drawn so that this light
is going like this this light is going
like this so you can see this path and
this path are roughly equal you might
you might equally why you you could have
thought of making this by by by doing it
more like you could have done it
like that so it's where now you can see
if we'd have made it like that this this
path is obviously longer than that path
right so I guess nowadays if you if you
did that experiment with a with a laser
which has got a very long coherence
length something like a frequency double
YAG laser this would work fine if you
tried doing this with an arc lamp or
something like that it wouldn't work
because this distance is much longer
than the coherence length of the source
so you're all used to the idea that
actually you don't see interference
fringes all the time in the in the
normal regular everyday life you don't
see interference fringes very much and
that's because in everyday life normally
the light is incoherent rather than
coherent so if you make if you make this
system here all the other micros an
interferometer we showed before and you
make it so that these paths are
accurately the same length then you
could get interference even with light
from a you can get interference even
from light from a tungsten lamp bulb and
and they look they produce really nice
beautiful fringes with different colors
maybe my students maybe seen these in
the back but anyway so here you can use
this to measure the optical path of some
object and yeah so this is yeah so this
is now again looking at this idea of
tilting something yeah I mentioned about
tilting with regard to the Michelson
interferometer I said if you tilt one of
those mirrors you would actually get you
would actually get the optical path
varying linearly with transverse
distance so if you actually looked at an
image of that interference what you
would see would be a series of fringes
and the spacing between the fringes
would increase this you check as you
increase the angle so this is the same
here we're getting the the offset
between these two paths these these two
beams by rotating this mirror slightly
so and then as you can see these two
beams arrive on a CCD camera it shows
here so you will actually in this case
you would get some variation in
intensity on this detector which you can
actually measure with the detector so it
would look something like this this
would now be then a function of X rather
than just being a function of the
optical path difference so so this bit
here I guess is it is explaining is
deriving that so we're adding two waves
together one is arriving normally the
other one is tilted through an angle
theta and so this this one here is going
to have a phase linear phase variation
as a function of X which will then
appear inside this cosine right so
finally what we get here then the the
value of the this bit here the dead cos
theta is that it's just this constant
here so all that does is it shifts sees
these fringes this way but normally what
if you were doing this experiment what
one thing you could do then is very
simply is to measure what this spatial
wave length of these fringes is and from
that you could for example if you knew
where all the other things you could
measure what this rotation theta was
couldn't you if you knew what the
rotation theta is then I don't know what
you could measure but you could measure
something probably
right so that so the Mac sender I
mentioned that this is like a
transmission interferometer the
Michelson is like a reflection
interferometer both of these sorts of
interferometers are used in microscopy
actually and so you can make an
interference microscope based on this
technique and if you could put in here
let's say I bought biological slide with
a bit of tissue or something on it and
you could use this to measure the
optical path in going through that that
object it's not trivial to do that you'd
normally today if you want to do it in a
microscope you'd have to place a
microscope objectives and things in here
and usually it's found that if you do
that that messes everything up so you
normally have to put the microscope
objectives in this one as well so you
end up really having to have two
microscopes and the whole thing becomes
very expensive so that sort of method
has really gone out of fashion although
many years ago Zeiss and people used to
make instruments like that the same goes
for a Michelson type interferometer you
can measure that for looking use that
put in a microscope for looking at
surfaces surface surface height changes
and so on okay so so this shows again
this how this comes about then you've
got this one wave which is coming along
this same angle here and so this
represents the phases this is a movie
did I just click it again and it'll do
something with it yes there we are so
okay so we think over it didn't go very
far anyway you probably is quick enough
to see it was a wave that's moving this
way but the important thing if I go back
and do it again and now okay so you all
got to watch this time
and you're see there's two things I want
you to watch one is the wave moving this
what a but more importantly look what
happens on on this line here where
you've got the camera placed ah right
yeah yeah yeah otherwise I think painful
is it have to be very quick
yes look at that okay so good so so this
this wave is is moving out here but you
can see if you look at where the
detector is it's as though the fringes
are the the phase is actually moving
continuously in one direction like that
right so
right so the point is that they're
efference has a constant phase because
it is normal on the on the vertical axis
so as the as the incident wave is
passing through the the the Peaks are
the locations where the the intensity
accumulates right so you end up getting
a peak whereas in the in the nulls it
means that as the wave is passing
through the phases are not lining up so
you end up getting a null that's how you
yeah yeah okay they yeah so I guess the
reference frame as you say is coming
along this way but of course that's got
fringes as well so the the pace of that
when they lack match up with the pace of
the other we would give the Maxima in
the yeah those locations where the the
peaks always arrive at the same time
right but a little bit below the peaks
will be actually out of phase so they
end up getting a null end of shelf
that's the end of that sounds like a
good place for a brighter day so let's
make a break unless anyone's got some
questions of course yeah Australia's
good question in Michelson
interferometer the object would have
affect the intensity according to
whatever phase delayed introduces but in
the magazine der interferometer it will
affect the spacing of the fringes is
that correct it will change the spacing
way we added this constant slope in the
signal beam searching the chosen the one
beam relative to another you can do in
either sort of interferometer so that's
not a fundamental difference between
in the two types oh so they I was
thinking when you have a tilted plane
are tilted mirror then you would measure
the spacing between fringes to find out
the object objects optical path length
but if you didn't have that then you
would just measure the the intensity
change in intensity to get the trapped
phase and then apply on wrapping
straight credit what you trying to find
out job phase because you're thinking of
it varying it's a spatially varying
quantity that you're trying yeah
measures yeah yeah so I mean very often
what you do of course is you actually
very often you want the fringes because
they give you something to actually look
at hmm you know so normally if you if
you don't do the tilting thing you just
what you'd see if the two paths were
equal would be let's say completely yeah
feature this yeah well yeah wouldn't it
yeah right
yeah and so let's say you put some
objects in there and we just got some
phase change yeah you you would just see
it's a mall change from that right right
which might not be very visible they see
yeah but for example then if you tilted
yeah then you would get some fringes and
you know sort of thing you might see of
course let's say you put in a disc of
material like this and you're trying to
measure the thickness of this disc of
material so what you would find is if it
was nicely uniformly thick you'd find
that you've got these regular sort of
fringes across here right whereas if it
wasn't then you might find that you get
fringes that have got some straight
something like this all right so by
looking at the the geometry of those
fringes you can quite easily measure the
the the changes in the the deformations
in this in the shape of it
yeah but - so you know and actually
change the the tilt it would change the
separation between these fringes and so
if you make the tilt very small then you
just effectively be looking at just a
single fringe you are you know fringes
really all right so then it would be
actually very difficult to to see the
the the changes in the thickness and s
you actually made you know very
sensitive measurements
okay so everyone back again
yeah I thought of something else perhaps
I could mention following on from this
diagram so in they in the lectures that
the system as you remember was arranged
to do this sort of thing so the idea
here is that the two parts now this path
and this part that roughly equal in size
in the old days when people were trying
to do this with non non coherent sources
then you had to be really careful and
for example this is a beam splitter
which is actually of course a lump of
glass maybe with some coating on it on
one side but you can see this this beam
has gone through this glass but this
beam hasn't gone through that glass and
so these are all things that you also
have to take into account and so in
order to correct for the optical effects
of say for instance that that slab of
glass there you might actually end up
putting a slab of glass in here some
sort of compensating piece of glass so
so that this that this would produce the
same dispersion in the two arms so there
was a lot of effort put into trying to
design these systems so that they would
work with broadband sources and so on
nowadays I guess most of the time we're
doing it with lasers and we don't have
to worry about any of these things but
if you look in any of the standard books
on on interferometry there are a number
of textbooks on interferometry that will
give you all the information if you want
to go more into it okay so now we're
going to change step slightly from
interference to the fraction and so
what's the difference between
interference and diffraction the answer
is really not a lot except that normally
interference
with we're talking about two beams
interfering whereas in diffraction we're
normally talking about lots of beams
interfering so you can think of the two
is really being effectively the same
same phenomenon actually and so we're
going to look at Huygens principle we're
then going to go on to the young Young's
interferometer so this this really does
show doesn't it
our interferometry is that or related to
diffraction because this let's just
really about diffraction and then we're
going to go to a more complicated case
of for an El diffraction okay so first
of all Huygens principle so I think at
the beginning of the lecture course
George said a bit about the history of
how Huygens came up with this idea quite
a long time ago wasn't it was even
before Newton or around
I think slightly before but around the
same time as Newton so when people were
still and really decided whether light
or it was a wave or a particle and of
course we still haven't decided so times
don't change really
but anyway so probably probably now we
could look in back we could say that
Newton was the first quantum optics but
anyway so but this was this was Huygens
idea he was saying that if you think of
a light beam a wave propagating in space
then at some moment in time this is a
wavefront the wavefront is a surface of
constant phase all right
and you could he came up with this model
for how light propagates where he
assumed there are all the points all the
points on this wavefront can be thought
of as being a source of what he called
secondary wavelets so each of these
points radiates like a spherical wave
and and then if you sum of those all up
and calculate what the what the effect
is at some distance in front here then
that would then predict what that wave
front what what the what the the wave
are done slightly later time so that's
the principle and so quite a simple idea
really and you can come up with a very
simple mathematical description of it
unfortunately it doesn't work
marvelously well and people put a lot of
effort into trying to improve the theory
and they came up with some rather you
know arbitrary they seem rather
arbitrary factors that they had to
include in order to get it to work
properly so I think nowadays we probably
look back at all this and think it was
just history and you know there are
better ways of really tackling the
problem but anyway from the very very
simple level though it's still a nice
simple theory for how you do that so if
you want to calculate what the field is
at this point here say you look at all
these waves that come from all these
points on the wave front add them all
together as phases of course and that
will give you the the field at that
point and there's it's showing how you
do that and then you do it you can do it
again and again and again and show how
the wave propagates okay so example a
hold in an opaque screen so I guess this
is pretty well the simplest example you
could probably possibly think of well no
I suppose the opaque screen we know
holding will be even simpler then you
wouldn't see anything but what we do is
we just have a very small hole in this
and so this is going to act like a
source that's going to radiate like a
spherical wave and what we'd see on this
screen that we place here you'd see that
that is virtually
we've been looking at it before anyway
we've done that before so there we are
there's the spherical wave and and then
we can nothing happened then ah each of
the IR right sorry I couldn't see
anything happening okay so this this is
the plane wave coming in and and now
something's come up
the the half of the hole is a delta
function did that just come up and now
out something else has come up this is
the spherical wave now and this is the
same expression we had for the spherical
wave in the previous lecture then so
remember this parabolic type
approximation we made for a spherical
wave this is called the free nail or
proximation but based based named after
the guy for now who I guess must have
come up with this in the first place and
you remember it's true it's going to be
approximately true if if if the x and y
a small compared with the distance L
right so you need that in order to apply
the binomial and right so there it is I
just mentioned again this this eye that
someone from MIT asked about earlier so
that I then but the bottom is saying
that this spherical wave is actually 90
degrees out of phase with the the wave
that arrives at this point so that's
quite interesting so it's you can think
of it being like a resonance sort of
effect I think and I guess this is
something that the Huygens principle
cannot really explain right this is why
you have to do more than Huygens but we
take it for granted in the class so we
don't get into this yeah okay yeah yes
so I think George is really saying that
eigen didn't actually have that hi there
so
so we're cleverer than he is okay so
that's so there we are there and oh now
we've got two holes
I knew something attack happen and the
spherical wave is now hidden behind here
partly more will come okay so now we've
got two hulls each of these is a
spherical wave and of course where we
get two of them there's going to be two
spherical waves which I think that's
good oh yes
another spherical wave on another
equation though there we are so these
are the equations of the what you see on
the screen for each of these two
spherical waves and if you look at them
you'll notice that see these are a
distance X naught and minus X naught
relative to the axis and you can see the
only difference in these expressions is
this sign plus and minus there so
without now what I add those together
and so here we're looking at some
observation point you've got to add
those things together yeah so of course
as it explains here if this distance we
look at this distance and this distance
if the path difference between those is
an exact number of wavelengths then
they're going to add up in phase if
they're if their exact number of half
wave lengths then it then they're
healing you're going to get destructive
interference and you're going to get a
black region so there they are there's
the distance is d1 and d2 this is the
expressions for those and again we can
use this for another approximation to
expand the square root and then d2 minus
d1 is going to be this minus this and
you see the only difference there's a
sign there George
and at and so yeah so there's going to
be 4 X naught X dashed over to L so that
one of the toes councils and you get
this alright so the optical path length
optical path difference is is this 2x
null x dashed over L right so X naught
is this this this distance here X dashed
is this distance here so both of those
are small quantities and L of course is
a big quantity so this is actually then
in terms of distance of course that you
know so distances at the top and the
distance at the bottom
this has got the right dimensions of
length for optical power right so as we
said that give constructive and this
destructive interference what you're
going to see is just fringes like this
so there's going to be positions where
these two add up in phase and positions
where they you get destructive
interference right so if if the optical
path difference is equal to lambda then
you if you put lambda equal to this
thing then you get that X dashed is
lambda L over 2 X naught right so this
this then tells you if you pay this end
lambda of course you'd have an N in here
so this distance here to 2 this distance
to there is given by this lambda L over
2 X naught so you can see the smaller X
naught it is the bigger X dashed is
smaller you make this the bigger you
make this
what happened
ah I think that's facing their right
okay so this is this is the that the the
spacing of the fringe is there yeah is
this lambda lambda L over 2 X naught
right okay and so this is going through
the maths of that so these are the two
waves we've added together the only
difference between them is this sign so
we can take out the other things as
factors and so these are the two times
with the sign difference so each of them
- I something plus e to the I something
gives cosine and so there's our final
answer what you see the intensity is the
modular square of this these are all
face terms so we don't worry about them
and I guess in this expression here we
don't worry about any of these other
things either these constants the only
thing that's important is is this this
term here and but we've seen to have
kept the four two squared I guess yeah
why we kept there anyway for cos squared
something and then we say that cos
squared of something is equal to 1 plus
cos 2 theta and so therefore our final
result then this is then our expression
very similar to what we had before of
course for the interferometry a
background with a cosine fringes on top
of it all right so here it here is it's
very similar to what we had before the M
here is is 1 and that of course is just
like with the case of the interferometer
that we described before you remember
we've got M equals 1 when the strength
of the object beam and the reference
beam were equal so in this case they're
equal aren't they because the two the
two slits in the screen are equal if you
made those two
slightly different width or something
then that would not be true
right then then you might end up with
something where we didn't have this
visibility of one okay so that's the
first example so that's that one really
what's showing how interferometry and
diffraction are really the same now
we're going to go on to some more about
diffraction and introduce the idea of a
transparency so all we're doing here is
we're introducing the concept of a
transparency that can change the the
magnitude and phase of a wave by a
transmitting through it and so this
shows here our transparency and you can
see here it shows that the the I don't
know what what this color whether it
means right so this means that yeah the
absorption of the screen and but it
might I guess also changed the phase it
might have some refractive index as well
yeah and only and it changes in
thickness as well right so so you have
to work out just like before the optical
path as the light goes through this
structure and it might change both the
amplitude and the phase of the light
right so we can think of this thin
transparency as just out having a sort
of multiplying effect if the if it's
thin enough so you can use your you can
use your Huygens type principle to say
that when the light has gone through
there it produces some wavefront on the
far side so okay so the assumptions the
thickness I've just said also the
features on the transparency what's that
mean what's that mean the features on
the transparency are smaller than lambda
ah right okay that's wild computer okay
right so yeah so this so so this
transparency can change the attenuation
right so the the black bits are going to
absorb the light but it can also change
the phase so phase delay and both of
these are going to be both of these
effects they're going to be dependent on
the material properties of this
transparency transparency of course and
also its thickness and as it points out
here in some cases you might be able to
think of these quantities the
attenuation and the phase delay as being
binary quantities they might only have
two different two different values for
example you might have black and white
all there might be a grayscale type
continuously varying type quantities and
and then the phase also you can think of
that
maybe it's it's continuously varying
phase change or it might have just two
values binary values or it might even
have multi multiple values maybe say
eight values you can code people often
use transparencies like this to code
binary numbers like up to some you know
basis of eight or something like that
okay so if if these assumptions are true
then according to Huygens principle this
the propagation through this all it's
going to do basically is change the the
this is going to change to this this is
going to change to this and you're not
going to get this affecting this very
much if this assumption of this
thickness being very thin is valid in
which case all it does all this
transparency does is multiplies
the field going in by the transmission
of the grating to give that the the
amplitude of the wave as it comes out
alright so g- is the wave is the
amplitude on the weighed g+ is the way
amplitude of the wave on the way out and
this transmits transmittance we call
this the amplitude transmittance of the
grating and you see it's a function of
position and as we just described it can
be broken up into an absorption term and
a phase term right and and as we just
said those two might either be they
could be binary or they could be
continuously variable right yeah so this
is saying how you can think of it as
being very similar to the to the Youngs
interferometer the two-slit experiment
but now you can see here is that we're
thinking of it has been many babes so so
you breaking this up into not not just
to two beams but many beams and each of
these is going to propagate but not
quite sure where this is trying to show
yeah okay so it's pretty well what I say
before so if this thing is thin enough
and then this point here is only going
to be affected by this one and not by
the others right so so then you will get
this simple multiplicative sort of
relationship okay so you can think of
diffraction then as being like
interference with many many beams okay
and
yeah so this is saying just like we said
before we put in this describing this
interference of these many many beams
now when we did the that the Young's
experiment each of the slits was
described by a delta function at a
particular position and now we're saying
that in general we can think of this as
not being just two slits but an integral
over many many apertures very small
apertures placed at these points x1 y1
and sorry the place at the points X Y
argot and then we're integrating over
all those points in order to get the
total effect of all those spherical
waves right so so this is now we're
saying that G G so this is G minus after
this
just saying that G the the this is this
is just the sifting theorem and now
we're going to stick that in there are
we yeah yeah yeah ok yeah yeah yeah okay
so
okay so this is what goes in then this
is so this is our our like our a single
point on the wavefront going in and this
gives rise to a wave that comes out a
spherical wave that comes out which is
of this form is the same as this
multiplied by the transmission of the
source and then this then radiates like
a spherical wave and then you're going
to get some distribution of amplitude on
this observation plane which is going to
be exactly the same as we have before
and we've got this again we've got this
parabolic phase function that comes from
the framel approximation so assumes that
x dashed why - just much smaller than
say okay so that's for a single point
and now we do now we integrate over all
of these and the final result for the
amplitude we'll see on the screen is
given by this expression then so this is
the incident wave this is the
transmission of the screen and this is
the bit that describes the propagation
of the spherical waves according to the
Fornell approximation okay so yeah so
this result is known as the fernell
diffraction integral or simply the fran
L integral and so we I mentioned earlier
that this approximation replacing these
the square root by the squared terms is
called the foreign an approximation all
right so all of these for now gets the
name of all of these things
okay yeah this is just pointing out that
this expression we've just written down
the form of this is exactly the form of
a convolution and so what that's a very
important result a convolution of course
means it's it's space invariant doesn't
it it means that you can write this it's
some it's something it's what goes in
convolve with something which is
independent of the coordinates and so if
you think of this as a convolution the
thing you have to convert is this thing
so you can think of this as as being
like like I won't always called the
point spread function or the amplitude
point spread function so if you put in
some amplitude and convolve it with this
then you'll get something out
okay so this is explaining what I just
said so this this is the sort of
behavior that you think of when you're
looking at you know in mechanical or
electrical systems you can think of a
system of being like a black box or in
this case a blue box and something goes
in and something comes out and we don't
have to worry too much about what's in
here but you can express what goes in of
course either in terms of some sort of
impulse response function or as we'll
come to in later lectures as some sort
of transfer function as type behavior
but the moment we do it in the spatial
domain so if we if we convolve this with
this point spread function then we'll
get this output like this okay now we go
on to do some examples of this and this
is probably the most important example
comes up all the time because you know
lots of optical components a circular so
lenses and so on
very often they've got this circular
form and so if this is comes up time and
time again and so all we've got to do
now then is we put in we say that they
are seek out our amplitude after the
light's gone through this this circle
circular aperture is one inside the
circle and zero outside the circle and
and so that for an our diffraction
integral that we have to work out for
this circular aperture it's just it's
just one inside and zero outside so it's
just oh no this is you just have to do
the integral over the inside so you did
this is just over the values of x and y
that are inside the bright part here
and now so unfortunately that integral
is a bit hard to do so we're not going
to do it at the moment in general but we
will do a very simple example a special
case of what happens along the axis so
if we put X and y both zero now we get
if we put X X - D cos no why - nique was
not the integral comes down to this and
and now that is an integral we can do
and it's an integral that probably a lot
of you have come come across before the
way to solve that is to go into polar
coordinates and so if you if you change
into polar coordinates you've got a
theta and a Phi a theater and a row part
the the feet apart is just an integral
of a constant so that just becomes 2 pi
and the integral of this other thing you
can now do integral of Rho e to the I of
Rho squared over something you can do
because the the row outside is the diff
differential of the bit inside so to
actually do that properly
you can make the substitution then make
the substitution that Z is equal to this
thing inside here and after you've made
that substitution you now just get
integral of e to the I Sai so it's
something that's something you can just
do and you get an answer so notice
ya notice that you get so this this
final result here this term here is just
the phase term as a function of Z so
that's just just what you'd get if it
was just a wave moving in free space and
and then these are the two terms notice
there are exactly the same form this is
an X of the I something and this is a
sign of the exactly the same thing okay
so so that's what we get as I say if you
if you go back to this if you wanted to
do this integral here it's a bit more
complicated and so we're not going to do
that at the moment but I guess
eventually we have to do something about
it
right so if we look at the that this
amplitude along the axis then you can
see it's got this sinusoidal sort of
behavior if you looked at the intensity
this would just go and you just get sine
squared or something sine squared as a
function of a constant over Zed so as it
points out here then it's going to have
peaks and nulls maxima and minima
corresponding so these particular values
of Zaid so you'll get the gain like a
sinusoidal variation in amplitude along
the axis right the other thing to note
is that where's it gone yeah notice that
this this said and this said cancelled
out this said here appeared when we made
this substitution and so these legs
finally canceled and the so the final
result has got no Z in it not no sorry
no Zed in the modulus part term so
you've got this series of fringes
which are of constant amplitude and
having a far away from this aperture you
go which might seem rather surprising
you might think that as you got closer
to the aperture it would get brighter
might you but it doesn't it seems I'm
just trying to think quickly why that
should be the case I'm sure it is it's
true not the sort of the by over two
faces disappeared it got that's why the
sign yeah that's true sorry I didn't
mention that yeah okay so when you put
in the limits of the integration then
you've got this minus this and this you
were replaced by side but of course we
need to get an extra eye to get the sign
so yes so the pie the pie over to phase
shift is also gone the other thing that
perhaps is worth mentioning also is that
when we got down to this integral of
course this is just this is just the
Fourier transform of one isn't it within
this region it's the Fourier transform
of a of a rect function and so you know
even if he looks at more complicated
example if it had this circular symmetry
whatever this amplitude transmittance of
this aperture of beam all we'd have to
do is to do the Fourier transform of
that amplitude transmittance of that
aperture as a function of this
coordinate Zhai right so that's a sort
of trick you can use to calculate these
things easily done right I think we're
in for the movies now so MATLAB has
calculated these things for us and when
I press the button it will go but I'm
not going to press it until you're ready
just in case it's very quick so this is
a big aperture right so now so what
you'll see is how the wave propagates
from that aperture this is actually
quite a big aperture so
don't expect to see a lot of diffraction
in this case when we come to this one
this is a small aperture and as we said
before the smaller is to start the more
it's going to spread right so you're
going to see a lot more effect on that
one and so you have to look quite
closely to see what happens as it
propagates so this is a radius of 40
lambda now you can see some sort of
fringes appearing and it becomes a like
like a bright halo around the edges and
if you look very closely you'll see
quite a lot of fine structure on this
and then the small one this is ten
wavelengths so this one changes a lot
quicker
you see it's doing quite drastic things
and now it by the time it gets to the
end you've even got a black spot in the
center right so the the fact that of
course we said before that the smaller
the aperture the more it spreads but
also the quicker it spreads so you can
see that actually this one has got a lot
further in the development of the
pattern than this one this one still
looks roughly similar to wet work what
it started as this one is looking very
very different
but it caused this the blinking spot
I've never heard of it called the
blinking spot but anyway well yeah but
I'm not even sure I I always think of
the Poisson spot as being what you get
from a on a pike object rather than for
a but maybe that's the arago spot but be
net just says the the same room yeah
yeah okay so now the rectangular
aperture the rectangular aperture
actually it turns out you can't get a
simple expression for what happens along
the axis of the rectangular case unlike
the circular case so that's probably why
George hasn't done it and so you can get
expressions in terms of Frontera
integrals for everywhere in fact for
this case so the square case then they
see what that does the big square yeah
you gain you can see you can see some
sort of fringes all of them by the by
the time it's got to the end here you
can see a bright lines around the edges
and also bright spots in the four
corners and then if we look at the small
square again it's changing more quickly
and you'll notice that we've developed
by then we've got quite strong sort of
streaking along these two axes here and
in fact so you see this final pattern
there that that is the diffraction
pattern of the of the square aperture
what does it say well we haven't done
friend off of diffraction yet have we
this is introduction to yeah this is an
example of Fraunhofer diffraction then
if you're a long way away you get what's
called Fraunhofer diffraction this is
what it looks like and as we're going to
show later this is very simply related
to the Fourier transform of the of the
structure that the
that we're going to look at okay yeah so
this case here if we eventually this one
would look like this and but we haven't
gone anywhere near far enough for it to
look like that but eventually it would
get like that the distance you have to
go depends on how big the aperture is
end of show